The 2011 work of Li, Wolynes, and Takada established that residue-residue contact energies, obtained by decomposing an all-atom force field with implicit solvation onto native residue pairs, are distributed approximately exponentially in single-domain proteins (down to about -15 kcal/mol), that contacts specific to one allosteric state are systematically weaker than contacts shared between states, and that these sequence-specific, nonlinear interaction features — built into the atomic-interaction-based coarse-grained (AICG) model — enable large-amplitude cracking motions relevant to allostery. The source identifies three open points about the contact-energy function itself: (i) the precise origin of the approximately exponential distribution law, which the authors noted is unclear but may reflect the selection temperature of heteropolymer design theory; (ii) how ligand and substrate binding energetics, deliberately omitted from the original decomposition to isolate sequence specificity, reshape contact-energy patterns and allosteric dynamics once incorporated; and (iii) the possibility, discussed in the supplementary material, of predicting residue contact energies by regression on atomic contact features. These three points remain unresolved.

Background

Allosteric proteins switch between conformations (for example an open and a closed state) in response to binding events, and this regulation underlies much of enzyme and signaling biology. Energy landscape theory describes a folded protein as a population on a funneled free-energy surface; the principle of minimal frustration states that evolution has selected sequences whose native interactions are mutually compatible, which is the physical basis of Go models — simplified native-centric models in which only residue pairs in contact in the native structure attract. Classical Go models assign uniform contact strengths, so they cannot represent sequence effects. The Miyazawa-Jernigan statistical potential, a knowledge-based table of pairwise residue contact energies inferred from the frequency of residue contacts in known structures, was the historical route to sequence-dependent energies. Li, Wolynes, and Takada (2011) instead obtained sequence-specific contact energies ε(i,j)\varepsilon(i,j) by an atomistic energy decomposition: for each native residue pair in a structure, they summed the AMBER force-field atom-atom interactions, including a generalized Born/surface-area (GB/SA) implicit-solvent term, yielding physically grounded residue-level energies. Surveying 40 single-domain proteins, they found that these contact energies follow an approximately exponential distribution down to a lower bound of about -15 kcal/mol, a law whose precise origin they could not explain, noting only that it may reflect the selection temperature — the effective temperature characterizing how strongly evolution samples sequences biased toward a stable fold in heteropolymer design theory (Pande, Grosberg, and Tanaka, 2000). For 71 allosteric proteins with two known structures, they further found that contacts formed in only one of the two states (state-specific contacts) are markedly weaker than contacts shared by both states, a sequence-design signature that facilitates conformational change, consistent with the local-frustration analysis of allosteric proteins by Ferreiro et al. (2011). Embedding these energies in the atomic-interaction-based coarse-grained (AICG) model — a Go-type model whose native contact strengths are weighted by the decomposed atomistic energies and whose parameters are set by a multiscale matching to all-atom simulations — improved predictions of native-state fluctuations and of the direction of conformational change, and simulations of adenylate kinase (AdK, a nucleotide kinase that interconverts open and closed forms) showed rare large-amplitude excursions toward the other state, achieved partly by sequence specificity and partly by nonlinearity of the contact potentials, which permits cracking, that is, local partial unfolding under strain. To isolate sequence effects the authors deliberately excluded all interactions with ligands and substrates from the energy decomposition, and a regression of contact energies on atomic contact features was only discussed in the supplementary material (SI Appendix, Fig. S13 and Table S4), not developed. The unresolved target is therefore a completed physical account of this sequence-specific contact-energy function: why it obeys the observed exponential statistics, how binding partners reshape it in functional contexts, and how it can be predicted cheaply enough to be used routinely in the multiscale protocol.

Problem Statement

Complete the physical account of the sequence-specific residue contact-energy function ε(i,j)\varepsilon(i,j) that underlies the atomistic-energy-decomposition-based AICG description of allosteric proteins, by addressing the open points about this function identified in Li, Wolynes, and Takada (2011): (1) Origin of the statistical law — explain the precise origin of the approximately exponential distribution of decomposed residue-residue contact energies observed in the native structures of the single-domain training set (extending down to the roughly -15 kcal/mol lower bound), which the source noted is unclear but may reflect the selection temperature or other principles of heteropolymer design and evolutionary stability selection. (2) Ligand and substrate modulation — address how ligand and substrate binding energetics, deliberately excluded from the original energy decomposition in order to isolate sequence specificity, reshape residue contact-energy patterns (including the state-specific versus shared contact asymmetry) and the resulting allosteric dynamics once incorporated into the sequence-specific contact-energy description of allosteric proteins. (3) Predictive parametrization — develop the possibility discussed in the source's SI Appendix (Fig. S13 and Table S4) of modeling residue contact energies by regression on atomic contact features (or an equivalent data-driven predictor), and explore its use for parametrizing the multiscale protocol more cheaply than direct all-atom energy decomposition.

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