Protein folding mechanisms and the multidimensional folding funnel

Protein Folding through Kinetic Discrimination

Proteins fold on a micros-ms time scale. However, the number of possible conformations of the polypeptide backbone is so large that random sampling would not allow the protein to fold within the lifetime of the universe, the Levinthal paradox. We show here that a protein chain can fold efficiently with high fidelity if on average native contacts survive longer than non-native ones, that is, if the dissociation rate constant for breakage of a contact is lower for native than for non-native interactions. An important consequence of this finding is that no pathway needs to be specified for a protein to fold. Instead, kinetic discrimination among formed contacts is a sufficient criterion for folding to proceed to the native state. Successful protein folding requires that productive contacts survive long enough to obtain a certain level of probability that other native contacts form before the first interacting unit dissociates. If native contacts survive longer than non-native ones, this prevents misfolding and provides the folding process with directionality toward the native state. If on average all contacts survive equally long, the protein chain is deemed to fold through random search through all possible conformations (i.e., the Levinthal paradox). A modest degree of cooperativity among the native contacts, that is, decreased dissociation rate next to neighboring contacts, shifts the required ratio of dissociation rates into a realistic regime and makes folding a stochastic process with a nucleation step. No kinetic discrimination needs to be invoked in regards to the association process, which is modeled as dependent on the diffusion rate of chain segments.

Folding time distributions as an approach to protein folding kinetics

A 27-residue lattice heteropolymer subject to Monte Carlo dynamics on a simple cubic lattice is studied over a range of temperatures. Folding time distributions are used to obtain information concerning the details of folding kinetics. The results are compared with those from methods based on mean force surfaces expressed in terms of a reduced set of variables and on a disconnectivity graph for the same system. A detailed analysis of the folding trajectories is given, and the importance of dead-end traps in determining the folding time is demonstrated. We show that the calculated folding kinetics can be modeled by a system of kinetic equations, with the essential rate constants determined from the Monte Carlo simulations and the resulting folding time distributions. The kinetic equations make possible an analysis of the variation of the importance of different channels with temperature. In particular, we show that the presence of intermediates may be masked in the folding time distributions, with the mean folding time being independent of the height of the barrier between the intermediates and collapsed globule state of the system. This and other results demonstrate that care has to be used in interpreting experimental folding data in terms of the underlying kinetics. Correspondingly, simulations are shown to have to satisfy certain requirements to obtain proper sampling of the dead-end traps.

Proteins: is the folding process dynamically encoded?

Recent findings for nano-structured polymeric systems support the existence of dynamic heterogeneities in glass-forming materials. The implications for structure formation in proteins are considered. The high speed and efficiency of the folding process might be explainable based on differences in the slow segmental mobility along the protein chain.

Denaturation of jack-bean urease by sodium n-dodecyl sulphate: A kinetic study below the critical micelle concentration

Kinetics of urease denaturation by anionic surfactant (sodium n-dodecyl sulphate, SDS) at concentrations below the critical micelle concentration (CMC) is investigated spectrophotometrically at neutral pH and the corresponding two-phase kinetic parameters of the process are estimated from a three-state reversible process using a binomial exponential relation based on the relaxation time method as: Using a prepared computer program, the experimental data are properly fitted into a binomial exponential relation, considering a two-phase denaturation pathway including a kinetically stable folded intermediate formed at SDS concentration of 1.1 mM. Forward and backward rate constants are estimated as: k(1)=0.2141+/-4.5 x 10(-3), k(2)=5.173 x 10(-3)+/-8.3 x 10(-5), k(-1)=0.09432+/-3.6 x 10(-4) and k(-2)=2.079 x 10(-3)+/-5.6 x 10(-5)s(-1) for the proposed mechanism. The rate-limiting step as well as the reaction coordinates in the denaturation mechanism are established. The mechanism involves formation of a kinetically stable folded native like intermediate through the electrostatic interactions. The intermediate was found to be more stable even than the native form (by about 9 kJmol(-1)) and still hexamer, because no loss of amplitude was observed. Electrophoresis experiments on the native and surfactant/urease complexes indicated a higher mobility for the kinetically folded native like intermediate.

Cooperativity and the origins of rapid, single-exponential kinetics in protein folding

The folding of naturally occurring, single-domain proteins is usually well described as a simple, single-exponential process lacking significant trapped states. Here we further explore the hypothesis that the smooth energy landscape this implies, and the rapid kinetics it engenders, arises due to the extraordinary thermodynamic cooperativity of protein folding. Studying Miyazawa-Jernigan lattice polymers, we find that, even under conditions where the folding energy landscape is relatively optimized (designed sequences folding at their temperature of maximum folding rate), the folding of protein-like heteropolymers is accelerated when their thermodynamic cooperativity is enhanced by enhancing the nonadditivity of their energy potentials. At lower temperatures, where kinetic traps presumably play a more significant role in defining folding rates, we observe still greater cooperativity-induced acceleration. Consistent with these observations, we find that the folding kinetics of our computational models more closely approximates single-exponential behavior as their cooperativity approaches optimal levels. These observations suggest that the rapid folding of naturally occurring proteins is, in part, a consequence of their remarkably cooperative folding.

Enhancing the stability and folding rate of a repeat protein through the addition of consensus repeats

Repeat proteins are constructed from a linear array of modular units, giving rise to an overall topology lacking long-range interactions. This suggests that stabilizing repeat modules based on consensus information might be added to a repeat protein domain, allowing it to be extended without altering its overall topology. Here we add consensus modules the ankyrin repeat domain from the Drosophila Notch receptor to investigate the structural tolerance to these modules, the relative thermodynamic stability of these hybrid proteins, and how alterations in the energy landscape influence folding kinetics. Insertions of consensus modules between repeats five and six of the Notch ankyrin domain have little effect on the far and near-UV CD spectra, indicating that neither secondary nor tertiary structure is dramatically altered. Furthermore, stable structure is maintained at increased denaturant concentrations in the polypeptides containing the consensus repeats, indicating that the consensus modules are capable of stabilizing much of the domain. However, insertion of the consensus repeats appears to disrupt cooperativity, producing a two-stage (three-state) unfolding transition in which the C-terminal repeats unfold at moderate urea concentrations. Removing the C-terminal repeats (Notch ankyrin repeats six and seven) restores equilibrium two-state folding and demonstrates that the high stability of the consensus repeats is propagated into the N-terminal, naturally occurring Notch ankyrin repeats. This stability increase greatly increases the folding rate, and suggests that the transition state ensemble may be repositioned in the chimeric consensus-stabilized proteins in response to local stability.

Exploring the relationship between funneled energy landscapes and two-state protein folding

It should take an astronomical time span for unfolded protein chains to find their native state based on an unguided conformational random search. The experimental observation that folding is fast can be rationalized by assuming that protein energy landscapes are sloped towards the native state minimum, such that rapid folding can proceed from virtually any point in conformational space. Folding transitions often exhibit two-state behavior, involving extensively disordered and highly structured conformers as the only two observable kinetic species. This study employs a simple Brownian dynamics model of "protein particles" moving in a spherically symmetrical potential. As expected, the presence of an overall slope towards the native state minimum is an effective means to speed up folding. However, the two-state nature of the transition is eradicated if a significant energetic bias extends too far into the non-native conformational space. The breakdown of two-state cooperativity under these conditions is caused by a continuous conformational drift of the unfolded proteins. Ideal two-state behavior can only be maintained on surfaces exhibiting large regions that are energetically flat, a result that is supported by other recent data in the literature (Kaya and Chan, Proteins: Struct Funct Genet 2003;52:510-523). Rapid two-state folding requires energy landscapes exhibiting the following features: (i) A large region in conformational space that is energetically flat, thus allowing for a significant degree of random sampling, such that unfolded proteins can retain a random coil structure; (ii) a trapping area that is strongly sloped towards the native state minimum.

Comparative all-atomic study of unfolding pathways for proteins chymotrypsin inhibitor 2 and barnase

The features of transition states and intermediates are important in the study on protein folding. However, transition states and intermediates could not be obviously identified from trajectories obtained by dynamic simulations. In this work, a different method to identify and characterize the transition states and intermediates by combining the root mean square deviation of C(alpha) atoms and the similarity factor Q to the native state is proposed. The unfolding processes based on all-atomic simulations for proteins chymotrypsin inhibitor 2 and barnase are studied, and the related transition states and intermediates are identified by observing an unfolding factor U = 1-F. Comparisons between the conformational cluster analysis and experimental results are also made. The various analyses on the unfolding behaviors indicate that our method can well define the transition states and intermediates, and the factor U (or F) can be used as a reaction coordinate of the folding and unfolding process. It is also found that three-state folding proteins might experience more complicated pathways and have more rugged energy landscapes than two-state folding proteins.

Energy landscapes and properties of biomolecules

Thermodynamic and dynamic properties of biomolecules can be calculated using a coarse-grained approach based upon sampling stationary points of the underlying potential energy surface. The superposition approximation provides an overall partition function as a sum of contributions from the local minima, and hence functions such as internal energy, entropy, free energy and the heat capacity. To obtain rates we must also sample transition states that link the local minima, and the discrete path sampling method provides a systematic means to achieve this goal. A coarse-grained picture is also helpful in locating the global minimum using the basin-hopping approach. Here we can exploit a fictitious dynamics between the basins of attraction of local minima, since the objective is to find the lowest minimum, rather than to reproduce the thermodynamics or dynamics.

A funneled energy landscape for cytochrome c directly predicts the sequential folding route inferred from hydrogen exchange experiments

Proteins fold through a variety of mechanisms. For a given protein, folding routes largely depend on the protein's stability and its native-state geometry, because the landscape is funneled. These ideas are corroborated for cytochrome c by using a coarse-grained topology-based model with a perfect funnel landscape that includes explicit modeling of the heme. The results show the importance of the heme as a nucleation site and explain the observed hydrogen exchange patterns of cytochrome c within the context of energy landscape theory.

UV Resonance Raman Determination of Polyproline II, Extended 2.51-Helix, and β-Sheet Ψ Angle Energy Landscape in Poly-l-Lysine and Poly-l-Glutamic Acid

UV resonance Raman (UVR) spectroscopy was used to examine the solution conformation of poly-l-lysine (PLL) and poly-l-glutamic acid (PGA) in their non-alpha-helical states. UVR measurements indicate that PLL (at pH = 2) and PGA (at pH = 9) exist mainly in a mixture of polyproline II (PPII) and a novel left-handed 2.5(1)-helical conformation, which is an extended beta-strand-like conformation with Psi approximately +170 degrees and Phi approximately -130 degrees . Both of these conformations are highly exposed to water. The energies of these conformations are very similar. We see no evidence of any disordered "random coil" states. In addition, we find that a PLL and PGA mixture at neutral pH is approximately 60% beta-sheet and contains PPII and extended 2.5(1)-helix conformations. The beta-sheet conformation shows little evidence of amide backbone hydrogen bonding to water. We also developed a method to estimate the distribution of Psi Ramachandran angles for these conformations, which we used to estimate a Psi Ramachandran angle energy landscape. We believe that these are the first experimental studies to give direct information on protein and peptide energy landscapes.

Thermodynamic prediction of protein neutrality

We present a simple theory that uses thermodynamic parameters to predict the probability that a protein retains the wild-type structure after one or more random amino acid substitutions. Our theory predicts that for large numbers of substitutions the probability that a protein retains its structure will decline exponentially with the number of substitutions, with the severity of this decline determined by properties of the structure. Our theory also predicts that a protein can gain extra robustness to the first few substitutions by increasing its thermodynamic stability. We validate our theory with simulations on lattice protein models and by showing that it quantitatively predicts previously published experimental measurements on subtilisin and our own measurements on variants of TEM1 beta-lactamase. Our work unifies observations about the clustering of functional proteins in sequence space, and provides a basis for interpreting the response of proteins to substitutions in protein engineering applications.

Conformational changes during the nanosecond-to-millisecond unfolding of ubiquitin

Steady-state and transient conformational changes upon the thermal unfolding of ubiquitin were investigated with nonlinear IR spectroscopy of the amide I vibrations. Equilibrium temperature-dependent 2D IR spectroscopy reveals the unfolding of the beta-sheet of ubiquitin through the loss of cross peaks formed between transitions arising from delocalized vibrations of the beta-sheet. Transient unfolding after a nanosecond temperature jump is monitored with dispersed vibrational echo spectroscopy, a projection of the 2D IR spectrum. Whereas the equilibrium study follows a simple two-state unfolding, the transient experiments observe complex relaxation behavior that differs for various spectral components and spans 6 decades in time. The transient behavior can be separated into fast and slow time scales. From 100 ns to 0.5 ms, the spectral features associated with beta-sheet unfolding relax in a sequential, nonexponential manner, with time constants of 3 micros and 80 micros. By modeling the amide I vibrations of ubiquitin, this observation is explained as unfolding of the less stable strands III-V of the beta-sheet before unfolding of the hairpin that forms part of the hydrophobic core. This downhill unfolding is followed by exponential barrier-crossing kinetics on a 3-ms time scale.

Optimal combination of theory and experiment for the characterization of the protein folding landscape of S6: how far can a minimalist model go?

The detailed characterization of the overall free energy landscape associated with the folding process of a protein is the ultimate goal in protein folding studies. Modern experimental techniques provide accurate thermodynamic and kinetic measurements on restricted regions of a protein landscape. Although simplified protein models can access larger regions of the landscape, they are oftentimes built on assumptions and approximations that affect the accuracy of the results. We present a new methodology that allows to combine the complementary strengths of theory and experiment for a more complete characterization of a protein folding landscape. We prove that this new procedure allows a simplified protein model to reproduce remarkably well (correlation coefficient > 0.9) all experimental data available on free energies differences upon single mutations for S6 ribosomal protein and two circular permutants. Our results confirm and quantify the hypothesis, recently formulated on the basis of experimental data, that the folding landscape of protein S6 is strongly affected by an atypical distribution of contact energies.

Barrier-limited, microsecond folding of a stable protein measured with hydrogen exchange: Implications for downhill folding

Folding experiments are conducted to test whether a covalently cross-linked coiled-coil folds so quickly that the process is no longer limited by a free-energy barrier. This protein is very stable and topologically simple, needing merely to "zipper up," while having an extrapolated folding rate of k(f) = 2 x 10(5) s(-1). These properties make it likely to attain the elusive "downhill folding" limit, at which a series of intermediates can be characterized. To measure the ultra-fast kinetics in the absence of denaturant, we apply NMR and hydrogen-exchange methods. The stability and its denaturant dependence for the hydrogen bonds in the central part of protein equal the values calculated for whole-molecule unfolding. Like-wise, their closing and opening rates indicate that these hydrogen bonds are broken and reformed in a single cooperative event representing the folding transition from the fully unfolded state to the native state. Additionally, closing rates for these hydrogen bonds agree with the extrapolated barrier-limited folding rate observed near the melting transition. Therefore, even in the absence of denaturant, where DeltaG(eq) approximately -6 kcal.mol(-1) (1 cal = 4.18 J) and tau(f) approximately 6 mus, folding remains cooperative and barrier-limited. Given that this prime candidate for downhill folding fails to do so, we propose that protein folding will remain barrier-limited for proteins that fold cooperatively.

UV Raman demonstrates that alpha-helical polyalanine peptides melt to polyproline II conformations

We examined the 204-nm UV Raman spectra of the peptide XAO, which was previously found by Shi et al.'s NMR study to occur in aqueous solution in a polyproline II (PPII) conformation. The UV Raman spectra of XAO are essentially identical to the spectra of small peptides such as ala(5) and to the large 21-residue predominantly Ala peptide, AP. We conclude that the non-alpha-helical conformations of these peptides are dominantly PPII. Thus, AP, which is highly alpha-helical at room temperature, melts to a PPII conformation. There is no indication of any population of intermediate disordered conformations. We continued our development of methods to relate the Ramachandran Psi-angle to the amide III band frequency. We describe a new method to estimate the Ramachandran Psi-angular distributions from amide III band line shapes measured in 204-nm UV Raman spectra. We used this method to compare the Psi-distributions in XAO, ala(5), the non-alpha-helical state of AP, and acid-denatured apomyoglobin. In addition, we estimated the Psi-angle distributions of peptide bonds which occur in non-alpha-helix and non-beta-sheet conformations in a small library of proteins.

Using protein folding rates to test protein folding theories

The fastest simple, kinetically two-state protein folds a million times more rapidly than the slowest. Here we review many recent theories of protein folding kinetics in terms of their ability to qualitatively rationalize, if not quantitatively predict, this fundamental experimental observation.

A flexible approach for understanding protein stability

A distance constraint model (DCM) is presented that identifies flexible regions within protein structure consistent with specified thermodynamic condition. The DCM is based on a rigorous free energy decomposition scheme representing structure as fluctuating constraint topologies. Entropy non-additivity is problematic for naive decompositions, limiting the success of heat capacity predictions. The DCM resolves non-additivity by summing over independent entropic components determined by an efficient network-rigidity algorithm. A minimal 3-parameter DCM is demonstrated to accurately reproduce experimental heat capacity curves. Free energy landscapes and quantitative stability-flexibility relationships are obtained in terms of global flexibility. Several connections to experiment are made.

An experimentally determined protein folding energy landscape

Energy landscapes have been used to conceptually describe and model protein folding but have been difficult to measure experimentally, in large part because of the myriad of partly folded protein conformations that cannot be isolated and thermodynamically characterized. Here we experimentally determine a detailed energy landscape for protein folding. We generated a series of overlapping constructs containing subsets of the seven ankyrin repeats of the Drosophila Notch receptor, a protein domain whose linear arrangement of modular structural units can be fragmented without disrupting structure. To a good approximation, stabilities of each construct can be described as a sum of energy terms associated with each repeat. The magnitude of each energy term indicates that each repeat is intrinsically unstable but is strongly stabilized by interactions with its nearest neighbors. These linear energy terms define an equilibrium free energy landscape, which shows an early free energy barrier and suggests preferred low-energy routes for folding.

Folding lambda-repressor at its speed limit

We show that the five-helix bundle lambda(6-85) can be engineered and solvent-tuned to make the transition from activated two-state folding to downhill folding. The transition manifests itself as the appearance of additional dynamics faster than the activated kinetics, followed by the disappearance of the activated kinetics when the bias toward the native state is increased. Our fastest value of 1 micros for the "speed" limit of lambda(6-85) is measured at low concentrations of a denaturant that smooths the free-energy surface. Complete disappearance of the activated phase is obtained in stabilizing glucose buffer. Langevin dynamics on a rough free-energy surface with variable bias toward the native state provides a robust and quantitative description of the transition from activated to downhill folding. Based on our simulation, we estimate the residual energetic frustration of lambda(6-85) to be delta(2) G approximately 0.64 k(2)T(2). We show that lambda(6-86), as well as very fast folding proteins or folding intermediates estimated to lie near the speed limit, provide a better rate-topology correlation than proteins with larger energetic frustration. A limit of beta > or = 0.7 on any stretching of lambda(6-85) barrier-free dynamics suggests that a low-dimensional free-energy surface is sufficient to describe folding.

Transition states for folding of circular-permuted proteins

To explore the role of entropy and chain connectivity in protein folding, a particularly interesting scheme, namely, the circular permutation, has been used. Recently, experimental observations showed that there are large differences in the folding mechanisms between the wild-type proteins and their circular permutants. These differences are strongly related to the change in the intrachain connectivity. Some results obtained by molecular dynamics simulations also showed a good agreement with the experimental findings. Here, we use a topology-based free-energy functional method to study the role of the chain connectivity in folding by comparing features of transition states of the wild-type proteins with those of their circular permutants. We concentrate our study on 3 small globular proteins, namely, the alpha-spectrin SH3 domain (SH3), the chymotrypsin inhibitor 2 (CI2), and the ribosomal protein S6, and obtain exciting results that are consistent with the available experimental and simulation results. A heterogeneity of the interaction energies between contacts for protein CI2 and for protein S6 is also introduced, which characterizes the strong interactions between contacts with long loops, as speculated from experiments for protein S6. The comparison between the folding nucleus of the wild-type proteins and those of their circular permutants indicates that chain connectivity affects remarkably the shapes of the energy profiles and thus the folding mechanism. Further comparisons between our theoretical calculated phi(th) values and the experimental observed phi(exp) values for the 3 proteins and their permutants show that our results are in good agreement with experimental ones and that correlations between them are high. These indicate that the free-energy functional method really provides a way to analyze the folding behavior of the circular-permuted proteins and therefore the folding mechanism of the wild-type proteins.

Stability and the evolvability of function in a model protein

Functional proteins must fold with some minimal stability to a structure that can perform a biochemical task. Here we use a simple model to investigate the relationship between the stability requirement and the capacity of a protein to evolve the function of binding to a ligand. Although our model contains no built-in tradeoff between stability and function, proteins evolved function more efficiently when the stability requirement was relaxed. Proteins with both high stability and high function evolved more efficiently when the stability requirement was gradually increased than when there was constant selection for high stability. These results show that in our model, the evolution of function is enhanced by allowing proteins to explore sequences corresponding to marginally stable structures, and that it is easier to improve stability while maintaining high function than to improve function while maintaining high stability. Our model also demonstrates that even in the absence of a fundamental biophysical tradeoff between stability and function, the speed with which function can evolve is limited by the stability requirement imposed on the protein.

Folding barrier in horse cytochrome c: support for a classical folding pathway

Native-state structures and conformations of ferrocytochrome c, nitrosylcytochrome c, and carbonmonoxycytochrome c are very similar. They are, however, immensely different from each other in terms of thermodynamic stability. The dramatic destabilization of ferrocytochrome c to the extent of 12 kcal mol(-1) produces no effect on the folding rate, and this is so in spite of the fact that all three test-tube variants fold in an apparent two-state manner. For all three proteins the folding barrier is early in time, sizable in energy, and is of the same magnitude (approximately 6.5 kcal mol(-1)). These results raise some challenges to the "new view" of protein folding. An early transition state, the search for which consumes most of the observed folding time, is suggested.

Imprint of evolution on protein structures

We attempt to understand the evolutionary origin of protein folds by simulating their divergent evolution with a three-dimensional lattice model. Starting from an initial seed lattice structure, evolution of model proteins progresses by sequence duplication and subsequent point mutations. A new gene's ability to fold into a stable and unique structure is tested each time through direct kinetic folding simulations. Where possible, the algorithm accepts the new sequence and structure and thus a "new protein structure" is born. During the course of each run, this model evolutionary algorithm provides several thousand new proteins with diverse structures. Analysis of evolved structures shows that later evolved structures are more designable than seed structures as judged by recently developed structural determinant of protein designability, as well as direct estimate of designability for selected structures by thermodynamic sampling of their sequence space. We test the significance of this trend predicted on lattice models on real proteins and show that protein domains that are found in eukaryotic organisms only feature statistically significant higher designability than their prokaryotic counterparts. These results present a fundamental view on protein evolution highlighting the relative roles of structural selection and evolutionary dynamics on genesis of modern proteins.

Discerning the Structure and Energy of Multiple Transition States in Protein Folding using ψ-Analysis

We quantify the degree to which folding occurs along a complex landscape with structurally distinct pathways using psi-analysis in combination with a protein engineering method that identifies native, non-covalent polypeptide interactions and their relative populations at the rate-limiting step. By probing the proximity of two specific partners, this method is extremely well-suited for comparison to theoretical simulations. Using ubiquitin as a model system, we detect individual pathways with site-resolved resolution, demonstrating that the protein folds through a native-like transition state ensemble with a common nucleus that contains heterogeneous features on its periphery. The consensus transition state topology has part of the major helix docked against four properly aligned beta-strands. However, structural heterogeneity exists in the transition state ensemble, wherein peripheral regions are differentially populated according to their relative stability. Pathway diversity reflects the variable order of formation of these peripheral elements, which radiate outward from the common nucleus. These results, which show only moderate agreement with traditional mutational phi-analysis, provide an extraordinarily detailed and quantitative description of protein folding.

Squeezed exponential kinetics to describe a nonglassy downhill folding as observed in a lattice protein model

We previously studied the so-called strange kinetics in the two-dimensional lattice HP model. To further study the strange kinetics, folding processes of a 27-mer cubic lattice protein model with Gō potential were investigated by simulating how the bundle of folding trajectories, consisting of a number of independent Monte Carlo simulations, evolves as the folding reaction proceeds, covering a wide range of temperature. Three realms of folding kinetics were observed depending on temperature. Although at temperatures where folding was two-state-like, the kinetics was conventional single exponential, we found that the time course data were well represented by a squeezed (or "shrunken") exponential function, exp [-(t/tau)beta] with beta > 1, at temperatures lower than the folding temperature, where folding was fastest and of a nonglassy downhill type. The squeezed exponential kinetics was found to pertain to the subdiffusion on the nonglassy downhill free energy surface and presents a marked contrast both to the single exponential kinetics and to the stretched exponential kinetics that was observed at lower temperatures where folding was also downhill but topological frustration came into effect. The observed temperature dependence of the folding kinetics suggests that some small single-domain proteins may follow the squeezed exponential kinetics at about the room temperature.

Model study of protein unfolding by interfaces

We study interface-induced protein unfolding on hydrophobic and polar interfaces by means of a two-dimensional lattice model and an exhaustive enumeration ground-state structure search, for a set of model proteins of length 20 residues. We compare the effects of the two types of interfaces, and search for criteria that influence the retention of a protein's native-state structure upon adsorption. We find that the unfolding proceeds by a large, sudden loss of native contacts. The unfolding at polar interfaces exhibits similar behavior to that at hydrophobic interfaces but with a much weaker interface coupling strength. Further, we find that the resistance of proteins to unfolding in our model is positively correlated with the magnitude of the folding energy in the native-state structure, the thermal stability (or energy gap) for that structure, and the interface energy for native-state adsorption. We find these factors to be of roughly equal importance.

Kinetics of the coil-to-helix transition on a rough energy landscape

The kinetics of folding of a fully atomic seven-residue polyalanine peptide in an implicit solvent are studied using molecular dynamics simulations. The use of an implicit solvent is found to dramatically increase the frustration of the energy landscape relative to simulations performed in an explicit solvent [Phys. Rev. Lett. 85, 2637 (2000)]. While the native state in both implicit and explicit solvent simulations is an alpha-helix, the kinetics of the coil-to-helix transition differ significantly. In contrast to the explicit solvent simulations, the native state in the implicit solvent simulations is not kinetically accessible at temperatures where it is thermodynamically stable and could not be brought into equilibrium with other conformational states. At temperatures where statistical equilibrium was achieved, the conformational diffusion folding mechanism, found earlier to be adequate for this peptide in an explicit solvent [Phys. Rev. Lett. 85, 2637 (2000)], is met with only limited success. Issues relating to the evaluation of the quality of implicit solvent models on the basis of thermodynamic criteria only are reexamined.

Fast protein folding kinetics

Proteins are complex molecules, yet their folding kinetics is often fast (microseconds) and simple, involving only a single exponential function of time (called two-state kinetics). The main model for two-state kinetics has been transition-state theory, where an energy barrier defines a slow step to reach an improbable structure. But how can barriers explain fast processes, such as folding? We study a simple model with rigorous kinetics that explains the high speed instead as a result of the microscopic parallelization of folding trajectories. The single exponential results from a separation of timescales; the parallelization of routes is high at the start of folding and low thereafter. The ensemble of rate-limiting chain conformations is different from in transition-state theory; it is broad, overlaps with the denatured state, is not aligned along a single reaction coordinate, and involves well populated, rather than improbable, structures.

Two-state folding over a weak free-energy barrier

We present a Monte Carlo study of a model protein with 54 amino acids that folds directly to its native three-helix-bundle state without forming any well-defined intermediate state. The free-energy barrier separating the native and unfolded states of this protein is found to be weak, even at the folding temperature. Nevertheless, we find that melting curves to a good approximation can be described in terms of a simple two-state system, and that the relaxation behavior is close to single exponential. The motion along individual reaction coordinates is roughly diffusive on timescales beyond the reconfiguration time for a single helix. A simple estimate based on diffusion in a square-well potential predicts the relaxation time within a factor of two.

Conformational rigidity in a lattice model of proteins

It is shown in this paper that some simulations of protein folding in lattice models, which use an incorrect implementation of the Monte Carlo algorithm, do not converge towards thermal equilibrium. I developed a rigorous treatment for protein folding simulation on a lattice model relying on the introduction of a parameter standing for the rigidity of the conformations. Its properties are discussed and its role during the folding process is elucidated. The calculation of thermal properties of small chains living on a two-dimensional lattice is performed and a Bortz-Kalos-Lebowitz scheme is implemented in the presented method in order to study kinetics of chains at very low temperature. The coefficients of the Arrhenius law obtained with this algorithm are found to be in excellent agreement with the value of the main potential barrier of the system. Finally, a scenario of the mechanisms, including the rigidity parameters, that guide a protein towards its native structure, at medium temperature, is given.

Analysis of the differences in the folding kinetics of structurally homologous proteins based on predictions of the gross features of residue contacts

It is a general notion that proteins with very similar three-dimensional structures would show very similar folding kinetics. However, recent studies reveal that the folding kinetic properties of some proteins contradict this thought (i.e., the members in a same protein family fold through different pathways). For example, it has been reported that some beta-proteins in the intracellular lipid-binding protein family fold through quite different pathways (Burns et al., Proteins 1998;33:107-118). Similar differences in folding kinetics are also observed in the members of the globin family (Nishimura et al., Nat Struct Biol 2000;7:679-686). In our study, we examine the possibility of predicting qualitative differences in folding kinetics of the intracellular lipid-binding proteins and two globin proteins (i.e., myoglobin and leghemoglobin). The problem is tackled by means of a contact map based on the average distance statistics between residues, the Average Distance Map (ADM), as constructed from sequence. The ADMs for the three proteins show overall similarity, but some local differences among maps are also observed. Our results demonstrate that some properties of the protein folding kinetics are consistent with local differences in the ADMs. We also discuss the general possibility of predicting folding kinetics from sequence information.

An atomically detailed study of the folding pathways of protein A with the stochastic difference equation

An algorithm is applied here to compute folding pathways of staphylococcal protein A, fragment B. Emphasis is on studies of the complete process, starting from an ensemble of fully denatured conformations and ending at the folded state. The stochastic difference equation algorithm is based on optimization of an action that makes it possible to use a large integration step. Motions with typical displacements that change rapidly on the size scale of the step are filtered out, providing numerically stable and approximate solutions. The present approach is unique in maintaining an atomically detailed picture while providing a systematic, controlled approximation to the classical equations of motion. Analysis of 130 trajectories suggests the following folding mechanism for protein A: At an early precollapse phase of the process, a few native hydrogen bonds form near the C terminus of the protein. The hydrogen bonds are formed mostly within the third helix. The next step is chain collapse that occurs in parallel to additional growth of secondary structure seeds. Therefore, the present study does not support a pure hydrophobic collapse, or substantial early formation of secondary structure. At the last step, native tertiary contacts are formed at the same time as the completion of the secondary structure elements. To a large extent, the process is parallel and not sequential. The early formation of the third helix of protein A, fragment B (in the calculation), is consistent with experimental data.

Fast-folding protein kinetics, hidden intermediates, and the sequential stabilization model

Do two-state proteins fold by pathways or funnels? Native-state hydrogen exchange experiments show discrete nonnative structures in equilibrium with the native state. These could be called hidden intermediates (HI) because their populations are small at equilibrium, and they are not detected in kinetic experiments. HIs have been invoked as disproof of funnel models, because funnel pictures appear to indicate (1) no specific sequences of events in folding; (2) a continuum, rather than a discrete ladder, of structures; and (3) smooth landscapes. In the present study, we solve the exact dynamics of a simple model. We find, instead, that the present microscopic model is indeed consistent with HIs and transition states, but such states occur in parallel, rather than along the single pathway predicted by the sequential stabilization model. At the microscopic level, we observe a huge multiplicity of trajectories. But at the macroscopic level, we observe two pathways of specific sequences of events that are relatively traditional except that they are in parallel, so there is not a single reaction coordinate. Using singular value decomposition, we show an accurate representation of the shapes of the model energy landscapes. They are highly complex funnels.

Diffusive dynamics of protein folding studied by molecular dynamics simulations of an off-lattice model

We report the results of a molecular dynamics study on the kinetic properties of a small off-lattice model of proteins. The model consists of a linear chain of monomers interacting via a number of potentials. These include hydrophobic, bond-angle, and torsion potentials. The ground-state conformation of the studied model is a beta-sheet motif. Molecular dynamics simulations focused on the time evolution of the reaction coordinate measuring the similarity of a given conformation with the native state. Folding time for the studied model is calculated following the diffusive-rate formula of Bryngelson and Wolynes [J. Phys. Chem. 93, 6902 (1989)] by using a computed separately configurational diffusion coefficient. Comparison of the folding time with the mean-first passage time obtained directly from folding simulations shows that the approximation depicting the dynamics of the reaction coordinate in protein folding as a diffusive motion on a free-energy landscape is quantitatively correct for the studied model.