Flexibly varying folding mechanism of a nearly symmetrical protein: B domain of protein A

Helical Content Correlations and Hydration Structures of the Folding Ensemble of the B Domain of Protein A

The B domain of protein A (BdpA), a small three-helix bundle, folds on a time scale of a few microseconds with heterogeneous native and unfolded states. It is widely used as a model for understanding protein folding mechanisms. In this work, we use structure-based models (SBMs) and atomistic simulations to comprehensively investigate how BdpA folding is associated with the formation of its secondary structure. The energy landscape visualization method (ELViM) was used to characterize the pathways that connect the folded and unfolded states of BdpA as well as the sets of structures displaying specific ellipticity patterns. We show that the native state conformational diversity is due mainly to the conformational variability of helix I. Helices I, II, and III occur in a weakly correlated manner, with Spearman's rank correlation coefficients of 0.1539 (I and II), 0.1259 (I and III), and 0.2561 (II and III). These results, therefore, suggest the highest cooperativity between helices II and III. Our results allow the clustering of partially folded structures of folding of the B domain of protein A on the basis of its secondary structure, paving the way to an understanding of environmental factors in the relative stability of the basins of the folding ensemble, which are illustrated by the structural dependency of the protein hydration structures, as computed with minimum-distance distribution functions.

Accurate prediction of protein folding mechanisms by simple structure-based statistical mechanical models

Recent breakthroughs in highly accurate protein structure prediction using deep neural networks have made considerable progress in solving the structure prediction component of the 'protein folding problem'. However, predicting detailed mechanisms of how proteins fold into specific native structures remains challenging, especially for multidomain proteins constituting most of the proteomes. Here, we develop a simple structure-based statistical mechanical model that introduces nonlocal interactions driving the folding of multidomain proteins. Our model successfully predicts protein folding processes consistent with experiments, without the limitations of protein size and shape. Furthermore, slight modifications of the model allow prediction of disulfide-oxidative and disulfide-intact protein folding. These predictions depict details of the folding processes beyond reproducing experimental results and provide a rationale for the folding mechanisms. Thus, our physics-based models enable accurate prediction of protein folding mechanisms with low computational complexity, paving the way for solving the folding process component of the 'protein folding problem'.

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.

Thermodynamics and folding landscapes of large proteins from a statistical mechanical model

Statistical mechanical models that afford an intermediate resolution between macroscopic chemical models and all-atom simulations have been successful in capturing folding behaviors of many small single-domain proteins. However, the applicability of one such successful approach, the Wako-Saitô-Muñoz-Eaton (WSME) model, is limited by the size of the protein as the number of conformations grows exponentially with protein length. In this work, we surmount this size limitation by introducing a novel approximation that treats stretches of 3 or 4 residues as blocks, thus reducing the phase space by nearly three orders of magnitude. The performance of the 'bWSME' model is validated by comparing the predictions for a globular enzyme (RNase H) and a repeat protein (IκBα), against experimental observables and the model without block approximation. Finally, as a proof of concept, we predict the free-energy surface of the 370-residue, multi-domain maltose binding protein and identify an intermediate in good agreement with single-molecule force-spectroscopy measurements. The bWSME model can thus be employed as a quantitative predictive tool to explore the conformational landscapes of large proteins, extract the structural features of putative intermediates, identify parallel folding paths, and thus aid in the interpretation of both ensemble and single-molecule experiments.

Folding with a protein's native shortcut network

A complex network approach to protein folding is proposed, wherein a protein's contact map is reconceptualized as a network of shortcut edges, and folding is steered by a structural characteristic of this network. Shortcut networks are generated by a known message passing algorithm operating on protein residue networks. It is found that the shortcut networks of native structures (SCN0s) are relevant graph objects with which to study protein folding at a formal level. The logarithm form of their contact order (SCN0_lnCO) correlates significantly with folding rate of two-state and nontwo-state proteins. The clustering coefficient of SCN0s (C_SCN0 ) correlates significantly with folding rate, transition-state placement and stability of two-state folders. Reasonable folding pathways for several model proteins are produced when C_SCN0 is used to combine protein segments incrementally to form the native structure. The folding bias captured by C_SCN0 is detectable in non-native structures, as evidenced by Molecular Dynamics simulation generated configurations for the fast folding Villin-headpiece peptide. These results support the use of shortcut networks to investigate the role protein geometry plays in the folding of both small and large globular proteins, and have implications for the design of multibody interaction schemes in folding models. One facet of this geometry is the set of native shortcut triangles, whose attributes are found to be well-suited to identify dehydrated intraprotein areas in tight turns, or at the interface of different secondary structure elements.

Unified understanding of folding and binding mechanisms of globular and intrinsically disordered proteins

Extensive experimental and theoretical studies have advanced our understanding of the mechanisms of folding and binding of globular proteins, and coupled folding and binding of intrinsically disordered proteins (IDPs). The forces responsible for conformational changes and binding are common in both proteins; however, these mechanisms have been separately discussed. Here, we attempt to integrate the mechanisms of coupled folding and binding of IDPs, folding of small and multi-subdomain proteins, folding of multimeric proteins, and ligand binding of globular proteins in terms of conformational selection and induced-fit mechanisms as well as the nucleation–condensation mechanism that is intermediate between them. Accumulating evidence has shown that both the rate of conformational change and apparent rate of binding between interacting elements can determine reaction mechanisms. Coupled folding and binding of IDPs occurs mainly by induced-fit because of the slow folding in the free form, while ligand binding of globular proteins occurs mainly by conformational selection because of rapid conformational change. Protein folding can be regarded as the binding of intramolecular segments accompanied by secondary structure formation. Multi-subdomain proteins fold mainly by the induced-fit (hydrophobic collapse) mechanism, as the connection of interacting segments enhances the binding (compaction) rate. Fewer hydrophobic residues in small proteins reduce the intramolecular binding rate, resulting in the nucleation–condensation mechanism. Thus, the folding and binding of globular proteins and IDPs obey the same general principle, suggesting that the coarse-grained, statistical mechanical model of protein folding is promising for a unified theoretical description of all mechanisms.

Thermodynamics of Protein Folding Studied by Umbrella Sampling along a Reaction Coordinate of Native Contacts

Spontaneous transitions between the native and non-native protein conformations are normally rare events that hardly take place in typical unbiased molecular dynamics simulations. It was recently demonstrated that such transitions can be well described by a reaction coordinate, Q, that represents the collective fraction of the native contacts between the protein atoms. Here we attempt to use this reaction coordinate to enhance the conformational sampling. We perform umbrella sampling simulations with biasing potentials on Q for two model proteins, Trp-Cage and BBA, using the CHARMM force field. Hamiltonian replica exchange is implemented in these simulations to further facilitate the sampling. The simulations appear to have reached satisfactory convergence, resulting in unbiased free energies as a function of Q. In addition to the native structure, multiple folded conformations are identified in the reconstructed equilibrium ensemble. Some conformations without any native contacts nonetheless have rather compact geometries and are stabilized by hydrogen bonds not present in the native structure. Whereas the enhanced sampling along Q reasonably reproduces the equilibrium conformational space, we also find that the folding of an α-helix in Trp-Cage is a slow degree of freedom orthogonal to Q and therefore cannot be accelerated by biasing the reaction coordinate. Overall, we conclude that whereas Q is an excellent parameter to analyze the simulations, it is not necessarily a perfect reaction coordinate for enhanced sampling, and better incorporation of other slow degrees of freedom may further improve this reaction coordinate.

Cooperativity and modularity in protein folding

A simple statistical mechanical model proposed by Wako and Saitô has explained the aspects of protein folding surprisingly well. This model was systematically applied to multiple proteins by Muñoz and Eaton and has since been referred to as the Wako-Saitô-Muñoz-Eaton (WSME) model. The success of the WSME model in explaining the folding of many proteins has verified the hypothesis that the folding is dominated by native interactions, which makes the energy landscape globally biased toward native conformation. Using the WSME and other related models, Saitô emphasized the importance of the hierarchical pathway in protein folding; folding starts with the creation of contiguous segments having a native-like configuration and proceeds as growth and coalescence of these segments. The Φ-values calculated for barnase with the WSME model suggested that segments contributing to the folding nucleus are similar to the structural modules defined by the pattern of native atomic contacts. The WSME model was extended to explain folding of multi-domain proteins having a complex topology, which opened the way to comprehensively understanding the folding process of multi-domain proteins. The WSME model was also extended to describe allosteric transitions, indicating that the allosteric structural movement does not occur as a deterministic sequential change between two conformations but as a stochastic diffusive motion over the dynamically changing energy landscape. Statistical mechanical viewpoint on folding, as highlighted by the WSME model, has been renovated in the context of modern methods and ideas, and will continue to provide insights on equilibrium and dynamical features of proteins.

Characterization of protein folding by a Φ-value calculation with a statistical-mechanical model

The Φ-value analysis approach provides information about transition-state structures along the folding pathway of a protein by measuring the effects of an amino acid mutation on folding kinetics. Here we compared the theoretically calculated Φ values of 27 proteins with their experimentally observed Φ values; the theoretical values were calculated using a simple statistical-mechanical model of protein folding. The theoretically calculated Φ values reflected the corresponding experimentally observed Φ values with reasonable accuracy for many of the proteins, but not for all. The correlation between the theoretically calculated and experimentally observed Φ values strongly depends on whether the protein-folding mechanism assumed in the model holds true in real proteins. In other words, the correlation coefficient can be expected to illuminate the folding mechanisms of proteins, providing the answer to the question of which model more accurately describes protein folding: the framework model or the nucleation-condensation model. In addition, we tried to characterize protein folding with respect to various properties of each protein apart from the size and fold class, such as the free-energy profile, contact-order profile, and sensitivity to the parameters used in the Φ-value calculation. The results showed that any one of these properties alone was not enough to explain protein folding, although each one played a significant role in it. We have confirmed the importance of characterizing protein folding from various perspectives. Our findings have also highlighted that protein folding is highly variable and unique across different proteins, and this should be considered while pursuing a unified theory of protein folding.

Building a minimum frustration framework for brain functions over long time scales

The minimum frustration principle (MFP) is a computational approach stating that, over the long time scales of evolution, proteins' free energy decreases more than expected by thermodynamical constraints as their amino acids assume conformations progressively closer to the lowest energetic state. This Review shows that this general principle, borrowed from protein folding dynamics, can also be fruitfully applied to nervous function. Highlighting the foremost role of energetic requirements, macromolecular dynamics, and above all intertwined time scales in brain activity, the MFP elucidates a wide range of mental processes from sensations to memory retrieval. Brain functions are compared with trajectories that, over long nervous time scales, are attracted toward the low-energy bottom of funnel-like structures characterized by both robustness and plasticity. We discuss how the principle, derived explicitly from evolution and selection of a funneling structure from microdynamics of contacts, is unlike other brain models equipped with energy landscapes, such as the Bayesian and free energy principles and the Hopfield networks. In summary, we make available a novel approach to brain function cast in a biologically informed fashion, with the potential to be operationalized and assessed empirically. © 2016 Wiley Periodicals, Inc.

Complexity of the folding transition of the B domain of protein A revealed by the high-speed tracking of single-molecule fluorescence time series

The equilibrium unfolding transition of the B domain of protein A (BdpA) was investigated by using single-molecule fluorescence spectroscopy based on line-confocal detection of fast-flowing samples. The method achieved the time resolution of 120 μs and the observation time of a few milliseconds in the single-molecule time-series measurements of fluorescence resonance energy transfer (FRET). Two samples of BdpA doubly labeled with donor and acceptor fluorophores, the first possessing fluorophores at residues 22 and 55 (sample 1) and the second at residues 5 and 55 (sample 2), were prepared. The equilibrium unfolding transition induced by guanidium chloride (GdmCl) was monitored by bulk measurements and demonstrated that the both samples obey the apparent two-state unfolding. In the absence of GdmCl, the single-molecule FRET measurements for the both samples showed a single peak assignable to the native state (N). The FRET efficiency for N shifts to lower values as the increase of GdmCl concentration, suggesting the swelling of the native state structure. At the higher concentration of GdmCl, the both samples convert to the unfolded state (U). Near the unfolding midpoint for sample 1, the kinetic exchange between N and U causes the averaging of the two states and the higher values of the relative fluctuation. The time series for different molecules in U showed slightly different FRET efficiencies, suggesting the apparent heterogeneity. Thus, the high-speed tracking of fluorescence signals from single molecules revealed a complexity and heterogeneity hidden in the apparent two-state behavior of protein folding.

Protein folding as a complex reaction: a two-component potential for the driving force of folding and its variation with folding scenario

The Helmholtz decomposition of the vector field of probability fluxes in a two-dimensional space of collective variables makes it possible to introduce a potential for the driving force of protein folding [Chekmarev, J. Chem. Phys. 139 (2013) 145103]. The potential has two components: one component (Φ) is responsible for the source and sink of the folding flow, which represent, respectively, the unfolded and native state of the protein, and the other (Ψ) accounts for the flow vorticity inherently generated at the periphery of the flow field and provides the canalization of the flow between the source and sink. Both components obey Poisson’s equations with the corresponding source/sink terms. In the present paper, we consider how the shape of the potential changes depending on the scenario of protein folding. To mimic protein folding dynamics projected onto a two-dimensional space of collective variables, the two-dimensional Müller and Brown potential is employed. Three characteristic scenarios are considered: a single pathway from the unfolded to the native state without intermediates, two parallel pathways without intermediates, and a single pathway with an off-pathway intermediate. To determine the probability fluxes, the hydrodynamic description of the folding reaction is used, in which the first-passage folding is viewed as a steady flow of the representative points of the protein from the unfolded to the native state. We show that despite the possible complexity of the folding process, the Φ-component is simple and universal in shape. The Ψ-component is more complex and reveals characteristic features of the process of folding. The present approach is potentially applicable to other complex reactions, for which the transition from the reactant to the product can be described in a space of two (collective) variables.

The role of high-dimensional diffusive search, stabilization, and frustration in protein folding

Proteins are polymeric molecules with many degrees of conformational freedom whose internal energetic interactions are typically screened to small distances. Therefore, in the high-dimensional conformation space of a protein, the energy landscape is locally relatively flat, in contrast to low-dimensional representations, where, because of the induced entropic contribution to the full free energy, it appears funnel-like. Proteins explore the conformation space by searching these flat subspaces to find a narrow energetic alley that we call a hypergutter and then explore the next, lower-dimensional, subspace. Such a framework provides an effective representation of the energy landscape and folding kinetics that does justice to the essential characteristic of high-dimensionality of the search-space. It also illuminates the important role of nonnative interactions in defining folding pathways. This principle is here illustrated using a coarse-grained model of a family of three-helix bundle proteins whose conformations, once secondary structure has formed, can be defined by six rotational degrees of freedom. Two folding mechanisms are possible, one of which involves an intermediate. The stabilization of intermediate subspaces (or states in low-dimensional projection) in protein folding can either speed up or slow down the folding rate depending on the amount of native and nonnative contacts made in those subspaces. The folding rate increases due to reduced-dimension pathways arising from the mere presence of intermediate states, but decreases if the contacts in the intermediate are very stable and introduce sizeable topological or energetic frustration that needs to be overcome. Remarkably, the hypergutter framework, although depending on just a few physically meaningful parameters, can reproduce all the types of experimentally observed curvature in chevron plots for realizations of this fold.

Folding pathway of a multidomain protein depends on its topology of domain connectivity

How do the folding mechanisms of multidomain proteins depend on protein topology? We addressed this question by developing an Ising-like structure-based model and applying it for the analysis of free-energy landscapes and folding kinetics of an example protein, Escherichia coli dihydrofolate reductase (DHFR). DHFR has two domains, one comprising discontinuous N- and C-terminal parts and the other comprising a continuous middle part of the chain. The simulated folding pathway of DHFR is a sequential process during which the continuous domain folds first, followed by the discontinuous domain, thereby avoiding the rapid decrease in conformation entropy caused by the association of the N- and C-terminal parts during the early phase of folding. Our simulated results consistently explain the observed experimental data on folding kinetics and predict an off-pathway structural fluctuation at equilibrium. For a circular permutant for which the topological complexity of wild-type DHFR is resolved, the balance between energy and entropy is modulated, resulting in the coexistence of the two folding pathways. This coexistence of pathways should account for the experimentally observed complex folding behavior of the circular permutant.

Frustration-induced protein intrinsic disorder

Spontaneous folding into a specific native structure is the most important property of protein to perform their biological functions within organisms. Spontaneous folding is understood on the basis of an energy landscape picture based on the minimum frustration principle. Therefore, frustration seemingly only leads to protein functional disorder. However, frustration has recently been suggested to have a function in allosteric regulation. Functional frustration has the possibility to be a key to our deeper understanding of protein function. To explore another functional frustration, we theoretically examined structural frustration, which is designed to induce intrinsic disorder of a protein and its function through the coupled folding and binding. We extended the Wako-Saitô-Muñoz-Eaton model to take into account a frustration effect. With the model, we analyzed the binding part of neuron-restrictive silencer factor and showed that designed structural frustration in it induces intrinsic disorder. Furthermore, we showed that the folding and the binding are cooperative in interacting with a target protein. The cooperativity enables an intrinsically disordered protein to exhibit a sharp switch-like folding response to binding chemical potential change. Through this switch-like response, the structural frustration may contribute to the regulation function of interprotein interaction of the intrinsically disordered protein.

Take home lessons from studies of related proteins

The 'Fold Approach' involves a detailed analysis of the folding of several topologically, structurally and/or evolutionarily related proteins. Such studies can reveal determinants of the folding mechanism beyond the gross topology, and can dissect the residues required for folding from those required for stability or function. While this approach has not yet matured to the point where we can predict the native conformation of any polypeptide chain in silico, it has been able to highlight, amongst others, the specific residues that are responsible for nucleation, pathway malleability, kinetic intermediates, chain knotting, internal friction and Paracelsus switches. Some of the most interesting discoveries have resulted from the attempt to explain differences between homologues.

Context and force field dependence of the loss of protein backbone entropy upon folding using realistic denatured and native state ensembles

The loss of conformational entropy is the largest unfavorable quantity affecting a protein's stability. We calculate the reduction in the number of backbone conformations upon folding using the distribution of backbone dihedral angles (ϕ,ψ) obtained from an experimentally validated denatured state model, along with all-atom simulations for both the denatured and native states. The average loss of entropy per residue is TΔS(BB)(U-N) = 0.7, 0.9, or 1.1 kcal·mol(-1) at T = 298 K, depending on the force field used, with a 0.6 kcal·mol(-1) dispersion across the sequence. The average equates to a decrease of a factor of 3-7 in the number of conformations available per residue (f = Ω(Denatured)/Ω(Native)) or to a total of f(tot) = 3(n)-7(n) for an n residue protein. Our value is smaller than most previous estimates where f = 7-20, that is, our computed TΔS(BB)(U-N) is smaller by 10-100 kcal mol(-1) for n = 100. The differences emerge from our use of realistic native and denatured state ensembles as well as from the inclusion of accurate local sequence preferences, neighbor effects, and correlated motions (vibrations), in contrast to some previous studies that invoke gross assumptions about the entropy in either or both states. We find that the loss of entropy primarily depends on the local environment and less on properties of the native state, with the exception of α-helical residues in some force fields.

From coarse-grained to atomic-level characterization of protein dynamics: transition state for the folding of B domain of protein A

Atomic-level molecular dynamics simulations are widely used for the characterization of the structural dynamics of proteins; however, they are limited to shorter time scales than the duration of most of the relevant biological processes. Properly designed coarse-grained models that trade atomic resolution for efficient sampling allow access to much longer time-scales. In-depth understanding of the structural dynamics, however, must involve atomic details. In this study, we tested a method for the rapid reconstruction of all-atom models from α carbon atom positions in the application to convert a coarse-grained folding trajectory of a well described model system: the B domain of protein A. The results show that the method and the spatial resolution of the resulting coarse-grained models enable computationally inexpensive reconstruction of realistic all-atom models. Additionally, by means of structural clustering, we determined the most persistent ensembles of the key folding step, the transition state. Importantly, the analysis of the overall structural topologies suggests a dominant folding pathway. This, together with the all-atom characterization of the obtained ensembles, in the form of contact maps, matches the experimental results well.

Statistical mechanics of protein allostery: roles of backbone and side-chain structural fluctuations

A statistical mechanical model of allosteric transition of proteins is developed by extending the structure-based model of protein folding to cases that a protein has two different native conformations. Partition function is calculated exactly within the model and free-energy surfaces associated with allostery are derived. In this paper, the model of allosteric transition proposed in a previous paper [Proc. Natl. Acad. Sci. U.S.A 134, 7775 (2010)] is reformulated to describe both fluctuation in side-chain configurations and that in backbone structures in a balanced way. The model is applied to example proteins, Ras, calmodulin, and CheY: Ras undergoes the allosteric transition between guanosine diphosphate (GDP)-bound and guanosine triphosphate (GTP)-bound forms, and the model results show that the GDP-bound form is stabilized enough to prevent unnecessary signal transmission, but the conformation in the GTP-bound state bears large fluctuation in side-chain configurations, which may help to bind multiple target proteins for multiple pathways of signaling. The calculated results of calmodulin show the scenario of sequential ordering in Ca(2+) binding and the associated allosteric conformational change, which are realized though the sequential appearing of pre-existing structural fluctuations, i.e., fluctuations to show structures suitable to bind Ca(2+) before its binding. Here, the pre-existing fluctuations to accept the second and third Ca(2+) ions are dominated by the side-chain fluctuation. In CheY, the calculated side-chain fluctuation of Tyr106 is coordinated with the backbone structural change in the β4-α4 loop, which explains the pre-existing Y-T coupling process in this protein. Ability of the model to explain allosteric transitions of example proteins supports the view that the large entropic effects lower the free-energy barrier of allosteric transition.

Simulation of chaperonin effect on protein folding: a shift from nucleation-condensation to framework mechanism

The iterative annealing mechanism (IAM) of chaperonin-assisted protein folding is explored in a framework of a well-established coarse-grained protein modeling tool, which enables the study of protein dynamics in a time-scale well beyond classical all-atom molecular mechanics. The chaperonin mechanism of action is simulated for two paradigm systems of protein folding, B domain of protein A (BdpA) and B1 domain of protein G (GB1), and compared to chaperonin-free simulations presented here for BdpA and recently published for GB1. The prediction of the BdpA transition state ensemble (TSE) is in perfect agreement with experimental findings. It is shown that periodic distortion of the polypeptide chains by hydrophobic chaperonin interactions can promote rapid folding and leads to a decrease in folding temperature. It is also demonstrated how chaperonin action prevents kinetically trapped conformations and modulates the observed folding mechanisms from nucleation-condensation to a more framework-like.

What lessons can be learned from studying the folding of homologous proteins?

The studies of the folding of structurally related proteins have proved to be a very important tool for investigating protein folding. Here we review some of the insights that have been gained from such studies. Our highlighted studies show just how such an investigation should be designed and emphasise the importance of the synergy between experiment and theory. We also stress the importance of choosing the right system carefully, exploiting the excellent structural and sequence databases at our disposal.

Entropic mechanism of large fluctuation in allosteric transition

A statistical mechanical model of allosteric transitions in proteins is developed by extending the structure-based model of protein folding to cases of multiple native conformations. The partition function is calculated exactly within the model and the free-energy surface reflecting allostery is derived. This approach is applied to an example protein, the receiver domain of the bacterial enhancer-binding protein NtrC. The model predicts the large entropy associated with a combinatorial number of preexisting transition routes. This large entropy lowers the free-energy barrier of the allosteric transition, which explains the large structural fluctuation observed in the NMR data of NtrC. The global allosteric transformation of NtrC is explained by the shift of preexisting distribution of conformations upon phosphorylation, but the local structural adjustment around the phosphorylation site is explained by the complementary induced-fit mechanism. Structural disordering accompanied by fluctuating interactions specific to two allosteric conformations underlies a large number of routes of allosteric transition.

Nearly symmetrical proteins: folding pathways and transition states

The folding pathways of the B domain of protein A have been the subject of many experimental and computational studies. Based on a statistical mechanical model, it has been suggested that the native state symmetry leads to multiple pathways, highly dependent on temperature and denaturant concentration. Experiments, however, have not confirmed this scenario. By considering four nearly symmetrical proteins, one of them being the above molecule, here we show that, if contact energies are properly taken into account, a different picture emerges from kinetic simulations of the above-mentioned model. This is characterized by a dominant folding pathway, which is consistent with the most recent experimental results. Given the simplicity of the model, we also report on a direct sampling of the transition state.

A Comparison of Three Perturbation Molecular Dynamics Methods for Modeling Conformational Transitions

Targeted, steered, and biased molecular dynamics (MD) are widely used methods for studying transition processes of biomolecules. They share the common feature of adding external perturbations along a conformational progress variable to guide the transition in a predefined direction in conformational space, yet differ in how these perturbations are applied. In the present paper, we report a comparison of these three methods on generating transition paths for two different processes: the unfolding of the B domain of protein A and a conformational transition of the catalytic domain of a Src kinase Lyn. Transition pathways were calculated with different simulation parameters including the choice of progress variable and the simulation length or biasing force constant. A comparison of the generated paths based on structural similarity finds that the three perturbation MD methods generate similar transition paths for a given progress variable in most cases. On the other hand, the path depends more strongly on the choice of progress variable used to move the system between the initial and final states. Potentials of mean force (PMF) were calculated starting from unfolding trajectories to estimate the relative probabilities of the paths. A lower PMF was found for the lowest biasing force constant with BMD.

Quantifying the structural requirements of the folding transition state of protein A and other systems

The B-domain of protein A is a small three-helix bundle that has been the subject of considerable experimental and theoretical investigation. Nevertheless, a unified view of the structure of the transition-state ensemble (TSE) is still lacking. To characterize the TSE of this surprisingly challenging protein, we apply a combination of psi analysis (which probes the role of specific side-chain to side-chain contacts) and kinetic H/D amide isotope effects (which measures hydrogen-bond content), building upon previous studies using mutational phi analysis (which probes the energetic influence of side-chain substitutions). The second helix is folded in the TSE, while helix formation appears just at the carboxy and amino termini of the first and third helices, respectively. The experimental data suggest a homogenous yet plastic TS with a native-like topology. This study generalizes our earlier conclusion, based on two larger alpha/beta proteins, that the TSEs of most small proteins achieve approximately 70% of their native state's relative contact order. This high percentage limits the degree of possible TS heterogeneity and requires a reevaluation of the structural content of the TSE of other proteins, especially when they are characterized as small or polarized.

Cooperativity, connectivity, and folding pathways of multidomain proteins

Multidomain proteins are ubiquitous in both prokaryotic and eukaryotic proteomes. Study on protein folding, however, has concentrated more on the isolated single domains of proteins, and there have been relatively few systematic studies on the effects of domain-domain interactions on folding. We here discuss this issue by examining human gammaD-crystallin, spore coat protein S, and a tandem array of the R16 and R17 domains of spectrin as example proteins by using a structure-based model of folding. The calculated results consistently explain the experimental data on folding pathways and effects of mutational perturbations, supporting the view that the connectivity of two domains and the distribution of domain-domain interactions in the native conformation are factors to determine kinetic and equilibrium properties of cooperative folding.

Predicting protein folding pathways at the mesoscopic level based on native interactions between secondary structure elements

Background

Since experimental determination of protein folding pathways remains difficult, computational techniques are often used to simulate protein folding. Most current techniques to predict protein folding pathways are computationally intensive and are suitable only for small proteins.

Results

By assuming that the native structure of a protein is known and representing each intermediate conformation as a collection of fully folded structures in which each of them contains a set of interacting secondary structure elements, we show that it is possible to significantly reduce the conformation space while still being able to predict the most energetically favorable folding pathway of large proteins with hundreds of residues at the mesoscopic level, including the pig muscle phosphoglycerate kinase with 416 residues. The model is detailed enough to distinguish between different folding pathways of structurally very similar proteins, including the streptococcal protein G and the peptostreptococcal protein L. The model is also able to recognize the differences between the folding pathways of protein G and its two structurally similar variants NuG1 and NuG2, which are even harder to distinguish. We show that this strategy can produce accurate predictions on many other proteins with experimentally determined intermediate folding states.

Conclusion

Our technique is efficient enough to predict folding pathways for both large and small proteins at the mesoscopic level. Such a strategy is often the only feasible choice for large proteins. A software program implementing this strategy (SSFold) is available at .

The complex folding pathways of protein A suggest a multiple-funnelled energy landscape

Folding proteins into their native states requires the formation of both secondary and tertiary structures. Many questions remain, however, as to whether these form into a precise order, and various pictures have been proposed that place the emphasis on the first or the second level of structure in describing folding. One of the favorite test models for studying this question is the B domain of protein A, which has been characterized by numerous experiments and simulations. Using the activation-relaxation technique coupled with a generic energy model (optimized potential for efficient peptide structure prediction), we generate more than 50 folding trajectories for this 60-residue protein. While the folding pathways to the native state are fully consistent with the funnel-like description of the free energy landscape, we find a wide range of mechanisms in which secondary and tertiary structures form in various orders. Our nonbiased simulations also reveal the presence of a significant number of non-native beta and alpha conformations both on and off pathway, including the visit, for a non-negligible fraction of trajectories, of fully ordered structures resembling the native state of nonhomologous proteins.

Mechanical unfolding and refolding pathways of ubiquitin

Mechanical unfolding and refolding of ubiquitin are studied by Monte Carlo simulations of a Gō model with binary variables. The exponential dependence of the time constants on the force is verified, and folding and unfolding lengths are computed, with good agreement with experimental results. Furthermore, the model exhibits intermediate kinetic states, as observed in experiments. Unfolding and refolding pathways and intermediate states, obtained by tracing single secondary structure elements, are consistent with simulations of previous all-atom models and with the experimentally observed step sizes.

Folding domain B of protein A on a dynamically partitioned free energy landscape

The B domain of staphylococcal protein A (BdpA) is a small helical protein that has been studied intensively in kinetics experiments and detailed computer simulations that include explicit water. The simulations indicate that BdpA needs to reorganize in crossing the transition barrier to facilitate folding its C-terminal helix (H3) onto the nucleus formed from helices H1 and H2. This process suggests frustration between two partially ordered forms of the protein, but recent varphi value measurements indicate that the transition structure is relatively constant over a broad range of temperatures. Here we develop a simplistic model to investigate the folding transition in which properties of the free energy landscape can be quantitatively compared with experimental data. The model is a continuation of the Muñoz-Eaton model to include the intermittency of contacts between structured parts of the protein, and the results compare variations in the landscape with denaturant and temperature to varphi value measurements and chevron plots of the kinetic rates. The topography of the model landscape (in particular, the feature of frustration) is consistent with detailed simulations even though variations in the varphi values are close to measured values. The transition barrier is smaller than indicated by the chevron data, but it agrees in order of magnitude with a similar alpha-carbon type of model. Discrepancies with the chevron plots are investigated from the point of view of solvent effects, and an approach is suggested to account for solvent participation in the model.

Universality and diversity of folding mechanics for three-helix bundle proteins

In this study we evaluate, at full atomic detail, the folding processes of two small helical proteins, the B domain of protein A and the Villin headpiece. Folding kinetics are studied by performing a large number of ab initio Monte Carlo folding simulations using a single transferable all-atom potential. Using these trajectories, we examine the relaxation behavior, secondary structure formation, and transition-state ensembles (TSEs) of the two proteins and compare our results with experimental data and previous computational studies. To obtain a detailed structural information on the folding dynamics viewed as an ensemble process, we perform a clustering analysis procedure based on graph theory. Moreover, rigorous p(fold) analysis is used to obtain representative samples of the TSEs and a good quantitative agreement between experimental and simulated Phi values is obtained for protein A. Phi values for Villin also are obtained and left as predictions to be tested by future experiments. Our analysis shows that the two-helix hairpin is a common partially stable structural motif that gets formed before entering the TSE in the studied proteins. These results together with our earlier study of Engrailed Homeodomain and recent experimental studies provide a comprehensive, atomic-level picture of folding mechanics of three-helix bundle proteins.

Searching for Multiple Folding Pathways of a Nearly Symmetrical Protein: Temperature Dependent Φ-Value Analysis of the B Domain of Protein A

The B domain of protein A (BdpA) is a popular paradigm for simulating protein folding pathways. The discrepancies between so many simulations and subsequent experimental testing may be attributable to the protein being highly symmetrical: changing experimental conditions could perturb the subtle interplay between the effects of symmetry in the native structure and the effects of asymmetry from specific interactions in a given sequence. If the protein folds via multiple pathways, perturbations, such as temperature, denaturant concentration, and mutation, should change the flux of micro pathways, leading to changes in the bulk properties of the transition state. We tested this hypothesis by conducting a Phi-analysis of BdpA as a function of temperature from 25.0 degrees C to 60.0 degrees C. The Phi-values had no significant dependence on temperature and the values at 55.0 degrees C (denaturing conditions) are very similar to those at 25.0 degrees C (folding conditions), indicating the structure of the transition state does not significantly change although the experimental conditions are considerably altered. The results suggest that BdpA folds via a single dominant folding pathway.

Rate determining factors in protein model structures

Previous research has shown a strong correlation of protein folding rates to the native state geometry, yet a complete explanation for this dependence is still lacking. Here we study the rate-geometry relationship with a simple statistical physics model, and focus on two classes of model geometries, representing ideal parallel and antiparallel structures. We find that the logarithm of the rate shows an almost perfect linear correlation with the "absolute contact order", but the slope depends on the particular class considered. We discuss these findings in the light of experimental results.

Downhill versus two-state protein folding in a statistical mechanical model

The authors address the problem of downhill protein folding in the framework of a simple statistical mechanical model, which allows an exact solution for the equilibrium and a semianalytical treatment of the kinetics. Focusing on protein 1BBL, a candidate for downhill folding behavior, and comparing it to the WW domain of protein PIN1, a two-state folder of comparable size, the authors show that there are qualitative differences in both the equilibrium and kinetic properties of the two molecules. However, the barrierless scenario which would be expected if 1BBL were a true downhill folder is observed only at low enough temperature.

Ising-like model for protein mechanical unfolding

The mechanical unfolding of proteins is studied by extending the Wako-Saitô-Muñoz-Eaton model. This model is generalized by including an external force, and its thermodynamics turns out to be exactly solvable. We consider two molecules, the 27th immunoglobulin domain of titin and protein PIN1. We determine equilibrium force-extension curves for the titin and study the mechanical unfolding of this molecule, finding good agreement with experiments. By using an extended form of the Jarzynski equality, we compute the free energy landscape of the PIN1 as a function of the molecule length.

Influence of temperature, friction, and random forces on folding of the B-domain of staphylococcal protein A: All-atom molecular dynamics in implicit solvent

The influences of temperature, friction, and random forces on the folding of protein A have been analyzed. A series of all-atom molecular dynamics folding simulations with the Amber ff99 potential and Generalized Born solvation, starting from the fully extended chain, were carried out for temperatures from 300 to 500 K, using (a) the Berendsen thermostat (with no explicit friction or random forces) and (b) Langevin dynamics (with friction and stochastic forces explicitly present in the system). The simulation temperature influences the relative time scale of the major events on the folding pathways of protein A. At lower temperatures, helix 2 folds significantly later than helices 1 and 3. However, with increasing temperature, the folding time of helix 2 approaches the folding times of helices 1 and 3. At lower temperatures, the complete formation of secondary and tertiary structure is significantly separated in time whereas, at higher temperatures, they occur simultaneously. These results suggest that some earlier experimental and theoretical observations of folding events, e.g., the order of helix formation, could depend on the temperature used in those studies. Therefore, the differences in temperature used could be one of the reasons for the discrepancies among published experimental and computational studies of the folding of protein A. Friction and random forces do not change the folding pathway that was observed in the simulations with the Berendsen thermostat, but their explicit presence in the system extends the folding time of protein A.