Reliable Computer Simulation Methods for Electrostatic Biomolecular Models Based on the Poisson–Boltzmann Equation

Abstract

The paper is concerned with the reliable numerical solution of a class of nonlinear interface problems governed by the Poisson–Boltzmann equation. Arising in electrostatic biomolecular models these problems typically contain measure-type source terms and their solution often exposes drastically different behaviour in different subdomains. The interface conditions reflect the requirement that the potential and its normal derivative must be continuous. In the first part of the paper, we discuss an appropriate weak formulation of the problem that guarantees existence and uniqueness of the generalized solution. In the context of the considered class of nonlinear equations, this question is not trivial and requires additional analysis, which is based on a special splitting of the problem into simpler subproblems whose weak solutions can be defined in standard Sobolev spaces. This splitting also suggests a rational numerical solution strategy and a way of deriving fully guaranteed error bounds. These bounds (error majorants) are derived for each subproblem separately and, finally, yield a fully computable majorant of the difference between the exact solution of the original problem and any energy-type approximation of it.The efficiency of the suggested computational method is verified in a series of numerical tests related to real-life biophysical systems.

ARGOS: An adaptive refinement goal‐oriented solver for the linearized Poisson–Boltzmann equation

“…It can be rigorously shown that under certain conditions on the regularity of the interface Γ there exists a unique potential φ that satisfies the integral formulation () for any test function v in a certain function space [22, 23]. By directly discretizing (), one obtains an approximation φ h of the exact potential φ .…”

“…This is in contrast to the functional type a posteriori error estimates developed in Ref. [23] or the residual‐based error indicator used, for example, in the fe‐manual routine of APBS, which aims at the reduction of the global (integral) energy norm of the error in the potential.…”

“…In the so‐called weak formulation 1 of the problem defining the regular part of the potential, this better behaved right hand side can be represented as a surface integral over the solute–solvent interface of a (polarization) charge density function, see Equation (21) in [21] and the interpretation after it. Thus, mathematically speaking, this right hand side can be interpreted as a certain charge density concentrated on the molecular surface (see Remarks 3.1 and 3.2 in [22] or Remarks 2.1 and 2.2 in [23]). We note that these splitting techniques can be applied in conjunction with any numerical procedure to solve the PBE, and in particular with the FD method.…”

Adaptive Goal-Oriented Solver for the Linearized Poisson- Boltzmann Equation

“…It can be rigorously shown that under certain conditions on the regularity of the interface Γ there exists a unique potential ϕ that satisfies the integral formulation (10) for any test function v in a certain function space. 39,40 By directly discretizing (10), one obtains an approximation ϕ h of the exact potential ϕ. Another widespread approach, from which both the numerical treatment and analysis of the LPBE (also the PBE) benefits, involves splitting the potential ϕ into two parts G and u, i.e., ϕ = G + u (see, e.g.…”

Weak formulations of the nonlinear Poisson-Boltzmann equation in biomolecular electrostatics

“…The fact that the regular component of a solution obtained by such a decomposition satisfies a weak formulation involving H 1 spaces means that this component can be numerically approximated by means of well studied methods, such as standard conforming finite elements. Besides, it also means that the duality approach for error estimation is applicable to obtain both a priori near-best approximation results and to compute guaranteed a posteriori error bounds, as done in [40,41]. Having a C 1 interface also has practical implications, since in this case it is easier to represent exactly with curved elements or isogeometric analysis.…”