Energetics of synchronisation for model flagella and cilia

TL;DR

This paper investigates the energetics of synchronization in flagella and cilia using three models, demonstrating that energy minimization influences their phase relationships and collective dynamics.

Contribution

It revisits Taylor's hypothesis by explicitly calculating flow fields in three models, showing energy minimization governs synchronization phases in flagella and cilia.

Findings

Energy minimization determines phase differences in waving sheets.

In-phase or opposite-phase motion depends on relative positions of spheres.

In-phase whirling of rods minimizes energy in cilia interactions.

Abstract

Synchronisation is often observed in the swimming of flagellated cells, either for multiple appendages on the same organism or between the flagella of nearby cells. Beating cilia are also seen to synchronise their dynamics. In 1951, Taylor showed that the observed in-phase beating of the flagella of co-swimming spermatozoa was consistent with minimisation of the energy dissipated in the surrounding fluid. Here we revisit Taylor's hypothesis for three models of flagella and cilia: (1) Taylor's waving sheets with both longitudinal and transverse modes, as relevant for flexible flagella; (2) spheres orbiting above a no-slip surface to model interacting flexible cilia; and (3) whirling rods above a no-slip surface to address the interaction of nodal cilia. By calculating the flow fields explicitly, we show that the rate of working of the model flagella or cilia is minimised in our three models for (1) a phase difference depending on the separation of the sheets and precise waving kinematics; (2) in-phase or opposite-phase motion depending on the relative position and orientation of the spheres; and (3) in-phase whirling of the rods. These results will be useful in future models probing the dynamics of synchronisation in these setups.