Narrow Escape Problem Research Articles

Trapping of particles diffusing in cylindrical cavity of arbitrary length and radius by two small absorbing disks on the cavity side wall: Narrow escape theory and beyond.

Narrow escape theory deals with the first passage of a particle diffusing in a cavity with small circular windows on the cavity wall to one of the windows. Assuming that (i) the cavity has no size anisotropy and (ii) all windows are sufficiently far away from each other, the theory provides an analytical expression for the particle mean first-passage time (MFPT) to one of the windows. This expression shows that the MFPT depends on the only global parameter of the cavity, its volume, independent of the cavity shape, and is inversely proportional to the product of the particle diffusivity and the sum of the window radii. Amazing simplicity and universality of this result raises the question of the range of its applicability. To shed some light on this issue, we study the narrow escape problem in a cylindrical cavity of arbitrary size anisotropy with two small windows arbitrarily located on the cavity side wall. We derive an approximate analytical solution for the MFPT, which smoothly goes from the conventional narrow escape solution in an isotropic cavity when the windows are sufficiently far away from each other to a qualitatively different solution in a long cylindrical cavity (the cavity length significantly exceeds its radius). Our solution demonstrates the mutual influence of the windows on the MFPT and shows how it depends on the inter-window distance. A key step in finding the solution is an approximate replacement of the initial three-dimensional problem by an equivalent one-dimensional one, where the particle diffuses along the cavity axis and the small absorbing windows are modeled by delta-function sinks. Brownian dynamics simulations are used to establish the range of applicability of our approximate approach and to learn what it means that the two windows are far away from each other.

The narrow escape problem of a chiral active particle (CAP): an optimal scheme.

We report a simulation study on the narrow escape kinetics of a chiral active particle (CAP) confined to a circular domain with a narrow escape opening. The study's main objective is to optimize the CAP's escape chances as a function of the relevant parameters, such as translational and rotational speeds of the CAP, domain size, etc. We identified three regimes in the escape kinetics, namely the noise-dominated regime, the optimal regime, and the chiral activity-dominated regime. In particular, the optimal regime is characterized by an escape scheme that involves a direct passage to the domain boundary at first and then a unidirectional drift along the boundary towards the exit. Furthermore, we propose a non-dimensionalization approach to optimize the escape performance across microorganisms with varying motile characteristics. Additionally, we explore the influence of the translational and rotational noise on the CAP's escape kinetics.

Asymptotic analysis of the narrow escape problem in general shaped domain with several absorbing necks

This paper considers the two-dimensional narrow escape problem in a domain which is composed of a relatively big head and several absorbing narrow necks. The narrow escape problem is to compute the mean first passage time (MFPT) of a Brownian particle traveling from inside the head to the end of the necks. The original model for MFPT is to solve a mixed Dirichlet–Neumann boundary value problem for the Poisson equation in the composite domain, and is computationally challenging. In this paper, we compute the MFPT by solving an equivalent Neumann–Robin type boundary value problem. By solving the new model, we obtain the three-term asymptotic expansion of the MFPT. We also conduct numerical experiments to show the accuracy of the high order expansion. As far as we know, this is the first result on high order asymptotic solution for narrow escape problem (NEP) in a general shaped domain with several absorbing neck windows. This work is motivated by Li (2014 J. Phys. A: Math. Theor. 47 505202), where the Neumann–Robin model was proposed to solve the NEP in a domain with a single absorbing neck.

Imperfect narrow escape problem.

We consider the kinetics of the imperfect narrow escape problem, i.e., the time it takes for a particle diffusing in a confined medium of generic shape to reach and to be adsorbed by a small, imperfectly reactive patch embedded in the boundary of the domain, in two or three dimensions. Imperfect reactivity is modeled by an intrinsic surface reactivity κ of the patch, giving rise to Robin boundary conditions. We present a formalism to calculate the exact asymptotics of the mean reaction time in the limit of large volume of the confining domain. We obtain exact explicit results in the two limits of large and small reactivities of the reactive patch, and a semianalytical expression in the general case. Our approach reveals an anomalous scaling of the mean reaction time as the inverse square root of the reactivity in the large-reactivity limit, valid for an initial position near the extremity of the reactive patch. We compare our exact results with those obtained within the "constant flux approximation"; we show that this approximation turns out to give exactly the next-to-leading-order term of the small-reactivity limit, and provides a good approximation of the reaction time far from the reactive patch for all reactivities, but not in the vicinity of the boundary of the reactive patch due to the above-mentioned anomalous scaling. These results thus provide a general framework to quantify the mean reaction times for the imperfect narrow escape problem.

Surface-particle interactions control the escape time of a particle from a nanopore-gated nanocavity system: a coarse grained simulation.

Nanopores and nanocavities are promising single molecule tools for investigating the behavior of individual molecules within confined spaces. For single molecule analysis, the total duration of time the analyte remains within the pore/cavity is highly important. However, this dwell time is ruled by a complex interplay among particle-surface interactions, external forces on the particle and Brownian diffusion, making the prediction of the dwell time challenging. Here, we show how the dwell time of an analyte in a nanocavity that is connected to the external environment by two nanopore gates depends on the sizes of the nanocavity/nanopore, as well as particle-wall interactions. For this purpose, we used a coarse-grained model that allowed us to simulate hundreds of individual analyte trajectories within a nanocavity volume. We found that by increasing the attraction between the particle and the wall, the diffusion process transforms from a usual 3D scenario (repulsive wall) to a 2D motion along the cavity surface (highly attractive wall). This results in a significant reduction of the average dwell time. Additionally, the comparison of our results with existing theories on narrow escape problem allowed us to quantify the reliability of theory derived for ideal conditions to geometries more similar to actual devices.

Narrow escape problem in the presence of the force field

This paper considers the narrow escape problem of a Brownian particle within a three‐dimensional Riemannian manifold under the influence of the force field. We compute an asymptotic expansion of mean sojourn time for Brownian particles. As a auxiliary result, we obtain the singular structure for the restricted Neumann Green's function which may be of independent interest.

Microbial narrow-escape is facilitated by wall interactions

Cells have evolved efficient strategies to probe their surroundings and navigate through complex environments. From metastatic spread in the body to swimming cells in porous materials, escape through narrow constrictions—a key component of any structured environment connecting isolated microdomains—is one ubiquitous and crucial aspect of cell exploration. Here, using the model microalgae Chlamydomonas reinhardtii, we combine experiments and simulations to achieve a tractable realization of the classical Brownian narrow-escape problem in the context of active confined matter. Our results differ from those expected for Brownian particles or leaking chaotic billiards and demonstrate that cell-wall interactions substantially modify escape rates and, under generic conditions, expedite spread dynamics.Received 4 October 2021Accepted 15 February 2022DOI:https://doi.org/10.1103/PhysRevResearch.4.L022029Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasBiological fluid dynamicsBiological movementMarine ecologySwimmingPhysical SystemsMicroswimmersBiological PhysicsInterdisciplinary PhysicsNonlinear Dynamics

Brownian dynamics simulations for the narrow escape problem in the unit sphere.

The narrow escape problem is a first-passage problem that concerns the calculation of the time needed for a Brownian particle to leave a domain with localized absorbing boundary traps, such that the measure of these traps is asymptotically small compared to the domain size. A common model for the mean first-passage time (MFPT) as a function of particle's starting location in a given domain with constant diffusivity is given by a Poisson partial differential equation subject to mixed Dirichlet-Neumann boundary conditions. The primary objective of this work is to perform direct numerical simulations of multiple particles undergoing Brownian motion in a three-dimensional spherical domain with boundary traps, compute MFPT values by averaging Brownian escape times, and compare these with explicit asymptotic results obtained previously by approximate solution of the Poisson problem. A close agreement of MFPT values is observed already at 10^{4} particle runs from a single starting point, providing a computational validation of the Poisson equation-based continuum model. Direct Brownian dynamics simulations are also used to study additional features of particle dynamics in narrow escape problems that cannot be captured in a continuum approach, such as average times spent by particles in a thin layer near the domain boundary, and effects of isotropic vs anisotropic near-boundary diffusion.

Distribution of first-reaction times with target regions on boundaries of shell-like domains

We study the probability density function (PDF) of the first-reaction times between a diffusive ligand and a membrane-bound, immobile imperfect target region in a restricted ‘onion-shell’ geometry bounded by two nested membranes of arbitrary shapes. For such a setting, encountered in diverse molecular signal transduction pathways or in the narrow escape problem with additional steric constraints, we derive an exact spectral form of the PDF, as well as present its approximate form calculated by help of the so-called self-consistent approximation. For a particular case when the nested domains are concentric spheres, we get a fully explicit form of the approximated PDF, assess the accuracy of this approximation, and discuss various facets of the obtained distributions. Our results can be straightforwardly applied to describe the PDF of the terminal reaction event in multi-stage signal transduction processes.

Narrow escape problem in two-shell spherical domains.

Intracellular transport in living cells is often spatially inhomogeneous with an accelerated effective diffusion close to the cell membrane and a ballistic motion away from the centrosome due to active transport along actin filaments and microtubules, respectively. Recently it was reported that the mean first passage time (MFPT) for transport to a specific area on the cell membrane is minimal for an optimal actin cortex width. In this paper, we ask whether this optimization in a two-compartment domain can also be achieved by passive Brownian particles. We consider a Brownian motion with different diffusion constants in the two shells and a potential barrier between the two, and we investigate the narrow escape problem by calculating the MFPT for Brownian particles to reach a small window on the external boundary. In two and three dimensions, we derive asymptotic expressions for the MFPT in the thin cortex and small escape region limits confirmed by numerical calculations of the MFPT using the finite-element method and stochastic simulations. From this analytical and numeric analysis, we finally extract the dependence of the MFPT on the ratio of diffusion constants, the potential barrier height, and the width of the outer shell. The first two are monotonous, whereas the last one may have a minimum for a sufficiently attractive cortex, for which we propose an analytical expression of the potential barrier height matching very well the numerical predictions.

Intercellular Communication as a Series of Narrow Escape Problems

Molecular communication is key for multicellular organisms. In plants, the exchange of nutrients and signals between cells is facilitated by tunnels called plasmodesmata. Such transport processes in complex geometries can be simulated using particle-based approaches, these, however, are computationally expensive. Here, we evaluate the narrow escape problem as a framework for describing intercellular transport. We introduce a volumetric adjustment factor for estimating escape times from non-spherical geometries. We validate this approximation against full 3D stochastic simulations and provide results for a range of cell sizes and diffusivities. We discuss how this approach can be extended using recent results on multiple trap problems to account for different plasmodesmata distributions with varying apertures.

Research Spotlights

The first of our two Research Spotlights articles in this issue gives a general framework and corresponding algorithm for computing accurate approximations to the spectral measures of self-adjoint operators, the key to which is the resolvent of the operator. As authors Matthew Colbrook, Andrew Horning, and Alex Townsend detail, the spectral measure is necessary in their applications of interest to give a more complete description of the operator and associated dynamics. Their article, “Computing Spectral Measures of Self-Adjoint Operators," begins with the formal mathematical description of the spectral measure and associated assumptions. Several applications that rely on the computation of spectral measure are highlighted throughout the article: for example, applications in particle and condensed matter physics are discussed in the context of a survey of previous work in estimation of spectral measure. The crux of the proposed approach, first formulated in section 4 and generalized to higher order kernels for improved accuracy in section 5, is to evaluate a smoothed approximation (i.e., defined through convolution with appropriate kernel) to the spectral measure by evaluating the resolvent, with the latter calculation akin to solving shifted systems of linear equations. The authors detail carefully and demonstrate graphically the challenges associated with designing a tractable and robust scheme. The discussion of algorithmic issues in section 6, including the ability of their approach to dynamically adjust to reach desired accuracy, also features use cases of their associated publicly available MATLAB code. Within the body of the article, the versatility of their proposed framework is well illustrated on differential, integral, and lattice operators. Readers may be interested in the suggestions in section 8 on possible further use cases for the new framework, such as in “understanding the behavior of large real-world networks and new random graph models." Our second article, “Optimization of the Mean First Passage Time in Near-Disk and Elliptical Domains in 2-D with Small Absorbing Traps," is coauthored by Sarafa A. Iyaniwura, Tony Wong, Colin B. Macdonald, and Michael J. Ward. Narrow escape or capture problems are those portrayed in the introduction as first passage time problems that describe the expected time for a Brownian particle to reach some absorbing set with small measure. Two of the applications in which such problems arise include the time it takes for a diffusing surface-bound molecule (the “particle” in this case) to reach a localized signaling region on the cell membrane and the time it takes for a predator to locate its prey. The authors define the “average MFPT" for a diffusion process to be the expected time for capture given a uniform distribution of starting points for the random walk. The optimal trap configurations for the average MFPT in geometries other than the disk had been unsolved and provided the impetus for the authors to investigate the question in the context of near-disk and elliptical domains. Through a combination of asymptotic analysis and numerical techniques (e.g., use of numerical quadrature and numerical time stepping for solving equations (3.4) and (4.3)), the authors design “hybrid asymptotic numerical" approaches to predicting optimal configurations of small stationary circular absorbing traps that minimize the average MFPT in these new domains. Though much of the paper is devoted to detailed derivations that will take some time for the reader to absorb, one can get some immediate appreciation for the results from the graphical illustrations in which the new results are compared against numerical PDE generated solutions. Extensions of the approach and remaining open problems are included in the last section for consideration by the reader.

Monte Carlo tracking drift-diffusion trajectories algorithm for solving narrow escape problems

Abstract This study deals with a narrow escape problem, a well-know difficult problem of evaluating the probability for a diffusing particle to reach a small part of a boundary far away from the starting position of the particle. A direct simulation of the diffusion trajectories would take an enormous computer simulation time. Instead, we use a different approach which drastically improves the efficiency of the diffusion trajectory tracking algorithm by introducing an artificial drift velocity directed to the target position. The method can be efficiently applied to solve narrow escape problems for domains of long extension in one direction which is the case in many practical problems in biology and chemistry. The algorithm is meshless both in space and time, and is well applied to solve high-dimensional problems in complicated domains. We present in this paper a detailed numerical analysis of the method for the case of a rectangular parallelepiped. Both stationary and transient diffusion problems are handled.

Competition between slow and fast regimes for extreme first passage times of diffusion * * The authors were supported by the National Science Foundation (Grant Nos. DMS-1944574, DMS-1814832, and DMS-1148230).

Many physical, chemical, and biological systems depend on the first passage time (FPT) of a diffusive searcher to a target. Typically, this FPT is much slower than the characteristic diffusion timescale. For example, this is the case if the target is small (the narrow escape problem) or if the searcher must escape a potential well. However, many systems depend on the first time a searcher finds the target out of a large group of searchers, which is the so-called extreme FPT. Since this extreme FPT vanishes in the limit of many searchers, the prohibitively slow FPTs of diffusive search can be negated by deploying enough searchers. However, the notion of ‘enough searchers’ is poorly understood. How can one determine if a system is in the slow regime (dominated by small targets or a deep potential, for example) or the fast regime (dominated by many searchers)? How can one estimate the extreme FPT in these different regimes? In this paper, we answer these questions by deriving conditions which ensure that a system is in either regime and finding approximations of the full distribution and all the moments of the extreme FPT in these regimes. Our analysis reveals the critical effect that initial searcher distribution and target reactivity can have on extreme FPTs.

Stochastic simulation algorithms for solving narrow escape diffusion problems by introducing a drift to the target

We suggest in this paper a new stochastic simulation algorithm for solving narrow escape problems governed by drift-diffusion-reaction equations of high dimension. The developed method drastically improves the efficiency of the diffusion trajectory tracking algorithm by introducing an artificial drift directed to the target position. The method is especially appropriate to solve narrow escape problems for domains of very long extension in one direction which is the case in many practical problems. We present simulation results for a diffusion transport problem of calculation of cathodoluminescence intensity, a diffusion flux of excitons to a threading dislocation, and the electron beam induced current in a semiconductor. The diffusion tracking algorithm is based on the random walk on spheres process. The method is meshless both in space and time, and is well applied to solve high-dimensional problems in complicated domains. The algorithms are based on tracking the trajectories of the diffusing particles exactly in accordance with the probabilistic distributions derived from the explicit representation of the relevant Green functions. They can be conveniently used not only for the solutions, but also for a direct calculation of fluxes to any part of the boundary without calculating the whole solution in the domain.

Mean escape time for randomly switching narrow gates in a steady flow

The escape of particles through a narrow absorbing gate in confined domains is a rich phenomenon in various systems in physics, chemistry and molecular biophysics. We consider the narrow escape problem in a bounded annular when the two gates randomly switch between different states with a switching rate k between the two gates. After briefly deriving the coupled partial differential equations for the escape time through two gates, we compute the mean escape time for particles escaping from the gates with different initial states. By numerical simulation under nonuniform boundary conditions, we quantify how narrow escape time is affected by the switching rate k between the two gates, arc length s between two gates, angular velocity w of the steady flow and diffusion coefficient D. We reveal that the mean escape time decreases with the switching rate k between the two gates, angular velocity w and diffusion coefficient D for fixed arc length, but takes the minimum when the two gates are evenly separated on the boundary for any given switching rate k between the two gates. In particular, we find that when the arc length size ε for the gates is sufficiently small, the average narrow escape time is approximately independent of the gate arc length size. We further indicate combinations of system parameters (regions located in the parameter space) such that the mean escape time is the longest or shortest. Our findings provide mathematical understanding for phenomena such as how ions select ion channels and how chemicals leak in annulus ring containers, when drift vector fields are present.

A fast solver for the narrow capture and narrow escape problems in the sphere

We present an efficient method to solve the narrow capture and narrow escape problems for the sphere. The narrow capture problem models the equilibrium behavior of a Brownian particle in the exterior of a sphere whose surface is reflective, except for a collection of small absorbing patches. The narrow escape problem is the dual problem: it models the behavior of a Brownian particle confined to the interior of a sphere whose surface is reflective, except for a collection of small patches through which it can escape.Mathematically, these give rise to mixed Dirichlet/Neumann boundary value problems of the Poisson equation. They are numerically challenging for two main reasons: (1) the solutions are non-smooth at Dirichlet-Neumann interfaces, and (2) they involve adaptive mesh refinement and the solution of large, ill-conditioned linear systems when the number of small patches is large.By using the Neumann Green's functions for the sphere, we recast each boundary value problem as a system of first-kind integral equations on the collection of patches. A block-diagonal preconditioner together with a multiple scattering formalism leads to a well-conditioned system of second-kind integral equations and a very efficient approach to discretization. This system is solved iteratively using GMRES. We develop a hierarchical, fast multipole method-like algorithm to accelerate each matrix-vector product. Our method is insensitive to the patch size, and the total cost scales with the number N of patches as O(Nlog⁡N), after a precomputation whose cost depends only on the patch size and not on the number or arrangement of patches. We demonstrate the method with several numerical examples, and are able to achieve highly accurate solutions with 100000 patches in one hour on a 60-core workstation. For that case, adaptive discretization of each patch would lead to a dense linear system with about 360 million degrees of freedom. Our preconditioned system uses only 13.6 million “compressed” degrees of freedom and a few dozen GMRES iterations.

Full distribution of first exit times in the narrow escape problem

In the scenario of the narrow escape problem (NEP) a particle diffuses in a finite container and eventually leaves it through a small ‘escape window’ in the otherwise impermeable boundary, once it arrives to this window and crosses an entropic barrier at the entrance to it. This generic problem is mathematically identical to that of a diffusion-mediated reaction with a partially-reactive site on the container’s boundary. Considerable knowledge is available on the dependence of the mean first-reaction time (FRT) on the pertinent parameters. We here go a distinct step further and derive the full FRT distribution for the NEP. We demonstrate that typical FRTs may be orders of magnitude shorter than the mean one, thus resulting in a strong defocusing of characteristic temporal scales. We unveil the geometry-control of the typical times, emphasising the role of the initial distance to the target as a decisive parameter. A crucial finding is the further FRT defocusing due to the barrier, necessitating repeated escape or reaction attempts interspersed with bulk excursions. These results add new perspectives and offer a broad comprehension of various features of the by-now classical NEP that are relevant for numerous biological and technological systems.

Locally and globally optimal configurations of N particles on the sphere with applications in the narrow escape and narrow capture problems.

Determination of optimal arrangements of N particles on a sphere is a well-known problem in physics. A famous example of such is the Thomson problem of finding equilibrium configurations of electrical charges on a sphere. More recently, however, similar problems involving other potentials and nonspherical domains have arisen in biophysical systems. Many optimal configurations have previously been computed, especially for the Thomson problem; however, few results exist for potentials that correspond to more applied problems. Here we numerically compute optimal configurations corresponding to the narrow escape and narrow capture problems in biophysics. We provide comprehensive tables of global energy minima for N≤120 and local energy minima for N≤65, and we exclude all saddle points. Local minima up to N=120 are available online.

Narrow escape problem for Brownian particles in a microsphere with internal circulation.

This paper considers the narrow escape problem of a Brownian particle within a two-dimensional domain with two escape windows and an internal circulation modeled by the flow within a Hill's vortex. To account for the spatially inhomogeneous flow within the domain, a Lagrangian study is undertaken using the complete equations of motion for a dense particle which is necessary to distinguish between the various regimes as the strength of the internal circulation is varied. For very low internal circulation the particle undergoes the conventional narrow escape problem, and agreement is good with the asymptotic expression. As the internal circulation is increased, regimes are identified with different scaling for the mean escape time. The potential application of this for drug delivery (were nanoparticles are encased in a microsphere) is discussed.