THE FORGOTTEN TOPOLOGICAL INDEX OF SOME DRUG STRUCTURES

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1 Acta Medica Mediterranea, 2016, 32: 579 THE FORGOTTEN TOPOLOGICAL INDEX OF SOME DRUG STRUCTURES WEI GAO 1*, MOHAMMAD REZA FARAHANI 2, LI SHI 3 1 School of Information Science and Technology, Yunnan Normal University, Kunming , China - 2 Department of Applied Mathematics, Iran University of Science and Technology, Tehran 16844, Iran - 3 Institute of Medical Biology, Chinese Academy of Medical Sciences and Peking Union Medical College, Kunming , China ABSTRACT A large number of previous drug experiments revealed that there are strong inherent connections between the drug s molecular structures and the bio-medical and pharmacology characteristics. The forgotten topological index is introduced to be applied into chemical compound and drug molecular structures, which is quite helpful for medical and pharmaceutical workers to test the biological features of new drugs. Such approaches are widely applied in developing areas which are not rich enough to afford the relevant equipment and chemical reagents. In this paper, using drug structure analysis and edge dividing technique, we determine the forgotten topological index of some molecular structures which appear widely in various classes of drugs. Key words: Theoretical medical, forgotten topological index, anti-cancer drug, polyphenylene dendrimer, Benzenoid System. Received February 15, 2016; Accepted March 02, 2016 Introduction In the rapid development of pharmacology, a large number of new drugs have been introduced each year. A lot of work is required to test the physical, chemical and biological properties of these new drugs, and this workload has become more and more cumbersome. Adequate equipment, reagents and human resources are needed to test the performance and the side effects of drugs. However, in poor, backward and lower income countries (like some countries in Southeast Asia, South America and Africa), there are insufficient funds to afford equipment and reagents which can be used to test the biochemical properties. Fortunately, previous research has revealed that chemical characteristics of drugs and their molecular structures are closely related. If we learn the indicators of these drugs according to the topological indices, the medical and pharmaceutical scientist can find it helpful to understand their medicinal properties, which influence the experimental defects. From this point of view, the technique of topological index computation is very suitable for developing countries in which they can obtain the medical and biological information of new drugs without chemical experiment conditions. In the medicine computation model, the structure of a drug is expressed as a graph, where each vertex expresses an atom and each edge represents a chemical bond between these atoms. Let G be a graph corresponding to a drug structure with atom (vertex) set V(G) and chemical bond (edge) set E(G). A topological index can be considered as a real-valued map ƒ: G + which maps each drug structure to certain real numbers. In the past decade, researchers introduced several indices, such as PI index, Wiener index, harmonic index and Zagreb index, to test the features of drug molecules. There were some articles to calculate these topological indices of special molecular graphs in chemical and pharmaceutical engineering (See Yan et al., (1-2),

2 580 Wei Gao, Mohammad Reza Farahani et Al Farahani et al., (3), Farahani and Gao (4), Gao and Shi (5), Gao and Wang (6-9), and Gao and Farahani [10] for more detail). The terminologies and notations used but not clearly explained in our paper can be found in Bondy and Mutry (11). Followed by the first and second Zagreb indices, Furtula and Gutman (12) defined the forgotten topological index (or, F-index) which is stated as where d(v) is the degree of vertex v in G (the number of vertex adjacent to vertex v) for any In Furtula and Gutman (12), the predictive ability of the forgotten topological index is similar to that of the first Zagreb index and for the entropy and acentric factor, and both of them yield correlation coefficients greater than De et al., (13) presented some basic properties of the forgotten topological index and showed that this index can enhance the physical-chemical applicability of the Zagreb index. Although there have been several advances in degree-based and distance-based indices of molecular graphs, the studies of the forgotten topological index for certain special drug structures are still largely limited. On the other hand, as critical and widespread drug structures, Dox-loaded micelle comprising PEG-PAsp block copolymer with chemically conjugated Dox SP[n], polyphenylene dendrimers D 2 [n] and D 4 [n], benzenoid systems and phenylenes, and polycyclic aromatic hydrocarbons are widely used in medical science and frequently appeared in drug structures. For these reasons, we give the exact expressions of forgotten topological index for these important drug structures. The rest of the paper is arranged as follows: first, we determine the forgotten topological index of SP[n]; then we consider the polyphenylene dendrimers D 2 [n] and D 4 [n] and obtain the corresponding results; third, we focus on the benzenoid systems and phenylenes and their forgotten topological indices are yielded; at last, we present the forgotten topological index of polycyclic aromatic hydrocarbons PAH n. Forgotten Topological Index of SP[n] Smart polymers, as a particular kind of macromolecules, present an impressive response to physiochemical transformation if their surroundings have been slightly disturbed, such as the changes in temperature, light, magnetic field, ionic disturbance, ph value, and so on (see Kroning (14), Marques et al., (15), and Nguyen and Broekmann (16) ). Hence, smart polymers can be regarded as stimuli responsive ones and environmentally responsive systems. These drug structures, as high-grade delivery systems, have been widely applied in biomedical fields, for example smart polymers with protein or nucleic acid delivery to intracellular targets similar to ribosome or nucleus in tissue engineering (see Duro-Castano (17), Chonkar et al., (18), Khandare and Calderon (19) and Hruby et al., (20) ). Polymeric micelles (for instance, Dox-conjugated PEG-b-poly (aspartate) (PEG PAsp) block copolymers, see Osada et al., (21) and Shanthi et al., (22) ), considered as a special class of smart polymer, are widely employed in delivering anti-cancer drugs. The Dox-loaded micelle containing PEGPAsp block copolymer with chemically conjugated Dox (its detail structure can be found in Figure 1) is a well-known family of smart polymer which can be efficacious as anthracycline anti-cancer antibiotic and utilized in a large classes of cancers. It is used as drug delivery carriers and excipients for enhancing the stability and time of drug retention. From this point of view, it has intensive anti-cancer activity and thus widely considered in pharmacology (for more details see Butt et al., (23) and Nishiyama and Kataoka (24) ). Figure 1: Dox-loaded micelle comprising PEG-PAsp block copolymer with chemically conjugated Dox SP[n]. Let n be the step of growth in this kind of polymer (see Figure 1). The structure of Doxloaded micelle containing PEGPAsp block copolymer with chemically conjugated Dox with n = 1, 2 and 3 are presented in Fig. 2, Fig. 3 and Fig. 4, respectively.

3 The forgotten topological index of some drug structures 581 Figure 2: The molecular structure of SP[1]. with chemically conjugated Dox. We obtain F(SP[n] = 744n+34. Proof. In terms of observing and computing, the edge set of (SP[n] can be divided into the following edge sub-sets. E 12 : d(u) =1, d(v)= 2; E 13 : d(u) =1, d(v)= 3; E 14 : d(u) =1, d(v)= 4; E 22 : d(u) =2, d(v)= 2; E 23 : d(u) =2, d(v)= 3; E 24 : d(u) =2, d(v)= 4; E 33 : d(u) =3, d(v)= 3; E 34 : d(u) =3, d(v)= 4; By means of further calculating, we infer E 12 =2n+1, E 13 =9n+1, E 14 =n, E 22 =5n+4, E 23 =18n- 1, E 24 =2n, E 33 =16n, E 44 =n. Therefore, according to the definition of forgotten topological index, we deduce Hence, we obtain the expected result. Figure 3: The molecular structure of SP[2]. Let (G) and (G) be the minimum and maximum degree of G. We divide edge set E(G) and vertex set V (G) into several partitions: for any i and j, (G) i,j (G), let E ij = {e=uv E(G) d(v) =i, d(u) =j}. The purpose of this section is to manifest the forgotten topological index of Dox-loaded micelle comprising PEG-PAsp block copolymer with chemically conjugated Dox. Theorem 1. Let SP[n] be the Dox-loaded micelle comprising PEG-PAsp block copolymer Forgotten Topological Index of Polyphenylene Dendrimers The purpose of this section is to present the forgotten topological index of polyphenylene dendrimers D 4 [n] and D 2 [n] with n which have frequently appeared in the structure of drugs. The kernel molecular part of D 4 [n] and D 2 [n] can be found in Figure 5. Figure 5: The kernel of D 4 [n] and D 2 [n], respectively.

4 582 Wei Gao, Mohammad Reza Farahani et Al Figure 4: The molecular structure of SP[3]. Furthermore, Figure 6 presents the D 2 [n] with three growth stages. Theorem 3. Theorem 2. F(D 2 [n]) = n Proof. In view of observation of polyphenylene dendrimers D 2 [n], we deduce that its edge set can be divided into three partitions: E 23 : d(u) =2 and d(v) = 3. Moreover, we deduce E 22 =56 2 n -48, E 23 =48 2 n -44, and E 36 = 36 2 n -35. Thus, using the definition of forgotten topological index, we infer Figure 6: Polyphenylene dendrimers D 2 [n] with three growth stages. The forgotten topological index of these drug structures are manifested in Theorem 2 and Theorem 3. F(D 4 [n]) = n Proof. In terms of observation of polyphenylene dendrimers D 4 [n], we get four partitions of its edge set:

5 The forgotten topological index of some drug structures 583 E 23 : d(u) =3 and d(v) = 3; E 34 : d(u) =3 and d(v) = 4. Additionally, we yield =, =, =, and =4. Then the result follows from the definitions of the forgotten topological index. Forgotten Topological Index of Benzenoid Systems and Phenylenes In this section we consider the forgotten topological index of benzenoid systems and phenylenes (for the structure see Rada et al., (25), Cyvin and Brunvoll (26), Pavlovic and Gutman (27), Yousefi-Azari et al., (28), and Xiao et al., (29) ). Note that coves, fjords, fissures, bays and lagoons are basic structural characteristics of the perimeter of the benzenoid systems which are seen in Figure 7. Figure 7: Classes of inlets occurring on the perimeter of a benzenoid system. We set S as a benzenoid system with n vertices, h hexagons and r inlets. Theorem 4 F(S)=8n+38h+38. Proof. By means of structure analysis of benzenoid system S, we get three partitions of its edge set: E 23 : d(u) =2 and d(v) = 3. Additionally, we yield E 22 =n-2h-r+2, E 23 = 2h and E 33 =3h-r-3. Hence, from the definitions of forgotten topological index, we yield For phenylene drug structure, we set PH as a phenylene with h hexagons and r inlets. The detailed structure is seen in Figure 8. Figure 8: A phenylene (PH) and its hexagonal squeeze (HS). Theorem 5. =. Proof. By observation of phenylene drug structure PH, we found that its edge set can be divided into the following three parts: E 23 : d(u) =2 and d(v) = 3. Additionally, we yield E 22 =2h-r+4, E 23 =2r and E 33 =6h-r-6. Then the result follows from the definitions of the forgotten topological index. Forgotten topological index of polycyclic aromatic hydrocarbons Polycyclic aromatic hydrocarbons PAH n is a family of hydrocarbon molecules which consists of cycles with length six (Benzene) and can be considered as small pieces of graphene sheets with the free valences of the dangling bonds saturated by H. Vice versa, a graphene sheet can be interpreted as an infinite PAH molecule. In this section, the first members of this hydrocarbon family are shown in Figure 9: let PAH 1 be the Benzene with six carbon (C) and six hydrogen (H) atoms, PAH 2 the Coronene with 24 carbon and twelve hydrogen atoms and PAH 3 the Circumcoronene with 54 carbon and eighteen hydrogen atoms. Figure 9: The structure of PAH n

6 584 Wei Gao, Mohammad Reza Farahani et Al By the analysis given by Farahani (30), we know that the edge set of PAH n can be divided into two parts: E 13 and E 33 such that E 13 =6n and E 33 =9n 2-3n. Our last result is obtained directly using the definition of the forgotten topological index. Theorem 6. F(PAH n )=162n 2 +6n. Conclusion Each year, a large number of new diseases are found due to the contentiously evolving viruses. This requires us to develop more new drugs to treat new diseases. Topological indices are introduced to test the medicinal properties of new drugs which is widely welcomed in developing areas. In this paper, by means of drug structure analysis and edge dividing technology, we presented the forgotten topological index of several drug structures. The result achieved in our paper illustrates the promising prospects of the application for pharmacological engineering. References 1) L. Yan, J. S. Li, W. Gao, Vertex PI index and Szeged index of certain special molecular graphs, The Open Biotechnology Journal, 2014, 8, ) L. Yan, W. Gao, J. S. Li, General harmonic index and general sum connectivity index of polyomino chains and nanotubes, Journal of Computational and Theoretical Nanoscience, 2015, 12(10), ) M. R. Farahani, W. Gao, M. R. R. Kanna, Indian Journal of Fundamental and Applied Life Sciences, 2015, 5 (S4), ) M. R. Farahani, W. Gao, The multiply version of Zagreb indices of a family of molecular graph "polycyclic aromatic hydrocarbons (PAHS)", Journal of Chemical and Pharmaceutical Research, 2015, 7(10), ) W. Gao, L. Shi, Szeged related indices of unilateral polyomino chain and unilateral hexagonal chain, IAENG International Journal of Applied Mathematics, 2015, 45(2), ) W. Gao, W. F. Wang, Second atom-bond connectivity index of special chemical molecular structures, Journal of Chemistry, Volume 2014, Article ID , 8 pages, 7) W. Gao, W. F. Wang, The vertex version of weighted wiener number for bicyclic molecular structures, Computational and Mathematical Methods in Medicine, Volume 2015, Article ID , 10 pages, 8) W. Gao, W. F. Wang, The eccentric connectivity polynomial of two classes of nanotubes, Chaos, Solitons & Fractals, In press, doi: /j.chaos ) W. Gao, W. F. Wang, Revised Szeged index and revised edge-szeged index of special chemical molecular structures, Journal of Interdisciplinary Mathematics, In press. 10) W. Gao, M. R. Farahani, Degree-based indices computation for special chemical molecular structures using edge dividing method, Applied Mathematics and Nonlinear Sciences, 2016, 1(1), ) J. A. Bondy, U. S. R. Mutry, Graph Theory, Spring, Berlin, ) B. Furtula, I. Gutman, A forgotten topological index, Journal of Mathematical Chemistry, 2015, 53(4), ) N. De, SK. MD. Abu Nayeem, A. Pal, F-index of some graph operations, manuscript. 14) A. Kroning, A. Furchner, D. Aulich, E. Bittrich, S. Rauch, P. Uhlmann, K. J. Eichhorn, M. Seeber, I. Luzinov, S. M. Kilbey, B. S. Lokitz, S.Minko, K. Hinrichs, In situ infrared ellipsometry for protein adsorption studies on ultrathin smart polymer brushes in aqueous environment, ACS Applied Materials& Interfaces, 2015, 7(23), ) N. N. Marques, A. M. da Silva Maia, R. C. Balaban, Development of dual-sensitive smart polymers by grafting chitosan with poly (N-isopropylacrylamide): an overview, Polimeros-Ciencia E Tecnologia, 2015, 25(3), ) T. M. H. Nguyen, P. Broekmann, Peter Smart hybrid polymers for advanced damascene electroplating: combination of superfill and leveling properties, Chemelectrochem, 2015, 2(8), ) A. Duro-Castano, J. Movellan, M. J. Vicent, Smart branched polymer drug conjugates as nanosized drug delivery systems, Biomaterials Science, 2015, 3(10), ) A. Chonkar, U. Nayak, N. Udupa, Smart polymers in nasal drug delivery, Indian Journal of Pharmaceutical Sciences, 2015: 77(4), ) J. Khandare, M. Calderon, Dendritic polymers for smart drug delivery applications, Nanoscale, 2015, 7(9), ) M. Hruby, S. K. Filippov, P. Stepanek, Smart polymers in drug delivery systems on crossroads: which way deserves following, European Polymer Journal, 2015, 21) K. Osada, R. J. Christie, K. Kataoka, Polymeric micelles from poly(ethylene glycol)- poly(amino acid) block copolymer for drug and gene delivery, Journal of The Royal Society Interface, 2009, 6, ) K. Shanthi, K. Vimala, D. Gopi, S. Kannan, Fabrication of a ph responsive DOX conjugated PEGylated palladium nanoparticle mediated drug delivery system: an in vitro and in vivo evaluation, RSC Advances, 2015, 5(56), ) A. M. Butt, A. M. C. Mohd, H. Katas, Synergistic effect of ph-responsive folate-functionalized poloxamer 407- TPGS-mixed micelles on targeted delivery of anticancer drugs, International Journal of Nanomedicine, 2015, 10, ) N. Nishiyama, K. Kataoka, Polymeric micelle drug carrier systems: PEG-PAsp(Dox) and second generation of micellar drugs, Polymer Drugs in the Clinical Stage, 2003, Volume 519 of the series Advances in Experimental Medicine and Biology,

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