Abstract
The Poisson–Boltzmann equation (PBE) is a nonlinear elliptic parametrized partial differential equation that arises in biomolecular modeling and is a fundamental tool for structural biology. It is used to calculate electrostatic potentials around an ensemble of fixed charges immersed in an ionic solution. Efficient numerical computation of the PBE yields a high number of degrees of freedom in the resultant algebraic system of equations, ranging from several hundred thousand to millions. Coupled with the fact that in most cases the PBE requires to be solved multiple times for a large number of system configurations, for example, in Brownian dynamics simulations or in the computation of similarity indices for protein interaction analysis, this poses great computational challenges to conventional numerical techniques. To accelerate such onerous computations, we suggest to apply the reduced basis method (RBM) and the (discrete) empirical interpolation method ((D)EIM) to the PBE with a special focus on simulations of complex biomolecular systems, which greatly reduces this computational complexity by constructing a reduced order model (ROM) of typically low dimension. In this study, we employ a simple version of the PBE for proof of concept and discretize the linearized PBE (LPBE) with a centered finite difference scheme. The resultant linear system is solved by the aggregation-based algebraic multigrid (AGMG) method at different samples of ionic strength on a three-dimensional Cartesian grid. The discretized LPBE, which we call the high-fidelity full order model (FOM), yields solution as accurate as other LPBE solvers. We then apply the RBM to the FOM. DEIM is applied to the Dirichlet boundary conditions which are nonaffine in the parameter (ionic strength), to reduce the complexity of the ROM. From the numerical results, we notice that the RBM reduces the model order from {mathcal {N}} = 2times 10^{6} to N = 6 at an accuracy of {mathcal {O}}(10^{-9}) and reduces the runtime by a factor of approximately 7600. DEIM, on the other hand, is also used in the offline-online phase of solving the ROM for different values of parameters which provides a speed-up of 20 for a single iteration of the greedy algorithm.
Highlights
Electrostatic interactions are important in biological processes such as molecular recognition, enzyme catalysis, and Communicated by Gabriel Wittum.15 Page 2 of 19 main groups of computational approaches which are used to model electrostatic interactions based on how the solvent is treated
We have presented a new, computationally efficient approach to solving the linearized PBE (LPBE) for varying parameter values occuring in biomolecular simulations
The true error between the reduced basis method (RBM) and the finite difference methods (FDM) is smaller than O(10−4), for all the parameter samples tested
Summary
Electrostatic interactions are important in biological processes such as molecular recognition, enzyme catalysis, and Communicated by Gabriel Wittum. These are not realistic because biomolecules have irregular shapes or geometries and charge distributions [6,7] This makes it necessary to apply numerical techniques to the PBE and the first of such methods were introduced in [8] where the electrostatic potential was determined at the active site of a protein (or enzyme). As a consequence of the nonaffine parameter dependence of these boundary conditions, we apply the (discrete) empirical interpolation method ((D)EIM) to reduce the resultant complexity in the reduced order model (ROM) during the online phase of the reduced basis method (RBM), see Sects. The novelty of this paper rests in the efficient construction of a low dimensional surrogate reduced order model (ROM) for the LPBE by the RBM and DEIM, whose solution is as accurate as those of popular PBE software packages, for example, the APBS. For the first time, the RBM has been applied to the LPBE in 3D for modeling of complex biomolecular systems, which are characterized by the presence of strong singularities generated by singular sources and subject to parametric nonaffine Dirichlet boundary conditions in the form of Yukawa potential
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