William A. Eaton

Folding dynamics and mechanism of Beta-hairpin formation

Protein chains coil into α-helices and β-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, α-helix formation has been extensively investigated in synthetic and natural peptides, including by nanosecond kinetic studies. In contrast, the mechanism of formation of β structures has not been studied experimentally. The minimal β-structure element is the β-hairpin, which is also the basic component of antiparallel β-sheets. Here we use a nanosecond laser temperature-jump apparatus to study the kinetics of folding a β-hairpin consisting of 16 amino-acid residues. Folding of the hairpin occurs in 6 µs at room temperature, which is about 30 times slower than the rate of α-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 β-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.

Probing the free-energy surface for protein folding with single-molecule fluorescence spectroscopy

Protein folding is inherently a heterogeneous process because of the very large number of microscopic pathways that connect the myriad unfolded conformations to the unique conformation of the native structure. In a first step towards the long-range goal of describing the distribution of pathways experimentally, Förster resonance energy transfer (FRET) has been measured on single, freely diffusing molecules. Here we use this method to determine properties of the free-energy surface for folding that have not been obtained from ensemble experiments. We show that single-molecule FRET measurements of a small cold-shock protein expose equilibrium collapse of the unfolded polypeptide and allow us to calculate limits on the polypeptide reconfiguration time. From these results, limits on the height of the free-energy barrier to folding are obtained that are consistent with a simple statistical mechanical model, but not with the barriers derived from simulations using molecular dynamics. Unlike the activation energy, the free-energy barrier includes the activation entropy and thus has been elusive to experimental determination for any kinetic process in solution.

Protein folding studied by single-molecule FRET.

A complete understanding of a protein-folding mechanism requires description of the distribution of microscopic pathways that connect the folded and unfolded states. This distribution can, in principle, be described by computer simulations and theoretical models of protein folding, but is hidden in conventional experiments on large ensembles of molecules because only average properties are measured. A long-term goal of single-molecule fluorescence studies is to time-resolve the structural events as individual molecules make transitions between folded and unfolded states. Although such studies are still in their infancy, the work till now shows great promise and has already produced novel and important information on current issues in protein folding that has been impossible or difficult to obtain from ensemble measurements.

Sickle Cell Hemoglobin Polymerization

Publisher Summary The chapter describes the understanding of the physics and physical chemistry of sickle cell hemoglobin polymerization in solutions and in red cells. The polymerization of sickle cell hemoglobin has probably become the best understood of all protein self-assembly systems. The structure of the hemoglobin S molecule, the structure of the various aggregated forms of hemoglobin S, and the structural analysis of the polymers are discussed in the chapter. The chapter discusses the thermodynamics of hemoglobin S polymerization, and includes a description of the nonideal behavior of concentrated hemoglobin S solutions and the effects of physiologically relevant variables, especially oxygen, and the presence of non-S hemoglobins on the polymerization process. Understanding the polymerization process is not only important for understanding the pathophysiology of sickle cell disease, but is critical to the major problem of developing a specific therapy that could be used in the treatment of patients. The kinetic and thermodynamic studies have played a major role by providing relevant and sensitive assays for potential therapeutic agents. The results of the thermodynamic and kinetic studies of solutions are used to explain various properties of cells, including morphological and rheological properties.

Kinetics of sickle hemoglobin polymerization: II. A double nucleation mechanism

A double nucleation mechanism for the polymerization of sickle hemoglobin is described. The mechanism accounts for all of the major kinetic observations: the appearance of a delay, the high concentration dependence of the delay time, and the stochastic behavior of slowly polymerizing samples in small volumes. The mechanism postulates that there are two pathways for polymer formation: polymerization is initiated by homogeneous nucleation in the solution phase, followed by nucleation of additional polymers on the surface of existing ones. This second pathway is called heterogeneous nucleation. Since the surface of polymers is continuously increasing with time, heterogeneous nucleation provides a mechanism for the extreme autocatalysis that is manifested as an apparent delay in the kinetic progress curves. In this mechanism, each spherulitic domain of polymers is considered to be initiated by a single homogeneous nucleation event. The mechanism explains the irreproducibility of the delay time for single domain formation as arising from stochastic fluctuations in the time at which the homogeneous nucleus for the first polymer is formed. Integration of the linearized rate equations that describe this model results in a simple kinetic form: A[cosh(Bt)-1] (Bishop & Ferrone, 1984). In the accompanying paper (Ferrone et al., 1985) it was shown that the initial 10 to 15% of progress curves, with delay times varying from a few milliseconds to over 10(5) seconds, is well fit by this equation. In this paper, we present an approximate statistical thermodynamic treatment of the equilibrium nucleation processes that shows how the nucleus sizes and nucleation equilibrium constants depend on monomer concentration. The equilibrium model results in expressions for B and B2A as a function of monomer concentration in terms of five adjustable parameters: the bimolecular addition rate of a monomer to the growing aggregate, the fraction of polymerized monomers that serve as heterogeneous nucleation sites, the free energy of intermolecular bonding within the polymer, and two parameters that describe the free energy change as a function of size for the bonding of the heterogeneous nucleus to a polymer surface. This model provides an excellent fit to the data for B and B2A as a function of concentration using physically reasonable parameters. The model also correctly predicts the time regime in which stochastic behavior is observed for polymerization in small volumes.

Characterizing the unfolded states of proteins using single-molecule FRET spectroscopy and molecular simulations

To obtain quantitative information on the size and dynamics of unfolded proteins we combined single-molecule lifetime and intensity FRET measurements with molecular simulations. We compared the unfolded states of the 64-residue, α/β protein L and the 66-residue, all-β cold-shock protein CspTm. The average radius of gyration (Rg) calculated from FRET data on freely diffusing molecules was identical for the two unfolded proteins at guanidinium chloride concentrations >3 M, and the FRET-derived Rg of protein L agreed well with the Rg previously measured by equilibrium small-angle x-ray scattering. As the denaturant concentration was lowered, the mean FRET efficiency of the unfolded subpopulation increased, signaling collapse of the polypeptide chain, with protein L being slightly more compact than CspTm. A decrease in Rg with decreasing denaturant was also observed in all-atom molecular dynamics calculations in explicit water/urea solvent, and Langevin simulations of a simplified representation of the polypeptide suggest that collapse can result from either increased interresidue attraction or decreased excluded volume. In contrast to both the FRET and simulation results, previous time-resolved small-angle x-ray scattering experiments showed no collapse for protein L. Analysis of the donor fluorescence decay of the unfolded subpopulation of both proteins gives information about the end-to-end chain distribution and suggests that chain dynamics is slow compared with the donor life-time of ≈2 ns, whereas the bin-size independence of the small excess width above the shot noise for the FRET efficiency distributions may result from incomplete conformational averaging on even the 1-ms time scale.

Single-Molecule Fluorescence Experiments Determine Protein Folding Transition Path Times

A Fraction of Folding An energy barrier has to be crossed as a protein transforms between folded and unfolded states. Molecular dynamic simulations have observed sharp transitions, with barrier crossing times of less than a microsecond, a fraction of the total folding time; however, this time range has been inaccessible to single-molecule experiments. Chung et al. (p. 981) described single-molecule fluorescence experiments that allowed measurement of the transition-path time for a fast-folding protein and to reduce the upper bound for a slow-folding protein. Although the folding rates differed by a factor of 10,000, the transition-path times differ by less than a factor of 5, pointing to energy landscape theory for the explanation. Quickly and slowly folding proteins take the same time to cross the barrier from the unfolded to the folded state. The transition path is the tiny fraction of an equilibrium molecular trajectory when a transition occurs as the free-energy barrier between two states is crossed. It is a single-molecule property that contains all the mechanistic information on how a process occurs. As a step toward observing transition paths in protein folding, we determined the average transition-path time for a fast- and a slow-folding protein from a photon-by-photon analysis of fluorescence trajectories in single-molecule Förster resonance energy transfer experiments. Whereas the folding rate coefficients differ by a factor of 10,000, the transition-path times differ by a factor of less than 5, which shows that a fast- and a slow-folding protein take almost the same time to fold when folding actually happens. A very simple model based on energy landscape theory can explain this result.

Kinetics of sickle hemoglobin polymerization. I. Studies using temperature-jump and laser photolysis techniques.

Using a combination of laser photolysis and temperature-jump techniques, the kinetics of hemoglobin S polymerization have been studied over a wide range of delay times (10(-3) to 10(5)s), concentrations (0.2 to 0.4 g/cm3) and temperatures (5 to 50 degrees C). A slow temperature-jump technique was used to induce polymerization in samples with delay times between 10(2) seconds and 10(5) seconds by heating a solution of completely deoxygenated hemoglobin S. For samples with shorter delay times, polymerization was induced by photodissociating the carbon monoxide complex in small volumes (10(-9) cm3) using a microspectrophotometer equipped with a cw argon ion laser. The photolysis technique is described in some detail because of its importance in studying hemoglobin S polymerization at physiological concentrations and temperatures. In order, to establish conditions for complete photodissociation with minimal laser heating, a series of control experiments on normal human hemoglobin was performed and theoretically modeled. The concentration dependence of the tenth time is found to decrease with increasing hemoglobin S concentration. In the range 0.2 to 0.3 g/cm3, the tenth time varies as the 36th power of the hemoglobin S concentration, while in the range 0.3 to 0.4 g/cm3 it decreases to 16th power. As the tenth times become shorter, the progress curves broaden, with the onset of polymerization becoming less abrupt. For tenth times greater than about 30 seconds, measurements with the laser photolysis technique on small volumes yield highly irreproducible tenth times, but superimposable progress curves, indicating stochastic behavior. The initial part of the progress curves from both temperature-jump and laser photolysis experiments is well fit with an equation for the concentration of polymerized monomer, delta (t) = A[cosh (Bt) -1], which results from integration of the linearized rate equations for the double nucleation mechanism described in the accompanying paper (Ferrone et al., 1985). The dependence of the parameters A and B on temperature and concentration is obtained from fitting over 300 progress curves. The rate B has a large concentration dependence, varying at 25 degrees C from about 10(-4) S-1 at 0.2 g/cm3 to about 100 s-1 at 0.4 g/cm3.

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.

Experimental tests of villin subdomain folding simulations.

We have used laser temperature-jump to investigate the kinetics and mechanism of folding the 35 residue subdomain of the villin headpiece. The relaxation kinetics are biphasic with a sub-microsecond phase corresponding to a helix-coil transition and a slower microsecond phase corresponding to overall unfolding/refolding. At 300 K, the folding time is 4.3(+/-0.6) micros, making it the fastest folding, naturally occurring protein, with a rate close to the theoretical speed limit. This time is in remarkable agreement with the prediction of 5 (+11,-3) micros by Zagrovic et al. from atomistic molecular dynamics simulations using an implicit solvent model. We test their prediction that replacement of the C-terminal phenylalanine residue with alanine will increase the folding rate by removing a transient non-native interaction. We find that the alanine substitution has no effect on the folding rate or on the equilibrium constant. Implications of this result for the validity of the simulated folding mechanism are discussed.