Novel Indices for the Topological Complexity of Molecules

Abstract

The development of molecular complexity measures is reviewed. Two novel sets of indices termed topological complexities are introduced proceeding from the idea that topological complexity increases with the overall connectivity of the molecular graph. The latter is assessed as the connectivity of all connected subgraphs in the molecular graph, including the graph itself. First-order, second-order, third-order, etc., topological complexities ⁱ TC are defined as the sum of the vertex degrees in the connected subgraphs with one, two, three, etc., edges, respectively. Zero-order complexity is also specified for the simplest subgraphs–the graph vertices. The overall topological complexity TC is then defined as the sum of the complexities of all orders. These new indices mirror the increase in complexity with the increase in the number of atoms and, at a constant number of atoms, with the increase in molecular branching and cyclicity. Topological complexities compare favorably to molecular connectivities of Kier and Hall, as demonstrated in detail for the classical QSPR test-the boiling points of alkanes. Related to the wide application of molecular connectivities to QSAR studies, a similar importance of the new indices is anticipated.     

The overall Wiener index--a new tool for characterization of molecular topology.

Recently, the concept of overall connectivity of a graph G, TC(G), was introduced as the sum of vertex degrees of all subgraphs of G. The approach of more detailed characterization of molecular topology by accounting for all substructures is extended here to the concept of overall distance OW(G) of a graph G, defined as the sum of distances in all subgraphs of G, as well as the sum of eth-order terms, (e)OW(G), with e being the number of edges in the subgraph. Analytical expressions are presented for OW(G) of several basic classes of graphs. The overall distance is analyzed as a measure of topological complexity in acyclic and cyclic structures. The potential usefulness of the components of this generalized Wiener index in QSPR/QSAR is evaluated by its correlation with a number of properties of C3-C8 alkanes and by a favorable comparison with models based on molecular connectivity indices.     

取代脂环烃物理化学性质的拓扑化学研究

为了探讨有机化合物的分子结构与理化性质的关系,作者在以前的工作基础上,基于一阶分子连通性拓扑指数构建了一个信息拓扑指数.将拓扑指数与取代脂环烃系列分子的气态标准生成热、气态标准熵、气态标准生成自由能、沸点、临界温度、临界压力、临界体积、汽化热、密度、热容及表面张力等十一种热力学性质及物理化学性质相关联,用一个通用公式对各类性质进行概括,然后用已有的实验数据与拓扑指数进行回归分析,得到一系列计算各性质的经验公式.回归结果发现,分子的理化性质与拓扑指数有较好的相关性,有5类性质的复相关系数大于0.99.用经验公式对各类性质进行重新计算,其结果与实验值比较符合,用实验值对计算值作图,发现各数据点紧靠对角线,说明误差较小.残差分布呈正态分布.     

Structural interpretation of the topological index. 2. The molecular connectivity index, the Kappa index, and the atom-type E-State index

The structural interpretation is extended to the topological indices describing cyclic structures. Three representatives of the topological index, such as the molecular connectivity index, the Kappa index, and the atom-type E-State index, are interpreted by mining out, through projection pursuit combining with a number theory method generating uniformly distributed directions on unit sphere, the structural features hidden in the spaces spanned by the three series of indices individually. Some interesting results, which can hardly be found by individual index, are obtained from the multidimensional spaces by several topological indices. The results support quantitatively the former studies on the topological indices, and some new insights are obtained during the analysis. The combinations of several molecular connectivity indices describe mainly three general categories of molecular structure information, which include degree of branching, size, and degree of cyclicity. The cyclicity can also be coded by the combination of chi cluster and path/cluster indices. The Kappa shape indices encode, in combination, significant information on size, the degree of cyclicity, and the degree of centralization/separation in branching. The size, branch number, and cyclicity information has also been mined out to interpret atom-type E-State indices. The structural feature such as the number of quaternary atoms is searched out to be an important factor. The results indicate that the collinearity might be a serious problem in the applications of the topological indices.     

Graph-based molecular alignment (GMA)

We describe a combined 2D/3D approach for the superposition of flexible chemical structures, which is based on recent progress in the efficient identification of common subgraphs and a gradient-based torsion space optimization algorithm. The simplicity of the approach is reflected in its generality and computational efficiency: the suggested approach neither requires precalculated statistics on the conformations of the molecules nor does it make simplifying assumptions on the topology of the molecules being compared. Furthermore, graph-based molecular alignment produces alignments that are consistent with the chemistry of the molecules as well as their general structure, as it depends on both the local connectivities between atoms and the overall topology of the molecules. We validate this approach on benchmark sets taken from the literature and show that it leads to good results compared to computationally and algorithmically more involved methods. The results suggest that, for most practical purposes, graph-based molecular alignment is a viable alternative to molecular field alignment with respect to structural superposition and leads to structures of comparable quality in a fraction of the time.     

拓扑指数与手性化合物构效关系研究

传统的拓扑指数仅考虑原子间的连接关系,不能区分手性信息．故本研究对Am指数与分子连接性指数进行了扩展,并将其用于3-苯基哌啶类手性化合物的构效关系研究．扩展后的拓扑指数构造的数学模型优于用传统的Am指数与分子连接性指数得到的模型．同时进行了变量的两两相关性测试,结果显示所选3个变量之间的相关性较小．通过交叉验证,证明手性拓扑指数所得到的数学模型是稳定的．并进一步运用人工神经网络,得到了满意的结果．     

Correlations between topological features and physicochemical properties of molecules

The article reviews in brief, thede novo group additivity approach and, at length, the different topological approaches to obtain predictive and internally consistent correlations between various properties and structural features of molecules. The stress has mainly been on the molecular connectivity method. A new rational scheme for nomenclature of connectivity indices of different orders and types is introduced. The concept of the perturbation connectivity parameter developed by us recently has been applied to obtain correlations for molar refraction, boiling point, molar volume, heat of atomisation, heat of combustion, heat of vaporisation, magnetic susceptibility and critical constants of alkanes, alcohols and alkylbenzenes. A comparative study of various approaches reveals that the present perturbation topological approach has an edge over other similar methods.     

Optimal characterization of structure for prediction of properties

Different topological and physicochemical parameters have been used to predict hydrophobicity (logP, octanol-water) of chemicals. We calculated a hydrogen bonding parameter (HB₁) and a large number of molecular connectivity and complexity indices for a diverse set of 382 molecules. It is known from earlier studies that topological indices (TIs) predict properties of congeneric sets reasonably well. Since HB₁ is an approximate quantifier of hydrogen bonding and has integral values, we used HB₁ to classify the diverse set into strongly and weakly hydrogen bonding subsets. In an attempt to examine the utility of Us in predicting properties of relatively similar groups of molecules, we carried out a correlation of logP with TIs for a subset (n = 139) of the original diverse set (n = 382) with a weak hydrogen bonding ability (HB₁ = 0). Results show that TIs give a better predictive model for the more homogeneous subset as compared to the diverse set of molecules.     

Unique description of chemical structures based on hierarchically ordered extended connectivities (HOC procedures). I. Algorithms for finding graph orbits and canonical numbering of atoms

An iterative algorithm is described for finding topological equivalence, ordering, and canonical numbering of vertexes (atoms) in molecular graphs. Like the Morgan algorithm, it is based on extended connectivities but: (i) the latter are used hierarchically, i. e., the discrimination in the next iteration is carried out only for the vertices having the same extended connectivities (ranks) at the previous iteration; (ii) at equal extended connectivities, additional discrimination is introduced by the ranks of adjacent vertices; (iii) there is no “best name” search; (iv) three levels of complexity of chemical structures are distinguished and handled by different procedures. Two schemes of application of HOC procedures are presented: one directed towards a fast canonical numbering for coding systems, and another one yielding levels of topological equivalence allowing a unique topological representation of the molecule with possible applications to similarity search, structure-activity correlations, etc.     

The hydrogen perturbation in molecular connectivity computations

A new algorithm for the delta(v) number, the basic parameter of molecular connectivity indices, is proposed. The new algorithm, which is centered on graph concepts like complete graphs and general graphs, encodes the information of the bonded hydrogen on different atoms through a perturbation parameter that makes use of no new graph concepts. The model quality of the new algorithm is tested with 13 properties of seven different classes of compounds, as well as with composite classes of compounds with the same property and with composite properties of the same class of compounds. Chosen properties and classes of compounds display different percentage of bonded hydrogen atoms, which allow a checking of the importance of this parameter. A comparison is drawn with previous results with zero contribution for the hydrogen perturbation as well as among results obtained by changing the number of compounds of a property but keeping constant the percentage of hydrogen atoms. Results underline the importance of the property as well as the importance of the number of compounds in determining the level of the hydrogen perturbation. Molecular connectivity terms are in some cases more critical than the combination of indices in detecting the perturbation introduced by the hydrogen atoms.     

Novel atomic-level-based AI topological descriptors: application to QSPR/QSAR modeling

Novel atomic level AI topological indexes based on the adjacency matrix and distance matrix of a graph is used to code the structural environment of each atomic type in a molecule. These AI indexes, along with Xu index, are successfully extended to compounds with heteroatoms in terms of novel vertex degree v(m), which is derived from the valence connectivity delta(v) of Kier-Hall to resolve the differentiation of heteroatoms in molecular graphs. The multiple linear regression (MLR) is used to develop the structure-property/activity models based on the modified Xu and AI indices. The efficiency of these indices is verified by high quality QSPR/QSAR models obtained for several representative physical properties and biological activities of several data sets of alcohols with a wide range of non-hydrogen atoms. The results indicate that the physical properties studied are dominated by molecular size, but other atomic types or groups have small influences dependent on the studied properties. Among all atomic types, -OH groups seem to be most important due to hydrogen-bonding interactions. On the contrary, -OH groups play a dominant role in biological activities studied, although molecular size is also an important factor. These results indicate that both Xu and AI indices are useful model parameters for QSPR/QSAR analysis of complex compounds.