Abstract
A Brownian particle with diffusion coefficient D is confined to a bounded domain Ω by a reflecting boundary, except for a small absorbing window \(\partial\Omega_a\). The mean time to absorption diverges as the window shrinks, thus rendering the calculation of the mean escape time a singular perturbation problem. In the three-dimensional case, we construct an asymptotic approximation when the window is an ellipse, assuming the large semi axis a is much smaller than \(|\Omega|^{1/3}\) (\(|\Omega|\) is the volume), and show that the mean escape time is \(E\tau\sim{\frac{|\Omega|}{2\pi Da}} K(e)\), where e is the eccentricity and \(K(\cdot)\) is the complete elliptic integral of the first kind. In the special case of a circular hole the result reduces to Lord Rayleigh's formula \(E\tau\sim{\frac{|\Omega|}{4aD}}\), which was derived by heuristic considerations. For the special case of a spherical domain, we obtain the asymptotic expansion \(E\tau={\frac{|\Omega|}{4aD}} [1+\frac{a}{R} \log \frac{R}{a} + O(\frac{a}{R})]\). This result is important in understanding the flow of ions in and out of narrow valves that control a wide range of biological and technological function. If Ω is a two-dimensional bounded Riemannian manifold with metric g and \(\varepsilon=|\partial\Omega_a|_g/|\Omega|_g\ll1\), we show that \(E\tau ={\frac{|\Omega|_g}{D\pi}}[\log{\frac{1}{\varepsilon}}+O(1)]\). This result is applicable to diffusion in membrane surfaces.
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References
B. Hille, Ionic Channels of Excitable Membranes, 2nd ed., Sinauer, Mass., 1992.
W. Im and B. Roux, Ion permeation and selectivity of ompf porin: A theoretical study based on molecular dynamics, brownian dynamics, and continuum electrodiffusion theory. J. Mol. Bio. 322(4):851–869 (2002).
W. Im and B. Roux, Ions and counterions in a biological channel: A molecular dynamics simulation of ompf porin from escherichia coliin an explicit membrane with 1 m kcl aqueous salt solution. J. Mol. Bio. 319(5):1177–1197 (2002).
B. Corry, M. Hoyles, T. W. Allen, M. Walker, S. Kuyucak and S. H. Chung, Reservoir boundaries in brownian dynamics simulations of ion channels. Biophys. J. 82:1975–1984 (2002).
S. Wigger-Aboud, M. Saraniti and R. S. Eisenberg, Self-consistent particle based simulations of three dimensional ionic solutions. Nanotech 3:443 (2003).
T. A. van der Straaten, J. Tang, R. S. Eisenberg, U. Ravaioli and N. R. Aluru, Three-dimensional continuum simulations of ion transport through biological ion channels: Effects of charge distribution in the constriction region of porin. J. Computational Electronics 1:335–340 (2002).
L. Dagdug, A. M. Berezhkovskii, S. Y. Shvartsman and G. H. Weiss, Equilibration in two chambers connected by a capillary. J. Chem. Phys. 119(23):12473–12478 (2003).
D. Holcman and Z. Schuss, Escape through a small opening: Receptor trafficking in a synaptic membrane. J. Stat. Phys. 117(5–6):975–1014 (2004).
D. Holcman, Z. Schuss, E. Korkotian, Calcium dynamics in denritic spines and spine motility Bio. J. 87:81–91 (2004).
E. Korkotian, D. Holcman and M. Segal, Dynamic regulation of spine-dendrite coupling in cultured hippocampal neurons. Euro. J. Neuroscience 20:2649–2663, (2004).
R. C. Malenka, J. A. Kauer, D. J. Perkel and R. A. Nicoll, The impact of postsynaptic calcium on synaptic transmission–its role in long-term potentiation. Trends Neurosci. 12(11):444–450 (1989).
D. Holcman and Z. Schuss, Stochastic chemical reactions in microdomains. J. Chem. Phys. 122:114710 (2005).
C. W. Gardiner, Handbook of Stochastic Methods, 2-nd edition, Springer, NY (1985).
Z. Schuss, Theory and Applications of Stochastic Differential Equations, Wiley Series in Probability and Statistics, Wiley, NY 1980.
H. L. F. von Helmholtz, Crelle, Bd. 7 (1860).
J. W. S. Baron Rayleigh, The Theory of Sound, Vol. 2, 2nd Ed., Dover, New York, 1945.
I. V. Grigoriev, Y. A. Makhnovskii, A. M. Berezhkovskii, and V. Y. Zitserman, Kinetics of escape through a small hole. J. Chem. Phys. 116(22):9574–9577 (2002).
J. D. Jackson, Classical Electrodymnics, 2nd Ed., Wiley, NY, 1975.
I. N. Sneddon, Mixed Boundary Value Problems in Potential Theory, Wiley, NY, 1966.
V. I. Fabrikant, Applications of Potential Theory in Mechanics, Kluwer, 1989.
V. I. Fabrikant, Mixed Boundary Value Problems of Potential Theory and Their Applications in Engineering, Kluwer, 1991.
A. I. Lur'e, Three-Dimensional Problems of the Theory of Elasticity, Interscience publishers, NY 1964.
S. S. Vinogradov, P. D. Smith and E. D. Vinogradova, Canonical Problems in Scattering and Potential Theory, Parts I and II, Chapman & Hall/CRC, 2002.
M. Dauge, Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions, Lecture Notes in Mathematics, 1341, Springer-Verlag, NY (1988).
V. A. Kozlov, V. G. Mazya and J. Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities, American Mathematical Society, Mathematical Surveys and Monographs, vol. 52, 1997.
V. A. Kozlov, J. Rossmann and V. G. Mazya, Spectral Problems Associated With Corner Singularities of Solutions of Elliptic Equations, Mathematical Surveys and Monographs, vol. 85, American Mathematical Society 2001.
R. G. Pinsky, Asymptotics of the principal eigenvalue and expected hitting time for positive recurrent elliptic operators in a domain with a small puncture. J. of Fun. Analysis 200(1):177–197 (2003).
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, (Translated by K. N. Trirogoff, L. W. Neustadt, ed.), John Wiley, 1962; also A.N. Kolmogorov, E.F. Mishchenko, L.S. Pontryagin, “On One Probability Optimal Control Problem”, Dokl. Acad. Nauk SSSR, 145(5):993–995 (1962).
A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Jones and Bartlett, Boston 1992.
B. Matkowsky and Z. Schuss, The exit problem for randomly perturbed dynhamical systems. SIAM J. Appl. Math. 33(12):365–382 (1977).
P. Hänngi, P. Talkner and M. Borkovec, 50 year after Kramers. Rev. Mod. Phys. 62:251 (1990).
M. Freidlin, Markov Processes And Differential Equations, Birkhauser Boston 2002.
W. D. Collins, On some dual series equations and their application to electrostatic problems for spheroidal caps. Proc. Cambridge Phil. Soc. 57:367–384 (1961).
W. D. Collins, Note on an electrified circular disk situated inside an earthed coaxial infinite hollow cylinder. Proc. Cambridge Phil. Soc. 57:623–627, (1961).
A. Singer, Z. Schuss and D. Holcman, Narrow Escape, Part III: Non-smooth domains and Riemann surfaces, (This journal).
P. R. Garabedian, Partial Differential Equations, Wiley, NY 1964.
T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer, NY, 1998.
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, NY 1972.
A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms, Volume 1, McGraw-Hill, NY, 1954.
G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 2000.
W. Magnus and F. Oberhettinger, Formulas and Theorems for the Special Functions of Mathematical Physics, Chelsea Publishing Company, NY, 1949.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, NY, 1973.
R. B. Kelman, Steady-State Diffusion Through a Finite Pore Into an Infinite Reservoir: an Exact Solution. Bull. of Math. Biop. 27:57–65 (1965).
R. S. Eisenberg, M. M. Kłosek and Z. Schuss, Diffusion as a chemical reaction: Stochastic trajectories between fixed concentrations. J. Chem. Phys. 102:1767–1780 (1995).
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Singer, A., Schuss, Z., Holcman, D. et al. Narrow Escape, Part I. J Stat Phys 122, 437–463 (2006). https://doi.org/10.1007/s10955-005-8026-6
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DOI: https://doi.org/10.1007/s10955-005-8026-6