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.     

Comparison between long-range interactions and contact order in determining the folding rate of two-state proteins: application of long-range order to folding rate prediction.

The contact order is believed to be an important factor for understanding protein folding mechanisms. In our earlier work, we have shown that the long-range interactions play a vital role in protein folding. In this work, we analyzed the contribution of long-range contacts to determine the folding rate of two-state proteins. We found that the residues that are close in space and are separated by at least ten to 15 residues in sequence are important determinants of folding rates, suggesting the presence of a folding nucleus at an interval of approximately 25 residues. A novel parameter "long-range order" has been proposed to predict protein folding rates. This parameter shows as good a relationship with the folding rate of two-state proteins as contact order. Further, we examined the minimum limit of residue separation to determine the long-range contacts for different structural classes. We observed an excellent correlation between long-range order and folding rate for all classes of globular proteins. We suggest that in mixed-class proteins, a larger number of residues can serve as folding nuclei compared to all-alpha and all-beta proteins. A simple statistical method has been developed to predict the folding rates of two-state proteins using the long-range order that produces an agreement with experimental results that is better or comparable to other methods in the literature.     

Hinge stiffness is a barrier to RNA folding

Cation-mediated RNA folding from extended to compact, biologically active conformations relies on a temporal balance of forces. The Mg^2 +-mediated folding of the Tetrahymena thermophila ribozyme is characterized by rapid nonspecific collapse followed by tertiary-contact-induced compaction. This article focuses on an autonomously folding portion of the Tetrahymena ribozyme, its P4-P6 domain, in order to probe one facet of the rapid collapse: chain flexibility. The time evolution of P4-P6 folding was followed by global and local measures as a function of Mg^2 + concentration. While all concentrations of Mg^2 + studied are sufficient to screen the charge on the helices, the rates of compaction and tertiary contact formation diverge as the concentration of Mg^2 + increases; collapse is greatly accelerated by Mg^2 +, while tertiary contact formation is not. These studies highlight the importance of chain stiffness to RNA folding; at 10 mM Mg^2 +, a stiff hinge limits the rate of P4-P6 folding. At higher magnesium concentrations, the rate-limiting step shifts from hinge bending to tertiary contact formation.     

Application of long-range order to predict unfolding rates of two-state proteins

Predicting the experimental unfolding rates of two-state proteins and models describing the unfolding rates of these proteins is quite limited because of the complexity present in the unfolding mechanism and the lack of experimental unfolding data compared with folding data. In this work, 25 two-state proteins characterized by Maxwell et al. (Protein Sci 2005;14:602–616) using a consensus set of experimental conditions were taken, and the parameter long-range order (LRO) derived from their three-dimensional structures were related with their experimental unfolding rates ln(k(u)). From the total data set of 30 proteins used by Maxwell et al. (Protein Sci 2005;14:602–616), five slow-unfolding proteins with very low unfolding rates were considered to be outliers and were not included in our data set. Except all beta structural class, LRO of both the all-alpha and mixed-class proteins showed a strong inverse correlation of r = -0.99 and -0.88, respectively, with experimental ln(k(u)). LRO shows a correlation of -0.62 with experimental ln(k(u)) for all-beta proteins. For predicting the unfolding rates, a simple statistical method has been used and linear regression equations were developed for individual structural classes of proteins using LRO, and the results obtained showed a better agreement with experimental results.     

Predicting residue-wise contact orders in proteins by support vector regression

Background  

The residue-wise contact order (RWCO) describes the sequence separations between the residues of interest and its contacting residues in a protein sequence. It is a new kind of one-dimensional protein structure that represents the extent of long-range contacts and is considered as a generalization of contact order. Together with secondary structure, accessible surface area, the B factor, and contact number, RWCO provides comprehensive and indispensable important information to reconstructing the protein three-dimensional structure from a set of one-dimensional structural properties. Accurately predicting RWCO values could have many important applications in protein three-dimensional structure prediction and protein folding rate prediction, and give deep insights into protein sequence-structure relationships.     

蛋白质折叠速率预测研究进展

蛋白质折叠速率预测是当今生物物理学最具挑战性的课题之一。近年来,该领域的研究取得了很大的进展,提出了许多经验参数,例如:接触序、长程序、总接触距离、链拓扑参数、二级结构含量、有效长度、螺旋参数、n-阶接触距离等。这些参数都和蛋白质的折叠速率有很好的相关性,基于这些参数的各种预测方法所得到的预测结果也与实验数据较好地吻合。     

Protein folding rates estimated from contact predictions

Folding rates of small single-domain proteins that fold through simple two-state kinetics can be estimated from details of the three-dimensional protein structure. Previously, predictions of secondary structure had been exploited to predict folding rates from sequence. Here, we estimate two-state folding rates from predictions of internal residue-residue contacts in proteins of unknown structure. Our estimate is based on the correlation between the folding rate and the number of predicted long-range contacts normalized by the square of the protein length. It is well known that long-range order derived from known structures correlates with folding rates. The surprise was that estimates based on very noisy contact predictions were almost as accurate as the estimates based on known contacts. On average, our estimates were similar to those previously published from secondary structure predictions. The combination of these methods that exploit different sources of information improved performance. It appeared that the combined method reliably distinguished fast from slow two-state folders.     

The folding process of acylphosphatase from Escherichia coli is remarkably accelerated by the presence of a disulfide bond

The acylphosphatase from Escherichia coli (EcoAcP) is the first AcP so far studied with a disulfide bond. A mutational variant of the enzyme lacking the disulfide bond has been produced by substituting the two cysteine residues with alanine (EcoAcP mutational variant C5A/C49A, mutEcoAcP). The native states of the two protein variants are similar, as shown by far-UV and near-UV circular dichroism and dynamic light-scattering measurements. From unfolding experiments at equilibrium using intrinsic fluorescence and far-UV circular dichroism as probes, EcoAcP shows an increased conformational stability as compared with mutEcoAcP. The wild-type protein folds according to a two-state model with a very fast rate constant (k_F^H2O = 72,600 s^− 1), while mutEcoAcP folds ca 1500-fold slower, via the accumulation of a partially folded species. The correlation between the hydrophobicity of the polypeptide chain and the folding rate, found previously in the AcP-like structural family, is maintained only when considering the mutant but not the wild-type protein, which folds much faster than expected from this correlation. Similarly, the correlation between the relative contact order and the folding rate holds only for mutEcoAcP. The correlation also holds for EcoAcP, provided the relative contact order value is recalculated by considering the disulfide bridge as an alternate path for the backbone to determine the shortest sequence separation between contacting residues. These results indicate that the presence of a disulfide bond in a protein is an important determinant of the folding rate and allows its contribution to be determined in quantitative terms.     

A simple measure of native-state topology and chain connectivity predicts the folding rates of two-state proteins with and without crosslinks

The folding rates of two-state proteins have been found to correlate with simple measures of native-state topology. The most prominent among these measures is the relative contact order (CO), which is the average CO, or localness, of all contacts in the native protein structure, divided by the chain length. Here, we test whether such measures can be generalized to capture the effect of chain crosslinks on the folding rate. Crosslinks change the chain connectivity and therefore also the localness of some of the native contacts. These changes in localness can be taken into account by the graph-theoretical concept of effective contact order (ECO). The relative ECO, however, the natural extension of the relative CO for proteins with crosslinks, overestimates the changes in the folding rates caused by crosslinks. We suggest here a novel measure of native-state topology, the relative logCO, and its natural extension, the relative logECO. The relative logCO is the average value for the logarithm of the CO of all contacts, divided by the logarithm of the chain length. The relative log(E)CO reproduces the folding rates of a set of 26 two-state proteins without crosslinks with essentially the same high correlation coefficient as the relative CO. In addition, it also captures the folding rates of eight two-state proteins with crosslinks.     

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.     

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.     

Local and non-local native topologies reveal the underlying folding landscape of proteins

Due to Plaxco, Simons, Baker and others, it is now well known that the two-state single domain protein folding rate is fairly well predicted from knowledge of the topology of the native structure. Plaxco et al found that the folding rates of two-state proteins correlate with the average degree to which native contacts are 'local' within the chain sequence: fast-folders usually have mostly local structures. Here, we dissected the native topology further by focusing on non-local and local contacts using lower and upper bounds of allowable sequence separation in computing the average contact order. We analyzed non-local and local contacts of 82 two-state proteins whose experimental folding rates span over six orders of magnitude. We observed that both the number of non-local contacts and the average sequence separation of non-local contacts (non-local CO) are both negatively correlated with the folding rate, showing that the non-local contacts dominate the barrier-crossing process. Surprisingly, the local contact orders of the proteins also correlate with the folding rates. However, this correlation shows a strong positive trend indicating the role of a diffusive search in the denatured basin.     

Predicting loop–helix tertiary structural contacts in RNA pseudoknots

Tertiary interactions between loops and helical stems play critical roles in the biological function of many RNA pseudoknots. However, quantitative predictions for RNA tertiary interactions remain elusive. Here we report a statistical mechanical model for the prediction of noncanonical loop–stem base-pairing interactions in RNA pseudoknots. Central to the model is the evaluation of the conformational entropy for the pseudoknotted folds with defined loop–stem tertiary structural contacts. We develop an RNA virtual bond-based conformational model (Vfold model), which permits a rigorous computation of the conformational entropy for a given fold that contains loop–stem tertiary contacts. With the entropy parameters predicted from the Vfold model and the energy parameters for the tertiary contacts as inserted parameters, we can then predict the RNA folding thermodynamics, from which we can extract the tertiary contact thermodynamic parameters from theory–experimental comparisons. These comparisons reveal a contact enthalpy (ΔH) of −14 kcal/mol and a contact entropy (ΔS) of −38 cal/mol/K for a protonated C⁺•(G–C) base triple at pH 7.0, and (ΔH = −7 kcal/mol, ΔS = −19 cal/mol/K) for an unprotonated base triple. Tests of the model for a series of pseudoknots show good theory–experiment agreement. Based on the extracted energy parameters for the tertiary structural contacts, the model enables predictions for the structure, stability, and folding pathways for RNA pseudoknots with known or postulated loop–stem tertiary contacts from the nucleotide sequence alone.     

BCL::Fold - De Novo Prediction of Complex and Large Protein Topologies by Assembly of Secondary Structure Elements

Computational de novo protein structure prediction is limited to small proteins of simple topology. The present work explores an approach to extend beyond the current limitations through assembling protein topologies from idealized α-helices and β-strands. The algorithm performs a Monte Carlo Metropolis simulated annealing folding simulation. It optimizes a knowledge-based potential that analyzes radius of gyration, β-strand pairing, secondary structure element (SSE) packing, amino acid pair distance, amino acid environment, contact order, secondary structure prediction agreement and loop closure. Discontinuation of the protein chain favors sampling of non-local contacts and thereby creation of complex protein topologies. The folding simulation is accelerated through exclusion of flexible loop regions further reducing the size of the conformational search space. The algorithm is benchmarked on 66 proteins with lengths between 83 and 293 amino acids. For 61 out of these proteins, the best SSE-only models obtained have an RMSD100 below 8.0 Å and recover more than 20% of the native contacts. The algorithm assembles protein topologies with up to 215 residues and a relative contact order of 0.46. The method is tailored to be used in conjunction with low-resolution or sparse experimental data sets which often provide restraints for regions of defined secondary structure.     

Conserved folding pathways of alpha-lactalbumin and lysozyme revealed by kinetic CD, fluorescence, NMR, and interrupted refolding experiments

In this report, it is shown by a combination of stopped-flow CD, fluorescence, and time-resolved NMR studies that the Ca^2 +-induced refolding of bovine α-lactalbumin (BLA) at constant denaturant concentration (4 M urea) exhibits triple-exponential kinetics. In order to distinguish between parallel folding pathways and a strictly sequential formation of the native state, interrupted refolding experiments were conducted. We show here that the Ca^2 +-induced refolding of BLA involves parallel pathways and the transient formation of a folding intermediate on the millisecond timescale. Our data furthermore suggest that the two structurally homologous proteins BLA and hen egg white lysozyme share a common folding mechanism. We provide evidence that the guiding role of long-range interactions in the unfolded state of lysozyme in mediating intersubdomain interactions during folding is replaced in the case of BLA by the Ca^2 +-binding site. Time-resolved NMR spectroscopy, in combination with fast ion release from caged compounds, enables the measurement of complex protein folding kinetics at protein concentrations as low as 100 μM and the concomitant detection of conformational transitions with rate constants of up to 8 s^− 1.     

Direct correlation between proteins' folding rates and their amino acid compositions: an ab initio folding rate prediction

Discovering the mechanism of protein folding, in molecular biology, is a great challenge. A key step to this end is to find factors that correlate with protein folding rates. Over the past few years, many empirical parameters, such as contact order, long-range order, total contact distance, secondary structure contents, have been developed to reflect the correlation between folding rates and protein tertiary or secondary structures. However, the correlation between proteins' folding rates and their amino acid compositions has not been explored. In the present work, we examined systematically the correlation between proteins' folding rates and their amino acid compositions for two-state and multistate folders and found that different amino acids contributed differently to the folding progress. The relation between the amino acids' molecular weight and degeneracy and the folding rates was examined, and the role of hydrophobicity in the protein folding process was also inspected. As a consequence, a new indicator called composition index was derived, which takes no structure factors into account and is merely determined by the amino acid composition of a protein. Such an indicator is found to be highly correlated with the protein's folding rate (r > 0.7). From the results of this work, three points of concluding remarks are evident. (1) Two-state folders and multistate folders have different rate-determining amino acids. (2) The main determining information of a protein's folding rate is largely reflected in its amino acid composition. (3) Composition index may be the best predictor for an ab initio protein folding rate prediction directly from protein sequence from the standpoint of practical application.     

Contact order dependent protein folding rates: kinetic consequences of a cooperative interplay between favorable nonlocal interactions and local conformational preferences

Physical mechanisms underlying the empirical correlation between relative contact order (CO) and folding rate among naturally occurring small single-domain proteins are investigated by evaluating postulated interaction schemes for a set of three-dimensional 27mer lattice protein models with 97 different CO values. Many-body interactions are constructed such that contact energies become more favorable when short chain segments sequentially adjacent to the contacting residues adopt native-like conformations. At a given interaction strength, this scheme leads to folding rates that are logarithmically well correlated with CO (correlation coefficient r = 0.914) and span more than 2.5 orders of magnitude, whereas folding rates of the corresponding Gō models with additive contact energies have much less logarithmic correlation with CO and span only approximately one order of magnitude. The present protein chain models also exhibit calorimetric cooperativity and linear chevron plots similar to that observed experimentally for proteins with apparent simple two-state folding/unfolding kinetics. Thus, our findings suggest that CO-dependent folding rates of real proteins may arise partly from a significant positive coupling between nonlocal contact favorabilities and local conformational preferences.     

A contact-waiting-time metric and RNA folding rates

Metrics for indirectly predicting the folding rates of RNA sequences are of interest. In this letter, we introduce a simple metric of RNA structural complexity, which accounts for differences in the energetic contributions of RNA base contacts toward RNA structure formation. We apply the metric to RNA sequences whose folding rates were previously determined experimentally. We find that the metric has good correlation (correlation coefficient: −0.95, p?0.01) with the logarithmically transformed folding rates of those RNA sequences. This suggests that the metric can be useful for predicting RNA folding rates. We use the metric to predict the folding rates of bacterial and eukaryotic group II introns. Future applications of the metric (e.g., to predict structural RNAs) could prove fruitful.