Statistical Thermodynamics of Protein Folding Pathways

Abstract

The funnel picture of protein folding holds that native structures are reached by a biased search down a rugged
free-energy landscape, but quantitative estimates of how much conformational entropy is actually lost along a
folding trajectory, and where in the landscape that loss is concentrated, remain scarce for realistic sequence
lengths. We applied statistical thermodynamic analysis, combining replica-exchange coarse-grained simulation with
Markov state model (MSM) reconstruction, to folding trajectories of 14 small single-domain proteins (35-76
residues) sampled at 12 temperatures between 280 K and 360 K, totalling 2.3 ms of aggregate simulation time
across 1,680 independent trajectories. Configurational entropy, estimated from the eigenvalue spectrum of the
200-state MSM transition matrix, fell from 6.8 ± 0.4 kcal/(mol K) in the unfolded ensemble to 1.9 ± 0.3 kcal/(mol K)
in the folded state across the protein set, a mean reduction of 72%, with 58% of the total entropy loss occurring
within a narrow free-energy window spanning the top 15% of the folding coordinate, consistent with a dominant,
entropy-limiting transition-state bottleneck rather than gradual funnel narrowing. Folding free-energy barriers,
computed from transition path theory applied to the MSMs, ranged from 2.1 to 5.7 kcal/mol and correlated with the
fraction of entropy lost at the bottleneck (Pearson r = 0.81, p < 0.001, n = 14), indicating that entropic bottleneck
sharpness, not merely landscape ruggedness, is a strong predictor of folding barrier height. Folding rates predicted
from barrier heights via transition-state theory reproduced experimental folding rates for the eight proteins with
available reference values within a factor of 3.1 on average (root-mean-square log-rate deviation of 0.61), and a
sensitivity analysis varying the MSM lag time between 20 ns and 200 ns changed estimated barrier heights by less
than 9%, indicating that the entropy-barrier correlation is robust to this key methodological choice. These results
provide a quantitative, sequence-length-independent account of where and how much conformational entropy is
lost during folding, and suggest that transition-state entropic bottleneck sharpness is a useful, computationally
accessible predictor of folding kinetics across structurally diverse small proteins.