Narrow Escape Problem Research Articles

Layer potential techniques for the narrow escape problem

The narrow escape problem consists in deriving the asymptotic expansion of the solution of a drift-diffusion equation with the Dirichlet boundary condition on a small absorbing part of the boundary and the Neumann boundary condition on the remaining reflecting boundaries. Using layer potential techniques, we rigorously find high-order asymptotic expansions of such solutions. The asymptotic formula explicitly exhibits the nonlinear interaction of many small absorbing targets. Based on the asymptotic theory for eigenvalue problems developed in Ammari et al. (2009) [3], we also construct high-order asymptotic formulas for the perturbation of eigenvalues of the Laplace and the drifted Laplace operators for mixed boundary conditions on large and small pieces of the boundary.

Narrow escape through a funnel and effective diffusion on a crowded membrane

Particles diffusing on a membrane crowded with obstacles have to squeeze between them through funnel-shaped narrow straits. The computation of the mean passage time through the straits is a new narrow escape problem that gives rise to new, hitherto unknown, behavior that we communicate here. The motion through the straits on the coarse scale of the narrow escape time is an effective diffusion with coefficient that varies nonlinearly with the density of obstacles. We calculate the coarse-grained diffusion coefficient on a planar lattice of circular obstacles and use it to estimate the density of obstacles on a neuronal membrane and in a model of a cytoplasm crowded by identical parallel circular rods.

The Narrow Escape Problem in a Flat Cylindrical Microdomain with Application to Diffusion in the Synaptic Cleft

The mean first passage time (MFPT) for a Brownian particle to reach a small target in cellular microdomains is a key parameter for chemical activation. Although asymptotic estimations of the MFPT are available for various geometries, these formulas cannot be applied to degenerated structures where one dimension is much smaller compared to the others. Here we study the narrow escape time problem for a Brownian particle to reach a small target located on the surface of a flat cylinder, where the cylinder height is comparable to the target size, and much smaller than the cylinder radius. When the cylinder is sealed, we estimate the MFPT for a Brownian particle to hit a small disk located centrally on the lower surface. For a laterally open cylinder, we estimate the conditional probability and the conditional MFPT to reach the small disk before exiting through the lateral opening. We apply our results to diffusion in the narrow synaptic cleft, and we compute the fraction and the mean time for neurotransmitters to find their specific receptors located on the postsynaptic terminal. Finally, we confirm our formulas with Brownian simulations.

The Mean Escape Time for a Narrow Escape Problem with Multiple Switching Gates

This article deals with the narrow escape problem when there are two gates which open alternatively in a random way. We set up the problem and carry out a rigorous asymptotic analysis to derive the mean escape time (MET) for a Brownian particle inside a domain to exit the domain through the switching gates. We show that the MET decreases as the switching rate between the gates increases, and we give upper and lower bounds for the decay rate. We then consider the case when there are multiple switching gates and derive the leading-order term of the asymptotic expansion of the MET.

A Pore−Cavity−Pore Device to Trap and Investigate Single Nanoparticles and DNA Molecules in a Femtoliter Compartment: Confined Diffusion and Narrow Escape

Spatial confinement from the nano- to the microscale is ubiquitous in nature. Striving to understand the behavior of nanoscale objects in confined domains we present a nanofluidic silicon device which consists of two stacked nanopores forming the in/outlets to a pyramidal cavity of micrometer dimensions (10 fL volume). Being electrically addressable, charged objects can be actively loaded into, trapped inside, and unloaded from the "pore-cavity-pore" (PCP) device. When operated passively, confined Brownian motion and the entropy barriers of the nanopores govern the behavior of nano-objects within the PCP device. We present measurements with single fluorescent nanoparticles as well as particle-ensembles and analyze their trajectories and residence times. Experimental data are compared to random walk simulations and analytical theories on confined diffusion and the Brownian escape of nano-objects across entropy barriers. Single particle data corroborate analytical solutions of the narrow escape problem, but ensemble measurements indicate crowding effects even at low particle concentrations. The utilization of the device to trap biomolecules is demonstrated for single λ-DNA molecules.

A pore-Cavity-Pore Device to Trap and Investigate Single Nano-Scale Objects in Femto-Liter Compartments: Confined Diffusion and Narrow Escape

Spatial confinement from the nano- to the micro-scale is ubiquitous in nature. Striving to emulate biological compartmentalization and to understand the fundamental physical behavior of molecules in confined domains, micro- and nano-structuring techniques have been used extensively to create artificial devices comprising liquid-filled compartments and channels. In addition, the development of robust solid-state structures, which allow for the observation and manipulation of single nano-scale objects is key for the realization of future lab-on-chip devices with improved functionality. Here we introduce an electrically addressable nano-fluidic silicon device that consists of two stacked nanopores forming the in/outlets to a pyramidal cavity of micrometer dimensions, i.e. femto-liter volume. The electrical properties of the PCP structure are investigated by impedance spectroscopy. Furthermore, we present a FEM simulation of the electric field inside the device. We then demonstrate how individual fluorescent nano-particles and DNA can be actively (by electrical means) and passively (entropically driven) loaded into, trapped inside, and unloaded from the ‘pore-cavity-pore’ (PCP) device. A fundamentally important problem in biology is the escape of nano-objects from a micro-domain through a small opening (narrow escape problem). Using the PCP device it is possible for the first time to obtain data on the narrow escape time under well-defined geometrical and experimental conditions. Single particle tracking and residence time data are presented and quantitatively compared to random walk simulations and analytical theories on confined diffusion and the narrow escape problem. Furthermore, we extend the escape studies towards polymeric analytes like DNA.

Diffusion laws in dendritic spines.

Dendritic spines are small protrusions on a neuronal dendrite that are the main locus of excitatory synaptic connections. Although their geometry is variable over time and along the dendrite, they typically consist of a relatively large head connected to the dendritic shaft by a narrow cylindrical neck. The surface of the head is connected smoothly by a funnel or non-smoothly to the narrow neck, whose end absorbs the particles at the dendrite. We demonstrate here how the geometry of the neuronal spine can control diffusion and ultimately synaptic processes. We show that the mean residence time of a Brownian particle, such as an ion or molecule inside the spine, and of a receptor on its membrane, prior to absorption at the dendritic shaft depends strongly on the curvature of the connection of the spine head to the neck and on the neck's length. The analytical results solve the narrow escape problem for domains with long narrow necks.

Extended narrow escape problem: Boundary homogenization-based analysis

Diffusion of particles in confined domains with absorbing spots on the otherwise reflecting boundaries is ubiquitous in nature and technology. Because of nonuniform boundary conditions, the problem of finding the mean first passage time (MFPT) of the particle to one of the spots is extremely complicated. We show how the difficulties can be overcome by means of boundary homogenization when the domain is a circular disk whose boundary contains n nonoverlapping identical absorbing arcs, which may occupy an arbitrary fraction of the boundary. We find the MFPT as a function of the fraction of the boundary occupied by the arcs (i) for n evenly spaced arcs and (ii) for two arcs arbitrarily located on the boundary. As the arc length tends to zero, our approximate solution reduces to the known asymptotic formula for the MFPT rigorously derived in studies devoted of the narrow escape problem.

Narrow escape and leakage of Brownian particles

Questions of flux regulation in biological cells raise a renewed interest in the narrow escape problem. The determination of a higher order asymptotic expansion of the narrow escape time depends on determining the singularity behavior of the Neumann Green's function for the Laplacian in a three-dimensional (3D) domain with a Dirac mass on the boundary. In addition to the usual 3D Coulomb singularity, this Green's function also has an additional weaker logarithmic singularity. By calculating the coefficient of this logarithmic singularity, we calculate the second term in the asymptotic expansion of the narrow escape time and in the expansion of the principal eigenvalue of the Laplace equation with mixed Dirichlet-Neumann boundary conditions, with small Dirichlet and large Neumann parts. We also determine the leakage flux of Brownian particles that diffuse from a source to an absorbing target on a reflecting boundary of a domain, if a small perforation is made in the reflecting boundary.

The narrow escape problem for diffusion in cellular microdomains

The study of the diffusive motion of ions or molecules in confined biological microdomains requires the derivation of the explicit dependence of quantities, such as the decay rate of the population or the forward chemical reaction rate constant on the geometry of the domain. Here, we obtain this explicit dependence for a model of a Brownian particle (ion, molecule, or protein) confined to a bounded domain (a compartment or a cell) by a reflecting boundary, except for a small window through which it can escape. We call the calculation of the mean escape time the narrow escape problem. This time diverges as the window shrinks, thus rendering the calculation a singular perturbation problem. Here, we present asymptotic formulas for the mean escape time in several cases, including regular domains in two and three dimensions and in some singular domains in two dimensions. The mean escape time comes up in many applications, because it represents the mean time it takes for a molecule to hit a target binding site. We present several applications in cellular biology: calcium decay in dendritic spines, a Markov model of multicomponent chemical reactions in microdomains, dynamics of receptor diffusion on the surface of neurons, and vesicle trafficking inside a cell.

Narrow Escape, Part II: The Circular Disk

We consider Brownian motion in a circular disk Ω, whose boundary \(\partial\Omega\) is reflecting, except for a small arc, \(\partial\Omega_a\), which is absorbing. As \(\varepsilon=|\partial\Omega_a|/|\partial \Omega|\) decreases to zero the mean time to absorption in \(\partial\Omega_a\), denoted \(E\tau\), becomes infinite. The narrow escape problem is to find an asymptotic expansion of \(E\tau\) for \(\varepsilon\ll1\). We find the first two terms in the expansion and an estimate of the error. The results are extended in a straightforward manner to planar domains and two-dimensional Riemannian manifolds that can be mapped conformally onto the disk. Our results improve the previously derived expansion for a general domain, \(E\tau = {\frac{|\Omega|}{D\pi}}\big[\log{\frac{1}{\varepsilon}}+O(1)\big],\) (\(D\) is the diffusion coefficient) in the case of a circular disk. We find that the mean first passage time from the center of the disk is \(E[\tau\,|\,{\boldmath x}(0)={\bf 0}]={\frac{R^2}{D}}\big[\log{\frac{1}{\varepsilon}}+ \log 2 +{ \frac{1}{4}} + O(\varepsilon)\big]\). The second term in the expansion is needed in real life applications, such as trafficking of receptors on neuronal spines, because \(\log{\frac{1}{\varepsilon}}\) is not necessarily large, even when e is small. We also find the singular behavior of the probability flux profile into \(\partial\Omega_a\) at the endpoints of \(\partial\Omega_a\), and find the value of the flux near the center of the window.

Narrow Escape, Part III: Non-Smooth Domains and Riemann Surfaces

We consider the narrow escape problem in two-dimensional Riemannian manifolds (with a metric g) with corners and cusps, in an annulus, and on a sphere. Specifically, we calculate the mean time it takes a Brownian particle diffusing in a domain Ω to reach an absorbing window when the ratio \({\varepsilon= {\frac{|\partial \Omega_a|_g}{|\partial \Omega|_g}}}\) between the absorbing window and the otherwise reflecting boundary is small. If the boundary is smooth, as in the cases of the annulus and the sphere, the leading term in the expansion is the same as that given in part I of the present series of papers, however, when it is not smooth, the leading order term is different. If the absorbing window is located at a corner of angle α, then \(E\tau = { \frac{|\Omega|_g}{\alpha D}}[\log{\frac{1}{\varepsilon}}+O(1)],\) if near a cusp, then \(E\tau\) grows algebraically, rather than logarithmically. Thus, in the domain bounded between two tangent circles, the expected lifetime is \(E\tau ={\frac{|\Omega|}{(d^{-1}-1)D}}(\frac{1}{\varepsilon} +O(1))\), where \(d<1\) is the ratio of the radii. For the smooth boundary case, we calculate the next term of the expansion for the annulus and the sphere. It can also be evaluated for domains that can be mapped conformally onto an annulus. This term is needed in real life applications, such as trafficking of receptors on neuronal spines, because \(\log{\frac{1}{\varepsilon}}\) is not necessarily large, even when \(\varepsilon = {\frac{|\partial \Omega_a|_g}{|\partial \Omega|_g}}\) is small. In these two problems there are additional parameters that can be small, such as the ratio δ of the radii of the annulus. The contributions of these parameters to the expansion of the mean escape time are also logarithmic. In the case of the annulus the mean escape time is \(E\tau = {\frac{|\Omega|_g}{\pi D} [\log\frac{1}{\varepsilon}+\frac 12\log\frac{1}{\delta}+O(1)]}\).