Binary Level Set Method for Variational Implicit Solvation Model

TL;DR

The binary level set method is applied to the Variational Implicit Solvent Model (VISM), which is a theoretical and computational tool to study biomolecular systems with complex topology and can be minimized by iteratively"flipping" thebinary level set function in a steepest descent fashion.

Abstract

In this article, we apply the binary level set method to the Variational Implicit Solvent Model (VISM), which is a theoretical and computational tool to study biomolecular systems with complex topology. Central in VISM is an effective free energy of all possible interfaces separating solutes (e.g., proteins) from solvent (e.g., water). Previously, such a functional is minimized numerically by the level set method to determine the stable equilibrium conformations and solvation free energies. We vastly improve the speed by applying the binary level set method, in which the interface is approximated by a binary level set function that only takes value $\pm 1$ on the solute/solvent region, leading to a discrete formulation of VISM energy. The surface area is approximated by convolution of an indicator function with a compactly supported kernel. The VISM energy can be minimized by iteratively "flipping" the binary level set function in a steepest descent fashion. Numerical experiments are performed to demonstrate the accuracy and performance of our method.

Figures (12)

• Figure 1: Explicit and implicit solvent model. The solute region, solvent region, and solute-solvent interface are denoted by $\Omega_m$, $\Omega_w$, and $\Gamma$
• Figure 2: Schematic view of a molecular system with implicit solvent. The atoms are located at $\br_i$ with charge $Q_i$ and LJ parameter $\sigma_i$ and $\varepsilon_i$. An interface $\Gamma$ (dashed line) separates the solvent region $\Omega_w$ from the solute region $\Omega_m$. In continuous level set method, $\Gamma$ is represented by the zero level set of a function. In the binary level set method, the computational domain is discretized in to grid cell. $\phi=-1$ for grid cells inside the solute region (white) and $\phi=1$ for grid cells in the solvent region (grey).
• Figure 3: Illustration of a scaled kernel centered at $\mathbf{x}_i$ and vanishing outside a sphere (dashed line). Black dots represent centers of grid cells in the solvent region $\Omega_w$ and circles represent the centers of grid cells in the solute region $\Omega_m$.
• Figure 4: Illustration of a tight initial surface (dashed line), a loose initial surface (dash-dotted line), and a VISM relaxed surface (solid line) $\Gamma$ surrounding the atoms (dots). For loose initial, we set $\phi=-1$ for all grid cells. The tight initial is the union of sphere of radius $\sigma_i$ centered at $\br_i$.
• Figure 5: log-log plot of the relative error versus the number of interval $n$ in one edge of the computational box. $n$ ranges from 20 to 200 with increments of 5. Each data point is an average of 6 spheres with random centers. $m$ is the slope of the line fitted by least-square. (L) sin-squared kernel and cos kernel with the same size. (R) sin-squared kernel with different size parameter $C$.