Kramers model with a power-law friction kernel: dispersed kinetics and dynamic disorder of biochemical reactions

Real-time single-molecule imaging of CaMKII-calmodulin interactions

The binding of calcium/calmodulin (CAM) to calcium/calmodulin-dependent protein kinase II (CaMKII) initiates an ATP-driven cascade that triggers CaMKII autophosphorylation. The autophosphorylation in turn increases the CaMKII affinity for CAM. Here, we studied the ATP dependence of CAM association with the actin-binding CaMKIIβ isoform using single-molecule total internal reflection fluorescence microscopy. Rhodamine-CAM associations/dissociations to surface-immobilized Venus-CaMKIIβ were resolved with 0.5 s resolution from video records, batch-processed with a custom algorithm. CAM occupancy was determined simultaneously with spot-photobleaching measurement of CaMKII holoenzyme stoichiometry. We show the ATP-dependent increase of the CAM association requires dimer formation for both the α and β isoforms. The study of mutant β holoenzymes revealed that the ATP-dependent increase in CAM affinity results in two distinct states. The phosphorylation-defective (T287.306-307A) holoenzyme resides only in the low-affinity state. CAM association is further reduced in the T287A holoenzyme relative to T287.306-307A. In the absence of ATP, the affinity of CAM for the T287.306-307A mutant and the wild-type monomer are comparable. The affinity of the ATP-binding impaired (K43R) mutant is even weaker. In ATP, the K43R holoenzyme resides in the low-affinity state. The phosphomimetic mutant (T287D) resides only in a 1000-fold higher-affinity state, with mean CAM occupancy of more than half of the 14-mer holoenzyme stoichiometry in picomolar CAM. ATP promotes T287D holoenzyme disassembly but does not elevate CAM occupancy. Single Poisson distributions characterized the ATP-dependent CAM occupancy of mutant holoenzymes. In contrast, the CAM occupancy of the wild-type population had a two-state distribution with both low- and high-affinity states represented. The low-affinity state was the dominant state, a result different from published in vitro assays. Differences in assay conditions can alter the balance between activating and inhibitory autophosphorylation. Bound ATP could be sufficient for CaMKII structural function, while antagonistic autophosphorylations may tune CaMKII kinase-regulated action-potential frequency decoding in vivo.

Everlasting impact of initial perturbations on first-passage times of non-Markovian random walks

Persistence, defined as the probability that a signal has not reached a threshold up to a given observation time, plays a crucial role in the theory of random processes. Often, persistence decays algebraically with time with non trivial exponents. However, general analytical methods to calculate persistence exponents cannot be applied to the ubiquitous case of non-Markovian systems relaxing transiently after an imposed initial perturbation. Here, we introduce a theoretical framework that enables the non-perturbative determination of persistence exponents of Gaussian non-Markovian processes with non stationary dynamics relaxing to a steady state after an initial perturbation. Two situations are analyzed: either the system is subjected to a temperature quench at initial time, or its past trajectory is assumed to have been observed and thus known. Our theory covers the case of spatial dimension higher than one, opening the way to characterize non-trivial reaction kinetics for complex systems with non-equilibrium initial conditions.

Conformational Transitions of Amyloid-β: A Langevin and Generalized Langevin Dynamics Simulation Study

The dynamics of conformational transitions of the disordered protein, amyloid-β, is studied via Langevin and generalized Langevin dynamics simulations. The transmission coefficient for the unfold-misfold transition of amyloid-β is calculated from multiple independent trajectories that originate at the transition state with different initial velocities and are directly correlated to Kramers and Grote-Hynes theories. For lower values of the frictional coefficient, a well-defined rate constant is obtained, whereas, for higher values, the transmission coefficient decays with time, indicating a breakdown of the Kramers and Grote-Hynes theories and the emergence of a dynamic disorder, which demonstrates the presence of multiple local minima in the misfolding potential energy surface. The calculated free energy profile describes a two-state transition of amyloid-β in the energy landscape. The transition path time distribution computed from these simulations is compared with the related experimental and theoretical results for the unfold-misfold transition of amyloid-β. The high free energy barrier for this transition confirms the misfolding of amyloid-β. These findings offer an insight into the dynamics of the unfold-misfold transition of this protein.

Heterogeneous kinetics of the loop formation of a single polymer chain in crowded and disordered media

The cytoplasmic volume of cells is occupied and crowded by a variety of macromolecules, such as proteins and cytoskeleton structures. Such diverse macromolecules make the cell cytoplasm not only structurally heterogeneous but also dynamically heterogeneous: Some macromolecules may diffuse freely inside cell cytoplasm at certain timescales while others hardly diffuse. Studies on the effects of the dynamic heterogeneity on reaction kinetics have been limited even though the effects of the crowdedness and structural heterogeneity were investigated extensively. In this study, we employ a simple model of mixtures of mobile and immobile matrix particles, tune the degree of dynamic heterogeneity by changing the fraction of immobile matrix particles, and investigate reaction kinetics in such heterogeneous media. We employ the loop formation of a single polymer chain as a model reaction and perform Langevin dynamics simulations. We find that the free-energy barrier of the loop formation is decreased as the systems become more crowded with matrix particles. But the free-energy barrier is not sensitive to the dynamic heterogeneity. As dynamic heterogeneity increases with an increase in the fraction of immobile matrix particles, however, the diffusivity of the system decreases significantly. The decrease in the diffusion (due to the dynamic heterogeneity) and the decrease in the free-energy barrier (due to the crowdedness) lead together to a complicated trend of the loop formation kinetics. As the volume fraction of immobile matrix particles reaches a critical value at the percolation transition, the reaction kinetics becomes significantly heterogeneous and the survival probability distribution of the chain loop formation becomes stretched-exponential. We also illustrate that the heterogeneous reaction rate near the percolation transition relates closely to the structures of local pores in which the polymer is located.

Passage through a sub-diffusing geometrical bottleneck

The usual Kramers theory of reaction rates in condensed media predict the rate to have an inverse dependence on the viscosity of the medium, η. However, experiments on ligand binding to proteins, performed long ago, showed the rate to have η^-ν dependence, with ν in the range of 0.4-0.8. Zwanzig [J. Chem. Phys. 97, 3587 (1992)] suggested a model in which the ligand has to pass through a fluctuating opening to reach the binding site. This fluctuating gate model predicted the rate to be proportional to η^-1/2. More recently, experiments performed by Xie et al. [Phys. Rev. Lett. 93, 180603 (2004)] showed that the distance between two groups in a protein undergoes not normal diffusion, but subdiffusion. Hence, in this paper, we suggest and solve a generalization of the Zwanzig model, viz., passage through an opening, whose size undergoes subdiffusion. Our solution shows that the rate is proportional to η^-ν with ν in the range of 0.5-1, and hence, the subdiffusion model can explain the experimental observations.

Fractional Viscosity Dependence of Reaction Kinetics in Glass-Forming Liquids

The diffusion of molecules in complex systems such as glasses and cell cytoplasm is slow, heterogeneous, and sometimes nonergodic. The effects of such intriguing diffusion on the kinetics of chemical and biological reactions remain elusive. In this Letter, we report that the kinetics of the polymer loop formation reaction in a Kob-Andersen (KA) glass forming liquid is influenced significantly by the dynamic heterogeneity. The diffusion coefficient D of a KA liquid deviates from the Stokes-Einstein relation at low temperatures and D shows a fractional dependence on the solvent viscosity η_{s}, i.e., D∼η_{s}^{-ξ_{D}} with ξ_{D}=0.85. The dynamic heterogeneity of a KA liquid affects the rate constant k_{rxn} of the loop formation and leads to the identical fractional dependence of k_{rxn} on η_{s} with k_{rxn}∼η_{s}^{-ξ} and ξ=ξ_{D}, contrary to reactions in dynamically homogeneous solutions where k_{rxn}∼η_{s}^{-1}.

Multiple CaMKII Binding Modes to the Actin Cytoskeleton Revealed by Single-Molecule Imaging

Localization of the Ca(2+)/calmodulin-dependent protein kinase II (CaMKII) to dendritic spine synapses is determined in part by the actin cytoskeleton. We determined binding of GFP-tagged CaMKII to tag-RFP-labeled actin cytoskeleton within live cells using total internal reflection fluorescence microscopy and single-molecule tracking. Stepwise photobleaching showed that CaMKII formed oligomeric complexes. Photoactivation experiments demonstrated that diffusion out of the evanescent field determined the track lifetimes. Latrunculin treatment triggered a coupled loss of actin stress fibers and the colocalized, long-lived CaMKII tracks. The CaMKIIα (α) isoform, which was previously thought to lack F-actin interactions, also showed binding, but this was threefold weaker than that observed for CaMKIIβ (β). The βE' splice variant bound more weakly than α, showing that binding by β depends critically on the interdomain linker. The mutations βT287D and αT286D, which mimic autophosphorylation states, also abolished F-actin binding. Autophosphorylation triggers autonomous CaMKII activity, but does not impair GluN2B binding, another important synaptic protein interaction of CaMKII. The CaMKII inhibitor tatCN21 or CaMKII mutations that inhibit GluN2B association by blocking binding of ATP (βK43R and αK42M) or Ca(2+)/calmodulin (βA303R) had no effect on the interaction with F-actin. These results provide the first rationale for the reduced synaptic spine localization of the αT286D mutant, indicating that transient F-actin binding contributes to the synaptic localization of the CaMKIIα isoform. The track lifetime distributions had a stretched exponential form consistent with a heterogeneously diffusing population. This heterogeneity suggests that CaMKII adopts different F-actin binding modes, which is most easily rationalized by multiple subunit contacts between the CaMKII dodecamer and the F-actin cytoskeleton that stabilize the initial weak (micromolar) monovalent interaction.

Fluctuating bottleneck model studies on kinetics of DNA escape from α-hemolysin nanopores

We have proposed a fluctuation bottleneck (FB) model to investigate the non-exponential kinetics of DNA escape from nanometer-scale pores. The basic idea is that the escape rate is proportional to the fluctuating cross-sectional area of DNA escape channel, the radius r of which undergoes a subdiffusion dynamics subjected to fractional Gaussian noise with power-law memory kernel. Such a FB model facilitates us to obtain the analytical result of the averaged survival probability as a function of time, which can be directly compared to experimental results. Particularly, we have applied our theory to address the escape kinetics of DNA through α-hemolysin nanopores. We find that our theoretical framework can reproduce the experimental results very well in the whole time range with quite reasonable estimation for the intrinsic parameters of the kinetics processes. We believe that FB model has caught some key features regarding the long time kinetics of DNA escape through a nanopore and it might provide a sound starting point to study much wider problems involving anomalous dynamics in confined fluctuating channels.

Stretching of single poly-ubiquitin molecules revisited: dynamic disorder in the non-exponential unfolding kinetics

A theoretical framework based on a generalized Langevin equation (GLE) with fractional Gaussian noise (fGn) and a power-law memory kernel is presented to describe the non-exponential kinetics of the unfolding of a single poly-ubiquitin molecule under a constant force [T.-L. Kuo, S. Garcia-Manyes, J. Li, I. Barel, H. Lu, B. J. Berne, M. Urbakh, J. Klafter, and J. M. Fernández, Proc. Natl. Acad. Sci. U.S.A. 107, 11336 (2010)]. Such a GLE-fGn strategy is made on the basis that the pulling coordinate variable x undergoes subdiffusion, usually resulting from conformational fluctuations, over a one-dimensional force-modified free-energy surface U(x, F). By using the Kramers' rate theory, we have obtained analytical formulae for the time-dependent rate coefficient k(t, F), the survival probability S(t, F) as well as the waiting time distribution function f(t, F) as functions of time t and force F. We find that our results can fit the experimental data of f(t, F) perfectly in the whole time range with a power-law exponent γ = 1/2, the characteristic of typical anomalous subdiffusion. In addition, the fitting of the survival probabilities for different forces facilitates us to reach rather reasonable estimations for intrinsic properties of the system, such as the free-energy barrier and the distance between the native conformation and the transition state conformation along the reaction coordinate, which are in good agreements with molecular dynamics simulations in the literatures. Although static disorder has been implicated in the original work of Kuo et al., our work suggests a sound and plausible alternative interpretation for the non-exponential kinetics in the stretching of poly-ubiquitin molecules, associated with dynamic disorder.

Kinetics of molecular transitions with dynamic disorder in single-molecule pulling experiments

Macromolecular transitions are subject to large fluctuations of rate constant, termed as dynamic disorder. The individual or intrinsic transition rates and activation free energies can be extracted from single-molecule pulling experiments. Here we present a theoretical framework based on a generalized Langevin equation with fractional Gaussian noise and power-law memory kernel to study the kinetics of macromolecular transitions to address the effects of dynamic disorder on barrier-crossing kinetics under external pulling force. By using the Kramers' rate theory, we have calculated the fluctuating rate constant of molecular transition, as well as the experimentally accessible quantities such as the force-dependent mean lifetime, the rupture force distribution, and the speed-dependent mean rupture force. Particular attention is paid to the discrepancies between the kinetics with and without dynamic disorder. We demonstrate that these discrepancies show strong and nontrivial dependence on the external force or the pulling speed, as well as the barrier height of the potential of mean force. Our results suggest that dynamic disorder is an important factor that should be taken into account properly in accurate interpretations of single-molecule pulling experiments.

Subdiffusion in hair bundle dynamics: the role of protein conformational fluctuations

The detection of sound signals in vertebrates involves a complex network of different mechano-sensory elements in the inner ear. An especially important element in this network is the hair bundle, an antenna-like array of stereocilia containing gated ion channels that operate under the control of one or more adaptation motors. Deflections of the hair bundle by sound vibrations or thermal fluctuations transiently open the ion channels, allowing the flow of ions through them, and producing an electrical signal in the process, eventually causing the sensation of hearing. Recent high frequency (0.1-10 kHz) measurements by Kozlov et al. [Proc. Natl. Acad. Sci. U.S.A. 109, 2896 (2012)] of the power spectrum and the mean square displacement of the thermal fluctuations of the hair bundle suggest that in this regime the dynamics of the hair bundle are subdiffusive. This finding has been explained in terms of the simple Brownian motion of a filament connecting neighboring stereocilia (the tip link), which is modeled as a viscoelastic spring. In the present paper, the diffusive anomalies of the hair bundle are ascribed to tip link fluctuations that evolve by fractional Brownian motion, which originates in fractional Gaussian noise and is characterized by a power law memory. The predictions of this model for the power spectrum of the hair bundle and its mean square displacement are consistent with the experimental data and the known properties of the tip link.

Two-point approximation to the Kramers problem with coloured noise

We present a method, founded on previous renewal approaches as the classical Wilemski-Fixman approximation, to describe the escape dynamics from a potential well of a particle subject to non-Markovian fluctuations. In particular, we show how to provide an approximated expression for the distribution of escape times if the system is governed by a generalized Langevin equation (GLE). While we show that the method could apply to any friction kernel in the GLE, we focus here on the case of power-law kernels, for which extensive literature has appeared in the last years. The method presented (termed as two-point approximation) is able to fit the distribution of escape times adequately for low potential barriers, even if conditions are far from Markovian. In addition, it confirms that non-exponential decays arise when a power-law friction kernel is considered (in agreement with related works published recently), which questions the existence of a characteristic reaction rate in such situations.

Enzyme dynamics point to stepwise conformational selection in catalysis

Recent data increasingly reveal that conformational dynamics are indispensable to enzyme function throughout the catalytic cycle, in substrate recruiting, chemical transformation, and product release. Conformational transitions may involve conformational selection and induced fit, which can be viewed as a special case in the catalytic network. NMR, X-ray crystallography, single-molecule FRET, and simulations clearly demonstrate that the free enzyme dynamics already encompass all the conformations necessary for substrate binding, preorganization, transition-state stabilization, and product release. Conformational selection and substate population shift at each step of the catalytic turnover can accommodate enzyme specificity and efficiency. Within such a framework, entropy can have a larger role in conformational dynamics than in direct energy transfer in dynamically promoted catalysis.

Rate theories for biologists

Some of the rate theories that are most useful for modeling biological processes are reviewed. By delving into some of the details and subtleties in the development of the theories, the review will hopefully help the reader gain a more than superficial perspective. Examples are presented to illustrate how rate theories can be used to generate insight at the microscopic level into biomolecular behaviors. An attempt is made to clear up a number of misconceptions in the literature regarding popular rate theories, including the appearance of Planck's constant in the transition-state theory and the Smoluchowski result as an upper limit for protein-protein and protein-DNA association rate constants. Future work in combining the implementation of rate theories through computer simulations with experimental probes of rate processes, and in modeling effects of intracellular environments so that theories can be used for generating rate constants for systems biology studies is particularly exciting.

A stationary-wave model of enzyme catalysis

An expression for the external force driving a system of two coupled oscillators in the condensed phase was derived in the frame of the Debye theory of solids. The time dependence and amplitude of the force is determined by the size of the cell embedding the coupled oscillators and its Debye temperature (theta(D)). The dynamics of the driven system of oscillators were followed in the two regimes of (a) low theta(D) and cell diameter, as a model of liquid water, and (b) large theta(D) and cell diameter, as a model of the core of a protein. The response in potential energy of the reference oscillator was computed for all possible values of the internal parameters of the system under investigation. For protein cores, the region in the parameter space of high maximum potential energy of the reference oscillator is considerably extended with respect to the corresponding simulation for water.

A lower bound to the survival probability and an approximate first passage time distribution for Markovian and non-Markovian dynamics in phase space

We derive a very general expression of the survival probability and the first passage time distribution for a particle executing Brownian motion in full phase space with an absorbing boundary condition at a point in the position space, which is valid irrespective of the statistical nature of the dynamics. The expression, together with the Jensen's inequality, naturally leads to a lower bound to the actual survival probability and an approximate first passage time distribution. These are expressed in terms of the position-position, velocity-velocity, and position-velocity variances. Knowledge of these variances enables one to compute a lower bound to the survival probability and consequently the first passage distribution function. As examples, we compute these for a Gaussian Markovian process and, in the case of non-Markovian process, with an exponentially decaying friction kernel and also with a power law friction kernel. Our analysis shows that the survival probability decays exponentially at the long time irrespective of the nature of the dynamics with an exponent equal to the transition state rate constant.

Single-molecule nonequilibrium periodic Mg2+-concentration jump experiments reveal details of the early folding pathways of a large RNA

The evolution of RNA conformation with Mg(2+) concentration ([Mg(2+)]) is typically determined from equilibrium titration measurements or nonequilibrium single [Mg(2+)]-jump measurements. We study the folding of single RNA molecules in response to a series of periodic [Mg(2+)] jumps. The 260-residue catalytic domain of RNase P RNA from Bacillus stearothermophilus is immobilized in a microfluidic flow chamber, and the RNA conformational changes are probed by fluorescence resonance energy transfer (FRET). The kinetics of population redistribution after a [Mg(2+)] jump and the observed connectivity of FRET states reveal details of the folding pathway that complement and transcend information from equilibrium or single-jump measurements. FRET trajectories for jumps from [Mg(2+)] = 0.01 to 0.1 mM exhibit two-state behavior whereas jumps from 0.01 mM to 0.4 mM exhibit two-state unfolding but multistate folding behavior. RNA molecules in the low and high FRET states before the [Mg(2+)] increase are observed to undergo dynamics in two distinct regions of the free energy landscape separated by a high barrier. We describe the RNA structural changes involved in crossing this barrier as a "hidden" degree of freedom because the changes do not alter the detected FRET value but do alter the observed dynamics. The associated memory prevents the populations from achieving their equilibrium values at the end of the 5- to 10-sec [Mg(2+)] interval, thereby creating a nonequilibrium steady-state condition. The capability of interrogating nonequilibrium steady-state RNA conformations and the adjustable period of [Mg(2+)]-jump cycles makes it possible to probe regions of the free energy landscape that are infrequently sampled in equilibrium or single-jump measurements.

Two-dimensional reaction free energy surfaces of catalytic reaction: effects of protein conformational dynamics on enzyme catalysis

We introduce a two-dimensional (2D) multisurface reaction free energy description of the catalytic cycle that explicitly connects the recently observed multi-time-scale conformational dynamics as well as dispersed enzymatic kinetics to the classical Michaelis-Menten equation. A slow conformational motion on a collective enzyme coordinate Q facilitates the catalytic reaction along the intrinsic reaction coordinate X, providing a dynamic realization of Pauling's well-known idea of transition-state stabilization. The catalytic cycle is modeled as transitions between multiple displaced harmonic wells in the XQ space representing different states of the cycle, which is constructed according to the free energy driving force of the cycle. Subsequent to substrate association with the enzyme, the enzyme-substrate complex under strain exhibits a nonequilibrium relaxation toward a new conformation that lowers the activation energy of the reaction, as first proposed by Haldane. The chemical reaction in X is thus enslaved to the down hill slow motion on the Q surface. One consequence of the present theory is that, in spite of the existence of dispersive kinetics, the Michaelis-Menten expression of the catalysis rate remains valid under certain conditions, as observed in recent single-molecule experiments. This dynamic theory builds the relationship between the protein conformational dynamics and the enzymatic reaction kinetics and offers a unified description of enzyme fluctuation-assisted catalysis.

Anomalous escape governed by thermal 1/f noise

We present an analytic study for subdiffusive escape of overdamped particles out of a cusp-shaped parabolic potential well which are driven by thermal, fractional Gaussian noise with a 1/omega 1-alpha power spectrum. This long-standing challenge becomes mathematically tractable by use of a generalized Langevin dynamics via its corresponding non-Markovian, time-convolutionless master equation: We find that the escape is governed asymptotically by a power-law whose exponent depends exponentially on the ratio of barrier height and temperature. This result is in distinct contrast to a description with a corresponding subdiffusive fractional Fokker-Planck approach, thus providing experimentalists an amenable testbed to differentiate between the two escape scenarios.

Exact analytical evaluation of time dependent transmission coefficient from the method of reactive flux for an inverted parabolic barrier

The paper demonstrates an elegant way of combining the normal mode analysis and the method of reactive flux to evaluate the time dependent transmission coefficient for a classical particle coupled to a set of harmonic oscillators, surmounting a one dimensional barrier. The author's analysis reproduces the results of Kohen and Tannor [J. Chem. Phys. 103, 6013 (1995)] and Bao [J. Chem. Phys. 124, 114103 (2006)]. Moreover the use of normal mode analysis has a better physical meaning.

Protein fluctuations and breakdown of time-scale separation in rate theories

A long-time fluctuation correlation function with a power-law form has been observed in recent single-molecule experiments by the Xie group. By analyzing the dynamics of an elastic network model (ENM) under white noise, we show that the observed long-time memory kernel can be explained by the discrepancy between the experimentally measured coordinate (or the coordinate directly coupled to protein function) and the minimum energy path of the system. Consequently, the dynamics of the measured collective coordinate has contributions from degrees of freedoms with a broad distribution of time scales. Our study also implies that the widely used ENM Hamiltonian should be viewed as a coarse-grained model of a protein over a rugged energy landscape. Large effective drag coefficients are needed to describe protein dynamics with the ENM's.

Approximate first passage time distribution for barrier crossing in a double well under fractional Gaussian noise

The distribution of waiting times, f(t), between successive turnovers in the catalytic action of single molecules of the enzyme beta-galactosidase has recently been determined in closed form by Chaudhury and Cherayil [J. Chem. Phys. 125, 024904 (2006)] using a one-dimensional generalized Langevin equation (GLE) formalism in combination with Kramers' flux-over-population approach to barrier crossing dynamics. The present paper provides an alternative derivation of f(t) that eschews this approach, which is strictly applicable only under conditions of local equilibrium. In this alternative derivation, a double well potential is incorporated into the GLE, along with a colored noise term representing protein conformational fluctuations, and the resulting equation transformed approximately to a Smoluchowski-type equation. f(t) is identified with the first passage time distribution for a particle to reach the barrier top starting from an equilibrium distribution of initial points, and is determined from the solution of the above equation using local boundary conditions. The use of such boundary conditions is necessitated by the absence of definite information about the precise nature of the boundary conditions applicable to stochastic processes governed by non-Markovian dynamics. f(t) calculated in this way is found to have the same analytic structure as the distribution calculated by the flux-over-population method.

Complex chemical kinetics in single enzyme molecules: Kramers’s model with fractional Gaussian noise

A model of barrier crossing dynamics governed by fractional Gaussian noise and the generalized Langevin equation is used to study the reaction kinetics of single enzymes subject to conformational fluctuations. The direct application of Kramers's flux-over-population method to this model yields analytic expressions for the time-dependent transmission coefficient and the distribution of waiting times for barrier crossing. These expressions are found to reproduce the observed trends in recent simulations and experiments.

Diffusive model of protein folding dynamics with Kramers turnover in rate

We study the folding kinetics of a three-helix bundle protein using a coarse polymer model. The folding dynamics can be accurately represented by one-dimensional diffusion along a reaction coordinate selected to capture the transition state. By varying the solvent friction, we show that position-dependent diffusion coefficients are determined by microscopic transitions on a rough energy landscape. A maximum in the folding rate at intermediate friction is explained by "Kramers turnover" in these microscopic dynamics that modulates the rate via the diffusion coefficient; overall folding remains diffusive even close to zero friction. For water friction, we find that the "attempt frequency" (or "speed limit") in a Kramers model of folding is about 2 micros-1, with an activation barrier of about 2kBT, and a folding transition path duration of approximately equal to 100 ns, 2 orders of magnitude less than the folding time of approximately equal to 10 micros.

Distance versus energy fluctuations and electron transfer in single protein molecules

Stochastic nature due to distance and energy fluctuations of single protein molecules involved in electron-transfer (ET) reactions is studied. Distance fluctuations have been assumed previously for causing the slow fluctuations in the ET rates between a donor-acceptor pair constrained to a native protein. Although the observed t(-1/2) power law can be derived using Langevin dynamics with a simple chain model, some discrepancies exist. The friction coefficient and the Rouse segment time constant deduced from experimental data are several orders of magnitude too large, even though the extracted force constant is reasonable. Therefore, questions are raised about the distance-fluctuation mechanism and the activationless ET hypothesis. As an alternative mechanism, we considered fluctuations in activation energy and analyzed the data from two different single protein experiments to determine spectral distribution of energy fluctuations.