Using a coarse-grained protein model, a molecular dynamics (MD) simulation of folding of one of the benchmark two-state folders, a beta-hairpin protein, has been performed. Each MD trajectory was divided into two parts - one to pass from an unfolded protein state to a native-like state by overcoming a free energy barrier that separates these states, and the other to explore the native-like states until the native state is attained . It has been found that the distributions of first-passage times (FPTs) for both segments of the trajectories are essentially single-exponential  . If the protein explores the native-like states for a time much longer than the time to overcome the free energy barrier,  the resulting FPT distribution may be approximately single-exponential, as is expected for two-state folders, but the mean FPT (MFPT) to reach the native state will be determined by the time to find the native state among the native-like ones. The Kramers rate formula to estimate the transition times from the unfolded to the native-like states shows that these times are in reasonable agreement with the corresponding times obtained in the simulations and may be far shorter than the MFPTs to reach the native state  .
Using a beta-hairpin protein as a representative example of two-state folders, we studied how the exploration of native-like states affects the folding kinetics . It has been found that the first-passage time (FPT) distributions are essentially single-exponential not only for the times to overcome the free energy barrier that separates unfolded and native-like states but also for the times to find the native state among the native-like ones . If the protein explores native-like states for a time much longer than the time to overcome the free energy barrier, which was found to be characteristic of high temperatures, the resulting FPT distribution to reach the native state remains close to exponential but the mean FPT (MFPT) is determined not by the height of the free energy barrier but by the time to explore native-like states. The mean time to overcome the free energy barrier is found to be in reasonable agreement with the Kramers rate formula and generally far shorter than the MFPT to reach the native state . The time to find the native state among native-like ones increases with temperature, which explains the known U-shape dependence of the MFPTs on temperature .
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