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State Key Laboratory of Electroanalytical Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun, People's Republic of China; and The Department of Chemistry and Department of Physics, State University of New York, Stony Brook, New York
Correspondence: Address reprint requests to Jin Wang, E-mail: jinwang{at}sprynet.com.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONSCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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6)), the kinetics is an exponential Poissonian process again. |
INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONSCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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posed the so-called Levinthal paradox 35 years ago. If protein folding proceeds with going through every possible state, then it takes cosmological time to reach the native state. In nature, protein folding finishes in millisecond to second. The recent energy landscape theory of protein folding (Bryngelson and Wolynes, 1989; Bryngelson et al., 1995; Chan and Dill, 1994; Abkevich et al., 1994; Wang et al., 1996) resolves the issue by assuming the underlying energy landscape is funneled toward the native state. Superimposed on that are the bumps and wiggles forming local traps. For folding to complete in biological timescale under physiological temperature (300 K), the slope of the funnel must be steep enough to overcome the local traps. The energy landscape theory is successful in explaining qualitatively and quantitatively many folding experiments (Bryngelson and Wolynes, 1989; Bryngelson et al., 1995; Chan and Dill, 1994; Abkevich et al., 1994; Wang et al., 1996).
Both theoretical and experimental investigations on folding and reaction kinetics show complex kinetics in different ranges of temperature (Bryngelson and Wolynes, 1989; Bryngelson et al., 1995; Chan and Dill, 1994; Abkevich et al., 1994; Wang et al., 1996; Gutin et al., 1996; Cieplak et al., 1999; Seno et al., 1998; Klimov and Thirumalai, 1998; Kaya and Chan, 2000, 2002; Itzhaki et al., 1995; Schuler et al., 2002; Lipman et al., 2003; Sabelko et al., 1999; Nguyen et al., 2003; Frauenfelder et al., 1988, 1991; Yang and Xie, 2002a,b). By varying the temperature, the underlying energy landscape structures can be probed in different levels, from the global to the local detail perspectives (Frauenfelder et al., 1988, 1991). The relationship between dynamics and functions of the biomolecules can be revealed. Although different theoretical approaches explain kinetic behavior within specific temperature ranges, the unified picture of kinetics of the whole temperature range seems still lacking. This is the purpose of the current study. Lately, the diffusive dynamics of folding is shown to have complex kinetics (Nguyen et al., 2003; Lee et al., 2003; Zhou et al., 2003; V. B. P. Leite, J. N. Onuchic, G. Stell, and J. Wang, unpublished results).
In this article, we study the kinetics in the whole temperature range. We show that the Poisson(exponential)-non-Poisson(non-exponential)-Poisson (exponential) kinetics emerge from high to low temperatures. This phenomena seems to be universal not only for protein folding, but might also exist in other biomolecular folding, biomolecular binding and reaction systems, electron transfer, viscous liquid, and glassy materials.
The current results of the study are also relevant to the single molecule studies where the mean of the observables is often unreliable due to the large statistical fluctuations (which are not smeared out by the number of molecules as in the bulk case), and cannot be used to accurately characterize the system. In general, fluctuations of the observables intrinsic for characterizing the system are obtained from the information on moments and distributions (Wang and Wolynes, 1995, 1999; Onuchic et al., 1999; Wang, 2003). The connections of the theory and simulations with the single molecule kinetic experiments can be made through the analysis of the long-time dynamic trajectories or multiple short-time runs where information on the mean, high-order moments and distributions, or histograms of the important observables can be extracted (Lu et al., 1998; Schenter et al., 1999; Moerner, 1996; Zhuang et al., 2000, 2002; Jia et al., 1999).
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONSCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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as an order parameter to represent the folding progress. The system is assumed to be in quasiequilibrium with respect to , 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., 1994; 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 
+ 1 available conformational states, one being the native state. A simplified version of the polypeptide chain energy is expressed as
 | (1) |
i 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, 1981) with the mean biasing toward the native state (Bryngelson and Wolynes, 1989; Bryngelson et al., 1995). 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, 1981; Bryngelson and Wolynes, 1989; Bryngelson et al., 1995). Note that the polymer connectivity is embodied in the entropy calculations,
 | (2) |
) is the average free energy for the polypeptide chain. T is a scaled temperature, + 1 is the number of conformational states of each residue, and 
and 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. 
and L are energy spreads of one- and two-body non-native interactions (roughness of the landscape). Note that the two-body energies L and L 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, 1989; Lee et al., 2003),
 | (3) |
, s) 
F()/T + log [D(, s)/D(, 0)]. In Eq. 3, s is the Laplace transform variable over time 
. 
is the Laplace transform of G(, ), the probability density function. G(, )d gives the probability for a polypeptide chain to stay between and + d at time . The value ni() is the initial condition for G(, ). D(, s) is the frequency-dependent diffusion coefficient (Bryngelson and Wolynes, 1989):
 | (4) |
() 1/ + (1 – 1/). The average 
...
R is taken over P(R, ), the probability distribution function of transition rate R from one state with order parameter to its neighboring states, which may have order parameters equal to 
, , or 
The explicit expression of P(R, ) can be found in Bryngelson and Wolynes (1989). The boundary conditions for Eq. 3 are set as a reflecting one at = i and an absorbing one at = f. The choice of an absorbing boundary condition at = 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 :
 | (5) |
In this work we mainly study the behavior of the first passage time (FPT) for the order parameter to reach 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.
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) is the distribution of the first passage time. When n = 1, the mean first passage time is given as
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, and by the observation that the distribution of the first passage time 
, where 
and 
are Laplace transforms of PFPT() and 
() (
), respectively, we can obtain the information of PFPT() by studying the behavior of 
Due to the fact that Eq. 5 is linear in G(, s), we can solve it with the numerical matrix-inversion technique. |
RESULTS |
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= 10 to match the physical scales. For simplicity we assume = L and = L. The ratio of the energy gap between the native state and the average of non-native states over the spread of non-native states, /, 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 / imply a steep folding funnel and small values of / imply a rough folding landscape. We set the initial distribution to be ni() = (–i), where i is set to be 0.05. In our calculations we set f = 0.95.
The mean first passage time (MFPT) for the folding process versus a scaled inverse temperature, T0/T (T0 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 /, ranging from the more funneled energy landscape to the more rough energy landscape. Notice that the energy gap and roughness of the landscape 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 /, and the MFPT reaches its minimum at temperature T0. 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).
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; Chan et al., 2004). The folding transition temperature Tf 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 Tf to the fastest folding temperature T0, with Tf/T0 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 Tf is higher than the fastest folding temperature T0. The ratio Tf/T0 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 Tf with respect to T0. This can be understood, since higher values of Tf means it is easier for folding to occur, and lower values of T0 means it is harder to turn over to the trap regime (T0 has the meaning of the kinetic dividing temperature of the low temperature trapping and high temperature for folding).
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; Chan et al., 2004), we provide an additional figure (Fig. 3) that plots mean first passage time versus Tf/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 T0 all below folding temperature Tf (since T0/T > 1). This implies that at high temperatures folding is slow and the folding time is shorter as the temperature decreases (above T0). Below folding temperature in between Tf and T0, folding process is speeding up relative to those at the temperatures higher than Tf. 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 Tf, indicating the local traps become more and more important in determining the folding rate near Tf. Fig. 3 explicitly shows the kinetic behavior discussed earlier (Plotkin and Onuchic, 2002a,b).
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2/2 versus inverse kinetic minimum (fastest folding) scaled temperature. We can see that at high temperatures (T0/T < 1 in this case), the ratio is a constant close to 2 (for example, the ratio is equal to 2.7 for / = 2.8). Notice that for the Poissonian process, n = n!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 Tk, the second-order moment ratio increases rapidly. Thus the value of the temperature Tk is a marker representing the onset of the large statistical fluctuations. Note that T0 < Tk < Tf. This implies that the complex kinetics occurs at a temperature Tk 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 (T0/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.
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2/2 versus inverse folding scaled temperature Tf/T. We can see above the folding transition temperature, the second moment ratio is close to 2. This implies Poissonian exponential kinetics. Below Tf and above Tk, 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 Tk. As the temperature drops even lower (Tf/T > 4.5 7.5), the second-order moment ratios drop rapidly to values close to 2, implying Poissonian exponential kinetics again.
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/). We found that in all the cases, T0 < Tk < Tf. Since Tk is the marker for complex kinetics, this leads to exponential kinetics at temperatures T > Tk. The non-exponential kinetics quickly emerges at temperatures T0 < T < Tk (and continue to temperatures at (1/61/4) T0 < T < T0). As the gap/roughness ratio increases (/), the ratio of folding temperature to kinetic transition temperature Tk (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 Tk < T < Tf for the exponential kinetics. In other words, it is relatively easier for folding (increasing Tf) and relatively harder for kinetic transition (decreasing Tk) to occur. Notice that Tk/T0 is almost a constant. T0 represents the onset of the rollover behavior in kinetics and Tk 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 Tk and T0 is almost a constant. The kinetic behavior around (Tk) is analogous to the glassy material at TA above the glass transition temperature where partial freezing occurs (Kirkpatrick and Wolynes, 1987; Kirkpatrick et al., 1989).
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/ = 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
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for large , 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 T0, 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.
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DISCUSSIONS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONSCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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6 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.
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CONCLUSIONS |
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONSCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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Submitted on March 12, 2004; accepted for publication June 28, 2004.
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