Thermodynamic limits of sperm swimming precision

Sperm swimming is crucial to fertilise the egg, in nature and in assisted reproductive technologies. Modelling the sperm dynamics involves elasticity, hydrodynamics, internal active forces, and out-of-equilibrium noise. Here we demonstrate experimentally the relevance of energy dissipation for sperm beating fluctuations. For each motile cell, we reconstruct the time-evolution of the two main tail's spatial modes, which together trace a noisy limit cycle characterised by a maximum level of precision $p_{max}$. Our results indicate $p_{max} \sim 10^2 s^{-1}$, remarkably close to the estimated precision of a dynein molecular motor actuating the flagellum, which is bounded by its energy dissipation rate according to the Thermodynamic Uncertainty Relation. Further experiments under oxygen deprivation show that $p_{max}$ decays with energy consumption, as it occurs for a single molecular motor. Both observations can be explained by conjecturing a high level of coordination among the conformational changes of dynein motors. This conjecture is supported by a theoretical model for the beating of an ideal flagellum actuated by a collection of motors, including a motor-motor nearest neighbour coupling of strength $K$: when $K$ is small the precision of a large flagellum is much higher than the single motor one. On the contrary, when $K$ is large the two become comparable.