Abstract
An adaptive finite element solver for the numerical calculation of the electrostatic coupling between molecules in a solvent environment is developed and tested. At the heart of the solver is a goal-oriented a posteriori error estimate for the electrostatic coupling, derived and implemented in the present work, that gives rise to an orders of magnitude improved precision and a shorter computational time as compared to standard finite difference solvers. The accuracy of the new solver ARGOS is evaluated by numerical experiments on a series of problems with analytically known solutions. In addition, the solver is used to calculate electrostatic couplings between two chromophores, linked to polyproline helices of different lengths and between the spike protein of SARS-CoV-2 and the ACE2 receptor. All the calculations are repeated by using the well-known finite difference solvers MEAD and APBS, revealing the advantages of the present finite element solver.
Highlights
Electrostatics is the key that allows proteins and other biological macromolecules to perform their multiple functions [1,2,3,4]
We present the electrostatic interaction between the receptor binding domain (RBD) of the spike protein of SARS-CoV-2 and the cell receptor ACE2 calculated with Adaptive Poisson–Boltzmann Solver (APBS) and Adaptive Refinement GoalOriented Solver [43] (ARGOS)
The electrostatic interaction computed on mesh refinement level (MRL) = 2 and 344,887 grid points is already accurate enough and within 0.1 % of the one computed on MRL = 5
Summary
Electrostatics is the key that allows proteins and other biological macromolecules to perform their multiple functions [1,2,3,4]. The discrete approximation of the (infinite) Coulombic self-interaction energy is not guaranteed to cancel out in the difference of the numerically computed electrostatic energies unless one uses exactly the same grids and redistribution of the point charges Another way to avoid the spurious grid artifact in the calculation of the energy difference is to apply proper splitting techniques for the full potential in the PBE. The FE method, which is employed in the present work, is the most successful for elliptic PDEs, since it combines geometrical flexibility and satisfactory convergence analysis with the ability to handle nonlinear problems involving interface jump conditions and nonsmooth source terms It enjoys a wide range of efficient iterative solvers for the resulting sparse linear systems, where the PBE is typically solved on unstructured finite element meshes.
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