We start the numerical calculations by setting *R*_{0} = 10^{9} s^{–1}, *N* = 100, and n = 10 to match the physical scales. For simplicity we assume d e = d L and D = DL. The ratio of the energy gap between the native state and the average of non-native states over the spread of non-native states, de/De, representing the energy bias or slope toward the native state relative to the spread (variance) or the fluctuations in energy of the polypeptide chain on the folding landscape, can be shown to be the controlling parameter in this problem. The large values of de/De imply a steep folding funnel and small values of de/De imply a rough folding landscape. We set the initial distribution to be *n*_{i}() = d(–_{i}), where _{i} is set to be 0.05. In our calculations we set _{f} = 0.95.

The mean first passage time (MFPT) t for the folding process versus a scaled inverse temperature, T_{0}/T (T_{0} is defined as the temperature of the minimum or optimal (fastest) mean first passage time), is plotted in Fig. 1 for several settings of the parameters de/De, ranging from the more funneled energy landscape to the more rough energy landscape. Notice that the energy gap d and roughness of the landscape De are dependent on both internal sequence compositions of the protein and external environments such as solvents, denaturants, pressure, etc. We have a U-shape curve for each fixed de/De, and the MFPT reaches its minimum at temperature T_{0}. At high temperatures, the MFPT is large although the diffusion process itself is fast (i.e., *D*(, *s*) is large). This long-time folding behavior is due to the instability of the native state. The MFPT is also large at low temperature, which indicates that the polypeptide chain is trapped in low-energy non-native states. The kinetic diffusion is very slow. This is in agreement with simulation studies and the experiments (this chevron rollover phenomena was first investigated and explained by Miller et al. (1992), Socci and Onuchic (1994, 1995), Socci et al. (1996), Gutin et al. (1996), Itzhaki et al. (1995), Cieplak et al. (1999), Seno et al. (1998), Klimov and Thirumalai (1998), Kaya and Chan (2000, 2002, 2003), Chan et al. (2004), Plotkin and Wolynes (1998), Plotkin and Onuchic (2002a,b), Zhou et al. (2003), and Nguyen et al. (2003).

In the study of chevron rollover, the relationship between thermodynamics and kinetics can be revealed. Recent extensive analyses have emphasized the relative positions of the point of fastest folding and the thermodynamic folding transition (Kaya and Chan, 2003; Chan et al., 2004). The folding transition temperature *T*_{f} can be obtained from either Eq. 2 or from the crossing of the mean first passage time for unfolding time with the mean first passage time for folding. Here we plot the relative ratio of folding transition temperature *T*_{f} to the fastest folding temperature *T*_{0}, with *T*_{f}/*T*_{0} as a function of the ratio of the energy gap to the roughness of the underlying folding energy landscape (Fig. 2). We notice that in general, the folding transition temperature *T*_{f} is higher than the fastest folding temperature *T*_{0}. The ratio *T*_{f}/*T*_{0} slightly increases as the folding landscape is more funneled relative to the local traps. This implies that the more funneled landscape leads to larger separation of *T*_{f} with respect to *T*_{0}. This can be understood, since higher values of *T*_{f} means it is easier for folding to occur, and lower values of *T*_{0} means it is harder to turn over to the trap regime (*T*_{0} has the meaning of the kinetic dividing temperature of the low temperature trapping and high temperature for folding).

In view of the recent theoretical results on chevron rollovers (Kaya and Chan, 2003; Chan et al., 2004), we provide an additional figure (Fig. 3) that plots mean first passage time versus *T*_{f}/*T*. Notice that the horizontal axis is equal to 1 at folding temperature. As we can see the kinetic minimum of the mean first passage time occurs at the temperatures *T*_{0} all below folding temperature *T*_{f} (since *T*_{0}/*T* > 1). This implies that at high temperatures folding is slow and the folding time is shorter as the temperature decreases (above *T*_{0}). Below folding temperature in between *T*_{f} and *T*_{0}, folding process is speeding up relative to those at the temperatures higher than *T*_{f}. This is in agreement with both experiments and theoretical investigations (Kaya and Chan, 2003; Chan et al., 2004). In fact, different sequences give different chevron rollover plots (Plotkin and Onuchic, 2002a,b; Kaya and Chan, 2003; Chan et al., 2004). As we can see clearly here, a rougher landscape (with smaller gap/roughness ratio) from a particular sequence tends to shift the kinetic curve and also the minimum or fastest folding time toward the temperature at *T*_{f}, indicating the local traps become more and more important in determining the folding rate near *T*_{f}. Fig. 3 explicitly shows the kinetic behavior discussed earlier (Plotkin and Onuchic, 2002a,b).

We also calculate higher-order moments for the FPT distribution. In Fig. 4, we show the behavior of the reduced second moment, t^{2}/t^{2} versus inverse kinetic minimum (fastest folding) scaled temperature. We can see that at high temperatures (*T*_{0}/*T* 1 in this case), the ratio is a constant close to 2 (for example, the ratio is equal to 2.7 for de/De = 2.8). Notice that for the Poissonian process, t^{n} = *n*!t^{n}. This implies an approximately underlying Poissonian statistics. The kinetic process can thus be shown as exponential (Wang, 2003; Lee et al., 2003; Zhou et al., 2003). On the other hand, as the temperature is lower than a specific temperature *T*_{k}, the second-order moment ratio increases rapidly. Thus the value of the temperature *T*_{k} is a marker representing the onset of the large statistical fluctuations. Note that *T*_{0} T_{k}T_{f}. This implies that the complex kinetics occurs at a temperature *T*_{k} below folding temperature but above the fastest folding temperature. The high-order moment domination implies that the mean first passage time (MFPT) becomes unreliable for characterizing the kinetics; that the distribution of the first passage time is, in general, needed; and that it develops a fattier tail than at exponential kinetics. This implies non-Poissonian statistics and therefore non-exponential kinetics. As the temperature drops even lower (*T*_{0}/*T* > 46, the range of values is due to different landscape parameters: gap/roughness ratios), the second-order moment ratios decrease rapidly to values close to 2. This implies an approximate Poissonian process and exponential kinetics again.

In Fig. 5 we draw a graph similar to that in Fig. 4 of the reduced second moment, t^{2}/t^{2} versus inverse folding scaled temperature *T*_{f}/*T*. We can see above the folding transition temperature, the second moment ratio is close to 2. This implies Poissonian exponential kinetics. Below *T*_{f} and above *T*_{k}, the second moment ratio is still close to 2. It implies again Poissonian exponential kinetics. The non-exponential kinetics only emerges when the temperature is below *T*_{k}. As the temperature drops even lower (*T*_{f}/*T* > 4.5 7.5), the second-order moment ratios drop rapidly to values close to 2, implying Poissonian exponential kinetics again.

In Fig. 6, we plot the ratio of *T*_{f}/*T*_{0}, *T*_{f}/*T*_{k}, and *T*_{k}/*T*_{0} versus gap/roughness ratio (de/De). We found that in all the cases, *T*_{0} T_{k}T_{f}. Since *T*_{k} is the marker for complex kinetics, this leads to exponential kinetics at temperatures *T* > *T*_{k}. The non-exponential kinetics quickly emerges at temperatures *T*_{0}T T_{k} (and continue to temperatures at (1/61/4) *T*_{0}T T_{0}). As the gap/roughness ratio increases (de/De), the ratio of folding temperature to kinetic transition temperature *T*_{k} (where kinetics is switched from exponential to non-exponential) increases. This indicates, as the underlying folding energy landscape is more funneled toward native state relative to the local traps, that there is a wider temperature window at *T*_{k} T T_{f} for the exponential kinetics. In other words, it is relatively easier for folding (increasing *T*_{f}) and relatively harder for kinetic transition (decreasing *T*_{k}) to occur. Notice that *T*_{k}/*T*_{0} is almost a constant. *T*_{0} represents the onset of the rollover behavior in kinetics and *T*_{k} represents the onset of the non-exponential kinetics—the transition from a "smoother" appearing folding landscape to a "rougher" or "bumpier" landscape, so that basins of attraction of part of the landscape start to be partially frozen or become traps. Notice that the ratio of *T*_{k} and *T*_{0} is almost a constant. The kinetic behavior around (*T*_{k}) is analogous to the glassy material at *T*_{A} above the glass transition temperature where partial freezing occurs (Kirkpatrick and Wolynes, 1987; Kirkpatrick et al., 1989).

In Fig. 7, the negative of the logarithm of the distribution of the first passage time in Laplace space *s* over 10 orders of magnitude in a Log-Log scale is plotted at an intermediate temperature (*T* = *T*_{0}/2 and de/De = 4.0). The fact the curve looks like a straight line implies that the distribution of first passage time in Laplace space is approximately a stretched exponential () which is the form of Laplace transform of the Lévy distribution in the time space. So we have

*u* lies between 0 and 1. From the asymptotic property of the Lévy distribution function we learn that

for large t, approaching power law distribution at long times. The power law exponent is linear with the slope of the above figure. It can be shown that power law exponent is a monotonically increasing function of the temperature below

*T*_{0}, implying that the tail of the FPT distribution becomes fattier (

*u* is smaller and the power law decay is slower) as the temperature drops lower.