From molecular connectivity indices to semiempirical connectivity terms: recent trends in graph theoretical descriptors

Correlation of the Molecular Cross-Sectional Area of Organic Monofunctional Compounds with Topological Descriptors

The molecular cross-sectional area (σ) has proved to be an interesting molecular measure not only in the field of adsorption phenomena on solids but also in biochemistry, physiology, or surfactant chemistry. The existing methods to estimate the cross-sectional areas are either not readily applicable or can only be applied to a limited number of compounds. The aim of this work was to describe a method, as general as possible, quick and easy to perform. To that end, the molecular cross-sectional areas were correlated with topological indices. The Emmett-Brunauer formula was used to calculate the reference cross-sectional areas (σEB) of 431 compounds. The correlations of the Wiener (W), hyper-Wiener (WW), Balaban (J), and Randić (χ) indices with σEB were compared for n-alkanes as well as branched and cyclic alkanes. Only the Wiener and hyper-Wiener indices correlated well with σEB, with the data being best fitted by power law regression curves. The lower degeneracy of the hyper-Wiener index did not translate into any significant gain of performance when correlated with σEB. Following the parsimony principle, the less complex Wiener index was thus selected to correlate with the σEB of compounds representing 31 other monofunctional and structural families. The integration of all the compound families into a single curve allowed a quick rough estimation of the cross-sectional areas. The specific reference equations σEB = qWp were determined for the 34 selected families, allowing the fast and reliable calculation of the cross-sectional area of any monofunctional compound.

Bridging the Monomer to Polymer Gap in Radical Polymer Design

Radical polymers bearing open-shell moieties at pendant sites exhibit unique redox and optoelectronic properties that are promising for many organic electronic applications. Nevertheless, gaps remain in relating the electronic properties of repeat units, which can be easily calculated, to the condensed-phase charge transport behaviors of these materials. To address this gap, we have performed the first quantum chemical study on a broad swathe of radical polymer design space that explicitly includes the coupling between polymer constraints and radical-mediated intramolecular charge transfer. For this purpose, a chemical space of 64 radical polymer chemistries was constructed based on varying backbone units, open-shell chemistries, and spacer units between the backbone and the radical groups. For each combination of backbone, radical, and spacer, comprehensive conformational sampling was used to calculate expected values of intrachain charge transport using several complementary metrics, including the end-to-end thermal Green's function, Delta-Wye transformed inverse resistance, and the Kirchhoff transport index. We observe that charge transport in radical polymers is primarily driven by the choice radical chemistry, which influences the optimal choice of backbone chemistry and spacer group that mediate radical alignment and avoid the formation of undesired trap states. Given the limited exploration of radical chemistries beyond the TEMPO radical for this class of materials, these findings suggest tremendous opportunities exist for synthetic exploration in radical polymers.

The Singularity of Four Kinds of Tricyclic Graphs

A singular graph G, defined when its adjacency matrix is singular, has important applications in mathematics, natural sciences and engineering. The chemical importance of singular graphs lies in the fact that if the molecular graph is singular, the nullity (the number of the zero eigenvalue) is greater than 0, then the corresponding chemical compound is highly reactive or unstable. By this reasoning, chemists have a great interest in this problem. Thus, the problem of characterization singular graphs was proposed and raised extensive studies on this challenging problem thereafter. The graph obtained by conglutinating the starting vertices of three paths Ps1, Ps2, Ps3 into a vertex, and three end vertices into a vertex on the cycle Ca1, Ca2, Ca3, respectively, is denoted as γ(a1,a2,a3,s1,s2,s3). Note that δ(a1,a2,a3,s1,s2)=γ(a1,a2,a3,s1,1,s2), ζ(a1,a2,a3,s)=γ(a1,a2,a3,1,1,s), φ(a1,a2,a3)=γ(a1,a2,a3,1,1,1). In this paper, we give the necessity and sufficiency that the γ−graph, δ−graph, ζ−graph and φ−graph are singular and prove that the probability that a randomly given γ−graph, δ−graph, ζ−graph or φ−graph being singular is equal to 325512,165256,4364, 2132, respectively. From our main results, we can conclude that such a γ−graph(δ−graph, ζ−graph, φ−graph) is singular if at least one cycle is a multiple of 4 in length, and surprisingly, the theoretical probability of these graphs being singular is more than half. This result promotes the understanding of a singular graph and may be promising to propel the solutions to relevant application problems.

Search for H-Bonded Motifs in Liquid Ethylene Glycol Using a Machine Learning Strategy

Trajectories of atomic positions derived from ab initio molecular dynamics (AIMD) simulations of H-bonded liquids contain a wealth of information on dominant structural motifs and recurrent patterns of association. Extracting this information from a detailed search of the trajectories over multiple time frames is, however, a daunting exercise. Here, we use a machine learning strategy based on the neural inspired approach of the self-organizing maps (SOM), a type of artificial neural network that uses unsupervised competitive learning, to analyze the AIMD trajectories of liquid ethylene glycol (EG). The objective was to find whether there are H-bonded fragments, of two or more H-bonded EG molecules, that are recurrent in the liquid and to identify them. The SOM represents a set of high-dimensional data mapped onto a two-dimensional, grid of neurons or nodes, while preserving the topological properties of the input space. We show here that clustering of the fragments by SOM in terms of the molecular conformation of the individual EG molecules of the fragment and their H-bond connectivity pattern facilitates the search for H-bonded motifs. Using this approach, we are able to identify a H-bonded cyclic dimer and a bifurcated H-bonded structure as recurring motifs that appear in the longer H-bonded fragments present in liquid EG.

Reverse Zagreb and Reverse Hyper-Zagreb Indices for Crystallographic Structure of Molecules

In the fields of chemical graph theory, topological index is a type of a molecular descriptor that is calculated based on the graph of a chemical compound. Topological indices help us collect information about algebraic graphs and give us mathematical approach to understand the properties of algebraic structures. With the help of topological indices, we can guess the properties of chemical compounds without performing experiments in wet lab. There are more than 148 topological indices in the literature, but none of them completely give all properties of under study compounds. Together, they do it to some extent; hence, there is always room to introduce new indices. In this paper, we present first and second reserve Zagreb indices and first reverse hyper-Zagreb indices, reverse GA index, and reverse atomic bond connectivity index for the crystallographic structure of molecules. We also present first and second reverse Zagreb polynomials and first and second reverse hyper-Zagreb polynomials for the crystallographic structure of molecules.

Topology-Based Analysis of OTIS (Swapped) Networks OKn and OPn

In the fields of chemical graph theory, topological index is a type of a molecular descriptor that is calculated based on the graph of a chemical compound. In this paper, M-polynomial OKn and OPn networks are computed. The M-polynomial is rich in information about degree-based topological indices. By applying the basic rules of calculus on M-polynomials, the first and second Zagreb indices, modified second Zagreb index, general Randić index, inverse Randić index, symmetric division index, harmonic index, inverse sum index, and augmented Zagreb index are recovered.

On molecular topological properties of diamond-like networks

The Randić (product) connectivity index and its derivative called the sum-connectivity index are well-known topological indices and both of these descriptors correlate well among themselves and with the π-electronic energies of benzenoid hydrocarbons. The general n connectivity of a molecular graph G is defined as [Formula: see text] and the n sum connectivity of a molecular graph G is defined as [Formula: see text], where the paths of length n in G are denoted by [Formula: see text] and the degree of each vertex vi is denoted by di. In this paper, we discuss third connectivity and third sum-connectivity indices of diamond-like networks and compute analytical closed results of these indices for diamond-like networks.

Improved Scaling of Molecular Network Calculations: The Emergence of Molecular Domains

The design of materials needed for the storage, delivery, and conversion of (re)useable energy is still hindered by the lack of new, hierarchical molecular screening methodologies that encode information on more than one length scale. Using a molecular network theory as a foundation, we show that to describe charge transport in disordered materials the network methodology must be scaled-up. We detail the scale-up through the use of adjacency lists and depth first search algorithms for during operations on the adjacency matrix. We consider two types of electronic acceptors, perylenediimide (PDI) and the fullerene derivative phenyl-C61-butyric acid methyl ester (PCBM), and we demonstrate that the method is scalable to length scales relevant to grain boundary and trap formations. Such boundaries lead to a decrease in the percolation ratio of PDI with system size, while the ratio for PCBM remains constant, further quantifying the stable, diverse transport pathways of PCBM and its success as a charge-accepting material.

A Simple Index for Characterizing Charge Transport in Molecular Materials

While advances in quantum chemistry have rendered the accurate prediction of band alignment relatively straightforward, the ability to forecast a noncrystalline, multimolecule system's conductivity possesses no simple computational form. Adapting the theory of classical resistor networks, we develop an index for quantifying charge transport in bulk molecular materials, without the requirement of crystallinity. The basic behavior of this index is illustrated through its application to simple lattices and clusters of common organic photovoltaic molecules, where it is shown to reproduce experimentally known performances for these materials. This development provides a quantitative computational means for determining a priori the bulk charge transport properties of molecular materials.

Two new atom centered fragment descriptors and scoring function enhance classification of antibacterial activity

Classification of pharmacologic activity of a chemical compound is an essential step in any drug discovery process. We develop two new atom-centered fragment descriptors (vertex indices)--one based solely on topological considerations without discriminating atom or bond types, and another based on topological and electronic features. We also assess their usefulness by devising a method to rank and classify molecules with regard to their antibacterial activity. Classification performances of our method are found to be superior compared to two previous studies on large heterogeneous data sets for hit finding and hit-to-lead studies even though we use much fewer parameters. It is found that for hit finding studies topological features (simple graph) alone provide significant discriminating power, and for hit-to-lead process small but consistent improvement can be made by additionally including electronic features (colored graph). Our approach is simple, interpretable, and suitable for design of molecules as we do not use any physicochemical properties. The singular use of vertex index as descriptor, novel range based feature extraction, and rigorous statistical validation are the key elements of this study.

SAR studies on β-cell K(ATP) channel openers

Recently Balaban has introduced a J-type G index for modeling those properties/activities where not only the "shape" but also size of the graph influences the property/activity of organic compounds acting as drugs. Here, for the first time G index has been used for modeling the pancreatic β-cell K(ATP) channel openers viz R/S-3,4-dihydro-2,2-dimethyl-6-halo-4-(substituted phenylaminocarbonylamino)-2H-1-benzopyrans. Modeling technology used in this paper has established that the referred activity is excellently modeled by using G as one of the correlating descriptors. The results are critically discussed using Ridge statistics.

Estimation of Anti-HIV Activity of HEPT Analogues Using MLR, ANN, and SVM Techniques

The present study deals with the estimation of the anti-HIV activity (log1/C) of a large set of 107 HEPT analogues using molecular descriptors which are responsible for the anti-HIV activity. The study has been undertaken by three techniques MLR, ANN, and SVM. The MLR model fits the train set with R²=0.856 while in ANN and SVM with higher values of R² = 0.850, 0.874, respectively. SVM model shows improvement to estimate the anti-HIV activity of trained data, while in test set ANN have higher R² value than those of MLR and SVM techniques. R_m² = metrics and ridge regression analysis indicated that the proposed four-variable model MATS5e, RDF080u, T(O⋯O), and MATS5m as correlating descriptors is the best for estimating the anti-HIV activity (log 1/C) present set of compounds.

Molecular topology as a novel approach for drug discovery

INTRODUCTION

Molecular topology (MT) has emerged in recent years as a powerful approach for the in silico generation of new drugs. One key part of MT is that, in the process of drug design/discovery, there is no need for an explicit knowledge of a drug's mechanism of action unlike other drug discovery methods.

AREAS COVERED

In this review, the authors introduce the topic by explaining briefly the most common methodology used today in drug design/discovery and address the most important concepts of MT and the methodology followed (QSAR equations, LDA, etc.). Furthermore, the significant results achieved, from this approach, are outlined and discussed.

EXPERT OPINION

The results outlined herein can be explained by considering that MT represents a new paradigm in the field of drug design. This means that it is not only an alternative method to the conventional methods, but it is also independent, that is, it represents a pathway to connect directly molecular structure with the experimental properties of the compounds (particularly drugs). Moreover, the process can be realized also in the reverse pathway, that is, designing new molecules from their topological pattern, what opens almost limitless expectations in new drugs development, given that the virtual universe of molecules is much greater than that of the existing ones.

Model of twelve properties of a set of organic solvents with graph-theoretical and/or experimental parameters

Twelve properties of a highly heterogeneous class of organic solvents have been modeled with a graph-theoretical molecular connectivity modified (MC) method, which allows to encode the core electrons and the hydrogen atoms. The graph-theoretical method uses the concepts of simple, general, and complete graphs, where these last types of graphs are used to encode the core electrons. The hydrogen atoms have been encoded by the aid of a graph-theoretical perturbation parameter, which contributes to the definition of the valence delta, delta(v), a key parameter in molecular connectivity studies. The model of the twelve properties done with a stepwise search algorithm is always satisfactory, and it allows to check the influence of the hydrogen content of the solvent molecules on the choice of the type of descriptor. A similar argument holds for the influence of the halogen atoms on the type of core electron representation. In some cases the molar mass, and in a minor way, special "ad hoc" parameters have been used to improve the model. A very good model of the surface tension could be obtained by the aid of five experimental parameters. A mixed model method based on experimental parameters plus molecular connectivity indices achieved, instead, to consistently improve the model quality of five properties. To underline is the importance of the boiling point temperatures as descriptors in these last two model methodologies.

Overcoming tumor drug resistance with C2-modified 10-deacetyl-7-propionyl cephalomannines: a QSAR study

The microtubule-stabilizing taxanes such as paclitaxel and docetaxel are the two most important anticancer drugs currently used in clinics for the treatment of various types of cancers. However, the major common drawbacks of these two drugs are drug resistance, neurotoxicity, substrate for drug transporter P-gp, cross-resistance with other chemotherapeutic agents, low oral bioavailability, and no penetration in the blood-brain barrier (BBB). These limitations have led to the search for new taxane derivatives with improved biological activity. In the present paper, we discuss the quantitative structure-activity relationship (QSAR) studies on a series of C2-modified 10-deacetyl-7-propionyl cephalomannines (IV) with respect to their binding affinities toward beta-tubulin and cytotoxic activities against both drug-sensitive and drug-resistant tumor cells, in which resistance is mediated through either P-gp overexpression or beta-tubulin mutation mechanisms, by the formulation of five QSARs. Hydrophobicity and molar refractivity of the substituents (pi(X) and MR(X)) are found to be the most important determinants for the activity. Parabolic correlations in terms of MR(X) (eqs 2 and 4 ) are encouraging examples in which the optimum values of MR(X) are well-defined. We believe that these two QSAR models may prove to be adequate predictive models that can help to provide guidance in design and synthesis, and subsequently yield very specific cephalomannine derivatives (IV) that may have high biological activities. On the basis of these two QSAR models, 10 cephalomannine analogues (IV-21 to IV-30) are suggested as potential synthetic targets. Internal (cross-validation (q(2)), quality factor (Q), Fischer statistics (F), and Y-randomization) and external validation tests have validated all the QSAR models.

Taxane analogues against lung cancer: a quantitative structure-activity relationship study

Lung cancer is the second most common cancer in both men (after prostate cancer) and women (after breast cancer). The microtubule-stabilizing taxane such as docetaxel is the only agent currently approved for both first- and second-line treatment of advanced non-small cell lung cancer. Although docetaxel has made significant progress in the treatment of lung cancers either using alone or in combination with various novel targeted agents, its use often results in various undesired side-effects. These limitations have led to the search for new taxane derivatives with fewer side-effects, superior pharmacological properties, and improved anticancer activity to maximize the induced benefits for lung cancer patients. Herein, four series of taxane derivatives were used to correlate their inhibitory activities against lung cancer cells with hydrophobic and steric descriptors to gain a better understanding of their chemical-biological interactions. A parabolic correlation with MR(Y) is the most encouraging example, in which the optimum value of this parameter is well defined. On the basis of this quantitative structure-activity relationship model, six compounds (3-23 to 3-28) are suggested as potential synthetic targets. Internal (cross-validation (q(2)), quality factor (Q), Fischer statistics (F ) and Y-randomization) and external validation tests have validated all the quantitative structure-activity relationship models.

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.

Application of molecular topology to the prediction of mosquito repellents of a group of terpenoid compounds

A topological-mathematical model based on multilinear regression analysis has been built to search new terpenoid actives as mosquito repellents. The structural depiction was performed using topological indices, and a four-variable model for the prediction of corrected repellent ratio (r (2) = 0.9672, Q (2) = 0.9435) was selected. The model was checked by cross-validation, internal validation, and randomization test. The results confirm its capability to predict the property analyzed. After carrying out a virtual screening upon such a model, new structures with potential repellent activity against mosquitoes are proposed.

QSAR study of 2-(1-Propylpiperidin-4-yl)-1H-benzimidazole-4-carboxamide as PARP inhibitors for treatment of cancer

Quantitative structure-activity relationship of the 2-(1-propylpiperidin-4-yl)-1H-benzimidazole-4-carboxamide as a potent inhibitor of poly(ADP-ribose) polymerase for cancer treatment was studied. A suitable set of molecular descriptors was calculated and the genetic algorithm was employed to select those descriptors that resulted in the best fitted models. Excellent results were obtained employing multiple linear regressions and critically discussed using a variety of statistical parameters. Furthermore, the model was validated using leave-one-out and leave-group-out cross-validation, external test set and chance correlation. A genetic algorithm-multiple linear regression model with seven selected descriptors was obtained. This model, with high statistical significance (R(2) = 0.935, Q(2)_(LOO)= 0.894, Q(2)_(LGO)= 0.875, F = 53.481), could be used to predict poly(ADP-ribose) polymerase inhibitor activity of the molecules.

Taxane analogues against breast cancer: a quantitative structure-activity relationship study

Breast cancer is the second leading cause of cancer death among women in the United States. Two taxane analogues, taxol and taxotere, are the most important antimitotic drugs currently in clinical use for the treatment of breast cancers. However, recent reports have indicated that the use of these drugs often results in various undesired side effects as well as multi-drug resistance. These limitations have led to the development of new taxane derivatives with fewer side effects, superior pharmacological properties, and improved anticancer activity to maximize the induced benefits for breast cancer patients. Herein, four series of taxane derivatives were used to correlate their inhibitory activities against breast cancer cells with their hydrophobic and steric properties in order to understand their chemical-biological interactions. The resulting QSARs show that the inhibitory activities of taxane analogues against breast cancers are mainly dependent either on their hydrophobicity or the hydrophobic/molar refractivity descriptor of their substituents. A parabolic correlation with MR(Y) is the most encouraging example, in which the optimum value of this parameter is well defined. We believe this correlation may prove to be an adequate predictive model that can help provide guidance in design and synthesis and subsequently yield highly specific compounds that may have high anti-breast-cancer activity with fewer side effects and superior pharmacological properties. On the basis of this QSAR model, five compounds are suggested as potential synthetic targets. Internal (cross-validation (LOO-q(2) and LMO-q(2)), quality factor (Q), Fischer statistics (F), and Y-randomization) and external validation tests have validated all the QSAR models.

Understanding tubulin/microtubule-taxane interactions: a quantitative structure-activity relationship study

For years, the microtubule-stabilizing agents paclitaxel and docetaxel (progenitors of the family of taxanes) have been the most successful anticancer drugs currently used in clinics. However, both drugs are associated with notorious side effects, drug resistance, and cross resistance with other chemotherapeutic agents. These limitations have led to the search for new drugs with improved biological activity. In the present paper, we discuss the interaction of taxanes with the tubulin/microtubule system by the formulation of 6 QSARs. Hydrophobicity of the substituents (pi) is found to be one of the most important determinants of the activity followed by steric parameters. Parabolic correlations (eqs 3 and 7) with B5 and pi are the most encouraging examples, where the optimum values of these parameters are well defined. We believe that these two QSARs may prove to be adequate predictive models that can help to provide guidance in design/synthesis and subsequently yield very specific compounds (IV and VIII) that may have high biological activities. On the basis of these two QSARs 3 and 7, 18 compounds (IV-12- IV-22 and VIII-16- VIII-22) are suggested as potential synthetic targets. Cross-validation, quality factor (Q), Fischer statistics (F), and Y-randomization tests have validated all the QSAR models.

Improvement of charge-transfer indices for multifunctional amino acids: application to lysozyme

Valence topological Charge-Transfer (CT) indices are applied to the calculation of pH at the isoelectric point (pI). The model is generalized for molecules with heteroatoms. The ability of the indices for the description of molecular charge distribution is established by comparing them with the pI of 21 amino acids. Linear correlation models are obtained. The CT indices improve multivariable regression equations for pI. The variance decreases by 95%. No superimposition of the corresponding G(k)-J(k) and G(k)(V)-J(k)(V) pairs is observed in most fits, which diminishes the risk of collinearity. The inclusion of heteroatoms in pi-electron system is beneficial for the description of pI, the because of either the role of the additional p orbitals provided by heteroatom or the role of steric factors in pi-electron conjugation. The use of CT/valence CT indices {G(k), J(k), G(k)(V), J(k)(V)} gives limited results for modelling amino acid pI. The inclusion of the number of acidic/basic groups improves the models. The effect is specially noticeable for amino acids with more than two functional groups. The fitting line for 21 amino acids is used to estimate lysozyme pI replacing (1 + Delta n/n(T)) with (M + Delta n)/n(T). The results for lysozyme fragments can estimate the pI of the whole protein with 1-13% error.

Investigation of DNA‐Binding Properties of Organic Molecules Using Quantitative Structure‐Activity Relationship (QSAR) Models

Due to the great potential of DNA as a receptor, many classes of synthetic and naturally occurring molecules exert their anticancer activities through DNA-binding. In the field of antitumor DNA-binding agents, a number of acridine and anthracycline derivatives are in the market as chemotherapeutic agents. However, the clinical application of such classes of compounds has encountered problems such as multi-drug resistance and secondary and/or collateral effects. Thus, there has been increasing interest in discovering and developing small molecules that are capable of DNA-binding, which will be expected to be used either in place of or in conjunction with, the existing compounds. The interest in the application of the QSAR paradigm has steadily increased in recent decades and we hope it may be useful in the design and development of DNA-binding molecules as new anticancer agents. In the present review, an attempt has been made to understand the DNA-binding properties of different compound series and discussed using 27 QSAR models, which reveal a number of interesting points. The most important determinants for the activity in these models are Hammett electronic (sigma and sigma+), hydrophobic, molar refractivity, and Sterimol width parameters.