Adapting Poisson-Boltzmann to the self-consistent mean field theory: application to protein side-chain modeling

Abstract

We present an extension of the self-consistent mean field theory for protein side-chain modeling in which solvation effects are included based on the Poisson-Boltzmann (PB) theory. In this approach, the protein is represented with multiple copies of its side chains. Each copy is assigned a weight that is refined iteratively based on the mean field energy generated by the rest of the protein, until self-consistency is reached. At each cycle, the variational free energy of the multi-copy system is computed; this free energy includes the internal energy of the protein that accounts for vdW and electrostatics interactions and a solvation free energy term that is computed using the PB equation. The method converges in only a few cycles and takes only minutes of central processing unit time on a commodity personal computer. The predicted conformation of each residue is then set to be its copy with the highest weight after convergence. We have tested this method on a database of hundred highly refined NMR structures to circumvent the problems of crystal packing inherent to x-ray structures. The use of the PB-derived solvation free energy significantly improves prediction accuracy for surface side chains. For example, the prediction accuracies for χ(1) for surface cysteine, serine, and threonine residues improve from 68%, 35%, and 43% to 80%, 53%, and 57%, respectively. A comparison with other side-chain prediction algorithms demonstrates that our approach is consistently better in predicting the conformations of exposed side chains.

Figure 1

Illustration of the definition of probabilist maps for Poisson-Boltzmann calculation. Let us consider a toy protein consisting of two amino acids, i and m, that can both adapt one of the two possible conformations, or rotamers. The two rotamers for amino acid i are assigned weights P(i, 1) and P(i, 2) such that P(i, 1) + P(i, 2) = 1. The corresponding multi-copy system is shown on the left, overlaid on a Cartesian grid. As the two conformations for amino acid i are independent realization of the side-chain position, the probability ρ_i(S) that amino acid i covers a site S is simply the weighted sum of the probabilities ρ_i1(S) and ρ_i2(S) of each of its conformations covering S (A). The same applies for amino acid m (B). As i and m co-exist in the protein, the probability γ_solv(S) that site S remains “uncovered,” i.e., accessible to solvent, is given by a product rule (C).

Figure 2

Effects of adding the electrostatic free energy as computed by PB in the SCMF functional free energy on side-chain prediction accuracy. The dihedral angles χ₁ and χ₂ are considered to be correctly predicted if their values are within 40° of their reference values in the native structures.

Figure 3

The accuracy with which the dihedral angles χ₁ of all residues in the DRESS database are predicted is plotted as a function of the accessibility of the residue to solvent, for the SCMF calculation with (continuous line) and without (dashed line) the PB solvation free energy.

Figure 4

The accuracy of SCMF-PB is compared to those of other side-chain prediction programs for a range of side-chain accessibility S (in %).

Figure 5

CPU time required for predicting the conformations of all side chains of a protein versus the number of residues of the protein on a log-log scale, for different methods. The slope of the fitted lines are 0.1, 0.6, 0.98, 1.0, 1.3, and 1.6 for SCMF-PB, TREEPACK, SCAP, OPUS, SCWRL4, and SCMF, respectively.