How Kinetics within the Unfolded State Affects Protein Folding: An Analysis Based on Markov State Models and an Ultra-Long MD Trajectory

Abstract

Understanding how kinetics in the unfolded state affects protein folding is a fundamentally important yet less well-understood issue. Here we employ three different models to analyze the unfolded landscape and folding kinetics of the miniprotein Trp-cage. The first is a 208 μs explicit solvent molecular dynamics (MD) simulation from D. E. Shaw Research containing tens of folding events. The second is a Markov state model (MSM-MD) constructed from the same ultralong MD simulation; MSM-MD can be used to generate thousands of folding events. The third is a Markov state model built from temperature replica exchange MD simulations in implicit solvent (MSM-REMD). All the models exhibit multiple folding pathways, and there is a good correspondence between the folding pathways from direct MD and those computed from the MSMs. The unfolded populations interconvert rapidly between extended and collapsed conformations on time scales ≤40 ns, compared with the folding time of ∼5 μs. The folding rates are independent of where the folding is initiated from within the unfolded ensemble. About 90% of the unfolded states are sampled within the first 40 μs of the ultralong MD trajectory, which on average explores ∼27% of the unfolded state ensemble between consecutive folding events. We clustered the folding pathways according to structural similarity into "tubes", and kinetically partitioned the unfolded state into populations that fold along different tubes. From our analysis of the simulations and a simple kinetic model, we find that, when the mixing within the unfolded state is comparable to or faster than folding, the folding waiting times for all the folding tubes are similar and the folding kinetics is essentially single exponential despite the presence of heterogeneous folding paths with nonuniform barriers. When the mixing is much slower than folding, different unfolded populations fold independently, leading to nonexponential kinetics. A kinetic partition of the Trp-cage unfolded state is constructed which reveals that different unfolded populations have almost the same probability to fold along any of the multiple folding paths. We are investigating whether the results for the kinetics in the unfolded state of the 20-residue Trp-cage is representative of larger single domain proteins.