Entropic barriers, transition states, funnels, and exponential protein folding kinetics: a simple model

This paper presents an analytically tractable model that captures the most elementary aspect of the protein folding problem, namely that both the energy and the entropy decrease as a protein folds. In this model, the system diffuses within a sphere in the presence of an attractive spherically symmetric potential. The native state is represented by a small sphere in the center, and the remaining space is identified with unfolded states. The folding temperature, the time-dependence of the populations, and the relaxation rate are calculated, and the folding dynamics is analyzed for both golf-course and funnel-like energy landscapes. This simple model allows us to illustrate a surprising number of concepts including entropic barriers, transition states, funnels, and the origin of single exponential relaxation kinetics.     

Characterization of the structure and dynamics of a near-native equilibrium intermediate in the unfolding pathway of an all beta-barrel protein

The structure and dynamics of equilibrium intermediate in the unfolding pathway of the human acidic fibroblast growth factor (hFGF-1) are investigated using a variety of biophysical techniques including multidimensional NMR spectroscopy. Guanidinium hydrochloride (GdnHCl)-induced unfolding of hFGF-1 proceeds with the accumulation of a stable intermediate state. The transition from the intermediate state to the unfolded state(s) is cooperative without the accumulation of additional intermediate(s). The intermediate state induced maximally in 0.96 m GdnHCl is found to be obligatory in the folding/unfolding pathway of hFGF-1. Most of the native tertiary structure interactions are preserved in the intermediate state. (1)H-(15)N chemical shift perturbation data suggest that the residues in the C-terminal segment including those located in the beta-strands IX, X, and XI undergo the most discernible structural change(s) in the intermediate state in 0.96 m GdnHCl. hFGF-1 in the intermediate state (0.96 m GdnHCl) does not bind to its ligand, sucrose octasulfate. Limited proteolytic digestion experiments and hydrogen-deuterium exchange monitored by (15)N heteronuclear single quantum coherence (HSQC) spectra show that the conformational flexibility of the protein in the intermediate state is significantly higher than in the native conformation. (15)N spin relaxation experiments show that many residues located in beta-strands IX, X, and XI exhibit conformational motions in the micro- to millisecond time scale. Analysis of (15)N relaxation data in conjunction with the amide proton exchange kinetics suggests that residues in the beta-strands II, VIII, and XII possibly constitute the stability core of the protein in the near-native intermediate state.     

Conformational effects in the reversed-phase liquid chromatography of ribonuclease A

This paper examines the reversed-phase liquid chromatographic behavior of ribonuclease A (RNase) using an n-butyl chemically bonded phase and a gradient of 10 mM H3PO4 and l-propanol. At a column temperature of 25 degrees C, a broad band followed by an overlapped late-eluting sharp peak is observed. As the temperature is raised, the sharp peak grows at the expense of the broad band until at 37 degrees C, only a single narrow-eluting band is found. Using an absorbance ratio of A288/A254, it is demonstrated that the broad band represents a folded or native state of RNase and the late-eluting band a denatured state. Based on postcolumn absorbance ratio changes in the denatured state as a function of time and the known behavior of the protein, reversible refolding or renaturation is proposed to take place in solution. RNase is denatured upon adsorbing to the bonded phase, and upon migration down the column, reversible refolding takes place in the mobile phase. The relaxation time for native state formation is assumed to be comparable to the time spent by RNase in the mobile phase. As temperature is raised, both the native and denatured states exist at equilibrium in solution, thus slowing the refolding process, until at 37 degrees C only the denatured peak appears. Changes in peak shape with flow rate provide further evidence for this model. The use of HCl or H2SO4 instead of H3PO4 yields similar results except that the temperature at which only the denatured peak is observed follows the order of salt stabilization of the native state.     

Heterogeneity even at the speed limit of folding: large-scale molecular dynamics study of a fast-folding variant of the villin headpiece

We have performed molecular dynamics simulations on a set of nine unfolded conformations of the fastest-folding protein yet discovered, a variant of the villin headpiece subdomain (HP-35 NleNle). The simulations were generated using a new distributed computing method, yielding hundreds of trajectories each on a time scale comparable to the experimental folding time, despite the large (10,000 atom) size of the simulation system. This strategy eliminates the need to assume a two-state kinetic model or to build a Markov state model. The relaxation to the folded state at 300 K from the unfolded configurations (generated by simulation at 373 K) was monitored by a method intended to reflect the experimental observable (quenching of tryptophan by histidine). We also monitored the relaxation to the native state by directly comparing structural snapshots with the native state. The rate of relaxation to the native state and the number of resolvable kinetic time scales both depend upon starting structure. Moreover, starting structures with folding rates most similar to experiment show some native-like structure in the N-terminal helix (helix 1) and the phenylalanine residues constituting the hydrophobic core, suggesting that these elements may exist in the experimentally relevant unfolded state. Our large-scale simulation data reveal kinetic complexity not resolved in the experimental data. Based on these findings, we propose additional experiments to further probe the kinetics of villin folding.     

Nanosecond dynamics of the single tryptophan reveals multi-state equilibrium unfolding of protein GB1

Unfolding of the immunoglobulin binding domain B1 of streptococcal protein G (GB1) was induced by guanidine hydrochloride (GdnHCl) and studied by circular dichroism, steady-state, and time-resolved fluorescence spectroscopy. The fluorescence methods employed the single tryptophan residue of GB1 as an intrinsic reporter. While the transitions monitored by circular dichroism and steady-state fluorescence coincided with each other, the transitions followed by dynamic fluorescence were markedly different. Specifically, fluorescence anisotropy data showed that a relaxation spectrum of tryptophan contained a slow motion with relaxation times of 9 ns in the native state and 4 ns in the unfolded state in 6 M GdnHCl. At intermediate GdnHCl concentrations of 3.8-4.2 M, however, the slow relaxation time increased to 18 ns. The fast nanosecond motion had an average time of 0.8 ns and showed no dependence on the formation of native structure. Overall, dynamic fluorescence revealed two preliminary stages in GB1 folding, which are equated with the formation of local structure in the beta(3)-strand hairpin and the initial collapse. Both stages exist without alpha-helix formation, i. e., before the appearance of any ordered secondary structure detectable by circular dichroism. Another stage in GB1 folding might exist at very low ( approximately 1 M) GdnHCl concentrations.     

Femtosecond and picosecond fluorescence of native bacteriorhodopsin and a nonisomerizing analog

The spectrally and temporally resolved fluorescence properties of native bacteriorhodopsin (bR) and bR reconstituted with a nonisomerizing analog of the retinal Schiff base (bR5.12) are examined. The first attempt to experimentally monitor the excited state relaxation processes in both type of pigments using ultrafast fluorescence spectroscopy is reported. The fluorescence is emitted from retinal molecules in an all-trans configuration. Substantial energy relaxation involves very fast intramolecular and intermolecular vibrational modes and these are shown to occur on a time scale faster than isomerization. The possible contribution of dielectric interaction between the retinal Schiff base and the protein environment for the excited state energy relaxation is discussed.     

Transient structure formation in unfolded acyl-coenzyme A-binding protein observed by site-directed spin labelling

Paramagnetic relaxation has been used to monitor the formation of structure in the folding peptide chain of guanidinium chloride-denatured acyl-coenzyme A-binding protein. The spin label (1-oxyl-2,2,5,5-tetramethyl-3-pyrroline-3-methyl)methanesulfonate (MTSL) was covalently bound to a single cysteine residue introduced into five different positions in the amino acid sequence. It was shown that the formation of structure in the folding peptide chain at conditions where 95% of the sample is unfolded brings the relaxation probe close to a wide range of residues in the peptide chain, which are not affected in the native folded structure. It is suggested that the experiment is recording the formation of many discrete and transient structures in the polypeptide chain in the preface of protein folding. Analysis of secondary chemical shifts shows a high propensity for alpha-helix formation in the C-terminal part of the polypeptide chain, which forms an alpha-helix in the native structure and a high propensity for turn formation in two regions of the polypeptide that form turns in the native structure. The results contribute to the idea that native-like structural elements form transiently in the unfolded state, and that these may be of importance to the initiation of protein folding.     

Exploring the Minimally Frustrated Energy Landscape of Unfolded ACBP

The unfolded state of globular proteins is not well described by a simple statistical coil due to residual structural features, such as secondary structure or transiently formed long-range contacts. The principle of minimal frustration predicts that the unfolded ensemble is biased toward productive regions in the conformational space determined by the native structure. Transient long-range contacts, both native-like and non-native-like, have previously been shown to be present in the unfolded state of the four-helix-bundle protein acyl co-enzyme binding protein (ACBP) as seen from both perturbations in nuclear magnetic resonance (NMR) chemical shifts and structural ensembles generated from NMR paramagnetic relaxation data. To study the nature of the contacts in detail, we used paramagnetic NMR relaxation enhancements, in combination with single-point mutations, to obtain distance constraints for the acid-unfolded ensemble of ACBP. We show that, even in the acid-unfolded state, long-range contacts are specific in nature and single-point mutations affect the free-energy landscape of the unfolded protein. Using this approach, we were able to map out concerted, interconnected, and productive long-range contacts. The correlation between the native-state stability and compactness of the denatured state provides further evidence for native-like contact formation in the denatured state. Overall, these results imply that, even in the earliest stages of folding, ACBP dynamics are governed by native-like contacts on a minimally frustrated energy landscape.     

Water and urea interactions with the native and unfolded forms of a beta-barrel protein

A fundamental understanding of protein stability and the mechanism of denaturant action must ultimately rest on detailed knowledge about the structure, solvation, and energetics of the denatured state. Here, we use (17)O and (2)H magnetic relaxation dispersion (MRD) to study urea-induced denaturation of intestinal fatty acid-binding protein (I-FABP). MRD is among the few methods that can provide molecular-level information about protein solvation in native as well as denatured states, and it is used here to simultaneously monitor the interactions of urea and water with the unfolding protein. Whereas CD shows an apparently two-state transition, MRD reveals a more complex process involving at least two intermediates. At least one water molecule binds persistently (with residence time >10 nsec) to the protein even in 7.5 M urea, where the large internal binding cavity is disrupted and CD indicates a fully denatured protein. This may be the water molecule buried near the small hydrophobic folding core at the D-E turn in the native protein. The MRD data also provide insights about transient (residence time <1 nsec) interactions of urea and water with the native and denatured protein. In the denatured state, both water and urea rotation is much more retarded than for a fully solvated polypeptide. The MRD results support a picture of the denatured state where solvent penetrates relatively compact clusters of polypeptide segments.     

Perturbations of the denatured state ensemble: modeling their effects on protein stability and folding kinetics.

By considering the denatured state of a protein as an ensemble of conformations with varying numbers of sequence-specific interactions, the effects on stability, folding kinetics, and aggregation of perturbing these interactions can be predicted from changes in the molecular partition function. From general considerations, the following conclusions are drawn: (1) A perturbation that enhances a native interaction in denatured state conformations always increases the stability of the native state. (2) A perturbation that promotes a non-native interaction in the denatured state always decreases the stability of the native state. (3) A change in the denatured state ensemble can alter the kinetics of aggregation and folding. (4) The loss (or increase) in stability accompanying two mutations, each of which lowers (or raises) the free energy of the denatured state, will be less than the sum of the effects of the single mutations, except in cases where both mutations affect the same set of partially folded conformations. By modeling the denatured state as the ensemble of all non-native conformations of hydrophobic-polar (HP) chains configured on a square lattice, it can be shown that the stabilization obtained from enhancement of native interactions derives in large measure from the avoidance of non-native interactions in the D state. In addition, the kinetic effects of fixing single native contacts in the denatured state or imposing linear gradients in the HH contact probabilities are found, for some sequences, to significantly enhance the efficiency of folding by a simple hydrophobic zippering algorithm. Again, the dominant mechanism appears to be avoidance of non-native interactions. These results suggest stabilization of native interactions and imposition of gradients in the stability of local structure are two plausible mechanisms involving the denatured state that could play a role in the evolution of protein folding and stability.     

Reversible unfolding of cytochrome c upon interaction with cardiolipin bilayers. 2. Evidence from phosphorus-31 NMR measurements.

31P NMR measurements were conducted to determine the structural and chemical environment of beef heart cardiolipin when bound to cytochrome c. 31P NMR line shapes infer that the majority of lipid remains in the bilayer state and that the average conformation of the lipid phosphate is not greatly affected by binding to the protein. An analysis of the spin-lattice (T1) relaxation times of hydrated cardiolipin as a function of temperature describes a T1 minimum at around 25 degrees C which leads to a correlation time for the phosphates in the lipid headgroup of 0.71 ns. The relaxation behavior of the protein-lipid complex was markedly different, showing a pronounced enhancement in the phosphorus spin-lattice relaxation rate. This effect of the protein increased progressively with increasing temperature, giving no indication of a minimum in T1 up to 75 degrees C. The enhancement in lipid phosphorus T1 relaxation was observed with protein in both oxidation states, being somewhat less marked for the reduced form. The characteristics of the T1 effects and the influence of the protein on other relaxation processes determined for the lipid phosphorus (spin-spin relaxation and longitudinal relaxation in the rotating frame) point to a strong paramagnetic interaction from the protein. A comparison with the relaxation behavior of samples spinning at the "magic angle" was also consistent with this mechanism. The results suggest that cytochrome c reversibly denatures on complexation with cardiolipin bilayers, such that the electronic ground state prevailing in the native structure of both oxidized and reduced protein can convert to high-spin states with greater magnetic susceptibility.     

Dynamics of folding and unfolding transition in a globular protein studied by time correlation functions from computer simulation

A method of calculating time correlation functions from records of computer simulated equilibrium conformational fluctuations in a globular protein is discussed. Use of the calculated time correlation function for discussions of dynamics of folding and unfolding transition in the two-dimensional lattice model of proteins. The time correlation functions can be approximated in general by a sum of two simple exponential terms. The relaxation time of the slower mode does not depend on the nature of the physical quantity with respect to which the time correlation function is calculated. This time characterizes the overall folding and unfolding transition. The relaxation time of the faster mode depends on the nature of the physical quantity and characterizes conformational fluctuations within each of the native and denatured states. The mechanism of a previously observed phenomenon of the acceleration of the folding and unfolding transition by short-range interactions is discussed.     

Conformational transitions of thioredoxin in guanidine hydrochloride

Spectral and hydrodynamic measurements of thioredoxin from Escherichia coli indicate that the compact globular structure of the native protein is significantly unfolded in the presence of guanidine hydrochloride concentrations in excess of 3.3 M at neutral pH and 25 degrees C. This conformational transition having a midpoint at 2.5 M denaturant is quantitatively reversible and highly cooperative. Stopped-flow measurements of unfolding in 4 M denaturant, observed with tryptophan fluorescence as the spectral probe, reveal a single kinetic phase having a relaxation time of 7.1 +/- 0.2 s. Refolding measurements in 2 M denaturant reveal three kinetic phases having relaxation times of 0.54 +/- 0.23, 14 +/- 6, and 500 +/- 130 s, accounting for 12 +/- 2%, 10 +/- 1%, and 78 +/- 3% of the observed change in tryptophan fluorescence. The dominant slowest phase is generated in the denatured state with a relaxation time of 42 s observed in 4 M denaturant. Both the slowest phase observed in refolding and the generation of the slowest phase in the denatured state have an activation enthalpy of 22 +/- 1 kcal/mol. These features of the slowest phase are compatible with an obligatory peptide isomerization of proline-76 to its cis isomer prior to refolding.     

Conformational propagation with prion-like characteristics in a simple model of protein folding

Protein refolding/misfolding to an alternative form plays an aetiologic role in many diseases in humans, including Alzheimer's disease, the systemic amyloidoses, and the prion diseases. Here we have discovered that such refolding can occur readily for a simple lattice model of proteins in a propagatable manner without designing for any particular alternative native state. The model uses a simple contact energy function for interactions between residues and does not consider the peculiarities of polypeptide geometry. In this model, under conditions where the normal (N) native state is marginally stable or unstable, two chains refold from the N native state to an alternative multimeric energetic minimum comprising a single refolded conformation that can then propagate itself to other protein chains. The only requirement for efficient propagation is that a two-faced mode of packing must be in the ground state as a dimer (a higher-energy state for this packing leads to less efficient propagation). For random sequences, these ground-state dimeric configurations tend to have more beta-sheet-like extended structure than almost any other sort of dimeric ground-state assembly. This implies that propagating states (such as for prions) are beta-sheet rich because the only likely propagating forms are beta-sheet rich. We examine the details of our simulations to see to what extent the observed properties of prion propagation can be predicted by a simple protein folding model. The formation of the alternative state in the present model shows several distinct features of amyloidogenesis and of prion propagation. For example, an analog of the phenomenon of conformationally distinct strains in prions is observed. We find a parallel between 'glassy' behavior in liquids and the formation of a propagatable state in proteins. This is the first report of simulation of conformational propagation using any heteropolymer model. The results imply that some (but not most) small protein sequences must maintain a sequence signal that resists refolding to propagatable alternative native states and that the ability to form such states is not limited to polypeptides (or reliant on regular hydrogen bonding per se) but can occur for other protein-like heteropolymers.     

Hydrophobic clustering in acid-denatured IL-2 and fluorescence of a Trp NH-pi H-bond

The single tryptophan at position 121 of human interleukin-2 (IL-2) can form an NH-pi hydrogen bond with Phe 117 involving the indole nitrogen and the benzene aromatic ring. At pH 5.5, this type of aromatic interaction results in a fluorescence quantum yield three-fold lower than that of a fully solvent exposed tryptophan. At pH 2.1, IL-2 forms a compact denatured state with twice the emission intensity of the native protein. Global analysis of time-resolved fluorescence emission at multiple emission wavelengths shows that native and acid-denatured IL-2 can be described by four decay components. The fractional amplitudes of the shortest sub-nanosecond lifetimes are higher in the native state, suggesting rapid quenching due to the NH-pi hydrogen bond. In the denatured state, longer lifetimes have greater fractional amplitudes, indicating a smaller population of hydrogen-bonded species. Electrostatic-dipolar relaxation of the tryptophan microenvironment upon excitation is greater in the native-state of IL-2 than the acid-denatured state. This suggests that acid-denaturation sequesters Trp 121 from polar residues, while maintaining an interaction with Phe 117. This is consistent with the model of secondary structure preservation and hydrophobic clustering in molten-globule intermediates.     

Effect of single amino acid replacements on the folding and stability of dihydrofolate reductase from Escherichia coli

The role of the secondary structure in the folding mechanism of dihydrofolate reductase from Escherichia coli was probed by studying the effects of amino acid replacements in two alpha helices and two strands of the central beta sheet on the folding and stability. The effects on stability could be qualitatively understood in terms of the X-ray structure for the wild-type protein by invoking electrostatic, hydrophobic, or hydrogen-bonding interactions. Kinetic studies focused on the two slow reactions that are thought to reflect the unfolding/refolding of two stable native conformers to/from their respective folding intermediates [Touchette, N. A., Perry, K. M., & Matthews, C. R. (1986) Biochemistry 25, 5445-5452]. Replacements at three different positions in helix alpha B selectively alter the relaxation time for unfolding while a single replacement in helix alpha C selectively alters the relaxation time for refolding. This behavior is characteristic of mutations that change the stability of the protein but do not affect the rate-limiting step. In striking contrast, replacements in strands beta F and beta G can affect both unfolding and refolding relaxation times. This behavior shows that these mutations alter the rate-limiting step in these native-to-intermediate folding reactions. It is proposed that the intermediates have an incorrectly formed beta sheet whose maturation to the structure found in the native conformation is one of the slow steps in folding.     

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.     

Exploring the Levinthal limit in protein folding

According to the thermodynamic hypothesis, the native state of proteins is uniquely defined by their amino acid sequence. On the other hand, according to Levinthal, the native state is just a local minimum of the free energy and a given amino acid sequence, in the same thermodynamic conditions, can assume many, very different structures that are as thermodynamically stable as the native state. This is the Levinthal limit explored in this work. Using computer simulations, we compare the interactions that stabilize the native state of four different proteins with those that stabilize three non-native states of each protein and find that the nature of the interactions is very similar for all such 16 conformers. Furthermore, an enhancement of the degree of fluctuation of the non-native conformers can be explained by an insufficient relaxation to their local free energy minimum. These results favor Levinthal’s hypothesis that protein folding is a kinetic non-equilibrium process.     

Protein unfolding transitions in an intrinsically unstable annexin domain: molecular dynamics simulation and comparison with nuclear magnetic resonance data

Unfolding transitions of an intrinsically unstable annexin domain and the unfolded state structure have been examined using multiple approximately 10-ns molecular dynamics simulations. Three main basins are observed in the configurational space: native-like state, compact partially unfolded or intermediate compact state, and the unfolded state. In the native-like state fluctuations are observed that are nonproductive for unfolding. During these fluctuations, after an initial loss of approximately 20% of the core residue native contacts, the core of the protein transiently completely refolds to the native state. The transition from the native-like basin to the partially unfolded compact state involves approximately 75% loss of native contacts but little change in the radius of gyration or core hydration properties. The intermediate state adopts for part of the time in one of the trajectories a novel highly compact salt-bridge stabilized structure that can be identified as a conformational trap. The intermediate-to-unfolded state transition is characterized by a large increase in the radius of gyration. After an initial relaxation the unfolded state recovers a native-like topology of the domain. The simulated unfolded state ensemble reproduces in detail experimental nuclear magnetic resonance data and leads to a convincing complete picture of the unfolded domain.