Search for a small egg by spermatozoa in restricted geometries

J. Yang, I. Kupka, Z. Schuss, D. Holcman*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)423-446
Number of pages24
JournalJournal of Mathematical Biology
Issue number2
StatePublished - 1 Aug 2016


  • Asymptotic estimates
  • Narrow escape
  • Numerical simulations
  • Random search process
  • Spermatozoa


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