Folding study of an Aib-rich peptide in DMSO by molecular dynamics simulations.

To evaluate the ability of molecular dynamics (MD) simulations using atomic force-fields to correctly predict stable folded conformations of a peptide in solution, we show results from MD simulations of the reversible folding of an octapeptide rich in alpha-aminoisobutyric acid (2-amino-2-methyl-propanoic acid, Aib) solvated in di-methyl-sulfoxide (DMSO). This solvent generally prevents the formation of secondary structure, whereas Aib-rich peptides show a high propensity to form secondary structural elements, in particular 3(10)- and alpha-helical structures. Aib is, moreover, achiral, so that Aib-rich peptides can form left- or right-handed helices depending on the overall composition of the peptide, the temperature, and the solvation conditions. This makes the system an interesting case to study the ensembles of peptide conformations as a function of temperature by MD simulation. Simulations involving the folding and unfolding of the peptide were performed starting from two initial structures, a right-handed alpha-helical structure and an extended structure, at three temperatures, 298 K, 340 K, and 380 K, and the results are compared with experimental nuclear magnetic resonance (NMR) data measured at 298 K and 340 K. The simulations generally reproduce the available experimental nuclear Overhauser effect (NOE) data, even when a wide range of conformations is sampled at each temperature. The importance of adequate statistical sampling in order to reliably interpret the experimental data is discussed.     

Conformational transition states of a beta-hairpin peptide between the ordered and disordered conformations in explicit water

The conformational transition states of a beta-hairpin peptide in explicit water were identified from the free energy landscapes obtained from the multicanonical ensemble, using an enhanced conformational sampling calculation. The beta-hairpin conformations were significant at 300 K in the landscape, and the typical nuclear Overhauser effect signals were reproduced, consistent with the previously reported experiment. In contrast, the disordered conformations were predominant at higher temperatures. Among the stable conformations at 300 K, there were several free energy barriers, which were not visible in the landscapes formed with the conventional parameters. We identified the transition states around the saddle points along the putative folding and unfolding paths between the beta-hairpin and the disordered conformations in the landscape. The characteristic features of these transition states are the predominant hydrophobic contacts and the several hydrogen bonds among the side-chains, as well as some of the backbone hydrogen bonds. The unfolding simulations at high temperatures, 400 K and 500 K, and their principal component analyses also provided estimates for the transition state conformations, which agreed well with those at 400 K and 500 K deduced from the current free energy landscapes at 400 K and 500 K, respectively. However, the transition states at high temperatures were much more widely distributed on the landscape than those at 300 K, and their conformations were different.     

Computer simulations of protein folding by targeted molecular dynamics

We have performed 128 folding and 45 unfolding molecular dynamics runs of chymotrypsin inhibitor 2 (CI2) with an implicit solvation model for a total simulation time of 0.4 microseconds. Folding requires that the three-dimensional structure of the native state is known. It was simulated at 300 K by supplementing the force field with a harmonic restraint which acts on the root-mean-square deviation and allows to decrease the distance to the target conformation. High temperature and/or the harmonic restraint were used to induce unfolding. Of the 62 folding simulations started from random conformations, 31 reached the native structure, while the success rate was 83% for the 66 trajectories which began from conformations unfolded by high-temperature dynamics. A funnel-like energy landscape is observed for unfolding at 475 K, while the unfolding runs at 300 K and 375 K as well as most of the folding trajectories have an almost flat energy landscape for conformations with less than about 50% of native contacts formed. The sequence of events, i.e., secondary and tertiary structure formation, is similar in all folding and unfolding simulations, despite the diversity of the pathways. Previous unfolding simulations of CI2 performed with different force fields showed a similar sequence of events. These results suggest that the topology of the native state plays an important role in the folding process.     

The effect of motional averaging on the calculation of NMR-derived structural properties.

The effect of motional averaging when relating structural properties inferred from nuclear magnetic resonance (NMR) experiments to molecular dynamics simulations of peptides is considered. In particular, the effect of changing populations of conformations, the extent of sampling, and the sampling frequency on the estimation of nuclear Overhauser effect (NOE) inter-proton distances, vicinal (3)J-coupling constants, and chemical shifts are investigated. The analysis is based on 50-ns simulations of a beta-heptapeptide in methanol at 298 K, 340 K, 350 K, and 360 K. This peptide undergoes reversible folding and samples a significant proportion of the available conformational space during the simulations, with at 298 K being predominantly folded and at 360 K being predominantly unfolded. The work highlights the fact that when motional averaging is included, NMR data has only limited capacity to distinguish between a single fully folded peptide conformation and various mixtures of folded and unfolded conformations. Proteins 1999;36:542-555.     

Comparison of the chemical and thermal denaturation of proteins by a two-state transition model

The conformational stabilities of eight proteins in terms of the free energy differences between the native "folded" state of the protein and its "unfolded" state were determined at 298 K by two methods: chemical denaturation at 298 K and extrapolation to 298 K of the thermal denaturation results at high temperature. The proteins were expressed in Escherichia coli from the Haemophilus influenzae and E. coli genes at different levels of expression, covered a molecular mass range from 13 to 37 kg mol(-1) per monomeric unit (some exhibiting unique structural features), and were oligomeric up to four subunits. The free energy differences were determined by application of a two-state transition model to the chemical and thermal denaturation results, ranged from 9.4 to 148 kJ mol(-1) at 298 K, and were found to be within the experimental uncertainties of both methods for all of the proteins. Any contributions from intermediate states detectable from chemical and thermal denaturation differences in the unfolding free energy differences in these proteins are within the experimental uncertainties of both methods.     

Equilibrium folding and unfolding pathways for a model protein

The folding–unfolding process of reduced bovine pancreatic trypsin inhibitor was investigated with an idealized model employing approximate free energies. The protein is regarded to consist of only C^α and C^β atoms. The backbone dihedral angles are the only conformational variables and are permitted to take discrete values at every 10°. Intraresidue energies consist of two terms: an empirical part taken from the observed frequency distributions of (?,ψ) and an additional favorable energy assigned to the native conformation of each residue. Interresidue interactions are simplified by assuming that there is an attractive energy operative only between residue pairs in close contact in the native structure. A total of 230,000 molecular conformations, with no atomic overlaps, ranging from the native state to the denatured state, are randomly generated by changing the sampling bias. Each conformation is classified according to its conformational energy, F; a conformational entropy, S(F) is estimated for each value of F from the number of samples. The dependence of S(F) on energy reveals that the folding–unfolding transition for this idealized model is an “all-or-none” type; this is attributable to the specific long-range interactions. Interresidue contact probabilities, averaged over samples representing various stages of folding, serve to characterize folding intermediates. Most probable equilibrium pathways for the folding–unfolding transition are constructed by connecting conformationally similar intermediates. The specific details obtained for bovine pancreatic trypsin inhibitor are as follows: (1) Folding begins with the appearance of nativelike medium-range contacts at a β-turn and at the α-helix. (2) These grow to include the native pair of interacting β-strands. This state includes intact regular secondary conformations, as well as the interstrand sheet contacts, and corresponds to an activated state with the highest free energy on the pathway. (3) Additional native long-range contacts are completely formed either toward the amino terminus or toward the carboxyl terminus. (4) In a final step, the missing contacts appear. Although these folding pathways for this model are not consistent with experimental reports, it does indicate multiple folding pathways. The method is general and can be applied to any set of calculated conformational energies and furthermore permits investigation of gross folding features.     

Fluorescence energy transfer indicates similar transient and equilibrium intermediates in staphylococcal nuclease folding

Fluorescence resonance energy transfer (FRET) is one of the few methods available to measure the rate at which a folding protein collapses. Using staphylococcal nuclease in which a cysteine residue was engineered in place of Lys64, permitted FRET measurements of the distance between the donor tryptophan 140 and 5-[[2-[(iodoacetyl)-amino]ethyl]amino]naphthalene-1-sulfonic acid-labeled Cys64. These measurements were undertaken on both equilibrium partially folded intermediates at low pH (A states), as well as transient intermediates during stopped-flow refolding. The results indicate that there is an initial collapse of the protein in the deadtime of the stopped-flow instrument, corresponding to a regain of approximately 60% of the native signal, followed by three slower transients. This is in contrast to circular dichroism measurements which show only 20-25% regain of the native secondary structure in the burst phase. Thus hydrophobic collapse precedes the formation of substantial secondary structure. The first two detected transient intermediate species have FRET properties essentially identical with those of the previously characterized equilibrium A state intermediates, suggesting similar structures between the equilibrium and transient intermediates.The effects of anions on the folding of acid-unfolded staphylococcal nuclease, and urea on the unfolding of the resulting A states, indicates that in folding the protein becomes compact prior to formation of major secondary structure, whereas in unfolding the protein expands prior to major loss of secondary structure. Comparison of the kinetics of refolding of staphylococcal nuclease, monitored by FRET, and for a proline-free variant, indicate that folding occurs via two partially folded intermediates leading to a native-like species with one (or more) proline residues in a non-native conformation. For the A states an excellent correlation between compactness measured by FRET, and compactness determined from small-angle X-ray scattering, was observed. Further, a linear relationship between compactness and free energy of unfolding was noted. Formation of soluble aggregates of the A states led to dramatic enhancement of the FRET, consistent with intermolecular fluorescence energy transfer.     

Thermodynamics and mechanism of alpha helix initiation in alanine and valine peptides

We used molecular dynamics simulations to study the folding/unfolding of one of turn of an alpha helix in Ac-(Ala)3-NHMe and Ac-(Val)3-NHMe. Using specialized sampling techniques, we computed free energy surfaces as functions of a conformational coordinate that corresponds to alpha helices at small values and to extended conformations at large values. Analysis of the peptide conformations populated during the simulations showed that alpha helices, reverse turns, and extended conformations correspond to minima on the free energy surfaces of both peptides. The free energy difference between alpha helix and extended conformations, determined from the equilibrium constants for helix unfolding, is approximately -1 kcal/mol for Ac-(Ala)3-NHMe and -5 kcal/mol for Ac-(Val)3-NHMe. The mechanism observed in our simulations, which includes reverse turns as important intermediates along the helix folding/unfolding pathway, is consistent with a mechanism proposed previously. Our results predict that both peptides (but especially the Ala peptide) have a much larger equilibrium constant for helix initiation than is predicted by the helix-coil transition theory with the host-guest parameters. We also predict a much greater difference in the equilibrium constants than the theory predicts. Insofar as helix initiation is concerned, our results suggest that the large difference between the helical propensities of Ala and Val cannot be explained by simple concepts such as side-chain rotamer restriction or unfavorable steric interactions. Rather, the origin of the difference appears to be quite complicated because it involves subtle differences in the solvation of the two peptides. The two peptides have similar turn-extended equilibria but very different helix-turn equilibria, and the difference in helical propensities reflects the fact that the helix-turn equilibrium strongly favors the turns in Ac-(Val)3-NHMe, while it favors the helices in Ac-(Ala)3-NHMe. We also computed thermodynamic decompositions of the free energy surfaces, and these revealed that the helix-turn equilibria are vastly different primarily because the changes in peptide-water interactions that accompany helix-to-turn conformational changes are qualitatively different for the two peptides.     

Using circular dichroism collected as a function of temperature to determine the thermodynamics of protein unfolding and binding interactions

Circular dichroism (CD) is an excellent spectroscopic technique for following the unfolding and folding of proteins as a function of temperature. One of its principal applications is to determine the effects of mutations and ligands on protein and polypeptide stability. If the change in CD as a function of temperature is reversible, analysis of the data may be used to determined the van't Hoff enthalpy and entropy of unfolding, the midpoint of the unfolding transition and the free energy of unfolding. Binding constants of protein-protein and protein-ligand interactions may also be estimated from the unfolding curves. Analysis of CD spectra obtained as a function of temperature is also useful to determine whether a protein has unfolding intermediates. Measurement of the spectra of five folded proteins and their unfolding curves at a single wavelength requires approximately 8 h.     

Effects of hydrated water on protein unfolding

The conformational stability of a protein in aqueous solution is described in terms of the thermodynamic properties such as unfolding Gibbs free energy, which is the difference in the free energy (Gibbs function) between the native and random conformations in solution. The properties are composed of two contributions, one from enthalpy due to intramolecular interactions among constituent atoms and chain entropy of the backbone and side chains, and the other from the hydrated water around a protein molecule. The hydration free energy and enthalpy at a given temperature for a protein of known three-dimensional structure can be calculated from the accessible surface areas of constituent atoms according to a method developed recently. Since the hydration free energy and enthalpy for random conformations are computed from those for an extended conformation, the thermodynamic properties of unfolding are evaluated quantitatively. The evaluated hydration properties for proteins of known transition temperature (Tm) and unfolding enthalpy (delta Hm) show an approximately linear dependence on the number of constituent heavy atoms. Since the unfolding free energy is zero at Tm, the enthalpy originating from interatomic interactions of a polypeptide chain and the chain entropy are evaluated from an experimental value of delta Hm and computed properties due to the hydrated water around the molecule at Tm. The chain enthalpy and entropy thus estimated are largely compensated by the hydration enthalpy and entropy, respectively, making the unfolding free energy and enthalpy relatively small. The computed temperature dependences of the unfolding free energy and enthalpy for RNase A, T4 lysozyme, and myoglobin showed a good agreement with the experimental ones.(ABSTRACT TRUNCATED AT 250 WORDS)     

Spectroscopic studies of structural changes in two beta-sheet-forming peptides show an ensemble of structures that unfold noncooperatively

A characterization of the conformation and stability of model peptide systems that form beta-sheets in aqueous solutions is considerably important in gaining insights into the mechanism of beta-sheet formation in proteins. We have characterized the conformation and equilibrium folding and unfolding of two 20-residue peptides whose NMR spectra suggest a three-stranded beta-sheet topology in aqueous solution: Betanova [Kortemme, T., Ramirez-Alvarado, M., and Serrano, L. (1998) Science 281, 253-256] and (D)P(D)P with d-Pro-Gly segments at the turns [Schenck, H. L., and Gellman, S. H. (1998) J. Am. Chem. Soc. 120, 4869-4870]. Both circular dichroism (CD) and infrared measurements indicate only 20-26% beta-sheet-like structure at 5 degrees C for Betanova and 42-59% beta-sheet for (D)P(D)P. For both peptides, the CD and infrared spectra change nearly linearly with increasing temperatures (or urea concentrations) and lack a sigmoidal signature characteristic of cooperative unfolding. Fluorescence resonance energy transfer (FRET) measurements between donor and acceptor molecules attached to the two ends confirm that Betanova is largely unstructured even at 10 degrees C; the average end-to-end distance estimated from FRET is closer to that of a random coil than a structured beta-sheet. In (D)P(D)P, the FRET results indicate a more compact structure that remains compact even at high temperatures (approximately 80 degrees C) or high urea concentrations (approximately 8 M). These results indicate that both these peptides access an ensemble of conformations at all temperatures or denaturant concentrations, with no significant free energy barrier separating the "folded" and "unfolded" conformations.     

OneG: a computational tool for predicting cryptic intermediates in the unfolding kinetics of proteins under native conditions

Understanding the relationships between conformations of proteins and their stabilities is one key to address the protein folding paradigm. The free energy change (ΔG) of unfolding reactions of proteins is measured by traditional denaturation methods and native hydrogen-deuterium (H/D) exchange methods. However, the free energy of unfolding (ΔG(U)) and the free energy of exchange (ΔG(HX)) of proteins are not in good agreement, though the experimental conditions of both methods are well matching to each other. The anomaly is due to any one or combinations of the following reasons: (i) effects of cis-trans proline isomerisation under equilibrium unfolding reactions of proteins (ii) inappropriateness in accounting the baselines of melting curves (iii) presence of cryptic intermediates, which may elude the melting curve analysis and (iv) existence of higher energy metastable states in the H/D exchange reactions of proteins. Herein, we have developed a novel computational tool, OneG, which accounts the discrepancy between ΔG(U) and ΔG(HX) of proteins by systematically accounting all the four factors mentioned above. The program is fully automated and requires four inputs: three-dimensional structures of proteins, ΔG(U), ΔG(U)(*) and residue-specific ΔG(HX) determined under EX2-exchange conditions in the absence of denaturants. The robustness of the program has been validated using experimental data available for proteins such as cytochrome c and apocytochrome b(562) and the data analyses revealed that cryptic intermediates of the proteins detected by the experimental methods and the cryptic intermediates predicted by the OneG for those proteins were in good agreement. Furthermore, using OneG, we have shown possible existence of cryptic intermediates and metastable states in the unfolding pathways of cardiotoxin III and cobrotoxin, respectively, which are homologous proteins. The unique application of the program to map the unfolding pathways of proteins under native conditions have been brought into fore and the program is publicly available at http://sblab.sastra.edu/oneg.html.     

Temperature dependence of the free energy landscape of the src-SH3 protein domain

The temperature dependence of the free energy landscape of the src-SH3 protein domain is investigated through fully atomic simulations in explicit solvent. Simulations are performed above and below the folding transition temperature, enabling an analysis of both protein folding and unfolding. The transition state for folding and unfolding, identified from the free energy surfaces, is found to be very similar, with structure in the central hydrophobic sheet and little structure throughout the rest of the protein. This is a result of a polarized folding (unfolding) mechanism involving early formation (late loss) of the central hydrophobic sheet at the transition state. Unfolding simulations map qualitatively well onto low-temperature free energy surfaces but appear, however, to miss important features observed in folding simulations. In particular, details of the folding mechanism involving the opening and closing of the hydrophobic core are not captured by unfolding simulations performed under strongly denaturing conditions. In addition, free energy surfaces at high temperatures do not display a desolvation barrier found at lower temperatures, involving the expulsion of water molecules from the hydrophobic core.     

Equilibrium denaturation of human growth hormone and its cysteine-modified forms

The equilibrium denaturation of human growth hormone (hGH) derived from heterologous gene expression in Escherichia coli was studied. Denaturation was measured by ultraviolet absorbance, intrinsic fluorescence, far ultraviolet circular dichroism, and size exclusion chromatography. The denaturation transitions obtained from each method of detection were coincident, indicating a two-state denaturation mechanism. The denaturation transitions were independent of the concentration of protein. The Gibbs free energy of unfolding is 14.5 +/- 1 kcal/mol. Human growth hormone contains two disulfide bridges between residues 53-165 (large loop) and 182-189 (small loop). The small loop was selectively reduced and cysteines alkylated with iodoacetic acid or iodoacetamide. The tetra-S-carbamidomethylated and tetra-S-carboxymethylated derivatives were also prepared. All S-alkylated hGH forms were indistinguishable from the native conformations in the absence of denaturant by far ultraviolet circular dichroism. The circular dichroism-detected equilibrium denaturation of each derivative was determined and the Gibbs free energy of unfolding of the tetra-S-modified forms was 5.3 +/- 0.5 kcal/mol and of the di-S-alkylated derivatives was 11.2 +/- 0.8 kcal/mol. These results for hGH are different than previously obtained results for bovine, ovine, and rat growth hormones. Stable equilibrium intermediates have been identified for these non-human species of growth hormone. The stable intermediates observed in the denaturation of reduced, alkylated hGH or nonhunam growth hormones are similar and characterized as compact, helical, lacking native-like tertiary structure, and having a tendency to aggregate. The apparent absence of intermediates in the folding of oxidized hGH is due to the relative instability of intermediates compared with their native structures. The hGH conformation is at least 5 kcal/mol more stable than the growth hormones from other species. Reduction and alkylation of the disulfide bridges of hGH diminish the stability differences between the native and intermediate states, such that the denaturation behavior is similar to the nonhuman growth hormones with well-populated intermediates. Most proteins do not demonstrate equilibrium folding intermediates presumably because intermediates are only marginally stable in conditions that disrupt the native state. The folding results with hGH and alkylated hGH substantiate this.     

Free energy landscape and folding mechanism of a beta-hairpin in explicit water: a replica exchange molecular dynamics study

The free energy landscape and the folding mechanism of the C-terminal beta-hairpin of protein G is studied by extensive replica exchange molecular dynamics simulations (40 replicas and 340 ns total simulation time), using the GROMOS96 force field and the SPC explicit water solvent. The study reveals that the system preferentially adopts a beta-hairpin structure at biologically important temperatures, and that the helix content is low at all temperatures studied. Representing the free energy landscape as a function of several types of reaction coordinates, four local minima corresponding to the folded, partially folded, molten globule, and unfolded states are identified. The findings suggest that the folding of the beta-hairpin occurs as the sequence: collapse of hydrophobic core --> formation of H-bond --> formation of the turn. Identifying the folded and molten globule states as the main conformations, the free energy landscape of the beta-hairpin is consistent with a two-state behavior with a broad transition state. The temperature dependence of the folding-unfolding transition is investigated in some detail. The enthalpy and entropy jumps at the folding transition temperature are found to be about three times lower than the experimental estimates, indicating that the folding-unfolding transition in silico is less cooperative than its in vitro counterpart.     

Molecular dynamics simulations of synthetic peptide folding

Because the time scale of protein folding is much greater than that of the widely used simulations of native structures, a detailed report of molecular dynamics simulations of folding has not been available. In this study, we Included the average solvent effect in the potential functions to simplify the calculation of the solvent effect and carried out long molecular dynamics simulations of the alanine-based synthetic peptides at 274 K. From either an extended or a randomly generated conformation, the simulations approached a helix-coil equilibrium in about 3 ns. The multiple minima problem did not prevent helix folding. The calculated helical ratio of Ac-AAQAAAAQAAAAQAAY-NH₂ was 47%, in good agreement with the circular dichroism measurement (about 50%). A helical segment with frayed ends was the most stable conformation, but the hydrophobic interaction favored the compact, distorted helix-turn-helix conformations. The transition between the two types of conformations occurred in a much larger time scale than helix propagation. The transient hydrogen bonds between the glutamine side chain and the backbone carbonyl group could reduce the free energy barrier of helix folding and unfolding. The substitution of a single alanine residue in the middle of the peptide with valine or glycine decreased the average helical ratio significantly, in agreement with experimental observations. © 1996 Wiley-Liss, Inc.     

Deconvoluting Protein (Un)folding Structural Ensembles Using X-Ray Scattering,Nuclear Magnetic Resonance Spectroscopy and Molecular Dynamics Simulation

The folding and unfolding of protein domains is an apparently cooperative process, but transient intermediates have been detected in some cases. Such (un)folding intermediates are challenging to investigate structurally as they are typically not long-lived and their role in the (un)folding reaction has often been questioned. One of the most well studied (un)folding pathways is that of Drosophila melanogaster Engrailed homeodomain (EnHD): this 61-residue protein forms a three helix bundle in the native state and folds via a helical intermediate. Here we used molecular dynamics simulations to derive sample conformations of EnHD in the native, intermediate, and unfolded states and selected the relevant structural clusters by comparing to small/wide angle X-ray scattering data at four different temperatures. The results are corroborated using residual dipolar couplings determined by NMR spectroscopy. Our results agree well with the previously proposed (un)folding pathway. However, they also suggest that the fully unfolded state is present at a low fraction throughout the investigated temperature interval, and that the (un)folding intermediate is highly populated at the thermal midpoint in line with the view that this intermediate can be regarded to be the denatured state under physiological conditions. Further, the combination of ensemble structural techniques with MD allows for determination of structures and populations of multiple interconverting structures in solution.     

(Un)Folding Mechanisms of the FBP28 WW Domain in Explicit Solvent Revealed by Multiple Rare Event Simulation Methods

We report a numerical study of the (un)folding routes of the truncated FBP28 WW domain at ambient conditions using a combination of four advanced rare event molecular simulation techniques. We explore the free energy landscape of the native state, the unfolded state, and possible intermediates, with replica exchange molecular dynamics. Subsequent application of bias-exchange metadynamics yields three tentative unfolding pathways at room temperature. Using these paths to initiate a transition path sampling simulation reveals the existence of two major folding routes, differing in the formation order of the two main hairpins, and in hydrophobic side-chain interactions. Having established that the hairpin strand separation distances can act as reasonable reaction coordinates, we employ metadynamics to compute the unfolding barriers and find that the barrier with the lowest free energy corresponds with the most likely pathway found by transition path sampling. The unfolding barrier at 300 K is ∼17 k_BT ≈ 42 kJ/mol, in agreement with the experimental unfolding rate constant. This work shows that combining several powerful simulation techniques provides a more complete understanding of the kinetic mechanism of protein folding.     

Molecular basis of co-operativity in protein folding.

The folding/unfolding transition of proteins is a highly co-operative process characterized by the presence of very few or no thermodynamically stable partially folded intermediate states. The purpose of this paper is to present a thermodynamic formalism aimed at describing quantitatively the co-operative folding behavior of proteins. In order to account for this behavior, a hierarchical algorithm aimed at evaluating the folding/unfolding partition function has been developed. This formalism defines the partition function in terms of multiple levels of interacting co-operative folding units. A co-operative folding unit is defined as a protein structural element that exhibits two-state folding/unfolding behavior. At the most fundamental level are those structural elements that behave co-operatively as a result of purely local interactions. Higher-order co-operative folding units are formed through interactions between different structural elements. The hierarchical formalism utilizes the crystallographic structure of the protein as a template to generate partially folded conformations defined in terms of co-operative folding units. The Gibbs free energy of those states and their corresponding statistical weights are then computed using experimental energetic parameters determined calorimetrically. This formalism has been applied to the case of myoglobin. It is shown that the hierarchical partition function correctly predicts the presence, energetics and co-operativity of the heat and cold denaturation transitions. The major contribution to the co-operative folding behavior arises from the solvent exposure of non-polar residues located in regions complementary to those that have undergone unfolding. This entropically uncompensated and energetically unfavorable solvent exposure characterizes all partially folded states but not the unfolded state, thus minimizing the population of partially folded intermediates throughout the folding/unfolding transition.