Advances in Mathematical Chemistry and Applications

An innovative theoretical mechanism leading to the classification of the stability of fullerene isomers is presented. This approach is based on the action of suitable topological potentials impacting on molecular stability, namely topological compactness and topological sphericality indexes, providing a fast and general ranking algorithm. Present results point out that electronic properties of sp² carbon systems are deeply rooted in the topology of their atomic network.

Similarity studies are important for chemistry and their applications range from the periodic table to the screening of large databases in the searching for new drugs. In this later case, it is assumed that similarity in molecular structure is related to similarity in reactivity. However, we state that structural formulas can be regarded as abstract representations emerging from the analysis of large amounts of data upon chemical reactivity. Hence, chemical formulas such as organic functions are not direct pictures of the atomic constitution of matter, but signs used to represent similarity in the reactivity of a class of substances. Therefore, reactivity, rather than molecular structure, becomes the fundamental feature of chemical substances. As reactivity is important, chemical identity is given by the relations substances establish with each other, giving place to a network of chemical reactions. We explore similarity in the network rather than in molecular structure. By characterising each substance in terms of the related ones, we show how Category Theory helps in this description. Afterwards, we study the similarity among substances using topological spaces, which leads us to concepts such as closure and neighbourhood, which formalise the intuition of things lying somewhere near around. The second focus of the chapter is the exploration of the potential of closure operators, and of topological closures in particular, as more general descriptors of chemical similarity. As we introduce the formalism, we develop a worked example, concerning the analysis of similarity among chemical elements regarding their ability to combine into binary compounds. The results show that several of the trends of chemical elements are found through the current approach.

One of the subjects of special interest in chemical structure handling is to be able to associate a unique quantitative value to each and every chemical compound. The job is not only huge from the stand point of the number of compounds known and to be known, but quite difficult as well from the angle of developing a suitable method. One of the common situations is to be able to discriminate isomeric structures where a large number of compounds having closely related structures for same number of atoms come into picture and this number grows very fast as the number of atoms increases. Getting quantitative descriptors having power of discriminating one compound from another is an important requirement for storage, retrieval and handling of chemical structures as well as for predicting molecular properties/activities. In this chapter, we review several molecular descriptors, mostly topological distance based, in the form of topological indices considering the connectivity aspect of molecular structures only, that have useful discriminative power.

Mendeleev periodic table of atoms is one of the most important principles in natural science. However, there is not such a thing for molecules. Here we propose three periodic tables for diatomic molecules, triatomic molecules and AH3 tetratomic molecules, respectively. The tables not only contain isolated molecules, but also the “virtual” diatomic molecules in polyatomic molecules. The form of these molecular periodic tables is analogous to that of Mendeleev periodic table. In the table, molecules are classified and arranged by their group number G, which is the number of valence electrons, and the periodic number P, which represents the size of molecules. Basic molecular properties, including bond length, binding energy, force constant, ionization potential, spin multiplicity, chemical reactivity, or bond angle, among others. change periodically with the tables. This periodicity originates from the shell-like electronic configurations of molecules. The periodic tables can be used to predict unknown properties of molecules, to understand the role of virtual molecules in polyatomic molecules, and to initiate new research fields such as the periodicity of aromatic compounds, clusters, or nanoparticles.

The very rapid growth in molecular sequence data from the daily accretion of large gene and protein sequencing projects have led to issues regarding viewing and analyzing the massive amounts of data. Graphical representation and numerical characterization of DNA, RNA and protein sequences have exhibited great potential to address these concerns. We review here in brief several different formulations of these representations and examples of applications to diverse problems based on what this author had presented at the Second Mathematical Chemistry Workshop of the Americas in Bogota, Colombia in 2010. In particular, we note several insights that were gained from such representations, and the applications to the bio-medicinal field.

A primary goal of quantitative structure-activity relationships (QSARs) and quantitative structure-property relationships (QSPRs) is to predict chemical activities from chemical structure. Chemical structure can be quantified in many ways resulting in hundreds, if not thousands, of measurements for every chemical. Chemical activities measures how the chemical interacts with other chemicals, e.g. toxicity, biodegradability, boiling point, and vapor pressure. Typically there are more chemical structure measurements than chemicals being measured, the so-called large-p, small-n problem. Here we review some of the statistical procedures that have been commonly used to explore these problems in the past and provide several examples of their use. Finally, we peek into the future to discuss two areas that we believe will see dramatically increased attention in the near future: model averaging and Bayesian techniques.

The importance of marine alga, namely Dunaliella tertiolecta, in toxicity determination of organics, inorganics, and mixtures, as well as for raw and treated industrial effluents is emphasized. Ultrastructural changes for metals below the toxic concentration in D. tertiolecta are also highlighted. Examples for synergistic, antagonistic and hormetic effects in case of exposure of D. tertiolecta to chemicals are given. In a case study, we focus on modeling the toxicity of selected phenols to D. tertiolecta. Quantitative structure-activity relationship (QSAR) methodology is employed to model the toxicities of phenolic compounds to D. tertiolecta using counterpropagation artificial neural networks (CP ANN). The endpoint for the toxicity determination is growth inhibition of algae exposed to chemicals in a batch system containing natural sea water enriched by the modified f/2 medium. Results reveal that QSAR methodology can be successfully applied to fill the data gap present in marine algal ecotoxicity data.

As Mycobacterium tuberculosis has become drug resistant, we need to design new anti-tuberculosis lead compounds. This review has focused on the application of various chemoinformatics methods that can be used in an attempt to search for potent and selective inhibitors against M. tuberculosis. One of the “rational” approaches towards designing such novel anti-bacterials is to develop and deploy QSAR of similar drug-like compounds, that helps in implementing and improvising the techniques and shares some newly identified potential anti-TB drug candidates. Here, we have also mentioned about some of the QSAR models developed and validated by various groups as well as our team for several derivatives showing anti-tuberculosis activity viz. fluoroquinolones, quinoxaline and nitrofuranyl amide derivatives etc. As the calculation of diverse physicochemical properties for such huge number of compounds is time consuming and also not costeffective, we have utilized molecular descriptors for regression modeling. Among different types of descriptors, the study has also been extended to understand the influence of each class of molecular descriptor for predicting structure-activity relationships, and the results indicate the preeminence of topological descriptors over other descriptor lessons. The methodologies described in this review are non specific and applicable to other syndromes also.

Systems Biology aims to define biological problems using the language of mathematics. With the advancement of high throughput technologies and their ever expanding capabilities to generate large scale “-omics” data, the basic goal of systems biology would be to integrate global data sets and develop a coherent understanding to the biological system under study. Biological interactions are highly complex where the components of the systems are connected in a highly intricate manner. Mathematical modeling plays a major role in capturing the dynamics of each and every component in the system, simplifying complex biological networks. In the field of infectious diseases, system level understanding is essential to gain valuable insights into the pathogenic processes. A thorough understanding of the perturbations in biological networks may aid in prioritizing of the drug targets. Schistosomiasis and Trypanosomiasis remain as the two neglected tropical diseases affecting human population worldwide. The central theme of this work revolves around developing a systems level understanding of the lipid metabolism of these two parasites, abstracting complex biological processes as a collection of interacting functions driven in time by a set of discrete biological events. An insight has been laid into the importance and application of systems biology which is emerging as an amalgamation of two important sciences “Mathematics” and “Biology”.

Owing to the evolution of resistant cells using standard antimicrobial methods, it is necessary for us to find alternative strategies to block the attack without exerting pressures on their duplicated structures and in some cases degradation of their Quorum Sensing signals is suggested to be more efficient. In this chapter, we give a brief introduction of quorum sensing basics. Then, we describe the computational approaches that are used to find molecules to inhibit the chemical signals. After that, we present the case studies using these computational methods. In one of the cases performed in E. coli with the autoinducer molecule indole, we elaborate how to evaluate and find molecules with the ability to degrade the indole. We present another case where biological molecules with potential ability to degrade the autoinducer 3-OH-PAME Quorum Sensing system in R. solanacearum were evaluated as potential quenchers.

Over the past couple of decades, we have arrived at the point where we are beginning to understand the mode of action of metaloenzymes and to use this knowledge for designing small organometallic catalysts; i.e. biomimetics. Quantum chemistry is an essential part of this process. However, due to methodical and technical limitations, quantum mechanical studies are limited to small models of these huge biosystems. This article attempts to bridge the fields of quantum chemistry and biochemistry by illustrating some of the basic mechanisms by which nature accomplishes catalysis. This chapter contains a critical discussion of the limitations of quantum mechanical methods, the pitfalls one can encounter along the way and the necessity for critical model evaluation.

Deciphering the folding mechanism of proteins is significant to comprehend their physiological functions. In this chapter, several significant and yet common factors of protein folding have been discussed: 1) Space limitation (confinement and macromolecular crowding). Proteins are confined and crowded in cellular circumstances, which facilitates the folding and enhances the stability through the entropic reduction of the unfolded states. 2) Solvent effects. The various solvation models have been described. Water is more than the environment, and can also participate in the folding by mediating the collapse of protein chains and searching for the native topologies along the free energy landscapes. 3) Pressure, temperature and pH. The high hydrostatic pressure induces the volume decrease, destroys the non-covalent interactions and increases the roughness of free energy landscape, which generally drives the equilibrium toward the unfolded states. pH modulates protein structure and dynamics through protonation/deprotonation of sidechains and sometimes causes the misfolding. Temperature changes alter the conformational dynamics but not the folding pathway. 4) Structural modifications (mutation, truncation/insertion and protonation/deprotonation). The mutations of key residues significantly alter the folding by distorting the cooperative interactions, which can result in the misfolding or aggregation; nonetheless, the rational design by mutations can be beneficial to protein folding. The proper truncations do not show obvious influences on protein structure and dynamics, and the loop insertions may reduce the unfolding free energy barrier and facilitate the unfolding kinetics. Protonation of key residues affects significantly the folding/unfolding equilibrium by altering the non-covalent interactions.

In the analysis of complex networks, the description of evolutionary processes, or investigations into dynamics on fitness or energy landscapes notions such as similarity, neighborhood, connectedness, or continuity of change appear in a natural way. These concepts are of an inherently topological nature. Nevertheless, the connection to the mathematical discipline of point set topology is rarely made in the literature, presumably because in most applications there is no natural object corresponding to an open or closed set. The link to textbook topology thus cannot be made in a straightforward manner. Many of the deep results of point set topology still remain valid, however, when open sets are abandoned and generalizations of the closure operator are used as the foundation of the mathematical theory. Here we survey some applications of such generalized point set topologies to chemistry and biology, providing an overview of the underlying mathematical structures.