Geometry-controlled kinetics.

It has long been appreciated that the transport properties of molecules can control reaction kinetics. This effect can be characterized by the time it takes a diffusing molecule to reach a target-the first-passage time (FPT). Determining the FPT distribution in realistic confined geometries has until now, however, seemed intractable. Here, we calculate this FPT distribution analytically and show that transport processes as varied as regular diffusion, anomalous diffusion, and diffusion in disordered media and fractals, fall into the same universality classes. Beyond the theoretical aspect, this result changes our views on standard reaction kinetics and we introduce the concept of 'geometry-controlled kinetics'. More precisely, we argue that geometry-and in particular the initial distance between reactants in 'compact' systems-can become a key parameter. These findings could help explain the crucial role that the spatial organization of genes has in transcription kinetics, and more generally the impact of geometry on diffusion-limited reactions.

Survival of an evasive prey.

We study the survival of a prey that is hunted by N predators. The predators perform independent random walks on a square lattice with V sites and start a direct chase whenever the prey appears within their sighting range. The prey is caught when a predator jumps to the site occupied by the prey. We analyze the efficacy of a lazy, minimal-effort evasion strategy according to which the prey tries to avoid encounters with the predators by making a hop only when any of the predators appears within its sighting range; otherwise the prey stays still. We show that if the sighting range of such a lazy prey is equal to 1 lattice spacing, at least 3 predators are needed in order to catch the prey on a square lattice. In this situation, we establish a simple asymptotic relation ln P(ev)(t) approximately (N/V)(2)ln P(imm)(t) between the survival probabilities of an evasive and an immobile prey. Hence, when the density rho = N/V of the predators is low, rho << 1, the lazy evasion strategy leads to the spectacular increase of the survival probability. We also argue that a short-sighting prey (its sighting range is smaller than the sighting range of the predators) undergoes an effective superdiffusive motion, as a result of its encounters with the predators, whereas a far-sighting prey performs a diffusive-type motion.

Survival probability of a particle in a sea of mobile traps: a tale of tails.

We study the long-time tails of the survival probability P(t) of an A particle diffusing in d-dimensional media in the presence of a concentration rho of traps B that move subdiffusively, such that the mean square displacement of each trap grows as tgamma with 0 < or = gamma < or =1. Starting from a continuous time random walk description of the motion of the particle and of the traps, we derive lower and upper bounds for P(t) and show that for gamma < or =2/(d+2) these bounds coincide asymptotically, thus determining asymptotically exact results. The asymptotic decay law in this regime is exactly that obtained for immobile traps. This means that for sufficiently subdiffusive traps, the moving A particle sees the traps as essentially immobile, and Lifshitz or trapping tails remain unchanged. For gamma >2/(d+2) and d< or =2 the upper and lower bounds again coincide, leading to a decay law equal to that of a stationary particle. Thus, in this regime the moving traps see the particle as essentially immobile. For d>2 , however, the upper and lower bounds in this gamma regime no longer coincide, and the decay law for the survival probability of the A particle remains ambiguous.

Single-molecule pulling experiments: when the stiffness of the pulling device matters.

Using Langevin modeling, we investigate the role of the experimental setup on the unbinding forces measured in single-molecule pulling experiments. We demonstrate that the stiffness of the pulling device, K(eff), may influence the unbinding forces through its effect on the barrier heights for both unbinding and rebinding processes. Under realistic conditions the effect of K(eff) on the rebinding barrier is shown to play the most important role. This results in a significant increase of the mean unbinding force with the stiffness for a given loading rate. Thus, in contrast to the phenomenological Bell model, we find that the loading rate (the multiplicative value K(eff)V, V being the pulling velocity) is not the only control parameter that determines the mean unbinding force. If interested in intrinsic properties of a molecular system, we recommend probing the system in the parameter range corresponding to a weak spring and relatively high loading rates where rebinding is negligible.

Number of distinct sites visited by a subdiffusive random walker.

The asymptotic mean number of distinct sites visited by a subdiffusive continuous-time random walker in two dimensions seems not to have been explicitly calculated anywhere in the literature. This number has been calculated for other dimensions for only one specific asymptotic behavior of the waiting time distribution between steps. We present an explicit derivation for two cases in all integer dimensions so as to formally complete a tableau of results. In this tableau we include the dominant as well as subdominant contributions in all integer dimensions. Other quantities that can be calculated from the mean number of distinct sites visited are also discussed.

Probing microscopic origins of confined subdiffusion by first-passage observables.

Subdiffusive motion of tracer particles in complex crowded environments, such as biological cells, has been shown to be widespread. This deviation from Brownian motion is usually characterized by a sublinear time dependence of the mean square displacement (MSD). However, subdiffusive behavior can stem from different microscopic scenarios that cannot be identified solely by the MSD data. In this article we present a theoretical framework that permits the analytical calculation of first-passage observables (mean first-passage times, splitting probabilities, and occupation times distributions) in disordered media in any dimensions. This analysis is applied to two representative microscopic models of subdiffusion: continuous-time random walks with heavy tailed waiting times and diffusion on fractals. Our results show that first-passage observables provide tools to unambiguously discriminate between the two possible microscopic scenarios of subdiffusion. Moreover, we suggest experiments based on first-passage observables that could help in determining the origin of subdiffusion in complex media, such as living cells, and discuss the implications of anomalous transport to reaction kinetics in cells.

First-passage times in complex scale-invariant media.

How long does it take a random walker to reach a given target point? This quantity, known as a first-passage time (FPT), has led to a growing number of theoretical investigations over the past decade. The importance of FPTs originates from the crucial role played by first encounter properties in various real situations, including transport in disordered media, neuron firing dynamics, spreading of diseases or target search processes. Most methods of determining FPT properties in confining domains have been limited to effectively one-dimensional geometries, or to higher spatial dimensions only in homogeneous media. Here we develop a general theory that allows accurate evaluation of the mean FPT in complex media. Our analytical approach provides a universal scaling dependence of the mean FPT on both the volume of the confining domain and the source-target distance. The analysis is applicable to a broad range of stochastic processes characterized by length-scale-invariant properties. Our theoretical predictions are confirmed by numerical simulations for several representative models of disordered media, fractals, anomalous diffusion and scale-free networks.

First-passage time distributions for subdiffusion in confined geometry.

In this Letter, we derive a relationship between the moments of the first-passage time for a random walk and the first-passage time density for subdiffusive processes modeled by continuous-time random walks. In particular, we show that the exact long-time behavior of the density depends only on the mean first-passage time of the corresponding normal diffusive process. In addition, we give explicit evaluations of the first-passage time distribution for general three-dimensional bounded domains. These results are relevant to systems involving anomalous diffusion in confinements.

Mesoscale engines by nonlinear friction.

A new approach to build mesoscopic-size engines that move translationally or rotationally and can perform useful functions such as the pulling of a cargo is presented. The approach is based on the transformation of internal vibrations of the moving object into directed motion, making use of the nonlinear properties of friction. This can be achieved by superimposing time-dependent external fields that break the spatial symmetry. The motion can be controlled and optimized by adjusting the system parameters.

The escape of a particle from a driven harmonic potential to an attractive surface.

We investigate theoretically the dynamics of a colloidal particle, trapped by optical tweezers, which gradually approaches an attractive surface with a constant velocity until it escapes the trap and jumps to the surface. We find that the height of the energy barrier in such a colloid-surface system follows the scaling DeltaE proportional, variant(z(0)(t)-const)(32) when the trap approaches the surface, z(0)(t) being the trap surface distance. Using this scaling we derive equations for the probability density function of the jump lengths, for the velocity dependence of its mean and most probable values, and for the variance. These can be used to extract the parameters of the particle-surface interaction from experimental data.

Field-induced dispersion in subdiffusion.

We discuss the response of continuous-time random walks to an oscillating external field within the generalized master equation approach. We concentrate on the time dependence of the two first moments of the walker's displacement. We show that for power-law waiting-time distributions with 0

From diffusion to anomalous diffusion: a century after Einstein's Brownian motion.

Einstein's explanation of Brownian motion provided one of the cornerstones which underlie the modern approaches to stochastic processes. His approach is based on a random walk picture and is valid for Markovian processes lacking long-term memory. The coarse-grained behavior of such processes is described by the diffusion equation. However, many natural processes do not possess the Markovian property and exhibit anomalous diffusion. We consider here the case of subdiffusive processes, which correspond to continuous-time random walks in which the waiting time for a step is given by a probability distribution with a diverging mean value. Such a process can be considered as a process subordinated to normal diffusion under operational time which depends on this pathological waiting-time distribution. We derive two different but equivalent forms of kinetic equations, which reduce to known fractional diffusion or Fokker-Planck equations for waiting-time distributions following a power law. For waiting time distributions which are not pure power laws one or the other form of the kinetic equation is advantageous, depending on whether the process slows down or accelerates in the course of time.

Saltatory drift in a randomly driven two-wave potential.

The dynamics of a classical particle in a one-dimensional, randomly driven potential is analysed both analytically and numerically. The potential considered here is composed of two identical spatially periodic saw-tooth-like components, one of which is externally driven by a random force. We show that under certain conditions the particle may travel against the averaged external force, performing a saltatory unidirectional drift with a constant velocity. Such a behaviour persists also in situations when the external force averages out to zero. We demonstrate that the physics behind this phenomenon stems from a particular behaviour of fluctuations in random force: upon reaching a certain level, random fluctuations exercise a locking function creating points of irreversibility which the particle cannot overpass. Repeated (randomly) in each cycle, this results in a saltatory unidirectional drift. This mechanism resembles the work of an escapement-type device in watches. Considering the overdamped limit, we propose simple analytical estimates for the particle's terminal velocity.

Actin-based motility: cooperative symmetry-breaking and phases of motion.

A microscopic model is proposed for the motility of a bead driven by the polymerization of actin filaments. The model exhibits a rich spectrum of behaviours similar to those observed in biomimetic experiments, which include spontaneous symmetry-breaking, various regimes of the bead's motion and correlations between the structure of the actin tail which propels the bead and the bead dynamics. The dependences of the dynamical properties (such as symmetry-breaking time, regimes of motion, mean velocity, and tail asymmetry) on the physical parameters (the bead radius and viscosity) agree well with the experimental observations. We find that most experimental observations can be reproduced taking into account only one type of filaments interacting with the bead: the detached filaments that push the bead. Our calculations suggest that the analysis of mean characteristics only (velocities, symmetry-breaking times, etc) does not always provide meaningful information about the mechanism of motility. The aim should be to obtain the corresponding distributions, which might be extremely broad and therefore not well represented by their mean only. Our findings suggest a simple coarse-grained description, which captures the main features obtained within the microscopic model.

Resonant activation in discrete systems.

The resonant activation phenomenon (RAP) in a discrete system is studied using the master equation formalism. We show that the RAP corresponds to a nonmonotonic behavior of the frequency dependent first passage time probability density function (PDF). An analytical expression for the resonant frequency is introduced, which, together with numerical results, helps understand the RAP behavior in the space spanned by the transition rates for the case of reflecting and absorbing boundary conditions. The limited range of system parameters for which the RAP occurs is discussed. We show that a minimum and a maximum in the mean first passage time can be obtained when both boundaries are absorbing. Relationships to some biological systems are suggested.

From deterministic dynamics to kinetic phenomena.

We investigate a one-dimensional Hamiltonian system that describes a system of particles interacting through short-range repulsive potentials. Depending on the particle mean energy epsilon the system demonstrates a spectrum of kinetic regimes, characterized by their transport properties ranging from ballistic motion to localized oscillations through anomalous diffusion regimes. We establish relationships between the observed kinetic regimes and the "thermodynamic" states of the system. The nature of heat conduction in the proposed model is discussed.

Translocation of a single-stranded DNA through a conformationally changing nanopore.

We investigate the translocation of a single-stranded DNA through a pore which fluctuates between two conformations, using coupled master equations. The probability density function of the first passage times of the translocation process is calculated, displaying a triple-, double-, or monopeaked behavior, depending on the interconversion rates between the conformations, the applied electric field, and the initial conditions. The cumulative probability function of the first passage times, in a field-free environment, is shown to have two regimes, characterized by fast and slow timescales. An analytical expression for the mean first passage time of the translocation process is derived, and provides, in addition to the interconversion rates, an extensive characterization of the translocation process. Relationships to experimental observations are discussed.

Catalytic reactions with bulk-mediated excursions: mixing fails to restore chemical equilibrium.

In this paper we analyze the effect of the bulk-mediated excursions (BME) of reactive species on the long-time behavior of the catalytic Langmuir-Hinshelwood-like A+B-->0 reactions in systems in which a catalytic plane (CP) is in contact with a liquid phase, containing concentrations of reactive particles. Such BME result from repeated particles desorption from the CP, subsequent diffusion in the liquid phase, and eventual readsorption on the CP away from the initial detachment point. This process leads to an effective superdiffusive transport along the CP. We consider both "batch" reactions, in which all particles of reactive species were initially adsorbed onto the CP, and reactions followed by a steady inflow of particles onto the CP. We show that for batch reactions the BME provide an effective mixing channel and here the mean-field-type behavior emerges. On the contrary, for reaction followed by a steady inflow of particles, we observe essential departures from the mean-field behavior and find that the mixing effect of the BME is insufficient to restore chemical equilibrium. We show that a steady state is established as t--> infinity, in which the limiting value of the mean coverage of the CP depends on the particles' diffusion coefficient in the bulk liquid phase, and that the spatial distributions of adsorbed particles are strongly correlated. Moreover, we show that the relaxation to such a steady state is a power-law function of time, in contrast to the exponential time dependence describing the approach to equilibrium in perfectly stirred systems.

Friction through dynamical formation and rupture of molecular bonds.

We introduce a model for friction in a system of two rigid plates connected by bonds (springs) and experiencing an external drive. The macroscopic frictional properties of the system are shown to be directly related to the rupture and formation dynamics of the microscopic bonds. Different regimes of motion are characterized by different rates of rupture and formation relative to the driving velocity. In particular, the stick-slip regime is shown to correspond to a cooperative rupture of the bonds. Moreover, the notion of static friction is shown to be dependent on the experimental conditions and time scales. The overall behavior can be described in terms of two Deborah numbers.