Predicting repeat protein folding kinetics from an experimentally determined folding energy landscape

The Notch ankyrin domain is a repeat protein whose folding has been characterized through equilibrium and kinetic measurements. In previous work, equilibrium folding free energies of truncated constructs were used to generate an experimentally determined folding energy landscape (Mello and Barrick, Proc Natl Acad Sci USA 2004;101:14102–14107). Here, this folding energy landscape is used to parameterize a kinetic model in which local transition probabilities between partly folded states are based on energy values from the landscape. The landscape‐based model correctly predicts highly diverse experimentally determined folding kinetics of the Notch ankyrin domain and sequence variants. These predictions include monophasic folding and biphasic unfolding, curvature in the unfolding limb of the chevron plot, population of a transient unfolding intermediate, relative folding rates of 19 variants spanning three orders of magnitude, and a change in the folding pathway that results from C‐terminal stabilization. These findings indicate that the folding pathway(s) of the Notch ankyrin domain are thermodynamically selected: the primary determinants of kinetic behavior can be simply deduced from the local stability of individual repeats.     

The role of protein homochirality in shaping the energy landscape of folding

The homochirality, or isotacticity, of the natural amino acids facilitates the formation of regular secondary structures such as alpha-helices and beta-sheets. However, many examples exist in nature where novel polypeptide topologies use both l- and d-amino acids. In this study, we explore how stereochemistry of the polypeptide backbone influences basic properties such as compactness and the size of fold space by simulating both lattice and all-atom polypeptide chains. We formulate a rectangular lattice chain model in both two and three dimensions, where monomers are chiral, having the effect of restricting local conformation. Syndiotactic chains with alternating chirality of adjacent monomers have a very large ensemble of accessible conformations characterized predominantly by extended structures. Isotactic chains on the other hand, have far fewer possible conformations and a significant fraction of these are compact. Syndiotactic chains are often unable to access maximally compact states available to their isotactic counterparts of the same length. Similar features are observed in all-atom models of isotactic versus syndiotactic polyalanine. Our results suggest that protein isotacticity has evolved to increase the enthalpy of chain collapse by facilitating compact helical states and to reduce the entropic cost of folding by restricting the size of the unfolded ensemble of competing states.     

Effects of confinement in chaperonin assisted protein folding: rate enhancement by decreasing the roughness of the folding energy landscape

Chaperonins, such as the GroE complex of the bacteria Escherichia coli, assist the folding of proteins under non-permissive folding conditions by providing a cavity in which the newly translated or translocated protein can be encapsulated. Whether the chaperonin cage plays a passive role in protecting the protein from aggregation, or an active role in accelerating folding rates, remains a matter of debate. Here, we investigate the role of confinement in chaperonin mediated folding through molecular dynamics simulations. We designed a substrate protein with an alpha/beta sandwich fold, a common structural motif found in GroE substrate proteins and confined it to a spherical hydrophilic cage which mimicked the interior of the GroEL/ES cavity. The thermodynamics and kinetics of folding were studied over a wide range of temperature and cage radii. Confinement was seen to significantly raise the collapse temperature, T(c), as a result of the associated entropy loss of the unfolded state. The folding temperature, T(f), on the other hand, remained unaffected by encapsulation, a consequence of the folding mechanism of this protein that involves an initial collapse to a compact misfolded state prior to rearranging to the native state. Folding rates were observed to be either accelerated or retarded compared to bulk folding rates, depending on the temperature of the simulation. Rate enhancements due to confinement were observed only at temperatures above the temperature T(m), which corresponds to the temperature at which the protein folds fastest. For this protein, T(m) lies above the folding temperature, T(f), implying that encapsulation alone will not lead to a rate enhancement under conditions where the native state is stable (T相似文献   

Protein folding in the landscape perspective: Chevron plots and non-arrhenius kinetics

We use two simple models and the energy landscape perspective to study protein folding kinetics. A major challenge has been to use the landscape perspective to interpret experimental data, which requires ensemble averaging over the microscopic trajectories usually observed in such models. Here, because of the simplicity of the model, this can be achieved. The kinetics of protein folding falls into two classes: multiple-exponential and two-state (single-exponential) kinetics. Experiments show that two-state relaxation times have “chevron plot” dependences on denaturant and non-Arrhenius dependences on temperature. We find that HP and HP+ models can account for these behaviors. The HP model often gives bumpy landscapes with many kinetic traps and multiple-exponental behavior, whereas the HP+ model gives more smooth funnels and two-state behavior. Multiple-exponential kinetics often involves fast collapse into kinetic traps and slower barrier climbing out of the traps. Two-state kinetics often involves entropic barriers where conformational searching limits the folding speed. Transition states and activation barriers need not define a single conformation; they can involve a broad ensemble of the conformations searched on the way to the native state. We find that unfolding is not always a direct reversal of the folding process. Proteins 30:2–33, 1998. © 1998 Wiley-Liss, Inc.     

Statistical mechanics of protein folding by exhaustive enumeration

It is hard to construct theories for the folding of globular proteins because they are large and complicated molecules having enormous numbers of nonnative conformations and having native states that are complicated to describe. Statistical mechanical theories of protein folding are constructed around major simplifying assumptions about the energy as a function of conformation and/or simplifications of the representation of the polypeptide chain, such as one point per residue on a cubic lattice. It is not clear how the results of these theories are affected by their various simplifications. Here we take a very different simplification approach where the chain is accurately represented and the energy of each conformation is calculated by a not unreasonable empirical function. However, the set of amino acid sequences and allowed conformations is so restricted that it becomes computationally feasible to examine them all. Hence we are able to calculate melting curves for thermal denaturation as well as the detailed kinetic pathway of refolding. Such calculations are based on a novel representation of the conformations as points in an abstract 12-dimensional Euclidean conformation space. Fast folding sequences have relatively high melting temperatures, native structures with relatively low energies, small kinetic barriers between local minima, and relatively many conformations in the global energy minimum's watershed. In contrast to other folding theories, these models show no necessary relationship between fast folding and an overall funnel shape to the energy surface, or a large energy gap between the native and the lowest nonnative structure, or the depth of the native energy minimum compared to the roughness of the energy landscape. Proteins 32:425–437, 1998. © 1998 Wiley-Liss, Inc.     

The energy landscape of modular repeat proteins: topology determines folding mechanism in the ankyrin family

Proteins consisting of repeating amino acid motifs are abundant in all kingdoms of life, especially in higher eukaryotes. Repeat-containing proteins self-organize into elongated non-globular structures. Do the same general underlying principles that dictate the folding of globular domains apply also to these extended topologies? Using a simplified structure-based model capturing a perfectly funneled energy landscape, we surveyed the predicted mechanism of folding for ankyrin repeat containing proteins. The ankyrin family is one of the most extensively studied classes of non-globular folds. The model based only on native contacts reproduces most of the experimental observations on the folding of these proteins, including a folding mechanism that is reminiscent of a nucleation propagation growth. The confluence of simulation and experimental results suggests that the folding of non-globular proteins is accurately described by a funneled energy landscape, in which topology plays a determinant role in the folding mechanism.     

Free energy landscape of protein folding in water: explicit vs. implicit solvent

The Generalized Born (GB) continuum solvent model is arguably the most widely used implicit solvent model in protein folding and protein structure prediction simulations; however, it still remains an open question on how well the model behaves in these large-scale simulations. The current study uses the beta-hairpin from C-terminus of protein G as an example to explore the folding free energy landscape with various GB models, and the results are compared to the explicit solvent simulations and experiments. All free energy landscapes are obtained from extensive conformation space sampling with a highly parallel replica exchange method. Because solvation model parameters are strongly coupled with force fields, five different force field/solvation model combinations are examined and compared in this study, namely the explicit solvent model: OPLSAA/SPC model, and the implicit solvent models: OPLSAA/SGB (Surface GB), AMBER94/GBSA (GB with Solvent Accessible Surface Area), AMBER96/GBSA, and AMBER99/GBSA. Surprisingly, we find that the free energy landscapes from implicit solvent models are quite different from that of the explicit solvent model. Except for AMBER96/GBSA, all other implicit solvent models find the lowest free energy state not the native state. All implicit solvent models show erroneous salt-bridge effects between charged residues, particularly in OPLSAA/SGB model, where the overly strong salt-bridge effect results in an overweighting of a non-native structure with one hydrophobic residue F52 expelled from the hydrophobic core in order to make better salt bridges. On the other hand, both AMBER94/GBSA and AMBER99/GBSA models turn the beta-hairpin in to an alpha-helix, and the alpha-helical content is much higher than the previously reported alpha-helices in an explicit solvent simulation with AMBER94 (AMBER94/TIP3P). Only AMBER96/GBSA shows a reasonable free energy landscape with the lowest free energy structure the native one despite an erroneous salt-bridge between D47 and K50. Detailed results on free energy contour maps, lowest free energy structures, distribution of native contacts, alpha-helical content during the folding process, NOE comparison with NMR, and temperature dependences are reported and discussed for all five models.     

High concentrations of viscogens decrease the protein folding rate constant by prematurely collapsing the coil

In several studies, viscogenic osmolytes have been suggested to decrease the folding rate constant of polypeptides by slowing their motion through the solvent. Here, we show that osmolytes may slow protein folding by prematurely collapsing the coil. At low or moderate concentrations of osmolytes (<30%), folding of the two-state protein CI2 becomes faster with increasing osmolyte concentrations, suggesting that the kinetics are governed by protein stability. However, at higher concentrations of osmolyte, the coil collapses in the dead-time of the refolding experiment, causing a dramatic drop in the folding rate. The collapsed state is non-native and appears to be different for different osmolytes.     

Local interactions and the optimization of protein folding

The role of local interactions in protein folding has recently been the subject of some controversy. Here we investigate an extension of Zwanzig's simple and general model of folding in which local and nonlocal interactions are represented by functions of single and multiple conformational degrees of freedom, respectively. The kinetics and thermodynamics of folding are studied for a series of energy functions in which the energy of the native structure is fixed, but the relative contributions of local and nonlocal interactions to this energy are varied over a broad range. For funnel shaped energy landscapes, we find that 1) the rate of folding increases, but the stability of the folded state decreases, as the contribution of local interactions to the energy of the native structure increases, and 2) the amount of native structure in the unfolded state and the transition state vary considerably with the local interaction strength. Simple exponential kinetics and a well-defined free energy barrier separating folded and unfolded states are observed when nonlocal interactions make an appreciable contribution to the energy of the native structure; in such cases a transition state theory type approximation yields reasonably accurate estimates of the folding rate. Bumps in the folding funnel near the native state, which could result from desolvation effects, side chain freezing, or the breaking of nonnative contacts, significantly alter the dependence of the folding rate on the local interaction strength: the rate of folding decreases when the local interaction strength is increased beyond a certain point. A survey of the distribution of strong contacts in the protein structure database suggests that evolutionary optimization has involved both kinetics and thermodynamics: strong contacts are enriched at both very short and very long sequence separations. Proteins 29:282–291, 1997. © 1997 Wiley-Liss, Inc.     

Molecular basis of cooperativity in protein folding IV. Core: A general cooperative folding model

The cooperative nature of the protein folding process is independent of the characteristic fold and the specific secondary structure attributes of a globular protein. A general folding/unfolding model should, therefore, be based upon structural features that transcend the peculiarities of α-helices, β-sheets, and other structural motifs found in proteins. The studies presented in this paper suggest that a single structural characteristic common to all globular proteins is essential for cooperative folding. The formation of a partly folded state from the native state results in the exposure to solvent of two distinct regions: (1) the portions of the protein that are unfolded; and (2) the “complementary surfaces,” located in the regions of the protein that remain folded. The cooperative character of the folding/unfolding transition is determined largely by the energetics of exposing complementary surface regions to the solvent. By definition, complementary regions are present only in partly folded states; they are absent from the native and unfolded states. An unfavorable free energy lowers the probability of partly folded states and increases the cooperativity of the transition. In this paper we present a mathematical formulation of this behavior and develop a general cooperative folding/unfolding model, termed the “complementary region” (CORE) model. This model successfully reproduces the main properties of folding/unfolding transitions without limiting the number of partly folded states accessible to the protein, thereby permitting a systematic examination of the structural and solvent conditions under which intermediates become populated. It is shown that the CORE model predicts two-state folding/unfolding behavior, even though the two-state character is not assumed in the model. © 1993 Wiley-Liss, Inc.     

A survey of flexible protein binding mechanisms and their transition states using native topology based energy landscapes

Many cellular functions rely on interactions between protein pairs and higher oligomers. We have recently shown that binding mechanisms are robust and owing to the minimal frustration principle, just as for protein folding, are governed primarily by the protein's native topology, which is characterized by the network of non-covalent residue-residue interactions. The detailed binding mechanisms of nine dimers, a trimer, and a tetramer, each involving different degrees of flexibility and plasticity during assembly, are surveyed here using a model that is based solely on the protein topology, having a perfectly funneled energy landscape. The importance of flexibility in binding reactions is manifested by the fly-casting effect, which is diminished in magnitude when protein flexibility is removed. Many of the grosser and finer structural aspects of the various binding mechanisms (including binding of pre-folded monomers, binding of intrinsically unfolded monomers, and binding by domain-swapping) predicted by the native topology based landscape model are consistent with the mechanisms found in the laboratory. An asymmetric binding mechanism is often observed for the formation of the symmetric homodimers where one monomer is more structured at the binding transition state and serves as a template for the folding of the other monomer. Phi values were calculated to show how the structure of the binding transition state ensemble would be manifested in protein engineering studies. For most systems, the simulated Phi values are reasonably correlated with the available experimental values. This agreement suggests that the overall binding mechanism and the nature of the binding transition state ensemble can be understood from the network of interactions that stabilize the native fold. The Phi values for the formation of an antibody-antigen complex indicate a possible role for solvation of the interface in biomolecular association of large rigid proteins.     

Probing the energy landscape of protein folding/unfolding transition states

Previous molecular dynamics (MD) simulations of the thermal denaturation of chymotrypsin inhibitor 2 (CI2) have provided atomic-resolution models of the transition state ensemble that is well supported by experimental studies. Here, we use simulations to further investigate the energy landscape around the transition state region. Nine structures within approximately 35 ps and 3 A C(alpha) RMSD of the transition state ensemble identified in a previous 498 K thermal denaturation simulation were quenched under the quasi-native conditions of 335 K and neutral pH. All of the structures underwent hydrophobically driven collapse in response to the drop in temperature. Structures less denatured than the transition state became structurally more native-like, while structures that were more denatured than the transition state tended to show additional loss of native structure. The structures in the immediate region of the transition state fluctuated between becoming more and less native-like. All of the starting structures had the same native-like topology and were quite similar (within 3.5 A C(alpha) RMSD). That the structures all shared native-like topology, yet diverged into either more or less native-like structures depending on which side of the transition state they occupied on the unfolding trajectory, indicates that topology alone does not dictate protein folding. Instead, our results suggest that a detailed interplay of packing interactions and interactions with water determine whether a partially denatured protein will become more native-like under refolding conditions.     

Folding-unfolding energy change of a simple sphere model protein and an energy landscape of the folding process

We have calculated the free energy of a spherical model of a protein or part of a protein generated in the way of protein folding. Two spherical models are examined; one is a homogeneous model consisting of only one residue type—hydrophobic. The other is a heterogeneous model consisting of two residue types—strong hydrophobic and weak hydrophobic. Both models show a folding transition state, and the latter model reproduces the trend of the experimental folded-unfolded energy change. The heterogeneous model suggests that in the folding process of a protein of more than 70 residues, a specific region of the protein folds first to form a stable region, then the other residues follow the folding process. The energy landscape of folding of a small protein is approximately a funnel model, whereas a flatter energy landscape is suggested for larger proteins of more than 55–70 residues. Proteins 33:408–416, 1998. © 1998 Wiley-Liss, Inc.     

Folding of a small helical protein using hydrogen bonds and hydrophobicity forces

A reduced protein model with five to six atoms per amino acid and five amino acid types is developed and tested on a three-helix-bundle protein, a 46-amino acid fragment from staphylococcal protein A. The model does not rely on the widely used Go approximation, which ignores non-native interactions. We find that the collapse transition is considerably more abrupt for the protein A sequence than for random sequences with the same composition. The chain collapse is found to be at least as fast as helix formation. Energy minimization restricted to the thermodynamically favored topology gives a structure that has a root-mean-square deviation of 1.8 A from the native structure. The sequence-dependent part of our potential is pairwise additive. Our calculations suggest that fine-tuning this potential by parameter optimization is of limited use.     

Hierarchical protein folding pathways: a computational study of protein fragments

We have previously presented a building block folding model. The model postulates that protein folding is a hierarchical top-down process. The basic unit from which a fold is constructed, referred to as a hydrophobic folding unit, is the outcome of combinatorial assembly of a set of "building blocks." Results obtained by the computational cutting procedure yield fragments that are in agreement with those obtained experimentally by limited proteolysis. Here we show that as expected, proteins from the same family give very similar building blocks. However, different proteins can also give building blocks that are similar in structure. In such cases the building blocks differ in sequence, stability, contacts with other building blocks, and in their 3D locations in the protein structure. This result, which we have repeatedly observed in many cases, leads us to conclude that while a building block is influenced by its environment, nevertheless, it can be viewed as a stand-alone unit. For small-sized building blocks existing in multiple conformations, interactions with sister building blocks in the protein will increase the population time of the native conformer. With this conclusion in hand, it is possible to develop an algorithm that predicts the building block assignment of a protein sequence whose structure is unknown. Toward this goal, we have created sequentially nonredundant databases of building block sequences. A protein sequence can be aligned against these, in order to be matched to a set of potential building blocks.     

A coarse-grained protein force field for folding and structure prediction

We have revisited the protein coarse-grained optimized potential for efficient structure prediction (OPEP). The training and validation sets consist of 13 and 16 protein targets. Because optimization depends on details of how the ensemble of decoys is sampled, trial conformations are generated by molecular dynamics, threading, greedy, and Monte Carlo simulations, or taken from publicly available databases. The OPEP parameters are varied by a genetic algorithm using a scoring function which requires that the native structure has the lowest energy, and the native-like structures have energy higher than the native structure but lower than the remote conformations. Overall, we find that OPEP correctly identifies 24 native or native-like states for 29 targets and has very similar capability to the all-atom discrete optimized protein energy model (DOPE), found recently to outperform five currently used energy models.     

Entropic barriers, transition states, funnels, and exponential protein folding kinetics: a simple model

This paper presents an analytically tractable model that captures the most elementary aspect of the protein folding problem, namely that both the energy and the entropy decrease as a protein folds. In this model, the system diffuses within a sphere in the presence of an attractive spherically symmetric potential. The native state is represented by a small sphere in the center, and the remaining space is identified with unfolded states. The folding temperature, the time-dependence of the populations, and the relaxation rate are calculated, and the folding dynamics is analyzed for both golf-course and funnel-like energy landscapes. This simple model allows us to illustrate a surprising number of concepts including entropic barriers, transition states, funnels, and the origin of single exponential relaxation kinetics.     

ALASKA: A Mathematica package for two-state kinetic analysis of protein folding reactions

A Mathematica package (ALASKA) has been developed to simplify the measurement of protein folding kinetics by analysis of ¹H NMR lineshape analysis. This package reads NMR data in ASCII format and can simulate an aromatic ¹ NMR spectrum with or without lineshape broadening from chemical exchange. We describe the analysis of a urea denaturation series of a fast-folding protein, the G46A/G48A variant of monomeric repressor.