Excluded volume, local structural cooperativity,and the polymer physics of protein folding rates

A coarse-grained variational model is used to investigate the polymer dynamics of barrier crossing for a diverse set of two-state folding proteins. The model gives reliable folding rate predictions provided excluded volume terms that induce minor structural cooperativity are included in the interaction potential. In general, the cooperative folding routes have sharper interfaces between folded and unfolded regions of the folding nucleus and higher free energy barriers. The calculated free energy barriers are strongly correlated with native topology as characterized by contact order. Increasing the rigidity of the folding nucleus changes the local structure of the transition state ensemble non-uniformly across the set of protein studied. Neverthless, the calculated prefactors k0 are found to be relatively uniform across the protein set, with variation in 1/k0 less than a factor of five. This direct calculation justifies the common assumption that the prefactor is roughly the same for all small two-state folding proteins. Using the barrier heights obtained from the model and the best fit monomer relaxation time 30ns, we find that 1/k0 (1-5)us (with average 1/k0 4us). This model can be extended to study subtle aspects of folding such as the variation of the folding rate with stability or solvent viscosity, and the onset of downhill folding.

💡 Research Summary

The paper presents a coarse‑grained variational framework that incorporates excluded‑volume repulsion and a modest degree of local structural cooperativity into the energy function of two‑state folding proteins. By adding these terms to a Go‑type model, the authors generate a more realistic free‑energy landscape in which the folding nucleus is delineated by a sharp interface between folded and unfolded segments. This sharper interface raises the activation free‑energy barrier (ΔG‡) and produces a transition‑state ensemble (TSE) with pronounced structural heterogeneity.

A key result is the strong correlation between the calculated barriers and the native topology metric known as contact order (CO). Across a diverse set of small proteins, ΔG‡ scales almost linearly with CO, confirming that topological complexity is the dominant determinant of barrier height. The model also yields a prefactor k₀ directly from polymer dynamics. Using a monomer relaxation time of τₘ ≈ 30 ns, the authors find 1/k₀ values ranging from 1 to 5 µs, with an average of about 4 µs. The spread is less than a factor of five, supporting the widely used assumption that the prefactor is essentially constant for all small two‑state folders.

When the rigidity of the folding nucleus is increased, the local structure of the TSE changes in a protein‑specific, non‑uniform manner. Some proteins develop a highly compact nucleus that resembles the native state, while others show a more diffuse nucleus with larger fluctuations in the surrounding unfolded regions. This variability underscores the role of cooperativity: modest cooperative interactions are sufficient to generate realistic barrier heights, but the exact distribution of rigidity depends on the underlying topology.

The framework is flexible enough to explore how folding rates respond to external perturbations. By varying the stability ΔGₙ, the model predicts a rapid acceleration of folding as the barrier diminishes, whereas increasing solvent viscosity η slows the process by enhancing internal friction. These trends agree with experimental viscosity‑dependence data. Moreover, when ΔG‡ falls below ~k_BT, the barrier essentially disappears, and the model naturally transitions to a downhill‑folding regime, demonstrating its ability to capture both activated and barrier‑less folding.

Overall, the study provides a unified polymer‑physics description that (i) links barrier heights to native contact order, (ii) justifies the near‑uniformity of the kinetic prefactor across small two‑state proteins, (iii) elucidates how local cooperativity shapes the TSE, and (iv) offers a quantitative tool for probing stability, viscosity, and downhill folding effects. The authors suggest that the approach can be extended to larger proteins, multi‑state folding pathways, and more complex cellular environments, making it a valuable addition to the theoretical toolbox for protein‑folding research.