Brownian search for targets hidden in cusp-like pockets: Progress and Applications
Applied Mathematics and Computational Biology, Ecole Normale Supérieure, 46 rue d'Ulm, 75005 Paris, France
Received: 24 October 2014
Revised: 5 November 2014
Published online: 15 December 2014
We report here recent progress in computing the search time for a stochastic particle to find a small target hidden in cusp-like pockets. The target is a small segment in dimension two, a small hole or a narrow ribbon in dimension three, placed at the end of a cusp. The asymptotic analysis of the diffusion equation reveals the role of the local geometry, and a mathematical difficulty comes from the boundary layer near the target. The methods are conformal mapping and matching asymptotic. We present applications in cell biology where cellular activation occurs when a diffusing particle finds a hidden site. This is the case during vesicular fusion initiated after a protein located between the vesicular and cell membranes binds to several diffusing calcium ions. Another example is a drug activation site located inside a deep molecular pocket. The analytical formulas clarify the role of small parameters.
© EDP Sciences, Springer-Verlag, 2014