Differential geometry based solvation model I: Eulerian formulation

Abstract

This paper presents a differential geometry based model for the analysis and computation of the equilibrium property of solvation. Differential geometry theory of surfaces is utilized to define and construct smooth interfaces with good stability and differentiability for use in characterizing the solvent-solute boundaries and in generating continuous dielectric functions across the computational domain. A total free energy functional is constructed to couple polar and nonpolar contributions to the salvation process. Geometric measure theory is employed to rigorously convert a Lagrangian formulation of the surface energy into an Eulerian formulation so as to bring all energy terms into an equal footing. By minimizing the total free energy functional, we derive coupled generalized Poisson-Boltzmann equation (GPBE) and generalized geometric flow equation (GGFE) for the electrostatic potential and the construction of realistic solvent-solute boundaries, respectively. By solving the coupled GPBE and GGFE, we obtain the electrostatic potential, the solvent-solute boundary profile, and the smooth dielectric function, and thereby improve the accuracy and stability of implicit solvation calculations. We also design efficient second order numerical schemes for the solution of the GPBE and GGFE. Matrix resulted from the discretization of the GPBE is accelerated with appropriate preconditioners. An alternative direct implicit (ADI) scheme is designed to improve the stability of solving the GGFE. Two iterative approaches are designed to solve the coupled system of nonlinear partial differential equations. Extensive numerical experiments are designed to validate the present theoretical model, test computational methods, and optimize numerical algorithms. Example solvation analysis of both small compounds and proteins are carried out to further demonstrate the accuracy, stability, efficiency and robustness of the present new model and numerical approaches. Comparison is given to both experimental and theoretical results in the literature. KeywordsGeneralized Poisson-Boltzmann equation; Biomolecular surface formation and evolution; Potential driving geometric flows; Solvation free energy; Multiscale models I IntroductionAmong the various components of molecular interactions, electrostatic interactions are of special importance 6,8,41,44,55,59,71, 142 -144 , 169 ,170 because of their long range and influence on polar or charged molecules -including water, aqueous ions, and amino or nucleic acids. Electrostatic interactions are ubiquitous for any system of charged or polar molecules, such as * Please address correspondence to Guowei Wei. wei@math.msu.edu. (proteins, nucleic acids, lipid bilayers, sugars, etc.) in their aqueous environment. Electrostatic solute-solvent interactions, therefore, are of central importance in analyzing molecular structure and modeling the intramolecular and intermolecular interactions of macromolecules in simulations. There are two types of solvation models: 131, 142 , 143 , 169 explicit ...

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

“…In comparison, Nicholls et al [39] reported a RMSE of 1.87 kcal/mol for the same set of molecules by the linear Poisson model and further, the RMSE was reduced to 1.71 ± 0.05 kcal/mol [39] by their explicit solvent approach, which is much more expensive. Additionally, the RMSE in the current work is only slightly greater than the one (RSME=1.76 kcal/mol) reported in [18], where the biomolecular surface is defined and generated from a nonlinear mean curvature flow equation so extra computational efforts are necessary. Therefore, FPDE transform-based molecular surface provides a relatively good foundation to predict the solvation energies for this set of molecules.…”

“…Since variational approaches have found their success in a variety of scientific and engineering fields [6, 16, 18–20, 45, 60, 62], a variational derivation of the PDE transform has also been presented [55, 76]. Here we briefly review the variational derivation of the PDE transform.…”

“…Piecewise-constant initial values are commonly used in our earlier work for molecular surface generation [5, 6, 18]. In this work, we set the piecewise-constant initial value as $$ufalse(\mathrm{boldr},0false)=true\{\begin{array}{cc}left0,& left\mathbf{r}\in {\cup }_{i}O({\mathrm{boldr}}_i,r_i);\\ left1,& left\text{otherwise},\end{array}$$ where r i and r i are the atomic coordinate and radius of atom i , respectively.…”

“…The union of all atomic spheres O ( r i , r i ) is denoted by ∪ i O ( r i , r i ). This type of piecewise-constant initial value is different from the initial value in our geometry flow based solvation model [18], where the two constants 0 and 1 are switched.…”

“…The latter component of solvation free energy can be modeled in many ways. One of the most commonly used model can be expressed as [18, 54] $$G_{\mathrm{normaln}\mathrm{normalp}}=\gamma \text{Area}+p\text{Vol}+{\int }_{{\Omega }_s}{\rho }_s{U}^{\mathrm{normala}\mathrm{normalt}\mathrm{normalt}\mathrm{normala}}d\mathrm{boldr},$$ where γ is the surface tension, p is the hydrodynamic pressure, ρ s is the solvent bulk density, Ω s denotes the solvent accessible region, and U att ( r ) is the solute-solvent van der Waals interaction potential at point r . The first term γArea is the surface energy, which describes the disruption of intermolecular and/or intramolecular bonds that occurs when the surface of a molecule is created in the solvent.…”