Entropy capacity determines protein folding

Search and study of the general principles that govern kinetics and thermodynamics of protein folding generate a new insight into the factors controlling this process. Here, based on the known experimental data and using theoretical modeling of protein folding, we demonstrate that there exists an optimal relationship between the average conformational entropy and the average energy of contacts per residue-that is, an entropy capacity-for fast protein folding. Statistical analysis of conformational entropy and number of contacts per residue for 5829 protein structures from four general structural classes (all-alpha, all-beta, alpha/beta, alpha+beta) demonstrates that each class of proteins has its own class-specific average number of contacts (class alpha/beta has the largest number of contacts) and average conformational entropy per residue (class all-alpha has the largest number of rotatable angles phi, psi, and chi per residue). These class-specific features determine the folding rates: alpha proteins are the fastest folding proteins, then follow beta and alpha+beta proteins, and finally alpha/beta proteins are the slowest ones. Our result is in agreement with the experimental folding rates for 60 proteins. This suggests that structural and sequence properties are important determinants of protein folding rates.     

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.     

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.     

Radius of gyration is indicator of compactness of protein structure

Search and study of the general principles that govern kinetics and thermodynamics of protein folding generate a new insight into the factors controlling this process. Statistical analysis of radii of gyration for 3769 protein structures from four general structural classes (all-alpha, all-beta, alpha/beta, alpha + beta) demonstrates that each class of proteins has its own class-specific radius of gyration, which determines compactness of protein structures: alpha-proteins have the largest radius of gyration. This indicates that they are less tightly packed than beta- and alpha + beta-proteins. Finally, alpha/beta-proteins are the most tightly packed proteins with the least radius of gyration. It should be underlined that radius of gyration normalized on the radius of gyration of ball with the same volume, is independent of the length in comparison with such parameters as compactness and number of contacts per residue.     

Optimal region of average side-chain entropy for fast protein folding

Search and study the general principles that govern kinetics and thermodynamics of protein folding generates new insight into the factors that control this process. Here, we demonstrate based on the known experimental data and using theoretical modeling of protein folding that side-chain entropy is one of the general determinants of protein folding. We show for proteins belonging to the same structural family that there exists an optimal relationship between the average side-chain entropy and the average number of contacts per residue for fast folding kinetics. Analysis of side-chain entropy for proteins that fold without additional agents demonstrates that there exists an optimal region of average side-chain entropy for fast folding. Deviation of the average side-chain entropy from the optimal region results in an anomalous protein folding process (prions, alpha-lytic protease, subtilisin, some DNA-binding proteins). Proteins with high or low side-chain entropy would have extended unfolded regions and would require some additional agents for complete folding. Such proteins are common in nature, and their structure properties have biological importance.     

Class-specific correlations between protein folding rate, structure-derived, and sequence-derived descriptors

Small single-domain proteins that fold by simple two-state kinetics have been shown to exhibit a wide variation in their folding rates. It has been proposed that folding mechanisms in these proteins are largely determined by the native-state topology, and a significant correlation between folding rate and measures of the average topological complexity, such as relative contact order (RCO), has been reported. We perform a statistical analysis of folding rate and RCO in all three major structural classes (alpha, beta, and alpha/beta) of small two-state proteins and of RCO in groups of analogous and homologous small single-domain proteins with the same topology. We also study correlation between folding rate and the average physicochemical properties of amino acid sequences in two-state proteins. Our results indicate that 1) helical proteins have statistically distinguishable, class-specific folding rates; 2) RCO accounts for essentially all the variation of folding rate in helical proteins, but for only a part of the variation in beta-sheet-containing proteins; and 3) only a small fraction of the protein topologies studied show a topology-specific RCO. We also report a highly significant correlation between the folding rate and average intrinsic structural propensities of protein sequences. These results suggest that intrinsic structural propensities may be an important determinant of the rate of folding in small two-state proteins.     

Predicting protein structure from long-range contacts

Short-range and long-range contacts are important in forming protein structure. The proteins can be grouped into four different structural classes according to the content and topology of alpha-helices and beta-strands, and there are all-alpha, all-beta, alpha/beta and alpha+beta proteins. However, there is much difference in statistical property for those classes of proteins. In this paper, we will discuss protein structure in the view of the relative number of long-range (short-range) contacts for each residue. We find the percentage of residues having a large number of long-range contacts in protein is small in all-alpha class of proteins, and large in all-beta class of proteins. However, the percentage of residues is almost the same in alpha/beta and alpha+beta classes of proteins. We calculate the percentage of residues having the number of long-range contacts greater than or equal to (>/=) N(L)=5, and 7 for 428 proteins. The average percentage is 13.3%, 54.8%, 41.4% and 37.0% for all-alpha, all-beta, alpha/beta and alpha+beta classes of proteins with N(L)=5, respectively. With N(L) increasing, the percentage decreases, especially for all-alpha class of proteins. In the meantime, the percentage of residues having the number of short-range contacts greater than or equal to N(S) (>/=N(S)) in protein samples is large for all-alpha class of proteins, and small for all-beta class of proteins, especially for large N(S). We also investigate the ability of amino residues in forming a large number of long-range and short-range contacts. Cys, Val, Ile, Tyr, Trp and Phe can form a large number of long-range contacts easily, and Glu, Lys, Asp, Gln, Arg and Asn can form a large number of long-range contacts, but with difficulty. We also discuss the relative ability in forming short-range contacts for 20 amino residues. Comparison with Fauchere-Pliska hydrophobicity scale and the percentage of residues having large number of long-range contacts is also made. This investigation can provide some insights into the protein structure.     

Influence of medium- and long-range interactions in different folding types of globular proteins

Recognition of protein fold from amino acid sequence is a challenging task. The structure and stability of proteins from different fold are mainly dictated by inter-residue interactions. In our earlier work, we have successfully used the medium- and long-range contacts for predicting the protein folding rates, discriminating globular and membrane proteins and for distinguishing protein structural classes. In this work, we analyze the role of inter-residue interactions in commonly occurring folds of globular proteins in order to understand their folding mechanisms. In the medium-range contacts, the globin fold and four-helical bundle proteins have more contacts than that of DNA-RNA fold although they all belong to all-alpha class. In long-range contacts, only the ribonuclease fold prefers 4-10 range and the other folding types prefer the range 21-30 in alpha/beta class proteins. Further, the preferred residues and residue pairs influenced by these different folds are discussed. The information about the preference of medium- and long-range contacts exhibited by the 20 amino acid residues can be effectively used to predict the folding type of each protein.     

Benchmarking of TASSER in the ab initio limit

A significant number of protein sequences in a given proteome have no obvious evolutionarily related protein in the database of solved protein structures, the PDB. Under these conditions, ab initio or template-free modeling methods are the sole means of predicting protein structure. To assess its expected performance on proteomes, the TASSER structure prediction algorithm is benchmarked in the ab initio limit on a representative set of 1129 nonhomologous sequences ranging from 40 to 200 residues that cover the PDB at 30% sequence identity and which adopt alpha, alpha + beta, and beta secondary structures. For sequences in the 40-100 (100-200) residue range, as assessed by their root mean square deviation from native, RMSD, the best of the top five ranked models of TASSER has a global fold that is significantly close to the native structure for 25% (16%) of the sequences, and with a correct identification of the structure of the protein core for 59% (36%). In the absence of a native structure, the structural similarity among the top five ranked models is a moderately reliable predictor of folding accuracy. If we classify the sequences according to their secondary structure content, then 64% (36%) of alpha, 43% (24%) of alpha + beta, and 20% (12%) of beta sequences in the 40-100 (100-200) residue range have a significant TM-score (TM-score > or = 0.4). TASSER performs best on helical proteins because there are less secondary structural elements to arrange in a helical protein than in a beta protein of equal length, since the average length of a helix is longer than that of a strand. In addition, helical proteins have shorter loops and dangling tails. If we exclude these flexible fragments, then TASSER has similar accuracy for sequences containing the same number of secondary structural elements, irrespective of whether they are helices and/or strands. Thus, it is the effective configurational entropy of the protein that dictates the average likelihood of correctly arranging the secondary structure elements.     

The nature of the free energy barriers to two-state folding

We introduce a simple procedure to analyze the temperature dependence of the folding and unfolding rates of two-state proteins. We start from the simple transition-state-like rate expression: k = D(eff)exp(-DeltaG(TS)/RT), in which upper and lower bounds for the intra-chain effective diffusion coefficient (D(eff)) are obtained empirically using the timescales of elementary processes in protein folding. From the changes in DeltaG(TS) as a function of temperature, we calculate enthalpies and heat capacities of activation, together with the more elusive entropies of activation. We then estimate the conformational entropy of the transition state by extrapolation to the temperature at which the solvation entropy vanishes by cancellation between polar and apolar terms. This approach is based on the convergence temperatures for the entropy of solvating apolar (approximately 385 K) and polar groups (approximately 335 K), the assumption that the structural properties of the transition state are somewhere in between the unfolded and folded states, and the established relationship between observed heat capacity and solvent accessibility.1 To circumvent the lack of structural information about transition states, we use the empirically determined heat capacities of activation as constraints to identify the extreme values of the transition state conformational entropy that are consistent with experiment. The application of this simple approach to six two-state folding proteins for which there is temperature-dependent data available in the literature provides important clues about protein folding. For these six proteins, we obtain an average equilibrium cost in conformational entropy of -4.3 cal x mol(-1)K(-1)per residue, which is in close agreement to previous empirical and computational estimates of the same quantity. Furthermore, we find that all these proteins have a conformationally diverse transition state, with more than half of the conformational entropy of the unfolded state. In agreement with predictions from theory and computer simulations, the transition state signals the change from a regime dominated by loss in conformational entropy to one driven by the gain in stabilization free energy (i.e., including protein interactions and solvation effects). Moreover, the height of the barrier is determined by how much stabilization free energy is realized at that point, which is related to the relative contribution of local versus non-local interactions. A remarkable observation is that the fraction of conformational entropy per residue that is present in the transition state is very similar for the six proteins in this study. Based on this commonality, we propose that the observed change in thermodynamic regime is connected to a change in the pattern of structure formation: from one driven by formation of pairwise interactions to one dominated by coupling of the networks of interactions involved in forming the protein core. In this framework, the barrier to two-state folding is crossed when the folding protein reaches a "critical native density" that allows expulsion of remaining interstitial water and consolidation of the core. The principle of critical native density should be general for all two-state proteins, but can accommodate different folding mechanisms depending on the particularities of the structure and sequence.     

Buried surface area,conformational entropy,and protein stability

In this paper, how to calculate the loss of conformational entropy owing to protein folding from data on protein stability, the disulfide bonding pattern and the buried surface area is suggested. The average values of the initiation constant and entropy loss per residue for the helix-coil transition, calculated according to the suggested approach, are in a good agreement with available data. It is shown that the loss of conformational entropy for proteins and protein fragments, thus calculated, can be quite accurately approximated by three straight lines, each representing the loss of conformational entropy in a given range of molecular weights. The first line corresponds to the formation of α helices, the second to proteins with molecular weights below 10,000, and the third to proteins with molecular weights above this value. The standard error varied from 1.3 kcal/mol, for the first region, to between ～3 and 5 kcal/mol for the third. Use of these approximating linear functions is proposed for the calculation of protein stabilities from x-ray data. Such calculations, performed for more than 20 proteins and their fragments, have led to at least qualitative agreement between the calculated stabilities and available data for more than 90% of the cases. While quantitative predictions are at the limit of the method's accuracy, predictions of the probabilities of proteins or their fragments to be stable are expected to give a correct answer in ～95% of the cases.     

Role of hydrophobic clusters and long-range contact networks in the folding of (alpha/beta)8 barrel proteins

Analysis on the three dimensional structures of (alpha/beta)(8) barrel proteins provides ample light to understand the factors that are responsible for directing and maintaining their common fold. In this work, the hydrophobically enriched clusters are identified in 92% of the considered (alpha/beta)(8) barrel proteins. The residue segments with hydrophobic clusters have high thermal stability. Further, these clusters are formed and stabilized through long-range interactions. Specifically, a network of long-range contacts connects adjacent beta-strands of the (alpha/beta)(8) barrel domain and the hydrophobic clusters. The implications of hydrophobic clusters and long-range networks in providing a feasible common mechanism for the folding of (alpha/beta)(8) barrel proteins are proposed.     

The folding type of a protein is relevant to the amino acid composition

The folding types of 135 proteins, the three-dimensional structures of which are known, were analyzed in terms of the amino acid composition. The amino acid composition of a protein was expressed as a point in a multidimensional space spanned with 20 axes, on which the corresponding contents of 20 amino acids in the protein were represented. The distribution pattern of proteins in this composition space was examined in relation to five folding types, alpha, beta, alpha/beta, alpha + beta, and irregular type. The results show that amino acid compositions of the alpha, beta, and alpha/beta types are located in different regions in the composition space, thus allowing distinct separation of proteins depending on the folding types. The points representing proteins of the alpha + beta and irregular types, however, are widely scattered in the space, and the existing regions overlap with those of the other folding types. A simple method of utilizing the "distance" in the space was found to be convenient for classification of proteins into the five folding types. The assignment of the folding type with this method gave an accuracy of 70% in the coincidence with the experimental data.     

Folding rate prediction using n-order contact distance for proteins with two- and three-state folding kinetics

It is a challenging task to understand the relationship between sequences and folding rates of proteins. Previous studies are found that one of contact order (CO), long-range order (LRO), total contact distance (TCD), chain topology parameter (CTP), and effective length (Leff) has a significant correlation with folding rate of proteins. In this paper, we introduce a new parameter called n-order contact distance (nOCD) and use it to predict folding rate of proteins with two- and three-state folding kinetics. A good linear correlation between the folding rate logarithm lnkf and nOCD with n=1.2, alpha=0.6 is found for two-state folders (correlation coefficient is -0.809, P-value<0.0001) and n=2.8, alpha=1.5 for three-state folders (correlation coefficient is -0.816, P-value<0.0001). However, this correlation is completely absent for three-state folders with n=1.2, alpha=0.6 (correlation coefficient is 0.0943, P-value=0.661) and for two-state folders with n=2.8, alpha=1.5 (correlation coefficient is -0.235, P-value=0.2116). We also find that the average number of contacts per residue Pm in the interval of m for two-state folders is smaller than that for three-state folders. The probability distribution P(gamma) of residue having gamma pairs of contacts fits a Gaussian distribution for both two- and three-state folders. We observe that the correlations between square radius of gyration S2 and number of residues for two- and three-state folders are both good, and the correlation coefficient is 0.908 and 0.901, and the slope of the fitting line is 1.202 and 0.795, respectively. Maybe three-state folders are more compact than two-state folders. Comparisons with nTCD and nCTP are also made, and it is found that nOCD is the best one in folding rate prediction.     

Signatures of co-translational folding

Global and co-translational protein folding may both occur in vivo, and understanding the relationship between these folding mechanisms is pivotal to our understanding of protein-structure formation. Within this study, over 1.5 million hydrophobic-polar sequences were classified based on their ability to attain a unique, but not necessarily minimal energy conformation through co-translational folding. The sequence and structure properties of the sets were then compared to elucidate signatures of co-translational folding. The strongest signature of co-translational folding is a reduced number of possible favorable contacts in the amino terminus. There is no evidence of fewer contacts, more local contacts, or less-compact structures. Co-translational folding produces a more compact amino- than carboxy-terminal region and an amino-terminal-biased set of core residues. In real proteins these signatures are also observed and found most strongly in proteins of the alpha/beta structural class of proteins (SCOP) where 71 % have an amino-terminal set of core residues. The prominence of co-translational features in experimentally determined protein structures suggests that the importance of co-translational folding is currently underestimated.     

Side-chain entropy effects on protein secondary structure formation

Loss of conformational entropy is one of the primary factors opposing protein folding. Both the backbone and side-chain of each residue in a protein will have their freedom of motion restricted in the final folded structure. The type of secondary structure of which a residue is part will have a significant impact on how much side-chain entropy is lost. Side-chain conformational entropies have previously been determined for folded proteins, simple models of unfolded proteins, alpha-helices, and a dipeptide model for beta-strands, but not for polyproline II (PII) helices. In this work, we present side-chain conformational estimates for the three regular secondary structure types: alpha-helices, beta-strands, and PII helices. Entropies are estimated from Monte Carlo computer simulations. Beta-strands are modeled as two structures, parallel and antiparallel beta-strands. Our data indicate that restraining a residue to the PII helix or antiparallel beta-strand conformations results in side-chain entropies equal to or higher than those obtained by restraining residues to the parallel beta-strand conformation. Side-chains in the alpha-helix conformation have the lowest side-chain entropies. The observation that extended structures retain the most side-chain entropy suggests that such structures would be entropically favored in unfolded proteins under folding conditions. Our data indicate that the PII helix conformation would be somewhat favored over beta-strand conformations, with antiparallel beta-strand favored over parallel. Notably, our data imply that, under some circumstances, residues may gain side-chain entropy upon folding. Implications of our findings for protein folding and unfolded states are discussed.     

Symmetric connectivity of secondary structure elements enhances the diversity of folding pathways

The influence of native connectivity of secondary structure elements (SSE) on folding is studied using coarse-grained models of proteins with mixed alpha and beta structure and the analysis of the structural database of wild-type proteins. We found that the distribution of SSE along a sequence determines the diversity of folding pathways. If alpha and beta SSE are localized in different parts of a sequence, the diversity of folding pathways is restricted. An even (symmetric) distribution of alpha and beta SSE with respect to sequence midpoint favors multiple folding routes. Simulations are supplemented by the database analysis of the distribution of SSE in wild-type protein sequences. On an average, two-thirds of wild-type proteins with mixed alpha and beta structure have symmetric distribution of alpha and beta SSE. The propensity for symmetric distribution of SSE is especially evident for large proteins with the number of SSE > or = 10. We suggest that symmetric SSE distribution in protein sequences may arise due to nearly random allocation of alpha and beta structure along wild-type sequences. The tendency of long sequences to misfold is perhaps compensated by the enhanced pathway diversity. In addition, folding pathways are shown to progress via hierarchic assembly of SSE in accordance with their proximity along a sequence. We demonstrate that under mild denaturation conditions folding and unfolding pathways are similar. However, the reversibility of folding/unfolding pathways is shown to depend on the distribution of SSE. If alpha and beta SSE are localized in different parts of a sequence, folding and unfolding pathways are likely to coincide.     

Rotamer strain energy in protein helices - quantification of a major force opposing protein folding

It is widely believed that the dominant force opposing protein folding is the entropic cost of restricting internal rotations. The energetic changes from restricting side-chain torsional motion are more complex than simply a loss of conformational entropy, however. A second force opposing protein folding arises when a side-chain in the folded state is not in its lowest-energy rotamer, giving rotameric strain. chi strain energy results from a dihedral angle being shifted from the most stable conformation of a rotamer when a protein folds. We calculated the energy of a side-chain as a function of its dihedral angles in a poly(Ala) helix. Using these energy profiles, we quantify conformational entropy, rotameric strain energy and chi strain energy for all 17 amino acid residues with side-chains in alpha-helices. We can calculate these terms for any amino acid in a helix interior in a protein, as a function of its side-chain dihedral angles, and have implemented this algorithm on a web page. The mean change in rotameric strain energy on folding is 0.42 kcal mol-1 per residue and the mean chi strain energy is 0.64 kcal mol-1 per residue. Loss of conformational entropy opposes folding by a mean of 1.1 kcal mol-1 per residue, and the mean total force opposing restricting a side-chain into a helix is 2.2 kcal mol-1. Conformational entropy estimates alone therefore greatly underestimate the forces opposing protein folding. The introduction of strain when a protein folds should not be neglected when attempting to quantify the balance of forces affecting protein stability. Consideration of rotameric strain energy may help the use of rotamer libraries in protein design and rationalise the effects of mutations where side-chain conformations change.     

Computational exploration of the network of sequence flow between protein structures

We investigate small sequence adjustments (of one or a few amino acids) that induce large conformational transitions between distinct and stable folds of proteins. Such transitions are intriguing from evolutionary and protein‐design perspectives. They make it possible to search for ancient protein structures or to design protein switches that flip between folds and functions. A network of sequence flow between protein folds is computed for representative structures of the Protein Data Bank. The computed network is dense, on an average each structure is connected to tens of other folds. Proteins that attract sequences from a higher than expected number of neighboring folds are more likely to be enzymes and alpha/beta fold. The large number of connections between folds may reflect the need of enzymes to adjust their structures for alternative substrates. The network of the Cro family is discussed, and we speculate that capacity is an important factor (but not the only one) that determines protein evolution. The experimentally observed flip from all alpha to alpha + beta fold is examined by the network tools. A kinetic model for the transition of sequences between the folds (with only protein stability in mind) is proposed. Proteins 2010. © 2009 Wiley‐Liss, Inc.     

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.