An empirical energy function for threading protein sequence through the folding motif

In this paper we present a new residue contact potantial derived by statistical analysis of protein crystal structures. This gives mean hydrophobic and pairwise contact energies as a function of residue type and distance interval. To test the accuracy of this potential we generate model structures by “threading” different sequences through backbone folding motifs found in the structural data base. We find that conformational energies calculated by summing contact potentials show perfect specificity in matching the correct sequences with each globular folding motif in a 161-protcin data set. They also identify correct models with the core folding motifs of heme-rythrin and immunoglobulin McPC603 V₁-do- main, among millions of alternatives possible when we align subsequences with α-helices and β-strands, and allow for variation in the lengths of intervening loops. We suggest that contact potentials reflect important constraints on nonbonded interaction in native proteins, and that “threading” may be useful for structure prediction by recognition of folding motif. © 1993 Wiley-Liss, Inc.     

Side-chain conformational entropy in protein folding.

An important, but often neglected, contribution to the thermodynamics of protein folding is the loss of entropy that results from restricting the number of accessible side-chain conformers in the native structure. Conformational entropy changes can be found by comparing the number of accessible rotamers in the unfolded and folded states, or by estimating fusion entropies. Comparison of several sets of results using different techniques shows that the mean conformational free energy change (T delta S) is 1 kcal.mol-1 per side chain or 0.5 kcal.mol-1 per bond. Changes in vibrational entropy appear to be negligible compared to the entropy change resulting from the loss of accessible rotamers. Side-chain entropies can help rationalize alpha-helix propensities, predict protein/inhibitor complex structures, and account for the distribution of side chains on the protein surface or interior.     

Consistency in structural energetics of protein folding and peptide recognition.

We report a new free energy decomposition that includes structure-derived atomic contact energies for the desolvation component, and show that it applies equally well to the analysis of single-domain protein folding and to the binding of flexible peptides to proteins. Specifically, we selected the 17 single-domain proteins for which the three-dimensional structures and thermodynamic unfolding free energies are available. By calculating all terms except the backbone conformational entropy change and comparing the result to the experimentally measured free energy, we estimated that the mean entropy gain by the backbone chain upon unfolding (delta Sbb) is 5.3 cal/K per mole of residue, and that the average backbone entropy for glycine is 6.7 cal/K. Both numbers are in close agreement with recent estimates made by entirely different methods, suggesting a promising degree of consistency between data obtained from disparate sources. In addition, a quantitative analysis of the folding free energy indicates that the unfavorable backbone entropy for each of the proteins is balanced predominantly by favorable backbone interactions. Finally, because the binding of flexible peptides to receptors is physically similar to folding, the free energy function should, in principle, be equally applicable to flexible docking. By combining atomic contact energies, electrostatics, and sequence-dependent backbone entropy, we calculated a priori the free energy changes associated with the binding of four different peptides to HLA-A2, 1 MHC molecule and found agreement with experiment to within 10% without parameter adjustment.     

CONFOLD: Residue‐residue contact‐guided ab initio protein folding

Predicted protein residue–residue contacts can be used to build three‐dimensional models and consequently to predict protein folds from scratch. A considerable amount of effort is currently being spent to improve contact prediction accuracy, whereas few methods are available to construct protein tertiary structures from predicted contacts. Here, we present an ab initio protein folding method to build three‐dimensional models using predicted contacts and secondary structures. Our method first translates contacts and secondary structures into distance, dihedral angle, and hydrogen bond restraints according to a set of new conversion rules, and then provides these restraints as input for a distance geometry algorithm to build tertiary structure models. The initially reconstructed models are used to regenerate a set of physically realistic contact restraints and detect secondary structure patterns, which are then used to reconstruct final structural models. This unique two‐stage modeling approach of integrating contacts and secondary structures improves the quality and accuracy of structural models and in particular generates better β‐sheets than other algorithms. We validate our method on two standard benchmark datasets using true contacts and secondary structures. Our method improves TM‐score of reconstructed protein models by 45% and 42% over the existing method on the two datasets, respectively. On the dataset for benchmarking reconstructions methods with predicted contacts and secondary structures, the average TM‐score of best models reconstructed by our method is 0.59, 5.5% higher than the existing method. The CONFOLD web server is available at http://protein.rnet.missouri.edu/confold/ . Proteins 2015; 83:1436–1449. © 2015 Wiley Periodicals, Inc.     

The energy landscape of modular repeat proteins: topology determines folding mechanism in the ankyrin family

Proteins consisting of repeating amino acid motifs are abundant in all kingdoms of life, especially in higher eukaryotes. Repeat-containing proteins self-organize into elongated non-globular structures. Do the same general underlying principles that dictate the folding of globular domains apply also to these extended topologies? Using a simplified structure-based model capturing a perfectly funneled energy landscape, we surveyed the predicted mechanism of folding for ankyrin repeat containing proteins. The ankyrin family is one of the most extensively studied classes of non-globular folds. The model based only on native contacts reproduces most of the experimental observations on the folding of these proteins, including a folding mechanism that is reminiscent of a nucleation propagation growth. The confluence of simulation and experimental results suggests that the folding of non-globular proteins is accurately described by a funneled energy landscape, in which topology plays a determinant role in the folding mechanism.     

Nucleation‐based prediction of the protein folding rate and its correlation with the folding nucleus size

The problem of protein self‐organization is in the focus of current molecular biology studies. Although the general principles are understood, many details remain unclear. Specifically, protein folding rates are of interest because they dictate the rate of protein aggregation which underlies many human diseases. Here we offer predictions of protein folding rates and their correlation with folding nucleus sizes. We calculated free energies of the transition state and sizes of folding nuclei for 84 proteins and peptides whose other parameters were measured at the point of thermodynamic equilibrium between their unfolded and native states. We used the dynamic programming method where each residue was considered to be either as folded as in its native state or completely disordered. The calculated and measured folding rates showed a good correlation at the temperature mid‐transition point (the correlation coefficient was 0.75). Also, we pioneered in demonstrating a moderate (‐0.57) correlation coefficient between the calculated sizes of folding nuclei and the folding rates. Predictions made by different methods were compared. The established good correlation between the estimated free energy barrier and the experimentally found folding rate of each studied protein/peptide indicates that our model gives reliable results for the considered data set. Proteins 2012; © 2012 Wiley Periodicals, Inc.     

Optimal ratio between the average conformational entropy and the average energy of interaction between residues for fast protein folding

Based on available experimental data and using a theoretical model of protein folding, we demonstrate that there is an optimal ratio between the average conformational entropy and the average contact energy per residue for fast protein folding. A statistical analysis of the conformational entropy and the number of contacts per residue for 5829 protein domains from four main classes (α, β, α/β, α+β) shows that each class has its own characteristic average number of contacts per residue and average conformational entropy per residue. These class-specific characteristics determine the protein folding rates: α-proteins are the fastest to fold, β-proteins are the second fastest, α+β-proteins are the third, and α/β-proteins are the last to fold.     

Secondary structure determines protein topology

Using a test set of 13 small, compact proteins, we demonstrate that a remarkably simple protocol can capture native topology from secondary structure information alone, in the absence of long-range interactions. It has been a long-standing open question whether such information is sufficient to determine a protein's fold. Indeed, even the far simpler problem of reconstructing the three-dimensional structure of a protein from its exact backbone torsion angles has remained a difficult challenge owing to the small, but cumulative, deviations from ideality in backbone planarity, which, if ignored, cause large errors in structure. As a familiar example, a small change in an elbow angle causes a large displacement at the end of your arm; the longer the arm, the larger the displacement. Here, correct secondary structure assignments (alpha-helix, beta-strand, beta-turn, polyproline II, coil) were used to constrain polypeptide backbone chains devoid of side chains, and the most stable folded conformations were determined, using Monte Carlo simulation. Just three terms were used to assess stability: molecular compaction, steric exclusion, and hydrogen bonding. For nine of the 13 proteins, this protocol restricts the main chain to a surprisingly small number of energetically favorable topologies, with the native one prominent among them.     

Predicting repeat protein folding kinetics from an experimentally determined folding energy landscape

The Notch ankyrin domain is a repeat protein whose folding has been characterized through equilibrium and kinetic measurements. In previous work, equilibrium folding free energies of truncated constructs were used to generate an experimentally determined folding energy landscape (Mello and Barrick, Proc Natl Acad Sci USA 2004;101:14102–14107). Here, this folding energy landscape is used to parameterize a kinetic model in which local transition probabilities between partly folded states are based on energy values from the landscape. The landscape‐based model correctly predicts highly diverse experimentally determined folding kinetics of the Notch ankyrin domain and sequence variants. These predictions include monophasic folding and biphasic unfolding, curvature in the unfolding limb of the chevron plot, population of a transient unfolding intermediate, relative folding rates of 19 variants spanning three orders of magnitude, and a change in the folding pathway that results from C‐terminal stabilization. These findings indicate that the folding pathway(s) of the Notch ankyrin domain are thermodynamically selected: the primary determinants of kinetic behavior can be simply deduced from the local stability of individual repeats.     

Contact order revisited: influence of protein size on the folding rate

Guided by the recent success of empirical model predicting the folding rates of small two-state folding proteins from the relative contact order (CO) of their native structures, by a theoretical model of protein folding that predicts that logarithm of the folding rate decreases with the protein chain length L as L(2/3), and by the finding that the folding rates of multistate folding proteins strongly correlate with their sizes and have very bad correlation with CO, we reexamined the dependence of folding rate on CO and L in attempt to find a structural parameter that determines folding rates for the totality of proteins. We show that the Abs_CO = CO x L, is able to predict rather accurately folding rates for both two-state and multistate folding proteins, as well as short peptides, and that this Abs_CO scales with the protein chain length as L(0.70 +/- 0.07) for the totality of studied single-domain proteins and peptides.     

Statistical analyses of protein folding rates from the view of quantum transition

Understanding protein folding rate is the primary key to unlock the fundamental physics underlying protein structure and its folding mechanism.Especially,the temperature dependence of the folding rate remains unsolved in the literature.Starting from the assumption that protein folding is an event of quantum transition between molecular conformations,we calculated the folding rate for all two-state proteins in a database and studied their temperature dependencies.The non-Arrhenius temperature relation for 16 proteins,whose experimental data had previously been available,was successfully interpreted by comparing the Arrhenius plot with the first-principle calculation.A statistical formula for the prediction of two-state protein folding rate was proposed based on quantum folding theory.The statistical comparisons of the folding rates for 65 two-state proteins were carried out,and the theoretical vs.experimental correlation coefficient was 0.73.Moreover,the maximum and the minimum folding rates given by the theory were consistent with the experimental results.     

More compact protein globules exhibit slower folding rates

We have demonstrated that, among proteins of the same size, alpha/beta proteins have on the average a greater number of contacts per residue due to their more compact (more "spherical") structure, rather than due to tighter packing. We have examined the relationship between the average number of contacts per residue and folding rates in globular proteins according to general protein structural class (all-alpha, all-beta, alpha/beta, alpha+beta). Our analysis demonstrates that alpha/beta proteins have both the greatest number of contacts and the slowest folding rates in comparison to proteins from the other structural classes. Because alpha/beta proteins are also known to be the oldest proteins, it can be suggested that proteins have evolved to pack more quickly and into looser structures.     

Surprisingly high correlation between early and late stages in non-two-state protein folding

We collected quantitative kinetic data on early and late stages of folding in non-two-state proteins from the literature, and studied the relationship between the kinetics of the two stages. There was a surprisingly high correlation between the rate constants of these stages. The correlation coefficient of the logarithmic rate constants was as high as 0.97, which could not be caused by chance. We also studied relationships of the logarithmic rate constants of the two stages with native three-dimensional structures represented by the residue-residue contact map. There were again surprisingly high correlations between the logarithmic rate constants and the number of non-local contact clusters obtained from the contact maps. Because the number of non-local contact clusters represents overall arrangement of substructures in a native protein, the results strongly suggested the importance of the arrangement of the substructures for the kinetics of both early and late stages of protein folding.     

Theory of kinetic partitioning in protein folding with possible applications to prions

This study focuses on the phenomenon of kinetic partitioning when a polypeptide chain has two ground-state conformations, one of which is kinetically more reachable than the other. We designed sequences for lattice model proteins with two different conformations of equal energy corresponding to the global energy minimum. Folding simulations revealed that one of these conformations was indeed much more kinetically accessible than the other. We found that the number and strength of local contacts in the ground-state conformation are the major factors that determine which conformation is reached faster; the greater the number of local contacts, the more kinetically reachable a conformation is. We present simple statistical–mechanical arguments to explain these findings. Our results may be relevant in explaining the phenomenology of such proteins as human plasminogen activator inhibitor-1 (PAI-1), photosystem II, and prions. Proteins 31:335–344, 1998. © 1998 Wiley-Liss, Inc.     

Sequence determinants of protein folding rates: positive correlation between contact energy and contact range indicates selection for fast folding

In comparison with intense investigation of the structural determinants of protein folding rates, the sequence features favoring fast folding have received little attention. Here, we investigate this subject using simple models of protein folding and a statistical analysis of the Protein Data Bank (PDB). The mean-field model by Plotkin and coworkers predicts that the folding rate is accelerated by stronger-than-average interactions at short distance along the sequence. We confirmed this prediction using the Finkelstein model of protein folding, which accounts for realistic features of polymer entropy. We then tested this prediction on the PDB. We found that native interactions are strongest at contact range l = 8. However, since short range contacts tend to be exposed and they are frequently formed in misfolded structures, selection for folding stability tends to make them less attractive, that is, stability and kinetics may have contrasting requirements. Using a recently proposed model, we predicted the relationship between contact range and contact energy based on buriedness and contact frequency. Deviations from this prediction induce a positive correlation between contact range and contact energy, that is, short range contacts are stronger than expected, for 2/3 of the proteins. This correlation increases with the absolute contact order (ACO), as expected if proteins that tend to fold slowly due to large ACO are subject to stronger selection for sequence features favoring fast folding. Our results suggest that the selective pressure for fast folding is detectable only for one third of the proteins in the PDB, in particular those with large contact order.     

Charge centers and formation of the protein folding core

Electrostatic interactions are important for protein folding. At low resolution, the electrostatic field of the whole molecule can be described in terms of charge center(s). To study electrostatic effects, the centers of positive and negative charge were calculated for 20 small proteins of known structure, for which hydrogen exchange cores had been determined experimentally. Two observations seem to be important. First, in all 20 proteins studied 30-100% of the residues forming hydrogen exchange core(s) were clustered around the charge centers. Moreover, in each protein more than half of the core sequences are located near the centers of charge. Second, the general architecture of all-alpha proteins from the set seems to be stabilized by interactions of residues surrounding the charge centers. In most of the alpha-beta proteins, either or both of the centers are located near a pair of consecutive strands, and this is even more characteristic for alpha/Beta and all-beta structures. Consecutive strands are very probable sites of early folding events. These two points lead to the conclusion that charge centers, defined solely from the structure of the folded protein may indicate the location of a protein's hydrogen exchange/folding core. In addition, almost all the proteins contain well-conserved continuous hydrophobic sequences of three or more residues located in the vicinity of the charge centers. These hydrophobic sequences may be primary nucleation sites for protein folding. The results suggest the following scheme for the order of events in folding: local hydrophobic nucleation, electrostatic collapse of the core, global hydrophobic collapse, and slow annealing to the native state. This analysis emphasizes the importance of treating electrostatics during protein-folding simulations.     

Water as a conformational editor in protein folding

As molecules approach one another in aqueous solution, desolvation free energy barriers to association are encountered. Experiments suggest these (de)solvation effects contribute to the free energy barriers separating the folded and unfolded states of protein molecules. To explore their influence on the energy landscapes of protein folding reactions, we have incorporated desolvation barriers into a semi-realistic, off-lattice protein model that uses a simplified physico-chemical force-field determined solely by the sequence of amino acids. Monte Carlo sampling techniques were used to study the effects on the thermodynamics and kinetics of folding of a number of systems, diverse in structure and sequence. In each case, desolvation barriers increase the stability of the native conformation and the cooperativity of the major folding/unfolding transition. The folding times of these systems are reduced significantly upon inclusion of desolvation barriers, demonstrating that the particulate nature of the solvent engenders a more defined route to the native fold.     

Shortening a loop can increase protein native state entropy

Protein loops are essential structural elements that influence not only function but also protein stability and folding rates. It was recently reported that shortening a loop in the AcP protein may increase its native state conformational entropy. This effect on the entropy of the folded state can be much larger than the lower entropic penalty of ordering a shorter loop upon folding, and can therefore result in a more pronounced stabilization than predicted by polymer model for loop closure entropy. In this study, which aims at generalizing the effect of loop length shortening on native state dynamics, we use all‐atom molecular dynamics simulations to study how gradual shortening a very long or solvent‐exposed loop region in four different proteins can affect their stability. For two proteins, AcP and Ubc7, we show an increase in native state entropy in addition to the known effect of the loop length on the unfolded state entropy. However, for two permutants of SH3 domain, shortening a loop results only with the expected change in the entropy of the unfolded state, which nicely reproduces the observed experimental stabilization. Here, we show that an increase in the native state entropy following loop shortening is not unique to the AcP protein, yet nor is it a general rule that applies to all proteins following the truncation of any loop. This modification of the loop length on the folded state and on the unfolded state may result with a greater effect on protein stability. Proteins 2015; 83:2137–2146. © 2015 Wiley Periodicals, Inc.     

Effect of mixed macromolecular crowding agents on protein folding

In cells, proteins fold and unfold in the presence of macromolecules with various sizes and shapes. Recent experiments by Liang and coworkers (J Biol Chem 2004;279:55109-55116; J Mol Biol 2006;364:469-482) show that protein refolding is enhanced by a mixture of two different crowding agents relative to the individual crowding agents and an optimal mixing ratio exists. Here, we present a theory that predicts the existence of an optimal mixing ratio. The theory is based on models for calculating the changes in the chemical potentials of the folded and unfolded states by a mixture of crowders. The existence of an optimal mixing ratio results from the dependences of these chemical-potential changes on crowder sizes and concentrations, which can be argued to be quite general. We further predict that, for any crowding agent, the stabilizing effect can be optimized both by varying the molecular weight and the mixing ratio of two species with different molecular weights.     

Sequence optimization for native state stability determines the evolution and folding kinetics of a small protein

Investigating the relative importance of protein stability, function, and folding kinetics in driving protein evolution has long been hindered by the fact that we can only compare modern natural proteins, the products of the very process we seek to understand, to each other, with no external references or baselines. Through a large-scale all-atom simulation of protein evolution, we have created a large diverse alignment of SH3 domain sequences which have been selected only for native state stability, with no other influencing factors. Although the average pairwise identity between computationally evolved and natural sequences is only 17%, the residue frequency distributions of the computationally evolved sequences are similar to natural SH3 sequences at 86% of the positions in the domain, suggesting that optimization for the native state structure has dominated the evolution of natural SH3 domains. Additionally, the positions which play a consistent role in the transition state of three well-characterized SH3 domains (by phi-value analysis) are structurally optimized for the native state, and vice versa. Indeed, we see a specific and significant correlation between sequence optimization for native state stability and conservation of transition state structure.