Quality meshing of implicit solvation models of biomolecular structures

Abstract

This paper describes a comprehensive approach to construct quality meshes for implicit solvation models of biomolecular structures starting from atomic resolution data in the Protein Data Bank (PDB). First, a smooth volumetric electron density map is constructed from atomic data using weighted Gaussian isotropic kernel functions and a two-level clustering technique. This enables the selection of a smooth implicit solvation surface approximation to the Lee-Richards molecular surface. Next, a modified dual contouring method is used to extract triangular meshes for the surface, and tetrahedral meshes for the volume inside or outside the molecule within a bounding sphere/box of influence. Finally, geometric flow techniques are used to improve the surface and volume mesh quality. Several examples are presented, including generated meshes for biomolecules that have been successfully used in finite element simulations involving solvation energetics and binding rate constants.

High-order fractional partial differential equation transform for molecular surface construction

“…Figure 8 shows the isosurfaces from Type III initial data at two different decay rates, κ = 1 and κ = 1/9. It is observed that κ = 1 gives a much better resolution of the molecular surface than κ = 1/9 does, this fact is consistent with the previous results [72]. …”

“…The third type of initial values is taken from the summation of Gaussian functions [8, 22, 27], $$ufalse(\mathrm{boldr},0false)=\sum _i\text{exp}false(-\kappa false(false|r-r_i{|}^2-r_i^{2}false)false).$$ In our simulation, we set the decay rate κ = 1 [72]. …”

“…The initial molecular surface can be obtained as the isosurface by setting u ( r , 0) = S , where S is a pre-designated isovalue of u , for example, u ( r , 0) = 1 [72, 76]. Although the two types of Gaussian initial data are able to offer smoother isosurfaces, they may still result in isosurfaces with singularities as demonstrated in the following sections.…”

Feature-preserving adaptive mesh generation for molecular shape modeling and simulation

“…An alternate way of defining the Gaussian kernel function is by putting the blobbyness and radius of each atom together as a constant, as formulated in the following equation [53]:…”

“…While the qualities of the interior tetrahedral meshes generated by LBIE are usually good, there are many "sharp" triangles on the surfaces due to the iso-contouring method used in LBIE, and hence many tetrahedra near the surfaces can be poorly shaped. A new version of LBIE has been introduced in [53] by applying surface and tetrahedral mesh smoothing techniques (such as edge contraction and weighted averaging). However, the tetrahedral mesh smoothing turns out to be slow and sometimes the algorithm fails.…”