Non-stochastic and stochastic linear indices of the ‘molecular pseudograph’s atom adjacency matrix’: application to ‘in silico’ studies for the rational discovery of new antimalarial compounds

Computational discovery of novel anthelmintic natural compounds from Agave Brittoniana trel. Spp. Brachypus

Helminth infections are a medical problem in the world nowadays. This report used bond-based 2D quadratic indices, a bond-level QuBiLs-MAS molecular descriptor family, and Linear Discriminant Analysis (LDA) to obtain a quantitative linear model that discriminates between anthelmintic and non-anthelmintic drug-like organic-compounds. The model obtained correctly classified 87.46% and 81.82% of the training and external data sets, respectively. The developed model was used in a virtual screening to predict the biological activity of all chemicals (19) previously obtained and chemically characterized by some authors of this report from Agave brittoniana Trel. spp. Brachypus. The model identified several metabolites (12) as possible anthelmintics, and a group of 5 novel natural products was tested in an in vitro assay against Fasciola hepatica (100% effectivity at 500 µg/mL). Finally, the two best hits were evaluated in vivo in bald/c mice and the same helminth parasite using a 25 mg/kg dose. Compound 8 (Karatavinoside A) showed an efficacy of 92.2% in vivo. It is important to remark that this natural compound exhibits similar-to-superior activity as triclabendazole, the best human fasciolicide available in the market against Fasciola hepatica, resulting in a novel lead scaffold with anti-helminthic activity. Keywords: TOMOCOMD-CARDD Software; QuBiLs-MAS, nonstochastic and stochastic bond-based quadratic indices; LDA-based QSAR model; Computational Screening, Anthelmintic Agent; Agave brittoniana Trel. spp. Brachypus, Fasciola hepatica.

QuBiLS-MAS, open source multi-platform software for atom- and bond-based topological (2D) and chiral (2.5D) algebraic molecular descriptors computations

BACKGROUND

In previous reports, Marrero-Ponce et al. proposed algebraic formalisms for characterizing topological (2D) and chiral (2.5D) molecular features through atom- and bond-based ToMoCoMD-CARDD (acronym for Topological Molecular Computational Design-Computer Aided Rational Drug Design) molecular descriptors. These MDs codify molecular information based on the bilinear, quadratic and linear algebraic forms and the graph-theoretical electronic-density and edge-adjacency matrices in order to consider atom- and bond-based relations, respectively. These MDs have been successfully applied in the screening of chemical compounds of different therapeutic applications ranging from antimalarials, antibacterials, tyrosinase inhibitors and so on. To compute these MDs, a computational program with the same name was initially developed. However, this in house software barely offered the functionalities required in contemporary molecular modeling tasks, in addition to the inherent limitations that made its usability impractical. Therefore, the present manuscript introduces the QuBiLS-MAS (acronym for Quadratic, Bilinear and N-Linear mapS based on graph-theoretic electronic-density Matrices and Atomic weightingS) software designed to compute topological (0-2.5D) molecular descriptors based on bilinear, quadratic and linear algebraic forms for atom- and bond-based relations.

RESULTS

The QuBiLS-MAS module was designed as standalone software, in which extensions and generalizations of the former ToMoCoMD-CARDD 2D-algebraic indices are implemented, considering the following aspects: (a) two new matrix normalization approaches based on double-stochastic and mutual probability formalisms; (b) topological constraints (cut-offs) to take into account particular inter-atomic relations; (c) six additional atomic properties to be used as weighting schemes in the calculation of the molecular vectors; (d) four new local-fragments to consider molecular regions of interest; (e) number of lone-pair electrons in chemical structure defined by diagonal coefficients in matrix representations; and (f) several aggregation operators (invariants) applied over atom/bond-level descriptors in order to compute global indices. This software permits the parallel computation of the indices, contains a batch processing module and data curation functionalities. This program was developed in Java v1.7 using the Chemistry Development Kit library (version 1.4.19). The QuBiLS-MAS software consists of two components: a desktop interface (GUI) and an API library allowing for the easy integration of the latter in chemoinformatics applications. The relevance of the novel extensions and generalizations implemented in this software is demonstrated through three studies. Firstly, a comparative Shannon's entropy based variability study for the proposed QuBiLS-MAS and the DRAGON indices demonstrates superior performance for the former. A principal component analysis reveals that the QuBiLS-MAS approach captures chemical information orthogonal to that codified by the DRAGON descriptors. Lastly, a QSAR study for the binding affinity to the corticosteroid-binding globulin using Cramer's steroid dataset is carried out.

CONCLUSIONS

From these analyses, it is revealed that the QuBiLS-MAS approach for atom-pair relations yields similar-to-superior performance with regard to other QSAR methodologies reported in the literature. Therefore, the QuBiLS-MAS approach constitutes a useful tool for the diversity analysis of chemical compound datasets and high-throughput screening of structure-activity data.

QuBiLs-MAS method in early drug discovery and rational drug identification of antifungal agents

The QuBiLs-MAS approach is used for the in silico modelling of the antifungal activity of organic molecules. To this effect, non-stochastic (NS) and simple-stochastic (SS) atom-based quadratic indices are used to codify chemical information for a comprehensive dataset of 2478 compounds having a great structural variability, with 1087 of them being antifungal agents, covering the broadest antifungal mechanisms of action known so far. The NS and SS index-based antifungal activity classification models obtained using linear discriminant analysis (LDA) yield correct classification percentages of 90.73% and 92.47%, respectively, for the training set. Additionally, these models are able to correctly classify 92.16% and 87.56% of 706 compounds in an external test set. A comparison of the statistical parameters of the QuBiLs-MAS LDA-based models with those for models reported in the literature reveals comparable to superior performance, although the latter were built over much smaller and less diverse datasets, representing fewer mechanisms of action. It may therefore be inferred that the QuBiLs-MAS method constitutes a valuable tool useful in the design and/or selection of new and broad spectrum agents against life-threatening fungal infections.

In vitro assessment of the acaricidal activity of computer-selected analogues of carvacrol and salicylic acid on Rhipicephalus (Boophilus) microplus

Rhipicephalus (Boophilus) microplus is a tick that causes huge economic losses in cattle. The indiscriminate use of acaricides has generated resistance to most compounds present on the market. This makes further investigation on other potential acaricides necessary, the in silico assay being an alternative to the design of new compounds. In the present study a biosilico assay was performed using TOMOCOMD-CARDD (TOpological MOlecular COMputer Design-Computer-Aided Rational Drug Design) and WEKA (Waikato Environment for Knowledge Analysis) software. Two carvacrol and four salicylic acid derivatives, synthesized by conventional methods and evaluated with the larval packet test on larvae of R. (B.) microplus were selected. All evaluated compounds presented acaricidal activity; however, ethyl 2-methoxybenzoate (91.8 ± 1.7 % mortality) and ethyl 2,5-dihydroxybenzoate (89.1 ± 1.6 % mortality) showed greater activity than salicylic acid. With regard to the carvacrol analogues, carvacrol acetate (67.8 ± 2.1 % mortality) and carvacrol methyl ether (71.7 ± 1.6 % mortality) also showed greater activity than carvacrol (35.9 ± 3.2 % mortality). TOMOCOMD-CARDD and WEKA software were helpful tools in the search for alternative structures with potential acaricidal activity on R. (B.) microplus.

Bond-based 2D quadratic fingerprints in QSAR studies: virtual and in vitro tyrosinase inhibitory activity elucidation

In this report, we show the results of quantitative structure-activity relationship (QSAR) studies of tyrosinase inhibitory activity, by using the bond-based quadratic indices as molecular descriptors (MDs) and linear discriminant analysis (LDA), to generate discriminant functions to predict the anti-tyrosinase activity. The best two models [Eqs (6) and (12)] out of the total 12 QSAR models developed here show accuracies of 93.51% and 91.21%, as well as high Matthews correlation coefficients (C) of 0.86 and 0.82, respectively, in the training set. The validation external series depicts values of 90.00% and 89.44% for these best two equations (6) and (12), respectively. Afterwards, a second external prediction data are used to perform a virtual screening of compounds reported in the literature as active (tyrosinase inhibitors). In a final step, a series of lignans is analysed using the in silico-developed models, and in vitro corroboration of the activity is carried out. An issue of great importance to remark here is that all compounds present greater inhibition values than Kojic acid (standard tyrosinase inhibitor: IC₅₀ = 16.67 μm). The current obtained results could be used as a framework to increase the speed, in the biosilico discovery of leads for the treatment of skin disorders.

Nucleotide's bilinear indices: novel bio-macromolecular descriptors for bioinformatics studies of nucleic acids. I. Prediction of paromomycin's affinity constant with HIV-1 Psi-RNA packaging region

A new set of nucleotide-based bio-macromolecular descriptors are presented. This novel approach to bio-macromolecular design from a linear algebra point of view is relevant to nucleic acids quantitative structure-activity relationship (QSAR) studies. These bio-macromolecular indices are based on the calculus of bilinear maps on Re(n)[b(mk)(x (m),y (m)):Re(n) x Re(n)-->Re] in canonical basis. Nucleic acid's bilinear indices are calculated from kth power of non-stochastic and stochastic nucleotide's graph-theoretic electronic-contact matrices, M(m)(k) and (s)M(m)(k), respectively. That is to say, the kth non-stochastic and stochastic nucleic acid's bilinear indices are calculated using M(m)(k) and (s)M(m)(k) as matrix operators of bilinear transformations. Moreover, biochemical information is codified by using different pair combinations of nucleotide-base properties as weightings (experimental molar absorption coefficient epsilon(260) at 260 nm and pH=7.0, first (Delta E(1)) and second (Delta E(2)) single excitation energies in eV, and first (f(1)) and second (f(2)) oscillator strength values (of the first singlet excitation energies) of the nucleotide DNA-RNA bases. As example of this approach, an interaction study of the antibiotic paromomycin with the packaging region of the HIV-1 Psi-RNA have been performed and it have been obtained several linear models in order to predict the interaction strength. The best linear model obtained by using non-stochastic bilinear indices explains about 91% of the variance of the experimental Log K (R=0.95 and s=0.08 x 10(-4)M(-1)) as long as the best stochastic bilinear indices-based equation account for 93% of the Log K variance (R=0.97 and s=0.07 x 10(-4)M(-1)). The leave-one-out (LOO) press statistics, evidenced high predictive ability of both models (q(2)=0.86 and s(cv)=0.09 x 10(-4)M(-1) for non-stochastic and q(2)=0.91 and s(cv)=0.08 x 10(-4)M(-1) for stochastic bilinear indices). The nucleic acid's bilinear indices-based models compared favorably with other nucleic acid's indices-based approaches reported nowadays. These models also permit the interpretation of the driving forces of the interaction process. In this sense, developed equations involve short-reaching (k<or=3), middle-reaching (4<k<9), and far-reaching (k=10 or greater) nucleotide's bilinear indices. This situation points to electronic and topologic nucleotide's backbone interactions control of the stability profile of paromomycin-RNA complexes. Consequently, the present approach represents a novel and rather promising way to theoretical-biology studies.

Study of peptide fingerprints of parasite proteins and drug-DNA interactions with Markov-Mean-Energy invariants of biopolymer molecular-dynamic lattice networks

Since the advent of Molecular Dynamics (MD) in biopolymers science with the study by Karplus et al. on protein dynamics, MD has become the by foremost well established, computational technique to investigate structure and function of biomolecules and their respective complexes and interactions. The analysis of the MD trajectories (MDTs) remains, however, the greatest challenge and requires a great deal of insight, experience, and effort. Here, we introduce a new class of invariants for MDTs based on the spatial distribution of Mean-Energy values ξ_k (L) on a 2D Euclidean space representation of the MDTs. The procedure forces one MD trajectory to fold into a 2D Cartesian coordinates system using a step-by-step procedure driven by simple rules. The ξ_k (L) values are invariants of a Markov matrix (¹ Π), which describes the probabilities of transition between two states in the new 2D space; which is associated to a graph representation of MDTs similar to the lattice networks (LNs) of DNA and protein sequences. We also introduce a new algorithm to perform phylogenetic analysis of peptides based on MDTs instead of the sequence of the polypeptide. In a first experiment, we illustrate this algorithm for 35 peptides present on the Peptide Mass Fingerprint (PMF) of a new protein of Leishmania infantum studied in this work. We report, by the first time, 2D Electrophoresis isolation, MALDI TOF Mass Spectroscopy characterization, and MASCOT search results for this PMF. In a second experiment, we construct the LNs for 422 MDTs obtained in DNA-Drug Docking simulations of the interaction of 57 anticancer furocoumarins with a DNA oligonucleotide. We calculated the respective ξ_k (L) values for all these LNs and used them as inputs to train a new classifier with Accuracy = 85.44% and 84.91% in training and validation respectively. The new model can be used as scoring function to guide DNA-Drug Docking studies in drug design of new coumarins for PUVA therapy. The new phylogenetics analysis algorithms encode information different from sequence similarity and may be used to analyze MDTs obtained in Docking or modeling experiments for any classes of biopolymers. The work opens new perspective on the analysis and applications of MD in polymer sciences.

A novel approach to predict aquatic toxicity from molecular structure

The main aim of the study was to develop quantitative structure-activity relationship (QSAR) models for the prediction of aquatic toxicity using atom-based non-stochastic and stochastic linear indices. The used dataset consist of 392 benzene derivatives, separated into training and test sets, for which toxicity data to the ciliate Tetrahymena pyriformis were available. Using multiple linear regression, two statistically significant QSAR models were obtained with non-stochastic (R2=0.791 and s=0.344) and stochastic (R2=0.799 and s=0.343) linear indices. A leave-one-out (LOO) cross-validation procedure was carried out achieving values of q2=0.781 (scv=0.348) and q2=0.786 (scv=0.350), respectively. In addition, a validation through an external test set was performed, which yields significant values of Rpred2 of 0.762 and 0.797. A brief study of the influence of the statistical outliers in QSAR's model development was also carried out. Finally, our method was compared with other approaches implemented in the Dragon software achieving better results. The non-stochastic and stochastic linear indices appear to provide an interesting alternative to costly and time-consuming experiments for determining toxicity.

Prediction of tyrosinase inhibition activity using atom-based bilinear indices

A set of novel atom-based molecular fingerprints is proposed based on a bilinear map similar to that defined in linear algebra. These molecular descriptors (MDs) are proposed as a new means of molecular parametrization easily calculated from 2D molecular information. The nonstochastic and stochastic molecular indices match molecular structure provided by molecular topology by using the kth nonstochastic and stochastic graph-theoretical electronic-density matrices, M(k) and S(k), respectively. Thus, the kth nonstochastic and stochastic bilinear indices are calculated using M(k) and S(k) as matrix operators of bilinear transformations. Chemical information is coded by using different pair combinations of atomic weightings (mass, polarizability, vdW volume, and electronegativity). The results of QSAR studies of tyrosinase inhibitors using the new MDs and linear discriminant analysis (LDA) demonstrate the ability of the bilinear indices in testing biological properties. A database of 246 structurally diverse tyrosinase inhibitors was assembled. An inactive set of 412 drugs with other clinical uses was used; both active and inactive sets were processed by hierarchical and partitional cluster analyses to design training and predicting sets. Twelve LDA-based QSAR models were obtained, the first six using the nonstochastic total and local bilinear indices and the last six with the stochastic MDs. The discriminant models were applied; globally good classifications of 99.58 and 89.96 % were observed for the best nonstochastic and stochastic bilinear indices models in the training set along with high Matthews correlation coefficients (C) of 0.99 and 0.79, respectively, in the learning set. External prediction sets used to validate the models obtained were correctly classified, with accuracies of 100 and 87.78 %, respectively, yielding C values of 1.00 and 0.73. This subset contains 180 active and inactive compounds not considered to fit the models. A simulated virtual screen demonstrated this approach in searching tyrosinase inhibitors from compounds never considered in either training or predicting series. These fitted models permitted the selection of new cycloartane compounds isolated from herbal plants as new tyrosinase inhibitors. A good correspondence between theoretical and experimental inhibitory effects on tyrosinase was observed; compound CA6 (IC(50)=1.32 microM) showed higher activity than the reference compounds kojic acid (IC(50)=16.67 microM) and L-mimosine (IC(50)=3.68 microM).

Estimation of ADME Properties in Drug Discovery: Predicting Caco-2 Cell Permeability Using Atom-Based Stochastic and Non-stochastic Linear Indices

The in vitro determination of the permeability through cultured Caco-2 cells is the most often-used in vitro model for drug absorption. In this report, we use the largest data set of measured P(Caco-2), consisting of 157 structurally diverse compounds. Linear discriminant analysis (LDA) was used to obtain quantitative models that discriminate higher absorption compounds from those with moderate-poorer absorption. The best LDA model has an accuracy of 90.58% and 84.21% for training and test set. The percentage of good correlation, in the virtual screening of 241 drugs with the reported values of the percentage of human intestinal absorption (HIA), was greater than 81%. In addition, multiple linear regression models were developed to predict Caco-2 permeability with determination coefficients of 0.71 and 0.72. Our method compares favorably with other approaches implemented in the Dragon software, as well as other methods from the international literature. These results suggest that the proposed method is a good tool for studying the oral absorption of drug candidates.

Unified QSAR approach to antimicrobials. Part 3: first multi-tasking QSAR model for input-coded prediction, structural back-projection, and complex networks clustering of antiprotozoal compounds

Several pathogen parasite species show different susceptibilities to different antiparasite drugs. Unfortunately, almost all structure-based methods are one-task or one-target Quantitative Structure-Activity Relationships (ot-QSAR) that predict the biological activity of drugs against only one parasite species. Consequently, multi-tasking learning to predict drugs activity against different species by a single model (mt-QSAR) is vitally important. In the two previous works of the present series we reported two single mt-QSAR models in order to predict the antimicrobial activity against different fungal (Bioorg. Med. Chem.2006, 14, 5973-5980) or bacterial species (Bioorg. Med. Chem.2007, 15, 897-902). These mt-QSARs offer a good opportunity (unpractical with ot-QSAR) to construct drug-drug similarity Complex Networks and to map the contribution of sub-structures to function for multiple species. These possibilities were unattended in our previous works. In the present work, we continue this series toward other important direction of chemotherapy (antiparasite drugs) with the development of an mt-QSAR for more than 500 drugs tested in the literature against different parasites. The data were processed by Linear Discriminant Analysis (LDA) classifying drugs as active or non-active against the different tested parasite species. The model correctly classifies 212 out of 244 (87.0%) cases in training series and 207 out of 243 compounds (85.4%) in external validation series. In order to illustrate the performance of the QSAR for the selection of active drugs we carried out an additional virtual screening of antiparasite compounds not used in training or predicting series; the model recognized 97 out of 114 (85.1%) of them. We also give the procedures to construct back-projection maps and to calculate sub-structures contribution to the biological activity. Finally, we used the outputs of the QSAR to construct, by the first time, a multi-species Complex Networks of antiparasite drugs. The network predicted has 380 nodes (compounds), 634 edges (pairs of compounds with similar activity). This network allows us to cluster different compounds and identify on average three known compounds similar to a new query compound according to their profile of biological activity. This is the first attempt to calculate probabilities of antiparasitic action of drugs against different parasites.

Atom-based stochastic and non-stochastic 3D-chiral bilinear indices and their applications to central chirality codification

Non-stochastic and stochastic 2D bilinear indices have been generalized to codify chemical structure information for chiral drugs, making use of a trigonometric 3D-chirality correction factor. In order to evaluate the effectiveness of this novel approach in drug design we have modeled the angiotensin-converting enzyme inhibitory activity of perindoprilate's sigma-stereoisomers combinatorial library. Two linear discriminant analysis models, using non-stochastic and stochastic linear indices, were obtained. The models had shown an accuracy of 95.65% for the training set and 100% for the external prediction set. Next the prediction of the sigma-receptor antagonists of chiral 3-(3-hydroxyphenyl)piperidines by multiple linear regression analysis was carried out. Two statistically significant QSAR models were obtained when non-stochastic (R(2)=0.953 and s=0.238) and stochastic (R(2)=0.961 and s=0.219) 3D-chiral bilinear indices were used. These models showed adequate predictive power (assessed by the leave-one-out cross-validation experiment) yielding values of q(2)=0.935 (s(cv)=0.259) and q(2)=0.946 (s(cv)=0.235), respectively. Finally, the prediction of the corticosteroid-binding globulin binding affinity of steroids set was performed. The obtained results are rather similar to most of the 3D-QSAR approaches reported so far. The validation of this method was achieved by comparison with previous reports applied to the same data set. The non-stochastic and stochastic 3D-chiral linear indices appear to provide a very interesting alternative to other more common 3D-QSAR descriptors.

Computational chemistry approach to protein kinase recognition using 3D stochastic van der Waals spectral moments

Three-dimensional (3D) protein structures now frequently lack functional annotations because of the increase in the rate at which chemical structures are solved with respect to experimental knowledge of biological activity. As a result, predicting structure-function relationships for proteins is an active research field in computational chemistry and has implications in medicinal chemistry, biochemistry and proteomics. In previous studies stochastic spectral moments were used to predict protein stability or function (González-Díaz, H. et al. Bioorg Med Chem 2005, 13, 323; Biopolymers 2005, 77, 296). Nevertheless, these moments take into consideration only electrostatic interactions and ignore other important factors such as van der Waals interactions. The present study introduces a new class of 3D structure molecular descriptors for folded proteins named the stochastic van der Waals spectral moments ((o)beta(k)). Among many possible applications, recognition of kinases was selected due to the fact that previous computational chemistry studies in this area have not been reported, despite the widespread distribution of kinases. The best linear model found was Kact = -9.44 degrees beta(0)(c) +10.94 degrees beta(5)(c) -2.40 degrees beta(0)(i) + 2.45 degrees beta(5)(m) + 0.73, where core (c), inner (i) and middle (m) refer to specific spatial protein regions. The model with a high Matthew's regression coefficient (0.79) correctly classified 206 out of 230 proteins (89.6%) including both training and predicting series. An area under the ROC curve of 0.94 differentiates our model from a random classifier. A subsequent principal components analysis of 152 heterogeneous proteins demonstrated that beta(k) codifies information different to other descriptors used in protein computational chemistry studies. Finally, the model recognizes 110 out of 125 kinases (88.0%) in a virtual screening experiment and this can be considered as an additional validation study (these proteins were not used in training or predicting series).

Bond-based 2D TOMOCOMD-CARDD approach for drug discovery: aiding decision-making in 'in silico' selection of new lead tyrosinase inhibitors

In this paper, we present a new set of bond-level TOMOCOMD-CARDD molecular descriptors (MDs), the bond-based bilinear indices, based on a bilinear map similar to those defined in linear algebra. These novel MDs are used here in Quantitative Structure-Activity Relationship (QSAR) studies of tyrosinase inhibitors, for finding functions that discriminate between the tyrosinase inhibitor compounds and inactive ones. In total 14 models were obtained and the best two discriminant functions (Eqs. 32 and 33) shown globally good classification of 91.00% and 90.17%, respectively, in the training set. The test set had accuracies of 93.33% and 88.89% for the models 32 and 33, correspondingly. A simulated virtual screening was also carried out to prove the quality of the determined models. In a final step, the fitted models were used in the biosilico identification of new synthesized tetraketones, where a good agreement could be observed between the theoretical and experimental results. Four compounds of the novel bioactive chemicals discovered as tyrosinase inhibitors: TK10 (IC(50) = 2.09 microM), TK11 (IC(50) = 2.61 microM), TK21 (IC(50) = 2.06 microM), TK23 (IC(50) = 3.19 microM), showed more potent activity than L-mimose (IC(50) = 3.68 microM). Besides, for this study a heterogeneous database of tyrosinase inhibitors was collected, and could be a useful tool for the scientist in the domain of tyrosinase enzyme researches. The current report could help to shed some clues in the identification of new chemicals that inhibits enzyme tyrosinase, for entering in the pipeline of drug discovery development.

Unified QSAR approach to antimicrobials. Part 2: Predicting activity against more than 90 different species in order to halt antibacterial resistance

There are many different kinds of pathogenic bacteria species with very different susceptibility profiles to different antibacterial drugs. One limitation of QSAR models is that they consider the biological activity of drugs against only one species of bacteria. In a previous paper, we developed a unified Markov model to describe the biological activity of different drugs tested in the literature against some antimicrobial species. Consequently, predicting the probability with which a drug is active against different species of bacteria with a single unified model is a goal of major importance. The work described here develops the unified Markov model to describe the biological activity of more than 70 drugs from the literature tested against 96 species of bacteria. We applied linear discriminant analysis (LDA) to classify drugs as active or inactive against the different tested bacterial species. The model correctly classified 199 out of 237 active compounds (83.9%) and 168 out of 200 inactive compounds (84%). Overall training predictability was 84% (367 out of 437 cases). Validation of the model was carried out using an external predicting series, with the model classifying 202 out of 243 (i.e., 83.13%) of the compounds. In order to show how the model functions in practice, a virtual screening was carried out and the model recognized as active 84.5% (480 out of 568) antibacterial compounds not used in the training or predicting series. The current study is an attempt to calculate within a unified framework the probabilities of antibacterial action of drugs against many different species.

Predicting antitrichomonal activity: A computational screening using atom-based bilinear indices and experimental proofs

Existing Trichomonas vaginalis therapies are out of reach for most trichomoniasis people in developing countries and, where available, they are limited by their toxicity (mainly in pregnant women) and their cost. New antitrichomonal agents are needed to combat emerging metronidazole-resistant trichomoniasis and reduce the side effects associated with currently available drugs. Toward this end, atom-based bilinear indices, a new TOMOCOMD-CARDD molecular descriptor, and linear discriminant analysis (LDA) were used to discover novel, potent, and non-toxic lead trichomonacidal chemicals. Two discriminant functions were obtained with the use of non-stochastic and stochastic atom-type bilinear indices for heteroatoms and H-bonding of heteroatoms. These atomic-level molecular descriptors were calculated using a weighting scheme that includes four atomic labels, namely atomic masses, van der Waals volumes, atomic polarizabilities, and atomic electronegativities in Pauling scale. The obtained LDA-based QSAR models, using non-stochastic and stochastic indices, were able to classify correctly 94.51% (90.63%) and 93.41% (93.75%) of the chemicals in training (test) sets, respectively. They showed large Matthews' correlation coefficients (C); 0.89 (0.79) and 0.87 (0.85), for the training (test) sets, correspondingly. The result of predictions on the 15% full-out cross-validation test also evidenced the robustness and predictive power of the obtained models. In addition, canonical regression analyses corroborated the statistical quality of these models (R(can) of 0.749 and of 0.845, correspondingly); they were also used to compute biological activity canonical scores for each compound. On the other hand, a close inspection of the molecular descriptors included in both equations showed that several of these molecular fingerprints are strongly interrelated with each other. Therefore, these models were orthogonalized using the Randić orthogonalization procedure. These classification functions were then applied to find new lead antitrichomonal agents and six compounds were selected as possible active compounds by computational screening. The designed compounds were synthesized and tested for in vitro activity against T. vaginalis. Out of the six compounds that were designed, and synthesized, three molecules (chemicals VA5-5a, VA5-5c, and VA5-12b) showed high to moderate cytocidal activity at the concentration of 10 microg/ml, other two compounds (VA5-8pre and VA5-8) showed high cytocidal and cytostatic activity at the concentration of 100 microg/ml and 10 microg/ml, correspondingly, and the remaining chemical (compound VA5-5e) was inactive at these assayed concentrations. Nonetheless, these compounds possess structural features not seen in known trichomonacidal compounds and thus can serve as excellent leads for further optimization of antitrichomonal activity. The LDA-based QSAR models presented here can be considered as a computer-assisted system that could potentially significantly reduce the number of synthesized and tested compounds and increase the chance of finding new chemical entities with antitrichomonal activity.

Unify QSAR approach to antimicrobials. Part 1: Predicting antifungal activity against different species

Most of up-to-date reported molecular descriptors encode only information about the molecular structure. In previous papers, we have extended stochastic descriptors to encode additional information such as target site, partition system, or biological species [Bioorg. Med. Chem. Lett.2005, 15, 551; Bioorg. Med. Chem. 2005, 13, 1119]. This work develops an unify Markov model to describe with a single linear equation the biological activity of 74 drugs tested in the literature against some of the fungi species selected from a list of 87 species (491 cases in total). The data were processed by linear discriminant analysis (LDA) classifying drugs as active or non-active against the different tested fungi species. The model correctly classifies 338 out of 368 active compounds (91.85%) and 89 out of 123 non-active compounds (72.36%). Overall training predictability was 86.97% (427 out of 491 compounds). Validation of the model was carried out by means of leave-species-out (LSO) procedure. After elimination step-by-step of all drugs tested against one specific species, we record the percentage of good classification of leave-out compounds (LSO-predictability). In addition, robustness of the model to the elimination of the compounds (LSO-robustness) was considered. This aspect was considered as the variation of the percentage of good classification of the modified model (Delta) in LSO with respect to the original one. Average LSO-predictability was 86.41+/-0.95% (average+/-SD) and Delta = -0.55%, being 6 the average number of drugs tested against each fungi species. Results for some of the 87 studied species were Candida albicans: 43 tested compounds, 100% of LSO-predictability, Delta = -3.49%; Candida parapsilosis 23, 100%, Delta = -0.86%; Aspergillus fumigatus 21, 95.20%, Delta = 0.05%; Microsporum canis 12, 91.60%, Delta = -2.84%; Trichophyton mentagrophytes 11, 100%, Delta = -0.51%; Cryptococcus neoformans 10, 90%, Delta = -0.90%. The present one is the first reported unify model that allows one predicting antifungal activity of any organic compound against a very large diversity of fungi pathogens.

Simple Stochastic Fingerprints Towards Mathematical Modeling in Biology and Medicine 2. Unifying Markov Model for Drugs Side Effects

Most of present mathematical models for biological activity consider just the molecular structure. In the present article we pretend extending the use of Markov chain models to define novel molecular descriptors, which consider in addition other parameters like target site or biological effect. Specifically, this mathematical model takes into consideration not only the molecular structure but the specific biological system the drug affects too. Herein, a general Markov model is developed that describes 19 different drugs side effects grouped in eight affected biological systems for 178 drugs, being 270 cases finally. The data was processed by linear discriminant analysis (LDA) classifying drugs according to their specific side effects, forward stepwise was fixed as strategy for variables selection. The average percentage of good classification and number of compounds used in the training/predicting sets were 100/95.8% for endocrine manifestations, (18 out of 18)/(13 out of 14); 90.5/92.3% for gastrointestinal manifestations, (38 out of 42)/(30 out of 32); 88.5/86.5% for systemic phenomena, (23 out of 26)/(17 out of 20); 81.8/77.3% for neurological manifestations, (27 out of 33)/(19 out of 25); 81.6/86.2% for dermal manifestations, (31 out of 38)/(25 out of 29); 78.4/85.1% for cardiovascular manifestation, (29 out of 37)/(24 out of 28); 77.1/75.7% for breathing manifestations, (27 out of 35)/(20 out of 26) and 75.6/75% for psychiatric manifestations, (31 out of 41)/(23 out of 31). Additionally a back-projection analysis (BPA) was carried out for two ulcerogenic drugs to prove in structural terms the physical interpretation of the models obtained. This article develops a mathematical model that encompasses a large number of drugs side effects grouped in specifics biological systems using stochastic absolute probabilities of interaction ((A)pi(k)(j)) by the first time.

Non-stochastic quadratic fingerprints and LDA-based QSAR models in hit and lead generation through virtual screening: theoretical and experimental assessment of a promising method for the discovery of new antimalarial compounds

In order to explore the ability of non-stochastic quadratic indices to encode chemical information in antimalarials, four quantitative models for the discrimination of compounds having this property were generated and statistically compared. Accuracies of 90.2% and 83.3% for the training and test sets, respectively, were observed for the best of all the models, which included non-stochastic quadratic fingerprints weighted with Pauling electronegativities. With a comparative purpose and as a second validation experiment, an exercise of virtual screening of 65 already-reported antimalarials was carried out. Finally, 17 new compounds were classified as either active/inactive ones and experimentally evaluated for their potential antimalarial properties on the ferriprotoporphyrin (FP) IX biocrystallization inhibition test (FBIT). The theoretical predictions were in agreement with the experimental results. In the assayed test compound C5 resulted more active than chloroquine. The current result illustrates the usefulness of the TOMOCOMD-CARDD strategy in rational antimalarial-drug design, at the time that it introduces a new family of organic compounds as starting point for the development of promising antimalarials.

Atom-based 3D-chiral quadratic indices. Part 2: Prediction of the corticosteroid-binding globulinbinding affinity of the 31 benchmark steroids data set

A quantitative structure-activity relationship (QSAR) study to predict the relative affinities of the steroid 'benchmark' data set to the corticosteroid-binding globulin (CBG) is described. It is shown that the 3D-chiral quadratic indices closely correlate with the measured CBG affinity values for the 31 steroids. The calculated descriptors were correlated with biological data through multiple linear regressions. Two statistically significant models were obtained when non-stochastic (R = 0.924 and s = 0.46) as well as stochastic (R = 0.929 and s = 0.46) 3D-chiral quadratic indices were used. A leave-one-out (LOO) approach to model validation is used here; the best results obtained in the cross-validation procedure with non-stochastic (q2 = 0.781) and stochastic (q2 = 0.735) 3D-chiral quadratic indices are better or similar to most of the 3D-QSAR approaches reported so far. These results support the idea that the 3D-chiral quadratic indices may be helpful in prediction of the corticosteroid-binding affinity for new compounds.

New ligand-based approach for the discovery of antitrypanosomal compounds

The antitrypanosomal activity of 10 already synthesized compounds was in silico predicted as well as in vitro and in vivo explored against Trypanosoma cruzi. For the computational study, an approach based on non-stochastic linear fingerprints to the identification of potential antichagasic compounds is introduced. Molecular structures of 66 organic compounds, 28 with antitrypanosomal activity and 38 having other clinical uses, were parameterized by means of the TOMOCOMD-CARDD software. A linear classification function was derived allowing the discrimination between active and inactive compounds with a confidence of 95%. As predicted, seven compounds showed antitrypanosomal activity (%AE>70) against epimastigotic forms of T. cruzi at a concentration of 100mug/mL. After an unspecific cytotoxic assay, three compounds were evaluated against amastigote forms of the parasite. An in vivo test was carried out for one of the studied compounds.

QSAR study for mycobacterial promoters with low sequence homology

The general belief is that quantitative structure-activity relationship (QSAR) techniques work only for small molecules and, protein sequences or, more recently, DNA sequences. However, with non-branched graph for proteins and DNA sequences the QSAR often have to be based on powerful non-linear techniques such as support vector machines. In our opinion, linear QSAR models based on RNA could be useful to assign biological activity when alignment techniques fail due to low sequence homology. The idea bases the high level of branching for the RNA graph. This work introduces the so-called Markov electrostatic potentials (k)xi(M) as a new class of RNA 2D-structure descriptors. Subsequently, we validate these molecular descriptors solving a QSAR classification problem for mycobacterial promoter sequences (mps), which constitute a very low sequence homology problem. The model developed (mps=-4.664.(0)xi(M)+0. 991.(1)xi(M)-2.432) was intended to predict whether a naturally occurring sequence is an mps or not on the basis of the calculated (k)xi(M) value for the corresponding RNA secondary structure. The RNA-QSAR approach recognises 115/135mps (85.2%) and 100% of control sequences. Average predictability and robustness were greater than 95%. A previous non-linear model predicts mps with a slightly higher accuracy (97%) but uses a very large parameter space for DNA sequences. Conversely, the (k)xi(M)-based RNA-QSAR encodes more structural information and needs only two variables.

Stochastic entropy QSAR for the in silico discovery of anticancer compounds: Prediction, synthesis, and in vitro assay of new purine carbanucleosides

A Markov model based QSAR is introduced for the rational selection of anticancer compounds. The model discriminates 90.3% of 226 structurally heterogeneous anticancer/non-anticancer compounds in training series. External validation series were used to validate the model; the 91.8% containing 85 compounds, not considered to fit the model, were correctly classified. The model developed is afterwards used in a simulation of a virtual search for anticancer compounds never considered either in training or in predicting series. The 87.7% of the 213 anticancer compounds used in this simulated search were correctly classified. The model also shows high values for specificity (0.89), sensitivity (0.91), and Mathews correlation coefficient (0.79). In addition, the present model compares better-to-similar with respect to other four models elsewhere reported if one takes into consideration 26 comparison parameters. Finally, we exemplify the use of the model in practice with the design of a new series of carbanucleosides. The compounds evaluated with the model were synthesized and experimentally assayed for their antitumor effects on the proliferation of murine leukemia cells (L1210/0) and human T-lymphocyte cells (CEM/0 and Molt4/C8). The more interesting activity was detected for the compound 5a with a predicted probability of 80.2% and IC(50) = 27.0, 27.2, and 29.4 microM, respectively, against the above-mentioned cellular lines. These values are comparable to those for the control compound Ara-A.

A novel non-stochastic quadratic fingerprints-based approach for the 'in silico' discovery of new antitrypanosomal compounds

A non-stochastic quadratic fingerprints-based approach is introduced to classify and design, in a rational way, new antitrypanosomal compounds. A data set of 153 organic chemicals, 62 with antitrypanosomal activity and 91 having other clinical uses, was processed by a k-means cluster analysis to design training and predicting data sets. Afterwards, a linear classification function was derived allowing the discrimination between active and inactive compounds. The model classifies correctly more than 93% of chemicals in both training and external prediction groups. The predictability of this discriminant function was also assessed by a leave-group-out experiment, in which 10% of the compounds were removed at random at each time and their activity predicted a posteriori. In addition, a comparison with models generated using four well-known families of 2D molecular descriptors was carried out. As an experiment of virtual lead generation, the present TOMOCOMD approach was finally satisfactorily applied on the virtual evaluation of 10 already synthesized compounds. The in vitro antitrypanosomal activity of this series against epimastigotes forms of Trypanosomal cruzi was assayed. The model was able to predict correctly the behaviour of these compounds in 90% of the cases.

Non-stochastic and stochastic linear indices of the molecular pseudograph’s atom-adjacency matrix: a novel approach for computational in silico screening and “rational” selection of new lead antibacterial agents

A novel approach (TOMOCOMD-CARDD) to computer-aided "rational" drug design is illustrated. This approach is based on the calculation of the non-stochastic and stochastic linear indices of the molecular pseudograph's atom-adjacency matrix representing molecular structures. These TOMOCOMD-CARDD descriptors are introduced for the computational (virtual) screening and "rational" selection of new lead antibacterial agents using linear discrimination analysis. The two structure-based antibacterial-activity classification models, including non-stochastic and stochastic indices, classify correctly 91.61% and 90.75%, respectively, of 1525 chemicals in training sets. These models show high Matthews correlation coefficients (MCC=0.84 and 0.82). An external validation process was carried out to assess the robustness and predictive power of the model obtained. These QSAR models permit the correct classification of 91.49% and 89.31% of 505 compounds in an external test set, yielding MCCs of 0.84 and 0.79, respectively. The TOMOCOMD-CARDD approach compares satisfactorily with respect to nine of the most useful models for antimicrobial selection reported to date. Finally, an in silico screening of 87 new chemicals reported in the anti-infective field with antibacterial activities is developed showing the ability of the TOMOCOMD-CARDD models to identify new lead antibacterial compounds.

New tyrosinase inhibitors selected by atomic linear indices-based classification models

In the present report, the use of the atom-based linear indices for finding functions that discriminate between the tyrosinase inhibitor compounds and inactive ones is presented. In this sense, discriminant models were applied and globally good classifications of 93.51% and 92.46% were observed for non-stochastic and stochastic linear indices best models, respectively, in the training set. The external prediction sets had accuracies of 91.67% and 89.44%. In addition, these fitted models were used in the screening of new cycloartane compounds isolated from herbal plants. A good behavior is shown between the theoretical and experimental results. These results provide a tool that can be used in the identification of new tyrosinase inhibitor compounds.

Linear indices of the 'macromolecular graph's nucleotides adjacency matrix' as a promising approach for bioinformatics studies. Part 1: prediction of paromomycin's affinity constant with HIV-1 psi-RNA packaging region

The design of novel anti-HIV compounds has now become a crucial area for scientists around the world. In this paper a new set of macromolecular descriptors (that are calculated from the macromolecular graph's nucleotide adjacency matrix) of relevance to nucleic acid QSAR/QSPR studies, nucleic acids' linear indices. A study of the interaction of the antibiotic Paromomycin with the packaging region of the HIV-1 psi-RNA has been performed as example of this approach. A multiple linear regression model predicted the local binding affinity constants [Log K (10(-4) M(-1))] between a specific nucleotide and the aforementioned antibiotic. The linear model explains more than 87% of the variance of the experimental Log K (R = 0.93 and s = 0.102 x 10(-4) M(-1)) and leave-one-out press statistics evidenced its predictive ability (q2 = 0.82 and s(cv) = 0.108 x 10(-4) M(-1)). The comparison with other approaches (macromolecular quadratic indices, Markovian Negentropies and 'stochastic' spectral moments) reveals a good behavior of our method.

A linear discrimination analysis based virtual screening of trichomonacidal lead-like compounds: Outcomes of in silico studies supported by experimental results

A computational (virtual) screening test to identify potential trichomonacidals has been developed. Molecular structures of trichomonacidal and non-trichomonacidal drugs were represented using stochastic and non-stochastic atom-based quadratic indices and a linear discrimination analysis (LDA) was trained to classify molecules regarding their antiprotozoan activity. Validation tests revealed that our LDA-QSAR models recognize at least 88.24% of trichomonacidal lead-like compounds and suggest using this methodology in virtual screening protocols. These classification functions were then applied to find new lead antitrichomonal compounds. In this connection, the biological assays of eight compounds, selected by computational screening using the present models, give good results (87.50% of good classification). In general, most of the compounds showed high activity against Trichomonas vaginalis at the concentration of 100 microg/ml and low cytotoxicity to this concentration. In particular, two heterocyclic derivatives (VA7-67 and VA7-69) maintained their efficacy at 10 microg/ml with an important trichomonacidal activity (100.00% of reduction), but it is remarkable that the compound VA7-67 did not show cytotoxic effects in macrophage cultivations. This result opens a door to a virtual study considering a higher variability of the structural core already evaluated, as well as of other chemicals not included in this study.

3D-chiral Atom, Atom-type, and Total Non-stochastic and Stochastic Molecular Linear Indices and their Applications to Central Chirality Codification

Non-stochastic and stochastic 2D linear indices have been generalized to codify chemical structure information for chiral drugs, making use of a trigonometric 3D-chirality correction factor. These descriptors circumvent the inability of conventional 2D non-stochastic [Y. Marrero-Ponce. J. Chem. Inf. Comp., Sci. l 44 (2004) 2010] and stochastic [Y. Marrero-Ponce, et al. Bioorg. Med. Chem., 13 (2005) 1293] linear indices to distinguish sigma-stereoisomers. In order to test the potential of this novel approach in drug design we have modelled the angiotensin-converting enzyme inhibitory activity of perindoprilate's sigma-stereoisomers combinatorial library. Two linear discriminant analysis models, using non-stochastic and stochastic linear indices, were obtained. The models showed an accuracy of 100% and 96.65% for the training set; and 88.88% and 100% in the external test set, respectively. Canonical regression analysis corroborated the statistical quality of these models (R(can) of 0.78 and of 0.77) and was also used to compute biology activity canonical scores for each compound. After that, the prediction of the sigma-receptor antagonists of chiral 3-(3-hydroxyphenyl)piperidines by linear multiple regression analysis was carried out. Two statistically significant QSAR models were obtained when non-stochastic (R2 = 0.982 and s = 0.157) and stochastic (R2 = 0.941 and s = 0.267) 3D-chiral linear indices were used. The predictive power was assessed by the leave-one-out cross-validation experiment, yielding values of q2 = 0.982 (s(cv) = 0.186) and q2 = 0.90 (s(cv) = 0.319), respectively. Finally, the prediction of the corticosteroid-binding globulin binding affinity of steroids set was performed. The best results obtained in the cross-validation procedure with non-stochastic (q2 = 0.904) and stochastic (q2 = 0.88) 3D-chiral linear indices are rather similar to most of the 3D-QSAR approaches reported so far. The validation of this method was achieved by comparison with previous reports applied to the same data set. The non-stochastic and stochastic 3D-chiral linear indices appear to provide an interesting alternative to other more common 3D-QSAR descriptors.

Atom, atom-type, and total nonstochastic and stochastic quadratic fingerprints: a promising approach for modeling of antibacterial activity

The TOpological MOlecular COMputer Design (TOMOCOMD-CARDD) approach has been introduced for the classification and design of antimicrobial agents using computer-aided molecular design. For this propose, atom, atom-type, and total quadratic indices have been generalized to codify chemical structure information. In this sense, stochastic quadratic indices have been introduced for the description of the molecular structure. These stochastic fingerprints are based on a simple model for the intramolecular movement of all valence-bond electrons. In this work, a complete data set containing 1006 antimicrobial agents is collected and presented. Two structure-based antibacterial activity classification models have been generated. The models (including nonstochastic and stochastic indices) classify correctly more than 90% of 1525 compounds in training sets. These models permit the correct classification of 92.28% and 89.31% of 505 compounds in an external test sets. The TOMOCOMD-CARDD approach, also, satisfactorily compares with respect to nine of the most useful models for antimicrobial selection reported to date. Finally, a virtual screening of 87 new compounds reported in the antiinfective field with antibacterial activities is developed showing the ability of the TOMOCOMD-CARDD models to identify new leads as antibacterial.

QSAR for anti-RNA-virus activity, synthesis, and assay of anti-RSV carbonucleosides given a unified representation of spectral moments, quadratic, and topologic indices

The unified representation of spectral moments, classic topologic indices, quadratic indices, and stochastic molecular descriptors show that all these molecular descriptors lie within the same family. Consequently, the same prior probability for a successful quantitative-structure-activity-relationship (QSAR) may be expected irrespective of which indices are selected. Herein, we used stochastic spectral moments as molecular descriptors to seek a QSAR using a database of 221 bioactive compounds previously tested against diverse RNA-viruses and 402 nonactive ones. The QSAR model thus obtained correctly classifies 90.9% of compounds in training. The model also correctly classifies a total of 87.9% of 207 compounds on additional external predicting series, 73 of them having anti-RNA-virus activity and 134 nonactive ones. In addition, all compounds were regrouped into five different subsets for leave-group-out studies: (1) anti-influenza, (2) anti-picornavirus, (3) anti-paramyxovirus, (4) anti-RSV/anti-influenza, and (5) broad range anti-RNA-virus activity. The model has retained overall accuracies of about 90% on these studies validating model robustness. Finally, we exemplify the practical use of the model with the discovery of compounds 124 and 128. These compounds presented MIC50 values=3.2 and 8 microg/mL against respiratory syncytial virus (RSV) respectively. Both compounds also have low cytotoxicity expressed by their Minimal Cytotoxic Concentrations >400 microg/mL for HeLa cells. The present approach represents an effort toward a formalization and application of molecular indices in bioorganic and medicinal chemistry.

Protein quadratic indices of the "macromolecular pseudograph's alpha-carbon atom adjacency matrix". 1. Prediction of Arc repressor alanine-mutant's stability

This report describes a new set of macromolecular descriptors of relevance to protein QSAR/QSPR studies, protein's quadratic indices. These descriptors are calculated from the macromolecular pseudograph's alpha-carbon atom adjacency matrix. A study of the protein stability effects for a complete set of alanine substitutions in Arc repressor illustrates this approach. Quantitative Structure-Stability Relationship (QSSR) models allow discriminating between near wild-type stability and reduced-stability A-mutants. A linear discriminant function gives rise to excellent discrimination between 85.4% (35/41)and 91.67% (11/12) of near wild-type stability/reduced stability mutants in training and test series, respectively. The model's overall predictability oscillates from 80.49 until 82.93, when n varies from 2 to 10 in leave-n-out cross validation procedures. This value stabilizes around 80.49% when n was > 6. Additionally, canonical regression analysis corroborates the statistical quality of the classification model (Rcanc = 0.72, p-level <0.0001). This analysis was also used to compute biological stability canonical scores for each Arc A-mutant. On the other hand, nonlinear piecewise regression model compares favorably with respect to linear regression one on predicting the melting temperature (tm)of the Arc A-mutants. The linear model explains almost 72% of the variance of the experimental tm (R = 0.85 and s = 5.64) and LOO press statistics evidenced its predictive ability (q2 = 0.55 and scv = 6.24). However, this linear regression model falls to resolve t(m) predictions of Arc A-mutants in external prediction series. Therefore, the use of nonlinear piecewise models was required. The tm values of A-mutants in training (R = 0.94) and test(R = 0.91) sets are calculated by piecewise model with a high degree of precision. A break-point value of 51.32 degrees C characterizes two mutants' clusters and coincides perfectly with the experimental scale. For this reason, we can use the linear discriminant analysis and piecewise models in combination to classify and predict the stability of the mutants' Arc homodimers. These models also permit the interpretation of the driving forces of such a folding process. The models include protein's quadratic indices accounting for hydrophobic (z1), bulk-steric (z2), and electronic (z3) features of the studied molecules. Preponderance of z1 and z3 over z2 indicates the higher importance of the hydrophobic and electronic side chain terms in the folding of the Arc dimer. In this sense, developed equations involve short-reaching (k < or = 3), middle- reaching (3 < k < or = 7) and far-reaching (k= 8 or greater) z1, 2, 3-protein's quadratic indices. This situation points to topologic/topographic protein's backbone interactions control of the stability profile of wild-type Arc and its A-mutants. Consequently, the present approach represents a novel and very promising way to mathematical research in biology sciences.