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5909 Journal of Intelligent & Fuzzy Systems 36 (2019) 5909–5918 DOI:10.3233/JIFS-181735 IOS Press Symmetric intuitionistic multiplicative aggregation operator for group decision making in intuitionistic multiplicative environments Chuan Yu Xu and Zhen Ming Ma∗ School of Mathematics and Statistics, Linyi University, Linyi 276005, China Abstract. Aggregation operators are indispensable to aggregate individual preference relations to a collective one in group decison making. Although some aggregation operators have been proposed in intuitionistic multiplicative environment, which can reflect our intuition more objectively, in this paper, it is pointed by an example that the existing aggregation operators generally can not aggregate individual intuitionistic multiplicative preference relations (IMPRs) into a collective one which limits the study of consensus of IMPRs in group decision making problems. Then, some new symmetric operations and symmetric intuitionistic multiplicative aggregation operator are proposed to overcome the above issue. At last, a procedure for group decision making problem with acceptable consensus based on the proposed operator is provided and an example illustrates the validity of the proposed method and comparison analyses are conducted. Keywords: Intuitionistic multiplicative preference relations, Symmetric intuitionistic multiplicative aggregation operator, Group decision making 1. Introduction Group decision making is one of the most common activities in our daily life, and decision information is usually subjectively provided by decision makers (DMs) over a set of objects considered. In group decision making procedure, in case the collection of finite alternatives is provided, DMs are required to express their preferences on the given collection of alternatives by the pairwise comparison methods, which are more accurate than non-pairwise comparison methods [23], where consis-tency of a preference relation, consensus of a group and aggregating individual preference relations into collective one are indispensable to obtain a collect result. ∗ Corresponding author. Zhen Ming Ma, School of Mathematics and Statistics, Linyi University, Linyi 276005, China. E-mail: zmma@whu.cn. The classical preference relations are generally devided into multiplicative preference relations [20] and fuzzy preference relations [15, 22], whose pairwise comparisons are single numerical values. Due to the increasing complexity of the socio-economic environment and the lack of knowledge or data about the problem domain, DMs have great difficulties in using the exact values to express their preference information about the alternatives or criteria. To circumvent this issue, intuitionistic fuzzy preference relations (IFPRs) [21, 25] are developed whose basic elements are intuitionistic fuzzy numbers [26]. Therefore, IFPRs are more convenient to describe the uncertainties of pairwise comparisons between alternatives in decision making problem. In recent years, many scholars have paid much attention to group decision making with IFPRs [4, 14, 24, 28], generalized intuitionistic fuzzy aggregation operators [4–10, 12, 13] and related measures [16, 17]. 1064-1246/19/$35.00 © 2019 – IOS Press and the authors. All rights reserved 5910 C.Y. Xu and Z.M. Ma / Symmetric intuitionistic multiplicative aggregation operator for group decision making It is pointed in Ref. [31] that the IFPRs use the 0.10.9 scale, which is a balanced scale to express the preference information. But in real life, the information are usually distributed asymmetrically, and the law of diminishing marginal utility is the common phenomenon in economics even in our daily life. Thus, Xia et al. [31] replaced the 0.1-0.9 scale in the IFPRs with the 1-9 scale and proposed the concept of IMPRs and some aggregation operators to aggregate these IMPRs. Xia and Xu [32] proposed some extended operations about the intuitionistic multiplicative information, and solved the group decision making problems based on IMPRs. Xu [27] derived the priority weight intervals from IMPRs by the intuitionistic multiplicative weighted geometic operator. Then, many scholars devote into group decision making with IMPRs [1, 19, 29, 33–35]. In spite of this, in Ref. [18], examples shows the operational laws in Refs. [31, 32] could be not closed and then they developed some new operational laws of intuitionistic multiplicative numbers (IMNs) based on which, some novel operators are proposed to aggregate the intuitionistic multiplicative preference information. However, we find in this paper that since different operators are used to aggregate the memberships and non-memberships of elements in IMPRs, the aggregated result of several individual IMPRs by operators in Refs. [18, 33] could not be an IMPR which is not reasonable and leads that it is not convenient to check and reach consensus of a group. To overcome this issue, in this paper, a novel symmetrical operational laws and symmetrical aggregation operator are proposed to aggregate the individual IMPRs into a collective IMPR in group decision making problems with consensus of a group invest-igated in intuitionistic multiplicative preference information. The remainder of the paper is organized as follows: Section 2 describes some basic concepts on the intuitionistic multiplicative sets (IMSs) and IMPRs. An example shows that the collective IMPR of several individual IMPRs by non-symmetrical aggregation operators is usually not an IMPR. In Section 3, symmetrical operational laws of IMNs are introduced. Based on the proposed operational laws, Section 4 provides a symmetrical aggregation operator and investigates some properties of the aggregation operator. In Section 5, a procedure for group decision making problem with acceptable consensus based on the proposed operator is given and a example illustrates the validity of the proposed method and comparison analyses are conducted. The main conclusions are drawn in Section 6. 2. Preliminaries To make the presentation self-contained, in what follows, we review some basic concepts on intuitionistic multiplicative sets. Deﬁnition 2.1. [31] Let X. be a fixed set. An intuitionistic multiplicative set (IMS) is defined as: D = {x, ρ (x) , σ (x) x ∈ X}, which assigns each element x a membership information ρ (x) and a nonmembership information σ (x), with the conditions: 0 < ρ (x) σ (x) ≤ 1, 1/9 ≤ ρ (x) , σ (x) ≤ 9. For convenience, let the pair (ρ (x) , σ (x)) be an intuitionistic multiplicative number. Deﬁnition 2.2. [31] Let X be a fixed set. Then, the intuitionistic multiplicative preference relation (IMPR) is defined as A = αij , where αij = n×n 9 ραij , σαij is an IMN, ραij ∈ S1/9 can be considered as the intensity degree that xi is preferred to xj , σαij ∈ 9 S1/9 can be considered as the intensity degree that xi is not preferred to xj , and both of them should satisfy the following conditions ραij = σαji , σαij = ραji and 0 ≤ ραij σαij ≤ 1, 1/9 ≤ ραij , σαij ≤ 9. To compare two IMNs, Xia et al. [31] gave the comparison laws as follows: Deﬁnition 2.3. [3, 31] For an IMN α = (ρα , σα ) , s (α) = ρα /σα is called the score function of α, and h (α) = ρα σα the accuracy function of α. The comparison laws are given as follows: (1) If s (α1 ) > s (α2 ), then α1 > α2 ; (2) If s (α1 ) = s (α2 ), then (a) If h (α1 ) > h (α2 ), then α1 > α2 ; (b) If h (α1 ) = h (α2 ), then α1 = α2 . In Refs. [31, 32], Xia and Xu proposed the following operational laws of IMNs. Deﬁnition 2.4. [31] Let α1 = ρα1 , σα1 , α2 = ρα2 , σα2 and α = (ρα , σα ) be three IMNs, and λ > 0, then ⎞ ⎛ 2ρα1 ρα2 , ⎜ 2+ρα1 2+ρα2 −ρα1 ρα2 ⎟ (1) α1 ⊗ α2 = ⎝ ⎠; (2) αλ = 1+2σα2 −1 2 2ραλ (1+2σα )λ −1 . , 2 (2+ρα )λ −ραλ 1+2σα1 Deﬁnition 2.5. [32] Let α1 = ρα1 , σα1 , α2 = ρα2 , σα2 and α = (ρα , σα ) be three IMNs, and λ > 0, γ > 0, then C.Y. Xu and Z.M. Ma / Symmetric intuitionistic multiplicative aggregation operator for group decision making ⎛ γρα1 ρα2 , ⎞ ⎛ 5911 ⎞ n ω ραii α ρα ⎜ γ+ρα1 γ+ρα2 −ρ 1 2 ⎟ (1) α1 ⊗ α2 = ⎝ ⎠; ⎜ ⎟ i=1 ⎜ n ωi n ωi , ⎟ ⎜ 1+ραi − ραi ⎟ IMWG(α1 , α2 , · · · , αn ) = ⎜ ⎟. i=1 ⎜ i=1 ⎟ ⎝ n ⎠ ωi 1+γσα2 −1 γ λ γρλ (2) αλ = (γ+ρ )αλ −ρλ , (1+γσγα ) −1 . α α 1+γσα1 1 + ραi −1 i=1 However, in Ref. [18], it is pointed that the above operation is not closed. Then, some new operational laws of IMNs are presented in next section. Deﬁnition 2.6. [18] Let α , α2 = = ρ , σ 1 α α 1 1 ρα2 , σα2 and α = (ρα , σα ) be three IMNs, and λ > 0, τ > 0, then ⎛ (1) α1 ⊗ α2 = ⎝ ⎛ 2 (2) αλ = ⎝9 9 log9 ρα1 ρα2 −log9 ρα1 log9 ρα2 +1 2 9 log9 ρα1 ρα2 +log9 ρα1 log9 ρα2 −1 2 1+log9 ρα 2 λ −1 1−2 ,9 1−log9 ρα 2 ⎞ , ⎠; λ ⎞ ⎠. Theorem 2.7. [18] Let αi = ραi , σαi be a colT lection of IMNs, and ω = ω1, ω2 , . . . , ωn be the weighting vector of αi with ωi ∈ [0, 1] and 1. Then ⎛ n 2 1+log9 ραi 2 n i=1 ωi ωi = ⎞ ⎜9 i=1 , ⎟ ⎟ ⎜ IMWG(α1 , α2 , · · · , αn ) = ⎜ ωi ⎟ . n 1+log9 ραi ⎠ ⎝ 1−2 2 9 i=1 Deﬁnition 2.8. [33] Let α , α2 = = ρ , σ 1 α α 1 1 ρα2 , σα2 and α = (ρα , σα ) be three IMNs, and λ > 0, τ > 0, then ρα1 ρα2 ρα1 +ρα2 +1 , α Theorem 2.9. [33] Let αi = ραi , σαi be a colT lection of IMNs, and ω = ω1, ω2 , . . . , ωn be the weighting vector of αi with ωi ∈ [0, 1] and 1. Then IMWG(α1 , α2 , · · · , αn )c =IMWG(αc1 , αc2 , · · · , αcn ), which is essential for investigating the collective IMPR in group decision making. Example 2.10. Let A1 , A2 be two IMPRs defined as: ⎞ ⎛ 1 5 1 (1, 1) 2, 25 , , 2 4 3 2 ⎜ ⎟ ⎟ ⎜ 2 1 8 1 ⎟ ⎜ 5 , 2 (1, 1) 2, 3 7, 3 ⎟ ⎜ ⎟, A1 = ⎜ 1 ⎜ 5, 1 (1, 1) 5, 17 ⎟ ⎟ ⎜ 3 4 3, 2 ⎠ ⎝ 1 1 8 1 (1, 1) 2, 2 3, 7 7, 5 ⎞ 1 (1, 1) 21 , 65 1, 23 , 3 7 ⎜ ⎟ ⎜ 6 1 1 2 ⎟ ⎟ ⎜ 5 , 2 (1, 1) 2, 15 4, 3 ⎟ ⎜ A2 = ⎜ ⎟ 1 ⎜ 2, 1 (1, 1) 43 , 1 ⎟ ⎟ ⎜ 3 5, 2 ⎠ ⎝ 2 1 3 (1, 3, 17 1, 1) , 3 4 4 ⎛ and ω = (0.5, 0.5). Then the aggregated A by IMWG operator is ⎛ ⎞ (1, 1) ⎜ (0.637, 1.119) ⎜ ⎜ ⎝ (1.006, 0.544) (0.894, 0.735) (0.422, 1.108) (0.206, 2.489) (1, 1) (0.248, 2.0) (2.0, 0.261) (0.439, 0.481) ⎟ ⎟ (1, 1) (2.433, 0.283) (0.452, 0.593) (0.234, 2.688) (1) α1 ⊗ α2 = ; ρα1 ρα2 + ρα1 + ρα2 λ (2) αλ = (1+σσ)αλ −σ λ , (1 + σα )λ − 1 . α The following example shows that the following property does not holds for the IMWG operator: n i=1 ωi = ⎟ (1.647, 0.440) ⎠ (1, 1) which is not an IMPR, similar case happens in Ref. [33]. Note that in group decision making, there are two methods to obtain the priority vector, one is to aggregate the individual priority vecto [18, 32] to obtain a collective priority vector and the other is to aggregate the individual preference relations to a collective ones, and then a corresp-onding priority vector. Generally, by non-symmetrical intuitionistic multiplicative aggreg-ation operators, the aggregated result by the latter method is not an IMPR in general 5912 C.Y. Xu and Z.M. Ma / Symmetric intuitionistic multiplicative aggregation operator for group decision making which can not induce a priority vector and the former method is not convenient to check the consensus of the IMPRs. However, we find that the symmetrical aggregation operators are suitable for the latter. 3. Symmetric intuitionistic multiplicative operation In order to overcome the shortcoming of the proposed IMWG operator in Refs. [18, 33], we introduce the symmetric operation and scalar symmetric operation on IMNs. Deﬁnition 3.1. Let α = ρ , σ , α2 = α α 1 1 1 ρα2 , σα2 and α = (ρα , σα ) be three IMNs, and λ > 0. Then we define (1) ⎛ α1 ⊗ α 2 = ⎞ (1+log9 ρα1 )(1+log9 ρα2 )−(1−log9 ρα1 )(1−log9 ρα2 ) ⎜ 9 (1+log9 ρα1 )(1+log9 ρα2 )+(1−log9 ρα1 )(1−log9 ρα2 ) , ⎟ ⎜ ⎟ ⎝ (1+log9 σα1 )(1+log9 σα2 )−(1−log9 σα1 )(1−log9 σα2 ) ⎠; 9 (1+log9 σα1 )(1+log9 σα2 )+(1−log9 σα1 )(1−log9 σα2 ) ⎞ ⎛ λ λ (1+log9 ρα ) −(1−log9 ρα ) ⎜ 9 (1+log9 ρα )λ +(1−log9 ρα )λ , ⎟ (2) αλ = ⎜ ⎝ (1+log9 σα )λ −(1−log9 σα )λ λ λ ( 9 1+log9 σα ) +(1−log9 σα ) ⎟. ⎠ Theorem 3.2. Let α1 = ρα1 , σα1 , α2 = ρα2 , σα2 and α = (ρα , σα ) be three IMNs. Then α1 ⊗ α2 and αλ (λ > 0) are also IMNs. Proof. Since 19 ≤ ραi , σαi ≤ 9 (i = 1, 2), then 0 ≤ log9 ρ9α ≤ 2 and 0 ≤ log9 9σαi ≤ 2 (i = 1, 2). We i have 1 ≤ ρα1 ⊗α2 9 (1+log9 ρα1 )(1+log9 ρα2 )−(1−log9 ρα1 )(1−log9 ρα2 ) =9 (1+log9 ρα1 )(1+log9 ρα2 )+(1−log9 ρα1 )(1−log9 ρα2 ) ≤ 9 1 ≤ σα1 ⊗α2 9 (1+log9 σα1 )(1+log9 σα2 )−(1−log9 σα1 )(1−log9 σα2 ) =9 (1+log9 σα1 )(1+log9 σα2 )+(1−log9 σα1 )(1−log9 σα2 ) ≤ 9. By Definition 2.1, for any IMNα = (ρα , σα ), it holds that log9 9σα + log9 ρ9α = log9 σα ρα ≤ 0, that is, 0 ≤ log9 9σα ≤ log9 ρ9α . Thus, we have 0 ≤ ρα1 ⊗α2 σα1 ⊗α2 (1+log9 ρα1 )(1+log9 ρα2 )−(1−log9 ρα1 )(1−log9 ρα2 ) =9 (1+log9 ρα1 )(1+log9 ρα2 )+(1−log9 ρα1 )(1−log9 ρα2 ) (1+log9 σα1 )(1+log9 σα2 )−(1−log9 σα1 )(1−log9 σα2 ) 9 (1+log9 σα1 )(1+log9 σα2 )+(1−log9 σα1 )(1−log9 σα2 ) ≤ 1. Therefore, α1 ⊗ α2 is an IMN. Since 19 ≤ ραi , σαi ≤ 9 (i = 1, 2) , it holds that 0 ≤ log9 ρ9α ≤ 2 i and 0 ≤ log9 9σαi ≤ 2 (i = 1, 2). For λ > 0, we (1+log9 ρα )λ −(1−log9 ρα )λ λ λ have 91 ≤ ραλ = 9 (1+log9 ρα ) +(1−log9 ρα ) ≤ 9 and 19 ≤ (1+log9 σα )λ −(1−log9 σα )λ λ λ σαλ = 9 (1+log9 σα ) +(1−log9 σα ) ≤ 9. Thus,ραλ σαλ = λ λ (1+log9 ρα ) −(1−log9 ρα ) (1+log9 σα )λ −(1−log9 σα )λ λ λ λ λ 9 (1+log9 ρα ) +(1−log9 ρα ) 9 (1+log9 σα ) +(1−log9 σα ) ≤ 1. Thus, αλ is an IMN. This completes the proof. 4. Symmetric intuitionistic multiplicative weighted geometric operator In this section, a novel aggregation operator is proposed based on the operational laws defined in Definition 3.1 to aggregate the intuitionistic multiplicative information. Deﬁnition 4.1. Let αi = (ραi , σαi )(i = 1, 2, · · · , n ) be a collection of IMNs. A symmetric intuitionistic multiplicative weighted geometric (SIMWG) operator is a mapping M n → M, such that SIMWG (α1 , α2 , · · · , αn ) = ⊗ni=1 αi ωi , where ω = (ω1 , ω2 , · · · , ωn )T is the weighting vector of αi with ωi ∈[0, 1] and n ωi = 1. In the i=1 case whereω1 = ω2 = · · · = ωn = n1 , the SIMWG operator reduces to the symmetric intuitionistic multiplicative geometric (SIMG) operator 1 SIMG (α1 , α2 , · · · , αn ) = ⊗ni=0 αi n . Based on the Definition 3.1 and Definition 4.1, the following theorems can be obtained. C.Y. Xu and Z.M. Ma / Symmetric intuitionistic multiplicative aggregation operator for group decision making Theorem 4.2. Let αi = ραi , σαi be a collection of T IMNs, and ω = ω1, ω2 , · · · , ωn be the weighting vector of αi with ωi ∈ [0, 1] and n i=1 ωi = 1. Then SIMWG(α1 , · · · , αn ) ⎛ n ωi n ωi ⎞ − 1+log ρ 1−log ρ ( 9 αi ) 9 αi ) ⎜ i=1 ( ⎟ i=1 ⎜ n ⎟ ω ωi ⎟ i n ⎜ ⎜ ⎟ ⎜ i=1 (1+log9 ραi ) +i=1 (1−log9 ραi ) ⎟ ⎜9 ,⎟ ⎟ =⎜ ωi n ωi ⎟ . ⎜ n ⎜ ⎟ − 1+log σ 1−log σ ( ( 9 αi ) 9 αi ) ⎜ ⎟ i=1 ⎜ i=1 ⎟ ωi n ωi ⎟ ⎜ n ⎝ (1+log9 σαi ) + (1−log9 σαi ) ⎠ i=1 9 i=1 (1) Proof. By using mathematical introduction on n: for n = 2, we have SIFWG(α1 , α2 ) = αω1 1 ⊗ αω2 2 ⎛ ω ω 1+log9 ρα1 1 − 1−log9 ρα1 1 ω1 ω 1+log9 ρα1 + 1+log9 ρα1 1 ⎞ ( ) ( ) ) ( ) ,⎟ ⎜9( ⎟ =⎜ ⎝ (1+log9 σα1 )ω1 −(1−log9 σα1 )ω1 ⎠ ⊗ ω1 ω1 9 (1+log9 σα1 ) +(1+log9 σα1 ) ⎛ ω2 ω2 ⎞ (1+log9 ρα2 ) −(1−log9 ρα2 ) ω ω ( ⎜ 9 1+log9 ρα2 ) 2 +(1+log9 ρα2 ) 2 , ⎟ ⎜ ⎟ ⎝ (1+log9 σα2 )ω2 −(1−log9 σα2 )ω2 ⎠ = ω2 ω2 9 (1+log9 σα2 ) +(1+log9 σα2 ) ⎛ ⎞ (1+log9 ρα1 )ω1 (1+log9 ρα2 )ω2 −(1−log9 ρα1 )ω1 (1−log9 ρα2 )ω2 ω ω ω ω ⎜ 9 (1+log9 ρα1 ) 1 (1+log9 ρα2 ) 2 +(1−log9 ρα1 ) 1 (1−log9 ρα2 ) 2 , ⎟ ⎜ ⎟ ⎝ (1+log9 σα1 )ω1 (1+log9 σα2 )ω2 −(1−log9 σα1 )ω1 (1−log9 σα2 )ω2 ⎠ ω1 ω2 ω1 ω2 9 (1+log9 σα1 ) (1+log9 σα2 ) +(1+log9 σα1 ) (1−log9 σα2 ) Suppose Equation (1) holds for n = k, that is, ωi k ωi ⎞ ⎛ k (1+log9 ραi ) − (1−log9 ραi ) ⎜ i=1 ⎟ i=1 ⎜ k ωi k ωi ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ i=1 (1+log9 ραi ) +i=1 (1−log9 ραi ) ⎟ ⎜9 ,⎟ ⎟ SIMWG(α) = ⎜ ωi k ωi ⎟ ⎜ k ⎜ (1+log9 σαi ) − (1−log9 σαi ) ⎟ ⎜ ⎟ ⎜ i=1 ⎟ i=1 ωi k ωi ⎟ ⎜ k ⎝ ⎠ (1+log9 σαi ) + (1−log9 σαi ) i=1 9 i=1 then, when n = k + 1, by the operational laws in Definition 3.1 we get SIMWG(α) = ⎛ ωi k ωi k 5913 ⎞ ( ) ( ) ⎟ ⎜ i=1 i=1 ⎜ k ωi k ωi ⎟ ⎟ ⎜ ⎟ ⎜ 1+log9 ραi ) + 1−log9 ραi ) ( ( ⎟ ⎜ i=1 i=1 ⎜9 ,⎟ ⎟ ⎜ ωi k ωi ⎟ ⊗ ⎜ k ⎜ (1+log9 σαi ) − (1−log9 σαi ) ⎟ ⎟ ⎜ ⎟ ⎜ i=1 i=1 ωi k ωi ⎟ ⎜ k ⎠ ⎝ (1+log9 σαi ) + (1−log9 σαi ) i=1 9 i=1 ωi k+1 ωi ⎞ ⎛ k+1 (1+log9 ραi ) − (1−log9 ραi ) ⎜ i=1 ⎟ i=1 ⎜ k+1 ωi k+1 ωi ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ i=1 (1+log9 ραi ) + i=1 (1−log9 ραi ) ⎟ ⎜9 ,⎟ ⎟ αk+1 = ⎜ ωi k+1 ωi ⎟ ⎜ k+1 ⎜ (1+log9 σαi ) − (1−log9 σαi ) ⎟ ⎜ ⎟ ⎜ i=1 ⎟ i=1 ωi k+1 ωi ⎟ ⎜ k+1 ⎝ ⎠ (1+log9 σαi ) + (1−log9 σαi ) i=1 9 i=1 that is, Theorem 4.2 holds for n = k + 1. Therefore, Theorem 4.2 holds for all n which completes the proof of the theorem. Particularly, when ραi σαi = 1 for i = 1, 2, · · · , n, we 1+log9 ραi − 1−log9 ραi have SIMWG(α1 , α2 , · · · , αn ) = 9a , 91a , where n ωi n ωi a= − 1+log9 ραi i=1 n i=1 1+log9 ραi ωi i=1 n + 1−log9 ραi 1−log9 ραi ωi which reduce i=1 to the aggregation operator on the 1-9 scale. Furthermore, the following properties are easily obtained. Proposition 4.3. Let αi (i = 1, 2, · · · , n) be a collection of IMNs. (1) If α1 = · · · = αn = (ρα , σα ), then, SIMWG (α1 , α2 , · · · , αn ) = ⊗ni=1 αωi = α; (2) If α∗i = ρα∗i , σα∗i (i = 1, 2 · · · n) is another collection of IMNs such that ραi ≤ ρα∗i and σαi ≥ σα∗i , for all i. Then SIMWG (α1 , α2 , · · · , αn ) ≤ ; SIMWG α∗1 , α∗2 , · · · , α∗n (3) Let α− = min and i ραi ,max i σαi + α = maxi ραi , mini σαi . Then, α− ≤ SIMWG (α1 , α2 , · · · , αn ) ≤ α+ . (4) (SIMWG(α1 , α2 , · · · , αn ))c = SIMWG(αc1 , αc2 , · · · , αcn ). 5914 C.Y. Xu and Z.M. Ma / Symmetric intuitionistic multiplicative aggregation operator for group decision making (k) Proof. (1) By Definition 3.1, we have (k) (k) (2) Sinceραi ≤ ρα∗i and σαi ≤ σα∗i , for all i, we have ωi ωi n (1+log9 ραi ) − i=1 n ωi (1+log9 ραi ) 9 i=1 (k) (k) i=1 ωi n + 1−log9 ραi i=1 ) ≤ ωi n i=1 n 9 i=1 1+log9 ρα∗ i n×n n − ωi i=1 n 1+log9 ρα∗ i + i=1 have known, before we aggregate and rank these preference values, a consensus checking is needed to make by the acceptable deviation measure among these IMPRs, without which it can lead unsatisfied or even incorrect results because there are unavoidable differences and even contradictions among the IMPRs provided by different DMs. (k) Deﬁnition 5.1. Let A(k) = αij and A(l) = n×n (l) αij be two IMPRs, given by two DMs ek ωi 1−log9 ρα∗ i ωi 1−log9 ρα∗ i and ωi n ωi n (1+log9 σαi ) − i=1 n ωi (1+log9 σαi ) 9 i=1 (1−log9 σαi ) i=1 ωi n + 1−log9 σαi ( ) i=1 n i=1 n 9 i=1 9 native yi is prior to the alternative yj and νij ∈ S1/9 denotes the intensity degree of that yi is not prior to the alternative yj . If the IMNs satisfy the con(k) (k) (k) (k) (k) (k) ditions μij = νji , νij = μji , μii = νii = 1 and μij νij ≤ 1, for i, j = 1, 2, · · · , n, then the individual preference information constructs the individual (k) (k) IMPRs A = αij ,k = 1, 2, · · · , p. As we (1−log9 ραi ) ( (k) 9 μij ∈ S1/9 denotes the intensity degree that the alter- SIMWG (α1 , α2 , · · · , αn ) = ⊗ni=1 αωi = 9log9 ρα , 9log9 σα = (ρα , σα ) . n (k) and yj , denoted by an IMN αij = (μij , νij ), where n×n ≥ ωi 1+log9 σα∗ i 1+log9 σα∗ i n − ωi i=1 n + i=1 and el by comparing the pairs of (yi , yj ) for i, j = 1, 2, · · · , n. Then we define the deviation measure between A(k) and A(l) as follows: ωi 1−log9 σα∗ i 1−log9 σα∗ i ωi . Therefore, by 2.3, we have SIMWG (α) Definition ∗ ∗ ≤ SIMWG α1 , α2 , · · · , α∗n . (3) and (4) are obvious, so we omit them. The above properties (1)-(3) of the SIMWG operator indicates the idempotency, monotonicity and boundedness of the proposed operator, so it can be considered as an average operator in intuitionistic multiplicative case; the property (4) of the SIMWG operator is necessary to guarantee the aggregated result of individual IMPRs is still an IMPR which is shown in the following section. 5. Group decision making based on the proposed SIMWG operator In this section, we utilize the proposed SIMWG operator above to group decision making with the intuitionistic multiplicative information which can be described as follows: Suppose that there are n alternatives yi (i = 1, 2, · · · , n) to be compared, and p DMs e1 , · · · , ep to give their preferences about these alternatives. They agree to express their appetites with the 1-9 scale, and the DM ek uses the 1-9 scale to provide his/her preferences about the alternatives yi D(A(k) , A(l) ) = 2 (k) (l) d(αij , αij ), n (n − 1) (2) ρα1 σα1 ln ρ + ln σ . (3) i 0.1, we need to modify these IMPRs to assure that the deviation measure is less than the given threshold value. The principle of modification is that the improved preference relations should not only satisfy the acceptability requirement but also preserve the initial preference information as much as possible. Thus, we assume that the modified individual IMPRs are denoted as B(k) , and then establish the following C.Y. Xu and Z.M. Ma / Symmetric intuitionistic multiplicative aggregation operator for group decision making p min D A(k) , B(k) k=1 ⎧ (1) < a, D B, B ⎪ ⎨ s.t. · · · ⎪ ⎩ D B, B(p) < a, 5915 The prominent characteristics of the proposed method are that (4) To get the ranking order of the alternatives, a detailed procedure of the approach is given: Step 1. Aggregate the individual IMPRs A(k) for k = 1, 2, · · · , p by the SIMWG operator to obtain the col lective IMPRA = αij n×n , where p αij = SIMWG α1ij , α2ij , · · · , αij ; Step 2. Check the acceptability of the deviation measures between the individual IMPRs A(k) and the collective one A for k = 1, 2, · · · , p. Concretely, calculate the deviation measures D(A(k) , A) between A(k) and A by Eq. (2). If the D(A(k) , A) < 0.1 for k = 1, 2, · · · , p, go to Step 4; otherwise, go to next step. Step 3. Compute with Eq. (4) and modify A(k) as B(k) for k = 1, 2, · · · , p, go to next step. Step 4. Utilize the SIMWG operator to obtain the average value αi of the alternative yi ; Step 5. Calculate the score function s(αi ) and the accuracy degree h(αi ) of αi , and obtain the ranking order of the alternatives according to s(αi )and h(αi ). (1) Based on the proposed SIMWG operator, the aggregated result is still an IMPR, that is, a collective IMPR can be provided which is convenient to investigate the consensus of a group in group decision making; (2) The modified individual IMPRs can accord with the principle of modification, that is, the improved IMPRs not only satisfy the acceptability requirement but also preserve the initial preference information as much as possible. Next, an example is used to illustrate the provided method. Example 5.2. Suppose that there is a group decision making problem involving the evaluation of four branch offices X = {y1 , y2 , y3 , y4 } in a company. An expert group is formed which consists of three DMs ek (k = 1, 2, 3) (whose weight vector is λ = (0.4, 0.3, 0.3) from each strategic decision area. These DMs ek (k = 1, 2, 3) provide their IMPRs (k) (k) A = αij , k = 1, 2, 3 over alternatives n×n yi (i = 1, 2, 3, 4), respectively, as follows: ⎞ ⎛ 1 5 1 (1, 1) 2, 25 , , 2 4 3 2 ⎜ ⎟ ⎜ 2 8 1 ⎟ ⎟ ⎜ 5 , 2 (1, 1) 2, 13 , 7 3 ⎟ ⎜ ⎟, A(1) = ⎜ 1 ⎜ 5, 1 (1, 1) 5, 17 ⎟ ⎟ ⎜ 3 4 3, 2 ⎠ ⎝ 1 1 8 1 2, 2 , , 5 (1, 1) 3 7 7 ⎞ ⎛ 1 (1, 1) 21 , 65 1, 23 , 3 7 ⎜ ⎟ ⎟ ⎜ 6 1 1 1 2 ⎜ 5 , 2 (1, 1) 2, 5 ,3 ⎟ 4 ⎜ ⎟ A(2) = ⎜ ⎟, 1 3 ⎟ ⎜ 2, 1 (1, , 2 1) , 1 ⎟ ⎜ 3 5 4 ⎠ ⎝ 1 2 1 3 (1, 1) 1, 3, 7 , 3 4 4 ⎞ ⎛ 1 (1, 1) 13 , 23 1, 35 , 3 3 ⎜ ⎟ ⎜ 3 1 6 ⎟ ⎜ 2 , 3 (1, 1) 2, 25 1, 7 ⎟ ⎜ (3) ⎟ A =⎜ ⎟. 2 1 ⎜ 3, 1 (1, 1) 3 , 2 ⎟ ⎟ ⎜ 5 5, 2 ⎠ ⎝ 6 1 (1, 3, 13 1) , 1 2, 7 3 We give the following detailed procedure: Fig. 1. Procedure of the proposed method Step 1 Aggregate the individual IMPRs A(k) for k = 1, 2, 3 by the SIMWG operator (1) to obtain the collective IMPR A, we get the collective IMPR as follows: 5916 C.Y. Xu and Z.M. Ma / Symmetric intuitionistic multiplicative aggregation operator for group decision making ⎛ (1, 1) (0.749, 0.809) (0.530, 0.932) (0.259, 2.574) ⎜ (0.809, 0.7493) ⎜ ⎜ ⎝ (0.932, 0.530) (1, 1) (2, 0.292) (0.292, 2) (1, 1) ⎞ ⎟ ⎟. (1.446, 0.387) ⎠ (0.650, 0.531) ⎟ (2.574, 0.259) (0.531, 0.650) (0.387, 1.446) (1, 1) Step 2 Calculate the deviation measures (k) (k) D(A , A) between A and A by Eq. (2). We get D(A(1) , A) = 0.139 > 0.1, D(A(2) , A) = 0.1078 > 0.1, D(A(3) , A) = 0.1373 > 0.1, that is, A(1) ,A(2) and A(3) are needed to be modified, go to next step. Step 3 Compute model (4), and we denote the individual IMPRs A(k) as B(k) ,k = 1, 2, 3 and collective IMPR as B by modifying their non-memberships as follows: 1 ⎞ ,2 2 ⎜ (0.498, 2) (1, 1) (2, 0.329) 8 , 0.749 ⎟ 7 ⎜ ⎟, B(1) = ⎜ ⎟ ⎝ 0.91, 41 (0.329, 2) (1, 1) (5, 0.182) ⎠ 1 8 ⎛ (1, 1) 2, 2 ⎛ 0.749, 7 (1, 1) 1 (2, 0.498) 1 4 , 0.91 (0.182, 5) , 1.158 (1, 0.889) ⎛ 2.513, 7 0.667, 4 0.862, 4 7 , 2.513 (1, 1) (1, 1) 1 , 1.5 (1, 0.861) 1 ⎞ 3, 3 (0.847, 1) 0.142, 3 (1, 1) ,3 ⎜ 1.5, 1 (1, 1) (2, 0.4) (1, 0.847) ⎟ 3 ⎜ B(3) = ⎜ 1 ⎟ ⎟, (1, 1) , 0.142 ⎠ ⎝ (0.861, 1) (0.4, 2) 3 1 1 3 3 B= ⎛ (1, 1) ⎜ (0.886, 0.749) ⎜ ⎝(0.889, 0.530) (0.749, 0.886) (0.531, 0.889) (0.259, 2.434) (1, 1) (2, 0.329) (0.329, 2) (1, 1) (2.434, 0.259) (0.750, 0.650) (0.217, 1.446) Here, we make a comparative analysis with another symmetrical intuitionistic multiplicative aggregation operator, that is, the IMWG operator proposed in Ref. [27]. A procedure is given as follows: Step 1 Aggregate the individual IMPRs A(k) for k = 1, 2, 3 by the IMWG operator s s ω ω ρ (l)i , σ (l)i ⎞ (0.650, 0.0.750) ⎟ ⎟. (1.446, 0.0.217) ⎠ (1, 1) Calculate the deviation measures (k) (k) and B by D(B , B) between B Eq.(2), we have D(B(1) , B) = 0.098 < 0.1, D(B(2) , B) = 0.099 < 0.1, D(B(3) , B) = 0.0968 < 0.1, that is, B(1) , B(2) ,B(3) and B(4) are of acceptable consensus, go to next step. Step 4 Utilize the SIMWG operator inEq. (1) with 1 1 1 weight vector w = 3 , 3 , 3 to obtain the average value βi of the alterna- αij l=1 αij in Ref. [30] to obtain the collective IMPR A, we get the collective IMPR ⎞ ⎛ ⎞ 1 ⎜ 1.158, 1 (1, 1) (2, 0.276) , 0.667 ⎟ 2 4 ⎜ (2) B =⎜ 3 ⎟ ⎟, (1, 1) , 0.862 ⎝ (0.889, 1) (0.276, 2) ⎠ 4 1 1 3 2 6. A comparative analysis l=1 (1, 1) 1 tive yi , we get β1 = (0.755,0.931) , β2 = (1.181,0.621), β3 = (0.663,0.955); Step 5 Calculate the score function, we have s(β1 ) = 0.811, s(β2 ) = 1.901, s(β3 ) = 0.695, that is, y2 y1 y3 . Thus, y2 is the best one. (1, 1) (0.771, 0.827) 0.574, 0.932 (0.304, 2.551) ⎜ (0.827, 0.7708) (1, 1) ⎜ ⎝ (0.932, 0.574) (0.302, 2.0) (2.0, 0.302) (0.696, 0.545) ⎟ (1, 1) ⎟. (1.256, 0.565) ⎠ (2.551, 0.304) (0.545, 0.696) (0.565, 1.256) (1, 1) Step 2 Calculate the deviation measures (k) (k) and A by D(A , A) between A Eq. (2), we have D(A(1) , A) = 0.146 > 0.1, D(A(2) , A) = 0.101 > 0.1, D(A(3) , A) = 0.1208 > 0.1, that is, A(1) ,A(2) and A(3) need to be modified, go to next step. Step 3 Compute model (4), and we denote the individual IMPRs A(k) as B(k) ,k = 1, 2, 3 and the collective IMPR as B by modifying their nonmemberships as follows: ⎛ (1, 1) 1 (2, 0.5) ⎜ (0.5, 2) (1, 1) ⎜ 1 ⎝ 0.805, 4 (0.283, 2) 1 8 B(1) = ⎜ ⎛ B (2) 2, 2 0.755, 7 1 ⎞ ,2 2 8 (2, 0.283) , 0.755 ⎟ 7 ⎟, ⎟ (1, 1) (5, 0.2) ⎠ 4 , 0.805 (0.2, 5) (1, 1) 1 , 0.881 (1, 0.694) 2.755, 7 0.667, 4 0.246, 4 (1, 1) 1 ⎞ , 2.755 1 ⎜ 0.881, 1 (1, 1) (2, 0.2) , 0.667 ⎟ 2 4 ⎜ =⎜ 3 ⎟ ⎟, (1, 1) , 0.246 ⎠ ⎝ (0.694, 1) (0.2, 2) 4 1 1 3 2 7 (1, 1) C.Y. Xu and Z.M. Ma / Symmetric intuitionistic multiplicative aggregation operator for group decision making ⎛ (1, 1) ⎜ 1.367, 1 3 ⎜ (3) B =⎜ ⎝ (0.934, 1) 1 ⎛ 1 , 1.367 (1, 0.934) (1, 1) (2, 0.4) (0.4, 2) (1, 1) 3 0.554, 13 1 3 , 2.540 ⎞ (1, 0.855) ⎟ ⎟ 1 3 , 0.554 ⎟, ⎠ 2.540, 3 (0.855, 1) (1, 1) (0.771, 0.802) (0.574, 0.805) (0.304, 2.365) ⎜ (0.802, 0.771) (1, 1) ⎝ (0.805, 0.574) (0.283, 2.0) B=⎜ (1, 1) ⎞ 5917 the future, we will focus on introducing the consistency of IMPRs based on the symmetrical operations in this paper and providing the method of checking and reaching consistency of IMPRs and consensus in GDM problems, simultaneously. (2.0, 0.283) (0.696, 0.755) ⎟ (1, 1) (2.365, 0.304) (0.755, 0.696) (0.289, 1.256) ⎟ (1.256, 0.289) ⎠ Acknowledgements (1, 1) Step 4 Utilize the IMWG operator to obtain the average value αi of the alternative yi , we have β1 = (0.783, 0.877) , β2 = (1.127, 0.617) ,β3 = (0.628, 0.986); Step 5 Calculate the score function, we have s(β1 ) = 0.893, s(β2 ) = 1.826, s(β3 ) = 0.637, that is, y2 y1 y3 . Thus, y2 is the best one. The result is the same as the ones obtained by the GDM method proposed in this paper, and this explains the validity of our GDM method. Furthermore, although the IMWG operator is more simple than the proposed one, the operation used to induce the IMWG operator in Ref. [30] is not always closed, thus it is not suitable for making a further study about the related results of IMPRs such as their consistency which is to be studied in the future. The authors would like to thank the editors and the anonymous reviewers for their insightful and constructive comments and suggestions that have led to this improved version of the paper. This research was supported by the NSF of Shandong Province (No. ZR2017MG027) National Training Program of Innovation and Entrepreneurship for Undergraduates (No. 201710452004). References [1] [2] [3] 7. Conclusions The present paper focused on the aggregations of IMPRs in the GDM problem including the following aspects: [4] (1) Symmetric operations and SIMWG operator were proposed to aggregate the individual IMPRs into a collective one which is convenient to investigate the consensus of a group in group decision making; (2) Associated with the proposed aggregation operator, a novel method of checking and reaching consensus is provided, which is simple and accords with the principle of modification, that is, the method can preserve the initial preference information as much as possible. 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Pedrycz, Intuitionistic multiplicative group analytic hierarchy process, its use in multicriteria group decision-making, IEEE Transactions on Cybernetics 48 (2018), 1950–1962. Linear and Multilinear Algebra ISSN: 0308-1087 (Print) 1563-5139 (Online) Journal homepage: http://www.tandfonline.com/loi/glma20 Inversion of conjugate-Toeplitz matrices and conjugate-Hankel matrices Zhaolin Jiang, Tin-Yau Tam & Yanfeng Wang To cite this article: Zhaolin Jiang, Tin-Yau Tam & Yanfeng Wang (2016): Inversion of conjugateToeplitz matrices and conjugate-Hankel matrices, Linear and Multilinear Algebra, DOI: 10.1080/03081087.2016.1182465 To link to this article: http://dx.doi.org/10.1080/03081087.2016.1182465 Published online: 11 May 2016. Submit your article to this journal Article views: 4 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=glma20 Download by: [Laurentian University] Date: 23 May 2016, At: 10:24 LINEAR AND MULTILINEAR ALGEBRA, 2016 http://dx.doi.org/10.1080/03081087.2016.1182465 Inversion of conjugate-Toeplitz matrices and conjugate-Hankel matrices Zhaolin Jianga , Tin-Yau Tamb and Yanfeng Wanga a Department of Mathematics, Linyi University, Linyi, P.R. China; b Department of Mathematics and Statistics, Downloaded by [Laurentian University] at 10:24 23 May 2016 Auburn University, Auburn, AL, USA ABSTRACT ARTICLE HISTORY The inverses of conjugate-Toeplitz (CT) and conjugate-Hankel (CH) matrices can be expressed by the Gohberg–Heinig type formula. We obtain an explicit inverse formula of CT matrix. Similarly, the formula and the decomposition of the inverse of a CH matrix are provided. Also the stability of the inverse formulas of CT and CH matrices are discussed. Examples are provided to verify the feasibility of the algorithms. Received 4 September 2015 Accepted 21 April 2016 KEYWORDS Conjugate-Toeplitz matrices; conjugate-Hankel matrices; inversion; stability COMMUNICATED BY C.-K. Li AMS SUBJECT CLASSIFICATIONS 15A09; 15A15; 15A69; 65F05 1. Introduction The family of Toeplitz matrices has important applications in various disciplines including image processing and signal processing. The inverse of an invertible Toeplitz matrix can be constructed using the last columns and entries of the original Toeplitz matrix. It was ﬁrst proposed by Trench [1] and constructed by Gohberg and Semencul [2] from the ﬁrst and last columns of T −1 , given that the ﬁrst component in the ﬁrst column is not zero. Labahn [3] obtained the formulas for the inverse of layered or striped Toeplitz matrices in terms of solutions of standard equations. In [4], the algorithm of Trench for the inversion of Toeplitz matrices is given with proof. Heinig and Rost [5] gave an inversion formula for every invertible Toeplitz matrix. The method gives the desired solution of fundamental equations, where the right-hand side of one of the equations is a shifted column of the Toeplitz matrix. Gohberg and Krupnik [6] devised that if the last entry of the ﬁrst column is nonzero, then T −1 can be recovered from its ﬁrst and second columns. Labahn and Ng modiﬁed this result in [7] and [8]. The Toeplitz matrix inverse was given in the form of Toeplitz Bezoutian of two columns in [9]. Lv and Huang [10] found a new Toeplitz matrix inversion formula in which the inverse can be expressed as a sum of products of circulant matrices and upper triangular Toeplitz matrices. The inverse was constructed using three columns of T −1 in [11]. In [12] and [13], the stability of the algorithms emerging from Toeplitz matrix inversion formulas is discussed. CONTACT Zhaolin Jiang jzh1208@sina.com; Yanfeng Wang © 2016 Informa UK Limited, trading as Taylor & Francis Group wyflyu@163.com Downloaded by [Laurentian University] at 10:24 23 May 2016 2 Z. JIANG ET AL. In [14,15], the authors introduced a generalization of Toeplitz matrices, called conjugate-Toeplitz (CT) matrices, and showed that certain properties of Toeplitz matrices could be extended to CT matrices. Gover and Barnett [16] introduced a corresponding algorithm for any strongly nonsingular CT matrix (i.e. all of its leading principal minors are nonzero). Algorithms for inverting CT matrices and solving CT systems of equations, using O(n2 ) ﬂops for matrices of order n, were given. The purpose of [17] is to show that some known properties of Toeplitz and CT matrices can be extended to rT matrices. In [18], an expression of the inverse of a conjugate-Toeplitz matrix is obtained. The necessary conditions of applying the generalized Trench algorithm for CT matrices are discussed. It is shown that there exist strongly invertible CT matrices for which the algorithm may not be applied. We propose two algorithms for computing the inverses of CT and CH matrices and their stability are discussed. Let c(x) = x̄ denote the complex conjugate of x. The following deﬁnitions can be found in [14,16]. Definition 1.1: An n × n matrix TC,n = [tij ] is CT if ti+1,j+1 = c(tij ) for all i, j, that is, ⎡ ⎢ ⎢ ⎢ TC,n = ⎢ ⎢ ⎣ t1 c(t0 ) c 2 (t−1 ) .. . t0 c(t−1 ) c 2 (t−2 ) .. . t2 c(t1 ) c 2 (t0 ) .. . ⎤ · · · tn−1 · · · c(tn−2 ) ⎥ ⎥ · · · c 2 (tn−3 ) ⎥ ⎥. ⎥ .. .. ⎦ . . c n−1 (t−n+1 ) c n−1 (t−n+2 ) c n−1 (t−n+3 ) · · · c n−1 (t0 ) An n × n matrix HC,n = [aij ] is CH if ai+1,j = c(ai,j+1 ) for all i, j, that is, ⎡ ⎢ ⎢ ⎢ HC,n = ⎢ ⎢ ⎣ a0 c(a1 ) c 2 (a2 ) .. . ··· ··· ··· . .. a1 c(a2 ) c 2 (a3 ) .. . an−2 c(an−1 ) c 2 (an ) .. . an−1 c(an ) c 2 (an+1 ) .. . ⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎦ c n−1 (an−1 ) c n−1 (an ) · · · c n−1 (a2n−3 ) c n−1 (a2n−2 ) 2. The inverse formula of CT matrix We provide inversion formula for CT matrix as a diﬀerence of two products of lower triangular CT, diagonal CT and upper triangular CT matrices, by taking advantage of the CT structure. Theorem 2.1: Let TC,n and TC,n−1 be invertible. Let x= z= x0 x1 . . . xn−1 T z−n+1 z−n+2 . . . z0 T y0 y−1 . . . y−n+1 , , w = wn−1 wn−2 . . . w0 ,y= (1) (2) −1 be the first column, the last column, the first row and the last row of TC,n , respectively. Then −1 −1 x0 = y0 , z0 = w0 , x0 z0 = 0. The inverses of TC,n and TC,n−1 are given by the following formulas: LINEAR AND MULTILINEAR ALGEBRA ⎡ ⎢ ⎢ −1 TC,n =⎢ ⎣ 0 c(x0 ) .. . x0 x1 .. . ... ... .. . 3 ⎡ ⎤ ⎤ y0 y−1 · · · y−n+1 ⎢ 0 c(y0 ) · · · c(y−n+2 ) ⎥ ⎥ ⎢ ⎥ ⎥ ⎥ ϒn ⎢ .. ⎥ . . .. .. ⎣ . ⎦ ⎦ 0 0 xn−1 c(xn−2 ) . . . c n−1 (x0 ) 0 0 c n−1 (y0 ) ⎤ ⎡ ⎤ ⎡ c(w1 ) 0 c(wn−1 ) · · · 0 ... 0 0 ⎥ ⎢ .. ⎥ ⎢ .. .. .. ⎢ . ⎥ ⎢ c(z−n+1 ) . . . . . 0⎥ . ⎥ ⎢ ⎥ ⎢ −⎢ n ⎢ ⎥ .. ⎥ .. . . . . . . n−1 ⎣0 ⎣ . . . c (wn−1 ) ⎦ . ⎦ . c(z−1 ) . . . c n−1 (z−n+1 ) 0 0 0 ··· 0 (3) Downloaded by [Laurentian University] at 10:24 23 May 2016 and ⎡ ⎢ ⎢ −1 =⎢ TC,n−1 ⎣ c(x0 ) c(x1 ) .. . 0 2 c (x0 ) .. . ... ... .. . 0 0 ⎡ ⎤ ⎤ c(y0 ) c(y−1 ) · · · c(y−n+2 ) ⎢ 0 ⎥ c 2 (y0 ) · · · c 2 (y−n+3 ) ⎥ ⎢ ⎥ ⎥ ⎥ ϒn−1 ⎢ .. ⎥ .. .. ⎣ . ⎦ ⎦ . . c(xn−2 ) c 2 (xn−3 ) . . . c n−1 (x0 ) 0 0 ⎤ ⎡ ⎡ 0 c(wn−1 ) · · · c(z−n+1 ) . . . ⎥ ⎢ ⎢ . . . .. .. .. .. −⎣ ⎦ n−1 ⎣ . c(z−1 ) . . . c n−1 (z−n+1 ) 0 c n−1 (y0 ) ⎤ c(w1 ) ⎥ .. ⎦, . c n−1 (wn−1 ) (4) where ϒn = diag (, c(), . . . , c n−1 ()), ϒn−1 = diag (c(), c 2 (), . . . , c n−1 ()) n = diag (c(∇), c 2 (∇), . . . , c n (∇)), n−1 = diag (c(∇), c 2 (∇), . . . , c n−1 (∇)) −1 M = 0, = t0 − R(c(TC,n−1 ))−1 Q = 0, ∇ = c n−1 (t0 ) − NTC,n−1 T R = t1 t2 · · · tn−1 , Q = c(t−1 ) c 2 (t−2 ) · · · c n−1 (t−n+1 ) , N = c n−1 (t−n+1 ) c n−1 (t−n+2 ) · · · c n−1 (t−1 ) , M = tn−1 c(tn−2 ) · · · c n−2 (t1 ) T . Proof: Partition TC,n as a block matrix in two ways: TC,n = t0 R Q c(TC,n−1 ) TC,n = TC,n−1 M n−1 N c (t0 ) , where R and N are 1 × (n − 1) matrices and Q and M are (n − 1) × 1 matrices. Since TC,n−1 is invertible, we have the following UL and LU factorizations according to the two partitions: TC,n = 1 R 0 c(TC,n−1 ) TC,n = TC,n−1 0 N 1 0 0 (c(TC,n−1 ))−1 −1 0 TC,n−1 0 ∇ 1 0 Q c(TC,n−1 ) TC,n−1 M . 0 1 (5) (6) 4 Z. JIANG ET AL. Equating the (1, 1) entries in (5) we have t0 = + R(c(TC,n−1 ))−1 Q, and equating the (n, n) entries in (6) we have −1 c n−1 (t0 ) = N(TC,n−1 )M + ∇. Downloaded by [Laurentian University] at 10:24 23 May 2016 By the determinant consideration, we conclude that , ∇ = 0 from (5) and (6) since TC,n −1 is nonsingular. The (1, 1) entry of TC,n is (t0 − R(c(TC,n−1 ))−1 Q)−1 = −1 . Similarly ∇ −1 −1 is the (n, n) entry of TC,n . Thus, x0 = y0 = −1 = (t0 − R(c(TC,n−1 ))−1 Q)−1 , (7) −1 )M)−1 . z0 = w0 = ∇ −1 = (c n−1 (t0 ) − N(TC,n−1 (8) In particular, x0 , z0 = 0. Taking inverse of (5), we have −1 TC,n = = 1 −(c(Tc,n−1 ))−1 Q 0 (c(Tc,n−1 −1 −(c(TC,n−1 ))−1 Q−1 0 0 (c(TC,n−1 ))−1 0 In−1 1 −R(c(TC,n−1 ))−1 0 (c(TC,n−1 ))−1 −1 0 0 c(TC,n−1 ) ))−1 −1 −−1 R(c(TC,n−1 ))−1 , 0 In−1 where In−1 denotes the (n − 1) × (n − 1) identity matrix. By the deﬁnitions of x, y in (1), we have ⎤ ⎡ ⎡ ⎤ x0 0 · · · 0 y0 y−1 · · · y−n+1 ⎥ ⎢ x1 ⎢0 ⎥ 0 ⎥ ⎢ ⎢ ⎥ −1 TC,n =⎢ . ⎥. ⎢ ⎥ .. −1 )) 0 (c(T ⎦ ⎣ .. ⎣ ⎦ C,n−1 I . I n−1 n−1 xn−1 0 Similarly, from (6) −1 = TC,n −1 M∇ −1 In−1 −TC,n−1 −1 0 ∇ −1 0 TC,n−1 0 ∇ and by the deﬁnitions of z and w in (2), we have ⎡ ⎤ z−n+1 ⎢ .. ⎥ T −1 0 ⎢ In−1 −1 . ⎥ TC,n = ⎢ ⎥ C,n−1 0 ∇ ⎣ z−1 ⎦ 0 0 ··· z0 Thus, −1 = xy + TC,n 0 In−1 and −1 TC,n = z∇w + In−1 0 −1 ∇ −1 −∇ −1 NTC,n−1 ⎡ ⎢ ⎢ ⎢ ⎣ In−1 wn−1 wn−2 c(TC,n−1 )−1 0 In−1 , In−1 −1 TC,n−1 In−1 0 , 0 ⎤ 0 .. ⎥ . ⎥ ⎥. 0⎦ · · · w0 (9) (10) LINEAR AND MULTILINEAR ALGEBRA 5 where x0 T , x = x0 x1 . . . xn−1 = −(c(TC,n−1 ))−1 Q−1 y = y0 y−1 . . . y−n+1 = y0 −−1 R(c(TC,n−1 ))−1 , −1 M∇ −1 −TC,n−1 T z = z−n+1 z−n+2 . . . z0 = , z0 −1 w = wn−1 wn−2 . . . w0 = −∇ −1 NTC,n−1 w0 . (11) (12) (13) (14) Downloaded by [Laurentian University] at 10:24 23 May 2016 Now let ⎡ 0 ··· ··· ⎢ .. ⎢1 . =⎢ ⎢ .. .. ⎣ . . 0 ··· 1 ⎤ ⎡ 0 1 ··· 0 ⎥ ⎥ ⎢ .. . . . . ⎥ ⎢. . . 0⎥ T ⎥. ⎥, = ⎢ ⎥ ⎢ .. .. ⎥ .. ⎦ ⎣ . 1⎦ . . 0 ··· ··· 0 0 0 ⎤ be the forward and backward shifts, respectively. Note that In−1 0 = , 0 In−1 In−1 0 T = 0 In−1 . (15) Since (c(A))−1 = c(A−1 ) for all nonsingular A, these equations together with (9) and (10) yield the following formula: −1 −1 − c(TC,n ) T = xy − c(z∇w) T . TC,n Since n = 0, we get −1 = TC,n n−1 n−1 k k k T k c k (z∇w)( k )T . c (xy)( ) − k=0 k=1 Rewrite it as a matrix product: −1 = x TC,n c(x) · · · ⎡ n−1 c n−1 (x) ⎢ ⎢ ϒn ⎢ ⎣ y c(y) T .. . ⎤ ⎥ ⎥ ⎥ ⎦ c n−1 (y)( n−1 )T ⎤ c(w) T ⎢ c 2 (w)( 2 )T ⎥ ⎢ ⎥ 2 2 n n c (z) · · · c (z) c(z) n ⎢ − ⎥. .. ⎣ ⎦ . ⎡ c n (w)( n )T This is precisely (3) and (4) follows as TC,n and TC,n−1 have the same form. 6 Z. JIANG ET AL. We remark that the inverses in (3) and (4) are expressed in terms of the ﬁrst and last columns of TC,n and TC,n−1 . This provides an algorithm to compute the inverse of TC,n and we show an example in Section 4. 3. The inverse formula of CH matrix We provide inversion formula for CH matrix by taking advantage of the CH structure and it leads to an algorithm for the computation of the inverse. Theorem 3.1: Let HC,n and HC,n−1 be invertible. Let Downloaded by [Laurentian University] at 10:24 23 May 2016 α= δ= α0 α1 . . . αn−1 T δ−n+1 δ−n+2 . . . δ0 T β−n+1 β−n+2 . . . β0 , , γ = γ0 γ1 . . . γn−1 ,β= (16) (17) −1 , respectively. Then be the last column, the first column, the first row and the last row of HC,n −1 −1 α0 = β0 , γ0 = δ0 , with α0 , γ0 = 0, and the inverses of HC,n and HC,n−1 are given by the following formulas: ⎤ β−n+1 · · · β−1 β0 ⎥ ⎢ ⎢ c(β−n+2 ) · · · c(β0 ) 0 ⎥ ⎥ ⎥ ⎢ ⎢ −1 =⎢ HC,n ⎥ ⎢ .. ⎥ .. n . . . . ⎦ ⎣ ⎣ . . . ⎦ . n−1 n−1 αn−1 c(αn−2 ) . . . c (α0 ) c (β0 ) 0 0 ⎡ ⎤ ⎡ ⎤ 0 0 ... 0 · · · c(γn−1 ) 0 c(γ1 ) ⎢ ⎥ ⎢ .. ⎥ .. .. .. ⎢ c(δ−n+1 ) . . . ⎥ ⎢ . . 0 . ⎥ . ⎥ n⎢ ⎥ (18) −⎢ ⎢ ⎢ n−1 ⎥ .. ⎥ .. . . . . . . ⎣ ⎣ c (γn−1 ) . . . . ⎦ . 0⎦ c(δ−1 ) . . . c n−1 (δ−n+1 ) 0 0 ··· 0 0 ⎡ and α0 α1 .. . 0 c(α0 ) .. . ... 0 0 ⎤ ⎡ ⎤ ⎤ ⎡ β−n+2 · · · β−1 β0 α0 0 ... 0 ⎥ ⎢ c(β−n+3 ) · · · c(β0 ) 0 ⎥ ⎢ α1 c(α0 ) 0 ⎥ ⎥ ⎢ ⎢ −1 HC,n−1 =⎢ . ⎥ .. .. ⎥ . n−1 ⎢ . .. . .. ⎦ ⎣ ⎣ .. . . . . ⎦ n−2 n−2 (α0 ) (β0 ) 0 0 αn−2 c(αn−3 ) . . . c c ⎡ ⎤ ⎤ ⎡ δ−n+1 γ1 0 ... 0 · · · γn−2 γn−1 ⎢ δ−n+2 c(δ−n+1 ) ⎥ ⎢ 0 c(γ2 ) · · · c(γn−1 ) 0 ⎥ ⎢ ⎥ ⎥ ⎢ −⎢ ⎥ ⎢ .. . .. ⎥ , .. n−1 . . . . . ⎣ ⎦ ⎣ . . . . . ⎦ . n−2 n−2 c(δ−2 ) . . . c (δ−n+1 ) (γn−1 ) 0 0 δ−1 c ⎡ (19) where n = diag ( , c( ) . . . , c n−1 ( )), n = diag (c( ), c 2 ( ) . . . , c n ( ), = an−1 − P(c(HC,n−1 ))−1 U, n−1 = diag ( , c( ), . . . , c n−2 ( )), n−1 = diag ( , c( ), . . . , c −1 = c n−1 (an−1 ) − GHC,n−1 W, n−2 ( )), LINEAR AND MULTILINEAR ALGEBRA T c(an ) c 2 (an+1 ) · · · c n−1 (a2n−2 ) , T G = c n−1 (an ) c n−1 (an+1 ) · · · c n−1 (a2n−2 ) , W = a0 c(a1 ) · · · c n−2 (an−2 ) . P= a0 a1 . . . an−2 , U = 7 Proof: The Hankel structure of HC,n allows us to represent HC,n in two ways as a 2 × 2 block matrix, namely P an−1 c(HC,n−1 ) U HC,n = , HC,n = W c n−1 (a HC,n−1 G n−1 ) . Downloaded by [Laurentian University] at 10:24 23 May 2016 Here P and G are 1 × (n − 1) matrices and U and W are (n − 1) × 1 matrices. Since HC,n−1 is invertible, we have the following factorizations of HC,n : HC,n = 1 c(HC,n−1 ) 0 HC,n = 0 HC,n−1 1 G 0 (c(HC,n−1 ))−1 0 P 0 −1 HC,n−1 0 1 c(HC,n−1 ) U W HC,n−1 1 0 0 , (20) . (21) Equating the (1, n) entries in (20) we have an−1 = + P(c(HC,n−1 ))−1 U, and equating the (n, 1) entries in (21) we have −1 )W + c n−1 (an−1 ) = G(HC,n−1 . From (20), (21), and the invertibility of HC,n−1 , we conclude that , = 0 are invertible. Furthermore, by taking inverses in (20) and (21), one sees that −1 is the (1, n) entry of −1 −1 and −1 is the (n, 1) entry of HC,n . Thus, HC,n γ0 = δ0 = −1 = (an−1 − P(c(HC,n−1 ))−1 U)−1 . (22) α0 = β 0 = −1 −1 = (c n−1 (an−1 ) − G(HC,n−1 )W)−1 , (23) In particular, α0 , γ0 = 0. Taking inverses of (20) and (21) yields −(c(HC,n−1 ))−1 U −1 = HC,n 0 × −1 HC,n = −1 −1 −1 0 In−1 − In−1 −1 P(c(H −1 C,n−1 )) −1 −1 −HC,n−1 W −1 0 (c(HC,n−1 ))−1 In−1 0 0 0 , −1 HC,n−1 0 − −1 GH −1 C,n−1 In−1 −1 0 . 8 Z. JIANG ET AL. By the deﬁnitions of α, β, δ and γ in (16) and (17), we have ⎡ ⎤ ⎤ 0 δ−n+1 ⎢ .. ⎥ ⎢ .. ⎥ 0 ⎢ . ⎥ ⎢ ⎥ −1 In−1 In−1 =⎢ . HC,n ⎢ ⎥ ⎥, −1 0 ⎣ 0 ⎦ (c(HC,n−1 )) ⎣ δ−1 ⎦ γ0 γ1 · · · γn−1 δ0 0 ··· 0 ⎤ ⎡ ⎡ ⎤ 0 · · · 0 α0 β−n+1 · · · β−1 β0 ⎢ ⎢ −1 α1 ⎥ 0 ⎥ ⎥ ⎢ ⎢ ⎥ 0 HC,n−1 −1 HC,n =⎢ ⎢ .. ⎥ .. ⎥ . 0 ⎣ In−1 ⎣ In−1 . ⎦ . ⎦ αn−1 0 ⎡ Downloaded by [Laurentian University] at 10:24 23 May 2016 Thus, −1 HC,n =δ γ + In−1 0 −1 =α β+ HC,n 0 In−1 (c(HC,n−1 ))−1 0 In−1 , (24) −1 In−1 0 , HC,n−1 (25) where α0 , −1 −HC,n−1 W −1 −1 β = β−n+1 β−n+2 . . . β0 = − −1 GHC,n−1 β0 , −1 −(c(HC,n−1 )) U −1 T = δ = δ−n+1 δ−n+2 . . . δ0 δ0 −1 P(c(HC,n−1 ))−1 . γ = γ0 γ1 . . . γn−1 = γ0 − α= α0 α1 . . . αn−1 T = (26) (27) , (28) (29) By (15), (24), and (25) we have the following formula: −1 −1 − c(HC,n ) = α β − c(δ γ ). HC,n −1 = Since n = 0, we get HC,n matrix product: n−1 k k k=0 c (α β) k − n−1 k k k=1 c (δ ⎡ −1 = α c(α) · · · n−1 c n−1 (α) HC,n ⎢ ⎢ n⎢ ⎣ γ ) k . Rewrite it as a β c(β) .. . ⎤ ⎥ ⎥ ⎥ ⎦ c n−1 (β) n−1 ⎤ c(γ ) ⎢ c 2 (γ ) 2 ⎥ ⎢ ⎥ ⎥. .. n⎢ ⎣ ⎦ . ⎡ − c(δ) 2 c 2 (δ) · · · n c n (δ) c n (γ ) n This is precisely (18) and (19) follows as HC,n and HC,n−1 have the same form. LINEAR AND MULTILINEAR ALGEBRA 9 4. Stability analysis We will analyse the stability of the inversion formulas given in Theorem 2.1 and Theorem 3.1. We present the error analysis of the explicit inversion formulas for CT and CH matrices in terms of vector norm · and its induced matrix norm · (there is no confusion in the context even though we use the same notation; see [19, p.292] for the deﬁnition of induced matrix norm). In particular, it is true for the 1-norm, ∞-norm and 2-norm, respectively. Theorem 4.1: Let > 0 and let x̂, ŷ, ẑ, ŵ be the corresponding numerical least squares −1 solutions of the linear systems for deriving (3). Denote by T̂C,n the inverse of T̂C,n . If x̂ − x ≤ x, ŷ − y ≤ y, ẑ − z ≤ z, ŵ − w ≤ w, Downloaded by [Laurentian University] at 10:24 23 May 2016 where · is a vector norm on Rn , then for any matrix norm · induced by the vector norm · , we have −1 −1 TC,n − T̂C,n ≤ ( 3 + 3 2 + 3)(||xy + |∇|zw). (30) −1 Proof: Rewrite the inverse formula for TC,n in (3) as TC,n = Lx ϒn Ry − Lz n Rw . Thus, −1 −1 ˆ n R̂w ι1 + ι2 . TC,n − T̂C,n ≤ Lx ϒn Ry − L̂x ϒ̂n R̂y + Lz n Rw − L̂z For the ﬁrst term ι1 , we have ι1 ≤ Lx ϒn Ry − L̂x ϒn Ry + L̂x ϒn Ry − L̂x ϒ̂n R̂y ≤ Lx − L̂x ϒn Ry + L̂x ϒn Ry − ϒ̂n R̂y ≤ xϒn Ry + L̂x − Lx + Lx ϒn Ry − ϒ̂n R̂y ≤ x||y + (1 + )xϒn Ry − ϒ̂n Ry + ϒ̂n Ry − ϒ̂n R̂y ≤ x||y + (1 + )x(||y + (1 + )||y) = ||( 3 + 3 2 + 3)xy. Similarly, ι2 ≤ |∇|( 3 + 3 2 + 3)zw. We have the desired result by summing the above two inequalities. Remark 1: Under the assumptions and notations of Theorem 4.1, we have: −1 −1 − T̂C,n ∞ ≤ ( 3 + 3 2 + 3)(||x∞ y∞ + |∇|z∞ w∞ ) TC,n (31) −1 −1 TC,n − T̂C,n 1 ≤ ( 3 + 3 2 + 3)(||x1 y1 + |∇|z1 w1 ) (32) A22 ≤ A1 A∞ (33) As for any square matrix A [19, p.313], we have from (30)–(32) that −1 −1 − T̂C,n 2 ≤ ( 3 + 3 2 + 3)(n||x2 y2 + (n − 1)|∇|z2 w2 ). TC,n (34) √ √ √ √ as x1 ≤ nx2 , y1 ≤ ny2 , z1 ≤ n − 1z2 , w1 ≤ n − 1w2 [19, p.279]. The upper bound for the 2-norm depends on n and it is much bigger when n is 10 Z. JIANG ET AL. large. Diﬀerent bounds suggest that the error analysis performed in terms of the 1-norm or ∞-norm can reveal the stability more precisely. Therefore, the formula presented in Theorem 2.1 is forward stable. Theorem 4.2: Let > 0 and let α̂, β̂, γ̂ , δ̂ be the corresponding numerical least squares −1 the inverse of solutions of the linear systems for deriving the formula (18). Denote by ĤC,n ĤC,n . If α̂ − α ≤ α, β̂ − β ≤ β, γ̂ − γ ≤ γ , δ̂ − δ ≤ δ, then for any matrix norm · induced by the vector norm · Downloaded by [Laurentian University] at 10:24 23 May 2016 −1 −1 HC,n − ĤC,n ≤ ( 3 + 3 2 + 3)(| |αβ + | |γ δ). −1 = Lα Proof: Rewrite the inverse formula for HC,n as HC,n −1 −1 HC,n − ĤC,n ≤ Lα n Rβ − L̂α ˆ n Rˆβ + Lδ n Rβ − Lδ (35) n Rγ . We obtain n Rγ − L̂δ ˆ n R̂γ κ1 + κ2 . For the ﬁrst term κ1 , we have κ1 ≤ Lα n Rβ − L̂α ≤ Lα − L̂α ≤ α n Rβ + L̂α n Rβ + L̂α n Rβ − L̂α ˆ n Rˆβ n Rβ − ˆ n R̂β n Rβ + L̂α − Lα + Lα ≤ α| |β + (1 + )α n Rβ − ˆ n R̂β n Rβ − ˆ n Rβ + ˆ n Rβ − ˆ n R̂β ≤ α| |β + (1 + )α(| |β + (1 + )| |β) = | |( 3 + 3 2 + 3)αβ. Similarly, κ2 ≤ | |( 3 + 3 2 + 3)γ δ. We obtain the desired result by summing the above two inequalities. Remark 2: Under the assumptions and notations of Theorem 4.2, we have: −1 −1 − ĤC,n 1 ≤ ( 3 + 3 2 + 3)(| |α1 β1 + | |γ 1 δ1 ) HC,n −1 −1 HC,n − ĤC,n ∞ ≤ ( 3 + 3 2 + 3)(| |α∞ β∞ + | |γ ∞ δ∞ ) (36) (37) Similar to the proof of (34) in Remark 1, we have from (33), (35)–(37) that −1 −1 HC,n − ĤC,n 2 ≤ ( 3 + 3 2 + 3)(n| |α2 β2 + (n − 1)| |γ 2 δ2 ). Therefore, (18) given in Theorem 3.1 is forward stable. 5. Examples We give two examples to demonstrate our main results. (38) LINEAR AND MULTILINEAR ALGEBRA 11 Example 5.1: Partition ⎡ i+1 −1+i ⎤ 2 −i 2 TC,3 = ⎣ 2+i 5 3+i 10 1−i 2 2−i 5 i 1+i 2 R t0 Q c(TC,2 ) ⎦= = TC,2 M N c 2 (t0 ) , where 1−i 2 2−i 5 2+i Q= 5 Downloaded by [Laurentian University] at 10:24 23 May 2016 c(TC,2 ) = i 1+i 2 3+i 10 1+i −i 2 , t0 = c 2 (t0 ) = 1+i 2+i 1−i 2 , R = −i 5 2 3+i 2−i T T , M = −1+i , N = 10 i . 5 2 , TC,2 = −1+i 2 Notice that TC,3 and TC,2 are invertible. Algorithm: Step 1 Compute (c(TC,2 ))−1 = 7i−1 5 −2i−4 5 −6i+8 5 i+7 5 −1 = , TC,2 −7i−1 5 2i−4 5 6i+8 5 −i+7 5 . Step 2 Find , ∇, x, y, z, w by (7), (8), and (11)–(14), respectively: x= = 1 3i 2 − 2 z = 52 − 5i2 1+3i 13−9i 5 , ∇ = 25 , 1 3i T , y = 12 − 3i2 − 35 + 5i 10 + 10 1 1 7i 13 9i T 3i , w = 10 + 10 5 − 5 10 + 10 −1 + 3i −1 − i 5 5i 2 − 2 13 9i 10 + 10 −1 by (3): Step 3 Compute TC,3 ⎡ ⎤ ⎡ 1+3i ⎤ 1 3i 0 0 0 0 2 − 2 5 −1 1 3i 1−3i TC,3 = ⎣ 5i − 35 0 ⎦⎣ 0 0 ⎦ 2 + 2 5 3i 1 i 3 1 3i 1+3i 0 0 10 + 10 − 5 − 5 2 − 2 5 ⎡ 1 3i 5 5i ⎤ − 3i − 1 − 2 2 2 2 1 3i ×⎣ 0 + −3i − 1⎦ 2 2 1 3i 0 0 2 − 2 ⎡ ⎤ ⎤ ⎡ 13+9i 0 0 0 0 0 25 13−9i 0 0 ⎦⎣ 0 − ⎣ 52 + 5i2 0 ⎦ 25 7i 1 5 5i 13+9i +5 2− 2 0 0 0 25 ⎤ ⎡ 5 3i 1 i−1 0 − 10 + 10 3i 1 ⎦ ×⎣ 0 0 10 + 10 ⎡ 0 1 3i 2 − 2 = ⎣ − 35 + 5i 1 3i 10 + 10 0 −1 + 3i 1+i −1 − i 0 5 5i ⎤ − 2 2 1 7i ⎦ . 5 − 5 13 9i 10 + 10 , . , 12 Z. JIANG ET AL. Example 5.2: Partition ⎡ HC,3 = ⎣ i+1 2 i −i −1+i 2 −1−i 2 −i 4 −1+i ⎤ 2 i ⎦= 4 i P a2 c(HC,2 ) U W HC,2 c 2 (a2 ) G = , where −1−i i −i −1+i 2 2 , HC,2 = −1−i −i i −1+i 2 4 2 4 −i i T P = 1+i , G = , U = −i i 2 4 4 Downloaded by [Laurentian University] at 10:24 23 May 2016 −1+i , 2 , a2 = c 2 (a2 ) = c(HC,2 ) = i ,W= 1+i 2 T i . Notice that HC,3 and HC,2 are invertible. Algorithm: Step 1 Compute (c(HC,2 ))−1 = Step 2 Find , i −2 − 2i −2 + 2i −4i −1 = , HC,2 −i −2 + 2i −2 − 2i 4i . , α, β, δ, γ by (22), (23), and (26)–(29), respectively: = = 33i−15 , 8 123+37i 44i+20 80−116i 8−216i T , β= α = − 219 − 92+144i − 44i+20 219 219 219 219 219 123+37i 92+144i 44i+20 116i−80 8−216i 44i+20 T δ= , γ = − 219 − 219 219 219 219 219 −1 Step 3 Compute HC,3 by (18): ⎡ − 44i+20 219 0 0 0 ⎤ ⎡ 33i−15 0 0 0 ⎤ 8 44i−20 −33i−15 ⎦⎣ 0 ⎦ 219 8 8−216i 44i+20 33i−15 80+116i 0 0 − 219 219 219 8 ⎡ 123+37i 92+144i 44i+20 ⎤ − − 219 219 219 44i−20 ⎦ × ⎣ − 92−144i 0 219 219 − 44i+20 0 0 219 ⎡ ⎤ ⎡ −33i−15 ⎤ 0 0 0 00 8 33i−15 ⎦ 0 0 ⎦⎣ − ⎣ 123−37i 0 0 219 8 92−144i 123+37i −33i−15 0 0 0 219 219 8 ⎤ ⎡ −116i−80 8+216i 0 219 219 × ⎣ 8−216i 0 0⎦ 219 −1 = ⎣ 80−116i HC,3 219 ⎡ 123+37i 219 = ⎣ 92+144i 219 − 20+44i 219 0 0 0 ⎤ − 92+144i − 20+44i 219 219 80−116i ⎦ . − 216+8i 219 219 −80+116i 219 8−216i 219 , . LINEAR AND MULTILINEAR ALGEBRA 13 Acknowledgements The authors are thankful to the referee for his/her careful reading of the manuscript. Disclosure statement No potential conﬂict of interest was reported by the authors. Funding The research was supported by the Development Project of Science & Technology of Shandong Province [grant number 2012GGX10115]; the AMEP of Linyi University, China. Downloaded by [Laurentian University] at 10:24 23 May 2016 References [1] Trench WF. An algorithm for the inversion of ﬁnite Toeplitz matrices. J. Soc. Indust. Appl. 1964;12:515–522. [2] Gohberg I, Semencul A. On the inversion of ﬁnite Toeplitz matrices and their continuous analogues. Mat. Issled. 1972;7:201–233. Russian. [3] Labahn G, Shalom T. Inversion of Toeplitz structured matrices using only standard equations. Linear Algebra Appl. 1994;207:49–70. [4] Zohar S. Toeplitz matrix inversion: the algorithm of W.F. Trench. J. Assoc. Comput. Mach. 16:592–610. [5] Heinig G, Rost K. 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Acta Mathematica Sinica, English Series Published online: December 16, 2016 DOI: 10.1007/s10114-016-5607-z Http://www.ActaMath.com Acta Mathematica Sinica, English Series © Springer-Verlag Berlin Heidelberg & The Editorial Office of AMS 2016 Norm Equalities and Inequalities for Three Circulant Operator Matrices Zhao Lin JIANG Yun Cheng QIAO1) Shu Dong WANG School of Sciences, Linyi University, Linyi 276005, P. R. China E-mail : jzh1208@sina.com lyuqyc@163.com 1743687469@qq.com Abstract In this paper, three new circulant operator matrices, scaled circulant operator matrices, diag-circulant operator matrices and retrocirculant operator matrices, are given respectively. Several norm equalities and inequalities for these operator matrices are proved. We show the special cases for norm equalities and inequalities, such as the usual operator norm and the Schatten p-norm. Pinching type inequality is also given for weakly unitarily invariant norms. These results are closely related to the nice structure of these special operator matrices. Furthermore, some special cases and specific examples are also considered. Keywords Operator matrix, norm, scaled circulant, diag-circulant, retrocirculant MR(2010) Subject Classification 1 15A60, 47A50, 47C05, 15B05 Introduction As is well-known, the circulant matrices [11, 27] play an important role in various ﬁelds, which are widely applied in image processing, signal processing, encoding, block ﬁltering design, graph theory, regular polygon solutions, control and system theory, network, Hermitean Hilbert transform and so on. Circulant graph is discussed as presented in [4, 13, 14, 17, 32, 37, 39, 43]. In [50], the classiﬁcation of primes of the doubly stochastic circulant matrices in control and system theory is explored. Guo and Huang [15] studied the existence and stability of periodic orbits in a ring network modelled by the delay diﬀerential equations. In [1], the authors estimated the Hölder norm of Hermitean Hilbert transform for Hölder continuous circulant (2 × 2) matrix functions. Zhang and Zhou [51] showed nested regular polygon solutions for planar 2N-body problems by using eigenvalues and eigenvectors of circulant matrices. Song et al. [44] researched shape feature description based on log-polar coordinates and symmetric circulant matrix. The authors [41, 52] studied low-density parity-check (LDPC) codes closed to cyclic structure. In [40], based on the properties of ω-circulant matrices, Narasimha showed that the linear convolution required in block ﬁltering can be decomposed into a sum of skew-cyclic convolutions. Such convolutions can be realized eﬃciently with half-length complex transforms when the signals are real. Received October 12, 2015, revised July 25, 2016, accepted September 19, 2016 Supported by National Natural Science Foundation of China (Grant No. 11671187) and the AMEP of Linyi University, China 1) Corresponding author 2 Jiang Z. L. et al. Lately, some mathematicians pay attention to the problems of norm estimates for operator matrices [6, 33], which are widely used in operator theory, mathematical physics, quantum information theory, and numerical analysis. In [28], several norm equalities and inequalities are proved for ω-circulant operator matrices. Jiang and Xu gave the special cases for norm equalities and inequalities, such as the usual operator norm and the Schatten p-norms. Pinching type inequality is also proposed for weakly unitarily invariant norms. Jiang and Hong [29] discussed the block imaginary circulant operator matrices and presented several norm equalities and inequalities for such matrices. In [46], the authors considered the norm equalities and inequalities for operator matrices. Based on the nice structures of circulant and skew-circulant operator matrices, they presented the pinching type inequalities for weakly unitarily invariant norms. Many works about norm equalities and inequalities of special operator matrices can be seen in [2, 7, 34, 35]. Some authors also focus on researching the norms of some circulant type matrices. For instance, Li et al. [38], gave four kinds of norms for circulant and left circulant matrix involving special numbers. Bose et al. [8] discussed the convergence in probability and the convergence in distribution of the spectral norms of scaled Toeplitz, circulant, reverse circulant, symmetric circulant and k-circulant matrices. Zhou and Jiang [53] proposed explicit formulae of spectral norms for g-circulant matrices. It is well known that the study of scaled circulant factor matrix is getting more and more actively. Jeﬀer and James [30] presented the scaled circulant factor matrix and gave the spectral set. In [20], the level-k scaled factor circulant matrix over any ﬁeld is introduced. Algorithms for ﬁnding the minimal polynomial of this matrices over any ﬁeld are presented by means of the algorithm for the Gröbner basis of the ideal in the polynomial ring. And two algorithms for ﬁnding the inverses of such matrices are also presented. A new algorithm for ﬁnding the inverse of a nonsingular scaled factor circulant matrix is presented by the Euclids algorithm. Extension is made to compute the group inverse and the Moore–Penrose inverse of the singular scaled factor circulant matrix in [21]. Jiang and Liu [22] presented the level-m scaled circulant factor matrix over the complex number ﬁeld and discussed its diagonalization and spectral decomposition and representation. Fast algorithms for calculating the inverse, self-reﬂective ginverse, group inverse and Moore–Penrose inverse of a scaled factor circulant matrix is presented by the fast algorithmfor computing polynomials in [23]. The scaled factor circulant matrices has been discussed in more detail [24–26]. Up to this day, some mathematicians pay attention to the product of diagonal and circulant matrices. For instance, in [47], Williamson published a note, in which he proved by means of complicated determinantal relations that the matrix ZA, which is the product of diagonal and circulant matrices, where A is the cyclic matrix whose ﬁrst row entries a0 , a1 , . . . , an−1 , and Z = diag(1, ω, ω 2 , . . . , ω n−1 ), satisﬁes the equation (ZA)n − Det(A)E = 0, where E is the unit matrix and ω is a primitive n-th root of unitary. Following it, Wegner [48] showed how this result can be obtained more simply. The product of diagonal and circulant matrices has been investigated in more detail [36, 45]. Recently, the investigation of retrocirculant matrix is becoming a hot topic. In [3], Aitken proposed that a retrocirculant is the product of a circulant and a certain permutation matrix P . And he determined the eigenvalues of the retrocirculants. In his proof, he used P Ω = ΩP where Norm Equalities and Inequalities for Three Circulant Operator Matrices 3 Ω is a certain unitary matrix. Then, Chao [10] provided a necessary and suﬃcient condition for a permutation matrix to commute with Ω. And he also determined the eigenvalues of many retrocirculants. In [42], Ronald gave the Moore–Penrose inverse of such a retrocirculant and showed that the nonzero eigenvalues of the Moore–Penrose inverse are the reciprocals of the nonzero eigenvalues of the retrocirculant. In 1981, Wang studied the properties of matrices of the form P (σ)A where σ is induced by an automorphism of an abelian group G and A is a group matrix, P (σ)A is a generalization of a retrocirculant. And also got the eigenvalues of P (σ)A in [49]. The retrocirculant matrices has been studied in more detail [5, 12, 31]. In this paper, we discuss general norm equalities and inequalities for scaled circulant operator matrices in Section 2 and Section 3. Moreover, the equality conditions in these norm inequalities are also given. Notice that we extend the research of [28], [29] and [46], these results are the special cases of our results (when d1 = d2 = · · · = dn−1 = 1, dn = ω = eiθ ; d1 = d2 = · · · = dn−1 = 1, dn = i; d1 = d2 = · · · = dn = 1 and d1 = d2 = · · · = dn−1 = 1, dn = −1). In Section 4 and Section 5, the norm equalities and inequalities for diag-circulant operator matrices are presented. And, we also show the equality conditions in these norm inequalities. In Section 6 and Section 7, we discuss the norm equalities and inequalities for retrocirculant operator matrices. When P is an n × n unit operator matrix, we discover that our results are the same as the research of Wathiq and Fuad [46]. 2 Norm Equalities for Scaled Circulant Operator Matrices We denote by B(H) the scaled circulant, the diag-circulant and the retrocirculant algebra of all bounded linear operators on a complex separable Hilbert space H. Let H (n) = n copies H denote the direct sum of n copies of H. If Aj,k j, k = 1, 2, . . . , n are operators in B(H), then operator matrix (or the partitioned operator) A = [Aj,k ] can be discussed in B(H (n) ), which is deﬁned by ⎛ ⎞ n A1k xk k=1 ⎜ ⎟ .. ⎜ ⎟ Ax = ⎜ ⎟ . ⎝ ⎠ n A x nk k k=1 for every vector x = (x1 · · · xn )T ∈ H (n) . If S1 , S2 , . . . , Sn are operators in B(H), we denote the n direct sum of them by j=1 Sj for the n × n block diagonal operator matrix, i.e., ⎞ ⎛ ⎜ ⎜ ⎜ Sj = ⎜ ⎜ ⎜ j=1 ⎝ n S1 ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠ S2 .. . Sn Thus, nj=1 Sj = max{Sj : j = 1, 2, . . . , n} and nj=1 Sj p = ( nj=1 Sj pp )1/p for n 1 ≤ p < ∞. In particular, j=1 S = n1/p Sp for 1 ≤ p < ∞. The pinching inequality for weakly unitarily invariant norms is one of the most useful Jiang Z. L. et al. 4 inequalities for operator matrices. It asserts that if A = [Ajk ], then n Ajj τ ≤ τ (A). (2.1) j=1 For the operator norm and the Schatten p-norms, the inequality (2.1) states that max{Ajj : j = 1, 2, . . . , n} ≤ A (2.2) and 1/p n Ajj pp ≤ Ap (2.3) j=1 for 1 ≤ p < ∞. It is known as [16] that for 1 < p < ∞, the equality in (2.3) holds if and only if A is block-diagonal, i.e., if and only if Ajk = 0, for j = k. The weakly unitarily invariant norm τ is deﬁned by τ (A) = τ (U AU ∗ ) for all A ∈ B(H) and all unitarily operator matrices U ∈ B(H). In the following, τ is used to denote the weakly unitarily invariant norm. Deﬁnition 2.1 If A0 , A1 , . . . , An−1 are operators in B(H), then the scaled circulant operator matrix A can be written as A = scacirc(A0 , A1 , . . . , An−1 ). It is the n × n partitioned matrix whose first row has entries A0 , A1 , . . . , An−1 and AR = RA, where ⎛ ⎞ 0 d1 I 0 ··· 0 ⎜ ⎟ ⎜ ⎟ 0 d2 I · · · 0 ⎜ 0 ⎟ ⎜ ⎟ .. .. . ⎜ .. ⎟ .. . , (2.4) R = DC = ⎜ . ⎟ . . . . ⎜ ⎟ ⎜ ⎟ ⎜ 0 0 0 · · · dn−1 I ⎟ ⎝ ⎠ dn I 0 0 ··· 0 n×n D = diag(d1 I, d2 I, . . . , dn I), | di |= 1, i = 1, 2, . . . , n and ⎛ ⎞ 0 I 0 ··· 0 ⎜ ⎟ ⎜ ⎟ 0 0 I · · · 0 ⎜ ⎟ ⎜ ⎟ .. ⎟ ⎜ .. .. .. . . C=⎜ . . . . . ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 0 0 ··· I ⎟ ⎝ ⎠ I 0 0 ··· 0 . n×n When d1 = d2 = · · · = dn = 1, A will be a circulant operator matrix in [46]. When d1 = d2 = · · · = dn−1 = 1 and dn = −1, A will be a skew circulant operator matrix in [46]. When d1 = d2 = · · · = dn−1 = 1 and dn = i, A will be a block imaginary circulant operator matrix in [29]. When d1 = d2 = · · · = dn−1 = 1 and dn = ω = eiθ , A will be an ω-circulant operator matrix in [28]. Theorem 2.2 Let A = scacirc(A0 , A1 , . . . , An−1 ) be a scaled circulant operator matrix defined in Definition 2.1. Then n−1 n−1 A0 + σ(A) = σ j=0 Ai (dω j )i i t=1 dt i=1 n−1 = n−1 σ A0 + j=0 Ai (dω j )i i t=1 dt i=1 Norm Equalities and Inequalities for Three Circulant Operator Matrices and n−1 A = diag(A0 , A0 , . . . , A0 ) + where ω = e 2πi n , i2 = −1, d = Ai i t=1 dt i=1 n n 5 Ri , t=1 dt = 0 and R is given in (2.4). Proof Let Δ = diag(δ1 I, δ2 I, . . . , δn I), where the elements δj of Δ can be computed by the recursion formula δj+1 = ddj δj , 1 ≤ j ≤ n and δn+1 = δ1 . Using the matrix Δ gives R = ΔdCΔ−1 and AR = RA iﬀ A(dΔCΔ−1 ) = dΔCΔ−1 A iﬀ Δ−1 AΔC = CΔ−1 AΔ. Thus Δ−1 AΔ is a block circulant matrix whose ﬁrst row is A1 d A2 d2 An−1 dn−1 A 0 , 1 . , 2 , . . . , n−1 t=1 dt t=1 dt t=1 dt It then follows that σ(A) = σ(Δ−1 AΔ), which is given on the above. Since Δ−1 AΔ is a block circulant matrix, n−1 Ai di −1 Δ AΔ = diag(A0 , A0 , . . . , A0 ) + C i. i d t=1 t i=1 Therefore, n−1 A = diag(A0 , A0 , . . . , A0 ) + i=1 Ai i t=1 dt Ri . Theorem 2.3 Let A0 , A1 , . . . , An−1 be any operators in B(H). Then, for every weakly unitarily invariant norm, we have n τ (A) = τ where ω = e 2πi n 2 , i = −1, d = n−1 A0 + n n k=1 Aj dj (k−1)j ω j d t t=1 j=1 , t=1 dt = 0 and |di | = 1, i = 1, 2, . . . , n. Specially, when d1 = d2 = · · · = dn = 1, Theorem 2.3 is the same as Theorem 1 in [46]. When d1 = d2 = · · · = dn−1 = 1 and dn = −1, Theorem 2.3 is the same as Theorem 2 in [46]. When d1 = d2 = · · · = dn−1 = 1 and dn = i, Theorem 2.3 is the same as Theorem 1 in [29]. If d1 = d2 = · · · = dn−1 = 1 and dn = ω = eiθ , then Theorem 2.3 agrees with Theorem 2.1 in [28]. 2πi Proof Let 1, ω, ω 2 , . . . , ω (n−1) be the n roots of unity with ω = e n . Now, let F = Fn ⊗ I, where I is a unit matrix, ⊗ is the Kronecker Product of matrix and ⎞ ⎛ 1 1 1 ··· 1 ⎟ ⎜ ⎟ ⎜ ω ω2 ··· ω n−1 ⎟ ⎜ 1 ⎟ 1 ⎜ ⎜ 1 ω2 ω4 ··· ω 2(n−1) ⎟ Fn = √ ⎜ . ⎟ ⎟ n⎜ . .. .. .. .. ⎟ ⎜ . ⎟ ⎜ . . . . . ⎠ ⎝ 1 ω n−1 ω 2(n−1) · · · ω (n−1)(n−1) n×n Jiang Z. L. et al. 6 Then it is easy to prove that F is a unitary operator in B(H). When |di | = 1, i = 1, 2, . . . , n, Δ is also a unitary operator in B(H). It is known that A1 d A2 d2 An−1 dn−1 Δ−1 . A = Δcirc A0 , 1 , 2 , . . . , n−1 t=1 dt t=1 dt t=1 dt Thus, every scaled circulant operator matrix is unitarily equivalent to a circulant operator matrix. A1 d A2 d2 An−1 dn−1 F ∗ Δ−1 AΔF = F ∗ circ A0 , 1 F , 2 , . . . , n−1 t=1 dt t=1 dt t=1 dt n n−1 = A0 + k=1 Aj dj (k−1)j ω j t=1 dt j=1 . From the invariance property of weakly unitarily invariant norms, we have n n−1 A0 + τ (A) = τ k=1 Aj dj (k−1)j ω j t=1 dt j=1 . Synthesize the norm equality in Theorem 2.1 to the usual operator norm and to the Schatten p-norms, we obtain the following corollary. Corollary 2.4 Let A0 , A1 , . . . , An−1 be any operators in B(H). Then n−1 Aj dj (k−1)j ω A = max A0 + j : k = 1, 2, . . . , n t=1 dt j=1 and p n−1 Aj dj (k−1)j A0 + ω j d p t=1 t j=1 k=1 n Ap = 1 p for 1 ≤ p < ∞. In particular (letting n = 2), we have ⎛ ⎞ A0 A1 ⎝ ⎠ = max d2 d1 A1 A0 and ⎛ A0 ⎝ d2 d1 A1 ⎞ ⎠ = A0 A1 p A0 + d2 A1 , A0 − d2 A1 d1 d1 p p d d2 2 A0 + A1 + A0 − A1 d1 d1 p 1 p p for 1 ≤ p < ∞. Specially, when d1 = d2 = · · · = dn = 1, Corollary 2.4 is the same as Corollary 1 in [46]. When d1 = d2 = · · · = dn−1 = 1 and dn = −1, Corollary 2.4 is the same as Corollary 2 in [46]. When d1 = d2 = · · · = dn−1 = 1 and dn = i, Corollary 2.4 is the same as Corollary 2 in [29]. When d1 = d2 = · · · = dn−1 = 1 and dn = ω = eiθ , Corollary 2.4 is the same as Corollary 2.2 in [28]. Norm Equalities and Inequalities for Three Circulant Operator Matrices 3 7 Pinching Type Inequalities for Scaled Circulant Operator Matrices In this section, our main aim is to discuss the pinching type inequalities for scaled circulant operator matrices. The pinching type inequality is one of the most important inequalities of operator matrices for weakly unitarily invariant norm. It asserts that if S = [Sjk ] (j, k = 1, 2, . . . , n), then τ ( nj=1 Sjj ) ≤ τ (S) ([9, 46]). Theorem 3.1 Let A = [Ajk ] be an operator matrix in B(H (n) ). Then, for every weakly unitarily invariant norm, one has n 1 τ n n−1 Q0 + k=1 Qj dj (k−1)j ω j t=1 dt j=1 ≤ τ (A), (3.1) where n Q0 = Ajj , j=1 n Q1 = δ1 A1n + Aj,j−1 , δ2 j=2 n δ1 A1,n−1 + A2,n + Q2 = Aj,j−2 , δ3 j=3 .. . Qn−2 = Qn−1 = ω=e 2πi n , i2 = −1, d = n−2 δ1 δn−1 δ1 δn n n n Aj,j−(n−2) , Aj,j+2 + j=1 j=n−1 n−1 Aj,j+1 + An,1 , (3.2) j=1 t=1 dt = 0 and |di | = 1, i = 1, 2, . . . , n. Specially, when d1 = d2 = · · · = dn = 1, Theorem 3.1 is the same as Theorem 3 in [46]. When d1 = d2 = · · · = dn−1 = 1 and dn = −1, Theorem 3.1 is the same as Theorem 4 in [46]. When d1 = d2 = · · · = dn−1 = 1 and dn = i, Theorem 3.1 is the same as Theorem 4 in [29]. When d1 = d2 = · · · = dn−1 = 1 and dn = ω = eiθ , Theorem 3.1 is the same as Theorem 3.1 in [28]. Proof Let Lk,n−k = [lrs ] be the n × n operator with ⎧ ⎨I, if r + s = k + 1 or r + s = k + n + 1; lrs = ⎩0, otherwise. Then it is easy to prove that ΔLk+1,k+n+1 is a unitary operator for all k = 1, 2, 3, . . . , n and n ΔLk+1,k+n+1 AL∗k+1,k+n+1 Δ∗ = scacirc(Q0 , Q1 , . . . , Qn−1 ) = Q, k=1 where n Q0 = Ajj , j=1 Jiang Z. L. et al. 8 n δ1 A1n + Q1 = Aj,j−1 , δ2 j=2 n Q2 = δ1 A1,n−1 + Q2,n + Aj,j−2 , δ3 j=3 .. . Qn−2 = δ1 δn−1 δ1 Qn−1 = δn n−2 n Aj,j−(n−2) , Aj,j+2 + j=1 j=n−1 n−1 Aj,j+1 + An,1 , j=1 and Δ is given in Theorem 2.2. From the proof of Theorem 2.3, we have n F ∗ (Δ∗ QΔ)F = n−1 Q0 + k=1 Qj dj (k−1)j . ω j d t t=1 j=1 By the invariance property of unitarily invariant norms and the triangle inequality, we have 1 τ n n n−1 Q0 + k=1 Qj dj (k−1)j ω j t=1 dt j=1 ≤ τ (A). Specializing the norm inequality (3.1) to the usual operator norm and to the Schatten p-norms [9, 46], we obtain the following corollaries. Corollary 3.2 Let A = [Ajk ] be an operator matrix in B(H (n) ). Then n−1 j Q d 1 j (k−1)j : k = 1, 2, . . . , n ≤ A max ω j Q0 + n d t t=1 j=1 and 1 n p n−1 j Q d j (k−1)j Q0 + ω j d p t t=1 j=1 k=1 n 1/p ≤ Ap for 1 ≤ p < ∞, where Qj is given in (3.2). When n = 2, Corollary 2.4 asserts that 1 d d 2 2 max Q0 + , Q0 − Q1 Q1 2 d1 d1 1 = max(A11 + A22 + A12 + A21 , A11 + A22 − A12 − A21 ) 2⎛ ⎞ A11 A12 ⎝ ⎠ ≤ A21 A22 and ⎛ A11 1 p p 1/p ⎝ ≤ (A11 + A22 + A12 + A21 p + A11 + A22 − A12 − A21 p ) 2 A21 ⎞ ⎠ A22 A12 p Norm Equalities and Inequalities for Three Circulant Operator Matrices 9 for 1 ≤ p < ∞. Specially, when d1 = d2 = · · · = dn = 1, Corollary 3.2 is the same as Corollary 3 in [46]. When d1 = d2 = · · · = dn−1 = 1 and dn = −1, Corollary 3.2 is the same as Corollary 4 in [46]. When d1 = d2 = · · · = dn−1 = 1 and dn = i, Corollary 3.2 is the same as Corollary 5 in [29]. When d1 = d2 = · · · = dn−1 = 1 and dn = ω = eiθ , Corollary 3.2 is the same as Corollary 3.2 in [28]. It should be mentioned here that the norm inequalities in Theorem 2.3 and Theorem 3.1 are sharp. This is demonstrated in the following proposition. Proposition 3.3 Let A0 , A1 , . . . , An−1 be any operators in B(H). If A = diag(A0 , A0 , . . . , Aj j A0 ) + n−1 j=1 j d R , then the inequality in Theorem 3.1 becomes an equality. t=1 t Specially, when d1 = d2 = · · · = dn = 1, Proposition 3.3 is the same as Proposition 1 in [46]. When d1 = d2 = · · · = dn−1 = 1 and dn = −1, Proposition 3.3 is the same as Proposition 1 in [46]. When d1 = d2 = · · · = dn−1 = 1 and dn = i, Proposition 3.3 is the same as Proposition 6 in [29]. When d1 = d2 = · · · = dn−1 = 1 and dn = ω = eiθ , Proposition 3.3 is the same as Proposition 3.3 in [28]. Aj j Proof Let A = diag(A0 , A0 , . . . , A0 ) + n−1 j=1 j dt R . Then it follows from Theorem 2.3 t=1 that n τ (A) = τ n−1 A0 + k=1 Aj dj (k−1)j ω j t=1 dt j=1 . Since Q0 = nA0 , Q1 = nAn−1 , Q2 = nAn−2 , . . . , Qn−2 = nA2 , and Qn−1 = nA1 , it follows that 1 τ n n Dk = τ (A), k=1 where D1 = n A0 + A1 d A2 d2 An−1 dn−1 + + · · · + n−1 d1 d1 d2 t=1 dt n−1 = n A0 + D2 = n A0 + Aj dj , j t=1 dt j=1 A1 d A2 d2 2 An−1 dn−1 n−1 ω+ ω + · · · + n−1 ω d1 d1 d2 t=1 dt n−1 = n A0 + Aj dj j ω , j d t t=1 j=1 .. . Dn = n A0 + A1 d n−1 A2 d2 2(n−1) An−1 dn−1 (n−1)(n−1) ω + ω + · · · + n−1 ω d1 d1 d2 t=1 dt n−1 = n A0 + Aj dj (n−1)j . ω j d t t=1 j=1 Proposition 3.4 Let A = [Ajk ] be an operator matrix in B(H n ), and let 1 < p < ∞. Then n n−1 Q dj Ap = n1 k=1 (Q0 + j=1 j j d ω (k−1)j )p if and only if A is a scaled circulant operator t=1 t Jiang Z. L. et al. 10 matrix. Specially, when d1 = d2 = · · · = dn = 1, Proposition 3.4 is the same as Proposition 2 in [46]. When d1 = d2 = · · · = dn−1 = 1 and dn = −1, Proposition 3.4 is the same as Proposition 2 in [46]. When d1 = d2 = · · · = dn−1 = 1 and dn = i, Proposition 3.4 is the same as Proposition 7 in [29]. When d1 = d2 = · · · = dn−1 = 1 and dn = ω = eiθ , Proposition 3.4 is the same as Proposition 3.4 in [28]. Proof In view of Proposition 3.3, it is suﬃcient to prove the “only if” part. Let ΔLk+1,k+n+1 Qj dj (k−1)j be as in the proof of Theorem 3.1. If Ap = n1 nk=1 (Q0 + n−1 )p , then it j=1 j dt ω t=1 follows from the proof of Theorem 3.1 that ΔL1,n+1 AL∗1,n+1 Δ−1 p = ΔL2,n+2 AL∗2,n+2 Δ−1 p = · · · = ΔLn,2n AL∗n,2n Δ−1 p = Ap and n−1 ΔLk+1,k+n+1 AL∗k+1,k+n+1 Δ−1 p = nAp . k=0 Now invoking Clarkson inequalities [18] for several operators in B(H), it follows that ΔL1,n+1 AL∗1,n+1 Δ−1 = ΔL2,n+2 AL∗2,n+2 Δ−1 = · · · = ΔLn,2n AL∗n,2n Δ−1 . Combining with the special structure of the matrix L, we obtain that A is a scaled circulant operator matrix. 4 Norm Equalities for Diag-circulant Operator Matrices Deﬁnition 4.1 If A1 , A2 , . . . , An are operators in B(H), then the diag-circulant operator matrix Γ can be written as Γ = diagω circ(A1 , A2 , . . . , An ) = ζcirc(A1 , A2 , . . . , An ), that is, ⎛ ⎞ A1 A2 A3 ··· An ⎜ ⎟ ⎜ ⎟ ωA1 ωA2 · · · ωAn−1 ⎟ ⎜ ωAn ⎜ ⎟ .. .. .. .. ⎜ ⎟ .. Γ=⎜ ⎟, . . . . . ⎜ ⎟ ⎜ n−2 ⎟ ⎜ ω A3 ω n−2 A4 ω n−2 A5 · · · ω n−2 A2 ⎟ ⎝ ⎠ ω n−1 A2 ω n−1 A3 ω n−1 A4 · · · ω n−1 A1 2πi where ζ = diag(I, ωI, ω 2 I, . . . , ω n−1 I), I is a unit matrix, ω = e n , i2 = −1. Theorem 4.2 Let A1 , A2 , . . . , An be any operators in B(H). Then, for every weakly unitarily invariant norm, we have n−1 n ω k(j−1) Aj τ (Γ) = τ k=0 , j=1 2πi where ω = e n , i2 = −1. Proof 2πi Let 1, ω, ω 2 , . . . , ω (n−1) be the n roots of unity with ω = e n . Now, let U = Un ⊗ I, where I is a unit matrix, ⊗ is the Kronecker Product of matrix and Un = ΛFn , where Λ = diag(1, ω, ω 2 , . . . , ω n−1 ) and Fn is the same as Theorem 2.3. Norm Equalities and Inequalities for Three Circulant Operator Matrices 11 Then it is easy to prove that U ∗ is a unitary operator in B(H). And ζ = diag(I, ωI, ω 2 I, . . . , ω n−1 I), I is a unit matrix, is also a unitary operator in B(H). Thus, n−1 U ∗ Γζ = n ω k(j−1) Aj k=0 . j=1 From the invariance property of weakly unitarily invariant norms, we deduce that n−1 n ω k(j−1) Aj τ (Γ) = τ k=0 . j=1 Synthesize the norm equality in Theorem 4.2 to the usual operator norm and to the Schatten p-norms, we obtain the following corollary. Corollary 4.3 Let A1 , A2 , . . . , An be any operators in B(H). Then we have n k(j−1) ω Aj : k = 1, 2, . . . , n Γ = max j=1 and n Γp = k=1 n p k(j−1) ω Aj 1 p p j=1 for 1 ≤ p < ∞. In particular (letting n = 2), we have ⎛ ⎞ A1 A2 ⎝ ⎠ = max(A1 − A2 , A1 + A2 ), ωA2 ωA1 and ⎛ A1 ⎝ ωA2 ⎞ 1 ⎠ = A1 − A2 pp + A1 + A2 pp p ωA1 A2 p for 1 ≤ p < ∞. Remark 4.4 Here we give some special cases of Corollary 4.3. (a) If A1 , A2 ∈ B(H), then ⎛ A1 ⎜ ⎜ ⎜ ωA2 ⎜ ⎜ ⎜ ω 2 A2 ⎜ .. ⎜ ⎜ . ⎝ ω n−1 A2 ⎞ A2 A2 ··· A2 ωA1 ωA2 ··· ωA2 2 ω A2 .. . 2 ω A1 .. . ··· .. . ω 2 A2 .. . ω n−1 A2 ω n−1 A2 ··· ω n−1 A1 = max(A1 + (n − 1)A2 , A1 − A2 ), ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ n×n Jiang Z. L. et al. 12 and ⎛ A1 ⎜ ⎜ ⎜ ωA2 ⎜ ⎜ ω 2 A ⎜ 2 ⎜ .. ⎜ ⎜ . ⎝ ω n−1 A2 ⎞ A2 A2 ··· A2 ωA1 ωA2 ··· ωA2 ω 2 A2 .. . ω 2 A1 .. . ··· .. . ω 2 A2 .. . ω n−1 A2 ω n−1 A2 ··· ω n−1 A1 ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ n×n p 1 = (A1 + (n − 1)A2 pp + (n − 1)A1 − A2 pp ) p for 1 ≤ p < ∞. (b) If A ∈ B(H), then ⎛ 0 A A ⎜ ⎜ 0 ωA ⎜ ωA ⎜ ⎜ ω 2 A ω2 A 0 ⎜ ⎜ . . .. ⎜ .. .. ⎜ . ⎝ n−1 n−1 n−1 ω A ω A ω A ⎛ 0 A A ⎜ ⎜ 0 ωA ⎜ ωA ⎜ ⎜ ω 2 2 A ω A 0 ⎜ ⎜ .. .. .. ⎜ ⎜ . . . ⎝ ω n−1 A ω n−1 A ω n−1 A ⎟ ⎟ ωA ⎟ ⎟ ω2 A ⎟ = (n − 1)A, ⎟ ⎟ .. ⎟ ⎟ . ⎠ 0 n×n ⎞ A ⎟ ⎟ ωA ⎟ ⎟ ⎟ 2 ω A ⎟ = 2(n − 1)Ap ⎟ .. ⎟ . ⎟ ⎠ 0 ⎞ ··· A ··· ··· .. . ··· ··· ··· ··· .. . ··· n×n p for 1 ≤ p < ∞. 5 Pinching Type Inequalities for Diag-circulant Operator Matrices Our main results in this section are pinching type inequalities for diag-circulant operator matrices. Theorem 5.1 Let A = [Ajk ] be an operator matrix in B(H (n) ). Then, for every weakly unitarily invariant norm, we have 1 τ n n−1 n ω k(j−1) Γj k=0 ≤ τ (A), j=1 where n Γ1 = Ajj , j=1 n Γ2 = A1n + Aj,j−1 , j=2 n Γ3 = A1,n−1 + A2,n + Aj,j−2 , j=3 (5.1) Norm Equalities and Inequalities for Three Circulant Operator Matrices 13 .. . n−2 Γn−1 = n Aj,j+2 + j=1 Aj,j−(n−2) , j=n−1 n−1 Γn = Aj,j+1 + An,1 (5.2) j=1 2πi and ω = e n , i2 = −1. Proof Let Lk+1,k+n+1 = [lrs ] be the n × n operator with ⎧ ⎨I, if r + s = k + 1 or r + s = k + n + 1; lrs = ⎩0, otherwise. Then it is easy to prove that ζLk+1,k+n+1 is a unitary operator for all k = 1, 2, 3, . . . , n and n ζLk+1,k+n+1 AL∗k+1,k+n+1 = diagω circ(Γ1 , Γ2 , . . . , Γn ) = Γ, k=1 where n Γ1 = Ajj , j=1 n Γ2 = A1n + Aj,j−1 , j=2 n Γ3 = A1,n−1 + A2,n + Aj,j−2 , j=3 .. . n−2 Γn−1 = n Aj,j+2 + j=1 Aj,j−(n−2) , j=n−1 n−1 Γn = Aj,j+1 + An,1 , j=1 and ζ is the same as Deﬁnition 4.1. From the proof of Theorem 4.2, we have ∗ n−1 n ω k(j−1) Γj . U Γζ = k=0 j=1 It follows from the invariance property of unitarily invariant norms and the triangle inequality, we have 1 τ n n−1 n ω k(j−1) Γj k=0 ≤ τ (A). j=1 Specializing the norm inequality (5.1) to the usual operator norm and to the Schatten p-norms [9, 46], we obtain the following corollary. Jiang Z. L. et al. 14 Corollary 5.2 Let A = [Ajk ] be an operator matrix in B(H (n) ). Then n 1 k(j−1) max ω Γj : k = 1, 2 . . . , n ≤ A n j=1 and 1 n n k=1 n p k(j−1) ω Γj 1/p ≤ Ap p j=1 for 1 ≤ p < ∞, where Γj is given in (5.2). The special case when n = 2 of Corollary 5.2 asserts that 1 max(Γ1 − Γ2 , Γ1 + Γ2 ) 2 1 = max(A11 + A22 − A12 − A21 , A11 + A22 + A12 + A21 ) 2⎛ ⎞ A11 A12 ⎝ ⎠ ≤ A21 A22 and ⎛ A11 1 p p 1/p ⎝ (A11 + A22 − A12 − A21 p + A11 + A22 + A12 + A21 p ) ≤ 2 A21 ⎞ ⎠ A22 A12 p for 1 ≤ p < ∞. It should be mentioned here that the norm inequalities in Theorem 4.2 and Theorem 5.1 are sharp. This is demonstrated in the following proposition. Proposition 5.3 Let A1 , A2 , . . . , An be any operators in B(H). If Γ = diagω circ(Γ1 , Γ2 , . . . , Γn ), then the inequality in Theorem 5.1 becomes an equality. Proof Let Γ = diagω circ(Γ1 , Γ2 , . . . , Γn ). Then it follows from Theorem 4.2 that n−1 n ω k(j−1) Aj τ (Γ) = τ k=0 . j=1 Since Γ1 = nA1 , Γ2 = nAn , Γ3 = nAn−1 , . . . , Γn−1 = nA3 , and Γn = nA2 , it follows that 1 τ n n−1 Dk = τ (Γ), k=0 where n D0 = n(A1 + A2 + A3 + · · · + An ) = n Aj , j=1 2 D1 = n(A1 + ωA2 + ω A3 + · · · + ω n n−1 (ω j−1 Aj ), An ) = n j=1 .. . Dn−1 = n(A1 + ω n−1 A2 + ω 2(n−1) A3 + · · · + ω (n−1)(n−1) An ) Norm Equalities and Inequalities for Three Circulant Operator Matrices n 15 (ω (n−1)(j−1) Aj ). =n j=1 Proposition 5.4 Let A = [Ajk ] be an operator matrix in B(H n ), and let 1 < p < ∞. Then n n 1 k(j−1) Ap = ω Γj n p j=1 k=0 if and only if A is a diag-circulant operator matrix. Proof In view of Proposition 5.4, it is suﬃcient to prove the “only if” part. Let ζLk+1,k+n+1 n−1 n be as in the proof of Theorem 5.1. If Ap = n1 k=0 ( j=1 ω k(j−1) Γj )p , then it follows from the proof of Theorem 5.1 that ζL1,n+1 AL∗1,n+1 p = ζL2,n+2 AL∗2,n+2 p = · · · = ζLn,2n AL∗n,2n p = Ap and n−1 ζLk+1,k+n+1 AL∗k+1,k+n+1 p = nAp . k=0 Now invoking the Clarkson inequalities [18] for several operators in B(H), it follows that ζL1,n+1 AL∗1,n+1 = ζL2,n+2 AL∗2,n+2 = · · · = ζLn,2n AL∗n,2n . Combining with the special structure of the matrix L, we get that A is a diag-circulant operator matrix. 6 Norm Equalities for Retrocirculant Operator Matrices Deﬁnition 6.1 ([19]) A matrix P ∈ Mn is called a permutation matrix if exactly one entry in each row and column is equal to 1, and all other entries are 0. Remark 6.2 ([19]) The determinant of a permutation matrix is ±1, so that permutation matrices are necessarily nonsingular, and P T = P −1 . Deﬁnition 6.3 is called a retrocirculant operator matrix if = PC, where P is an n × n symmetric permutation operator matrix and P 2 = In , P = P ⊗ I, I is a unit operator matrix, C = circ(A1 , A2 , . . . , An ) is a circulant operator matrix [46] and A1 , A2 , . . . , An are operators in B(H). Theorem 6.4 Let A1 , A2 , . . . , An be any operators in B(H). Then, for every weakly unitarily invariant norm, we have n n ω k(1−j) Aj τ( ) = τ k=1 2πi where ω = e n , i2 = −1, and , j=1 is given in Definition 6.3. Specially, when P is an n × n unit operator matrix, this result is the same as Theorem 3 in [46]. Proof It is obvious that P is a unitary operator matrix in B(H), which is given in Deﬁnition 6.3. Thus, from the invariance property of weakly unitarily invariant norms, we have τ ( ) = τ (PC) = τ (C) = τ (circ(A1 , A2 , . . . , An )), Jiang Z. L. et al. 16 where and C are given in Deﬁnition 6.3. Following [46], we know that n n ω k(1−j) Aj τ (circ(A1 , A2 , . . . , An )) = τ k=1 . j=1 Consequently, n n ω k(1−j) Aj τ( ) = τ k=1 . j=1 Synthesize the norm equality in Theorem 6.4 to the usual operator norm and to the Schatten p-norms, we obtain the following corollary. Corollary 6.5 Let A1 , A2 , . . . , An be any operators in B(H). Then we have n k(1−j)Aj = max ω : k = 0, 1, . . . , n − 1 j=1 and n−1 n p = k=0 ω p k(1−j)Aj 1 p p j=1 for 1 ≤ p < ∞. Specially, when P is an n × n unit operator matrix, Corollary 6.5 is the same as Corollary 1 in [46]. 7 Pinching Type Inequalities for Retrocirculant Operator Matrices Our main results in this section are pinching type inequalities for retrocirculant operator matrices. Theorem 7.1 Let A = [Ajk ] be an operator matrix in B(H (n) ). Then, for every weakly unitarily invariant norm, we have 1 τ n n−1 n ω k(1−j) Gj k=0 ≤ τ ( ), (7.1) j=1 where n G1 = Ajj , j=1 n G2 = A1n + Aj,j−1 , j=2 n G3 = A1,n−1 + A2,n + Aj,j−2 , j=3 .. . n−2 Gn−1 = n Aj,j+2 + j=1 Aj,j−(n−2) , j=n−1 n−1 Gn = Aj,j+1 + An,1 , j=1 (7.2) Norm Equalities and Inequalities for Three Circulant Operator Matrices 2πi and ω = e n , i2 = −1, and 17 is given in Definition 6.3. Specially, when P is an n × n unit. operator matrix, Theorem 7.1 is the same as Theorem 3 in [46]. Proof From [46], we deduce that n−1 1 τ n n ≤ τ (circ(A1 , A2 , . . . , An )). ω k(1−j) Gj j=1 k=0 By Theorem 6.4, we get τ ( ) = τ (circ(A1 , A2 , . . . , An )). Thus, 1 τ n n−1 n ≤ τ ( ). ω k(1−j) Gj j=1 k=0 Specializing the norm inequality (7.1) to the usual operator norm and to the Schatten p-norms, we obtain the following corollary. Corollary 7.2 Let A = [Ajk ] be an operator matrix in B(H (n) ). Then n 1 k(1−j) max ω Gj : k = 0, 1, . . . , n − 1 ≤ A n j=1 and n−1 n 1 n k=0 ω k(1−j) p Gj 1/p ≤ Ap p j=1 for 1 ≤ p < ∞, where Gj is given in (7.2). Specially, when P is an n × n unit operator matrix, Corollary 7.2 is the same as Corollary 3 in [46]. It should be mentioned here that the norm inequalities in Theorem 6.4 and Theorem 7.1 are sharp. This is demonstrated in the following proposition. Proposition 7.3 Let A1 , A2 , . . . , An be any operators in B(H). If ity in Theorem 7.1 becomes an equality. = PC, then the inequal- Specially, when P is an n × n unit operator matrix, Proposition 7.3 is the same as Proposition 1 in [46]. Proof Let = PC. Then it follows from Theorem 6.4 that n−1 n ω k(1−j) Aj τ( ) = τ k=0 . j=1 Since G1 = nA1 , G2 = nAn , G3 = nAn−1 , . . . , Gn−1 = nA3 , and Gn = nA2 , it follows that 1 τ n n−1 Dk = τ (circ(A1 , A2 , . . . , An )) = τ ( ), k=0 where n D0 = n[A1 + A2 + A3 + · · · + An ] = n Aj , j=1 Jiang Z. L. et al. 18 n D1 = n[A1 + ω n−1 An + ω n−2 An−1 + · · · + ωA2 ] = n (ω (n−1)(1−j) Aj ), j=1 .. . n 2 Dn−1 = n[A1 + ωAn + ω An−1 + · · · + ω n−1 An ] = n (ω 1−j Aj ). j=1 Proposition 7.4 Let A = [Ajk ] be an operator matrix in B(H n ), and let 1 < p < ∞. Then n−1 n 1 k(1−j) ω Gj Ap = n p j=1 k=0 if and only if A is a retrocirculant operator matrix. Specially, when P is an n × n unit operator matrix, Proposition 7.4 is the same as Proposition 2 in [46]. Proof In view of Proposition 7.3, it is suﬃcient to prove the “only if” part. Let Lk+1,k+n+1 be n k(1−j) as in the proof of Theorem 7.1. If Ap = n1 n−1 Gj )p , then it follows from k=0 ( j=1 ω the proof of Theorem 7.1 that L1,n+1 AL∗1,n+1 p = L2,n+2 AL∗2,n+2 p = · · · = Ln,2n AL∗n,2n p = Ap and n−1 Lk+1,k+n+1 AL∗k+1,k+n+1 p = nAp . k=0 Now invoking Clarkson inequalities [18] for several operators in B(H), it follows that L1,n+1 AL∗1,n+1 = L2,n+2 AL∗2,n+2 = · · · = Ln,2n AL∗n,2n . 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Comput., 233, 582–587 (2014) Applied Mathematics and Computation 277 (2016) 1–9 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc Analysis of the structured perturbation for the BSCCB linear system✩ Zhao-Lin Jiang a,b, Xia Tang a,b,∗ a b Department of Mathematics, Linyi University, Linyi 276005, PR China School of Mathematical Sciences, Shandong Normal University, Jinan 250014, PR China a r t i c l e i n f o a b s t r a c t Keywords: BSCCB linear system Structured perturbation Relative error Optimal backward perturbation In this paper, based on block style spectral decomposition of the block skew circulant with circulant blocks (BSCCB) matrix, the structure perturbation is discussed, which includes the condition number and relative error of the BSCCB linear system. Then the optimal backward perturbation bound of the BSCCB linear system is analyzed. Simultaneously, the algorithm for the optimal backward perturbation bound is presented. At the end of the paper, a numerical example is provided to verify the effectiveness of the algorithm. © 2016 Elsevier Inc. All rights reserved. 1. Introduction Skew circulant and circulant matrices have important applications in various disciplines including signal processing, image processing, communications, preconditioner and solving Toeplitz matrix problems in [1–10]. Davis [11], Jiang and Zhou [12] have put on ﬁrm basis with circulant and skew circulant matrices. In [13], the authors showed the different operators on linear vector space that are isomorphic to the algebra of n × n complex skew circulant matrices. Norm equalities and inequalities for operator matrices are considered in [14]. In [15], explicit inverse matrices of Tribonacci skew circulant type matrices are studied. The authors consider the determinants and inverses of generalized Lucas skew circulant type matrices in [16]. The BSCCB matrix is an extension of skew circulant and circulant matrices and we believe the BSCCB linear system can be used in those ﬁelds as well. Rigal and Gaches [17] discussed posteriori analysis of the compatibility of a computed solution to the uncertain data of a linear system by some new theorems generalizing a result of Oettli and Prager. The bound of the optimal backward perturbation for a block circulant linear system has been obtained by Liu and Guo in [18]. The optimal backward perturbation bounds for undetermined systems were proposed by J.G Sun and Z Sun in [19]. The block skew circulant with circulant blocks matrix with the ﬁrst row (s11 , . . . , s1m , s21 , . . . , s2m , . . . , sn1 , . . . , snm ) is meant by a square matrix of the form ⎛ S1 ⎜−Sn ⎜ . S=⎜ ⎜ .. ⎝−S 3 −S2 ✩ ∗ S2 S1 .. . ··· −S3 ··· S2 .. . −Sn ··· Sn−1 ··· .. . S1 −Sn ⎞ Sn Sn−1 ⎟ .. ⎟ ⎟ . ⎟, S2 ⎠ S1 Project is supported by National Natural Science Foundation of China (Grant no.11301251, 11301252) and the AMEP of Linyi University, China. Corresponding author. Tel.: +86 18754929620. E-mail address: tangxia0910@sina.cn (X. Tang). http://dx.doi.org/10.1016/j.amc.2015.12.030 0096-3003/© 2016 Elsevier Inc. All rights reserved. (1) 2 Z.-L. Jiang, X. Tang / Applied Mathematics and Computation 277 (2016) 1–9 and for any k = 1, 2, . . . , n, ⎛ sk1 ⎜skm ⎜ . Sk = ⎜ ⎜ .. ⎝s k3 sk2 sk2 sk1 .. . ··· sk3 ··· sk2 .. . skm ··· sk(m−1) ··· .. . sk1 skm skm ⎞ sk(m−1) ⎟ .. ⎟ ⎟ . ⎟. sk2 ⎠ sk1 The matrix is denoted by BSCCB(s11 , . . . , s1m , . . . , sn1 , . . . , snm ). 2. The block style spectral decomposition of the BSCCB matrix Based on Kronecker products in [12], the BSCCB matrix S can be decomposed as S= n (k−1 ⊗ Sk ), (2) k=1 where is an n × n matrix with the following form ⎛ 0 ⎜0 ⎜ . =⎜ ⎜ .. ⎝0 −1 1 0 .. . ··· 0 0 1 .. . 0 ··· ⎞ ··· ··· .. . 0 0 0 0⎟ .. ⎟ ⎟ . ⎟. 1⎠ 0 By using the Eqs. (10) and (11) in [20], the style spectral decomposition of the matrix is = Q 0 Q T , (3) here Q is an orthogonal matrix. When n is even, ⎛ φ1 ⎜ 0 = ⎜ ⎝ ⎞ φ2 .. ⎟ ⎟, ⎠ . φ 2n When n is odd, ⎛ φ1 ⎜ ⎜ 0 = ⎜ ⎜ ⎝ φj = ⎞ φ2 .. . φ n−1 2 cos θ j − sin θ j ⎟ ⎟ ⎟, ⎟ ⎠ sin θ j , cos θ j −1 θj = 2j − 1 π, n j= 1,2, ... , n2 , 1,2, ... , n−1 2 , n is even. n is odd. Take the Eqs. (2) and (3) into consideration, then S= n (k−1 ⊗ Sk ) = k=1 = n n T (Q k−1 0 Q ) ⊗ Sk k=1 (Q ⊗ Im )(k−1 ⊗ Sk )(Q T ⊗ Im ) 0 k=1 = (Q ⊗ Im ) n (k−1 ⊗ Sk )(Q T ⊗ Im ). 0 k=1 Obviously, Q ⊗ Im is also an orthogonal matrix. (4) Z.-L. Jiang, X. Tang / Applied Mathematics and Computation 277 (2016) 1–9 3 The matrix Sk can be expressed as Sk = m skl ϒ l−1 , (5) l=1 and Y is a square matrix of order m with the following form ⎛ 0 ⎜0 ⎜. ϒ =⎜ ⎜ .. ⎝0 1 1 0 .. . ··· 0 ··· ··· .. . 0 0 0 1 .. . 0 ··· ⎞ 0 0⎟ .. ⎟ ⎟ . ⎟. 1⎠ 0 Based on the Eqs. (2.5) and (2.6) in [18], the style spectral decomposition of the matrix Y is ϒ = J ϒ0 J T , (6) where J is an orthogonal matrix. When m is even, ⎛ υ1 ⎜ ϒ0 = ⎜ ⎝ υ2 .. ⎞ ⎟ ⎟. ⎠ . υ m2 −1 0 υ m2 = 0 , υj = 1 cos θh − sin θh 2h m sin θh − 1. , θh = π , h = 1, 2, . . . , cos θh m 2 When m is odd, ⎛ υ1 ⎜ ⎜ ϒ0 = ⎜ ⎜ ⎝ ⎞ υ2 .. . υ m−1 2 cos θh − sin θh υh = ⎟ ⎟ ⎟, ⎟ ⎠ sin θh , cos θh 1 θh = 2h π , h = 1, 2, . . . , m m−1 2 From Eqs. (4), (5) and (6), we obtain m n S = QJ skl k−1 ⊗ ϒ0l−1 Q TJ , 0 (7) k=1 l=1 and here Q J = (Q ⊗ Im )(In ⊗ J ), In and Im are identity matrices with order n and m. So Eq. (7) is the block style spectral decomposition of the matrix S. 3. Analysis of the structured perturbation 3.1. Condition number and relative error of BSCCB linear system Consider linear system Sx = b, where S is the matrix deﬁned in (1). We will discuss the problem at two different conditions. 1° n is even , ⎛ 11 m n ⎜ skl k−1 ⊗ ϒ0l−1 = ⎜ 0 ⎝ k=1 l=1 ⎞ 22 .. ⎟ ⎟, ⎠ . tt and pp = m n k=1 l=1 skl φ pk−1 ⊗ ϒ0l−1 , t = n , p = 1, 2, . . . , t. 2 4 Z.-L. Jiang, X. Tang / Applied Mathematics and Computation 277 (2016) 1–9 2° n is odd ⎛ 11 m n ⎜ skl k−1 ⊗ ϒ0l−1 = ⎜ 0 ⎝ ⎞ .. ⎟ ⎟, ⎠ . tt k=1 l=1 ˜ and pp = m n skl φ pk−1 ⊗ ϒ0l−1 , t= k=1 l=1 ˜ = m n n−1 , 2 p = 1, 2, . . . , t. (−1 )k−1 skl ϒ0l−1 . k=1 l=1 Let δi (i = 1, 2, . . . , n ) and j ( j = 1, 2, . . . , m ) are eigenvalues of matrix and Y respectively, then the eigenvalues of S are obtained (refer to [12,21]) λi j = m n skl δik−1 l−1 . j k=1 l=1 Lemma 1. S is invertible if and only if f (δi , f (δi , j ) = λi j = m n j ) = 0 (i = 1, 2, . . . , n, j = 1, 2, . . . , m ), where skl δik−1 l−1 . j (8) k=1 l=1 Let σi j = | f (δi , j )|, i = 1, 2, . . . , n, j = 1, 2, . . . , m. κ= (9) max{σi j } . min{σi j } Theorem 1. If S = BSCCB(s11 , . . . , s1m , . . . , sn1 , . . . , snm ), then σ11 , . . . , σ1m , σ21 , . . . , σ2m , . . . , σn1 , . . . , σnm are the singular values of the matrix S. Proof. S∗ is the conjugate transpose of S, thus ⎛ S1∗ ⎜ S2∗ ⎜ . S∗ = ⎜ ⎜ .. ⎝S∗ n−1 Sn∗ −Sn∗ S1∗ .. . ∗ Sn−2 ∗ Sn−1 ··· ··· .. . ··· ··· −S3∗ −S4∗ .. . S1∗ S2∗ ⎞ −S2∗ −S3∗ ⎟ .. ⎟ ⎟ . ⎟. −Sn∗ ⎠ S1∗ S is a normal matrix as SS∗ = S∗ S. From Theorem 2.5.4 in [22], matrix S is a unitarily diagonalizable matrix. Then U ∗ SU = = diag(λ11 , . . . , λ1m , . . . , λn1 , . . . , λnm ), (10) where U ∈ Mmn is a unitary matrix and λi j (i = 1, 2, . . . , n, j = 1, 2, . . . , m ) are the eigenvalues of matrix S. Take the conjugate transpose at both sides of the Eq. (10), U ∗ S∗U = ∗ = diag(λ11 , . . . , λ1m , . . . , λn1 , . . . , λnm ), then U ∗ (S∗ S )U− = diag(|λ11 |2 , . . . , |λ1m |2 , . . . , |λn1 |2 , . . . , |λnm |2 ). For any i = 1, 2, . . . , n, j = 1, 2, . . . , m, |λij |2 are the eigenvalues of the matrix S∗ S. From the deﬁnition of singular values, the singular values of S are 1 σi j (S ) = λi j (S∗ S ) 2 = |λi j |. Recall the Eqs. (8) and (9), the proof is completed. Z.-L. Jiang, X. Tang / Applied Mathematics and Computation 277 (2016) 1–9 5 From the deﬁnition of the spectral norm of matrix, we can get S2 = max [λi j (S∗ S )] 2 = max{σi j }. 1≤i≤n 1≤i≤n 1 1≤ j≤m 1≤ j≤m Let S be the perturbation of the coeﬃcient matrix S and b be the perturbation of the vector b, S = BSCCB(τ s11 , . . . , τ s1m , . . . , τ sn1 , . . . , τ snm ) with the following form ⎛ S1 ⎜−Sn S = ⎜ ⎝ ... −S2 ⎞ Sn Sn−1 ⎟ .. ⎟ ⎠, . Sn−1 ··· S1 .. . ··· ··· .. . −Sn S1 and for any k = 1, 2, . . . , n, ⎛ τ sk1 ⎜τ skm Sk = ⎜ ⎝ ... τ sk2 τ sk(m−1) ··· τ sk1 .. . ··· ··· .. . τ skm τ skm ⎞ τ sk(m−1) ⎟ ⎟. .. ⎠ . τ sk1 Now let b = b + b, S = S + S, f (δi , j) = m n b = τ b, (skl + τ skl )δik−1 l−1 . j k=1 l=1 If m n |τ skl | < min {σi j }, 1≤i≤n k=1 l=1 then 1≤ j≤m m n (skl + τ skl )δik−1 l−1 f (δi , j ) = j k=1 l=1 n m m n k−1 l−1 ≥ skl δi − |τ skl ||δi |k−1 | j |l−1 j k=1 l=1 k=1 l=1 ≥ min {σi j } − 1≤i≤n 1≤ j≤m m n |τ skl | > 0, k=1 l=1 S is an invertible matrix via Lemma 1. Let σmin = min {σi j }, η = 1≤i≤n m n |τ skl |. k=1 l=1 1≤ j≤m By Sx = b, S x = b, we obtain S−1 τ b + S−1 (S − S )x, x−x= S−1 b − S−1 b = τ b2 S − S2 x2 + , σmin − η σmin − η S − S2 S − S2 τ b2 S 2 x − x2 τ b2 ≤ + ≤ + , x 2 (σmin − η )x2 σmin − η σmin − η b2 S 2 S−1 2 τ b2 + S−1 2 S − S2 x2 ≤ x − x2 ≤ where S2 = max {σi j }. 1≤i≤n 1≤ j≤m S2 = | − 1| S − S = S is a BSCCB matrix apparently, and S − S − S2 = S − S2 . So S − S2 ≤ max 1≤i≤n 1≤ j≤m m n k=1 l=1 |τ skl ||δi |k−1 | j |l−1 = m n k=1 l=1 |τ skl | = η. 6 Z.-L. Jiang, X. Tang / Applied Mathematics and Computation 277 (2016) 1–9 Theorem 2. Let S, S, b, η, σmin are deﬁned as above. If η < σ min , then the bound of relative error of the BSCCB linear system is σmax η x − x2 τ b2 ≤ + , x 2 σmin − η b2 σmax (11) σmax = S2 . (12) where 3.2. Optimal backward perturbation bound of the BSCCB linear system Let x be approximate solution to Sx = b and let ≡ {(S, b)|(S + S ) x = b + b}, ζ ( inf S, b, x) ≡ (S,b)∈ (S + S ) x = b + b, which equals to (S, b) x −1 = b − S x. Due to [17], we can get b − S x 2 ζ ( ( · is unitary invaritant norm ). x) = 1 + x22 Let x be an approximate solution to Sx = b, S is deﬁned in (1), ≡ {(S, b)|(S + S ) x = b + b, S is a BSCCB matrix}, ζ ( x) ≡ inf (S,b∈) {S, bF }. x − b. So = φ (as S = 0 is a BSCCB matrix), b = (S + S ) ζ 2 ( x) = inf (S,b)∈ {S2F + S x + S x − b2F }. Since S2F = mn m n (τ skl )2 , k=1 l=1 and S = Q J m n τ skl k−1 ⊗ ϒ0l−1 Q TJ . 0 k=1 l=1 The question will be analyzed in two different situations, 1° n is even ⎛ 2 ⎞ τ 11 2 T . ⎝ ⎠Q J .. Sx + S x − bF = x + S x − b Q J τ tt F ⎛ 2 ⎞⎛ ( 0 ) ⎞ τ 11 x1 ⎝ ⎜ . ⎠⎝ .. ⎟ .. = − r 0 . ⎠ τ tt x (0 ) t F Z.-L. Jiang, X. Tang / Applied Mathematics and Computation 277 (2016) 1–9 7 ⎛ 2 ⎞ m n ( 0 ) k−1 l−1 τ s φ ⊗ ϒ x kl 1 0 1 ⎜ k=1 l=1 ⎟ ⎜ ⎟ ⎜ ⎟ . .. = ⎜ − r ⎟ 0 ⎜ ⎟ n m ⎝ ⎠ ( 0 ) k−1 l−1 k=1 l=1 τ skl φt ⊗ ϒ0 xt F 2 = M (τ s11 , . . . , τ s1m , . . . , τ sn1 , . . . , τ snm )T − r0 , F where r0 = Q TJ (b − S x ), Q TJ x = x1(0) ⎛ ··· ⎞ xt(0) T , M = (M1 , M2 , . . . , Mn ), M1,k,1 · · · M1,k,m .. ⎠, M k−1 .. Mk = ⎝ ... ⊗ ϒ0l−1 x(p0) , p,k,l = φ p . . Mt,k,1 · · · Mt,k,m n t = , p = 1, 2, . . . , t, k = 1, 2, . . . , n, l = 1, 2, . . . , m. 2 2° n is odd ⎛ 2 ⎞ τ 11 .. ⎜ ⎟ 2 T . ⎜ ⎟ Sx + Sx − bF = Q J ⎝ Q J x + Sx − b ⎠ τ tt τ ˜ F ⎛ 2 ⎞⎛ (0) ⎞ τ 11 x1 ⎜ ⎜ . ⎟⎜ .. ⎟ .. ⎜ ⎟ ⎟⎜ . ⎟ − r0 = ⎠⎝ (0) ⎠ ⎝ τ x tt t−1 (0 ) τ ˜ xt F ⎛ n m 2 ⎞ τ s φ k−1 ⊗ ϒ0l−1 x1(0) ⎜ k=1 l=1 kl 1 ⎟ ⎜ ⎟ . ⎜ ⎟ .. ⎜ ⎟ ⎜ n m ⎟ = ⎜ ( 0 ) ⎟ − r0 τ skl φtk−1 ⊗ ϒ0l−1 xt−1 ⎜ ⎟ ⎜ k=1 l=1 ⎟ ⎜ ⎟ ⎝ ⎠ m n ( 0 ) l−1 (−1 )k−1 τ skl ϒ0 xt k=1 l=1 F 2 = M (τ s11 , . . . , τ s1m , . . . , τ sn1 , . . . , τ snm )T − r0 , F where r0 = Q TJ (b − S x ), M= T QM x = x1(0) M1 , M2 , . . . , Mn−1 , m ··· (0 ) xt−1 xt(0) (−1 )k−1 ϒ0l−1 xt(0) , l=1 M p,k,l = φ pk−1 ⊗ ϒ0l−1 x(p0) , Let t= n+1 , p = 1, . . . , t, 2 T , ⎛ M1,k,1 .. ⎝ Mk = . Mt,k,1 k = 1, . . . , n − 1, ⎛ 2 ⎞ τ s11 . 2 ⎝ ⎠ .. g(τ s11 , . . . , τ snm ) = mn (τ skl ) + M − r0 , k=1 l=1 τs m n nm and then ∂g = 0, ∂τ skl ··· .. . ··· F ⎞ M1,k,m .. ⎠, . Mt,k,m l = 1, . . . , m. (13) 8 Z.-L. Jiang, X. Tang / Applied Mathematics and Computation 277 (2016) 1–9 which equals to ⎛ τ s11 ⎞ (2mnImn + 2MT M )⎝ ... ⎠ − 2MT r0 = 0, τ snm 2 ∂ g = 2mnImn + 2MT M > 0. ∂ (τ skl )2 As g is a convex function about (τ s11 , . . . , τ snm ), so the point of the minimal value is ⎛ ⎞ τ s11 ⎝ .. ⎠ = (mnImn + MT M )−1 MT r0 . . τ snm Substituting it back into (13), we obtain the next theorem. Theorem 3. The square of the bound about the optimal backward perturbation of the BSCCB linear system is 2 ζ 2 (x ) = mnr0 M (mnImn + MT M )−2 MT r0 + M (mnImn + MT M )−1 MT − Imn r0 F . Let M = U V T be the singular value decomposition of M, U and V are real orthogonal matrices, = diag(σ1 , . . . , σnm ), σ j ≥ 0( j = 1, 2, . . . , nm ), then 2 ζ 2 (x ) = mnr0T U V T (mnImn + 2 )−2V U T r0 + U V T (mnImn + 2 )−1V U T − Imn r0 F 2 = mnr1T (mnImn + 2 )−2 r1 + (mnImn + 2 )−1 − Imn r1 F = mnr1T (mnImn + 2 )−2 r1 + m2 n2 r1T (mnImn + 2 )−2 r1 = r1T diag(β1 , β2 , . . . , βmn )r1 , where r1 = U T r0 , β j = mnσ j 2 +m2 n2 (mn+σ j 2 )2 = mn , mn+σ j 2 j = 1, 2, . . . , mn. σ 2 j Remark 1. As σ j 2 ≤ M2F = mnx̂22 , then 1 + x̂22 ≥ 1 + mn can be obtained, hence mn 1 ≥ . 1+x̂22 mn+σ j 2 From what we analyzed above, the following algorithm can be obtained. 4. Numerical example A simple numerical example is presented to verify the conclusion presented in this paper. Suppose that n = 3, m = 2 in the following example. If the coeﬃcient matrix of the BSCCB linear system is S = BSCCB(7, 8, 6, 2, 5, 9 ), and the constant vector b = (3, 5, 7, 2, 6, 9 )T . Now, three perturbations are given S1 = BSCCB(0.2, 0.1, 0.05, 0, 0.025, 0.15 ), b1 = (0.05, 0.3, 0.15, 0.2, 0, 0.07 )T , S2 = BSCCB(0.03, 0.015, 0.038, −0.02, 0.05, 0.045 ), b2 = (−0.04, 0, 0.025, −0.035, 0.015, −0.055 )T , S3 = BSCCB(0.003, 0.002, 0.0025, −0.0045, 0.002, −0.007 ), b3 = (0.003, −0.0035, 0.0015, −0.005, 0.0025, 0.001 )T . From the equation Ŝx̂ = b̂, the approximate solutions of Sx = b are ⎛ ⎞ 0.2976 ⎜−0.5172⎟ ⎜ 0.0673 ⎟ ⎟ x=⎜ ⎜−0.1179⎟, ⎝ ⎠ 0.6586 0.1771 ⎛ ⎞ 0.2687 ⎜−0.4881⎟ ⎜ 0.0387 ⎟ ⎟ x̂1 = ⎜ ⎜−0.0683⎟, ⎝ ⎠ 0.6640 0.1787 ⎛ ⎞ 0.2853 ⎜−0.5058⎟ ⎜ 0.0732 ⎟ ⎟ x̂2 = ⎜ ⎜ −0.1241 ⎟, ⎝ ⎠ 0.6726 0.1565 ⎛ ⎞ 0.2978 ⎜−0.5173⎟ ⎜ 0.0689 ⎟ ⎟ x̂3 = ⎜ ⎜−0.1195⎟, ⎝ ⎠ 0.6602 0.1756 where x is the solution of Sx = b and x̂i (i = 1, 2, 3 ) is the solution of (S + Si )x = b + bi , (i = 1, 2, 3 ), respectively. Z.-L. Jiang, X. Tang / Applied Mathematics and Computation 277 (2016) 1–9 9 Algorithm 1 step 1 Form the style spectral decomposition of the matrix and ϒ , = Q 0 Q T , ϒ = J ϒ0 J T . step 2 Form the block style spectral decomposition of the BSCCB matrix. step 3 Compute r = b − Sx̂. step 4 Compute r0 = Q TJ r. step 5 Compute ⎛ (0 ) ⎞ x1 ⎜ .. ⎟ ⎜ . ⎟ T Q J x̂ = ⎜ (0) ⎟. ⎝xt−1 ⎠ xt(0) step 6 Form M. step 7 Compute the singular value decomposition of M. step 8 Compute ζ 2 (x̂ ). Table 1 The numerical comparison for two kinds of perturbation analysis. Case 0 Case 1 Case 2 Case 3 0 0.0771 0.0341 0.0034 κ ζ1 (x̂ ) ζ2 (x̂ ) 7.5056 7.2431 7.4303 7.5049 0 0.3103 0.1020 0.0074 0 0.7834 0.2773 0.0247 Based on the Algorithm 1, we obtain the Table 1, where max{σ } is the relative error of the BSCCB linear system, κ = min{σ i j} is the condition number, ζ1 (x̂)= b−Sx̂2 and ζ2 (x̂ ) can be obtained from the Algorithm. ij 1+x̂2 2 From the numerical example, the accuracy of the conclusion and the effectiveness of the algorithm are veriﬁed. References [1] D. Bertaccini, M.K. Ng, Skew-circulant preconditioners for systems of lmf-based ode codes, Numerical Analysis and Its Applications, Lecture Notes in Computer Science, 1988, 2001, pp. 93–101. [2] R.-H. Chan, X.-Q. Jin, Circulant and skewcirculant preconditioners for skew-hermitian type toeplitz systems, BIT Numer. Math. 31 (1991) 632–646. [3] R.-H. Chan, K.P. Ng, Toeplitz preconditioners for hermitian toeplitz systems, Linear Algebra Appl. 190 (1993) 181–208. [4] T. 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Applied Mathematics and Computation 274 (2016) 220–228 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc Explicit group inverse of an innovative patterned matrix Zhao-lin Jiang, Dan-dan Wang∗ Department of Mathematics, Linyi University, Linyi 276005, PR China a r t i c l e i n f o Keywords: Patterned matrix RFPL-Toeplitz matrix Group inverse Singularity a b s t r a c t In this paper, we present an innovative patterned matrix, RFPL-Toeplitz matrix, is neither the extension of Toeplitz matrix nor its special case. We show that the group inverse of this new patterned matrix can be represented as the sum of products of lower and upper triangular Toeplitz matrices. First, the explicit expression of the group inverse of an RFPL-Toeplitz matrix is obtained. Second, the decomposition of the group inverse is given. Finally, an example demonstrates availability of the two methods for the group inverse. © 2015 Elsevier Inc. All rights reserved. 1. Introduction As is well-known, Toeplitz matrix family are also patterned matrix family, have important applications in various disciplines including the elliptic Dirichlet-periodic boundary value problems [1], sinc discretizations of partial and ordinary differential equations [2–7], signal processing [8], numerical analysis [8], system theory [8], etc. Citations of a large number of results have been made in the books of Heining and Rost [9] and of Iohvidov [10]. Mukherjee and Maiti [11] showed that Toeplitz matrices often arise in applications in econometrics, statistics, psychometrics, multichannel ﬁltering, structural engineering, reﬂection seismology, etc., and it is desirable to have techniques which exploit their special structure. Possible applications of the results related to their determinant, inverse, and eigenvalue problem are suggested. It is an ideal research area and hot issue for generalized inverses of Toeplitz matrix. In [12], some methods of characterizing and computing generalized inverses of Toeplitz matrices over ﬁelds and over rings with the extended Rao condition were presented. In [13], Heinig exhibited that the reﬂexive generalized inverses of Toeplitz mosaic matrices are Bezoutians. The group inverse of a structured matrix was studied in [14]. In [15], Diao and Wei discussed the structured perturbations of group inverse. In [16], Wei and Diao showed that the group inverse of a real singular Toeplitz matrix can be represented as the sum of products of lower and upper triangular Toeplitz matrices. In [17], the Moore–Penrose inverse for a matrix bordered by a row and a column was obtained by Hartwig. In [18], Xu presented the Moore–Penrose inverses of Toeplitz matrices can be represented as a sum of products of lower and upper triangular Toeplitz matrices. In [19], Heinig and Hellinger showed that the Moore–Penrose inverses of Hankel matrices were generalized Bezoutians. In [20], based on Bezoutian representations of A+ , the fast algorithms for obtaining the Moore–Penrose inverse of a square Toeplitz matrix A was given by Heinig and Hellinger. In [21], Adukov discussed a generalized inversion of ﬁnite rank Hankel operators and Hankel or Toeplitz operators with block matrices having ﬁnitely many rows. In [22], Wei modiﬁed the algorithm to obtain the Moore–Penrose inverse of a rank-deﬁcient Toeplitz matrix. In [23], the displacement rank of the Drazin inverse was studied by Diao et al. The explicit Drazin inverse of singular Toeplitz matrix was presented in [24]. In [25], the representations for the Drazin inverse of 2 × 2 block matrices was considered by Li and Wei. ∗ Corresponding author. Tel.: +86 15628607659. E-mail address: wdd1703@sina.com (D.-d. Wang). http://dx.doi.org/10.1016/j.amc.2015.11.021 0096-3003/© 2015 Elsevier Inc. All rights reserved. Z.-l. Jiang, D.-d. Wang / Applied Mathematics and Computation 274 (2016) 220–228 221 Deﬁnition 1. The row ﬁrst-plus-last (RFPL) Toeplitz matrix with the ﬁrst row (t0 , t−1 , t−2 , . . . , t2−n , t1−n ) and the ﬁrst column (t0 , t1 , t2 , . . . , tn−2 , tn−1 )T is meant a square matrix of the form ⎛ t0 ⎜ ⎜ t1 ⎜ ⎜ ⎜ t2 ⎜ . ⎜ . ⎜ . ⎜ ⎝ tn−2 tn−1 t−1 t−2 t0 + t1−n t−1 t1 + t2−n .. . t0 + t1−n ··· .. . .. . .. . .. . ··· t1 + t2−n .. . tn−3 + t−2 tn−3 + t−2 tn−2 + t−1 ⎞ t2−n .. . .. . .. . t1−n t2−n .. . t0 + t1−n t1 + t2−n t−1 t0 + t1−n t−2 ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , (1) n×n denoted by TRF PL [ f r(t0 , t−1 , t−2 , . . . , t2−n , t1−n ); f c(t0 , t1 , t2 , . . . , tn−2 , tn−1 )T ]. It is particular that the RFPL-Toeplitz matrix is neither the extension of Toeplitz matrix nor its special case and it is a total new patterned matrix. For an RFPL-Toeplitz matrix TRFPL in (1) can be represented as following three splits: (i) ⎛ t0 + t1−n ⎜ ⎜t1 + t2−n ⎜ 1 ⎜ TRF PL = t +t t0 + t1−n ⎜ ⎜ 2 . 3−n ⎝ . . tn−1 + t0 ⎛ t0 + t1−n ... .. . t1 + t2−n .. . tn−2 + t1 t0 + t1−n .. . ··· 0 t0 + t1−n ⎜ 0 t−1 t0 + t1−n t−2 t−1 0 .. . 0 0 .. . 0 t0 + t1−n .. . 0 ⎜ ⎜ ×⎜ ⎜ ⎝ ⎛ t1 + t2−n ... .. . t2 + t3−n .. . tn−1 + t0 t1 + t2−n .. . ··· t1−n ⎜ ⎜t2−n ⎜ 1 ⎜ − t t0 + t1−n ⎜ ⎜ 3−n ⎝ .. . t0 ... ... .. . .. . ... 0 0 0 0 .. . .. . t1 + t2−n 0 ⎞ t1−n t2−n .. . t−1 t0 + t1−n ⎞ 0 .. . t0 + t1−n ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 0 0 0 .. . .. . t2 + t3−n 0 ⎞⎛ t0 + t1−n ⎟⎜ 0 ⎟ 0 .. . t1 + t2−n ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎠⎝ ... ... .. . .. . ... 0 0 t−1 t−2 0 .. . 0 0 .. . 0 t−1 .. . 0 ⎛ 0 ... .. . 0 ⎞ t1−n ⎟ .. ⎟ ⎟ . ⎟. t−2 t−1 ⎟ ⎠ (2) (ii) ⎛ t0 ⎜ ⎜ t1 1⎜ TRF PL = ⎜ t t0 ⎜ ⎜ .2 ⎝ . ⎛ ... .. . 0 t0 t1 .. . . tn−1 tn−2 0 ⎜0 t−1 0 t−2 t−1 0 0 .. . 0 0 .. . 0 ⎜ ⎜ ×⎜0 ⎜. ⎝ .. t0 .. . ··· ... ... .. . .. . ... ⎞⎛ t0 ⎟⎜ 0 0 ⎟⎜ ⎟⎜ ⎟ 0 0 ⎟⎜ ⎜. .. ⎟ ⎠⎝ .. 0 0 0 .. . .. . t1 . t0 t−1 0 t2−n .. . t−1 t1−n .. . ··· 0 .. . 0 t0 .. . 0 ⎛ ⎜ 3−n ⎝ .. . t0 t1−n ... .. . t−2 t−1 ⎞ t1−n t1−n ⎜ t2−n ⎟ ⎜t 2−n .. ⎟ ⎟ ⎜ ⎜ . ⎟ + ⎜t ⎟ ⎠ 0 ... ... .. . .. . ... t−1 t0 0 ⎞ 0 t1−n ⎜ t2−n ⎟ ⎜ t1 .. ⎟ ⎟ 1⎜ . ⎟− ⎜ t ⎟ t0 ⎜ ⎜ .2 ⎠ ⎝ . t−1 . t0 t n−1 0 0 .. . .. . t2−n 0 ⎞⎛ 0 ⎟⎜0 0 ⎟ ⎟⎜ ⎟⎜0 0 ⎟⎜ ⎜. .. ⎟ ⎠⎝ .. . t1−n 0 0 t1 .. . 0 .. . ··· tn−2 0 1 0 0 0 .. . 0 1 .. . 0 ... ... .. . .. . ... ⎞ 0 0 0 .. . .. . t1 0⎟ ⎟ ⎟ ⎟ 0⎟ .. ⎟ ⎠ . 0 ⎞ 0 0⎟ .. ⎟ ⎟ . ⎟. ⎟ ⎠ 0 1 (3) 222 Z.-l. Jiang, D.-d. Wang / Applied Mathematics and Computation 274 (2016) 220–228 (iii) ⎛ t0 + t1−n ⎜t1 + t2−n ⎜t + t 2 3−n TRF PL = ⎜ ⎜ . ⎝ .. tn−1 + t0 t−1 t0 + t1−n t1 + t2−n .. . tn−2 + t−1 t−2 t−1 t0 + t1−n .. . tn−3 + t−2 ⎞ ··· ··· ··· .. . ··· t1−n t2−n ⎟ ⎟ t3−n ⎟ − α eT , 1 ⎟ .. ⎠ . t0 + t1−n (4) where α = (t1−n t2−n t3−n . . . t0 )T , eT1 = (1 0 0 . . . 0). So (4) can be expressed as TRF PL = T − α eT1 , T is a Toeplitz matrix. Obviously, every RFPL-Toeplitz matrix TRFPL can be expressed as a sum of a Toeplitz matrix and a matrix with rank one. Deﬁnition 2 [16]. For an n × n singular matrix M, there exists a unique matrix X = MD , the Drazin inverse [26] of M satisﬁes the following equations XMX = X, MX = XM, Mv+1 X = Mv , where v is the index of M. The index of the matrix M is the smallest nonnegative integer v such that rank(Mv ) = rank(Mv+1 ). When the index of M is equal to one, the Drazin inverse is called group inverse, denoted by M . In particular, if M is nonsingular, then M = M−1 . 2. The group inverse formula of TRF PL [ f r(t0 , t−1 , t−2 , . . . , t2−n , t1−n ); f c(t0 , t1 , t2 , . . . , tn−2 , tn−1 )T ] In this section, we give two algorithms for the group inverse of an RFPL-Toeplitz matrix. We get the explicit expression and the decomposition of the group inverse of an RFPL-Toeplitz matrix. Theorem 1. For an RFPL-Toeplitz matrix TRFPL deﬁned in Deﬁnition 1, dividing TRFPL to blocks t0 c TRF PL = bT , T1 (5) where bT = (t−1 , t−2 , t−3 . . . t1−n ), bT denotes the transpose of b, c = (t1 , t2 , t3 . . . tn−1 )T and ⎛ t0 + t1−n t−1 ⎜ ⎜ t1 + t2−n ⎜ .. T1 = ⎜ . ⎜ ⎜ ⎝ t0 + t1−n t1 + t2−n .. . tn−3 + t−2 tn−3 + t−2 tn−2 + t−1 t−2 .. . .. . .. . ··· ··· .. . .. . t2−n .. . t0 + t1−n t1 + t2−n t−1 t0 + t1−n t−2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (n−1)×(n−1) is a real singular Toeplitz matrix. Assume the matrices T1 and TRFPL have index one. For the ease of the presentation, we consider the following notation. Let g = T1 c, hT = bT T1 , z = t0 − bT T1 c, s = (I − T1 T1 )c, rT = bT (I − T1 T1 ), α = bT (I − T1 T1 )c, β = 1 + bT (T1 )2 c, γ = hT g = bT (T1 )2 c, δ = bT (T1 )3 c = hT T1 g, q = T1 g = (T1 )2 c, pT = hT T1 = bT (T1 )2 . 2 ) = rank(T ) if and only if β = 0, z = 0 and s = r = 0, in which case 1. rank(TRFPL ) = rank(TRF 1 PL TRF = PL hT Ng Ng hT N , N 1 1 where N = T1 − β T1 ghT − β ghT T1 + δ T gh . β2 2 ) = rank(T ) + 1 if and only if z = 0, and at least one of s and r is zero vector, in which case 2. rank(TRFPL ) = rank(TRF 1 PL TRF = PL 1 z 1 z ( βz s − g) 1 z ( βz rT − hT ) T1 + 1z ghT − 1z (T1 + βz I)grT − 1z shT (T1 + βz I) . Z.-l. Jiang, D.-d. Wang / Applied Mathematics and Computation 274 (2016) 220–228 223 2 ) = rank(T ) + 2 if and only if α = 0, in which case 3. rank(TRFPL ) = rank(TRF 1 PL TRF = PL 1 T 0 αr 1 T1 − α1 grT − α1 shT − αz2 srT αs . Proof. For an RFPL-Toeplitz matrix TRFPL , we have TRF PL = where ⎛ 1 ⎜0 ⎜. In−1 = ⎜ ⎜ .. ⎝0 0 bT T1 = 0 1 .. . 0 0 ... ... .. . ··· ... t0 c 0 T1 bT 1 0 In−1 c t0 0 1 In−1 , 0 ⎞ 0 0 .. . 1 0 0 0⎟ .. ⎟ ⎟ . .⎟ ⎠ 0 1 (n−1)×(n−1) By using the property of group inverse(see (3) and (4) in [16]), we have TRF = PL 0 1 0 In−1 T1 bT c t0 0 1 In−1 . 0 (6) 2 ) = rank(T ), it follows from (3) and the ﬁrst result of Lemma 1 in [16] that 1. If rank(TRFPL ) = rank(TRF 1 PL TRF = PL 0 1 0 In−1 hT Ng Ng = N hT N Ng hT Ng 0 1 In−1 0 hT N , N where 1 N = T1 − T ghT − β 1 1 β ghT T1 + δ T gh . β2 2 ) = rank(T ) + 1, it follows from (3) and the second result of Lemma 1 in [16] that 2. If rank(TRFPL ) = rank(TRF 1 PL TRF = PL 0 In−1 1 z 1 z = 1 z 1 z A 1 0 ( βz s − g) ( βz rT − hT ) 1 z ( βz s − g) 1 z ( βz rT − hT ) 0 1 In−1 0 , A where 1 T 1 1 β β gh − (T1 + I)grT − shT (T1 + I). z z z z z A = T1 + 2 ) = rank(T ) + 2, it follows from (3) and the third result of Lemma 1 in [16] that 3. If rank(TRFPL ) = rank(TRF 1 PL TRF = PL 0 In−1 = 1 0 T1 − α1 grT − α1 shT − αz2 srT αs 1 T 0 αr 0 1 In−1 0 1 T 0 αr 1 T1 − α1 grT − α1 shT − αz2 srT αs 1 . Theorem 2. For an RFPL-Toeplitz matrix TRFPL deﬁned in Deﬁnition 1, we denote that (xT , x0 ) and (yT , y0 )T are the ﬁrst row vector and , respectively. Then x0 = y0 and the group inverse TRF can be expressed by ﬁrst column vector of the group inverse TRF PL PL TRF = PL x0 y T1 − gxT − yhT − xT , λgh − μspT − μqrT − κ yxT T where g, h, s, p, q, r, α , β , δ , z are deﬁned in Theorem 1 and x0 , y0 , λ, μ, κ , xT , y are deﬁned in the following table: (7) 224 Z.-l. Jiang, D.-d. Wang / Applied Mathematics and Computation 274 (2016) 220–228 λ μ κ δ β2 δ β2 0 0 rank(T1 )+1 1 z 1 z 1 z 0 rank(T1 )+2 0 0 0 z Case 2 rank(TRF PL ) = rank(TRF PL ) x0 = y0 1 rank(T1 ) 2 3 Proof. It follows from the ﬁrst result of Theorem 1 that hT Ng Ng TRF = PL hT N N = xT x0 y T1 − β1 ghT − β1 ghT T1 − βδ2 ghT 2γ , γ2 where xT = hT N, y = Ng and x0 = hT Ng = δ(1 − β + β 2 ) = βδ2 . Then gxT = ghT N δ T γ T δ·γ T gh − gh + gh β β β2 1 δ = ghT T1 − 2 ghT . β β = ghT T1 − It also get that yhT = 1 T1 ghT − β δ T gh . β2 Therefore N =T1 − 1 T ghT − β 1 1 β =T1 − gxT − yhT − ghT T1 + δ T gh . β2 θ T gh β2 If choose λ = βρ2 , μ = 0 and κ = 0, then it is easy to obtain TRF = PL xT x0 y T1 − gxT − yhT − βσ2 ghT x0 y T1 − gxT − yhT − = xT . λghT − uqrT − uspT − kyxT Using the second result of Theorem 1, we derive g gx = z T z and 1 yh = z T Thus β T β z TRF = PL = s − g hT = 1 z β z ghT , z grT − 2 β z2 shT − ghT . z T1 + 1z ghT − 1z 1 β T r − hT z z T1 + βz I grT − 1z shT ( ( ) ) (T1 + βz I) xT x0 y T1 − 1z gxT − 1z T1 grT − zβ2 grT − 1z shT T1 − zβ2 shT x0 y T1 − gxT − yhT − 1z ghT − 1z lrT − 1z spT x0 y T1 − gxT − yhT − = = 1 β ( s − g) z z = r − hT xT if we select λ = μ = 1z , κ = 0. xT , λgh − μqrT − μspT − κ yxT T xT y hT N 1 β T r − hT z z 1 T ( αr Ng ) 1 z 1 ( βz s − g) αs Z.-l. Jiang, D.-d. Wang / Applied Mathematics and Computation 274 (2016) 220–228 225 Finally, using the third result of Theorem 1, we have gxT = 1 α Therefore 1 grT , yhT = 1 T T1 − α1 grT − α1 shT − αz2 srT αr x0 y T1 − gxT − yhT − zyxT x0 y T1 − gxT − yhT − 1 = = shT . 0 αs TRF = PL α xT xT , λghT − μqrT − μspT − κ yxT if we choose λ = μ = 0, κ = z. Theorem 3. Using the following notations of the vectors: x = (xn−1 , . . . , x1 )T , h = (hn−1 , . . . , h1 )T , r = (rn−1 , . . . , r1 )T , p = ( pn−1 , . . . , p1 )T , y = (y−n+1 , . . . , y−1 )T , g = (g−n+1 , . . . , g−1 )T , s = (s−n+1 , . . . , s−1 )T , q = (q−n+1 , . . . , q−1 )T . Let TRFPL be deﬁned in Deﬁnition 1. If TRFPL fulﬁlls the condition of Theorem 1, then the group inverse TRF of TRFPL can be expressed in PL the following formula: TRF =κ( PL 3 − μ( where ⎛ 1 10 11 − 0 ⎜y−n+1 ⎜y ⎜ −n+2 1 =⎜ ⎝ ... ⎛ 4 − y−1 0 −1 g−1 −1 ⎜0 ⎜ . ⎜ 7 = ⎜ .. ⎝0 0 ⎛ 0 ⎜0 ⎜. ⎜ 9 = ⎜ .. ⎝0 0 12 13 + ... ... ··· .. . ... ... ... ··· .. . ... −1 ⎛ 5 y−n+1 .. . y−2 0 0 x1 .. . xn−2 ⎜g−n+1 ⎜g ⎜ −n+2 5 =⎜ ⎝ ... )− 0 0 0 ⎜ x1 ⎜x ⎜ 2 3 =⎜ ⎝ ... xn−1 ⎛ 2 g−n+1 .. . g−2 hn−1 −1 .. . 0 0 hn−1 0 .. . 0 0 hn−2 hn−1 .. . ... ... 6 − 3 14 ⎞ ⎞ 0 0 0 .. . x1 ... ... .. . ··· ... 17 4 − 8 )+ 6] 8 18 , ⎛ 0 ⎜0 ⎜. ⎜ 2 = ⎜ .. ⎝0 0 ⎛ λ[( 5 − I) 9 − xn−1 0 .. . 0 0 xn−2 xn−1 .. . 0 0 ... ... .. . ... ... ⎞ x1 x1 ⎟ .. ⎟ ⎟ . ⎟, xn−1 ⎠ 0 ⎞ 0 0⎟ ⎟ 0⎟, .. ⎟ .⎠ 0 0 ⎜0 ⎜. ⎜ 4 = ⎜ .. ⎝0 0 y−1 0 .. . 0 0 y−2 y−1 .. . 0 0 ... ... .. . ... ... y−n+1 y−n+2 ⎟ .. ⎟ ⎟ . ⎟, y−1 ⎠ 0 0 0 0 .. . 0 0⎟ ⎟ 0 ⎟, .. ⎟ . ⎠ −1 0 ⎜0 ⎜. ⎜ 6 = ⎜ .. ⎝0 0 g−1 0 .. . 0 0 g−2 g−1 .. . 0 0 ... ... .. . ... ... g−n+1 g−n+2 ⎟ .. ⎟ ⎟ . ⎟, g−1 ⎠ 0 0 ⎜ h1 ⎜h ⎜ 2 8 =⎜ ⎝ ... hn−1 0 0 h1 .. . hn−2 ... ... ... .. . ... 0 0 0 .. . h1 0 0⎟ ⎟ 0⎟, .. ⎟ .⎠ 0 0 0 ... ... ... .. . ... g−n+1 ... ... .. . ··· ... 16 0 0⎟ ⎟ 0⎟, .. ⎟ .⎠ 0 y−n+1 7 + 1 15 − 0 0 0 .. . ··· ··· ··· .. . ··· hn−2 hn−1 .. . 0 0 2 + ⎞ ⎞ h1 h2 ⎟ .. ⎟ ⎟ . ⎟, hn−1 ⎠ −1 ⎞ h1 h2 ⎟ .. ⎟ ⎟ . ⎟, hn−1 ⎠ 0 ⎛ ⎛ ⎛ 0 ⎜q−n+1 ⎜q ⎜ −n+2 10 = ⎜ ⎝ ... q−1 q−n+1 .. . q−2 ⎞ ⎞ 0 0 0 .. . q−n+1 ⎞ 0 0⎟ ⎟ 0⎟ .. ⎟ .⎠ 0 226 Z.-l. Jiang, D.-d. Wang / Applied Mathematics and Computation 274 (2016) 220–228 ⎛ ⎞ ⎛ 0 ⎜0 ⎜. ⎜ 11 = ⎜ .. ⎝0 0 rn−1 0 .. . 0 0 rn−2 rn−1 .. . 0 0 ... ... .. . ... ... rn−1 ⎠ 0 0 ⎜0 ⎜. ⎜ 13 = ⎜ .. ⎝0 0 q−1 0 .. . 0 0 q−2 q−1 .. . 0 0 ... ... .. . ··· ... q−n+1 q−n+2 ⎟ .. ⎟ ⎟ . ⎟, q−1 ⎠ 0 ⎜s−n+1 ⎜s ⎜ −n+2 14 = ⎜ ⎝ ... 0 ⎜0 ⎜. ⎜ 15 = ⎜ .. ⎝0 0 pn−1 0 .. . 0 0 pn−2 pn−1 .. . 0 0 p1 p2 ⎟ .. ⎟ ⎟ . ⎟, pn−1 ⎠ 0 0 ⎜0 ⎜. ⎜ 17 = ⎜ .. ⎝0 0 s−1 0 .. . 0 0 ⎛ ⎛ ⎛ 0 ⎜ r1 ⎜r ⎜ 2 12 = ⎜ ⎝ ... rn−1 ⎞ ⎛ ⎞ ... ... .. . ··· ... ... ... .. . ··· ... s−2 s−1 .. . 0 0 r1 r2 ⎟ .. ⎟ ⎟ . ⎟, ⎞ s−n+1 s−n+2 ⎟ .. ⎟ ⎟ . ⎟, s−1 ⎠ 0 0 0 r1 .. . rn−2 ... ... ··· .. . ... ⎞ 0 0 0 .. . r1 0 0⎟ ⎟ 0⎟, .. ⎟ .⎠ 0 s−1 s−n+1 .. . s−2 ... ... ··· .. . ... s−n+1 0 ⎜ p1 ⎜ p ⎜ 2 16 = ⎜ ⎝ ... pn−1 0 0 p1 .. . pn−2 ... ... ··· .. . ... 0 0 0 .. . p1 0 ⎛ ⎛ x0 ⎜0 ⎜0 ⎜ 18 = ⎜ ⎝ ... 0 0 0 0 x0 + x 1 x2 .. . xn−1 0 0 x0 + x1 .. . xn−2 0 0 0 .. . ··· ··· ··· .. . ··· ⎞ 0 0⎟ ⎟ 0⎟ .. ⎟ .⎠ 0 ⎞ 0 0⎟ ⎟ 0⎟, .. ⎟ .⎠ 0 ⎞ 0 0 ⎟ ⎟ 0 ⎟. .. ⎟ . ⎠ x0 + x 1 Proof. For an RFPL-Toeplitz matrix TRFPL , we have −1 TRF PL = where (TRF PL )T , ⎛ −1 ⎜0 ⎜ . =⎜ ⎜ .. ⎝0 1 f = −1 ... ... .. . ··· ... 0 0 .. . 1 0 ··· 0 ⎛ 0 ⎜0 ⎜. Jn−1 = ⎜ ⎜ .. ⎝0 1 ⎛ 0 ⎜0 ⎜. −1 =⎜ ⎜ .. ⎝0 1 (8) 0 1 .. . 0 0 0 ⎞ 1 0⎟ .. ⎟ ⎟ = f .⎟ Jn−1 0⎠ 0 n×n 1×(n−1) 1 , 0 (9) Jn−1 , e (10) , ⎞ 0 0 .. . 1 0 ··· ··· .. . ··· ··· 0 1 .. . 0 0 1 0⎟ .. ⎟ ⎟ , .⎟ ⎠ 0 0 (n−1)×(n−1) 0 0 .. . 1 0 ... ... .. . ··· ... 0 1 .. . 0 0 1 0⎟ .. ⎟ ⎟ = 0 .⎟ 1 0⎠ 1 n×n ··· 1 ⎞ and e= 0 0 1×(n−1) . By using (5) and the property of group inverse (see (3) and (4) in [16]), we have TRF = PL −1 (TRF PL )T . (11) It follows from (6)–(8) and Theorem 2 that TRF = PL Jn−1 x f + Jn−1 GT Jn−1 Jn−1 x (x0 + ex) f + (y + eG )Jn−1 x0 + ex T T , (12) Z.-l. Jiang, D.-d. Wang / Applied Mathematics and Computation 274 (2016) 220–228 227 where G = T1 − xgT − hyT − hgT λT − psT μT − rqT μT − xyT κ T . From (4), (8), (9) and the property of group inverse, we have the recurrence relation TRF = PL i+1, j+1 ( ) (TRF PL )i, j + + xi , (TRF PL )i, j + , j = 1, (13) j = 1, where =(xi g− j − g−n+i xn− j ) + (hi y− j − y−n+i hn− j ) + λ(hi g− j − g−n+i hn− j ) + μ( pi s− j − s−n+i pn− j ) + μ(ri q− j − q−n+i rn− j ) + κ(xi y− j − y−n+i xn− j ). It follows from (10) we can easily prove the formula of the group inverse TRF . PL 3. Numerical example In this section, an example demonstrates availability of the two algorithms for the group inverse. Example 1. TRFPL is a 5 × 5 matrix as following ⎛0 ⎜0 TRF PL = ⎜0 ⎝ 0 0 1 −1 0 0 1 0 1 −1 0 0 ⎞ −1 0⎟ 0 ⎟, ⎠ 1 −1 0 0 1 −1 0 dividing TRFPL to blocks TRF PL = where T1 = t0 c bT , T1 −1 1 0 0 0 −1 1 0 0 0 −1 1 1 0 0 −1 , t0 = 0, bT = (1, 0, 0 − 1), c = (0, 0, 0, 0)T . 2 ) = rank(T ) = rank(T 2 ). Then T It is easy to get rank(TRFPL ) = rank(TRF RFPL and T1 are singular matrices with index one. 1 PL 1 Method 1. By Theorem 1, we obtain x = ( − 0.1250, 0.1250, 03750, −0.3750, 0)T , y = (0, 0, 0, 0, 0)T , s = 0, r = (0.5, 0, 0.5, −0.5)T It is easy to check that z = t0 − bT T1 c = 0. By using the ﬁrst result of Theorem 2. We can obtain the group inverse TRF of TRFPL PL ⎛0 ⎜0 ⎜0 TRF = PL ⎝ 0 0 −0.125 −0.375 0.375 0.125 −0.125 0.125 −0.125 −0.375 0.375 0.125 0.375 0.125 −0.125 −0.375 0.375 ⎞ −0.375 0.375 ⎟ 0.125 ⎟. ⎠ −0.125 −0.375 Method 2. By using Theorem 3, we can get x0 = 0, λ = βδ2 = 0, μ = 0, κ = 0, g = T1 c = 0. It is easy to get the group inverse of TRFPL . ⎛−1 ⎞⎛ 0 0 0 −1 0 0 0 0 ⎟⎜0 0 ⎟⎜0 ⎠⎝ 0 0 −1 0 −0.125 0 0 0 0 ⎜0 + ⎜0 ⎝ 0 0 0 −0.375 0.375 0.125 −0.125 0 0 −0.375 0.375 0.125 0 0 0 −0.375 0.375 0 0 ⎟ 0 ⎟ ⎠ 0 −0.375 ⎜0 = ⎜0 ⎝ 0 0 −0.125 −0.375 0.375 0.125 −0.125 0.125 −0.125 −0.375 0.375 0.125 0.375 0.125 −0.125 −0.375 0.375 −0.375 0.375 ⎟ 0.125 ⎟. ⎠ −0.125 −0.375 ⎜0 ⎜0 TRF = − PL ⎝ 0 0 ⎛0 ⎛0 0 −1 0 0 0 0 0 −1 0 0 ⎞ ⎞ 0.125 −0.125 0 0 0 0.375 0.125 −0.125 0 0 ⎞ −0.375 0.375 ⎟ 0.125 ⎟ ⎠ −0.125 0 228 Z.-l. Jiang, D.-d. Wang / Applied Mathematics and Computation 274 (2016) 220–228 Acknowledgment We would like to thank Dr. Yimin Wei for his useful discussions on this paper. This project is supported by the Development Project of Science & Technology of Shandong Province (Grant no. 2012GGX10115), the National Natural Science Foundation of China (Grant nos. 11301251, 11301252) and the AMEP of Linyi University, China. References [1] Z.-Z. Bai, G.-Q. Li, L.-Z. 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European Journal of Medicinal Chemistry 136 (2017) 165e183 Contents lists available at ScienceDirect European Journal of Medicinal Chemistry journal homepage: http://www.elsevier.com/locate/ejmech Research paper Design, synthesis and antimicrobial evaluation of novel benzimidazole-incorporated sulfonamide analogues Hui-Zhen Zhang a, b, 1, Shi-Chao He a, 1, Yan-Jun Peng a, Hai-Juan Zhang b, Lavanya Gopala a, 2, Vijai Kumar Reddy Tangadanchu a, 3, Lin-Ling Gan c, **, Cheng-He Zhou a, * a Institute of Bioorganic & Medicinal Chemistry, Key Laboratory of Applied Chemistry of Chongqing Municipality, School of Chemistry and Chemical Engineering, Southwest University, Chongqing 400715, China b School of Pharmacy, Linyi University, Linyi 276000, China c School of Pharmacy, Chongqing Medical and Pharmaceutical College, Chongqing 401331, China a r t i c l e i n f o a b s t r a c t Article history: Received 11 December 2016 Received in revised form 24 April 2017 Accepted 30 April 2017 Available online 2 May 2017 A novel series of benzimidazole-incorporated sulfonamide analogues were designed and synthesized with an effort to overcome the increasing antibiotic resistance. Compound 5c gave potent activities against Gram-positive bacteria and fungi, and 2,4-dichlorobenzyl derivative 5g showed good activities against Gram-negative bacteria. Both of these two active molecules 5c and 5g could effectively intercalate into calf thymus DNA to form compoundDNA complex respectively, which might block DNA replication to exert their powerful antimicrobial activity. Molecular docking experiments suggested that compounds 5c and 5g could insert into base-pairs of DNA hexamer duplex by the formation of hydrogen bonds with guanine of DNA. The transportation behavior of these highly active compounds by human serum albumin (HSA) demonstrated that the electrostatic interactions played major roles in the strong association of active compounds with HSA, and which was also conﬁrmed by the full geometry calculation optimizations. © 2017 Elsevier Masson SAS. All rights reserved. Keywords: Benzimidazole Sulfonamide analogues Antibacterial Antifungal Calf thymus DNA HSA 1. Introduction The antibiotic resistance which was accelerated by the use and misuse of antimicrobial drugs has been a major global challenge for public health. During the past decades, a dramatic increase in human-pathogenic bacteria worldwide was seen due to resistance to one or multiple antibiotics. More and more infections caused by resistant microorganisms fail to respond to conventional treatment, and in some cases, even last resort antibiotics have lost their power. Recently, the World Health Organization titled a recent world health day as “Combat drug resistance: no action today means no * Corresponding author. ** Corresponding author. E-mail addresses: ganlinling2012@163.com (L.-L. Gan), zhouch@swu.edu.cn (C.-H. Zhou). 1 These two authors contributed equally to this work. 2 Postdoctoral fellow from Sri Venkateswara University, India. 3 Postdoctoral researcher from CSIR-Indian Institute of Chemical Technology, Hyderabad 500007, India. http://dx.doi.org/10.1016/j.ejmech.2017.04.077 0223-5234/© 2017 Elsevier Masson SAS. All rights reserved. cure tomorrow”, and triggered an increase in research activity, and several promising strategies have been developed to restore treatment options against infections by resistant bacterial pathogens [1]. Sulfonamides as artiﬁcial antifolic agents have been widely used for the prevention and cure of bacterial infections in biological systems and recently have evoked high favor in biology and medicine because of their diverse pharmacological activities [2] including antibacterial [3], antifungal [4], antiviral [5], antitumor [6], anti-inﬂammatory [7], and carbonic anhydrase inhibitors [8]. As the analogues of aminobenzoic acid, sulfonamides could compete with it to availably prevent the synthesis of nucleic acids and proteins, and then inhibit the growth of various microorganisms. Furthermore, sulfonamide compounds have attracted increasing research in supramolecular chemistry because they could combine the features of different fragments through the coordination of phenylamino and sulfonyl amino groups [9]. Particularly, Ag-sulfadiazine has been signiﬁcantly employed in burn therapy, which is better than the free ligand or AgNO3. Up to now, numerous sulfonamides bearing aromatic heterocycles such 166 H.-Z. Zhang et al. / European Journal of Medicinal Chemistry 136 (2017) 165e183 as isoxazole, thiazole, pyridazine and pyrimidine have been successfully developed and used in clinic like sulfadiazine, sulfachlorpyridazine, sulfathiazole and sulﬁsoxazole with excellent antimicrobial activities. This has stimulated considerable efforts towards the synthesis and development of completely new structural sulfonamide derivatives with excellent activity, broad spectrum and low toxicity. Our previous efforts have identiﬁed that the introduction of aromatic heterocycles such as imidazole [10], triazole [11], tetrazole [12], thiazole [13] and benzene-fused azoles like benzimidazole [14] and benzotriazole [15] into target molecules could highly improve the antimicrobial activities. 1,2,3-Triazole sulfonamide WXL-1 containing 2,4-diﬂuorobenzyl group was almost 32-fold more potent than precursor sulfonamide against Pseudomonas aeruginosa and Shigella dysenteriae [16]. The sulfonamide ZHZ1 bearing 2-methyl-5-nitroimidazole fragment displayed almost equivalent anti-P. aeruginosa efﬁciency to Chloromycin (MIC ¼ 16 mg/mL) (Fig. 1), and further research found that this compound could effectively intercalate into calf thymus DNA to form compoundDNA complex which might block DNA replication to exert their powerful antimicrobial activities. Moreover, the transportation behavior of this compound by human serum albumin (HSA) showed that electrostatic interactions played major roles in the strong association of compound ZHZ-1 and HSA [17]. In view of such stimulating properties and as an extension of our studies on the development of sulfonamide azoles, the methylene moiety as the bioisostere of amino group was introduced into the sulfonamide fragment with the aim to investigate the effect on antimicrobial activity [18]. Benzimidazole ring as the fused heterocycle of benzene and imidazole was structurally similar to purine, and its derivatives could compete with purine to inhibit the synthesis of nucleic acids and proteins inside the bacterial cell wall, and then kill the bacterial strains or inhibit their growth [19]. Additionally, ﬂuoro- and chloro-substituted benzyl groups were introduced into the target molecules in order to improve the pharmacological properties by enhancing the rate of absorption and the transportation of drugs in vivo [20]. The newly synthesized sulfonamide analogues were screened for their antibacterial and antifungal activities in vitro. DNA as one of the most important targets has attracted growing interest in investigating the interaction of small molecules with DNA to explore the possible antimicrobial action mechanism [21]. The interactions of most active compounds with calf thymus DNA were evaluated by UVevis absorption spectroscopy on a molecular level to explore their probable antimicrobial mechanisms. Moreover, molecular docking studies were employed to reconﬁrm the interaction behaviors between the prepared compounds and DNA hexamer duplex [22]. Additionally, the transportation behavior of the highly active molecules by human serum albumin (HSA) was Fig. 1. Structures of some antimicrobial sulfonamides. investigated by ﬂuorescence spectroscopy to preliminarily study their absorption, distribution and metabolism [23]. 2. Results and discussion 2.1. Chemistry The target benzimidazole-incorporated sulfonamide analogues were prepared from commercial acetanilide and chlorosulfonic acid. Their synthetic routes were outlined in Scheme 1. Acetanilide was reacted with chlorosulfonic acid to produce N-protected sulfonyl chloride 2, and then further treated by sodium sulﬁte and sodium bicarbonate to give p-acetylamino benzenesulﬁnic acid sodium salt 3. The latter was subsequently reacted with chloromethyl benzimidazole to afford benzimidazole-incorporated sulfonamide analogue 4. The N-alkylation of benzimidazole ring of compound 4 with halobenzyl halides in acetonitrile at 70 C using potassium carbonate as base respectively produced the target aralkyl benzimidazole sulfonamide analogues 5ah in 52.1e73.7% yields which showed that the substituents exhibited effect on the formation of target compounds to some extent. Generally, the chlorobenzyl halides gave higher yields than ﬂuorobenzyl ones. In this series, it was found that the strong electron-withdrawing 3-F group gave the lowest yield (52.1%), while 3-Cl substituted one provided the highest yield of 73.7%. Moreover, compound 4 was also reacted with a series of alkyl bromides to produce compounds 6ai with a large difference in yields of 24.2e71.2%. Furthermore, the N-alkylation of compound 4 with carbazole alkyl bromide was successfully performed to give carbazole alkylated benzimidazole sulfonamide analogue 7 in yield of 72.0%. Generally, it was thought that the presence of protons on the nitrogen atom of the sulfonamide skeleton was favorable for the bioactivity, and thus compounds 5e7 were further transformed into the deprotected sulfonamide analogues 8e10 in ethanol in the presence of sodium hydroxide in order to explore their effect on the bioactivities. 2.2. Analysis of spectra All the new compounds were characterized by IR, 1H NMR, 13C NMR, MS and HRMS spectra. Their spectral analyses were consistent with the assigned structures and listed in the experimental section. The mass spectra for benzimidazole compounds gave a major fragment of [MþH]þ according to their molecular formula. 2.2.1. IR spectra In IR spectra, all the synthesized benzimidazole sulfonamide analogues 4e7 gave broad absorption in 33693352 cm1 which indicated the presence of NH group of amide moiety, whereas two broad absorption between 3332 and 3307 and 32093185 cm1 were attributed to the NH group in compounds 8e10. The characteristic C¼N bands of benzimidazole ring in all benzimidazole derivatives appeared in the region between 1694 and 1639 cm1. All the other absorption bands were also observed at the expected regions. 2.2.2. 1H NMR spectra In 1H NMR spectra, compounds 4e7 gave singlets at 2.12e2.10 ppm assigned to the CH3 protons linked to the amide moiety. The singlets at 5.19e5.07 ppm of compounds 5e7 were assigned to the CH2 protons linked to benzimidazole ring, while the N-CH2 protons of alkyl chain in compound 6 gave lower shift signals at 4.35e4.21 ppm. The substitution of alkyl group by halobenzyl moiety to yield compound 5 led to downﬁeld shifts of N-CH2 H.-Z. Zhang et al. / European Journal of Medicinal Chemistry 136 (2017) 165e183 167 Scheme 1. Synthesis of benzimidazole sulfonamide analogues 4e10. Reagents and conditions: (i) chlorosulfonic acid, 0 C; (ii) sodium sulﬁte, sodium bicarbonate, 80 C; (iii) chloromethyl benzimidazole, acetone, 50 C; (iv) substituted halobenzyl halide, potassium carbonate, acetonitrile, 70 C; (v) alkyl bromide, potassium carbonate, acetonitrile, 70 C; (vi) 9-(5-bromopentyl)-9H-carbazole, potassium carbonate, acetonitrile, 70 C; (vii) 2 mol/L NaOH, ethanol, reﬂux. protons (5.63e5.56 ppm), which were higher than those of CH2 protons attached C2-benzimidazole ring because of the strong electron-withdrawing ability of halophenyl moieties and nitrogen atom in benzimidazole ring. The corresponding deprotected compounds 8 and 9 gave upﬁeld shifts of SO2-CH2 protons down to 4.90e4.87 and 5.03e4.88 ppm due to the absence of acetyl group respectively. Moreover, the peaks for the 3,5-H protons in the sulfonamide analogue ring in the protected compounds 4e7 appeared at d 7.76e7.67 ppm, whereas deprotection led to an upﬁeld shift to 6.61e6.58 ppm. In addition, all the other aromatic and aliphatic protons appeared at the appropriate chemical shifts and integral values. 2.2.3. 13C NMR spectra The 13C NMR spectral analyses were in accordance with the assigned structures. No large differences were found in the 13C NMR chemical shifts for the carbonyl carbon in compounds 5e7 (d 169.7e169.6 ppm). The signals for the phenyl 3,5-C in the sulfonamide analogue skeletons in derivatives 5e7 were observed at 119.0e118.9 ppm, respectively. It was noticeable that the deprotection of these compounds to compounds 8e10 resulted in upﬁeld 13 C shifts (5.9e5.8 ppm) of these two carbons because of the absence of the strong electron-withdrawing acetyl group. The signals at 56.3e54.5 ppm in compounds 4e10 were assigned to the methylene carbon which was connected with sulfonyl group. For compounds 5 and 8, the methylene carbon linked to the benzimidazole ring was appeared at 50.2e45.1 ppm. All the other carbons gave 13C peaks at the expected regions. 2.3. Biological activity The in vitro antimicrobial screening for all the synthesized compounds was evaluated against four Gram-positive bacteria (Staphylococcus aureus ATCC 6538, Methicillin-resistant Staphylococcus aureus N315 (MRSA), Micrococcus luteus ATCC 4698 and Bacillus subtilis ATCC 21216), four Gram-negative bacteria (Escherichia coli ATCC 8099, Pseudomonas aeruginosa ATCC 27853, Bacillus typhi and Bacillus proteus ATCC 13315) and ﬁve fungi (Candida albicans ATCC 76615, Candida mycoderma, Candida utilis, Saccharomyces cerevisia and Aspergillus ﬂavus) using two fold serial dilution technique recommended by National Committee for Clinical Laboratory Standards (NCCLS) with the positive control of clinically antimicrobial drugs Chloromycin, Norﬂoxacin and Fluconazole [24]. The values of ClogP, a partition coefﬁcient as a kind of measurement for hydrophobicity/lipophilicity, were calculated using ChemDraw Ultra 10.0 software integrated with Cambridge Software (Cambridge Soft Corporation). The antibacterial and antifungal data as well as ClogP values were depicted in Tables 1 and 2. 2.3.1. Antibacterial activity The in vitro antibacterial screening demonstrated that some of the newly synthesized compounds could effectively inhibit the growth of all the tested microorganisms and exhibit broad antimicrobial spectrum and potent antibacterial activity. Compound 4 168 H.-Z. Zhang et al. / European Journal of Medicinal Chemistry 136 (2017) 165e183 Table 1 ClogP values and antibacterial data as MIC (mg/mL) for compounds 4e10.a,b,c Compds 4 5a 5b 5c 5d 5e 5f 5g 5h 6a 6b 6c 6d 6e 6f 6g 6h 6i 7 8a 8b 8c 8d 8e 8f 8g 8h 9a 9b 9c 9d 9e 9f 9g 9h 9i 10 Chloromycin Norﬂoxacin ClogP 1.16 3.08 3.08 3.08 3.65 3.65 3.65 4.37 4.25 1.70 2.23 3.29 3.82 4.35 4.87 5.93 6.99 10.16 6.05 2.89 2.89 2.89 3.46 3.46 3.46 4.18 4.06 1.51 2.04 3.10 3.63 4.16 4.68 5.74 6.80 9.97 5.86 1.09 0.58 Gram-Positive bacteria Gram-Negative bacteria S. aureus MRSA B. subtilis M. luteus B. proteus E. coli P. aeruginosa B. typhi 32 512 512 4 32 32 512 64 64 128 128 32 512 8 64 512 512 128 256 64 512 16 16 32 256 8 512 64 256 512 128 512 512 512 512 512 512 8 8 128 128 256 16 32 128 512 128 512 128 64 128 512 512 512 512 512 512 512 64 128 512 32 512 256 128 128 128 128 512 512 512 512 512 512 512 128 16 1 32 64 512 16 32 128 256 16 128 32 16 64 128 256 128 512 512 512 512 16 512 512 16 512 8 32 512 64 32 512 512 512 512 512 512 512 32 32 2 128 32 512 64 64 64 128 128 256 128 256 32 256 512 256 512 512 512 128 32 512 512 64 256 8 64 256 128 256 256 512 128 512 512 256 256 512 8 1 64 128 256 32 512 128 256 32 128 64 64 256 256 256 256 512 512 512 512 256 256 128 256 256 256 128 512 64 256 512 256 512 512 512 512 512 512 32 4 64 256 128 32 32 64 128 8 256 256 128 256 512 64 512 512 256 512 512 4 128 256 128 32 64 16 512 64 64 512 512 512 512 512 512 512 64 16 1 32 128 256 256 16 256 512 16 512 32 128 4 512 512 256 512 512 512 16 512 512 16 16 256 256 32 512 64 256 512 128 512 512 512 512 512 512 16 1 128 64 128 16 16 128 64 4 512 64 32 32 512 512 512 512 512 512 64 16 512 16 16 512 512 128 64 128 32 512 512 512 512 512 512 512 64 32 1 a Minimal inhibitory concentrations were determined by micro broth dilution method for microdilution plates. S. aureus, Staphylococcus aureus (ATCC25923); MRSA, Methicillin-Resistant Staphylococcus aureus (N315); B. subtilis, Bacillus subtilis; M. luteus, Micrococcus luteus (ATCC4698); B. proteus, Bacillus proteus (ATCC13315); E. coli, Escherichia coli (JM109); P. aeruginosa, Pseudomonas aeruginosa; B. typhi, Bacillus typhi; C. albicans, Candida albicans (ATCC76615); C. mycoderma, Candida mycoderma; C. utilis, Candida utilis; S. cerevisia, Saccharomyces cerevisia; A. ﬂavus, Aspergillus ﬂavus. c ClogP values were calculated by ChemDraw Ultra 10.0. b exhibited moderate to good activity against the tested bacteria (MIC ¼ 32e128 mg/mL) in comparison with clinical drugs. Table 1 displayed the signiﬁcant effects of the substituents on the benzene ring on the biological activity. Noticeably, in this series of halobenzyl benzimidazole sulfonamide analogues, compound 5c with 4-ﬂuorobenzyl group gave the best activities against the tested Gram-positive bacteria with MIC values of 4e64 mg/mL. Particularly, its anti-S. aureus (MIC ¼ 4 mg/mL) activity was 2-fold more potent to clinical Chloromycin and Norﬂoxacin, and for B. subtilis strains, it also showed two times more active than Chloromycin (MIC ¼ 16 mg/mL). Especially, this compound displayed equivalent inhibition activity against MRSA to Chloromycin (MIC ¼ 16 mg/mL). The replacement of 4-ﬂuorobenzyl moiety by 2,4-dichlorobenzyl group, which yielded compound 5g, resulted in good bioactivities against Gram-negative bacteria with MIC values ranging from 4 to 32 mg/mL. Compound 5g showed eight times higher activity (MIC ¼ 4 mg/mL) than Chloromycin against B. typhi. However, the substitution of 4-ﬂuorobenzyl fragment in compound 5c by 3,4-dichlorobenzyl group which gave derivative 5h was not beneﬁcial for the antibacterial activities. In comparison with halobenzyl compounds 5ah, most of alkyl derivatives 6ai exerted relatively lower activities in inhibiting the growth of the tested strains. Compound 6c with pentyl group gave good anti-P. aeruginosa activity with MIC value of 4 mg/mL which was 4-fold more potent than Chloromycin. The replacement of pentyl chain by heptyl group, which yielded compound 6e, resulted in strong activity towards the tested S. aureus strains (MIC ¼ 8 mg/ mL). Additionally, when the alkyl substituents were extended to decyl, dodecyl and octadecyl groups, compounds 6fi gave relatively weaker inhibitory activity. These results suggested that the length of alkyl chain possessed remarkable effects on biological activities. The short alkyl chains with poor lipophilicity or long alkyl chains with poor hydrophilicity in these compounds might make them unfavorable for being delivered to the binding sites. Some of the deprotected halobenzyl benzimidazole sulfonamide analogues 8ah exerted relatively better activity than the corresponding protected ones to some extent in inhibiting the growth of the tested bacteria. Specially, compound 8f with 4-chlorobenzyl moiety was 32-fold or 16-fold more potent than its precursor against B. subtilis or M. luteus strains (MIC ¼ 8 and 8 mg/mL, respectively). 2,4-Dichlorobenzyl benzimidazole sulfonamide analogue 8g displayed equivalent bioactivity against S. aureus to Chloromycin and Norﬂoxacin with MIC value of 8 mg/mL. However, the deprotection of alkyl compounds 6ai, which yielded H.-Z. Zhang et al. / European Journal of Medicinal Chemistry 136 (2017) 165e183 Table 2 Antifungal data as MIC (mg/mL) for compounds 4e10.a,b,c Compds C. albicans C. mycoderma C. utilis S. cerevisiae A. ﬂavus 4 5a 5b 5c 5d 5e 5f 5g 5h 6a 6b 6c 6d 6e 6f 6g 6h 6i 7 8a 8b 8c 8d 8e 8f 8g 8h 9a 9b 9c 9d 9e 9f 9g 9h 9i 10 Fluconazole 32 256 256 64 32 512 128 512 64 64 64 256 512 512 512 512 512 512 512 128 128 32 16 512 256 16 64 256 64 512 512 512 64 512 512 512 512 1 64 512 512 64 512 256 32 512 512 256 16 16 512 512 512 512 512 512 32 8 512 512 512 512 16 512 512 512 4 512 512 512 256 512 512 64 16 4 16 32 512 16 512 512 512 128 512 64 64 32 512 128 512 16 512 512 512 4 512 128 512 512 32 2 256 64 64 512 512 512 512 8 512 512 512 8 32 512 32 64 512 128 64 128 32 512 16 64 512 512 8 512 512 512 32 8 8 32 512 256 64 256 256 512 32 128 512 64 512 512 256 64 64 16 128 64 256 64 8 256 512 32 16 64 128 32 512 256 512 512 512 512 128 32 256 256 32 16 128 4 512 32 8 256 512 512 512 512 512 512 128 256 a Minimal inhibitory concentrations were determined by micro broth dilution method for microdilution plates. b S. aureus, Staphylococcus aureus (ATCC25923); MRSA, Methicillin-Resistant Staphylococcus aureus (N315); B. subtilis, Bacillus subtilis; M. luteus, Micrococcus luteus (ATCC4698); B. proteus, Bacillus proteus (ATCC13315); E. coli, Escherichia coli (JM109); P. aeruginosa, Pseudomonas aeruginosa; B. typhi, Bacillus typhi; C. albicans, Candida albicans (ATCC76615); C. mycoderma, Candida mycoderma; C. utilis, Candida utilis; S. cerevisia, Saccharomyces cerevisia; A. ﬂavus, Aspergillus ﬂavus. c ClogP values were calculated by ChemDraw Ultra 10.0. compounds 9ai, led to weak antibacterial activities. Much research has reported that the introduction of carbazole heterocycle is helpful to improve antimicrobial potency [18]. In our work, it was found that the carbazole incorporated sulfonamide analogue 7 possessed considerable potentiality with MIC value of 16 mg/mL. Its corresponding deprotected compound 10 gave moderate activity against B. subtilis and B. typhi strains. 2.3.2. Antifungal activity The antifungal evaluation in vitro displayed that some prepared benzimidazole sulfonamide analogues exhibited good bioactivities against the tested fungal strains. Compound 4 showed moderate to good antifungal activities with MIC values ranging from 16 to 128 mg/mL. In the series of sulfonamide analogues 5ah, compound 5c bearing 4-ﬂuorobenzyl group exerted the relatively best activities in inhibiting the growth of all the tested fungal strains. The replacement of 4-ﬂuorobenzyl moiety by 2-chlorobenzyl fragment, which yielded compound 5d, resulted in good activity against Fluconazole-insensitive A. ﬂavus (MIC ¼ 8 mg/mL). Moreover, 3,4- 169 dichlorobenzyl substituted compound 5g gave MIC value of 16 mg/mL against A. ﬂavus, which was 16 times more active than reference drug. The length of aliphatic chain exhibited obvious effects on antifungal activity. The suitable length of alkyl chain to exert the best antifungal efﬁcacy was observed to be (CH2)2 moiety, and the propyl derivative 6b generally gave better activity in contrast with other alkyl derivatives with shorter or longer chain length. It displayed comparable inhibitory activity against S. cerevisiae to reference (MIC ¼ 16 mg/mL). Compound 6f containing octyl group was more effective than the reference at 8 mg/mL concentration. Additionally, the deprotected compound 9b also gave good activity in inhibiting the growth of the tested fungal with MIC values ranging from 4 to 64 mg/mL. The deprotected compound 8a bearing 2-ﬂuorobenzyl fragment exhibited good activities against C. mycoderma, C. utilis and S. cerevisiae with MIC values of 8, 4 and 8 mg/mL, respectively. Moreover, its anti-A. ﬂavus activity was 8-fold more potent than the reference drug (MIC ¼ 256 mg/mL). Notably, compound 8g with 2,4dichlorobenzyl moiety showed much stronger activities against C. utilis and A. ﬂavus (MIC ¼ 2 and 4 mg/mL) than Fluconazole (MIC ¼ 8 and 256 mg/mL). 2.3.3. Effect of ClogP values on antimicrobial activity Hydrophobic/lipophilic properties possessed an important role in exerting biological activity [25]. The ClogP values as one of the most important factors have been extensively employed to predict the bioactivity of target molecules. The calculated liposome/water partition coefﬁcients (ClogP) for all newly prepared compounds were shown in Table 1. The results demonstrated that compounds with lower values of ClogP showed better antimicrobial activities, and these compounds possessed comparable ClogP values to the reference drugs with equivalent potency. As shown in Table 1, the ClogP values of compounds 6aei generally increased with the increasing length of alkyl groups, and the enhancement of the antimicrobial activities was observed in compounds 6aee, but the bioactivities were decreased in compounds 6fei. These might be explained by the possibility that higher lipophilic compounds were unfavorable for being delivered to the binding sites in organism, and manifested the signiﬁcant role of suitable lipophilicity in drug design. 2.4. Interactions with calf thymus DNA DNA is the informational molecule encoding the genetic instructions and has been widely researched for the advisable design and development of efﬁcient drugs. Calf thymus DNA has always been employed as a model because of its biological importance and commercial available properties. The binding behavior of compound 5c (exerting good inhibition against Gram-positive bacteria and fungal strains) and 5g (displaying good inhibition against Gram-negative bacteria strains) with calf thymus DNA was studied to explore the possible antimicrobial mechanism of action on a molecular level in vitro with neutral red (NR) dye as a spectral probe using UVevis spectroscopic methods [26]. 2.4.1. Absorption spectra of DNA in the presence of compounds 5c and 5g The absorption spectroscopy as one of the most important techniques is extensively employed in DNA-binding studies. Generally, it is considered that hypochromism and hyperchromism are vital spectral features to distinguish changes in the DNA doublehelical structure. As reported, hyperchromism was generated from the breakage of the DNA duplex secondary structure, while hypochromism was originated from the stabilization of the DNA duplex 170 H.-Z. Zhang et al. / European Journal of Medicinal Chemistry 136 (2017) 165e183 by either an intercalation binding mode or the electrostatic effects of small molecules [27]. The observed hypochromism intensively recommended a close proximity of the aromatic chromophore to the DNA bases, which might be due to the strong interaction between the electronic states of intercalating chromophore and that of the DNA base. With a ﬁxed concentration of DNA, the UVevis absorption spectra were recorded with an increasing amount of compounds 5c and 5g. As shown in Fig. 2 and Fig. 3, the UVevis spectra showed that the maximum absorption of DNA (at 260 nm) displayed a proportional increase with increase in the concentration of compounds 5c and 5g. Meanwhile, the absorption value for the measured values of the 5ceDNA or 5geDNA complex was slightly greater than the simply sum of free DNA and free compound 5c or 5g, which was observed in the inset of Figs. 2 and 3. These indicated a weak hyperchromic effect existed between DNA and compound 5c or 5g. These demonstrated that a weak hypochromic effect existed between DNA and compound 5c or 5g. Moreover, the intercalation of the aromatic chromophore of compound 5c or 5g into the helix and the strong overlap of p-p* states in the large pconjugated system with the electronic states of DNA bases were consistent with the observed spectral changes [28]. On the basis of the variations in the absorption spectra of DNA upon binding to 5c or 5g, equation (1) can be utilized to calculate the binding constant (K). A0 xC xC 1 ¼ þ 0 K½Q x x x x AA DC C DC C (1) Fig. 2. UV absorption spectra of DNA with different concentrations of compound 5c (pH ¼ 7.4, T ¼ 293 K). Inset: comparison of absorption at 260 nm between the 5ceDNA complex and the sum values of free DNA and free compound 5c. c(DNA) ¼ 1.28 105 mol/L, and c(compound 5c) ¼ 0e1.6 105 mol/L for curves aeg respectively at increment 0.2 105. Fig. 3. UV absorption spectra of DNA with different concentrations of compound 5g (pH ¼ 7.4, T ¼ 293 K). Inset: comparison of absorption at 260 nm between the 5geDNA complex and the sum values of free DNA and free compound 5g. c(DNA) ¼ 1.28 105 mol/L, and c(compound 5g) ¼ 0e1.6 105 mol/L for curves aeg respectively at increment 0.2 105. H.-Z. Zhang et al. / European Journal of Medicinal Chemistry 136 (2017) 165e183 The plot of A0/(A-A0) versus 1/[compound 5c(or 5g)] is constructed by using the absorption titration data and linear ﬁtting (Supporting Information: Fig. S1 or Fig. S2), yielding the binding constant, K ¼ 1.26 104 or 1.52 104 L/mol, R ¼ 0.999 or 0.999, SD ¼ 0.04 or 0.09 respectively (R is the correlation coefﬁcient. SD is standard deviation). 2.4.2. Absorption spectra of NR interaction with DNA Neutral Red (NR) as a planar phenazine dye is structurally 171 similar to other planar dyes acridine, thiazine and xanthene. It has been displayed that the binding of NR with DNA is intercalation binding type. Therefore, NR was used as a spectral probe to investigate the binding mode of 5c or 5g with DNA in this work. The absorption spectra of the NR dye upon the addition of DNA were showed in Fig. S3 (Supporting Information). It was apparent that the absorption peak of the NR at around 460 nm gave gradual decrease with the increasing concentration of DNA, and a new band at around 530 nm developed. This was because of the formation of Fig. 4. UV Absorption spectra of the competitive reaction between 5c and neutral red with DNA. c(DNA) ¼ 1.28 105 mol/L, c(NR) ¼ 2 105 mol/L, and c(compound 5c) ¼ 0e4.8 105 mol/L for curves aei respectively at increment 0.6 105. (Inset) Absorption spectra of the system with the increasing concentration of 5c in the wavelength range of 260e285 nm absorption spectra of competitive reaction between compound 5c and NR with DNA. Fig. 5. UV Absorption spectra of the competitive reaction between 5g and neutral red with DNA. c(DNA) ¼ 1.28 105 mol/L, c(NR) ¼ 2 105 mol/L, and c(compound 5g) ¼ 0e4.8 105 mol/L for curves aei respectively at increment 0.6 105. (Inset) Absorption spectra of the system with the increasing concentration of 5g in the wavelength range of 260e285 nm absorption spectra of competitive reaction between compound 5g and NR with DNA. 172 H.-Z. Zhang et al. / European Journal of Medicinal Chemistry 136 (2017) 165e183 the new DNAeNR complex. An isosbestic point at 504 nm provided evidence of DNAeNR complex formation. 2.4.3. Absorption spectra of competitive interaction of compound 5c or 5g and NR with DNA As showed in Fig. 4 or Fig. 5, the competitive binding between NR and 5c (or 5g) with DNA was observed in the absorption spectra. With the increasing concentration of compound 5c or 5g, an apparent intensity increase was observed around 275 nm. Compared with the absorption around 275 nm of NReDNA complex, the absorbance at the same wavelength exhibited the reverse process (inset of Fig. 4 or Fig. 5). These various spectral changes were consistent with the intercalation of compound 5c or 5g into DNA by substituting for NR in the DNAeNR complex. As depicted above, although compounds 5c and 5g displayed different bioactivities, both of them could intercalate into calf thymus DNA to form compoundDNA complex which might block DNA replication to exert their powerful antimicrobial activities. 2.5. Molecular docking of compound 5c or 5g with DNA hexamer duplex Molecular docking study as a useful method is widely employed to investigate the binding modes of small molecules to DNA. Up to now, the full-length 3D structure of calf thymus DNA is not available in Protein Data Bank (PDB). Since calf thymus DNA is B-DNA, the CT-DNA sequence d(CGATCG)2 (PDB code: 3FT6) was chosen as receptor model. In our research, molecular docking study was performed between compound 5c or 5g and DNA hexamer duplex to understand the binding model. The docking mode with the lowest binding free energy (4.01 kJ mol1 for 5c or 4.26 kJ mol1 for 5g) is shown in Fig. 6 and Fig. 7 or Fig. 8 and Fig. 9. The results demonstrated that the hydrogen atom connected to the nitrogen atom and oxygen atom in the sulfonyl group of compounds 5c and 5g formed two hydrogen bonds with the guanine of DNA, thus preventing the formation of hydrogen bond between cytosine in DNA. This kind of interaction resulted in decreased stability of DNA, and therefore inhibited its physiological function. All these suggested that the simulation results were in accordance with the above spectral experiment results. 2.6. Interactions of compound 5c or 5g with HSA 2.6.1. UVevis absorption spectral study UVevis absorption measurement as operational method is applicable to explore the structural change of protein and to identify the complex formation. In our binding experiment, UVevis absorption spectroscopic method was employed to evaluate the binding behaviors between compound 5c or 5g and HSA. As shown in Fig. S4 or Fig. S5 (Supporting Information), the absorption peak observed at 278 nm was attributed to the aromatic rings in Tryptophan (Trp-214), Tyrosine (Tyr-411) and Phenylalanine (Phe) residues in HSA. With the addition of compound 5c or 5g, the peak intensity increased, indicating that compound 5c or 5g could interact with HSA and the peptide strands of HSA were extended [29]. Fig. 8. Molecular modeling of compound 5g and DNA hexamer duplex (PDB: 3FT6). The dashed lines represent the hydrogen bonding interactions between compound 5g and DNA hexamer duplex (Total binding score is 5.59). Fig. 6. Molecular modeling of compound 5c and DNA hexamer duplex (PDB: 3FT6). The dashed lines represent the hydrogen bonding interactions between compound 5c and DNA hexamer duplex (Total binding score is 5.04). Fig. 7. Stereoview of the conformation of compound 5c intercalated to DNA hexamer duplex to form compound 5ceDNA complex. Fig. 9. Stereoview of the conformation of compound 5g intercalated to DNA hexamer duplex to form compound 5geDNA complex. H.-Z. Zhang et al. / European Journal of Medicinal Chemistry 136 (2017) 165e183 2.6.2. Fluorescence quenching mechanism Fluorescence spectroscopy is a favorable method to investigate the interactions of small molecules with HSA. The ﬂuorescence intensity of Trp-214 may change when HSA interacts with other small molecules, which could be reﬂected in the ﬂuorescence spectra of HSA in the UV region [30]. The effect of compound 5c or 5g on the ﬂuorescence intensity to HSA at 293 K was shown in Fig. 11 or Fig. 12. It was obvious that HSA had a strong ﬂuorescence emission with a peak at 348 nm owing to the single Try-214 residue. The intensity of this characteristic broad emission band regularly decreased with the increased concentrations of 173 compound 5c or 5g. In Fig. 10 or Fig. 11, the black line showed the only emission spectrum of compound 5c or 5g, which indicated that compound 5c or 5g did not possess signiﬁcant ﬂuorescence features, and therefore the effect of compound 5c or 5g on ﬂuorescence of HSA would be negligible at the excitation wavelength (295 nm) [31]. The ﬂuorescence quenching data can be analyzed by the wellknown Stern-Volmer equation [32]: F0 ¼ 1 þ KSV ½Q ¼ 1 þ Kq t0 ½Q F (2) Fig. 10. Emission spectra of HSA in the presence of various concentrations of compound 5c. c(HSA) ¼ 1.0 105 mol/L; c(compound 5c)/(105 mol/L), aem: from 0.0 to 1.2 at increments of 0.2; black line shows the emission spectrum of compound 5c only; T ¼ 293 K, lex ¼ 295 nm. Fig. 11. Emission spectra of HSA in the presence of various concentrations of compound 5g. c(HSA) ¼ 1.0 105 mol/L; c(compound 5g)/(105 mol/L), aem: from 0.0 to 1.2 at increments of 0.2; black line shows the emission spectrum of compound 5g only; T ¼ 293 K, lex ¼ 295 nm. 174 H.-Z. Zhang et al. / European Journal of Medicinal Chemistry 136 (2017) 165e183 Fig. 12. Van't Hoff plots of the 5ceHSA system. The Stern-Volmer plots of HSA in the presence of compound 5c or 5g at different concentrations and temperatures could be calculated and were showed in Fig. S6 or Fig. S7 (Supporting Information). The values of KSV and Kq for the interaction of compound 5c or 5g with HSA at different temperatures were showed in Table 3. The KSV values were inversely correlated with the temperature, which indicated that the ﬂuorescence quenching of HSA might be initiated by the formation of compoundeHSA complex rather than dynamic collisions. The Kq values obtained at different temperatures were in 1012 L/mol s1 (Table 3), which far exceeded the diffusion controlled rate constants of various quenchers with a biopolymer (2.0 1010 L/mol s1), and indicated that the quenching was not initiated by the dynamic diffusion process but occurred in the statically formation of compoundeHSA complex [31]. 2.6.3. Binding constant and site For a static quenching process, the data could be described by the Modiﬁed Stern-Volmer equation [33]: F0 DF ¼ 1 1 1 þ fa Ka ½Q fa (3) The modiﬁed Stern-Volmer plots were showed in Fig. S8 or Fig. S9 (Supporting Information) and the calculated results were depicted in Table 4. When small molecules bind to a set of equivalent sites on a macromolecule, the equilibrium binding constants and the numbers of binding sites can also be calculated according to the Scatchard equation [34]: . r D ¼ nKb rKb f (4) The Scatchard plots were shown in Fig. S10 or Fig. S11 (Supporting Information) and the Kb and n were listed in Table 4. The modiﬁed Stern-Volmer and Scatchard plots for the compoundeHSA system at different temperatures were given in Table 4. The decreased trend of Ka and Kb with increased temperatures was in accordance with KSV's depended on temperatures. The value of the binding site n was approximately 1, which showed one high afﬁnity binding site, was present in the interaction of compound 5c or 5g with HSA. The results also showed that the binding constants were moderate and the effects of temperatures were not signiﬁcant, thus both compounds 5c and 5g might be stored and carried by this protein. 2.6.4. Binding mode and thermodynamic parameters Generally, there are four types of non-covalent interactions including hydrogen bonds, van der Waals forces, electrostatic interactions and hydrophobic bonds, which play substantial roles in small molecules binding to proteins [35]. The thermodynamic parameters enthalpy (DH) and entropy (DS) change of binding reaction are the main evidence for conﬁrming the interactions between small molecules and protein. If the DH does not vary signiﬁcantly over the studied temperatures range, then its value and DS can be Table 3 Stern-Volmer quenching constants for the interaction of compounds 5c and 5g with HSA at various temperatures. pH T (K) Ksv (L mol1) 5c 7.4 a b 273 293 313 5g 4 3.26 10 2.20 104 1.83 104 R is the correlation coefﬁcient. S.D. is standard deviation. Kq (L mol1 S1) 5c 4 5.25 10 4.08 104 3.02 104 Ra 5g 12 5.09 10 3.44 1012 2.86 1012 12 8.20 10 6.38 1012 4.72 1012 S.D.b 5c 5g 5c 5g 0.975 0.997 0.996 0.996 0.999 0.998 0.049 0.011 0.011 0.030 0.007 0.011 H.-Z. Zhang et al. / European Journal of Medicinal Chemistry 136 (2017) 165e183 175 Table 4 Binding constants and sites of 5ceHSA and 5geHSA systems at pH ¼ 7.4. T (K) Modiﬁed Stern-Volmer method 273 293 313 Scatchard method 104 Ka (L/mol) R 5c 5g 5c 5g 5c 5.43 3.69 3.58 1.11 1.08 1.06 0.997 0.999 0.998 0.197 0.552 0.258 0.211 0.217 0.355 104 Kb (L/mol) R 5g 5c 5g 5c 5g 5c 5g 5c 5g 0.197 0.552 0.258 7.86 5.84 4.16 6.56 5.93 4.21 0.998 0.999 0.999 0.998 0.999 0.999 0.114 0.108 0.013 0.029 0.020 0.015 0.97 0.95 1.14 1.07 1.06 1.18 S.D. evaluated from the van't Hoff equation: DH ln K ¼ RT þ DS R n Table 5 Thermodynamic parameters of 5ceHSA and 5geHSA systems at different temperatures. (5) In order to explain the binding model between compound and HSA, the thermodynamic parameters were calculated from the van't Hoff plots. The DH was estimated from the slope of the van't Hoff relationship (Fig. 12 or Fig. 13). The free energy change (DG) was then calculated from the following equation: DG ¼ DH T DS S.D. (6) The values of DH, DG and DS were summarized in Table 5. The negative values of free energy DG of the interaction between compound 5c or 5g and HSA suggested that the binding process was spontaneous, and the negative values of DH indicated that the binding was mainly enthalpy-driven and involved an exothermic reaction, the DS was unfavorable for it. A positive DS value is frequently taken as a typical evidence for hydrophobic interaction, which was consistent with the above discussion. Therefore, DH < 0 and DS > 0 obtained in this case indicated that the electrostatic interactions played an important role in the binding of compound 5c or 5g to HSA [36]. The above experiments displayed that compounds 5c and 5g could effectively interact with HSA, thereby causing hypochromic effect of ultraviolet spectroscopy. When the electronic transfer T(K) 273 293 313 DH (kJ mol1) DG (kJ mol1) DS (J mol1 K1) 5c 5g 5c 5g 5c 5g 1.757 0.820 24.310 25.542 26.774 21.139 22.042 24.116 82.610 74.429 occurred between compound 5c or 5g and HSA, it caused energy transfer without radiation, and therefore resulted in the quenching of ﬂuorescence spectrum. Further molecular electrostatic potentiality for the compounds 5c and 5g was investigated by full geometry optimizations of the studied systems which was performed by using the B3LYP functional with 6-31G* basis set. Calculation presented in this work was carried out by the GAUSSIAN 09 program package. The results manifested that the nucleophilic effect of carbonyl and sulfonyl groups (Fig. 14, red region) in compound 5c or 5g might induce an electrostatic effect with positive electricity of Lys199 in HSA. Therefore, compound 5c or 5g with HSA might interact by electrostatic interactions [37]. This result was also evidenced by the value of enthalpy change (DH) and entropy change (DS) from the van't Hoff equation, which was accordant with the literature (when DH < 0 and DS > 0, the main force is electrostatic interaction). Fig. 13. Van't Hoff plots of the 5geHSA system. 176 H.-Z. Zhang et al. / European Journal of Medicinal Chemistry 136 (2017) 165e183 4. Experimental 4.1. General methods Melting points were recorded on Xe6 melting point apparatus and uncorrected. TLC analysis was done using pre-coated silica gel plates. FT-IR spectra were carried out on Bruker RFS100/S spectrophotometer (Bio-Rad, Cambridge, MA, USA) using KBr pellets in the 400e4000 cm1 range. 1H NMR and 13C NMR spectra were recorded on a Bruker AV 600 spectrometer using TMS as an internal standard. The following abbreviations were used to designate aryl groups: Bim ¼ benzimidazolyl, Ph ¼ phenyl. The chemical shifts were reported in parts per million (ppm), the coupling constants (J) were expressed in hertz (Hz) and signals were described as singlet (s), doublet (d), triplet (t) as well as multiplet (m). The mass spectra were recorded on LCMSe2010A and HRMS. All chemicals and solvents were commercially available and were used without further puriﬁcation. 4.1.1. Synthesis of 4-acetamidobenzene-1-sulfonyl chloride (2) 4-Acetamidobenzene-1-sulfonyl chloride 2 was prepared according to the literature procedure, starting from acetaniline (5.002 g, 0.037 mol) and chlorosulfonic acid (14 mL). Yield: 81.6%; mp: 138e140 C. (literature mp: 142e144 C) [17]. 4.1.2. Synthesis of sodium 4-acetamidobenzenesulﬁnate (3) A mixture of compound 2 (4.343 g, 18.6 mmol), sodium sulﬁte (3.486 g, 27.7 mmol) and sodium bicarbonate (2.322 g, 27.6 mmol) was stirred in water at 80 C. After the reaction was completed (monitored by TLC, eluent, methanol/ethyl acetate, 1/2, V/V), the system was washed by ethyl acetate, and the water phase was collected and evaporated to afford compound 3 as white solid. Fig. 14. Electrostatic potential of compounds 5c and 5g. 3. Conclusion In conclusion, a novel series of benzimidazole-incorporated sulfonamide analogues have been successfully prepared starting from commercially available acetanilide. All the new compounds were conﬁrmed by 1H NMR, 13C NMR, IR, MS and HRMS spectra. Among these sulfonamide analogues, compound 5c bearing 4ﬂuorobenzyl group gave potent activities against Gram-positive bacteria and fungal strains (MIC ¼ 4e64 mg/mL), and 2,4dichlorobenzyl derivative 5g showed good activities against Gram-negative bacteria with MIC values ranging from 4 to 32 mg/ mL. These results manifested that compounds 5c and 5g should be worthy to be further investigated as potential antimicrobial agents. Further research demonstrated that the two active molecules 5c and 5g could effectively intercalate into calf thymus DNA to form the compoundDNA complex, which might block DNA replication to exert their powerful antimicrobial activity. Molecular docking experiments suggested that compounds 5c and 5g could intercalate into base-pairs of DNA hexamer duplex by the formation of hydrogen bonds with guanine of DNA. The binding research demonstrated that HSA could effectively store and carry compounds 5c and 5g by electrostatic interactions, and which was also conﬁrmed by the full geometry calculation optimizations. All these results opened up a promising starting point to optimize the structures of sulfonamide benzimidazoles as potent antimicrobial agents. 4.1.3. Synthesis of N-(4-((1H-benzo[d]imidazol-2-yl) methylsulfonyl)phenyl)acetamide (4) A suspension of compound 3 (3.959 g, 17.9 mmol), chloromethyl benzimidazole (2.992 g, 17.8 mmol) and tetrabutylammonium iodide (0.010 g) was stirred in acetone (30 mL) at 50 C. After the reaction was completed (monitored by TLC, eluent, acetone/petroleum ether, 1/1, V/V), the reaction system was ﬁltered, and the residue was collected and washed with water to give compound 4 as yellow solid. Yield: 60%; mp: 155e156 C; IR (KBr) n: 3429 (BimNH), 3359 (NH), 3024 (aromatic CH), 3024 (CH2), 1681 (C¼N), 1589, 1543 (aromatic frame), 737 cm1; 1H NMR (600 MHz, DMSOd6) d: 2.10 (s, 3H, COCH3), 4.91 (s, 2H, Bim-CH2), 7.20e7.19 (m, 2H, Bim-6,7-H), 7.53 (d, 2H, J ¼ 6.0 Hz, Bim-5,8-H), 7.69 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.75 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 10.39 (s, H, NHCOCH3), 12.54 (s, H, Bim-1-H) ppm; 13C NMR (150 MHz, DMSOd6) d: 24.7, 54.6, 115.7, 117.1, 119.9, 123.0, 123.4, 129.0, 130.0, 137.9, 139.0, 145.8, 152.8, 169.8 ppm. 4.1.4. Synthesis of N-(4-((1-(2-ﬂuorobenzyl)-1H-benzo[d]imidazol2-yl)methylsulfonyl)phenyl)acetamide (5a) A suspension of compound 4 (0.492 g, 1.49 mmol) and potassium carbonate (0.425 g, 3.07 mmol) was stirred in acetonitrile (30 mL) at 50 C. After 0.5 h, 1-(chloromethyl)-2-ﬂuorobenzene (0.342 g, 2.37 mmol) was added and the reaction system was stirred at 70 C continuously. After the reaction was completed (monitored by TLC, eluent, acetone/petroleum ether, 1/1, V/V), the solvent was removed and the residue was exacted with chloroform (3 20 mL), dried over anhydrous sodium sulfate and puriﬁed by silica gel column chromatography (eluent, acetone/petroleum ether, 1/1, V/V) to afford compound 5a as yellow solid. Yield: 54%; H.-Z. Zhang et al. / European Journal of Medicinal Chemistry 136 (2017) 165e183 mp: 215e217 C; IR (KBr) n: 3360 (NH), 3037 (aromatic CH), 2994 (CH2), 1687 (C¼N), 1584, 1531 (aromatic frame), 739 cm1; 1H NMR (600 MHz, DMSO-d6) d: 2.11 (s, 3H, COCH3), 5.14 (s, 2H, BimCH2), 5.63 (s, 2H, 2-FPh-CH2), 6.93e6.90 (t, H, J ¼ 9.0 Hz, 2-FPh-5H), 7.11e7.09 (t, H, J ¼ 6.0 Hz, 2-FPh-4-H), 7.24e7.16 (m, 4H, Bim5,6,7,8-H), 7.35e7.32 (m, H, 2-FPh-6-H), 7.60e7.59 (m, H, 2-FPh-3H), 7.74 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.77 (d, 2H, J ¼ 6.0 Hz, Ph-2,6H), 10.41 (s, H, NHCOCH3) ppm; 13C NMR (150 MHz, DMSO-d6) d: 24.6, 49.1, 54.6, 111.4, 116.0, 119.0, 119.90, 122.7, 123.5, 123.7, 125.1, 129.5, 129.9, 130.3, 132.5, 135.6, 142.8, 144.0, 144.8, 160.6, 169.7 ppm; TOF-MS (m/z): 460 [MþNa]þ; HRMS (TOF) calcd. for C23H20FN3O3SNa [MþNa]þ, 460.1102; found, 460.1105. 4.1.5. Synthesis of N-(4-((1-(3-ﬂuorobenzyl)-1H-benzo[d]imidazol2-yl)methylsulfonyl)phenyl)acetamide (5b) Compound 5b was prepared according to the procedure described for compound 5a, starting from compound 4 (0.210 g, 0.638), 1-(chloromethyl)-3-ﬂuorobenzene (0.216 g, 1.505 mmol) and potassium carbonate (0.187 g, 1.355 mmol). The pure product 5b was obtained as yellow solid. Yield: 52.1%; mp: 221e224 C; IR (KBr) n: 3362 (NH), 3038 (aromatic CH), 2993 (CH2), 1689 (C¼N), 1582, 1533 (aromatic frame), 736 cm1; 1H NMR (600 MHz, DMSOd6) d: 2.11 (s, 3H, COCH3), 5.16 (s, 2H, Bim-CH2), 5.61 (s, 2H, 3-FPhCH2), 6.98e6.94 (m, 2H, 3-FPh-2,6-H), 7.11e7.08 (m, H, 3-FPh-4-H), 7.20e7.18 (m, 2H, Bim-6,7-H), 7.30e7.29 (m, H, Bim-8-H), 7.37e7.33 (m, H, 3-FPh-5-H), 7.61e7.59 (m, H, Bim-5-H), 7.76 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.77 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 10.41 (s, H, NHCOCH3) ppm; 13C NMR (150 MHz, DMSO-d6) d: 24.7, 47.1, 54.5, 111.6, 114.3, 114.8, 118.9, 119.9, 122.7, 123.4, 123.5, 130.0, 131.0, 132.6, 135.6, 139.9, 142.9, 144.0, 144.8, 162.8, 169.7 ppm; TOF-MS (m/z): 460 [MþNa]þ; HRMS (TOF) calcd. for C23H20FN3O3SNa [MþNa]þ, 460.1102; found, 460.1101. 4.1.6. Synthesis of N-(4-((1-(4-ﬂuorobenzyl)-1H-benzo[d]imidazol2-yl)methylsulfonyl)phenyl)acetamide (5c) Compound 5c was prepared according to the procedure depicted for compound 5a, starting from compound 4 (0.201 g, 0.610 mmol), 1-(chloromethyl)-4-ﬂuorobenzene (0.176 g, 1.225 mmol) and potassium carbonate (0.174 g, 1.260 mmol). The pure product 5c was obtained as yellow solid. Yield: 67.7%; mp: 199e201 C; IR (KBr) n: 3364 (NH), 3037 (aromatic CH), 2995 (CH2), 1687 (C¼N), 1585, 1534 (aromatic frame), 740 cm1; 1H NMR (600 MHz, DMSO-d6) d: 2.11 (s, 3H, COCH3), 5.14 (s, 2H, Bim-CH2), 5.56 (s, 2H, 4-FPh-CH2), 7.19e7.12 (m, 6H, 4-FPh-2,3,5,6-H, Bim-6,7H), 7.30e7.29 (m, H, Bim-8-H), 7.59e7.58 (m, H, Bim-5-H), 7.75 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.77 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 10.40 (s, H, NHCOCH3) ppm; 13C NMR (150 MHz, DMSO-d6) d: 24.7, 47.0, 54.6, 111.7, 115.9, 119.0, 119.9, 122.6, 123.4, 129.5, 130.0, 132.6, 133.1, 135.6, 142.9, 143.9, 144.8, 162.1, 169.7 ppm; TOF-MS (m/z): 460 [MþNa]þ; HRMS (TOF) calcd. for C23H20FN3O3SNa [MþNa]þ, 460.1102; found, 460.1104. 4.1.7. Synthesis of N-(4-((1-(2-chlorobenzyl)-1H-benzo[d]imidazol2-yl)methylsulfonyl)phenyl)acetamide (5d) Compound 5d was prepared according to the experimental procedure described for compound 5a, starting from compound 4 (0.271 g, 0.824 mmol), 1-chloro-2-(chloromethyl)benzene (0.200 g, 1.242 mmol) and potassium carbonate (0.180 g, 1.260 mmol). The pure product 5d was obtained as yellow solid. Yield: 66.7%; mp: 158e160 C; IR (KBr) n: 3367 (NH), 3039 (aromatic CH), 2993 (CH2), 1689 (C¼N), 1587, 1535 (aromatic frame), 738 cm1; 1H NMR (600 MHz, DMSO-d6) d: 2.11 (s, 3H, COCH3), 5.11 (s, 2H, Bim-CH2), 5.62 (s, 2H, 2-ClPh-CH2), 6.52 (d, H, J ¼ 6.0 Hz, 2-ClPh-6-H), 7.22e7.17 (m, 4H, Bim-5,6,7,8-H), 7.32e7.30 (t, H, J ¼ 6.0 Hz, 2-ClPh4-H), 7.53 (d, H, J ¼ 6.0 Hz, 2-ClPh-5-H), 7.63 (d, H, J ¼ 6.0 Hz, 2- 177 ClPh-3-H), 7.73 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.77 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 10.41 (s, H, NHCOCH3) ppm; 13C NMR (150 MHz, DMSO-d6) d: 24.7, 45.5, 54.7, 111.4, 119.0, 120.0, 122.8, 123.7, 127.9, 128.4, 129.8, 129.9, 130.0, 132.1, 132.4, 134.0, 135.6, 142.9, 144.2, 144.9, 169.7 ppm; TOF-MS (m/z): 477 [MþNa]þ; HRMS (TOF) calcd. for C23H20ClN3O3SNa [MþNa]þ, 476.0806; found, 476.0805. 4.1.8. Synthesis of N-(4-((1-(3-chlorobenzyl)-1H-benzo[d]imidazol2-yl)methylsulfonyl)phenyl)acetamide (5e) Compound 5e was prepared according to the procedure described for compound 5a, starting from compound 4 (0.239 g, 0.726 mmol), 1-chloro-3-(chloromethyl)benzene (0.176 g, 1.089 mmol) and potassium carbonate (0.156 g, 1.130 mmol). The pure product 5e was obtained as yellow solid. Yield: 73.7%; mp: 205e207 C; IR (KBr) n: 3365 (NH), 3039 (aromatic CH), 2997 (CH2), 1686 (C¼N), 1589, 1536 (aromatic frame), 739 cm1; 1H NMR (600 MHz, DMSO-d6) d: 2.11 (s, 3H, COCH3), 5.13 (s, 2H, Bim-CH2), 5.64 (s, 2H, 3-ClPh-CH2), 6.48 (d, H, J ¼ 6.0 Hz, 3-ClPh-6-H), 7.23e7.17 (m, 4H, Bim-5,6,7,8-H), 7.39e7.33 (t, H, J ¼ 6.0 Hz, 3-ClPh5-H), 7.45 (d, H, J ¼ 6.0 Hz, 3-ClPh-4-H), 7.57 (d, H, J ¼ 6.0 Hz, 3ClPh-2-H), 7.74 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.77 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 10.41 (s, H, NHCOCH3) ppm; 13C NMR (150 MHz, DMSO-d6) d: 34.6, 49.9, 54.7, 111.4, 119.0, 120.0, 122.8, 123.3, 123.4, 128.4, 129.7, 129.9, 130.0, 132.2, 132.5, 133.9, 135.6, 142.9, 144.4, 144.9, 169.7 ppm; TOF-MS (m/z): 477 [MþNa]þ; HRMS (TOF) calcd. for C23H20ClN3O3SNa [MþNa]þ, 476.0806; found, 476.0803. 4.1.9. Synthesis of N-(4-((1-(4-chlorobenzyl)-1H-benzo[d]imidazol2-yl)methylsulfonyl)phenyl)acetamide (5f) Compound 5f was prepared according to the procedure described for compound 5a, starting from compound 4 (0.287 g, 0.872 mmol), 1-chloro-4-(chloromethyl)benzene (0.254 g, 1.570 mmol) and potassium carbonate (0.182 g, 1.319 mmol). The pure product 5f was obtained as yellow solid. Yield: 68.1%; mp: 208e210 C; IR (KBr) n: 3366 (NH), 3038 (aromatic CH), 2996 (CH2), 1687 (C¼N), 1588, 1537 (aromatic frame), 738 cm1; 1H NMR (600 MHz, DMSO-d6) d: 2.11 (s, 3H, COCH3), 5.15 (s, 2H, Bim-CH2), 5.59 (s, 2H, 4-ClPh-CH2), 7.14 (d, 2H, J ¼ 6.0 Hz, 4-ClPh-2,6-H), 7.21e7.17 (m, 2H, Bim-6,7-H), 7.29e7.27 (m, H, Bim-8-H), 7.36 (d, 2H, J ¼ 6.0 Hz, 4-ClPh-3,5-H), 7.60e7.59 (m, H, Bim-5-H), 7.75 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.77 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 10.41 (s, H, NHCOCH3) ppm; 13C NMR (150 MHz, DMSO-d6) d: 24.7, 47.0, 54.6, 111.7, 119.0, 119.8, 122.7, 123.5, 129.0, 129.3, 130.0, 132.5, 132.6, 135.5, 135.9, 142.8, 143.9, 144.8, 169.7 ppm; TOF-MS (m/z): 477 [MþNa]þ; HRMS (TOF) calcd. for C23H20ClN3O3SNa [MþNa]þ, 476.0806; found, 476.0809. 4.1.10. Synthesis of N-(4-((1-(2,4-dichlorobenzyl)-1H-benzo[d] imidazol-2-yl)methylsulfonyl)phenyl) acetamide (5g) Compound 5g was prepared according to the procedure described for compound 5a, starting from compound 4 (0.247 g, 0.751 mmol), 2,4-dichloro-1-(chloromethyl)benzene (0.185 g, 0.931 mmol) and potassium carbonate (0.207 g, 1.500 mmol). The pure product 5g was obtained as white solid. Yield: 59.4%; mp: 166e168 C; IR (KBr) n: 3363 (NH), 3039 (aromatic CH), 2993 (CH2), 1689 (C¼N), 1585, 1531 (aromatic frame), 740 cm1; 1H NMR (600 MHz, DMSO-d6) d: 2.12 (s, 3H, COCH3), 5.15 (s, 2H, Bim-CH2), 5.61 (s, 2H, 2,4-Cl2Ph-CH2), 6.49 (d, H, J ¼ 6.0 Hz, 2,4-Cl2Ph-6-H), 7.28e7.21 (m, 4H, Bim-5,6,7,8-H), 7.64 (d, H, J ¼ 6.0 Hz, 2,4-Cl2Ph-5H), 7.69 (s, H, 2,4-Cl2Ph-3-H), 7.73 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.77 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 10.41 (s, H, NHCOCH3) ppm; 13C NMR (150 MHz, DMSO-d6) d: 24.7, 45.2, 54.7, 111.4, 119.0, 120.1, 122.9, 123.8, 128.1, 129.4, 129.6, 129.9, 132.4, 133.1, 133.3, 133.4, 135.5, 178 H.-Z. Zhang et al. / European Journal of Medicinal Chemistry 136 (2017) 165e183 142.9, 144.3, 144.9, 169.7 ppm; ESI-MS (m/z): 511 [MþNa]þ; HRMS (ESI) calcd. for C23H19Cl2N3O3SNa [MþNa]þ, 510.0416; found, 510.0425. 4.1.11. Synthesis of N-(4-((1-(3,4-dichlorobenzyl)-1H-benzo[d] imidazol-2-yl)methylsulfonyl)phenyl) acetamide (5h) Compound 5h was prepared according to the procedure described for compound 5a, starting from compound 4 (0.180 g, 0.547 mmol), 3,4-dichloro-1-(chloromethyl)benzene (0.157 g, 0.803 mmol) and potassium carbonate (0.114 g, 0.826 mmol). The pure product 5h was obtained as yellow solid. Yield: 61.0%; mp: 167e169 C; IR (KBr) n: 3365 (NH), 3038 (aromatic CH), 2995 (CH2), 1687 (C¼N), 1585, 1535 (aromatic frame), 741 cm1; 1H NMR (600 MHz, DMSO-d6) d: 2.11 (s, 3H, COCH3), 5.19 (s, 2H, Bim-CH2), 5.61 (s, 2H, 3,4-Cl2Ph-CH2), 7.04 (d, H, J ¼ 6.0 Hz, 3,4-Cl2Ph-6-H), 7.29e7.19 (m, 4H, Bim-5,6,7,8-H), 7.44 (s, H, 3,4-Cl2Ph-2-H), 7.57 (d, H, J ¼ 6.0 Hz, 3,4-Cl2Ph-5-H), 7.75 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.77 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 10.41 (s, H, NHCOCH3) ppm; 13C NMR (150 MHz, DMSO-d6) d: 24.7, 46.6, 54.5, 111.6, 118.9, 119.9, 122.8, 123.6, 127.7, 129.5, 129.9, 130.6, 131.2, 131.7, 132.6, 135.4, 138.2, 142.9, 144.0, 144.8, 169.6 ppm; ESI-MS (m/z): 511 [MþNa]þ; HRMS (ESI) calcd. for C23H19Cl2N3O3SNa [MþNa]þ, 510.0416; found, 510.0424. 4.1.12. Synthesis of N-(4-((1-ethyl-1H-benzo[d]imidazol-2-yl) methylsulfonyl)phenyl)acetamide (6a) Pure compound 6a was obtained in process of synthesizing compound 5a as yellow oil. Yield: 52.8%; mp: >250 C; IR (KBr) n: 3360 (NH), 3041 (aromatic CH), 2990 (CH2), 1691 (C¼N), 1587, 1500 (aromatic frame), 738 cm1; 1H NMR (600 MHz, DMSO-d6) d: 1.34e1.32 (t, 3H, J ¼ 6.0 Hz, CH2CH3), 2.10 (s, 3H, COCH3), 4.35e4.31 (m, 2H, CH2CH3), 5.10 (s, 2H, Bim-CH2), 7.21e7.19 (t, H, J ¼ 6.0 Hz, Bim-6-H), 7.29e7.26 (t, H, J ¼ 9.0 Hz, Bim-7-H), 7.59e7.55 (m, 2H, Bim-5,8-H), 7.72 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.77 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 10.41 (s, H, NHCOCH3) ppm; 13C NMR (150 MHz, DMSOd6) d: 15.2, 24.7, 39.2, 54.6, 111.2, 118.9, 119.7, 122.4, 123.2, 129.9, 132.6, 135.2, 142.8, 143.2, 144.8, 169.6 ppm; ESI-MS (m/z): 358 [MþH]þ; HRMS (ESI) calcd. for C18H20N3O3S [MþH]þ, 358.1220; found, 358.1223. 4.1.13. Synthesis of N-(4-((1-propyl-1H-benzo[d]imidazol-2-yl) methylsulfonyl)phenyl)acetamide (6b) Pure compound 6b was obtained in process of synthesizing compound 5a as yellow solid. Yield: 51.2%; mp: 116e118 C; IR (KBr) n: 3362 (NH), 3037 (aromatic CH), 2991 (CH2), 1687 (C¼N), 1589, 1508 (aromatic frame), 739 cm1; 1H NMR (600 MHz, DMSOd6) d: 0.88e0.86 (t, 3H, J ¼ 6.0 Hz, CH2CH2CH3), 1.77e1.71 (m, 2H, CH2CH2CH3), 2.10 (s, 3H, COCH3), 4.24e4.21 (t, 2H, J ¼ 9.0 Hz, CH2CH2CH3), 5.13 (s, 2H, Bim-CH2), 7.24e7.22 (t, H, J ¼ 6.0 Hz, Bim6-H), 7.30e7.28 (t, H, J ¼ 6.0 Hz, Bim-7-H), 7.58 (d, H, J ¼ 6.0 Hz, Bim-8-H), 7.63 (d, H, J ¼ 6.0 Hz, Bim-5-H), 7.71 (d, 2H, J ¼ 6.0 Hz, Ph3,5-H), 7.77 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 10.44 (s, H, NHCOCH3) ppm; 13C NMR (150 MHz, DMSO-d6) d: 15.2, 22.1, 24.6, 39.2, 54.5, 111.2, 119.0, 119.7, 122.4, 123.2, 129.9, 132.6, 135.2, 142.8, 143.2, 144.7, 169.6 ppm; ESI-MS (m/z): 372 [MþH]þ; HRMS (ESI) calcd. for C19H22N3O3S [MþH]þ, 372.1376; found, 372.1385. 4.1.14. Synthesis of N-(4-((1-pentyl-1H-benzo[d]imidazol-2-yl) methylsulfonyl)phenyl)acetamide (6c) Pure compound 6c was obtained in process of synthesizing compound 5a as yellow solid. Yield: 28.1%; mp: 94e96 C; IR (KBr) n: 3359 (NH), 3030 (aromatic CH), 2994 (CH2), 1680 (C¼N), 1589, 1491 (aromatic frame), 741 cm1; 1H NMR (600 MHz, DMSO-d6) d: 0.86e0.84 (t, 3H, J ¼ 6.0 Hz, (CH2)4CH3), 1.31e1.26 (m, 4H, (CH2)2(CH2)2CH3), 1.72e1.67 (m, 2H, CH2CH2(CH2)2CH3), 2.10 (s, 3H, COCH3), 4.24e4.21 (t, 2H, J ¼ 9.0 Hz, CH2(CH2)3CH3), 5.08 (s, 2H, Bim-CH2), 7.20e7.18 (t, H, J ¼ 6.0 Hz, Bim-6-H), 7.28e7.25 (t, H, J ¼ 9.0 Hz, Bim-7-H), 7.57e7.55 (t, 2H, J ¼ 6.0 Hz, Bim-5,8-H), 7.71 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.76 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 10.40 (s, H, NHCOCH3) ppm; 13C NMR (150 MHz, DMSO-d6) d: 14.3, 22.3, 24.6, 29.2, 29.4, 44.2, 54.7, 111.3, 118.9, 119.7, 122.4, 123.2, 129.9, 132.6, 135.5, 142.6, 143.5, 144.8, 169.7 ppm; ESI-MS (m/z): 401 [MþH]þ; HRMS (ESI) calcd. for C21H26N3O3S [MþH]þ, 400.1685; found, 400.1693. 4.1.15. Synthesis of N-(4-((1-hexyl-1H-benzo[d]imidazol-2-yl) methylsulfonyl)phenyl)acetamide (6d) Pure compound 6d was obtained in process of synthesizing compound 5a as yellow oil. Yield: 47.5%; mp: 108e110 C; IR (KBr) n: 3352 (NH), 3037 (aromatic CH), 2999 (CH2), 1687 (C¼N), 1589, 1493 (aromatic frame), 738 cm1; 1H NMR (600 MHz, DMSO-d6) d: 0.86e0.84 (t, 3H, J ¼ 6.0 Hz, (CH2)5CH3), 1.30e1.25 (m, 6H, (CH2)2(CH2)3CH3), 1.70e1.65 (m, 2H, CH2CH2(CH2)3CH3), 2.10 (s, 3H, COCH3), 4.24e4.21 (t, 2H, J ¼ 9.0 Hz, CH2(CH2)4CH3), 5.08 (s, 2H, Bim-CH2), 7.21e7.18 (t, H, J ¼ 9.0 Hz, Bim-6-H), 7.28e7.25 (t, H, J ¼ 9.0 Hz, Bim-7-H), 7.57e7.55 (t, 2H, J ¼ 6.0 Hz, Bim-5,8-H), 7.71 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.76 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 10.41 (s, H, NHCOCH3) ppm; 13C NMR (150 MHz, DMSO-d6) d: 14.3, 22.5, 24.6, 26.3, 29.6, 31.3, 44.2, 54.6, 111.3, 118.9, 119.6, 122.4, 123.2, 129.9, 132.6, 135.5, 142.6, 143.4, 144.8, 169.6 ppm; ESI-MS (m/z): 415 [MþH]þ; HRMS (ESI) calcd. for C22H28N3O3S [MþH]þ, 414.1846; found, 414.1855. 4.1.16. Synthesis of N-(4-((1-heptyl-1H-benzo[d]imidazol-2-yl) methylsulfonyl)phenyl)acetamide (6e) Compound 6e was obtained as yellow solid. Yield: 24.2%; mp: 102e103 C; IR (KBr) n: 3354 (NH), 3032 (aromatic CH), 2994 (CH2), 1685 (C¼N), 1592, 1498 (aromatic frame), 738 cm1; 1H NMR (600 MHz, DMSO-d6) d: 0.86e0.83 (t, 3H, J ¼ 9.0 Hz, (CH2)6CH3), 1.27e1.23 (m, 8H, (CH2)2(CH2)4CH3), 1.70e1.66 (m, 2H, CH2CH2(CH2)4CH3), 2.10 (s, 3H, COCH3), 4.23e4.21 (t, 2H, J ¼ 6.0 Hz, CH2(CH2)5CH3), 5.08 (s, 2H, Bim-CH2), 7.21e7.18 (t, H, J ¼ 9.0 Hz, Bim-6-H), 7.27e7.25 (t, H, J ¼ 6.0 Hz, Bim-7-H), 7.57e7.55 (t, 2H, J ¼ 6.0 Hz, Bim-5,8-H), 7.70 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.76 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 10.40 (s, H, NHCOCH3) ppm; 13C NMR (150 MHz, DMSO-d6) d: 14.4, 22.5, 24.6, 26.6, 28.8, 29.7, 31.6, 44.1, 54.6, 111.3, 118.9, 119.7, 122.4, 123.2, 129.9, 132.6, 135.5, 142.7, 143.4, 144.8, 169.6 ppm; ESI-MS (m/z): 429 [MþH]þ; HRMS (ESI) calcd. for C23H30N3O3S [MþH]þ, 428.2002; found, 428.2010. 4.1.17. Synthesis of N-(4-((1-octyl-1H-benzo[d]imidazol-2-yl) methylsulfonyl)phenyl)acetamide (6f) Pure compound 6f was obtained in process of synthesizing compound 5a as yellow solid. Yield: 71.2%; mp: 97e99 C; IR (KBr) n: 3357 (NH), 3038 (aromatic CH), 2996 (CH2), 1685 (C¼N), 1590, 1498 (aromatic frame), 739 cm1; 1H NMR (600 MHz, DMSO-d6) d: 0.86e0.83 (t, 3H, J ¼ 9.0 Hz, (CH2)7CH3), 1.27e1.23 (m, 10H, (CH2)2(CH2)5CH3), 1.69e1.66 (m, 2H, CH2CH2(CH2)5CH3), 2.10 (s, 3H, COCH3), 4.23e4.21 (t, 2H, J ¼ 6.0 Hz, CH2(CH2)6CH3), 5.08 (s, 2H, Bim-CH2), 7.21e7.18 (t, H, J ¼ 9.0 Hz, Bim-6-H), 7.28e7.25 (t, H, J ¼ 9.0 Hz, Bim-7-H), 7.57e7.55 (t, 2H, J ¼ 6.0 Hz, Bim-5,8-H), 7.70 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.76 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 10.41 (s, H, NHCOCH3) ppm; 13C NMR (150 MHz, DMSO-d6) d: 14.4, 22.5, 24.6, 26.6, 29.0, 29.1, 29.7, 31.7, 44.1, 54.6, 111.3, 118.9, 119.7, 122.4, 123.2, 129.9, 132.6, 135.5, 142.7, 143.4, 144.8, 169.6 ppm; ESI-MS (m/ z): 443 [MþH]þ; HRMS (ESI) calcd. for C24H32N3O3S [MþH]þ, 442.2159; found, 442.2168. H.-Z. Zhang et al. / European Journal of Medicinal Chemistry 136 (2017) 165e183 4.1.18. Synthesis of N-(4-((1-decyl-1H-benzo[d]imidazol-2-yl) methylsulfonyl)phenyl)acetamide (6g) Compound 6g was synthesized according to the experimental procedure reported for compound 5a, starting from compound 4 (0.407 g, 1.237 mmol), 1-bromononane (0.410 g, 1.856 mmol) and potassium carbonate (0.256 g, 1.856 mmol). The crude product 6g was obtained as yellow solid. Yield: 47.7%; mp: 78e80 C; IR (KBr) n: 3352 (NH), 3034 (aromatic CH), 2996 (CH2), 1687 (C¼N), 1590, 1505 (aromatic frame), 741 cm1; 1H NMR (600 MHz, DMSO-d6) d: 0.86e0.84 (t, 3H, J ¼ 6.0 Hz, (CH2)9CH3), 1.26e1.22 (m, 14H, (CH2)2(CH2)7CH3), 1.70e1.66 (m, 2H, CH2CH2(CH2)7CH3), 2.10 (s, 3H, COCH3), 4.23e4.21 (t, 2H, J ¼ 6.0 Hz, CH2(CH2)8CH3), 5.08 (s, 2H, Bim-CH2), 7.21e7.18 (t, H, J ¼ 9.0 Hz, Bim-6-H), 7.27e7.25 (t, H, J ¼ 6.0 Hz, Bim-7-H), 7.57e7.55 (t, 2H, J ¼ 6.0 Hz, Bim-5,8-H), 7.70 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.76 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 10.41 (s, H, NHCOCH3) ppm; 13C NMR (150 MHz, DMSO-d6) d: 14.4, 22.5, 24.6, 26.6, 29.1, 29.4, 29.4, 29.6, 31.7, 44.1, 54.7, 111.3, 118.9, 119.7, 122.4, 123.2, 129.9, 132.7, 135.6, 142.7, 143.4, 144.8, 169.6 ppm; ESIMS (m/z): 471 [MþH]þ; HRMS (ESI) calcd. for C26H36N3O3S [MþH]þ, 470.2472; found, 470.2476. 4.1.19. Synthesis of N-(4-((1-dodecyl-1H-benzo[d]imidazol-2-yl) methylsulfonyl)phenyl)acetamide (6h) Prepared according to the general procedure described for compound 5a, starting from compound 4 (0.363 g, 1.103 mmol), 1bromododecane (0.412 g, 1.655 mmol) and potassium carbonate (0.228 g, 1.655 mmol), the pure compound 6h (0.243 g) was obtained as yellow solid. Yield: 44.3%; mp: 135e137 C; IR (KBr) n: 3357 (NH), 3038 (aromatic CH), 2991 (CH2), 1687 (C¼N), 1590, 1508 (aromatic frame), 736 cm1; 1H NMR (600 MHz, DMSO-d6) d: 0.86e0.83 (t, 3H, J ¼ 9.0 Hz, (CH2)11CH3), 1.26e1.22 (m, 18H, (CH2)2(CH2)9CH3), 1.69e1.67 (m, 2H, CH2CH2(CH2)9CH3), 2.10 (s, 3H, COCH3), 4.24e4.21 (t, 2H, J ¼ 9.0 Hz, CH2(CH2)10CH3), 5.08 (s, 2H, Bim-CH2), 7.21e7.18 (t, H, J ¼ 9.0 Hz, Bim-6-H), 7.27e7.25 (t, H, J ¼ 6.0 Hz, Bim-7-H), 7.56 (d, 2H, J ¼ 6.0 Hz, Bim-5,8-H), 7.71 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.76 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 10.41 (s, H, NHCOCH3) ppm; 13C NMR (150 MHz, DMSO-d6) d: 14.4, 22.5, 24.6, 26.6, 29.1, 29.2, 29.4, 29.4, 29.5, 29.5, 29.7, 31.8, 44.2, 54.7, 111.3, 118.9, 119.7, 122.4, 123.2, 129.9, 132.7, 135.6, 142.7, 143.4, 144.8, 169.6 ppm; ESI-MS (m/z): 499 [MþH]þ; HRMS (ESI) calcd. for C28H40N3O3S [MþH]þ, 498.2785; found, 498.2793. 4.1.20. Synthesis of N-(4-((1-octadecyl-1H-benzo[d]imidazol-2-yl) methylsulfonyl)phenyl)acetamide (6i) Compound 6i was synthesized according to the experimental procedure reported for compound 5a, starting from compound 4 (0.350 g, 1.064 mmol), 1-bromooctadecane (0.495 g, 1.486 mmol) and potassium carbonate (0.220 g, 1.596 mmol). The crude product was obtained and puriﬁed via silica gel column chromatography (eluent, acetone/petroleum ether, 1/1, V/V) to give pure compound 6i (0.328 g) as yellow solid. Yield: 53.0%; mp: 143e145 C; IR (KBr) n: 3356 (NH), 3034 (aromatic CH), 2997 (CH2), 1689 (C¼N), 1593, 1501 (aromatic frame), 738 cm1; 1H NMR (600 MHz, DMSO-d6) d: 0.85e0.83 (t, 3H, J ¼ 6.0 Hz, (CH2)17CH3), 1.26e1.22 (m, 30H, (CH2)2(CH2)15CH3), 1.69e1.66 (m, 2H, CH2CH2(CH2)15CH3), 2.10 (s, 3H, COCH3), 4.23e4.21 (t, 2H, J ¼ 6.0 Hz, CH2(CH2)16CH3), 5.07 (s, 2H, Bim-CH2), 7.20e7.17 (t, H, J ¼ 9.0 Hz, Bim-6-H), 7.26e7.24 (t, H, J ¼ 6.0 Hz, Bim-7-H), 7.56 (d, 2H, J ¼ 6.0 Hz, Bim-5,8-H), 7.70 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.76 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 10.41 (s, H, NHCOCH3) ppm; 13C NMR (150 MHz, DMSO-d6) d: 14.3, 22.5, 24.6, 26.6, 29.1, 29.2, 29.4, 29.4, 29.5, 29.5, 29.6, 31.8, 44.1, 54.7, 111.2, 118.9, 119.7, 122.3, 123.1, 129.9, 132.7, 135.6, 142.8, 143.4, 144.8, 169.6 ppm; ESI-MS (m/z): 583 [MþH]þ; HRMS (ESI) calcd. for C34H52N3O3S [MþH]þ, 582.3724; found, 582.3727. 179 4.1.21. Synthesis of N-(4-((1-(5-(9H-carbazol-9-yl)pentyl)-1Hbenzo[d]imidazol-2-yl)methylsulfonyl) phenyl)acetamide (7) Prepared the same way as the general procedure described for compound 5a, starting from compound 4 (0.176 g, 0.535 mmol), 9(5-bromopentyl)-9H-carbazole (0.342 g, 1.082 mmol) and potassium carbonate (0.152 g, 1.101 mmol), pure compound 7 (0.216 g) was synthesized as white solid. Yield: 72.0%; mp: 118e120 C; IR (KBr) n: 3369 (NH), 3042 (aromatic CH), 2984 (CH2), 1694 (C¼N), 1591, 1525 (aromatic frame), 741 cm1; 1H NMR (600 MHz, DMSOd6) d: 1.46e1.32 (m, 2H, carbazole-(CH2)2CH2), 1.75e1.71 (m, 2H, carbazole-CH2CH2), 1.82e1.78 (m, 2H, carbazole-(CH2)3CH2), 2.10 (s, 3H, COCH3), 4.20e4.18 (t, H, J ¼ 6.0 Hz, carbazole-CH2), 4.39e4.36 (t, H, J ¼ 9.0 Hz, carbazole-(CH2)4CH2), 5.00 (s, 2H, Bim-CH2), 7.24e7.18 (m, 4H, Bim-5,6,7,8-H), 7.48e7.42 (m, 3H, carbazole4,5,10-H), 7.57e7.53 (m, 3H, carbazole-3,11,12-H), 7.67 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.76 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 8.15e8.13 (d, 2H, carbazole-6,9-H), 10.39 (s, H, NHCOCH3) ppm; 13C NMR (150 MHz, DMSO-d6) d: 24.5, 24.7, 28.7, 29.5, 42.7, 44.1, 54.6, 109.7, 111.3, 119.0, 119.1, 119.7, 120.7, 122.3, 122.6, 123.2, 126.1, 129.9, 132.6, 135.6, 140.5, 142.7, 143.4, 144.8, 169.6 ppm; ESI-MS (m/z): 566 [MþH]þ; HRMS (ESI) calcd. for C33H33N4O3S [MþH]þ, 565.2268; found, 565.2271. 4.1.22. Synthesis of 4-((1-(2-ﬂuorobenzyl)-1H-benzo[d]imidazol-2yl)methylsulfonyl)aniline (8a) To a solution of compound 5a (0.300 g, 0.685 mmol) in ethanol 15 mL was added 0.4 mL 2 mol/L sodium hydroxide solution. The mixture was reﬂuxed for 10 h (monitored by TLC, eluent, acetone/ petroleum ether, 1/1, V/V). After cooling to the room temperature, the solvent was removed in vacuo to give the deprotected compound 8a as yellow solid. Yield: 97.4%; mp: 205e207 C; IR (KBr) n: 3318, 3207 (NH), 3058 (aromatic CH), 2988 (CH2), 1647 (C¼N), 1587, 1506 (aromatic frame), 739 cm1; 1H NMR (600 MHz, DMSOd6) d: 4.92 (s, 2H, Bim-CH2), 5.57 (s, 2H, 2-FPh-CH2), 6.21 (s, 2H, NH2), 6.60 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 6.91e6.89 (t, H, J ¼ 6.0 Hz, 2FPh-5-H), 7.10e7.08 (t, H, J ¼ 6.0 Hz, 2-FPh-4-H), 7.24e7.19 (m, 4H, Bim-5,6,7,8-H), 7.34e7.31 (m, H, 2-FPh-6-H), 7.37 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 7.62e7.61 (m, H, 2-FPh-3-H) ppm; 13C NMR (150 MHz, DMSO-d6) d: 49.1, 55.2, 111.3, 113.1, 116.0, 119.9, 122.6, 123.4, 123.6, 123.8, 125.1, 129.5, 130.3, 130.5, 135.6, 142.9, 144.5, 154.6, 160.4 ppm; ESI-MS (m/z): 418 [MþNa]þ; HRMS (ESI) calcd. for C21H18FN3O2SNa [MþNa]þ, 418.0996; found, 418.1005. 4.1.23. Synthesis of 4-((1-(3-ﬂuorobenzyl)-1H-benzo[d]imidazol-2yl)methylsulfonyl)aniline (8b) Compound 8b was prepared according to the experimental procedure described for compound 8a, starting from compound 5b (0.095 g, 0.217 mmol) and 0.2 mL 2 mol/L sodium hydroxide solution. The pure product 8b was obtained as yellow solid. Yield: 95.3%; mp: 212e214 C; IR (KBr) n: 3314, 3209 (NH), 3061 (aromatic CH), 2993 (CH2), 1649 (C¼N), 1588, 1501 (aromatic frame), 738 cm1; 1H NMR (600 MHz, DMSO-d6) d: 4.95 (s, 2H, Bim-CH2), 5.57 (s, 2H, 3-FPh-CH2), 6.21 (s, 2H, NH2), 6.61 (d, 2H, J ¼ 6.0 Hz, Ph3,5-H), 6.97e6.92 (m, 2H, 3-FPh-2,6-H), 7.11e7.08 (m, H, 3-FPh-4H), 7.20e7.18 (m, 2H, Bim-6,7-H), 7.31e7.30 (m, H, Bim-8-H), 7.37e7.33 (m, H, 3-FPh-5-H), 7.39 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 7.63e7.61 (m, H, Bim-5-H), 10.41 (s, H, NHCOCH3) ppm; 13C NMR (600 MHz, DMSO-d6) d: 47.1, 55.2, 111.5, 113.1, 114.3, 114.9, 119.8, 122.6, 123.4, 123.8, 130.5, 131.1, 135.6, 139.9, 139.9, 142.9, 144.5, 154.6, 162.8 ppm; ESI-MS (m/z): 418 [MþNa]þ; HRMS (ESI) calcd. for C21H18FN3O2SNa [MþNa]þ, 418.0996; found, 418.1004. 4.1.24. Synthesis of 4-((1-(4-ﬂuorobenzyl)-1H-benzo[d]imidazol-2yl)methylsulfonyl)aniline (8c) Compound 8c was prepared according to the procedure 180 H.-Z. Zhang et al. / European Journal of Medicinal Chemistry 136 (2017) 165e183 depicted for compound 8a, starting from compound 5c (0.140 g, 0.320 mmol) and 0.2 mL 2 mol/L sodium hydroxide solution. The pure product 5h was obtained as yellow solid. Yield: 95.2%; mp: 224e226 C; IR (KBr) n: 3307, 3200 (NH), 3069 (aromatic CH), 2997 (CH2), 1654 (C¼N), 1594, 1507 (aromatic frame), 740 cm1; 1H NMR (600 MHz, DMSO-d6) d: 4.93 (s, 2H, Bim-CH2), 5.52 (s, 2H, 4FPh-CH2), 6.21 (s, 2H, NH2), 6.60 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.19e7.12 (m, 6H, 4-FPh-2,3,5,6-H, Bim-6,7-H), 7.31e7.30 (m, H, Bim-8-H), 7.39 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 7.61e7.60 (m, H, Bim-5H) ppm; 13C NMR (150 MHz, DMSO-d6) d: 46.9, 55.3, 111.6, 113.1, 115.9, 119.8, 122.6, 123.3, 123.8, 129.5, 130.6, 133.2, 135.6, 143.0, 144.4, 154.6, 162.0 ppm; ESI-MS (m/z): 418 [MþNa]þ; HRMS (ESI) calcd. for C21H18FN3O2SNa [MþNa]þ, 418.0996; found, 418.1000. 4.1.25. Synthesis of 4-((1-(2-chlorobenzyl)-1H-benzo[d]imidazol-2yl)methylsulfonyl)aniline (8d) Compound 8d was prepared according to the experimental procedure reported for compound 8a, starting from compound 5d (0.148 g, 0.325 mmol) and 0.2 mL 2 mol/L sodium hydroxide solution. The pure product 5i was obtained as yellow solid. Yield: 97.5%; mp: 184e186 C; IR (KBr) n: 3311, 3207 (NH), 3062 (aromatic CH), 2989 (CH2), 1659 (C¼N), 1588, 1501 (aromatic frame), 741 cm1; 1H NMR (600 MHz, DMSO-d6) d: 4.88 (s, 2H, Bim-CH2), 5.56 (s, 2H, 2-ClPh-CH2), 6.22 (s, 2H, NH2), 6.50 (d, H, J ¼ 6.0 Hz, 2ClPh-6-H), 6.60 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.24e7.17 (m, 4H, Bim5,6,7,8-H), 7.32e7.29 (t, H, J ¼ 6.0 Hz, 2-ClPh-4-H), 7.35 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 7.53 (d, H, J ¼ 6.0 Hz, 2-ClPh-5-H), 7.65 (d, H, J ¼ 6.0 Hz, 2-ClPh-3-H) ppm; 13C NMR (150 MHz, DMSO-d6) d: 45.3, 56.3, 111.3, 113.1, 120.0, 122.7, 123.5, 123.6, 128.0, 128.4, 129.8, 130.0, 130.5, 132.1, 134.1, 135.7, 142.9, 144.7, 154.6 ppm; ESI-MS (m/z): 435 [MþNa]þ; HRMS (ESI) calcd. for C21H18ClN3O2SNa [MþNa]þ, 434.0700; found, 434.0712. 4.1.26. Synthesis of 4-((1-(3-chlorobenzyl)-1H-benzo[d]imidazol-2yl)methylsulfonyl)aniline (8e) Pure compound 8e was obtained in process of synthesizing compound 8a as yellow solid. Yield: 93.7%; mp: 237e239 C; IR (KBr) n: 3314, 3202 (NH), 3067 (aromatic CH), 2993 (CH2), 1657 (C¼N), 1585, 1503 (aromatic frame), 740 cm1; 1H NMR (600 MHz, DMSO-d6) d: 4.92 (s, 2H, Bim-CH2), 5.59 (s, 2H, 3-ClPh-CH2), 6.21 (s, 2H, NH2), 6.61 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.07 (d, H, J ¼ 6.0 Hz, 3ClPh-6-H), 7.24e7.18 (m, 4H, Bim-5,6,7,8-H), 7.29e7.26 (t, H, J ¼ 9.0 Hz, 3-ClPh-5-H), 7.35 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 7.51 (d, H, J ¼ 6.0 Hz, 3-ClPh-4-H), 7.52 (d, H, J ¼ 6.0 Hz, 3-ClPh-2-H) ppm; 13C NMR (150 MHz, DMSO-d6) d: 50.2, 56.3, 111.2, 113.3, 119.5, 122.8, 123.7, 123.8, 127.8, 128.3, 129.7, 130.1, 130.4, 132.0, 134.2, 135.8, 142.9, 144.8, 154.7 ppm; ESI-MS (m/z): 435 [MþNa]þ; HRMS (ESI) calcd. for C21H18ClN3O2SNa [MþNa]þ, 434.0700; found, 434.0712. 4.1.27. Synthesis of 4-((1-(4-chlorobenzyl)-1H-benzo[d]imidazol-2yl)methylsulfonyl)aniline (8f) Compound 8f was prepared according to the experimental procedure described for compound 8a, starting from compound 5f (0.150 g, 0.329 mmol) and 0.2 mL 2 mol/L sodium hydroxide solution. The pure product 8f was obtained as yellow solid. Yield: 90.4%; mp: 223e225 C; IR (KBr) n: 3319, 3206 (NH), 3064 (aromatic CH), 2989 (CH2), 1654 (C¼N), 1587, 1505 (aromatic frame), 739 cm1; 1H NMR (600 MHz, DMSO-d6) d: 4.93 (s, 2H, Bim-CH2), 5.54 (s, 2H, 4-ClPh-CH2), 6.21 (s, 2H, NH2), 6.61 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.13 (d, 2H, J ¼ 6.0 Hz, 4-ClPh-2,6-H), 7.20e7.16 (m, 2H, Bim-6,7-H), 7.30e7.28 (m, H, Bim-8-H), 7.36 (d, 2H, J ¼ 6.0 Hz, 4ClPh-3,5-H), 7.38 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 7.62e7.61 (m, H, Bim-5-H) ppm; 13C NMR (150 MHz, DMSO-d6) d: 47.0, 55.2, 111.6, 113.1, 119.8, 122.6, 123.4, 123.7, 129.0, 129.3, 130.6, 132.7, 135.6, 136.0, 143.0, 144.5, 154.6 ppm; ESI-MS (m/z): 413 [MþH]þ; HRMS (ESI) calcd. for C21H19ClN3O2S [MþH]þ, 412.0881; found, 412.0885. 4.1.28. Synthesis of 4-((1-(2,4-dichlorobenzyl)-1H-benzo[d] imidazol-2-yl)methylsulfonyl)aniline (8g) Compound 8g was prepared according to the experimental procedure depicted for compound 8a, starting from compound 5g (0.198 g, 0.405 mmol) and 0.2 mL 2 mol/L sodium hydroxide solution. The pure product 8g was obtained as yellow solid. Yield: 93.4%; mp: 216e218 C; IR (KBr) n: 3329, 3201 (NH), 3066 (aromatic CH), 2993 (CH2), 1639 (C¼N), 1593, 1500 (aromatic frame), 736 cm1; 1H NMR (600 MHz, DMSO-d6) d: 4.91 (s, 2H, Bim-CH2), 5.55 (s, 2H, 2,4-Cl2Ph-CH2), 6.21 (s, 2H, NH2), 6.47 (d, H, J ¼ 6.0 Hz, 2,4-Cl2Ph-6-H), 6.60 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.29e7.18 (m, 4H, Bim-5,6,7,8-H), 7.34 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 7.65 (d, H, J ¼ 6.0 Hz, 2,4-Cl2Ph-5-H), 7.70 (s, H, 2,4-Cl2Ph-3-H) ppm; 13C NMR (150 MHz, DMSO-d6) d: 45.1, 55.2, 111.3, 113.1, 120.0, 122.8, 123.5, 123.6, 128.1, 129.4, 129.6, 130.5, 133.0, 133.3, 133.4, 135.5, 142.9, 144.8, 154.6 ppm; ESI-MS (m/z): 447 [MþH]þ; HRMS (ESI) calcd. for C21H18Cl2N3O2S [MþH]þ, 446.0491; found, 446.0500. 4.1.29. Synthesis of 4-((1-(3,4-dichlorobenzyl)-1H-benzo[d] imidazol-2-yl)methylsulfonyl)aniline (8h) Compound 8h was prepared according to the experimental procedure reported for compound 8a, starting from compound 5h (0.140 g, 0.287 mmol) and 0.2 mL 2 mol/L sodium hydroxide solution. The pure product 8h was obtained as yellow solid. Yield: 94.7%; mp: 235e237 C; IR (KBr) n: 3324, 3205 (NH), 3061 (aromatic CH), 2988 (CH2), 1642 (C¼N), 1594, 1503 (aromatic frame), 740 cm1; 1H NMR (600 MHz, DMSO-d6) d: 5.03 (s, 2H, Bim-CH2), 5.62 (s, 2H, 3,4-Cl2Ph-CH2), 6.25 (s, 2H, NH2), 6.66 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.09 (d, H, J ¼ 6.0 Hz, 3,4-Cl2Ph-6-H), 7.26e7.23 (m, 2H, Bim-6,7-H), 7.34e7.33 (m, H, Bim-8-H), 7.44 (d, 2H, J ¼ 6.0 Hz, Ph2,6-H), 7.49 (s, H, 3,4-Cl2Ph-2-H), 7.62 (d, H, J ¼ 6.0 Hz, 3,4-Cl2Ph-5H), 7.68e7.66 (m, H, Bim-5-H) ppm; 13C NMR (150 MHz, DMSO-d6) d: 46.6, 56.1, 111.5, 113.1, 119.9, 122.7, 123.5, 123.8, 127.7, 129.5, 130.5, 130.6, 131.2, 131.7, 135.5, 138.2, 143.0, 144.6, 154.5 ppm; ESI-MS (m/ z): 469 [MþNa]þ; HRMS (ESI) calcd. for C21H17Cl2N3O2SNa [MþNa]þ, 468.0311; found, 468.0314. 4.1.30. Synthesis of 4-((1-ethyl-1H-benzo[d]imidazol-2-yl) methylsulfonyl)aniline (9a) Prepared according to the general procedure described for 8a, starting from compound 6a (0.135 g, 0.378 mmol) and 0.2 mL 2 mol/L sodium hydroxide solution, pure compound 9a was obtained as yellow solid. Yield: 90.8%; mp: 232e234 C; IR (KBr) n: 3332, 3208 (NH), 3061 (aromatic CH), 2988 (CH2), 1675 (C¼N), 1589, 1493 (aromatic frame), 740 cm1; 1H NMR (600 MHz, DMSOd6) d: 1.32e1.30 (t, 3H, J ¼ 6.0 Hz, CH2CH3), 4.30e4.27 (m, 2H, CH2CH3), 4.90 (s, 2H, Bim-CH2), 6.19 (s, 2H, NH2), 6.60 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.21e7.19 (t, H, J ¼ 6.0 Hz, Bim-6-H), 7.27e7.25 (t, H, J ¼ 6.0 Hz, Bim-7-H), 7.35 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 7.58e7.56 (t, 2H, J ¼ 6.0 Hz, Bim-5,8-H) ppm; 13C NMR (150 MHz, DMSO-d6) d: 15.2, 39.1, 55.2, 111.1, 113.1, 119.7, 122.3, 123.1, 123.9, 130.5, 135.2, 143.0, 143.8, 154.5 ppm; ESI-MS (m/z): 316 [MþH]þ; HRMS (ESI) calcd. for C16H18N3O2S [MþH]þ, 316.1114; found, 316.1121. 4.1.31. Synthesis of 4-((1-propyl-1H-benzo[d]imidazol-2-yl) methylsulfonyl)aniline (9b) Prepared according to the general procedure described for 8a, starting from compound 6b (0.122 g, 0.329 mmol) and 0.2 mL 2 mol/L sodium hydroxide solution, pure compound 9b was obtained as yellow solid. Yield: 97.2%; mp: 237e239 C; IR (KBr) n: 3327, 3205 (NH), 3051 (aromatic CH), 2993 (CH2), 1684 (C¼N), 1595, 1487 (aromatic frame), 737 cm1; 1H NMR (600 MHz, DMSO- H.-Z. Zhang et al. / European Journal of Medicinal Chemistry 136 (2017) 165e183 d6) d: 0.87e0.84 (t, 3H, J ¼ 9.0 Hz, CH2CH2CH3), 1.75e1.69 (m, 2H, CH2CH2CH3), 4.17e4.15 (t, 2H, J ¼ 6.0 Hz, CH2CH2CH3), 4.89 (s, 2H, Bim-CH2), 6.19 (s, 2H, NH2), 6.58 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.20e7.18 (t, H, J ¼ 6.0 Hz, Bim-6-H), 7.26e7.24 (t, H, J ¼ 6.0 Hz, Bim7-H), 7.33 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 7.58 (d, 2H, J ¼ 6.0 Hz, Bim5,8-H) ppm; 13C NMR (150 MHz, DMSO-d6) d: 11.4, 22.9, 45.5, 55.3, 111.3, 113.1, 119.7, 122.2, 123.0, 123.9, 130.5, 135.6, 142.8, 144.1, 154.5 ppm; ESI-MS (m/z): 330 [MþH]þ; HRMS (ESI) calcd. for C17H20N3O2S [MþH]þ, 330.1271; found, 330.1277. 4.1.32. Synthesis of 4-((1-pentyl-1H-benzo[d]imidazol-2-yl) methylsulfonyl)aniline (9c) Compound 9c was prepared according to the procedure depicted for compound 8a, starting from compound 6c (0.042 g, 0.105 mmol) and 0.2 mL 2 mol/L sodium hydroxide solution. The product 9c (0.033 g) was obtained as yellow solid. Yield: 89.1%; mp: 233e235 C; IR (KBr) n: 3316, 3200 (NH), 3046 (aromatic CH), 2987 (CH2), 1680 (C¼N), 1587, 1494 (aromatic frame), 741 cm1; 1H NMR (600 MHz, DMSO-d6) d: 0.86e0.84 (t, 3H, J ¼ 6.0 Hz, (CH2)4CH3), 1.31e1.24 (m, 4H, (CH2)2(CH2)2CH3), 1.72e1.67 (m, 2H, CH2CH2(CH2)2CH3), 4.19e4.16 (t, 2H, J ¼ 9.0 Hz, CH2(CH2)3CH3), 4.88 (s, 2H, Bim-CH2), 6.18 (s, 2H, NH2), 6.59 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.20e7.18 (t, H, J ¼ 6.0 Hz, Bim-6-H), 7.27e7.24 (t, H, J ¼ 9.0 Hz, Bim7-H), 7.33 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 7.58e7.54 (m, 2H, Bim-5,8H) ppm; 13C NMR (150 MHz, DMSO-d6) d: 14.3, 22.3, 28.8, 29.3, 44.0, 55.3, 111.2, 113.1, 119.7, 122.3, 123.0, 123.8, 130.5, 135.6, 142.8, 144.0, 154.5 ppm; ESI-MS (m/z): 358 [MþH]þ; HRMS (ESI) calcd. for C19H24N3O2S [MþH]þ, 358.1584; found, 358.1587. 4.1.33. Synthesis of 4-((1-hexyl-1H-benzo[d]imidazol-2-yl) methylsulfonyl)aniline (9d) Compound 9d was prepared according to the procedure depicted for compound 8a, starting from compound 6d (0.134 g, 0.324 mmol) and 0.2 mL 2 mol/L sodium hydroxide solution. The product 9d (0.108 g) was obtained as yellow solid. Yield: 94.2%; mp: 238e240 C; IR (KBr) n: 3319, 3201 (NH), 3042 (aromatic CH), 2986 (CH2), 1672 (C¼N), 1583, 1494 (aromatic frame), 740 cm1; 1H NMR (600 MHz, DMSO-d6) d: 0.86e0.84 (t, 3H, J ¼ 6.0 Hz, (CH2)5CH3), 1.26e1.25 (m, 6H, (CH2)2(CH2)3CH3), 1.70e1.66 (m, 2H, CH2CH2(CH2)3CH3), 4.19e4.16 (t, 2H, J ¼ 9.0 Hz, CH2(CH2)4CH3), 4.88 (s, 2H, Bim-CH2), 6.20 (s, 2H, NH2), 6.59 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.20e7.18 (t, H, J ¼ 6.0 Hz, Bim-6-H), 7.27e7.24 (t, H, J ¼ 9.0 Hz, Bim7-H), 7.33 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 7.58e7.54 (m, 2H, Bim-5,8H) ppm; 13C NMR (150 MHz, DMSO-d6) d: 14.3, 22.5, 26.3, 29.6, 31.3, 44.1, 55.3, 111.2, 113.1, 119.7, 122.2, 123.0, 123.9, 130.5, 135.6, 142.9, 144.0, 154.5 ppm; ESI-MS (m/z): 373 [MþH]þ; HRMS (ESI) calcd. for C20H26N3O2S [MþH]þ, 372.1740; found, 372.1744. 4.1.34. Synthesis of 4-((1-heptyl-1H-benzo[d]imidazol-2-yl) methylsulfonyl)aniline (9e) Compound 9e was synthesized according to the experimental procedure reported for compound 8a, starting from compound 6e (0.079 g, 0.185 mmol) and 0.2 mL 2 mol/L sodium hydroxide solution. The desired compound 9e (0.067 g) was obtained as yellow solid. Yield: 94.4%; mp: 219219 C; IR (KBr) n: 3316, 3196 (NH), 3044 (aromatic CH), 2981 (CH2), 1675 (C¼N), 1579, 1482 (aromatic frame), 738 cm1; 1H NMR (600 MHz, DMSO-d6) d: 0.86e0.84 (t, 3H, J ¼ 6.0 Hz, (CH2)6CH3), 1.27e1.23 (m, 8H, (CH2)2(CH2)4CH3), 1.69e1.67 (m, 2H, CH2CH2(CH2)4CH3), 4.19e4.16 (t, 2H, J ¼ 9.0 Hz, CH2(CH2)5CH3), 4.88 (s, 2H, Bim-CH2), 6.19 (s, 2H, NH2), 6.58 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.20e7.18 (t, H, J ¼ 6.0 Hz, Bim-6-H), 7.27e7.24 (t, H, J ¼ 9.0 Hz, Bim-7-H), 7.33 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 7.58e7.54 (m, 2H, Bim-5,8-H) ppm; 13C NMR (150 MHz, DMSO-d6) d: 14.4, 22.5, 26.6, 28.8, 29.6, 31.6, 44.1, 55.3, 111.2, 113.1, 119.7, 122.2, 123.0, 123.9, 130.4, 135.6, 142.8, 144.0, 154.5 ppm; ESI-MS (m/z): 387 181 [MþH]þ; HRMS (ESI) calcd. for C21H28N3O2S [MþH]þ, 386.1897; found, 386.1898. 4.1.35. Synthesis of 4-((1-octyl-1H-benzo[d]imidazol-2-yl) methylsulfonyl)aniline (9f) Compound 9f was synthesized according to the experimental procedure reported for compound 8a, starting from compound 6f (0.288 g, 0.652 mmol) and 0.4 mL 2 mol/L sodium hydroxide solution. The desired compound 9f (0.256 g) was obtained as yellow solid. Yield: 98.5%; mp: 213e215 C; IR (KBr) n: 3319, 3203 (NH), 3040 (aromatic CH), 2985 (CH2), 1670 (C¼N), 1584, 1478 (aromatic frame), 736 cm1; 1H NMR (600 MHz, DMSO-d6) d: 0.86e0.83 (t, 3H, J ¼ 9.0 Hz, (CH2)7CH3), 1.26e1.23 (m, 10H, (CH2)2(CH2)5CH3), 1.69e1.67 (m, 2H, CH2CH2(CH2)5CH3), 4.19e4.16 (t, 2H, J ¼ 9.0 Hz, CH2(CH2)6CH3), 4.88 (s, 2H, Bim-CH2), 6.19 (s, 2H, NH2), 6.58 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.20e7.18 (t, H, J ¼ 6.0 Hz, Bim-6-H), 7.27e7.24 (t, H, J ¼ 9.0 Hz, Bim-7-H), 7.33 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 7.58e7.54 (m, 2H, Bim-5,8-H) ppm; 13C NMR (150 MHz, DMSO-d6) d: 14.4, 22.5, 26.6, 29.0, 29.1, 29.6, 31.6, 44.1, 55.3, 111.2, 113.1, 119.7, 122.2, 123.0, 123.9, 130.4, 135.6, 142.8, 144.0, 154.5 ppm; ESI-MS (m/ z): 401 [MþH]þ; HRMS (ESI) calcd. for C22H30N3O2S [MþH]þ, 400.2053; found, 400.2055. 4.1.36. Synthesis of 4-((1-decyl-1H-benzo[d]imidazol-2-yl) methylsulfonyl)aniline (9g) Compound 9g was prepared according to the experimental procedure reported for compound 8a, starting from compound 6g (0.235 g, 0.500 mmol) and 0.4 mL 2 mol/L sodium hydroxide solution. The desired compound 9g (0.179 g) was obtained as yellow solid. Yield: 85.1%; mp: 214e216 C; IR (KBr) n: 3309, 3192 (NH), 3038 (aromatic CH), 2977 (CH2), 1672 (C¼N), 1589, 1485 (aromatic frame), 739 cm1; 1H NMR (600 MHz, DMSO-d6) d: 0.86e0.84 (t, 3H, J ¼ 6.0 Hz, (CH2)9CH3), 1.26e1.23 (m, 14H, (CH2)2(CH2)7CH3), 1.69e1.66 (m, 2H, CH2CH2(CH2)7CH3), 4.19e4.16 (t, 2H, J ¼ 9.0 Hz, CH2(CH2)8CH3), 4.87 (s, 2H, Bim-CH2), 6.19 (s, 2H, NH2), 6.59 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.20e7.18 (t, H, J ¼ 6.0 Hz, Bim-6-H), 7.26e7.24 (t, H, J ¼ 6.0 Hz, Bim-7-H), 7.33 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 7.58e7.54 (m, 2H, Bim-5,8-H) ppm; 13C NMR (150 MHz, DMSO-d6) d: 14.4, 22.5, 26.6, 29.1, 29.4, 29.4, 29.6, 31.7, 44.1, 55.3, 111.2, 113.1, 119.7, 122.2, 123.0, 123.9, 130.4, 135.6, 142.8, 144.0, 154.5 ppm; ESIMS (m/z): 429 [MþH]þ; HRMS (ESI) calcd. for C24H34N3O2S [MþH]þ, 428.2366; found, 428.2373. 4.1.37. Synthesis of 4-((1-dodecyl-1H-benzo[d]imidazol-2-yl) methylsulfonyl)aniline (9h) Compound 9h was obtained as yellow solid. Yield: 95.1%; mp: 208e210 C; IR (KBr) n: 3313, 3192 (NH), 3031 (aromatic CH), 2977 (CH2), 1670 (C¼N), 1589, 1482 (aromatic frame), 736 cm1; 1H NMR (600 MHz, DMSO-d6) d: 0.86e0.84 (t, 3H, J ¼ 6.0 Hz, (CH2)11CH3), 1.26e1.23 (m, 18H, (CH2)2(CH2)9CH3), 1.69e1.67 (m, 2H, CH2CH2(CH2)9CH3), 4.19e4.16 (t, 2H, J ¼ 9.0 Hz, CH2(CH2)10CH3), 4.87 (s, 2H, Bim-CH2), 6.18 (s, 2H, NH2), 6.59 (d, 2H, J ¼ 6.0 Hz, Ph3,5-H), 7.20e7.18 (t, H, J ¼ 6.0 Hz, Bim-6-H), 7.26e7.24 (t, H, J ¼ 6.0 Hz, Bim-7-H), 7.33 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 7.58e7.54 (m, 2H, Bim-5,8-H) ppm; 13C NMR (150 MHz, DMSO-d6) d: 14.4, 22.5, 26.6, 29.1, 29.2, 29.4, 29.4, 29.4, 29.5, 29.6, 31.8, 44.1, 55.3, 111.2, 113.1, 1197, 122.2, 123.0, 123.9, 130.4, 135.6, 142.8, 144.0, 154.5 ppm; ESI-MS (m/z): 457 [MþH]þ; HRMS (ESI) calcd. for C26H38N3O2S [MþH]þ, 456.2679; found, 456.2683. 4.1.38. Synthesis of 4-((1-octadecyl-1H-benzo[d]imidazol-2-yl) methylsulfonyl)aniline (9i) Compound 9i was obtained as yellow solid. Yield: 76.4%; mp: 186e188 C; IR (KBr) n: 3310, 3185 (NH), 3025 (aromatic CH), 2981 (CH2), 1675 (C¼N), 1589, 1476 (aromatic frame), 738 cm1; 1H 182 H.-Z. Zhang et al. / European Journal of Medicinal Chemistry 136 (2017) 165e183 NMR (600 MHz, DMSO-d6) d: 0.86e0.83 (t, 3H, J ¼ 9.0 Hz, (CH2)17CH3), 1.26e1.22 (m, 30H, (CH2)2(CH2)15CH3), 1.69e1.67 (m, 2H, CH2CH2(CH2)15CH3), 4.19e4.16 (t, 2H, J ¼ 9.0 Hz, CH2(CH2)16CH3), 4.87 (s, 2H, Bim-CH2), 6.19 (s, 2H, NH2), 6.58 (d, 2H, J ¼ 6.0 Hz, Ph-3,5-H), 7.20e7.17 (t, H, J ¼ 9.0 Hz, Bim-6-H), 7.26e7.23 (t, H, J ¼ 9.0 Hz, Bim-7-H), 7.32 (d, 2H, J ¼ 6.0 Hz, Ph-2,6-H), 7.58e7.53 (m, 2H, Bim-5,8-H) ppm; 13C NMR (150 MHz, DMSO-d6) d: 14.4, 22.5, 26.6, 29.1, 29.2, 29.4, 29.4, 29.5, 29.5, 29.6, 31.8, 44.1, 55.3, 111.1, 113.1, 119.7, 122.2, 123.0, 123.9, 130.4, 136.6, 142.8, 144.0, 154.5 ppm; ESI-MS (m/z): 541 [MþH]þ; HRMS (ESI) calcd. for C32H50N3O2S [MþH]þ, 540.3618; found, 540.3627. 4.1.39. Synthesis of 4-((1-(5-(9H-carbazol-9-yl)pentyl)-1H-benzo [d]imidazol-2-yl)methylsulfonyl)aniline (10) Compound 10 was prepared according to the experimental procedure reported for compound 8a, starting from compound 7 (0.184 g, 0.327 mmol) and 0.2 mL 2 mol/L sodium hydroxide solution. The desired compound 10 (0.156 g) was obtained as yellow solid. Yield: 91.8%; mp: 188e190 C; IR (KBr) n: 3329, 3209 (NH), 3055 (aromatic CH), 2995 (CH2), 1681 (C¼N), 1597, 1485 (aromatic frame), 744 cm1; 1H NMR (600 MHz, DMSO-d6) d: 1.36e1.31 (m, 2H, carbazole-(CH2)2CH2), 1.73e1.70 (m, 2H, carbazole-CH2CH2), 1.81e1.78 (m, 2H, carbazole-(CH2)3CH2), 4.15e4.12 (t, H, J ¼ 9.0 Hz, carbazole-CH2), 4.38e4.36 (t, H, J ¼ 6.0 Hz, carbazole-(CH2)4CH2), 4.80 (s, 2H, Bim-CH2), 6.18 (s, 2H, NH2), 6.58 (d, 2H, J ¼ 6.0 Hz, Ph3,5-H), 7.23e7.17 (m, 4H, Bim-5,6,7,8-H), 7.32 (d, 2H, J ¼ 6.0 Hz, Ph2,6-H), 7.47e7.42 (m, 3H, carbazole-4,5,10-H), 7.57e7.55 (m, 3H, carbazole-3,11,12-H), 8.15e8.13 (d, 2H, carbazole-6,9-H) ppm; 13C NMR (150 MHz, DMSO-d6) d: 24.6, 28.7, 29.4, 42.7, 44.0, 55.3, 109.7, 111.2, 1131, 119.1, 119.6, 120.7, 122.2, 122.5, 123.0, 123.8, 126.1, 130.4, 135.6, 140.5, 142.8, 143.9, 154.5 ppm; ESI-MS (m/z): 524 [MþH]þ; HRMS (ESI) calcd. for C31H31N4O2S [MþH]þ, 523.2162; found, 523.2170. 4.2. Antibacterial and antifungal assays Minimal inhibitory concentration (MIC, mg/mL) is deﬁned as the lowest concentration of target compounds that completely inhibit the growth of bacteria, by means of standard two-fold serial dilution method in 96-well microtest plates according to the National Committee for Clinical Laboratory Standards (NCCLS). The tested microorganism strains were provided by the School of Pharmaceutical Sciences, Southwest University and the College of Pharmacy, Third Military Medical University. Chloromycin, Norﬂoxacin and Fluconazole, were used as control drugs. DMSO with inoculation bacterial not medicine was used as positive control to ensure that the solvent had no effect on bacteria growth. All the bacteria and fungi growth was monitored visually and spectrophotometrically, and the experiments were performed in triplicate. The MIC values in mg/mL were summarized in Tables 1 and 2 4.2.1. Antibacterial assays The prepared compounds 4e10 were evaluated for their antibacterial activities against Gram-positive bacteria (S. aureus ATCC 6538, Methicillin-resistant Staphylococcus aureus N315 (MRSA), M. luteus and B. subtilis ATCC 21216), Gram-negative bacteria (E. coli ATCC 8099, P. aeruginosa ATCC 27853, B. typhi and B. proteus ATCC 13315). The bacterial suspension was adjusted with sterile saline to a concentration of 1 105 CFU. Initially the compounds were dissolved in DMSO to prepare the stock solutions, then the tested compounds and reference drugs were prepared in MuellereHinton broth (Guangdong huaikai microbial sci. & tech co., Ltd, Guangzhou, Guangdong, China) to obtain the required concentrations of 512, 256, 128, 64, 32, 16, 8, 4, 2, 1, 0.5 mg/mL. These dilutions were inoculated and incubated at 37 C for 24 h. 4.2.2. Antifungal assays The newly synthesized compounds 4e10 were evaluated for their antifungal activities against C. albicans ATCC 76615, A. fumigatus ATCC 96918, C. utilis, S. cerevisia and A. ﬂavus. A spore suspension in sterile distilled water was prepared from one day old culture of the fungi growing on Sabouraud agar (SA) media. The ﬁnal spore concentration was 1e5 103 spore mL1. From the stock solutions of the tested compounds and reference antifungal drug Fluconazole, dilutions in sterile RPMI 1640 medium (Neuronbc Laboraton Technology CO., Ltd, Beijing, China) were made resulting in eleven wanted concentrations (0.5e512 mg/mL) of each tested compound. These dilutions were inoculated and incubated at 35 C for 24 h. 4.3. Molecular docking All the docking studies were carried out using Surﬂex-Docking 2.0 on a window 7 workstation. The crystal structure of DNA (PDB entry 3FT6) was downloaded from the protein data bank and used for docking studies. Water molecules were removed from protein PDB ﬁles, and hydrogen atoms were added. The 3D structures of compounds 5c and 5g were ﬁrst built using SurﬂexDocking 2.0 sketch followed by energy minimization. At last we used the Surﬂex-docking program to automatically dock the drugs into the binding pockets of DNA. In the docking process, 20 conformations were obtained, and among them the lowest free energy solution (or the highest total score) was chosen for our compounds modeling. To conﬁrm our docking experiment reliable, we redocked the native ligand into the 3FT6 (Supporting Information: Fig. S12), the RMSD value is 0.68 (<2), suggesting the methodology is reliable. Acknowledgments This work was partially supported by National Natural Science Foundation of China (Nos. 21672173 and 21372186), the Research Fund for International Young Scientists from International (Regional) Cooperation and Exchange Program of NSFC (No. 81650110529), Chongqing Special Foundation for Postdoctoral Research Proposal (Xm2016039), and Fundamental Research Funds for the Central Universities (XDJK2016E058), Doctoral Scientiﬁc Research Foundation of Linyi University (No. LYDX2016BS030). Appendix A. Supplementary data Supplementary data related to this article can be found at http:// dx.doi.org/10.1016/j.ejmech.2017.04.077. References [1] M.F. Chellat, L. Ragu z, R. Riedl, Angew. Chem. Int. Ed. 55 (2016) 2e30. [2] S.C. He, P. Jeyakkumar, S.R. Avula, X.L. Wang, H.Z. Zhang, C.H. Zhou, Sci. Sin. Chim. 46 (2016) 823e847. [3] N. Srivastava, A. Kumar, Eur. J. Med. Chem. 67 (2013) 464e468. [4] J. Lal, S.K. Gupta, D. Thavaselvam, D.D. Agarwal, Eur. J. Med. 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Send Orders for Reprints to reprints@benthamscience.ae Medicinal Chemistry, 2019, 15, 1-14 1 RESEARCH ARTICLE Design and Synthesis of Novel Sulfonamide-Derived Triazoles and Bioactivity Exploration Shi-Chao He1,2, Hui-Zhen Zhang1,*, Hai-Juan Zhang1, Qing Sun1 and Cheng-He Zhou2* 1 School of Pharmacy, Linyi University, Linyi 276000, P.R China, 2School of Chemistry and Chemical Engineering, Southwest University, Chongqing 400715, P.R China Abstract: Objective: Due to the incidence of resistance, a series of sulfonamide-derived 1,2,4triazoles were synthesized and evaluated. ARTICLE HISTORY Received: July 16, 2018 Revised: October 29, 2018 Accepted: October 30, 2018 DOI: 10.2174/1573406414666181106124852 Results: In vitro antimicrobial evaluation found that 2-chlorobenzyl sulfonamide 1,2,4-triazole 7c exhibited excellent antibacterial activities against MRSA, B. subtilis, B. typhi and E. coli with MIC values of 0.02−0.16 µmol/mL, which were comparable or even better than Chloromycin. The preliminary mechanism suggested that compound 7c could effectively bind with DNA, and also it could bind with human microsomal heme through hydrogen bonds in molecular docking. Computational chemical studies were performed on compound 7c to understand the structural features that are essential for activity. Additionally, compound 7c could generate a small amount of reactive oxygen species (ROS). Conclusion: Compound 7c could serve as a potential clinical antimicrobial candidate. Keywords: Antibacterial, antifungal, cytotoxicity, sulfonamides, triazole. 1. INTRODUCTION In the past few decades, the incidence of systemic microbial infections has rapidly increased and become a major threat to public health. Various synthetic drugs such as sulfonamides, quinolones, azoles and so on are available throughout the world, however, the overuse of anti-infective drugs for the prevention and treatment of diseases has accelerated the dramatic growing emergence of drug resistance. Especially, the occurrence of multi-drug resistant bacteria and fungi has become a severe problem in both community and hospital-acquired infections. Therefore, the discovery of novel structurally antimicrobial agents with good pharmacological profiles and excellent activity toward resistant strains is highly desirable [1]. Sulfonamides as important artificial antimicrobial agents have been widely used in the clinic since 80 years ago. Currently, sulfonamide compounds have attracted increasing attention due to various biological activities [2] such as antimicrobial [3-5], anticancer [6] and so on [7, 8]. As already reported, sulfonamides could compete with aminobenzoic acid to affect the synthesis of nucleic acid, and then interrupt the growth of microorganisms. Up to now, numerous sulfonamide derivatives containing aromatic rings have been successfully marketed and widely employed in the treatment *Address correspondence to this author at the School of Pharmacy, Linyi University, Linyi 276000, P.R China; Tel: +86-539-7258637; Fax: +86-5397258637; E-mail: zhanghuizhen@lyu.edu.cn 1573-4064/19 $58.00+.00 of infections Fig. (1) [9]. Significantly, supramolecular Agsulfadiazine used in burn therapy exhibited a better therapeutic effect than the free ligand or AgNO3 Fig. (1) [10]. Recently, numerous efforts have been devoted to the further development of the novel sulfonamide derivatives with high activity, broad antimicrobial spectrum and low toxicity [1114]. 1,2,4-Triazole derivatives are known as an important type of poly-nitrogen electron-rich heterocyclic compounds with excellent safety profiles, favorable pharmacokinetic characteristics and the capability of forming hydrogen bonds [15, 16]. Therefore, the introduction of 1,2,4-triazole fragment is beneficial to improve the binding capacity with biomolecular targets and increase water solubility of target compounds. So far, a plenty of predominant triazole-based drugs have been successfully developed and prevalently used in antimicrobial field, such as Fluconazole, Itraconazole, Voriconazole, Posaconazole, Efinaconazole and Terconazole etc [17]. Notably, it is commonly considered that the 1,2,4-triazole ring in Fluconazole can efficiently coordinate with the Fe2+ in human microsomal heme protein to restrain the biosynthesis of ergosterol, thus inhibiting the growth of fungi [18]. The above observations strongly suggest that 1,2,4-triazole derivatives possess large potentiality as novel antimicrobial agents. Therefore, the exploration of newly structural molecules with sulfonamide nucleus and 1,2,4-triazole moiety has become one of the predominant directions [19]. Inspired by these observations, a series of novel sulfonamide-derived 1,2,4-triazoles were designed from the following respects Fig. (1). © 2019 Bentham Science Publishers 2 Medicinal Chemistry, 2019, Vol. 15, No. 00 He et al. Fig. (1). Structures of sulfonamide-derived clinical drugs. (a) As was reported, tertiary amino moiety as the bioisostere of tertiary alcohol fragment could be employed for the drug design to regulate physicochemical properties of biomolecules and interact with various enzymes and receptors in biological systems to exert bioactivities [20]. (b) Various halobenzyl groups were employed into the target compounds because numerous works has shown that the incorporation of halobenzyl moieties was beneficial to improve biological and pharmacological properties by enhancing the rate of absorption and transport of drugs in vivo [21]. (c) Ethylene chain could modulate the molecular flexibility, which might be helpful to improve molecular binding ability with the microbial targets [20]. Based on the above considerations, a series of novel sulfonamide-derived 1,2,4-triazoles were synthesized in two ways. The in vitro antibacterial and antifungal activities were evaluated against four Gram-positive bacteria, four Gramnegative bacteria and five fungi. The interaction of the bioactive molecule with calf thymus DNA was performed and molecular computational studies were employed to investigate the possible antibacterial mechanism. Moreover, the cytotoxicity, ROS and docking with human microsomal heme of active compound were also investigated to explore its further biological activity [22, 23]. 2. EXPERIMENTAL 2.1. General Methods 2.1.1. Synthesis of 4-acetamidobenzene-1-sulfonyl Chloride (2) The desired 4-acetamidobenzene-1-sulfonyl chloride 2 was synthesized as white solid [11]. Yield: 90%; mp: 139– 140ºC. (literature mp: 142–144ºC) 2.1.2. Synthesis of N- (4-sulfamoylphenyl) acetamide (3) The N- (4-sulfamoylphenyl) acetamide 3 was obtained as white solid [11]. Yield: 82%; mp: 212–214ºC. (literature mp: 219–220ºC). 2.1.3. Synthesis of N- (4-(N- (2-chlorobenzyl) sulfamoyl) phenyl) acetamide (4a) A mixture of compound 3 (5.094 g, 23.8 mmol) and potassium carbonate (2.324 g, 16.8 mmol) was stirred in acetone (60 mL) at 50ºC. 1-Chloro-2- (chloromethyl)benzene (3.821 g, 23.7 mmol) was added and the reaction system was stirred at 70ºC continuously after 0.5 h. When the reaction was completed (monitored by TLC, eluent, chloroform/ethyl acetate, 5/1, V/V), the system was filtered and washed with methanol (3 × 25 mL). Compound 4a was obtained and purified as white solid by silica gel column chromatography (eluent, chloroform/ethyl acetate, 10/1, V/V). Yield: 17%; mp: 180–182ºC; IR (KBr) ν: 3301 (N−H), 3107, 3062 (aromatic C−H), 2924, 2856 (aliphatic C−H), 1678 (C=O), 1592, 1530 (aromatic frame), 1333, 1158, 756 cm-1; 1H NMR (600 MHz, DMSO-d6) δ: 10.31 (s, 1H, NHCOCH3), 8.08 (t, J = 6.1 Hz, 1H, SO2NH), 7.77–7.69 (apparent s, 4H, Ph-H), 7.44–7.42 (m, 1H, 2-ClPh-3-H), 7.38 (d, J = 7.3 Hz, 1H, 2ClPh-6-H), 7.32–7.26 (m, 2H, 2-ClPh-4,5-H), 4.03 (d, J = 6.2 Hz, 2H, 2-ClPh-CH2), 2.09 (s, 3H, COCH3) ppm; 13C NMR (151 MHz, DMSO-d6) δ: 169.4, 143.3, 135.4, 134.6, 132.6, 130.2, 129.5, 129.4, 128.1, 127.6, 119.1, 44.1, 24.6 ppm. 2.1.4. Synthesis of N- (4- (N- (4-fluorobenzyl) sulfamoyl) phenyl) acetamide (4b) Compound 4b was prepared according to the procedure described for compound 4a starting from compound 3 (3.005 g, 14.0 mmol) and 1-(chloromethyl)-4-fluorobenzene (2.003 g, 13.9 mmol). The pure product 4b was obtained as white solid. Yield: 7%; mp: 196–198ºC; IR (KBr) ν: 3304 (N−H), 3100, 3048 (aromatic C−H), 2930, 2859 (aliphatic C−H), 1672 (C=O), 1590, 1531 (aromatic frame), 1343, 1163, 748, 611 cm-1; 1H NMR (600 MHz, DMSO-d6) δ: 10.30 (s, 1H, NHCOCH3), 8.02 (t, J = 6.2 Hz, 1H, SO2NH), 7.74 (d, J = 8.8 Hz, 2H, Ph-2,6-H), 7.71 (d, J = 8.8 Hz, 2H, Ph-3,5-H), 7.27 (dd, J = 7.9, 5.9 Hz, 2H, 4-FPh-3,5-H), 7.10 (t, J = 8.8 Hz, 2H, 4-FPh-2,6-H), 3.94 (d, J = 6.2 Hz, 2H, 4-FPh-CH2), 2.09 (s, 3H, COCH3) ppm; 13C NMR (151 MHz, DMSO-d6) Bioactivity Exploration of Sulfonamide-Derived Triazoles δ: 169.4, 161.8, 143.2, 134.9, 134.5, 130.1, 128.1, 119.1, 115.5, 45.8, 24.6 ppm. 2.1.5. Synthesis of N- (4- (N- (4-chlorobenzyl) sulfamoyl) phenyl) acetamide (4c) Compound 4c was prepared according to the procedure described for compound 4a starting from compound 3 (5.108 g, 23.8 mmol) and 1-chloro-4-(chloromethyl)benzene (3.124 g, 19.4 mmol). The pure product 4c was obtained as white solid. Yield: 17%; mp: 189–191ºC; IR (KBr) ν: 3304 (N−H), 3103, 3040 (aromatic C−H), 2928, 2857 (aliphatic C−H), 1670 (C=O), 1593, 1530 (aromatic frame), 1328, 1154, 784 cm-1; 1H NMR (600 MHz, DMSO-d6) δ: 10.30 (s, 1H, NHCOCH3), 8.05 (t, J = 6.4 Hz, 1H, SO2NH), 7.74 (d, J = 8.8 Hz, 2H, Ph-2, 6-H), 7.71 (d, J = 8.8 Hz, 2H, Ph-3,5-H), 7.34 (d, J = 8.3 Hz, 2H, 4-ClPh-3,5-H), 7.26 (d, J = 8.4 Hz, 2H, 4-ClPh-2,6-H), 3.95 (d, J = 6.4 Hz, 2H, 4-ClPh-CH2), 2.09 (s, 3H, COCH3) ppm; 13C NMR (151 MHz, DMSO-d6) δ: 169.4, 143.2, 137.4, 134.8, 132.2, 129.9, 128.6, 128.1, 119.1, 45.8, 24.6 ppm. Medicinal Chemistry, 2019, Vol. 15, No. 00 3 for 8 h. When the reaction was completed (monitored by TLC, eluent, chloroform/ethyl acetate, 5/1, V/V), the residue was extracted with chloroform (3 × 25 mL). Compound 5a was prepared and purified as white solid. Yield: 82%; mp: 90–92ºC; IR (KBr) ν: 3371 (N−H), 3108, 3060 (aromatic C−H), 2929, 2857 (aliphatic C−H), 1703 (C=O), 1591, 1529, 1495 (aromatic frame), 1334, 1156, 752, 548 cm-1; 1H NMR (600 MHz, DMSO-d6) δ: 10.38 (s, 1H, NHCOCH3), 7.81 (apparent s, 4H, Ph-H), 7.51 (dd, J = 7.0, 2.3 Hz, 1H, 2ClPh-3-H), 7.45 (dd, J = 7.2, 2.0 Hz, 1H, 2-ClPh-4-H), 7.36– 7.33 (m, 2H, 2-ClPh-5,6-H), 4.45 (s, 2H, 4-FPh-CH2), 3.49 (t, J = 7.3 Hz, 2H, NCH2CH2), 3.30 (t, J = 7.3 Hz, 2H, NCH2CH2), 2.11 (s, 3H, COCH3) ppm; 13C NMR (151 MHz, DMSO-d6) δ: 169.7, 144.0, 134.6, 132.9, 130.7, 130.3, 129.9, 128.8, 128.4, 127.8, 119.3, 50.7, 49.06, 30.4, 24.6 ppm; HRMS (TOF) found, m/z 444.9978 [M + H] +, calcd for C17H19BrClN2O3S: 444.9983. 2.1.9. Synthesis of N- (4- (N- (2-bromoethyl)-N- (4fluorobenzyl) sulfamoyl) phenyl) acetamide (5b) Compound 4d was prepared according to the procedure described for compound 4a starting from compound 3 (3.007 g, 14.0 mmol) and 2,4-dichloro-1-(chloromethyl)benzene (3.076 g, 14.4 mmol). The pure product 4d was obtained as white solid. Yield: 3%; mp: 233–235ºC; IR (KBr) ν: 3300 (N−H), 3112, 3054 (aromatic C−H), 2925, 2858 (aliphatic C−H), 1678 (C=O), 1593, 1528 (aromatic frame), 1338, 1153, 763 cm-1; 1H NMR (600 MHz, DMSO-d6) δ: 10.39 (s, 1H, NHCOCH3), 7.81 (apparent s, 4H, Ph-H), 7.62 (s, 1H, SO2NH), 7.50–7.39 (m, 3H, 2,4-Cl2Ph-3,5,6-H), 4.31 (s, 2H, 2,4-Cl2Ph-CH2), 2.07 (s, 3H, COCH3) ppm; 13C NMR (151 MHz, DMSO-d6) δ: 169.4, 144.6, 140.5, 133.9, 130.8, 130.1, 130.0, 129.4, 119.1, 115.6, 115.3, 45.8, 24.6 ppm. Compound 5b was prepared according to the procedure described for compound 5a starting from compound 4b (0.307 g, 1.0 mmol) and 1,2-dibromoethane (0.434 g, 2.3 mmol). The pure product 5b was obtained as white solid. Yield: 74%; mp: 132–134ºC; IR (KBr) ν: 3295 (N−H), 3102, 3047 (aromatic C−H), 2933, 2865 (aliphatic C−H), 1671 (C=O), 1590, 1536, 1509 (aromatic frame), 1340, 1160, 835, 748, 611 cm-1; 1H NMR (600 MHz, DMSO-d6) δ: 10.37 (s, 1H, NHCOCH3), 7.81 (apparent s, 4H, Ph-H), 7.45–7.28 (m, 2H, 4-FPh-2,6-H), 7.18 (t, J = 8.4 Hz, 2H, 4-FPh-3,5-H), 4.31 (s, 2H, 4-FPh-CH2), 3.43–3.39 (t, 2H, NCH2CH2), 3.27–3.23 (t, 2H, NCH2CH2), 2.10 (s, 3H, COCH3) ppm; 13C NMR (151 MHz, DMSO-d6) δ: 169.5, 161.4, 143.9, 132.7, 130.8, 130.7, 128.7, 119.3, 115.8, 50.0, 49.1, 30.4, 24.6 ppm; HRMS (TOF) found, m/z 429.0273 [M + H]+, calcd for C17H19BrFN2O3S: 429.0278. 2.1.7. Synthesis of N- (4- (N- (3,4-dichlorobenzyl) sulfamoyl) phenyl) acetamide (4e) 2.1.10. Synthesis of N- (4- (N- (2-bromoethyl)-N- (4chlorobenzyl) sulfamoyl) phenyl) acetamide (5c) Compound 4e was prepared according to the procedure described for compound 4a starting from compound 3 (3.011 g, 14.1 mmol) and 3,4-dichloro-1-(chloromethyl)benzene (2.709 g, 13.9 mmol). The pure product 4e was obtained as white solid. Yield: 4%; mp: 183–185ºC; IR (KBr) ν: 3305 (N−H), 3103, 3049 (aromatic C−H), 2926, 2857 (aliphatic C−H), 1679 (C=O), 1592, 1529 (aromatic frame), 1369, 1155, 748 cm-1; 1H NMR (600 MHz, DMSO-d6) δ: 10.29 (s, 1H, NHCOCH3), 8.11 (t, J = 5.7 Hz, 1H, SO2NH), 7.73 (d, J = 8.9 Hz, 2H, Ph-2, 6-H), 7.70–7.68 (m, 2H, Ph-3,5-H), 7.53 (d, J = 8.3 Hz, 1H, 2,3-Cl2Ph-5-H), 7.43 (d, J = 1.9 Hz, 1H, 2,3-Cl2Ph-2-H), 7.23 (dd, J = 8.3, 2.0 Hz, 1H, 2,3-Cl2Ph-6H), 3.99 (d, J = 5.6 Hz, 2H, 2,3-Cl2Ph-CH2), 2.09 (s, 3H, COCH3) ppm; 13C NMR (151 MHz, DMSO-d6) δ: 169.4, 143.3, 139.6, 134.7, 131.3, 130.8, 130.1, 129.9, 128.3, 128.1, 119.0, 45.3, 24.6 ppm. Compound 5c was prepared according to the procedure described for compound 5a starting from compound 4c (0.515 g, 1.5 mmol) and 1,2-dibromoethane (0.361 g, 1.9 mmol). The pure product 5c was obtained as yellow syrup. Yield: 70%; IR (KBr) ν: 3364 (N−H), 3104, 3042 (aromatic C−H), 2928, 2857 (aliphatic C−H), 1702 (C=O), 1594, 1528, 1492 (aromatic frame), 1326, 1153, 784, 611 cm-1; 1H NMR (600 MHz, DMSO-d6) δ: 10.38 (s, 1H, NHCOCH3), 7.82 (apparent s, 4H, Ph-H), 7.43–7.40 (m, 2H, 4-ClPh-2,6-H), 7.37–7.34 (m, 2H, 4-ClPh-3,5-H), 4.33 (s, 2H, 4-FPh-CH2), 3.43 (t, J = 7.3 Hz, 2H, NCH2 CH2), 3.27 (t, J = 7.3 Hz, 2H, NCH2CH2), 2.10 (s, 3H, COCH3) ppm; 13C NMR (151 MHz, DMSO-d6) δ: 169.5, 144.0, 132.8, 132.6, 130.5, 128.9, 128.7, 128.5, 119.3, 51.6, 49.1, 30.4, 24.6 ppm; HRMS (TOF) found, m/z 444.9981 [M + H]+, calcd for C17H19BrClN2O3S: 444.9983. 2.1.8. Synthesis of N- (4- (N- (2-bromoethyl)-N- (2-chlorobenzyl) sulfamoyl) phenyl) acetamide (5a) 2.1.11. Synthesis of N- (4- (N- (2-bromoethyl)-N- (2,4-dichlorobenzyl) sulfamoyl) phenyl) acetamide (5d) The mixture of 1,2-dibromoethane (0.340 g, 1.8 mmol), compound 4a (0.513 g, 1.5 mmol) and potassium carbonate (0.759 g, 5.5 mmol) was reacted in acetone (50 mL) at 50ºC Compound 5d was prepared according to the procedure described for compound 5a starting from compound 4d (0.110 g, 0.3 mmol) and 1,2-dibromoethane (0.081 g, 0.4 2.1.6. Synthesis of N- (4- (N- (2, 4-dichlorobenzyl) sulfamoyl) phenyl) acetamide (4d) 4 Medicinal Chemistry, 2019, Vol. 15, No. 00 mmol). The pure product 5d was obtained as a white solid. Yield: 89%; mp: 164–166ºC; IR (KBr) ν: 3300 (N−H), 3113, 3055 (aromatic C−H), 2926, 2857 (aliphatic C−H), 1670 (C=O), 1594, 1538, 1502, 1474 (aromatic frame), 1343, 1159, 1095, 763 cm-1; 1H NMR (600 MHz, DMSO-d6) δ: 10.38 (s, 1H, NHCOCH3), 7.87–7.79 (apparent s, 4H, Ph-H), 7.41–7.36 (m, 1H, 2,4-Cl2Ph-3-H), 7.24 (dd, J = 8.7 Hz, 1H, 2,4-Cl2Ph-5-H), 7.19–7.11 (m, 1H, 2,4-Cl2Ph-6-H), 4.36 (m, 4H, 2,4-Cl2Ph-CH2, NCH2 CH2), 3.31 (m, 2H, NCH2CH2), 2.12 (s, 3H, COCH3) ppm; 13C NMR (151 MHz, DMSO-d6) δ: 169.5, 144.1, 135.8, 133.3, 132.0, 130.3, 128.9, 128.8, 128.4, 127.5, 119.4, 52.2, 50.0, 49.8, 24.6 ppm; HRMS (TOF) found, m/z 478.9597 [M + H]+, calcd for C17H18BrCl2N2O3S: 478.9593. 2.1.12. Synthesis of N- (4- (N- (2-bromoethyl)-N- (3,4dichlorobenzyl) sulfamoyl) phenyl) acetamide (5e) Compound 5e was prepared according to the procedure described for compound 5a starting from compound 4e (0.150 g, 0.4 mmol) and 1,2-dibromoethane (0.125 g, 0.7 mmol). The pure product 5e was obtained as yellow syrup. Yield: 79%; IR (KBr) ν: 3318 (N−H), 3105, 3054 (aromatic C−H), 2925, 2858 (aliphatic C−H), 1678 (C=O), 1591, 1530, 1470 (aromatic frame), 1368, 1156, 746, 612 cm-1; 1H NMR (600 MHz, DMSO-d6) δ: 10.38 (s, 1H, NHCOCH3), 7.81 (apparent s, 4H, Ph-H), 7.62 (d, J = 8.3 Hz, 1H, 2,3-Cl2Ph-4H), 7.52 (d, J = 2.0 Hz, 1H, 2,3-Cl2Ph-5-H), 7.34 (dd, J = 8.3, 2.0 Hz, 1H, 2,3-Cl2Ph-6-H), 4.34 (s, 2H, 2,3-Cl2PhCH2), 3.47 (t, J = 7.1 Hz, 2H, NCH2 CH2), 3.36 (t, J = 6.8 Hz, 2H, NCH2 CH2), 2.10 (s, 3H, COCH3) ppm; 13C NMR (151 MHz, DMSO-d6) δ: 169.6, 144.1, 138.8, 132.5, 131.5, 131.1, 130.7, 130.5, 128.8, 128.8, 119.3, 51.1, 50.6, 30.6, 24.6 ppm; HRMS (TOF) found, m/z 478.9589 [M + H]+, calcd for C17H18BrCl2N2O3S: 478.9593. 2.1.13. Synthesis of N- (4- (N- (2- (1H-1, 2,4-triazol-1-yl) ethyl)-n-benzylsulfamoyl) phenyl) acetamide (6a) The acetone solution of compound 11 (0.063 g, 1.5 mmol) and potassium carbonate (0.026 g, 0.2 mmol) was reacted at 70ºC for 0.5 h. The benzyl chloride (0.024 g, 1.5 mmol) was added at room temperature, and then the mixture was stirred at 60ºC. When the reaction came to the end (monitored by TLC, eluent, chloroform/methanol, 15/1, V/V), the residue was extracted with chloroform (3 × 20 mL). Compound 6a was prepared and purified as white solid. Yield: 67%; mp: 179–181ºC; IR (KBr) ν: 3302 (N−H), 3093, 3040, 3003 (aromatic C−H), 2942, 2869, 2802 (aliphatic C−H), 1691 (C=O), 1592, 1541, 1512 (aromatic frame), 1317, 1151, 992, 732 cm-1; 1H NMR (600 MHz, DMSO-d6) δ: 10.33 (s, 1H, NHCOCH3), 8.28 (s, 1H, TRA C5-H), 7.84 (s, 1H, TRA C3-H), 7.77 (d, J = 8.3 Hz, 2H, Ph2,6-H), 7.73 (d, J = 8.6 Hz, 2H, Ph-3,5-H), 7.36–7.26 (m, 2H, NCH2Ph-2,6-H), 7.14 (m, 3H, NCH2Ph-3,4,5-H), 4.32 (s, 2H, NCH2Ph), 4.21 (t, J = 6.2 Hz, 2H, NCH2 CH2), 3.51 (t, J = 6.2 Hz, 2H, NCH2 CH2), 2.10 (s, 3H, COCH3) ppm; 13 C NMR (151 MHz, DMSO-d6) δ: 169.5, 151.9, 144.7, 132.3, 131.2, 131.1, 128.7, 124.9, 123.7, 123.6, 119.2, 48.2, 48.0, 46.4, 24.6 ppm; HRMS (TOF) found, m/z 400.1435 [M + H]+, calcd for C19H22N5O3S: 400.1438. He et al. 2.1.14. Synthesis of N- (4- (N- (2- (1H-1, 2,4-triazol-1-Yl) ethyl)-N- (2-fluorobenzyl) sulfamoyl) phenyl) acetamide (6b) Compound 6b was synthesized according to the experimental procedure reported for compound 6a, starting from compound 11 (0.066 g, 0.2 mmol) and 1-(chloromethyl)-2fluorobenzene (0.030 g, 0.2 mmol). The pure product 6b was obtained as white solid. Yield: 81%; mp: 175–177ºC; R (KBr) ν: 3303 (N−H), 3094, 3044, 3004 (aromatic C−H), 2870, 2802 (aliphatic C−H), 1692 (C=O), 1592, 1541 (aromatic frame), 1343, 1149, 760, 609 cm-1 ; 1H NMR (600 MHz, DMSO-d6) δ: 10.34 (s, 1H, NHCOCH3), 8.28 (s, 1H, TRA C5-H), 7.84 (s, 1H, TRA C3-H), 7.77 (d, J = 8.8 Hz, 2H, Ph-2,6-H), 7.73 (d, J = 8.7 Hz, 2H, Ph-3,5-H), 7.34 (dd, J = 6.6 Hz, 1H, 2-FPh-3-H), 7.28 (t, J = 7.4 Hz, 1H, 2-FPh4-H), 7.14 (t, J = 8.1 Hz, 2H, 2-FPh-5,6-H), 4.32 (s, 2H, 2FPh-CH2), 4.21 (t, J = 6.4 Hz, 2H, NCH2CH2), 3.51 (t, J = 6.4 Hz, 2H, NCH2CH2), 2.10 (s, 3H, COCH3) ppm; 13C NMR (151 MHz, DMSO-d6) δ: 169.3, 156.4, 148.3, 148.2, 141.2, 127.3, 126.9, 125.2, 118.9, 118.0, 111.8, 108.7, 99.0, 55.6, 37.6, 35.2, 26.8 ppm; HRMS (TOF) found, m/z 418.1347 [M + H]+, calcd for C19H20FN5O3S: 418.1344. 2.1.15. Synthesis of N- (4-(N- (2- (1h-1, 2,4-triazol-1-Yl) ethyl)-N- (2-chlorobenzyl) sulfamoyl) phenyl) acetamide (6c) A solution of 1H-1,2,4-triazole (0.096 g, 1.4 mmol) and potassium carbonate (0.233 g, 1.7 mmol) was reacted in acetonitrile (25 mL) at 70ºC for 0.5 h. Compound 5a (0.463 g, 1.4 mmol) was added to the mixture at room temperature, and then the system was stirred at 70ºC. When the reaction was completed (monitored by TLC, eluent, chloroform/methanol, 20/1, V/V), the residue extracted with chloroform (3 × 20 mL). Compound 6c was provided and purified as white solid. Yield: 56%; mp: 187–189ºC; R (KBr) ν: 3301 (N−H), 3103, 3043 (aromatic C−H), 2865, 2792 (aliphatic C−H), 1689 (C=O), 1590, 1532 (aromatic frame), 1341, 1163, 802, 611 cm-1; 1H NMR (600 MHz, DMSO-d6) δ: 10.39 (s, 1H, NHCOCH3), 8.28 (s, 1H, TRA C5-H), 7.84 (s, 1H, TRA C3-H), 7.81 (d, J = 9.1 Hz, 2H, Ph-2,6-H), 7.78 (d, J = 9.1 Hz, 2H, Ph-3,5-H), 7.41 (d, J = 8.0 Hz, 1H, 2ClPh-3-H), 7.33–7.26 (m, 3H, 2-ClPh-4,5,6-H), 4.34 (s, 2H, 2-ClPh-CH2), 4.20 (t, J = 6.4 Hz, 2H, NCH2CH2), 3.55 (t, J = 6.4 Hz, 2H, NCH2CH2), 2.10 (s, 3H, COCH3) ppm; 13C NMR (151 MHz, DMSO-d6) δ: 169.6, 151.9, 144.7, 144.0, 134.2, 132.8, 132.0, 130.4, 129.9, 129.8, 128.9, 127.7, 119.3, 50.5, 48.6, 48.2, 24.6 ppm; HRMS (TOF) found, m/z 434.1043 [M + H]+, calcd for C19H20ClN5O3S: 434.1048. 2.1.16. Synthesis of N- (4- (N- (2- (1h-1, 2,4-triazol-1-yl) ethyl)-N- (4-fluorobenzyl) sulfamoyl) phenyl) acetamide (6d) Compound 6d was synthesized according to the experimental procedure reported for compound 6c, starting from compound 5b (0.390 g, 0.9 mmol) and 1H-1, 2,4 -triazole (0.065 g, 0.9 mmol). The pure product 6d was obtained as white solid. Yield: 78%; mp: 177–179ºC; R (KBr) ν: 3303 (N−H), 3097, 3036 (aromatic C−H), 2996, 2944, 2873, 2804 (aliphatic C−H), 1692 (C=O), 1591, 1543, 1509 (aromatic frame), 1315, 1149, 797, 610 cm-1; 1H NMR (600 MHz, DMSO-d6) δ: 10.37 (s, 1H, NHCOCH3), 8.24 (s, 1H, TRA Bioactivity Exploration of Sulfonamide-Derived Triazoles C5-H), 7.85 (s, 1H, TRA C3-H), 7.79 (d, J = 8.9 Hz, 2H, Ph2,6-H), 7.76 (d, J = 8.9 Hz, 2H, Ph-3,5-H), 7.26 (d, J = 8.5 Hz, 2H, 4-FPh-2,6-H), 7.13 (d, J = 8.5 Hz, 2H, 4-FPh-3,5H), 4.22 (s, 2H, 4-FPh-CH2), 4.15 (t, J = 6.4 Hz, 2H, NCH2CH2), 3.46 (t, J = 6.4 Hz, 2H, NCH2CH2), 2.10 (s, 3H, COCH3) ppm; 13C NMR (151 MHz, DMSO-d6) δ: 169.5, 161.3, 151.9, 144.7, 143.9, 132.4, 130.6, 128.7, 119.3, 115.8, 115.7, 51.7, 49.1, 48.0, 24.9 ppm; HRMS (TOF) found, m/z 418.1347 [M + H]+, calcd for C19H20FN5O3S: 418.1344. 2.1.17. Synthesis of N- (4- (N- (2-(1h-1, 2,4-triazol-1-yl) ethyl)-N- (4-chlorobenzyl) sulfamoyl) phenyl) acetamide (6e) Compound 6e was synthesized according to the experimental procedure reported for compound 6c, starting from compound 5c (0.390 g, 0.8 mmol) and 1H-1,2,4-triazole (0.064 g, 0.9 mmol). The pure product 6e was obtained as white solid. Yield: 62%; mp: 195–197ºC; R (KBr) ν: 3300 (N−H), 3096, 3036 (aromatic C−H), 2997, 2942, 2870, 2804 (aliphatic C−H), 1690 (C=O), 1591, 1543, 1515 (aromatic frame), 1317, 1151, 799, 610 cm-1; 1H NMR (600 MHz, DMSO-d6) δ: 10.37 (s, 1H, NHCOCH3), 8.25 (s, 1H, TRA C5-H), 7.84 (s, 1H, TRA C3-H), 7.78 (apparent s, 4H, Ph-H), 7.37 (d, J = 8.3 Hz, 2H, 4-ClPh-3,5-H), 7.24 (d, J = 8.3 Hz, 2H, 4-ClPh-2,6-H), 4.23 (s, 2H, 4-ClPh-CH2), 4.16 (t, J = 6.4 Hz, 2H, NCH2 CH2), 3.47 (t, J = 6.4 Hz, 2H, NCH2CH2), 2.10 (s, 3H, COCH3) ppm; 13C NMR (151 MHz, DMSO-d6) δ: 169.5, 151.9, 144.7, 143.9, 136.0, 132.8, 132.35, 130.4, 128.9, 128.7, 119.3, 51.8, 49.1, 48.1, 24.6 ppm; HRMS (TOF) found, m/z 434.1045 [M + H]+, calcd for C19H20ClN5O3S: 434.1048. 2.1.18. Synthesis of N- (4-(N- (2-(1h-1, 2,4-triazol-1-yl) ethyl)-N(2,4-dichlorobenzyl) sulfamoyl) phenyl) acetamide (6f) Compound 6f was synthesized according to the experimental procedure reported for compound 6c, starting from compound 5d (0.457 g, 0.9 mmol) and 1H-1,2,4 -triazole (0.106 g, 1.54 mmol). The pure product 6f was obtained as white solid. Yield: 69%; mp: 199–201ºC; R (KBr) ν: 3306 (N−H), 3096, 3035 (aromatic C−H), 2999, 2971, 2804 (aliphatic C−H), 1694 (C=O), 1591, 1542, 1514 (aromatic frame), 1341, 1152, 792, 611 cm-1; 1H NMR (600 MHz, DMSO-d6) δ: 10.37 (s, 1H, NHCOCH3), 8.30 (s, 1H, TRA C5-H), 7.82 (s, 1H, TRA C3-H), 7.80 (apparent s, 4H, Ph-H), 7.54 (s, 1H, 2,4-Cl2Ph-3-H), 7.37 (d, 1H, 2,4-Cl2Ph-5-H)7.33 (d, 1H, 2,4-Cl2Ph-6-H), 4.31 (s, 2H, 2,4-Cl2Ph-CH2), 4.23 (t, J = 6.2 Hz, 2H, NCH2CH2), 3.55 (t, J = 6.1 Hz, 2H, NCH2CH2), 2.11 (s, 1H, COCH3) ppm; 13C NMR (151 MHz, DMSO-d6) δ: 169.6, 151.9, 144.7, 144.1, 133.7, 133.6, 133.4, 131.8, 129.3, 128.9, 127.9, 119.3, 118.5, 50.2, 48.8, 48.3, 24.6 ppm; HRMS (TOF) found, m/z 468.0653 [M + H]+, calcd for C19H19Cl2N5O3S: 468.0658. 2.1.19. Synthesis of N- (4-(N- (2- (1h-1, 2,4-triazol-1-yl) Ethyl)-N- (3,4-Dichlorobenzyl) Sulfamoyl) Phenyl) Acetamide (6g) Compound 6g was synthesized according to the experimental procedure reported for compound 6c, starting from compound 5e (0.193 g, 0.4 mmol) and 1H-1,2,4 -triazole (0.051 g, 0.7 mmol). The pure product 6g was obtained as white solid. Yield: 73%; mp: 195–197ºC; R (KBr) ν: 3394 Medicinal Chemistry, 2019, Vol. 15, No. 00 5 (N−H), 3084, 3018 (aromatic C−H), 2960, 2923, 2887 (aliphatic C−H), 1687 (C=O), 1590, 1542, 1511 (aromatic frame), 1332, 1156, 828, 612 cm-1; 1H NMR (600 MHz, DMSO-d6) δ: 10.38 (s, 1H, NHCOCH3), 8.29 (s, 1H, TRA C5-H), 7.83 (s, 1H, TRA C3-H), 7.79 (apparent s, 4H, Ph-H), 7.56 (d, J = 8.3 Hz, 1H, 2,3-Cl2Ph-4-H), 7.35 (d, J = 2.0 Hz, 1H, 2,3-Cl2Ph-5-H), 7.20 (dd, J = 8.3, 2.0 Hz, 1H, 2,3-Cl2Ph6-H), 4.24 (s, 2H, 2,3-Cl2Ph-CH2), 4.23 (t, J = 6.2 Hz, 2H, NCH2CH2), 3.52 (t, J = 6.3 Hz, 2H, NCH2CH2), 2.11 (s, 1H, COCH3) ppm; 13C NMR (151 MHz, DMSO-d6) δ: 169.5, 162.9, 151.8, 144.7, 144.1, 138.4, 131.5, 131.0, 130.3, 128.8, 128.7, 127.3, 119.3, 51.4, 49.1, 48.1, 24.6 ppm; HRMS (TOF) found, m/z 468.0656 [M + H]+, calcd for C19H19Cl2N5O3S: 468.0658. 2.1.20. Synthesis of N- (2- (1h-1, 2,4-triazol-1-yl) ethyl)-4amino-N-benzylbenzenesulfonamide (7a) 0.5 mL 2 mol/L sodium hydroxide solution was added to a solution of compound 6a in ethanol 15 mL. The mixture was refluxed for 10 h (monitored by TLC, eluent, acetone/petroleum ether, 1/1, V/V). After cooling to the room temperature, the solvent was evaporated and the residue was treated with water (30 mL) and extracted with chloroform (3 × 30 mL). The organic layers were combined, dried over anhydrous sodium sulfate and concentrated in vacuo to give the deprotected compound 7a as white solid. Yield: 92%; mp: 154–156ºC; R (KBr) ν: 3453, 3365 (N−H), 3003 (aromatic C−H), 2924, 2856 (aliphatic C−H), 1592, 1503 (aromatic frame), 1324, 1151, 733, 550 cm-1 ; 1H NMR (600 MHz, CDCl3) δ: 7.84 (s, 1H, TRA C5-H), 7.77 (s, 1H, TRA C3-H), 7.61 (d, J = 8.4 Hz, 2H, Ph-2,6-H), 7.28 (m, 3H, NCH2Ph-2,5,6-H), 7.17–7.11 (m, 2H, NCH2Ph-3,4-H), 6.71 (d, J = 8.5 Hz, 2H, Ph-3,5-H), 4.18 (t, J = 6.0 Hz, 2H, NCH2CH2), 4.08 (s, 2H, NCH2Ph), 3.43 (t, J = 6.3 Hz, 2H, NCH2CH2) ppm; 13C NMR (151 MHz, CDCl3) δ: 152.1, 150.9, 143.7, 135.8, 129.5, 128.8, 128.6, 128.1, 126.7, 114.2, 53.8, 50.8, 47.7 ppm; HRMS (TOF) found, m/z 358.1335 [M + H]+, calcd for C17H19N5O2S: 358.1332. 2.1.21. Synthesis of N- (2- (1h-1, 2,4-triazol-1-yl) ethyl)-4amino-N- (2-fluorobenzyl) benzenesulfonamide (7b) Compound 7b was obtained as white solid. Yield: 90%; mp: 161–163ºC; IR (KBr) ν: 3453, 3367 (N−H), 3109 (aromatic C−H), 2950, 2924, 2854 (aliphatic C−H), 1592, 1502 (aromatic frame), 1117, 616 cm-1 ; 1H NMR (600 MHz, CDCl3) δ: 7.91 (s, 1H, TRA C5-H), 7.74 (s, 1H, TRA C3-H), 7.57 (d, J = 8.5 Hz, 2H, Ph-2,6-H), 7.23 (m, 2H, 2-FPh-3,4H), 7.08 (m, 1H, 2-FPh-6-H), 6.98 (t, J = 9.1 Hz, 1H, 2-FPh5-H), 6.69 (d, J = 8.5 Hz, 2H, Ph-3,5-H), 4.27 (t, J = 6.2 Hz, 2H, NCH2CH2), 4.20 (s, 2H, 2-FPh-CH2), 3.46 (t, J = 6.2 Hz, 2H, NCH2CH2) ppm; 13C NMR (151 MHz, CDCl3) δ: 160.0, 152.1, 150.9, 143.7, 131.1, 129.9, 129.9, 129.6, 124.6, 115.6, 115.4, 114.2, 50.8, 48.9, 48.1 ppm; HRMS (TOF) found, m/z 376.1242 [M + H]+, calcd for C17H18FN5O2S: 376.1238. 2.1.22. Synthesis of N- (2- (1h-1, 2,4-triazol-1-yl) ethyl)-42-chlorobenzyl) benzenesulfonamide (7c) Compound 7c was obtained as light brown syrup. Yield: 91%; IR (KBr) ν: 3453, 3366 (N−H), 3128, 3034 (aromatic C−H), 2914, 2859 (aliphatic C−H), 1591, 1504 (aromatic frame), 1326, 1153, 881, 555 cm-1; 1H NMR (600 MHz, 6 Medicinal Chemistry, 2019, Vol. 15, No. 00 DMSO-d6) δ: 8.28 (s, 1H, TRA C5-H), 7.84 (s, 1H, TRA C3 H), 7.49 (d, J = 8.7 Hz, 2H, Ph-2,6-H), 7.40 (dd, J = 5.8, 3.5 Hz, 1H, 2-ClPh-3-H), 7.33–7.24 (m, 3H, 2-ClPh-4,5,6-H), 6.66 (d, J = 8.7 Hz, 2H, Ph-3,5-H), 4.24 (s, 2H, 2-ClPhCH2), 4.19 (t, J = 6.5 Hz, 2H, NCH2 CH2), 3.47 (t, J = 6.5 Hz, 2H, NCH2 CH2) ppm; 13C NMR (151 MHz, DMSO-d6) δ: 153.8, 151.9 144.7, 134.6, 132.7, 130.3, 129.8, 129.7, 129.6, 127.68, 123.0, 113.4, 50.5, 49.1, 48.3 ppm; HRMS (TOF) found, m/z 414.0766 [M + Na]+, calcd for C17H18ClN5NaO2S: 414.0762. 2.1.23. Synthesis of N- (2-(1h-1, 2,4-triazol-1-yl) ethyl)-4amino-N- (4-fluorobenzyl) benzenesulfonamide (7d) Compound 7d was obtained as white solid. Yield: 92%; mp: 144–146ºC; R (KBr) ν: 3450, 3316 (N−H), 3056 (aromatic C−H), 2966, 2906, 2859 (aliphatic C−H), 1594, 1506 (aromatic frame), 1322, 1150, 733, 551 cm-1; 1H NMR (600 MHz, DMSO-d6) δ: 8.23 (s, 1H, TRA C5-H), 7.85 (s, 1H, TRA C3-H), 7.46 (d, J = 8.7 Hz, 2H, Ph-2,6-H), 7.24 (m, 2H, 4-FPh-2,6-H), 7.12 (m, 2H, 4-FPh-3,5-H), 6.64 (d, J = 8.7 Hz, 2H, Ph-3,5-H), 6.09 (s, 2H, NH2), 4.14 (t, J = 6.5 Hz, 2H, NCH2CH2), 4.12 (s, 2H, 4-FPh-CH2), 3.38 (t, J = 6.5 Hz, 2H, NCH2 CH2) ppm; 13C NMR (151 MHz, DMSO-d6) δ: 161.2, 153.7, 151.8, 144.6, 133.4, 130.6, 130.5, 129.6, 115.6, 113.3, 51.8, 49.0, 48.2 ppm; HRMS (TOF) found, m/z 376.1254 [M + H]+, calcd for C17H18FN5O2S: 376.1238. 2.1.24. Synthesis of N- (2- (1h-1, 2,4-triazol-1-yl) ethyl)-4amino-N- (4-chlorobenzyl) benzenesulfonamide (7e) Compound 7e was obtained as white solid. Yield: 91%; mp: 148–152ºC; R (KBr) ν: 3479, 3382 (N−H), 3132, 3101 (aromatic C−H), 2961, 2922, 2857 (aliphatic C−H), 1594, 1503 (aromatic frame), 1318, 1145, 875, 546 cm-1; 1H NMR (600 MHz, CDCl3) δ: 7.91 (s, 1H, TRA C5-H), 7.78 (s, 1H, TRA C3-H), 7.59 (d, J = 8.4 Hz, 2H, Ph-2,6-H), 7.24 (d, J = 8.1 Hz, 2H, 4-ClPh-3,5-H), 7.04 (d, J = 8.0 Hz, 2H, 4-ClPh2,6-H), 6.71 (d, J = 8.4 Hz, 2H, Ph-3,5-H), 4.24 (t, J = 6.1 Hz, 2H, NCH2CH2), 4.01 (s, 2H, 4-ClPh-CH2), 3.42 (t, J = 6.1 Hz, 2H, NCH2CH2) ppm; 13C NMR (151 MHz, CDCl3) δ: 152.2, 151.0, 144.2, 134.3, 134.1, 129.8, 129.5, 129.0, 126.4, 114.2, 53.1, 49.1, 47.6 ppm; HRMS (TOF) found, m/z 414.0765 [M + Na] +, calcd for C17H18ClN5NaO2S: 414.0762. 2.1.25. Synthesis of N- (2- (1h-1,2,4-triazol-1-yl) ethyl)-4amino-N- (2,4-dichlorobenzyl) benzenesulfonamide (7f) Compound 7f was obtained as yellow syrup. Yield: 94%; R (KBr) ν: 3430 (N−H), 3136 (aromatic C−H), 2958, 2925, 2858 (aliphatic C−H), 1599, 1515 (aromatic frame), 1099, 805, 469 cm-1; 1H NMR (600 MHz, DMSO-d6) δ: 8.29 (s, 1H, TRA C5-H), 7.82 (s, 1H, TRA C3-H), 7.53 (d, J = 1.8 Hz, 1H, 2,4-Cl2Ph-3-H), 7.49 (d, J = 8.6 Hz, 2H, Ph-2,6-H), 7.37–7.28 (m, 2H, 2,4-Cl2Ph-5,6-H), 6.67 (d, J = 8.5 Hz, 2H, Ph-3,5-H), 6.13 (s, 2H, NH2), 4.23 (t, J = 6.2 Hz, 2H, NCH2CH2), 4.21 (s, 2H, 2,4-Cl2Ph-CH2), 3.47 (t, J = 6.2 Hz, 2H, NCH2 CH2) ppm; 13C NMR (151 MHz, DMSO-d6) δ: 153.9, 151.8, 144.7, 134.0, 133.6, 133.2, 131.7, 129.8, 129.2, 127.8, 122.7, 113.4, 50.3, 48.8, 48.4 ppm; HRMS (TOF) found, m/z 448.0377 [M + Na]+, calcd for C17H17Cl2N5NaO2S: 448.0372. He et al. 2.1.26. Synthesis of N- (2- (1h-1, 2,4-triazol-1-Yl) ethyl)-4amino-N- (3,4-dichlorobenzyl) benzenesulfonamide (7g) Compound 7g was obtained as white solid. Yield: 89%; mp: 157–159ºC; IR (KBr) ν: 3458, 3330 (N−H), 3145 (aromatic C−H), 2964, 2920, 2863 (aliphatic C−H), 1592, 1502 (aromatic frame), 1323, 1151, 874, 548 cm-1; 1H NMR (600 MHz, CDCl3) δ: 7.95 (s, 1H, TRA C5-H), 7.80 (s, 1H, TRA C3-H), 7.58 (d, J = 8.5 Hz, 2H, Ph-2,6-H), 7.32 (d, J = 8.2 Hz, 1H, 3,4-Cl2Ph-5-H), 7.11 (s, 1H, 3,4-Cl2Ph-2-H), 6.93 (d, J = 8.1 Hz, 1H, 3,4-Cl2Ph-6-H), 6.71 (d, J = 8.5 Hz, 2H, Ph-3,5-H), 4.29 (t, J = 6.0 Hz, 2H, NCH2CH2), 3.97 (s, 2H, 3,4-Cl2Ph-CH2), 3.45 (t, J = 6.0 Hz, 2H, NCH2CH2) ppm; 13C NMR (151 MHz, CDCl3) δ: 152.2, 151.0, 144.2, 137.8, 134.3, 134.1, 129.8, 129.5, 129.0, 128.5, 126.4, 114.2, 53.1, 49.1, 47.6 ppm; HRMS (TOF) found, m/z 426.0557 [M + H]+, calcd for C17H17Cl2N5O2S: 426.0553. 2.1.27. Synthesis of tert-butyl (4-acetamidophenyl) sulfonyl) carbamate (8) To solution of the compound 3 (0.214 g, 1.1 mmol) in dichloromethane (20 mL) at 0oC was added triethylamine (0.15 mL, 1.1 mmol), followed by 4-(dimethylamino)-pyridin (0.010 g, 0.08 mmol). The di-tertbutyl dicarbonate (0.327 g, 1.5 mmol) was added to the reaction mixture and stirred at 0 o C for 0.5 h, and then transferred the system to room temperature stirring for another 13 h. After the reaction came to the end (monitored by TLC, eluent, chloroform/methanol = 15/1, V/V), concentrated and chromatography (eluent, dichloromethane/methanol = 4/1, V/V) afforded the intermediates 8 as white solid. Yield: 71%; mp: 137–139ºC; IR (KBr) ν: 3358 (N−H), 3117 (aromatic C−H), 2980, 2934, 2854 (aliphatic C−H), 1676 (C=O), 1592, 1530 (aromatic frame), 1247, 1151, 739, 613 cm-1; 1H NMR (600 MHz, DMSO-d6) δ: 11.47 (s, 1H, SO2NH), 10.38 (s, 1H, NHCOCH3), 7.80 (apparent s, 4H, Ph-H), 2.09 (s, 3H, COCH3), 1.29 (s, 9H, Boc-H) ppm; 13C NMR (151 MHz, DMSO-d6) δ: 169.6, 150.3, 144.1, 133.6, 129.1, 118.8, 82.5, 28.0, 24.6 ppm; HRMS (TOF) found, m/z 315.1006 [M + H]+, calcd for C13H19N2O5S: 315.1009. 2.1.28. Synthesis of tert-butyl (4-acetamidophenyl) sulfonyl) (tert-butyl) carbamate (9) Compound 9 was obtained as white solid from procedure for the preparation of compound 8. Yield: 28%; mp: 130– 132 oC; IR (KBr) ν: 3302 (N−H), 3120, 3059 (aromatic C−H), 2982, 2934 (aliphatic C−H), 1675 (C=O), 1596, 1542 (aromatic frame), 1345, 1143, 623 cm-1; 1H NMR (600 MHz, DMSO-d6) δ: 10.36 (s, 1H, NHCOCH3), 7.89 (d, J = 8.8 Hz, 2H, Ph-2,6-H), 7.79 (d, J = 8.8 Hz, 2H, Ph-3,5-H), 2.09 (s, 3H, COCH3), 1.45 (s, 9H, N(CH3)3), 1.40 (s, 9H, Boc-H) ppm; 13C NMR (151 MHz, DMSO-d6) δ: 168.9, 155.3, 143.9, 133.2, 128.9, 118.0, 80.4, 49.8, 28.9, 24.3 ppm; HRMS (TOF) found, m/z 371.1638 [M + H]+, calcd for C17H27N2O5S: 371.1635. 2.1.29. Synthesis of tert-butyl (4-acetamidophenyl) sulfonyl) (2-bromoethyl) carbamate (10) The compound 8 (3.314 g, 10.5 mmol) and potassium carbonate (2.170 g, 15.7 mmol) in acetonitrile (100 mL) was stirred at 60oC for 2 h, followed by the addition of 1,2dibromoethane (4.0 mL, 19.1 mmol), and then the reaction Bioactivity Exploration of Sulfonamide-Derived Triazoles system was stirred at 70 oC for about 10 h. After the reaction came to the end (monitored by TLC, eluent, dichloromethane/methanol = 15: 1, V/V), the solvent was evaporated and the residue was extracted with ethyl acetate (3 × 50 mL). The organic extracts were collected and then dried over anhydrous sodium sulfate and purified by silica gel column chromatography (eluent, petroleum ether/ethyl acetate = 4/1, V/V) to afford white solid of intermediate 10. Yield: 64%; mp: 150–152 oC; IR (KBr) ν: 3314 (N−H), 3106, 3048 (aromatic C−H), 2982, 2934 (aliphatic C−H), 1674 (C=O), 1596, 1534 (aromatic frame), 1352, 1159, 731,565 cm-1; 1H NMR (600 MHz, CDCl3) δ: 7.86 (d, J = 8.7 Hz, 2H, Ph-2,6-H), 7.68 (d, J = 8.3 Hz, 2H, Ph-3,5-H), 4.16 (t, J = 7.4 Hz, 2H, NCH2CH2), 3.59 (t, J = 7.4 Hz, 2H, NCH2CH2), 2.12 (s, 3H, COCH3), 1.38 (s, 9H, Boc-H) ppm; 13C NMR (151 MHz, CDCl3) δ: 168.7, 150.6, 142.8, 134.2, 129.4, 118.8, 85.1, 47.7, 28.9, 27.9, 24.7 ppm; HRMS (TOF) found, m/z 421.0429 [M + H]+, calcd for C15H22BrN2O5S: 421.0427. 2.1.30. Synthesis of N- (4-(N- (2- (1h-1, 2,4-triazol-1-yl) ethyl) sulfamoyl) phenyl) acetamide (11) Compound 11 was synthesized according to the experimental procedure reported for compound 6a, starting from compound 10 (0.420g, 1.0 mmol) and 1H-1, 2,4-triazole (0.069 g, 1.0 mmol). The pure product 11 was obtained as white solid. Yield: 82%; mp: 155–157ºC; IR (KBr) ν: 3303 (N−H), 3032 (aromatic C−H), 2995, 2924, 2875, 2812 (aliphatic C−H), 1687 (C=O), 1594, 1547, 1512 (aromatic frame), 1323, 1161, 751, 579 cm-1; 1H NMR (600 MHz, DMSO-d6) δ: 10.30 (s, 1H, NHCOCH3), 8.43 (s, 1H, TRA C5-H), 7.94 (s, 1H, TRA C3-H), 7.75 (d, J = 8.5 Hz, 2H, Ph2,6-H), 7.71 (s, 1H, SO2NH), 7.69 (d, J = 8.5 Hz, 1H, Ph3,5-H), 4.22 (t, J = 6.1 Hz, 2H, NCH2 CH2), 3.13 (t, J = 6.1 Hz, 2H, NCH2 CH2), 2.09 (s, 3H, COCH3) ppm; 13C NMR (151 MHz, DMSO-d6) δ: 169.4, 150.0 144.9, 143.4, 134.2, 128.1, 119.2, 49.0, 42.4, 24.6 ppm; HRMS (TOF) found, m/z 309.3447 [M + H]+, calcd for C12H12N5O3S: 309.3442. 2.1.31. Synthesis of tert-butyl (2- (1h-1, 2,4-triazol-1-yl) ethyl)((4-acetamidophenyl) sulfonyl) carbamate (12) The by-product 12 was obtained as white solid from procedure for the preparation of compound 11. Yield: 18%; mp: 147–149ºC; IR (KBr) ν: 3432 (N−H), 3121, 3034 (aromatic C−H), 299, 2935, 2803 (aliphatic C−H), 1728 (C=O), 1594, 1541, 1510 (aromatic frame), 1357, 1162, 633 cm-1; 1 H NMR (600 MHz, DMSO-d6) δ: 10.40 (s, 1H, NHCOCH3), 8.46 (s, 1H, TRA C5-H), 7.98 (s, 1H, TRA C3-H), 7.82 (d, J = 9.0 Hz, 2H, Ph-2,6-H), 7.79 (d, J = 8.8 Hz, 2H, Ph-3,5-H), 4.49 (t, J = 5.9 Hz, 2H, NCH2 CH2), 4.16 (t, J = 5.9 Hz, 2H, NCH2CH2), 2.10 (s, 3H), 1.21 (s, 9H, Boc-H) ppm; 13C NMR (151 MHz, DMSO-d6) δ: 169.6, 152.0, 150.5, 144.8, 144.5, 133.0, 129.5, 118.7, 84.5, 48.8, 46.6, 27.8, 24.6 ppm; HRMS (TOF) found, m/z 409.1425 [M + H]+, calcd for C17H23N5O5S: 409.1420. 2.1.32. Synthesis of N- (2- (1h-1, 2,4-triazol-1-yl) ethyl)-4aminobenzenesulfonamide (13) The white solid of target product 13 was obtained according to the experimental procedure reported for compound 7a, starting from compound 11 (0.052g, 0.2 mmol). Yield: 98%; mp: 161–163ºC; IR (KBr) ν: 3446, 3355 (N−H), 3114 (aro- Medicinal Chemistry, 2019, Vol. 15, No. 00 7 matic C−H), 2955, 2923, 2853 (aliphatic C−H), 1596, 1503 (aromatic frame), 1299, 1145, 688 cm-1; 1H NMR (600 MHz, DMSO-d6) δ: 8.41 (s, 1H, TRA C5-H), 7.94 (s, 1H, TRA C3 H), 7.39 (d, J = 8.5 Hz, 2H, Ph-2,6-H), 7.33 (s, 1H, SO2NH), 6.61 (d, J = 8.6 Hz, 2H, Ph-3,5-H), 5.94 (s, 2H, NH2), 4.20 (t, J = 6.2 Hz, 2H, NCH2CH2), 3.07 (t, J = 6.1 Hz, 2H, NCH2CH2) ppm; 13C NMR (151 MHz, DMSO-d6) δ: 153.1, 151.9, 144.8, 128.9, 125.5, 113.2, 48.9, 42.4 ppm; HRMS (TOF) found, m/z 290.0689 [M + H]+, calcd for C10H13FN5O2S: 290.2967. 3. RESULTS AND DISCUSSION 3.1. Chemistry The synthetic routes of sulfonamide-derived 1,2,4triazoles were outlined in Scheme 1. During the procedure, the N-protected sulfonyl chloride 2 was prepared with superior yield of 91% by the reaction of acetanilide with chlorosulfonic acid, and it was further treated by ammonium hydroxide to provide p-acetylaminobenzene sulfonamide 3 in 82% yield. Subsequent N-alkylation of compound 3 was performed with a series of halobenzyl halides to afford the secondary amine derivatives 4a–e with yields of 3%−17%. Intermediates 4a–e further reacted with 1,2-dibromoethane to obtain the tertiary amine sulfonamides 5a–e in good yields of 70%–89%. Sulfonamide triazole intermediates 6a–e were efficiently obtained by the N-alkylation of compounds 5a–e with 1,2,4-triazole in 56%–78% yields. To improve the yield of the target compounds, the second synthetic route was designed by using t-butyloxy carbonyl (Boc) group to protect one hydrogen atom of sulfonamide 3. Thus the Boc-protected sulfonamide 8 and the by-product 9 were achieved from p-acetylaminobenzene sulfonamide 3 in yields of 71% and 28%, respectively, and then compound 8 reacted with 1,2-dibromoethane to afford tert-N-sulfonamide 10 in yield of 64%. Subsequently, compound 10 experienced nucleophilic substitution to conveniently afford the desired sulfonamide intermediate 11 in a high yield of 82% and Boc deprotected by-product 12 at a low yield of 18%. Further Nalkylation of secondary amine 11 with halobenzyl halides produced intermediates 6a–g with yields ranging from 67% to 81%. Finally, intermediates 6a–g and 11 were further transformed into the deprotected sulfonamide derivatives 7ag and 13 in ethanol in order to explore their influence on the bioactivity. 3.2. Analysis of Spectra For the IR, 1H NMR and 13C NMR spectra, all the absorption bands were observed at the expected regions. The presence of acetylamino moiety of sulfonamide compounds 4−6 and 8−12 in IR spectra was confirmed by the broad absorption in 3394−3295 cm-1 for NH group and 1728−1670 cm-1 for characteristic C=O bands, whereas in 1H NMR spectra, the CH3 protons linked to the amide moiety were observed singlets at 2.07–2.12 ppm, and in 13C NMR spectra, the carbonyl carbon was found at δ 169.7–168.7 ppm. 3.3. Biological Activity The in vitro antibacterial and antifungal screening of the sulfonamide compounds was explored normally [24]. The 8 Medicinal Chemistry, 2019, Vol. 15, No. 00 He et al. Reagents and conditions: (i) chlorosulfonic acid, 0ºC; (ii) ammonium hydroxide, 0ºC; (iii) halobenzyl halide, potassium carbonate, acetone, 50−70ºC; (iv) 1,2dibromoethane, potassium carbonate, acetone, 50ºC; (v) 1,2,4-triazole, potassium carbonate, acetonitrile, 60−70ºC; (vi) 2 mol/L NaOH, ethanol, reflux; (vii) di-tertbutyl dicarbonate, 4-(dimethylamino)-pyridine, triethylamine, dichloromethane, 0ºC−r.t.; (viii) 1,2-dibromoethane, potassium carbonate, acetonitrile, 60−70ºC. Scheme 1 Synthesis of sulfonamide-derived 1,2,4-triazoles 4−13. antibacterial and antifungal data were displayed in Table 1 and Table 2. 3.3.1. Antibacterial Activity As shown in Table 1, the sulfonamides 5a−e exhibited some activity against the tested bacterial strains with MIC values ranging from 0.27 to 1.19 µmol/mL. Some important effects of substituents on the benzene ring on biological activity were observed. Non-substituted derivative 6a possessed relatively weaker antibacterial activity in comparison with halogen substituted compounds 6b−g, but it exerted stronger anti-E. coli (DH52) activity (MIC = 0.04 µmol/mL) than Chloromycin. Among halobenzyl sulfonamide derivatives, sulfonamide 6d bearing 4-fluorobenzyl group showed better activities than compounds 6b and 6e with 2-fluorobenzyl and 4-chlorobenzyl groups, however, it displayed relatively lower potencies in inhibiting the growth of the tested strains than other substituted compounds. The 2-chlorobenzyl compound 6c displayed equivalent inhibitory potency to compounds 6f and 6g with dichlorobenzyl group against the tested strains. Especially, replacement of 2-chlorobenzyl moiety with 2,4-dichlorobenzyl group, which generated compound 6f, resulted in good inhibitory potency against Gram-negative S. dysenteriae with MIC value of 0.27 µmol/mL. Most of deprotected compounds 7a−g exerted relatively superior activities in inhibiting the growth of the tested bacteria to the corresponding protected ones to some extent, and compound 13 without halobenzyl group showed lower biological activities than halobenzyl contained ones. Noticeably, the 1,2,4-triazole-based sulfonamide 7c bearing 2-chlorobenzyl moiety showed the strongest inhibition towards Gram-negative E. coli (DH52 and JM109) strains (MIC= 0.02 µmol/mL), which was 5-fold more potent than Chloromycin. Additionally, it displayed equivalent inhibitory potency against MRSA to Chloromycin (MIC = 0.16 µmol/mL). This indicated that compound 7c could be further studied as potential novel antibacterial agents. The compounds 7b and 7c bearing chlorobenzyl moieties exerted relatively better activity than the dichlorobenzyl group substituted ones 7f and 7g, which might probably be attributed to change of the electronic distribution and physicochemical properties, thereby affecting the absorption, distribution and metabolism of the bioactive molecules. Moreover, sulfonamide 8 with Boc moiety exhibited high inhibitory activities against S. aureus and E. coli DH52 strains with MIC values of 0.09 µmol/mL. The deprotected sulfonamide triazole 11 displayed higher activity than the Boc-protected compound 12 to some extent. These results suggested that the Boc group possessed remarkable effects on biological activities. The good hydrophilicity of Boc moiety in these compounds might make it easy for them be delivered to the binding sites. 3.3.2. Antifungal Activity As depicted in Table 2, the in vitro antifungal data indicated that some target sulfonamide-derived 1,2,4-triazoles displayed moderate inhibitory potencies against the tested Bioactivity Exploration of Sulfonamide-Derived Triazoles Medicinal Chemistry, 2019, Vol. 15, No. 00 9 Antibacterial data as MIC (µmol/mL) for compounds 5–13a,b, c. Table 1. Gram-Positive Bacteria Gram-Negative Bacteria Compds MRSA S. aureus B. subtilis M. luteus B. typhi E. coli (DH52) E. coli (JM109) S. dysenteriae 5a 0.57 0.29 0.57 0.29 0.57 1.15 1.15 1.15 5b 1.19 1.19 1.19 1.19 0.60 0.60 0.30 1.19 5c 1.15 0.57 0.57 0.57 1.15 0.57 1.15 0.57 5d 1.07 1.07 0.53 1.07 0.53 0.27 1.07 1.07 5e 0.27 1.07 1.07 0.53 1.07 0.27 0.53 1.07 6a 0.64 0.64 1.28 1.28 1.28 0.04 0.64 1.28 6b 0.61 1.23 1.23 1.23 1.23 1.23 0.61 1.23 6c 0.29 0.59 0.29 0.59 1.18 0.29 0.59 1.18 6d 0.61 0.61 0.31 1.23 0.61 0.61 1.23 0.61 6e 1.18 0.59 0.59 0.59 0.59 1.18 1.18 1.18 6f 0.27 0.27 0.27 1.10 1.10 0.27 0.55 0.27 6g 1.10 1.10 0.27 0.55 1.10 0.27 1.10 1.10 7a 1.43 1.43 1.43 1.43 1.43 1.43 1.43 1.43 7b 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 7c 0.16 0.65 0.04 0.65 0.08 0.02 0.02 0.65 7d 1.36 1.36 0.68 1.36 0.68 0.68 1.36 1.36 7e 0.65 0.33 0.33 0.65 0.33 0.02 0.08 0.65 7f 1.20 0.60 0.04 0.30 0.60 0.30 1.20 0.60 7g 1.20 1.20 0.60 1.20 0.60 1.20 0.60 1.20 8 0.18 0.09 0.18 0.18 0.18 0.09 0.18 0.18 9 0.69 0.64 1.38 0.69 1.38 1.38 1.38 1.38 10 0.61 0.61 1.22 0.61 1.22 0.30 1.22 1.22 11 1.66 0.83 1.66 1.66 1.66 1.66 0.83 1.66 12 1.25 1.25 1.25 1.25 1.25 0.31 0.63 1.25 13 0.48 1.92 1.92 0.48 0.96 1.92 1.92 1.92 Chloromycin 0.05 0.05 0.10 0.02 0.10 0.10 0.10 0.05 Norfloxacin 0.03 0.01 0.01 0.01 0.01 0.003 0.003 0.05 a Minimal inhibitory concentrations were determined by micro broth dilution method for microdilution plates. S. aureus, Staphylococcus aureus (ATCC25923); MRSA, Methicillin-Resistant Staphylococcus aureus (N315); B. subtilis, Bacillus subtilis; M. luteus, Micrococcus luteus (ATCC4698); B. proteus, Bacillus proteus (ATCC13315); E. coli, Escherichia coli (JM109); P. aeruginosa, Pseudomonas aeruginosa; B. typhi, Bacillus typhi. c ClogP values were calculated by ChemDraw Ultra 10.0. b Table 2. Antifungal data as MIC (µmol/mL) for compounds 5–13d. Compds C. albicans C. mycoderma C. utilis S. cerevisiae A. flavus 5a 1.19 0.30 1.19 1.19 1.19 5b 0.57 1.15 0.57 1.15 0.29 Table 2. contd… 10 Medicinal Chemistry, 2019, Vol. 15, No. 00 He et al. Compds C. albicans C. mycoderma C. utilis S. cerevisiae A. flavus 5c 0.29 0.14 1.15 0.29 0.29 5d 0.53 0.53 1.07 1.07 0.53 5e 0.53 0.27 1.07 0.13 0.53 6a 1.23 1.23 0.61 1.23 1.23 6b 0.59 0.29 1.18 1.18 0.59 6c 0.29 0.15 1.18 0.29 0.29 6d 1.10 0.55 0.55 0.55 1.10 6e 0.27 1.10 1.10 1.10 0.27 6f 1.28 0.64 1.28 1.28 1.28 6g 1.23 0.15 1.23 1.23 1.23 7a 1.36 0.68 0.17 1.36 0.68 7b 0.08 0.08 0.33 0.16 0.33 7c 0.08 0.02 0.33 0.65 0.33 7d 0.04 1.20 1.20 1.20 1.20 7e 0.60 1.20 1.20 0.30 1.20 7f 1.43 1.43 1.43 1.43 1.43 7g 0.34 1.36 0.68 1.36 1.36 8 0.18 0.05 0.09 0.18 0.18 9 1.38 0.35 0.04 0.69 1.38 10 1.22 0.61 0.30 1.22 1.22 11 1.66 0.05 0.83 1.66 1.66 12 1.25 0.63 0.63 1.25 1.25 13 1.92 1.92 0.96 0.96 1.92 Fluconazole 0.003 0.01 0.03 0.05 0.84 d C. albicans, Candida albicans (ATCC76615); C. mycoderma, Candida mycoderma; C. utilis, Candida utilis; S. cerevisia, Saccharomyces cerevisia; A. flavus, Aspergillus flavus. fungal strains. Sulfonamide intermediates 5a−e showed moderate to good antifungal activities with MIC values ranging from 0.27 to 1.19 µmol/mL, and notably compound 5e with a 3,4-dichlorobenzyl group exhibited significant inhibitory activity against S. cerevisiae with MIC value of 0.13 µmol/mL, which was equipotent to Fluconazole. Among the halobenzyl sulfonamide 1,2,4-triazoles 6a−g and 7a−g, compounds 6a and 7a without halogen atom showed low activities in inhibiting the growth of all the tested fungal strains. Moreover, 2-fluorobenzyl and 2chlorobenzyl substituted compounds 6b and 6c gave low MIC value of 0.15 µmol/mL against C. mycoderma, which was better than compound 6d with 4-fluorobenzyl group. Intriguingly, 2-chlorobenzyl derivative 7c without acetyl group exerted better anti-A. flavus activity (MIC = 0.33 µmol/mL) than Fluconazole (MIC = 0.84 µmol/mL), and it also exhibited equivalent activity against C. mycoderma in comparison with Fluconazole (MIC = 0.02 µmol/mL). Compound 7d with 4-fluorobenzyl group showed better inhibitory against C. utilis (MIC = 0.17 µmol/mL) than 2- fluorobenzyl derivative 7b, whereas the latter exhibited better activity against C. albicans than compound 7d. Sulfonamide derivatives 7c and 7e−g with chlorine atom had relatively better antifungal activity than fluorinated ones 7b and 7d, which indicated that the replacement of fluoro substituent with chloro moiety was beneficial for the antifungal activity. Compound 8 with a Boc moiety showed higher inhibitory activity against Fluconazole-insensitive A. flavus with the MIC value of 0.18 µmol/mL than Fluconazole (MIC = 0.84 µmol/mL). The t-butyl-derived by-product 9 exhibited excellent inhibitory efficiency against C. utilis strains with MIC value of 0.04 µmol/mL, which was superior to Fluconazole (MIC = 0.03 µmol/mL). Sulfonamide 1,2,4-triazole 12 showed higher antifungal activity in comparison with the corresponding Boc-deprotected one 11, which might be attributed to the improved water solubility. As mentioned above, it was demonstrated that the antimicrobial efficacies might be closely related to acetyl frag- Bioactivity Exploration of Sulfonamide-Derived Triazoles Medicinal Chemistry, 2019, Vol. 15, No. 00 11 work studied the ROS level using DCFH-DA as a fluorescence probe and imaged by fluorescence microscope [27, 28]. To some extent, the amount of ROS produced in MCF-7 cells was positively correlated with anticancer activity. However, excessive ROS might cause cellular injury and thus lead to cell death and DNA damage. As shown in Fig. (3), the generated amount of ROS increased with the increasing concentration of compound 7c, thereby the anticancer activity also increased. Especially, compound 7c showed lower ROS generation than hydrogen peroxide (H2O2) at a concentration of 100 µg/mL (safe dose to normal human cells). Therefore, the newly synthesized compound 7c exhibited better safety than the reference by downregulating ROS generation. 6. INTERACTIONS WITH CALF THYMUS DNA Fig. (2). Cytotoxic assay of target compound 7c on human breast cancer cell line (MCF-7) by CCK-8 Kit. Each data bar represents an average of three parallels, and error bars indicate one standard deviation from the mean (Blank: PBS). ment, Boc moiety and different halobenzyl groups to some extent. 4. CELL TOXICITY The highly active compound 7c was further investigated for its cytotoxic properties on human breast cancer cell lines (MCF-7) by using the Cell-Counting Kit-8 (CCK-8) [25]. The phosphate buffered saline (PBS) was selected as a positive control (blank). Compound 7c was dissolved in a mixture of ethanol and water to prepare the stock solutions, which then was diluted by PBS to obtain the required concentrations of 6.25, 12.5, 25.0, 50.0 and 100.0 µg/mL, respectively. The CKK-8 assay showed that compound 7c had high cytotoxicity against MCF-7 with an IC50 value of 6.33 µg/mL Fig. (2). With the increase of the concentration of the compound 7c, the cell viability decreased. 5. EVALUATION OF ROS GENERATION IN MCF-7 CELLS Although the relationship between antimicrobial mechanism and cell toxicity had not been fully elucidated, this Calf thymus deoxyribonucleic acid (DNA) was employed to study the interaction behavior of compound 7c with it to explore the possible antimicrobial mechanism [29, 30]. In the experiments, the UV-vis spectra gave a proportional increase and slight red shift with the increasing concentration of compound 7c and a fixed concentration of DNA Fig. (4). Besides, the sum value of free DNA and free compound 7c was a little greater than the absorption value of 7c–DNA complex (the inset of Fig. 4), which demonstrated that a weak hypochromic effect is present between compound 7c and DNA. The spectral changes might agree with the intercalation of compound 7c into the helix and the overlap of π-π* states of DNA bases [31, 32]. The binding constant (K) was calculated using equation 1. The plot of A0/(A-A0) versus 1/[compound 7c] was obtained by using the absorption titration data and linear fitting Fig. (5), K = 1.14 × 104 L/mol, R = 0.999, SD = 0.08 (R is the correlation coefficient. SD is standard deviation). 7. MOLECULAR MODELING STUDIES Docking study could offer more insights into understanding the interactions of sulfonamide-derived 1,2,4-triazoles and the structural features of the active site of protein. The crystal structure data of human microsomal heme (PDB code: 2HI4) was obtained from the Protein Data Bank (PDB), which was a representative target to investigate the antibacterial mechanism of 1,2,4-triazole derivatives. Target compound 7c was selected to dock into the heme (the hemo- Fig. (3). Compound 7c-induced ROS formation in MCF-7 cells and H2O2 as positive control. 12 Medicinal Chemistry, 2019, Vol. 15, No. 00 He et al. As shown in Fig. (6), it showed the detailed binding mode between the active sites of human microsomal heme and the most active derivative 7c bearing a 2-chlorobenzyl group. The predominant intermolecular force of hydrogen bond was labeled by a dashed line to understand the interaction of the complex. The analyses of hydrogen bond interaction confirmed that amino acid residue Ala 317 played a relatively important role in binding potency. The backbone of Ala 317 was inclined to form a hydrogen bond with one of the oxygen atoms at -SO2NR in compound 7c, and the distance was 2.70 Å. The result demonstrated that compound 7c could act with the heme protein through hydrogen bonds. 8. COMPUTATIONAL CHEMICAL STUDIES Fig. (4). UV absorption spectra of DNA with different concentrations of compound 7c (pH = 7.4, T = 293 K). Inset: comparison of absorption at 260 nm between the 7c–DNA complex and the sum values of free DNA and free compound 7c. c(DNA) = 4.85 × 10-5 mol/L, and c(compound 7c) = 0–1.17 × 10-5 mol/L for curves a–g respectively at increment 0.167 × 10-5. Fig. (5). The plot of A0/(A-A0) versus 1/[compound 7c]. globin complex Fe2+ of heme protein) to form a coordination bond with the nitrogen atom in the 1,2,4-triazole ring, which made the hemoglobin lose its chance of binding to oxygen and inhibited the lanolin sterol 14α site demethylation reaction, thereby the growth of the microorganism was inhibited [33]. In an attempt to understand the further possible mechanism of the excellent biological activity and low toxicity of active compound 7c bearing 2-chlorobenzyl groups, computational chemical assays were done and shown in Fig. (7). Meanwhile, molecular electrostatic potential (MEP) and atomic polar tensor (APT) charges had been performed to explore the electrostatic binding characteristic through the surface and atomic level of molecule, respectively Fig. (7(b– c)). This MEP map was generated for a selection of compound at the neutral state via the B3LYP/6-31C* theoretical computation [34, 35], and the MEP surface gave an indication of the charged surface area and hydrophilicity of compound Fig. (7(b)). The results revealed that compound 7c possessed more negative charge regions (in red) on the oxygen and nitrogen atoms of -SO2NR than nitrogen atoms of 1,2,4-triazole ring (in yellow). The oxygen and nitrogen atoms of SO2NR were electronically available with their lone pairs, which were probably oriented toward the outer part of the molecule and therefore were accessible to interact with their surroundings. On the other hand, the APT atomic charges of compound 7c via the Gaussian 09 theoretical calculations were showed in Fig. (7c). It was found that the APT charge of oxygen and nitrogen atoms in -SO2NR moiety was lower than other atoms; especially the nitrogen atom had the lowest APT charge. This data demonstrated that the oxygen and nitrogen atoms in -SO2NR moiety were most likely to form hydrogen bonds. Because of the π−π conjugative effect and steric hindrance, the nitrogen lone pairs in SO2NR moiety were buried in the structure of compound 7c, which was favorable for forming hydrogen bonds. It might indicate that the capability Fig. (6). (a) Molecular modeling of compound 7c docked into the binding site of human microsomal heme (PDB: 2HI4). The dashed line represent the hydrogen bonding interactions between compound 7c and heme; (b) Three-dimensional conformation of compound 7c docked in heme; (c) Stereoview of the conformation of compound 7c intercalated to heme to form 7c–heme complex. Bioactivity Exploration of Sulfonamide-Derived Triazoles Medicinal Chemistry, 2019, Vol. 15, No. 00 13 Fig. (7). (a) Structure of active compound 7c; (b) Electrostatic potential of compound 7c; (c) APT atomic charges of compound 7c. of forming hydrogen bond was related with the nitrogen atoms of -SO2NR moiety. These results were in agreement with the binding mode obtained from above docking study. CONSENT FOR PUBLICATION CONCLUSION CONFLICT OF INTEREST In this work, a series of sulfonamide–derived 1,2,4triazoles were successfully synthesized in two ways starting from commercial commercial acetanilide and chlorosulfonic acid. All the structures of the new compounds were characterized by IR, 1H NMR, 13C NMR and HRMS. The antimicrobial evaluation in vitro revealed that some target compounds showed effective antibacterial activity with suitable ClogP values, and even displayed equipotent or superior activities to the reference drugs. Structure-activity relationships demonstrated that 1,2,4-triazole and sulfonyl fragments exerted a significant influence on biological activity. Noticeably, 2-chlorobenzyl group substituted compound 7c exhibited particularly strong antibacterial activity against most of the tested bacterial and fungal strains (MIC = 0.02– 0.65 µmol/mL) and also displayed an objective cancer toxicity against MCF-7 cells by evaluating cell toxicity and ROS generation. The interaction of active compound 7c with calf thymus DNA evidenced that it could properly bind with calf thymus DNA, which might be a factor to exert its powerful bioactivity. Molecular docking indicated that compound 7c could act with the residue of Ala 317 in human microsomal heme protein through hydrogen bonds. The MEP surface and APT atomic charges studies indicated the capability of hydrogen bond formation with one of the oxygen atoms in sulfonyl moiety, which were in agreement with the binding mode obtained from the above docking study. All these results sufficiently indicated that it should be a promising starting point to optimize the structures of sulfonamide-derived 1,2,4-triazoles as potential clinical antimicrobial candidates. The authors declare no conflict of interest, financial or otherwise. ETHICS APPROVAL AND CONSENT TO PARTICIPATE Not applicable. HUMAN AND ANIMAL RIGHTS No Animals/Humans were used for studies that are the basis of this research. Not applicable. ACKNOWLEDGMENTS This work was partially supported by the National Natural Science Foundation of China (No. 21672173), the Research Fund for International Young Scientists from International (Regional) Cooperation and Exchange Program of NSFC (No. 81650110529), Shandong Provincial Natural Science Foundation (No. ZR2017PB001) and Doctoral Scientific Research Foundation of Linyi University (No. LYDX2016BS030). SUPPLEMENTARY DATA Supplementary material is available on the publisher’s website along with the published article. REFERENCES [1] [2] [3] [4] [5] [6] Chellat, M.F.; Raguž, L.; Riedl, R. Targeting antibiotic resistance. Angew. Chem. Int. Ed., 2016, 55, 6600-6626. He, S.C.; Jeyakkumar, P.; Avula, S.R.; Wang, X.L.; Zhang, H.Z.; Zhou, C.H. Recent advance in sulfonamide-based medicinal chemistry. Sci. Sin: Chim., 2016, 46, 823-847 (in Chinese). Srivastava, N.; Kumar, A. 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DISCLAIMER: The above article has been published in Epub (ahead of print) on the basis of the materials provided by the author. The Editorial Department reserves the right to make minor modifications for further improvement of the manuscript. PMID: 30398118 Z. Kristallogr. NCS 2021; 236(4): 819–820 Zhi Wei Ning, Lin Ling Gan, Shu Jing Zhou, Jie Tian, Yu Hui Du and Hui Zhen Zhang* Crystal structure of N1,N2-bis(2-ﬂuorobenzyl) benzene-1,2-diamine,C20H18F2N2 Table : Data collection and handling. Crystal: Size: Wavelength: μ: Diffractometer, scan mode: θmax, completeness: N(hkl)measured, N(hkl)unique, Rint: Criterion for Iobs, N(hkl)gt: N(param)reﬁned: Programs: https://doi.org/10.1515/ncrs-2021-0094 Received March 17, 2021; accepted March 30, 2021; published online April 9, 2021 Abstract C20H18F2N2, monoclinic, P21/c (no. 14), a = 12.4233(11) Å, b = 7.3805(7) Å, c = 18.9531(16) Å, β = 104.109(3)°, V = 104.109(3) Å3, Z = 4, Rgt(F ) = 0.0464, wRref(F 2) = 0.1201, T = 296(2) K. CCDC no.: 2074208 The molecular structure is shown in the figure. Table 1 contains crystallographic data and Table 2 contains the list of the atoms including atomic coordinates and displacement parameters. Source of material A mixture of o-phenylene diamine (0.541 g, 5 mmol), 1-bromopropane (0.615 g, 5 mmol) and potassium carbonate (0.828 g, 6 mmol) was stirred in ethanol at room temperature. After the reaction was completed (monitored *Corresponding author: Hui Zhen Zhang, School of Pharmacy, Linyi University, Linyi, 276000, P. R. China, E-mail: zhanghuizhen@lyu.edu.cn Zhi Wei Ning, Shu Jing Zhou, Jie Tian and Yu Hui Du, School of Pharmacy, Linyi University, Linyi, 276000, P. R. China Lin Ling Gan, Chongqing Engineering Research Center of Pharmaceutical Sciences, School of Pharmacy, Chongqing Medical and Pharmaceutical College, Chongqing, 401331, P. R. China Open Access. © 2021 Zhi Wei Ning et al., published by De Gruyter. International License. Colourless prism . × . × . mm Mo Kα radiation (. Å) . mm− SMART, φ and ω .°, >% , , . Iobs >  σ(Iobs),   Bruker [], SHELX [, ], Olex [] by TLC, eluent, dichloromethane/petroleum ether, 1/1, v/v), the solvent was removed and the residue was extracted with dichloromethane (3 × 40 mL), dried over anhydrous sodium sulfate and puriﬁed by silica gel column chromatography (eluent, dichloromethane/petroleum ether, 1/1, v/v) to afford the white solid. Yield, 645 mg, 40%. The title compound was dissolved in dichloromethane, and the solvent was evaporated slowly at room temperature. After three days, colorless crystals were obtained. Experimental details Data reduction was carried out using SAINT+ and SADABS [1]. The structure was determined by intrinsic phasing routines in the SHELXT program [2] and reﬁned by fullmatrix least-squares methods in SHELXL [3] by using Olex 2 [4]. All of the hydrogen atoms were placed in the calculated positions. Comment Phenylamine derivatives are an important class of organic compounds, which have a wide range of applications in medicine and chemical industry [5, 6]. For instance, phenylamines are widely used as antimicrobial drugs, anti-inﬂammatory analgesics and antilipemic drugs in the ﬁeld of medicine, and as insecticides and fungicides in the ﬁeld of pesticides [7]. Additionally, phenylamines can This work is licensed under the Creative Commons Attribution 4.0 820 Z.W. Ning et al.: Crystal structure of C20H18F2N2 Table : Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å). Atom x y z Uiso*/Ueq C Ha C H C H C H C Hb C C HA HB C C H C H C H C H C C HA HB C C C H C H C H C H Fa FAb F N HA N HA . () . . () . . () . . () . . () . . () . () . . . () . () . . () . . () . . () . . () . () . . . () . () . () . . () . . () . . () . . () . () . () . () . () . () . () . () . . () . . () . . () . . () . . () . () . . . () . () . . () . . () . . () . . () . () . . . () . () . () . . () . . () . . () . . () . () . () . () . () . () . () . () . . () . . () . . () . . () . . () . () . . . () . () . . () . . () . . () . . () . () . . . () . () . () . . () . . () . . () . . () . () . () . () . () . () . () . () .* . () .* . () .* . () .* . () .* . () . () .* .* . () . () .* . () .* . () .* . () .* . () . () .* .* . () . () . () .* . () .* . () .* . () .* . () . () . () . () . ()* . () . ()* a Occupancy: .(), b Occupancy: .(). also be used to synthesize some important medicinal intermediates and other chemicals with important application value in organic synthesis [8]. Although N1,N2-bis(2-ﬂuorobenzyl)benzene-1,2-diamine is a known compound, its crystal structure has not been reported yet. Therefore, in consideration of the importance of this compound it was synthesized form commercial material, and also its crystal structure will be reported. There is one molecule in the asymmetric unit (see the Figure). The bond lengths and angles in the title molecule are in normal ranges. In a word, the title compound was synthesized under mild conditions, which can be readily used as a key intermediate to develop new drugs, novel reagents and advanced materials. Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission. Research funding: This work was supported by Shandong Provincial Natural Science Foundation (No. ZR2020QB167) and Key Research and Development project of Shandong Province (No. 2019GSF108216). Conﬂict of interest statement: The authors declare no conﬂicts of interest regarding this article. References 1. Bruker. SAINT+, version 7.60A (Includes XPREP and SADABS); Bruker AXS Inc.: Madison, WI, USA, 2009. 2. Sheldrick G. M. SHELXTL – integrated space-group and crystalstructure determination. Acta Crystallogr. 2015, A71, 3–8. 3. Sheldrick G. M. Crystal structure reﬁnement with SHELXL. Acta Crystallogr. 2015, C71, 3–8. 4. Dolomanov O. V., Bourhis L. J., Gildea R. J., Howard J. A. K., Puschmann H. OLEX2: a complete structure solution, reﬁnement and analysis program. J. Appl. Crystallogr. 2009, 42, 339–341. 5. Lezama J. O. G., Iriarte A. G., Domínguez R. E., Robles N. L. Study of the structural and conformational properties of ﬂuorosubstituted thioacetanilide derivatives. J. Mol. Struct. 2020, 1222, 128768. 6. Giese S., Klimov K., Mikeházi A., Kelemen Z., Frost D. S., Steinhauer S., Müller P., Nyulászi L., Müller C. 2-(Dimethylamino)phosphinine: a phosphorus-containing aniline derivative. Angew. Chem. Int. Ed. 2020, 60, 3581–3586. 7. Saad M. M. I., Fumio M. Inﬂuence of pesticides and neuroactive amines on cAMP levels of two-spotted spider mite (Acari: Tetranychidae). Insect Biochem. 2016, 19, 715–722. 8. Kesavan S., Kumar D. R., Baynosa M. L., Shim J. J. Potentiodynamic formation of diaminobenzene ﬁlms on an electrochemically reduced graphene oxide surface: determination of nitrite in water samples. Mater. Sci. Eng. C 2018, 85, 97–106. Received: 1 June 2020 Revised: 11 November 2020 Accepted: 14 November 2020 DOI: 10.1002/ptr.6966 RESEARCH ARTICLE Fucoidan mitigated diabetic nephropathy through the downregulation of PKC and modulation of NF-κB signaling pathway: in vitro and in vivo investigations Jingge Xu1 | Yan Wang1 | Zhen Wang1,2 | Lanping Guo2 | Xinpeng Li1 1 College of Pharmacy, Linyi University, Linyi, Shandong, China 2 National Resources Center of Chinese Material Medica, China Academy of Chinese Medical Sciences, Beijing, China The persistence of hyperglycemia and oxidative stress in diabetic patients ultimately leads to diabetic nephropathy (DN). In this study, we investigated the effect of sulfated polysaccharides (SPS) extracted from Laminaria japonica in relieving DN symptoms. To induce the diabetic model, normal rats were kept on a high-sugar, high-fat Correspondence Xinpeng Li, College of Pharmacy, College of Life Sciences, Linyi University, Linyi 276000, China. Email: lixinpeng513@126.com diet, then they were injected with streptozocin. Groups of these rats were later treated with SPS and/or protein kinase C (PKC) inhibitor. The analyses performed herein demonstrate that although diabetes significantly decreases the body weights of rats, SPS and inhibitor treatments increase these weights, as well as the ratios of Funding information Key project at central government level: The ability establishment of sustainable use for valuable Chinese medicine resources, Grant/ Award Number: 2060302; Natural Science Foundation of Shandong Province, Grant/ Award Number: ZR2019BD055 renal to total body weight. Serum biochemical analyses indicate that blood urea nitrogen and serum creatinine levels gradually decrease in the SPS group. In addition, DN symptoms are substantially relieved by SPS and/or inhibitor treatments, as evidenced by histopathological analyses. Changes in the expressions of PKC-α, PKC-β, Pselectin, nuclear factor kappa B (NF-κB), and p65, detected by immunohistochemistry and western blot assessments, show that SPS regulates diabetic nephropathy via the PKC/NF-κB pathway. KEYWORDS Laminaria japonica polysaccharides, diabetic nephropathy, histopathological analysis, PKC/ NF-κB 1 | I N T RO DU CT I O N kidney (Faria, Faria, MAB, Silva, & Duarte, 2013; Vaziri & RodríguezIturbe, 2006). Diabetic nephropathy (DN), a highly prevalent complication of diabe- Laminaria japonica is an edible brown algae that is rich in polysac- tes mellitus, is the primary cause of end-stage and chronic kidney dis- charides, including the sulfated polysaccharide (SPS), fucoidan. Con- eases (CKD; Mitsuo et al., 2019; Panyang et al., 2019). It is also sidering their anti-inflammatory and anti-oxidation effects, SPSs responsible for the development of end-stage renal disease extracted from Laminaria japonica are commonly used to clinically (Wu et al., 2018), a progressive and irreversible malady that is charac- treat CKD. As the main medicinal ingredients of CKD drugs, SPSs terized by initial ultrafiltration, proteinuria, mesangial matrix expan- have shown high therapeutic efficiency (Wang et al., 2019). Moreover, sion, interstitial fibrosis, basement membrane thickening, and renal they have a good effect on DN (Leiro, Castro, Arranz, & Lamas, 2007; cell damage. Although DN affects 20–30% of diabetic patients, its Zhang, Zhang, Tang, & Mao, 2020). However, despite the effective- pathogenesis remains unclear. However, available data indicate that ness of SPS (Hong, 2017; Wang et al., 2015), their mechanism of chronic inflammation and oxidative stress play an important role in action is not well understood. Available in vivo studies suggest that the development of nephropathy. The frequent combination of diabe- SPS-mediated DN inhibition is influenced by the TGF-β signaling path- tes and hypertension promotes DN pathogenesis due to the effect of way (Chen et al., 2015; Li et al., 2017b). In diabetic patients, high glu- the latter disease in causing oxidative stress and inflammation in the cose levels promote the activation of the PKC/NF-κB pathway, Phytotherapy Research. 2020;1–12. wileyonlinelibrary.com/journal/ptr © 2020 John Wiley & Sons, Ltd. 1 2 XU ET AL. resulting in the formation of PKC, a molecule that plays an important The experiments were performed on male Wistar rats (220 ± 20 role in TGF-β signal transduction (Min et al., 2009a). Moreover, exces- g, SPFII Certificate) bought from Service Biotechnology Co., Ltd. sive glucose instigates oxidative stress, which also leads to higher (Wuhan, China). All procedures conducted in this study had been pre- PKC expressions in diabetes patients (Min et al., 2009b). When oxida- viously approved by the Animal Ethics Committee of the School of tive stress increases, renal lesions appear and inflammatory symptoms Pharmacy at Linyi University. start to show (Lezoualc & Behl, 1998). Based on previous studies, natural SPS-containing products extracted from marine plants (e.g., the petroleum ether fraction of Toona sinensis Roem seed extracts 2.2 | Extraction of the SPS (Mohammed, Abbas, Badi, et al., 2020; Wan, Xiao, Wei, et al., 2015) and Gum Arabic extract of Acacia Senegal) can act on PKC/NF-κB or The polysaccharides were obtained by water extraction and alcohol TGF-β signaling pathways in DN. Considering that SPS can also inhibit precipitation, as described previously (Li et al., 2017). Briefly, the Lami- PKC expression in the diabetic rats to relieve oxidative stress (Yu, naria japonica plant was cut and boiled in water (100 C) for 2 hr, then Zhang, Cui, et al., 2014), they are possibly involved in the signaling the extract was filtered through diatomite and concentrated. Subse- pathway of DN. However, further studies are needed to confirm that quently, alginate was removed by magnesium chloride and ethanol. the onset and progression of DN are mediated by PKC/NF-κB. The supernatant was concentrated again by centrifugation then puri- In light of these, our study investigates the role of the PKC/NF- fied overnight using a dialysis membrane (molecular weight κB signaling pathway in the mechanism of SPS-induced DN ameliora- cutoff = 3,500 Da) against pure water. Finally, the SPSs were precipi- tion in STZ-induced DN (Wang et al., 2015). Furthermore, we vali- tated by adding four volumes of 95% ethanol. dated our in vivo findings by studying the effect of varied concentration of SPS on the PKC/NF-κB signaling pathway in HGinduced NRK-52E cells. 2.3 | of SPS 2 The total carbohydrate content in SPS was determined according to MATERIALS AND METHODS | Determination of the chemical composition the phenol-sulfuric acid method, using L-fucose as the standard 2.1 | Reagents and animals (Masuko et al., 2005). Meanwhile, the uronic acid content was assessed based on the carbazole colorimetry method, using D- Dried whole Laminaria japonica plants were purchased from a market glucuronic acid as the standard (Bitter & Muir, 1962). The sulfate and in the city of Linyi, Shandong province, China. Streptozotocin (STZ), L-fucose contents were examined according to the gelatin-barium bisindolylmaleimide (PKC inhibitor, GF 109203X), and the monosac- chloride and L-cysteine hydrochloride methods, using potassium sul- charide were purchased from Sigma–Aldrich (St. Louis, MO). All other fate and L-fucose as standards, respectively (Dische & Shettles, 1948; chemicals and reagents were obtained from general commercial Kawai, Seno, & Anno, 1969). A high-performance gel permeation sources and used without prior treatment, unless otherwise specified. chromatography system equipped with a TSK-G3000 column The antibodies were purchased from PeproTech, Inc. (Rocky Hill, NJ), (300 mm × 7.8 mm) and a refractive index detector was used to and they include PKC-α (Rabbit Polyclonal, catalog number: determine the molecular weights of the SPS components (Komatsu, 21991-1-AP, WB: 1:1,000, ICH: 1:100), PKC-β (Rabbit Polyclonal, cat- Takahata, Tanaka, Ishimitsu, & Okada, 1993). The composition of the alog number: 12919-1-AP, WB: 1:1,000), p65 (Rabbit Polyclonal, cata- neutral sugar (monosaccharide) was estimated by high-performance log number: 10745-1-AP, WB: 1:2,000, ICH: 1:100), NF-κB1 (Rabbit liquid chromatography (Honda et al., 1989). Polyclonal, catalog number: 14220-1-AP, WB: 1:1,000), P-selection (Mouse monoclonal, catalog number: 60322-1-Ig, WB: 1:2,000), and β-actin (Mouse monoclonal, catalog number: 60008-1-Ig, WB: 2.4 | In vivo experiments' model and grouping 1:5,000). When used in WB or IHC, the antibodies were diluted one thousand or one hundred times, respectively, with a primary anti-dil- Male Wistar rats were randomly divided into four groups: normal glu- uent. Biotin-conjugated cose (NG), STZ-induced DN, STZ-induced diabetic nephropathy + SPS affinipure goat Anti-Mouse IgG (H+L; catalog number: SA00004-1) (DN + SPS), and STZ-induced diabetic nephropathy + inhibitor (DN + and Biotin-conjugated affinipure goat Anti-Rabbit IgG (H+L; catalog IG) groups, with 20 rats in each group. All rats were subjected to number: SA00004-2) were provided by Solarbio (Beijing, China). light/dark cycles (12:12 hr) for 1 week and fed with a high-sugar, SOD (catalog number: BC0170), GSH (catalog number: BC1175), high-fat diet for 4 weeks in order to establish a Type 2 diabetic model. MDA (catalog number: BC0020), ABTS (catalog number: BC4770), On the fifth week, STZ was injected into DN, DN + IG, and DN + SPS and Catalase (catalog number: BC0200) assay kits were purchased group rats, while saline was given to the NG rats (Lin, Yun, Dong, Li, & from Solarbio (Beijing, China). The rat nuclear factor-κB (NF-κB) Xiong, 2017; Shao, Wei, Tong, & Ying, 2014). The feeding regime of ELISA Kit (catalog number: RA20170) was obtained from Bioswamp rats in all groups was common. The PKC inhibitor, GF 109203X, was (Wuhan, China). injected daily into DN + IG rats, starting three days before modeling The secondary antibodies including 3 XU ET AL. and ending three days after. Upon terminating the PKC injections, the The viability of the cultured cells was measured using the rats were fed for 5 weeks, and on the 10th week, samples of their 3-(4,5-dimethyl-2-thiazolyl)-2,5-diphenyl-2-H-tetrazolium urine were collected and analyzed for protein content, so as to assess (MTT) assay, and the optimum SPS concentration was determined. bromide renal injury. The DN + IG and DN + SPS groups were gavage-fed with SPS (200 mg kg−1) for 80 days, starting on the 6th week. Meanwhile, the NG and DN groups were given saline by gastric administration. 2.9 In vitro experiments' model and grouping | The cells were cultured for 72 hr in six-well plates, and they were 2.5 | Assessment of renal function divided into five groups: normal glucose (NG), high glucose (HG), high glucose + SPS (HG+SPS), high glucose + inhibitor (HG+IG), and high Every week, the rats were weighed, and their blood glucose levels glucose + inhibitor + SPS (HG + IG + SPS) groups. were tested. Rat urine was collected and measured (volume) on the 8th, 12th, 16th, and 20th weeks, using a metabolic cage. The urine samples collected on the 10th and 20th weeks were analyzed for the 2.10 | Western blot analysis total protein content. At the end of the experiment, the body weights of rats were determined, then the kidneys were removed and washed Western blot analyses were conducted in order to express/quantify with PBS before being weighed. An automated chemistry analyzer proteins in the kidneys of rats and in NRK-52E cells, as detailed in a (Synchron CX5, Beckman Coulter, Brea, CA) was used to assess the previous study (Li et al., 2019). When removed, each kidney (or NRK- BUN and Scr levels in the plasma of the rats. 52E cells) was washed with ice-cold phosphate-buffered saline, cut into small pieces, and suspended in 200 μl NP-40 lysis buffer. The mixture was centrifuged at 4 C and 12,000g for 15 min, then the 2.6 | Histologic examination and immunohistochemistry supernatant was collected and analyzed using the BCA Protein Assay Kit to determine the total protein concentration. After adding 5× loading buffer, the protein sample was boiled at 99 C for 10 min, then Upon removal, the right kidney was immediately fixed in 4% formalde- the denatured proteins were loaded and separated by 12% SDS- hyde. After routine paraffin embedding, the sample was cut into 5-μm PAGE. Subsequently, the sample was electro-transferred onto a poly- sections and stained with hematoxylin and eosin and periodic acid-sil- vinylidene fluoride membrane, blocked in 5% milk TBST at room tem- ver-methenamine (PASM) for the detection of histopathological alter- perature for 1 hr, then incubated overnight with primary antibodies at ations and morphological changes in the glomerular mesangium and 4 C. Next, the membrane was incubated with horseradish peroxidase- basement membrane (Lu et al., 2018; Yang & Tang, 2017). Masson conjugated secondary antibody at room temperature for 1 hr. Then, staining was used to estimate kidney fibrosis (Singh, 1964), while the membrane was analyzed by chemiluminescence, using β-actin as a immunohistochemistry was used to determine the expressions of loading control, to visualize the proteins. PKC-α and p65. 2.11 2.7 | | Statistical analysis Cell cultures for in vitro experiments All quantitative data presented in this text are expressed as mean ± NRK-52E cells were purchased from the Institute of Basic Medicine at SD. Statistical analysis was carried out using Graphpad Prism 8.0 and the Chinese Academy of Medical Sciences (Beijing, China). The cell variations among groups were assessed by one-way analysis of vari- lines were maintained in Dulbecco's Modified Eagle's Medium (Gibco, ance (ANOVA) followed by Dunnett's Test. Meanwhile, nonparametric NY; glucose: 5.5 mmol L−1) supplemented with 10% fetal bovine data were evaluated using the Mann-Whitney U test. Differences serum. The cells were grown at 37 C in a humidified water-saturated were considered to be statistically significant and very significant at atmosphere containing 5% CO2. p < .05 (*) and p < .01 (**), respectively. 2.8 | Cytotoxicity assay and SPS concentration screening 3 When 90% confluency was reached, the cells were transferred to | RE SU LT S 3.1 | Chemical composition of the sulfated polysaccharides 96-well plates, with 2 × 103 cells in each well. Following 24-hr starvation in a serum-free medium, some cells were treated with SPS −1 (10–1,280 μg ml ) and cultured for 96 hr. Other cells were cultured for 72 hr with 30 mmol L −1 −1 glucose and 50, 100, or 200 μg ml SPS. The molecular weight of SPS was found to be 9554 Da (Mw; Figure 1a). SPS are composed of 30.1% sulfate**, 25.46% fucose, and 50.77% total sugars (Figure 1b). Analysis of the monosaccharide 4 XU ET AL. F I G U R E 1 (a) Determination of SPS molecular weight by high performance liquid chromatography with gel permeation chromatography; (b) determination of peak time of neutral sugar standard by HPLC; (c) determination of peak time of neutral sugar in SPS by HPLC; (d) chemical composition and neutral sugar composition of SPS; Fuc, fucose; Gal, galactose; HPLC, high-performance liquid chromatography; Man, mannose, Glc, glucose; Rha, rhamnose; Xyl, xylose, Rib: ribose; SPS, sulfated polysaccharide [Colour figure can be viewed at wileyonlinelibrary.com] shows that fucose is the most abundant component, followed by 3.2.2 | Variations in blood glucose levels galactose (0.35 mole ratio, compared to fucose), mannose (0.04), rhamnose (0.08), xylose (0.03), and glucose (0.36; Figure 1c). During the 4 weeks preceding the experiment, the rats were kept on a high-sugar, high-fat diet, in order to develop insulin resistance. Rats from all groups, except the NG group, died late in the young-aged 3.2 Effect of SPS on DN | STZ-induced period. DN rats showed the highest rate of mortality (50%), followed by DN + SPS (30%) and DN + IG (20%) rats. On the 3.2.1 | Renal index variations 5th week, rats in groups other than the normal group were injected with STZ. Blood analyses showed that the glucose levels in these rats The renal index (RI = renal weight × 100/body weight) is an important were higher than 16.5 mmol L−1, the prescribed value of the DN parameter used to assess renal function. As shown in Figure 2a, the model. Although these levels fluctuated within a narrow range, DN group exhibits significantly higher RI values than the NG group. throughout the experiment, they were always consistent with hyper- The renal indexes of the DN + IG and DN + SPS groups are substan- glycemia conditions (Figure 2b). The concentrations of blood glucose tially lower than those of the DN group, with no appreciable differ- detected on the 40th day were found to be significantly different ence among the three groups. from those of normal rats (Figure 2c). XU ET AL. F I G U R E 2 (a) Ratio of kidney weight to total body weight in each group of rats, measured at the end of the experiment (##p < .01 vs. the NG group; **p < .01 vs. the DN group); (b) variations in blood glucose levels before SPS and IG administration; (c) blood glucose levels in rats of all groups, measured on the fortieth day after modeling (**p < .01 vs. the NG group); variations in the body weights of (d) NG and DN, (e) DN and DN + IG, (f) DN and DN + SPS, and (g) DN + IG and DN + SPS rats during the experiment. DN, diabetic nephropathy; NG, normal glucose; SPS, sulfated polysaccharide 5 6 3.2.3 XU ET AL. | Body weight variations 3.2.6 | Histologic examination results Before the experiment, all rats weighed between 240 and 250 g. After No pathological changes in the glomeruli, renal tubules, and renal 4 weeks of feeding (high-sugar, high-fat diet), the rat weights interstitium of NG group rats were observed. However, the diabetic increased significantly. Upon establishing the diabetic model, changes model group showed inflammatory infiltration. The DN + IG and DN + in rat body weights became different in each group. Unlike the NG SPS groups, on the other hand, presented kidney histologies similar to rats whose weights showed little variation, DN rats lost weight that of the NG group (Figure 3e). As for the cellular morphology, NG (Figure 2d). Meanwhile, DN + IG rats had higher body weights than rats showed normal nuclear arrangement and uniform cytoplasm, the DN rats (Figure 2e). The body weights of DN + SPS rats were sim- whereas the cells of DN rats had disordered nuclei. Similarly, the skel- ilar to those of model rats in the early stage; however, in the late stage etal muscle fibers in the former group were found to be orderly of the experiment, the weights gradually increased (Figure 2f). Rats in arranged, while those in the latter group were disordered and atro- the DN + IG and DN + SPS groups presented similar body weights phic. The DN + IG and DN + SPS groups exhibited ordered arrange- (Figure 2g). ments of skeletal muscle fibers, similar to the DN group, with no significant nuclear increase and inward migration, and no obvious inflammatory cell infiltration (Figure 3f). Glomerular sclerosis, atrophy 3.2.4 | protein Changes in urine volume and urinary of glomeruli, and hypertrophy of few glomeruli were observed by light microscopy, after PASM staining. As shown in Figure 3g, glycogen deposition in the DN group was significantly enhanced, compared to Once the diabetic model was established, DN rats started drinking the NG group. However, treatment with PKC inhibitor or SPS appre- more water per day than the other rats. On average, three DN rats ciably reduced the elevated levels of deposited glycogen. Finally, Mas- consumed about 800 ml of water per day. The cages in which son staining experiments showed that the increased number of renal these rats were kept were lined with padding whose humidity was collagen fibers observed in DN rats was substantially diminished upon higher than that of the padding used in other group cages. An ani- the administration of SPS or PKC inhibitor (Figure 3h). Figure 3g mal metabolic cage was used to monitor the 24-hr urine volume depicts the evaluation of these pathological results. released from rats on the 8th, 12th, 16th, and 20th weeks. Figure 3a shows that DN rats excreted significantly more urine than the other groups. As the experiment proceeded, the urine vol- 3.3 Effect of SPS on DN | umes of the DN + SPS and DN + IG groups gradually decreased and approached the volumes of the normal group. To monitor the PKC-α and p65 expressions in kidneys were determined by immu- excretion of urinary protein and evaluate renal injury, the ratio of nohistochemistry (Figure 4c,d). These expressions were found to be urine protein/creatinine was also measured on the 10th and 20th increased in DN rats and decreased in DN+IG and DN+SPS rats, weeks (Figure 3b). The results indicate that after 10 weeks of compared to the normal group. Similarly, western blot analyses indi- experimentation, the DN and DN + SPS groups exhibited minimal cate that PKC-α was significantly expressed in the DN group; how- differences. Similarly, the NG and DN + IG groups presented insig- ever, the level of this protein was relatively low in the SPS group. nificant differences in the ratio of protein to creatinine on the The expressions of inflammatory factors, P-selectin, NF-κB, and 10th week. However, at the end of the experiment, the ratio deter- p65, also showed appreciable variations (Figure 4a,b). mined for the DN + SPS group was found to be much lower than that of the DN group. 3.4 3.2.5 | Changes in BUN and Scr levels 3.4.1 Effect of SPS on NRK-52E cells in vitro | | Cytotoxicity of SPS Scr and BUN are important indicators of renal function. These indi- The toxicity of SPS towards NRK-52E cells was determined by analyz- cators were measured on the 8th, 12th, 16th, and 20th weeks of ing the viability of cells treated with varying concentrations of the experimentation (Figure 3c,d). The results indicate that even though polysaccharides (10–1,000 μg ml−1), using the MTT viability assay. the BUN levels of rats in different groups were similar on the 8th The results illustrated in Figure 5a show that SPS is nontoxic, and week, they were quite different on the 12th, 16th, and 20th weeks. thus, it can be safely used for pharmacological screening. Meanwhile, significant variations were observed between the Scr levels of the normal and DN groups, starting from the 8th week. However, the Scr variations between DN + SPS, DN + IG, and NG 3.4.2 | SPS concentration screening groups were found to be statistically insignificant. It should be noted that Scr levels differed greatly between different rats in the Compared to the NG group, the HG group treated with a high con- DN group. centration of glucose showed greater cell viability. SPS treatment 7 XU ET AL. FIGURE 3 Legend on next page. 8 XU ET AL. F I G U R E 4 (a) Detection of related protein expressions by western blotting; (b) statistical analysis of the expression of related proteins (#p < .05, ##p < .01 vs. the NG group; *p < .05, **p < .01 vs. the DN group, n = 3); (c) detection of PKC-α and p65 expressions in the kidneys of rats by immunohistochemistry; (d) statistical analysis of the expression of related proteins (#p < .05, ##p < .01 vs. the NG group; *p < .05, **p < .01 vs. the DN group, n = 3). DN, diabetic nephropathy; NG, normal glucose [Colour figure can be viewed at wileyonlinelibrary.com] 4 | DI SCU SSION effectively reduced the viability of cells, particularly at higher concentrations (200 μg ml−1) where the viability value is similar to that To ensure the development of a diabetic model, a high-sugar, high-fat of the normal group. Based on these results, the SPS concentration diet was fed to all rats, except those in the normal group. This diet of 200 μg ml −1 was chosen as the optimum concentration for the in vitro experiments (Figure 5b). increases the body weights of the rats; however, STZ administration significantly reduces DN rat weights, with no appreciable effect on rats in other groups (just slight fluctuations). The state of hyperglycemia in DN, DN+IG, and DN+SPS groups was regularly confirmed by 3.4.3 | Effect of SPS on NRK-52E cells the analysis of the blood glucose levels. Blood biochemical analyses demonstrate that on the 16th week, the kidneys of DN rats begin to Western blot analyses indicate that PKC-α was significantly expressed show pathological changes, resulting in significantly increased BUN in the HG group; however, the level of this protein in the SPS group and Scr values (Kirtane et al., 2005; Perrone, Madias, & Levey, 1992). was relatively low. The expressions of inflammatory factors, P- These values are maintained within the normal range in the DN+IG selectin, NF-κB, and p65, also showed appreciable variations and DN+SPS groups. Therefore, the PKC inhibitor and SPS are both (Figure 5c,d). capable of protecting the kidneys of diabetic rats. F I G U R E 3 (a) Changes in the 24-hr urine volumes of rats in each group, measured on the 8th, 12th, 16th, and 20th weeks (**p < .01 vs. the DN group); (b) Ratio of urine volume to creatinine change, measured on the 16th and 20th weeks (##p < .01 vs. the NG group; *p < .05, **p < .01 vs. the DN group); Variations in (c) BUN (##p < .01 vs. the NG group; *p < .05, **p < .01 vs. the DN group) and (d) Scr levels (##p < .01 vs. the NG group; **p < .01 vs. the DN group) at the end of the experiment; Observation of (e) kidney injury and (f) skeletal muscle injury in rats by hematoxylin and eosin staining; Observation of rat kidneys in each group by (g) PASM and (h) Masson staining; (i) Pathological experiment score table. DN, diabetic nephropathy; PASM, periodic acid-silver-methenamine [Colour figure can be viewed at wileyonlinelibrary.com] 9 XU ET AL. F I G U R E 5 (a) Cytotoxicity test of SPS by MTT; (b) Screening of SPS action concentration (##p < .01 vs. the NG group; **p < .01 vs. the HG group, n = 6); (c) detection of related protein expressions by western blotting; (d) statistical analysis of the expressions of related proteins (#p < .05, ##p < .01 vs. the NG group; *p < .05, **p < .01 vs. the HG group, repeated three times). HG, high glucose; MTT, 3-(4,5-dimethyl-2-thiazolyl)2,5-diphenyl-2-H-tetrazolium bromide; NG, normal glucose; SPS, sulfated polysaccharide In order to evaluate renal function in diabetic rats, the excreted the kidneys of diabetic rats; however, the expressions of these pro- urine was collected and analyzed on specific dates. The analyses teins are significantly reduced under the effect of the PKC inhibitor showed that diabetic rats produced more urine than normal rats. and SPS. These results were confirmed by western blot analyses However, after 16 weeks of experimentation, the urine volumes of showing that the PKC/NF-κB pathway might play an important role in SPS rats were found to be appreciably lower than those of diabetic the development of DN. The similarity between the effects of SPS rats. The difference becomes more substantial with time. This sug- and PKC inhibitor further proves that the PKC signaling pathway gests that the renal protection effect of SPS is gradual, not instanta- might be involved in the mechanism of DN treatment by SPS. neous. In addition to volume measurements, the collected urine was analyzed for protein and Scr contents (Figure 3). To verify the results of in vivo experiments, the effect of SPS on high-glucose-induced NRK-52E cells was assessed in vitro. The Pathological analyses of the kidney were performed at the end of obtained results demonstrate that higher concentrations of SPS have the experiment. These analyses indicated that the onset of diabetes a stronger effect on the DN model cells (within the investigated range significantly changed the structure of kidney skeletal muscles, of concentration). Moreover, SPS significantly decreased PKC expres- resulting in the appearance of podocyte fusion. Moreover, diabetes sion in these cells, as evidenced by western blot analyses. The use of aggravates the kidney by inciting inflammation and fibrosis, both of the PKC inhibitor significantly reduces the expressions of PKC and which are typical DN symptoms (Zhu et al., 2016). Immunohistochem- the downstream inflammatory mediating factor NF-κB, compared to istry assessments indicate that PKC and p65 are highly expressed in the HG group. Considering that the effect of SPS on high-glucose- 10 XU ET AL. induced NRK-52E cells is similar to that of the PKC inhibitor, and that concentration. The free radical inhibition rate reached 85% both species are capable of inhibiting the expression of downstream (Figure 6a). Meanwhile, the in vivo antioxidant activity test showed inflammatory factors, it is suggested that SPS might alleviate DN that the DN+SPS and DN+IG groups exhibit significantly higher symptoms via the PKC/NF-κB pathway. SOD inhibition rates and catalase activity than the DN group Although the DN-inhibition activity of SPS has been previously (Figure 6b,c). Based on the GSH test, the activity of the DN+IG reported, the mechanism of action remains unclear (Chen group was higher than that of the DN group, but the DN+SPS group et al., 2015; Wang et al., 2015). The available studies suggest that did not show an obvious difference (Figure 6d). The DN+SPS and the NF-κB signaling pathway might be implicated in the therapeutic DN+IG groups are significantly better than the DN group in terms effect of SPS; however, the upstream regulatory factors are not of MDA content (Figure 6e). Overall, the tests conducted in vivo known. According to previous research, diabetic microvascular com- and in vitro indicate that the antioxidant activity of SPS is relatively plications are caused by elevated oxidative stress (Anjaneyulu & good, and that this activity is important for the inhibition of DN. Chopra, 2004). The excessive production of reactive oxygen species To assess the mechanism of DN-inhibition by SPS, we compared activates the transduction pathways involved in the development of the activity of SPS to that of the PKC inhibitor (Juan, Chuang, Long, diabetic microangiopathy by promoting the gene expression of Lin, & Huang, 2009). Pathological and blood biochemical analyses con- inflammatory factors, proinflammatory factors, and adhesion fac- firmed that the inhibition of PKC expression in the kidneys of rats tors, which ultimately accelerates renal damage (Tan, Forbes, & alleviates the symptoms of DN, such as inflammation. NF-κB and p65 Cooper, 2007). Herein, we tested the antioxidant activity of SPS expressions were also reduced by PKC inhibition upon the administra- in vitro and in vivo. The in vitro experiments demonstrate a gradual tion of SPS or the inhibitor. This was confirmed by ELISA testing increase (Figure 6f). in antioxidant activity with the increasing SPS F I G U R E 6 (a) Detection of ABTS free radical inhibition rate of SPS in vitro; (b) SOD inhibition percentage in rat kidneys (##p < .01 vs. the NG group; **p < .01 vs. the DN group, n ≥ 6); (c) detection of catalase activity in the kidneys of rats (#p < .05 vs. the NG group; *p < .05 vs. the DN group, n ≥ 6); (d) GSH (##p < .01 vs. the NG group; **p < .01 vs. the DN group, n ≥ 6) and (e) MDA contents in kidneys (##p < .01 vs. the NG group; **p < .01 vs. the DN group, n ≥ 6); (f) NF-κB detection in kidneys by ELISA (#p < .05 vs. the NG group; *p < .05, **p < .01 vs. the DN group, n ≥ 6). DN, diabetic nephropathy; NG, normal glucose 11 XU ET AL. In a nutshell, our findings indicate that SPS could suppress NF-κB signaling by down-regulating PKC. Therefore, it is safe to infer that the PKC/NF-κB signaling pathway might be involved in the development of DN which also means that both NF-κB and PKC could be seen as potential therapeutic targets for the treatment/management of DN. ACKNOWLEDGMENTS This work was supported by grants from the Natural Science Foundation of Shandong Province (No. ZR2019BD055). It is a key project at the central government level that establishes the sustainable use of valuable Chinese medicinal resources (2060302). We thank LetPub (www.letpub.com) for its linguistic assistance during the preparation of this manuscript. CONF LICT OF IN TE RE ST The authors declare there is no conflict of interest. ORCID Xinpeng Li https://orcid.org/0000-0003-1165-4723 RE FE R ENC E S Anjaneyulu, M., & Chopra, K. (2004). Quercetin attenuates thermal hyperalgesia cold allodynia in STZ-induced diabetic rats. Indian Journal of Experimental Biology, 42, 766–769. Bitter, T., & Muir, H. M. (1962). 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Phytotherapy Research. 2020;1–12. https://doi.org/10.1002/ptr.6966 Physiologia Plantarum 2019 © 2019 Scandinavian Plant Physiology Society, ISSN 0031-9317 A new insight into the mechanism for cytosolic lipid droplet degradation in senescent leaves Chaoyue Zhanga , Yuangang Qub , Yuji Lianb , Mingyu Chapmanc , Navid Chapmanc , Jie Xina , Huawei Xina* and Lin Liua* a College of Pharmacy, Linyi University, Linyi 276005, China b College of Life Sciences, Linyi University, Linyi 276005, China c Department of Chemistry, University of Rhode Island, Kingston, RI 02881, USA Correspondence *Corresponding authors, e-mail: xinhuawei@lyu.edu.cn; liulin@lyu.edu.cn Received 2 July 2019; revised 4 August 2019 doi:10.1111/ppl.13039 Leaf senescence involves lipid droplet (LD) degradation that produces toxic fatty acids, but little is known about how the toxic metabolites are isolated from the rest of the cellular components. Our ultramicroscopic characterization of cytosolic LD degradation in central vacuole-absent cells and central vacuole-containing cells of senescent watermelon leaves demonstrated two degradation pathways: the small vacuole-associated pathway and the central vacuole-associated pathway. This provided an insight into the subcellular mechanisms for the isolation of the fatty acids derived from LDs. The central vacuole-containing cells, including mesophyll cells and vascular parenchyma cells, adopted the central vacuole-associated pathway, indicated by the presence of LDs in the central vacuole, which is believed to play a crucial role in scavenging toxic metabolites. The central vacuole-absent intermediary cells, where senescence caused the occurrence of numerous small vacuoles, adopted the small vacuole-associated pathway, evidenced by the occurrence of LDs in the small vacuoles. The assembly of organelles, including LDs, small vacuoles, mitochondria and peroxisome-like organelles, occurred in the central vacuole-absent intermediary cell in response to leaf senescence. Introduction Lipid droplets (LDs) are organelles that consist of a core of neural lipids surrounded by a phospholipid monolayer and a variety of surface-associated proteins (Murphy 2012). LDs form on membranes of endoplasmic reticula, chloroplasts or thylakoids by accumulating neutral lipids between the two leaﬂets of the membranes and then pinch off into the cytosol or chloroplast stroma (Martin and Parton 2006, Partridge and Murphy 2009, Chapman et al. 2012, Thiam et al. 2013, Ohsaki et al. 2014, Thiam and Beller 2017). LDs located in the cytosol are called cytosolic LDs, and those inside the chloroplast are called plastoglobules (Murphy 2012). Cytosolic LDs in this study are simply called LDs. Abbreviations – LD, lipid droplet; SDP1, SUGAR DEPENDENT1. Physiol. Plant. 2019 Leaf senescence involves the genetically controlled degradation of cellular components and the subsequent conversion of the products into movable molecules that are reused by growing parts of the parent plant (Buchanan-Wollaston 1997, Yoshida 2003, Guo 2013, Jibran et al. 2013, Zhang and Zhou 2013). The occurrence of massive LD degradation in senescent leaves is repeatedly suggested by the upregulated expression of lipases (Nam 1997, Andersson et al. 2004, Buchanan-Wollaston et al. 2005, Gregersen and Holm 2007). LD degradation is a highly complex process, during which neutral lipids stored in the organelle are broken down to produce fatty acids (Deruyffelaere et al. 2015, 2018, D’Andrea 2016, Pyc et al. 2017, Farquharson 2018). Fatty acids are toxic to other cell components if they are free in the cytosol (Wenzel and Hale 1978, Kunz et al. 2009, Yao et al. 2011, Plötz et al. 2016, Huang 2018). Therefore, LD degradation must involve concurrent separation of toxic fatty acids from the rest of the cellular components so that the fatty acids do not have any chances to damage the cell function before they are eventually converted to movable sugars. Autophagy in plant cells is a major catabolic pathway whereby cytoplasmic constituents, including LDs, are delivered to the vacuole for degradation. Enhancement of autophagy-related genes required for nutrient recycling has been demonstrated in senescent leaves (Doelling et al. 2002), suggesting increased activity of autophagy in response to senescence. The role of autophagic vacuoles in LD degradation has been demonstrated in Arabidopsis thaliana leaves during starvation, by the presence of LDs in vacuoles and the colocalization of LDs with autophagic markers (Fan et al. 2019). Our ultramicroscopic characterization of cytosolic LD dynamics in senescent watermelon leaves demonstrated two pathways for LD degradation, the small vacuole-associated pathway and the central vacuole-associated pathway, which provided an insight into the mechanism for the isolation of the toxic fatty acids. Materials and methods Plant materials Seeds of Citrullus lanatus (the cultivar Jingxin 1) were sown in a nursery bed in a greenhouse and then transplanted in the ﬁeld fertilized with livestock manure in late April. During June and July, samples were collected from young, mature and senescent leaves from plants aged 70–100 days. The senescent leaves were determined by their position on the stem as well as their decrease in chlorophyll content. Chlorophyll measurement Based on the method of Arnon (1949), chlorophylls were extracted with acetone, and measured at an optical density of 663 and 645 nm with a Hitachi UH5300 Spectrophotometer (Hitachi High-Technology). Transmission electron microscopy Segments from leaf blades were cut from plants into cold primary ﬁxative (composed of 2.5% glutaraldehyde and 0.5% paraformaldehyde in 0.1 M sodium cacodylate buffer). After 30 min, leaf segments were trimmed to remove edge material damaged during initial preparation. Trimmed segments were then ﬁxed in fresh primary ﬁxative for 4 h. After rinsing three times in the same buffer, the segments were post-ﬁxed in 1% osmium tetroxide (in 0.1 M sodium cacodylate buffer) for 4 h. Subsequently, dehydration was done via an ethanol series. Ethanol in leaf tissues was completely replaced by acetone before the tissues were embedded in Embed-812 resin (EMS). The embedded samples were sectioned at 60 nm thickness with an Ultracut R microtome (Leica). The ultrathin sections were stained with 5% uranyl acetate for 30 min, followed by an incubation in a 0.1% lead citrate solution for 5 min and were observed under a Jeol 1220 transmission electron microscope (JEOL). Histochemistry Identiﬁcation of LDs was performed using a method based on their reaction with osmium tetroxide to produce a brownish color (Jones 2002). Leaf samples were prepared as described above in the transmission electron microscopy paragraph, sectioned at 1 μm thickness with an Ultracut R microtome (Leica) and observed under a light microscope (CX23, Olympus). LDs would show a brownish color. Identiﬁcation of storage proteins was performed with toluidine blue O (Sigma-Aldrich). Samples were prepared as described in transmission electron microscopy, sectioned at 1 μm thickness, stained with toluidine blue O and observed under the light microscope (CX23, Olympus). Identiﬁcation of starch grains was done with I2 -KI2 based on the property of starch that it reacts with I2 to produce a blue color (Wang 1986). Tannins were identiﬁed with FeCl3 based on the property of tannins that they react with this chemical to produce a blue-green color (Wang 1986). Samples were treated with FeCl3 before dehydration and embedment. Results Histochemical identiﬁcation of storage reserves in leaf blades Watermelon stems produce leaves at different developmental stages, including young, mature and senescent leaves. During development, leaves were different in leaf blade size and chlorophyll contents (Fig. 1A–C, Table 1). The leaf blade consisted of epidermis, mesophyll and vascular tissues (Fig. 1D). Histochemistry was done with leaves before and after senescence. Starch grains were identiﬁed with I2 -KI in mesophyll cells of leaf blades before senescence (Fig. 1E), while cytosolic LDs were identiﬁed with Sudan III and osmium tetroxide in mesophyll and vascular tissues of senescent leaf blades (Fig. 1F). Mesophyll cells, ordinary companion Physiol. Plant. 2019 Fig. 1. Legend on next Page. Physiol. Plant. 2019 Table 1. Chlorophyll contents in fresh leaf blades of Citrullus lanatus. Developmental stage of leaf Chlorophyll a content (mg g−1 ) Chlorophyll b content (mg g−1 ) Total chlorophyll content (mg g−1 ) Young Mature Senescent 0.5 ± 0.05 0.67 ± 0.01 0.34 ± 0.03 0.15 ± 0.02 0.19 ± 0.01 0.12 ± 0.01 0.65 ± 0.07 0.87 ± 0.02 0.46 ± 0.04 cells and vascular parenchyma cells contained a large central vacuole (Fig. 1E,F). LDs were present in the large central vacuole of these cell types during leaf senescence (Fig. 1F). Intermediary cells, the highly specialized companion cells, did not contain a central vacuole (Fig. 1E,F). The central vacuole-absent intermediary cells of senescent leaves accumulated LDs, indicated by the brownish color after being treated with osmium tetroxide or Sudan III (Fig. 1F). Plastoglobules in chloroplasts of senescent leaves were prominent (Fig. 1F). In addition, no accumulation of starch and storage proteins was identiﬁed with I2 -KI and toluidine blue O, respectively, in leaf blades before and after senescence (Fig. 1E,F). Nor accumulation of tannins was identiﬁed with FeCl3 in leaf blades at different developmental stages (Fig. 1E,F). Therefore, LDs were the only form of storage reserves that were identiﬁed histochemically in senescent leaves (Fig. 1F). LDs in central vacuole-containing mesophyll cells Mesophyll cells contained a large central vacuole (Fig. 1D–F). Comparative observation revealed that leaf senescence caused an increase in size of cytosolic LDs in these central vacuole-containing cells. LDs were approximately 0.2 μm in diameter before leaf senescence (Fig. 2A), whereas they were over 1 μm in senescent leaves (Fig. 2B). The LDs were spherical or irregularly shaped in senescent leaves (Fig. 2B,C). In addition, the LDs became less electron dense in response to leaf senescence (Fig. 2B,C). They had the same electron density as plastoglobules in chloroplasts in the same mesophyll cell (Fig. 2C). Leaf senescence caused LD degradation in the central vacuole. The vacuolar degradation was indicated by the presence of LDs in the center of the large central vacuole in senescent leaves (Fig. 2B), in contrast to LDs in leaves before senescence, which were not associated with the large central vacuole, but away from it (Fig. 2A). LDs were frequently found in the periphery of the central vacuole (Fig. 2C). LDs in central vacuole-containing vascular parenchyma cells Leaf senescence led to the presence of cytosolic LDs in vascular parenchyma cells, including ordinary vascular parenchyma cells and ordinary companion cells (Fig. 3B–D). LDs appeared as black inclusions and had irregular shapes, quite different from typical LDs that were spherical or elliptical (Fig. 2B). However, these black inclusions were identiﬁed as LDs rather than storage proteins or tannins by means of histochemistry. LDs were seldom found in these vascular parenchyma cells before senescence (Fig. 3A), but they accumulated in senescent leaves (Fig. 3B–D), indicating that LD accumulation was a result of leaf senescence. Similar to what happened in mesophyll cells, leaf senescence caused LD degradation in the central vacuoles of the vascular tissue. The vacuolar degradation was evidenced by the occurrence of LDs in the center or periphery of the central vacuole (Fig. 3B–D). The remnants of LDs in the central vacuole was visible. LDs were also present in smaller vacuoles (Fig. 3D). LDs in central vacuole-absent cells Intermediary cells did not possess a large central vacuole except small vacuoles and were therefore central vacuole-absent cells. Small vacuoles were much more abundant in senescent leaves than in leaves before senescence (Fig. 3A,B). It indicated that leaf senescence caused the marked increase of small vacuoles. Comparative observation of the central vacuole-absent intermediary cells before and after leaf senescence Fig. 1. Leaf blades and histochemical identiﬁcation of storage reserves. (A–C) Photos of a young leaf blade (A), a mature leaf blade (B) and a senescent leaf blade (C). (D) A portion of the cross section of the young leaf blade, showing epidermis, mesophyll and vascular tissues. Leaf tissue sections (1 μm thick) were stained with toluidine blue O. (E) A portion of the cross section of the mature leaf blade, showing the presence of starch grains in chloroplasts (arrows). Leaf tissues were treated with osmium tetroxide, I2 -KI and FeCl3 before dehydration and, after being sectioned, leaf tissue sections (1 μm thick) were stained with toluidine blue O. (F) A portion of cross section of a senescent leaf blade. Leaf tissues were treated with osmium tetroxide and FeCl3 before dehydration, and leaf tissue sections (1 μm thick) were stained with toluidine blue O. Large LDs (thick arrows) are present in the periphery of the central vacuole of mesophyll cells. Plastoglobules (thin arrows) are abundant. The central vacuole-absent intermediary cell has a brown appearance, indicative of the presence of a lot of small LDs. CC, companion cell; EP, epidermis; IC, intermediary cell; MC, mesophyll cell; PA, palisade cell; S, starch grain; SC, sponge cell; SE, sieve element; VB, vascular bundle; VE, vessel element; VP, vascular parenchyma cell. Physiol. Plant. 2019 Fig. 2. Cytosolic LDs in mesophyll cells of leaves before and after senescence. (A) A portion of the mesophyll cell of the young leaf, showing two cytosolic LDs (arrows) that are present between chloroplasts and plasma membrane. White arrows indicate plastoglobules in the chloroplast. (B) A portion of the mesophyll cell of the senescent leaf, showing a large spherical LD in the central vacuole. (C) A portion of the mesophyll cell of the senescent leaf, showing LDs in the periphery of the central vacuole. CH, chloroplast; G, Golgi apparatus; IC, intermediary cell; LD, lipid droplet; MC, mesophyll cell; PG, plastoglobule; S, starch grain; V, vacuole. (Fig. 3A,B) showed that senescence induced the occurrence of numerous black inclusions. These black inclusions were identiﬁed as LDs rather than tannins or storage proteins by means of histochemistry, as their shapes and color were too irregular and different from typical LDs. Therefore, leaf senescence induced the occurrence of LDs with irregular appearances in the central vacuole-absent vascular parenchyma cells. These cytosolic LDs had much higher electron density than plastoglobules in chloroplasts in mesophyll cells of the same leaf. Leaf senescence caused LD degradation in the central vacuole-absent cells. It was noticeable that the LD degradation was small vacuole-associated, as indicated by the presence of LDs inside the small vacuoles (Figs 4 and 5). Cytosolic LDs came in close contact with small vesicles or cup-shaped autophagosome-like structures at ﬁrst and, as LD degradation was in progress, the LDs were present in small vacuoles (Figs 4 and 5). The remaining Physiol. Plant. 2019 of LDs was visible in the small vacuoles, indicative of a close interaction between LDs and the small vacuoles. Apparently, the central vacuole-absent intermediary cells adopted the small vacuole-associated pathway for LD degradation. Senescence induced gathering of different types of organelles. Assembly of LDs, small vacuoles and mitochondria was present (Figs 4 and 5). The small vesicles or autophagosome-like structures came in close contact with the LDs, and the coupled organelles were close to the mitochondrion (Fig. 4). As seen in Fig. 4, the LDs could even produce a protrusion toward the mitochondrion. More complex aggregation of LDs, small vacuoles or autophagosome-like structures, and organelles similar to peroxisomes in appearance also occurred (Fig. 5). The peroxisome-like organelle had a dense matrix surrounded by a single unit membrane. LDs and the peroxisome-like organelle were inside the small vacuole. Fig. 3. Cytosolic LDs in vascular tissues of senescent leaves. (A) The phloem of the mature leaf, showing the absence of cytosolic LDs in vacuole-containing vascular parenchyma cells and vacuole-absent intermediary cells. (B) The phloem of the senescent leaf, showing dark cytosolic LDs (thin arrows) in vascular parenchyma cells, ordinary companion cells, intermediary cells and sieve elements. The vascular parenchyma cells and ordinary companion cells contain a large central vacuole, while the intermediary cells contain numerous small vacuoles (thick arrows) instead of a large central vacuole. LDs (thin arrows) are much more abundant in the central vacuole-absent intermediary cells. (C) Magniﬁcation of the vascular parenchyma cell in (B), highlighting dark LDs (arrows) in the central vacuole. (D) The vascular parenchyma cell of the senescent leaf, showing LDs (thin arrows) in the central vacuole. The remaining of an LD can be seen in the central vacuole (short thick arrow). CC, companion cell; IC, intermediary cell; MC, mesophyll cell; N, nucleus; SE, sieve element; V, vacuole; VP, vascular parenchyma. Discussion We demonstrated an increase of cytosolic LDs in watermelon leaves in response to leaf senescence. It is known that senescence leads to the breakdown of membrane lipids to produce fatty acids, which are converted to neutral lipids and stored in LDs, accountable for the increase of LDs in senescent leaves (Murphy 2012). The conversion is of signiﬁcance, as fatty acids are toxic if they are free in the cell (Wenzel and Hale 1978, Kunz et al. 2009, Yao et al. 2011, Plötz et al. 2016, Huang 2018). For example, accumulation of free fatty acids in Arabidopsis leaves caused by the impaired function of the peroxisomal ABC-transporter 1 or the core 𝛽-oxidation enzyme keto-acyl-thiolase 2 has been shown to produce dramatic membrane damage (Kunz et al. 2009). It has been demonstrated that the mechanisms of fatty acid sequestration in leaves during senescence involve triacylglycerol synthesis and LD accumulation (Kunz et al. 2009, Fan et al. 2017). Characterization of Arabidopsis mutants defective in SUGAR DEPENDENT1 (SDP1) triacylglycerol lipase and PEROXISOMAL ABC TRANSPORTER 1 has shown that the disruption of SDP1 increases the triacylglycerol accumulation in LDs and decreases free tatty acids in leaves, protecting the cell against lipotoxicity and oxidative damage (Fan et al. 2017). Therefore, LDs play an important role in the detoxiﬁcation of fatty acids via lowering the free fatty acid level during leaf senescence. Our ultramicroscopy demonstrated that cytosolic LD degradation caused by senescence in watermelon leaves followed two different subcellular pathways: the small vacuole-associated pathway (Fig. 6A) and the central vacuole-associated pathway (Fig. 6B). Which pathway the cell adopted was dependent upon the cell type. Central vacuole-absent cell types took the small vacuole-associated pathway, while central vacuole-containing cell types followed the central vacuole-associated pathway. LDs came into contact with cup-shaped autophagosome-like structures before they entered small vacuoles in senescent watermelon leaves. It was Physiol. Plant. 2019 Fig. 4. Cytosolic LDs, autophagosome-like structures, mitochondria and small vacuoles in the central vacuole-absent intermediary cell of senescent leaves. LDs have a dark appearance. The remaining of an LD is present in a small vacuole (short thick arrow). The gathering of an LD, autophagosome-like structures (short thin arrows) and a mitochondrion can be seen. The autophagosome-like structures come in contact with the LD, which produces a protrusion (long thin arrow) toward the mitochondrion. IC, intermediary cell; LD, lipid droplet; M, mitochondrion; MC, mesophyll cell; PG, plastoglobule; SV, small vacuole. obvious that the membranous structures played a role in the vacuolar degradation of LDs. A possibility was that LDs were recognized by the autophagosome and then delivered to small vacuoles for degradation. More work is needed to prove the cup-shaped membranous structures to be autophagosomes. Vacuoles are lytic compartments with various functions, including a crucial role in detoxiﬁcation by scavenging toxic molecules (Marty 1999). The process where cellular components are delivered to the vacuole for vacuolar catabolism is called autophagy (Doelling et al. 2002). Vacuolar degradation of LDs has been demonstrated in senescent cucumber leaves (Ge et al. 2016), Arabidopsis leaves under adverse conditions (Fan et al. 2019) and cotyledons of germinating oil seeds (Tzen and Huang 1992, Huang 2018). Lipases located in vacuoles have been identiﬁed (Carter et al. 2004), such as Atg15 (Epple et al. 2001), which is required for degradation of autophagic bodies and accelerates the degradation of LDs through lipolysis (Maeda et al. 2015, Ramya and Rajasekharan 2016). The vacuolar autophagy of LDs, which has been demonstrated in Arabidopsis leaves by the colocalization of LDs with autophagic markers and the presence of LDs in vacuoles, is termed lipophagy, Physiol. Plant. 2019 Fig. 5. Cytosolic LDs, peroxisome-like structures, autophagosome-like structures, mitochondria and small vacuoles in the central vacuole-absent intermediary cell of senescent leaves. There are a lot of small vacuoles and cytosolic LDs in the intermediary cell. Some small vacuoles contain remaining of LDs (thick arrows). Cup-shaped autophagosome-like structures (thin arrows) are in association with LDs. The inset highlights the gathering of an LD, small vacuoles and a peroxisome-like organelle. CH, chloroplast; IC, intermediary cell; LD, lipid droplet; M, mitochondrion; MC, mesophyll cell; PG, plastoglobule; PO, peroxisome-like organelle; SV, small vacuole. a process morphologically resembling microlipophagy and requiring the core components of the macroautophagic machinery (Fan et al. 2019). Basal autophagy is responsible for triacylglycerol synthesis and inducible autophagy accountable for LD degradation (Fan et al. 2019). Through vacuolar degradation, the toxic fatty acids derived from the process are isolated within the vacuole, and therefore damages to the rest of the cell components could be prevented. The protection is necessary for the cell to have more time to fulﬁll the eventual conversion of fatty acids into movable sugars that are reused by the vigorously growing parts of the parent plant. We also demonstrated the assembly of LDs, small vacuoles, mitochondria and peroxisome-like organelles in the central vacuole-absent intermediary cells during leaf senescence. Reasonably, the assembly could facilitate physical interaction between organelles and therefore the LD degradation and the conversion of the derived fatty acids to movable metabolites (Fig. 6C). The assembly reminded us of the physical interaction between LDs and glyoxysomes in germinating oil seeds. During germination of oil seeds, LD–glyoxysome Fig. 6. Diagram of the pathways for cytosolic LD degradation in central vacuole-absent and central vacuole-containing cells of senescent watermelon leaves. (A) The central vacuole-associated pathway in central vacuole-containing cells. The cytosolic LD is delivered into the central vacuole for degradation. (B) The small vacuole-associated pathway in central vacuole-absent cells. The cytosolic LD is in contact with the autophagosome-like structure, which helps to deliver the LD into the small vacuole for degradation. (C) The pathway for conversion of fatty acids to glucose (Thompson et al. 1998). Fatty acids derived from cytosolic LDs are delivered into the peroxisome, where they are converted to succinate via 𝛽-oxidation and glyoxylate cycle. The succinate is then transferred into the mitochondrion and converted to malate through the tricarboxylic acid cycle therein. The malate is transported from the mitochondrion to the cytosol, where it is converted to glucose via gluconeogenesis. AS, autophagosome-like structure; FA, fatty acid; LD, lipid droplet; M, mitochondrion; P, peroxisome; SV, small vacuole; V, vacuole. contact has been repeatedly demonstrated (Frederick et al. 1968, Vigil 1970, Cui et al. 2016). Through such physical contact, SDP1, a lipase, is delivered from the glyoxysome to the LD surface, resulting in the metabolism of neutral lipids (Thazar-Poulot et al. 2015). The toxic fatty acids derived from the lipase activity are then imported into the glyoxysome through transporter proteins (Cui et al. 2016). 𝛽-Oxidation within the glyoxysome metabolizes the fatty acids to acetyl-CoAs, which enter the glyoxylate cycle and are converted to non-toxic succinate (Thompson et al. 1998). The succinate is then transferred to the mitochondrion, where it is converted to malate through the tricarboxylic acid cycle. The malate enters gluconeogenesis in the cytosol to produce glucose (Thompson et al. 1998, Graham 2008, Yang and Benning 2018). In the central vacuole-absent cell of senescent watermelon leaves, the gathering of organelles could facilitate the conversion of fatty acids to sugars via improving the transfer of fatty acids from small vacuoles to peroxisomes, and of succinate from peroxisomes to mitochondria. However, there is more work to do to verify the hypothesis. Author contributions C.Z. and L.L. were responsible for the transmission electron microscopy, histochemical identiﬁcation of storage reserves and manuscript writing. Y.Q. was responsible for the chlorophyll measurement. J.X. was responsible for tannin identiﬁcation. Y.L. took part in laboratory work. H.X., M.C. and N.C. took part in the writing of this manuscript and laboratory work. 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Plant. 2019 Micron 99 (2017) 1–8 Contents lists available at ScienceDirect Micron journal homepage: www.elsevier.com/locate/micron Ultramicroscopy reveals a layer of multiply folded membranes around the tannin-accumulating vacuole in honeysuckle petal trichomes Shengnan Qu a , Navid Chapman b , Zhengyan Xia a , Mingxiao Feng c,d , Shangcai Feng a , Zhen Wang a , Lin Liu a,∗ a College of Pharmacy, Linyi University, Linyi 276000, China Department of Chemistry, University of Rhode Island, Kingston, RI 02881, USA c Department of Life Science, Linyi University, Linyi 276000, China d Department of Biological Sciences, University of South Carolina, Columbia, SC 29208, USA1 b a r t i c l e i n f o Article history: Received 12 January 2017 Received in revised form 23 March 2017 Accepted 24 March 2017 Available online 27 March 2017 Keywords: Membrane proliferation Tannin-accumulating vacuole Trichome Petal Honeysuckle a b s t r a c t Transmission electron microscopy was used to reveal a layer of multiply folded membranes that closely surrounded the tannin-accumulating vacuole in cells of honeysuckle petal trichomes. A huge amount of tannins were deposited in the peripheral region and the center of the vacuole. The proliﬁc membranes extended to the tannins deposited along the vacuole periphery. It was difﬁcult to distinguish the vacuole membrane, and it seemed as if it was the layer of multiply folded membranes that separated the vacuole lumen from the cytoplasm. In addition, there were also membrane assemblies in the cytoplasm away from the vacuole, which were continuous with the proliferated membranes bordering the vacuole. Therefore, the tannin-accumulating vacuole was in close association with a very large network of proliferated membranes. The occurrence of such a layer of multiply folded membranes around the tannin-accumulating vacuole might be a structural strategy for improvement of the efﬁciency of vacuolar accumulation of tannins. © 2017 Published by Elsevier Ltd. 1. Introduction A vacuole is an intracellular compartment enclosed by a membrane, which is called tonoplast in plant cells. The tonoplast is a very dynamic membrane and can take many forms to enhance the interaction between the vacuole and the cytoplasm (Reisen et al., 2005; Wiltshire and Collings, 2009). Vacuoles have various biological functions (Matile, 1978; Marty, 1999). In addition to serving as a storage site for metabolites (Ryan and Walker-Simmons, 1983; Kurkdjian and Guern, 1989; Chrispeels, 1991; Pham and Roberts, 1991), vacuoles play important roles in autophagy (Wittenbach et al., 1982; Boller and Wiemken, 1986; Barceló et al., 1991; Liu, 2016), turgor generation (Marty, 1999), and interactions with environmental factors (Wink, 1988, 1993; Bhattacharya et al., 2010). Vacuoles in vascular plants often accumulate tannins in large quantity (Mueller and Beckman, 1976; Parham and Kaustinen, 1977; Mueller and Greenwood, 1978; Hutzler et al., 1998; Kondo and Kawashima, 2000; Kefeli et al., 2003; Liu, 2012) to provide ∗ Corresponding author. 1 E-mail address: liulin@lyu.edu.cn (L. Liu). Current address. http://dx.doi.org/10.1016/j.micron.2017.03.013 0968-4328/© 2017 Published by Elsevier Ltd. protection against herbivores and pathogens (Blackman, 2000; Kefeli et al., 2003; Forkner et al., 2004), UV radiation (Lovelock et al., 1992), drought (Georgieva et al., 2010), and extreme temperature (Stefanowska et al., 2002). Tannins are oligomers or polymers rich in phenolic groups, and can be subdivided into two categories, hydrolysable tannins and condensed tannins, also called proanthocyanidins. Hydrolysable tannins are molecules with a polyol as a central core, the hydroxyl groups of which are partially or totally esteriﬁed with phenolic groups like gallic acids. The typical site for the synthesis and deposition of hydrolysable tannins has been suggested to be cell walls (Grundhöfer et al., 2001). Condensed tannins are oligomers or polymers of ﬂavonoid units, including epicatechin (extension unit) and catechin (terminal unit), linked by carbon–carbon bonds that are not susceptible to hydrolysis (Winkel-Shirley, 2001; Dixon, 2005; Liu et al., 2013). Epicatechins are formed from anthocyanidins by the action of anthocyanidin reductase (Xie et al., 2003; Dixon et al., 2005). A typical characteristic of tannins is that they can complex with other molecules including proteins, based on hydrophobic interactions and hydrogen bonding. The tannin’s phenolic group donates the hydrogen that forms strong hydrogen bonds with the protein’s carboxyl group, causing precipitation of proteins. Perhaps because of this property, massive accumulation of tannins is always 2 S. Qu et al. / Micron 99 (2017) 1–8 found in vacuoles or cell surfaces. In these spaces tannins are prevented from interfering with plant metabolism, for they do not have free access to functional proteins in the cytoplasm. Only after cell breakdown do tannins have opportunities to meet and form complexes with proteins and produce metabolic effects. Apparently, vacuoles play a vital role in the control of tannin activities. The monomers or precursors of vacuolar tannins are thought to be synthesized somewhere outside the vacuole and then transported into the vacuole lumen. In other words, the vacuolar accumulation of tannins is a very complex process which requires the cooperation of other organelles (Dixon, 2005; Zhao et al., 2010; Brillouet et al., 2013). Therefore, a close association with other organelles involved in tannin synthesis would facilitate the vacuolar tannin accumulation. In this paper we report our discovery that a layer of multiply-folded membranes surrounded the tanninaccumulating vacuole in cells of honeysuckle petal trichomes. Petals and leaf blades were cut into small pieces (<1 mm × 1 mm) and then ﬁxed in 2% glutaraldehyde in phosphate buffer for 4 or 12 h at room temperature. Three concentration levels of buffer were applied, 0.05 M, 0.1 M, and 0.2 M, pH 6.8. After tissues were washed 6 times in distilled water to remove excess phosphate ions, a secondary ﬁxation was carried out in 1% osmium tetroxide in corresponding phosphate buffer at pH 6.8 for 2 h at room temperature. Then dehydration was performed through a graded ethanol series. After replacement of ethanol by pure acetone, tissues were embedded in Embed-812 resin. Embedded tissues were sectioned at 60 nm thick on a Leica ultramicrotome. Sections were stained with 1% aqueous uranyl acetate for 10 min at room temperature in dark, and then with lead citrate solution for 1 min. Observation and photography were performed under a Philips Tecnai 12 transmission electron microscope. 2. Materials and methods 2.3. Histochemistry of tannins 2.1. Plant materials Small pieces of petals and leaf blades were immersed in 2.5% glutaraldehyde in phosphate buffer for 4 h at room temperature. After being washed in buffer, two different treatments were followed. First, based on the theory that osmium tetroxide reacts rapidly with tannins containing o-dihydroxy groups to give very stable chelate complexes, which give rise to blue, brown, or black colors (Nielson and Grifﬁth, 1978), a portion of aldehyde-ﬁxed tissues were immersed in 1% osmium tetroxide in phosphate buffer at pH 6.8 for 2 h. Second, based on the theory that tannins react with ferric chloride to produce blue, brown, or black, depending upon their Plants of honeysuckle (Lonicera japonica) and persimmon (Diospyros kaki) were grown in the mountainous region of Linyi, a place with four distinct seasons in Shandong Province, China. Leaf blades and petals of ﬂowers that just began to open were collected from honeysuckle plants in June. Leaf blades of persimmon were collected in June and July. Leaf samples were used as control to ensure the observation from petal glandular trichome was true. 2.2. Transmission electron microscopy Fig. 1. Light microscopy of honeysuckle petal. (A) A glandular trichome consisting of a 4-cell stalk and a multicellular head (arrow). (B) Cells in the head of the glandular trichome contain a large vacuole that contains tannins, which react with osmium tetroxide and ferric chloride. Treated with osmium tetroxide. (C) and (D) Different types of cells in the petal are highly vacuolated, and contain some starch grains (arrows) but no tannins. Treated with osmium tetroxide. EP, epidermis; ME, mesophyll cell; NGT, non-glandular trichome; T, tannins; V, vacuole; VB, vascular bundle. S. Qu et al. / Micron 99 (2017) 1–8 3 origin (Chen et al., 2003), a portion of tissues were immersed in 5% ferric chloride solution for 4 h. Tissues were dehydrated in a graded ethanol series followed by pure acetone, and then embedded in Embed-812 resin. Embedded samples were sectioned at 1.5 ␮m, and sections were observed under the light microscope. 3. Results 3.1. Histochemical localization of tannins in honeysuckle petals Petals of honeysuckle form a tubule at the lower portion, while the upper portion is divided into two parts, upper and lower lips. The outer or abaxial surface is covered with a large number of trichomes, including glandular and nonglandular types. The nonglandular trichome is an elongated single cell, while the glandular trichome consists of a uniseriate stalk and a multicellular head (Fig. 1A). Each cell in the head was found to contain a large central vacuole that occupies most of the cell volume (Fig. 1B). The vacuole was demonstrated to contain substances that react strongly with osmium tetroxide or ferric chloride, indicating the existence of tannin accumulation. Mesophyll cells, epidermal cells, and most vascular parenchyma cells of the petal contained a large central vacuole. The central vacuole revealed a negative reaction with osmium tetroxide or ferric chloride, indicating its lack of tannin accumulation (Fig. 1C and D). Therefore, the glandular trichome was tannin-accumulating tissue, while the mesophyll, vascular bundle, and epidermis did not accumulate tannins. 3.2. Ultramicroscopic observation of tannin-accumulating vacuoles in glandular trichomes The vacuole in the glandular trichome accumulated a large quantity of tannins, which were highly electron-dense due to their strong osmiophilicity. The electron-dense tannins were deposited along the periphery and in the central region of the vacuole. The spaces between these electron-dense deposits were ﬁlled with substances that were less electron-dense (Fig. 2). The tannin-containing vacuole in the glandular trichome was quite distinct in morphology. As seen in Figs. 3–5, the tannin-containing vacuole was closely surrounded by a layer of multiply-folded membranes. The proliﬁc membranes were found all over the surface of the vacuole. Each membrane was about 10 nm Fig. 2. Vacuolar accumulation of tannins in the glandular trichome. Electron-dense tannins are deposited along the vacuole periphery (arrows) or suspended in the central area. T, tannin. Fig. 3. Proliﬁc membranes around the tannin-accumulating vacuole in the glandular trichome. (A) A portion of a cell, showing that the tannin-accumulating vacuole’s peripheral region is electron-dense. (B) A higher magniﬁcation of the squared area in A, showing proliﬁc membranes (long arrows) that are in close association with electron-dense tannin deposit in the vacuole periphery. Note circular cross sections (short arrows). CP, cytoplasm; CU, cuticle; CW, cell wall; T, tannins; V, vacuole. in thickness. The proliferated membranes extended to the tannin deposited in the vacuole periphery. The electron density of the proliﬁc membranes was much lower than that of the tannin deposit. The membranes away from the tannin deposit did not show clear images due to weak contrast, while the images of those membranes close to or embedded within the tannin deposit became clearer because of stronger contrast. It was difﬁcult to distinguish the vacuole membrane from the proliﬁc membranes. As a result, it seemed as if it was the layer of multiply folded membranes that separated the vacuole lumen from the cytoplasm. In addition, another type of membranous structure was also present in the tannin-producing cells. As seen in Figs. 5 and 6, there were assemblies of membranes which were closely arranged, and the area of membrane aggregation was more electron-dense than the cytoplasm, similar to the area of the membrane proliferation around the tannin-accumulating vacuole. The membrane of these structures looked similar to the vastly proliferated membrane bordering the vacuole. The membrane units of these assemblies were also less electron-dense than tannin deposits and seemed to be negatively stained. These membrane assemblies a certain distance away from the vacuole were continuous with the membranes that closely surrounded the vacuole. Apparently, the tannin-accumulating vacuole was in close association with a huge network of proliferated membranes. In order to prove that the proliferation of membranes was a fact rather than an artifact, vacuoles in the mesophylls of honeysuckle petals and leaf blades and vacuoles in the mesophyll and vascular parenchyma of persimmon leaf blades were also examined. These tissues were treated the exactly same way as the petal trichomes. The vacuole in mesophyll cells of honeysuckle petals 4 S. Qu et al. / Micron 99 (2017) 1–8 Fig. 4. Proliﬁc membranes around the tannin-accumulating vacuole in the glandular trichome. (A) A portion of a cell, showing that the tannin-accumulating vacuole’s peripheral region is more electron-dense than the inner region and cytoplasm. (B) A higher magniﬁcation of the squared area in A, showing proliﬁc membranes (arrows) around the vacuole. CP, cytoplasm; CW, cell wall; T, tannins; V, vacuole. Fig. 5. Proliﬁc membranes around the vacuole and membrane assemblies away from the vacuole in glandular trichome. (A) There are proliﬁc membranes (arrows) around the vacuole and membrane assemblies (square) in the cytoplasm. The later are continuous with the former. (B) A higher magniﬁcation of the squared region in A, showing membrane assemblies (arrow). CP, cytoplasm; P, plastid; T, tannins; V, vacuole. and leaves did not accumulate tannins. The tannin-free vacuole had a distinct tonoplast, and there was no proliferation of membranes bordering the vacuole (Fig. 7). The vacuole in the mesophyll and vascular parenchyma of persimmon leaf blades accumulated a large amount of tannins. Tannins were densely deposited along the inner surface of the tonoplast, forming an electron-dense layer (Fig. 8). The tannin-accumulating vacuole was not bordered by proliﬁc membranes, quite different to the tannin-accumulating vacuole in honeysuckle petal. Apparently, the occurrence of the proliferation of membranes bordering the vacuole in honeysuckle petal was a fact rather than an artifact. 4. Discussion The results of this research demonstrated a layer of multiply folded membranes between the vacuole lumen and the cytoplasm in tannin-accumulating cells of honeysuckle petal trichomes. It was hard to distinguish the vacuole membrane from the proliﬁc membranes, implying that they were very close. Besides, there were also other membrane assemblies a little away from the vacuole in the cytoplasm, which were continuous with the proliﬁc membranes bordering the vacuole. Therefore, the vacuole was actually in close association with a large network of membranes. Careful sample treatments and comparative examinations of vacuoles in mesophylls of honeysuckle petals and leaves as well as persimmon leaves ensured the occurrence of the proliferation of membranes around the tannin-accumulating vacuole in honeysuckle petal trichomes was a fact rather than an artifact. It is remarkable that no Fig. 6. Proliﬁc membranes around the vacuole and membrane assemblies away from the vacuole in glandular trichome. The membrane assemblies in the cytoplasm (arrows) are continuous with the proliﬁc membranes (the two arrows to the upper left) around the vacuole. CP, cytoplasm; CW, cell wall; VPR, vacuole peripheral region. S. Qu et al. / Micron 99 (2017) 1–8 5 Fig. 7. Tannin-free vacuoles in honeysuckle petal and leaf. (A) A portion of a petal mesophyll cell, containing a large central vacuole that does not accumulate tannins. (B) A higher magniﬁcation of the squared region in A, showing absence of proliﬁc membranes around the vacuole. Arrows indicate the tonoplast. (C) A portion of a leaf mesophyll cell, containing a large central vacuole that does not accumulate tannins. (D) A higher magniﬁcation of the squared region in C, showing absence of proliﬁc membranes around the vacuole. Arrows indicate the tonoplast. CH, chloroplast; CW, cell wall; ER, endoplasmic reticulum; P, plastid; V, vacuole. Fig. 8. Tannin-accumulating vacuoles in persimmon leaves. (A) A vascular parenchyma cell, containing a large central vacuole where tannins are deposited densely along the inner surface of the tonoplast. (B) A portion of a mesophyll cell, containing a large central vacuole where tannins are deposited densely along the inner surface of the tonoplast. (C) A higher magniﬁcation of the squared region in B, showing the absence of proliﬁc membranes around the vacuole. CH, chloroplast; CW, cell wall; S, starch; T, tannins; V, vacuole. 6 S. Qu et al. / Micron 99 (2017) 1–8 identiﬁcation of such proliferation of membranes bordering vacuoles has been reported thus far, even though it is common for a vacuole to accumulate tannins in plants. This may imply that the proliferation of membranes around the vacuole was quite rare and only present in a minority of plants. The proliﬁc membranes bordering the vacuole in honeysuckle petals were somewhat similar to the proliferated smooth endoplasmic reticulum membranes demonstrated in the cytoplasm in plant cells (Sennerby-Forsse et al., 1987; Ferrero et al., 2015) and named as organized smooth endoplasmic reticulum (Snapp et al., 2003). The tremendous increase in the number and extension of smooth endoplasmic reticulum is thought to be an adaptation to their varied functions that require elevated levels of endoplasmic reticulum resident enzymes (Boutanaev et al., 2015; Ferrero et al., 2015; Storbeck et al., 2015). Interestingly, overexpression of endoplasmic reticulum located enzymes can induce proliferation of endoplasmic reticulum (Jingami et al., 1987; Wright et al., 1988; Ferrero et al., 2015), indicating a close relationship between the membrane proliferation and the elevated level of the membrane resident proteins. In addition to serving as the carrier of enzymes, the smooth endoplasmic reticulum is also found to be the origin of vacuoles (Viotti et al., 2013), even though they are usually thought to be formed by the fusion of multiple vesicles derived from Golgi apparatus (Marty, 1999). The proliﬁc membranes bordering the tannin-accumulating vacuole in honeysuckle petals were presumed to be organized smooth endoplasmic reticulum. The proliferation of endoplasmic reticulum around the tannin-accumulating vacuole suggested there might be a close functional cooperation between the two organelles. The vacuole surrounded by the proliﬁc membranes in honeysuckle petals accumulated a huge amount of tannins, implying that the occurrence of the proliﬁc membranes around the tannincontaining vacuole might be associated with vacuolar tannin accumulation. Although no chemical analysis was performed to determine their nature, the vacuolar tannins were presumed to be condensed tannins, based on the knowledge that condensed tannins are stored mainly in vacuoles, while cell walls are the typical site for the synthesis and deposition of hydrolysable tannins (Grundhöfer et al., 2001). The proliferation of membranes around the tanninaccumulating vacuole was really meaningful, considering that the synthesis of monomers or precursors of vacuolar tannins might take place somewhere outside the vacuole (Dixon, 2005; Zhao et al., 2010; Brillouet et al., 2013) and hence cooperation between the vacuole and other organelles might be necessary. There are several conclusions or assumptions about the origin of vacuolar tannins, which can be summarized as follows. First, a multienzymatic complex responsible for ﬂavonoid biosynthesis is found to be bonded to the cytosolic surface of endoplasmic reticulum (Wagner and Hrazdina, 1984; Hrazdina et al., 1987; Burbulis and Winkel-Shirley, 1999; Saslowsky and Winkel-Shirley, 2001). The biological synthesis of condensed tannins, which are members of the ﬂavonoid family (Dixon, 2005; Liu et al., 2013), is also supposed to take place in endoplasmic reticulum (Brillouet et al., 2013). After being produced on the cytosolic surface of endoplasmic reticulum, tannins are transported into the interior of endoplasmic reticulum, which undergo fragmentation to form tannin-containing vesicles that fuse with large vacuoles and release cargos therein, resulting in vacuolar accumulation (Parham and Kaustinen, 1977; Zobel, 1986; Rao, 1988). Second, the biosynthesis of tannin monomers is found to happen on the cytosolic surface of endoplasmic reticulum, whereas polymerization occurs inside the vacuole (Zhao et al., 2010). Third, anthocyanidin reductase, the enzyme responsible for the synthesis of epicatechin which is the main monomer of condensed tannins, has been demonstrated to be located in the cytosol (Zhu et al., 2014), indicating that the monomer is synthesized in the cytosol and subsequently transported to endoplasmic reticulum or vacuoles for polymerization. Polymerization could not take place in the cytosol, otherwise tannins would bind and precipitate proteins and therefore interfere with cell metabolism (Hagerman and Butler, 1981). Fourth, tannins are produced by a tannin-producing organelle that is named tannosome and is found to be originated from the thylakoid inside chloroplasts (Brillouet et al., 2013), which are the source of all plant aromatic compounds through the shikimate pathway (Herrmann, 1995) and are capable of synthesizing ﬂavonoid compounds (Saunders and McClure, 1976; Zaprometov and Nikolaeva, 2003). The tannin-containing tannosomes are subsequently shuttled into the vacuole. Apparently, endoplasmic reticulum is thought to be a very important organelle that is involved in vacuolar accumulation of tannins as the carrier of enzymes for the synthesis of tannins or tannin monomers. In addition, endoplasmic reticulum also carry ﬂavonoid transporters required for vacuolar tannin accumulation (Zhao and Dixon, 2009; Brillouet et al., 2013). Some ﬂavonoid transporters, identiﬁed via characterizations of mutants at molecular level (Shirley et al., 1995; Winkel-Shirley, 2001; Abrahams et al., 2002; Lepiniec et al., 2006; Brillouet et al., 2013), have been demonstrated to be located on endoplasmic reticulum and vacuole membranes (Abrahams et al., 2002; Lepiniec et al., 2006; Marinova et al., 2007a,b; Zhao and Dixon, 2009; Brillouet et al., 2013). Putative P-type ATPases to support ﬂavonoid transporters have also been identiﬁed and demonstrated to be located on endoplasmic reticulum as well as vacuole membranes (Baxter et al., 2005; Verweij et al., 2008). The transport form of epicatechin, the extension unit of condensed tannin chains, has already been identiﬁed as epicatechin 3 -Oglucoside (Pang et al., 2008, 2013). Therefore, the proliferation of membranes (presumed to be organized smooth endoplasmic reticulum as discussed above) around the vacuole in honeysuckle petals could greatly increase the capacity of accommodation for more enzymes and transporters responsible for the synthesis and transport of tannins or tannin monomers. The fact that the proliﬁc membranes and the tonoplast were very close could be favorable for the transport of tannins or tannin monomers from the large network of membranes to the vacuole. In one word, the occurrence of proliﬁc membranes around the tannin-accumulating vacuole might be a strategy to facilitate the vacuolar accumulation of tannins. In conclusion, there was a layer of multiply folded membranes between the tannin-accumulating vacuole and the cytoplasm in honeysuckle petal trichomes. There were also membrane assemblies in the cytoplasm a little far away from the vacuole, which were continuous with the proliﬁc membranes around the vacuole. 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Bot. 57, 801–810. 2018 International Conference on Social Sciences, Education and Management (SOCSEM 2018) A Case Study on Acquisition of the Sentence Final Particle “MA” in a Chinese-speaking Child Zhang Di 1, 2, Wu Xiangyan1, *, Han Yuying1 1 Cognitive Science and Linguistic Research Center, School of Foreign Language, Linyi University, Linyi, Shandong, China 2 Institute of Chinese Studies, University Tunku Abdul Rahman, Kuala Lumpur, Malaysia Keywords: Child Language; Particles; Cognitive Development; Corpus Abstract: The development of particle is an important part of children’s language acquisition. This study selected a child’s long-term tracing corpus (one year old to three and a half years old), and examined the development of the child’s sentence final particle (SFP) “MA” in detail. By analyzing the development of “MA” of all ages, the quantitative analysis data of the sentence final particle “MA” on the age group can be obtained, and the particle’s acquisition stage from the perspective of cognitive development is analyzed. 1. Introduction In recent years, the international community began to focus on the development of preschool children’s language. Mood is a kind of grammatical category that expresses the complex subjective feelings of human beings. The means of expressing the tone are intonation, sentence change, modal particle and other tone composition (modal adverb, auxiliary verb and interjection), and modal particle is the most basic form of Chinese tone. In the language of Chinese children, modal particles appear very early, and take mistakes rarely. Why are children so sensitive to the perception of modal particles? This is worthy of our in-depth study. However, the development of children’s modal particles has not been given enough attention. Furthermore, the development of the whole language system of children’s language has not been favored by scholars. This is not related to the fact that the ontology of modal speech has not developed. This study attempts to reveal the mechanism of children’s modal particle acquisition by referring to the research results of the predecessors in the process of children’s language development, hoping to promote the further development of children’s other modal particles the study. First, the paper of Li Yuming and Chen Qianrui’s (1998) which using the Chinese question system as a material, investigates the problem of language understanding and language development of Chinese children by using the method of case experiment and longitudinal observation. In the group experiment, the author used the “Xiao hong mao zhang de hao kan MA?” to examine the particles “MA”, the results show that different questionnaires have no effect on understanding in this age group. In the case study, the particle “MA” occurs at the age of 2 years. When children grow up to 2 years and 11 months, the negative form of “MA” question appears, and these questions are less doubtful, or even no interrogative sentence. Finally, this study concludes that a positive form of a non-questioning occurs before a negative form of a non-question. And in general, a sentence where a child has a question is asked before a sentence that is no doubt. Second, Kong Lingda, a professor of Anhui Normal University, his two graduate students Qian Yijun (2003) and Li Huimin (2005), respectively, choose the children’s language acquisition about particles as the title of graduation thesis. They used the case method sub-age population surveys and case follow-up survey which discuss the development of the children’s modal system. But their longitudinal research corpus is made up of two children’s corpora. In their study, “MA” occurs in 2 years and 6 months. They examined the object which the particles “MA” and “BA”’s tone attached, revealed the characteristics and processes of children how to learn the particles “MA”, and discussed some relevant issues. To sum up, we believe that in the case of vertical investigation, it Copyright © (2018) Francis Academic Press, UK 531 should be a continuous track of a child. This thesis studies the corpus which is a child’s continuous track in order to make up for the lack of corpus collection. “How can the particle ‘MA’ is learned by children early?” and “What the characteristics do the children process?” These questions are worthy of our further study. 2. The Usage of the SFP “MA” This research takes one year old boy as an object with the study video records in the natural state uninterrupted once a week until two and a half years. The aim is to collect a large number of corpora and investigate the acquisition of the children’s particles “MA” as well as analyze the development of the SFP“MA” of the children’s language acquisition by tracking this stage’s video survey. The corpus of this study is from a boy (a year and three months) who was born in Linyi city, China. In the video, the boy, his parents, his sister and his grandmother take turns to speak. Their utterances are transcribed into CHAT corpus and annotated according to CHILDES standards. The corpus is analyzed by CLAN program (XieNan & ZhangDi, 2017). During the 119 times of recording, the modal particle “MA” in the corpus has appeared 1512 times in all, and that was where the corpus of this study derived from. From some previous studies, mood expressing human beings’ complicated emotion belongs to grammatical category. In this study, we study the mood that can be divided into three different subcategories. It contains declaration mood, imperative mood, and interrogative mood. “MA” can be used in imperative, declaration and interrogative moods. (Li, 2005; Zhang, 2001). The use of “MA” can be divided as follows. “MA” is regarded as an imperative particle which can express advice, requests, orders, and urges. Examples are provided below. Rang wo kankan MA. Let me see MA. ‘Please let me see it.’ This sentence is an imperative sentence. “MA” in this sentence indicates that the speaker’s disagreement, possibly combined with indignation or impatience at the hearer’s opposite point of view and trying to persuade the hearer. To some extent, “MA” in this example expresses the meaning of the request. Chapppell (1991) claims that “MA” is to remind the listener that the entire proposition is obvious or self-evident from the preceding discussion or from their common cultural knowledge. An example is provided below. zhe bu shi tutu MA? This is not tutu MA? ‘Is this tutu?’ This sentence is a declaration sentence. “MA” in this sentence indicates that the speaker’s view combining with some doubts or the affirmation of the speaker. “MA” in interrogative mood often indicates that the speaker does not know the answer to a question and he/she wants the speaker to answer it. The example will be shown below. Guan shang men MA? Close the door MA? ‘Do you want to close the door?’ This sentence is an interrogative sentence. “MA” in this sentence indicates the speaker’s doubts. The speaker expects to get the answer from the questioner, and sometimes the usage of “MA” makes the sentence with containing the idea which the speaker wants to get some advice from the hearer. 3. The Findings of the SFP “MA” in the Corpus “MA” appeared for the age of 1; 10;25 in the corpus. The conversation will be provided in. 532 *MOT: hao le BA *CHI: Hao le. *CHI: Ke yi MA? OK BA OK LE. Can MA? ‘Are you OK?’ ‘OK.’ ‘Is this OK?’ In this context, the above sentence shows interrogative mood. The child wanted to wash his hands, then the child’s mother asked the boy whether he wanted to wash his hands and agreed the child to wash his hands. And after the child washed his hands, he asked his mother whether his hands was clean or dirty. In this sentence, the use of “MA” indicates that the child is not sure about the factuality. “MA” accounts for 10.49% of sentence final particles. There are 1848 sentences produced by WMX with the final ending “MA”. Among them, there are 1279 sentences expressing interrogative mood, 427 sentences expressing declarative mood and 142 sentences expressing imperative mood. The Figure 4 shown below illustrates the percentages of the three different moods with “MA”. 4. The Explanation for the Acquisition of the SFP“MA” Through the analysis of the SFP “MA” in the child’s language, we can see that the SFP “MA” frequency of the child is increased after 2;03 indicating that the child learns the SFP “MA” before 2;03. How does the child learn to use this particle? Children learn the language through a lot of methods. The development of children’s language is inseparable from adult’s language demonstration. In the early childhood, imitation is one of the important method which children use to learn a language. The process of learning SFPs have experienced from mechanical imitation to selective imitation. We have removed the direct imitation of the sentence in the time of statistical corpus, since this mechanical imitation is not the speech output of the child. Children learn the particle meaning by using of indirect, selective imitation mainly. They also will take some strategies when they learn to use particles. The first step is learning from the simple strategy. The same tone can be expressed by different grammatical forms, for example, the usage of “MA” in an interrogative sentence of judging right or wrong not only can express the interrogative, but also through the repeated questioning the structure of affirmative and negative to express. When a child acquires a language form, it inhibits the acquisition of other forms and delays the acquisition of another forms of expressing the tone. Second, there is a tail strategy. The tone is attached to the end of the sentence where is the most likely place to attract the attention of children. The usage of the end of the strategy is conducive to the acquisition of particles. At present, there are two theories about children’s language acquisition mechanism. Tomasello (2003) as the representative of the basis of the development of syntactic theory. The theory confirms that the development of children’s language is the process of the organic unity of biological germination, the process of social and cultural evolution and the individual process. It indicates the unity of language acquisition and language cognitive process confirming the general field of language acquisition. But this theory denies the existence of universal grammar, and holds that the development of language should be placed in the framework of cognitive and social competence. Chomsky (1965) as the representative of the congenital module theory. This theory establishes the language processing and accesses to the field of particularity. It holds the view that the language is a special instinct and different from other animals. People have a natural language knowledge which exists in the human brain in the form of universal grammar. We can not avoid the processing modularity of the language symbol system. We find that the rapid expansion of the frequency and usage of surge learning particle “MA” in children’s language only through point view of imitating and without pragmatic reasoning. Children’s acquisition of language must have a mechanism to support, but this usage of innate things in the end is the universal grammar and the general cognitive mechanism is also worth more detailed and in-depth study. Children’s particles appeared early and developed rapidly, indicating that the acquisition of children’s particles is mainly communicative function Driven, that is, pragmatic first (Halliday, 2007). At the same time, the role of cognitive development is indelible, so we think that children learn particles are the result of pragmatic and cognitive synthesis. According 533 to the children’s language acquisition process of pragmatic characteristics, social context and language input on the acquisition of children’s modal particles is particularly important. 5. Conclusions Children with their physiology get development, at the same time, mental development has also been important. They began to come out from the self-centered world, take the initiative to the outside world to express their physical needs and maintain their growth for food, clothing. Before children can express their will in the imperative sentence, or even in the pre-language stage, they use the eyes, gestures body language to express their demands. If children want to meet their own needs, only it is not enough by a potential gestures or the body language. They also need language to help them. Psychological needs ask for children become more and more intense in their growth process. When their own ideas can not get satisfaction, their minds will have doubts, and express their own questions in the language. So the emergence of the interrogative sentence is prompted and then the SFP“MA” expressing the interrogative tone comes into being. The present study investigates a Mandarin speaking child WMX’s acquisition of the SFP “MA” from 1;03 to 3;06. The main aim of the study is to discuss the availability of the SFP “MA” from semantic and syntactic perspectives. Based on WMX’s corpus, the present study finds that the first emergence of the SFP “MA” is at 1;10;25, and most of “MA” indicate interrogative mood. This thesis explains why WMX early acquires “MA” from semantic and syntactic perspectives. In semantic view, “MA” stands for different degrees, that is to say, “MA” marks low degree. When children acquire “MA”, it means children could use his words to express his doubt. We assume that there is an acquisition mechanism in children’s mind. When they are growing, they gradually acquire it. Acknowledgements This research is a periodic result of a Shandong Social Science Planning and Research Project (No. 16CZWJ27) and is sponsored by “the International Cooperation Training Projects of Excellent Key Teachers of Shandong Higher Education Institutions”. References [1] Chao, Y.R. (1968). A Grammar of Spoken Chinese. Berkeley: University of California Press. [2] Chappell, H. (1991). Strategies for the Assertion of Obviousness and Disagreement in Mandarin: a Semantic Study of the Modal Particle “ME”. Australian Journal of Linguistics. 3, 39-65. [3] Leung, W. (2008). Promising Approaching for the Analysis of Sentence-final Particles in Cantonese: the Case of [aa3]. Asian Social Science. 4, 74. [4] Bing, L. (2006). Chinese Final Particles and the Syntax of the Periphery. Yichang: LOT. [5] Xiaolu, Y. (2011). CP and the Left Periphery in Early Child Mandarin Chinese: The Case of Sentence Final Particles. Paper Presented at Glow in Asia VIII. [6] Levinson, S. (2001). Pragmatics. 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E-mail: wuxiangyan@lyu.edu.cn. 535 zwwx@overseaen.com http://www.overseaen.net 2019 年 10 月 Tel:+86-551-65690811 65690812 ISSN 1009-5039 Overseas English 海外英语 Overseas English 海外英语 Irrational Narration in The Black Cat ZHU Fei-ran (Linyi University, Linyi 276000, China) Abstract: Edgar Allan Poe, a writer of American Romantic period, is a forerunner in the use of irrational narration, especially in his dark tales, opening up a new approach for literature. Whether from narration perspective, plot or theme, The Black Cat, one of his masterpiece, is the typical manifestation of this narration. Key words: Edgar Allan Poe; the Black Cat; irrational narration 中图分类号：I106 文献标识码：A 文章编号：1009-5039(2019)19-0186-01 Edgar Allan Poe is honored as one of the important roles of American Romantic movement, leaving a large number of classic works. While most writers are advocating rational narration, Poe employs irrational narration, which brings a brand new reading ex⁃ perience and opens up a unique writing style of American writers. The Black Cat is the concentrated embodiment of this narration. When readers appreciate this novel from elementary to profound, the narration perspective is conspicuous, plots further, and eventu⁃ ally potential themes will be pondered. Hence the paper is sup⁃ posed to analyze irrational narration in the Black Cat from the three aspects above. Firstly, in this work, the author abandons traditional omni⁃ scient perspective and adopts the first person narration; therefore, “reliability is difficult to measure, primarily because of the immedi⁃ acy of the narration and because of the lack of other voices and in⁃ formation.”[1]187 The protagonist, described in the Black Cat, is the psychopath whose moral values are absolutely distorted, recounting the story of his abuse and cats' revenge. Through this internalvisual-angle, readers merely rely on the limited perspective of the narrator, impossible to recognize the true thoughts of other charac⁃ ters, resulting in strong subjectivity as well as one-sidedness. The identity of psychopath makes it difficult for him to narrate calmly and logically. What's more, Poe empowers the protagonist enough rights to speak out freely, not interfering to regulate his eccentric narration, which adds lots of irrational factors to the story. There is no doubt that readers find it hard to believe in this morbid, selfcentered narrator. Owing to subjectivity and unreliability, the first person narra⁃ tion is the most dramatic, making the story irrational. Secondly, "although the term of supernaturalism is not perfect⁃ ly established until the middle of the 19th century, Poe's tales in⁃ voke the supernatural they never exploit, or rather, Poe's tale are ir⁃ rational, concerning with perversity, monomania, and obsession re⁃ lated to an ego-directed mysticism. "[2]51 In the Black Cat, Poe ap⁃ plies supernatural factors and as a consequence, the plots present a non-linear causality. In western culture, black cats are inherently mysterious and always regarded as witches in disguise. They are symbols of death and evil which bring destruction through supernatural powers. Af⁃ ter the protagonist maltreated Pluto, an array of supernatural hap⁃ pened continually, ultimately resulting in his overthrow. On the night of the very day he hung Pluto, there was a confla⁃ gration in his residence. When everything had gone in smoke, a partition remained mysteriously, with a cat's relief on it and a noose around its neck. Later another black cat came from nowhere, rather similar to Pluto in appearance and temperament, except for the white mark on its breast which gradually formed the shape of gallows. At length, the cat enticed the protagonist to murder his wife and consigned him to the hangman with informing voice. In the light of biology, it is unlikely for a cat which, being entombed for four days, managed to survive and cried horribly; thus readers have to attribute it to the employing of supernaturalism. These fantastic and unrealistic plots, like karma, aggravate the cats' revenge, which entangles readers in a contradiction be⁃ tween believing and not believing, and adds mysterious fantasy of the novel. Eventually, Edgar Allan Poe expresses the theme of evil hu⁃ man nature from Freudian personality theory. Due to melancholy personality and keen intuition, Poe believes that evil is hidden in human beings so that everyone is prone to commit crime and selfdestruction. Unlike Hawthorne,“Poe deeply probes into the field of human unconsciousness to discover the evil part of human na⁃ ture.”[3]14Actually, the protagonist's abnormal behaviors aim to ful⁃ fill the primitive impulse of id.“According to Freud, the id is the instinctual, irrational and unconscious part of the personality”[3]16, which drives people to do evil to satisfy the dark side of human mind. Initially, the protagonist's consciousness is dominant so that he behaves himself normally. Later, he becomes an alcoholic which easily shows the tendency to break away from reason and retreat in⁃ to id. Due to the temptation of alcoholism, the evil hidden inside was completely exposed with the consequence that his disposition has undergone a dramatic alteration. The protagonist becomes (下转第207页) 收稿日期： 2019-06-11 修回日期： 2019-06-20 基金项目：校级重点，省级大学生创新创业项目“爱伦坡的非理性叙事与美国梦研究” 的阶段性成果，项目编号: 201910452015 作者简介：朱斐然(1997—) ，女，山东成武人，本科在读，研究方向为美国文学。 186 一一一一一一一一中外文学文化研究本栏目责任编辑：王力 2019 年 10 月 Overseas English 海外英语 interest have become increasingly prominent. Immersed in the so⁃ cial conditions, she comments“how deep might be the romance in the lives of some of those who elbowed me daily in the busy streets of the town in which I resided”(Elizabeth, 1992: 1). Deeply per⁃ ceiving the clash between the capitalists and workers, which is brought by the Industrial Revolution, she is gradually keeping a watchful eye on the lives of people in Manchester. Manchester, functioning as an epitome, plays a pivotal role in Gaskell’s process of writing and her depictions of cities and towns. To be precise, she creates the other distinguished place called Mil⁃ ton that is situated in the north of England in North and South. Her touch with the working class impels her to act on the appalling con⁃ ditions of those miserable workers, who are suffering a lot in their daily life in North and South. The seventh chapter delineates Man⁃ chester’s status quo from the perspective of Margaret. Urban air polluted by black smoke, trimly houses and streets, textile workers operating like machines, and discursive people in the streets consti⁃ tute a picture peculiar to this emerging industrial city. From the geographical view, Elizabeth Gaskell depicts experience of people in different places, and the ubiquitous influence of the Industrial Revolution on people in different stratums. (上接第186页) the horror of human heart. With the help of irrational narration, Poe's tales, with the shock from the soul, are distinctive and unpar⁃ alleled, far different from any of his contemporaries. He has been hailed as the“American's first authentic neurotic genius", becom⁃ ing one of the pioneers of genuine American literature and trailblaz⁃ ers of world literature. increasingly irritable and cannot control his behaviors. Things that used to be fine are now disgusting or even intolerable to him. As a result, he uncontrollably gouged out the cat's eye, then hung it, and ultimately hacked his wife to death. In addition, Poe deliberately sets the final episode in the gloomy cellar to inspire evil of the protagonist. The French philoso⁃ pher Gaston Bachelard mentioned in Poetics of Space that space is not a container for filling objects, but a habitation of human con⁃ sciousness. If the attic is a rational area, the cellar, an underground force, is the existence of human irrationality and unconsciousness, which induces inner desires and impulses. Just like Eve in the Garden of Eden stealing forbidden fruits, if an individual's id is totally irresistible, he or she will be dominat⁃ ed by irrational force, desperately eager to do criminal acts. In conclusion, Edger Allan Poe applies irrational narration in the Black Cat during perspective exploiting, plot-knitting and theme probing, which enables him to penetrate into the conscious⁃ ness and subconsciousness of human beings, thus readers aware of 本栏目责任编辑：王力 4 Conclusion Elizabeth Gaskell conducts a subtle writing of rustic village and bustling city in North and South, revealing impacts of different locations on shaping characters’psychological process and charac⁃ ters will also elevate the aesthetic arts of geography vice versa. Dif⁃ ferent places in North and South demonstrate Elizabeth Gaskell’s competence in manipulating geographical spaces and her sense of geographical space. References： [1] Gaskell, Elizabeth. North and South [M]. London: Penguin Books Ltd, 1994. [2] Gaskell, Elizabeth. Mary Barton [M]. London: Penguin Books Ltd, 1992. [通联编辑：王力] References： [1] Andrea Schwenke Wyile. Expanding the View of First-person Narration[J].Children’ s Literature in Education,1999,30(3):187. [2] 李雪琴 . 埃德加·爱伦·坡短篇中的哥特风格[D]. 四川：四川师范大学,2006:51. [3] 陈琛 . 从精神分析角度研究埃德加·爱伦·坡哥特式小说中的心理畸形人物[D]. 沈阳：沈阳师范大学 . [4] 唐文 . 权利·死亡·荒诞——对约瑟夫·海勒黑色幽默小说的阐释[M]. 上海：上海译文出版社,2016. [5] 加斯东·巴什拉 . 空间的诗学[M]. 上海：上海译文出版社, 2013:77-82. [通联编辑：王力] 一一一一一一一中外文学文化研究 207 53 54 教育心理微信对大学生人际交往的影响及应对措施赵慧慧（临沂大学摘要侯延绪庄冬文山东·临沂 276000）微信因其注册方便、功能强大、操作简单等特点，成为一款超过九亿人使用的手机应用。大学生作为微信最活跃的使用群体，微信以其独特的网络社交模式和传播格局影响着他们的人际交往方式。文章从积极和消极两个方面分析微信对大学生人际交往产生的影响，并就如何规避微信的消极影响提出应对措施。关键词微信人际交往影响中图分类号： G645 建议文献标识码： A DOI:10.16400/j.cnki.kjdkz.2018.10.079 Impact of WeChat on the Interpersonal Communication of College Students ZHAO Huihui, HOU Yanxu, ZHUANG Dongwen (Linyi University, Linyi, Shandong 276000) Abstract With its convenience of registration, powerful functions and simplicity in operation, WeChat has become a mobile application that more than 0.9 billion people are using. As the most active group of WeChat users, the WeChat affects their interpersonal communication with its unique network social mode and communication pattern. This article analyzes the influence of WeChat on College Students' interpersonal communication from positive and negative aspects, and puts forward some suggestions on how to avoid the negative influence of WeChat. Keywords WeChat; interpersonal communication; influence; advice 0 引言微信不仅因其便利性受到广泛欢迎，更因其功能性影响了人朋友间的双向了解。朋友圈中的点赞功能也拉近了朋友间的距离。另外，通过合理利用微信中漂流瓶、摇一摇、附近的人等们的社交方式。微信已不仅仅是一种通讯工具，更是一个将现实功能，大学生还可能会结识到不同职业、不同年龄、不同爱好甚世界与虚拟世界连接起来的平台。大学生热衷于接触新事物，追至不同国籍的微信好友，使大学生在维系原有交际圈的同时突赶潮流，微信的种种优势使其在该群体中备受青睐。微信已渗入破了地域限制，拓展了交际圈，从而弥补了现实生活中社交有大学生生活的方方面面，继而影响了他们的人际交往行为。 1 微信对大学生人际交往行为的积极影响限的问题，其人际交往对象也随之多元化。 1.3 综合提升了大学生的社交能力大学生人际交往行为的塑造离不开与他人的交往互动。 1.1 人际沟通更加便捷高效当代大学生面临着考试、技能提升、毕业、家庭经济、择业当大学生的现实人际关系得到进一步加强、网络社交渠道得以等多重压力，对沟通的便捷性需求较多，微信极大地迎合了这一点。在 WIFI 免费覆盖的高校校园内，大学生几乎能够随时进一步拓展时，他们能从每一段关系中感受到他人对自身的评价与态度，不断进行自我完善。借助于微信这个双向影响平随地使用微信与他人取得联系。而且，相比于传统的通信工台，大学生会逐渐理解和掌握一定的社会道德规范、价值观念具，如电话、短信等，微信不收取任何通信费用，只需在超出流等，并以此作为准则来不断调整自己的行为方式，从而使自己量套餐的情况下，交少量的流量费，大大降低了社交成本，使沟的社交能力得到综合性提升，人际交往行为逐渐散发出独特的通更加便捷高效。此外，微信语音对讲功能与传统的文字信息编入相互结人格魅力。 2 微信对大学生人际交往行为的消极影响合，大学生在不同的情境下可以选用不同的聊天方式，实现了 2.1 弱化了学生的现实人际交往人际交往“得体”化。与较为熟悉的同辈或家人用语音对讲功微信虽然为大学生提供了宽松自主的人际交往环境，但过能会显得更加亲昵，联系也随之变得更加密切。而直观的文字度依赖微信与他人交流则必然会减少现实人际交往。研究发显得更加有客观性和庄重感，更适合与不熟悉的长辈、教师进行沟通。现，大学生同学之间、师生之间、家人之间大都通过微信联系，这在一定程度上会减少他们现实中的人际交往行为。而且，微 1.2 拓展了大学生的交际圈信自身也带有一定的娱乐性，如小游戏，朋友圈、微信订阅号等，微信的互动性、匿名性完全迎合了现代大学生的猎奇心很多闲暇时间较多的大学生慢慢养成机不离手的坏习惯，甚至理，使他们的人际交往得到了前所未有的扩大与发展。大学生成为“手机控”、 “微信控”。尽管微信功能丰富，但难以复制现的微信好友大部分来源于现实生活中的人际关系，通过发送文字、图片、视频、朋友圈等功能，与朋友进行双向分享，加深了与实交流中的真实情感体验，久而久之，现实生活中的人际交往就会变得冷漠而疏远，学生的现实人际交往的能力也会减弱。 2018 年 / 第 29 期 / 10 月（中）165 51 教育心理 2.2 微信交友存在安全隐患随着微时代的到来，越来越多的大学生们，通过在朋友圈发辅导员应与学生勤交流，谨防学生上当受骗等恶性事件的发生。另外，辅导员还可以利用微信向学生传导社会正能量，宣送照片、分享定位、摇一摇、查找附近的人、允许陌生人看十张照扬正确的学习和交往方式，引导学生理性地讨论各种社会热点片等来拓展社交，却在不经意间泄露了自己的隐私，降低了个人问题；对于学生中自发形成的话题，要善于把握时机参与学生信息的安全性。在虚拟网络上，大学生群体自我保护意识较为的讨论中，深化与学生的互动交流引导大学生树立正确的社薄弱，易轻信他人，容易受到犯罪分子的引诱而上当受骗。 2.3 容易诱发学生的不良社交行为交认知。 3.3 开展丰富多彩的校园活动微信提供的是一个虚拟平台，一定程度上打破了传统的地丰富多彩的校园文化活动，特别是一些与学生专业相关的域、社会制度、意识形态的约束。大学生心智还未成熟，自身辨团体协作类的活动，可以丰富大学生的课余生活，加深学生对别是非的能力较弱，但探索欲较强，并乐于关注社会上的热点自己所学行业领域的感性认知，开阔眼界，提高他们的专业素问题。他们的世界观、人生观、价值观很容易受到网络信息的影响，从而诱发不良行为，如在微信上发布有关传销的信息，导养以及判断能力，并在现实活动的过程中锻炼他们的人际交往能力。通过积极参与活动，学生可以感受现实生活的美好，从致更多人受到不良影响。而减少花费在虚拟世界中的时间，找回真正的社交生活。总之，微信是一把双刃剑，它既能丰富大学生的沟通手段、 3 高校规避微信消极影响的措施微信给人们的日常生活提供便利的同时，也出现许多不同拓展人脉、提升其社交能力，也会造成诸如弱化现实人际交往、程度的问题。这种情况在接受能力强但辨识能力较弱的大学生群体中更加凸显，因此如何规避微信的消极影响，引导大学社交行为不当等问题。高校是大学生生活和学习的场所，理应有责任通过各种方式引导学生正确使用微信，做到趋利避害，生的健康成长成为高校无法忽视的问题。使之真正成为学生人际交往过程中的有利工具。 3.1 开展安全教育，提高防范意识课题项目：临沂大学大学生创新创业训练计划项目资助学校应大力开展有关网络安全的课程及相关的讲座等，提 “微信对大学生人际交往行为的影响研究——以临沂大学为例” 高大学生的自我防范意识和自我保护的能力。通过开设思想教育课程，教育学生用正确、理智的态度分析网络信息，培养学（项目编号： 201610452097）生的网络道德意识。另外，学校还可以利用微信平台开办学生参考文献一对一心理咨询服务，真正了解大学生心理发展和实际需要。 [1] 陈军.微信对大学生人际交往的影响[J].新闻世界.2015(01):59-60. [2] 李春霞.微信对大学生人际交往影响的调查——以粤西地区高校为例.长春: 大学学报,2017(4):47-50. [3] 徐斐斐.大学生微信用户人际交往的使用与满足研究[D].山东大学,2014. 通过微信公众号等形式，针对学生的特点和学校热点问题定期向学生推送各种信息，引导学生关注身边的事物，防止过度依赖微信。张琰, 孙亮.大学生微信社交对其现实人际交往的影响研究 [J]. 中小企业管理与科技(下旬刊),2015(9):183-184. [5] 乔木. 现代网络社交工具对大学生人际关系的影响及对策研究 [D]. 成都理 [4] 3.2 发挥辅导员的导向作用，指引学生树立正确的社交认知辅导员是大学生群体中的关键意见领袖，应发挥其导向作工大学,2012. 用，引导学生对一些错误或虚假的网络信息建立正确的认识。（上接第 145 页）同时“储蓄存款和商业银行”一课的结尾，高中政治教师在换而变幻无穷。因此，高中政治教师在熟练掌握其他常用的高中政治课堂结尾方法的基础上，再运用此新途径设计课堂结归纳总结活期储蓄和定期储蓄的异同后，让学生了解自己的家尾，定能很好地解决高中政治课堂结尾方法缺少多样性、教师庭财产是如何分配二者，还有没有比储蓄存款更好的投资理与学生间缺乏互动性的问题。财。如此设计，教师既使学生把课堂上新习得的知识实践于生活，懂得如何以投资者身份去选择理财，又为下节“股票、债券参考文献和保险”的学习做好铺垫。 [1] [2] 2 结语综上所述，在一节高中政治课堂结尾，教师只要开动脑筋，单一常见的归纳总结法也会因课堂结尾“四要素”间的彼此替 166 [3][5][7][9] 李冲锋.教学技能应用指导 [M].华东师范大学出版社,2007. [4][8] 郭芬云.课的导入与结束策略[M].北京师范大学出版社,2010. [6] 2018 年 / 第 29 期 / 10 月（中） 52 胡田庚.中学思想政治教学设计与案例研究[M].科学出版社,2012. 王彦才,郭翠菊.现代教师教学技能 [M].北京师范大学出版社,2010. 任志鸿.高中优秀教案政治必修 1 配人教版[M].南方出版社,2010. 42 43 44 第1期文化创新比较研究传媒文化微信对大学生人际交往行为的影响 —— —基于临沂大学在校生的个案研究赵登九，庄冬文（临沂大学，山东临沂 276000 ）摘要 : 移动互联、应用为王的时代，微信凭借其强大的用户基数当之无愧成为了最具影响力的社交应用之一。作为微信用户群体中重要的一部分，大学生在人际交往方面也受到了微信较大的影响。本文以临沂大学在校大学生为研究对象，通过线上问卷调查的形式，针对微信对在校大学生人际交往产生的影响进行研究和总结。关键词 : 微信；大学生；人际交往；影响中图分类号：G645 文献标识码：A 文章编号：2096-4110 （2018 ）01 （a ）-0077-03 新时代，以智能手机为载体的微信网络交际软件护意识等情况；第四部分是微信使用的性别差异问题。占据了我们大部分时间，为我们的生活带来了诸多便本次调查对象为临沂大学理工类和文史类大一到大四利，尤其是大学生一样的年轻人群利用微信交际软件学生，共发放问卷 568 份，回收有效问卷 556 份，有效更为突出。 2011 年以来，微信的应用功能不断创新，它问卷率为 97.89% 。已不是单纯的交际软件，同时还具备了支付、理财、交费等功能，它已逐步取代了短信、打电话。微信作为时 2 研究结果及其分析下最为流行的媒体，强烈吸引着大学生群体的目光，对 2.1 临沂大学在校生使用微信的基本情况他们的学习和生活也起到了日益重要的影响。本文将（1 ）调查样本的基本信息。通过调查问卷的形式对临沂大学学生使用微信进行人本次调查尽量做到大范围全覆盖，使调查结果具际交往的情况进行调查，进一步探索微信对当代大学有说服力。在临沂大学全校园进行了广泛采样，从学生生的人际交往产生的影响。的不同性别、专业、年级等方面进行采样调查，这样确保了调查样本的代表性。在 556 份有效样本中，男性占 51.7% ，女性占 48.3% ，男女比例基本相当，其中文史类 1 研究方法与样本选择学生占 60.3% ，理工类学生占 39.7% 。而在年级分布中，本研究采取抽样和网上发放调查问卷的方法，对以大二、大三学生为主，所占比例分别为 52.2% 和临沂大学在校学生使用微信及其人际交往的情况进行 20.2% ，大一和大四学生相对较小，所占比例分别为调查，并运用问卷星软件进行数据分析，从不同角度和 12.6% 和 15.0% 。层面研究微信对大学生人际交往行为的影响。（2 ）大学生使用微信的具体情况调查问卷分为四部分。第一部分为调查样本的基本信息，包括被调查学生的性别、年级和专业类别；第表1 二部分是关于大学生使用微信的基本情况，调查内容大学生使用微信的比例分布使用情况人数包括大学生使用微信的频率、最常使用的微信功能及比例（% ）未曾使用（终止访问） 14 2.5 使用微信联系最多的对象；第三部分主要调查微信对曾经使用，现在不用 86 15.5 大学生现实的人际交往产生的影响，主要包括通过微正在使用 456 82.0 信进行交往的方式，与陌生人互动的情况，个人隐私保 [基金项目 ] 临沂大学大学生创新创业训练计划项目资表2 —以临助 “微信对大学生人际交往行为的影响研究—— 沂大学为例 ”（项目编号：201610452097 ）。 [作者简介 ] 赵登九 (1996-) ，男，湖北十堰人，本科在读。庄冬文（1968- ），女，山东莒南人，硕士，副教授，研究方向：语言教学，语篇分析。大学生使用微信的频率使用频率人数比例（% ）基本不使用 19 3.4 偶尔使用 115 20.7 经常使用 279 50.2 几乎不离手 143 25.7 77 48 传媒文化文化创新比较研究第1期如表 1 、2 所示，八成以上（82% ）的大学生正在使如表 5 所示，大多学生认为微信对其人际交往具用微信，其中超过一半（50.2% ）的学生经常使用微信，有积极的影响。 49.6% 的学生认定微信使人际交往成本 1/4 以上（25.7% ）的大学生几乎不离手。由此可见，大学变低，47.5% 的同学认为微信使联系更迅速、交往更便生使用微信的人数与频率都较高。捷。认为沟通更加密切和结识新朋友，拓宽人脉的比例分别为 38.3% 和 22.7% ；只有 18.9% 的同学认为没有影表3 大学生最常使用的微信功能比例响，还有极少数同学（1.4% ）认为微信恶化或解除部分微信功能（限选三项）人数比例（% ）文字信息 364 65.5 语音对讲 403 72.5 图片及视频发送 242 43.6 原有熟人关系。（3 ）微信使用中个人隐私问题。朋友圈 333 59.9 附近的人 28 5.0 摇一摇 13 2.3 非常不好，完全没有隐私 5 0.9 漂流瓶 12 2.2 非常好，隐私完全不会被泄露 25 4.5 表6 大学生对于微信在维护个人隐私上的评价评价人数比例（% ）游戏中心 32 5.8 比较差，隐私只会被熟人看到 38 6.8 收发 QQ 离线消息、邮件 193 34.7 比较好，隐私只会被熟人看到 242 43.5 支付功能 210 37.8 一般，隐私有时会被泄露 276 49.6 其他 11 2.0 表4 表7 大学生使用微信联系最多的对象在隐私设置中开启的功能联系对象（限选三项）人数比例（% ）在隐私设置中开启的功能人数比例（% ）同学 263 47.3 允许陌生人查看十张照片 36 6.5 朋友 392 70.5 不看他（她）的照片 29 5.2 家人亲属 426 76.7 通讯录黑名单 51 9.2 老师 296 53.2 朋友圈黑名单 70 12.6 陌生人 23 4.1 可通过微信号搜索到我 148 26.6 其他 19 3.4 可通过手机号搜索到我 231 41.6 可通过 QQ 号搜索到我 249 44.8 向我推荐通讯录朋友 268 48.2 功能是：语音对讲、文字信息和朋友圈，所占比例分别向我推荐 QQ 好友 305 54.9 为 72.5% 、65.5% 和 59.9% 。大学生使用微信联系最多的加我为朋友时需验证 421 75.8 系统默认，自己没有设置 64 11.5 表 3 、表 4 显示，大学生在微信中最常使用的 3 个为家人亲属、朋友和老师，所占比例分别为 76.7% 、 70.5% 和 53.2% ，很少有人（4.1% ）与陌生人联系。表 6 、表 7 显示，49.6% 的大学生认为微信保密功 2.2 使用微信对临沂大学学生的人际交往的影响能一般；43.5% 的学生认为隐私会被熟人看到；还有（1 ）大学生使用微信交往的方式。 0.9%的学生认为非常不好，完全没有隐私。因此，在微信调查显示，大学生主要通过文字聊天、语音聊天、隐私设置功能中，75.8% 的大学生设置加我为朋友时需验证。这说明大多学生具有一定的个人隐私保护意识。朋友圈、群聊、视频聊天 5 种方式进行交往。其中，文字 2.3 大学生微信使用中的性别差异聊天、语音聊天及朋友圈是主要的交往方式，所占比例（1 ）微信功能使用上的差异。分别为 82.2% ，63.3% 及 44.8% 。群聊和视频聊天也占有相当的比例，分别为 36.9% 和 22.3% 。表8 （2 ）微信对学生人际交往的影响。表5 人数微信功能男女文字信息 49.2 50.5 语音视频通信 81.9 83.4 比例（% ）朋友圈 49.6 69.8 37.8 15.2 微信对大学生个体人际交往的影响影响不同性别最常使用的微信功能（% ）恶化或解除部分原有熟人关系 8 1.4 附近的人没有作用 105 18.9 摇一摇 40.2 19.8 漂流瓶 30.5 10.1 结识新朋友，拓宽人脉 126 22.7 沟通更加亲密 213 38.3 游戏中心 27.6 9.8 47.1 52.7 58.2 69.6 联系更迅速，交往更便捷 264 47.5 收发 QQ 离线消息、邮件人际交往成本变低 276 49.6 支付功能 78 49 第1期文化创新比较研究传媒文化如表 8 所示，在微信功能的使用上，女生比男生更第二，当前，大学生的人际交往受微信影响十分深倾向于使用文字信息、语音视频通信、朋友圈，收发 QQ 远。在大学生自己看来，微信对他们的影响总体是良好离线消息与邮件、微信支付等功能，而男生则更倾向于的，微信增加了他们与认识的身边人的联系，强化了已使用附近的人、摇一摇、漂流瓶、游戏中心等功能。有的社交网络。通过使用微信，提高了大学生的交际效率，（2 ）微信好友的构成及与陌生人的交往。更加节省的交际成本，进一步促进了朋友之间的感情。第三，微信使用中的性别差异问题。微信使用中存表9 不同性别微信好友的构成（% ）在性别差异，男生和女生在最常使用的微信功能、微信好友构成男女亲密的朋友或伴侣 88.3 94.2 父母或其他亲人 46.5 58.2 普通同学 82.2 82.6 此次调查研究尚存在诸多不足，一是调查范围局老师 11.5 13.5 限。只对临沂大学 568 名在校生作为调查样本，而且采通过微信功能结交到的新朋友 12.3 7.2 用的是线上问卷调查的方式，样本虽有一定的代表性，好友的来源、构成及与陌生人的互动等方面都存在不同程度的差异。但仍有所欠缺。二是调查时间局限。调查本应运用访谈如表 9 所示，亲密的朋友或伴侣、父母或其他亲法进行个案的研究，对个案进行深入研究才能进一步人、普通朋友及老师在女性微信好友构成中的比例均促进论题的完整和科学，因此，本文的结果和结论尚存高于男性；而在男性好友构成中，通过微信功能结交到在一定的局限性，尚待加强。在今后的课题研究中，笔的新朋友的比例明显高于女性，分别为 12.3% 和 7.2% 。者将创新研究方法和手段，以期有效全面地对有关问这反映出女性具有较高的隐私保护及自我保护意识。题进行深入剖析和探讨，得到更为理想结论。对“不同性别的大学生是否在微信中添加过陌生人”这一问题的调查同样证明了这一点。在被调查对象中，有参考文献 56.6% 的男性，39.8% 的女性选择“是”。在微信中添加陌 [1] 郭燕荣 , 麻文斌 . 试析微信对于大学生人际交往的生人的比例男性明显高于女性，而没有添加过陌生人影响 [J]. 社科纵横 ,2014,29(6):110-115. 的女性比例明显高于男性，分别为 60.2% 和 43.4% 。 [2] Lin, C. A . Looking back ：The Contribution of Blumler and Katz ’s Uses and Mass Communication to 3 结语 Communication Research [J]. Journal of Broadcasting 微信改变和影响着大学生的交往方式和习惯。本＆ Electronic Media, 1996(40) ：574-581. 次研究主要以临沂大学在校生为研究对象，通过实证 [3] 聂磊，傅翠晓，程丹 . 微信朋友圈：社会网络视角下调查以及分析，本研究得出了如下结论：的虚拟社区 [J]. 新闻记者，2013 （5 ）: 71-75. 第一，当前，大学生使用微信的情况。微信已成为 [4] Bonebrake, K. College Students ’ Internet Use, Rela- 大学生生活的重要组成部分，并且使用频率较高；他们 tionship Formation, and Personality Correlates [J].Cy- 主要通过语音对讲、文字信息、朋友圈等微信功能进行 ber Psychology & Behavior, 2002(6): 48-55. 人际交往；交往的人群以家人、亲属、朋友和老师为主， —论人的延伸 [5] （加）马歇尔·麦克卢汉 . 理解媒介—— 对于陌生人的联系则相对比较少。 [M]. 北京：商务印书馆，2000. 参考文献参考文献是在学术研究过程中，对某一著作或论文的整体的参考或借鉴。征引过的文献在注释中已注明，不再出现于文后参考文献中。按照字面的意思，参考文献是文章或著作等写作过程中参考过的文献。然而，按照 GB/T7714-2015 《信息与文献参考文献著录规则》”的定义，文后参考文献是指：“为撰写或编辑论文和著作而引用的有关文献信息资源。根据《中国学术期刊（光盘版）检索与评价数据规范（试行）》和《中国高等学校社会科学学报编排规范（修订版）》的要求，很多刊物对参考文献和注释作出区分，将注释规定为“对正文中某一内容作进一步解释或补充说明的文字”，列于文末并与参考文献分列或置于当页脚地。 79 50 45 46 47