Current Computer - Aided Drug Design - Volume 7, Issue 2, 2011

Understanding how drugs cause the desired effect on the organism is of utmost importance for drug design. This is not only a scientific motto, but also an economic issue which pharmaceutical companies have comprehended quite well. Understanding the relationship between the structure of a substance and its biological activity is critical for drug discovery. A wealth of knowledge has been generated to grasp such relationships mainly from quantum chemical and mathematical description of molecules. Due to the large number of chemical substances with potential biological activity, which are normally collected in real or virtual libraries, the first step in the search for new drugs is to screen the chemical libraries which may lead to new drugs with less side effects and desirable therapeutic activity. Screening large libraries with quantum chemical tools normally can be very demanding on computer resources. In contrast, the use of mathematical characterization of molecules allows screening within a reasonable time and with low computational costs. This kind of relationship between mathematics and chemistry is one of the interests of the field of theoretical chemistry currently known as Mathematical Chemistry. Due to its interaction with other disciplines like biochemistry, the omics sciences, drug design, predictive toxicology, and development of new materials, mathematical chemistry has become an interesting area of research, with scientific meetings spread all over the world. This issue of Current Computer-Aided Drug Design (CCADD) gathers several of the manuscripts presented at the First Mathematical Chemistry Workshop of the Americas (1MCWA), held in Bogota (Colombia) at the campus of the Universidad de los Andes (August 21-22, 2009). The meeting was sponsored by the Universidad de los Andes (Colombia), the Universidad de Pamplona (Colombia), the Natural Resources Research Institute of the University of Minnesota Duluth (USA),) and the International Society of Mathematical Chemistry. Guillermo Restrepo (Universidad de Pamplona), Jose L. Villaveces (Universidad de los Andes) and Subhash C. Basak (University of Minnesota) were the chairpersons of the event. The 1MCWA was the first international workshop on Mathematical Chemistry organized in Latin America and its aim was to spread the scope of events on Mathematical Chemistry organized by Subhash C. Basak and collaborators using two Indo-US forums i.e. Indo-US Workshop on Mathematical Chemistry (http://gisdata.nrri.umn.edu/pers/pm/indous6/general.htm) and Indo-US Lecture Series on Discrete Mathematical Chemistry (http://www.nrri.umn.edu/indouslecture). The 1MCWA was also the continuation of a series of courses/events on Mathematical Chemistry organized by Guillermo Restrepo: 7th North-eastern Symposium of Mathematics (Colombia, 2009); 9th WSEAS International Conference on Computers (Greece, 2005); Quantum Similarity (Colombia, 2005); Mathematical Chemistry: Periodicity (Colombia, 2004); and Mathematical Chemistry (Colombia, 2001). The six papers collected in the CCADD issue cover different themes of current interest for the mathematical chemistry community as well as for the field of drug design. In the first paper, “On Molecular Graph Comparison”, Jenny A. Melo and Edgar Daza review some distance and similarity coefficients developed to quantify similarity between molecular structures characterized by molecular graphs. This manuscript can be, therefore, linked to the previous CCADD issue on “Chemo-Bioinformatics Based Mathematical Descriptors and their Applications in Computational Drug Design”, Volume 6, Number 4, December 2010, where seven papers cover applications of graph theoretical characterization of molecules to predicting molecular stability, screening of large chemical libraries, computational drug design, predicting anti-HIV activity of substances and characterization of drug-DNA interactions. The second paper, “Chemotopology: Beyond Neighbourhoods”, by Guillermo Restrepo and Heber Mesa reviews the chemotopological method, a procedure which combines classification results with topology to study chemical sets. The authors describe the mathematical basis of the method and illustrate its applications by using examples of chemical sets like amino acids, benzimidazoles and steroids, among others. “Quantitative Structure-Activity Relationships for Anticancer Activity of 2-Phenylindoles Using Mathematical Molecular Descriptors” is the title of the third paper, where Subhash C. Basak, Qianhong Zhu and Denise Mills developed reliable models for predicting activity of 89 phenylindole derivatives against breast cancer. This paper is an instance of how easily computed mathematical descriptors from the chemical graph theory are a good option for estimating biological activities of potential drugs. In the fourth manuscript, “Comparison of QSARs and Characterization of Structural Basis of Bioactivity Using Partial Order Theory and Formal Concept Analysis: A Case Study with Mutagenicity”, Guillermo Restrepo, Subhash C. Basak and Denise Mills use elements of order theory to assess the performance of different QSAR models developed to predict mutagenicity of 95 aromatic and heteroaromatic amines. In so doing, they moot order theory as a versatile tool to compare QSAR methodologies, in general, on the basis of their particular statistics. The manuscript also introduces a novel data analysis technique, Formal Concept Analysis, which can be considered as an approach to relate molecular frameworks with biological activity for the treated property: mutagenicity. Andres Bernal and Edgar Daza turn more biological when treating mathematically chemical networks in the fifth paper of this issue: “Metabolic Networks: Beyond the Graph”. They describe the metabolic network as a complex network of chemical reactions and point out how wrong conclusions may arise from a naive graph interpretation of such networks. As a constructive critique, the authors discuss how the use of other kinds of mathematical structures i.e. hypergraphs, is more suitable to deal with metabolic networks.....

Since the last half of the nineteenth century, molecular graphs have been present in several branches of chemistry. When used for molecular structure representation, they have been compared after mapping the corresponding graphs into mathematical objects. However, direct molecular comparison of molecular graphs is a research field less explored. The goal of this mini-review is to show some distance and similarity coefficients which were proposed to directly compare molecular graphs or which could be useful to do so.

We have shown in several papers the importance of using topology, particularly set-point topology, to deal with chemical questions related to the concept of similarity. The procedure developed has been called “chemotopology” and it has been applied to different chemical sets e.g. chemical elements, benzimidazoles, sterorids, amino acids and hydrides. The idea behind chemotopology is to run a hierarchical cluster analysis study on a set of objects characterised by different attributes. From this study a dendrogram is obtained, which gathers similarity neighbourhoods for the set of objects. By using a mathematical characterisation of a dendrogram it is possible to select a collection of objects' neighbourhoods which in turn become a basis for a topology. With this basis at hand different properties of subsets of objects can be calculated, all of them related to the concept of similarity e.g. closures, derived sets, boundaries, interiors and exteriors. We have also shown the chemical meaning of each one of these properties. In this manuscript, we review the foundations of the chemotopological method as well as its different applications to chemical sets. By means of examples we illustrate how the method can be used as a versatile tool for drug discovery. We also study the relationship between the topologies generated from dendrograms of a given set of objects and the dendrograms that can be obtained for particular topologies on the set of objects.

Calculated atom pairs (APs) and a set of 369 topological indices (TIs) calculated by POLLY, Triplet, and Molconn-Z software were used to develop QSARs for anticancer activity of a group of 2-phenyl indoles. The TIs included both topostructural (TS) and topochemical (TC) indices. Results show that ridge regression using TS indices, TC indices, and atom pairs produced high-quality models for the prediction of anticancer activity of a set of 89 phenylindole derivatives. Quality of QSARs derived in this paper is comparable or superior to both CoMFA and other statistical models reported for 2-phynylindoles in the earlier published literature. Easily calculated molecular descriptors like TIs and APs used in this paper may find application in the QSAR and in silico prediction of bioactivity of new phenylindole derivatives.

Fifteen quantitative structure-activity relationship (QSAR) models developed by various authors for the prediction of mutagenicity of aromatic and heteroaromatic amines were analyzed and thirteen of them, based on 95 amines, were compared using their respective statistics and order theory (Hasse Diagram Technique, HDT) to obtain an ordering of QSAR models. The technique of Formal Concept Analysis (FCA) was applied to the set of 95 amines to extract concepts and, in general, knowledge about the relationship between structural attributes and mutagenicity. HDT may be useful as a general tool for the comparison of different classes of QSAR models. FCA turns out to be a novel mathematical technique for seeking for relationships between molecular structure and activity.

Drugs are devised to enter into the metabolism of an organism in order to produce a desired effect. From the chemical point of view, cellular metabolism is constituted by a complex network of reactions transforming metabolites one in each other. Knowledge on the structure of this network could help to develop novel methods for drug design, and to comprehend the root of known unexpected side effects. Many large-scale studies on the structure of metabolic networks have been developed following models based on different kinds of graphs as the fundamental image of the reaction network. Graphs models, however, comport wrong assumptions regarding the structure of reaction networks that may lead into wrong conclusions if they are not taken into account. In this article we critically review some graph-theoretical approaches to the analysis of centrality, vulnerability and modularity of metabolic networks, analyzing their limitations in estimating these key network properties, consider some proposals explicit or implicitly based on directed hypergraphs regarding their ability to overcome these issues, and review some recent implementation improvements that make the application of these models in increasingly large networks a viable option.

There are at least two significant applications of partial order theory in chemistry: Ranking methods and substances' properties prediction. In both cases, a set of objects is endowed with a partial order relation e.g. “more polluting than”, “can be obtained from”, “more reactive than” etc. The couple of set and partial order relation is known in mathematics as a partially ordered set (poset). Ranking methods, such as the Hasse diagram technique, lead to a partial order where several incomparabilities (lack of order) appear between pairs of objects. This phenomenon is quite common in ranking studies, and it often is circumvented by a combination of object features leading to a total order. However, such a combination introduces subjectivities and bias in the ranking process. Here a step-by-step procedure is shown to turn incomparabilities into comparabilities taking into account all the possible bias by a linear combination of features. In such a manner, it is possible to predict how probable it is to obtain a particular total order from a given poset. Similarly, it is possible to calculate the needed bias over certain attributes to obtain a particular total order. An example application is shown where substances are ranked according to their bioconcentration factor and biodegradation potential. Another application of partial order theory to chemistry has to do with the prediction of properties for a set of substances related in a (preferably systematic) chemical fashion. A customary relation is “can be obtained from”; if such a relation is set up for a given molecular structure e.g. benzene, and all its substituted derivatives (say chlorinated ones) are considered, then the set of benzene and its chlorinated derivatives are partially ordered. Taking advantage of the poset generated, different methods can be applied to predict properties of the substances considered in the poset. Such methods include the poset-average, cluster expansion, and splinoid methods. In this paper we discuss each one of these methods, its advantages and disadvantages and we outline its applicability to estimate cooperative free energies of hemoglobins with different degree of oxygenation.

Molecular docking has become an increasingly important tool for drug discovery. In this review, we present a brief introduction of the available molecular docking methods, and their development and applications in drug discovery. The relevant basic theories, including sampling algorithms and scoring functions, are summarized. The differences in and performance of available docking software are also discussed. Flexible receptor molecular docking approaches, especially those including backbone flexibility in receptors, are a challenge for available docking methods. A recently developed Local Move Monte Carlo (LMMC) based approach is introduced as a potential solution to flexible receptor docking problems. Three application examples of molecular docking approaches for drug discovery are provided.