Fig 1.
Sperm chemotaxis in external flow.
(A) Simulated, three-dimensional concentration field c(r) of chemoattractant released from a freely-rotating, spherical egg (yellow sphere) suspended in unsteady shear flow as a model of small-scale turbulence. An exemplary simulated sperm cell (trajectory in red) finds the elongated concentration filament by chance and subsequently ‘surfs’ along the filament by chemotaxis. (B) Same as (A), but for the prototypical idealization of simple shear flow vext(r) = αy ex accounting for convection and co-rotation by the external flow. We obtain a generic form of the concentration filament, Eq (1), and characterize surfing along the filament analytically as a damped oscillation. Parameters correspond to sea urchin A. punctuala, assuming continuous release of chemoattractant at constant rate for an exposure time tmax = 6 min. Constant shear rate α = 0.17 s−1 in (B), corresponding to root-mean-square shear rate of (A). Same color-code for concentration in (A) and (B), but different level sets. We use a generic theoretical description of helical sperm chemotaxis, see Methods and materials for details (helix radius r0 = 7 μm not visible at length-scale of figure). Rotating views of the 3D images are provided as S1 and S2 Movies.
Fig 2.
Sperm-egg-encounter probability displays maximum as function of shear rate in simulations for sea urchin sperm at physiological flow rates.
Probability Psperm:egg(α) that a single sperm cell finds an egg as function of external shear rate α. Simulations account for flow-induced distortion of concentration fields into long filaments as well as convection and co-rotation of sperm cells by the flow (green triangles, mean ± SD). Without co-rotation results change only marginally (blue circles). Simulation results agree with predictions from our theory of filament surfing (red, presented below). Without sperm chemotaxis, the encounter probability is virtually zero (<10−5, black). Our theory has a single fit parameter, the flux of sperm cells arriving at the filament, jout = 0.063 m−2s−1. This value matches in magnitude the limit jout = ρegg vh/4 = 0.04 m−2s−1 for a ballistic swimmer with random initial conditions, see Sec E in S1 Text for details. Parameters as in Fig 1B.
Fig 3.
Comparison to experiment at moderate shear rates and short exposure time.
Fertilization probability Pfert(α) that an egg becomes fertilized as function of external shear rate α from previous experiments with red abalone H. rufescens gametes in a Taylor-Couette chamber (filled gray triangles: with chemotaxis, open gray triangles: inhibited chemotaxis; for α = 0 s−1 a different experimental protocol was used) [20] and our corresponding simulations (filled blue circles: with chemotaxis, open blue circles: without chemotaxis, mean ± SD). We find reasonable agreement using a single fit parameter, fertilizability pf ≈ 60%, which characterizes the fraction of sperm-egg encounters that result in successful fertilization, see Eq (5). From the experimental protocol, we estimate a background concentration cbg ∼ 4 nM of chemoattractant. While our theory of filament surfing does not directly apply due to this high background concentration, a near-field estimate (red line) yields a similar decay of fertilization probability as function of shear rate α. The single fit parameter of the theory, jout = 4.8 ⋅ 103 m−2s−1, is again consistent with the limit jout = ρegg vh/4 = 7.5 ⋅ 103 m−2s−1 of a ballistic swimmer with random initial conditions (note the the higher value of jout compared to Fig 2 due to higher egg density). For simplicity, simulations do not account for co-rotation of sperm cells, see Fig A in S1 Text for results with co-rotation.
Fig 4.
Fertilization in strong flows and high egg density.
Previous measurements of fertilization probability Pfert(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 Pfert(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 [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 Pfert rapidly drops above a characteristic flow strength α > 100 s−1, which is consistent with a scale estimate α = 2πvh/(0.1regg) (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 Psperm:egg(t) = 1 − exp(−qt) and rate
). The fertilizability pf = 10% is obtained from an independent experiment [25], see Fig B in S1 Text. From the experimental protocol, we estimate a high background concentration cbg = 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.
Fig 5.
Proposed mechanism explaining optimal flow strength for sperm chemotaxis.
Egg cells (yellow) release signaling molecules (blue) that guide sperm cells of marine species with external fertilization (red). External flows (black arrows) stretch concentration gradients into millimeter-long filaments. If sperm cells encounter such filaments, they can “surf” by chemotaxis towards the egg. In strong flows, however, sperm cells may fail to follow the filament after encounter, because the effective diameter of filaments is too small. Additionally, in very long filaments, sperm cell may not reach the egg within the sperm-egg exposure time (which is set by the lifetime of the smallest eddies for turbulent flow). Thus, it is not the total volume of the chemoattractant plume that determines fertilization success, but the geometric shape of filaments. The competition between increasing filament length, which favors sperm-egg encounters, and decreasing filament diameter, which jeopardizes filament surfing, sets an optimal flow strength that maximizes sperm-egg encounters. The optimal flow strength predicted by our theory matches physiological flow strengths in typical habitats.