Materials and Methods
- The Complex Kinetics of Protein Folding in Wide Temperature Ranges

In this study, we use the fraction of native conformations {rho} as an order parameter to represent the folding progress. The system is assumed to be in quasiequilibrium with respect to {rho}, and the states are kinetically locally connected. This is a good approximation when folding has a definite free energy barrier between non-native and native states. It might not be as good for the downhill case, where there is no barrier. The results we obtain in this article, however, seem not to be influenced qualitatively by the kinetic connectivity assumption. It has been shown that in the global kinetic connectivity case, similar kinetic behavior occurs (Saven et al., 1994Go; J. Wang, W. M. Huang, H. Y. Lu, and E. K. Wang, 2004, unpublished results; Wang et al., 1996). We therefore go ahead with the local kinetic connectivity—the case for diffusion on the underlying folding energy landscape in this article.

In this model there are N residues in a polypeptide chain. For each residue there are {nu} + 1 available conformational states, one being the native state. A simplified version of the polypeptide chain energy is expressed as

,where the summation indices i and j are labels for amino-acid residues, and ai is the state of ith residue. The three terms represent the one-body potential, two-body interactions for nearest-neighbor residues in sequence, and interactions for residues close in space but not in sequence, respectively. Due to the sequence heterogeneity, the energies and interactions can be approximated by random variables of Gaussian distributions (Derrida, 1981Go) with the mean biasing toward the native state (Bryngelson and Wolynes, 1989Go; Bryngelson et al., 1995Go). Along with the assumption that energies for different configurations are uncorrelated, one can easily generate an energy landscape with roughness characterized by the spreads of these probability distributions and with the mean biasing toward the native state (funneled landscape). Using a microcanonical ensemble analysis, the average free energy and thermodynamic properties of the polypeptide chain can be obtained (Derrida, 1981Go; Bryngelson and Wolynes, 1989Go; Bryngelson et al., 1995Go). Note that the polymer connectivity is embodied in the entropy calculations,
F({rho}) is the average free energy for the polypeptide chain. T is a scaled temperature, n + 1 is the number of conformational states of each residue, and d {epsilon} and d L are energy differences between the native and average non-native states for one- and two-body interactions (energy gap biased to native state), respectively. D{epsilon} and DL are energy spreads of one- and two-body non-native interactions (roughness of the landscape). Note that the two-body energiesd L and DL include contributions from the second and third terms in Eq. 1. The last term in F is the configurational entropy contribution.

The kinetic process along the above free-energy landscape is approximated via the use of Metropolis rate dynamics. Using continuous-time random walks, the generalized Fokker-Planck diffusion equation in the Laplace-transformed space can be obtained (Bryngelson and Wolynes, 1989Go; Lee et al., 2003Go),

,where U({rho}, s) {equiv}F({rho})/T + log [D({rho}, s)/D({rho}, 0)]. In Eq. 3, s is the Laplace transform variable over time t. is the Laplace transform of G({rho}, t), the probability density function. G({rho}, {tau})d{rho} gives the probability for a polypeptide chain to stay between {rho} and {rho} + d{rho} at time {tau}. The value ni({rho}) is the initial condition for G({rho}, t). D({rho}, s) is the frequency-dependent diffusion coefficient (Bryngelson and Wolynes, 1989Go):
,where l({rho}) {equiv} 1/n  + (1 – 1/n ){rho}. The average <...>R is taken over P(R, {rho}), the probability distribution function of transition rate R from one state with order parameter {rho} to its neighboring states, which may have order parameters equal to r -(1/N), {rho}, or  r +(1/N). The explicit expression of P(R, {rho}) can be found in Bryngelson and Wolynes (1989)Go. The boundary conditions for Eq. 3 are set as a reflecting one at {rho} = {rho}i and an absorbing one at {rho} = {rho}f. The choice of an absorbing boundary condition at {rho} = {rho}f facilitates our calculation for the first passage time and its distribution.

One can rewrite Eq. 3 in its integral-equation representation by integrating it twice over {rho}:


In this work we mainly study the behavior of the first passage time (FPT) for the order parameter to reach {rho}f. This FPT characterizes the folding time. By taking the derivatives with respect to s in Eq. 5 and taking the limit of s = 0, we can iteratively obtain the moments of the first passage time.

 where the PFPT(t) is the distribution of the first passage time. When n = 1, the mean first passage time is given as
We can also solve the integral Eq. 5 directly for , and by the observation that the distribution of the first passage time , where and are Laplace transforms of PFPT(t) and å(t) (å(t) = ∫ {rho}i{rho}f G(r , t)dr), respectively, we can obtain the information of PFPT({tau}) by studying the behavior of Due to the fact that Eq. 5 is linear in G({rho}, s), we can solve it with the numerical matrix-inversion technique.

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