Molecular Skin Surface-Based Transformation Visualization between Biological Macromolecules

Abstract

Molecular skin surface (MSS), proposed by Edelsbrunner, is a C² continuous smooth surface modeling approach of biological macromolecules. Compared to the traditional methods of molecular surface representations (e.g., the solvent exclusive surface), MSS has distinctive advantages including having no self-intersection and being decomposable and transformable. For further promoting MSS to the field of bioinformatics, transformation between different MSS representations mimicking the macromolecular dynamics is demanded. The transformation process helps biologists understand the macromolecular dynamics processes visually in the atomic level, which is important in studying the protein structures and binding sites for optimizing drug design. However, modeling the transformation between different MSSs suffers from high computational cost while the traditional approaches reconstruct every intermediate MSS from respective intermediate union of balls. In this study, we propose a novel computational framework named general MSS transformation framework (GMSSTF) between two MSSs without the assistance of union of balls. To evaluate the effectiveness of GMSSTF, we applied it on a popular public database PDB (Protein Data Bank) and compared the existing MSS algorithms with and without GMSSTF. The simulation results show that the proposed GMSSTF effectively improves the computational efficiency and is potentially useful for macromolecular dynamic simulations.

Figure 1

Examples of the molecular skin surfaces with PDB numbers: 1XD7, 1J5F, 114D, 101D, 1AIE, and 133D.

Figure 2

Transformation from a caffeine MSS to a protein MSS.

Figure 3

The red line shows the MSS defined by two weighted points marked in thick blue (B). All green circles represent $\sqrt{\text{conv}\left(B\right)}$.

Figure 4

The flowchart of constructing a MSS in ℝ³.

Figure 5

Superimposition of two Voronoi complexes constructed by two weighted point sets under SIGP assumption. The resulting intermediate Voronoi complex (right-most) has degenerate Voronoi cells such as the marked Voronoi vertex which is surrounded by four Voronoi regions.

Figure 6

The linear interpolation between source vertices (a, b) and target vertices (c, d, and e) produces a transforming Delaunay cell which is a triangular prism.

Figure 7

An example of partial molecular movement.

Figure 8

Superimposition of two Voronoi complexes constructed by two weighted point sets belonging to one “molecule” with partial movement. While this simple four-atom “molecule” only moves one of its atoms, the resulting intermediate Voronoi complex (right-most) has degenerate Voronoi cells such as the Voronoi vertex v_B, which is an intersection at an endpoint of an edge.

Figure 9

In (a), two Voronoi polygons overlap to form a new intermediate Voronoi polygon. In (b), a Voronoi polygon intersects a Voronoi edge and forms another Voronoi edge. In (c), two Voronoi edges intersect at an intermediate Voronoi vertex. In (d), two Voronoi edges overlap to form an intermediate Voronoi edge.

Figure 10

Computational speed comparison of using KVA with/without GMSSTF.

Figure 11

Computational speed comparison of using CSA with/without GMSSTF.

Figure 12

Computational speed comparison on GPU implementation based on Chavent's algorithm with/without GMSSTF.