Accurate Boundary Integral Formulations for the Calculation of Electrostatic Forces with an Implicit-Solvent Model.

An accurate force calculation with the Poisson-Boltzmann equation is challenging, as it requires the electric field on the molecular surface. Here we present a calculation of the electric field on the solute-solvent interface that is exact for piecewise linear variations of the potential and analyze four different alternatives to compute the force using a boundary element method. We performed a verification exercise for two cases: the isolated and two interacting molecules. Our results suggest that the boundary element method outperforms the finite difference method, as the latter needs a much finer mesh than in solvation energy calculations to get acceptable accuracy in the force, whereas the same surface mesh as in a standard energy calculation is appropriate for the boundary element method. Among the four evaluated alternatives of force calculation, we saw that the most accurate one is based on the Maxwell stress tensor. However, for a realistic application, like the barnase-barstar complex, the approach based on variations of the energy functional, which is less accurate, gives equivalent results. This analysis is useful toward using the Poisson-Boltzmann equation for force calculations in applications where high accuracy is key, for example, to feed molecular dynamics models or to enable the study of the interaction between large molecular structures, like viruses adsorbed onto substrates.

Quantitative electrostatic force tomography for virus capsids in interaction with an approaching nanoscale probe.

Electrostatic interactions are crucial for the assembly, disassembly and stability of proteinaceous viral capsids. Moreover, at the molecular scale, elucidating the organization and structure of the capsid proteins in response to an approaching nanoprobe is a major challenge in biomacromolecular research. Here, we report on a generalized electrostatic model, based on the Poisson-Boltzmann equation, that quantifies the subnanometric electrostatic interactions between an AFM tip and a proteinaceous capsid from molecular snapshots. This allows us to describe the contributions of specific amino acids and atoms to the interaction force. We show validation results in terms of total electrostatic forces with previous semi-empirical generalized models at available length scales (d > 1 nm). Then, we studied the interaction of the Zika capsid with conical and spherical AFM tips in a tomography-type analysis to identify the most important residues and atoms, showing the localized nature of the interaction. This method can be employed for the interpretation of force microscopy experiments in fundamental virological characterization and in diverse nanomedicine applications, where specific regions of the protein cages are aimed to electrostatically interact with molecular sized functionalized inhibitors, or tailoring protein-cage functional properties for nucleic acid delivery.

Predicting the Orientation of Adsorbed Proteins Steered with Electric Fields Using a Simple Electrostatic Model.

Under the most common experimental conditions, the adsorption of proteins to solid surfaces is a spontaneous process that leads to a rather compact layer of randomly oriented molecules. However, controlling such orientation is critically important for the development of catalytic surfaces. In this regard, the use of electric fields is one of the most promising alternatives. Our work is motivated by experimental observations that show important differences in catalytic activity of a trypsin-covered surface, which depended on the applied potential during the adsorption. Even though adsorption results from the combination of several processes, we were able to determine that (under the selected conditions) mean-field electrostatics play a dominant role, determining the orientation and yielding a difference in catalytic activity. We simulated the electrostatic potential numerically, using an implicit-solvent model based on the linearized Poisson-Boltzmann equation. This was implemented in an extension of the code PyGBe that included an external electric field, and rendered the electrostatic component of the solvation free energy. Our model (extensions available at the Github repository) allowed estimating the overall affinity of the protein with the surface, and their most likely orientation as a function of the potential applied. Our results show that the active sites of trypsin are, on average, more exposed when the electric field is negative, which agrees with the experimental results of catalytic activity, and confirm the premise that electrostatic interactions can be used to control the orientation of adsorbed proteins.

Towards optimal boundary integral formulations of the Poisson-Boltzmann equation for molecular electrostatics.

The Poisson-Boltzmann equation offers an efficient way to study electrostatics in molecular settings. Its numerical solution with the boundary element method is widely used, as the complicated molecular surface is accurately represented by the mesh, and the point charges are accounted for explicitly. In fact, there are several well-known boundary integral formulations available in the literature. This work presents a generalized expression of the boundary integral representation of the implicit solvent model, giving rise to new forms to compute the electrostatic potential. Moreover, it proposes a strategy to build efficient preconditioners for any of the resulting systems, improving the convergence of the linear solver. We perform systematic benchmarking of a set of formulations and preconditioners, focusing on the time to solution, matrix conditioning, and eigenvalue spectrum. We see that the eigenvalue clustering is a good indicator of the matrix conditioning, and show that they can be easily manipulated by scaling the preconditioner. Our results suggest that the optimal choice is problem-size dependent, where a simpler direct formulation is the fastest for small molecules, but more involved second-kind equations are better for larger problems. We also present a fast Calderón preconditioner for first-kind formulations, which shows promising behavior for future analysis. This work sets the basis towards choosing the most convenient boundary integral formulation of the Poisson-Boltzmann equation for a given problem.

Efficient mesh refinement for the Poisson-Boltzmann equation with boundary elements.

The Poisson-Boltzmann equation is a widely used model to study electrostatics in molecular solvation. Its numerical solution using a boundary integral formulation requires a mesh on the molecular surface only, yielding accurate representations of the solute, which is usually a complicated geometry. Here, we utilize adjoint-based analyses to form two goal-oriented error estimates that allow us to determine the contribution of each discretization element (panel) to the numerical error in the solvation free energy. This information is useful to identify high-error panels to then refine them adaptively to find optimal surface meshes. We present results for spheres and real molecular geometries, and see that elements with large error tend to be in regions where there is a high electrostatic potential. We also find that even though both estimates predict different total errors, they have similar performance as part of an adaptive mesh refinement scheme. Our test cases suggest that the adaptive mesh refinement scheme is very effective, as we are able to reduce the error one order of magnitude by increasing the mesh size less than 20% and come out to be more efficient than uniform refinement when computing error estimations. This result sets the basis toward efficient automatic mesh refinement schemes that produce optimal meshes for solvation energy calculations.

Free Energies of the Disassembly of Viral Capsids from a Multiscale Molecular Simulation Approach.

Molecular simulations of large biological systems, such as viral capsids, remains a challenging task in soft matter research. On one hand, coarse-grained (CG) models attempt to make the description of the entire viral capsid disassembly feasible. On the other hand, the permanent development of novel molecular dynamics (MD) simulation approaches, like enhanced sampling methods, attempt to overcome the large time scales required for such simulations. Those methods have a potential for delivering molecular structures and properties of biological systems. Nonetheless, exploring the process on how a viral capsid disassembles by all-atom MD simulations has been rarely attempted. Here, we propose a methodology to analyze the disassembly process of viral capsids from a free energy perspective, through an efficient combination of dynamics using coarse-grained models and Poisson-Boltzmann simulations. In particular, we look at the effect of pH and charge of the genetic material inside the capsid, and compute the free energy of a disassembly trajectory precalculated using CG simulations with the SIRAH force field. We used our multiscale approach on the Triatoma virus (TrV) as a test case, and find that even though an alkaline environment enhances the stability of the capsid, the resulting deprotonation of the genetic material generates a Coulomb-type electrostatic repulsion that triggers disassembly.

A Boundary-Integral Approach for the Poisson-Boltzmann Equation with Polarizable Force Fields.

Implicit-solvent models are widely used to study the electrostatics in dissolved biomolecules, which are parameterized using force fields. Standard force fields treat the charge distribution with point charges; however, other force fields have emerged which offer a more realistic description by considering polarizability. In this work, we present the implementation of the polarizable and multipolar force field atomic multipole optimized energetics for biomolecular applications (AMOEBA), in the boundary integral Poisson-Boltzmann solver PyGBe. Previous work from other researchers coupled AMOEBA with the finite-difference solver APBS, and found difficulties to effectively transfer the multipolar charge description to the mesh. A boundary integral formulation treats the charge distribution analytically, overlooking such limitations. This becomes particularly important in simulations that need high accuracy, for example, when the quantity of interest is the difference between solvation energies obtained from separate calculations, like happens for binding energy. We present verification and validation results of our software, compare it with the implementation on APBS, and assess the efficiency of AMOEBA and classical point-charge force fields in a Poisson-Boltzmann solver. We found that a boundary integral approach performs similarly to a volumetric method on CPU. Also, we present a GPU implementation of our solver. Moreover, with a boundary element method, the mesh density to correctly resolve the electrostatic potential is the same for standard point-charge and multipolar force fields. Finally, we saw that for binding energy calculations, a boundary integral approach presents more consistent results than a finite difference approximation for multipolar force fields. © 2019 Wiley Periodicals, Inc.

Regional and physician specialty-associated variations in the medical management of atherosclerotic renal-artery stenosis.

For people enrolled in Cardiovascular Outcomes in Renal Atherosclerotic Lesions (CORAL), we sought to examine whether variation exists in the baseline medical therapy of different geographic regions and if any variations in prescribing patterns were associated with physician specialty. Patients were grouped by location within the United States (US) and outside the US (OUS), which includes Canada, South America, Europe, South Africa, New Zealand, and Australia. When comparing US to OUS, participants in the US took fewer anti-hypertensive medications (1.9 ± 1.5 vs. 2.4 ± 1.4; P < .001) and were less likely to be treated with an angiotensin-converting enzyme inhibitor or angiotensin II receptor blocker (46% vs. 62%; P < .001), calcium channel antagonist (37% vs. 58%; P < .001), and statin (64% vs. 75%; P < .05). In CORAL, the identification of variations in baseline medical therapy suggests that substantial opportunities exist to improve the medical management of patients with atherosclerotic renal-artery stenosis.