arXiv subjects

Christopher D. Cooper

Publications and source records attributed to Christopher D. Cooper.

15 recordsLinked to original sources

arXiv2025

A Self-Adjusting FEM-BEM Coupling Scheme for the Nonlinear Poisson-Boltzmann Equation

The Poisson-Boltzmann equation is widely used to model molecular electrostatics; however, it is usually solved in linearised form because the sinh nonlinearity is challenging, limiting its applicability in highly charged systems such as nucleic acids. This work presents a solution method for the nonlinear Poisson-Boltzmann equation based on a coupled finite/boundary element scheme that automatically finds an optimal relaxation parameter, ensuring fast and reliable convergence of the nonlinear solver without user intervention. We validated our solver against APBS for a spherical cavity, and used RNA-based structures to perform a thorough study of the different algorithmic choices, and to test our implementation. We found that the best alternative to solve the Poisson-Boltzmann equation was using a Newton-Raphson method where the nonlinearity was gradually introduced with a cubic approximation in the first iteration. Newton-Raphson was also the best method to find the optimal relaxation factor, reducing the number of iterations by 40%. Including other optimisation techniques, we were able to obtain a 1.37x speed-up with respect to the best hand-picked relaxation factor for 1HC8 (molecule with highest charge in our tests), avoiding any trial-and-error process to find the relaxation factor.

physics.comp-ph↗

arXiv2024

An Investigation of Physics Informed Neural Networks to solve the Poisson-Boltzmann Equation in Molecular Electrostatics

Physics-informed neural networks (PINN) is a machine learning (ML)-based method to solve partial differential equations that has gained great popularity due to the fast development of ML libraries in the last few years. The Poisson-Boltzmann equation (PBE) is widely used to model mean-field electrostatics in molecular systems, and in this work we present a detailed investigation of the use of PINN to solve the PBE. Starting from a multidomain PINN for the PBE with an interface, we assess the importance of incorporating different features into the neural network architecture. Our findings indicate that the most accurate architecture utilizes input and output scaling layers, a random Fourier features layer, trainable activation functions, and a loss balancing algorithm. The accuracy of our implementation is of the order of 10$^{-2}$ -- $10^{-3}$, which is similar to previous work using PINN to solve other differential equations. We also explore the possibility of incorporating experimental information into the model, and discuss challenges and future work, especially regarding the nonlinear PBE. Along with this manuscript, we are providing an open-source implementation to easily perform computations from a PDB file. We hope this work will motivate application scientists into using PINN to study molecular electrostatics, as ML technology continues to evolve at a high pace.

physics.chem-ph↗

arXiv2024

Some challenges of diffused interfaces in implicit-solvent models

The standard Poisson-Boltzmann model for molecular electrostatics assumes a sharp variation of the permittivity and salt concentration along the solute-solvent interface. The discontinuous field parameters are not only difficult numerically, but also are not a realistic physical picture, as it forces the dielectric constant and ionic strength of bulk in the near-solute region. An alternative to alleviate some of these issues is to represent the molecular surface as a diffuse interface, however, this also presents challenges. In this work we analysed the impact of the shape of the interfacial variation of the field parameters in solvation and binding energy. However we used a hyperbolic tangent function ($\tanh(k_p x)$) to couple the internal and external regions, our analysis is valid for other definitions. Our methodology was based on a coupled finite element (FEM) and boundary element (BEM) scheme that allowed us to have a special treatment of the permittivity and ionic strength in a bounded FEM region near the interface, while maintaining BEM elsewhere. Our results suggest that the shape of the function (represented by $k_p$) has a large impact on solvation and binding energy. We saw that high values of $k_p$ induce a high gradient on the interface, to the limit of recovering the sharp jump when $k_p\to\infty$, presenting a numerical challenge where careful meshing is key. Using the FreeSolv database to compare with molecular dynamics, our calculations indicate that an optimal value of $k_p$ for solvation energies was around 3. However, more challenging binding free energy tests make this conclusion more difficult, as binding showed to be very sensitive to small variations of $k_p$. In that case, optimal values of $k_p$ ranged from 2 to 20.

physics.chem-ph↗

arXiv2023

Coupling finite and boundary element methods to solve the Poisson--Boltzmann equation for electrostatics in molecular solvation

The Poisson--Boltzmann equation is widely used to model electrostatics in molecular systems. Available software packages solve it using finite difference, finite element, and boundary element methods, where the latter is attractive due to the accurate representation of the molecular surface and partial charges, and exact enforcement of the boundary conditions at infinity. However, the boundary element method is limited to linear equations and piecewise constant variations of the material properties. In this work, we present a scheme that couples finite and boundary elements for the Poisson--Boltzmann equation, where the finite element method is applied in a confined {\it solute} region, and the boundary element method in the external {\it solvent} region. As a proof-of-concept exercise, we use the simplest methods available: Johnson--N\'ed\'elec coupling with mass matrix and diagonal preconditioning, implemented using the Bempp-cl and FEniCSx libraries via their Python interfaces. We showcase our implementation by computing the polar component of the solvation free energy of a set of molecules using a constant and a Gaussian-varying permittivity. We validate our implementation against the finite difference code APBS (to 0.5\%), and show scaling from protein G B1 (955 atoms) up to immunoglobulin G (20\,148 atoms). For small problems, the coupled method was efficient, outperforming a purely boundary integral approach. For Gaussian-varying permittivities, which are beyond the applicability of boundary elements alone, we were able to run medium to large sized problems on a single workstation. Development of better preconditioning techniques and the use of distributed memory parallelism for larger systems remains an area for future work. We hope this work will serve as inspiration for future developments for molecular electrostatics with implicit solvent models.

physics.comp-ph↗

arXiv2023

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 piece-wise 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 than 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 towards 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.

physics.chem-ph↗

arXiv2022

High-productivity, high-performance workflow for virus-scale electrostatic simulations with Bempp-Exafmm

Biomolecular electrostatics is key in protein function and the chemical processes affecting it. Implicit-solvent models via the Poisson-Boltzmann (PB) equation provide insights with less computational cost than atomistic models, making large-system studies -- at the scale of viruses -- accessible to more researchers. Here we present a high-productivity and high-performance linear PB solver based on Exafmm, a fast multipole method library, and Bempp, a Galerkin boundary element method package. The workflow integrates an easy-to-use Python interface with optimized computational kernels, and can be run interactively via Jupyter notebooks, for faster prototyping. Our results show the capability of the software, confirm code correctness, and assess performance with between 8,000 and 2 million elements. Showcasing the power of this interactive computing platform, we study the conditioning of two variants of the boundary integral formulation with just a few lines of code. Mesh-refinement studies confirm convergence as $1/N$, for $N$ boundary elements, and a comparison with results from the trusted APBS code using various proteins shows agreement. Our binding energy calculations using 9 various complexes align with the results from using five other grid-based PB solvers. Performance results include timings, breakdowns, and computational complexity. Exafmm offers evaluation speeds of just a few seconds for tens of millions of points, and $\mathcal{O}(N)$ scaling. The trend observed in our performance comparison with APBS demonstrates the advantage of Bempp-Exafmm in applications involving larger structures or requiring higher accuracy. Computing the solvation free energy of a Zika virus, represented by 1.6 million atoms and 10 million boundary elements, took 80-min runtime on a single compute node (dual 20-core).

physics.comp-ph↗

arXiv2022

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 an spontaneous process that leads to a rather compact layer of randomly oriented molecules. Due to the importance of this process for the development of catalytic surfaces, a number of existing computational and experimental approaches try to predict and control the orientation of such molecules. However, and despite their own advantages, these tend to be either too expensive computationally, or oversimplified, undermining their ability to predict the most appropriate experimental conditions to maximize the catalytic activity of adsorbed proteins. To address this current need, we present an efficient computational approach to model the behavior of proteins near surfaces in the presence of an external electric field, based on continuum electrostatics. Our model can not only estimate the overall affinity of the protein with the surface, but also their most likely orientation as a function of the potential applied. In this way, a rational selection of the potential can be performed to maximize the accessibility of the protein's active site to the solvent. The model relies on the Poisson-Boltzmann equation and was implemented in an extension of the code PyGBe that includes an external electric field, and renders the electrostatic component of the solvation free energy. To demonstrate the feasibility of this technique, we investigate the adsorption of trypsin onto a carbon electrode under potentiostatic conditions both numerically and experimentally. We found that even though the adsorption process is largely dominated by hydrophobic effects, the orientation of trypsin can be controlled through an external potential, influencing the position of the active sites, and resulting in an important change in the catalytic activity of the surface.

cond-mat.soft↗

arXiv2021

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.

math.NA↗

arXiv2020

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

The Poisson-Boltzmann equation is a widely used model to study the 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 allows 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\%. This result sets the basis towards efficient automatic mesh refinement schemes that produce optimal meshes for solvation energy calculations.

math.NA↗

arXiv2020

Computational nanoplasmonics in the quasistatic limit for biosensing applications

This work uses the long-wavelength limit to compute LSPR response of biosensors, expanding the open-source PyGBe code to compute the extinction cross-section of metallic nanoparticles in the presence of any target for sensing. The target molecule is represented by a surface mesh, based on its crystal structure. PyGBe is research software for continuum electrostatics, written in Python with computationally expensive parts accelerated on GPU hardware, via PyCUDA. It is also accelerated algorithmically via a treecode that offers O(N log N) computational complexity. These features allow PyGBe to handle problems with half a million boundary elements or more. Using a model problem consisting of an isolated silver nanosphere in an electric field, our results show grid convergence as 1/N, and accurate computation of the extinction cross-section as a function of wavelength (compared with an analytical solution). For a model of a sensor-analyte system, consisting of a spherical silver nanoparticle and a set of bovine serum albumin (BSA) proteins, our results again obtain grid convergence as 1/N (with respect to the Richardson extrapolated value). Computing the LSPR response as a function of wavelength in the presence of BSA proteins captures a red-shift of 0.5 nm in the resonance frequency due to the presence of the analytes at 1-nm distance. The final result is a sensitivity study of the biosensor model, obtaining the shift in resonance frequency for various distances between the proteins and the nanoparticle. All results in this paper are fully reproducible, and we have deposited in archival data repositories all the materials needed to run the computations again and re-create the figures. PyGBe is open source under a permissive license and openly developed. Documentation is available at http://barbagroup.github.io/pygbe/docs/.

physics.comp-ph↗

arXiv2020

A Simple Electrostatic Model for the Hard-Sphere Solute Component of Nonpolar Solvation

We propose a new model for estimating the free energy of forming a molecular cavity in a solvent, by assuming this energy is dominated by the electrostatic energy associated with creating the static (interface) potential inside the cavity. The new model approximates the cavity-formation energy as that of a shell capacitor: the inner, solute-shaped conductor is held at the static potential, and the outer conductor (at the first solvation shell) is held at zero potential. Compared to cavity energies computed using free-energy pertubation with explicit-solvent molecular dynamics, the new model exhibits surprising accuracy (Mobley test set, RMSE 0.45 kcal/mol). Combined with a modified continuum model for solute-solvent van der Waals interactions, the total nonpolar model has RMSE of 0.55 kcal/mol on this test set, which is remarkable because the two terms largely cancel. The overall nonpolar model has a small number of physically meaningful parameters and compares favorably to other published models of nonpolar solvation. Finally, when the proposed nonpolar model is combined with our solvation-layer interface condition (SLIC) continuum electrostatic model, which includes asymmetric solvation-shell response, we predict solvation free energies with an RMS error of 1.35 kcal/mol relative to experiment, comparable to the RMS error of explicit-solvent FEP (1.26 kcal/mol). Moreover, all parameters in our model have a clear physical meaning, and employing reasonable temperature dependencies yields remarkable correlation with solvation entropies.

physics.chem-ph↗

arXiv2018

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 AMOEBA, in the boundary integral Poisson-Boltzmann solver \texttt{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. 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-Botlzmann solver. We found that a boundary integral approach performs similarly to a volumetric method on CPU, however, it presents an important speedup when ported to the GPU. Moreover, with a boundary element method, the mesh density to correctly resolve the electrostatic potential is the same for stardard point-charge and multipolar force fields. Finally, we saw that polarizability plays an important role to consider cooperative effects, for example, in binding energy calculations.

physics.comp-ph↗

arXiv2015

Probing protein orientation near charged nanosurfaces for simulation-assisted biosensor design

Protein-surface interactions are ubiquitous in biological processes and bioengineering, yet are not fully understood. In biosensors, a key factor determining the sensitivity and thus the performance of the device is the orientation of the ligand molecules on the bioactive device surface. Adsorption studies thus seek to determine how orientation can be influenced by surface preparation. In this work, protein orientation near charged nanosurfaces is obtained under electrostatic effects using the Poisson-Boltzmann equation, in an implicit-solvent model. Sampling the free energy for protein GB1D4' at a range of tilt and rotation angles with respect to the charged surface, we calculated the probability of the protein orientations and observed a dipolar behavior. This result is consistent with published experimental studies and combined Monte Carlo and molecular dynamics simulations using this small protein, validating our method. More relevant to biosensor technology, antibodies such as immunoglobulin G are still a formidable challenge to molecular simulation, due to their large size. We obtained the probability distribution of orientations for the iso-type IgG2a at varying surface charge and salt concentration. This iso-type was not found to have a preferred orientation in previous studies, unlike the iso-type IgG1 whose larger dipole moment was assumed to make it easier to control. We find that the preferred orientation of IgG2a can be favorable for biosensing with positive surface charge of 0.05C/m$^{2}$ or higher and 37mM salt concentration. The results also show that local interactions dominate over dipole moment for this protein. Improving immunoassay sensitivity may thus be assisted by numerical studies using our method (and open-source code), guiding changes to fabrication protocols or protein engineering of ligand molecules to obtain more favorable orientations.

physics.bio-ph↗

arXiv2015

Poisson-Boltzmann model for protein-surface electrostatic interactions and grid-convergence study using the PyGBe code

Interactions between surfaces and proteins occur in many vital processes and are crucial in biotechnology: the ability to control specific interactions is essential in fields like biomaterials, biomedical implants and biosensors. In the latter case, biosensor sensitivity hinges on ligand proteins adsorbing on bioactive surfaces with a favorable orientation, exposing reaction sites to target molecules. Protein adsorption, being a free-energy-driven process, is difficult to study experimentally. This paper develops and evaluates a computational model to study electrostatic interactions of proteins and charged nanosurfaces, via the Poisson-Boltzmann equation. We extended the implicit-solvent model used in the open-source code PyGBe to include surfaces of imposed charge or potential. This code solves the boundary integral formulation of the Poisson-Boltzmann equation, discretized with surface elements. PyGBe has at its core a treecode-accelerated Krylov iterative solver, resulting in O(N log N) scaling, with further acceleration on hardware via multi-threaded execution on \gpu s. It computes solvation and surface free energies, providing a framework for studying the effect of electrostatics on adsorption. We then derived an analytical solution for a spherical charged surface interacting with a spherical molecule, then completed a grid-convergence study to build evidence on the correctness of our approach. The study showed the error decaying with the average area of the boundary elements, i.e., the method is O(1/N), which is consistent with our previous verification studies using PyGBe. We also studied grid-convergence using a real molecular geometry (protein GB1D4'), in this case using Richardson extrapolation (in the absence of an analytical solution) and confirmed the O(1/N) scaling in this case.

physics.comp-ph↗

arXiv2013

A biomolecular electrostatics solver using Python, GPUs and boundary elements that can handle solvent-filled cavities and Stern layers

The continuum theory applied to bimolecular electrostatics leads to an implicit-solvent model governed by the Poisson-Boltzmann equation. Solvers relying on a boundary integral representation typically do not consider features like solvent-filled cavities or ion-exclusion (Stern) layers, due to the added difficulty of treating multiple boundary surfaces. This has hindered meaningful comparisons with volume-based methods, and the effects on accuracy of including these features has remained unknown. This work presents a solver called PyGBe that uses a boundary-element formulation and can handle multiple interacting surfaces. It was used to study the effects of solvent-filled cavities and Stern layers on the accuracy of calculating solvation energy and binding energy of proteins, using the well-known APBS finite-difference code for comparison. The results suggest that if required accuracy for an application allows errors larger than about 2%, then the simpler, single-surface model can be used. When calculating binding energies, the need for a multi-surface model is problem-dependent, becoming more critical when ligand and receptor are of comparable size. Comparing with the APBS solver, the boundary-element solver is faster when the accuracy requirements are higher. The cross-over point for the PyGBe code is in the order of 1-2% error, when running on one GPU card (NVIDIA Tesla C2075), compared with APBS running on six Intel Xeon CPU cores. PyGBe achieves algorithmic acceleration of the boundary element method using a treecode, and hardware acceleration using GPUs via PyCuda from a user-visible code that is all Python. The code is open-source under MIT license.

physics.comp-ph↗