Nanosecond lasers were used to measure the rate of conformational changes in myoglobin after ligand dissociation at ambient temperatures. At low solvent viscosities the rate is independent of viscosity, but at high viscosities it depends on approximately the inverse first power of the viscosity. Kramers theory for unimolecular rate processes can be used to explain this result if the friction term is modified to include protein as well as solvent friction. The theory and experiment suggest that the dominant factor in markedly reducing the rate of conformational changes in myoglobin at low temperatures (less than 200 K) is the very high viscosity (greater than 10(7) centipoise) of the glycerol-water solvent. That is, at low temperatures conformational substates may not be "frozen" so much as "stuck."

After photodissociation of carbon monoxide bound to myoglobin, the protein relaxes to the deoxy equilibrium structure in a quake-like motion. Investigation of the proteinquake and of related intramolecular equilibrium motions shows that states and motions have a hierarchical glasslike structure.The dynamic aspects of proteins have been studied extensively in recent years and a picture of ever increasing complexity has emerged. To bring some order into the complexity, we have introduced a model that classifies states and motions (1). In the present paper, we describe the model and its experimental basis in more detail. STATES, SUBSTATES, AND MOTIONSWe consider myoglobin (Mb), an oxygen storage protein, consisting of 153 amino acids, with molecular weight of 17,900 and approximate dimensions of2.5 x 4.4 x 4.4 nm (2). Embedded in the protein matrix is a heme group with a central iron atom, which binds small ligands such as dioxygen (02) or carbon monoxide (CO) reversibly. Thus, two states are involved in the function of Mb, deoxyMb and liganded Mb (e.g., MbCO). In the liganded state, the heme is planar and the iron has spin 0 and lies close to the mean heme plane. In the unliganded state, the heme group is domed, the iron has spin 2 and lies =0.5 A away from the mean heme plane, and the globin structure differs somewhat from the liganded one (3).A protein molecule in a particular state can assume a very large number of conformational substates (CS) (4-6). Different substates have the same overall structure, but they differ in details; they perform the same function, but with different rates. equilibrium (10). Return to equilibrium occurs through a proteinquake: the released strain energy is dissipated through waves [phonons (11) or solitons (12)1 and through the propagation of a deformation (2, 3). HIERARCHY OF SUBSTATESThe experiments described in the next section imply that the proteinquake released by photodissociation of MbCO propagates sequentially: Fig. 2, consequently is much more complex than we originally anticipated (4).The valley in the top diagram of Fig. 2a represents one state, say MbCO. MbCO can exist in a large number of conformational substates, CS1, separated by high barriers.Each valley in the first tier is structured into substates (CS2) with smaller barriers. The furcation continues through two more tiers, with decreasing barrier heights. The dynamic

Ligand binding to heme proteins is studied by using flash photolysis over wide ranges in time (100 ns-1 ks) and temperature (10-320 K). Below about 200 K in 75% glycerol/water solvent, ligand rebinding occurs from the heme pocket and is nonexponential in time. The kinetics is explained by a distribution, g(H), of the enthalpic barrier of height H between the pocket and the bound state. Above 170 K rebinding slows markedly. Previously we interpreted the slowing as a "matrix process" resulting from the ligand entering the protein matrix before rebinding. Experiments on band III, an inhomogeneously broadened charge-transfer band near 760 nm (approximately 13,000 cm-1) in the photolyzed state (Mb*) of (carbonmonoxy)myoglobin (MbCO), force us to reinterpret the data. Kinetic hole-burning measurements on band III in Mb* establish a relation between the position of a homogeneous component of band III and the barrier H. Since band III is red-shifted by 116 cm-1 in Mb* compared with Mb, the relation implies that the barrier in relaxed Mb is 12 kJ/mol higher than in Mb*. The slowing of the rebinding kinetics above 170 K hence is caused by the relaxation Mb*----Mb, as suggested by Agmon and Hopfield [(1983) J. Chem. Phys. 79, 2042-2053]. This conclusion is supported by a fit to the rebinding data between 160 and 290 K which indicates that the entire distribution g(H) shifts. Above about 200 K, equilibrium fluctuations among conformational substates open pathways for the ligands through the protein matrix and also narrow the rate distribution. The protein relaxations and fluctuations are nonexponential in time and non-Arrhenius in temperature, suggesting a collective nature for these protein motions. The relaxation Mb*----Mb is essentially independent of the solvent viscosity, implying that this motion involves internal parts of the protein. The protein fluctuations responsible for the opening of the pathways, however, depend strongly on the solvent viscosity, suggesting that a large part of the protein participates. While the detailed studies concern MbCO, similar data have been obtained for MbO2 and CO binding to the beta chains of human hemoglobin and hemoglobin Zürich. The results show that protein dynamics is essential for protein function and that the association coefficient for binding from the solvent at physiological temperatures in all these heme proteins is governed by the barrier at the heme.

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Elucidating the mechanism of folding of polynucleotides depends on accurate estimates of free energy surfaces and a quantitative description of the kinetics of structure formation. Here, the kinetics of hairpin formation in single-stranded DNA are measured after a laser temperature jump. The kinetics are modeled as configurational diffusion on a free energy surface obtained from a statistical mechanical description of equilibrium melting profiles. The effective diffusion coefficient is found to be strongly temperaturedependent in the nucleation step as a result of formation of misfolded loops that do not lead to subsequent zipping. This simple system exhibits many of the features predicted from theoretical studies of protein folding, including a funnel-like energy surface with many folding pathways, trapping in misfolded conformations, and non-Arrhenius folding rates.H airpin loops are ubiquitous in single-stranded DNA and RNA. Knowing the time scales and mechanism of formation of these loops is an essential first step toward understanding the folding problem. Although the stability of hairpin loops and the kinetics of hairpin formation have been a subject of intense investigation for over 30 years (1-5), our understanding of the kinetics is limited. In particular, there is no simple physical model that describes in a consistent way both the thermodynamics and kinetics of hairpin formation. With the exception of some early work on the helix-to-coil transition, in which the kinetics were described in terms of a statistical mechanical kinetic ''zipper'' model (6, 7), the kinetics of hairpin-to-coil transition have been described more recently in terms of a two-state system with Arrhenius temperature dependence for the rates of hairpin formation and unwinding (5, 8). The equilibrium dynamics of hairpins, obtained from fluctuation correlation spectroscopy measurements of hairpins labeled with fluorescent donor and acceptor pairs (8, 9), have revealed a number of kinetic features that are not easily explained within the framework of a simple two-state analysis. First, the data of Libchaber and coworkers show that the rate coefficient corresponding to the closing of hairpins has a non-Arrhenius temperature dependence (8). Second, they report a puzzling result in which the apparent activation energy for forming hairpins with poly(dA) loops increases as the loop size increases. Third, Klenerman and coworkers report stretched exponential kinetics at temperatures well below the melting temperature (9). These observations suggest a failure of the simplest two-state analysis and require a modification of even the more rigorous kinetic ''zipper'' model in the form in which it was applied to helix-coil kinetics (6).Here we present a model for the dynamics of hairpins that is consistent with many of the apparently anomalous kinetic observations. The dynamics are described as configurational diffusion along a free energy profile that we calculate from a statistical mechanical ''zipper'' model that describes the equilibrium melt...

Absorption spectroscopy with nanosecond time resolution shows that myoglobin undergoes conformational relaxation on the same time scale as geminate rebinding of carbon monoxide. Ligand rebinding following photodissociation of the heme-CO complex was measured from the amplitude of the average difference spectrum, while conformational changes were measured from changes in the detailed shape of the Soret spectra of the deoxyhemes. Experiments in which the solvent viscosity was varied between 1 and 300 cP and the temperature between 268 and 308 K were analyzed by fitting the multiwavelength kinetic data with both empirical and molecular models. Novel numerical techniques were employed in fitting the data, including the use of singular value decomposition to remove the effects of temperature and solvent on the spectra and of a Monte Carlo method to overcome the multiple minimum problem in searching parameter space. The molecular model is the minimal model that incorporates all of the major features of myoglobin kinetics at ambient temperatures, including a fast and slow rebinding conformation and two geminate states for each conformation. The results of fitting the kinetic data with this model indicate that the geminate-rebinding rates for the two conformations differ by at least a factor of 100. The differences between the spectra of the two conformations generated from the fits are similar to the differences between those of the R and T conformations of hemoglobin. In modeling the data, the dependence of the rates on temperature and viscosity was parametrized using a modification of Kramers theory which includes the contributions of both protein and solvent to the friction. The rate of the transition from the fast to the slow rebinding conformation is found to be inversely proportional to the viscosity when the viscosity exceeds about 30 cP and nearly viscosity independent at low viscosity. The viscosity dependence at high viscosities suggests that the two conformations differ by the global displacement of protein atoms on the proximal side of the heme observed by X-ray crystallography. We suggest that the conformational change observed in our experiments corresponds to the final portion of the nonexponential conformational relaxation recently observed by Anfinrud and co-workers, which begins on a picosecond time scale. Furthermore, extrapolation of our data to temperatures near that of the solvent glass transition suggests that this conformational relaxation may very well be the one postulated by Frauenfelder and co-workers to explain the decrease in the rate of geminate rebinding with increasing temperature above 180 K.

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A statistical mechanical "zipper" model is applied to describe the equilibrium melting of short DNA hairpins with poly(dT) loops ranging from 4 to 12 bases in the loop. The free energy of loop formation is expressed in terms of the persistence length of the chain. This method provides a new measurement of the persistence length of single-stranded DNA, which is found to be approximately 1.4 nm for poly(dT) strands in 100 mM NaCl. The free energy of the hairpin relative to the random coil state is found to scale with the loop size with an apparent exponent of > or = 7, much larger than the exponent of approximately 1.5-1.8 expected from considerations of loop entropy alone. This result indicates a strong dependence of the excess stability of the hairpins, from stacking interactions of the bases within the loop, on the size of the loop. We interpret this excess stability as arising from favorable hydrophobic interactions among the bases in tight loops and which diminish as the loops get larger. Free energy profiles along a generalized reaction coordinate are calculated from the equilibrium zipper model. The transition state for hairpin formation is identified as an ensemble of looped conformations with one basepair closing the loop, and with a lower enthalpy than the random coil state. The equilibrium model predicts apparent activation energy of approximately -11 kcal/mol for the hairpin closing step, in remarkable agreement with the value obtained from kinetics measurements.