Vladimir P. Voloshin

An algorithm for three‐dimensional Voronoi S‐network

The paper presents an algorithm for calculating the three‐dimensional Voronoi–Delaunay tessellation for an ensemble of spheres of different radii (additively‐weighted Voronoi diagram). Data structure and output of the algorithm is oriented toward the exploration of the voids between the spheres. The main geometric construct that we develop is the Voronoi S‐network (the network of vertices and edges of the Voronoi regions determined in relation to the surfaces of the spheres). General scheme of the algorithm and the key points of its realization are discussed. The principle of the algorithm is that for each determined site of the network we find its neighbor sites. Thus, starting from a known site of the network, we sequentially find the whole network. The starting site of the network is easily determined based on certain considerations. Geometric properties of ensembles of spheres of different radii are discussed, the conditions of applicability and limitations of the algorithm are indicated. The algorithm is capable of working with a wide variety of physical models, which may be represented as sets of spheres, including computer models of complex molecular systems. Emphasis was placed on the issue of increasing the efficiency of algorithm to work with large models (tens of thousands of atoms). It was demonstrated that the experimental CPU time increases linearly with the number of atoms in the system, O(n).

Volumetric Properties of Hydrated Peptides: Voronoi–Delaunay Analysis of Molecular Simulation Runs

The study of hydration, folding, and interaction of proteins by volumetric measurements has been promoted by recent advances in the development of highly sensitive instrumentations. However, the separation of the measured apparent volumes into contributions from the protein and the hydration water, V(app) = V(int) + ΔV, is still challenging, even with the detailed microscopic structural information from molecular simulations. By the examples of the amyloidogenic polypeptides hIAPP and Aβ42 in aqueous solution, we analyze molecular dynamics simulation runs for different temperatures, using the Voronoi-Delaunay tessellation method. This method allows a parameter free determination of the intrinsic volume V(int) of complex solute molecules without any additional assumptions. For comparison, we also use fused sphere calculations, which deliver van der Waals and solute accessible surface volumes as special cases. The apparent volume V(app) of the solute molecules is calculated by different approaches, using either a traditional distance based selection of hydration water or the construction of sequential Voronoi shells. We find an astonishing coincidence with the predictions of a simple empirical approach, which is based on experimentally determined amino acid side chain contributions (Biophys. Chem.1999, 82, 35). The intrinsic volumes of the polypeptides are larger than their apparent volumes and also increase with temperature. This is due to a negative contribution of the hydration water ΔV to the apparent volume. The absolute value of this contribution is less than 10% of the intrinsic volume for both molecules and decreases with temperature. Essential volumetric differences between hydration water and bulk water are observed in the nearest neighborhood of the solute only, practically in the first two Delaunay sublayers of the first Voronoi shell. This also helps to understand the pressure dependence of the partial molar volumes of proteins.

Calculation of the volumetric characteristics of biomacromolecules in solution by the Voronoi-Delaunay technique

Recently a simple formalism was proposed for a quantitative analysis of interatomic voids inside a solute molecule and in the surrounding solvent. It is based on the Voronoi-Delaunay tessellation of structures, obtained in molecular simulations: successive Voronoi shells are constructed, starting from the interface between the solute molecule and the solvent, and continuing to the outside (into the solvent) as well as into the interior of the molecule. Similarly, successive Delaunay shells, consisting of Delaunay simplexes, can also be constructed. This technique can be applied to interpret volumetric data, obtained, for example, in studies of proteins in aqueous solution. In particular, it allows replacing qualitatively and descriptively introduced properties by strictly defined quantities, such as the thermal volume, by the boundary voids. The extension and the temperature behavior of the boundary region, its structure and composition are discussed in detail, using the example of a molecular dynamics model of an aqueous solution of the human amyloid polypeptide, hIAPP. We show that the impact of the solute on the local density of the solvent is short ranged, limited to the first Delaunay and the first Voronoi shell around the solute. The extra void volume, created in the boundary region between solute and solvent, determines the magnitude and the temperature dependence of the apparent volume of the solute molecule.

Structural and thermodynamic properties of different phases of supercooled liquid water

Computer simulation results are reported for a realistic polarizable potential model of water in the supercooled region. Three states, corresponding to the low density amorphous ice, high density amorphous ice, and very high density amorphous ice phases are chosen for the analyses. These states are located close to the liquid-liquid coexistence lines already shown to exist for the considered model. Thermodynamic and structural quantities are calculated, in order to characterize the properties of the three phases. The results point out the increasing relevance of the interstitial neighbors, which clearly appear in going from the low to the very high density amorphous phases. The interstitial neighbors are found to be, at the same time, also distant neighbors along the hydrogen bonded network of the molecules. The role of these interstitial neighbors has been discussed in connection with the interpretation of recent neutron scattering measurements. The structural properties of the systems are characterized by looking at the angular distribution of neighboring molecules, volume and face area distribution of the Voronoi polyhedra, and order parameters. The cumulative analysis of all the corresponding results confirms the assumption that a close similarity between the structural arrangement of molecules in the three explored amorphous phases and that of the ice polymorphs I(h), III, and VI exists.

The Voronoi–Delaunay approach for the free volume analysis of a packing of balls in a cylindrical container

Abstract The paper describes an approach for free volume analysis of packing of balls confined in a cylinder. The generalized Voronoi diagram is used as an underlying data structure. Two problems are addresses. One is an efficient construction of the confined Voronoi diagram inside a cylindrical boundary. The second is the analysis of the Voronoi network to study a distribution of empty spaces (voids) in the system. The approach is implemented in three-dimensional space and tested on the models of disordered packed bed of 300 balls in cylinders of different radii. The models were obtained by using the Monte Carlo method. The analysis of bottle-necks of channels between balls (and between balls and the boundary) and spatial distribution of the pores was performed.

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.

Disentangling Volumetric and Hydrational Properties of Proteins

We used molecular dynamics simulations of a typical monomeric protein, SNase, in combination with Voronoi-Delaunay tessellation to study and analyze the temperature dependence of the apparent volume, Vapp, of the solute. We show that the void volume, VB, created in the boundary region between solute and solvent, determines the temperature dependence of Vapp to a major extent. The less pronounced but still significant temperature dependence of the molecular volume of the solute, VM, is essentially the result of the expansivity of its internal voids, as the van der Waals contribution to VM is practically independent of temperature. Results for polypeptides of different chemical nature feature a similar temperature behavior, suggesting that the boundary/hydration contribution seems to be a universal part of the temperature dependence of Vapp. The results presented here shine new light on the discussion surrounding the physical basis for understanding and decomposing the volumetric properties of proteins and biomolecules in general.

Fast Calculation of the Empty Volume in Molecular Systems by the Use of Voronoi-Delaunay Subsimplexes

The calculation of the occupied and empty volume in an ensemble of overlapping spheres is not a simple task in general. There are different analytical and numerical methods, which have been developed for the treatment of specific problems, for example the calculation of local intermolecular voids or ‒ vice versa ‒ of the volume of overlapping atoms. A very efficient approach to solve these problems is based on the Voronoi-Delaunay subsimplexes, which are special triangular pyramids defined at the intersection of a Voronoi polyhedron and Delaunay simplex. The subsimplexes were proposed in a paper [1] (Sastry S.et al., Phys. Rev. E, vol.56, 5524–5532, 1997) for the calculation of the cavity volume in simple liquids. Later, the subsimplexes were applied for the treatment of the union of strongly overlapping spheres [2] (Voloshin V.P. et al., Proc. of the 8th ISVD, 170–176, 2011). In this article we discuss wider applications of subsimplexes for the calculation of the occupied and empty volumes of different structural units, selected in molecular systems. In particular, we apply them to Voronoi and Delaunay shells, defined around a solute, as well as their intersection. It opens a way to calculate the components of the partial molar volume of a macromolecule in solution, what is important for the interpretation of experimental volumetric data for protein solutions. The method is illustrated by the application to molecular dynamics models of a hIAPP polypeptide molecule in water at different temperatures.

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.

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.