In the companion paper, we presented a set of induced dipole interaction models using four types of screening functions, which include the Applequist (no screening), the Thole linear, the Thole exponential model, and the Thole Tinker-like (another form of exponential screening function) functions. In this work, we evaluate the performance of polarizability models using large set of amino acid analog pairs that are frequently observed in protein structures as a benchmark. For each amino acid pair we calculated quantum mechanical interaction energies at the MP2/aug-cc-pVTZ//MP2/6-311++G(d,p) level with the basis set superposition error (BSSE) correction and compared them with molecular mechanics results. Encouragingly, all the polarizable models significantly outperform the additive F94 and F03 models (mimicking AMBER ff94/ff99 and ff03 force fields, respectively) in reproducing the BSSE-corrected quantum mechanical interaction energies. Particularly, the root-mean-square errors (RMSE) for three Thole models in Set A (where the 1–2 and 1–3 interactions are turned off and all 1–4 interactions are included) are 1.456, 1.417 and 1.406 kcal/mol for Model AL (Thole Linear), Model AE (Thole exponential) and Model AT (Thole Tinker-like), respectively. In contrast, the RMSE are 3.729 and 3.433 kcal/mol for F94 and F03 models, respectively. A similar trend was observed for the average unsigned errors (AUE), which are 1.057, 1.025, 1.011, 2.219 and 2.070 kcal/mol for AL, AE, AT, F94/ff99 and F03, respectively. Analyses based on the trend line slopes indicate that the two fixed charge models substantially underestimate the relative strengths of non-charge-charge interactions by 24% (F03) and 35% (F94), respectively, whereas the four polarizable models over-estimate the relative strengths by 5% (AT), 3% (AL, AE) and 13% (AA), respectively. Agreement was further improved by adjusting the van der Waals parameters. Judging from the notably improved accuracy in comparison to the fixed charge models, the polarizable models are expected to form the foundation for the development of high quality polarizable force fields for protein and nucleic acid simulations.

We implemented and optimized seven finite-difference solvers for the full nonlinear Poisson-Boltzmann equation in biomolecular applications, including four relaxation methods, one conjugate gradient method, and two inexact Newton methods. The performance of the seven solvers was extensively evaluated with a large number of nucleic acids and proteins. Worth noting is the inexact Newton method in our analysis. We investigated the role of linear solvers in its performance by incorporating the incomplete Cholesky conjugate gradient and the geometric multigrid into its inner linear loop. We tailored and optimized both linear solvers for faster convergence rate. In addition, we explored strategies to optimize the successive over-relaxation method to reduce its convergence failures without too much sacrifice in its convergence rate. Specifically we attempted to adaptively change the relaxation parameter and to utilize the damping strategy from the inexact Newton method to improve the successive over-relaxation method. Our analysis shows that the nonlinear methods accompanied with a functional-assisted strategy, such as the conjugate gradient method and the inexact Newton method, can guarantee convergence in the tested molecules. Especially the inexact Newton method exhibits impressive performance when it is combined with highly efficient linear solvers that are tailored for its special requirement.

A well-behaved physics-based all-atom scoring function for protein structure prediction is analyzed with several widely used all-atom decoy sets. The scoring function, termed AMBER/Poisson-Boltzmann (PB), is based on a refined AMBER force field for intramolecular interactions and an efficient PB model for solvation interactions. Testing on the chosen decoy sets shows that the scoring function, which is designed to consider detailed chemical environments, is able to consistently discriminate all 62 native crystal structures after considering the heteroatom groups, disulfide bonds, and crystal packing effects that are not included in the decoy structures. When NMR structures are considered in the testing, the scoring function is able to discriminate 8 out of 10 targets. In the more challenging test of selecting near-native structures, the scoring function also performs very well: for the majority of the targets studied, the scoring function is able to select decoys that are close to the corresponding native structures as evaluated by ranking numbers and backbone Calpha root mean square deviations. Various important components of the scoring function are also studied to understand their discriminative contributions toward the rankings of native and near-native structures. It is found that neither the nonpolar solvation energy as modeled by the surface area model nor a higher protein dielectric constant improves its discriminative power. The terms remaining to be improved are related to 1-4 interactions. The most troublesome term is found to be the large and highly fluctuating 1-4 electrostatics term, not the dihedral-angle term. These data support ongoing efforts in the community to develop protein structure prediction methods with physics-based potentials that are competitive with knowledge-based potentials.

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