  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.
