Implementation of the Voronoi-Delaunay method for analysis of intermolecular voids

A. Laganà et al. (Eds.): ICCSA 2004, LNCS 3045, pp. 217–226, 2004. © Springer-Verlag Berlin Heidelberg 2004

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

A.V. Anikeenko1, M.G. Alinchenko1 ,V.P. Voloshin1, N.N. Medvedev1, M.L. Gavrilova2, and P. Jedlovszky3

1 Institute of Chemical Kinetics and Combustion SB RAS, Novosibirsk, Russia [email protected]

2 Department of Computer Science, University of Calgary, Calgary, AB, Canada 3 Department of Colloid Chemistry, Eötvös Loránd University, Hungary

Abstract. 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.

1 Introduction

The Voronoi-Delaunay approach is well applicable to structural analysis of monatomic systems (computer models of simple liquids, amorphous solids, crystals, packings of balls). Geometrically these systems are represented as an ensemble of discrete points or spheres of equal radius, and the original mathematical premises of the method [8,28] are applicable for structural analysis of such systems [7,19,27]. Applying the method to molecular systems (molecular liquids, solutions, polymers, biological molecules) requires a modification of this classic data structure. The molecular systems usually consist of atoms of various radii; in addition, atoms in a molecule are connected via chemical bonds whose lengths are usually shorter than the sum of the atomic radii. Thus from a mathematical point of view a molecular system is an ensemble of balls of different radii some of which are partially overlapped.

One of the common problems in molecular systems analysis is determination of a region of space assigned to an atom [8,9]. The classical Voronoi polyhedron in 3D is suitable for this purpose in the systems of equal atoms but fails for a general case because its construction neglects atomic radii. It is well known that this problem can be solved using the additively weighted Voronoi diagram [20], where a measure of the distance between a point of space and a center of the i-th atom is defined as y=x+Wi, where y is an Euclidean distance between a pint in space and the center of the atom, and Wi is a weight of an i-th atom. This measure has a simple physical

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interpretation. If value Wi is taken as a radius of i-th atom Ri then measure y represents a shortest Euclidean distance to its surface. Due to this fact this region is referred to as the Voronoi S-region in physics [4]. Note that Voronoi S-region can be defined not only for spheres, but also for physical bodies of other shape [14].

The next important physical problem is investigation of voids between molecules (cavities, pockets, channels). It differs from the calculations of region assigned to a given atom: any interatomic void is associated with a group of atoms, not with a single atom. In the case of monatomic systems the classical Delaunay simplexes are used for presentation of voids [6,25-27]. For molecular systems the Deluanay S-simplexes which present a dual constructions for the Voronoi S-region tessellation can be used analogously [4,17,18]. The Delaunay S-simplex is determined by centers of four atoms which are incidental to a common vertex of the Voronoi S-regions. This quadruplet of atoms gives the elementary (simplicial) cavity. Any complex void between atoms can be composed of such simplicial cavities. Connectivity of such simplexes can be studied using the Voronoi S-network, which is a network of edges and vertexes of the all Voronoi S-regions in the system. Interatomic voids can be analyzed using the Voronoi S-network and S-simplexes in frame of the ideology developed in this classical approach.

Despite the fact that the Voronoi-Delaunay constructions are well-studies, they are do not used to their full potential in physics or molecular biology, especially in three-dimensional environments. One of the reasons is the complexity of methodical and technical implementation of the method. In particular, after calculation of the Voronoi S-network and Deluanay S-simplexes one needs to use them to reveal voids and calculate their characteristics. In this paper, we address this problem and try to explain how the method can be implemented in order to be useful and make applications of Voronoi-Delaunay method more practical for physical applications. We also demonstrate that it can be an efficient method for studying large-scale 3D models.

2 Main Stages of Implementation of the 3D Model

2.1 Basic Geometric Concepts of the Voronoi-Delaunay Method

An initial construction of the approach is the Voronoi S-region: a region of space all points of which are closer to the surface of a given ball than to the surfaces of the other balls of the system, see Fig 1. For balls of the same size this region coincides obviously with the classical Voronoi polyhedron defined for the atomic centers, Fig.1a. However, the Voronoi S-region is not a polyhedron in general, for balls of different size its faces are pieces of hyperboloids (Fig.1b). The most important peculiarity of the S-region is that it determines naturally a region of space assigned to a given ball (molecule). Jointly, the edges and vertices of Voronoi S-regions form the Voronoi S-network of a given system (thick lines in Fig.1c).

It is known that Voronoi S-regions constructed for all atoms of the system form a partitioning which covers the space without overlapping and gaps [5,9,11,12,18,20]. This Voronoi S-tesselation divides the atoms of the system, similarly to the classical Voronoi tessellation, into the quadruplets of atoms (the Delaunay S-simplexes), representing elementary cavities between the atoms (see Fig. 2).

Implementation of the Voronoi-Delaunay Method for Analysis 219

a) b) c)

Fig. 1. 2D-illustration of the Voronoi S-regions in systems of atoms of equal and different radii.

a) b) c)

Fig. 2. The Delaunay S-simplexes (thick lines) for configurations in Fig.1. They can coincide with the classical Delaunay simplexes as in (a) and (b) or be different (c) which depends on radii of atoms. Thin lines show the Voronoi S-network.

The set of all vertices and edges of the Voronoi S-regions determines the Voronoi S-network. Each vertex (site) of the Voronoi S-network is the center of the interstitial sphere, which corresponds to one of the Delaunay S-simplexes. Each edge (bond) is a fairway passing through the bottleneck between three atoms from one site to the neighboring one.

2.2 Basic Data Structure for Representation of the Voronoi S-Network

To work with any network we should know coordinates of the network sites and their connectivity. In addition, to calculate characteristics of voids one needs radii of interstitial spheres and radii of bottle-necks. Let array_D contain the coordinates of network sites. An order in which the sites are recorded in this array defines the numbering of the sites. Let array_Ri contain the radii of interstitial spheres. Each sphere corresponds to one of the sites of the network. Array DD establishes the connectivity of network sites. By that it determines the bonds of the network. Each bond defines the bottleneck between a pair of sites. It is useful to have a special array_Rb which contains the radii for all bottlenecks; they are needed for analysis of complex voids. Finally, explicit information is desirable for work with S-simplexes. To this end, the simplest way is to create array_DA representing a table of incidence of the network sites and the numbers of atoms relating to the corresponding sites. All this information is calculated at calculation of the Voronoi S-network. The empty

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volume of all S-simplexes should be calculated and recorded in an array_Ve. This information is enough to start the analysis of voids.

Note the problem of overlapping atoms can be solved easily for analysis of voids. Since each bond of the Voronoi S-network is a locus of the points equidistant from the surfaces of the nearest three balls, this locus does not change if we decrease (or increase) the radii of the balls by the same value d. Similarly, the site (the common vertex of the Voronoi S-regions) does not change its position if the radii of the corresponding four balls are changed by the same value. Thus we construct a reduced system by decreasing the radii of all balls of the initial system by some constant value d to avoid the overlapping of atoms. Then we construct the Voronoi S-network using the algorithm for constructing the S-network for the system of non-overlapping balls.

The required arrays that determine the Voronoi S-network (the coordinates of network sites, the table of connectivity, the table of the incidence of sites and atoms) fully coincide for the initial and reduced systems. The values of the radii of interstitial spheres and bottlenecks for the initial system are obviously different from the corresponding values of the reduced system on the constant value d.

2.3 Determination of Interatomic Voids on the Voronoi S-Network

Empty space in a 3D-system of atoms is a complex singly connected system confined by spherical surfaces of atoms. Any interatomic void to be distinguished is a part of this system and it depends on the detection criterion. A physical way of defining voids is through the value of the radius of a probe (test sphere), which can be located in a given void. The number, size and morphology of the voids depend on the probe radius: some most spacious cavities represent voids for large probe, and almost the entire interatomic space is accessible for small one.

A simple but important characteristic of interatomic space is a set of interstitial spheres. These spheres represent real empty volume between atoms. The values of their radii indicate the scale of voids in a system. A more comprehensive analysis of voids and interatomic channels requires knowing the system of bottlenecks, i.e., the analysis of the bonds of the Voronoi S-network. If a probe can be moved along the bond then the network sites at the ends of the bond are sure to be also accessible for this probe [18]. Thus, the regions accessible for a given probe can be found by distinguishing the bonds whose bottleneck radius exceeds a given value, see Fig.3. The clusters consisting of these bonds represent the fairways (skeletons) of the regions along which a given probe can be moved. Distinguishing bonds on the Voronoi S-network using bottlenecks is called Rb-coloring of the network [26,16,18].

Representation of voids by clusters of colored bonds is highly descriptive for illustrating the locations of complex voids inside the model. However, to perform a deeper physical analysis of the voids, their volumes should be calculated. This can be performed with the help of the Delaunay S-simplexes. Knowing the sites of the Voronoi S-network involved in a given cluster, we know all S-simplexes composing this void. The union of the empty volumes of these S-simplexes provides the “body” of the void to be found, Fig. 3 (right). The rest of the empty space in the model is inaccessible for such probe. The proposed representation of voids provides a quantitative basis for analyzing the various characteristics of voids

Implementation of the Voronoi-Delaunay Method for Analysis 221

Fig. 3. Left: The Voronoi S-network (thick lines) and Delaunay S-simplexes (thin lines) for a molecular system. Each bond of the network is a fairway passing through the bottleneck between atoms. Right: Voids in the system accessible for a probe shown as a disk between figures. The fairways of the voids (thick lines) are clusters of the Voronoi S-network bonds which bottleneck radii is greater then radius of the probe.

2.4 Representation of Voids by Spherocylinders and Calculation of Empty Volume

The voids accessible for relatively large probes (i.e. of the order of atomic size) are more interesting for physicists since these voids might play an important role in the mechanism of the diffusion of small molecules. The radii of the probes used fall in the range between 1.0Å and 1.6Å. Preliminary analysis shows that for molecular systems voids corresponding to such probes are usually rather compact. (A complex, branching structure of the voids starts to appear for considerably smaller probes). Although the shape of the compact voids is not simple, their main characteristics can be described by just a few parameters, such as their length, width and orientation, implying that these voids can be represented by bodies of rather simple shape in order to make their detailed analysis mathematically feasible.

For our analysis, we suggest to represent voids by spherocylinders [1] (i.e., cylinders covered by hemispheres of the same radius at the two basic circular faces). We calculate these parameters directly, by means of the “inertia tensor” of the void instead of artificially “fitting” them. The inertia tensor of a void is calculated using the cluster of bonds and sites on the Voronoi S-network representing a given void. The fictitious “mass”, equal to the value of the empty volume of the corresponding Delaunay S-simplex, is assigned to each site of the cluster. Thus, the volume of the void is concentrated on the S-network sites, and hence the continuous body of complex shape of the void is represented by a system of a finite number of “massive” points, for which the inertia tensor can be readily calculated. The axis along which the principal value of the inertia tensor is minimal indicates the direction of the largest extension of the void. It is taken as the axis of the required spherocylinder. To calculate the length of the spherocylinder L (i.e., the length of its axis in the cylindrical part), all the sites of the cluster are projected to this axis, and the mean square deviation of these projections from the centre of the fictitious mass of the cluster (lying always on this axis) are calculated. Finally, the radius R of the spherocylinder is unambiguously determined from the condition of the equality of the volumes of the spherocylinder and the void. This condition can be written simply as

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)1(,3

42

+= RLRVvoid π

where Vvoid is the volume of the void, determined as the sum of the empty volumes of the composing Delaunay S-simplexes.

At the first glance, the empty volume inside simplexes seems to be easily calculated analytically as the volume of the whole simplex minus the volume of the parts of its own atoms composing this simplex. However, the simplex often involves “alien” atoms assigned to other simplexes [23,18]. It is more difficult to take into account the volume occupied by these atoms. Moreover, there can be several alien atoms incoming in a given simplex. Additional problems arise for the case of overlapping atoms, which should be taken into account to correctly compute the empty volume. Thus, it is rather difficult to derive an analytical formula for calculation of the empty simplex volume. This, however, can be done numerically. To this end, we fill the simplex with sampling points (randomly or regularly) and determine the fraction of points outside the atoms. Implementation of this idea can be rather efficient because the list of atoms that can enter in the simplex is readily defined by the Voronoi S-tessellation. Note that under certain conditions some Delaunay S-simplexes can cover each other. Such covering of S-simplexes can result in error during calculation of the volume of complex voids. Fortunately, this possibility can be ignored for molecular systems. To verify this, we had compared the sum of the volumes of all Delaunay S-simplexes with total volume of the model. The difference was found is negligibly small for our models (hundredth of a percent).

3 The Improved Algorithm for 3D Voronoi S-Network Calculation

There were several attempts to implement an algorithm for additively weighted Voronoi construction. However a detailed investigation of the problem has been made only for the case of 2D [9,11,12,20]. The 3D applications are restricted to the Voronoi S-regions (additively weighted Voronoi cell) [10,22]. In our earlier papers [4] we used our previous version of the algorithm for the Voronoi S-network calculation, but it was not very efficient for large models. A specific algorithm for numerical calculations of S-network for straight lines and spherocylinders was realized in [14], which can be also applied to spherical particles but it is much slower. Here we present our method which is efficient for large models and specialized for investigation of voids in complex molecular systems.

The main idea is simple and based on technique proposed many years ago for calculation of the 3D Voronoi polyhedra [24,15], where starting from a Voronoi vertex (site), the neighboring sites are calculated consecutively for every face of the polyhedron. A difference now is only that the other formulas for calculation of the coordinates of sites are used, see e.g. [9,18,20]. To calculate a new site we involve in calculation only limited number of atoms in the neighborhood of a given site. If we know these neighbours, the CPU time for calculation of a site does not depend on a total number of atoms in the model. Using linked-list based structure, which establishes a correspondence among coordinates and numbers of atoms, we immediately reestablish atoms which are close to a given point (e.g. to a site of the network) [2]. This improvement makes construction of Voronoi network much faster.

Implementation of the Voronoi-Delaunay Method for Analysis 223

Fig. 4. Left: CPU time for Voronoi S-network computation as a function of the number of atoms in the model. Right: The profile of the fraction of empty volume across the simulated lipid bilayer along the membrane normal axis Z. Dashed vertical lines show the division of the system into three separate regions according to the behavior of this profile

For illustration of efficiency of the algorithm we carried out some tests (Fig. 4 left). A PC with Intel processor P4 with 1700 MHz and RAM 256 was used. Three types of models were tested: dense non-crystalline packing of equal balls (curve 1), dense disordered packing of balls with radii 1 and 0.5 in fraction of one-to-one (curve 2), and a molecular system on the base of the model of lipid bilayer in water (curve 3). Starting configurations of the models for all types contained about 10000 atoms in boxes with periodic boundary conditions. Enlargement of the models was made by replication of starting configuration into bigger box according the periodic boundary condition. Calculation of the Voronoi S-network implies creation of the arrays D, DD, DA, Ri, Rb and their recording in hard disk of the computer. Different types of models demonstrate different CPU time (what depends of the structure of models), but all of them demonstrate clear lineal dependence on number of atoms.

4 Application of Model to Analysis of Lipid Bilayers

We illustrate application of the method to a computer model of the fully hydrated DMPC bilayer as obtained from a recent all-atom Monte Carlo simulation, Fig. 5. Each of the two membrane layers contain 25 DMPC molecules, described by the CHARMM22 force field optimized for proteins and phospholipid molecules and the bilayer is hydrated by 2033 water molecules. The sample analyzed consists of 1000 independent configurations, each of them saved after performing 105 new Monte Carlo.

In analyzing the distribution and properties of the voids in the model we have first determined the fraction of the empty space across the membrane. The resulting profile along the membrane normal axis Z is shown in Fig. 4 (right). As seen, three different membrane regions can be clearly distinguished according to the behaviour of this profile. Region 1, in the middle of the membrane, is characterized by a relatively large fraction of the empty volume, which is considerably lower in the adjacent region 2. Finally, in region 3, located apart from the lipid bilayer, the fraction of the empty space is the highest in the entire system. These regions, marked also in Fig. 4 (right), roughly coincide with the region of the hydrophobic lipid tails, the dense region of the hydrated zwitterionic headgroups and the region of bulk-like water, respectively.

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Fig. 5. 3D configuration of the DMPC lipid bilayer in a box with periodic boundary conditions.

5 Experimental Analysis

In the analysis, we have determined the voids using different values of the probe radius between 1.0Å and 1.6Å, with an increment of 0.1Å. In this way, the criterion of void detection has been varied in the analyses, allowing us a more reliable characterization of the properties of the voids. For quantitative characterization of the voids we depicture them as spherocylinders, see section 2.5. In the following, characteristics of the spherocylinders (i.e., length, radius, volume and orientation) are studied as a function of the probe radius in three separate regions of the membrane.

The dependence of the mean values of the length L of the spherocylinders on the probe radius in the three membrane regions are shown in Fig. 6 (left). As is seen, the observed length of the voids is clearly different in the three different parts of the membrane, and this difference is preserved for all probe radii used, showing that this finding is independent from the void detection criterion. It should be noted that the longest spherocylinders, and hence the most elongated voids are found in region 1, i.e., at the middle of the membrane, whereas the largest fraction of the empty volume occurs in the region of bulk-like water (region 3, see Fig.6 (left)). This finding indicates that the empty volume is distributed considerably more uniformly in the aqueous region than in the hydrocarbon phase of the bilayer.

Fig. 6. An average length L of the spherocylinders representing the voids (left) and an average cosine of the angle α formed by the bilayer normal axis Z with the main axis of the spherocylinders representing the voids (right), as a function of the probe radius. Squares: region 1 (hydrocarbon tails), circles: region 2 (headgroups), triangles: region 3 (bulk-like water).

Implementation of the Voronoi-Delaunay Method for Analysis 225

In analyzing the orientation of the voids we have calculated the mean cosine of the angle α formed by the main axis of the spherocylinder and the bilayer normal axis Z. Isotropic orientation of spherocylinders results in the mean cosine value of 0.5, whereas for preferential orientations perpendicular and parallel to the plane of bilayer the inequalities cosα>0.5 and cosα<0.5, respectively, hold. The dependence of the mean value of cosα on the probe radius in the three separate membrane regions is shown in Fig. 6(right). Observe that the mean cosine value is clearly larger than 0.5 in regions 1 and 2, being larger in region 1 than 2 for all probe sizes used. Thus, the preferred orientation of the pores is clearly perpendicular to the plane of the membrane in the region of the hydrophobic tails, and also, in a smaller extent, in the region of the headgroups. This finding reflects the fact that the arrangement of the voids located between the lipid tails follows the preferred arrangement of these tails.

6 Conclusion

Application of the Voronoi–Delaunay method to molecular system modeling and analysis are discussed in the paper. The methodological aspects of Voronoi network consturtion in 3D, determination of intermolecular voids, and calculation of some physical characteristics of voids are presented. The method illustrated on the experimental analysis of the Monte Carlo models of fully hydrated DMPC bilayer. It provides a detailed overview of pitfalls and a handy solution to analysis of general behaviour of the interatomic voids in such systems.

Acknowledgements. This work was supported by the RFFI (grant 01-03-32903 and 04-03-32283), INTAS (grant 01-0067), OTKA (grant F038187) and CRDF (grant NO-008-X1), University of Calgary Starter Grant, and NSERC Grant.

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