Fast alternating direction implicit schemes for geometric flow equations and nonlinear poisson equation in biomolecular solvation analysis

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University of Alabama Libraries

The present work introduces new alternating direction implicit (ADI) methods to solve potential driven geometric flow partial differential equations (PDEs) for biomolecular surface generation and the nonlinear Poisson equations for electrostatic analysis. For solving potential driven geometric flow PDEs, an extra factor is usually added to stabilize the explicit time integration. However, there are two existing ADI schemes based on a scaled form, which involves nonlinear cross derivative terms that have to be evaluated explicitly. This affects the stability and accuracy of these ADI schemes. To overcome these difficulties, we propose a new ADI algorithm based on the unscaled form so that cross derivatives are not involved. Central finite differences are employed to discretize the nonhomogenous diffusion process of the geometric flow. The proposed ADI algorithm is validated through benchmark examples with analytical solutions, reference solutions, or literature results. Moreover, quantitative indicators of a biomolecular surface, including surface area, surface-enclosed volume, and solvation free energy, are analyzed for various proteins. The proposed ADI method is found to be unconditionally stable and more accurate than the existing ADI schemes in all tests of biomolecular surface generation. The proposed ADI schemes have also been applied in solving the nonlinear Poisson equation for electrostatic solvation analysis. Compared with the existing biconjugate gradient iterative solver, the ADI scheme is more efficient. The CPU time cost is validated through the solvation analysis of an one atom Kirkwood model and a set of 17 small molecules whose experimental measurements are available. Additionally, application of the proposed ADI scheme is considered for electrostatic solvation analysis of a set of 19 proteins. The proposed ADI scheme enables the use of a large time increment in the steady state simulation so that the proposed ADI algorithm is efficient for biomolecular surface generation and solvation analysis.

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Mathematics, Biomedical engineering, Bioinformatics