Gustavo A. Arteca

Shape group studies of molecular similarity: relative shapes of Van der Waals and electrostatic potential surfaces of nicotinic agonists

Abstract In drug design, the usual strategy involves characterizing and comparing the shapes of molecules. We apply a simple method to accomplish this goal: determining the symmetry-independent shape groups (homology groups of algebraic topology) of a molecular surface. In this paper, we have adapted the method to describing the interrelation between Van der Waals and electrostatic potential surfaces. We describe rigorously the shape features in a series of molecules by using specific ranges of electrostatic potential over a Van der Waals surface. We consider a series of four nicotinic agonists as an example and discuss their expected activities as potential drugs on the basis of the shape similarities found.

A method for the characterization of foldings in protein ribbon models

The ribbon model of chain macromolecules is a useful tool for analyzing some of the large-scale shape features of these complex systems. Up to now, the ribbon model has been used mostly to produce graphical displays, which are usually analyzed by visual inspection. In this work we suggest a computational method for characterizing automatically, in a concise and algebraic fashion, some of the important shape features of these ribbon models. The procedure is based on a graph-theoretical and knot-theoretical characterization of three well-defined projections of a space curve associated with the ribbon. The labeled graphs can be characterized by the handedness of the crossovers in the ribbon that are the vertices of the graph. The method can be used to provide a fully algebraic representation of the changes occurring when a molecule, such as a protein, undergoes conformational rearrangements (folding), as well as to provide a shape comparison for a pair of related molecular ribbons. This algebraic representation is well suited for easy storage, retrieval, and computer manipulation of the information on the ribbons shape. Illustrative examples of the method are provided.

Approximate calculation of physical properties of enclosed central field quantum systems

It is shown that certain formulas satisfied by exact eigenfunctions are very useful to obtain good approximations for the values of physical properties of enclosed central field quantum systems, even though one uses rather poor trial functions. The properties studied in this work (under the assumption of the validity of a very simple model) are the hyperfine splitting and the pressure of the system.

Implementing knot-theoretical characterization methods to analyze the backbone structure of proteins: Application to CTF L7/L12 and carboxypeptidase A inhibitor proteins

In this work we apply a recently developed method for characterizing the shape of the tertiary structure of proteins. The approach is based on a combination of graph- and knot-theoretical characterizations of Cartesian projections of the space curve describing the protein backbone. The proposed technique reduces the essential shape features to a topologically based code formed by a sequence of knot symbols and polynomials. These polynomials are topological invariants that describe the overcrossing and knotting patterns of curves derived from the molecular space curve. These descriptors are algorithmically computed. The procedure is applied to describe the structure of the carboxy terminal fragment of the L7/L12 chloroplast ribosomal protein (CTF L7/L12) and the potato carboxypeptidase A inhibitor protein (PCI), which has a set of three disulfide bridges. In the former case, we describe the proteins shape features in terms of its alpha-helices, and a backbone simplified by considering helices without internal structure. An extension of the methodology to describe disulfide bridges is discussed and applied to PCI. Changes in the knot-theoretical characterization due to possible uncertainties in the resolution of the X-ray structure, as well as the inclusion of low-frequency motions of the backbone, are also discussed.

A new method for summation of divergent power series

A new method for summation of divergent power series is developed. It only requires the knowledge of the form of both the small and large λ‐power expansion (λ being the perturbation parameter) and few coefficients of one of them to yield excellent results. Convergence is proved for a simple two‐level model, and reasonable arguments are given for more complex and interesting models. The method is quite general and contains some resummation techniques reported previously as particular cases. The anharmonic, mean square, displacement function, the ground‐state eigenvalue of the quantum‐mechanical anharmonic oscillator, and the ground‐state energy of the hydrogen atom in a magnetic field calculated in this way are shown to be of striking accuracy in the whole range of the perturbation parameter.

Heuristic lipophilicity potential for computer-aided rational drug design

In this contribution we suggest a heuristic molecular lipophilicitypotential (HMLP), which is a structure-based technique requiring noempirical indices of atomic lipophilicity. The input data used in thisapproach are molecular geometries and molecular surfaces. The HMLP is amodified electrostatic potential, combined with the averaged influences fromthe molecular environment. Quantum mechanics is used to calculate theelectron density function ρ(r) and the electrostatic potential V(r), andfrom this information a lipophilicity potential L(r) is generated. The HMLPis a unified lipophilicity and hydrophilicity potential. The interactions ofdipole and multipole moments, hydrogen bonds, and charged atoms in amolecule are included in the hydrophilic interactions in this model. TheHMLP is used to study hydrogen bonds and water–octanol partitioncoefficients in several examples. The calculated results show that the HMLPgives qualitatively and quantitatively correct, as well as chemicallyreasonable, results in cases where comparisons are available. Thesecomparisons indicate that the HMLP has advantages over the empiricallipophilicity potential in many aspects. The HMLP is a three-dimensional andeasily visualizable representation of molecular lipophilicity, suggested asa potential tool in computer-aided three-dimensional drug design.

Modeling lipophilicity from the distribution of electrostatic potential on a molecular surface

SummaryMolecular lipophilicity L is represented as a function of four surface electrostatic potential descriptors: L=f(BF+,BF-,BR+,BR-). Each B descriptor is computed from the products of elements of molecular surface area, Δsi, and the molecular electrostatic potential (MEP), V(ri), at the center of an area element: B = ∑i Δi V(ri). Octanol-water partition coefficients (Pow) are correlated with these four surface-MEP descriptors: log Pow=c0+c1BF++c2BF-+c3BR++c4BR-. Good correlations are obtained for homologous series of aliphatic alcohols, amines and acids, as well as for a set of aromatic compounds with various functional groups. Within this approach, we find that the molecular fragment contributions of surface-MEP descriptors to log P are approximately additive. We have computed the values for the following fragments:-CH2-,-CH3,-COOH,-OH and-NH2. These contributions can be used to estimate the molecular lipophilicity and partition coefficients of new compounds, without additional quantum-mechanical calculations. The proposed approach provides a reasonably accurate tool that can be useful in quantitative structure-activity relations for computer-aided rational drug design. More importantly, the correlation model is conceptually simpler than previous work in the literature and can be improved systematically.

Shape group theory of van der Waals surfaces

In this article we present a method for the study of shapes of general, asymmetric van der Waals surfaces. The procedure is simple to apply and it consists of two steps. First, the surface is decomposed into spherical domains, according to the interpenetration of the van der Waals atomic spheres. Each domain defines a topological object that is either a 2-manifold or some truncated 2-manifold. Second, we compute the homology groups for all the objects into which the surface is divided. These groups are topological and homotopical invariants of the domains, hence they remain invariant to conformational changes that preserve the essential features of these domains of decomposition. In particular, these homology groups do not depend explicitly on the molecular symmetry. Major rearrangements of the nuclear configurations, however, do alter the decomposition into spherical domains, and the corresponding variation of the homology groups can be followed easily under conformational rearrangements. We discuss a partitioning of the metric internal configuration spaceM into shape regions of van der Waals surfaces, which allows one to identify those rearrangements which introduce an essential change in shape and to distinguish them from those which do not alter the fundamental shape of the molecular surface. The dependence of the shape group partitioning ofM on the symmetry under permutation of nuclear changes is discussed briefly, considering a simple illustrative example.

SUMMATION OF STRONGLY DIVERGENT PERTURBATION SERIES

A new method for summing strongly divergent perturbation series is presented. It is based on the change of the power series into a convergent sequence by means of an order‐dependent mapping obtained from a simple scaling relation. The perturbation expansions for a one‐dimensional integral and for the ground states of the anharmonic oscillator and of the linear confining potential model are accurately summed in the most unfavorable strong‐coupling limit.

Variations in chain compactness and topological complexity uncover folding processes in the relaxation dynamics of unfolded in vacuo lysozyme

Chain collapse and the formation of a near-native tertiary structure are believed to be two key features controlling the progress of a protein folding transition. In this work, we study the interrelation between these two properties along computer-simulated relaxation trajectories of unfolded in vacuo lysozyme. Large-scale molecular shape transitions are monitored within a space defined by two discriminating descriptors of chain compactness and entanglement (or “topological”) complexity. For the system studied here, results indicate that successful refolding into native-like conformers requires a balance between polymer collapse and a topologically “correct” organization of chain loops. Although no single factor dominates the relaxation paths, compactization appears to be a necessary condition for near-native refolding. Whenever initial collapse is limited or absent, we find a “derailed” folding path with high configurational frustration. We also show that disulfide-reduced lysozyme unfolds differently, y...