Superposition of molecules: Electron density fitting by application of fourier transforms

Molecular similarity measures

Molecular similarity is a pervasive concept in chemistry. It is essential to many aspects of chemical reasoning and analysis and is perhaps the fundamental assumption underlying medicinal chemistry. Dissimilarity, the complement of similarity, also plays a major role in a growing number of applications of molecular diversity in combinatorial chemistry, high-throughput screening, and related fields. How molecular information is represented, called the representation problem, is important to the type of molecular similarity analysis (MSA) that can be carried out in any given situation. In this work, four types of mathematical structure are used to represent molecular information: sets, graphs, vectors, and functions. Molecular similarity is a pairwise relationship that induces structure into sets of molecules, giving rise to the concept of chemical space. Although all three concepts - molecular similarity, molecular representation, and chemical space - are treated in this chapter, the emphasis is on molecular similarity measures. Similarity measures, also called similarity coefficients or indices, are functions that map pairs of compatible molecular representations that are of the same mathematical form into real numbers usually, but not always, lying on the unit interval. This chapter presents a somewhat pedagogical discussion of many types of molecular similarity measures, their strengths and limitations, and their relationship to one another. An expanded account of the material on chemical spaces presented in the first edition of this book is also provided. It includes a discussion of the topography of activity landscapes and the role that activity cliffs in these landscapes play in structure-activity studies.

An efficient and accurate molecular alignment and docking technique using ab initio quality scoring

An accurate and efficient molecular alignment technique is presented based on first principle electronic structure calculations. This new scheme maximizes quantum similarity matrices in the relative orientation of the molecules and uses Fourier transform techniques for two purposes. First, building up the numerical representation of true ab initio electronic densities and their Coulomb potentials is accelerated by the previously described Fourier transform Coulomb method. Second, the Fourier convolution technique is applied for accelerating optimizations in the translational coordinates. In order to avoid any interpolation error, the necessary analytical formulas are derived for the transformation of the ab initio wavefunctions in rotational coordinates. The results of our first implementation for a small test set are analyzed in detail and compared with published results of the literature. A new way of refinement of existing shape based alignments is also proposed by using Fourier convolutions of ab initio or other approximate electron densities. This new alignment technique is generally applicable for overlap, Coulomb, kinetic energy, etc., quantum similarity measures and can be extended to a genuine docking solution with ab initio scoring.

Toward High Throughput 3D Virtual Screening Using Spherical Harmonic Surface Representations

Searching chemical databases for possible drug leads is often one of the main activities conducted during the early stages of a drug development project. This article shows that spherical harmonic molecular shape representations provide a powerful way to search and cluster small-molecule databases rapidly and accurately. Our clustering results show that chemically meaningful clusters may be obtained using only low order spherical harmonic expansions. Our database search results show that using low order spherical harmonic shape-based correlation techniques could provide a practical and efficient way to search very large 3D molecular databases, hence leading to a useful new approach for high throughput 3D virtual screening. The approach described is currently being extended to allow the rapid search and comparison of arbitrary combinations of molecular surface properties.

TGSA-Flex: Extending the capabilities of the Topo-Geometrical superposition algorithm to handle flexible molecules

In this work, an extension of the already studied Topo-Geometrical Superposition Approach (TGSA) is presented. TGSA, a general-purpose, fast, automatic, and user-intuitive three-dimensional molecular alignment procedure, was originally designed to superpose rigid molecules simply based on atomic numbers, molecular coordinates, and connectivity. The algorithm is further developed to enable handling rotations around single bonds; in this way, common structural features, which were not properly aligned due to conformational causes, can be brought together, thus improving the molecular similarity picture of the final alignment. The present procedure, implemented in Fortran 90 and named TGSA-Flex, is deeply detailed and tested over four molecular sets: amino acids, nordihydroguaiaretic acid (NDGA) derivatives, HIV-1 protease inhibitors, and 1-[2-hydroxyethoxy)methyl]-6-(phenylthio)thymine (HEPT) derivatives. TGSA-Flex performance is evaluated by means of computational time, number of superposed atoms (also comparing it with respect to the rigid approach), and index of fit between the compared structures.

Multiple flexible alignment with SEAL: a study of molecules acting on the colchicine binding site

An extension of the steric and electrostatic alignment alignment (SEAL) method (MultiSEAL) is described that allows the overlay of multiple molecules and conformations. The method is well-suited for the systematic study of possible alignments, also revealing information about the conformational energies associated with a given overlay. It has been tested on three examples: angiotensin II antagonists, 5-HT3 antagonists, and dopaminergic compounds. The utility of the method is further demonstrated in an analysis of molecules that putatively bind to the colchicine site of tubulin. On the basis of its overlay with colchicine, allocolchicine, 2-methoxy-5-(2',3',4'-trimethoxyphenyl)tropone, and combretastatin A-4, it appears that 2-methoxyestradiol (2-ME) is unlikely to fit the colchine site properly. The weak antimitotic activity of 2-ME may be explained by its partial fit in the site.

Interconversion between 3D molecular representations: some macromolecular applications of spherical harmonic-Bessel expansions about an arbitrary center

An accurate interconversion of three-dimensional molecular spatial information between two alternative orthogonal representations is demonstrated. These alternative descriptions are useful for exhaustive six-dimensional grid searches of empirical energy functions and for estimating the contents of asymmetric units in sparsely packed, non-centrosymmetric crystalline arrays. To illustrate the fidelity of this interconversion, it is implemented to perform an exhaustive grid search in six-dimensional relative rotational and translational space (phase space) with the spherical harmonic-Bessel representation. This search is an extension of Crowther's crystallographic fast rotational overlap function from Patterson space to direct, three-dimensional space. The search uses a three-dimensional spherical harmonic expansion about a single center to represent a probe molecule's shape, charge and van der Waals parameters and numerous complete expansions about single centers to represent the complementary shape, electrostatic potential and van der Waals potential of several target sites about the complementary molecule. The rapid calculation of unique, complete expansions about numerous sites on a grid is made possible by application of the Fourier convolution theorem. Expansions from nearby grid points can be used to reconstruct the same molecular shape accurately, if this shape is contained entirely within the spherical zone of expansion about each grid point. This arbitrariness for the choice of origin, and the accuracy of interconversion between three-dimensional spherical harmonic-Bessel and Fourier representations, suggest a method for describing the contents of non-centrosymmetric, sparsely packed crystals. Incomplete Fourier representations (diffraction amplitudes) provide complex spherical harmonic-Bessel coefficients for a crystal packed with symmetry-enforced, non-centrosymmetrically arranged, non-overlapping spherical expansion zones.