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.