Kernel modeling for molecular surfaces using a uniform solution

Abstract

In this paper, a rational Bézier surface is proposed as a uniform approach to modeling all three types of molecular surfaces (MS): the van der Waals surface (vdWS), solvent accessible surface (SAS) and solvent excluded surface (SES). Each molecular surface can be divided into molecular patches, which can be defined by their boundary arcs. The solution consists of three steps: topology modeling, boundary modeling and surface modeling. Firstly, using a weighted $\alpha$-shape, topology modeling creates two networks to describe the neighboring relationship of the molecular atoms. Secondly, boundary modeling derives all boundary arcs from the networks. Thirdly, surface modeling constructs all three types of molecular surfaces patch-by-patch, based on the networks and the boundary arcs. For an SES, the singularity is specially treated to avoid self-intersections. Instead of approximation, this proposed solution can produce precise shapes of molecular surfaces. Since rational Bézier representation is much simpler than a trimmed non-uniform rational B-spline surface (NURBS), computational load can be significantly saved when dealing with molecular surfaces. It is also possible to utilize the hardware acceleration for tessellation and rendering of a rational Bézier surface. CAGD kernel modelers typically use NURBSs as a uniform representation to handle different types of free-form surface. This research indicates that rational Bézier representation, more specifically, a bi-cubic or 2×4 rational Bézier surface, is sufficient for kernel modeling of molecular surfaces and related applications.

Introduction

Biological molecules like proteins are important for all biological organisms. In order to understand the molecular functions, molecular structures are intensively studied at different levels. Since molecules interact at their surfaces, an understanding of the molecular surfaces (MS) can be useful for studying these interactions. Molecular surfaces can be used in molecular visualization and analysis, function predictions and drug design. Three types of molecular model have been proposed: the van der Waals surface (vdWS), solvent accessible surface (SAS), and solvent excluded surface (SES). Different methods have been proposed to model these surfaces.

The vdWS [1], [2] is used to describe a molecule based on its atoms of van der Waals radius. It can be defined as the union of all portions of every atomic sphere surface that is not occluded by neighboring atoms (Fig. 1(a)). Several methods have been developed to model vdWSs [3], [4], [5]. A simple application of a vdWS is to compute the molecular volume, which can be described as the volume enclosed by the vdWS. Adams [6] discussed the calculation of the volume and area of vdWSs.

Although the vdWS is a reasonable model for molecules, it does not address the issue of whether or not an atom is accessible to the solvent environment. The SAS [2], [7] was proposed, taking into account the effect of the solvent. It is described as the surface created by the center of a probe rolling over the entire vdWS. This can be considered as a van der Waals surface whose atomic radii have been extended by the probe radius (Fig. 1(b)). Hence, the modeling of an SAS can be similar to that of a vdWS [8], [9], [10], [11], [12]. Calculations of the volumes and areas of SASs were studied in [13], [14], [15], [16], [17].

The SES [18] is another description of the molecular surfaces which also takes into account the solvent. It, however, is defined as the surface traced out by the inward-facing surface of a probe (Fig. 1 (c)). The SES consists of contact surfaces and re-entrant surfaces. Contact surfaces are the parts of the vdWS that can be touched by a probe and the re-entrant surfaces are the inward-facing parts of the probe atom when it is in touch with more than one atom. An SES is a smooth molecular surfaces except at the singular points, which will be elaborated later. As proposed by Connolly [8], an SES consists of three types of region: concave spherical patches, convex spherical patches and saddle-shaped toroidal patches (briefly saddle patches). Various methods have been proposed to model SESs [19], [20], [21], [22].

The three different types of molecular surfaces are all important for structure and function analysis. Often, different methods are used to model these different types of molecular surfaces. For example, Richard’s method [18] is used to model an SAS and Connolly’s method [23] is used to model an SES. As such, often different data structures are required to model different surfaces. To have compatible and uniform molecular surface representation, simple polygonal representation [9], [24], [25], [26], [27] is used through approximation and tessellation. Such an approximation method may produce an inaccurate representation, which may cause problems, especially for dynamics surface simulation. Besides, tessellation can easily create millions of triangular meshes. A rational Bézier surface is more cost-effective in terms of computing compared to a trimmed non-uniform rational B-spline surface (trimmed NURBS) [28], [29]. With a rational Bézier surface, there is also the advantage of creating an adaptive and crack-free tessellation model when needed. Furthermore, the hardware support for Bézier surfaces has been developed. For example, NVIDIA’s GeForce3 supports the tessellation of Bézier surfaces with OpenGL extensions for Bézier surface evaluation (both rational and non-rational) [30].

Connolly’s method [23] modeled an SES as a collection of analytical patches without handling singularities. CAGD is used to describe various surfaces including the molecular surfaces. Bajaj et al. [28], [29] used a trimmed NURBS (not a NURBS) to describe the molecular surfaces with a control mesh and a set of trimmed curves. A rational Bézier surface is more convenient in modeling and rendering with potential hardware acceleration. A piecewise polynomial Bernstein–Bézier (BB) spline function was used by Zhao [31] for molecular surface representation. Theoretically, this approach can only produce an approximate representation. For accurate molecular surface representation, a rational Bézier surface is necessary.

The objectives of this research are as follows:

• To develop a uniform approach to modeling all three types of molecular surfaces.

• To have an accurate representation of molecular surfaces instead of approximation.

• To design an effective and robust method for molecular surface representation, using a low-degree rational Bézier surface, thus avoiding a computationally expensive NURBS.

The proposed research has several applications:

• To design a kernel modeler for molecular surfaces using a uniform rational Bézier representation, and therefore uniform data structure.

• To tell exactly how many rational Bézier patches there are in the molecular surfaces, and thus to calculate the surface areas, etc.

• To speed up the rendering process using graphics hardware.

• To support level of detail through subdivision of the control meshes of the rational Bézier represented molecular surfaces.