Accurate solution of multi‐region continuum biomolecule electrostatic problems using the linearized Poisson–Boltzmann equation with curved boundary elements

Abstract

We present a boundary-element method (BEM) implementation for accurately solving problems in biomolecular electrostatics using the linearized Poisson-Boltzmann equation. Motivating this implementation is the desire to create a solver capable of precisely describing the geometries and topologies prevalent in continuum models of biological molecules. This implementation is enabled by the synthesis of four technologies developed or implemented specifically for this work. First, molecular and accessible surfaces used to describe dielectric and ion-exclusion boundaries were discretized with curved boundary elements that faithfully reproduce molecular geometries. Second, we avoided explicitly forming the dense BEM matrices and instead solved the linear systems with a preconditioned iterative method (GMRES), using a matrix compression algorithm (FFTSVD) to accelerate matrix-vector multiplication. Third, robust numerical integration methods were employed to accurately evaluate singular and near-singular integrals over the curved boundary elements. Finally, we present a general boundary-integral approach capable of modeling an arbitrary number of embedded homogeneous dielectric regions with differing dielectric constants, possible salt treatment, and point charges. A comparison of the presented BEM implementation and standard finite-difference techniques demonstrates that for certain classes of electrostatic calculations, such as determining absolute electrostatic solvation and rigid-binding free energies, the improved convergence properties of the BEM approach can have a significant impact on computed energetics. We also demonstrate that the improved accuracy offered by the curved-element BEM is important when more sophisticated techniques, such as non-rigid-binding models, are used to compute the relative electrostatic effects of molecular modifications. In addition, we show that electrostatic calculations requiring multiple solves using the same molecular geometry, such as charge optimization or component analysis, can be computed to high accuracy using the presented BEM approach, in compute times comparable to traditional finitedifference methods.

A quasi-static continuum model describing interactions between plasmons and non-absorbing biomolecules

“…41 and extended here to the quasi-static regime for studying plasmon-biomolecule interactions. Then, we will give a quick introduction to precorrected-FFT technique and introduce the procedure intended for the efficient treatment of frequency-sweep analysis.…”

“…41 that can exactly represent the underlying geometry and plasmonic particles are discretized into flat triangles. Techniques for integrating Laplace Green's function and its normal derivative over curved elements and flat triangles are discussed in Refs.…”

“…Similarly, a surface's exterior equation contains contributions from the surface, its parent, and its siblings. 41 In order to derive BEM matrix elements, we define a tree describing topology of the surfaces. Fig.…”

“…41 In order to compute all the interactions, the following block matrix-vector multiplications are performed for every node in the tree. Assume that current node corresponds to the region X. Denote its parent region by W, sibling regions by S i , and child regions by Y i .…”

“…41 Here, we will use continuum modeling to describe the interactions between plasmons and biomolecules. Biomolecules are represented by their molecular surfaces which allow us for taking into account the influence of exact binding conformations as well as structural differences between different proteins on the response of plasmonic nanoparticles.…”