Small single-domain proteins fold with apparent two-state thermodynamics and kinetics, yet the physical origin of the free-energy barrier and of the cooperativity that makes folding all-or-none remains only phenomenologically understood. Bai (2006) showed within Zwanzig's solvable folding model that a locally uphill potential of mean force (from desolvation and the correction of non-native interactions) combined with a many-body cooperative stabilization of the native state can reproduce calorimetric two-state behavior, linear chevron plots, heat and cold denaturation, non-Arrhenius kinetics, and hidden folding intermediates. The open problem is to give this picture a microscopic, quantitative foundation: derive the required cooperative many-body stabilization and the uphill-barrier contributions from atomic-level interactions, and determine their magnitude, temperature, and sequence/topology dependence. Whether folding cooperativity reflects evolutionary selection for specific native structures remains, in the source, a concluding speculation offered without a proposed test.

Background

Small single-domain proteins (roughly fewer than 100 amino acids) usually give the appearance of two-state folding: thermal melting shows a single cooperative transition in which the calorimetric enthalpy matches the van't Hoff enthalpy (ΔHcal/ΔHvH≈1\Delta H_{\mathrm{cal}}/\Delta H_{\mathrm{vH}}\approx 1, the signature of an all-or-none process), and the logarithm of the observed folding/unfolding relaxation rate varies linearly with denaturant concentration (a linear chevron plot). The dominant theoretical framework, the energy-landscape or funnel picture, describes folding as diffusion on a potential of mean force (PMF — the free energy projected onto a coarse coordinate such as the number of formed native contacts) that slopes downhill toward the native state; in this picture the free-energy barrier arises mainly from the loss of chain configurational entropy. A solvable reference is Zwanzig's 1995 model, in which each of NN residues is either native or in one of γ\gamma non-native conformations with energy E(P)=P⋅UE(P)=P\cdot U (PP the number of non-native residues), so the barrier comes entirely from the degeneracy of non-native states; native-centric Gō models, which include only pairwise interactions present in the native structure, embody the same assumption.

Two bodies of evidence strain this entropy-only barrier picture. First, simulations of hydrophobic-cluster formation in explicit water indicate that expelling solvent from buried surfaces can itself cost free energy along the folding coordinate (a desolvation barrier), consistent with the large positive activation enthalpy observed for folding at room temperature. Second, high-resolution structures of 'hidden' partially unfolded intermediates — locally unfolded forms detected, for example, by native-state hydrogen exchange, an NMR method that reads out transient opening of specific segments under otherwise native conditions — reveal non-native interactions that must be corrected during folding, which should also create uphill segments on the PMF.

Bai (2006) incorporated these ingredients into Zwanzig's framework by modifying the energy to E(P)=P⋅U−ε δP0E(P)=P\cdot U-\varepsilon\,\delta_{P0}, where δP0\delta_{P0} is a Kronecker delta and ε\varepsilon is a many-body cooperative stabilization acting only when the segment is fully native, and by taking UU slightly negative so that the PMF is locally uphill toward the transition state. With parameters of order N=20N=20, γ=2\gamma=2, ε=16\varepsilon=16 (model units), the model reproduces apparent two-state calorimetry and linear chevrons for short segments; a stepwise sequence of cooperative segment formations resolves the speed problem for 100-residue chains (the Levinthal paradox — the counting argument that an unconstrained conformational search would take astronomically long); and a quadratic temperature dependence U(T)=U0+b(T−T0)+c(T−T0)2U(T)=U_0+b(T-T_0)+c(T-T_0)^2 yields heat and cold denaturation with non-Arrhenius kinetics. Bai speculated that the cooperative, hidden-intermediate character of small-protein folding could be a consequence of evolutionary selection for highly specific native structures; the speculation was offered without a proposed test.

Several gaps are inherent in this phenomenological picture as described in the published abstract: the microscopic origin of ε\varepsilon is not established (desolvation and specific side-chain packing are candidates, but the term is inserted by hand rather than derived from atomic-level interactions); the model omits chain connectivity, excluded volume, and explicit intrachain geometry; the magnitude, temperature, and sequence dependence of the desolvation and non-native-correction barriers are undetermined, so the generality of the locally uphill PMF within the small single-domain proteins to which the model is applied is unknown; the evolutionary speculation is offered without a proposed test; and the quadratic U(T)U(T) is a pragmatic fit not validated against transfer free-energy data (the thermodynamics of transferring groups between solvent environments).

Problem Statement

Give the uphill-PMF/cooperative-stabilization picture of small-protein folding a quantitative microscopic foundation. Specifically, starting from atomic-level interactions (explicit solvent, side-chain packing, and intrachain interactions in a stated force field, or equivalent experimental transfer thermodynamics), determine: (1) the physical origin and predicted magnitude of the many-body cooperative stabilization of native and intermediate states — the ε\varepsilon term that is required phenomenologically to reproduce apparent two-state calorimetry and linear chevron kinetics, but is currently inserted by hand; and (2) the magnitude, temperature dependence, and sequence and topology dependence of the desolvation and non-native-correction contributions to the locally uphill potential of mean force, including the solvent-mediated temperature dependence modeled phenomenologically by U(T)U(T), and whether these quantitatively reproduce the observed cooperativity ratios, positive activation enthalpies, linear chevrons, heat and cold denaturation, non-Arrhenius kinetics, and hidden-intermediate populations for the small single-domain proteins to which the model is applied, in realistic chains with connectivity and excluded volume (extension to larger chains only through the source's stepwise-segment construction). These components form one scientific objective: replacing the phenomenological parameters of the 2006 model by derived, experimentally validated microscopic quantities. The objective is a microscopic account that predicts folding-barrier energetics and cooperative signatures (cooperativity ratios, chevron behavior, activation enthalpies, non-Arrhenius kinetics, denaturation behavior, hidden-intermediate populations) from stated atomic ingredients for small single-domain proteins, with quantitative validation against experimental two-state observables. Bai's concluding speculation — that the cooperative, hidden-intermediate character of small-protein folding could be the consequence of evolutionary selection for highly specific native structures — was offered without a proposed test and is not part of the objective.

References

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