Investigation of routes and funnels in protein folding by free energy functional methods

Potential Self-Peptide Inhibitors of the SARS-CoV-2 Main Protease

The SARS-CoV-2 main protease (Mpro) plays an essential role in viral replication, cleaving viral polyproteins into functional proteins. This makes Mpro an important drug target. Mpro consists of an N-terminal catalytic domain and a C-terminal α-helical domain (MproC). Previous studies have shown that peptides derived from a given protein sequence (self-peptides) can affect the folding and, in turn, the function of that protein. Since the SARS-CoV-1 MproC is known to stabilize its Mpro and regulate its function, we hypothesized that SARS-CoV-2 MproC-derived self-peptides may modulate the folding and the function of SARS-CoV-2 Mpro. To test this, we studied the folding of MproC in the presence of various self-peptides using coarse-grained structure-based models and molecular dynamics simulations. In these simulations of MproC and one self-peptide, we found that two self-peptides, the α1-helix and the loop between α4 and α5 (loop4), could replace the equivalent native sequences in the MproC structure. Replacement of either sequence in full-length Mpro should, in principle, be able to perturb Mpro function albeit through different mechanisms. Some general principles for the rational design of self-peptide inhibitors emerge: The simulations show that prefolded self-peptides are more likely to replace native sequences than those which do not possess structure. Additionally, the α1-helix self-peptide is kinetically stable and once inserted rarely exchanges with the native α1-helix, while the loop4 self-peptide is easily replaced by the native loop4, making it less useful for modulating function. In summary, a prefolded α1-derived peptide should be able to inhibit SARS-CoV-2 Mpro function.

Biotinylation Eliminates the Intermediate State of Top7 Designed with an HIV-1 Epitope

Broadly neutralizing antibodies against HIV-1 are rare with the 2F5 antibody being one of the most protective. Insertion of an antibody epitope into a stable and small protein scaffold overcomes many of the obstacles found to produce antibodies. However, the design leads to grafting of epitopes that may cause protein aggregation. Here, I investigated the 2F5 epitope grafted into the Top7 as the scaffold in which the resulting immunoreactive protein precipitates along the storage time, as opposed to its completely soluble biotinylated version. Molecular dynamics showed that biotinylation eliminates the intermediate state of the scaffold-epitope Top7-2F5 by switching a noncooperative to a cooperative folding. The aggregation propensity of the Top7-designed proteins is examined in light of thermodynamic cooperativity and kinetic traps along the decreasing depth of the intermediate ensemble in the free energy landscape. This protocol may predict stable and soluble scaffold-epitopes with the purpose of composing novel therapeutic and diagnostic platforms.

High Energy Channeling and Malleable Transition States: Molecular Dynamics Simulations and Free Energy Landscapes for the Thermal Unfolding of Protein U1A and 13 Mutants

The spliceosome protein U1A is a prototype case of the RNA recognition motif (RRM) ubiquitous in biological systems. The in vitro kinetics of the chemical denaturation of U1A indicate that the unfolding of U1A is a two-state process but takes place via high energy channeling and a malleable transition state, an interesting variation of typical two-state behavior. Molecular dynamics (MD) simulations have been applied extensively to the study of two-state unfolding and folding of proteins and provide an opportunity to obtain a theoretical account of the experimental results and a molecular model for the transition state ensemble. We describe herein all-atom MD studies including explicit solvent of up to 100 ns on the thermal unfolding (UF) of U1A and 13 mutants. Multiple MD UF trajectories are carried out to ensure accuracy and reproducibility. A vector representation of the MD unfolding process in RMSD space is obtained and used to calculate a free energy landscape for U1A unfolding. A corresponding MD simulation and free energy landscape for the protein CI2, well known to follow a simple two state folding/unfolding model, is provided as a control. The results indicate that the unfolding pathway on the MD calculated free energy landscape of U1A shows a markedly extended transition state compared with that of CI2. The MD results support the interpretation of the observed chevron plots for U1A in terms of a high energy, channel-like transition state. Analysis of the MDUF structures shows that the transition state ensemble involves microstates with most of the RRM secondary structure intact but expanded by ~14% with respect to the radius of gyration. Comparison with results on a prototype system indicates that the transition state involves an ensemble of molten globule structures and extends over the region of ~1-35 ns in the trajectories. Additional MDUF simulations were carried out for 13 U1A mutants, and the calculated φ-values show close accord with observed results and serve to validate our methodology.

Small Neutral Crowding Solute Effects on Protein Folding Thermodynamic Stability and Kinetics

Molecular crowding is a ubiquitous phenomenon in biological systems, with significant consequences on protein folding and stability. Small compounds, such as the osmolyte trimethylamine N-oxide (TMAO), can also present similar effects. To analyze the effects arising from these solute-like molecules, we performed a series of crowded coarse-grained folding simulations. Two well-known proteins were chosen, CI2 and SH3, modeled by the alpha-carbon-structure-based model. In the simulations, the crowding molecules were represented by low-sized neutral atom beads in different concentrations. The results show that a low level of the volume fraction occupied by neutral agents can change protein stability and folding kinetics for the two systems. However, the kinetics were shown to be unaffected in their respective folding temperatures. The faster kinetics correlates with changes in the folding route for systems at the same temperature, where non-native contacts were shown to be relevant. The transition states of the two systems with and without crowders are similar. In the case of SH3, there are differences in the structuring of two strands, which may be associated with the increase in its folding rate, in addition to the destabilization of the denatured ensemble. The present study also detected a crossover in the thermodynamic stability behavior, previously observed experimentally and theoretically. As the temperature increases, crowders change from destabilizing to stabilizing agents.

Protein folding from heterogeneous unfolded state revealed by time-resolved X-ray solution scattering

One of the most challenging tasks in biological science is to understand how a protein folds. In theoretical studies, the hypothesis adopting a funnel-like free-energy landscape has been recognized as a prominent scheme for explaining protein folding in views of both internal energy and conformational heterogeneity of a protein. Despite numerous experimental efforts, however, comprehensively studying protein folding with respect to its global conformational changes in conjunction with the heterogeneity has been elusive. Here we investigate the redox-coupled folding dynamics of equine heart cytochrome c (cyt-c) induced by external electron injection by using time-resolved X-ray solution scattering. A systematic kinetic analysis unveils a kinetic model for its folding with a stretched exponential behavior during the transition toward the folded state. With the aid of the ensemble optimization method combined with molecular dynamics simulations, we found that during the folding the heterogeneously populated ensemble of the unfolded state is converted to a narrowly populated ensemble of folded conformations. These observations obtained from the kinetic and the structural analyses of X-ray scattering data reveal that the folding dynamics of cyt-c accompanies many parallel pathways associated with the heterogeneously populated ensemble of unfolded conformations, resulting in the stretched exponential kinetics at room temperature. This finding provides direct evidence with a view to microscopic protein conformations that the cyt-c folding initiates from a highly heterogeneous unfolded state, passes through still diverse intermediate structures, and reaches structural homogeneity by arriving at the folded state.

The Role of Electrostatics and Folding Kinetics on the Thermostability of Homologous Cold Shock Proteins

Understanding which aspects contribute to the thermostability of proteins is a challenge that has persisted for decades, and it is of great relevance for protein engineering. Several types of interactions can influence the thermostability of a protein. Among them, the electrostatic interactions have been a target of particular attention. Aiming to explore how this type of interaction can affect protein thermostability, this paper investigated four homologous cold shock proteins from psychrophilic, mesophilic, thermophilic, and hyperthermophilic organisms using a set of theoretical methodologies. It is well-known that electrostatics as well as hydrophobicity are key-elements for the stabilization of these proteins. Therefore, both interactions were initially analyzed in the native structure of each protein. Electrostatic interactions present in the native structures were calculated with the Tanford-Kirkwood model with solvent accessibility, and the amount of hydrophobic surface area buried upon folding was estimated by measuring both folded and extended structures. On the basis of Energy Landscape Theory, the local frustration and the simplified alpha-carbon structure-based model were modeled with a Debye-Hückel potential to take into account the electrostatics and the effects of an implicit solvent. Thermodynamic data for the structure-based model simulations were collected and analyzed using the Weighted Histogram Analysis and Stochastic Diffusion methods. Kinetic quantities including folding times, transition path times, folding routes, and Φ values were also obtained. As a result, we found that the methods are able to qualitatively infer that electrostatic interactions play an important role on the stabilization of the most stable thermophilic cold shock proteins, showing agreement with the experimental data.

Atomistic simulations indicate the functional loop-to-coiled-coil transition in influenza hemagglutinin is not downhill

Influenza hemagglutinin (HA) mediates viral entry into host cells through a large-scale conformational rearrangement at low pH that leads to fusion of the viral and endosomal membranes. Crystallographic and biochemical data suggest that a loop-to-coiled-coil transition of the B-loop region of HA is important for driving this structural rearrangement. However, the microscopic picture for this proposed "spring-loaded" movement is missing. In this study, we focus on understanding the transition of the B loop and perform a set of all-atom molecular dynamics simulations of the full B-loop trimeric structure with the CHARMM36 force field. The free-energy profile constructed from our simulations describes a B loop that stably folds half of the postfusion coiled coil in tens of microseconds, but the full coiled coil is unfavorable. A buried hydrophilic residue, Thr59, is implicated in destabilizing the coiled coil. Interestingly, this conserved threonine is the only residue in the B loop that strictly differentiates between the group 1 and 2 HA molecules. Microsecond-scale constant temperature simulations revealed that kinetic traps in the structural switch of the B loop can be caused by nonnative, intramonomer, or intermonomer β-sheets. The addition of the A helix stabilized the postfusion state of the B loop, but introduced the possibility for further β-sheet structures. Overall, our results do not support a description of the B loop in group 2 HAs as a stiff spring, but, rather, it allows for more structural heterogeneity in the placement of the fusion peptides during the fusion process.

Molecular dynamics simulations of lysozyme-lipid systems: probing the early steps of protein aggregation

Using the molecular dynamics simulation, the role of lipids in the lysozyme transition into the aggregation-competent conformation has been clarified. Analysis of the changes of lysozyme secondary structure upon its interactions with the model bilayer membranes composed of phosphatidylcholine and its mixtures with phosphatidylglycerol (10, 40, and 80 mol%) within the time interval of 100 ns showed that lipid-bound protein is characterized by the increased content of β-structures. Along with this, the formation of protein-lipid complexes was accompanied by the increase in the gyration radius and the decrease in RMSD of polypeptide chain. The results obtained were interpreted in terms of the partial unfolding of lysozyme molecule on the lipid matrix, with the magnitude of this effect being increased with increasing the fraction of anionic lipids. Based on the results of molecular dynamics simulation, a hypothetical model of the nucleation of lysozyme amyloid fibrils in a membrane environment was suggested.

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.

As Simple As Possible, but Not Simpler: Exploring the Fidelity of Coarse-Grained Protein Models for Simulated Force Spectroscopy

Mechanical unfolding of a single domain of loop-truncated superoxide dismutase protein has been simulated via force spectroscopy techniques with both all-atom (AA) models and several coarse-grained models having different levels of resolution: A Gō model containing all heavy atoms in the protein (HA-Gō), the associative memory, water mediated, structure and energy model (AWSEM) which has 3 interaction sites per amino acid, and a Gō model containing only one interaction site per amino acid at the Cα position (Cα-Gō). To systematically compare results across models, the scales of time, energy, and force had to be suitably renormalized in each model. Surprisingly, the HA-Gō model gives the softest protein, exhibiting much smaller force peaks than all other models after the above renormalization. Clustering to render a structural taxonomy as the protein unfolds showed that the AA, HA-Gō, and Cα-Gō models exhibit a single pathway for early unfolding, which eventually bifurcates repeatedly to multiple branches only after the protein is about half-unfolded. The AWSEM model shows a single dominant unfolding pathway over the whole range of unfolding, in contrast to all other models. TM alignment, clustering analysis, and native contact maps show that the AWSEM pathway has however the most structural similarity to the AA model at high nativeness, but the least structural similarity to the AA model at low nativeness. In comparison to the AA model, the sequence of native contact breakage is best predicted by the HA-Gō model. All models consistently predict a similar unfolding mechanism for early force-induced unfolding events, but diverge in their predictions for late stage unfolding events when the protein is more significantly disordered.

NTL9 Folding at Constant pH: The Importance of Electrostatic Interaction and pH Dependence

The folding process of the N-terminal domain of ribosomal protein L9 (NTL9) was investigated at constant-pH computer simulations. Evaluation of the role of electrostatic interaction during folding was carried out by including a Debye-Hückel potential into a Cα structure-based model (SBM). In this study, the charges of the ionizable residues and the electrostatic potential are susceptible to the solution conditions, such as pH and ionic strength, as well as to the presence of charged groups. Simulations were performed under different pHs, and the results were validated by comparing them with experimental values of pKa and with denaturation experiment data. Also, the free energy profiles, Φ-values, and folding routes were calculated for each condition. It was shown how charges vary along the folding under different pH, which is subject to different scenarios. This study reveals how simplified models can capture essential physical features, reproducing experimental results, and presenting the role of electrostatic interactions before, during, and after the transition state.

Geometrical Frustration in Interleukin-33 Decouples the Dynamics of the Functional Element from the Folding Transition State Ensemble

Interleukin-33 (IL-33) is currently the focus of multiple investigations into targeting pernicious inflammatory disorders. This mediator of inflammation plays a prevalent role in chronic disorders such as asthma, rheumatoid arthritis, and progressive heart disease. In order to better understand the possible link between the folding free energy landscape and functional regions in IL-33, a combined experimental and theoretical approach was applied. IL-33 is a pseudo- symmetrical protein composed of three distinct structural elements that complicate the folding mechanism due to competition for nucleation on the dominant folding route. Trefoil 1 constitutes the majority of the binding interface with the receptor whereas Trefoils 2 and 3 provide the stable scaffold to anchor Trefoil 1. We identified that IL-33 folds with a three-state mechanism, leading to a rollover in the refolding arm of its chevron plots in strongly native conditions. In addition, there is a second slower refolding phase that exhibits the same rollover suggesting similar limitations in folding along parallel routes. Characterization of the intermediate state and the rate limiting steps required for folding suggests that the rollover is attributable to a moving transition state, shifting from a post- to pre-intermediate transition state as you move from strongly native conditions to the midpoint of the transition. On a structural level, we found that initially, all independent Trefoil units fold equally well until a Q_CA of 0.35 when Trefoil 1 will backtrack in order to allow Trefoils 2 and 3 to fold in the intermediate state, creating a stable scaffold for Trefoil 1 to fold onto during the final folding transition. The formation of this intermediate state and subsequent moving transition state is a result of balancing the difficulty in folding the functionally important Trefoil 1 onto the remainder of the protein. Taken together our results indicate that the functional element of the protein is geometrically frustrated, requiring the more stable elements to fold first, acting as a scaffold for docking of the functional element to allow productive folding to the native state.

Conformational frustration in calmodulin-target recognition

Calmodulin (CaM) is a primary calcium (Ca(2+) )-signaling protein that specifically recognizes and activates highly diverse target proteins. We explored the molecular basis of target recognition of CaM with peptides representing the CaM-binding domains from two Ca(2+) -CaM-dependent kinases, CaMKI and CaMKII, by employing experimentally constrained molecular simulations. Detailed binding route analysis revealed that the two CaM target peptides, although similar in length and net charge, follow distinct routes that lead to a higher binding frustration in the CaM-CaMKII complex than in the CaM-CaMKI complex. We discovered that the molecular origin of the binding frustration is caused by intermolecular contacts formed with the C-domain of CaM that need to be broken before the formation of intermolecular contacts with the N-domain of CaM. We argue that the binding frustration is important for determining the kinetics of the recognition process of proteins involving large structural fluctuations.

Detecting selection on protein stability through statistical mechanical models of folding and evolution

The properties of biomolecules depend both on physics and on the evolutionary process that formed them. These two points of view produce a powerful synergism. Physics sets the stage and the constraints that molecular evolution has to obey, and evolutionary theory helps in rationalizing the physical properties of biomolecules, including protein folding thermodynamics. To complete the parallelism, protein thermodynamics is founded on the statistical mechanics in the space of protein structures, and molecular evolution can be viewed as statistical mechanics in the space of protein sequences. In this review, we will integrate both points of view, applying them to detecting selection on the stability of the folded state of proteins. We will start discussing positive design, which strengthens the stability of the folded against the unfolded state of proteins. Positive design justifies why statistical potentials for protein folding can be obtained from the frequencies of structural motifs. Stability against unfolding is easier to achieve for longer proteins. On the contrary, negative design, which consists in destabilizing frequently formed misfolded conformations, is more difficult to achieve for longer proteins. The folding rate can be enhanced by strengthening short-range native interactions, but this requirement contrasts with negative design, and evolution has to trade-off between them. Finally, selection can accelerate functional movements by favoring low frequency normal modes of the dynamics of the native state that strongly correlate with the functional conformation change.

Comparison of structure determination methods for intrinsically disordered amyloid-β peptides

Intrinsically disordered proteins (IDPs) represent a new frontier in structural biology since the primary characteristic of IDPs is that structures need to be characterized as diverse ensembles of conformational substates. We compare two general but very different ways of combining NMR spectroscopy with theoretical methods to derive structural ensembles for the disease IDPs amyloid-β 1-40 and amyloid-β 1-42, which are associated with Alzheimer's Disease. We analyze the performance of de novo molecular dynamics and knowledge-based approaches for generating structural ensembles by assessing their ability to reproduce a range of NMR experimental observables. In addition to the comparison of computational methods, we also evaluate the relative value of different types of NMR data for refining or validating the IDP structural ensembles for these important disease peptides.

SIMS: a hybrid method for rapid conformational analysis

Proteins are at the root of many biological functions, often performing complex tasks as the result of large changes in their structure. Describing the exact details of these conformational changes, however, remains a central challenge for computational biology due the enormous computational requirements of the problem. This has engendered the development of a rich variety of useful methods designed to answer specific questions at different levels of spatial, temporal, and energetic resolution. These methods fall largely into two classes: physically accurate, but computationally demanding methods and fast, approximate methods. We introduce here a new hybrid modeling tool, the Structured Intuitive Move Selector (sims), designed to bridge the divide between these two classes, while allowing the benefits of both to be seamlessly integrated into a single framework. This is achieved by applying a modern motion planning algorithm, borrowed from the field of robotics, in tandem with a well-established protein modeling library. sims can combine precise energy calculations with approximate or specialized conformational sampling routines to produce rapid, yet accurate, analysis of the large-scale conformational variability of protein systems. Several key advancements are shown, including the abstract use of generically defined moves (conformational sampling methods) and an expansive probabilistic conformational exploration. We present three example problems that sims is applied to and demonstrate a rapid solution for each. These include the automatic determination of “active” residues for the hinge-based system Cyanovirin-N, exploring conformational changes involving long-range coordinated motion between non-sequential residues in Ribose-Binding Protein, and the rapid discovery of a transient conformational state of Maltose-Binding Protein, previously only determined by Molecular Dynamics. For all cases we provide energetic validations using well-established energy fields, demonstrating this framework as a fast and accurate tool for the analysis of a wide range of protein flexibility problems.

Microsecond folding experiments and simulations: a match is made

For the past two decades, protein folding experiments have been speeding up from the second or millisecond time scale to the microsecond time scale, and full-atom simulations have been extended from the nanosecond to the microsecond and even millisecond time scale. Where the two meet, it is now possible to compare results directly, allowing force fields to be validated and refined, and allowing experimental data to be interpreted in atomistic detail. In this perspective we compare recent experiments and simulations on the microsecond time scale, pointing out the progress that has been made in determining native structures from physics-based simulations, refining experiments and simulations to provide more quantitative underlying mechanisms, and tackling the problems of multiple reaction coordinates, downhill folding, and complex underlying structure of unfolded or misfolded states.

Native-based simulations of the binding interaction between RAP74 and the disordered FCP1 peptide

By dephosphorylating the C-terminal domain (CTD) of RNA polymerase II (Pol II), the Transcription Factor IIF (TFIIF)-associating CTD phosphatase (FCP1) performs an essential function in recycling Pol II for subsequent rounds of transcription. The interaction between FCP1 and TFIIF is mediated by the disordered C-terminal tail of FCP1, which folds to form an α-helix upon binding the RAP74 subunit of TFIIF. The present work reports a structure-based simulation study of this interaction between the folded winged-helix domain of RAP74 and the disordered C-terminal tail of FCP1. The comparison of measured and simulated chemical shifts suggests that the FCP1 peptide samples 40-60% of its native helical structure in the unbound disordered ensemble. Free energy calculations suggest that productive binding begins when RAP74 makes hydrophobic contacts with the C-terminal region of the FCP1 peptide. The FCP1 peptide then folds into an amphipathic helix by zipping up the binding interface. The relative plasticity of FCP1 results in a more cooperative binding mechanism, allows for a greater diversity of pathways leading to the bound complex, and may also eliminate the need for "backtracking" from contacts that form out of sequence.

Polymer uncrossing and knotting in protein folding, and their role in minimal folding pathways

We introduce a method for calculating the extent to which chain non-crossing is important in the most efficient, optimal trajectories or pathways for a protein to fold. This involves recording all unphysical crossing events of a ghost chain, and calculating the minimal uncrossing cost that would have been required to avoid such events. A depth-first tree search algorithm is applied to find minimal transformations to fold [Formula: see text], [Formula: see text], [Formula: see text], and knotted proteins. In all cases, the extra uncrossing/non-crossing distance is a small fraction of the total distance travelled by a ghost chain. Different structural classes may be distinguished by the amount of extra uncrossing distance, and the effectiveness of such discrimination is compared with other order parameters. It was seen that non-crossing distance over chain length provided the best discrimination between structural and kinetic classes. The scaling of non-crossing distance with chain length implies an inevitable crossover to entanglement-dominated folding mechanisms for sufficiently long chains. We further quantify the minimal folding pathways by collecting the sequence of uncrossing moves, which generally involve leg, loop, and elbow-like uncrossing moves, and rendering the collection of these moves over the unfolded ensemble as a multiple-transformation "alignment". The consensus minimal pathway is constructed and shown schematically for representative cases of an [Formula: see text], [Formula: see text], and knotted protein. An overlap parameter is defined between pathways; we find that [Formula: see text] proteins have minimal overlap indicating diverse folding pathways, knotted proteins are highly constrained to follow a dominant pathway, and [Formula: see text] proteins are somewhere in between. Thus we have shown how topological chain constraints can induce dominant pathway mechanisms in protein folding.

The dominant folding route minimizes backbone distortion in SH3

Energetic frustration in protein folding is minimized by evolution to create a smooth and robust energy landscape. As a result the geometry of the native structure provides key constraints that shape protein folding mechanisms. Chain connectivity in particular has been identified as an essential component for realistic behavior of protein folding models. We study the quantitative balance of energetic and geometrical influences on the folding of SH3 in a structure-based model with minimal energetic frustration. A decomposition of the two-dimensional free energy landscape for the folding reaction into relevant energy and entropy contributions reveals that the entropy of the chain is not responsible for the folding mechanism. Instead the preferred folding route through the transition state arises from a cooperative energetic effect. Off-pathway structures are penalized by excess distortion in local backbone configurations and contact pair distances. This energy cost is a new ingredient in the malleable balance of interactions that controls the choice of routes during protein folding.

A semi-analytical description of protein folding that incorporates detailed geometrical information

Much has been done to study the interplay between geometric and energetic effects on the protein folding energy landscape. Numerical techniques such as molecular dynamics simulations are able to maintain a precise geometrical representation of the protein. Analytical approaches, however, often focus on the energetic aspects of folding, including geometrical information only in an average way. Here, we investigate a semi-analytical expression of folding that explicitly includes geometrical effects. We consider a Hamiltonian corresponding to a Gaussian filament with structure-based interactions. The model captures local features of protein folding often averaged over by mean-field theories, for example, loop contact formation and excluded volume. We explore the thermodynamics and folding mechanisms of beta-hairpin and alpha-helical structures as functions of temperature and Q, the fraction of native contacts formed. Excluded volume is shown to be an important component of a protein Hamiltonian, since it both dominates the cooperativity of the folding transition and alters folding mechanisms. Understanding geometrical effects in analytical formulae will help illuminate the consequences of the approximations required for the study of larger proteins.

Composition-based effective chain length for prediction of protein folding rates

Folding rate prediction is a useful way to find the key factors affecting folding kinetics of proteins. Structural information is more or less required in the present prediction methods, which limits the application of these methods to various proteins. In this work, an "effective length" is defined solely based on the composition of a protein, namely, the number of specific types of amino acids in a protein. A physical theory based on a minimalist model is employed to describe the relation between the folding rates and the effective length of proteins. Based on the resultant relationship between folding rates and effective length, the optimal sets of amino acids are found through the enumeration over all possible combinations of amino acids. This optimal set achieves a high correlation (with the coefficient of 0.84) between the folding rates and the optimal effective length. The features of these amino acids are consistent with our model and landscape theory. Further comparisons between our effective length and other factors are carried out. The effective length is physically consistent with structure-based prediction methods and has the best predictability for folding rates. These results all suggest that both entropy and energetics contribute importantly to folding kinetics. The ability to accurately and efficiently predict folding rates from composition enables the analysis of the kinetics for various kinds of proteins. The underlying physics in our method may be helpful to stimulate further understanding on the effects of various amino acids in folding dynamics.

Insights into protein folding mechanisms from large scale analysis of mutational effects

Protein folding mechanisms are probed experimentally using single-point mutant perturbations. The relative effects on the folding (phi-values) and unfolding (1 - phi) rates are used to infer the detailed structure of the transition-state ensemble (TSE). Here we analyze kinetic data on > 800 mutations carried out for 24 proteins with simple kinetic behavior. We find two surprising results: (i) all mutant effects are described by the equation: DeltaDeltaG(double dagger)(unfold)=0.76DeltaDeltaG(eq) +/- 1.8kJ/mol. Therefore all data are consistent with a single phi-value (0.24) with accuracy comparable to experimental precision, suggesting that the structural information in conventional phi-values is low. (ii) phi-values change with stability, increasing in mean value and spread from native to unfolding conditions, and thus cannot be interpreted without proper normalization. We eliminate stability effects calculating the phi-values at the mutant denaturation midpoints; i.e., conditions of zero stability (phi(0)). We then show that the intrinsic variability is phi(0) = 0.36 +/- 0.11, being somewhat larger for beta-sheet-rich proteins than for alpha-helical proteins. Importantly, we discover that phi(0)-values are proportional to how many of the residues surrounding the mutated site are local in sequence. High phi(0)-values correspond to protein surface sites, which have few nonlocal neighbors, whereas core residues with many tertiary interactions produce the lowest phi(0)-values. These results suggest a general mechanism in which the TSE at zero stability is a broad conformational ensemble stabilized by local interactions and without specific tertiary interactions, reconciling phi-values with many other empirical observations.

Residue-specific analysis of frustration in the folding landscape of repeat beta/alpha protein apoflavodoxin

Flavodoxin adopts the common repeat beta/alpha topology and folds in a complex kinetic reaction with intermediates. To better understand this reaction, we analyzed a set of Desulfovibrio desulfuricans apoflavodoxin variants with point mutations in most secondary structure elements by in vitro and in silico methods. By equilibrium unfolding experiments, we first revealed how different secondary structure elements contribute to overall protein resistance to heat and urea. Next, using stopped-flow mixing coupled with far-UV circular dichroism, we probed how individual residues affect the amount of structure formed in the experimentally detected burst-phase intermediate. Together with in silico folding route analysis of the same point-mutated variants and computation of growth in nucleation size during early folding, computer simulations suggested the presence of two competing folding nuclei at opposite sides of the central beta-strand 3 (i.e., at beta-strands 1 and 4), which cause early topological frustration (i.e., misfolding) in the folding landscape. Particularly, the extent of heterogeneity in folding nuclei growth correlates with the in vitro burst-phase circular dichroism amplitude. In addition, phi-value analysis (in vitro and in silico) of the overall folding barrier to apoflavodoxin's native state revealed that native-like interactions in most of the beta-strands must form in transition state. Our study reveals that an imbalanced competition between the two sides of apoflavodoxin's central beta-sheet directs initial misfolding, while proper alignment on both sides of beta-strand 3 is necessary for productive folding.

An estimate of the numbers and density of low-energy structures (or decoys) in the conformational landscape of proteins

BACKGROUND

The conformational energy landscape of a protein, as calculated by known potential energy functions, has several minima, and one of these corresponds to its native structure. It is however difficult to comprehensively estimate the actual numbers of low energy structures (or decoys), the relationships between them, and how the numbers scale with the size of the protein.

METHODOLOGY

We have developed an algorithm to rapidly and efficiently identify the low energy conformers of oligo peptides by using mutually orthogonal Latin squares to sample the potential energy hyper surface. Using this algorithm, and the ECEPP/3 potential function, we have made an exhaustive enumeration of the low-energy structures of peptides of different lengths, and have extrapolated these results to larger polypeptides.

CONCLUSIONS AND SIGNIFICANCE

We show that the number of native-like structures for a polypeptide is, in general, an exponential function of its sequence length. The density of these structures in conformational space remains more or less constant and all the increase appears to come from an expansion in the volume of the space. These results are consistent with earlier reports that were based on other models and techniques.

Macromolecular crowding modulates folding mechanism of alpha/beta protein apoflavodoxin

Protein dynamics in cells may be different from those in dilute solutions in vitro, because the environment in cells is highly concentrated with other macromolecules. This volume exclusion because of macromolecular crowding is predicted to affect both equilibrium and kinetic processes involving protein conformational changes. To quantify macromolecular crowding effects on protein folding mechanisms, we investigated the folding energy landscape of an alpha/beta protein, apoflavodoxin, in the presence of inert macromolecular crowding agents, using in silico and in vitro approaches. By means of coarse-grained molecular simulations and topology-based potential interactions, we probed the effects of increased volume fractions of crowding agents (phi(c)) as well as of crowding agent geometry (sphere or spherocylinder) at high phi(c). Parallel kinetic folding experiments with purified Desulfovibro desulfuricans apoflavodoxin in vitro were performed in the presence of Ficoll (sphere) and Dextran (spherocylinder) synthetic crowding agents. In conclusion, we identified the in silico crowding conditions that best enhance protein stability, and discovered that upon manipulation of the crowding conditions, folding routes experiencing topological frustrations can be either enhanced or relieved. Our test-tube experiments confirmed that apoflavodoxin's time-resolved folding path is modulated by crowding agent geometry. Macromolecular crowding effects may be a tool for the manipulation of protein-folding and function in living cells.

Folding, stability and shape of proteins in crowded environments: experimental and computational approaches

How the crowded environment inside cells affects folding, stability and structures of proteins is a vital question, since most proteins are made and function inside cells. Here we describe how crowded conditions can be created in vitro and in silico and how we have used this to probe effects on protein properties. We have found that folded forms of proteins become more compact in the presence of macromolecular crowding agents; if the protein is aspherical, the shape also changes (extent dictated by native-state stability and chemical conditions). It was also discovered that the shape of the macromolecular crowding agent modulates the folding mechanism of a protein; in addition, the extent of asphericity of the protein itself is an important factor in defining its folding speed.

Quantitative criteria for native energetic heterogeneity influences in the prediction of protein folding kinetics

Energy landscape theory requires that the protein-folding mechanism is generally globally directed or funneled toward the native state. The collective nature of transition state ensembles further suggests that sufficient averaging of the native interactions can occur so that the knowledge of the native topology may suffice for predicting the mechanism. Nevertheless, while simple homogeneously weighted native topology-based models predict the folding mechanisms for many proteins, for other proteins knowledge of the native topology, by itself, seems not to suffice in determining the folding mechanism. Simulations of proteins with differing topologies reveal that the failure of homogeneously weighted topology-based models can, however, be completely understood within the framework of a funneled energy landscape and can be quantified by comparing the fluctuation of entropy cost for forming contacts to the expected fluctuations in contact energy. To be precise, we find the transition state ensembles of proteins with all-alpha topologies, which are more uniform in the specific entropy cost of contact formation, have transition state ensembles that are more readily perturbed by differences in energetic weights than are the transition state ensembles of proteins with significant amounts of beta-structure, where the specific entropy costs of contact formation are more widely distributed. This behavior is consistent with a random-field Ising model analogy that follows from the free energy functional approach to folding.

SOD1-associated ALS: a promising system for elucidating the origin of protein-misfolding disease

Amyotropic lateral sclerosis (ALS) is a neurodegenerative disease linked to misfolding and aggregation of the homodimeric enzyme superoxide dismutase (SOD1). In contrast to the precursors of other neurodegenerative diseases, SOD1 is a soluble and simple-to-study protein with immunoglobulin-like structure. Also, there are more than 120 ALS-provoking SOD1 mutations at the disposal for detailed elucidation of the disease-triggering factors at molecular level. In this article, we review recent progress in the characterization of the folding and assembly pathway of the SOD1 dimer and how this is affected by ALS-provoking mutations. Despite the diverse nature of these mutations, the results offer so far a surprising simplicity. The ALS-provoking mutations decrease either protein stability or net repulsive charge: the classical hallmarks for a disease mechanism triggered by association of non-native protein. In addition, the mutant data identifies immature SOD1 monomers as the species from which the cytotoxic pathway emerges, and point at compromised folding cooperativity as a key disease determinant. The relative ease by which these data can be obtained makes SOD1 a promising model for elucidating also the origin of other neurodegenerative diseases where the precursor proteins are structurally more elusive.

Folding of proteins with an all-atom Go-model

The Go-like potential at a residual level has been successfully applied to the folding of proteins in many previous works. However, taking into consideration more detailed structural information in the atomic level, the definition of contacts used in these traditional Go-models may not be suitable for all-atom simulations. Here, in this work, we develop a rational definition of contacts considering the screening effect in the crowded intramolecular environment. In such a scheme, a large amount of screened atom pairs are excluded and the number of contacts is decreased compared to the case of the traditional definition. These contacts defined by such a new definition are compatible with the all-atom representation of protein structures. To verify the rationality of the new definition of contacts, the folding of proteins CI2 and SH3 is simulated by all-atom molecular dynamics simulations. A high folding cooperativity and good correlation of the simulated Phi-values with those obtained experimentally, especially for CI2, are found. This suggests that the all-atom Go-model is improved compared to the traditional Go-model. Based on the comparison of the Phi-values, the roles of side chains in the folding are discussed, and it is concluded that the side-chain structures are more important for local contacts in determining the transition state structures. Moreover, the relations between side chain and backbone orderings are also discussed.

Distinct unfolding and refolding pathways of ribonuclease a revealed by heating and cooling temperature jumps

Heating and cooling temperature jumps (T-jumps) were performed using a newly developed technique to trigger unfolding and refolding of wild-type ribonuclease A and a tryptophan-containing variant (Y115W). From the linear Arrhenius plots of the microscopic folding and unfolding rate constants, activation enthalpy (DeltaH(#)), and activation entropy (DeltaS(#)) were determined to characterize the kinetic transition states (TS) for the unfolding and refolding reactions. The single TS of the wild-type protein was split into three for the Y115W variant. Two of these transition states, TS1 and TS2, characterize a slow kinetic phase, and one, TS3, a fast phase. Heating T-jumps induced protein unfolding via TS2 and TS3; cooling T-jumps induced refolding via TS1 and TS3. The observed speed of the fast phase increased at lower temperature, due to a strongly negative DeltaH(#) of the folding-rate constant. The results are consistent with a path-dependent protein folding/unfolding mechanism. TS1 and TS2 are likely to reflect X-Pro(114) isomerization in the folded and unfolded protein, respectively, and TS3 the local conformational change of the beta-hairpin comprising Trp(115). A very fast protein folding/unfolding phase appears to precede both processes. The path dependence of the observed kinetics is suggestive of a rugged energy protein folding funnel.

An analytical study of the interplay between geometrical and energetic effects in protein folding

Analytical studies have several advantages for an understanding of the mechanisms of protein folding such as the interplay between geometrical and energetic effects. In this paper, we introduce a Gaussian filament with a C(alpha) structure-based (Go) potential as a new theoretical scheme based on a Hamiltonian approach. This model takes into account geometrical information in a realistic fashion without the need of phenomenological descriptions. In order to make this model more appropriate for comparison with protein folding simulations and experiments, we introduce a many-body interaction into the potential term to enhance cooperativity. We apply our new analytical model to a beta-hairpin-type peptide and compare our results with a molecular dynamics simulation of a structure-based model.

Experimental free energy surface reconstruction from single-molecule force spectroscopy using Jarzynski's equality

We used the atomic force microscope to manipulate and unfold individual molecules of the titin I27 domain and reconstructed its free energy surface using Jarzynski's equality. The free energy surface for both stretching and unfolding was reconstructed using an exact formula that relates the nonequilibrium work fluctuations to the molecular free energy. In addition, the unfolding free energy barrier, i.e., the activation energy, was directly obtained from experimental data for the first time. This Letter demonstrates that Jarzynski's equality can be used to analyze nonequilibrium single-molecule experiments, and to obtain the free energy surfaces for molecular systems, including interactions for which only nonequilibrium work can be measured.

Folding behavior of ribosomal protein S6 studied by modified Gō-like model

Recent experimental and theoretical studies suggest that, although topology is the determinant factor in protein folding, especially for small single-domain proteins, energetic factors also play an important role in the folding process. The ribosomal protein S6 has been subjected to intensive studies. A radical change of the transition state in its circular permutants has been observed, which is believed to be caused by a biased distribution of contact energies. Since the simplistic topology-only Gō-like model is not able to reproduce such an observation, we modify the model by introducing variable contact energies between residues based on their physicochemical properties. The modified Gō-like model can successfully reproduce the Phi-value distributions, folding nucleus, and folding pathways of both the wild-type and circular permutants of S6. Furthermore, by comparing the results of the modified and the simplistic models, we find that the hydrophobic effect constructs the major force that balances the loop entropies. This may indicate that nature maintains the folding cooperativity of this protein by carefully arranging the location of hydrophobic residues in the sequence. Our study reveals a strategy or mechanism used by nature to get out of the dilemma when the native structure, possibly required by biological function, conflicts with folding cooperativity. Finally, the possible relationship between such a design of nature and amyloidosis is also discussed.

Folding of S6 structures with divergent amino acid composition: pathway flexibility within partly overlapping foldons

Studies of circular permutants have demonstrated that the folding reaction of S6 from Thermus thermophilus (S6(T)) is malleable and responds in an ordered manner to changes of the sequence separation between interacting residues: the S6(T) permutants retain a common nucleation pattern in the form of a two-strand-helix motif that can be recruited from different parts of the structure. To further test the robustness of the two-strand-helix nucleus we have here determined the crystal structure and folding reaction of an evolutionary divergent S6 protein from the hyperthermophilic bacterium Aquifex aeolicus (S6(A)). Although the overall topology of S6(A) is very similar to that of S6(T) the architecture of the hydrophobic core is radically different by containing a large proportion of stacked Phe side-chains. Despite this disparate core composition, the folding rate constant and the kinetic m values of S6(A) are identical to those of S6(T). The folding nucleus of S6(A) is also found to retain the characteristic two-strand-helix motif of the S6(T) permutants, but with a new structural emphasis. The results suggest that the protein folding reaction is linked to topology only in the sense that the native-state topology determines the repertoire of accessible nucleation motifs. If the native structure allows several equivalent ways of recruiting a productive nucleus the folding reaction is free to redistribute within these topological constraints.

Quantifying Polypeptide Conformational Space: Sensitivity to Conformation and Ensemble Definition

Quantifying the density of conformations over phase space (the conformational distribution) is needed to model important macromolecular processes such as protein folding. In this work, we quantify the conformational distribution for a simple polypeptide (N-mer polyalanine) using the cumulative distribution function (CDF), which gives the probability that two randomly selected conformations are separated by less than a "conformational" distance and whose inverse gives conformation counts as a function of conformational radius. An important finding is that the conformation counts obtained by the CDF inverse depend critically on the assignment of a conformation's distance span and the ensemble (e.g., unfolded state model): varying ensemble and conformation definition (1 --> 2 A) varies the CDF-based conformation counts for Ala(50) from 10(11) to 10(69). In particular, relatively short molecular dynamics (MD) relaxation of Ala(50)'s random-walk ensemble reduces the number of conformers from 10(55) to 10(14) (using a 1 A root-mean-square-deviation radius conformation definition) pointing to potential disconnections in comparing the results from simplified models of unfolded proteins with those from all-atom MD simulations. Explicit waters are found to roughen the landscape considerably. Under some common conformation definitions, the results herein provide (i) an upper limit to the number of accessible conformations that compose unfolded states of proteins, (ii) the optimal clustering radius/conformation radius for counting conformations for a given energy and solvent model, (iii) a means of comparing various studies, and (iv) an assessment of the applicability of random search in protein folding.

Multiple routes lead to the native state in the energy landscape of the beta-trefoil family

In general, the energy landscapes of real proteins are sufficiently well designed that the depths of local energetic minima are small compared with the global bias of the native state. Because of the funneled nature of energy landscapes, models that lack energetic frustration have been able to capture the main structural features of the transition states and intermediates found in experimental studies of both small and large proteins. In this study we ask: Are the experimental differences in folding mechanisms among members of a particular structural family due to local topological constraints that deviate from the tertiary fold common to the family? The beta-trefoil structural family members IL-1beta, hisactophilin, and acidic/basic FGFs were chosen to address this question. It has been observed that the topological landscape of the beta-trefoils allows for the population of diverse, geometrically disconnected routes that provide energetically similar but structurally distinct ways for this family to fold. Small changes in topology or energetics can alter the preferred route. Taken together, these results indicate that the global fold of the beta-trefoil family determines the energy landscape but that the routes accessed on that landscape might differ as a result of functional requirements of the individual family members.

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

The folding pathway of the B domain of protein A is the pathway most intensively studied by computer simulations. Recent systematic measurement of Phi values by Sato et al., however, has shown that none of the published computational predictions is consistent with the detailed features of the experimentally observed folding mechanism. In this article we use a statistical mechanical model of folding to show that sensitive dependence of multiple transition state ensembles on temperature and the denaturant concentration is the key to resolving the inconsistency among simulations and the experiment. Such sensitivity in multiple transition state ensembles is a natural consequence of symmetry-breaking in a nearly symmetrical protein.

Folding pathway dependence on energetic frustration and interaction heterogeneity for a three-dimensional hydrophobic protein model

Monte Carlo simulations of a hydrophobic protein model of 40 monomers in the cubic lattice are used to explore the effect of energetic frustration and interaction heterogeneity on its folding pathway. The folding pathway is described by the dependence of relevant conformational averages on an appropriate reaction coordinate, pfold, defined as the probability for a given conformation to reach the native structure before unfolding. We compare the energetically frustrated and heterogeneous hydrophobic potential, according to which individual monomers have a higher or lower tendency to form contacts unspecifically depending on their hydrophobicities, to an unfrustrated homogeneous Go-type potential with uniformly attractive native interactions and neutral non-native interactions (called Go1 in this study), and to an unfrustrated heterogeneous potential with neutral non-native interactions and native interactions having the same energy as the hydrophobic potential (called Go2 in this study). Folding kinetics are slowed down dramatically when energetic frustration increases, as expected and previously observed in a two-dimensional model. Contrary to our previous results in two dimensions, however, it appears that the folding pathway and transition state ensemble can be significantly dependent on the energy function used to stabilize the native structure. The sequence of events along the reaction coordinate, or the order along this coordinate in which different regions of the native conformation become structured, turns out to be similar for the hydrophobic and Go2 potentials, but with analogous events tending to occur at lower pfold values in the first case. In particular, the transition state obtained from the ensemble around pfold = 0.5 is more structured for the hydrophobic potential. For Go1, not only the transition state ensemble but the order of events itself is modified, suggesting that interaction heterogeneity, in addition to energetic frustration, can have significant effects on the folding mechanism, most likely by modifying the probability of different contacts in the unfolded state, the starting point for the folding reaction. Although based on a simple model, these results provide interesting insight into how sequence-dependent switching between folding pathways might occur in real proteins.

Topological frustration and the folding of interleukin-1 beta

The cytokine, interleukin-1beta (IL-1beta), adopts a beta-trefoil fold. It is known to be much slower folding than similarly sized proteins, despite having a low contact order. Proteins are sufficiently well designed that their folding is not dominated by local energetic traps. Therefore, protein models that encode only the folded structure and are energetically unfrustrated (Gō-type), can capture the essentials of the folding routes. We investigate the folding thermodynamics of IL-1beta using such a model and molecular dynamics (MD) simulations. We develop an enhanced sampling technique (a modified multicanonical method) to overcome the sampling problem caused by the slow folding. We find that IL-1beta has a broad and high free energy barrier. In addition, the protein fold causes intermediate unfolding and refolding of some native contacts within the protein along the folding trajectory. This "backtracking" occurs around the barrier region. Complex folds like the beta-trefoil fold and functional loops like the beta-bulge of IL-1beta can make some of the configuration space unavailable to the protein and cause topological frustration.