## Alexei Cheviakov & Michael Ward on Narrow Escape Problems

#### Fast Breaking Papers Commentary, June 2011

Article: AN ASYMPTOTIC ANALYSIS OF THE MEAN FIRST PASSAGE TIME FOR NARROW ESCAPE PROBLEMS: PART II: THE SPHERE
Authors: |

Alexei Cheviakov and Michael Ward talk with ScienceWatch.com and answer a few questions about this month's Fast Breaking Paper paper in the field of Mathematics.

**Why do you think your paper is highly
cited?**

We believe that the paper is highly cited for several factors. From a biophysical viewpoint, the problem is very topical and there has been much recent work in the biophysics community by the research groups of David Holcman and of Olivier Benichou in calculating the mean first passage time for diffusing particles to reach small distant target sites.

From a mathematical viewpoint, the problem is essentially equivalent to determining the rate of heat loss through a given number of small windows on the boundary of an almost leakproof container. Although the problem we studied is very easy to formulate, a rather intricate analysis was needed to determine an explicit asymptotic formula for how the heat loss depends on the specific locations of the absorbing windows.

The new specific approximate formula that we derived for the mean first passage formula can readily be used by biophysicists in an explicit scaling law. However, in addition, our result shows that the problem of determining the mean first passage time is very closely connected to the well-known and long-standing mathematical problem (called the Fekete point problem) relating to the determination of the minimal energy configuration of repelling Coulomb particles on the surface of a sphere. The problem that we studied, and the results obtained, should have some appeal to two diverse scientific communities (biophysics, and mathematics [approximation theory]).

Coauthor Michael Ward.

Moreover, the mathematical method of matched asymptotic expansions used in the paper is also rather general and can perhaps be carried over to many new related problems.

**Does it describe a new discovery, methodology, or
synthesis of knowledge?**

It does describe a number of new results, as outlined below. In particular, it offers a general framework for which narrow escape problems in three dimensional domains can be considered.

**Would you summarize the significance of your paper
in layman's terms?**

Narrow escape problems considered in the paper are important in various areas involving passage of particles that undergo Brownian motion through membranes with holes. In a biophysical context, it gives the expected time for a Brownian particle to reach some distant target site, which then initiates some specific biological function.

Our paper contains new results on how to compute times needed for particles to escape from domains with small traps (holes) on the boundary. In particular, a higher-order term has been derived which depends on overall spatial configuration of traps on the boundary of the domain. Global optimization has been performed for N traps on the sphere to find optimal locations that minimize escape times. Optimal trap locations were shown to be the same locations of N repelling electrons on the boundary of a sphere (at least for not-very-large N).

The results are also related to heat transfer rates. Very roughly speaking, one can think of optimal locations of N traps on a sphere as an arrangement of N identical windows on the boundary of a spherical room that makes the room cool down in the quickest way.

**How did you become involved in this research, and
how would you describe the particular challenges, setbacks, and
successes that you've encountered along the way?**

Our interest in this line of inquiry was triggered through reading many interesting papers relating to the narrow escape problem by David Holcman's research group (together with Zeev Schuss from Tel Aviv University and Amit Singer from Princeton).

These outstanding scientists are strongly connected to the biophysical applications of the theory. Given that we are, primarily, theoretical applied mathematicians who advance and develop semi-rigorous mathematical techniques, our main goal was to show how a systematic asymptotic method could be powerfully used to extract approximate, but explicit, formulae for the mean first passage time in rather general situations.

**Where do you see your research leading in the
future?**

The future directions include possible applications to specific processes in cell biology, whereby more biologically realistic scenarios are taken into account. Such processes include intermittent directed motion, binding-unbinding events on sticky surfaces, and subdiffusion-type behavior due to the crowded cytoplasm. From a mathematical viewpoint, we are also interested in considering domains with a large number of surface or volume traps, and to see how our results relate to those that can be obtained with homogenization theory.

**Alexei Cheviakov, M.Sc., Ph.D.
College of Arts & Science
University of Saskatchewan
Saskatoon, SK, Canada**

**Michael Ward, B.Sc., Ph.D.
Department of Mathematics
University of British Columbia
Vancouver, BC, Canada**

**Dr. Ronny Straube
Max Planck Institute for Dynamics of Complex Technical Systems
Magdeburg,
Germany**Web

KEYWORDS: NARROW ESCAPE, MEAN FIRST PASSAGE TIME, MATCHED ASYMPTOTIC EXPANSIONS, SURFACE NEUMANN GREEN'S FUNCTIONS, DISCRETE VARIATIONAL PROBLEM, LOGARITHMIC SWITCHBACK TERMS, EIGENVALUE, DIFFUSION, PERTURBATION, WINDOWS, DOMAINS, TRAPS.