96 research outputs found
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 widepsread. 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, which can not be identified solely by the MSD
data. In this paper we present a theoretical framework which permits to
calculate analytically 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 which 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.Comment: 21 pages, 3 figures. Submitted versio
Geometry-controlled kinetics
It has long been appreciated that transport properties 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). Although essential
to quantify the kinetics of reactions on all time scales, determining the FPT
distribution was deemed so far intractable. Here, we calculate analytically
this FPT distribution and show that transport processes as various as regular
diffusion, anomalous diffusion, diffusion in disordered media and in fractals
fall into the same universality classes. Beyond this theoretical aspect, this
result changes the views on standard reaction kinetics. More precisely, we
argue that geometry can become a key parameter so far ignored in this context,
and introduce the concept of "geometry-controlled kinetics". These findings
could help understand the crucial role of spatial organization of genes in
transcription kinetics, and more generally the impact of geometry on
diffusion-limited reactions.Comment: Submitted versio
Diffusion-limited reactions and mortal random walkers in confined geometries
Motivated by the diffusion-reaction kinetics on interstellar dust grains, we
study a first-passage problem of mortal random walkers in a confined
two-dimensional geometry. We provide an exact expression for the encounter
probability of two walkers, which is evaluated in limiting cases and checked
against extensive kinetic Monte Carlo simulations. We analyze the continuum
limit which is approached very slowly, with corrections that vanish
logarithmically with the lattice size. We then examine the influence of the
shape of the lattice on the first-passage probability, where we focus on the
aspect ratio dependence: Distorting the lattice always reduces the encounter
probability of two walkers and can exhibit a crossover to the behavior of a
genuinely one-dimensional random walk. The nature of this transition is also
explained qualitatively.Comment: 18 pages, 16 figure
Occupation times of random walks in confined geometries: From random trap model to diffusion limited reactions
We consider a random walk in confined geometry, starting from a site and
eventually reaching a target site. We calculate analytically the distribution
of the occupation time on a third site, before reaching the target site. The
obtained distribution is exact, and completely explicit in the case or
parallepipedic confining domains. We discuss implications of these results in
two different fields: The mean first passage time for the random trap model is
computed in dimensions greater than 1, and is shown to display a non-trivial
dependence with the source and target positions ; The probability of reaction
with a given imperfect center before being trapped by another one is also
explicitly calculated, revealing a complex dependence both in geometrical and
chemical parameters
Geometrical organization of solutions to random linear Boolean equations
The random XORSAT problem deals with large random linear systems of Boolean
variables. The difficulty of such problems is controlled by the ratio of number
of equations to number of variables. It is known that in some range of values
of this parameter, the space of solutions breaks into many disconnected
clusters. Here we study precisely the corresponding geometrical organization.
In particular, the distribution of distances between these clusters is computed
by the cavity method. This allows to study the `x-satisfiability' threshold,
the critical density of equations where there exist two solutions at a given
distance.Comment: 20 page
Non-Markovian polymer reaction kinetics
Describing the kinetics of polymer reactions, such as the formation of loops
and hairpins in nucleic acids or polypeptides, is complicated by the structural
dynamics of their chains. Although both intramolecular reactions, such as
cyclization, and intermolecular reactions have been studied extensively, both
experimentally and theoretically, there is to date no exact explicit analytical
treatment of transport-limited polymer reaction kinetics, even in the case of
the simplest (Rouse) model of monomers connected by linear springs. We
introduce a new analytical approach to calculate the mean reaction time of
polymer reactions that encompasses the non-Markovian dynamics of monomer
motion. This requires that the conformational statistics of the polymer at the
very instant of reaction be determined, which provides, as a by-product, new
information on the reaction path. We show that the typical reactive
conformation of the polymer is more extended than the equilibrium conformation,
which leads to reaction times significantly shorter than predicted by the
existing classical Markovian theory.Comment: Main text (7 pages, 5 figures) + Supplemantary Information (13 pages,
2 figures
Mean first-passage times of non-Markovian random walkers in confinement
The first-passage time (FPT), defined as the time a random walker takes to
reach a target point in a confining domain, is a key quantity in the theory of
stochastic processes. Its importance comes from its crucial role to quantify
the efficiency of processes as varied as diffusion-limited reactions, target
search processes or spreading of diseases. Most methods to determine the FPT
properties in confined domains have been limited to Markovian (memoryless)
processes. However, as soon as the random walker interacts with its
environment, memory effects can not be neglected. Examples of non Markovian
dynamics include single-file diffusion in narrow channels or the motion of a
tracer particle either attached to a polymeric chain or diffusing in simple or
complex fluids such as nematics \cite{turiv2013effect}, dense soft colloids or
viscoelastic solution. Here, we introduce an analytical approach to calculate,
in the limit of a large confining volume, the mean FPT of a Gaussian
non-Markovian random walker to a target point. The non-Markovian features of
the dynamics are encompassed by determining the statistical properties of the
trajectory of the random walker in the future of the first-passage event, which
are shown to govern the FPT kinetics.This analysis is applicable to a broad
range of stochastic processes, possibly correlated at long-times. Our
theoretical predictions are confirmed by numerical simulations for several
examples of non-Markovian processes including the emblematic case of the
Fractional Brownian Motion in one or higher dimensions. These results show, on
the basis of Gaussian processes, the importance of memory effects in
first-passage statistics of non-Markovian random walkers in confinement.Comment: Submitted version. Supplementary Information can be found on the
Nature website :
http://www.nature.com/nature/journal/v534/n7607/full/nature18272.htm
Enhanced reaction kinetics in biological cells
The cell cytoskeleton is a striking example of "active" medium driven
out-of-equilibrium by ATP hydrolysis. Such activity has been shown recently to
have a spectacular impact on the mechanical and rheological properties of the
cellular medium, as well as on its transport properties : a generic tracer
particle freely diffuses as in a standard equilibrium medium, but also
intermittently binds with random interaction times to motor proteins, which
perform active ballistic excursions along cytoskeletal filaments. Here, we
propose for the first time an analytical model of transport limited reactions
in active media, and show quantitatively how active transport can enhance
reactivity for large enough tracers like vesicles. We derive analytically the
average interaction time with motor proteins which optimizes the reaction rate,
and reveal remarkable universal features of the optimal configuration. We
discuss why active transport may be beneficial in various biological examples:
cell cytoskeleton, membranes and lamellipodia, and tubular structures like
axons.Comment: 10 pages, 2 figure
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 last decade1. 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 to
determine the FPT properties in confining domains have been limited to
effective 1D geometries, or for space dimensions larger than one only to
homogeneous media1. Here we propose a general theory which allows one to
accurately evaluate the mean FPT (MFPT) in complex media. Remarkably, this
analytical approach provides a universal scaling dependence of the MFPT on both
the volume of the confining domain and the source-target distance. This
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 emblematic models of disordered media,
fractals, anomalous diffusion and scale free networks.Comment: Submitted version. Supplementary Informations available on Nature
websit
Kinetics of active surface-mediated diffusion in spherically symmetric domains
We present an exact calculation of the mean first-passage time to a target on
the surface of a 2D or 3D spherical domain, for a molecule alternating phases
of surface diffusion on the domain boundary and phases of bulk diffusion. We
generalize the results of [J. Stat. Phys. {\bf 142}, 657 (2011)] and consider a
biased diffusion in a general annulus with an arbitrary number of regularly
spaced targets on a partially reflecting surface. The presented approach is
based on an integral equation which can be solved analytically. Numerically
validated approximation schemes, which provide more tractable expressions of
the mean first-passage time are also proposed. In the framework of this minimal
model of surface-mediated reactions, we show analytically that the mean
reaction time can be minimized as a function of the desorption rate from the
surface.Comment: Published online in J. Stat. Phy
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