We show that diffusion can play an important role in proteinfolding kinetics. We explicitly calculate the diffusion coefficient of protein folding in a lattice model. We found that diffusion typically is configuration-or reaction coordinate-dependent. The diffusion coefficient is found to be decreasing with respect to the progression of folding toward the native state, which is caused by the collapse to a compact state constraining the configurational space for exploration. The configuration-or position-dependent diffusion coefficient has a significant contribution to the kinetics in addition to the thermodynamic free-energy barrier. It effectively changes (increases in this case) the kinetic barrier height as well as the position of the corresponding transition state and therefore modifies the folding kinetic rates as well as the kinetic routes. The resulting folding time, by considering both kinetic diffusion and the thermodynamic folding free-energy profile, thus is slower than the estimation from the thermodynamic free-energy barrier with constant diffusion but is consistent with the results from kinetic simulations. The configuration-or coordinate-dependent diffusion is especially important with respect to fast folding, when there is a small or no free-energy barrier and kinetics is controlled by diffusion. Including the configurational dependence will challenge the transition state theory of protein folding. The classical transition state theory will have to be modified to be consistent. The more detailed folding mechanistic studies involving phi value analysis based on the classical transition state theory also will have to be modified quantitatively.phi value analysis ͉ spatial-dependent diffusion ͉ transition state theory ͉ Monte Carlo simulations S tudying the kinetics of protein folding is key to understanding the fundamental underlying mechanism. Levinthal posed the so-called Levinthal paradox in 1969 (1). If protein folding proceeds with every possible state, then it takes cosmological time to reach the native state. In nature, protein folding is completed on a time scale from milliseconds to seconds. The recent energy landscape theory of protein folding (2-6) resolves the issue by assuming the underlying energy landscape is funneled toward the native state. Superimposed on the funnel are the bumps and wiggles that form local traps. For folding to be completed in a biological time scale under physiological temperature (300 K), the slope of the funnel must be steep enough to overcome the local traps. Energy landscape theory is successful in qualitatively and quantitatively explaining many folding experiments (2-7).Both theoretical and experimental investigations on folding and reaction kinetics have explored kinetics in different ranges of temperature and other environmental conditions (2-26). By varying environmental conditions, the underlying energy landscape structures can be probed in different levels, from locally to globally detailed perspectives (19,20). The relationship between the dynamics a...

The energy landscape approach has played a fundamental role in advancing our understanding of protein folding. Here, we quantify protein folding energy landscapes by exploring the underlying density of states. We identify three quantities essential for characterizing landscape topography: the stabilizing energy gap between the native and nonnative ensembles δE, the energetic roughness ΔE, and the scale of landscape measured by the entropy S. We show that the dimensionless ratio between the gap, roughness, and entropy of the system Λ ¼ δE∕ðΔE ffiffiffiffiffi ffi 2S p Þ accurately predicts the thermodynamics, as well as the kinetics of folding. Large Λ implies that the energy gap (or landscape slope towards the native state) is dominant, leading to more funneled landscapes. We investigate the role of topological and energetic roughness for proteins of different sizes and for proteins of the same size, but with different structural topologies. The landscape topography ratio Λ is shown to be monotonically correlated with the thermodynamic stability against trapping, as characterized by the ratio of folding temperature versus trapping temperature. Furthermore, Λ also monotonically correlates with the folding kinetic rates. These results provide the quantitative bridge between the landscape topography and experimental folding measurements.energy landscape theory | biomolecular dynamics U nderstanding how the amino acid sequence (i.e., primary structure) of each protein enables the native three-dimensional structure to be reached is one of the major challenges in molecular biophysics. In 1969, the arguments of Levinthal led to the suggestion of an apparent kinetic paradox (1). That is, if proteins were to randomly explore all possible states, cosmological timescales would be required for each protein to find the folded configuration. However, naturally occurring proteins fold in milliseconds to seconds. Protein folding theory has resolved this paradox by demonstrating that the underlying energy landscape is "funneled" towards the native state (2-9), however, local traps may be encountered during folding. To ensure that the folding occurs on biologically relevant timescales, the steepness of the protein folding funnel should be large, compared with the roughness due to local traps. Although this theory has provided the conceptual framework for interpreting folding experiments, both qualitatively and quantitatively (2, 5-18), it has yet to be explicitly demonstrated how the shape of the underlying landscape governs the thermodynamic stability and speed of folding, as measured experimentally (19). Here, we meet this challenge by quantifying the landscape topography and establishing the connection between the thermodynamics and kinetics of protein folding.Naturally selected proteins differ from random sequences in that they fold into unique three-dimensional functional configurations. This indicates that the information necessary to fold is embedded in the amino acid sequence. This intrinsic information manifests in the fo...

The energy landscape theory has been an invaluable theoretical framework in the understanding of biological processes such as protein folding, oligomerization, and functional transitions. According to the theory, the energy landscape of protein folding is funneled toward the native state, a conformational state that is consistent with the principle of minimal frustration. It has been accepted that real proteins are selected through natural evolution, satisfying the minimum frustration criterion. However, there is evidence that a low degree of frustration accelerates folding. We examined the interplay between topological and energetic protein frustration. We employed a Cα structure-based model for simulations with a controlled nonspecific energetic frustration added to the potential energy function. Thermodynamics and kinetics of a group of 19 proteins are completely characterized as a function of increasing level of energetic frustration. We observed two well-separated groups of proteins: one group where a little frustration enhances folding rates to an optimal value and another where any energetic frustration slows down folding. Protein energetic frustration regimes and their mechanisms are explained by the role of non-native contact interactions in different folding scenarios. These findings strongly correlate with the protein free-energy folding barrier and the absolute contact order parameters. These computational results are corroborated by principal component analysis and partial least square techniques. One simple theoretical model is proposed as a useful tool for experimentalists to predict the limits of improvements in real proteins.