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

Electrostatic Contribution of Surface Charge Residues to the Stability of a Thermophilic Protein: Benchmarking Experimental and Predicted pKa Values

Optimization of the surface charges is a promising strategy for increasing thermostability of proteins. Electrostatic contribution of ionizable groups to the protein stability can be estimated from the differences between the pKa values in the folded and unfolded states of a protein. Using this pKa-shift approach, we experimentally measured the electrostatic contribution of all aspartate and glutamate residues to the stability of a thermophilic ribosomal protein L30e from Thermococcus celer. The pKa values in the unfolded state were found to be similar to model compound pKas. The pKa values in both the folded and unfolded states obtained at 298 and 333 K were similar, suggesting that electrostatic contribution of ionizable groups to the protein stability were insensitive to temperature changes. The experimental pKa values for the L30e protein in the folded state were used as a benchmark to test the robustness of pKa prediction by various computational methods such as H++, MCCE, MEAD, pKD, PropKa, and UHBD. Although the predicted pKa values were affected by crystal contacts that may alter the side-chain conformation of surface charged residues, most computational methods performed well, with correlation coefficients between experimental and calculated pKa values ranging from 0.49 to 0.91 (p<0.01). The changes in protein stability derived from the experimental pKa-shift approach correlate well (r = 0.81) with those obtained from stability measurements of charge-to-alanine substituted variants of the L30e protein. Our results demonstrate that the knowledge of the pKa values in the folded state provides sufficient rationale for the redesign of protein surface charges leading to improved protein stability

A Continuum Poisson-Boltzmann Model for Membrane Channel Proteins

Membrane proteins constitute a large portion of the human proteome and perform a variety of important functions as membrane receptors, transport proteins, enzymes, signaling proteins, and more. The computational studies of membrane proteins are usually much more complicated than those of globular proteins. Here we propose a new continuum model for Poisson-Boltzmann calculations of membrane channel proteins. Major improvements over the existing continuum slab model are as follows: 1) The location and thickness of the slab model are fine-tuned based on explicit-solvent MD simulations. 2) The highly different accessibility in the membrane and water regions are addressed with a two-step, two-probe grid labeling procedure, and 3) The water pores/channels are automatically identified. The new continuum membrane model is optimized (by adjusting the membrane probe, as well as the slab thickness and center) to best reproduce the distributions of buried water molecules in the membrane region as sampled in explicit water simulations. Our optimization also shows that the widely adopted water probe of 1.4 {\AA} for globular proteins is a very reasonable default value for membrane protein simulations. It gives an overall minimum number of inconsistencies between the continuum and explicit representations of water distributions in membrane channel proteins, at least in the water accessible pore/channel regions that we focus on. Finally, we validate the new membrane model by carrying out binding affinity calculations for a potassium channel, and we observe a good agreement with experiment results.Comment: 40 pages, 6 figures, 5 table

Poisson-Boltzmann Calculations of Nonspecific Salt Effects on Protein-Protein Binding Free Energies

The salt dependence of the binding free energy of five protein-protein hetero-dimers and two homo-dimers/tetramers was calculated from numerical solutions to the Poisson-Boltzmann equation. Overall, the agreement with experimental values is very good. In all cases except one involving the highly charged lactoglobulin homo-dimer, increasing the salt concentration is found both experimentally and theoretically to decrease the binding affinity. To clarify the source of salt effects, the salt-dependent free energy of binding is partitioned into screening terms and to self-energy terms that involve the interaction of the charge distribution of a monomer with its own ion atmosphere. In six of the seven complexes studied, screening makes the largest contribution but self-energy effects can also be significant. The calculated salt effects are found to be insensitive to force-field parameters and to the internal dielectric constant assigned to the monomers. Nonlinearities due to high charge densities, which are extremely important in the binding of proteins to negatively charged membrane surfaces and to nucleic acids, make much smaller contributions to the protein-protein complexes studied here, with the exception of highly charged lactoglobulin dimers. Our results indicate that the Poisson-Boltzmann equation captures much of the physical basis of the nonspecific salt dependence of protein-protein complexation

Exploring a coarse-grained distributive strategy for finite-difference Poisson–Boltzmann calculations

We have implemented and evaluated a coarse-grained distributive method for finite-difference Poisson–Boltzmann (FDPB) calculations of large biomolecular systems. This method is based on the electrostatic focusing principle of decomposing a large fine-grid FDPB calculation into multiple independent FDPB calculations, each of which focuses on only a small and a specific portion (block) of the large fine grid. We first analyzed the impact of the focusing approximation upon the accuracy of the numerical reaction field energies and found that a reasonable relative accuracy of 10−3 can be achieved when the buffering space is set to be 16 grid points and the block dimension is set to be at least (1/6)3 of the fine-grid dimension, as in the one-block focusing method. The impact upon efficiency of the use of buffering space to maintain enough accuracy was also studied. It was found that an “optimal” multi-block dimension exists for a given computer hardware setup, and this dimension is more or less independent of the solute geometries. A parallel version of thedistributive focusing method was also implemented. Given the proper settings, the distributive method was able to achieve respectable parallel efficiency with tested biomolecular systems on a loosely connected computer cluster

Acceleration of Linear Finite-Difference Poisson-Boltzmann Methods on Graphics Processing Units

Electrostatic interactions play crucial roles in biophysical processes such as protein folding and molecular recognition. Poisson-Boltzmann equation (PBE)-based models have emerged as widely used in modeling these important processes. Though great efforts have been put into developing efficient PBE numerical models, challenges still remain due to the high dimensionality of typical biomolecular systems. In this study, we implemented and analyzed commonly used linear PBE solvers for the ever-improving graphics processing units (GPU) for biomolecular simulations, including both standard and preconditioned conjugate gradient (CG) solvers with several alternative preconditioners. Our implementation utilizes standard Nvidia CUDA libraries cuSPARSE, cuBLAS, and CUSP. Extensive tests show that good numerical accuracy can be achieved given that the single precision is often used for numerical applications on GPU platforms. The optimal GPU performance was observed with the Jacobi-preconditioned CG solver, with a significant speedup over standard CG solver on CPU in our diversified test cases. Our analysis further shows that different matrix storage formats also considerably affect the efficiency of different linear PBE solvers on GPU, with the diagonal format best suited for our standard finite-difference linear systems. Further efficiency may be possible with matrix-free operations and integrated grid stencil setup specifically tailored for the banded matrices in PBE-specific linear systems.Comment: 5 figures, 2 table