We study a reaction-diffusion process that involves two species of atoms, immobile and diffusing. We assume that initially only immobile atoms, uniformly distributed throughout the entire space, are present. Diffusing atoms are injected at the origin by a source which is turned on at time t=0. When a diffusing atom collides with an immobile atom, the two atoms form an immobile stable molecule. The region occupied by molecules is asymptotically spherical with radius growing as t^{1/d} in d>=2 dimensions. We investigate the survival probability that a diffusing atom has not become a part of a molecule during the time interval t after its injection and the probability density of such a particle. We show that asymptotically the survival probability (i) saturates in one dimension, (ii) vanishes algebraically with time in two dimensions (with exponent being a function of the dimensionless flux and determined as a zero of a confluent hypergeometric function), and (iii) exhibits a stretched exponential decay in three dimensions.
Fractional Brownian motion is a generalised Gaussian diffusive process that is found to describe numerous stochastic phenomena in physics and biology. Here we introduce a multi-dimensional fractional Brownian motion (FBM) defined as a superposition of conventional FBM for each coordinate in analogy to multi-dimensional Brownian motion, and study its first passage properties. Starting from the well-established first passage time statistics of one-dimensional FBM and the associated approximation schemes, we explore the first passage time behaviour of multi-dimensional FBM and compare these resul…
We study the efficiency of a two-channel reaction between two walkers on a finite one-dimensional periodic lattice. The walkers perform a combination of synchronous and asynchronous jumps on the lattice and react instantaneously when they meet at the same site (first channel) or upon position exchange (second channel). We develop a method based on a conditional first-passage problem to obtain exact results for the mean number of time steps needed for the reaction to take place as well as for higher order moments. Previous results obtained in the framework of a difference equation approach are …
We consider the first-passage problem for $N$ identical independent particles that are initially released uniformly in a finite domain $Ω$ and then diffuse toward a reactive area $Γ$, which can be part of the outer boundary of $Ω$ or a reaction centre in the interior of $Ω$. For both cases of perfect and partial reactions, we obtain the explicit formulas for the first two moments of the fastest first-passage time (fFPT), i.e., the time when the first out of the $N$ particles reacts with $Γ$. Moreover, we investigate the full probability density of the fFPT. We discuss a significant role of the…
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. 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 fun…
We consider a sequential cascade of molecular first-reaction events towards a terminal reaction centre in which each reaction step is controlled by diffusive motion of the particles. The model studied here represents a typical reaction setting encountered in diverse molecular biology systems, in which, e.g., a signal transduction proceeds via a series of consecutive "messengers": the first messenger has to find its respective immobile target site triggering a launch of the second messenger, the second messenger seeks its own target site and provokes a launch of the third messenger and so on, r…
Synaptic transmission between neurons is governed by a cascade of stochastic reaction-diffusion events that lead to calcium-induced vesicle release of neurotransmitter. Since experimental measurements of such systems are challenging due their nanometer and sub-millisecond scale, numerical simulations remain the principal tool for studying calcium dependent synaptic vesicle fusion, despite limitations of time-consuming calculations. In this paper we develop an analytical solution to rapidly explore dynamical stochastic reaction-diffusion problems, based on first-passage times. This is the first…
For two molecules to react they first have to meet. Yet, reaction times are rarely on par with the first-passage times that govern such molecular encounters. A prime reason for this discrepancy is stochastic transitions between reactive and non-reactive molecular states, which results in effective gating of product formation and altered reaction kinetics. To better understand this phenomenon we develop a unifying approach to gated reactions on networks. We first show that the mean and distribution of the gated reaction time can always be expressed in terms of ungated first-passage and return t…
We study the survival probability and the corresponding first passage time density of fractional Brownian motion confined to a two-dimensional open wedge domain with absorbing boundaries. By analytical arguments and numerical simulation we show that in the long time limit the first passage time density scales as t**{-1+pi*(2H-2)/(2*Theta)} in terms of the Hurst exponent H and the wedge angle Theta. We discuss this scaling behaviour in connection with the reaction kinetics of FBM particles in a one-dimensional domain.
The classical theory of chemical reactions can be understood in terms of diffusive barrier crossing, where the rate of a reaction is determined by the inverse of the mean first passage time (FPT) to cross a free energy barrier. Whenever a few reaction events suffice to trigger a response or the energy barriers are not high, the mean first passage time alone does not suffice to characterize the kinetics, i.e., the kinetics do not occur on a single time-scale. Instead, the full statistics of the FPT are required. We present a spectral representation of the FPT statistics that allows us to unders…
We study reaction-diffusion processes with concentration-dependent diffusivity. First, we determine the decay of the concentration in the single-species and two-species diffusion-controlled annihilation processes. We then consider two natural inhomogeneous realizations. The two-species annihilation process is investigated in the situation when the reactants are initially separated, namely each species occupies a half space. In particular, we determine the growth law of the width of the reaction zone. The single-species annihilation process is studied in the situation when the spatially localiz…
Text-book concepts of diffusion- versus kinetic-control are well-defined for reaction-kinetics involving macroscopic concentrations of diffusive reactants that are adequately described by rate-constants -- the inverse of the mean-first-passage-time to the reaction-event. In contradistinction, an open important question is whether the mean-first-passage-time alone is a sufficient measure for biochemical reactions that involve nanomolar reactant concentrations. Here, using a simple yet generic, exactly-solvable model we study the conspiratory effect of diffusion and chemical reaction-limitations…
Most biochemical reactions in living cells rely on diffusive search for target molecules or regions in a heterogeneous overcrowded cytoplasmic medium. Rapid re-arrangements of the medium constantly change the effective diffusivity felt locally by a diffusing particle and thus impact the distribution of the first-passage time to a reaction event. Here, we investigate the effect of these dynamic spatio-temporal heterogeneities onto diffusion-limited reactions. We describe a general mathematical framework to translate many results for ordinary homogeneous Brownian motion to heterogeneous diffusio…
The path decomposition expansion represents the propagator of the irreversible reaction as a convolution of the first-passage, last-passage and rebinding time probability densities. Using path integral technique, we give an elementary, yet rigorous, proof of the path decomposition expansion of the Green's functions describing the non-reactive case and the irreversible reaction of an isolated pair of molecules. To this end, we exploit the connection between boundary value problems and interaction potential problems with $δ$- and $δ'$-function perturbation. In particular, we employ a known exact…
Stochastic restart may drastically reduce the expected run time of a computer algorithm, expedite the completion of a complex search process, or increase the turnover rate of an enzymatic reaction. These diverse first-passage-time (FPT) processes seem to have very little in common but it is actually quite the other way around. Here we show that the relative standard deviation associated with the FPT of an optimally restarted process, i.e., one that is restarted at a (non-zero) rate which brings the mean FPT to a minimum, is always unity. We interpret, further generalize, and discuss this findi…
Many out of equilibrium phenomena, such as diffusion-limited reactions or target search processes, are controlled by first-passage events. So far the general determination of the mean first-passage time (FPT) to a target in confinement has left aside aging processes, involved in contexts as varied as glassy dynamics, tracer diffusion in biological membranes or transport of cold atoms in optical lattices. Here we consider general non-Markovian scale-invariant processes in arbitrary dimension, displaying aging, and demonstrate that all the moments of the FPT obey universal scalings with the conf…
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 aspec…
Stochastic synthesis-degradation processes: first-passage properties and connections with resetting
Pritzker School of Molecular Engineering, University of Chicago（芝加哥大学普里茨克分子工程学院）；Instituto de Física, Universidad Nacional Autónoma de México（墨西哥国立自治大学物理研究所）
Persistence, defined as the probability that a fluctuating signal has not reached a threshold up to a given observation time, plays a crucial role in the theory of random processes. It quantifies the kinetics of processes as varied as phase ordering, reaction diffusion or interface relaxation dynamics. The fact that persistence can decay algebraically with time with non trivial exponents has triggered a number of experimental and theoretical studies. However, general analytical methods to calculate persistence exponents cannot be applied to the ubiquitous case of non-Markovian systems relaxing…
The first-passage time (FPT), i.e., the moment when a stochastic process reaches a given threshold value for the first time, is a fundamental mathematical concept with immediate applications. In particular, it quantifies the statistics of instances when biomolecules in a biological cell reach their specific binding sites and trigger cellular regulation. Typically, the first-passage properties are given in terms of mean first-passage times. However, modern experiments now monitor single-molecular binding-processes in living cells and thus provide access to the full statistics of the underlying …
We investigate restricted diffusion in a bounded domain towards a small partially reactive target in three- and higher-dimensional spaces. We propose a simple explicit approximation for the principal eigenvalue of the Laplace operator with mixed Robin-Neumann boundary conditions. This approximation involves the harmonic capacity and the surface area of the target, the volume of the confining domain, the diffusion coefficient and the reactivity. The accuracy of the approximation is checked by using a finite-elements method. The proposed approximation determines also the mean first-reaction time…
We briefly review the principles, mathematical bases, numerical shortcuts and applications of fast random walk (FRW) algorithms. This Monte Carlo technique allows one to simulate individual trajectories of diffusing particles in order to study various probabilistic characteristics (harmonic measure, first passage/exit time distribution, reaction rates, search times and strategies, etc.) and to solve the related partial differential equations. The adaptive character and flexibility of FRWs make them particularly efficient for simulating diffusive processes in porous, multiscale, heterogeneous, …
How much time does it take two molecules to react? If a reaction occurs upon contact, the answer to this question boils down to the classic first-passage time problem: find the time it takes the two molecules to meet. However, this is not always the case as molecules switch stochastically between reactive and non-reactive states. The reaction is then said to be "gated" by the internal states of the molecules involved which could have a dramatic influence on kinetics. A unified, continuous-time, approach to gated reactions on networks was presented in [Phys. Rev. Lett. 127, 018301, 2021]. Here,…
This paper provides a theoretical framework of deriving the forward and backward Feynman-Kac equations for the distribution of functionals of the path of a particle undergoing both diffusion and chemical reaction. Very general forms of the equations are obtained. Once given the diffusion type and reaction rate, a specific forward or backward Feynman-Kac equation can be obtained. The listed in the paper include the ones for normal/anomalous diffusions and reactions with linear/nonlinear rates. Using the derived equations, we also study the occupation time in half-space, the first passage time t…
We present an exact calculation of the mean first-passage time to a small target on the surface of a 2D or 3D spherical domain, for a molecule performing surface-mediated diffusion. This minimal model of interfacial reactions, which explicitly takes into account the combination of surface and bulk diffusion, shows the importance of correlations induced by the coupling of the switching dynamics to the geometry of the confinement, ignored so far. Interestingly, our results show that, in the context of interfacial systems in confinement, the reaction time can be minimized as a function of the des…
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 tractab…
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 di…
Freidlin-Wentzell theory of large deviations can be used to compute the likelihood of extreme or rare events in stochastic dynamical systems via the solution of an optimization problem. The approach gives exponential estimates that often need to be refined via calculation of a prefactor. Here it is shown how to perform these computations in practice. Specifically, sharp asymptotic estimates are derived for expectations, probabilities, and mean first passage times in a form that is geared towards numerical purposes: they require solving well-posed matrix Riccati equations involving the minimize…
Dynamics of a particle diffusing in a confinement can be seen a sequence of bulk-diffusion-mediated hops on the confinement surface. Here, we investigate the surface hopping propagator that describes the position of the diffusing particle after a prescribed number of encounters with that surface. This quantity plays the central role in diffusion-influenced reactions and determines their most common characteristics such as the propagator, the first-passage time distribution, and the reaction rate. We derive explicit formulas for the surface hopping propagator and related quantities for several …
Calcium (Ca) plays an important role in regulating various cellular processes. In a variety of cell types, Ca signaling occurs within microdomains where channels deliver localized pulses of Ca which activate a nearby collection of Ca-sensitive receptors. The small number of channels involved ensures that the signaling process is stochastic. The aggregate response of several thousand of these microdomains yields a whole-cell response which dictates the cell behavior. Here, we study analytically the statistical properties of a population of these microdomains in response to a trigger signal. We …
We present an efficient Monte Carlo method to simulate reaction-diffusion processes with spatially varying particle annihilation or transformation rates as it occurs for instance in the context of motor-driven intracellular transport. Like Green's function reaction dynamics and first-passage time methods, our algorithm avoids small diffusive hops by propagating sufficiently distant particles in large hops to the boundaries of protective domains. Since for spatially varying annihilation or transformation rates the single particle diffusion propagator is not known analytically, we present an alg…
A Universal Control Budget for First-Passage Kinetics
Center for Systems Biology Dresden（德累斯顿系统生物学中心）；Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；Max Planck Institute of Molecular Cell Biology and Genetics（马克斯·普朗克分子细胞生物学与遗传学研究所）；Department of Physics, Graduate School of Science, The University of Tokyo（东京大学理学研究科物理学系）
Ageing in systems without detailed balance is studied in bosonic contact and pair-contact processes with Levy diffusion. In the ageing regime, the dynamical scaling of the two-time correlation function and two-time response function is found and analysed. Exact results for non-equilibrium exponents and scaling functions are derived. The behaviour of the fluctuation-dissipation ratio is analysed. A passage time from the quasi-stationary regime to the ageing regime is defined, in qualitative agreement with kinetic spherical models and p-spin spherical glasses.
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 o…
Starting from a continuous time random walk (CTRW) model of particles that may evanesce as they walk, our goal is to arrive at macroscopic integro-differential equations for the probability density for a particle to be found at point r at time t given that it started its walk from r_0 at time t=0. The passage from the CTRW to an integro-differential equation is well understood when the particles are not evanescent. Depending on the distribution of stepping times and distances, one arrives at standard macroscopic equations that may be "normal" (diffusion) or "anomalous" (subdiffusion and/or sup…
The paper addresses Brownian motion in the logarithmic potential with time-dependent strength, $U(x,t) = g(t) \log(x)$, subject to the absorbing boundary at the origin of coordinates. Such model can represent kinetics of diffusion-controlled reactions of charged molecules or escape of Brownian particles over a time-dependent entropic barrier at the end of a biological pore. We present a simple asymptotic theory which yields the long-time behavior of both the survival probability (first-passage properties) and the moments of the particle position (dynamics). The asymptotic survival probability,…
Mean first passage times of higher-dimensional velocity jump processes
California State University at Northridge（北岭加州州立大学）；University of California at Los Angeles（加利福尼亚大学洛杉矶分校）；University of Notre Dame（圣母大学）；University of Alberta（阿尔伯塔大学）
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 Markov…
We first derive the Hamilton-Jacobi theory underlying continuous-time Markov processes, and then use the construction to develop a variational algorithm for estimating escape (least improbable or first passage) paths for a generic stochastic chemical reaction network that exhibits multiple fixed points. The design of our algorithm is such that it is independent of the underlying dimensionality of the system, the discretization control parameters are updated towards the continuum limit, and there is an easy-to-calculate measure for the correctness of its solution. We consider several applicatio…
Complex energy landscapes often arise in biological systems, e.g. for protein folding, biochemical reactions or intracellular transport processes. Their physical effects are often reflected in the first-passage times arising from these energy landscapes. However, their calculation is notoriously challenging and it is often difficult to identify the most relevant features of a given energy landscape. Here we show how this can be achieved by coarse-graining the Fokker-Planck equation to a master equation and decomposing its first-passage times in an iterative process. We apply this method to the…
The Doi formalism treats a reaction-diffusion process as a quantum many-body problem. We use this second quantized formulation as a starting point to derive a numerical scheme for simulating $X\to 0$ reaction-diffusion processes, following a well-established time discretization procedure. In the case of a reaction zone localized in the configuration space, this formulation provides also a systematic way of designing an optimized, multiple time step algorithm, spending most of the computation time to sample the configurations where the reaction is likely to occur.
In this work we consider a simple random walk embedded in a generic branched structure and we find a close-form formula to calculate the hitting time $H\left(i,f\right)$ between two arbitrary nodes $i$ and $j$. We then use this formula to obtain the set of hitting times $\left\{ H\left(i,f\right)\right\} $ for combs and their expectation values, namely the mean-first passage time $\left( \mbox{MFPT}_{f} \right)$, where the average is performed over the initial node while the final node $f$ is given, and the global mean-first passage time $\left( \mbox{GMFPT} \right)$, where the average is perf…
Certain biochemical reactions can only be triggered after binding of a sufficient number of particles to a specific target region such as an enzyme or a protein sensor. We investigate the distribution of the reaction time, i.e., the first instance when all independently diffusing particles are bound to the target. When each particle binds irreversibly, this is equivalent to the first-passage time of the slowest (last) particle. In turn, reversible binding to the target renders the problem much more challenging and drastically changes the distribution of the reaction time. We derive the exact s…
We study the dynamics of a Brownian particle in Morse potential under thermal fluctuations, modeled by Gaussian white noise whose amplitude depends on absolute temperature. Dynamics of such a particle is investigated by numerically integrating the corresponding Langevin equation. From the mean first passage time (escape time), we study the dependence of Kramer's rate on temperature and viscosity of the medium. An approximate expression for the reaction rate is found by solving differential equation for the mean first passage time. The expression shows a temperature dependent pre-factor for the…
