Decomposition of a protein solution into Voronoi shells and Delaunay layers Kim A.V 1 , Voloshin V.P. 1 , Medvedev N.N. 1,2 1 Institute of Chemical Kinetics and Combustion, SB RAS, 630090 Novosibirsk, Russia 2 Novosibirsk State University, Novosibirsk, Russia e-mail: nikmed@kinetics.nsc.ru Geiger A. 3 3 Physikalische Chemie, Technische Universität Dortmund, 44221 Dortmund, Germany e-mail: alfons.geiger@udo.edu Abstract— A simple formalism is proposed for a quantitative analysis of interatomic voids inside and outside of a molecule in solution. It can be applied for the interpretation of volumetric data, obtained in studies of protein folding in water. The method is based on the Voronoi-Delaunay tessellation of molecular-dynamic models of solutions. It is suggested to select successive Voronoi shells, starting from the interface between the solute molecule and the solvent, and continuing to the outside (into the solvent) as well as into the inner of the molecule. Similarly, successive Delaunay layers, consisting of Delaunay simplexes, can also be calculated. Geometrical properties of the selected shells and layers are discussed. The behavior of inner and outer voids is discussed by the example of a molecular-dynamic model of an aqueous solution of the polypeptide hIAPP. Index terms—Voronoi diagram, solvation shell, molecular dynamics of solutions, Voronoi cells, Delaunay simplexes. I. INTRODUCTION The investigation of biological molecules in aqueous solution is an important problem of molecular biology. In particular, it is important to understand the mechanism of protein folding in water at different temperatures and pressures. Volumetric experiments are used for this study. The change of the volume of the solution, induced by the addition of a protein molecule is measured [1]. The influence of temperature and pressure induces changes of the volumetric properties, both inside the solute molecule, at its boundary, and also in the surrounding water. The knowledge of these contributions to the volume of the solution helps to validate propositions about the occurring conformational changes. However, using only experimental data, it is very difficult to separate these contributions. Computer simulations help to solve this problem. Models of the solutions are generated usually by molecular dynamic simulations, see for example [2]. The next step is the analysis of the models: detection and characterization of interatomic voids and local densities. There are very different approaches used for the analysis of voids in atomic and molecular systems. Some of them were developed for the investigation of the empty space between the atoms in liquids and glasses [3-5], granular matters and colloids [6,7], polymers and membranes [8,9]. Others are specialized to study cavities and pockets in large biological molecules [10-12]. Solvation shells [13,14] and the boundary region between proteins are also studied [15,16]. Consecutive shells, consisting of Voronoi cells, were used for the analysis of the density of hydration shells around polypeptides in [17]. However, we are not aware of articles, where the voids both inside and in the surroundings of a solute molecule were analyzed. Such investigations should be made by a single-stage method for all regions of the solution. Fortunately, there is no necessity to develop a new method for such a work. At present, there is no doubt, that the most suitable and general method for the selection and analysis of voids and the local density in molecular system is an approach, which is based on Voronoi diagrams (the Voronoi -Delaunay method) [18,19]. In this work, we present a simple technique for the decomposition of the Voronoi-Delaunay tessellation into shells (layers) related with the solute. It allows to characterize voids (local density) both inside and outside the solute molecule. II. VORONOI-DELAUNAY TESSELLATION OF A SOLUTION Fig.1 shows a two-dimensional illustration of a solution model and its Voronoi-Delaunay tessellation. Remember, the size of the atoms should be taken into account, if one studies interatomic voids [3,20,21]. This means that the Voronoi tessellation should be calculated allowing for the surface of the atoms. Thus we should deal with S-tessellation [22,23] (additively weighted [18]), instead of the ordinary Voronoi tessellation (related with the atomic centers). In this case we make a correct assignment of the empty space to a given atom, i.e. we include all points of space, which are closer to the surface of a given atom, than to the surfaces of all other atoms of the system. A simpler variant, which considers the atomic surfaces, is the well- known power or radical tessellation [18,21,24]. In this case the assignment of the empty space to individual atoms is not quite physical, but it is easier to implement. The known complexities of the S-tessellation (theoretically possible disconnectivity of the tessellation and overlapping of Delaunay simplexes in some cases [19,23,25] do not arise for our molecular systems, where the size difference of the atoms is rather small (usually less than a factor of 2). In addition, these peculiarities of the S-tessellation can be easily taken in to account at the calculation of the tessellation. The molecules of the solvent (usually water molecules) are considered as uniform spheres, as it is usually done in structure analyses of computer models of water and water solutions. Note, the specific features of the interaction 2012 Ninth International Symposium on Voronoi Diagrams in Science and Engineering 978-0-7695-4724-4/12 $26.00 © 2012 IEEE DOI 10.1109/ISVD.2012.18 95