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* Departamento de Física—Instituto de Biociências, Letras e Ciências Exatas, Universidade Estadual Paulista, São José do Rio Preto, Brazil; Department of Physics, Center for Theoretical Biological Physics, University of California at San Diego, La Jolla, California; Department of Chemistry, State University of New York at Stony Brook, Stony Brook, New York; and State Key Laboratory of Electroanalytical Chemistry, Changchun Institute of Applied Chemistry of Chinese Academy of Sciences, Changchun, People's Republic of China
Correspondence: Address reprint requests to Vitor B. P. Leite, E-mail: vleite{at}df.ibilce.unesp.br
Correspondence: Address reprint requests to Jin Wang, E-mail: jin.wang.1{at}stonybrook.edu.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMODEL AND METHODSRESULTSDISCUSSIONS AND CONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMODEL AND METHODSRESULTSDISCUSSIONS AND CONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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; Baldwin, 1995; Laurents et al., 1988). The possibility that multiple paths leading to folding may exist has led to an alternative statistical approach to this problem (Bryngelson et al., 1995; Wolynes et al., 1995; Chan and Dill, 1994; Abkevich et al., 1994; Wang et al, 1996).
The resulting energy-landscape theory looks at a coarse-grained-level picture of folding and tries to uncover the general principles governing its mechanisms. In general, the energy landscape is rough due to the many different possible competing interactions among the amino acid residues. On the other hand, the landscape of real proteins has evolved toward a funnel, shaped with a gradient leading toward the native folded basin. There are in general many possible paths leading to folding. A small number of discrete pathways emerge only when the landscape becomes rough and local traps become significant (Wang et al., 1996). The predictions of the funneled-landscape picture of folding have been confirmed and verified by many lattice and off-lattice simulations (Chan and Dill, 1994; Abkevich et al., 1994; Klimov and Thirumalai, 1998; Socci and Onuchic, 1994, 1995; Chahine et al., 2002; Kaya and Chan, 2000, 2002, 2003a,b,c; Boczko and Brooks, 1995). They are also in good qualitative and quantitative agreement with protein-folding thermodynamic and kinetic experiments (Itzhaki et al., 1995; Huang and Oas, 1995; Eaton et al., 1997, 2000; Garcia-Mira et al., 2002; Sabelko et al., 1999; Nguyen et al., 2003, Kuhlman et al., 1998).
The landscape theory of folding defines a temperature where the native protein structure becomes stable, Tf. It also defines a glassy temperature, Tg, where the landscape becomes rough compared with the bias toward a native folded state, and trapping into local minimum occurs. The ratio Tf/Tg defines how good a folder a protein is, i.e., the energy gap should be larger than the underlying roughness of the landscape (Goldstein et al., 1992; Chan and Dill, 1994; Chan et al., 2004; Abkevich et al., 1994; Klimov and Thirumalai, 1998). The cooperativity in the physical interactions can influence the ratio of Tf/Tg significantly (Chan et al., 2004).
There have been theoretical studies on average measurable variables of the folding kinetics (mean first-passage times, MFPT, or average rates) (Bryngelson and Wolynes, 1989; Saven et al., 1994; Wang et al., 1996; Shakhnovich et al., 1989; Gutin et al., 1996; Seno et al., 1998; Cieplak et al., 1999; Klimov and Thirumalai, 1998; Socci et al., 1996). Although the average-value description of the kinetics is important, it can sometimes miss important features of folding dynamics that are crucial for uncovering the fundamental mechanisms. For different sequence compositions or external conditions (physical properties, temperatures, and solvent conditions) the progress of folding can probe different parts of the energy landscape and detect different local barriers. Therefore many possible timescales may coexist, and the kinetics can become nonexponential or multiexponential. In bulk measurements, it is often difficult to distinguish whether the observed nonexponential kinetics is intrinsic or due to the inhomogeneous distribution of single exponential processes. With recent technological advances, however, single-molecule detections have become possible (Moerner, 1996; Lu et al., 1998; Deniz et al., 2000). Single molecules are sensitive probes to the local environments and therefore an ideal tool to understand the structures of the energy landscape of the proteins (Wang and Wolynes, 1995, 1999; Onuchic et al., 1999; Wang, 2003a,b; Schenter et al., 1999). Lately, a number of remarkable initial single-molecule folding experiments have been undertaken (Zhuang et al., 2000, 2002; Jia et al., 1999; Deniz et al., 2000; Schuler et al., 2002; Lipman et al., 2003) but their interpretation is nontrivial. Statistical fluctuations are intrinsic to single molecules, since they are not weighted down by the number of the molecules as they are in the bulk, and they can be directly measured. Single-molecule data are essentially sequences of on-and-off spikes as a function of time. Therefore, to determine whether they are associated with exponential decay is a challenging task.
In this article, we investigate the kinetics of folding based on the moments or fluctuations of the FPTs of folding and dynamic collapse, with respect to temperature and physical properties such as hydrophobicity. From these studies, we are able to determine the way in which the exponential or nonexponential nature of the folding kinetics will emerge as a function of these conditions. Furthermore, we obtain information about the dependence of the distribution of folding times upon various conditions. This gives the full dynamic picture of the folding instead of just the MFPT. The lattice simulation presented is able to establish a link among the energy-landscape theory of protein folding, simulations, and single-molecule folding experiments (Wang and Wolynes, 1995, 1999; Onuchic et al., 1999; Berezhkovskii et al., 1999; Wang, 2003a,b; Lee et al., 2003; Zhou et al., 2003; Gopich et al., 2003a,b; Zhuang et al., 2000, 2002; Jia et al., 1999; Deniz et al., 2000; Schuler et al., 2002; Lipman et al., 2003).
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MODEL AND METHODS |
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; Socci and Onuchic, 1994, 1995). The protein is modeled by a 27-length polymer chain (27-mer) on a three-dimensional cubic lattice. The 27-mer lattice is a renormalized (or reduced) description of small globular proteins. We use a Monte Carlo algorithm with standard polymer local moves—i.e., end, corner flip, and 90° crankshaft (Socci and Onuchic, 1994, 1995). The energy for the heteropolymer is given by
 | (1) |
):
We use the previously-designed sequence ABABBBBBABBABABAAABBAAAAAAB (Shakhnovich and Gutin, 1993). In this study only one good folding sequence is used. Effects of sequence variations are not analyzed. Monte Carlo sampling with a local move set is used to determine the density of states and to define the kinetics of the model (Abkevich et al., 1994; Socci and Onuchic, 1994, 1995; Chahine et al., 2002). The density of states is determined as a function of energy E, number of native nonbonded contacts Q, and the total number of nonbonded contacts Z. Within the microcanonical ensemble, the free energy of the system can be obtained, and the complete thermodynamics can be determined. Four phases are typically found—i.e., the noncompact unfolded states (so-called random-coil states); compact unfolded states; trapping states; and the native state. The transition temperatures Tf (folding) and Tg (local trapping) are determined.
We can measure the MFPT to reach the native state for the designed sequence starting from random-coil configurations. By repeating the dynamic Monte Carlo simulations with different initial conditions, one can obtain information about the statistical distributions of these folding times and therefore determine the high-order moments of the FPT. The moments of first-passage time (
) of events for folding and collapse are generally given by
 | (2) |
is simply given by
 | (3) |
The ratio of the nth moment and the mean to the nth power,
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RESULTS |
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; Kaya and Chan, 2000, 2002; 2003c), the mean first-passage time (MFPT) as a function of temperature has a U-shaped dependence, with higher values on both the high and low temperature ranges and a kinetic minimum near the folding temperature. We found that above this temperature, the distribution of first-passage times (FPT) is nearly Poissonian, indicating single-exponential kinetics. At the high temperatures, due to the large thermo motions, the different kinetic paths sense roughly similar barriers since the local details are smeared or washed out by the thermo motions. Only the dominant barriers matter. So there are multiple parallel paths, all experiencing similar barriers, resulting in single-exponential kinetics. On the other hand, when the temperature drops below the temperature of the kinetic minimum, the distribution of FPT deviates from Poissonian significantly, indicating nonexponential kinetics and multiple traps. In certain temperature regimes, the distribution at long times approaches a power law. This indicates that the fluctuation in FPT is significant and deviates from the mean. In other words, the FPT is non-self-averaging and the distribution has fatty tails. This indicates the existence of intermittence, where rare events can give a significant contribution to the folding event. Due to the ruggedness of the underlying energy landscape at low temperatures, specific discrete paths are themselves different from each other and give distinct contributions to the kinetics. The full distribution of FPT, not only its mean, is needed to characterize the whole system dynamics. At even lower temperatures, the kinetics is back to Poissonian, indicating exponential kinetics and the domination of a single trap. By increasing (or decreasing) hydrophobicity, the distribution deviates less (or more) from Poissonian in the same temperature range.
Folding
The results for the folding FPT moments (second, third, and fourth moments) and their ratios as well as their derivatives with respect to temperature are shown in Fig. 1 (low hydrophobicity) and Fig. 2 (high hydrophobicity). For our sequence, at low hydrophobicity, Tg = 1.2 and Tf = 1.6 (Socci and Onuchic, 1994, 1995). The kinetic minimum of folding 
is near 1.8. For T > 
Rn = n!; for 
Rn/n! > 1; and Rn increases significantly. For high hydrophobicity, Tg = 1.0 and Tf = 1.3, the kinetic minimum is near 1.5, and one can see a similar behavior: for 
Rn = n! and for Rn/n! > 1, but Rn does not increase as quickly as for low hydrophobicity.
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, this model behavior is not similar to that of many real proteins. It corresponds to a situation in which folding rate would decrease when native stability is increased from the transition midpoint. The underlying reason for this critical shortcoming is because the two-letter models studied here are not sufficiently cooperative, with Tf/Tg too small to mimic that of real apparent two-state proteins. Although the simulation results presented are useful, these limitations of the model might have significant impact on their relationship with experiments. The cooperativity can play an important role here in solving this discrepancy.
The simulations are in agreement with analytical calculations as well as detailed models (Lee et al., 2003; Zhou et al., 2003) and with other ways of calculating 
This indicates that the ratios of moments or the fluctuations in first-passage times are efficient and sensitive ways to identify the kinetic transition temperatures and the effect of fluctuations. In Figs. 1 and 2 one can see that the derivative of Rn/n! provides an effective parameter for locating As we can see, the sensitivity on the kinetic fluctuations increases rapidly as the order of the moments increases. Although high-order moments will be better indicators than low-order moments, the data requirement is much greater. In principle, there should be a balance in extracting the useful information from the trajectories in the simulations and experiments.
Note that the ratios of the moments will become important at the single-molecule level. In the bulk experiments, the high-order moments are significantly suppressed by the large number of molecules in the sample (on the order of 108 molecules). Furthermore, it is often hard to distinguish the intrinsic dynamics from the observed nonexponential kinetics, since the same data could result from either inhomogeneous distributions of the different protein molecules with exponential kinetics or homogeneous distribution of the intrinsic underlying nonexponential kinetics.
Collapse
Using methods similar to those for the folding simulations, all simulations were repeated to determine the collapse times. For this study, the collapse time is taken to be at the point when the final state for the collapse is achieved—i.e., when the system reaches the nonbonded contacts Z = 25—instead of at the native configuration for the folded state. In the results below we have the same sequence in the same two limits we had before. Now the deviations on Rn are expected to appear when the system reaches the kinetic minimum for collapse 
For favorable collapsed states, starting with random-coil conformations, the uncollapsed fraction decays exponentially. However if the glassy trapping regime is reached, the dynamics becomes slow and one observes slow trajectories and nonexponential decays.
For low hydrophobicity, Tg = 1.2, 
; and for high hydrophobicity, 
The simulation results of Rn/n! and their corresponding derivatives with respect to temperature as a function of temperature are shown in Figs. 3 and 4. Roughly for 
Rn/n! = 1, and for 
Rn/n! > 1. Note that Rn starts to increase above Tg (at T = 1.6 for low hydrophobicity, and at T = 1.3 for high hydrophobicity), a possible indication of kinetic glassy transition temperature TA postulated in the theories of the short-range random first-order phase transitions in glasses (Kirkpatrick et al., 1989). This is an indication that deviation in Rn is a sensitive probe of the dynamics of collapse events (see Figs. 3 b and 4 b for the derivative part).
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Very low temperature results
The above results on kinetics and fluctuations on folding are valid for temperatures between the folding and the glassy trapping temperatures. At even lower temperature (see Fig. 5), in the case of collapse in the low hydrophobicity limit, Rn drops. At temperature 0.7 we obtain 
which is quite close to 2! (the high temperature limit of Poisson single-exponential kinetics). The exact numbers are hard to obtain, because at low temperatures a significant percentage of runs do not reach the collapsed state (they reach the Monte Carlo step simulation limit first, 1.0 x 109). Still, the results suggest that the ratios at very low temperature behave much as they do in the high temperature limit, as predicted in analytical calculation (
) (Wang, 2004a). Due to the finite size of our systems, kinetics is now controlled by a single barrier.
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In Fig. 6, all kinetic regimes are shown through a plot of the survival population of uncollapsed runs as a function of time. At high temperature (T = 2.0), the system follows an exponential decay. In this regime, multiple diffusive pathways for collapse are available and they are roughly equivalent, yielding a single-exponential characteristic time. At temperatures around the relevant transition temperatures (T = Tg = 1.0), fluctuations in are at their highest, diffusive paths are not equivalent, and the system exhibits intermittence. One can clearly see that a single exponential does not fit the survival probability. Even a fit to a stretched exponential is not sufficient and the system shows a power-law behavior over three decades at times longer than MFPT. At very low temperature (T = 0.7), a single path dominates the process. All three regimes are in agreement with the theoretical predictions (Wang, 2004a).
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This kind of kinetic behavior in the fluctuation of the kinetic folding times can be understood physically as follows: At very high temperature, due to the large thermal motion and kinetic energy, the folding has multiple parallel trajectories that cause an average barrier, and therefore the process is a single-exponential one. When the temperature is reduced, the folding event encounters different barriers, and the kinetics becomes multiexponential. When the temperature becomes even lower (below the glassy trapping temperature), the system has only limited states to explore (the others are frozen), and the kinetics is controlled by escapes from a single dominant deep trap. Thus the process becomes a single-exponential one again.
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DISCUSSIONS AND CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONMODEL AND METHODSRESULTSDISCUSSIONS AND CONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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; Nguyen et al., 2003) observed a "strange" kinetic behavior in which kinetics varies from single-exponential to nonexponential transitions as the temperature decreases. This corresponds to the previous investigation (Kaya and Chan, 2002, 2003c) and our current model prediction in the relatively high temperature regime (the physiological temperature regime), where a change from single-exponential to nonexponential kinetic behavior is observed. Indeed, previous (Kaya and Chan, 2002, 2003c) and current investigations provide evidence that these kinetic features are robust across several different models. This temperature-dependence of the protein dynamics can also be probed in ligand binding experiments. Frauenfelder et al. (1988, 1991) have observed this behavior for the binding kinetics of CO to myoglobin. Sadqi et al. (2003) observed a dynamical slowdown of the hydrophobic collapse upon temperature change. They also observed that, at higher temperatures, as the protein becomes increasingly compact due to a stronger hydrophobic force, there is a slowdown of the dynamics of collapse. This dynamic hydrophobic effect is a high-temperature analog of the dynamic glass transition predicted by theory. In our collapse studies, we also see a similar complex behavior of the kinetics when the temperature changes. To study the role of hydrophobic collapse in folding and its link with the experiments (Sadqi et al., 2003), we plot the evolution of survival probability or the fraction of uncollapse in time. In Fig. 7, we show, under the condition of high hydrophobicity, that the kinetics (the rate) derived from the survival probability at the optimal temperature (T 
1.5) is the fastest (measured by the slope of the curve). The kinetics becomes slower when the temperature either increases or decreases beyond the optimal temperature. This is consistent with kinetic slowdown at higher temperatures observed by Sadqi et al. (2003) in the folding experiments (Kaya and Chan, 2000, 2002, 2003c). We also see similar behavior in the low hydrophobicity case in Fig. 8, except that the optimal temperature is at 2.0. In general, the average kinetics is slower in the low hydrophobicity case than in the high hydrophobicity case, because of the weaker hydrophobic driving force.
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,b; Yang et al., 2003), reaction dynamics also experiences nonexponential kinetic behavior and the distribution of kinetics follows an approximate power law. The theoretical framework presented here provides a microscopic foundation to interpret these experiments. Other single-molecule experiments also show similar behavior (Zhuang et al., 2000, 2002; Jia et al., 1999; Schuler et al., 2002; Lipman et al., 2003). In future studies, it will be important to explore these kinetic fluctuations upon changing experimental conditions such as temperature and denaturant solvents (which would be equivalent to changing the hydrophobic contents of the system modeled with low and high hydrophobicity in this study). One of the advantages of this approach is that it provides a link among theory, simulations, and experiments. In the theoretical approach, the first-passage time and its statistical properties can be easily obtained from the simulations following the procedures outlined in this article. In the experiments, information about FPT properties can be obtained from the kinetic folding trajectories (for example, fluorescence). This will stimulate the current round, and the next, of single-molecule experiments and even more detailed simulations, for study of the full range of kinetic behavior.
More simulations at the microscopic level, which will provide the detailed mechanistic features, are also necessary. The only limiting step is that the statistics of the kinetics requires long-time trajectories. Therefore some intermediate coarse-grained description, for example at the residue-residue level, may be needed to supplement the fully atomistic studies.
This approach proposed for the folding kinetics is quite general. It is not necessarily limited to folding studies. In fact, it is a framework that can be generalized and extended to other systems such as RNA folding, biomolecule binding, viscous liquids, and glassy materials. In fact, such kinetic investigations have already been carried out using an energy landscape framework for viscous liquids and glassy materials (Kirkpatrick et al., 1989; Xia and Wolynes, 2001) and binding of biomolecules (Wang et al., 2004b).
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ACKNOWLEDGEMENTS |
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Work at the University of California at San Diego was supported by National Science Foundation grants (INT-0000419 and MCB-0084797). G.S. was supported by the Division of Chemical Sciences, Office of the Basic Energy Sciences, Office of Energy Research, U.S. Department of Energy. V.B.P.L. was supported by Fundacão de Amparo à Pesquisa do Estado de São Paulo (Brazil) and a grant from the Conselho Nacional de Desenvolvimento Cientificoe Technológico-National Science Foundation International Program (INT-0000419).
Submitted on May 19, 2004; accepted for publication July 26, 2004.
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The same notation as in 
The same notation of numbers as in 
) high temperature, T = 2.0; (
) at transition temperature T = Tg = 1.0; and (
) very low temperature, T = 0.7. The lines are the exponential fits for each temperature. At high and very low temperature the fits appropriately describe the data (continuum lines). At the transition temperature one can clearly see the nonexponential behavior (compare with the dashed line). In the long-time limit between t = 106 and 109, the survival probability is nearly a straight line in the log-log plot implying a power law dependence on time.
