Chain length is the main determinant of the folding rate for proteins with three-state folding kinetics

We demonstrate that chain length is the main determinant of the folding rate for proteins with the three-state folding kinetics. The logarithm of their folding rate in water (k(f)) strongly anticorrelates with their chain length L (the correlation coefficient being -0.80). At the same time, the chain length has no correlation with the folding rate for two-state folding proteins (the correlation coefficient is -0.07). Another significant difference of these two groups of proteins is a strong anticorrelation between the folding rate and Baker's "relative contact order" for the two-state folders and the complete absence of such correlation for the three-state folders.     

Local secondary structure content predicts folding rates for simple,two-state proteins

Many single-domain proteins exhibit two-state folding kinetics, with folding rates that span more than six orders of magnitude. A quantity of much recent interest for such proteins is their contact order, the average separation in sequence between contacting residue pairs. Numerous studies have reached the surprising conclusion that contact order is well-correlated with the logarithm of the folding rate for these small, well-characterized molecules. Here, we investigate the physico-chemical basis for this finding by asking whether contact order is actually a composite number that measures the fraction of local secondary structure in the protein; viz. turns, helices, and hairpins. To pursue this question, we calculated the secondary structure content for 24 two-state proteins and obtained coefficients that predict their folding rates. The predicted rates correlate strongly with experimentally determined rates, comparable to the correlation with contact order. Further, these predicted folding rates are correlated strongly with contact order. Our results suggest that the folding rate of two-state proteins is a function of their local secondary structure content, consistent with the hierarchic model of protein folding. Accordingly, it should be possible to utilize secondary structure prediction methods to predict folding rates from sequence alone.     

Kinetically robust monomeric protein from a hyperthermophile

Equilibrium and kinetic studies were carried out under denaturation conditions to clarify the energetic features of the high stability of a monomeric protein, ribonuclease HII, from a hyperthermophile, Thermococcus kodakaraensis (Tk-RNase HII). Guanidine hydrochloride (GdnHCl)-induced unfolding and refolding were measured with circular dichroism at 220 nm, and heat-induced denaturation was studied with differential scanning calorimetry. Both GdnHCl- and heat-induced denaturation are very reversible. It was difficult to obtain the equilibrated unfolding curve of Tk-RNase HII below 40 degrees C, because of the remarkably slow unfolding. The two-state unfolding and refolding reactions attained equilibrium at 50 degrees C after 2 weeks. The Gibbs energy change of GdnHCl-induced unfolding (DeltaG(H(2)O)) at 50 degrees C was 43.6 kJ mol(-1). The denaturation temperature in the DSC measurement shifted as a function of the scan rate; the denaturation temperature at a scan rate of 90 degrees C h(-1) was higher than at a scan rate of 5 degrees C h(-1). The unfolding and refolding kinetics of Tk-RNase HII were approximated as a first-order reaction. The ln k(u) and ln k(r) values depended linearly on the denaturant concentration between 10 and 50 degrees C. The DeltaG(H(2)O) value obtained from the rate constant in water using the two-state model at 50 degrees C, 44.5 kJ mol(-1), was coincident with that from the equilibrium study, 43.6 kJ mol(-1), suggesting the two-state folding of Tk-RNase HII. The values for the rate constant in water of the unfolding for Tk-RNase HII were much smaller than those of E. coli RNase HI and Thermus thermophilus RNase HI, which has a denaturation temperature similar to that of Tk-RNase HII. In contrast, little difference was observed in the refolding rates among these proteins. These results indicate that the stabilization mechanism of monomeric protein from a hyperthermophile, Tk-RNase HII, with reversible two-state folding is characterized by remarkably slow unfolding.     

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.     

High concentrations of viscogens decrease the protein folding rate constant by prematurely collapsing the coil

In several studies, viscogenic osmolytes have been suggested to decrease the folding rate constant of polypeptides by slowing their motion through the solvent. Here, we show that osmolytes may slow protein folding by prematurely collapsing the coil. At low or moderate concentrations of osmolytes (<30%), folding of the two-state protein CI2 becomes faster with increasing osmolyte concentrations, suggesting that the kinetics are governed by protein stability. However, at higher concentrations of osmolyte, the coil collapses in the dead-time of the refolding experiment, causing a dramatic drop in the folding rate. The collapsed state is non-native and appears to be different for different osmolytes.     

Acidic conditions stabilise intermediates populated during the folding of Im7 and Im9

The helical bacterial immunity proteins Im7 and Im9 have been shown to fold via kinetic mechanisms of differing complexity, despite having 60 % sequence identity. At pH 7.0 and 10 degrees C, Im7 folds in a three-state mechanism involving an on-pathway intermediate, while Im9 folds in an apparent two-state transition. In order to examine the folding mechanisms of these proteins in more detail, the folding kinetics of both Im7 and Im9 (at 10 degrees C in 0.4 M sodium sulphate) have been examined as a function of pH. Kinetic modelling of the folding and unfolding data for Im7 between pH 5.0 and 8.0 shows that the on-pathway intermediate is stabilised by more acidic conditions, whilst the native state is destabilised. The opposing effect of pH on the stability of these states results in a significant population of the intermediate at equilibrium at pH 6.0 and below. At pH 7.0, the folding and unfolding kinetics for Im9 can be fitted adequately by a two-state model, in accord with previous results. However, under acidic conditions there is a clear change of slope in the plot of the logarithm of the folding rate constant versus denaturant concentration, consistent with the population of one or more intermediate(s) early during folding. The kinetic data for Im9 at these pH values can be fitted to a three-state model, where the intermediate ensemble is stabilised and the native state destabilised as the pH is reduced, rationalising previous results that showed that an intermediate is not observed experimentally at pH 7.0. The data suggest that intermediate formation is a general step in immunity protein folding and demonstrate that it is necessary to explore a wide range of refolding conditions in order to show that intermediates do not form in the folding of other small, single-domain proteins.     

Folding thermodynamics and kinetics of the leucine-rich repeat domain of the virulence factor Internalin B

Although the folding of alpha-helical repeat proteins has been well characterized, much less is known about the folding of repeat proteins containing beta-sheets. Here we investigate the folding thermodynamics and kinetics of the leucine-rich repeat (LRR) domain of Internalin B (InlB), an extracellular virulence factor from the bacterium Lysteria monocytogenes. This domain contains seven tandem leucine-rich repeats, of which each contribute a single beta-strand that forms a continuous beta-sheet with neighboring repeats, and an N-terminal alpha-helical capping motif. Despite its modular structure, InlB folds in an equilibrium two-state manner, as reflected by the identical thermodynamic parameters obtained by monitoring its sigmoidal urea-induced unfolding transition by different spectroscopic probes. Although equilibrium two-state folding is common in alpha-helical repeat proteins, to date, InlB is the only beta-sheet-containing repeat protein for which this behavior is observed. Surprisingly, unlike other repeat proteins exhibiting equilibrium two-state folding, InlB also folds by a simple two-state kinetic mechanism lacking intermediates, aside from the effects of prolyl isomerization on the denatured state. However, like other repeat proteins, InlB also folds significantly more slowly than expected from contact order. When plotted against urea, the rate constants for the fast refolding and single unfolding phases constitute a linear chevron that, when fitted with a kinetic two-state model, yields thermodynamic parameters matching those observed for equilibrium folding. Based on these kinetic parameters, the transition state is estimated to comprise 40% of the total surface area buried upon folding, indicating that a large fraction of the native contacts are formed in the rate-limiting step to folding.     

Osmolyte effects on kinetics of FKBP12 C22A folding coupled with prolyl isomerization

Unfolding and refolding kinetics of human FKBP12 C22A were monitored by fluorescence emission over a wide range of urea concentration in the presence and absence of protecting osmolytes glycerol, proline, sarcosine and trimethylamine-N-oxide (TMAO). Unfolding is well described by a mono-exponential process, while refolding required a minimum of two exponentials for an adequate fit throughout the urea concentration range considered. The bi-exponential behavior resulted from complex coupling between protein folding, and prolyl isomerization in the denatured state in which the urea-dependent rate constant for folding was greater than, equal to, and less than the rate constants for prolyl isomerization within the urea concentration range of zero to five molar. Amplitudes and the observed folding and unfolding rate constants were fitted to a reversible three-state model composed of two sequential steps involving the native state and a folding-competent denatured species thermodynamically linked to a folding-incompetent denatured species. Excellent agreement between thermodynamic parameters for FKBP12 C22A folding calculated from the kinetic parameters and those obtained directly from equilibrium denaturation assays provides strong support for the applicability of the mechanism, and provides evidence that FKBP12 C22A folding/unfolding is two-state, with prolyl isomer heterogeneity in the denatured ensemble. Despite the chemical diversity of the protecting osmolytes, they all exhibit the same kinetic behavior of increasing the rate constant of folding and decreasing the rate constant for unfolding. Osmolyte effects on folding/unfolding kinetics are readily explained in terms of principles established in understanding osmolyte effects on protein stability. These principles involve the osmophobic effect, which raises the Gibbs energy of the denatured state due to exposure of peptide backbone, thereby increasing the folding rate. This effect also plays a key role in decreasing the unfolding rate when, as is often the case, the activated complex exposes more backbone than is exposed in the native state.     

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.     

Solvation effects and driving forces for protein thermodynamic and kinetic cooperativity: how adequate is native-centric topological modeling?

What energetic and solvation effects underlie the remarkable two-state thermodynamics and folding/unfolding kinetics of small single-domain proteins? To address this question, we investigate the folding and unfolding of a hierarchy of continuum Langevin dynamics models of chymotrypsin inhibitor 2. We find that residue-based additive Gō-like contact energies, although native-centric, are by themselves insufficient for protein-like calorimetric two-state cooperativity. Further native biases by local conformational preferences are necessary for protein-like thermodynamics. Kinetically, however, even models with both contact and local native-centric energies do not produce simple two-state chevron plots. Thus a model protein's thermodynamic cooperativity is not sufficient for simple two-state kinetics. The models tested appear to have increasing internal friction with increasing native stability, leading to chevron rollovers that typify kinetics that are commonly referred to as non-two-state. The free energy profiles of these models are found to be sensitive to the choice of native contacts and the presumed spatial ranges of the contact interactions. Motivated by explicit-water considerations, we explore recent treatments of solvent granularity that incorporate desolvation free energy barriers into effective implicit-solvent intraprotein interactions. This additional feature reduces both folding and unfolding rates vis-à-vis that of the corresponding models without desolvation barriers, but the kinetics remain non-two-state. Taken together, our observations suggest that interaction mechanisms more intricate than simple Gō-like constructs and pairwise additive solvation-like contributions are needed to rationalize some of the most basic generic protein properties. Therefore, as experimental constraints on protein chain models, requiring a consistent account of protein-like thermodynamic and kinetic cooperativity can be more stringent and productive for some applications than simply requiring a model heteropolymer to fold to a target structure.     

Preorganized secondary structure as an important determinant of fast protein folding

The folding and unfolding kinetics of the B-domain of staphylococcal protein A, a small three-helix bundle protein, were probed by NMR. The lineshape of a single histidine resonance was fit as a function of denaturant to give folding and unfolding rate constants. The B-domain folds extremely rapidly in a two-state manner, with a folding rate constant of 120,000 s-1, making it one of the fastest-folding proteins known. Diffusion-collision theory predicts folding and unfolding rate constants that are in good agreement with the experimental values. The apparent rate constant as a function of denaturant ('chevron plot') is predicted within an order of magnitude. Our results are consistent with a model whereby fast-folding proteins utilize a diffusion-collision mechanism, with the preorganization of one or more elements of secondary structure in the unfolded protein.     

Solvent-tuning the collapse and helix formation time scales of lambda(6-85)*

The lambda(6-85)(*) pseudo-wild type of lambda repressor fragment is a fast two-state folder (k(f) approximately 35 microsec(-1) at 58 degrees C). Previously, highly stable lambda(6-85)(*) mutants with k(f) > 30 microsec(-1) have been engineered to fold nearly or fully downhill. Stabilization of the native state by solvent tuning might also tune lambda(6-85)(*) away from two-state folding. We test this prediction by examining the folding thermodynamics and kinetics of lambda(6-85)(*) in a stabilizing solvent, 45% by weight aqueous ethylene glycol at -28 degrees C. Detection of kinetics by circular dichroism at 222 nm (sensitive to helix content) and small angle X-ray scattering (measuring the radius of gyration) shows that refolding from guanidine hydrochloride denatured conditions exhibits very different time scales for collapse and secondary structure formation: the two processes become decoupled. Collapse remains a low-barrier activated process, while the fastest of several secondary structure formation time scales approaches the downhill folding limit. Two-state folding of lambda(6-85)(*) is not a robust process.     

Rapid folding with and without populated intermediates in the homologous four-helix proteins Im7 and Im9

The kinetics and thermodynamics of the folding of the homologous four-helix proteins Im7 and Im9 have been characterised at pH 7.0 and 10 degrees C. These proteins are 60 % identical in sequence and have the same three-dimensional structure, yet appear to fold by different kinetic mechanisms. The logarithm of the folding and unfolding rates of Im9 change linearly as a function of urea concentration and fit well to an equation describing a two-state mechanism (with a folding rate of 1500 s-1, an unfolding rate of 0. 01 s-1, and a highly compact transition state that has approximately 95 % of the native surface area buried). By contrast, there is clear evidence for the population of an intermediate during the refolding of Im7, as indicated by a change in the urea dependence of the folding rate and the presence of a significant burst phase amplitude in the refolding kinetics. Under stabilising conditions (0.25 M Na2SO4, pH 7.0 and 10 degrees C) the folding of Im9 remains two-state, whilst under similar conditions (0.4 M Na2SO4, pH 7.0 and 10 degrees C) the intermediate populated during Im7 refolding is significantly stabilised (KUI=125). Equilibrium denaturation experiments, under the conditions used in the kinetic measurements, show that Im7 is significantly less stable than Im9 (DeltaDeltaG 9.3 kJ/mol) and the DeltaG and m values determined accord with those obtained from the fit to the kinetic data. The results show, therefore, that the population of an intermediate in the refolding of the immunity protein structure is defined by the precise amino acid sequence rather than the global stability of the protein. We discuss the possibility that the intermediate of Im7 is populated due to differences in helix propensity in Im7 and Im9 and the relevance of these data to the folding of helical proteins in general.     

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.     

Thermodynamics and kinetics of protein folding: an evolutionary perspective

This article appeals to an evolutionary model which postulates that primordial proteins were described by small polypeptide chains which (i) lack disulfide bridges, and (ii) display slow folding rates with multi-state kinetics, to determine relations between structural properties of proteins and their folding kinetics. We parameterize the energy landscape of proteins in terms of thermodynamic activation variables. The model studies evolutionary changes in these thermodynamic parameters, and we invoke relations between these activation variables and structural properties of the protein to predict the following correspondence between protein structure and folding kinetics. 1. Proteins with inter- and intra-chain disulfide bridges: large variability in both folding rates and stability of intermediates, multi-state kinetics. 2. Proteins which lack inter and intra-chain disulfide bridges. 2.1 Single-domain chains: fast folding rates; unstable intermediates; two-state kinetics. 2.2 Multi-domain monomers: intermediate rates; metastable intermediates; multi-state kinetics. 2.3 Multi-domain oligomers: slow rates; metastable intermediates; multi-state kinetics. The evolutionary model thus provides a kinetic characterization of one important subfamily of proteins which we describe by the following properties: Folding dynamics of single-domain proteins which lack disulfide bridges are described by two-state kinetics. Folding rate of this class of proteins is positively correlated with the thermodynamic stability of the folded state.     

Charge-charge interactions in the denatured state influence the folding kinetics of ribonuclease Sa

Gaining a better understanding of the denatured state ensemble of proteins is important for understanding protein stability and the mechanism of protein folding. We studied the folding kinetics of ribonuclease Sa (RNase Sa) and a charge-reversal variant (D17R). The refolding kinetics are similar, but the unfolding rate constant is 10-fold greater for the variant. This suggests that charge-charge interactions in the denatured state and the transition state ensembles are more favorable in the variant than in RNase Sa, and shows that charge-charge interactions can influence the kinetics and mechanism of protein folding.     

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.     

The dynamical contact order: Protein folding rate parameters based on quantum conformational transitions

Protein folding is regarded as a quantum transition between the torsion states of a polypeptide chain. According to the quantum theory of conformational dynamics, we propose the dynamical contact order (DCO) defined as a characteristic of the contact described by the moment of inertia and the torsion potential energy of the polypeptide chain between contact residues. Consequently, the protein folding rate can be quantitatively studied from the point of view of dynamics. By comparing theoretical calculations and experimental data on the folding rate of 80 proteins, we successfully validate the view that protein folding is a quantum conformational transition. We conclude that (i) a correlation between the protein folding rate and the contact inertial moment exists; (ii) multi-state protein folding can be regarded as a quantum conformational transition similar to that of two-state proteins but with an intermediate delay. We have estimated the order of magnitude of the time delay; (iii) folding can be classified into two types, exergonic and endergonic. Most of the two-state proteins with higher folding rate are exergonic and most of the multi-state proteins with low folding rate are endergonic. The folding speed limit is determined by exergonic folding.     

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.