Narrow escape problem

Narrow escape problem

The narrow escape problem is an ubiquitous problem in biology, biophysics and cellular biology.

The formulation is the following: a Brownian particle (ion, molecule, or protein) is confined to a bounded domain (a compartment or a cell) by a reflecting boundary, except for a small window through which it can escape. The narrow escape problem is that of calculating the mean escape time. This time diverges as the window shrinks, thus rendering the calculation a singular perturbation problem.



The motion of a particle is described by the Smoluchowski limit of the Langevin equation:

dX_t=\sqrt{2D} \, dB_t+\frac{1}{\gamma}F(x)

where D is the diffusion coefficient of the particle, γ is the friction coefficient per unit of mass, F(x) the force per unit of mass, and Bt is a Brownian motion.

Mean first passage time

A common question is to estimate the mean sojourn time of a particle diffusing in a bounded domain Ω before it escapes through a small absorbing window \partial\Omega_a in its boundary \partial\Omega. The time is estimated asymptotically in the limit \varepsilon= \frac{\partial\Omega_a}{\partial\Omega} \ll 1

The probability density function (pdf) pε(x,t) is the probability of finding the particle at position x at time t.

The pdf satisfies the Fokker–Planck equation

 \frac{\partial}{\partial t}p_{\varepsilon}(x,t)=D \Delta p_{\varepsilon}(x,t)-\frac{1}{\gamma}\nabla ( p_\varepsilon (x,t) F(x))

with initial condition

p_\varepsilon (x,0) = \rho_0(x) \,

and mixed Dirichlet–Neumann boundary conditions (t > 0)

p_\varepsilon (x,t) = 0\text{ for }x \in \partial\Omega_a
D\frac{\partial}{\partial n}p_\varepsilon (x,t) - \frac{p_\varepsilon (x,t)}{\gamma} F(x)\cdot n(x)=0 \text{ for }x \in \partial \Omega - \partial\Omega_a

The function

 u_\varepsilon (y) = \int_\Omega \int_0^\infty p_\varepsilon (x,t y) \, dt \, dx

represents the mean sojourn time of particle, conditioned on the initial position y. It is the solution of the boundary value problem

D\Delta u_\varepsilon (y) + \frac{1}{\gamma}F(y)\cdot\nabla u_{\varepsilon}(y) = -1
u_\varepsilon (y) = 0\text{ for }y \in \partial\Omega_a
\frac{\partial u_\varepsilon (y)}{\partial n} = 0\text{ for }y \in \partial\Omega_r

The solution depends on the dimension of the domain. For a particle diffusing on a disk

u_\varepsilon (y)=\frac{A}{\pi D}\ln\frac{1}{\varepsilon}+O(1)

where A is the surface of the domain. The function u_{\epsilon}(y)does not depend on the initial position y, except for a small boundary layer near due to the asymptotic form. The first order term matters in dimension 2. For a circular disk of radius R, the mean escape time of a particle starting in the center is

E(\tau | x(0)=0 ) = \frac{R^2}{D}\left(\log\left(\frac{1}{\varepsilon}\right) + \log 2 + \frac{1}{4}+O(\varepsilon)\right).

The escape time averaged with respect to a uniform initial distribution of the particle is given by

E(\tau ) = \frac{R^2}{D}\left(\log\left(\frac{1}{\varepsilon}\right) + \log 2 + \frac{1}{8} + O(\varepsilon)\right).

The geometry of the small opening can affect the escape time: if the absorbing window is located at a corner of angle α , then

E\tau = \frac{|\Omega|}{\alpha D} \left[\log \frac{1}{\varepsilon} +O(1)\right].

More surprising, near a cusp in a two dimensional domain, the escape time Eτ grows algebraically, rather than logarithmically: in the domain bounded between two tangent circles, the escape time is

 E\tau = \frac{|\Omega|}{(d-1)D} \left(\frac{1}{\varepsilon} + O(1) \right),

where d > 1 is the ratio of the radii. Finally, when the domain is an annulus, the escape time to a small opening located on the inner circle involves a second parameter which is \beta = {\frac{R_1}{R_2}} < 1, the ratio of the inner to the outer radii, the escape time, averaged with respect to a uniform initial distribution, is

E\tau = \frac{(R_2^2-R_1^2)}D\left[\log \frac{1}{\varepsilon} +
\log 2 + 2\beta^2 \right]  +\frac{1}{2}\frac{R_2^2}{1-\beta^2}\log\frac{1}{\beta}- \frac{1}{4}R_2^2 +

This equation contains two terms of the asymptotic expansion of Eτ and is the angle of the absorbing boundary. The case β close to 1 remains open, and for general domains, the asymptotic expansion of the escape time remains an open problem. So does the problem of computing the escape time near a cusp point in three-dimensional domains. For Brownian motion in a field of force

 F(x)\neq 0 \,

the gap in the spectrum is not necessarily between the first and the second eigenvalues, depending on the relative size of the small hole and force barriers the particle has to overcome in order to escape. The escape stream is not necessarily Poissonian.

Biological Applications

Stochastic chemical reactions in microdomains [1]

The forward rate of chemical reactions is the reciprocal of the narrow escape time, which generalizes the classical Smoluchowski formula for Brownian particles located in an infinite medium. A Markov description can be used to estimate the binding and unbinding to a small number of sites.


  1. ^ * Holcman D, Schuss Z., Stochastic chemical reactions in microdomains, J Chem Phys. 2005 Mar 15;122(11):114710.
  • Z. Schuss, A. Singer, and D. Holcman The narrow escape problem for diffusion in cellular microdomains Proc Natl Acad Sci U S A. 2007;104(41):16098–103.
  • Singer A, Schuss Z, Holcman D."Narrow escape and leakage of Brownian particles. " Phys Rev E Stat Nonlin Soft Matter Phys. 2008 78:051111.
  • M. J. Ward, S. Pillay, A. Peirce, and T. Kolokolnikov An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part I: Two-Dimensional Domains
  • Singer A, Schuss Z, Holcman D, et al.,Narrow escape, part I JOURNAL OF STATISTICAL PHYSICS : 122 : 3 Pages: 437–463 FEB 2006
  • Singer A, Schuss Z, Holcman D, et al.,Narrow escape, part II JOURNAL OF STATISTICAL PHYSICS : 122 : 3 Pages: 465–489 FEB 2006
  • Singer A, Schuss Z, Holcman D, et al.,Narrow escape, part III JOURNAL OF STATISTICAL PHYSICS : 122 : 3 Pages: 491–509 FEB 2006
  • Holcman D, Schuss Z," Diffusion escape through a cluster of small absorbing windows" JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL : 41: 15: 155001 2008

External links

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Attempts to escape Oflag IV-C — For a complete list, see List of attempts to escape Oflag IV C. Prisoners made numerous attempts to escape Oflag IV C, one of the most famous German Army prisoner of war camps for officers in World War II. Between 30 and 36 (German/Allied… …   Wikipedia

  • Bridge and torch problem — The bridge and torch problem (also known as The Midnight Train cite web|title=MURDEROUS MATHS BRAINBENDERS|url=|accessdate=2008 02 08] and Dangerous crossing cite web|title=Some simple and not so… …   Wikipedia

  • The Problem Child — Infobox Book | name = The Problem Child title orig = translator = image caption = author = Michael Buckley illustrator = Peter Ferguson cover artist = Peter Ferguson country = United States language = English series = The Sisters Grimm series… …   Wikipedia

  • animal learning — ▪ zoology Introduction       the alternation of behaviour as a result of individual experience. When an organism can perceive and change its behaviour, it is said to learn.       That animals can learn seems to go without saying. The cat that… …   Universalium

  • Conditional preservation of the saints — The Five Articles of Remonstrance Conditional election Unlimited atonement Total depravity …   Wikipedia

  • History of Deportivo de La Coruña — HistoryThe very beginning1906 was the year when Deportivo de la Sala Calvet was founded, and the club s first ever president was Luis Cornide. Records indicate that back then the team consisted of Salvador Fojón, Venancio Deus, Juan Long, Ángel… …   Wikipedia

  • The Spiderwick Chronicles — This article is about the novel series. For the feature film adaptation, see The Spiderwick Chronicles (film). The Spiderwick Chronicles Beyond the Spiderwick Chronicles Cover of The Spiderwick Chronicles Boxed Set, 2004 T …   Wikipedia

  • List of characters in Fables — This is a list of characters in Fables , a fictional fantasy comic series for mature readers published by DC Comics. New York FablesBigby Wolfnow White The Cubs Snow and Bigby s seven children are a rowdy, unpredictable bunch of hybrids that seem …   Wikipedia

  • Comanche history — For a summary of Comanche history see Comanche. Comanche territory c.1850 Forming a part of the Eastern Shoshone linguistic group in southeastern Wyoming who moved on to the buffalo Plains around 1500 AD (based on glottochronological estimations) …   Wikipedia

  • George Anson's voyage around the world — While Great Britain was at war with Spain in 1740, Commodore George Anson led a squadron of eight ships on a mission to disrupt or capture Spain s Pacific possessions. Returning to England in 1744 by way of China and thus completing a… …   Wikipedia