Alexey V. Anikeenko

Implementation of the Voronoi-Delaunay Method for Analysis of Intermolecular Voids

Voronoi diagram and Delaunay tessellation have been used for a long time for structural analysis of computer simulation of simple liquids and glasses. However the method needs a generalization to be applicable to molecular and biological systems. Crucial points of implementation of the method for analysis of intermolecular voids in 3D are discussed in this paper. The main geometrical constructions – the Voronoi S-network and Delaunay S-simplexes, are discussed. The Voronoi network “lies” in the empty spaces between molecules and represents a “navigation map” for intermolecular voids. The Delaunay S-simplexes determine the simplest interatomic cavities and serve as building blocks for composing complex voids. An algorithm for the Voronoi S-network calculation is illustrated on example of lipid bilayer model.

Shapes of Delaunay Simplexes and Structural Analysis of Hard Sphere Packings

In this chapter we apply a computational geometry technique to investigate the structure of packings of hard spheres. The hard sphere model is the base for understanding the structure of many physical matters: liquids, solids, colloids and granular materials. The structure analysis is based on the concept of the Voronoi Diagram (Voronoi-Delaunay tessellation), which is well known in mathematics and physics. The Delaunay simplexes are used as the main instrument for this work. They define the simplest structural elements in the three-dimensional space. A challenging problem is to relate geometrical characteristics of the simplexes (e.g. their shape) with structural properties of the packing. In this chapter we review our recent results related to this problem. The presented outcome may be of interest to both mathematicians and physicists. The idea of structural analysis of atomic systems, which was first proposed in computational physics, is a subject for further mathematical development. On the other hand, physicists, chemists and material scientists, who are still using traditional methods for structure characterization, have an opportunity to learn more about this new technique and its implementation. We present the analysis of hard sphere packings with different densities. Our method permits to tackle a renowned physical problem: to reveal a geometrical principle of disordered packings. The proposed analysis of Delaunay simplexes can also be applied to structural investigation of other molecular systems.

A novel delaunay simplex technique for detection of crystalline nuclei in dense packings of spheres

The paper presents a new approach for revealing regions (nuclei) of crystalline structures in computer models of dense packings of spherical atoms using the Voronoi-Delaunay method. A simplex Delaunay, comprised of four atoms, is a simplest element of the structure. All atomic aggregates in an atomic structure consist of them. A shape of the simplex and the shape of its neighbors are used to determine whether the Delaunay simplex belongs to a given crystalline structure. Characteristics of simplexes defining their belonging to FCC and HCP structures are studied. Possibility to use this approach for investigation of other structures is demonstrated. In particular, polytetrahedral aggregates of atoms untypical for crystals are discussed. Occurrence and growth of regions in FCC and HCP structures is studied on an example of homogeneous nucleation of the Lennard-Jones liquid. Volume fraction of these structures in the model during the process of crystallization is calculated.

An Algorithm for the Calculation of Volume and Surface of Unions of Spheres. Application for Solvation Shells

A simple algorithm for the calculation of the volume and surface area of a union of spheres of different radii is presented. It is based on the ideas published in S. Sastry et al, Phys. Rev. E, v.56, pp.5524-5532, 1997 [1], where they computed volume and surface of interatomic voids in simple liquids. They proposed to work with the intersection of Delaunay simplexes and the corresponding Voronoi polyhedra. Analytical formulas for volume and surface area were derived for the atoms occupying this region. This could be achieved without explicit calculation of multiple intersections of the overlapping atoms. We have implemented such ideas for the calculation of the occupied volume and its surface inside the polyhedra defined by power Voronoi diagram. This allows calculating the required values for spheres with different radii. Simple analytical formulas are also valid in this case. We applied our algorithm to the calculation of the solvation shell volume for complex solute molecules in molecular dynamics models of solutions. A comparison of our program with the available implementation of the certified algorithm for unions of spheres by F. Cazals et al. (ACM Trans. Math. Soft. 38 (1), 2011) [2] shows coincidence of the results.

Critical densities in hard sphere packings. Delaunay simplex analysis.

A large set of computer models (more then 200 models) of hard sphere packings with packing fraction eta between 0.52 - 0.72 is examined. Every packing consist of 10.000 identical spheres in the model box with periodic boundary conditions. Delaunay simplexes (quadruples of mutually closest spheres) with shape resembling to perfect tetrahedron or quartoctahedron are studied. Fraction of such simplexes is studied as a function of packing density. Structure changes at the transition from disordered to crystalline phase are discussed. A limited packing fraction of the non-crystalline packing is estimated as 0.6455plusmn0.0015. The ratio of tetrahedral to quartoctahedral simplexes (T/Q) in the packing at this density provided to be close to 2/3. We pay attention to one more critical interval of density at around eta=0.665 plusmn0.005. At this density the crystalline nuclei which were in the packing run into unified crystal and the ratio T/Q reaches a crystalline value 1/2.

Hydration Shells in Voronoi Tessellations

An interesting property of the Voronoi tessellation is studied in the context of its application to the analysis of hydration shells in computer simulation of solutions. Namely the shells around a randomly chosen cell in a Voronoi tessellation attract extra volume from outside. There is a theoretical result which says that the mean volume of the first shell around a randomly chosen cell is greater than the anticipated value. The paper investigates this phenomenon for Voronoi tessellations constructed for computer models of point patterns with different variability of the Voronoi cell volumes (Poisson point process, RSA systems of hard spheres and molecular dynamics models of water). It analyzes also the subsequent shells, and proposes formulas for the mean shell volumes for all shell numbers. The obtained results are of value in calculations of the contribution of hydration of water to the “apparent” volume of the solutes.

Application of Procrustes Distance to Shape Analysis of Delaunay Simplexes

The concept of Procrustes distance is applied to the shape analysis of the Delaunay simplexes. Procrustes distance provides a measure of coincidence of two point sets {xi} and {yi}, i=1..N. For this purpose the variance of point deviations is calculated at the optimal superposition of the sets. It allows to characterize the shape proximity of a given simplex to shape of a reference one, e.g. to the shape of the regular tetrahedron. This approach differs from the method used in physics, where the variations of edge lengths are calculated in order to characterize the simplex shape. We compare both methods on an example of structure analysis of dense packings of hard spheres. The method of Procrustes distance reproduces known structural results; however, it allows to distinguish more details because it deals with simplex vertices, which define the simplex uniquely, in contrast to simplex edges.

Establishment of Constraints on Amyloid Formation Imposed by Steric Exclusion of Globular Domains

In many disease-related and functional amyloids, the amyloid-forming regions of proteins are flanked by globular domains. When located in close vicinity of the amyloid regions along the chain, the globular domains can prevent the formation of amyloids because of the steric repulsion. Experimental tests of this effect are few in number and non-systematic, and their interpretation is hampered by polymorphism of amyloid structures. In this situation, modeling approaches that use such a clear-cut criterion as the steric tension can give us highly trustworthy results. In this work, we evaluated this steric effect by using molecular modeling and dynamics. As an example, we tested hybrid proteins containing an amyloid-forming fragment of Aβ peptide (17-42) linked to one or two globular domains of GFP. Searching for the shortest possible linker, we constructed models with pseudo-helical arrangements of the densely packed GFPs around the Aβ amyloid core. The molecular modeling showed that linkers of 7 and more residues allow fibrillogenesis of the Aβ-peptide flanked by GFP on one side and 18 and more residues when Aβ-peptide is flanked by GFPs on both sides. Furthermore, we were able to establish a more general relationship between the size of the globular domains and the length of the linkers by using analytical expressions and rigid body simulations. Our results will find use in planning and interpretation of experiments, improvement of the prediction of amyloidogenic regions in proteins, and design of new functional amyloids carrying globular domains.