Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule

Langevin formulation for single-file diffusion

We introduce a stochastic equation for the microscopic motion of a tagged particle in the single-file model. This equation provides a compact representation of several of the system's properties such as fluctuation-dissipation and linear-response relations, achieved by means of a diffusion noise approach. Most importantly, the proposed Langevin equation reproduces quantitatively the three temporal regimes and the corresponding time scales: ballistic, diffusive, and subdiffusive.

Correlation between square of electron tunneling matrix element and donor-acceptor distance in fluctuating protein media

Correlation between fluctuations of the square of electron tunneling matrix element T_DA² and the donor-acceptor distance R_DA in the electron transfer (ET) reaction from bacteriopheophytin anion to the primary quinone of the reaction center in the photosynthetic bacteria Rhodobacter sphaeroides is investigated by a combined study of molecular dynamics simulations of the protein conformation fluctuation and quantum chemical calculations. We adopted two kinds of R_DA ; edge-to-edge distance R_EE and center-to-center distance R_CC . The value of T_DA² distributed over more than 5 orders of magnitude and the fluctuation of the value of R_DA distributed over more than 1.8 Å for the 10⁶ instantaneous conformations of 1 ns simulation. We made analysis of the time-averaged correlation step by step as follows. We divide the 10⁶ simulation data into 1000/t parts of small data set to obtain the averaged data points of <T_DA²> _t and <R_EE > _t or <R_CC > _t . Plotting the 1000/t sets of log₁₀ <T_DA²> _t as a function of <R_EE > _t or <R_CC > _t , we made a principal coordinate analysis for these distributions. The slopes <β_E > _t and <β_C > _t of the primary axis are very large at small value of t and they are decreased considerably as t becomes large. The ellipticity for the distribution of <T_DA²> _tvs <R_EE > _t which can be a measure for the degree of correlation became very small when t is large, while it does not hold for the distribution of <T_DA²> _tvs <R_CC > _t . These results indicate that only the correlation between <T_DA²> _t and <R_EE > _t for large t satisfies the well-known linear relation ("Dutton law"), although the slope is larger than the original value 1.4 Å^-1. Based on the present result, we examined the analysis of the dynamic disorder by means of the single-molecule spectroscopy by Xie and co-workers with use of the "Dutton law".

Fractional Langevin equation: overdamped, underdamped, and critical behaviors

The dynamical phase diagram of the fractional Langevin equation is investigated for a harmonically bound particle. It is shown that critical exponents mark dynamical transitions in the behavior of the system. Four different critical exponents are found. (i) alpha_{c}=0.402+/-0.002 marks a transition to a nonmonotonic underdamped phase, (ii) alpha_{R}=0.441... marks a transition to a resonance phase when an external oscillating field drives the system, and (iii) alpha_{chi_{1}}=0.527... and (iv) alpha_{chi_{2}}=0.707... mark transitions to a double-peak phase of the "loss" when such an oscillating field present. As a physical explanation we present a cage effect, where the medium induces an elastic type of friction. Phase diagrams describing over and underdamped regimes, with or without resonances, show behaviors different from normal.

Protein dynamics from a NMR perspective: networks of coupled rotators and fractional Brownian dynamics

Nuclear magnetic resonance (NMR) has proven to be the most valuable tool for investigating internal dynamics of proteins. In this perspective, the interpretation of NMR relaxation data eventually relies on a model of the motions. In this article, we propose to compare two radically different approaches that aim at describing internal dynamics in proteins. It is shown that the correlation functions predicted by a network of coupled rotators can be interpreted in terms of a heuristic approach based on fractional Brownian dynamics for each of the vectors in the network. Our results are interpreted in terms of the probability distributions of relaxation modes in both processes, the median of which turns out to be the relevant quantity for the comparison of both models.

Subdiffusion in peptides originates from the fractal-like structure of configuration space

Molecular dynamics simulation of oligopeptide chains reveals configurational subdiffusion at equilibrium extending from 10(-12) to 10(-8) s. Trap models, involving a random walk with a distribution of waiting times, cannot account for the subdiffusion, which is found rather to arise from the fractal-like structure of the accessible configuration space.

Memory effects in the asymptotic diffusive behavior of a classical oscillator described by a generalized Langevin equation

We investigate the memory effects present in the asymptotic dynamics of a classical harmonic oscillator governed by a generalized Langevin equation. Using Laplace analysis together with Tauberian theorems we derive asymptotic expressions for the mean values, variances, and velocity autocorrelation function in terms of the long-time behavior of the memory kernel and the correlation function of the random force. The internal and external noise cases are analyzed. A simple criterion to determine if the diffusion process is normal or anomalous is established.

Critical exponent of the fractional Langevin equation

We investigate the dynamical phase diagram of the fractional Langevin equation and show that critical exponents mark dynamical transitions in the behavior of the system. For a free and harmonically bound particle the critical exponent alpha(c)=0.402+/-0.002 marks a transition to a nonmonotonic underdamped phase. The critical exponent alpha(R)=0.441... marks a transition to a resonance phase, when an external oscillating field drives the system. Physically, we explain these behaviors using a cage effect, where the medium induces an elastic type of friction. Phase diagrams describing the underdamped, the overdamped and critical frequencies of the fractional oscillator, recently used to model single protein experiments, show behaviors vastly different from normal.

Analysis of the entire sequence of a single photon experiment on a flavin protein

The large amount of statistical data collected by single biomolecule experiments often demonstrates complex and non-Markovian relaxation over many time scales. Analyzing and interpreting these data is a major challenge because of the inherently statistical noise and the lack of definite theoretical descriptions or computer simulations on biologically relevant time scales. This paper reports one of the first complete sequence analyses of a single photon experiment on the flavin protein to determine an underlying physical picture for protein motions on the millisecond to second regimes. The robustness of Bayesian information analysis combined with the nonparametric maximum entropy method (MEM) incorporates all available information of the single-molecule data sequence and maximizes our ability to test the legitimacy of possible models. Our analysis of the experimental data is consistent with the stochastic Gaussian diffusion model where the slow protein motions are modeled as a collection of over-damped diffusive normal modes and reveals non-universal and distinct dynamic features that are specific for protein functions.

Anomalous lineshapes and aging effects in two-dimensional correlation spectroscopy

Multitime correlation functions provide useful probes for the ensembles of trajectories underlying the stochastic dynamics of complex systems. These can be obtained by measuring their optical response to sequences of ultrashort optical pulse. Using the continuous time random walk model for spectral diffusion, we analyze the signatures of anomalous relaxation in two-dimensional four wave mixing signals. Different models which share the same two point joint probability distribution show markedly different lineshapes and may be distinguished. Aging random walks corresponding to waiting time distributions with diverging first moment show dependence of 2D lineshapes on initial observation time, which persist for long times.

COCIS: Markov processes in single molecule fluorescence

This article examines the current status of Markov processes in single molecule fluorescence. For molecular dynamics to be described by a Markov process, the Markov process must include all states involved in the dynamics and the FPT distributions out of those states must be describable by a simple exponential law. The observation of non-exponential first-passage time distributions or other evidence of non-Markovian dynamics is common in single molecule studies and offers an opportunity to expand the Markov model to include new dynamics or states that improve understanding of the system.

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.

Subdiffusive target problem: survival probability

The asymptotic survival probability of a spherical target in the presence of a single subdiffusive trap or surrounded by a sea of subdiffusive traps in a continuous Euclidean medium is calculated. In one and two dimensions the survival probability of the target in the presence of a single trap decays to zero as a power law and as a power law with logarithmic correction, respectively. The target is thus reached with certainty, but it takes the trap an infinite time on average to do so. In dimensions higher than two a single trap may never reach the target and so the survival probability is finite and, in fact, does not depend on whether the traps move diffusively or subdiffusively. When the target is surrounded by a sea of traps, on the other hand, its survival probability decays as a stretched exponential in all dimensions (with a logarithmic correction in the exponent for d=2). A trap will therefore reach the target with certainty, and will do so in a finite time. These results may be directly related to enzyme binding kinetics on DNA in the crowded cellular environment.

Dielectric dispersion interpretation of single enzyme dynamic disorder, spectral diffusion, and radiative fluorescence lifetime

A formulation based on measurable dielectric dispersion of enzymes is developed to estimate fluctuations in electrostatic interaction energy on time scales as long as milliseconds to seconds at a local site in enzymes. Several single molecule experimental obsevations occur on this time scale, currently unreachable by real time computational trajectory simulations. We compare the experimental results on the autocorrelation function of the fluctuations of catalysis rate with the calculations using the dielectric dispersion formulation. We also discuss the autocorrelation functions of the fluorescence lifetime and of spectral diffusion. We use a previously derived relation between the observables and the electric field fluctuations and calculate the latter using dielectric dispersion data for the proteins and the Onsager regression hypothesis.

Anomalous relaxation and dielectric response

It is shown that all the known experimental (quasi)stationary dielectric response functions of glassy media can be derived from a standard generalized Langevin description of overdamped torsional dipole oscillators in trapping potentials with random orientations under some minimal assumptions. The non-Markovian theory obeys the fluctuation-dissipation theorem and the Onsager regression theorem. Moreover, it displays no aging on the time scale of the dielectric response, all in assumption of local thermal (quasi)equilibrium. Aging might come from jumping among metastable traps. It occurs on a quite different time scale which is not related to the principal dielectric response. We put the old phenomenological theory of Cole and Cole, Davidson and Cole, and others on a firm basis within a stochastic, thermodynamically consistent approach.

Dynamic disorder in single-molecule Michaelis-Menten kinetics: The reaction-diffusion formalism in the Wilemski-Fixman approximation

Single-molecule equations for the Michaelis-Menten [Biochem. Z. 49, 333 (1913)] mechanism of enzyme action are analyzed within the Wilemski-Fixman [J. Chem. Phys. 58, 4009 (1973); 60, 866 (1974)] approximation after the effects of dynamic disorder--modeled by the anomalous diffusion of a particle in a harmonic well--are incorporated into the catalytic step of the reaction. The solution of the Michaelis-Menten equations is used to calculate the distribution of waiting times between successive catalytic turnovers in the enzyme beta-galactosidase. The calculated distribution is found to agree qualitatively with experimental results on this enzyme obtained at four different substrate concentrations. The calculations are also consistent with measurements of correlations in the fluctuations of the fluorescent light emitted during the course of catalysis, and with measurements of the concentration dependence of the randomness parameter.

When does the Michaelis-Menten equation hold for fluctuating enzymes?

Enzymes are dynamic entities: both their conformation and catalytic activity fluctuate over time. When such fluctuations are relatively fast, it is not surprising that the classical Michaelis-Menten (MM) relationship between the steady-state enzymatic velocity and the substrate concentration still holds. However, recent single-molecule experiments have shown that this is the case even for an enzyme whose catalytic activity fluctuates on the 10(-4)-10 s range. The purpose of this paper is to examine various scenarios in which slowly fluctuating enzymes would still obey the MM relationship. Specifically, we consider (1) the quasi-static condition (e.g., the conformational fluctuation of the enzyme-substrate complex is much slower than binding, catalysis, and the conformational fluctuations of the free enzyme), (2) the quasi-equilibrium condition (when the substrate dissociation is much faster than catalysis, irrespective of the time scales or amplitudes of conformational fluctuations), and (3) the conformational-equilibrium condition (when the dissociation and catalytic rates depend on the conformational coordinate in the same way). For each of these scenarios, the physical meaning of the apparent Michaelis constant and catalytic rate constant is provided. Finally, as an example, the theoretical analysis of a recent single-molecule enzyme assay is considered in light of the perspectives presented in this paper.

Information theoretical approach to single-molecule experimental design and interpretation

We use Shannon's definition of information to develop a theory to predict a photon-counting-based single-molecule experiment's ability to measure the desired property. We treat three phenomena that are commonly measured on single molecules: spectral fluctuations of a solvatochromic dye; assignment of the azimuthal dipole angle; determination of a distance by fluorescence resonant energy transfer using Förster's theory. We consider the effect of background and other "imperfections" on the measurement through the decrease in information.

Polymer translocation through a nanopore: a showcase of anomalous diffusion

The translocation dynamics of a polymer chain through a nanopore in the absence of an external driving force is analyzed by means of scaling arguments, fractional calculus, and computer simulations. The problem at hand is mapped on a one-dimensional anomalous diffusion process in terms of the reaction coordinate s (i.e., the translocated number of segments at time t ) and shown to be governed by a universal exponent alpha=2(2nu+2-gamma(1), where nu is the Flory exponent and gamma(1) is the surface exponent. Remarkably, it turns out that the value of alpha is nearly the same in two and three dimensions. The process is described by a fractional diffusion equation which is solved exactly in the interval 0<s<N with appropriate boundary and initial conditions. The solution gives the probability distribution of translocation times as well as the variation with time of the statistical moments <s(t) and <s2(t)-<s(t)>2, which provide a full description of the diffusion process. The comparison of the analytic results with data derived from extensive Monte Carlo simulations reveals very good agreement and proves that the diffusion dynamics of unbiased translocation through a nanopore is anomalous in its nature.

Theory of the statistics of kinetic transitions with application to single-molecule enzyme catalysis

Single-molecule spectroscopy can monitor transitions between two microscopic states when these transitions are associated with the emission of photons. A general formalism is developed for obtaining the statistics of such transitions from a microscopic model when the dynamics is described by master or rate equations or their continuum analog, multidimensional reaction-diffusion equations. The focus is on the distribution of the number of transitions during a fixed observation time, the distribution of times between transitions, and the corresponding correlation functions. It is shown how these quantities are related to each other and how they can be explicitly calculated in a straightforward way for both immobile and diffusing molecules. Our formalism reduces to renewal theory when the monitored transitions either go to or originate from a single state. The influence of dynamics slow compared with the time between monitored transitions is treated in a simple way, and the probability distributions are expressed in terms of Mandel-type formulas. The formalism is illustrated by a detailed analysis of the statistics of catalytic turnovers of enzymes. When the rates of conformational changes are slower than the catalytic rates which are in turn slower than the binding relaxation rate, (1) the mean number of turnovers is shown to have the classical Michaelis-Menten form, (2) the correlation function of the number of turnovers is a direct measure of the time scale of catalytic rate fluctuations, and (3) the distribution of the time between consecutive turnovers is determined by the steady-state distribution.

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.

Anomalous diffusion induced by a Mittag-Leffler correlated noise

We introduce a Mittag-Leffler correlated random force leading to anomalous diffusion. Starting from a generalized Langevin equation, and using Laplace analysis we derive exact expressions for the mean values, variances and diffusion coefficient for a free particle in terms of generalized Mittag-Leffler functions and its derivatives. The asymptotic behavior of these quantities are obtained, from which the anomalous diffusion behavior of the particle is displayed.

Probing anomalous relaxation by coherent multidimensional optical spectroscopy

We propose to study the origin of algebraic decay of two-point correlation functions observed in glasses, proteins, and quantum dots by their nonlinear response to sequences of ultrafast laser pulses. Power-law spectral singularities and temporal relaxation in two-dimensional correlation spectroscopy signals are predicted for a continuous time random walk model of stochastic spectral jumps in a two-level system with a power-law distribution of waiting times psi(t) approximately t;{-alpha-1}. Spectroscopic signatures of stationary ensembles for 1<alpha<2 and aging effects in nonstationary ensembles with 0<alpha<1 are identified.

Model of Fluorescence Intermittency in Single Enzymes

The intermittent emission of fluorescent light from single enzymes, quantum dots, and other nanoscale systems is often characterized by statistical correlations in the emitted signal. A one-dimensional model of such correlations in enzymes, based on a model of protein conformational fluctuations developed by Kou and Xie (Phys. Rev. Lett. 2004, 93, 180603), is formulated in the present paper in terms of the dynamics of a particle moving stochastically between "on" and "off" states under the action of fractional Gaussian noise. The model yields predictions for the short and long time behavior of the following quantities: the time correlation function, C(t), of the fluctuations of the signal intensity, the distribution, f(t), of time intervals between intensity fluctuations, and the Mandel parameter, Q(t), describing the extent of bunching or anti-bunching in the signal. At short times, C(t) and f(t) are found to decay exponentially, while, at long times, they are found to decay as power laws, the exponents being functions solely of the nature of the temporal correlations in the noise. The results are in good qualitative agreement with results from single-molecule experiments on fluorescence intermittency in the enzyme cholesterol oxidase carried out by Xie and co-workers (Science 1998, 282, 1877). The Mandel parameter, Q(t), for this model is positive at short and long times, indicating super-Poisson statistics in these limits, consistent with bunching of the fluorescent signal.

Central limit theorem for anomalous scaling due to correlations

We derive a central limit theorem for the probability distribution of the sum of many critically correlated random variables. The theorem characterizes a variety of different processes sharing the same asymptotic form of anomalous scaling and is based on a correspondence with the Lévy-Gnedenko uncorrelated case. In particular, correlated anomalous diffusion is mapped onto Lévy diffusion. Under suitable assumptions, the nonstandard multiplicative structure used for constructing the characteristic function of the total sum allows us to determine correlations of partial sums exclusively on the basis of the global anomalous scaling.

Escape driven by alpha-stable white noises

We explore the archetype problem of an escape dynamics occurring in a symmetric double well potential when the Brownian particle is driven by white Lévy noise in a dynamical regime where inertial effects can safely be neglected. The behavior of escaping trajectories from one well to another is investigated by pointing to the special character that underpins the noise-induced discontinuity which is caused by the generalized Brownian paths that jump beyond the barrier location without actually hitting it. This fact implies that the boundary conditions for the mean first passage time (MFPT) are no longer determined by the well-known local boundary conditions that characterize the case with normal diffusion. By numerically implementing properly the set up boundary conditions, we investigate the survival probability and the average escape time as a function of the corresponding Lévy white noise parameters. Depending on the value of the skewness beta of the Lévy noise, the escape can either become enhanced or suppressed: a negative asymmetry parameter beta typically yields a decrease for the escape rate while the rate itself depicts a non-monotonic behavior as a function of the stability index alpha that characterizes the jump length distribution of Lévy noise, exhibiting a marked discontinuity at alpha=1. We find that the typical factor of 2 that characterizes for normal diffusion the ratio between the MFPT for well-bottom-to-well-bottom and well-bottom-to-barrier-top no longer holds true. For sufficiently high barriers the survival probabilities assume an exponential behavior versus time. Distinct non-exponential deviations occur, however, for low barrier heights.

Structural relaxation in complex liquids: Non-Markovian dynamics in a bistable potential

The time correlation function C(t) identical with <x(0)x(t)> of the distance fluctuations of a particle moving in a bistable potential under the action of fractional Gaussian noise (fGn) is calculated from a Smoluchowski-type equation derived from a generalized Langevin equation (GLE). The time derivative of this function, dC(t)dt, is compared with data from optical Kerr effect measurements of liquid crystal dynamics in the vicinity of the isotropic-to-nematic transition, which are related to the time derivative of an orientational correlation function. A number of characteristic features of the experimental decay curves, including short and intermediate time power law behavior and long time exponential relaxation, are qualitatively reproduced by the analytical calculations, even though the latter do not explicitly treat orientational degrees of freedom. The GLE formalism with fGn was, in fact, originally proposed as a model of protein conformational fluctuations, so the present results suggest that it may also serve more generally as a model of structural relaxation in complex condensed phase media.

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.

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.

Generalized Langevin equation with fractional derivative and long-time correlation function

We investigate the motion of a particle governed by a generalized Langevin equation with fractional derivative, nonlocal dissipative force, and long-time correlation function. We derive general expressions for the variances with a linear external force. We also analyze their asymptotic behaviors for the power-law correlation function without external force.

Hydration dependence of the mass fractal dimension and anomalous diffusion of vibrational energy in proteins

Vibrational dynamics of proteins and energy flow depend on protein geometry as well as interactions of a protein molecule with the surrounding solvent. We compute the mass fractal dimension D of proteins ranging from 100 to over 10,000 amino acids comparing values for the bare protein with those computed when buried and hydration waters are included in the calculation. Including water in the calculation increases D by about 0.3 to 2.87 on average above D computed for the dehydrated protein. The mass fractal dimension of proteins that are partially unfolded by molecular dynamics (MD) simulation is also computed and found to vary little when the radius of gyration changes within about 10% of that for the Protein Data Bank structure. MD simulations of vibrational energy diffusion in proteins reveal that the exponent characterizing anomalous diffusion of vibrational energy does not change much with hydration, which is seen to be due to an increase in the spectral dimension with hydration by a factor similar to the increase in D.

Coherence and correlations in multitime quantum measurements of stochastic quantum trajectories

Quantum effects on multitime distributions and correlation functions of single objects, stemming from both the dynamics and repeated measurements, are calculated for a driven harmonic system using a superoperator generating functional formalism. Marked differences between multipoint observables associated with classical and quantum measurements are identified. The effects of quantum collapse and measurement resolution are discussed.

Current and universal scaling in anomalous transport

Anomalous transport in tilted periodic potentials is investigated within the framework of the fractional Fokker-Planck dynamics and the underlying continuous time random walk. The analytical solution for the stationary, anomalous current is obtained in closed form. We derive a universal scaling law for anomalous diffusion occurring in tilted periodic potentials. This scaling relation is corroborated with precise numerical studies covering wide parameter regimes and different shapes for the periodic potential, being either symmetric or ratchetlike.

Chain dynamics and power-law distance fluctuations of single-molecule systems

Chain-dynamics-induced distance fluctuations between any two points in a finite chain with or without cross links are investigated. This model leads to three regimes of temporal behavior for distance autocorrelation: (i) initial flat time dependence, (ii) t(-alpha) power law, and (iii) long-time exponential decay. For an ideal Rouse chain with frequency-independent friction, alpha = 1/2. The span of the characteristic power-law behavior of a long chain could be reduced significantly with the presence of cross links.

Anomalous continuous-time random-walk spectral diffusion in coherent third-order optical response

Recursive relations are derived for nonlinear optical response functions of a two-level chromophore with stochastic frequency fluctuations described by a continuous-time random walk. Stationary ensembles are constructed and signatures of anomalous relaxation in the photon echo signal are discussed for a two state jump modulation with a power law waiting time density function psi(t) approximately t(-alpha-1). Stretched exponential decay of the photon echo signal is predicted for 0<alpha<1 and power law asymptotics for 1<alpha<2.

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

Kramers' model for the rate of chemical reaction is generalized to explain the phenomena of dispersed kinetics and dynamic disorder in biochemical reactions, by incorporating the newly observed power-law friction kernel into the generalized Langevin equation for a one-dimensional reaction ordinate. This new model accounts for time scale overlap between conformational and chemical dynamics, and quantitatively describes the multi-exponential kinetics and memory effects of fluctuating rate constants, which have been revealed by recent single-molecule experiments.

Fluctuating enzymes: lessons from single-molecule studies

Recent single-molecule enzymology measurements with improved statistics have demonstrated that a single enzyme molecule exhibits large temporal fluctuations of the turnover rate constant at a broad range of time scales (from 1 ms to 100 s). The rate constant fluctuations, termed as dynamic disorder, are associated with fluctuations of the protein conformations observed on the same time scales. We discuss the unique information extractable from these experiments and the reconciliation of these observations with ensemble-averaged Michaelis-Menten equation. A theoretical model based on the generalized Langevin equation (GLE) treatment of Kramers' barrier crossing problem for chemical reactions accounts naturally for the observation of dynamic disorder and highly dispersed kinetics.

Multiple time scale dynamics of distance fluctuations in a semiflexible polymer: A one-dimensional generalized Langevin equation treatment

Time-dependent fluctuations in the distance x(t) between two segments along a polymer are one measure of its overall conformational dynamics. The dynamics of x(t), modeled as the coordinate of a particle moving in a one-dimensional potential well in thermal contact with a reservoir, is treated with a generalized Langevin equation whose memory kernel K(t) can be calculated from the time-correlation function of distance fluctuations C(t) identical with x(0)x(t). We compute C(t) for a semiflexible continuum model of the polymer and use it to determine K(t) via the GLE. The calculations demonstrate that C(t) is well approximated by a Mittag-Leffler function and K(t) by a power-law decay on time scales of several decades. Both functions depend on a number of parameters characterizing the polymer, including chain length, degree of stiffness, and the number of intervening residues between the two segments. The calculations are compared with the recent observation of a nonexponential C(t) and a power law K(t) in the conformational dynamics within single molecule proteins [Min et al., Phys. Rev. Lett. 94, 198302 (2005)].

Simple measure of memory for dynamical processes described by a generalized Langevin equation

Memory effects are a key feature in the description of the dynamical systems governed by the generalized Langevin equation, which presents an exact reformulation of the equation of motion. A simple measure for the estimation of memory effects is introduced within the framework of this description. Numerical calculations of the suggested measure and the analysis of memory effects are also applied for various model physical systems as well as for the phenomena of "long time tails" and anomalous diffusion.