Dai, Wei. First passage times and relaxation times of unfolded proteins and the funnel model of protein folding. Retrieved from https://doi.org/doi:10.7282/T3JH3PB4
DescriptionProtein folding has been a challenging puzzle for decades but it is still not fully understood. One important way to gain insights of the mechanism is to study how kinetics in the unfolded state affects protein folding. The answer to this fundamental issue hinges on the time scale to equilibrate the unfolded state and the energy landscape of the unfolded state. We construct Markov state models (MSMs) of several mini-proteins to study the kinetics of their unfolded state ensemble and find that the folding kinetics are two-state even though there are multiple folding pathways with nonuniform barriers, which are the direct consequences of rapid mixing within the unfolded state. Also, we introduce a time integral of a proper correlation function, namely relaxation time, to characterize the time scale of equilibration within the unfolded state. However, the mean first passage times (MFPTs) between different regions of the unfolded state are observed to be orders of magnitude longer than the folding time. This seeming paradox is solved by the derivation of a simple relation that shows the mean first passage time to any state is equal to the relaxation time of that state divided by its equilibrium population. This simple relation explains why MFPTs among unfolded states can be very long but the energy landscape can still be smooth (minimally frustrated). As a matter of fact, when the folding kinetics is two-state, all of the unfolded state relaxation times are faster than the folding time. This result supports the well-established funnel-like energy landscape picture and resolves an apparent contradiction between this model and the recently proposed kinetic hub model of protein folding. Markov state model is a powerful tool but we seek for alternative ways of studying kinetics when MSM does not work very well. For example, diffusion maps of dimensionality reduction and discrete transition-based reweighting analysis method, are very useful in determining a geometrical measure that preserves intrinsic dynamics and in fully utilizing enhanced sampling simulation data.