Transition states and the meaning of Phi-values in protein folding kinetics

What is the mechanism of two-state protein folding? The rate-limiting step is typically explored through a Phi-value, which is the mutation-induced change in the transition state free energy divided by the change in the equilibrium free energy of folding. Phi-values ranging from 0 to 1 have been interpreted as meaning the transition state is denatured-like (0), native-like (1) or in-between. But there is no classical interpretation for the experimental Phi-values that are negative or >1. Using a rigorous method to identity transition states via an exact lattice model, we find that nonclassical Phi-values can arise from parallel microscopic flow processes, such as those in funnel-shaped energy landscapes. Phi < 0 results when a mutation destabilizes a slow flow channel, causing a backflow into a faster flow channel. Phi > 1 implies the reverse: a backflow from a fast channel into a slow one. Using a 'landscape mapping' method, we find that Phi correlates with the acceleration/deceleration of folding induced by mutations, rather than with the degree of nativeness of the transition state.     

Visualization of Protein Folding Funnels in Lattice Models

Protein folding occurs in a very high dimensional phase space with an exponentially large number of states, and according to the energy landscape theory it exhibits a topology resembling a funnel. In this statistical approach, the folding mechanism is unveiled by describing the local minima in an effective one-dimensional representation. Other approaches based on potential energy landscapes address the hierarchical structure of local energy minima through disconnectivity graphs. In this paper, we introduce a metric to describe the distance between any two conformations, which also allows us to go beyond the one-dimensional representation and visualize the folding funnel in 2D and 3D. In this way it is possible to assess the folding process in detail, e.g., by identifying the connectivity between conformations and establishing the paths to reach the native state, in addition to regions where trapping may occur. Unlike the disconnectivity maps method, which is based on the kinetic connections between states, our methodology is based on structural similarities inferred from the new metric. The method was developed in a 27-mer protein lattice model, folded into a 3×3×3 cube. Five sequences were studied and distinct funnels were generated in an analysis restricted to conformations from the transition-state to the native configuration. Consistent with the expected results from the energy landscape theory, folding routes can be visualized to probe different regions of the phase space, as well as determine the difficulty in folding of the distinct sequences. Changes in the landscape due to mutations were visualized, with the comparison between wild and mutated local minima in a single map, which serves to identify different trapping regions. The extension of this approach to more realistic models and its use in combination with other approaches are discussed.     

Folding funnels and conformational transitions via hinge-bending motions

In this article we focus on presenting a broad range of examples illustrating low-energy transitions via hinge-bending motions. The examples are divided according to the type of hinge-bending involved; namely, motions involving fragments of the protein chains, hinge-bending motions involving protein domains, and hinge-bending motions between the covalently unconnected subunits. We further make a distinction between allosterically and nonallosterically regulated proteins. These transitions are discussed within the general framework of folding and binding funnels. We propose that the conformers manifesting such swiveling motions are not the outcome of “induced fit” binding mechanism; instead, molecules exist in an ensemble of conformations that are in equilibrium in solution. These ensembles, which populate the bottoms of the funnels,a priori contain both the “open” and the “closed” conformational isomers. Furthermore, we argue that there are no fundamental differences among the physical principles behind the folding and binding funnels. Hence, there is no basic difference between funnels depicting ensembles of conformers of single molecules with fragment, or domain motions, as compared to subunits in multimeric quaternary structures, also showing such conformational transitions. The difference relates only to the size and complexity of the system. The larger the system, the more complex its corresponding fused funnel(s). In particular, funnels associated with allosterically regulated proteins are expected to be more complicated, because allostery is frequently involved with movements between subunits, and consequently is often observed in multichain and multimolecular complexes. This review centers on the critical role played by flexibility and conformational fluctuations in enzyme activity. Internal motions that extend over different time scales and with different amplitudes are known to be essential for the catalytic cycle. The conformational change observed in enzyme-substrate complexes as compared to the unbound enzyme state, and in particular the hinge-bending motions observed in enzymes with two domains, have a substantial effect on the enzymatic catalytic activity. The examples we review span the lipolytic enzymes that are particularly interesting, owing to their activation at the water-oil interface; an allosterically controlled dehydrogenase (lactate dehydrogenase); a DNA methyltransferase, with a covalently-bound intermediate; large-scale flexible loop motions in a glycolytic enzyme (TIM); domain motion in PGK, an enzyme which is essential in most cells, both for ATP generation in aerobes and for fermentation in anaerobes; adenylate kinase, showing large conformational changes, owing to their need to shield their catalytic centers from water; a calcium-binding protein (calmodulin), involved in a wide range of cellular calcium-dependent signaling; diphtheria toxin, whose large domain motion has been shown to yield “domain swapping” the hexameric glutamate dehydrogenase, which has been studied both in a thermophile and in a mesophile; an allosteric enzyme, showing subunit motion between the R and the T states (aspartate transcarbamoylase), and the historically well-studied lac represoor. Nonallosteric subunit transitions are also addressed with some examples (aspartate receptor andBamHI endonuclease). Hence, using this enzyme-catalysis-centered discussion, we address energy funnel landscapes of large-scale conformational transitions, rather than the faster, quasi-harmonic, thermal fluctuations.     

Realistic protein-protein association rates from a simple diffusional model neglecting long-range interactions, free energy barriers, and landscape ruggedness

We develop a simple but rigorous model of protein-protein association kinetics based on diffusional association on free energy landscapes obtained by sampling configurations within and surrounding the native complex binding funnels. Guided by results obtained on exactly solvable model problems, we transform the problem of diffusion in a potential into free diffusion in the presence of an absorbing zone spanning the entrance to the binding funnel. The free diffusion problem is solved using a recently derived analytic expression for the rate of association of asymmetrically oriented molecules. Despite the required high steric specificity and the absence of long-range attractive interactions, the computed rates are typically on the order of 10(4)-10(6) M(-1) sec(-1), several orders of magnitude higher than rates obtained using a purely probabilistic model in which the association rate for free diffusion of uniformly reactive molecules is multiplied by the probability of a correct alignment of the two partners in a random collision. As the association rates of many protein-protein complexes are also in the 10(5)-10(6) M(-1) sec(-1) range, our results suggest that free energy barriers arising from desolvation and/or side-chain freezing during complex formation or increased ruggedness within the binding funnel, which are completely neglected in our simple diffusional model, do not contribute significantly to the dynamics of protein-protein association. The transparent physical interpretation of our approach that computes association rates directly from the size and geometry of protein-protein binding funnels makes it a useful complement to Brownian dynamics simulations.     

Folding funnels, binding funnels, and protein function.

Folding funnels have been the focus of considerable attention during the last few years. These have mostly been discussed in the general context of the theory of protein folding. Here we extend the utility of the concept of folding funnels, relating them to biological mechanisms and function. In particular, here we describe the shape of the funnels in light of protein synthesis and folding; flexibility, conformational diversity, and binding mechanisms; and the associated binding funnels, illustrating the multiple routes and the range of complexed conformers. Specifically, the walls of the folding funnels, their crevices, and bumps are related to the complexity of protein folding, and hence to sequential vs. nonsequential folding. Whereas the former is more frequently observed in eukaryotic proteins, where the rate of protein synthesis is slower, the latter is more frequent in prokaryotes, with faster translation rates. The bottoms of the funnels reflect the extent of the flexibility of the proteins. Rugged floors imply a range of conformational isomers, which may be close on the energy landscape. Rather than undergoing an induced fit binding mechanism, the conformational ensembles around the rugged bottoms argue that the conformers, which are most complementary to the ligand, will bind to it with the equilibrium shifting in their favor. Furthermore, depending on the extent of the ruggedness, or of the smoothness with only a few minima, we may infer nonspecific, broad range vs. specific binding. In particular, folding and binding are similar processes, with similar underlying principles. Hence, the shape of the folding funnel of the monomer enables making reasonable guesses regarding the shape of the corresponding binding funnel. Proteins having a broad range of binding, such as proteolytic enzymes or relatively nonspecific endonucleases, may be expected to have not only rugged floors in their folding funnels, but their binding funnels will also behave similarly, with a range of complexed conformations. Hence, knowledge of the shape of the folding funnels is biologically very useful. The converse also holds: If kinetic and thermodynamic data are available, hints regarding the role of the protein and its binding selectivity may be obtained. Thus, the utility of the concept of the funnel carries over to the origin of the protein and to its function.     

Microscopic reversibility of protein folding in molecular dynamics simulations of the engrailed homeodomain

The principle of microscopic reversibility states that at equilibrium the number of molecules entering a state by a given path must equal those exiting the state via the same path under identical conditions or, in structural terms, that the conformations along the two pathways are the same. There has been some indirect evidence indicating that protein folding is such a process, but there have been few conclusive findings. In this study, we performed molecular dynamics simulations of an ultrafast unfolding and folding protein at its melting temperature to observe, on an atom-by-atom basis, the pathways the protein followed as it unfolded and folded within a continuous trajectory. In a total of 0.67 micros of simulation in water, we found six transient denaturing events near the melting temperature (323 and 330 K) and an additional refolding event following a previously identified unfolding event at a high temperature (373 K). In each case, unfolding and refolding transition state ensembles were identified, and they agreed well with experiment on the basis of a comparison of S and Phi values. On the basis of several structural properties, these 13 transition state ensembles agreed very well with each other and with four previously identified transition states from high-temperature denaturing simulations. Thus, not only were the unfolding and refolding transition states part of the same ensemble, but in five of the seven cases, the pathway the protein took as it unfolded was nearly identical to the subsequent refolding pathway. These events provide compelling evidence that protein folding is a microscopically reversible process. In the other two cases, the folding and unfolding transition states were remarkably similar to each other but the paths deviated.     

Folding and binding cascades: dynamic landscapes and population shifts

Whereas previously we have successfully utilized the folding funnels concept to rationalize binding mechanisms (Ma B, Kumar S, Tsai CJ, Nussinov R, 1999, Protein Eng 12:713-720) and to describe binding (Tsai CJ, Kumar S, Ma B, Nussinov R, 1999, Protein Sci 8:1181-1190), here we further extend the concept of folding funnels, illustrating its utility in explaining enzyme pathways, multimolecular associations, and allostery. This extension is based on the recognition that funnels are not stationary; rather, they are dynamic, depending on the physical or binding conditions (Tsai CJ, Ma B, Nussinov R, 1999, Proc Natl Acad Sci USA 96:9970-9972). Different binding states change the surrounding environment of proteins. The changed environment is in turn expressed in shifted energy landscapes, with different shapes and distributions of populations of conformers. Hence, the function of a protein and its properties are not only decided by the static folded three-dimensional structure; they are determined by the distribution of its conformational substates, and in particular, by the redistributions of the populations under different environments. That is, protein function derives from its dynamic energy landscape, caused by changes in its surroundings.     

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

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

Assessing the energy landscape of CAPRI targets by FunHunt

RosettaDock has repeatedly created high-resolution structures of protein complexes in the CAPRI experiment, thanks to the explicit modeling of conformational changes of the monomers at the side chain level. These models can be selected based on their energy. During the search for the lowest-energy model, RosettaDock samples a deep funnel around the native orientation, but additional funnels may appear in the energy landscape, especially in cases where backbone conformational changes occur upon binding. We have previously developed FunHunt, a Support Vector Machine-based classifier that distinguishes the energy funnels around the native orientation from other funnels in the energy landscape. Here we assess the ability of FunHunt to help in model selection in the CAPRI experiment. For all of 12 recent CAPRI targets, FunHunt clearly identifies a near-native funnel in comparison to the funnel around the lowest energy model identified by the RosettaDock global search protocol. FunHunt is also able to choose a near-native orientation among models submitted by predictor groups, demonstrating its general applicability for model selection. This suggests that FunHunt will be a valuable tool in coming CAPRI rounds for the selection of models, and for the definition of regions that need further refinement with restricted backbone flexibility.     

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.     

Polymer principles and protein folding.

This paper surveys the emerging role of statistical mechanics and polymer theory in protein folding. In the polymer perspective, the folding code is more a solvation code than a code of local phipsi propensities. The polymer perspective resolves two classic puzzles: (1) the Blind Watchmaker's Paradox that biological proteins could not have originated from random sequences, and (2) Levinthal's Paradox that the folded state of a protein cannot be found by random search. Both paradoxes are traditionally framed in terms of random unguided searches through vast spaces, and vastness is equated with impossibility. But both processes are partly guided. The searches are more akin to balls rolling down funnels than balls rolling aimlessly on flat surfaces. In both cases, the vastness of the search is largely irrelevant to the search time and success. These ideas are captured by energy and fitness landscapes. Energy landscapes give a language for bridging between microscopics and macroscopics, for relating folding kinetics to equilibrium fluctuations, and for developing new and faster computational search strategies.     

Direct Observation of Parallel Folding Pathways Revealed Using a Symmetric Repeat Protein System

Although progress has been made to determine the native fold of a polypeptide from its primary structure, the diversity of pathways that connect the unfolded and folded states has not been adequately explored. Theoretical and computational studies predict that proteins fold through parallel pathways on funneled energy landscapes, although experimental detection of pathway diversity has been challenging. Here, we exploit the high translational symmetry and the direct length variation afforded by linear repeat proteins to directly detect folding through parallel pathways. By comparing folding rates of consensus ankyrin repeat proteins (CARPs), we find a clear increase in folding rates with increasing size and repeat number, although the size of the transition states (estimated from denaturant sensitivity) remains unchanged. The increase in folding rate with chain length, as opposed to a decrease expected from typical models for globular proteins, is a clear demonstration of parallel pathways. This conclusion is not dependent on extensive curve-fitting or structural perturbation of protein structure. By globally fitting a simple parallel-Ising pathway model, we have directly measured nucleation and propagation rates in protein folding, and have quantified the fluxes along each path, providing a detailed energy landscape for folding. This finding of parallel pathways differs from results from kinetic studies of repeat-proteins composed of sequence-variable repeats, where modest repeat-to-repeat energy variation coalesces folding into a single, dominant channel. Thus, for globular proteins, which have much higher variation in local structure and topology, parallel pathways are expected to be the exception rather than the rule.     

Kinetic partitioning mechanism governs the folding of the third FnIII domain of tenascin-C: evidence at the single-molecule level

Statistical mechanics and molecular dynamics simulations proposed that the folding of proteins can follow multiple parallel pathways on a rugged energy landscape from unfolded state en route to their folded native states. Kinetic partitioning mechanism is one of the possible mechanisms underlying such complex folding dynamics. Here, we use single-molecule atomic force microscopy technique to directly probe the multiplicity of the folding pathways of the third fibronectin type III domain from the extracellular matrix protein tenascin-C (TNfn3). By stretching individual (TNfn3)₈ molecules, we forced TNfn3 domains to undergo mechanical unfolding and refolding cycles, allowing us to directly observe the folding pathways of TNfn3. We found that, after being mechanically unraveled and then relaxed to zero force, TNfn3 follows multiple parallel pathways to fold into their native states. The majority of TNfn3 fold into the native state in a simple two-state fashion, while a small percentage of TNfn3 were found to be trapped into kinetically stable folding intermediate states with well-defined three-dimensional structures. Furthermore, the folding of TNfn3 was also influenced by its neighboring TNfn3 domains. Complex misfolded states of TNfn3 were observed, possibly due to the formation of domain-swapped dimeric structures. Our studies revealed the ruggedness of the folding energy landscape of TNfn3 and provided direct experimental evidence that the folding dynamics of TNfn3 are governed by the kinetic partitioning mechanism. Our results demonstrated the unique capability of single-molecule AFM to probe the folding dynamics of proteins at the single-molecule level.     

Population of on-pathway intermediates in the folding of ubiquitin

The role that intermediate states play in protein folding is the subject of intense investigation and in the case of ubiquitin has been controversial. We present fluorescence-detected kinetic data derived from single and double mixing stopped-flow experiments to show that the F45W mutant of ubiquitin (WT*), a well-studied single-domain protein and most recently regarded as a simple two-state system, folds via on-pathway intermediates. To account for the discrepancy we observe between equilibrium and kinetic stabilities and m-values, we show that the polypeptide chain undergoes rapid collapse to an intermediate whose presence we infer from a fast lag phase in interrupted refolding experiments. Double-jump kinetic experiments identify two direct folding phases that are not associated with slow isomerisation reactions in the unfolded state. These two phases are explained by kinetic partitioning which allows molecules to reach the native state from the collapsed state via two possible competing routes, which we further examine using two destabilised ubiquitin mutants. Interrupted refolding experiments allow us to observe the formation and decay of an intermediate along one of these pathways. A plausible model for the folding pathway of ubiquitin is presented that demonstrates that obligatory intermediates and/or chain collapse are important events in restricting the conformational search for the native state of ubiquitin.     

Computational assessment of folding energy landscapes in heterodimeric coiled coils

The coiled coil structural motif consists of alpha helices supercoiling around each other to form staggered knobs‐into‐holes packing. Such structures are deceptively simple, especially as they often can be described with parametric equations, but are known to exist in various conformations. Even the simplest systems, consisting of 2 monomers, can assemble into a wide range of states. They can form canonical as well as noncanonical coiled coils, be parallel or antiparallel, where helices associate with different degrees of shift, tilt, and rotation. Here, we investigate the energy landscape of heterodimeric coiled coils by carrying out de novo folding simulations starting from amino acid sequence. We folded a diverse set of 22 heterodimers and demonstrate that the approach is capable of identifying the atomic details in the experimental structure in the majority of cases. Our methodology also enables exploration of alternative states that can be accessible in solution beyond the experimentally determined structure. For many systems, we observe folding energy landscapes with multiple energy minima and several isoenergetic states. By comparing coiled coils from single domains and those extracted from larger proteins, we find that standalone coiled coils have deeper energy wells at the experimentally determined conformation. By folding the competing homodimeric states in addition to the heterodimers, we observe that the structural specificity towards the heteromeric state is often small. Taken together, our results demonstrate that de novo folding simulations can be a powerful tool to characterize structural specificity of coiled coils when coupled to assessment of energy landscapes.     

On the degradation of forest ecosystems by extreme events: Statistical Model Checking of a hybrid model

In this paper, we study the vulnerability of forest ecosystems perturbed by extreme events, such as those arising from climate change. To investigate the complex interactions between the biological dynamics of the forest and the climatic activity, we construct an original hybrid model, obtained by coupling a continuous reaction–diffusion system, which describes the spatio-temporal dynamics of the forest ecosystem, with a discrete probabilistic process, which models the possible occurrences of extreme events. Properties of ecological interest are considered: invariance of the persistence equilibrium, attraction to the extinction equilibrium and emergence of degraded states. Those properties of the hybrid model are verified through an extension of the Statistical Model Checking framework. We establish the existence of a threshold above which the persistence equilibrium of the forest ecosystem is compromised and give a numerical assessment of this threshold in terms of the probability and intensity of extreme events. We also present non-trivial parameter conditions for which the forest ecosystem converges to a degraded savanna-like state.     

A Parallel Implementation of the Wuchty Algorithm with Additional Experimental Filters to More Thoroughly Explore RNA Conformational Space

We present new modifications to the Wuchty algorithm in order to better define and explore possible conformations for an RNA sequence. The new features, including parallelization, energy-independent lonely pair constraints, context-dependent chemical probing constraints, helix filters, and optional multibranch loops, provide useful tools for exploring the landscape of RNA folding. Chemical probing alone may not necessarily define a single unique structure. The helix filters and optional multibranch loops are global constraints on RNA structure that are an especially useful tool for generating models of encapsidated viral RNA for which cryoelectron microscopy or crystallography data may be available. The computations generate a combinatorially complete set of structures near a free energy minimum and thus provide data on the density and diversity of structures near the bottom of a folding funnel for an RNA sequence. The conformational landscapes for some RNA sequences may resemble a low, wide basin rather than a steep funnel that converges to a single structure.     

Repeat-protein folding: new insights into origins of cooperativity, stability, and topology

Although our understanding of globular protein folding continues to advance, the irregular tertiary structures and high cooperativity of globular proteins complicates energetic dissection. Recently, proteins with regular, repetitive tertiary structures have been identified that sidestep limitations imposed by globular protein architecture. Here we review recent studies of repeat-protein folding. These studies uniquely advance our understanding of both the energetics and kinetics of protein folding. Equilibrium studies provide detailed maps of local stabilities, access to energy landscapes, insights into cooperativity, determination of nearest-neighbor interaction parameters using statistical thermodynamics, relationships between consensus sequences and repeat-protein stability. Kinetic studies provide insight into the influence of short-range topology on folding rates, the degree to which folding proceeds by parallel (versus localized) pathways, and the factors that select among multiple potential pathways. The recent application of force spectroscopy to repeat-protein unfolding is providing a unique route to test and extend many of these findings.     

Protein folding: binding of conformationally fluctuating building blocks via population selection.

Here we review different aspects of the protein folding literature. We present a broad range of observations, showing them to be consistent with a general hierarchical protein folding model. In such a model, local relatively stable, conformationally fluctuating building blocks bind through population selection, to yield the native state. The model includes several components: (1) the fluctuating building blocks that constitute local minima along the polypeptide chain, which even if unstable still possess higher population times than all alternate conformations; (2) the landscape around the bottom of the funnels; (3) the consideration that protein folding involves intramolecular recognition; (4) similar landscapes are observed for folding and for binding, and that (5) the landscape is dynamic, changing with the conditions. The model considers protein folding to be guided by native interactions. The reviewed literature includes the effects of changing the conditions, intermediates and kinetic traps, mutations, similar topologies, fragment complementation experiments, fragments and pathways, focusing on one specific well-studied example, that of the dihydrofolate reductase, chaperones, and chaperonines, in vivo vs. in vitro folding, still using the dihydrofolate example, amyloid formation, and molecular "disorder". These are consistent with the view that binding and folding are similar events, with the differences stemming from different stabilities and hence population times.     

Parallel protein-unfolding pathways revealed and mapped

Theoretical studies of protein folding suggest that multiple folding pathways should exist, but there is little experimental evidence to support this. Here we demonstrate changes in the flux between different transition states on parallel folding pathways, resulting in unprecedented upward curvature in the denaturant-dependent unfolding kinetics of a beta-sandwich protein. As denaturant concentration increases, the highly compact transition state of one pathway becomes destabilized and the dominant flux of protein molecules shifts toward another pathway with a less structured transition state. Furthermore, point mutations alter the relative accessibility of the pathways, allowing the structure of two transition states on separate, direct folding pathways to be mapped by systematic Phi-value analysis. It has been suggested that pathways with diffuse rather than localized transition states are evolutionarily selected to prevent misfolding, and indeed we find that the transition state favored at high concentrations of denaturant is more polarized than the physiologically relevant one.