There is a simple physical explanation of the Poisson(exponential)-power law(non-exponential)-Poisson(exponential) transition in folding kinetics: At high temperatures (higher than Tk/T case), there are multiple parallel kinetic paths leading toward the folded state and every path sees roughly similar-sized barriers (the barrier is smaller compared with kT). The highest barrier dominates and determines the kinetics since the smaller barrier can be easily overcome due to the large thermal motions. The resulting kinetics is single-exponential and the process is Poissonian. At lower temperature below Tk (T0/TkT0/T 46 in this case), more and more traps become important. One can expand the Gaussian-like density of states near a frozen one, resulting in the linearization in energy of the exponential. The density of states in this temperature range thus becomes exponentially distributed. Since the first passage time is exponentially dependent on the barrier (the energy), then the distribution of first passage time follows a power law () for certain low temperature regimes at long times. The tail of the distribution of the first passage time becomes fattier as the temperature decreases. The rare kinetic events can play an important role. Specific kinetic path might be dominating. At very low temperature (lower then T0/T > 46 in this case), only very few states are kinetically accessible. The kinetics therefore is dominated with the energy barrier in the deepest valley the system is trapped into, resulting to exponential process and Poissonian statistics again.
Four temperature scales appear in this study, all representing different scales or levels of the landscape. Tf is the folding transition temperature. The average kinetics characterized by the mean first passage time has a U-shape dependence on temperature, with the fastest time at temperature T0. The fluctuations of the kinetics measured by the ratios of the second order moment to the square of the first-order moment has a bell-shape dependence on temperature, with the turning point at high temperature side at Tk. At very low temperatures, T0/T > 46, the system is frozen (to single traps). It is important to notice that the point of the chevron rollover (the fast folding time), T0 is related but different in value from the onset of the complex kinetics (exponential-non-exponential transition), Tk. In other words, although the turning point of the average kinetics is at T0, the turning point of the fluctuations of the kinetics or the transition from exponential to non-exponential kinetics is at Tk (on the high temperature side). Since Tk > T0 (in fact Tf > Tk > T0 > T0/(46)), the fluctuations in kinetics first sense the traps or bumps of the landscape. In other words, the fluctuations are more sensitive toward the shape of the landscape. They can be used to probe the underlying structure.
It is worth mentioning that although we focus on the study of the protein folding problem in this article, the approach we use here is very general for treating problems with barrier crossings on a complex energy landscape. In fact, several experiments on folding (Sabelko et al., 1999; Nguyen et al., 2003), binding (Frauenfelder et al., 1988, 1991), and reaction-conformational dynamics (Yang and Xie, 2002a,b) already show the existence of complex kinetics in different temperature ranges. With the rapid advances in the experiments as well as the computational power, the dynamic trajectories (long-time or multiple short-time) can be obtained and analyzed relatively easily compared with those done before (V. B. P. Leite, J. N. Onuchic, G. Stell, and J. Wang, unpublished results). It will be interesting to see the comparisons with the analytical results obtained here in the wide temperature ranges to reveal the intrinsic features and topology of the underlying energy landscape.