Equilibrium folding and unfolding pathways for a model protein

The folding–unfolding process of reduced bovine pancreatic trypsin inhibitor was investigated with an idealized model employing approximate free energies. The protein is regarded to consist of only C^α and C^β atoms. The backbone dihedral angles are the only conformational variables and are permitted to take discrete values at every 10°. Intraresidue energies consist of two terms: an empirical part taken from the observed frequency distributions of (?,ψ) and an additional favorable energy assigned to the native conformation of each residue. Interresidue interactions are simplified by assuming that there is an attractive energy operative only between residue pairs in close contact in the native structure. A total of 230,000 molecular conformations, with no atomic overlaps, ranging from the native state to the denatured state, are randomly generated by changing the sampling bias. Each conformation is classified according to its conformational energy, F; a conformational entropy, S(F) is estimated for each value of F from the number of samples. The dependence of S(F) on energy reveals that the folding–unfolding transition for this idealized model is an “all-or-none” type; this is attributable to the specific long-range interactions. Interresidue contact probabilities, averaged over samples representing various stages of folding, serve to characterize folding intermediates. Most probable equilibrium pathways for the folding–unfolding transition are constructed by connecting conformationally similar intermediates. The specific details obtained for bovine pancreatic trypsin inhibitor are as follows: (1) Folding begins with the appearance of nativelike medium-range contacts at a β-turn and at the α-helix. (2) These grow to include the native pair of interacting β-strands. This state includes intact regular secondary conformations, as well as the interstrand sheet contacts, and corresponds to an activated state with the highest free energy on the pathway. (3) Additional native long-range contacts are completely formed either toward the amino terminus or toward the carboxyl terminus. (4) In a final step, the missing contacts appear. Although these folding pathways for this model are not consistent with experimental reports, it does indicate multiple folding pathways. The method is general and can be applied to any set of calculated conformational energies and furthermore permits investigation of gross folding features.     

Analyzing the effect of homogeneous frustration in protein folding

The energy landscape theory has been an invaluable theoretical framework in the understanding of biological processes such as protein folding, oligomerization, and functional transitions. According to the theory, the energy landscape of protein folding is funneled toward the native state, a conformational state that is consistent with the principle of minimal frustration. It has been accepted that real proteins are selected through natural evolution, satisfying the minimum frustration criterion. However, there is evidence that a low degree of frustration accelerates folding. We examined the interplay between topological and energetic protein frustration. We employed a C_α structure‐based model for simulations with a controlled nonspecific energetic frustration added to the potential energy function. Thermodynamics and kinetics of a group of 19 proteins are completely characterized as a function of increasing level of energetic frustration. We observed two well‐separated groups of proteins: one group where a little frustration enhances folding rates to an optimal value and another where any energetic frustration slows down folding. Protein energetic frustration regimes and their mechanisms are explained by the role of non‐native contact interactions in different folding scenarios. These findings strongly correlate with the protein free‐energy folding barrier and the absolute contact order parameters. These computational results are corroborated by principal component analysis and partial least square techniques. One simple theoretical model is proposed as a useful tool for experimentalists to predict the limits of improvements in real proteins.Proteins 2013; 81:1727–1737. © 2013 Wiley Periodicals, Inc.     

Reliable protein folding on complex energy landscapes: the free energy reaction path

A theoretical framework is developed to study the dynamics of protein folding. The key insight is that the search for the native protein conformation is influenced by the rate r at which external parameters, such as temperature, chemical denaturant, or pH, are adjusted to induce folding. A theory based on this insight predicts that 1), proteins with complex energy landscapes can fold reliably to their native state; 2), reliable folding can occur as an equilibrium or out-of-equilibrium process; and 3), reliable folding only occurs when the rate r is below a limiting value, which can be calculated from measurements of the free energy. We test these predictions against numerical simulations of model proteins with a single energy scale.     

Mechanisms of the multiphasic kinetics in the folding and unfolding of globular proteins

The multiphasic kinetics of the protein folding and unfolding processes are examined for a “cluster model” with only two thermodynamically stable macroscopic states, native (N) and denatured (D), which are essentially distributions of microscopic states. The simplest kinetic schemes consistent with the model are: N-(fast) → I-(slow) → D for unfolding and N ← (fast)-D₂ ← (slow)-D₁ for refolding. The fast phase during the unfolding process can be visualized as the redistribution of the native population N to I within its free energy valley. Then, this population crosses over the free energy barrier to the denatured state D in the slow phase. Therefore, the macrostate I is a kinetic intermediate which is not stable at equilibrium. For the refolding process, the initial equilibrium distribution of the denatured state D appears to be separated into D₁ and D₂ in the final condition because of the change in position of the free energy barrier. The fast refolding species D₂ is due to the “leak” from the broadly distributed D state, while the rest is the slow refolding species D₁, which must overpass the free energy barrier to reach N. At an early stage of the folding process the amino acid chain is considered to be composed of several locally ordered regions, which we call clusters, connected by random coil chain parts. Thus, the denatured state contains different sizes and distributions of clusters depending on the external condition. A later stage of the folding process is the association of smaller clusters. The native state is expressed by a maximum-size cluster with possible fluctuation sites reflecting this association. A general discussion is given of the correlation between the kinetics and thermodynamics of proteins from the overall shape of the free energy function. The cluster model provides a conceptual link between the folding kinetics and the structural patterns of globular proteins derived from the X-ray crystallographic data.     

Structure of a partially unfolded form of Escherichia coli dihydrofolate reductase provides insight into its folding pathway

Proteins frequently fold via folding intermediates that correspond to local minima on the conformational energy landscape. Probing the structure of the partially unfolded forms in equilibrium under native conditions can provide insight into the properties of folding intermediates. To elucidate the structures of folding intermediates of Escherichia coli dihydrofolate reductase (DHFR), we investigated transient partial unfolding of DHFR under native conditions. We probed the structure of a high‐energy conformation susceptible to proteolysis (cleavable form) using native‐state proteolysis. The free energy for unfolding to the cleavable form is clearly less than that for global unfolding. The dependence of the free energy on urea concentration (m‐value) also confirmed that the cleavable form is a partially unfolded form. By assessing the effect of mutations on the stability of the partially unfolded form, we found that native contacts in a hydrophobic cluster formed by the F‐G and Met‐20 loops on one face of the central β‐sheet are mostly lost in the partially unfolded form. Also, the folded region of the partially unfolded form is likely to have some degree of structural heterogeneity. The structure of the partially unfolded form is fully consistent with spectroscopic properties of the near‐native kinetic intermediate observed in previous folding studies of DHFR. The findings suggest that the last step of the folding of DHFR involves organization in the structure of two large loops, the F‐G and Met‐20 loops, which is coupled with compaction of the rest of the protein.     

Proline peptide isomerization and protein folding

The unfolding-refolding of proteins is a cooperative process and, as judged by equilibrium properties, occurs in one step involving the native,N, and the unfoldedU, conformational states. Kinetic studies have shown that the denatured protein exists as a mixture of slow-(U)_Sand fast-(U)_Frefolding forms produced by proline peptidecis-trans isomerization. Proline residues inU _Fare in the same configuration as in the native protein while they are in non-native configuration inU _S. For protein folding to occur quicklyU _Smust be converted intoU _F. The fact that the equilibrium and kinetic properties of are the same as those found for prolinecis-trans isomerization taken together with the absence of slow phase in the kinetics of refolding of a protein devoid of proline, support this view. However, the absence of a linear correlation between half-time of reactivation of denatured enzymes and their proline-contents, as well as the dissimilarities in the kinetic properties of in unfolding and refolding experiments are not consistent with the model. Conformational energy calculation and experimental results on refolding of proteins suggest that some proline residues are non-essential. They will not block protein folding even in wrong isomeric form. The native-like folded structure with incorrect proline isomers will serve as intermediate state(s) in which these prolines will more readily isomerize to the correct isomeric form. The picture becomes more complex when one considers the consequence ofcis-trans isomerism of non-proline residues on protein folding.     

Electrostatic interactions in the denatured state and in the transition state for protein folding: effects of denatured state interactions on the analysis of transition state structure

The development of electrostatic interactions during the folding of the N-terminal domain of the ribosomal protein L9 (NTL9) is investigated by pH-dependent rate equilibrium free energy relationships. We show that Asp8, among six acidic residues, is involved in non-native, electrostatic interactions with K12 in the transition state for folding as well as in the denatured state. The perturbed native state pK(a) of D8 (pK(a) = 3.0) appears to be maintained through non-native interactions in both the transition state and the denatured state. Mutational effects on the stability of the transition state for protein (un)folding are often analyzed in respect to change in ground states. Thus, the interpretation of transition state analysis critically depends on an understanding of mutational effects on both the native and denatured state. Increasing evidence for structurally biased denatured states under physiological conditions raises concerns about possible denatured state effects on folding studies. We show that the structural interpretation of transition state analysis can be altered dramatically by denatured state effects.     

Selection of optimal variants of Gō-like models of proteins through studies of stretching

The Gō-like models of proteins are constructed based on the knowledge of the native conformation. However, there are many possible choices of a Hamiltonian for which the ground state coincides with the native state. Here, we propose to use experimental data on protein stretching to determine what choices are most adequate physically. This criterion is motivated by the fact that stretching processes usually start with the native structure, in the vicinity of which the Gō-like models should work the best. Our selection procedure is applied to 62 different versions of the Gō model and is based on 28 proteins. We consider different potentials, contact maps, local stiffness energies, and energy scales—uniform and nonuniform. In the latter case, the strength of the nonuniformity was governed either by specificity or by properties related to positioning of the side groups. Among them is the simplest variant: uniform couplings with no i, i + 2 contacts. This choice also leads to good folding properties in most cases. We elucidate relationship between the local stiffness described by a potential which involves local chirality and the one which involves dihedral and bond angles. The latter stiffness improves folding but there is little difference between them when it comes to stretching.     

Kinetic analysis of local structure formations in protein folding

The protein folding process is described by a cluster model based on the assumption that local structures or clusters are formed at an early stage in different regions of the polypeptide chain. Possible local structural elements in a globular protein are helices, bends, and hydrophobic cores whose formation is presumably determined by the interaction with the environment. Thus the tendency of local structure formation is expressed by a surface free energy of the cluster, which is assigned to the interface between the cluster and its environment. The probability of finding the chain of N residues with k clusters and m residues in the cluster is represented by a cluster distribution map. The cluster model exhibits a distinct two-state-like equilibrium transition, which can be seen on this map as well-separated native and denatured populations at the midpoint of the transition. The native population is localized at k ≈ 1 and m ≈ N, while the position of the denatured population can vary significantly depending on the surface free energy of the cluster. If the surface free energy is strong, the denatured population is localized near k = 0 and m = 0. On the other hand, if the surface free energy is weak, the denatured population is localized at high k and m values. The dynamics of the cluster model are treated as a stochastic process involving the transition from a state (k,m) to one of its six neighbors. The transition probability for each transition is determined by the free energy difference between two states; thus no activation process is assumed. However, the conversion of the two macrostates, native and denatured populations, involves the free energy activation due to the cooperative interaction of the macrosystem. The dynamics are analyzed by following the time evolution of the population profile on the cluster distribution map. Kinetic schemes are proposed to describe the multistep mechanism of protein folding and unfolding.     

Single-Molecule Studies of the Im7 Folding Landscape

Under appropriate conditions, the four-helical Im7 (immunity protein 7) folds from an ensemble of unfolded conformers to a highly compact native state via an on-pathway intermediate. Here, we investigate the unfolded, intermediate, and native states populated during folding using diffusion single-pair fluorescence resonance energy transfer by measuring the efficiency of energy transfer (or proximity or P ratio) between pairs of fluorophores introduced into the side chains of cysteine residues placed in the center of helices 1 and 4, 1 and 3, or 2 and 4. We show that while the native states of each variant give rise to a single narrow distribution with high P values, the distributions of the intermediates trapped at equilibrium (denoted I^eqm) are fitted by two Gaussian distributions. Modulation of the folding conditions from those that stabilize the intermediate to those that destabilize the intermediate enabled the distribution of lower P value to be assigned to the population of the unfolded ensemble in equilibrium with the intermediate state. The reduced stability of the I^eqm variants allowed analysis of the effect of denaturant concentration on the compaction and breadth of the unfolded state ensemble to be quantified from 0 to 6 M urea. Significant compaction is observed as the concentration of urea is decreased in both the presence and absence of sodium sulfate, as previously reported for a variety of proteins. In the presence of Na₂SO₄ in 0 M urea, the P value of the unfolded state ensemble approaches that of the native state. Concurrent with compaction, the ensemble displays increased peak width of P values, possibly reflecting a reduction in the rate of conformational exchange among iso-energetic unfolded, but compact conformations. The results provide new insights into the initial stages of folding of Im7 and suggest that the unfolded state is highly conformationally constrained at the outset of folding.     

Modeling the role of disulfide bonds in protein folding: Entropic barriers and pathways

The role of disulfide bonds in directing protein folding is studied using lattice models. We find that the stability and the specificity of the disulfide bond interactions play quite different roles in the folding process: Under some conditions, the stability decreases the overall rate of folding; the specificity, however, by yielding a simpler connectivity of intermediates, always increases the rate of folding. This conclusion is intimately related to the selection mechanism entailed by entropic driving forces, such as the loop formation probability, and entropic barriers separating the native and the many native-like metastable states. The folding time is found to be a minimum for a certain range of the effective disulfide bond interaction. Examination of a model, which allows for the formation of disulfide bonded intermediates, suggests that folding proceeds via a threestage multiple pathways kinetics. We show that there are pathways to the native state involving only native-like intermediates, as well as those that are mediated by nonnative intermediates. These findings are interpreted in terms of the appropriate energy landscape describing the barriers connecting low energy conformations. The consistency of our conclusions with several experimental studies is also discussed. © 1995 Wiley-Liss, Inc.     

The statistics of unique native states for random peptides

Given a probability distribution from which the energy spectrum of a random peptide is to be sampled, we derive a general expression for the probability that such a peptide will fold to a unique native state and for the probability distribution of the native energy. This latter result allows us to localize the energy of folding based on model parameters and is one advantage of our formulation. Evidence from both the lattice theory of proteins and protein threading experiments suggest that the energy spectrum for the compact states of a peptide chain is Gaussian in form. For this reason we have derived from the more general framework the specific formulas that apply in the Gaussian case, where one requires only the number of states and the variance of the Gaussian distribution in order to apply the theory. This simplicity allows us to perform calculations that we compare with calculations previously made by others based on statistical thermodynamics. We find qualitative agreement, but a significant correction to prior estimates of folding probability derived from the Gaussian assumption is necessary. © 1996 John Wiley & Sons, Inc.     

Iterative annealing mechanism explains the functions of the GroEL and RNA chaperones

Molecular chaperones are ATP‐consuming machines, which facilitate the folding of proteins and RNA molecules that are kinetically trapped in misfolded states. Unassisted folding occurs by the kinetic partitioning mechanism according to which folding to the native state, with low probability as well as misfolding to one of the many metastable states, with high probability, occur rapidly. GroEL is an all‐purpose stochastic machine that assists misfolded substrate proteins to fold. The RNA chaperones such as CYT‐19, which are ATP‐consuming enzymes, help the folding of ribozymes that get trapped in metastable states for long times. GroEL does not interact with the folded proteins but CYT‐19 disrupts both the folded and misfolded ribozymes. The structures of GroEL and RNA chaperones are strikingly different. Despite these differences, the iterative annealing mechanism (IAM) quantitatively explains all the available experimental data for assisted folding of proteins and ribozymes. Driven by ATP binding and hydrolysis and GroES binding, GroEL undergoes a catalytic cycle during which it samples three allosteric states, T (apo), R (ATP bound), and R^″ (ADP bound). Analyses of the experimental data show that the efficiency of the GroEL–GroES machinery and mutants is determined by the resetting rate k _{R ″ → T} , which is largest for the wild‐type (WT) GroEL. Generalized IAM accurately predicts the folding kinetics of Tetrahymena ribozyme and its variants. Chaperones maximize the product of the folding rate and the steady‐state native state fold by driving the substrates out of equilibrium. Neither the absolute yield nor the folding rate is optimized.     

Kinetics of protein folding: Nucleation mechanism,time scales,and pathways

The kinetics and thermodynamics of protein folding is investigated using low friction Langevin simulation of minimal continuum mode of proteins. We show that the model protein has two characteristic temperatures: (a) T_θ, at which the chain undergoes a collapse transition from an extended conformation; (b) T_f(< T_θ), at which a finite size first-order transition to the folded state takes place. The kinetics of approach to the native state from initially denatured conformations is probed by several novel correlation functions. We find that the overall kinetics of approach to the native conformation occurs via a three-stage multiple pathway mechanism. The initial stage, characterized by a series of local dihedral angle transitions, eventually results in the compaction of the protein. Subsequently, the molecule acquires native-like structures during the second stage of folding. The final stage of folding involves activated transitions from one of the native-like structures to the native conformation. The first two stages are characterized by a multiplicity of pathways while relatively few paths are involved in the final stage. A detailed analysis of the dynamics of individual trajectories reveals a novel picture of protein folding. We find that afraction of the initial population reaches the native conformation without the formation of any detectable intermediates. This pathway is associated with a nucleation mechanism, i.e., once a critical number of tertiary contacts are established then the native state is reached rapidly. The remaining fraction of molecules become trapped in misfolded structures (stabilized by incorrect tertiary contacts). The slow folding involves transitions over barriers from these structures to the native conformation. The theoretical predictions are compared with recent experiments that probe protein folding kinetics by hydrogen exchange labeling technique. © 1995 John Wiley & Sons, Inc.     

Salt contribution to RNA tertiary structure folding stability

Accurate quantification of the ionic contribution to RNA folding stability could greatly enhance our ability to understand and predict RNA functions. Recently, motivated by the potential importance of ion correlation and fluctuation in RNA folding, we developed the tightly bound ion (TBI) model. Extensive experimental tests showed that the TBI model can lead to better treatment of multivalent ions than the Poisson-Boltzmann equation. In this study, we use the model to quantify the contribution of salt (Na⁺ and Mg²⁺) to the RNA tertiary structure folding free energy. Folding of the RNA tertiary structure often involves intermediates. We focus on the folding transition from an intermediate state to the native state, and compute the electrostatic folding free energy of the RNA. Based on systematic calculations for a variety of RNA molecules, we derive a set of formulas for the electrostatic free energy for tertiary structural folding as a function of the sequence length and compactness of the RNA and the Na⁺ and Mg²⁺ concentrations. Extensive comparisons with experimental data suggest that our model and the extracted empirical formulas are quite reliable.     

On the mechanism of protein fold‐switching by a molecular sensor

Alternate frame folding (AFF) is a mechanism by which conformational change can be engineered into a protein. The protein structure switches from the wild‐type fold (N) to a circularly‐permuted fold (N′), or vice versa, in response to a signaling event such as ligand binding. Despite the fact that the two native states have similar structures, their interconversion involves folding and unfolding of large parts of the molecule. This rearrangement is reported by fluorescent groups whose relative proximities change as a result of the order–disorder transition. The nature of the conformational change is expected to be similar from protein to protein; thus, it may be possible to employ AFF as a general method to create optical biosensors. Toward that goal, we test basic aspects of the AFF mechanism using the AFF variant of calbindin D_9k. A simple three‐state model for fold switching holds that N and N′ interconvert through the unfolded state. This model predicts that the fundamental properties of the switch—calcium binding affinity, signal response (i.e., fluorescence change upon binding), and switching rate—can be controlled by altering the relative stabilities of N and N′. We find that selectively destabilizing N or N′ changes the equilibrium properties of the switch (binding affinity and signal response) in accordance with the model. However, kinetic data indicate that the switching pathway does not require whole‐molecule unfolding. The rate is instead limited by unfolding of a portion of the protein, possibly in concert with folding of a corresponding region. Proteins 2010. © 2010 Wiley‐Liss, Inc.     

Detailed folding structures of M-lycotoxin-Hc1a and its mutageneses using 2D HP model

Although the hydrophobic-polar (HP) model was proposed a decade ago, it applies almost to no real-case study because of its intense computation. In this study, a 2D HP model was applied to study the folding structures of M-lycotoxin-Hc1a, an antimicrobial peptide, in order to get full pictures of its numerous folding structures. The normalised hydrophobicity index was used to convert M-lycotoxin-Hc1a and its six mutageneses into HP sequences, and then the 2D HP model was used to compute all the possible folding structures (3²⁴ = 282,429,536,481), and finally the normalised hydrophobicity index was used to distinguish the native state. The results showed that M-lycotoxin-Hc1a had 6 and 138 folding structures at their native state with the minimal energy of ? 13 at pH 2 and pH 7 when glycine served as hydrophobic amino acid. When glycine serves as polar amino acid, M-lycotoxin-Hc1a had 12 and 54 folding structures at their native state with the minimal energy of ? 12 and ? 13 at pH 2 and pH 7, respectively. This study advanced the knowledge on how to apply the HP model to real-life study, and how the mutageneses influenced the folding structures of M-lycotoxin-Hc1a, their native states and minimal energy at different pH levels.     

A novel approach for large-scale polypeptide folding based on elastic networks using continuous optimization

We present a new computationally efficient method for large-scale polypeptide folding using coarse-grained elastic networks and gradient-based continuous optimization techniques. The folding is governed by minimization of energy based on Miyazawa-Jernigan contact potentials. Using this method we are able to substantially reduce the computation time on ordinary desktop computers for simulation of polypeptide folding starting from a fully unfolded state. We compare our results with available native state structures from Protein Data Bank (PDB) for a few de-novo proteins and two natural proteins, Ubiquitin and Lysozyme. Based on our simulations we are able to draw the energy landscape for a small de-novo protein, Chignolin. We also use two well known protein structure prediction software, MODELLER and GROMACS to compare our results. In the end, we show how a modification of normal elastic network model can lead to higher accuracy and lower time required for simulation.