Fig 4. Fertilization in strong flows and high egg density.

Previous measurements of fertilization probability P_fert(ϵ) for sea urchin S. purpuratus at strong turbulence, characterized by density-normalized dissipation rate ϵ (filled gray triangles) [25, 38] and our corresponding simulations P_fert(α) as function of shear rate α (open blue circles, mean ± SD) match well, using a single fit parameter a = 0.075 that relates dissipation rate ϵ and typical shear rate α (using the known relationship $\alpha \left(ϵ\right)=a\sqrt{ϵ/\nu }$ [24, 26]). Both simulation and experiment are well captured by a minimal theory of a ballistic swimmer in simple shear flow (red), see Sec A in S1 Text. Fertilization probability P_fert rapidly drops above a characteristic flow strength α > 100 s⁻¹, which is consistent with a scale estimate α = 2πv_h/(0.1r_egg) (vertical dotted line). At these high shear rates, active swimming becomes negligible compared to convection. The case of low shear rates is well described by the limit case of a ballistic swimmer in the absence of flow α = 0 s⁻¹ (dotted horizontal line, Eq (5) with P_sperm:egg(t) = 1 − exp(−qt) and rate $q=\pi r_{\text{egg}}^2{v}_h{\rho }_{\text{egg}}$). The fertilizability p_f = 10% is obtained from an independent experiment [25], see Fig B in S1 Text. From the experimental protocol, we estimate a high background concentration c_bg = 500 − 4000 nM of chemoattractant, which renders sperm chemotaxis ineffective. Corresponding results for simulations with co-rotation are shown in Fig C in S1 Text.