Fig. 4

Simulations reproduce the beat and steering dynamics. a Stroboscopic view of experimental (red) and simulated (blue) beat pattern using an active semi-flexible filament and anisotropic drag force. Time interval between snapshots (fading lines) is Δt = 2 ms. Simulation parameters: κ = 1.9 nN μm², T ₁ ~ 0.65 nN μm, T ₂ = 0.15T ₁, ψ = 2.26, ω_o = 30 Hz, L = 41 μm, ξ_⊥/ξ_∥ = 1.81, ξ _∥ = 0.69 fNs μm⁻² and λ/L = 0.65 (Supplementary Movie 1). Scale bar represents 10 µm. b Representative simulation of flagellar beat with a second-harmonic amplitude T ₂ = 0.3T ₁. The mid-piece is aligned for visualization. The time interval between snapshots is Δt = 4 ms. The red thick line shows the non-symmetric trajectory of the flagellum tip. Scale bar represents 10 µm. c Rotation velocity Ω vs. normalized wavelength λ. Note that Ω has been normalized to the second-harmonic torque amplitude. The inset shows that Ω scales linearly with T* = T ₂/T ₁. d Average curvature < C > vs. phase ψ of the second-harmonic torque. Note that the curvature has been normalized by T* = T ₂/T ₁. e Simulated sperm trajectory resulting from a slowly changing phase ψ over time (phase indicated by the color of the trajectory). By modulating the phase, sperm swim on curvilinear paths (Supplementary Movie 2). Scale bar represents 20 µm. f Average dissipated power P _d (blue) vs. generated power P _g (red) in simulations, and average dissipated power measured in experiments (gray lines). The simulated dissipated power shows good agreement with the experimental results. Of note, power is relocated along the flagellum