Is it always possible to distinguish two- and three-state systems by evaluating the van't Hoff enthalpy?

Many small globular proteins are traditionally classified as thermodynamical two-state systems, i.e., the protein is either in the native, active state (folded) or in the denatured state (unfolded). We challenge this view and show that there may exist (protein) systems for which a van't Hoff analysis of experimental data cannot determine whether the system corresponds to two or three thermodynamical states when only temperatures in a narrow temperature region around the transition are considered. We generalize a widely employed two-state protein folding model to include a third, transition state. For this three-state system we systematically study the deviation of the calorimetric enthalpy (heat of transition) from the van't Hoff enthalpy, a measure of the two-stateness of a transition. We show that under certain conditions the heat capacity of the three-state system can be almost indistinguishable from the heat capacity for the two-state system over a broad temperature interval. The consequence may be that some three-state (or even more than three-states) systems have been misinterpreted as two-state systems when the conclusion is drawn solely upon the van't Hoff enthalpy. These findings are important not only for proteins, but also for the interpretation of thermodynamical systems in general.     

Modeling protein density of states: additive hydrophobic effects are insufficient for calorimetric two-state cooperativity

A well-established experimental criterion for two-state thermodynamic cooperativity in protein folding is that the van't Hoff enthalpy DeltaH(vH) around the transition midpoint is equal, or very nearly so, to the calorimetric enthalpy DeltaH(cal) of the entire transition. This condition is satisfied by many small proteins. We use simple lattice models to provide a statistical mechanical framework to elucidate how this calorimetric two-state picture may be reconciled with the hierarchical multistate scenario emerging from recent hydrogen exchange experiments. We investigate the feasibility of using inverse Laplace transforms to recover the underlying density of states (i.e., enthalpy distribution) from calorimetric data. We find that the constraint imposed by DeltaH(vH)/DeltaH(cal) approximately 1 on densities of states of proteins is often more stringent than other "two-state" criteria proposed in recent theoretical studies. In conjunction with reasonable assumptions, the calorimetric two-state condition implies a narrow distribution of denatured-state enthalpies relative to the overall enthalpy difference between the native and the denatured conformations. This requirement does not always correlate with simple definitions of "sharpness" of a transition and has important ramifications for theoretical modeling. We find that protein models that assume capillarity cooperativity can exhibit overall calorimetric two-state-like behaviors. However, common heteropolymer models based on additive hydrophobic-like interactions, including highly specific two-dimensional Gō models, fail to produce proteinlike DeltaH(vH)/DeltaH(cal) approximately 1. A simple model is constructed to illustrate a proposed scenario in which physically plausible local and nonlocal cooperative terms, which mimic helical cooperativity and environment-dependent hydrogen bonding strength, can lead to thermodynamic behaviors closer to experiment. Our results suggest that proteinlike thermodynamic cooperativity may require a cooperative interplay between local and nonlocal interactions. The prospect of using calorimetric data to constrain Z-scores of knowledge-based potentials is discussed.     

Equilibrium thermal transitions of collagen model peptides

The folding of collagen in vitro is very slow and presents difficulties in reaching equilibrium, a feature that may have implications for in vivo collagen function. Peptides serve as good model systems for examining equilibrium thermal transitions in the collagen triple helix. Investigations were carried out to ascertain whether a range of synthetic triple-helical peptides of varying sequences can reach equilibrium, and whether the triple helix to unfolded monomer transition approximates a two-state model. The thermal transitions for all peptides studied are fully reversible given sufficient time. Isothermal experiments were carried out to obtain relaxation times at different temperatures. The slowest relaxation times, on the order of 10-15 h, were observed at the beginning of transitions, and were shown to result from self-association limited by the low concentration of free monomers, rather than cis-trans isomerization. Although the fit of the CD equilibrium transition curves and the concentration dependence of T(m) values support a two-state model, the more rigorous comparison of the calorimetric enthalpy to the van't Hoff enthalpy indicates the two-state approximation is not ideal. Previous reports of melting curves of triple-helical host-guest peptides are shown to be a two-state kinetic transition, rather than an equilibrium transition.     

Polymer principles of protein calorimetric two-state cooperativity

The experimental calorimetric two-state criterion requires the van't Hoff enthalpy DeltaH(vH) around the folding/unfolding transition midpoint to be equal or very close to the calorimetric enthalpy DeltaH(cal) of the entire transition. We use an analytical model with experimental parameters from chymotrypsin inhibitor 2 to elucidate the relationship among several different van't Hoff enthalpies used in calorimetric analyses. Under reasonable assumptions, the implications of these DeltaH(vH)'s being approximately equal to DeltaH(cal) are equivalent: Enthalpic variations among denatured conformations in real proteins are much narrower than some previous lattice-model estimates, suggesting that the energy landscape theory "folding to glass transition temperature ratio" T(f) /T(g) may exceed 6.0 for real calorimetrically two-state proteins. Several popular three-dimensional lattice protein models, with different numbers of residue types in their alphabets, are found to fall short of the high experimental standard for being calorimetrically two-state. Some models postulate a multiple-conformation native state with substantial pre-denaturational energetic fluctuations well below the unfolding transition temperature, or predict a significant post-denaturational continuous conformational expansion of the denatured ensemble at temperatures well above the transition point, or both. These scenarios either disagree with experiments on protein size and dynamics, or are inconsistent with conventional interpretation of calorimetric data. However, when empirical linear baseline subtractions are employed, the resulting DeltaH(vH)/DeltaH(cal)'s for some models can be increased to values closer to unity, and baseline subtractions are found to correspond roughly to an operational definition of native-state conformational diversity. These results necessitate a re-assessment of theoretical models and experimental interpretations.     

Protein folding cooperativity: basic insights from minimalist models

Basic concepts about two-state, cooperative protein folding and its relation to first-order phase transitions are reviewed. Minimalist models capable of reproducing the required free energy barrier between folded and unfolded macroscopic states are described. A significantly more restrictive "calorimetric" criterion is also discussed, which is based on direct comparison between model and experimental heat capacities with additional assumptions about conformational enthalpy variation in the unfolded state.     

The Folding Trajectory of RNase H Is Dominated by Its Topology and Not Local Stability: A Protein Engineering Study of Variants that Fold via Two-State and Three-State Mechanisms

Proteins can sample a variety of partially folded conformations during the transition between the unfolded and native states. Some proteins never significantly populate these high-energy states and fold by an apparently two-state process. However, many proteins populate detectable, partially folded forms during the folding process. The role of such intermediates is a matter of considerable debate. A single amino acid change can convert Escherichia coli ribonuclease H from a three-state folder that populates a kinetic intermediate to one that folds in an apparent two-state fashion. We have compared the folding trajectories of the three-state RNase H and the two-state RNase H, proteins with the same native-state topology but altered regional stability, using a protein engineering approach. Our data suggest that both versions of RNase H fold through a similar trajectory with similar high-energy conformations. Mutations in the core and the periphery of the protein affect similar aspects of folding for both variants, suggesting a common trajectory with folding of the core region followed by the folding of the periphery. Our results suggest that formation of specific partially folded conformations may be a general feature of protein folding that can promote, rather than hinder, efficient folding.     

Criteria for downhill protein folding: calorimetry, chevron plot, kinetic relaxation, and single-molecule radius of gyration in chain models with subdued degrees of cooperativity

Recent investigations of possible downhill folding of small proteins such as BBL have focused on the thermodynamics of non-two-state, "barrierless" folding/denaturation transitions. Downhill folding is noncooperative and thermodynamically "one-state," a phenomenon underpinned by a unimodal conformational distribution over chain properties such as enthalpy, hydrophobic exposure, and conformational dimension. In contrast, corresponding distributions for cooperative two-state folding are bimodal with well-separated population peaks. Using simplified atomic modeling of a three-helix bundle-in a scheme that accounts for hydrophobic interactions and hydrogen bonding-and coarse-grained C(alpha) models of four real proteins with various degrees of cooperativity, we evaluate the effectiveness of several observables at defining the underlying distribution. Bimodal distributions generally lead to sharper transitions, with a higher heat capacity peak at the transition midpoint, compared with unimodal distributions. However, the observation of a sigmoidal transition is not a reliable criterion for two-state behavior, and the heat capacity baselines, used to determine the van't Hoff and calorimetric enthalpies of the transition, can introduce ambiguity. Interestingly we find that, if the distribution of the single-molecule radius of gyration were available, it would permit discrimination between unimodal and bimodal underlying distributions. We investigate kinetic implications of thermodynamic noncooperativity using Langevin dynamics. Despite substantial chevron rollovers, the relaxation of the models considered is essentially single-exponential over an extended range of native stabilities. Consistent with experiments, significant deviations from single-exponential behavior occur only under strongly folding conditions.     

Distinguishing between two-state and three-state models for ubiquitin folding

Conflicting results exist regarding whether the folding of mammalian ubiquitin at 25 degrees C is a simple, two-state kinetic process or a more complex, three-state process with a defined kinetic intermediate. We have measured folding rate constants up to about 1000 s(-1) using conventional rapid mixing methods in single-jump, double-jump, and continuous-flow modes. The linear dependence of folding rates on denaturant concentration and the lack of an unaccounted "burst-phase" change for the fluorescence signal indicate that a two-state folding model is adequate to describe the folding pathway. This behavior also is seen for folding in the presence of the stabilizing additives 0.23 M sodium sulfate and 1 M sodium chloride. These results stress the need for caution in interpreting deviations from ideal two-state "chevron" behavior when folding is heterogeneous or folding rate constants are near the detection limit.     

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.     

pH dependent unfolding characteristics of DLC8 dimer: Residue level details from NMR

Environment dependence of folding and unfolding of a protein is central to its function. In the same vein, knowledge of pH dependence of stability and folding/unfolding is crucial for many biophysical equilibrium and kinetic studies designed to understand protein folding mechanisms. In the present study we investigated the guanidine induced unfolding transition of dynein light chain protein (DLC8), a cargo adaptor of the dynein complex in the pH range 7-10. It is observed that while the protein remains a dimer in the entire pH range, its stability is somewhat reduced at alkaline pH. Global unfolding features monitored using fluorescence spectroscopy revealed that the unfolding transition of DLC8 at pH 7 is best described by a three-state model, whereas, that at pH 10 is best described by a two-state model. Chemical shift perturbations due to pH change provided insights into the corresponding residue level structural perturbations in the DLC8 dimer. Likewise, backbone (15)N relaxation measurements threw light on the corresponding motional changes in the dimeric protein. These observations have been rationalized on the basis of expected changes with increasing pH in the protonation states of the titratable residues on the structure of the protein. These, in turn provide an explanation for the change from three-state to two-state guanidine induced unfolding transition as the pH is increased from 7 to 10. All these results exemplify and highlight the role of environment vis-à-vis the sequence and structure of a given protein in dictating its folding/unfolding characteristics.     

Nonideality and protein thermal denaturation

We studied the thermal denaturation of eglin c by using CD spectropolarimetry and differential scanning calorimetry (DSC). At low protein concentrations, denaturation is consistent with the classical two-state model. At concentrations greater than several hundred microM, however, the calorimetric enthalpy and the midpoint transition temperature increase with increasing protein concentration. These observations suggested the presence of intermediates and/or native state aggregation. However, the transitions are symmetric, suggesting that intermediates are absent, the DSC data do not fit models that include aggregation, and analytical ultracentrifugation (AUC) data show that native eglin c is monomeric. Instead, the AUC data show that eglin c solutions are nonideal. Analysis of the AUC data gives a second virial coefficient that is close to values calculated from theory and the DSC data are consistent with the behavior expected for nonideal solutions. We conclude that the concentration dependence is caused by differential nonideality of the native and denatured states. The nondeality arises from the high charge of the protein at acid pH and is exacerbated by low buffer concentrations. Our conclusion may explain differences between van't Hoff and calorimetric denaturation enthalpies observed for other proteins whose behavior is otherwise consistent with the classical two-state model.     

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 rates and low-entropy-loss routes of two-state proteins

We develop a simple model for computing the rates and routes of folding of two-state proteins from the contact maps of their native structures. The model is based on the graph-theoretical concept of effective contact order (ECO). The model predicts that proteins fold by "zipping up" in a sequence of small-loop-closure events, depending on the native chain fold. Using a simple equation, with a few physical rate parameters, we obtain a good correlation with the folding rates of 24 two-state folding proteins. The model rationalizes data from Phi-value analysis that have been interpreted in terms of delocalized or polarized transition states. This model indicates how much of protein folding may take place in parallel, not along a single reaction coordinate or with a single transition state.     

Simulation and experiment conspire to reveal cryptic intermediates and a slide from the nucleation-condensation to framework mechanism of folding

There is a change from three-state to two-state kinetics of folding across the homeodomain superfamily of proteins as the mechanism slides from framework to nucleation-condensation. The tendency for framework folding in this family correlates with inherent helical propensity. The cellular myeloblastis protein (c-Myb) falls in the mechanistic transition region. An earlier, preliminary report of protein engineering experiments and molecular dynamics simulations (MD) showed that the folding mechanism for this protein has aspects of both the nucleation-condensation and framework models. In the more in-depth analysis of the MD trajectories presented here, we find that folding may be attributed to both of these mechanisms in different regions of the protein. The folding of the loop, middle helix, and turn is best described by nucleation-condensation, whereas folding of the N and C-terminal helices may be described by the framework model. Experimentally, c-Myb folds by apparent two-state kinetics, but the MD simulations predict that the kinetics hide a high-energy intermediate. We stabilized this hypothetical folding intermediate by deleting a residue (P174) in the loop between its second and third helices, and the mutant intermediate is long-lived in the simulations. Equilibrium and kinetic experiments demonstrate that folding of the DeltaP174 mutant is indeed three-state. The presence and shape of the intermediate observed in the simulations were confirmed by small angle X-ray scattering experiments.     

Thermal unfolding mechanism of lipocalin-type prostaglandin D synthase

Lipocalin-type prostaglandin (PG) D synthase (L-PGDS) is a dual-functioning protein in the lipocalin family, acting as a PGD(2)-synthesizing enzyme and as an extracellular transporter for small lipophilic molecules. We earlier reported that denaturant-induced unfolding of L-PGDS follows a four-state pathway, including an activity-enhanced state and an inactive intermediate state. In this study, we investigated the thermal unfolding mechanism of L-PGDS by using differential scanning calorimetry (DSC) and CD spectroscopy. DSC measurements revealed that the thermal unfolding of L-PGDS was a completely reversible process at pH 4.0. The DSC curves showed no concentration dependency, demonstrating that the thermal unfolding of L-PGDS involved neither intermolecular interaction nor aggregation. On the basis of a simple two-state unfolding mechanism, the ratio of van't Hoff enthalpy (DeltaH(vH)) to calorimetric enthalpy (DeltaH(cal)) was below 1, indicating the presence of an intermediate state (I) between the native state (N) and unfolded state (U). Then, statistical thermodynamic analyses of a three-state unfolding process were performed. The heat capacity curves fit well with a three-state process; and the estimated transition temperature (T(m)) and enthalpy change (DeltaH(cal)) of the N<-->I and I<-->U transitions were 48.2 degrees C and 190 kJ.mol(-1), and 60.3 degrees C and 144 kJ.mol(-1), respectively. Correspondingly, the thermal unfolding monitored by CD spectroscopy at 200, 235 and 290 nm revealed that L-PGDS unfolded through the intermediate state, where its main chain retained the characteristic beta-sheet structure without side-chain interactions.     

Calorimetric and spectroscopic investigation of the helix-to-coil transition of the self-complementary deoxyribonucleotide ATGCAT

Differential scanning calorimetry and temperature-dependent uv spectroscopy are used to thermodynamically characterize the double-strand to single-strand transition of the self-complementary deoxyribo-oligonucleotide ATGCAT. The calorimetric experiments provide a value of 33.6 kcal (mol of double strand)^?1 for the transition between 10 and 90° C. In conjunction with available temperature-dependent nmr data (which reveals terminal base pair fraying), we attempt to define specifically those interactions to which the calorimetrically measured enthalpy change refers.Values of ΔH_V.H. (van 't Hoff enthalpy change) are derived from the spectroscopic and calorimetric data and compared with the ΔH obtained directly from the calorimetric experiment. This comparison reveals that the part of the thermally-induced transition that occurs between 10 and 90°C is well represented by a two-state process. It is noted that in assessing the applicability of the two-state model it is best to compare the ΔH_cal. with ΔH_V.H. obtained from the calorimetric rather than the spectroscopic data.     

The fluorescence detected guanidine hydrochloride equilibrium denaturation of wild-type staphylococcal nuclease does not fit a three-state unfolding model

A three-state equilibrium unfolding of a protein can be difficult to detect if two of the states fail to differ in some easily measurable way. It has been unclear whether staphylococcal nuclease unfolds in a two-state fashion, with only the native and denatured states significantly populated at equilibrium, or in a three-state manner, with a well-populated intermediate. Since equilibrium unfolding experiments are commonly used to determine protein stability and the course of denaturation are followed by changes in the fluorescence which has difficulty in distinguishing various states, this is a potential problem for many proteins. Over the course of twenty years we have performed more than one hundred guanidine hydrochloride equilibrium denaturations of wild-type staphylococcal nuclease; to our knowledge, a number of denaturations unrivaled in any other protein system. A careful examination of the data from these experiments shows no sign of the behavior predicted by a three-state unfolding model. Specifically, a three-state unfolding should introduce a slight, but characteristic, non-linearity to the plot of stability versus denaturant concentration. The average residuals from this large number of repeated experiments do not show the predicted behavior, casting considerable doubt on the likelihood of a three-state unfolding for the wild-type protein. The methods used for analysis here could be applied to other protein systems to distinguish a two-state from a three-state denaturation.     

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.