Transition paths, diffusive processes, and preequilibria of protein folding

Abstract

Fundamental relationships between the thermodynamics and kinetics of protein folding were investigated using chain models of natural proteins with diverse folding rates by extensive comparisons between the distribution of conformations in thermodynamic equilibrium and the distribution of conformations sampled along folding trajectories. Consistent with theory and single-molecule experiment, duration of the folding transition paths exhibits only a weak correlation with overall folding time. Conformational distributions of folding trajectories near the overall thermodynamic folding/unfolding barrier show significant deviations from preequilibrium. These deviations, the distribution of transition path times, and the variation of mean transition path time for different proteins can all be rationalized by a diffusive process that we modeled using simple Monte Carlo algorithms with an effective coordinate-independent diffusion coefficient. Conformations in the initial stages of transition paths tend to form more nonlocal contacts than typical conformations with the same number of native contacts. This statistical bias, which is indicative of preferred folding pathways, should be amenable to future single-molecule measurements. We found that the preexponential factor defined in the transition state theory of folding varies from protein to protein and that this variation can be rationalized by our Monte Carlo diffusion model. Thus, protein folding physics is different in certain fundamental respects from the physics envisioned by a simple transition-state picture. Nonetheless, transition state theory can be a useful approximate predictor of cooperative folding speed, because the height of the overall folding barrier is apparently a proxy for related rate-determining physical properties. P rotein folding is an intriguing phenomenon at the interface of physics and biology. In the early days of folding kinetics studies, folding was formulated almost exclusively in terms of mass-action rate equations connecting the folded, unfolded, and possibly, one or a few intermediate states (1,2). With the advent of site-directed mutagenesis, the concept of free energy barriers from transition state theory (TST) (3) was introduced to interpret mutational data (4), and subsequently, it was adopted for the Φ-value analysis (5). Since the 1990s, the availability of more detailed experimental data (6), in conjunction with computational development of coarse-grained chain models, has led to an energy landscape picture of folding (7)(8)(9)(10)(11)(12)(13)(14)(15). This perspective emphasizes the diversity of microscopic folding trajectories, and it conceptualizes folding as a diffusive process (16-25) akin to the theory of Kramers (26).For two-state-like folding, the transition path (TP), i.e., the sequence of kinetic events that leads directly from the unfolded state to the folded state (27, 28), constitutes only a tiny fraction of a folding trajectory that spends most of the time diffusing, seemingly unproductively, in the vicinity of the ...

The loop hypothesis: contribution of early formed specific non-local interactions to the determination of protein folding pathways

“…Furthermore, Chan et al (Chan et al 2011) presented a set of coarse-grained models for real proteins that can successfully capture the experimental contact order-related folding rates. However, in the recent detailed simulation of the folding transition paths of a similar set of model proteins (mentioned above in section A-d), Zhang and Chan (2013) found that the transition paths of folding have more non-local contact than typical conformations in the folding quasipreequilibrium with the same number of native contacts. Fersht (Fersht 2000) also asserts that the correlation cannot exclude the role of some tertiary interactions in formation of the extended nucleus.…”

“…Juraszky et al (Juraszek and Bolhuis 2006) found that in the folding of the small protein, i.e., the WW cage, 80 % of the pathways are initiated by NLI while secondary structures appear later. Zhang and Chan (2013) performed explicitchain simulations using coarse grained chain models of natural proteins and computed the transient distributions of conformations sampled along the trajectories of many molecules. They found that conformations in the initial phases of the faster pathways were enriched with NLIs.…”

Sampling large conformational transitions: adenylate kinase as a testing ground

“…The problem of correctly discounting erroneous crossing events makes TST ill-suited for systems with highly diffusive dynamics (high entropic barriers) where recrossing happens many times during a single reactive transition [83,84]. Kramers' theory, which is related to TST and can be derived from variational TST in the diffusion limit [85], provides a scheme for deriving rate expressions under the assumption of Langevin dynamics along a suitable reaction coordinate.…”

The mean shape of transition and first-passage paths

“…The experimental advances created theoretical interest in transition paths and led to intense simulation activities [14][15][16] as well as the development of analytic approaches [5,17,18]. In this work, we present a theoretical framework for transition paths involving a combination of the backward Fokker-Planck equation, the forward Fokker-Planck equation, and the renewal equation approach, and use it to derive the mean shape of transition paths.…”