Time-reversal symmetric component of the flagellar beat enhances the translational and rotational velocities of isolated Chlamydomonas flagella

The beating of cilia and flagella is essential to perform many important biological functions, including generating fluid flows on the cell surface or propulsion of micro-organisms. In this work, we analyze the motion of isolated and demembranated flagella from green algae Chlamydomonas reinhardtii, which act as ATP-driven micro-swimmers. The waveform of the Chlamydomonas beating flagella has an asymmetric waveform that is known to involve the superposition of a static component, corresponding to a fixed, intrinsic curvature, and a dynamic wave component traveling in the base-to-tip direction at the fundamental beat frequency, plus higher harmonics. Here, we demonstrate that these modes are not sufficient to reproduce the observed flagella waveforms. We find that two extra modes play an essential role to describe the motion: first, a time-symmetric mode, which corresponds to a global oscillation of the axonemal curvature, and second, a secondary tip-to-base wave component at the fundamental frequency that propagates opposite to the dominant base-to-tip wave, albeit with a smaller amplitude. Although the time-symmetric mode cannot, by itself, contribute to propulsion (scallop theorem), it does enhance the translational and rotational velocities of the flagellum by approximately a factor of 2. This mode highlights a long-range coupled on/off activity of force-generating dynein motors and can provide further insight into the underling biology of the ciliary beat.