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Graphical Processing Unit accelerated Poisson equation solver and its application for calculation of single ion potential in ion-channels

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Poisson and Poisson-Boltzmann equations (PE and PBE) are widely used in molecular modeling to estimate the electrostatic contribution to the free energy of a system. In such applications, PE often needs to be solved multiple times for a large number of system configurations. This can rapidly become a highly demanding computational task. To accelerate such calculations we implemented a graphical processing unit (GPU) PE solver described in this work. The GPU solver performance is compared to that of our central processing unit (CPU) implementation of the solver. During the performance analysis the following three characteristics were studied: (1) precision associated with the modeled system discretization on the grid, (2) numeric precision associated with the floating point representation of real numbers (this is done via comparison of calculations with single precision (SP) and double precision (DP)), and (3) execution time. Two types of example calculations were carried out to evaluate the solver performance: (1) solvation energy of a single ion and a small protein (lysozyme), and (2) a single ion potential in a large ion-channel (α-hemolysin). In addition, influence of various boundary condition (BC) choices was analyzed, to determine the most appropriate BC for the systems that include a membrane, typically represented by a slab with the dielectric constant of low value. The implemented GPU PE solver is overall about 7 times faster than the CPU-based version (including all four cores). Therefore, a single computer equipped with multiple GPUs can offer a computational power comparable to that of a small cluster. Our calculations showed that DP versions of CPU and GPU solvers provide nearly identical results. SP versions of the solvers have very similar behavior: in the grid scale range of 1-4 grids/Å the difference between SP and DP versions is less than the difference stemming from the system discretization. We found that for the membrane protein, the use of a focusing technique with periodic boundary conditions in rough grid provides significantly better results than using a focusing technique with the electric potential set to zero at the boundaries.
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