Applications of fuzzy theory to spatially resolved analysis of solids

Fuzzy theory is applied to compare response profiles and surfaces in spatially resolved materials analysis. Different options are proposed for the type of membership function, the definition of spreads, and specially defined similarity values. These similarity values are expressed analytically, thus permitting the development of a fast computer program. A special normalization in the calculation of the similarity values allows a set of distributions to be arranged hierarchically by clustering; the distributions are displayed as a dendrogram. The fuzzy techniques are also used to judge the homogeneity of a solid with respect to the distribution of chemical elements on its surface. Depth profiles from Rutherford back-scattering spectrometry and results of surface analysis by spark-source mass spectrometry are investigated as practical examples.     

Critical point representations of electron density maps for the comparison of benzodiazepine-type ligands

A procedure for the comparison of three-dimensional electron density distributions is proposed for similarity searches between pharmacological ligands at various levels of crystallographic resolution. First, a graph representation of molecular electron density distributions is generated using a critical point analysis approach. Pairwise as well as multiple comparisons between the obtained graphs of critical points are then carried out using a Monte Carlo/simulated annealing technique, and results are compared with genetic algorithm solutions.     

Similarity analysis in two and three dimensions using lattice animals and polycubes

The quantification and analysis of molecular similarity are fundamental problems of both theoretical and applied chemistry. The continuum similarity problem of planar domains with Jordan curve boundaries can be discretized and quantified using interior filling animals (square cell configurations). A similar approach is applicable to the continuum similarity problem of formal molecular bodies enclosed by contour surfaces, where interior filling polycubes provide a method for discretization and quantification of molecular similarity in three dimensions. This technique leads to resolution based similarity measures (RBSMs), suitable for automatic, non-visual evaluation of the degree of similarity between shapes of general objects, in particular, of molecular charge distributions, or fused sphere Van der Waals surfaces. Using the framework of the RBSM method, the polycube method of chirality quantification is extended to the quantification of approximate symmetry of molecular electron distributions.     

Automatic search for maximum similarity between molecular electrostatic potential distributions

Summary A new computer program has been developed to automatically obtain the relative position of two molecules in which the similarity between molecular electrostatic-potential distributions is greatest. These distributions are considered in a volume around the molecules, and the similarity is measured by the Spearman rank coefficient. The program has been tested using several pairs of molecules: water vs. water; phenylethylamine and phenylpropylamine vs. benzylamine; and methotrexate vs. dihydrofolic acid.     

Analysis of data fusion methods in virtual screening: theoretical model

This paper presents a theoretical model of how data fusion can be used to combine the results of multiple similarity searches of chemical databases. The model is based on frequency distributions of similarity values that are fused using a multiple integration over regions defined by the particular fusion rule that is being applied. For pairwise fusion, the resulting double integrals are straightforward to evaluate for simple model distributions. Similarity values for recovered-active and recovered-nonactive frequency distributions are independently modeled using a constant background, linearly biased terms, and a first-order correlated term. The model shows that two standard fusion rules can give performance enhancements in some cases but that the results of fusion are dependent on many factors that, taken together, can lead to seemingly inconsistent levels of enhancement.     

Extending molecular similarity to energy surfaces: Boltzmann similarity measures and indices

Maxwell-Boltzmann statistics provides the adequate mathematical background allowing to define similarity measures involving molecular energy surfaces and electrostatic potential maps. Boltzmann similarity measures are described and various illustrative examples are used to show the practical viability of the theory. A new molecular similarity index is also presented. Finally, hybrid measures involving Boltzmann and density distributions are defined.     

Introduction of the conditional correlated Bernoulli model of similarity value distributions and its application to the prospective prediction of fingerprint search performance

A statistical approach named the conditional correlated Bernoulli model is introduced for modeling of similarity scores and predicting the potential of fingerprint search calculations to identify active compounds. Fingerprint features are rationalized as dependent Bernoulli variables and conditional distributions of Tanimoto similarity values of database compounds given a reference molecule are assessed. The conditional correlated Bernoulli model is utilized in the context of virtual screening to estimate the position of a compound obtaining a certain similarity value in a database ranking. Through the generation of receiver operating characteristic curves from cumulative distribution functions of conditional similarity values for known active and random database compounds, one can predict how successful a fingerprint search might be. The comparison of curves for different fingerprints makes it possible to identify fingerprints that are most likely to identify new active molecules in a database search given a set of known reference molecules.     

Structural diversity of small molecule libraries

A novel method for assessing structural diversity is presented. Maximum common subgraph identity is used as the measure of similarity between two chemical structures. A conditional probability treatment of similarity distributions for libraries of chemical structures is used to define diversity. This evaluation method together with the evaluation of traditional physicochemical properties is used to assess a large number of chemical libraries and to understand structural differences between these.     

Balancing the influence of molecular complexity on fingerprint similarity searching

Differences in molecular complexity and size are known to bias the evaluation of fingerprint similarity. For example, complex molecules tend to produce fingerprints with higher bit density than simple ones, which often leads to artificially high similarity values in search calculations. We introduce here a variant of the Tversky coefficient that makes it possible to modulate or eliminate molecular complexity effects when evaluating fingerprint similarity. This has enabled us to study in detail the role of molecular complexity in similarity searching and the relationship between reference and active database compounds. Balancing complexity effects leads to constant distributions of similarity values for reference and database molecules, independent of how compound contributions are weighted. When searching for active compounds with varying complexity, hit rates can be optimized by modulating complexity effects, rather than eliminating them, and adjusting relative compound weights. For reference molecules and active database compounds having different complexity, preferred parameter settings are identified.     

Similarity measures based on Gaussian-type promolecular electron density models: alignment of small rigid molecules

Various molecular similarity measures (overlap, Coulomb, kinetic, electrostatic energy) and similarity indices (Carbó, Hodgkin-Richards, Kulczynski, Shape Tanimoto) are applied to the superposition of 3D promolecular electron density (PED) distributions. The original aspect of the paper lies in the consideration of smoothed PEDs, which allow to decrease the number of local solutions to a superposition problem, together with the use of the less common kinetic and electrostatic energy similarity measures. Results are obtained for a family of five rigid endothiapepsin ligands that were already considered in previous applications, based on graph representations of their PED. In the present work, it is observed that the use of smoothed PED and the kinetic similarity measure, together with the Kulczynski or Shape Tanimoto index, performed the best to align molecules of different sizes.     

MEPSIM: A computational package for analysis and comparison of molecular electrostatic potentials

Summary MEPSIM is a computational system which allows an integrated computation, analysis, and comparison of molecular electrostatic potential (MEP) distributions. It includes several modules. Module MEPPLA supplies MEP values for the points of a grid defined on a plane which is specified by a set of three points. The results of this program can easily be converted into MEP maps using third-parties graphical software. Module MEPMIN allows to find automatically the MEP minima of a molecular system. It supplies the cartesian coordinates of these minima, their values, and all the geometrical relationships between them (distances, angles, and dihedral angles). Module MEPCOMP computes a similarity coefficient between the MEP distributions of two molecules and finds their relative position that maximizes the similarity. Module MEPCONF performs the same process as MEPCOMP, considering not only the relative position of both molecules but also a conformational degree of freedom of one of them. The most recently developed module, MEPPAR, is another modification of MEPCOMP in order to compute the MEP similarity between two molecules, but only taking into account a particular plane. The latter module is particularly useful to compare MEP distributions generated by systems of aromatic rings. MEPSIM can use several wavefunction computation approaches to obtain MEP distributions. MEPSIM has a menu type interface to simplify the following tasks: creation of input files from output files of external programs (GAUSSIAN and AMPAC/MOPAC), setting the parameters for the current computation, and submitting jobs to the batch queues of the computer. MEPSIM has been coded in FORTRAN and its current version runs on VMS/VAX computers.     

Calculating the knowledge-based similarity of functional groups using crystallographic data

A knowledge-based method for calculating the similarity of functional groups is described and validated. The method is based on experimental information derived from small molecule crystal structures. These data are used in the form of scatterplots that show the likelihood of a non-bonded interaction being formed between functional group A (the `central group') and functional group B (the `contact group' or `probe'). The scatterplots are converted into three-dimensional maps that show the propensity of the probe at different positions around the central group. Here we describe how to calculate the similarity of a pair of central groups based on these maps. The similarity method is validated using bioisosteric functional group pairs identified in the Bioster database and Relibase. The Bioster database is a critical compilation of thousands of bioisosteric molecule pairs, including drugs, enzyme inhibitors and agrochemicals. Relibase is an object-oriented database containing structural data about protein-ligand interactions. The distributions of the similarities of the bioisosteric functional group pairs are compared with similarities for all the possible pairs in IsoStar, and are found to be significantly different. Enrichment factors are also calculated showing the similarity method is statistically significantly better than random in predicting bioisosteric functional group pairs.     

Quantitative comparison of molecular electrostatic potential distributions from several semiempirical and ab initio wave functions

A quantitative comparative analysis of molecular electrostatic potential (MEP) distributions generated from different wave functions was carried out. Wave functions were computed by using MNDO, AMl, STO-3G, 3-21G, 4-31G, 6-31G, 4-31G*, 6-31G*, and 6-31G** methods. Ten different compounds, which include usual atoms and groups of biomolecules, such as hydroxyl, carbonyl, amine, amide, imine, double and triple bonds, and heteroaromatic rings, were studied. For each compound, MEP values in the points of a common 3-D grid were computed; thereafter, the similarity between each pair of MEP distributions generated by different methods was assessed. Similarities were measured using the Spearman rank correlation coefficient. A similarity matrix was obtained for each compound. Similarity matrices were averaged and a hierarchical cluster analysis was carried out to classify the different quantum chemical methods. In the compounds studied, the main conclusion is the negligible difference between the pattern of MEP distributions generated from all split valence basis sets (with and without polarization functions). © 1993 John Wiley & Sons, Inc.     

Charge distributions in polyacetylene chains containing a positively charged defect

We present a comparative study of the AIM, CHELPG, GAPT, MK, Mulliken, NPA, and RESP charge distributions associated with a positively charged soliton on increasingly large trans‐polyacetylene chains, at HF, MP2, and DFT levels of theory. The charge storage in the soliton‐bearing systems is explored in detail, including charge magnitude, charge separation, charge alternation, and chain length effects. The grouping of the charge distributions at a given level of theory, as well as the sensitivity of a given charge distribution to the inclusion of electron correlation in its computation, are investigated using similarity analysis. Several of the charge definitions have been applied for the first time for charged soliton‐bearing systems, and there are substantial differences between the charge distributions for the charged and neutral systems. Thus, AIM charges are no longer one of the largest charge values, the AIM charges can be in counterphase with other definitions, and the GAPT charges for neutral systems are quite different from the GAPT charges for charged systems, e.g., the magnitudes of the GAPT charges are anomalously large and increase with the size of the charged system. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

From Shape Similarity to Shape Complementarity: Toward a Docking Theory

Formal relations between similarity and docking are analyzed, and a general docking theory is proposed for colored mixtures of multivariate distributions. X and Y being two colored mixtures with given marginal distributions, their shape complementarity coefficient is defined as the lower bound of the variance of (X–Y)· (X-Y), taken over the set of joint distributions of X and Y. The docking is performed via minimization of the shape complementarity coefficient for all translations and rotations of the mixtures. The properties of the docking criterion are derived, and are shown to satisfy the practical requirements encountered in molecular shape analysis.     

Analysis of data fusion methods in virtual screening: similarity and group fusion

In a recent companion paper we have related the operation of simple data fusion rules used in virtual screening to a multiple integral formalism. In this paper we extend these ideas to the analysis of data fusion methods applied to real data. We examine several cases of similarity fusion using different coefficients and different representations and consider the reasons for positive or negative results in terms of the similarity distributions. Results are obtained using the SUM-, MAX- MIN-, and CombMNZ-fusion rules. We also develop a customized fusion rule, which provides an estimate of the optimal possible result for fusing multiple searches of a specific database; this shows that similarity fusion can, in principle, achieve retrieval enhancements even if this is not achieved in practice with current fusion rules. The methods are extended to analyze the comparatively successful results of group fusion with multiple actives, and we provide a rationale for the observed superiority of the MAX-rule over the SUM-rule in this context.