Reduced C(beta) statistical potentials can outperform all-atom potentials in decoy identification

Abstract

We developed a series of statistical potentials to recognize the native protein from decoys, particularly when using only a reduced representation in which each side chain is treated as a single C(beta) atom. Beginning with a highly successful all-atom statistical potential, the Discrete Optimized Protein Energy function (DOPE), we considered the implications of including additional information in the all-atom statistical potential and subsequently reducing to the C(beta) representation. One of the potentials includes interaction energies conditional on backbone geometries. A second potential separates sequence local from sequence nonlocal interactions and introduces a novel reference state for the sequence local interactions. The resultant potentials perform better than the original DOPE statistical potential in decoy identification. Moreover, even upon passing to a reduced C(beta) representation, these statistical potentials outscore the original (all-atom) DOPE potential in identifying native states for sets of decoys. Interestingly, the backbone-dependent statistical potential is shown to retain nearly all of the information content of the all-atom representation in the C(beta) representation. In addition, these new statistical potentials are combined with existing potentials to model hydrogen bonding, torsion energies, and solvation energies to produce even better performing potentials. The ability of the C(beta) statistical potentials to accurately represent protein interactions bodes well for computational efficiency in protein folding calculations using reduced backbone representations, while the extensions to DOPE illustrate general principles for improving knowledge-based potentials.

Figure 1.

Reference state used by DOPE for the case where the reference sphere has radius a = 24 Å and the cutoff distance for computing nonbonded interactions is assumed to be at least 48 Å. In actual applications, it is likely that a significantly shorter cutoff distance is used. Then, this function must be normalized differently as the longer distance information is irrelevant (see Appendix).

Figure 2.

(A) The four-basin system used to define DOPE-Back. The basins called β and PPII are very common in β-sheet structures. The α_R basin contains helical geometries. The basin termed ɛ comprises the rest of the Ramachandran map. (B) The more conventional five-basin Ramachandran map. The ɛ-basin from A is the union of γ and α_L basins as well as the portion of α_R basin that corresponds to turn geometries.

Figure 3.

Reference states used in (A) DOPE-N1, (B) DOPE-N2, (C) DOPE-N3, and (D) DOPE-N4. Superimposed on these reference states is the DOPE-like spherical reference state obtained by matching the maxima of the distributions. The inequivalence between these distributions emphasizes the utility of the new local reference states introduced by DOPE_NN.

Figure 4.

The DOPE, DOPE_NN, and DOPE-Back statistical potentials all yield strikingly different interaction energies. (A) Local interaction energies for DOPE_NN, (B) nonlocal interaction energies for DOPE_NN compared with DOPE, (C) interaction energies of DOPE-Back compared with DOPE. All interactions are between two arginine α-carbons.

Figure 5.

Probability distributions of the correlation coefficient between DOPE and DOPE-C_β vs. DOPE-Back and DOPE-C_β-Back over all 306 decoy sets. To illustrate the geometric meaning of particular correlation coefficients, correlations between all-atom and reduced β-carbon DOPE-Back statistical potentials are taken from the Baker decoy set for the proteins 1ubx, 1ail, and 1bq9.