Estimation of protein folding free energy barriers from calorimetric data by multi-model Bayesian analysis

The Wako-Saitô-Muñoz-Eaton Model for Predicting Protein Folding and Dynamics

Despite the recent advances in the prediction of protein structures by deep neutral networks, the elucidation of protein-folding mechanisms remains challenging. A promising theory for describing protein folding is a coarse-grained statistical mechanical model called the Wako-Saitô-Muñoz-Eaton (WSME) model. The model can calculate the free-energy landscapes of proteins based on a three-dimensional structure with low computational complexity, thereby providing a comprehensive understanding of the folding pathways and the structure and stability of the intermediates and transition states involved in the folding reaction. In this review, we summarize previous and recent studies on protein folding and dynamics performed using the WSME model and discuss future challenges and prospects. The WSME model successfully predicted the folding mechanisms of small single-domain proteins and the effects of amino-acid substitutions on protein stability and folding in a manner that was consistent with experimental results. Furthermore, extended versions of the WSME model were applied to predict the folding mechanisms of multi-domain proteins and the conformational changes associated with protein function. Thus, the WSME model may contribute significantly to solving the protein-folding problem and is expected to be useful for predicting protein folding, stability, and dynamics in basic research and in industrial and medical applications.

Is Protein Folding a Thermodynamically Unfavorable, Active, Energy-Dependent Process?

The prevailing current view of protein folding is the thermodynamic hypothesis, under which the native folded conformation of a protein corresponds to the global minimum of Gibbs free energy G. We question this concept and show that the empirical evidence behind the thermodynamic hypothesis of folding is far from strong. Furthermore, physical theory-based approaches to the prediction of protein folds and their folding pathways so far have invariably failed except for some very small proteins, despite decades of intensive theory development and the enormous increase of computer power. The recent spectacular successes in protein structure prediction owe to evolutionary modeling of amino acid sequence substitutions enhanced by deep learning methods, but even these breakthroughs provide no information on the protein folding mechanisms and pathways. We discuss an alternative view of protein folding, under which the native state of most proteins does not occupy the global free energy minimum, but rather, a local minimum on a fluctuating free energy landscape. We further argue that ΔG of folding is likely to be positive for the majority of proteins, which therefore fold into their native conformations only through interactions with the energy-dependent molecular machinery of living cells, in particular, the translation system and chaperones. Accordingly, protein folding should be modeled as it occurs in vivo, that is, as a non-equilibrium, active, energy-dependent process.

Folding Free Energy Surfaces from Differential Scanning Calorimetry

Protein folding/unfolding processes involve a large number of weak, non-covalent interactions and are more appropriately described in terms of the movement of a point representing protein conformation in a plot of internal free energy versus conformational degrees of freedom. While these energy landscapes have an astronomically large number of dimensions, it has been shown that many relevant aspects of protein folding can be understood in terms of their projections onto a few relevant coordinates. Remarkably, such low-dimensional free energy surfaces can be obtained from experimental DSC data using suitable analytical models. Here, we describe the experimental procedures to be used to obtain the high-quality DSC data that are required for free-energy surface analysis.

Predicting and Simulating Mutational Effects on Protein Folding Kinetics

Mutational perturbations of protein structures, i.e., phi-value analysis, are commonly employed to probe the extent of involvement of a particular residue in the rate-determining step(s) of folding. This generally involves the measurement of folding thermodynamic parameters and kinetic rate constants for the wild-type and mutant proteins. While computational approaches have been reasonably successful in understanding and predicting the effect of mutations on folding thermodynamics, it has been challenging to explore the same on kinetics due to confounding structural, energetic, and dynamic factors. Accordingly, the frequent observation of fractional phi-values (mean of ~0.3) has resisted a precise and consistent interpretation. Here, we describe how to construct, parameterize, and employ a simple one-dimensional free energy surface model that is grounded in the basic tenets of the energy landscape theory to predict and simulate the effect of mutations on folding kinetics. As a proof of principle, we simulate one-dimensional free energy profiles of 806 mutations from 24 different proteins employing just the experimental destabilization as input, reproduce the relative unfolding activation free energies with a correlation of 0.91, and show that the mean phi-value of 0.3 essentially corresponds to the extent of stabilization energy gained at the barrier top while folding.

A Disordered Loop Mediates Heterogeneous Unfolding of an Ordered Protein by Altering the Native Ensemble

The high flexibility of long disordered or partially structured loops in folded proteins allows for entropic stabilization of native ensembles. Destabilization of such loops could alter the native ensemble or promote alternate conformations within the native ensemble if the ordered regions themselves are held together weakly. This is particularly true of downhill folding systems that exhibit weak unfolding cooperativity. Here, we combine experimental and computational methods to probe the response of the native ensemble of a helical, downhill folding domain PDD, which harbors an 11-residue partially structured loop, to perturbations. Statistical mechanical modeling points to continuous structural changes on both temperature and mutational perturbations driven by entropic stabilization of partially structured conformations within the native ensemble. Long time-scale simulations of the wild-type protein and two mutants showcase a remarkable conformational switching behavior wherein the parallel helices in the wild-type protein sample an antiparallel orientation in the mutants, with the C-terminal helix and the loop connecting the helices displaying high flexibility, disorder, and non-native interactions. We validate these computational predictions via the anomalous fluorescence of a native tyrosine located at the interface of the helices. Our observations highlight the role of long loops in determining the unfolding mechanisms, sensitivity of the native ensembles to mutational perturbations and provide experimentally testable predictions that can be explored in even two-state folding systems.

Tunable order-disorder continuum in protein-DNA interactions

DNA-binding protein domains (DBDs) sample diverse conformations in equilibrium facilitating the search and recognition of specific sites on DNA over millions of energetically degenerate competing sites. We hypothesize that DBDs have co-evolved to sense and exploit the strong electric potential from the array of negatively charged phosphate groups on DNA. We test our hypothesis by employing the intrinsically disordered DBD of cytidine repressor (CytR) as a model system. CytR displays a graded increase in structure, stability and folding rate on increasing the osmolarity of the solution that mimics the non-specific screening by DNA phosphates. Electrostatic calculations and an Ising-like statistical mechanical model predict that CytR exhibits features of an electric potential sensor modulating its dimensions and landscape in a unique distance-dependent manner, while DNA plays the role of a non-specific macromolecular chaperone. Accordingly, CytR binds its natural half-site faster than the diffusion-controlled limit and even random DNA conforming to an electrostatic-steering binding mechanism. Our work unravels for the first time the synergistic features of a natural electrostatic potential sensor, a novel binding mechanism driven by electrostatic frustration and disorder, and the role of DNA in promoting distance-dependent protein structural transitions critical for switching between specific and non-specific DNA-binding modes.

Position-, disorder-, and salt-dependent diffusion in binding-coupled-folding of intrinsically disordered proteins

Successful extensions of protein-folding energy landscape theory to intrinsically disordered proteins' (IDPs') binding-coupled-folding transition can enormously simplify this biomolecular process into diffusion along a limited number of reaction coordinates, and the dynamics subsequently is described by Kramers' rate theory. As the critical pre-factor, the diffusion coefficient D has direct implications on the binding kinetics. Here, we employ a structure-based model (SBM) to calculate D in the binding-folding of an IDP prototype. We identify a strong position-dependent D during binding by applying a reaction coordinate that directly measures the fluctuations in a Cartesian configuration space. Using the malleability of the SBM, we modulate the degree of conformational disorder in an isolated IDP and determine complex effects of intrinsic disorder on D varying for different binding stages. Here, D tends to increase with disorder during initial binding but shows a non-monotonic relationship with disorder in terms of a decrease-followed-by-increase in D during the late binding stage. The salt concentration, which correlates with electrostatic interactions via Debye-Hückel theory in our SBM, also modulates D in a stepwise way. The speeding up of diffusion by electrostatic interactions is observed during the formation of the encounter complex at the beginning of binding, while the last diffusive binding dynamics is hindered by non-native salt bridges. Because D describes the diffusive speed locally, which implicitly reflects the roughness of the energy landscape, we are eventually able to portray the binding energy landscape, including that from IDPs' binding, then to binding with partial folding, and finally to rigid docking, as well as that under different environmental salt concentrations. Our theoretical results provide key mechanistic insights into IDPs' binding-folding, which is internally conformation- and externally salt-controlled with respect to diffusion.

Protein Folding Cooperativity and Thermodynamic Barriers of the Simplest β-Sheet Fold: A Survey of WW Domains

Theory and experiments have shown that microsecond folding proteins exhibit characteristic thermodynamic properties that reflect the limited cooperativity of folding over marginal barriers (downhill folding). Those studies have mostly focused on proteins with large α-helical contents and small size, which tend to be the fastest folders. A key open question is whether such properties are also present in the fastest all-β proteins. We address this issue by investigating the unfolding thermodynamics of a collection of WW domains as representatives of the simplest β-sheet fold. WW domains are small microsecond folders, although they do not fold as fast as their α-helical counterparts. In previous work on the NEDD4-WW4 domain, we reported deviations from two-state thermodynamics that were less apparent and thus suggestive of an incipient downhill scenario. Here we investigate the unfolding thermodynamics of four other WW domains (NEDD4-WW3, YAP65-WW1(L30K), FBP11-WW1, and FBP11-WW2) by performing all of the thermodynamic tests for downhill folding that have been previously developed on α-helical proteins. This set of five WW domains shares low sequence identity and include examples from two specificity classes, thus providing a comprehensive survey. Thermodynamic analysis of the four new WW domains consistently reveals all of the properties of downhill folding equilibria, which are in all cases more marked than what we found before in NEDD4-WW4. Our results show that fast-folding all-β proteins do share limited cooperativity and gradual unfolding thermodynamics with fast α-helical proteins and suggest that the free energy barrier to folding of natural proteins is mostly determined by size and fold topology and much less by the specific amino acid sequence.

Roles of conformational disorder and downhill folding in modulating protein-DNA recognition

Transcription factors are thought to efficiently search for their target DNA site via a combination of conventional 3D diffusion and 1D diffusion along the DNA molecule mediated by non-specific electrostatic interactions. This process requires the DNA-binding protein to quickly exchange between a search competent and a target recognition mode, but little is known as to how these two binding modes are encoded in the conformational properties of the protein. Here, we investigate this issue on the engrailed homeodomain (EngHD), a DNA-binding domain that folds ultrafast and exhibits a complex conformational behavior consistent with the downhill folding scenario. We explore the interplay between folding and DNA recognition using a coarse-grained computational model that allows us to manipulate the folding properties of the protein and monitor its non-specific and specific binding to DNA. We find that conformational disorder increases the search efficiency of EngHD by promoting a fast gliding search mode in addition to sliding. When gliding, EngHD remains loosely bound to DNA moving linearly along its length. A partially disordered EngHD also binds more dynamically to the target site, reducing the half-life of the specific complex via a spring-loaded mechanism. These findings apply to all conditions leading to partial disorder. However, we also find that at physiologically relevant temperatures EngHD is well folded and can only obtain the conformational flexibility required to accelerate 1D diffusion when it folds/unfolds within the downhill scenario (crossing a marginal free energy barrier). In addition, the conformational flexibility of native downhill EngHD enables its fast reconfiguration to lock into the specific binding site upon arrival, thereby affording finer control of the on- and off-rates of the specific complex. Our results provide key mechanistic insights into how DNA-binding domains optimize specific DNA recognition through the control of their conformational dynamics and folding mechanism.

When fast is better: protein folding fundamentals and mechanisms from ultrafast approaches

Protein folding research stalled for decades because conventional experiments indicated that proteins fold slowly and in single strokes, whereas theory predicted a complex interplay between dynamics and energetics resulting in myriad microscopic pathways. Ultrafast kinetic methods turned the field upside down by providing the means to probe fundamental aspects of folding, test theoretical predictions and benchmark simulations. Accordingly, experimentalists could measure the timescales for all relevant folding motions, determine the folding speed limit and confirm that folding barriers are entropic bottlenecks. Moreover, a catalogue of proteins that fold extremely fast (microseconds) could be identified. Such fast-folding proteins cross shallow free energy barriers or fold downhill, and thus unfold with minimal co-operativity (gradually). A new generation of thermodynamic methods has exploited this property to map folding landscapes, interaction networks and mechanisms at nearly atomic resolution. In parallel, modern molecular dynamics simulations have finally reached the timescales required to watch fast-folding proteins fold and unfold in silico. All of these findings have buttressed the fundamentals of protein folding predicted by theory, and are now offering the first glimpses at the underlying mechanisms. Fast folding appears to also have functional implications as recent results connect downhill folding with intrinsically disordered proteins, their complex binding modes and ability to moonlight. These connections suggest that the coupling between downhill (un)folding and binding enables such protein domains to operate analogically as conformational rheostats.

On the polymer physics origins of protein folding thermodynamics

A remarkable feature of the spontaneous folding of many small proteins is the striking similarity in the thermodynamics of the folding process. This process is characterized by simple two-state thermodynamics with large and compensating changes in entropy and enthalpy and a funnel-like free energy landscape with a free-energy barrier that varies linearly with temperature. One might attribute the commonality of this two-state folding behavior to features particular to these proteins (e.g., chain length, hydrophobic/hydrophilic balance, attributes of the native state) or one might suspect that this similarity in behavior has a more general polymer-physics origin. Here we show that this behavior is also typical for flexible homopolymer chains with sufficiently short range interactions. Two-state behavior arises from the presence of a low entropy ground (folded) state separated from a set of high entropy disordered (unfolded) states by a free energy barrier. This homopolymer model exhibits a funneled free energy landscape that reveals a complex underlying dynamics involving competition between folding and non-folding pathways. Despite the presence of multiple pathways, this simple physics model gives the robust result of two-state thermodynamics for both the cases of folding from a basin of expanded coil states and from a basin of compact globule states.

Interplay between the folding mechanism and binding modes in folding coupled to binding processes

Proteins that fold upon binding to their partners exhibit complex binding behavior such as induced-fit. Using molecular simulations we find that the ability to bind via induced-fit requires the disordered protein to fold downhill. Thus induced-fit appears as a manifestation of a conformational rheostat.

Toward a quantitative description of microscopic pathway heterogeneity in protein folding

Experimentally consistent statistical modeling of protein folding thermodynamics reveals unprecedented complexity with numerous parallel folding routes in five different proteins.

Imprints of function on the folding landscape: functional role for an intermediate in a conserved eukaryotic binding protein

In the computational characterization of single domain protein folding, the effective free energies of numerous microstates are projected onto few collective degrees of freedom that in turn serve as well-defined reaction coordinates. In this regard, one-dimensional (1D) free energy profiles are widely used mainly for their simplicity. Since folding and functional landscapes are interlinked, how well can these reduced representations capture the structural and dynamic features of functional states while being simultaneously consistent with experimental observables? We investigate this issue by characterizing the folding of the four-helix bundle bovine acyl-CoA binding protein (bACBP), which exhibits complex equilibrium and kinetic behaviours, employing an Ising-like statistical mechanical model and molecular simulations. We show that the features of the 1D free energy profile are sufficient to quantitatively reproduce multiple experimental observations including millisecond chevron-like kinetics and temperature dependence, a microsecond fast phase, barrier heights, unfolded state movements, the intermediate structure and average ϕ-values. Importantly, we find that the structural features of the native-like intermediate (partial disorder in helix 1) are intricately linked to a unique interplay between packing and electrostatics in this domain. By comparison with available experimental data, we propose that this intermediate determines the promiscuous functional behaviour of bACBP that exhibits broad substrate specificity. Our results present evidence to the possibility of employing the statistical mechanical model and the resulting 1D free energy profile to not just understand folding mechanisms but to even extract features of functionally relevant states and their energetic origins.

Consistent Treatment of Hydrophobicity in Protein Lattice Models Accounts for Cold Denaturation

The hydrophobic effect stabilizes the native structure of proteins by minimizing the unfavorable interactions between hydrophobic residues and water through the formation of a hydrophobic core. Here, we include the entropic and enthalpic contributions of the hydrophobic effect explicitly in an implicit solvent model. This allows us to capture two important effects: a length-scale dependence and a temperature dependence for the solvation of a hydrophobic particle. This consistent treatment of the hydrophobic effect explains cold denaturation and heat capacity measurements of solvated proteins.

A Method for Extracting the Free Energy Surface and Conformational Dynamics of Fast-Folding Proteins from Single Molecule Photon Trajectories

Single molecule fluorescence spectroscopy holds the promise of providing direct measurements of protein folding free energy landscapes and conformational motions. However, fulfilling this promise has been prevented by technical limitations, most notably, the difficulty in analyzing the small packets of photons per millisecond that are typically recorded from individual biomolecules. Such limitation impairs the ability to accurately determine conformational distributions and resolve sub-millisecond processes. Here we develop an analytical procedure for extracting the conformational distribution and dynamics of fast-folding proteins directly from time-stamped photon arrival trajectories produced by single molecule FRET experiments. Our procedure combines the maximum likelihood analysis originally developed by Gopich and Szabo with a statistical mechanical model that describes protein folding as diffusion on a one-dimensional free energy surface. Using stochastic kinetic simulations, we thoroughly tested the performance of the method in identifying diverse fast-folding scenarios, ranging from two-state to one-state downhill folding, as a function of relevant experimental variables such as photon count rate, amount of input data, and background noise. The tests demonstrate that the analysis can accurately retrieve the original one-dimensional free energy surface and microsecond folding dynamics in spite of the sub-megahertz photon count rates and significant background noise levels of current single molecule fluorescence experiments. Therefore, our approach provides a powerful tool for the quantitative analysis of single molecule FRET experiments of fast protein folding that is also potentially extensible to the analysis of any other biomolecular process governed by sub-millisecond conformational dynamics.

Slow proton transfer coupled to unfolding explains the puzzling results of single-molecule experiments on BBL, a paradigmatic downhill folding protein

A battery of thermodynamic, kinetic, and structural approaches has indicated that the small α-helical protein BBL folds-unfolds via the one-state downhill scenario. Yet, single-molecule fluorescence spectroscopy offers a more conflicting view. Single-molecule experiments at pH 6 show a unique half-unfolded conformational ensemble at mid denaturation, whereas other experiments performed at higher pH show a bimodal distribution, as expected for two-state folding. Here we use thermodynamic and laser T-jump kinetic experiments combined with theoretical modeling to investigate the pH dependence of BBL stability, folding kinetics and mechanism within the pH 6-11 range. We find that BBL unfolding is tightly coupled to the protonation of one of its residues with an apparent pKa of ~ 7. Therefore, in chemical denaturation experiments around neutral pH BBL unfolds gradually, and also converts in binary fashion to the protonated species. Moreover, under the single-molecule experimental conditions (denaturant midpoint and 279 K), we observe that proton transfer is much slower than the ~ 15 microseconds folding-unfolding kinetics of BBL. The relaxation kinetics is distinctly biphasic, and the overall relaxation time (i.e. 0.2-0.5 ms) becomes controlled by the proton transfer step. We then show that a simple theoretical model of protein folding coupled to proton transfer explains quantitatively all these results as well as the two sets of single-molecule experiments, including their more puzzling features. Interestingly, this analysis suggests that BBL unfolds following a one-state downhill folding mechanism at all conditions. Accordingly, the source of the bimodal distributions observed during denaturation at pH 7-8 is the splitting of the unique conformational ensemble of BBL onto two slowly inter-converting protonation species. Both, the unprotonated and protonated species unfold gradually (one-state downhill), but they exhibit different degree of unfolding at any given condition because the native structure is less stable for the protonated form.

Replica exchange molecular dynamics simulations of an α/β-type small acid soluble protein (SASP)

Small acid soluble proteins (SASPs) of α/β-type play a major role in the resistance of spore DNAs to external assaults. It has been found that α/β-type SASP exhibits intrinsic disorder on isolation, but it acquires a defined native state upon binding to DNA. This disorder to order transition is not yet understood. Other questions related to the role of the thermodynamics and structure of the individual protein in the complex formation remain elusive. Characterization of the unbound state of α/β-type SASP in experiments could be a challenging problem because of the heterogeneous nature of the ensemble. Here, computer simulations can help gain more insights into the unbound state of α/β-type SASP. In the present work, by using replica exchange molecular dynamics (REMD), we simulated an α/β-type SASP on isolation with an implicit solvent. We found that α/β-type SASP undergoes a continuous phase transition with a small free energy barrier, a common feature of intrinsically disordered proteins (IDPs). Additionally, we detected the presence of residual α-helical structures at local level and a high degree of plasticity in the chain which can contribute to the fast disorder to order transition by reducing the fly-casting mechanism.

Predictions from an Ising-like Statistical Mechanical Model on the Dynamic and Thermodynamic Effects of Protein Surface Electrostatics

Charged residues on the surface of a protein are known hot-spots for post-translational modification, protein/ligand-binding, and tuning conformational stabilities. Recent experimental evidence points to the fact that surface electrostatics can also modulate thermodynamic barriers and hence folding mechanisms. To probe for this behavior across different proteins, we develop a novel version of the Wako-Saitô-Muñoz-Eaton (WSME) model in which we include an electrostatic potential term in the energy function while simplifying the treatment of solvation free energy. Both of the energy terms are obtained by quantitatively fitting the model to differential scanning calorimetry (DSC) experiments that carry critical information on the protein partition function. We characterize four sets of structural/functional homologues (HEWL/BLA, CspB, engrailed, α-spectrin) either by fitting the experimental data of a single domain in the homologous set and predicting the conformational behavior of the rest with the same set of parameters or by performing semiblind predictions. The model with the added electrostatic term is able to successfully reproduce the order of thermodynamic stabilities and relaxation rates of most of the homologues. In parallel, we predict diverse conformational features including a wide range of thermodynamic barriers (∼9-40 kJ/mol), broad native ensembles in helical proteins, structured unfolded states and intermediates, rugged folding landscapes, and further provide an independent protein-specific estimate of the folding speed limit at 298 K (1/(7-300 μs)). Our results are evidence that protein surface electrostatics can be tailored to not only engineer stabilities but also folding mechanisms and the ruggedness of the underlying landscape.

Fitting in a complex χ(2) landscape using an optimized hypersurface sampling

Fitting a data set with a parametrized model can be seen geometrically as finding the global minimum of the χ(2) hypersurface, depending on a set of parameters {P(i)}. This is usually done using the Levenberg-Marquardt algorithm. The main drawback of this algorithm is that despite its fast convergence, it can get stuck if the parameters are not initialized close to the final solution. We propose a modification of the Metropolis algorithm introducing a parameter step tuning that optimizes the sampling of parameter space. The ability of the parameter tuning algorithm together with simulated annealing to find the global χ(2) hypersurface minimum, jumping across χ(2){P(i)} barriers when necessary, is demonstrated with synthetic functions and with real data.