Exploring folding free energy landscapes using computational protein design

Recent advances in computational protein design have allowed exciting new insights into the sequence dependence of protein folding free energy landscapes. Whereas most previous studies have examined the sequence dependence of protein stability and folding kinetics by characterizing naturally occurring proteins and variants of these proteins that contain a small number of mutations, it is now possible to generate and characterize computationally designed proteins that differ significantly from naturally occurring proteins in sequence and/or structure. These computer-generated proteins provide insights into the determinants of protein structure, stability and folding, and make it possible to disentangle the properties of proteins that are the consequence of natural selection from those that reflect the fundamental physical chemistry of polypeptide chains.     

Osmolyte-induced protein folding free energy changes

Upon addition of protecting osmolyte to an aqueous solution of an intrinsically unstructured protein, spectral observables are often seen to change in a sigmoid fashion as a function of increasing osmolyte concentration. Commonly, such data are analyzed using the linear extrapolation model (LEM), a method that defines a scale from 0%-100% folded species at each osmolyte concentration by means of extending pre- and post-folding baselines into the transition region. Defining the 0%-100% folding scale correctly for each osmolyte is an important part of the analysis, leading to evaluation of the fraction of folded protein existing in the absence of osmolytes. In this study, we used reduced and carboxyamidated RNase T1 (RCAM-T1) as an intrinsically unstructured protein, and determined the thermodynamic stability of RCAM-T1 induced by naturally occurring osmolytes. Because the folded fraction of the protein population determined by experiments of thermal and urea-induced denaturation is nonzero in the absence of osmolytes at 15 degrees C, the commonly used LEM can lead to false values of DeltaG[stackD-->N0] for protein folding due to the arbitrary assumption that the protein is 100% unfolded in the presence of buffer alone. To correct this problem, titration of the protein solution with urea and extrapolating back to zero urea concentration gives the spectral value for 100% denatured protein. With fluorescence as the observable we redefine F/F0 to F/F0extrap = 1.0 and require that the denatured-state baseline have this value as its intercept. By so doing, the 0%-100% scale-corrected DeltaG[D-->N0] values of RCAM-T1 folding in the presence of various osmolytes are then found to be identical, with small error, demonstrating that DeltaG[D-->N0] is independent of the osmolytes used. Such a finding is an important step in validating this quantity derived from the LEM as having the properties expected of an authentic thermodynamic parameter. The rank order of osmolyte efficacies in stabilizing RCAM-T1 is sarcosine > sucrose > sorbitol > proline > betaine > glycerol.     

Simulating RNA folding kinetics on approximated energy landscapes

We present a general computational approach to simulate RNA folding kinetics that can be used to extract population kinetics, folding rates and the formation of particular substructures that might be intermediates in the folding process. Simulating RNA folding kinetics can provide unique insight into RNA whose functions are dictated by folding kinetics and not always by nucleotide sequence or the structure of the lowest free-energy state. The method first builds an approximate map (or model) of the folding energy landscape from which the population kinetics are analyzed by solving the master equation on the map. We present results obtained using an analysis technique, map-based Monte Carlo simulation, which stochastically extracts folding pathways from the map. Our method compares favorably with other computational methods that begin with a comprehensive free-energy landscape, illustrating that the smaller, approximate map captures the major features of the complete energy landscape. As a result, our method scales to larger RNAs. For example, here we validate kinetics of RNA of more than 200 nucleotides. Our method accurately computes the kinetics-based functional rates of wild-type and mutant ColE1 RNAII and MS2 phage RNAs showing excellent agreement with experiment.     

Protein folding: the free energy surface

Quantitative models and experiments are revealing how the folding free energy surface of a protein is sculpted by sequence and environment. The sometimes conflicting demands of folding, structure and function determine which folding pathways, if any, dominate. Recent advances include experimental estimates of diffusive barrier-crossing times, the observation of ultrafast folders amenable to full-atom simulation, the use of thermodynamic tuning and nonconservative mutations to probe 'hidden' parts of the free energy surface, and a complete microscopic theory of 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.     

Calculation of the free energy and cooperativity of protein folding

Calculation of the free energy of protein folding and delineation of its pre-organization are of foremost importance for understanding, predicting and designing biological macromolecules. Here, we introduce an energy smoothing variant of parallel tempering replica exchange Monte Carlo (REMS) that allows for efficient configurational sampling of flexible solutes under the conditions of molecular hydration. Its usage to calculate the thermal stability of a model globular protein, Trp cage TC5b, achieves excellent agreement with experimental measurements. We find that the stability of TC5b is attained through the coupled formation of local and non-local interactions. Remarkably, many of these structures persist at high temperature, concomitant with the origin of native-like configurations and mesostates in an otherwise macroscopically disordered unfolded state. Graph manifold learning reveals that the conversion of these mesostates to the native state is structurally heterogeneous, and that the cooperativity of their formation is encoded largely by the unfolded state ensemble. In all, these studies establish the extent of thermodynamic and structural pre-organization of folding of this model globular protein, and achieve the calculation of macromolecular stability ab initio, as required for ab initio structure prediction, genome annotation, and drug design.     

Cunning simplicity of protein folding landscapes

Funnel-like landscapes are widely used to visualize protein folding. It might seem that any funnel-like energy landscape helps to avoid the 'Levinthal paradox', i.e. to avoid sampling the impossibly large number of conformations for a folding protein. This cunning suggestion, reinforced by beautiful drawings of the energy funnels, stimulated some simple models of protein folding; one of them [D.J. Bicout and A. Szabo (2000) Protein Sci., 9, 452-465] is especially straightforward and instructive. A thorough analysis of this strict funnel model (which does not consider a nucleation of phase separation in the course of folding) shows that it cannot provide a simultaneous explanation for both major features observed for protein folding: (i) folding within non-astronomical time, and (ii) co-existence of the native and the unfolded states during the folding process. On the contrary, the nucleation mechanism of protein folding can account for both these major features simultaneously.     

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.     

Exploring structures in protein folding funnels with free energy functionals: the denatured ensemble

We discuss the formulation of free energy functionals that describe the formation of structure in partially folded proteins. These free energy functionals take into account the inhomogeneous nature of contact energies, chain entropy and cooperative contributions reflecting the many body character of some folding forces like hydrophobicity, but do not directly account for non-native contacts because they assume the validity of the minimal frustration principle. We show how the free energy functionals can be used to interpret experiments on partially folded proteins that probe the fractional occupancy of specific local structures. In particular, we study the hydrogen protection factors in lysozyme studied in transient experiments by Gladwin and Evans and by Nash and Jonas using equilibrium pressure denaturation and the NMR order parameters measured by Dobson and Kim for the homologous protein alpha-lactalbumin.     

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.     

Non-linear rate-equilibrium free energy relationships and Hammond behavior in protein folding

Non-linear rate-equilibrium relationships upon mutation or changes in solvent conditions are frequently observed in protein folding reactions and are usually interpreted in terms of Hammond behavior. Here we first give a general overview over the concept of transition state movements in chemical reactions and discuss its application to protein folding. We then show examples for genuine Hammond behavior and for apparent transition state movements caused by other effects like changes in the rate-limiting step of the folding reaction or ground state effects, i.e. structural changes in either the native state or the unfolded state. These examples show that apparent transition state movements can easily be mistaken for Hammond behavior. We describe experimental tests using self- and cross-interaction parameters to distinguish between structural changes in a single transition state following Hammond behavior and apparent transition state movements caused by other effects.     

Exploring structures in protein folding funnels with free energy functionals: the transition state ensemble

We use free energy functionals that account for the partial ordering of residues in the transition state ensemble to characterize the free energy surfaces for fast folding proteins. We concentrate on chymotrypsin inhibitor and lambda-repressor. We show how the explicit cooperativity that can arise from many body forces, such as side-chain ordering or hydrophobic surface burial, determines the crossover from folding with a large delocalized nucleus and the specific small classical nucleus of the type envisioned in nucleation growth scenarios. We compare the structural correlations present in the transition state ensemble obtained from free energy functionals with those inferred from experiment using extrathermodynamic free energy relations for folding time obtained via protein engineering kinetics experiments. We also use the free energy functionals to examine both the size of barriers and multidimensional representations of the free energy profiles in order to address the question of appropriate reaction coordinates for folding.     

Comparing the energy landscapes for native folding and aggregation of PrP

Protein sequences are evolved to encode generally one folded structure, out of a nearly infinite array of possible folds. Underlying this code is a funneled free energy landscape that guides folding to the native conformation. Protein misfolding and aggregation are also a manifestation of free-energy landscapes. The detailed mechanisms of these processes are poorly understood, but often involve rare, transient species and a variety of different pathways. The inherent complexity of misfolding has hampered efforts to measure aggregation pathways and the underlying energy landscape, especially using traditional methods where ensemble averaging obscures important rare and transient events. We recently studied the misfolding and aggregation of prion protein by examining 2 monomers tethered in close proximity as a dimer, showing how the steps leading to the formation of a stable aggregated state can be resolved in the single-molecule limit and the underlying energy landscape thereby reconstructed. This approach allows a more quantitative comparison of native folding versus misfolding, including fundamental differences in the dynamics for misfolding. By identifying key steps and interactions leading to misfolding, it should help to identify potential drug targets. Here we describe the importance of characterizing free-energy landscapes for aggregation and the challenges involved in doing so, and we discuss how single-molecule studies can help test proposed structural models for PrP aggregates.     

Calculation of mutational free energy changes in transition states for protein folding

Recent advances in experimental and computational methods have made it possible to determine with considerable accuracy the structures whose formation is rate limiting for the folding of some small proteins-the transition state ensemble, or TSE. We present a method to analyze and validate all-atom models of such structures. The method is based on the comparison of experimental data with the computation of the change in free energy of the TSE resulting from specific mutations. Each mutation is modeled individually in all members of an ensemble of transition state structures using a method originally developed to predict mutational changes in the stability of native proteins. We first apply this method to six proteins for which we have determined the TSEs with a technique that uses experimental mutational data (Phi-values) as restraints in the structure determination and find a highly significant correlation between the calculated free energy changes and those derived from experimental kinetic data. We then use the procedure to analyze transition state structures determined by molecular dynamics simulations of unfolding, again finding a high correlation. Finally, we use the method to estimate changes in folding rates of several hydrophobic core mutants of Fyn SH3. Taken together, these results show that the procedure developed here is a tool of general validity for analyzing, assessing, and improving the quality of the structures of transition states for protein folding.     

Rough energy landscapes in protein folding: dimeric E. coli Trp repressor folds through three parallel channels

The folding mechanism of the dimeric Escherichia coli Trp repressor (TR) is a kinetically complex process that involves three distinguishable stages of development. Following the formation of a partially folded, monomeric ensemble of species, within 5 ms, folding to the native dimer is controlled by three kinetic phases. The rate-limiting step in each phase is either a non-proline isomerization reaction or a dimerization reaction, depending on the final denaturant concentration. Two approaches have been employed to test the previously proposed folding mechanism of TR through three parallel channels: (1) unfolding double-jump experiments demonstrate that all three folding channels lead directly to native dimer; and (2) the differential stabilization of the transition state for the final step in folding and the native dimer, by the addition of salt, shows that all three channels involve isomerization of a dimeric species. A refined model for the folding of Trp repressor is presented, in which all three channels involve a rapid dimerization reaction between partially folded monomers followed by the isomerization of the dimeric intermediates to yield native dimer. The ensemble of partially folded monomers can be captured at equilibrium by low pH; one-dimensional proton NMR spectra at pH 2.5 demonstrate that monomers exist in two distinct, slowly interconverting conformations. These data provide a potential structural explanation for the three-channel folding mechanism of TR: random association of two different monomeric forms, which are distinguished by alternative packing modes of the core dimerization domain and the DNA-binding, helix-turn-helix, domain. One, perhaps both, of these packing modes contains non-native contacts.     

Electrostatic free energy landscapes for DNA helix bending

Nucleic acids are highly charged polyanionic molecules; thus, the ionic conditions are crucial for nucleic acid structural changes such as bending. We use the tightly bound ion theory, which explicitly accounts for the correlation and ensemble effects for counterions, to calculate the electrostatic free energy landscapes for DNA helix bending. The electrostatic free energy landscapes show that DNA bending energy is strongly dependent on ion concentration, valency, and size. In a Na⁺ solution, DNA bending is electrostatically unfavorable because of the strong charge repulsion on backbone. With the increase of the Na⁺ concentration, the electrostatic bending repulsion is reduced and thus the bending becomes less unfavorable. In contrast, in an Mg²⁺ solution, ion correlation induces a possible attractive force between the different parts of the helical strands, resulting in bending. The electrostatically most favorable and unfavorable bending directions are toward the major and minor grooves, respectively. Decreasing the size of the divalent ions enhances the electrostatic bending attraction, causing an increased bending angle, and shifts the most favorable bending to the direction toward the minor groove. The microscopic analysis on ion-binding distribution reveals that the divalent ion-induced helix bending attraction may come from the correlated distribution of the ions across the grooves in the bending direction.