A mammalian sperm cell has a head of roughly radius and swims at in a fluid with the…

PHYSICS · Multi-part · Fermi University Physics · Biophysics & Medical Physics · Physical foundations of the living cell · Physical foundations of the living cell

Forces, Machines and Signals, Ch Life at low Reynolds number

Problem

A mammalian sperm cell has a head of roughly a=2 μma = 2\,\mathrm{\mu m} radius and swims at v=150 μm s−1v = 150\,\mathrm{\mu m\,s^{-1}} in a fluid with the viscosity and density of water (η=1.0×10−3 Pa s\eta = 1.0\times10^{-3}\,\mathrm{Pa\,s}, ρ=1.0×103 kg m−3\rho = 1.0\times10^{3}\,\mathrm{kg\,m^{-3}}). Approximate the head as a sphere of the cell's mass density, taken equal to water's. ((a)) Estimate its Reynolds number and state whether it lives in the Stokes limit. ((b)) Estimate the stopping time τm=2ρa2/9η\tau_m = 2\rho a^2/9\eta and the coasting distance vτmv\tau_m once the tail stops beating. ((c)) Egg-derived chemoattractant diffuses with D=1.0×10−9 m2 s−1D = 1.0\times10^{-9}\,\mathrm{m^2\,s^{-1}}. Estimate the Peclet number Pe=va/D\mathrm{Pe} = va/D and comment on whether the sperm cell's own swimming much changes the rate at which it encounters the attractant compared with diffusion alone.

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