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.     

Comparing the folding and misfolding energy landscapes of phosphoglycerate kinase

Partitioning of polypeptides between protein folding and amyloid formation is of outstanding pathophysiological importance. Using yeast phosphoglycerate kinase as model, here we identify the features of the energy landscape that decide the fate of the protein: folding or amyloidogenesis. Structure formation was initiated from the acid-unfolded state, and monitored by fluorescence from 10 ms to 20 days. Solvent conditions were gradually shifted between folding and amyloidogenesis, and the properties of the energy landscape governing structure formation were reconstructed. A gradual transition of the energy landscape between folding and amyloid formation was observed. In the early steps of both folding and misfolding, the protein searches through a hierarchically structured energy landscape to form a molten globule in a few seconds. Depending on the conditions, this intermediate either folds to the native state in a few minutes, or forms amyloid fibers in several days. As conditions are changed from folding to misfolding, the barrier separating the molten globule and native states increases, although the barrier to the amyloid does not change. In the meantime, the native state also becomes more unstable and the amyloid more stable. We conclude that the lower region of the energy landscape determines the final protein structure.     

Mapping folding energy landscapes with theory and experiment

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 and all-atom simulations provide a way to obtain accurate thermodynamic and kinetic measurements, but they are oftentimes restricted to probe limited regions of a protein landscape. Although simplified protein models can access larger regions of the landscape, they are built on assumptions and approximations that can affect the accuracy of the results. We review here recent promising approaches that allow to combine the complementary strengths of theory and experiment for a more complete characterization of a protein folding landscape at multiple resolutions. Recent results and possible applications are discussed.     

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.     

The folding funnel landscape for the peptide Met-enkephalin

We study the free energy landscape of the small peptide Met-enkephalin. Our data were obtained from a generalized-ensemble Monte Carlo simulation taking the interactions among all atoms into account. We show that the free energy landscape resembles that of a funnel, indicating that this peptide is a good folder. Our work demonstrates that the energy landscape picture and folding concept, developed in the context of simplified protein models, can also be used to describe the folding in more realistic models.     

Evolutionary computer programming of protein folding and structure predictions

In order to understand the mechanism of protein folding and to assist the rational de-novo design of fast-folding, non-aggregating and stable artificial enzymes it is very helpful to be able to simulate protein folding reactions and to predict the structures of proteins and other biomacromolecules. Here, we use a method of computer programming called "evolutionary computer programming" in which a program evolves depending on the evolutionary pressure exerted on the program. In the case of the presented application of this method on a computer program for folding simulations, the evolutionary pressure exerted was towards faster finding deep minima in the energy landscape of protein folding. Already after 20 evolution steps, the evolved program was able to find deep minima in the energy landscape more than 10 times faster than the original program prior to the evolution process.     

A systematic study of the vibrational free energies of polypeptides in folded and random states

Molecular vibrations, especially low frequency motions, may be used as an indication of the rigidity or the flatness of the protein folding energy landscape. We have studied the vibrational properties of native folded as well as random coil structures of more than 60 polypeptides. The picture we obtain allows us to perceive how and why the energy landscape progressively rigidifies while still allowing potential flexibility. Compared with random coil structures, both alpha-helices and beta-hairpins are vibrationally more flexible. The vibrational properties of loop structures are similar to those of the corresponding random coil structures. Inclusion of an alpha-helix tends to rigidify peptides and so-called building blocks of the structure, whereas the addition of a beta-structure has less effect. When small building blocks coalesce to form larger domains, the protein rigidifies. However, some folded native conformations are still found to be vibrationally more flexible than random coil structures, for example, beta(2)-microglobulin and the SH3 domain. Vibrational free energy contributes significantly to the thermodynamics of protein folding and affects the distribution of the conformational substates. We found a weak correlation between the vibrational folding energy and the protein size, consistent with both previous experimental estimates and theoretical partition of the heat capacity change in protein folding.     

Protein energy landscape exploration with structure-based models

Exploring the multi-dimensional energy landscape of a large protein in detail is a computational challenge. Such investigations may include analysis of multiple folding pathways, rate constants for important conformational transitions, locating intermediate states populated during folding, estimating energetic and entropic barriers that separate populated basins, and visualising a high-dimensional surface. The complexity of the landscape can be simplified through coarse-grained structure-based models (SBMs). These widely used coarse-grained representations of proteins provide a minimalist approximation to the free energy landscape, which subsumes the folding behaviour of many single-domain proteins. Here we describe the combination of SBMs with discrete path sampling (DPS), and show how this approach can provide details of the landscape and folding pathways. Combining SBMs and DPS provides an efficient framework for sampling the protein free energy landscape and for calculating various kinetic and thermodynamic quantities.     

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.     

A knowledge-based move set for protein folding

The free energy landscape of protein folding is rugged, occasionally characterized by compact, intermediate states of low free energy. In computational folding, this landscape leads to trapped, compact states with incorrect secondary structure. We devised a residue-specific, protein backbone move set for efficient sampling of protein-like conformations in computational folding simulations. The move set is based on the selection of a small set of backbone dihedral angles, derived from clustering dihedral angles sampled from experimental structures. We show in both simulated annealing and replica exchange Monte Carlo (REMC) simulations that the knowledge-based move set, when compared with a conventional move set, shows statistically significant improved ability at overcoming kinetic barriers, reaching deeper energy minima, and achieving correspondingly lower RMSDs to native structures. The new move set is also more efficient, being able to reach low energy states considerably faster. Use of this move set in determining the energy minimum state and for calculating thermodynamic quantities is discussed.     

Increasing stability reduces conformational heterogeneity in a protein folding intermediate ensemble

A multi-site, time-resolved fluorescence resonance energy transfer methodology has been used to study structural heterogeneity in a late folding intermediate ensemble, IL, of the small protein barstar. Four different intra-molecular distances have been measured within the structural components of IL. The IL ensemble is shown to consist of different sub-populations of molecules, in each of which one or more of the four distances are native-like and the remaining distances are unfolded-like. In very stable conditions that favor formation of IL, all four distances are native-like in most molecules. In less stable conditions, one or more distances are unfolded-like. As stability is decreased, the proportion of molecules with unfolded-like distances increases. Thus, the results show that protein folding intermediates are ensembles of different structural forms, and they demonstrate that conformational entropy increases as structures become less stable. These observations provide direct experimental evidence in support of a basic tenet of energy landscape theory for protein folding, that available conformational space, as represented by structural heterogeneity in IL, becomes restricted as the stability is increased. The results also vindicate an important prediction of energy landscape theory, that different folding pathways may become dominant under different folding conditions. In more stable folding conditions, uniformly native-like compactness is achieved during folding to IL, whereas in less stable conditions, uniformly native-like compactness is achieved only later during the folding of IL to N.     

Hierarchic finite level energy landscape model: to describe the refolding kinetics of phosphoglycerate kinase

One of the most intriguing predictions of energy landscape models is the existence of non-exponential protein folding kinetics caused by hierarchical structures in the landscapes. Here we provide the strongest evidence so far of such hierarchy and determine the time constants and weights of the kinetic components of the suggested hierarchic energy landscape. To our knowledge, the idea of hierarchical folding energy barriers has never been tested over such a broad timescale. Refolding of yeast phosphoglycerate kinase was initiated from the guanidine-unfolded state by stopped-flow or manual mixing and monitored by tryptophan fluorescence from 1 ms to 15 min. The strategy to build a model that describes folding of yeast phosphoglycerate kinase was to start from the simplest paradigm and modify it stepwise to the necessary minimal extent after repeated comparisons with the experiments. We made no a priori assumptions about the folding landscape. The result was a hierarchic finite level landscape model that quantitatively describes the refolding of yeast phosphoglycerate kinase from 1 ms to 15 min. The early steps of the folding process happen in the upper region of the landscape, where the surface has a hierarchic structure. This leads to stretched kinetics in the early phase of the folding. The lower region of the energy landscape is dominated by a trap that reflects the accumulation of molten globule intermediate state. From this intermediate, the protein can reach the global energy minimum corresponding to the native state through a cross-barrier folding step.     

The Dominant Folding Route Minimizes Backbone Distortion in SH3

Energetic frustration in protein folding is minimized by evolution to create a smooth and robust energy landscape. As a result the geometry of the native structure provides key constraints that shape protein folding mechanisms. Chain connectivity in particular has been identified as an essential component for realistic behavior of protein folding models. We study the quantitative balance of energetic and geometrical influences on the folding of SH3 in a structure-based model with minimal energetic frustration. A decomposition of the two-dimensional free energy landscape for the folding reaction into relevant energy and entropy contributions reveals that the entropy of the chain is not responsible for the folding mechanism. Instead the preferred folding route through the transition state arises from a cooperative energetic effect. Off-pathway structures are penalized by excess distortion in local backbone configurations and contact pair distances. This energy cost is a new ingredient in the malleable balance of interactions that controls the choice of routes during protein folding.     

Concerted kinetic folding of a multidomain ribozyme with a disrupted loop-receptor interaction

The free energy landscape for the folding of large, multidomain RNAs is rugged, and kinetically trapped, misfolded intermediates are a hallmark of RNA folding reactions. Here, we examine the role of a native loop-receptor interaction in determining the ruggedness of the energy landscape for folding of the Tetrahymena ribozyme. We demonstrate a progressive smoothing of the energy landscape for ribozyme folding as the strength of the loop-receptor interaction is reduced. Remarkably, with the most severe mutation, global folding is more rapid than for the wild-type ribozyme and proceeds in a concerted fashion without the accumulation of long-lived kinetic intermediates. The results demonstrate that a complex interplay between native tertiary interactions, divalent ion concentration, and non-native secondary structure determines the ruggedness of the energy landscape. Furthermore, the results suggest that kinetic folding transitions involving large regions of highly structured RNAs can proceed in a concerted fashion, in the absence of significant stable, preorganized tertiary structure.     

Conformational heterogeneity of an equilibrium folding intermediate quantified and mapped by scanning mutagenesis

It is challenging to experimentally define an energy landscape for protein folding that comprises multiple partially unfolded states. Experimental results are often ambiguous as to whether a non-native state is conformationally homogeneous. Here, we tested an approach combining systematic mutagenesis and a Br?nsted-like analysis to reveal and quantify conformational heterogeneity of folding intermediate states. Using this method, we resolved an otherwise apparently homogeneous equilibrium folding intermediate of Borrelia burgdorferi OspA into two conformationally distinct species and determined their relative populations. Furthermore, we mapped the structural differences between these intermediate species, which are consistent with the non-native species that we previously proposed based on native-state hydrogen exchange studies. When treated as a single state, the intermediate ensemble exhibited fractional Phi-values for mutations and Hammond-type behaviors that are often observed for folding transition states. We found that a change in relative population of the two species within the intermediate ensemble explains these properties well, suggesting that fractional Phi-values and Hammond-type behaviors exhibited by folding intermediates and transition states may arise more often from conformational heterogeneity than from a single partial structure. Our results are consistent with the presence of multiple minima in a rugged energy landscape predicted from theoretical studies. The method described here provides a promising means to probe a complex folding energy landscape.     

Ruggedness in the Free Energy Landscape Dictates Misfolding of the Prion Protein

Experimental determination of the key features of the free energy landscapes of proteins, which dictate their adeptness to fold correctly, or propensity to misfold and aggregate and which are modulated upon a change from physiological to aggregation-prone conditions, is a difficult challenge. In this study, sub-millisecond kinetic measurements of the folding and unfolding of the mouse prion protein reveal how the free energy landscape becomes more complex upon a shift from physiological (pH 7) to aggregation-prone (pH 4) conditions. Folding and unfolding utilize the same single pathway at pH 7, but at pH 4, folding occurs on a pathway distinct from the unfolding pathway. Moreover, the kinetics of both folding and unfolding at pH 4 depend not only on the final conditions but also on the conditions under which the processes are initiated. Unfolding can be made to switch to occur on the folding pathway by varying the initial conditions. Folding and unfolding pathways appear to occupy different regions of the free energy landscape, which are separated by large free energy barriers that change with a change in the initial conditions. These barriers direct unfolding of the native protein to proceed via an aggregation-prone intermediate previously identified to initiate the misfolding of the mouse prion protein at low pH, thus identifying a plausible mechanism by which the ruggedness of the free energy landscape of a protein may modulate its aggregation propensity.     

Energetics of the native energy landscape of a two-domain calcium sensor protein: distinct folding features of the two domains

The protein folding energy landscape allows a thorough understanding of the protein folding problem which in turn helps in understanding various aspects of biological functions. Characterizing the cooperative unfolding units and the intermediates along the folding funnel of a protein is a challenging task. In this paper, we investigated the native energy landscape of EhCaBP, a calcium sensor, belonging to the same EF-hand superfamily as calmodulin. EhCaBP is a two-domain EF-hand protein consisting of two EF-hands in each domain and binding to four Ca2+ cations. Native-state hydrogen exchange (HX) was used to assess the folding features of the landscape and also to throw light on the structure-folding function paradigm of calcium sensor proteins. HX measurements under the EX2 regime provided the thermodynamic information about the protein folding events under native conditions. HX studies revealed that the unfolding of EhCaBP is not a two-state process. Instead, it proceeds through cooperative units. The C-terminal domain exhibits less denaturant dependence than the N-terminal domain, suggesting that the former is dominated by local fluctuations. It is interesting to note that the N- and C-terminal domains of EhCaBP have distinct folding features. In fact, these observed differences can regulate the domain-dependent target recognition of two-domain Ca2+ sensor proteins.     

Toward minimalist models of larger proteins: a ubiquitin-like protein

Our recently developed off-lattice bead model capable of simulating protein structures with mixed alpha/beta content has been extended to model the folding of a ubiquitin-like protein and provides a means for examining the more complex kinetics involved in the folding of larger proteins. Using trajectories generated from constant-temperature Langevin dynamics simulations and sampling with the multiple multi-histogram method over five-order parameters, we are able to characterize the free energy landscape for folding and find evidence for folding through compact intermediates. Our model reproduces the observation that the C-terminus loop structure in ubiquitin is the last to fold in the folding process and most likely plays a spectator role in the folding kinetics. The possibility of a productive metastable intermediate along the folding pathway consisting of collapsed states with no secondary structure, and of intermediates or transition structures involving secondary structural elements occurring early in the sequence, is also supported by our model. The kinetics of folding remain multi-exponential below the folding temperature, with glass-like kinetics appearing at T/T(f) approximately 0.86. This new physicochemical model, designed to be predictive, helps validate the value of modeling protein folding at this level of detail for genomic-scale studies, and motivates further studies of other protein topologies and the impact of more complex energy functions, such as the addition of solvation forces.