A GPU-accelerated Direct-sum Boundary Integral Poisson-Boltzmann Solver

In this paper, we present a GPU-accelerated direct-sum boundary integral method to solve the linear Poisson-Boltzmann (PB) equation. In our method, a well-posed boundary integral formulation is used to ensure the fast convergence of Krylov subspace based linear algebraic solver such as the GMRES. The molecular surfaces are discretized with flat triangles and centroid collocation. To speed up our method, we take advantage of the parallel nature of the boundary integral formulation and parallelize the schemes within CUDA shared memory architecture on GPU. The schemes use only $11N+6N_c$ size-of-double device memory for a biomolecule with $N$ triangular surface elements and $N_c$ partial charges. Numerical tests of these schemes show well-maintained accuracy and fast convergence. The GPU implementation using one GPU card (Nvidia Tesla M2070) achieves 120-150X speed-up to the implementation using one CPU (Intel L5640 2.27GHz). With our approach, solving PB equations on well-discretized molecular surfaces with up to 300,000 boundary elements will take less than about 10 minutes, hence our approach is particularly suitable for fast electrostatics computations on small to medium biomolecules

An Adapative Treecode-accelerated Boundary Integral Solver for Computing the Electrostatics of a Biomolecule

The Poisson-Boltzmann equation (PBE) is a widely-used model in the calculation of electrostatic potential for solvated biomolecules. PBE is an interface problem defined in the whole space with the interface being a molecular surface of a biomolecule, and has been solved numerically by finite difference, finite element, and boundary integral methods. Unlike the finite difference and finite element methods, the boundary integral method works directly over the whole space without approximating the whole space problem into an artificial boundary value problem. Hence, it is expected to solve PBE in higher accuracy. However, so far, it was only applied to a linear PBE model. Recently, a solution of PBE was split into three component functions. One of them, G, is a known function that collects all the singularity points of PBE so that the other two components become continuously twice differentiable within the protein and solvent regions. Such an approach has led to efficient PBE finite element solvers. This provided motivation to study the application of this solution decomposition to the development of a new boundary integral algorithm for solving PBE. Reformulating the interface problem of $\Psi$ into a boundary integral equation is nontrivial because the involved flux interface condition is discontinuous. Development of a fast numerical algorithm for solving the resulted boundary integral equation is an attractive research topic. In this masters thesis, we focus on one key step of our new boundary integral algorithm: how to solve for the second component function $\Psi$ of the PBE solution by a boundary integral method. This work becomes important by itself because the sum of $\Psi$ with $G$ gives the solution of the Poisson dielectric model for the case of a biomolecule in water. In this project, we obtain the new boundary integral equation and develop an adaptive treecode-accelerated boundary integral algorithm. We then program the new algorithm in Fortran and make various numerical tests to validate our new algorithm and program package. In particular, numerical tests performed against analytic models verify the effectiveness of the solver, and comparisons to experimental data verify its accuracy for real-world applications. In this way, it is demonstrated that this solver and solution decomposition can compute the electrostatics of a biomolecule in water with high numerical accuracy

Improvements to the APBS biomolecular solvation software suite

The Adaptive Poisson-Boltzmann Solver (APBS) software was developed to solve the equations of continuum electrostatics for large biomolecular assemblages that has provided impact in the study of a broad range of chemical, biological, and biomedical applications. APBS addresses three key technology challenges for understanding solvation and electrostatics in biomedical applications: accurate and efficient models for biomolecular solvation and electrostatics, robust and scalable software for applying those theories to biomolecular systems, and mechanisms for sharing and analyzing biomolecular electrostatics data in the scientific community. To address new research applications and advancing computational capabilities, we have continually updated APBS and its suite of accompanying software since its release in 2001. In this manuscript, we discuss the models and capabilities that have recently been implemented within the APBS software package including: a Poisson-Boltzmann analytical and a semi-analytical solver, an optimized boundary element solver, a geometry-based geometric flow solvation model, a graph theory based algorithm for determining p$K_a$ values, and an improved web-based visualization tool for viewing electrostatics

Biomolecular electrostatics with continuum models: a boundary integral implementation and applications to biosensors

The implicit-solvent model uses continuum electrostatic theory to represent the salt solution around dissolved biomolecules, leading to a coupled system of the Poisson-Boltzmann and Poisson equations. This thesis uses the implicit-solvent model to study solvation, binding and adsorption of proteins. We developed an implicit-solvent model solver that uses the boundary element method (BEM), called PyGBe. BEM numerically solves integral equations along the biomolecule-solvent interface only, therefore, it does not need to discretize the entire domain. PyGBe accelerates the BEM with a treecode algorithm and runs on graphic processing units. We performed extensive verification and validation of the code, comparing it with experimental observations, analytical solutions, and other numerical tools. Our results suggest that a BEM approach is more appropriate than volumetric based methods, like finite-difference or finite-element, for high accuracy calculations. We also discussed the effect of features like solvent-filled cavities and Stern layers in the implicit-solvent model, and realized that they become relevant in binding energy calculations. The application that drove this work was nano-scale biosensors-- devices designed to detect biomolecules. Biosensors are built with a functionalized layer of ligand molecules, to which the target molecule binds when it is detected. With our code, we performed a study of the orientation of proteins near charged surfaces, and investigated the ideal conditions for ligand molecule adsorption. Using immunoglobulin G as a test case, we found out that low salt concentration in the solvent and high positive surface charge density leads to favorable orientations of the ligand molecule for biosensing applications. We also studied the plasmonic response of localized surface plasmon resonance (LSPR) biosensors. LSPR biosensors monitor the plasmon resonance frequency of metallic nanoparticles, which shifts when a target molecule binds to a ligand molecule. Electrostatics is a valid approximation to the LSPR biosensor optical phenomenon in the long-wavelength limit, and BEM was able to reproduce the shift in the plasmon resonance frequency as proteins approach the nanoparticle