For teleost fish fertilisation, sperm must proceed through a small opening on the egg surface, referred to as the micropyle. In this paper, we have used boundary element simulations to explore whether the hydrodynamic attraction between sperm and a fish egg can be a sperm guidance cue. Hydrodynamical egg-sperm interactions alone do not increase the chances of an egg encounter, nor do they induce surface swimming for virtual turbot fish sperm across smooth spheres with a diameter of 1mm, which is representative of a turbot fish egg. When a repulsive surface force between the virtual turbot sperm and the egg is introduced, as motivated by surface charge and van-der-Waals interactions for instance, we find that extended surface swimming of the virtual sperm across a model turbot egg occurs, but ultimately the sperm escapes from the egg. This is due to the small exit angle of the scattering associated with the initial sperm-egg interaction at the egg surface, leading to a weak drift away from the egg, in combination with a weak hydrodynamical attraction between both gametes, though the latter is not sufficient to prevent eventual escape. The resulting transience is not observed experimentally but is a detailed quantitative difference between theory and observation in that stable surface swimming is predicted for eggs with radii larger than about 1.8mm. Regardless, the extended sperm swimming trajectory across the egg constitutes a two-dimensional search for the micropyle and thus the egg is consistently predicted to provide a guidance cue for sperm once they are sufficiently close. In addition, the observation that the virtual turbot sperm swims stably next to a flat plane given repulsive surface interactions, but does not swim stably adjacent to a turbot-sized egg, which is extremely large by sperm-lengthscales, also highlights that the stability of sperm swimming near a boundary is very sensitive to geometry.

The Hakubi Center for Advanced Research Kyoto University Kyoto 606 8501 Japan; Research Institute for Mathematical Sciences Kyoto University Kyoto 606 8502 Japan

University of South Bohemia in Ceske Budejovice Faculty of Fisheries and Protection of Waters South Bohemian Research Center of Aquaculture and Biodiversity of Hydrocenoses Research Institute of Fish Culture and Hydrobiology Czech Republic

Wolfson Centre for Mathematical Biology Mathematical Institute University of Oxford Oxford OX2 6GG UK

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