From QSAR models of drugs to complex networks: state-of-art review and introduction of new Markov-spectral moments indices.

Quantitative Structure-Activity/Property Relationships (QSAR/QSPR) models have been largely used for different kind of problems in Medicinal Chemistry and other Biosciences as well. Nevertheless, the applications of QSAR models have been restricted to the study of small molecules in the past. In this context, many authors use molecular graphs, atoms (nodes) connected by chemical bonds (links) to represent and numerically characterize the molecular structure. On the other hand, Complex Networks are useful in solving problems in drug research and industry, developing mathematical representations of different systems. These systems move in a wide range from relatively simple graph representations of drug molecular structures (molecular graphs used in classic QSAR) to large systems. We can cite for instance, drug-target interaction networks, protein structure networks, protein interaction networks (PINs), or drug treatment in large geographical disease spreading networks. In any case, all complex networks have essentially the same components: nodes (atoms, drugs, proteins, microorganisms and/or parasites, geographical areas, drug policy legislations, etc.) and links (chemical bonds, drug-target interactions, drug-parasite treatment, drug use, etc.). Consequently, we can use the same type of numeric parameters called Topological Indices (TIs) to describe the connectivity patterns in all these kinds of Complex Networks irrespective the nature of the object they represent and use these TIs to develop QSAR/QSPR models beyond the classic frontiers of drugs small-sized molecules. The goal of this work, in first instance, is to offer a common background to all the manuscripts presented in this special issue. In so doing, we make a review of the most used software and databases, common types of QSAR/QSPR models, and complex networks involving drugs or their targets. In addition, we review both classic TIs that have been used to describe the molecular structure of drugs and/or larger complex networks. In second instance, we use for the first time a Markov chain model to generalize Spectral moments to higher order analogues coined here as the Stochastic Spectral Moments TIs of order k (πk). Lastly, we report for the first time different QSAR/QSPR models for different classes of networks found in drug research, nature, technology, and social-legal sciences using πk values. This work updates our previous reviews Gonzalez-Diaz et al. Curr Top Med Chem. 2007; 7(10): 1015-29 and Gonzalez-Diaz et al. Curr Top Med Chem. 2008; 8(18):1676-90. It has been prepared in response to the kind invitation of the editor Prof. AB Reitz in commemoration of the 10th anniversary of this journal in 2010.

New Markov-Shannon Entropy models to assess connectivity quality in complex networks: from molecular to cellular pathway, Parasite-Host, Neural, Industry, and Legal-Social networks.

Graph and Complex Network theory is expanding its application to different levels of matter organization such as molecular, biological, technological, and social networks. A network is a set of items, usually called nodes, with connections between them, which are called links or edges. There are many different experimental and/or theoretical methods to assign node-node links depending on the type of network we want to construct. Unfortunately, the use of a method for experimental reevaluation of the entire network is very expensive in terms of time and resources; thus the development of cheaper theoretical methods is of major importance. In addition, different methods to link nodes in the same type of network are not totally accurate in such a way that they do not always coincide. In this sense, the development of computational methods useful to evaluate connectivity quality in complex networks (a posteriori of network assemble) is a goal of major interest. In this work, we report for the first time a new method to calculate numerical quality scores S(L(ij)) for network links L(ij) (connectivity) based on the Markov-Shannon Entropy indices of order k-th (θ(k)) for network nodes. The algorithm may be summarized as follows: (i) first, the θ(k)(j) values are calculated for all j-th nodes in a complex network already constructed; (ii) A Linear Discriminant Analysis (LDA) is used to seek a linear equation that discriminates connected or linked (L(ij)=1) pairs of nodes experimentally confirmed from non-linked ones (L(ij)=0); (iii) the new model is validated with external series of pairs of nodes; (iv) the equation obtained is used to re-evaluate the connectivity quality of the network, connecting/disconnecting nodes based on the quality scores calculated with the new connectivity function. This method was used to study different types of large networks. The linear models obtained produced the following results in terms of overall accuracy for network reconstruction: Metabolic networks (72.3%), Parasite-Host networks (93.3%), CoCoMac brain cortex co-activation network (89.6%), NW Spain fasciolosis spreading network (97.2%), Spanish financial law network (89.9%) and World trade network for Intelligent & Active Food Packaging (92.8%). In order to seek these models, we studied an average of 55,388 pairs of nodes in each model and a total of 332,326 pairs of nodes in all models. Finally, this method was used to solve a more complicated problem. A model was developed to score the connectivity quality in the Drug-Target network of US FDA approved drugs. In this last model the θ(k) values were calculated for three types of molecular networks representing different levels of organization: drug molecular graphs (atom-atom bonds), protein residue networks (amino acid interactions), and drug-target network (compound-protein binding). The overall accuracy of this model was 76.3%. This work opens a new door to the computational reevaluation of network connectivity quality (collation) for complex systems in molecular, biomedical, technological, and legal-social sciences as well as in world trade and industry.

From chemical graphs in computer-aided drug design to general Markov-Galvez indices of drug-target, proteome, drug-parasitic disease, technological, and social-legal networks.

Complex Networks are useful in solving problems in drug research and industry, developing mathematical representations of different systems. These systems move in a wide range from relatively simple graph representations of drug molecular structures to large systems. We can cite for instance, drug-target protein interaction networks, drug policy legislation networks, or drug treatment in large geographical disease spreading networks. In any case, all these networks have essentially the same components: nodes (atoms, drugs, proteins, microorganisms and/or parasites, geographical areas, drug policy legislations, etc.) and edges (chemical bonds, drug-target interactions, drug-parasite treatment, drug use, etc.). Consequently, we can use the same type of numeric parameters called Topological Indices (TIs) to describe the connectivity patterns in all these kinds of Complex Networks despite the nature of the object they represent. The main reason for this success of TIs is the high flexibility of this theory to solve in a fast but rigorous way many apparently unrelated problems in all these disciplines. Another important reason for the success of TIs is that using these parameters as inputs we can find Quantitative Structure-Property Relationships (QSPR) models for different kind of problems in Computer-Aided Drug Design (CADD). Taking into account all the above-mentioned aspects, the present work is aimed at offering a common background to all the manuscripts presented in this special issue. In so doing, we make a review of the most common types of complex networks involving drugs or their targets. In addition, we review both classic TIs that have been used to describe the molecular structure of drugs and/or larger complex networks. Next, we use for the first time a Markov chain model to generalize Galvez TIs to higher order analogues coined here as the Markov-Galvez TIs of order k (MGk). Lastly, we illustrate the calculation of MGk values for different classes of networks found in drug research, nature, technology, and social-legal sciences.