Narrow escape problem explained

The narrow escape problem^{[1] [2]} is a ubiquitous problem in biology, biophysics and cellular biology.

The mathematical 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.^{[3] [4] [5] [6] [7] [8] [9]}

When escape is even more stringent due to severe geometrical restrictions at the place of escape, the narrow escape problem becomes the dire strait problem.^{[10] [11]}

The narrow escape problem was proposed in the context of biology and biophysics by D. Holcman and Z. Schuss,^[12] and later on with A.Singer and led to the narrow escape theory in applied mathematics and computational biology.^{[13] [14] [15]}

Formulation

The motion of a particle is described by the Smoluchowski limit of the Langevin equation:^{[16] [17]} $$dX\_t\; =\; \backslash sqrt\; \backslash ,\; dB\_t\; +\; \backslash frac\; F(x)\; \backslash ,\; dt,$$where

D is the diffusion coefficient of the particle, \gamma is the friction coefficient per unit of mass, F(x) the force per unit of mass, and B _{t} is a Brownian motion.

Mean first passage time and the Fokker-Planck equation

A common question is to estimate the mean sojourn time of a particle diffusing in a bounded domain

\Omega before it escapes through a small absorbing window \partial\Omega _{a} in its boundary \partial\Omega . The time is estimated asymptotically in the limit $\backslash varepsilon\; =\; \backslash frac$ \ll 1

The probability density function (pdf)

p _{\varepsilon} (x,t) is the probability of finding the particle at position x at time t .

The pdf satisfies the Fokker–Planck equation:$$\backslash frac\; p\_(x,t)\; =\; D\; \backslash Delta\; p\_(x,t)-\backslash frac\backslash nabla\; (p\_\backslash varepsilon\; (x,t)\; F(x))$$with initial condition$$p\_\backslash varepsilon\; (x,0)\; =\; \backslash rho\_0(x)\; \backslash ,$$and mixed Dirichlet–Neumann boundary conditions (

t > 0 )$$p\_\backslash varepsilon\; (x,t)\; =\; 0\backslash textx\; \backslash in\; \backslash partial\backslash Omega\_a$$$$D\backslash fracp\_\backslash varepsilon\; (x,t)\; -\; \backslash frac\; F(x)\backslash cdot\; n(x)=0\; \backslash textx\; \backslash in\; \backslash partial\; \backslash Omega\; -\; \backslash partial\backslash Omega\_a$$

The function$$u\_\backslash varepsilon\; (y)\; =\; \backslash int\_\backslash Omega\; \backslash int\_0^\backslash infty\; p\_\backslash varepsilon\; (x,t\; y)\; \backslash ,\; dt\; \backslash ,\; dx$$represents the mean sojourn time of particle, conditioned on the initial position

y . It is the solution of the boundary value problem

$$D\backslash Delta\; u\_\backslash varepsilon\; (y)\; +\; \backslash fracF(y)\backslash cdot\backslash nabla\; u\_(y)\; =\; -1$$$$u\_\backslash varepsilon\; (y)\; =\; 0\backslash texty\; \backslash in\; \backslash partial\backslash Omega\_a$$$$\backslash frac\; =\; 0\backslash texty\; \backslash in\; \backslash partial\backslash Omega\_r$$

The solution depends on the dimension of the domain. For a particle diffusing on a two-dimensional disk$$u\_\backslash varepsilon\; (y)=\backslash frac\backslash ln\backslash frac+O(1),$$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 the absorbing boundary 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$$E(\backslash tau\; |\; x(0)=0)\; =\; \backslash frac\backslash left(\backslash log\backslash left(\backslash frac\backslash right)\; +\; \backslash log\; 2\; +\; \backslash frac\; +\; O(\backslash varepsilon)\backslash right).$$

The escape time averaged with respect to a uniform initial distribution of the particle is given by$$E(\backslash tau)\; =\; \backslash frac\backslash left(\backslash log\backslash left(\backslash frac\backslash right)\; +\; \backslash log\; 2\; +\; \backslash frac\; +\; O(\backslash varepsilon)\backslash right).$$

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

\alpha , then:

$$E\backslash tau\; =\; \backslash frac$$

\left[\log \frac{1}{\varepsilon} +O(1)\right].

More surprising, near a cusp in a two dimensional domain, the escape time

E\tau grows algebraically, rather than logarithmically: in the domain bounded between two tangent circles, the escape time is:$$E\backslash tau\; =\; \backslash frac$$ \left(\frac + O(1) \right), where 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 = { (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:$$E\backslash tau\; =\; \backslash fracD\backslash left[\backslash log\; \backslash frac\{1\}\{\backslash varepsilon\}\; +\; \backslash log\; 2\; +\; 2\backslash beta^2\; \backslash right]\; +\; \backslash frac\; \backslash frac\; \backslash log\backslash frac-\; \backslash fracR\_2^2\; +\; O(\backslash varepsilon,\backslash beta^4)R\_2^2.$$

This equation contains two terms of the asymptotic expansion of

E\tau and 2 \epsilon is the angle of the absorbing boundary. The case \beta 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 $$F(x)\; \backslash neq\; 0$$ the gap in the spectrum is not necessarily small between the first and the second eigenvalues, depending on the relative size of the small hole and the force barriers, the particle has to overcome in order to escape. The escape stream is not necessarily Poissonian.

Analytical results

A theorem that relates the Brownian motion escape problem to a (deterministic) partial differential equation problem is the following.

Here

\partial _{n} : = n ⋅ \nabla is the derivative in the direction n , the exterior normal to \partial\Omega. Moreover, the average of the variance can be calculated from the formula$$\backslash bar\; :=\; \backslash frac\; \backslash int\_\; v(x)\; dx=\; \backslash frac\; \backslash int\_\; T^2(x)\; dx\; =:\; T^2$$

The first part of the theorem is a classical result, while the average variance was proved in 2011 by Carey Caginalp and Xinfu Chen.^{[18] [19] [20]}

The escape time has been the subject of a number of studies using the small gate as an asymptotically small parameter. The following closed form result gives an exact solution that confirms these asymptotic formulae and extends them to gates that are not necessarily small.

Another set of results concerns the probability density of the location of exit.

That is, for any (Borel set)

\gamma\subset\partial\Omega , the probability that a particle, starting either at the origin or uniformly distributed in \Omega , exhibiting Brownian motion in \Omega , reflecting when it hits \partial\Omega\setminus\Gamma , and escaping once it hits \Gamma , ends up escaping from \gamma is$$P(\backslash gamma)\; =\; \backslash int\_\; \backslash bar(y)\; dS\_y$$where dS _{y} is the surface element of \partial\Omega at y\in \partial\Omega .

Simulations of Brownian motion escape

In simulation there is a random error due to the statistical sampling process. This error can be limited by appealing to the central limit theorem and using a large number of samples. There is also a discretization error due to the finite size approximation of the step size in approximating the Brownian motion. One can then obtain empirical results as step size and gate size vary. Using the exact result quoted above for the particular case of the circle, it is possible to make a careful comparison of the exact solution with the numerical solution.^{[21] [22]} This illuminates the distinction between finite steps and continuous diffusion. A distribution of exit locations was also obtained through simulations for this problem.

Biological applications

Stochastic chemical reactions in microdomains

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.^[23]

External links

• Applied Mathematics and Computational Biology in Ecole Normale Superieure, Paris
• Carey Caginalp publications and lectures http://www.pitt.edu/~careycag/
• Comptes Rendus paper http://www.pitt.edu/~careycag/paper1.pdf
• ARMA paper http://www.pitt.edu/~careycag/paper2.pdf

Notes and References

1. Schuss . Z. . Singer . A. . Holcman . D. . The narrow escape problem for diffusion in cellular microdomains . Proceedings of the National Academy of Sciences . Proceedings of the National Academy of Sciences USA . 104 . 41 . 2007-09-27 . 0027-8424 . 10.1073/pnas.0706599104 . 16098–16103. 17901203 . 1994903 . 2007PNAS..10416098S . free.
2. D Holcman, Z Schuss, The narrow escape problem SIAM Review 56 (2), 213-257 (2014)
3. Singer . A. . Schuss . Z. . Holcman . D. . Narrow escape and leakage of Brownian particles . Physical Review E . American Physical Society (APS) . 78 . 5 . 2008-11-14 . 1539-3755 . 10.1103/physreve.78.051111 . 051111. 19113099 . 0808.2288 . 2008PhRvE..78e1111S . 8739640 .
4. 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
5. Holcman . D . Schuss . Z . Diffusion escape through a cluster of small absorbing windows . Journal of Physics A: Mathematical and Theoretical . IOP Publishing . 41 . 15 . 2008-04-02 . 1751-8113 . 10.1088/1751-8113/41/15/155001 . 155001. 2008JPhA...41o5001H . 4179599 .
6. Holcman, D., & Schuss, Z. (2015). Stochastic Narrow Escape in Molecular and Cellular Biology: Analysis and Applications. Springer.
7. Cheviakov . Alexei F. . Ward . Michael J. . Straube . Ronny . An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part II: The Sphere . Multiscale Modeling & Simulation . Society for Industrial & Applied Mathematics (SIAM) . 8 . 3 . 2010 . 1540-3459 . 10.1137/100782620 . 836–870. 11858/00-001M-0000-0013-908F-6 . free .
8. Cheviakov . Alexei F. . Zawada . Daniel . Narrow-escape problem for the unit sphere: Homogenization limit, optimal arrangements of large numbers of traps, and the N² conjecture . Physical Review E . American Physical Society (APS) . 87 . 4 . 2013-04-22 . 1539-3755 . 10.1103/physreve.87.042118 . 042118. 23679384 . 2013PhRvE..87d2118C .
9. Coombs . Daniel . Straube . Ronny . Ward . Michael . Diffusion on a Sphere with Localized Traps: Mean First Passage Time, Eigenvalue Asymptotics, and Fekete Points . SIAM Journal on Applied Mathematics . Society for Industrial & Applied Mathematics (SIAM) . 70 . 1 . 2009 . 0036-1399 . 10.1137/080733280 . 302–332. 11858/00-001M-0000-0013-9335-3 . free .
10. D. Holcman Z. Schuss, The dire strait time, SIAM Multiscale Modeling and simulations, 10(4), 1204–1231.
11. Holcman . D . Schuss . Z . Control of flux by narrow passages and hidden targets in cellular biology . Reports on Progress in Physics . IOP Publishing . 76 . 7 . 2013-06-20 . 0034-4885 . 10.1088/0034-4885/76/7/074601 . 074601. 23787818 . 2013RPPh...76g4601H . 2102724 .
12. Holcman . D. . Schuss . Z. . Escape Through a Small Opening: Receptor Trafficking in a Synaptic Membrane . Journal of Statistical Physics . Springer Science and Business Media LLC . 117 . 5–6 . 2004 . 0022-4715 . 10.1007/s10955-004-5712-8 . 975–1014. 2004JSP...117..975H . 6324415 .
13. Singer . A. . Schuss . Z. . Holcman . D. . Eisenberg . R. S. . Narrow Escape, Part I . Journal of Statistical Physics . Springer Science and Business Media LLC . 122 . 3 . 2006-01-20 . 0022-4715 . 10.1007/s10955-005-8026-6 . 437–463. 2006JSP...122..437S . 14014727 . math-ph/0412048 .
14. Singer . A. . Schuss . Z. . Holcman . D. . Narrow Escape, Part II: The Circular Disk . Journal of Statistical Physics . Springer Science and Business Media LLC . 122 . 3 . 2006-01-20 . 0022-4715 . 10.1007/s10955-005-8027-5 . 465–489. math-ph/0412050 . 2006JSP...122..465S . 15765954 .
15. Singer . A. . Schuss . Z. . Holcman . D. . Narrow Escape, Part III: Non-Smooth Domains and Riemann Surfaces . Journal of Statistical Physics . Springer Science and Business Media LLC . 122 . 3 . 2006-01-20 . 0022-4715 . 10.1007/s10955-005-8028-4 . 491–509. 2006JSP...122..491S . 12317568 .
16. Z. Schuss, Theory and Applications of Stochastic Differential Equations (Wiley Series in Probability and Statistics - (1980)
17. Z. Schuss, Theory and Applications of Stochastic Processes. An Analytical Approach. Series: Applied Mathematical Sciences, Vol. 170.
18. Caginalp. Carey . Chen. Xinfu . 2011-02-01 . Analytical and numerical results for first escape time in 2D. Comptes Rendus Mathématique. en . 349 . 3–4 . 191–194 . 10.1016/j.crma.2010.11.024. 1631-073X.
19. Chen. Xinfu . Caginalp. Carey. 2012-01-01. Analytical and Numerical Results for an Escape Problem. Archive for Rational Mechanics and Analysis. en. 203. 1. 329–342 . 10.1007/s00205-011-0455-6 . 2012ArRMA.203..329C . 32394342. 1432-0673.
20. Book: Caginalp, Carey. Analytical and numerical results on escape. University of Pittsburgh. 2011. B. Phil. Thesis.
21. Hughes. Aoife . Morris. Richard. Tomkins. Melissa. 2020-03-31. PyEscape: A narrow escape problem simulator package for Python. Journal of Open Source Software. en . 5. 47 . 2072 . 2020JOSS....5.2072H . 10.21105/joss.02072. 2475-9066. free.
22. Hughes. Aoife. Faulkner. Christine. Morris. Richard J.. Tomkins. Melissa . 2021 . Intercellular communication as a series of narrow escape problems. IEEE Transactions on Molecular, Biological and Multi-Scale Communications. 7. 2. 89–93. 10.1109/TMBMC.2021.3083719. 2332-7804. free.
23. Holcman . D. . Schuss . Z. . Stochastic chemical reactions in microdomains . The Journal of Chemical Physics . AIP Publishing . 122 . 11 . 2005-03-15 . 0021-9606 . 10.1063/1.1849155 . 114710. 15836246 . math-ph/0412089 . 2005JChPh.122k4710H . 845444 .