Towards a consistent modeling of protein thermodynamic and kinetic cooperativity: how applicable is the transition state picture to folding and unfolding?

To what extent do general features of folding/unfolding kinetics of small globular proteins follow from their thermodynamic properties? To address this question, we investigate a new simplified protein chain model that embodies a cooperative interplay between local conformational preferences and hydrophobic burial. The present four-helix-bundle 55mer model exhibits protein-like calorimetric two-state cooperativity. It rationalizes native-state hydrogen exchange observations. Our analysis indicates that a coherent, self-consistent physical account of both the thermodynamic and kinetic properties of the model leads naturally to the concept of a native state ensemble that encompasses considerable conformational fluctuations. Such a multiple-conformation native state is seen to involve conformational states similar to those revealed by native-state hydrogen exchange. Many of these conformational states are predicted to lie below native baselines commonly used in interpreting calorimetric data. Folding and unfolding kinetics are studied under a range of intrachain interaction strengths as in experimental chevron plots. Kinetically determined transition midpoints match well with their thermodynamic counterparts. Kinetic relaxations are found to be essentially single-exponential over an extended range of model interaction strengths. This includes the entire unfolding regime and a significant part of a folding regime with a chevron rollover, as has been observed for real proteins that fold with non-two-state kinetics. The transition state picture of protein folding and unfolding is evaluated by comparing thermodynamic free energy profiles with actual kinetic rates. These analyses suggest that some chevron rollovers may arise from an internal frictional effect that increasingly impedes chain motions with more native conditions, rather than being caused by discrete deadtime folding intermediates or shifts of the transition state peak as previously posited.     

Engineering strand-specific DNA nicking enzymes from the type IIS restriction endonucleases BsaI, BsmBI, and BsmAI

To determine the extent to which protein folding rates and free energy landscapes have been shaped by natural selection, we have examined the folding kinetics of five proteins generated using computational design methods and, hence, never exposed to natural selection. Four of these proteins are complete computer-generated redesigns of naturally occurring structures and the fifth protein, called Top7, has a computer-generated fold not yet observed in nature. We find that three of the four redesigned proteins fold much faster than their naturally occurring counterparts. While natural selection thus does not appear to operate on protein folding rates, the majority of the designed proteins unfold considerably faster than their naturally occurring counterparts, suggesting possible selection for a high free energy barrier to unfolding. In contrast to almost all naturally occurring proteins of less than 100 residues but consistent with simple computational models, the folding energy landscape for Top7 appears to be quite complex, suggesting the smooth energy landscapes and highly cooperative folding transitions observed for small naturally occurring proteins may also reflect the workings of natural selection.     

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.     

The highly cooperative folding of small naturally occurring proteins is likely the result of natural selection

To illuminate the evolutionary pressure acting on the folding free energy landscapes of naturally occurring proteins, we have systematically characterized the folding free energy landscape of Top7, a computationally designed protein lacking an evolutionary history. Stopped-flow kinetics, circular dichroism, and NMR experiments reveal that there are at least three distinct phases in the folding of Top7, that a nonnative conformation is stable at equilibrium, and that multiple fragments of Top7 are stable in isolation. These results indicate that the folding of Top7 is significantly less cooperative than the folding of similarly sized naturally occurring proteins, suggesting that the cooperative folding and smooth free energy landscapes observed for small naturally occurring proteins are not general properties of polypeptide chains that fold to unique stable structures but are instead a product of natural selection.     

Solvation effects and driving forces for protein thermodynamic and kinetic cooperativity: how adequate is native-centric topological modeling?

What energetic and solvation effects underlie the remarkable two-state thermodynamics and folding/unfolding kinetics of small single-domain proteins? To address this question, we investigate the folding and unfolding of a hierarchy of continuum Langevin dynamics models of chymotrypsin inhibitor 2. We find that residue-based additive Gō-like contact energies, although native-centric, are by themselves insufficient for protein-like calorimetric two-state cooperativity. Further native biases by local conformational preferences are necessary for protein-like thermodynamics. Kinetically, however, even models with both contact and local native-centric energies do not produce simple two-state chevron plots. Thus a model protein's thermodynamic cooperativity is not sufficient for simple two-state kinetics. The models tested appear to have increasing internal friction with increasing native stability, leading to chevron rollovers that typify kinetics that are commonly referred to as non-two-state. The free energy profiles of these models are found to be sensitive to the choice of native contacts and the presumed spatial ranges of the contact interactions. Motivated by explicit-water considerations, we explore recent treatments of solvent granularity that incorporate desolvation free energy barriers into effective implicit-solvent intraprotein interactions. This additional feature reduces both folding and unfolding rates vis-à-vis that of the corresponding models without desolvation barriers, but the kinetics remain non-two-state. Taken together, our observations suggest that interaction mechanisms more intricate than simple Gō-like constructs and pairwise additive solvation-like contributions are needed to rationalize some of the most basic generic protein properties. Therefore, as experimental constraints on protein chain models, requiring a consistent account of protein-like thermodynamic and kinetic cooperativity can be more stringent and productive for some applications than simply requiring a model heteropolymer to fold to a target structure.     

Equilibrium collapse and the kinetic 'foldability' of proteins.

An important element of protein folding theory has been the identification of equilibrium parameters that might uniquely distinguish rapidly folding polypeptide sequences from those that fold slowly. One such parameter, termed sigma, is a dimensionless, equilibrium measure of the coincidence of chain compaction and folding that is predicted to be an important determinant of relative folding kinetics. To test this prediction and improve our understanding of the putative relationship between nonspecific compaction of the unfolded state and protein folding kinetics, we have used small-angle X-ray scattering and circular dichroism spectroscopy to measure the sigma of five well-characterized proteins. Consistent with theoretical predictions, we find that near-perfect coincidence of the unfolded state contraction and folding (sigma approximately 0) is associated with the rapid kinetics of these naturally occurring proteins. We do not, however, observe any significant correlation between sigma and either the relative folding rates of these proteins or the presence or absence of well-populated kinetic intermediates. Thus, while sigma approximately 0 may be a necessary condition to ensure rapid folding, differences in sigma do not account for the wide range of rates and mechanisms with which naturally occurring proteins fold.     

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

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

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.     

Criteria for downhill protein folding: calorimetry, chevron plot, kinetic relaxation, and single-molecule radius of gyration in chain models with subdued degrees of cooperativity

Recent investigations of possible downhill folding of small proteins such as BBL have focused on the thermodynamics of non-two-state, "barrierless" folding/denaturation transitions. Downhill folding is noncooperative and thermodynamically "one-state," a phenomenon underpinned by a unimodal conformational distribution over chain properties such as enthalpy, hydrophobic exposure, and conformational dimension. In contrast, corresponding distributions for cooperative two-state folding are bimodal with well-separated population peaks. Using simplified atomic modeling of a three-helix bundle-in a scheme that accounts for hydrophobic interactions and hydrogen bonding-and coarse-grained C(alpha) models of four real proteins with various degrees of cooperativity, we evaluate the effectiveness of several observables at defining the underlying distribution. Bimodal distributions generally lead to sharper transitions, with a higher heat capacity peak at the transition midpoint, compared with unimodal distributions. However, the observation of a sigmoidal transition is not a reliable criterion for two-state behavior, and the heat capacity baselines, used to determine the van't Hoff and calorimetric enthalpies of the transition, can introduce ambiguity. Interestingly we find that, if the distribution of the single-molecule radius of gyration were available, it would permit discrimination between unimodal and bimodal underlying distributions. We investigate kinetic implications of thermodynamic noncooperativity using Langevin dynamics. Despite substantial chevron rollovers, the relaxation of the models considered is essentially single-exponential over an extended range of native stabilities. Consistent with experiments, significant deviations from single-exponential behavior occur only under strongly folding conditions.     

Explicit-chain model of native-state hydrogen exchange: implications for event ordering and cooperativity in protein folding

Native-state hydrogen exchange experiments on several proteins have revealed partially unfolded conformations with diverse stabilities. These equilibrium observations have been used to support kinetic arguments that folding proceeds via a sequential "pathway." This interpretative logic is evaluated here by analyzing the relationship between thermodynamic behavior and folding kinetics in a class of simplified lattice protein models. The chain models studied have varying degrees of cooperative interplay (coupling) between local helical conformational preference and favorable nonlocal interactions. When model cooperativity is high, as native conditions are weakened, "isotherms" of free energy of exchange for residues belonging to the same helix merge together before global unfolding. The point of merger depends on the model energetic favorability of the helix. This trend is similar to the corresponding experimental observations. Kinetically, we find that the ordering of helix formation in the very last stage of native core assembly tends to follow the stabilities of their converged isotherms. In a majority (but not all) of folding trajectories, the final assembly of helices that are thermodynamically more stable against exchange precedes that of helices that are less stable against exchange. These model features are in partial agreement with common experimental interpretations. However, the model results also underscore the ensemble nature of the folding process: the kinetics of helix formation is not a discrete, strictly "all-or-none" process as that envisioned by certain non-explicit-chain models. Helices generally undergo many cycles of partial formation and dissolution before their conformations are fixed in the final assembly stage of folding, a kinetic stage that takes up only approximately 2% of the average folding time in the present model; and the ordering of the helices' final assembly in some trajectories can be different from the dominant ordering stipulated by the exchange isotherms.     

Thermodynamics and kinetics of protein folding: an evolutionary perspective

This article appeals to an evolutionary model which postulates that primordial proteins were described by small polypeptide chains which (i) lack disulfide bridges, and (ii) display slow folding rates with multi-state kinetics, to determine relations between structural properties of proteins and their folding kinetics. We parameterize the energy landscape of proteins in terms of thermodynamic activation variables. The model studies evolutionary changes in these thermodynamic parameters, and we invoke relations between these activation variables and structural properties of the protein to predict the following correspondence between protein structure and folding kinetics. 1. Proteins with inter- and intra-chain disulfide bridges: large variability in both folding rates and stability of intermediates, multi-state kinetics. 2. Proteins which lack inter and intra-chain disulfide bridges. 2.1 Single-domain chains: fast folding rates; unstable intermediates; two-state kinetics. 2.2 Multi-domain monomers: intermediate rates; metastable intermediates; multi-state kinetics. 2.3 Multi-domain oligomers: slow rates; metastable intermediates; multi-state kinetics. The evolutionary model thus provides a kinetic characterization of one important subfamily of proteins which we describe by the following properties: Folding dynamics of single-domain proteins which lack disulfide bridges are described by two-state kinetics. Folding rate of this class of proteins is positively correlated with the thermodynamic stability of the folded state.     

Probing possible downhill folding: native contact topology likely places a significant constraint on the folding cooperativity of proteins with approximately 40 residues

Experiments point to appreciable variations in folding cooperativity among natural proteins with approximately 40 residues, indicating that the behaviors of these proteins are valuable for delineating the contributing factors to cooperative folding. To explore the role of native topology in a protein's propensity to fold cooperatively and how native topology might constrain the degree of cooperativity achievable by a given set of physical interactions, we compared folding/unfolding kinetics simulated using three classes of native-centric C^α chain models with different interaction schemes. The approach was applied to two homologous 45-residue fragments from the peripheral subunit-binding domain family and a 39-residue fragment of the N-terminal domain of ribosomal protein L9. Free-energy profiles as functions of native contact number were computed to assess the heights of thermodynamic barriers to folding. In addition, chevron plots of folding/unfolding rates were constructed as functions of native stability to facilitate comparison with available experimental data. Although common Gō-like models with pairwise Lennard-Jones-type interactions generally fold less cooperatively than real proteins, the rank ordering of cooperativity predicted by these models is consistent with experiment for the proteins investigated, showing increasing folding cooperativity with increasing nonlocality of a protein's native contacts. Models that account for water-expulsion (desolvation) barriers and models with many-body (nonadditive) interactions generally entail higher degrees of folding cooperativity indicated by more linear model chevron plots, but the rank ordering of cooperativity remains unchanged. A robust, experimentally valid rank ordering of model folding cooperativity independent of the multiple native-centric interaction schemes tested here argues that native topology places significant constraints on how cooperatively a protein can fold.     

Contact order dependent protein folding rates: kinetic consequences of a cooperative interplay between favorable nonlocal interactions and local conformational preferences

Physical mechanisms underlying the empirical correlation between relative contact order (CO) and folding rate among naturally occurring small single-domain proteins are investigated by evaluating postulated interaction schemes for a set of three-dimensional 27mer lattice protein models with 97 different CO values. Many-body interactions are constructed such that contact energies become more favorable when short chain segments sequentially adjacent to the contacting residues adopt native-like conformations. At a given interaction strength, this scheme leads to folding rates that are logarithmically well correlated with CO (correlation coefficient r = 0.914) and span more than 2.5 orders of magnitude, whereas folding rates of the corresponding Gō models with additive contact energies have much less logarithmic correlation with CO and span only approximately one order of magnitude. The present protein chain models also exhibit calorimetric cooperativity and linear chevron plots similar to that observed experimentally for proteins with apparent simple two-state folding/unfolding kinetics. Thus, our findings suggest that CO-dependent folding rates of real proteins may arise partly from a significant positive coupling between nonlocal contact favorabilities and local conformational preferences.     

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.     

Role of unfolded state heterogeneity and en-route ruggedness in protein folding kinetics

In order to improve our understanding of the physical bases of protein folding, there is a compelling need for better connections between experimental and computational approaches. This work addresses the role of unfolded state conformational heterogeneity and en-route intermediates, as an aid for planning and interpreting protein folding experiments. The expected kinetics were modeled for different types of energy landscapes, including multiple parallel folding routes, preferential paths dominated by one primary folding route, and distributed paths with a wide spectrum of microscopic folding rate constants. In the presence of one or more preferential routes, conformational exchange among unfolded state populations slows down the observed rates for native protein formation. We find this to be a general phenomenon, taking place even when unfolded conformations interconvert much faster than the "escape" rate constants to folding. Dramatic kinetic deceleration is expected in the presence of an increasing number of folding-incompetent unfolded conformations. This argues for the existence of parallel folding paths involving several folding-competent unfolded conformations, during the early stages of protein folding. Deviations from single-exponential behavior are observed for unfolded conformations exchanging at comparable rates or more slowly than folding events. Analysis of the effect of en-route (on-path) intermediate formation and landscape ruggedness on folding kinetics leads to the following unexpected conclusions: (1) intermediates, which often retard native state formation, may in some cases accelerate folding, and (2) rugged landscapes, usually associated with stretched exponentials, display single-exponential behavior in the presence of late high-friction paths.     

Protein folding in high-dimensional spaces: hypergutters and the role of nonnative interactions

We explore the consequences of very high dimensionality in the dynamical landscape of protein folding. Consideration of both typical range of stabilizing interactions, and folding rates themselves, leads to a model of the energy hypersurface that is characterized by the structure of diffusive "hypergutters" as well as the familiar "funnels". Several general predictions result: 1), intermediate subspaces of configurations will always be visited; 2), specific but nonnative interactions may be important in stabilizing these low-dimensional diffusive searches on the folding pathway, as well as native interactions; 3), sequential barriers will commonly be found, even in "two-state" proteins; 4), very early times will show characteristic departures from single-exponential kinetics; and 5), contributions of nonnative interactions to Phi-values and "Chevron plots" are calculable, and may be significant. The example of a three-helix bundle is treated in more detail as an illustration. The model also shows that high-dimensional structures provide conceptual relations between different models of protein folding. It suggests that kinetic strategies for fast folding may be encoded rather generally in nonnative as well as in native interactions. The predictions are related to very recent findings in experiment and simulation.     

Changes of protein folding pathways by circular permutation. Overlapping nuclei promote global cooperativity

The evolved properties of proteins are not limited to structure and stability but also include their propensity to undergo local conformational changes. The latter, dynamic property is related to structural cooperativity and is controlled by the folding-energy landscape. Here we demonstrate that the structural cooperativity of the ribosomal protein S6 is optimized by geometric overlap of two competing folding nuclei: they both include the central beta-strand 1. In this way, folding of one nucleus catalyzes the formation of the other, contributing to make the folding transition more concerted overall. The experimental evidence is provided by an extended set of circular permutations of S6 that allows quantitative analysis of pathway plasticity at the level of individual side chains. Because similar overlap between competing nuclei also has been discerned in other proteins, we hypothesize that the coupling of several small nuclei into extended "supernuclei" represents a general principle for propagating folding cooperativity across large structural distances.     

Modulation of Folding Kinetics of Repeat Proteins: Interplay between Intra- and Interdomain Interactions

Repeat proteins have unique elongated structures that, unlike globular proteins, are quite modular. Despite their simple one-dimensional structure, repeat proteins exhibit intricate folding behavior with a complexity similar to that of globular proteins. Therefore, repeat proteins allow one to quantify fundamental aspects of the biophysics of protein folding. One important feature of repeat proteins is the interfaces between the repeating units. In particular, the distribution of stabilities within and between the repeats was previously suggested to affect their folding characteristics. In this study, we explore how the interface affects folding kinetics and cooperativity by investigating two families of repeat proteins, namely, the Ankyrin and tetratricopeptide repeat proteins, which differ in the number of interfacial contacts that are formed between their units as well as in their folding behavior. By using simple topology-based models, we show that modulating the energetic strength of the interface relative to that of the repeat itself can drastically change the protein stability, folding rate, and cooperativity. By further dissecting the interfacial contacts into several subsets, we isolated the effects of each of these groups on folding kinetics. Our study highlights the importance of interface connectivity in determining the folding behavior.