The calculation of rate coefficients is a discipline of nonlinear science of importance to much of physics, chemistry, engineering, and biology. Fifty years after Kramers' seminal paper on thermally activated barrier crossing, the authors report, extend, and interpret much of our current understanding relating to theories of noise-activated escape, for which many of the notable contributions are originating from the communities both of physics and of physical chemistry. Theoretical as well as numerical approaches are discussed for single- and many-dimensional metastable systems (including fields) in gases and condensed phases. The role of many-dimensional transition-state theory is contrasted with Kramers' reaction-rate theory for moderate-to-strong friction; the authors emphasize the physical situation and the close connection between unimolecular rate theory and Kramers' work for weakly damped systems. The rate theory accounting for memory friction is presented, together with a unifying theoretical approach which covers the whole regime of weak-to-moderate-to-strong friction on the same basis (turnover theory). The peculiarities of noise-activated escape in a variety of physically different metastable potential configurations is elucidated in terms of the mean-first-passage-time technique. Moreover, the role and the complexity of escape in driven systems exhibiting possibly multiple, metastable stationary nonequilibrium states is identified. At lower temperatures, quantum tunneling effects start to dominate the rate mechanism. The early quantum approaches as well as the latest quantum versions of Kramers' theory are discussed, thereby providing a description of dissipative escape events at all temperatures. In addition, an attempt is made to discuss prominent experimental work as it relates to Kramers' reaction-rate theory and to indicate the most important areas for future research in theory and experiment.

Reaction-rate theory: fifty years after Kramers

Recent experiments, advances in theory, and analogies to other complex systems such as glasses and spin glasses yield insight into protein dynamics. The basis of the understanding is the observation that the energy landscape is complex: Proteins can assume a large number of nearly isoenergetic conformations (conformational substates). The concepts that emerge from studies of the conformational substates and the motions between them permit a quantitative discussion of one simple reaction, the binding of small ligands such as carbon monoxide to myoglobin.

https://science.sciencemag.org/content/sci/254/5038/1598.full.pdf

The energy landscapes and motions of proteins.

Fractional dynamics has experienced a firm upswing during the past few years, having been forged into a mature framework in the theory of stochastic processes. A large number of research papers developing fractional dynamics further, or applying it to various systems have appeared since our first review article on the fractional Fokker–Planck equation (Metzler R and Klafter J 2000a, Phys. Rep. 339 1–77). It therefore appears timely to put these new works in a cohesive perspective. In this review we cover both the theoretical modelling of sub- and superdiffusive processes, placing emphasis on superdiffusion, and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function. We also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems.

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The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics

Because proteins are central to cellular function, researchers have sought to uncover the secrets of how these complex macromolecules execute such a fascinating variety of functions. Although static structures are known for many proteins, the functions of proteins are governed ultimately by their dynamic character (or 'personality'). The dream is to 'watch' proteins in action in real time at atomic resolution. This requires addition of a fourth dimension, time, to structural biology so that the positions in space and time of all atoms in a protein can be described in detail.

http://www.ifm.liu.se/edu/coursescms/TFKE40/literature/KernD_Nature07_Review.pdf

Dynamic personalities of proteins.

The dynamics of a folded globular protein (bovine pancreatic trypsin inhibitor) have been studied by solving the equations of motion for the atoms with an empirical potential energy function. The results provide the magnitude, correlations and decay of fluctuations about the average structure. These suggest that the protein interior is fluid-like in that the local atom motions have a diffusional character.

Dynamics of folded proteins

Myoglobin rebinding of carbon monoxide and dioxygen after photodissociation has been observed in the temperature range between 40 and 350 K. A system was constructed that records the change in optical absorption at 436 nm smoothly and without break between 2 musec and 1 ksec. Four different rebinding processes have been found. Between 40 and 160 K, a single process is observed. It is not exponential in time, but approximately given by N(t) = (1 + t/to)-n, where to and n are temperature-dependent, ligand-concentration independent, parameters. At about 170 K, a second and at 200 K, a third concentration-independent process emerge. At 210 K, a concentration-dependent process sets in. If myoglobin is embedded in a solid, only the first three can be seen, and they are all nonexponential. In a liquid glycerol-water solvent, rebinding is exponential. To interpret the data, a model is proposed in which the ligand molecule, on its way from the solvent to the binding site at the ferrous heme iron, encounters four barriers in succession. The barriers are tentatively identified with known features of myoglobin. By computer-solving the differential equation for the motion of a ligand molecule over four barriers, the rates for all important steps are obtained. The temperature dependences of the rates yield enthalpy, entropy, and free-energy changes at all barriers. The free-energy barriers at 310 K indicate how myoglobin achieves specificity and order. For carbon monoxide, the heights of these barriers increase toward the inside; carbon monoxide consequently is partially rejected at each of the four barriers. Dioxygen, in contrast, sees barriers of about equal height and moves smoothly toward the binding site. The entropy increases over the first two barriers, indicating a rupturing of bonds or displacement of residues, and then smoothly decreases, reaching a minimum at the binding site. The magnitude of the decrease over the innermost barrier implies participation of heme and/or protein. The nonexponential rebinding observed at low temperatures and in solid samples implies that the innermost barrier has a spectrum of activation energies. The shape of the spectrum has been determined; its existence can be explained by assuming the presence of many conformational states for myoglobin. In a liquid at temperatures above about 230 K, relaxation among conformational states occurs and rebinding becomes exponential.

Dynamics of ligand binding to myoglobin

Carbon monoxide bound to myoglobin can be photodissociated with high quantum yield The subsequent rebinding can be followed optically In the temperature range between 40 and 200 K, rebinding can be described by a function of the form ${t}_{0}$, where $n$ and $H(t)={(1+\frac{t}{{t}_{0}})}^{\ensuremath{-}n}$ are temperature-dependent parameters This behavior can be explained by assuming the existence of an activation-energy spectrum; the form of this spectrum is determined The energy spectrum may be due to the existence of myoglobin conformers

Activation Energy Spectrum of a Biomolecule: Photodissociation of Carbonmonoxy Myoglobin at Low Temperatures

Rebinding of carbon monoxide to the beta chain of hemoglobin after photodissociation by a laser flash is intramolecular below about 200 K. Above 25 K, rebinding occurs via classical over-the-barrier motion; below, quantum-mechanical tunneling dominates. Both are described by an energy spectrum peaked at Epeak=4.0 kilojoules per mole. The barrier width d(E), determined from the energy dependence of the tunneling rate, depends on barrier height, d(E) approximately 0.05 nanometer X (E/Epeak) 1.5.

http://science.sciencemag.org/content/sci/192/4243/1002.full.pdf

Tunneling in ligand binding to heme proteins.

Rebinding of carbon monoxide to myoglobin and to cytochrome P-450 after removal by a light flash occurs down to 50 degrees K for myoglobin and 25 degrees K for cytochrome P-450 in glycerol-water solution. Above 240 degrees K the reaction is second order; between 240 degrees and 200 degrees K the rebinding becomes exponential and independent of the carbon monoxide concentration. Below 150 degrees K the reaction follows a power law and is approximately 10(3) times faster for cytochrome P-450 than for myoglobin.

Dynamics of carbon monoxide binding by heme proteins.

Many transient phenomena extend over more than two orders of magnitude in time. Such processes are most efficiently observed with systems that possess logarithmic time bases. A digital analyzer is described that records smoothly and in one sweep transient processes over more than eight orders of magnitude in time. A typical sweep covers the time range from 2 μsec to 5 min.

K. W. Beeson

Papers

Dynamics of ligand binding to myoglobin

Activation Energy Spectrum of a Biomolecule: Photodissociation of Carbonmonoxy Myoglobin at Low Temperatures

Tunneling in ligand binding to heme proteins.

Dynamics of carbon monoxide binding by heme proteins.

Transient analyzer with logarithmic time base