Theoretical study of a landscape of protein folding-unfolding pathways. Folding rates at midtransition

A structural perspective of compensatory evolution

The study of molecular evolution is important because it reveals how protein functions emerge and evolve. Recently, several types of studies indicated that substitutions in molecular evolution occur in a compensatory manner, whereby the occurrence of a substitution depends on the amino acid residues at other sites. However, a molecular or structural basis behind the compensation often remains obscure. Here, we review studies on the interface of structural biology and molecular evolution that revealed novel aspects of compensatory evolution. In many cases structural studies benefit from evolutionary data while structural data often add a functional dimension to the study of molecular evolution.

Protein folding as flow across a network of folding-unfolding pathways. 1. The mid-transition case

Prediction of protein folding rates and folding nuclei is an important problem of protein science. Most of the previously proposed models for protein folding in vitro are based on the nucleation mechanism of this process. Our model considering protein folding as a flow arising in a network of folding-unfolding pathways at a coarse-grained free-energy landscape was described a few years ago, along with an algorithm for calculation of protein folding rates. Here we extend our approach and describe in detail a mathematically strict algorithm for calculating the "folding nuclei", arising as bottlenecks of the flow. Although the proposed physical theory uses no adjustable parameters, its results are in good agreement with experiment. This paper presents (i) the general theory and (ii) the results for the simplest case, i.e., folding/unfolding at the midpoint of thermodynamic equilibrium between the native and unfolded states of a protein; results for "in-water" conditions, i.e., for the case when no denaturant is added and the native state of a protein is much more stable than the unfolded one, will be described in the next paper of the series.

Is protein folding rate dependent on number of folding stages? Modeling of protein folding with ferredoxin-like fold

Statistical analysis of protein folding rates has been done for 84 proteins with available experimental data. A surprising result is that the proteins with multi-state kinetics from the size range of 50-100 amino acid residues (a.a.) fold as fast as proteins with two-state kinetics from the same size range. At the same time, the proteins with two-state kinetics from the size range 101-151 a.a. fold faster than those from the size range 50-100 a.a. Moreover, it turns out unexpectedly that usually in the group of structural homologs from the size range 50-100 a.a., proteins with multi-state kinetics fold faster than those with two-state kinetics. The protein folding for six proteins with a ferredoxin-like fold and with a similar size has been modeled using Monte Carlo simulations and dynamic programming. Good correlation between experimental folding rates, some structural parameters, and the number of Monte Carlo steps has been obtained. It is shown that a protein with multi-state kinetics actually folds three times faster than its structural homologs.

KineticDB: a database of protein folding kinetics

We propose here KineticDB, a systematically compiled database of protein folding kinetics, which contains about 90 unique proteins. The main goal of the KineticDB is to provide users with a diverse set of protein folding rates determined experimentally. The search for determinants of protein folding is still in progress, aimed at obtaining a new understanding of the folding process. Comparison with experimental protein folding rates has been the main tool for validation of both theoretical models and empirical relationships during the last 10 years. It is, therefore, necessary to provide a researcher with as much data as possible in a simple and easy-to-use way. At present, the KineticDB contains the results of folding kinetics measurements of single-domain proteins and separate protein domains as well as short peptides without disulfide bonds. It includes data on about 90 unique proteins and many mutants that have been systematically accumulated over the last 10 years and is the largest collection of protein folding kinetic data presented as a database. The KineticDB is available at http://kineticdb.protres.ru/db/index.pl.

Excluded volume, local structural cooperativity, and the polymer physics of protein folding rates

A coarse-grained variational model is used to investigate the polymer dynamics of barrier crossing for a diverse set of two-state folding proteins. The model gives reliable folding rate predictions provided excluded volume terms that induce minor structural cooperativity are included in the interaction potential. In general, the cooperative folding routes have sharper interfaces between folded and unfolded regions of the folding nucleus and higher free energy barriers. The calculated free energy barriers are strongly correlated with native topology as characterized by contact order. Increasing the rigidity of the folding nucleus changes the local structure of the transition state ensemble nonuniformly across the set of proteins studied. Nevertheless, the calculated prefactors k(0) are found to be relatively uniform across the protein set, with variation in 1/k(0) less than a factor of 5. This direct calculation justifies the common assumption that the prefactor is roughly the same for all small two-state folding proteins. Using the barrier heights obtained from the model and the best-fit monomer relaxation time 30 ns, we find that 1/k(0) approximately 1-5 mus (with average 1/k(0) approximately 4 micros). This model can be extended to study subtle aspects of folding such as the variation of the folding rate with stability or solvent viscosity and the onset of downhill folding.

Using Stochastic Roadmap Simulation to Predict Experimental Quantities in Protein Folding Kinetics: Folding Rates and Phi-Values

This paper presents a new method for studying protein folding kinetics. It uses the recently introduced Stochastic Roadmap Simulation (SRS) method to estimate the transition state ensemble (TSE) and predict the rates and the Phi-values for protein folding. The new method was tested on 16 proteins, whose rates and Phi-values have been determined experimentally. Comparison with experimental data shows that our method estimates the TSE much more accurately than an existing method based on dynamic programming. This improvement leads to better folding-rate predictions. We also compute the mean first passage time of the unfolded states and show that the computed values correlate with experimentally determined folding rates. The results on Phi-value predictions are mixed, possibly due to the simple energy model used in the tests. This is the first time that results obtained from SRS have been compared against a substantial amount of experimental data. The results further validate the SRS method and indicate its potential as a general tool for studying protein folding kinetics.

Denaturation of jack-bean urease by sodium n-dodecyl sulphate: A kinetic study below the critical micelle concentration

Kinetics of urease denaturation by anionic surfactant (sodium n-dodecyl sulphate, SDS) at concentrations below the critical micelle concentration (CMC) is investigated spectrophotometrically at neutral pH and the corresponding two-phase kinetic parameters of the process are estimated from a three-state reversible process using a binomial exponential relation based on the relaxation time method as: Using a prepared computer program, the experimental data are properly fitted into a binomial exponential relation, considering a two-phase denaturation pathway including a kinetically stable folded intermediate formed at SDS concentration of 1.1 mM. Forward and backward rate constants are estimated as: k(1)=0.2141+/-4.5 x 10(-3), k(2)=5.173 x 10(-3)+/-8.3 x 10(-5), k(-1)=0.09432+/-3.6 x 10(-4) and k(-2)=2.079 x 10(-3)+/-5.6 x 10(-5)s(-1) for the proposed mechanism. The rate-limiting step as well as the reaction coordinates in the denaturation mechanism are established. The mechanism involves formation of a kinetically stable folded native like intermediate through the electrostatic interactions. The intermediate was found to be more stable even than the native form (by about 9 kJmol(-1)) and still hexamer, because no loss of amplitude was observed. Electrophoresis experiments on the native and surfactant/urease complexes indicated a higher mobility for the kinetically folded native like intermediate.

A framework for describing topological frustration in models of protein folding

In a natively folded protein of moderate or larger size, the protein backbone may weave through itself in complex ways, raising questions about what sequence of events might have to occur in order for the protein to reach its native configuration from the unfolded state. A mathematical framework is presented here for describing the notion of a topological folding barrier, which occurs when a protein chain must pass through a hole or opening, formed by other regions of the protein structure. Different folding pathways encounter different numbers of such barriers and therefore different degrees of frustration. A dynamic programming algorithm finds the optimal theoretical folding path and minimal degree of frustration for a protein based on its natively folded configuration. Calculations over a database of protein structures provide insights into questions such as whether the path of minimal frustration might tend to favor folding from one or from many sites of folding nucleation, or whether proteins favor folding around the N terminus, thereby providing support for the hypothesis that proteins fold co-translationally. The computational methods are applied to a multi-disulfide bonded protein, with computational findings that are consistent with the experimentally observed folding pathway. Attention is drawn to certain complex protein folds for which the computational method suggests there may be a preferred site of nucleation or where folding is likely to proceed through a relatively well-defined pathway or intermediate. The computational analyses lead to testable models for protein folding.

Loop-closure events during protein folding: rationalizing the shape of Phi-value distributions

In the past years, the folding kinetics of many small single-domain proteins has been characterized by mutational Phi-value analysis. In this article, a simple, essentially parameter-free model is introduced which derives folding routes from native structures by minimizing the entropic loop-closure cost during folding. The model predicts characteristic folding sequences of structural elements such as helices and beta-strand pairings. Based on few simple rules, the kinetic impact of these structural elements is estimated from the routes and compared to average experimental Phi-values for the helices and strands of 15 small, well-characterized proteins. The comparison leads on average to a correlation coefficient of 0.62 for all proteins with polarized Phi-value distributions, and 0.74 if distributions with negative average Phi-values are excluded. The diffuse Phi-value distributions of the remaining proteins are reproduced correctly. The model shows that Phi-value distributions, averaged over secondary structural elements, can often be traced back to entropic loop-closure events, but also indicates energetic preferences in the case of a few proteins governed by parallel folding processes.

Cooperativity in two-state protein folding kinetics

We present a solvable model that predicts the folding kinetics of two-state proteins from their native structures. The model is based on conditional chain entropies. It assumes that folding processes are dominated by small-loop closure events that can be inferred from native structures. For CI2, the src SH3 domain, TNfn3, and protein L, the model reproduces two-state kinetics, and it predicts well the average Phi-values for secondary structures. The barrier to folding is the formation of predominantly local structures such as helices and hairpins, which are needed to bring nonlocal pairs of amino acids into contact.

Quantifying the roughness on the free energy landscape: entropic bottlenecks and protein folding rates

The prediction of protein folding rates and mechanisms is currently of great interest in the protein folding community. A close comparison between theory and experiment in this area is promising to advance our understanding of the physical-chemical principles governing the folding process. The delicate interplay of entropic and energetic/enthalpic factors in the protein free energy regulates the details of this complex reaction. In this article, we propose the use of topological descriptors to quantify the amount of heterogeneity in the configurational entropy contribution to the free energy. We apply the procedure to a set of 16 two-state folding proteins. The results offer a clean and simple theoretical explanation for the experimentally measured folding rates and mechanisms, in terms of the intrinsic entropic roughness along the populated folding routes on the protein free energy landscape.

Outlining folding nuclei in globular proteins

Our theoretical approach for prediction of folding/unfolding nuclei in three-dimensional protein structures is based on a search for free energy saddle points on networks of protein unfolding pathways. Under some approximations, this search is performed rapidly by dynamic programming and results in prediction of Phi values, which can be compared with those found experimentally. In this study, we optimize some details of the model (specifically, hydrogen atoms are taken into account in addition to heavy atoms), and compare the theoretically obtained and experimental Phi values (which characterize involvement of residues in folding nuclei) for all 17 proteins, where Phi values are now known for many residues. We show that the model provides good Phi value predictions for proteins whose structures have been determined by X-ray analysis (the average correlation coefficient is 0.65), with a more limited success for proteins whose structures have been determined by NMR techniques only (the average correlation coefficient is 0.34), and that the transition state free energies computed from the same model are in a good anticorrelation with logarithms of experimentally measured folding rates at mid-transition (the correlation coefficient is -0.73).

Prediction of folding rates and transition-state placement from native-state geometry

A variety of experimental and theoretical studies have established that the folding process of monomeric proteins is strongly influenced by the topology of the native state. In particular, folding times have been shown to correlate well with the contact order, a measure of contact locality. Our investigation focuses on identifying additional topologic properties that correlate with experimentally measurable quantities, such as folding rates and transition-state placement, for both two- and three-state folders. The validation against data from 40 experiments shows that a particular topological property that measures the interdependence of contacts, termed cliquishness or clustering coefficient, can account with statistically significant accuracy both for the transition state placement and especially for folding rates. The observed correlations can be further improved by optimally combining the distinct topological information captured by cliquishness and contact order.

The topomer search model: A simple, quantitative theory of two-state protein folding kinetics

Most small, single-domain proteins fold with the uncomplicated, single-exponential kinetics expected for diffusion on a smooth energy landscape. Despite this energetic smoothness, the folding rates of these two-state proteins span a remarkable million-fold range. Here, we review the evidence in favor of a simple, mechanistic description, the topomer search model, which quantitatively accounts for the broad scope of observed two-state folding rates. The model, which stipulates that the search for those unfolded conformations with a grossly correct topology is the rate-limiting step in folding, fits observed rates with a correlation coefficient of approximately 0.9 using just two free parameters. The fitted values of these parameters, the pre-exponential attempt frequency and a measure of the difficulty of ordering an unfolded chain, are consistent with previously reported experimental constraints. These results suggest that the topomer search process may dominate the relative barrier heights of two-state protein-folding reactions.

Elastic properties of proteins: insight on the folding process and evolutionary selection of native structures

We carry out a theoretical study of the vibrational and relaxation properties of naturally occurring proteins with the purpose of characterizing both the folding and equilibrium thermodynamics. By means of a suitable model, we provide a full characterization of the spectrum and eigenmodes of vibration at various temperatures by merely exploiting the knowledge of the protein native structure. It is shown that the rate at which perturbations decay at the folding transition correlates well with experimental folding rates. This validation is carried out on a list of about 30 two-state folders. Furthermore, the qualitative analysis of residues mean square displacements (shown to reproduce crystallographic data accurately) provides a reliable and statistically accurate method to identify crucial folding sites/contacts. This novel strategy is validated against clinical data for human immunodeficiency virus type 1 (HIV-1) protease. Finally, we compare the spectra and eigenmodes of vibration of natural proteins against randomly generated compact structures and regular random graphs. The comparison reveals a distinctive enhanced flexibility of natural structures accompanied by slow relaxation times at the folding temperature. The fact that these properties are connected intimately to the presence and assembly of secondary motifs hints at the special criteria adopted by evolution in the selection of viable folds.

Folding of circular permutants with decreased contact order: general trend balanced by protein stability

To examine the influence of contact order and stability on the refolding rate constant for two-state proteins, we have analysed the folding kinetics of the small beta-alpha-beta protein S6 and two of its circular permutants with relative contact orders of 0.19, 0.15 and 0.12. Data reveal a small but significant increase of the refolding rate constant (log k(f)) with decreasing contact order. At the same time, the decreased contact order is correlated to losses in global stability and alterations of the folding nucleus. When the differences in stability are accounted for by addition of Na2SO4 or by comparison of the folding kinetics at the transition mid-point, the dependence between log k(f) and contact order becomes stronger and follows the general correlation for two-state proteins. The observation emphasizes the combined action of topology and stability in controlling the rate constant of protein folding.