title: Search for a small egg by spermatozoa in restricted geometries.
abstract: The search by swimmers for a small target in a bounded domain is ubiquitous in cellular biology, where a prominent case is that of the search by spermatozoa for an egg in the uterus. This is one of the severest selection processes in animal reproduction. We present here a mathematical model of the search, its analysis, and numerical simulations. In the proposed model the swimmers' trajectories are rectilinear and the speed is constant. When a trajectory hits an obstacle or the boundary, it is reflected at a random angle and continues the search with the same speed. Because hitting a small target by a trajectory is a rare event, asymptotic approximations and stochastic simulations are needed to estimate the mean search time in various geometries. We consider searches in a disk, in convex planar domains, and in domains with cusps. The exploration of the parameter space for spermatozoa motion in different uterus geometries leads to scaling laws for the search process.
authors.affiliation: Applied Mathematics and Computational Biology, Ecole Normale Supérieure, IBENS, 46 rue d'Ulm, 75005, Paris, France.|School of Life and Environmental Sciences, Guilin University of Electronic Technology, Guilin, 541004, Guangxi, China.
authors.forename: J
authors.initials: J
authors.lastname: Yang
authors.affiliation: Applied Mathematics and Computational Biology, Ecole Normale Supérieure, IBENS, 46 rue d'Ulm, 75005, Paris, France.
authors.forename: I
authors.initials: I
authors.lastname: Kupka
authors.affiliation: Department of Mathematics, Tel-Aviv University, 69978, Tel-Aviv, Israel.
authors.forename: Z
authors.initials: Z
authors.lastname: Schuss
authors.affiliation: Applied Mathematics and Computational Biology, Ecole Normale Supérieure, IBENS, 46 rue d'Ulm, 75005, Paris, France. david.holcman@ens.fr.|Mathematical Institute, University of Oxford, Andrew Wiles Building, Woodstock Rd, Oxford, OX2 6GG, UK. david.holcman@ens.fr.
authors.forename: D
authors.initials: D
authors.lastname: Holcman
mesh_terms: D000818:Animals; D055592:Biophysical Phenomena; D005260:Female; D006801:Humans; D008297:Male; D008954:Models, Biological; D010063:Ovum; D013094:Spermatozoa / Q000502:physiology*; D014599:Uterus / Q000033:anatomy & histology
keywords: Asymptotic estimates; Narrow escape; Numerical simulations; Random search process; Spermatozoa