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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.

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We study the problem of a target search by a Brownian particle subject to stochastic resetting to a pair of sites. The mean search time is minimized by an optimal resetting rate which does not vary smoothly, in contrast with the well-known single site case, but exhibits a discontinuous transition as the position of one resetting site is varied while keeping the initial position of the particle fixed, or vice versa. The discontinuity vanishes at a "liquid-gas" critical point in position space. This critical point exists provided that the relative weight m of the further site is comprised in the interval [2.9028...,8.5603...]. When the initial position is a random variable that follows the resetting point distribution, a discontinuous transition also exists for the optimal rate as the distance between the resetting points is varied, provided that m exceeds the critical value m_{c}=6.6008.... This setup can be mapped onto an intermittent search problem with switching diffusion coefficients and represents a minimal model for the study of distributed resetting.

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The geometry of ion and metabolite channels in membranes of biological cells and organelles is usually far from that of a regular right cylinder. Rather, the channels have complex shapes that are characterized by the so-called vestibules and constriction zones which play roles of molecular filters determining the channel selectivity. In the present paper we discuss several channel structures with varying radius that approximate most of the cases found in nature, specifically, channels of smoothly varying radius and channels composed of multiple cylindrical sections of different lengths and radii including channels containing very thin circular constrictions. We consider diffusive transport of electrically neutral molecules driven by the concentration gradient and derive analytical expressions for the diffusion resistance - the integral parameter that describes steady-state transport properties of membrane channels.

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Recent progress in biophysics (for example, in studies of chemical sensing and spatiotemporal cell-signaling) poses new challenges to statistical theory of trapping of single diffusing particles. Here we deal with one of them, namely, trapping kinetics of single particles diffusing in a half-space bounded by a reflecting flat surface containing an absorbing circular disk. This trapping problem is essentially two-dimensional and the question of the angular dependence of the kinetics on the particle starting point is highly nontrivial. We propose an approximate approach to the problem that replaces the absorbing disk by an absorbing hemisphere of a properly chosen radius. This replacement makes the problem angular-independent and essentially one-dimensional. After the replacement one can find an exact solution for the particle propagator (Green's function) that allows one to completely characterize the kinetics. Extensive testing of the theoretical predictions based on the absorbing hemisphere approximation against three-dimensional Brownian dynamics simulations shows excellent agreement between the analytical and simulation results when the particle starts sufficiently far away from the disk. Our approach fails and the angular dependence of the kinetics is important when the distance of the particle starting point from the disk center is comparable with the disk radius. However, even when the initial distance is only two disk radii, the maximum relative error of the theoretical predictions is about 10%. The relative error rapidly decreases as the initial distance increases.

Assuntos

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Being motivated by recent progress in nanopore sensing, we develop a theory of the effect of large analytes, or blockers, trapped within the nanopore confines, on diffusion flow of small solutes. The focus is on the nanopore diffusion resistance which is the ratio of the solute concentration difference in the reservoirs connected by the nanopore to the solute flux driven by this difference. Analytical expressions for the diffusion resistance are derived for a cylindrically symmetric blocker whose axis coincides with the axis of a cylindrical nanopore in two limiting cases where the blocker radius changes either smoothly or abruptly. Comparison of our theoretical predictions with the results obtained from Brownian dynamics simulations shows good agreement between the two.

Assuntos

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We focus on the derivation of a general position-dependent effective diffusion coefficient to describe two-dimensional (2D) diffusion in a narrow and smoothly asymmetric channel of varying width under a transverse gravitational external field, a generalization of the symmetric channel case using the projection method introduced earlier by Kalinay and Percus [P. Kalinay and J. K. Percus, J. Chem. Phys. 122, 204701 (2005)10.1063/1.1899150]. To this end, we project the 2D Smoluchowski equation into an effective one-dimensional generalized Fick-Jacobs equation in the presence of constant force in the transverse direction. The expression for the diffusion coefficient given in Eq. (34) is our main result. This expression is a more general effective diffusion coefficient for narrow 2D channels in the presence of constant transverse force, which contains the well-known previous results for a symmetric channel obtained by Kalinay, as well as the limiting cases when the transverse gravitational external field goes to zero and infinity. Finally, we show that diffusivity can be described by the interpolation formula proposed by Kalinay, D_{0}/[1+(1/4)w^{'2}(x)]^{-η}, where spatial confinement, asymmetry, and the presence of a constant transverse force can be encoded in η, which is a function of channel width (w), channel centerline, and transverse force. The interpolation formula also reduces to well-known previous results, namely, those obtained by Reguera and Rubi [D. Reguera and J. M. Rubi, Phys. Rev. E 64, 061106 (2001)10.1103/PhysRevE.64.061106] and by Kalinay [P. Kalinay, Phys. Rev. E 84, 011118 (2011)10.1103/PhysRevE.84.011118].

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We study diffusion of a Brownian particle in a two-dimensional periodic channel of abruptly alternating width. Our main result is a simple approximate analytical expression for the particle effective diffusivity, which shows how the diffusivity depends on the geometric parameters of the channel: lengths and widths of its wide and narrow segments. The result is obtained in two steps: first, we introduce an approximate one-dimensional description of particle diffusion in the channel, and second, we use this description to derive the expression for the effective diffusivity. While the reduction to the effective one-dimensional description is standard for systems of smoothly varying geometry, such a reduction in the case of abruptly changing geometry requires a new methodology used here, which is based on the boundary homogenization approach to the trapping problem. To test the accuracy of our analytical expression and thus establish the range of its applicability, we compare analytical predictions with the results obtained from Brownian dynamics simulations. The comparison shows excellent agreement between the two, on condition that the length of the wide segment of the channel is equal to or larger than its width.

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We study trapping of particles diffusing in a two-dimensional rectangular chamber by a binding site located at the end of a rectangular sleeve. To reach the site a particle first has to enter the sleeve. After that it has two options: to come back to the chamber or to diffuse to the site where it is trapped. The main result of the present work is a simple expression for the mean particle lifetime as a function of its starting position and geometric parameters of the system. This expression is obtained by an approximate reduction of the initial two-dimensional problem to the effective one-dimensional one which can be solved with relative ease. Our analytical predictions are tested against the results obtained from Brownian dynamics simulations. The test shows excellent agreement between the two for a wide range of the geometric parameters of the system.

Assuntos

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A theoretical framework based on using the Frenet-Serret moving frame as the coordinate system to study the diffusion of bounded Brownian point-like particles has been recently developed [L. Dagdug et al., J. Chem. Phys. 145, 074105 (2016)]. Here, this formalism is extended to a variable cross section tube with a helix with constant torsion and curvature as a mid-curve. For the sake of clarity, we will divide this study into two parts: one for a helical tube with a constant cross section and another for a helical tube with a variable cross section. For helical tubes with a constant cross section, two regimes need to be considered for systematic calculations. On the one hand, in the limit when the curvature is smaller than the inverse of the helical tube radius R, the resulting coefficient is that obtained by Ogawa. On the other hand, we also considered the limit when torsion is small compared to R, and to the best of our knowledge, the expression thus obtained has not been previously reported in the literature. In the more general case of helical tubes with a variable cross section, we also had to limit ourselves to small variations of R. In this case, we obtained one of the main contributions of this work, which is an expression for the diffusivity dependent on R', torsion, and curvature that consistently reduces to the well-known expressions within the corresponding limits.

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Axial diffusion in channels and tubes of smoothly-varying geometry can be approximately described as one-dimensional diffusion in the entropy potential with a position-dependent effective diffusion coefficient, by means of the modified Fick-Jacobs equation. In this work, we derive analytical expressions for the position-dependent effective diffusivity for two-dimensional asymmetric varying-width channels, and for three-dimensional curved midline tubes, formed by straight walls. To this end, we use a recently developed theoretical framework using the Frenet-Serret moving frame as the coordinate system (2016 J. Chem. Phys. 145 074105). For narrow tubes and channels, an effective one-dimensional description reducing the diffusion equation to a Fick-Jacobs-like equation in general coordinates is used. From this last equation, one can calculate the effective diffusion coefficient applying Neumann boundary conditions.

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We developed a theoretical framework to study the diffusion of Brownian point-like particles in bounded geometries in two and three dimensions. We use the Frenet-Serret moving frame as the coordinate system. For narrow tubes and channels, we use an effective one-dimensional description reducing the diffusion equation to a Fick-Jacobs-like equation. From this last equation, we can calculate the effective diffusion coefficient applying Neumann boundary conditions. On one hand, for channels with a straight axis our theoretical approximation for the effective coefficient does coincide with the reported in the literature [D. Reguera and J. M. Rubí, Phys. Rev. E 64, 061106 (2001) and P. Kalinay and J. K. Percus, ibid. 74, 041203 (2006)]. On the other hand, for tubes with a straight axis and circular cross-section our analytical expression does not coincide with the reported by Rubí and Reguera and by Kalinay and Percus, although it is practically identical.

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This paper deals with diffusion of point particles in linearly corrugated two-dimensional channels. Such geometry allows one to obtain an approximate analytical expression that gives the particle effective diffusivity as a function of the geometric parameters of the channel. To establish its accuracy and the range of applicability, the expression is tested against Brownian dynamics simulation results. The test shows that the expression works very well for long channel periods, but fails when the period is not long enough compared to the minimum width of the channel. To fix this deficiency, we propose a simple empirical correction to the analytical expression. The resulting corrected expression for the effective diffusivity is in excellent agreement with the simulation results for all values of the channel period.

Assuntos

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This paper focuses on trapping of diffusing particles by a sphere with an absorbing cap of arbitrary size on the otherwise reflecting surface. We approach the problem using boundary homogenization which is an approximate replacement of non-uniform boundary conditions on the surface of the sphere by an effective uniform boundary condition with appropriately chosen effective trapping rate. One of the main results of our analysis is an expression for the effective trapping rate as a function of the surface fraction occupied by the absorbing cap. As the cap surface fraction increases from zero to unity, the effective trapping rate increases from that for a small absorbing disk on the otherwise reflecting sphere to infinity which corresponds to a perfectly absorbing sphere. The obtained expression for the effective trapping rate is applied to find the rate constant describing trapping of diffusing particles by an absorbing cap on the surface of the sphere. Finally, we find the capacitance of a metal cap of arbitrary size on a dielectric sphere using the relation between the capacitance and the rate constant of the corresponding diffusion-limited reaction. The relative error of our approximate expressions for the rate constant and the capacitance is less than 5% over the entire range of the cap surface fraction from zero to unity.

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We study trapping of diffusing particles by a cylindrical surface formed by rolling a flat surface, containing alternating absorbing and reflecting stripes, into a tube. For an arbitrary stripe orientation with respect to the tube axis, this problem is intractable analytically because it requires dealing with non-uniform boundary conditions. To bypass this difficulty, we use a boundary homogenization approach which replaces non-uniform boundary conditions on the tube wall by an effective uniform partially absorbing boundary condition with properly chosen effective trapping rate. We demonstrate that the exact solution for the effective trapping rate, known for a flat, striped surface, works very well when this surface is rolled into a cylindrical tube. This is shown for both internal and external problems, where the particles diffuse inside and outside the striped tube, at three orientations of the stripe direction with respect to the tube axis: (a) perpendicular to the axis, (b) parallel to the axis, and

Assuntos

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A covariant description of diffusion of point-size Brownian particles in bounded geometries is presented. To this end, we provide a formal theoretical framework using differential geometry. We propose a coordinate transformation to map the boundaries of a general two-dimensional channel into a straight channel in a non-Cartesian geometry. The new shape of the boundaries naturally suggests a reduction to one dimension. As a consequence of this coordinate transformation, the Fick equation with boundary conditions transforms as a generalized Fick-Jacobs-like equation, in which the leading-order term is equivalent to the Fick-Jacobs approximation. The expression for the effective diffusion coefficient derived here depends on the position and the derivatives of the channel's width and centerline. Finally, we validate our analytic predictions for the effective diffusion coefficients for two periodic channels.

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This paper deals with transport of point Brownian particles in a cylindrical tube with dead ends in the presence of laminar flow of viscous fluid in the cylindrical part of the tube (Poiseuille flow). It is assumed that the dead ends are identical and are formed by spherical cavities connected to the cylindrical part of the tube by narrow necks. The focus is on the effective velocity and diffusivity of the particles as functions of the mean flow velocity and geometric parameter of the tube. Entering a dead end, the particle interrupts its propagation along the tube axis. Later it returns, and the axial motion continues. From the axial propagation point of view, the particle entry into a dead end and its successive return to the flow is equivalent to the particle reversible binding to the tube wall. The effect of reversible binding on the transport parameters has been previously studied assuming that the particle survival probability in the bound state decays as a single exponential. However, this is not the case when the particle enters a dead end, since escape from the dead end is a non-Markovian process. Our analysis of the problem consists of two steps: First, we derive expressions for the effective transport parameters in the general case of non-Markovian binding. Second, we find the effective velocity and diffusivity by substituting into these expressions known results for the moments of the particle lifetime in the dead end [L. Dagdug, A. M. Berezhkovskii, Yu. A. Makhnovskii, and V. Yu. Zitserman, J. Chem. Phys. 127, 224712 (2007)]. To check the accuracy of our theory, we compare its predictions with the values of the effective velocity and diffusivity obtained from Brownian dynamics simulations. The comparison shows excellent agreement between the theoretical predictions and numerical results.

Assuntos

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In this work, we derive a general effective diffusion coefficient to describe the two-dimensional (2D) diffusion in a narrow and smoothly asymmetric channel of varying width, embedded on a curved surface, in the simple diffusion of non-interacting, point-like particles under no external field. To this end, we extend the generalization of the Kalinay-Percus' projection method [J. Chem. Phys. 122, 204701 (2005); Phys. Rev. E 74, 041203 (2006)] for the asymmetric channels introduced in [L. Dagdug and I. Pineda, J. Chem. Phys. 137, 024107 (2012)], to project the anisotropic two-dimensional diffusion equation on a curved manifold, into an effective one-dimensional generalized Fick-Jacobs equation that is modified according to the curvature of the surface. For such purpose we construct the whole expansion, writing the marginal concentration as a perturbation series. The lowest order in the perturbation parameter, which corresponds to the Fick-Jacobs equation, contains an additional term that accounts for the curvature of the surface. We explicitly obtain the first-order correction for the invariant effective concentration, which is defined as the correct marginal concentration in one variable, and we obtain the first approximation to the effective diffusion coefficient analogous to Bradley's coefficient [Phys. Rev. E 80, 061142 (2009)] as a function of the metric elements of the surface. In a straightforward manner, we study the perturbation series up to the nth order, and derive the full effective diffusion coefficient for two-dimensional diffusion in a narrow asymmetric channel, with modifications according to the metric terms. This expression is given as D(ξ)=D(0)/w'(ξ)√(g(1)/g(2)){arctan[√(g(2)/g(1))(y(0)'(ξ)+w'(ξ)/2)]-arctan[√(g(2)/g(1))(y(0)'(ξ)-w'(ξ)/2)]}, which is the main result of our work. Finally, we present two examples of symmetric surfaces, namely, the sphere and the cylinder, and we study certain specific channel configurations on these surfaces.

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The mean lifetime of a particle diffusing in a cylindrical cavity with a circular absorbing spot on the cavity wall is studied analytically as a function of the spot radius, its location on the wall, the particle initial position, and the cavity shape determined by its length and radius. When the spot radius tends to zero our formulas for the mean lifetime reduce to the result given by the solution of the narrow escape problem, according to which the mean lifetime is proportional to the ratio of the cavity volume to the spot radius and is independent of the cavity shape, the spot location on the cavity wall, and the particle starting point, assuming that this point is not too close to the spot. When the spot radius is not small enough, the asymptotic narrow escape formula for the mean lifetime fails, and one should use more general formulas derived in the present study. To check the accuracy and to establish the range of applicability of the formulas, we compare our theoretical predictions with the results of Brownian dynamics simulations.

Assuntos