Minimalist models for protein folding and design

Protein folding research during the past decade has emphasized the dominant role of native state topology in determining the speed and mechanism of folding for small proteins; this has been illustrated by simulations using minimalist protein models. The advantages of minimalist protein models lie in their ability to rapidly collect meaningful statistics about folding pathways and kinetics, their ease of characterization with coarse-grained order parameters and their concentration on the essential physics of the problem to connect with experimental observables for a target protein. The maturation of experimental protein folding has driven the need for more quantitative protein simulations to better understand the balance between sequence details and fold topology. In the past year, we have seen the emergence of more complex minimalist models, ranging from all-atom Gō potentials to coarse-grained bead models in which Gō interactions are replaced or supplemented by more physically motivated potentials. The reduced computational cost at the coarse-grained level of abstraction will potentially enable both folding studies on a genomic scale and systematic application in protein design.     

Influence of denatured and intermediate states of folding on protein aggregation

We simulate the aggregation thermodynamics and kinetics of proteins L and G, each of which self-assembles to the same alpha/beta [corrected] topology through distinct folding mechanisms. We find that the aggregation kinetics of both proteins at an experimentally relevant concentration exhibit both fast and slow aggregation pathways, although a greater proportion of protein G aggregation events are slow relative to those of found for protein L. These kinetic differences are correlated with the amount and distribution of intrachain contacts formed in the denatured state ensemble (DSE), or an intermediate state ensemble (ISE) if it exists, as well as the folding timescales of the two proteins. Protein G aggregates more slowly than protein L due to its rapidly formed folding intermediate, which exhibits native intrachain contacts spread across the protein, suggesting that certain early folding intermediates may be selected for by evolution due to their protective role against unwanted aggregation. Protein L shows only localized native structure in the DSE with timescales of folding that are commensurate with the aggregation timescale, leaving it vulnerable to domain swapping or nonnative interactions with other chains that increase the aggregation rate. Folding experiments that characterize the structural signatures of the DSE, ISE, or the transition state ensemble (TSE) under nonaggregating conditions should be able to predict regions where interchain contacts will be made in the aggregate, and to predict slower aggregation rates for proteins with contacts that are dispersed across the fold. Since proteins L and G can both form amyloid fibrils, this work also provides mechanistic and structural insight into the formation of prefibrillar species.     

Analyses of the folding properties of ferredoxin‐like fold proteins by means of a coarse‐grained Gō model: Relationship between the free energy profiles and folding cores

The folding mechanisms of proteins with multi‐state transitions, the role of the intermediate states, and the precise mechanism how each transition occurs are significant on‐going research issues. In this study, we investigate ferredoxin‐like fold proteins which have a simple topology and multi‐state transitions. We analyze the folding processes by means of a coarse‐grained Gō model. We are able to reproduce the differences in the folding mechanisms between U1A, which has a high‐free‐energy intermediate state, and ADA2h and S6, which fold into the native structure through two‐state transitions. The folding pathways of U1A, ADA2h, S6, and the S6 circular permutant, S6_p54‐55, are reproduced and compared with experimental observations. We show that the ferredoxin‐like fold contains two common regions consisting folding cores as predicted in other studies and that U1A produces an intermediate state due to the distinct cooperative folding of each core. However, because one of the cores of S6 loses its cooperativity and the two cores of ADA2h are tightly coupled, these proteins fold into the native structure through a two‐state mechanism. Proteins 2014; 82:954–965. © 2013 Wiley Periodicals, Inc.     

The protein folding 'speed limit'

How fast can a protein possibly fold? This question has stimulated experimentalists to seek fast folding proteins and to engineer them to fold even faster. Proteins folding at or near the speed limit are prime candidates for all-atom molecular dynamics simulations. They may also have no free energy barrier, allowing the direct observation of intermediate structures on the pathways from the unfolded to the folded state. Both experimental and theoretical approaches predict a speed limit of approximately N/100micros for a generic N-residue single-domain protein, with alpha proteins folding faster than beta or alphabeta. The predicted limits suggest that most known ultrafast folding proteins can be engineered to fold more than ten times faster.     

Temperature-dependent folding pathways of Pin1 WW domain: an all-atom molecular dynamics simulation of a Gō model

We study the folding thermodynamics and kinetics of the Pin1 WW domain, a three-stranded beta-sheet protein, by using all-atom (except nonpolar hydrogens) discontinuous molecular dynamics simulations at various temperatures with a Gō model. The protein exhibits a two-state folding kinetics near the folding transition temperature. A good agreement between our simulations and the experimental measurements by the Gruebele group has been found, and the simulation sheds new insights into the structure of transition state, which is hard to be straightforwardly captured in experiments. The simulation also reveals that the folding pathways at approximately the transition temperature and at low temperatures are much different, and an intermediate state at a low temperature is predicted. The transition state of this small beta-protein at its folding transition temperature has a well-established hairpin 1 made of beta1 and beta2 strands while its low-temperature kinetic intermediate has a formed hairpin 2 composed of beta2 and beta3 strands. Theoretical results are compared with other simulation results as well as available experimental data. This study confirms that specific side-chain packing in an all-atom Gō model can yield a reasonable prediction of specific folding kinetics for a given protein. Different folding behaviors at different temperatures are interpreted in terms of the interplay of entropy and enthalpy in folding process.     

Structural analysis of kinetic folding intermediates for a TIM barrel protein, indole-3-glycerol phosphate synthase, by hydrogen exchange mass spectrometry and Gō model simulation

The structures of partially folded states appearing during the folding of a (βα)₈ TIM barrel protein, the indole-3-glycerol phosphate synthase from Sulfolobus solfataricus (sIGPS), was assessed by hydrogen exchange mass spectrometry (HX-MS) and Gō model simulations. HX-MS analysis of the peptic peptides derived from the pulse-labeled product of the sub-millisecond folding reaction from the urea-denatured state revealed strong protection in the (βα)₄ region, modest protection in the neighboring (βα)_1-3 and (βα)₅β₆ segments and no significant protection in the remaining N and C-terminal segments. These results demonstrate that this species is not a collapsed form of the unfolded state under native-favoring conditions nor is it the native state formed via fast-track folding. However, the striking contrast of these results with the strong protection observed in the (βα)_2-5β₆ region after 5 s of folding demonstrates that these species represent kinetically distinct folding intermediates that are not identical as previously thought. A re-examination of the kinetic folding mechanism by chevron analysis of fluorescence data confirmed distinct roles for these two species: the burst-phase intermediate is predicted to be a misfolded, off-pathway intermediate, while the subsequent 5 s intermediate corresponds to an on-pathway equilibrium intermediate. Comparison with the predictions using a C^α Gō model simulation of the kinetic folding reaction for sIGPS shows good agreement with the core of the structure offering protection against exchange in the on-pathway intermediate(s). Because the native-centric Gō model simulations do not explicitly include sequence-specific information, the simulation results support the hypothesis that the topology of TIM barrel proteins is a primary determinant of the folding free energy surface for the productive folding reaction. The early misfolding reaction must involve aspects of non-native structure not detected by the Gō model simulation.     

Relationship between energy distribution and fold stability: Insights from molecular dynamics simulations of native and mutant proteins

Most proteins must fold to a well-defined structure with a minimal stability to perform their function. Here we use a simple, molecular dynamics-based, energy decomposition approach to map the principal energetic interactions in a set of proteins representative of different folds. This work involves the all-atom simulation and analysis of the native structures and mutants of five different proteins representative of an all-alpha (yACPB, Protein A), all-beta (SH3), and a mixed alpha/beta fold (Proteins G and L). Given a certain structure, a native sequence and a set of mutants, we show that our model discriminates the ability of a mutation to yield a more or less stable protein, in agreement with experimental data, catching the principal energetic determinants of protein stabilization. Our approach identifies the interaction determinants responsible to define a fold and shows that mutations can either modulate the strength of pair-wise coupling between residues important for folding, or modify the profile of the principal interactions. Furthermore, we address the question of how to evaluate the fitness of a sequence to a given structure by comparing the information contained in the energy map, which recapitulates the chemistry of the sequence, to that contained in the contact map, which recapitulates the fold topology. The results show that the better fit between the energetic properties of the sequence and the fold topology corresponds to a higher stabilization of the protein. We discuss the relevance of these observations to the analysis of protein designability and to the rational evolution of new sequences.     

The Limited Role of Nonnative Contacts in the Folding Pathways of a Lattice Protein

Models of protein energetics that neglect interactions between amino acids that are not adjacent in the native state, such as the Gō model, encode or underlie many influential ideas on protein folding. Implicit in this simplification is a crucial assumption that has never been critically evaluated in a broad context: Detailed mechanisms of protein folding are not biased by nonnative contacts, typically argued to be a consequence of sequence design and/or topology. Here we present, using computer simulations of a well-studied lattice heteropolymer model, the first systematic test of this oft-assumed correspondence over the statistically significant range of hundreds of thousands of amino acid sequences that fold to the same native structure. Contrary to previous conjectures, we find a multiplicity of folding mechanisms, suggesting that Gō-like models cannot be justified by considerations of topology alone. Instead, we find that the crucial factor in discriminating among topological pathways is the heterogeneity of native contact energies: The order in which native contacts accumulate is profoundly insensitive to omission of nonnative interactions, provided that native contact heterogeneity is retained. This robustness holds over a surprisingly wide range of folding rates for our designed sequences. Mirroring predictions based on the principle of minimum frustration, fast-folding sequences match their Gō-like counterparts in both topological mechanism and transit times. Less optimized sequences dwell much longer in the unfolded state and/or off-pathway intermediates than do Gō-like models. For dynamics that bridge unfolded and unfolded states, however, even slow folders exhibit topological mechanisms and transit times nearly identical with those of their Gō-like counterparts. Our results do not imply a direct correspondence between folding trajectories of Gō-like models and those of real proteins, but they do help to clarify key topological and energetic assumptions that are commonly used to justify such caricatures.     

Ubiquitin folds via a flip-twist-lock mechanism

To perform specific functional activities, the majority of proteins should fold into their distinct three-dimensional conformations. However, the biologically active conformation of a protein is generally found to be marginally stable than the other conformations that the chain can adopt. How a protein finds its native conformation from its post-synthesis unfolded structure in a complex conformational landscape is the unsolved question that still drives the protein folding community. Here, we report the folding mechanism of a globular protein, ubiquitin, from its chemically denatured state using all-atom molecular dynamics simulations. From the kinetic analysis of the simulated trajectories we show that the folding process can be described by the hydrophobic collapse mechanism, initiated by the “dewetting transition”, and subsequently assisted by the origination of an N-terminal folding nucleus, and finally supported by a native salt-bridge interaction between K11 and E34. We show that ubiquitin folds via an intermediate. Finally, we confirm the presence of “biological water” and explain its role to the folding process.     

Folding and unfolding of gammaTIM monomers and dimers

Kinetic simulations of the folding and unfolding of triosephosphate isomerase (TIM) from yeast were conducted using a single monomer gammaTIM polypeptide chain that folds as a monomer and two gammaTIM chains that fold to the native dimer structure. The basic protein model used was a minimalist Gō model using the native structure to determine attractive energies in the protein chain. For each simulation type--monomer unfolding, monomer refolding, dimer unfolding, and dimer refolding--thirty simulations were conducted, successfully capturing each reaction in full. Analysis of the simulations demonstrates four main conclusions. First, all four simulation types have a similar "folding order", i.e., they have similar structures in intermediate stages of folding between the unfolded and folded state. Second, despite this similarity, different intermediate stages are more or less populated in the four different simulations, with 1), no intermediates populated in monomer unfolding; 2), two intermediates populated with beta(2)-beta(4) and beta(1)-beta(5) regions folded in monomer refolding; 3), two intermediates populated with beta(2)-beta(3) and beta(2)-beta(4) regions folded in dimer unfolding; and 4), two intermediates populated with beta(1)-beta(5) and beta(1)-beta(5) + beta(6) + beta(7) + beta(8) regions folded in dimer refolding. Third, simulations demonstrate that dimer binding and unbinding can occur early in the folding process before complete monomer-chain folding. Fourth, excellent agreement is found between the simulations and MPAX (misincorporation proton alkyl exchange) experiments. In total, this agreement demonstrates that the computational Gō model is accurate for gammaTIM and that the energy landscape of gammaTIM appears funneled to the native state.     

Matching simulation and experiment: a new simplified model for simulating protein folding.

Simulations of simplified protein folding models have provided much insight into solving the protein folding problem. We propose here a new off-lattice bead model, capable of simulating several different fold classes of small proteins. We present the sequence for an alpha/beta protein resembling the IgG-binding proteins L and G. The thermodynamics of the folding process for this model are characterized using the multiple multihistogram method combined with constant-temperature Langevin simulations. The folding is shown to be highly cooperative, with chain collapse nearly accompanying folding. Two parallel folding pathways are shown to exist on the folding free energy landscape. One pathway contains an intermediate--similar to experiments on protein G, and one pathway contains no intermediates-similar to experiments on protein L. The folding kinetics are characterized by tabulating mean-first passage times, and we show that the onset of glasslike kinetics occurs at much lower temperatures than the folding temperature. This model is expected to be useful in many future contexts: investigating questions of the role of local versus nonlocal interactions in various fold classes, addressing the effect of sequence mutations affecting secondary structure propensities, and providing a computationally feasible model for studying the role of solvation forces in protein folding.     

Cooperativity,smooth energy landscapes and the origins of topology-dependent protein folding rates

The relative folding rates of simple, single-domain proteins, proteins whose folding energy landscapes are smooth, are highly dispersed and strongly correlated with native-state topology. In contrast, the relative folding rates of small, Gō-potential lattice polymers, which also exhibit smooth energy landscapes, are poorly dispersed and insignificantly correlated with native-state topology. Here, we investigate this discrepancy in light of a recent, quantitative theory of two-state folding kinetics, the topomer search model. This model stipulates that the topology-dependence of two-state folding rates is a direct consequence of the extraordinarily cooperative equilibrium folding of simple proteins. We demonstrate that traditional Gō polymers lack the extreme cooperativity that characterizes the folding of naturally occurring, two-state proteins and confirm that the folding rates of a diverse set of Gō 27-mers are poorly dispersed and effectively uncorrelated with native state topology. Upon modestly increasing the cooperativity of the Gō-potential, however, significantly increased dispersion and strongly topology-dependent kinetics are observed. These results support previous arguments that the cooperative folding of simple, single-domain proteins gives rise to their topology-dependent folding rates. We speculate that this cooperativity, and thus, indirectly, the topology-rate relationship, may have arisen in order to generate the smooth energetic landscapes upon which rapid folding can occur.     

Protein topology affects the appearance of intermediates during the folding of proteins with a flavodoxin-like fold

The topology of a native protein influences the rate with which it is formed, but does topology affect the appearance of folding intermediates and their specific role in kinetic folding as well? This question is addressed by comparing the folding data recently obtained on apoflavodoxin from Azotobacter vinelandii with those available on all three other alpha-beta parallel proteins the kinetic folding mechanism of which has been studied, i.e. Anabaena apoflavodoxin, Fusarium solani pisi cutinase and CheY. Two kinetic folding intermediates, one on-pathway and the other off-pathway, seem to be present during the folding of proteins with an alpha-beta parallel, also called flavodoxin-like, topology. The on-pathway intermediate lies on a direct route from the unfolded to the native state of the protein involved. The off-pathway intermediate needs to unfold to allow the production of native protein. Available simulation data of the folding of CheY show the involvement of two intermediates with characteristics that resemble those of the two intermediates experimentally observed. Apparently, protein topology governs the appearance and kinetic roles of protein folding intermediates during the folding of proteins that have a flavodoxin-like fold.     

Knots in rings. The circular knotted protein Momordica cochinchinensis trypsin inhibitor-II folds via a stable two-disulfide intermediate

The aim of this work was to elucidate the oxidative folding mechanism of the macrocyclic cystine knot protein MCoTI-II. We aimed to investigate how the six-cysteine residues distributed on the circular backbone of the reduced unfolded peptide recognize their correct partner and join up to form a complex cystine-knotted topology. To answer this question, we studied the oxidative folding of the naturally occurring peptide using a range of spectroscopic methods. For both oxidative folding and reductive unfolding, the same disulfide intermediate species was prevalent and was characterized to be a native-like two-disulfide intermediate in which the Cys1-Cys18 disulfide bond was absent. Overall, the folding pathway of this head-to-tail cyclized protein was found to be similar to that of linear cystine knot proteins from the squash family of trypsin inhibitors. However, the pathway differs in an important way from that of the cyclotide kalata B1, in that the equivalent two-disulfide intermediate in that case is not a direct precursor of the native protein. The size of the embedded ring within the cystine knot motif appears to play a crucial role in the folding pathway. Larger rings contribute to the independence of disulfides and favor an on-pathway native-like intermediate that has a smaller energy barrier to cross to form the native fold. The fact that macrocyclic proteins are readily able to fold to a complex knotted structure in vitro in the absence of chaperones makes them suitable as protein engineering scaffolds that have remarkable stability.     

First principles prediction of protein folding rates

Experimental studies have demonstrated that many small, single-domain proteins fold via simple two-state kinetics. We present a first principles approach for predicting these experimentally determined folding rates. Our approach is based on a nucleation-condensation folding mechanism, where the rate-limiting step is a random, diffusive search for the native tertiary topology. To estimate the rates of folding for various proteins via this mechanism, we first determine the probability of randomly sampling a conformation with the native fold topology. Next, we convert these probabilities into folding rates by estimating the rate that a protein samples different topologies during diffusive folding. This topology-sampling rate is calculated using the Einstein diffusion equation in conjunction with an experimentally determined intra-protein diffusion constant. We have applied our prediction method to the 21 topologically distinct small proteins for which two-state rate data is available. For the 18 beta-sheet and mixed alpha-beta native proteins, we predict folding rates within an average factor of 4, even though the experimental rates vary by a factor of approximately 4 x 10(4). Interestingly, the experimental folding rates for the three four-helix bundle proteins are significantly underestimated by this approach, suggesting that proteins with significant helical content may fold by a faster, alternative mechanism. This method can be applied to any protein for which the structure is known and hence can be used to predict the folding rates of many proteins prior to experiment.     

Roles of physical interactions in determining protein-folding mechanisms: molecular simulation of protein G and alpha spectrin SH3

Protein-folding mechanisms of two small globular proteins, IgG binding domain of protein G and alpha spectrin SH3 domain are investigated via Brownian dynamics simulations with a model made of coarse-grained physical energy functions responsible for sequence-specific interactions and weak Gō-like energies. The folding pathways of alpha spectrin SH3 are known to be mainly controlled by the native topology, while protein G folding is anticipated to be more sensitive to the sequence-specific effects than native topology. We found in the folding of protein G that the C terminal beta hairpin is formed earlier and is rigid, once ordered, in the presence of an intact C terminal turn. The alpha helix is found to exhibit repeated partial formations/deformations during folding and to be stabilized via the tertiary contact with preformed beta sheets. This predicted scenario is fully consistent with experimental phi value data. Moreover, we found that the folding route is critically affected when the hydrophobic interaction is excluded from physical energy terms, suggesting that the hydrophobicity critically contributes to the folding propensity of protein G. For the folding of alpha spectrin SH3, we found that the distal beta hairpin and diverging turn are parts formed early, fully in harmony with previous results of simple Gō-like and experimental analysis, supporting that the folding route of SH3 domain is robust and coded by the native topology. The hybrid method provides useful tools for analyzing roles of physical interactions in determining folding mechanisms.     

Accelerating all-atom protein folding simulations through reduced dihedral barriers

Protein folding requires extensive changes of backbone and sidechain dihedral angles, whose energy barriers constitute obstacles for the folding kinetics. Folding of small proteins is furthermore thought to be path-independent. Here, we propose that time-consuming all-atom protein folding simulations may be accelerated through a reduction of the dihedral barriers of the force field. In order to investigate this hypothesis, we performed various folding simulations of two small proteins. We report an acceleration towards smaller root-mean-square deviations from the native protein structure using our proposed method.     

An evolutionary strategy for all-atom folding of the 60-amino-acid bacterial ribosomal protein l20

We have investigated an evolutionary algorithm for de novo all-atom folding of the bacterial ribosomal protein L20. We report results of two simulations that converge to near-native conformations of this 60-amino-acid, four-helix protein. We observe a steady increase of "native content" in both simulated ensembles and a large number of near-native conformations in their final populations. We argue that these structures represent a significant fraction of the low-energy metastable conformations, which characterize the folding funnel of this protein. These data validate our all-atom free-energy force field PFF01 for tertiary structure prediction of a previously inaccessible structural family of proteins. We also compare folding simulations of the evolutionary algorithm with the basin-hopping technique for the Trp-cage protein. We find that the evolutionary algorithm generates a dynamic memory in the simulated population, which leads to faster overall convergence.     

Simulating protein evolution in sequence and structure space

Naturally occurring proteins comprise a special subset of all plausible sequences and structures selected through evolution. Simulating protein evolution with simplified and all-atom models has shed light on the evolutionary dynamics of protein populations, the nature of evolved sequences and structures, and the extent to which today's proteins are shaped by selection pressures on folding, structure and function. Extensive mapping of the native structure, stability and folding rate in sequence space using lattice proteins has revealed organizational principles of the sequence/structure map important for evolutionary dynamics. Evolutionary simulations with lattice proteins have highlighted the importance of fitness landscapes, evolutionary mechanisms, population dynamics and sequence space entropy in shaping the generic properties of proteins. Finally, evolutionary-like simulations with all-atom models, in particular computational protein design, have helped identify the dominant selection pressures on naturally occurring protein sequences and structures.