Shan Zhao

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  Pseudo-time coupled smooth interface models for biomolecular solvation calculations
  Shan Zhao
  Recently, we have introduced a differential geometry based model, the minimal molecular surface, to characterize the dielectric boundary between biomolecules and the surrounding aqueous environment. The mean curvature flow is used to minimize a surface free energy functional to drive the surface formation and evolution. More recently, several potential driven geometric flow models have been introduced in the literature for the analysis and computation of the equilibrium property of solvation, by appropriately coupling polar and nonpolar contributions in the free energy functional. The solvent-solute interface is usually treated as a sharp interface with discontinuous dielectric profile in a Lagrangian formulation, while in an Eulerian formulation a smeared interface model with continuous dielectric profile provides a convenient setting for solvation calculations. In the present study, we further extend the smeared interface model by considering a generalized nonlinear Poisson-Boltzmann (PB) equation in order to account for the salt effect. A new pseudo-time coupling between the surface geometric flows and electrostatic PB potential is introduced. Such a coupling allows for a fast numerical solution of governing nonlinear partial differential equations. Example solvation analysis of both small compounds and proteins are carried out to examine the proposed models and numerical approaches. Numerical results are compared to the experimental measurements and to those obtained by using other theoretical methods in the literature.
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  Operator splitting ADI schemes for nonlinear biomolecular solvation models
  Shan Zhao

  Recently, we have introduced a pseudo-time coupled nonlinear partial differential equation (PDE) model for biomolecular solvation analysis. Based on a free energy optimization, a boundary value system is derived to couple a nonlinear Poisson-Boltzmann (NPB) equation for electrostatic potential with a generalized Laplace-Beltrami (GLB) equation defining the biomolecular surface. By introducing a pseudo-time in both processes, a more efficient coupling is achieved through the steady state solution of two nonlinear parabolic PDEs. For the GLB equation, the pseudo-transient continuation is attained via a potential driven geometric flow PDE, which defines a smooth biomolecular surface to characterize the dielectric boundary between biomolecules and the surrounding aqueous environment. The resulting smooth dielectric profile, however, introduce some instability issue in solving the time-dependent NPB equation. This motivates us to develop an operator splitting alternating direction implicit (ADI) scheme, in which the nonlinear instability is completely avoided through analytical integration. To speed up the computation of molecular surface, a new fully implicit ADI scheme is developed too for solving the geometric flow equation. Unconditional stability can be realized by both ADI schemes in solving unsteady NPB and GLB equations separately. In solving a coupled system for real biomolecules and chemical compounds, the proposed numerical schemes are found to be conditionally stable. Nevertheless, the time stability can be maintained by using very large time increments, so that the present biomolecular simulation becomes much faster.