Advances in Mathematical Chemistry and Applications

Mathematical chemistry or more accurately discrete mathematical chemistry had a tremendous growth spurt in the second half of the twentieth century and the same trend is continuing now. This growth was fueled primarily by two major factors: 1) Novel applications of discrete mathematical concepts to chemical and biological systems, and 2) Availability of high speed computers and associated software whereby hypothesis driven as well as discovery oriented research on large data sets could be carried out in a timely manner. This led to the development of not only a plethora of new concepts, but also to various useful applications to such important areas as drug discovery, protection of human as well as ecological health, and chemoinformatics. Following the completion of the Human Genome Project in 2003, discrete mathematical methods were applied to the “omics” data to develop descriptors relevant to bioinformatics, toxicoinformatics, and computational biology. This chapter will discuss the major milestones in the development of concepts of mathematical chemistry, mathematical proteomics as well as their important applications in chemobioinformatics with special reference to the contributions of Basak and coworkers.

We give some basic mathematical ideas of partially ordered sets (posets), which frame into the mathematical way of thinking illustrated in the Erlangen Programme by Felix Klein. The programme entails extracting relevant variables to study, symbolising them and relating them through a function. We show several examples where the mathematical way of thinking, restricted to partial orders, is found in chemistry. The examples are: Geoffroy’s affinity table, benzene’s structure, posetic predictive methods, multicriteria situations and derivation of concepts. Finally we question the ranking process by showing how it disregards its underlying, and not always recognised, posetic nature.

Graph theory based descriptors of molecular structure play important role in QSPR/ QSAR models. This chapter reviews some attempts to optimize the characterization of molecular structure via an integrated representation that accounts in a systemic manner for the contributions of all substructures. In its simplest version this approach counts the subgraphs of all sizes, the resulted single number being shown to be a very sensitive measure of structural complexity. The most complete version builds (i) an ordered set of counts of subgraphs of increasing number of edges, (ii) weights each subgraph with the value of selected graph-invariant, building a weighted ordered set, and (iii) sums up all the subgraph contributions to produce the overall value of the graph-invariant. The invariants tested include vertex degrees, vertex distances, and the graph non-adjacency numbers, the corresponding overall topological indices being called overall connectivity, overall Wiener, overall Zagreb and overall Hosoya indices. Their properties are analyzed in detail in acyclic and cyclic graphs. It is shown that they all are reliable measures of molecular structural complexity, increasing in value with the basic complexifying patterns of branching and cyclicity of molecular skeleton. The structure-property models derived for 10 physicochemical properties of alkane compounds show considerable improvement compared to models derived from molecular connectivity indices. The latest extension of these ideas is demonstrated with extended connectivities, walk counts, and Bourgas indices, the latter of which are the first integrated measures of graph complexity and vertex centrality.

The four connectivity matrices are presented: the vertex-product-connectivity matrix, the edge-product-connectivity matrix, the vertex-sum-connectivity matrix and the edge-sum-connectivity matrix. The half-sum of their matrix elements are the corresponding connectivity indices: the vertex-product-connectivity index, the edgeproduct- connectivity index, vertex-sum-connectivity index and the edge-sumconnectivity index. The suitability of these four forms of connectivity indices in developing structure-property relationships is illustrated on four sets of alkanes for 14 experimental physico-chemical properties. Their polynomials and spectra are also given. The method used for constructing polynomials of the connectivity matrices considered is the Le Verrier-Fadeev-Frame method, that has been modified by Balasubramanian and Živković.

The fingerprints that are widely used for similarity-based virtual screening typically encode the presence or absence of fragments, without any indication as to their relative importance. This chapter discusses the use of weighted fingerprints, where each fragment is associated with a weight denoting its degree of importance in quantifying the degree of similarity between a reference structure and a database structure. Extensive studies using the World of Molecular Bioactivity and MDL Drug Data Report databases show that weighting fragments according to their frequency of occurrence within a molecule can increase the effectiveness of screening, but that this is not the case when fragments are weighted according to their frequency of occurrence within a database.

MOLGEN 5.x combines the efficiency of the molecular generator MOLGEN 3.5 and the flexibility of MOLGEN 4.x. To achieve this, the software was reimplemented based on a totally new concept. The most visible new features are fuzzy molecular formula input and explicit use of atom state patterns. We describe the version MOLGEN 5.0 of this new series.

In this paper, we review classical and recent developments on comparability graphs. Also, we demonstrate that comparability graphs are useful to analyze molecular graphs by presenting classical and new results. In fact, it turns out that the underlying model is quite general and, hence, could be used to analyze any kind of network data.

This chapter deals with the use of molecular topology (MT) in the selection and design of new drugs. After an introduction of the actual methods used for drug design, the basic concepts of MT are defined, including examples of calculation of topological indices, which are numerical descriptors of molecular structures. The goal is making this calculation familiar to the potential students and allowing a straightforward comprehension of the topic. Finally, the achievements obtained in this field are detailed, so that the reader can figure out the great interest of this approach.

A rudimentary treatment of density functional theory (DFT) is presented in this article. Various global and local reactivity descriptors are defined within the broad framework of conceptual DFT. A theory of chemical reactivity is developed in terms of these descriptors and the associated electronic structure principles.

In the chapter we present a short overview of QSAR (Quantitative Structure- Activity Relationship) modeling. The QSAR paradigm grounds on an assumption that properties of a compound depend on its chemical structure. In its final form a QSAR model is expressed as a mathematical relationship between molecular structure and property. A model is built on existing knowledge, i.e., on a set of compounds with known structures and known properties. The QSAR models are widely used in rational drug design and in the environmental toxicology. As examples we present a case study of QSAR modeling in searching for new anti-tuberculosis drugs and the predictions of five toxicological endpoints with the internet available program CAESAR.

A review of methods employed for the assessment of druglikeness using 2D structural and atom type descriptors is presented. These methods are classified as Druglike Filters (DLFs) and Druglike Indices (DLIs), depending on the characterization of druglikeness, using known drug and non-drug databases. The DLF methods specify a set of rules based on calculated property distributions, whereas the DLI methods aim to assess druglikeness through a single number derived from multiple descriptors. A review of ranges calculated from property profiles of known drugs is given, along with a careful re-assessment for twenty five descriptors based on an analysis of a recent drug database. A discussion of future direction for the development and utility of these approaches is presented.

Diseases caused by parasites have an overwhelming impact on public health throughout the world, particularly in the tropics and subtropics. Malaria and leishmaniasis are two such widely known neglected parasitic diseases. The current global situation indicates more than one million deaths from these two diseases every year despite several efforts by WHO to combat them. Vectors for carrying and transmitting these parasites are arthropods. Use of insect repellents is a vital countermeasure in reducing these arthropod-related diseases. However, despite access to many available drugs for treatment of these diseases, their growing resistance poses serious concerns and necessitates development of novel countermeasures. The present chapter discusses how the in silico methodologies can be utilized to develop pharmacophore models to identify novel antimalarials, antileishmanial, and insect repellents. The models presented in this chapter not only provided important molecular insights to better understand the “interaction pharmacophores” but also guided generation of templates for virtual screening of compound databases to identify novel bioactive agents. The pharmacophore models presented here demonstrated a new computational approach for organizing molecular characteristics that were both statistically and mechanistically significant for potent activity and useful for identification of novel analogues as well.

This chapter is for those in the field of mathematical chemistry who would like to practice their skills on other than normal molecules; for colleagues in the physics community with a curiosity about periodicities of particles from molecules to strings; and for specialists in informatics. Similarities are shown to exist in the constructions of periodic systems for five orders of particles.