Dynamics of intramolecular contact formation in polypeptides: distance dependence of quenching rates in a room-temperature glass.

Quenching of the triplet state of tryptophan by cysteine is an important new tool for measuring the rate of forming a specific contact between amino acids in a polypeptide chain. To determine the length scale associated with this contact, tryptophan was embedded in a room-temperature glass containing a high concentration of cysteine. The decay of the triplet population is extended in time, consistent with a rate coefficient that decreases exponentially with distance. Solving the diffusion equation with this distant-dependent rate reproduces the observed bimolecular rates in water and shows that quenching at low viscosities takes place less than or similar to A from van der Waals contact between the tryptophan and cysteine.

Two-state expansion and collapse of a polypeptide.

The initial phase of folding for many proteins is presumed to be the collapse of the polypeptide chain from expanded to compact, but still denatured, conformations. Theory and simulations suggest that this collapse may be a two-state transition, characterized by barrier-crossing kinetics, while the collapse of homopolymers is continuous and multi-phasic. We have used a laser temperature-jump with fluorescence spectroscopy to measure the complete time-course of the collapse of denatured cytochrome c with nanosecond time resolution. We find the process to be exponential in time and thermally activated, with an apparent activation energy approximately 9 k(B)T (after correction for solvent viscosity). These results indicate that polypeptide collapse is kinetically a two-state transition. Because of the observed free energy barrier, the time scale of polypeptide collapse is dramatically slower than is predicted by Langevin models for homopolymer collapse.

Fast kinetics and mechanisms in protein folding.

This review describes how kinetic experiments using techniques with dramatically improved time resolution have contributed to understanding mechanisms in protein folding. Optical triggering with nanosecond laser pulses has made it possible to study the fastest-folding proteins as well as fundamental processes in folding for the first time. These include formation of alpha-helices, beta-sheets, and contacts between residues distant in sequence, as well as overall collapse of the polypeptide chain. Improvements in the time resolution of mixing experiments and the use of dynamic nuclear magnetic resonance methods have also allowed kinetic studies of proteins that fold too fast (greater than approximately 10(3) s-1) to be observed by conventional methods. Simple statistical mechanical models have been extremely useful in interpreting the experimental results. One of the surprises is that models originally developed for explaining the fast kinetics of secondary structure formation in isolated peptides are also successful in calculating folding rates of single domain proteins from their native three-dimensional structure.

Measuring the rate of intramolecular contact formation in polypeptides.

Formation of a specific contact between two residues of a polypeptide chain is an important elementary process in protein folding. Here we describe a method for studying contact formation between tryptophan and cysteine based on measurements of the lifetime of the tryptophan triplet state. With tryptophan at one end of a flexible peptide and cysteine at the other, the triplet decay rate is identical to the rate of quenching by cysteine. We show that this rate is also close to the diffusion-limited rate of contact formation. The length dependence of this end-to-end contact rate was studied in a series of Cys-(Ala-Gly-Gln)(k)-Trp peptides, with k varying from 1 to 6. The rate decreases from approximately 1/(40 ns) for k = 1 to approximately 1/(140 ns) for k = 6, approaching the length dependence expected for a random coil (n(-3/2)) for the longest peptides.

Two-state expansion and collapse of a polypeptide.

The initial phase of folding for many proteins is presumed to be the collapse of the polypeptide chain from expanded to compact, but still denatured, conformations. Theory and simulations suggest that this collapse may be a two-state transition, characterized by barrier-crossing kinetics, while the collapse of homopolymers and random heteropolymers is continuous and multi-phasic. A new rapid-mixing flow technique has been used to resolve the late stages of polypeptide collapse, at time scales >/=45 microseconds. We have used a laser temperature-jump with fluorescence spectroscopy to resolve the complete time-course of the collapse of denatured cytochrome c with nanosecond time resolution. We find the process to be exponential in time and thermally activated, with an apparent activation energy approximately 9 k(B)T (after correction for solvent viscosity). These results indicate that polypeptide collapse is kinetically a two-state transition. Because of the observed free energy barrier, the time scale of polypeptide collapse is dramatically slower than is predicted by Langevin models for homopolymer collapse.

Compactness of the denatured state of a fast-folding protein measured by submillisecond small-angle x-ray scattering.

Time-resolved small-angle x-ray scattering was used to measure the radius of gyration of cytochrome c after initiation of folding by a pH jump. Submillisecond time resolution was obtained with a microfabricated diffusional mixer and synchrotron radiation. The results show that the protein first collapses to compact denatured structures before folding very fast to the native state.

A simple model for calculating the kinetics of protein folding from three-dimensional structures.

An elementary statistical mechanical model was used to calculate the folding rates for 22 proteins from their known three-dimensional structures. In this model, residues come into contact only after all of the intervening chain is in the native conformation. An additional simplifying assumption is that native structure grows from localized regions that then fuse to form the complete native molecule. The free energy function for this model contains just two contributions-conformational entropy of the backbone and the energy of the inter-residue contacts. The matrix of inter-residue interactions is obtained from the atomic coordinates of the three-dimensional structure. For the 18 proteins that exhibit two-state equilibrium and kinetic behavior, profiles of the free energy versus the number of native peptide bonds show two deep minima, corresponding to the native and denatured states. For four proteins known to exhibit intermediates in folding, the free energy profiles show additional deep minima. The calculated rates of folding the two-state proteins, obtained by solving a diffusion equation for motion on the free energy profiles, reproduce the experimentally determined values surprisingly well. The success of these calculations suggests that folding speed is largely determined by the distribution and strength of contacts in the native structure. We also calculated the effect of mutations on the folding kinetics of chymotrypsin inhibitor 2, the most intensively studied two-state protein, with some success.

Is cooperative oxygen binding by hemoglobin really understood?

The enormous success of structural biology challenges the physical scientist. Can biophysical studies provide a truly deeper understanding of how a protein works than can be obtained from static structures and qualitative analysis of biochemical data? We address this question in a case study by presenting the key concepts and experimental results that have led to our current understanding of cooperative oxygen binding by hemoglobin, the paradigm of structure function relations in multisubunit proteins. We conclude that the underlying simplicity of the two-state allosteric mechanism could not have been demonstrated without novel physical experiments and a rigorous quantitative analysis.

A statistical mechanical model for beta-hairpin kinetics.

Understanding the mechanism of protein secondary structure formation is an essential part of the protein-folding puzzle. Here, we describe a simple statistical mechanical model for the formation of a beta-hairpin, the minimal structural element of the antiparallel beta-pleated sheet. The model accurately describes the thermodynamic and kinetic behavior of a 16-residue, beta-hairpin-forming peptide, successfully explaining its two-state behavior and apparent negative activation energy for folding. The model classifies structures according to their backbone conformation, defined by 15 pairs of dihedral angles, and is further simplified by considering only the 120 structures with contiguous stretches of native pairs of backbone dihedral angles. This single sequence approximation is tested by comparison with a more complete model that includes the 2(15) possible conformations and 15 x 2(15) possible kinetic transitions. Finally, we use the model to predict the equilibrium unfolding curves and kinetics for several variants of the beta-hairpin peptide.

Folding dynamics and mechanism of beta-hairpin formation.

Protein chains coil into alpha-helices and beta-sheet structures. Knowing the timescales and mechanism of formation of these basic structural elements is essential for understanding how proteins fold. For the past 40 years, alpha-helix formation has been extensively investigated in synthetic and natural peptides, including by nanosecond kinetic studies. In contrast, the mechanism of formation of beta structures has not been studied experimentally. The minimal beta-structure element is the beta-hairpin, which is also the basic component of antiparallel beta-sheets. Here we use a nanosecond laser temperature-jump apparatus to study the kinetics of folding a beta-hairpin consisting of 16 amino-acid residues. Folding of the hairpin occurs in 6 micros at room temperature, which is about 30 times slower than the rate of alpha-helix formation. We have developed a simple statistical mechanical model that provides a structural explanation for this result. Our analysis also shows that folding of a beta-hairpin captures much of the basic physics of protein folding, including stabilization by hydrogen bonding and hydrophobic interactions, two-state behaviour, and a funnel-like, partially rugged energy landscape.

Laser temperature jump study of the helix<==>coil kinetics of an alanine peptide interpreted with a 'kinetic zipper' model.

The kinetics of the helix<==>coil transition of an alanine-based peptide following a laser-induced temperature jump were monitored by the fluorescence of an N-terminal probe, 4-(methylamino)benzoic acid (MABA). This probe forms a peptide hydrogen bond to the helix backbone, which changes its fluorescence quantum yield. The MABA fluorescence intensity decreases in a single exponential relaxation, with relaxation times that are weakly temperature dependent, exhibiting a maximum value of approximately 20 ns near the midpoint of the melting transition. We have developed a new model, the kinetic version of the equilibrium 'zipper' model for helix<==>coil transitions to explain these results. In this 'kinetic zipper' model, an enormous reduction in the number of possible species results from the assumption that each molecule contains either no helical residues or a single contiguous region of helix (the single-sequence approximation). The decay of the fraction of N-terminal residues that are helical, calculated from numerical solutions of the kinetic equations which describe the model, can be approximately described by two exponential relaxations having comparable amplitudes. The shorter relaxation time results from rapid unzipping (and zipping) of the helix ends in response to the temperature jump, while the longer relaxation time results from equilibration of helix-containing and non-helix-containing structures by passage over the nucleation free energy barrier. The decay of the average helix content is dominated by the slower process. The model therefore explains the experimental observation that relaxation for the N-terminal fluorescent probe is approximately 8-fold faster than that for the infrared probe of Williams et al. [(1996) Biochemistry 35, 691-697], which measures the average helix content, but does not account for the absence of observable amplitude for the slow relaxation in the fluorescence experiments (<10% slow phase). If we assume that the activation barrier for the coil-->helix rate is purely entropic, the model can also explain the maximum in the temperature dependence of the relaxation time for the fluorescent probe. Parameters that best reproduce the melting curves and the ratio of relaxation times predict a value of the cooperativity parameter sigma which is approximately 3-fold larger than previously reported values obtained from fitting equilibrium data only. The helix growth rate of approximately 10(8) s-1 that reproduces the experimental relaxation times is approximately 100-fold slower than those observed in molecular dynamics simulations. These parameters can be used to simulate the kinetically cooperative formation of a helix from the all-coil state.

Can a two-state MWC allosteric model explain hemoglobin kinetics?

We have analyzed the nanosecond-millisecond kinetics of ligand binding and conformational changes in hemoglobin. The kinetics were determined from measurements of precise time-resolved optical spectra following nanosecond photodissociation of the heme-carbon monoxide complex. To fit the data, it was necessary to extend the two-state allosteric model of Monod, Wyman, and Changeux (MWC) to include geminate ligand rebinding and nonexponential tertiary relaxation within the R quaternary structure. Considerable simplification of the model is obtained by using a linear free energy relation for the rates of quaternary transitions, and by incorporating concepts from recent studies on the physics of geminate rebinding and conformational changes in myoglobin. The model, described by 85 coupled differential equations, quantitatively explains a demanding set of complex kinetic data. Moreover, with the same set of kinetic parameters it simultaneously fits the equilibrium data on ligand binding and the distribution of ligation states. The present results, together with those from single-crystal oxygen binding studies, indicate that the two-state MWC allosteric model has survived its most critical tests.

Submillisecond protein folding kinetics studied by ultrarapid mixing.

An ultrarapid-mixing continuous-flow method has been developed to study submillisecond folding of chemically denatured proteins. Turbulent flow created by pumping solutions through a small gap dilutes the denaturant in tens of microseconds. We have used this method to study cytochrome c folding kinetics in the previously inaccessible time range 80 micros to 3 ms. To eliminate the heme-ligand exchange chemistry that complicates and slows the folding kinetics by trapping misfolded structures, measurements were made with the imidazole complex. Fluorescence quenching due to excitation energy transfer from the tryptophan to the heme was used to monitor the distance between these groups. The fluorescence decrease is biphasic. There is an unresolved process with tau < 50 micros, followed by a slower, exponential process with tau = 600 micros at the lowest denaturant concentration (0.2 M guanidine hydrochloride). These kinetics are interpreted as a barrier-free, partial collapse to the new equilibrium unfolded state at the lower denaturant concentration, followed by slower crossing of a free energy barrier separating the unfolded and folded states. The results raise several fundamental issues concerning the dynamics of collapse and barrier crossings in protein folding.

Allosteric effectors do not alter the oxygen affinity of hemoglobin crystals.

In solution, the oxygen affinity of hemoglobin in the T quaternary structure is decreased in the presence of allosteric effectors such as protons and organic phosphates. To explain these effects, as well as the absence of the Bohr effect and the lower oxygen affinity of T-state hemoglobin in the crystal compared to solution, Rivetti C et al. (1993a, Biochemistry 32:2888-2906) suggested that there are high- and low-affinity subunit conformations of T, associated with broken and unbroken intersubunit salt bridges. In this model, the crystal of T-state hemoglobin has the lowest possible oxygen affinity because the salt bridges remain intact upon oxygenation. Binding of allosteric effectors in the crystal should therefore not influence the oxygen affinity. To test this hypothesis, we used polarized absorption spectroscopy to measure oxygen binding curves of single crystals of hemoglobin in the T quaternary structure in the presence of the "strong" allosteric effectors, inositol hexaphosphate and bezafibrate. In solution, these effectors reduce the oxygen affinity of the T state by 10-30-fold. We find no change in affinity (< 10%) of the crystal. The crystal binding curve, moreover, is noncooperative, which is consistent with the essential feature of the two-state allosteric model of Monod J, Wyman J, and Changeux JP (1965, J Mol Biol 12:88-118) that cooperative binding requires a change in quaternary structure. Noncooperative binding by the crystal is not caused by cooperative interactions being masked by fortuitous compensation from a difference in the affinity of the alpha and beta subunits. This was shown by calculating the separate alpha and beta subunit binding curves from the two sets of polarized optical spectra using geometric factors from the X-ray structures of deoxygenated and fully oxygenated T-state molecules determined by Paoli M et al. (1996, J Mol Biol 256:775-792).

Submillisecond kinetics of protein folding.

New experimental methods permit observation of protein folding and unfolding on the previously inaccessible nanosecond-microsecond timescale. These studies are beginning to establish times for the elementary motions in protein folding - secondary structure and loop formation, local hydrophobic collapse, and global collapse to the compact denatured state. They permit an estimate of about one microsecond for the shortest time in which a protein can possibly fold.

A multiparameter analysis of sickle erythrocytes in patients undergoing hydroxyurea therapy.

During 24 weeks of hydroxyurea treatment, we monitored red blood cell (RBC) parameters in three patients with sickle cell disease, including F-cell and F-reticulocyte profiles, distributions of delay times for intracellular polymerization, sickle erythrocyte adherence to human umbilical vein endothelial cells in a laminar flow chamber, RBC phthalate density profiles, mean corpuscular hemoglobin concentration and cation content, reticulocyte mean corpuscular hemoglobin concentration, 1H-nuclear magnetic resonance transverse relaxation rates of packed RBCs, and plasma membrane lateral and rotational mobilities of band 3 and glycophorins. Hydroxyurea increases the fraction of cells with sufficiently long delay times to escape the microcirculation before polymerization begins. Furthermore, high pretreatment adherence to human umbilical vein endothelial cells of sickle RBCs decreased to normal after only 2 weeks of hydroxyurea treatment, preceding the increase in fetal hemoglobin levels. The lower adhesion of sickle RBCs to endothelium would facilitate escape from the microcirculation before polymerization begins. Hydroxyurea shifted several biochemical and biophysical parameters of sickle erythrocytes toward values observed with hemoglobin SC disease, suggesting that hydroxyurea moderates sickle cell disease toward the milder, but still clinically significant, hemoglobin SC disease. The 50% reduction in sickle crises documented in the Multicenter Study of Hydroxyurea in Sickle Cell Disease is consistent with this degree of erythrocyte improvement.

Diffusion-limited contact formation in unfolded cytochrome c: estimating the maximum rate of protein folding.

How fast can a protein fold? The rate of polypeptide collapse to a compact state sets an upper limit to the rate of folding. Collapse may in turn be limited by the rate of intrachain diffusion. To address this question, we have determined the rate at which two regions of an unfolded protein are brought into contact by diffusion. Our nanosecond-resolved spectroscopy shows that under strongly denaturing conditions, regions of unfolded cytochrome separated by approximately 50 residues diffuse together in 35-40 microseconds. This result leads to an estimate of approximately (1 microsecond)-1 as the upper limit for the rate of protein folding.

Fast events in protein folding.

Understanding how proteins fold is one of the central problems in biochemistry. A new generation of kinetic experiments has emerged to investigate the mechanisms of protein folding on the previously inaccessible submillisecond time scale. These experiments provide the first glimpse of processes such as secondary structure formation, local hydrophobic collapse, global collapse to compact denatured states, and fast barrier crossings to the native state. Key results are summarized and discussed in terms of the statistical energy landscape theory of protein folding.