Estimation of protein folding rate from Monte Carlo simulations and entropy capacity

The problem of protein self-organization is one of the most important problems of molecular biology nowadays. Despite the recent success in the understanding of general principles of protein folding, details of this process are yet to be elucidated. Moreover, the prediction of protein folding rates has its own practical value due to the fact that aggregation directly depends on the rate of protein folding. The time of folding has been calculated for 67 proteins with known experimental data at the point of thermodynamic equilibrium between unfolded and native states using a Monte Carlo model where each residue is considered to be either folded as in the native state or completely disordered. The times of folding for 67 proteins which reach the native state within the limit of 10(8) Monte Carlo steps are in a good correlation with the experimentally measured folding rate at the mid-transition point (the correlation coefficient is -0.82). Theoretical consideration of a capillarity model for the process of protein folding demonstrates that the difference in the folding rate for proteins sharing more spherical and less spherical folds is the result of differences in the conformational entropy due to a larger surface of the boundary between folded and unfolded phases in the transition state for proteins with more spherical fold. The capillarity model allows us to predict the folding rate at the same level of correlation as by Monte Carlo simulations. The calculated model entropy capacity (conformational entropy per residue divided by the average contact energy per residue) for 67 proteins correlates by about 78% with the experimentally measured folding rate at the mid-transition point.     

Importance of native-state topology for determining the folding rate of two-state proteins

Understanding the relationship between amino acid sequences and folding rate of proteins is a challenging task similar to protein folding problem. In this work, we have analyzed the relative importance of protein sequence and structure for predicting the protein folding rates in terms of amino acid properties and contact distances, respectively. We found that the parameters derived with protein sequence (physical-chemical, energetic, and conformational properties of amino acid residues) show very weak correlation (|r| < 0.39) with folding rates of 28 two-state proteins, indicating that the sequence information alone is not sufficient to understand the folding rates of two-state proteins. However, the maximum positive correlation obtained for the properties, number of medium-range contacts, and alpha-helical tendency reveals the importance of local interactions to initiate protein folding. On the other hand, a remarkable correlation (r varies from -0.74 to -0.88) has been obtained between structural parameters (contact order, long-range order, and total contact distance) and protein folding rates. Further, we found that the secondary structure content and solvent accessibility play a marginal role in determining the folding rates of two-state proteins. Multiple regression analysis carried out with the combination of three properties, beta-strand tendency, enthalpy change, and total contact distance improved the correlation to 0.92 with protein folding rates. The relative importance of existing methods along with multiple-regression model proposed in this work will be discussed. Our results demonstrate that the native-state topology is the major determinant for the folding rates of two-state proteins.     

A statistical model for predicting protein folding rates from amino acid sequence with structural class information

Prediction of protein folding rates from amino acid sequences is one of the most important challenges in molecular biology. In this work, I have related the protein folding rates with physical-chemical, energetic and conformational properties of amino acid residues. I found that the classification of proteins into different structural classes shows an excellent correlation between amino acid properties and folding rates of two- and three-state proteins, indicating the importance of native state topology in determining the protein folding rates. I have formulated a simple linear regression model for predicting the protein folding rates from amino acid sequences along with structural class information and obtained an excellent agreement between predicted and experimentally observed folding rates of proteins; the correlation coefficients are 0.99, 0.96 and 0.95, respectively, for all-alpha, all-beta and mixed class proteins. This is the first available method, which is capable of predicting the protein folding rates just from the amino acid sequence with the aid of generic amino acid properties and structural class information.     

Analysis and prediction of protein folding rates using quadratic response surface models

Understanding the relationship between amino acid sequences and folding rates of proteins is an important task in computational and molecular biology. In this work, we have systematically analyzed the composition of amino acid residues for proteins with different ranges of folding rates. We observed that the polar residues, Asn, Gln, Ser, and Lys, are dominant in fast folding proteins whereas the hydrophobic residues, Ala, Cys, Gly, and Leu, prefer to be in slow folding proteins. Further, we have developed a method based on quadratic response surface models for predicting the folding rates of 77 two- and three-state proteins. Our method showed a correlation of 0.90 between experimental and predicted protein folding rates using leave-one-out cross-validation method. The classification of proteins based on structural class improved the correlation to 0.98 and it is 0.99, 0.98, and 0.96, respectively, for all-alpha, all-beta, and mixed class proteins. In addition, we have utilized Baysean classification theory for discriminating two- and three-state proteins, which showed an accuracy of 90%. We have developed a web server for predicting protein folding rates and it is available at http://bioinformatics.myweb.hinet.net/foldrate.htm.     

Prediction of protein folding rates from primary sequences using hybrid sequence representation

The ability to predict protein folding rates constitutes an important step in understanding the overall folding mechanisms. Although many of the prediction methods are structure based, successful predictions can also be obtained from the sequence. We developed a novel method called prediction of protein folding rates (PPFR), for the prediction of protein folding rates from protein sequences. PPFR implements a linear regression model for each of the mainstream folding dynamics including two-, multi-, and mixed-state proteins. The proposed method provides predictions characterized by strong correlations with the experimental folding rates, which equal 0.87 for the two- and multistate proteins and 0.82 for the mixed-state proteins, when evaluated with out-of-sample jackknife test. Based on in-sample and out-of-sample tests, the PPFR's predictions are shown to be better than most of other sequence only and structure-based predictors and complementary to the predictions of the most recent sequence-based QRSM method. We show that simultaneous incorporation of several characteristics, including the sequence, physiochemical properties of residues, and predicted secondary structure provides improved quality. This hybridized prediction model was analyzed to reveal the complementary factors that can be used in tandem to predict folding rates. We show that bigger proteins require more time for folding, higher helical and coil content and the presence of Phe, Asn, and Gln may accelerate the folding process, the inclusion of Ile, Val, Thr, and Ser may slow down the folding process, and for the two-state proteins increased beta-strand content may decelerate the folding process. Finally, PPFR provides strong correlation when predicting sequences with low similarity.     

Folding kinetics of a naturally occurring helical peptide: implication of the folding speed limit of helical proteins

The folding mechanism and dynamics of a helical protein may strongly depend on how quickly its constituent alpha-helices can fold independently. Thus, our understanding of the protein folding problem may be greatly enhanced by a systematic survey of the folding rates of individual alpha-helical segments derived from their parent proteins. As a first step, we have studied the relaxation kinetics of the central helix (L9:41-74) of the ribosomal protein L9 from the bacterium Bacillus stearothermophilus , in response to a temperature-jump ( T-jump) using infrared spectroscopy. L9:41-74 has been shown to exhibit unusually high helicity in aqueous solution due to a series of side chain-side chain interactions, most of which are electrostatic in nature, while still remaining monomeric over a wide concentration range. Thus, this peptide represents an excellent model system not only for examining how the folding rate of naturally occurring helices differs from that of the widely studied alanine-based peptides, but also for estimating the folding speed limit of (small) helical proteins. Our results show that the T-jump induced relaxation rate of L9:41-74 is significantly slower than that of alanine-based peptides. For example, at 11 degrees C its relaxation time constant is about 2 micros, roughly seven times slower than that of SPE(5), an alanine-rich peptide of similar chain length. In addition, our results show that the folding rate of a truncated version of L9:41-74 is even slower. Taken together, these results suggest that individual alpha-helical segments in proteins may fold on a time scale that is significantly slower than the folding time of alanine-based peptides. Furthermore, we argue that the relaxation rate of L9:41-74 measured between 8 and 45 degrees C provides a realistic estimate of the ultimate folding rate of (small) helical proteins over this temperature range.     

Predicting protein folding rates using the concept of Chou's pseudo amino acid composition

One of the most important challenges in computational and molecular biology is to understand the relationship between amino acid sequences and the folding rates of proteins. Recent works suggest that topological parameters, amino acid properties, chain length and the composition index relate well with protein folding rates, however, sequence order information has seldom been considered as a property for predicting protein folding rates. In this study, amino acid sequence order was used to derive an effective method, based on an extended version of the pseudo-amino acid composition, for predicting protein folding rates without any explicit structural information. Using the jackknife cross validation test, the method was demonstrated on the largest dataset (99 proteins) reported. The method was found to provide a good correlation between the predicted and experimental folding rates. The correlation coefficient is 0.81 (with a highly significant level) and the standard error is 2.46. The reported algorithm was found to perform better than several representative sequence-based approaches using the same dataset. The results indicate that sequence order information is an important determinant of protein folding rates.     

Optimum folding pathways of proteins: their determination and properties

We develop a dynamic optimization technique for determining optimum folding pathways of proteins starting from different initial configurations. A coarse-grained Go model is used. Forces acting on each bead are (i) the friction force, (ii) forces from bond length constraints, (iii) excluded volume constraints, and (iv) attractive forces between residue pairs that are in contact in the native state. An objective function is defined as the total attractive energy between nonbonded residues, which are neighbors in the native state. The objective function is minimized over all feasible paths, satisfying bond length and excluded volume constraints. The optimization problem is nonconvex and contains a large number of constraints. An augmented Lagrangian method with a penalty barrier function was used to solve the problem. The method is applied to a 36-residue protein, chicken villin headpiece. Sequences of events during folding of the protein are determined for various pathways and analyzed. The relative time scales are compared and scaled according to experimentally measured events. Formation times of the helices, turn, and the loop agree with experimental data. We obtain the overall folding time of the protein in the range of 600 ns-1.2 micros that is smaller than the experimental result of 4-5 micros, showing that the optimal folding times that we obtain may be possible lower bounds. Time dependent variables during folding and energies associated with short- and long-range interactions between secondary structures are analyzed in modal space using Karhunen-Loeve expansion.     

Temperature effects on the nucleation mechanism of protein folding and on the barrierless thermal denaturation of a native protein

Recently, the authors proposed a kinetic model for the nucleation mechanism of protein folding where a protein was treated as a heteropolymer with all the bonds and bond angles equal and constant. As a crucial idea of the model, an overall potential around a cluster of native residues in which a protein residue performs a chaotic motion is considered to be a combination of three potentials: effective pairwise, average dihedral, and confining. The overall potential as a function of the distance from the cluster center has a double well shape which allows one to determine the rates with which the cluster emits and absorbs residues by using a first passage time analysis. One can then develop a kinetic theory for the nucleation mechanism of protein folding and evaluate the protein folding time. In the present paper we evaluate the optimal temperature at which the protein folding time is the shortest. A method is also proposed to determine the temperature dependence of the folding time without carrying out the time consuming calculations for a series of temperatures. Using Taylor series expansions in the formalism of the first passage time analysis, one can calculate the temperature dependence of the cluster emission and absorption rates in the vicinity of some temperature T(0) if they are known at T(0). Thus one can evaluate the protein folding time t(f) at any other temperature T in the vicinity of T(0) at which the folding time t(f) is known. We also present a model for the thermal denaturation of a protein occurring via the decay of the native structure of the protein. Due to a sufficiently large temperature increase or decrease, the rate with which a cluster of native residues within a protein emits residues becomes larger than the absorption rate in the whole range of cluster sizes up to the size of the whole protein. This leads to the unfolding of the protein in a barrierless way, i.e., as spinodal decomposition. Knowing the cluster emission and absorption rates as functions of temperature and cluster size, one can find the threshold temperatures of cold and hot barrierless denaturation as well as the corresponding unfolding times. Both proposed methods are illustrated by numerical calculations for two model proteins, one consisting of 124 amino acids, the other consisting of 2500 residues. The first one roughly mimicks a bovine pancreatic ribonuclease while the second one is a representative of the largest proteins which are extremely difficult to study by straightforward Monte Carlo or molecular dynamics simulations.     

Direct measurement of barrier heights in protein folding

A serious limitation in the study of protein folding reactions resides in the lack of experimental methods to measure the absolute height of the free energy barrier. This is particularly unfortunate given that if folding barriers are small, as theory predicts, it might be possible to resolve folding mechanisms directly. Here we explore the performance of a recently developed method to extract folding barriers from equilibrium differential scanning calorimetry (DSC) experiments. To test the method, we compare the thermodynamic barrier heights for 15 proteins obtained from available DSC data with the folding rates measured in kinetic experiments. The correlation between these two parameters is very good (r2 = 0.9) and has a slope consistent with the same energy scale. These results confirm that it is possible to measure free energy barriers for natural proteins from equilibrium DSC experiments. Furthermore, the measured barrier heights are small (<8 RT), in general, and marginal or nonexisting for fast-folding proteins.     

Hierarchy of folding and unfolding events of protein G, CI2, and ACBP from explicit-solvent simulations

The study of the mechanism which is at the basis of the phenomenon of protein folding requires the knowledge of multiple folding trajectories under biological conditions. Using a biasing molecular-dynamics algorithm based on the physics of the ratchet-and-pawl system, we carry out all-atom, explicit solvent simulations of the sequence of folding events which proteins G, CI2, and ACBP undergo in evolving from the denatured to the folded state. Starting from highly disordered conformations, the algorithm allows the proteins to reach, at the price of a modest computational effort, nativelike conformations, within a root mean square deviation (RMSD) of approximately 1 A?. A scheme is developed to extract, from the myriad of events, information concerning the sequence of native contact formation and of their eventual correlation. Such an analysis indicates that all the studied proteins fold hierarchically, through pathways which, although not deterministic, are well-defined with respect to the order of contact formation. The algorithm also allows one to study unfolding, a process which looks, to a large extent, like the reverse of the major folding pathway. This is also true in situations in which many pathways contribute to the folding process, like in the case of protein G.     

Contact pair dynamics during folding of two small proteins: chicken villin head piece and the Alzheimer protein beta-amyloid

The folding of an extended protein to its unique native state requires establishment of specific, predetermined, often distant, contacts between amino acid residue pairs. The dynamics of contact pair formation between various hydrophobic residues during folding of two different small proteins, the chicken villin head piece (HP-36) and the Alzheimer protein beta-amyloid (betaA-40), are investigated by Brownian dynamics (BD) simulations. These two proteins represent two very different classes-HP-36 being globular while betaA-40 is nonglobular, stringlike. Hydropathy scale and nonlocal helix propensity of amino acids are used to model the complex interaction potential among the various amino acid residues. The minimalistic model we use here employs a connected backbone chain of atoms of equal size while an amino acid is attached to each backbone atom as an additional atom of differing sizes and interaction parameters, determined by the characteristics of each amino acid. Even for such simple models, we find that the low-energy structures obtained by BD simulations of both the model proteins mimic the native state of the real protein rather well, with a best root-mean-square deviation of 4.5 A for HP-36. For betaA-40 (where a single well-defined structure is not available), the simulated structures resemble the reported ensemble rather well, with the well-known beta-bend correctly reproduced. We introduce and calculate a contact pair distance time correlation function, C(P) (ij)(t), to quantify the dynamical evolution of the pair contact formation between the amino acid residue pairs i and j. The contact pair time correlation function exhibits multistage dynamics, including a two stage fast collapse, followed by a slow (microsecond long) late stage dynamics for several specific pairs. The slow late stage dynamics is in accordance with the findings of Sali et al. Analysis of the individual trajectories shows that the slow decay is due to the attempt of the protein to form energetically more favorable pair contacts to replace the less favorable ones. This late stage contact formation is a highly cooperative process, involving participation of several pairs and thus entropically unfavorable and expected to face a large free energy barrier. This is because any new pair contact formation among hydrophobic pairs will require breaking of several contacts, before the favorable ones can be formed. This aspect of protein folding dynamics is similar to relaxation in glassy liquids, where also alpha relaxation requires highly cooperative process of hopping. The present analysis suggests that waiting time for the necessary pair contact formation may obey the Poissonian distribution. We also study the dynamics of Forster energy transfer during folding between two tagged amino acid pairs. This dynamics can be studied by fluorescence resonance energy transfer (FRET). It is found that suitably placed donor-acceptor pairs can capture the slow dynamics during folding. The dynamics probed by FRET is predicted to be nonexponential.     

Improved theoretical description of protein folding kinetics from rotations in the phase space of relevant order parameters

A method is introduced to construct a better approximation for the reaction coordinate for protein folding from known order parameters. The folding of a two-state off-lattice alpha helical Go-type protein is studied using molecular dynamics simulations. Folding times are computed directly from simulation, as well as theoretically using an equation derived by considering Brownian-type dynamics for the putative reaction coordinate. Theoretical estimates of the folding time using the number of native contacts (Qn) as a reaction coordinate were seen to differ quite significantly from the true folding time of the protein. By considering the properties of the bimodal free energy surface of this protein as a function of Qn and another relevant coordinate for folding Q (the total number of contacts), we show that by introducing a rotation in the phase space of the order parameters Q and Qn, we can construct a new reaction coordinate q that leads to a fivefold improvement in the estimate of the folding rate. This new coordinate q, resulting from the rotation, lies along the line connecting the unfolded and folded ensemble minima of the free energy map plotted as a function of the original order parameters Q and Qn. Possible reasons for the remaining discrepancy between the folding time computed theoretically and from folding simulations are discussed.     

Protein folding kinetics: barrier effects in chemical and thermal denaturation experiments

Recent experimental work on fast protein folding brings about an intriguing paradox. Microsecond-folding proteins are supposed to fold near or at the folding speed limit (downhill folding), but yet their folding behavior seems to comply with classical two-state analyses, which imply the crossing of high free energy barriers. However, close inspection of chemical and thermal denaturation kinetic experiments in fast-folding proteins reveals systematic deviations from two-state behavior. Using a simple one-dimensional free energy surface approach we find that such deviations are indeed diagnostic of marginal folding barriers. Furthermore, the quantitative analysis of available fast-kinetic data indicates that many microsecond-folding proteins fold downhill in native conditions. All of these proteins are then promising candidates for an atom-by-atom analysis of protein folding using nuclear magnetic resonance.1 We also find that the diffusion coefficient for protein folding is strongly temperature dependent, corresponding to an activation energy of approximately 1 kJ.mol-1 per protein residue. As a consequence, the folding speed limit at room temperature is about an order of magnitude slower than the approximately 1 micros estimates from high-temperature T-jump experiments. Our analysis is quantitatively consistent with the available thermodynamic and kinetic data on slow two-state folding proteins and provides a straightforward explanation for the apparent fast-folding paradox.     

Ultrafast folding of a computationally designed Trp-cage mutant: Trp2-cage

Miniproteins provide useful model systems for understanding the principles of protein folding and design. These proteins also serve as useful test cases for theories of protein folding, and their small size and ultrafast folding kinetics put them in a regime of size and time scales that is now becoming accessible to molecular dynamics simulations. Previous estimates have suggested the "speed limit" for folding is on the order of 1 mus. Here a computationally designed mutant of the 20-residue Trp-cage miniprotein, Trp2-cage, is presented. The Trp2-cage has greater stability than the parent and folds on the ultrafast time scale of 1 mICROs at room temperature, as determined from infrared temperature-jump experiments.     

Engineering a beta-sheet protein toward the folding speed limit

Recent experimental studies have shown that alpha-helical proteins can approach the folding "speed limit", where folding switches from an activated to a downhill process in free energy. beta-sheet proteins are generally thought to fold more slowly than helix bundles. However, based on studies of hairpins, folding should still be able to approach the microsecond time scale. Here we demonstrate how the hPin1 WW domain, a triple-stranded beta-sheet protein with a sharp thermodynamic melting transition, can be engineered toward the folding "speed limit" without a significant loss in thermal denaturation cooperativity.     

How does the first water shell fold proteins so fast?

First shells of hydration and bulk solvent play a crucial role in the folding of proteins. Here, the role of water in the dynamics of proteins has been investigated using a theoretical protein-solvent model and a statistical physics approach. We formulate a hydration model where the hydrogen bonds between water molecules pertaining to the first shell of the protein conformation may be either mainly formed or broken. At thermal equilibrium, hydrogen bonds are formed at low temperature and are broken at high temperature. To explore the solvent effect, we follow the folding of a large sampling of protein chains, using a master-equation evolution. The dynamics shows a clear mechanism. Above the glass-transition temperature, a large ratio of chains fold very rapidly into the native structure irrespective of the temperature, following pathways of high transition rates through structures surrounded by the solvent with broken hydrogen bonds. Although these states have an infinitesimal probability, they act as strong dynamical attractors and fast folding proceeds along these routes rather than pathways with small transition rates between configurations of much higher equilibrium probabilities. At a given low temperature, a broad jump in the folding times is observed. Below this glass temperature, the pathways where hydrogen bonds are mainly formed become those of highest rates although with conformational changes of huge relaxation times. The present results reveal that folding obeys a double-funnel mechanism.     

Engineering the protein folding landscape in gram-negative bacteria

Gram-negative bacteria, especially Escherichia coli, are often the preferred hosts for recombinant protein production because of their fast doubling times, ability to grow to high cell density, propensity for high recombinant protein titers and straightforward protein purification techniques. The utility of simple bacteria in such studies continues to improve as a result of an ever-increasing body of knowledge regarding their native protein biogenesis machinery. From translation on the ribosome to interaction with cytosolic accessory factors to transport across the inner membrane into the periplasmic space, cellular proteins interact with many different types of cellular machinery and each interaction can have a profound effect on the protein folding process. This review addresses key aspects of cellular protein folding, solubility and expression in E. coli with particular focus on the elegant biological machinery that orchestrates the transition from nascent polypeptide to folded, functional protein. Specifically highlighted are a variety of different techniques to intentionally alter the folding environment of the cell as a means to understand and engineer intracellular protein folding and stability.     

A new folding paradigm for repeat proteins

The folding/unfolding transitions of a series of designed consensus tetratricopeptide repeat proteins are quantitatively described by the classical one-dimensional Ising model, which thus represents a new folding paradigm for repeat proteins. Moreover, for the first time for any protein, a theoretical model predicts the folding/unfolding transition midpoint and the width of the transition.     

Oscillatory molecular driving force for protein folding at high concentration: a molecular simulation

This paper presents a Langevin dynamics simulation that suggests a novel way to fold protein at high concentration, a fundamental issue in neurodegenerative diseases in vivo and the production of recombinant proteins in vitro. The simulation indicates that the folding of a coarse-grained beta-barrel protein at high concentration follows the "collapse-rearrangement" mechanism but it yields products of various forms, including single proteins in the native, misfolded, and uncollapsed forms and protein aggregates. Misfolded and uncollapased proteins are the "nucleus" of the aggregates that also encapsulate some correctly folded proteins (native proteins). An optimum hydrophobic interaction strength (epsilon*(p)) between the hydrophobic beads of the model protein, which results from a compromise between the kinetics of collapse and rearrangement, is identified for use in increasing the rate of folding over aggregating. Increased protein concentration hinders the structural transitions in both collapse and rearrangement and thus favors aggregation. A new method for protein folding at high concentration is proposed, which uses an oscillatory molecular driving force (epsilon*(p)) to promote the dissociation of aggregates in the low epsilon*(p) regime while promoting folding at a high epsilon*(p). The advantage of this method in enhancing protein folding while depressing aggregation is illustrated by a comparison with the methods based on direct dilution or applying a denaturant gradient.