Phase-dependent forcing and synchronization in the three-sphere model of Chlamydomonas

The green alga Chlamydomonas swims with synchronized beating of its two flagella, and is experimentally observed to exhibit run-and-tumble behaviour similar to bacteria. Recently, we studied a simple hydrodynamic three-sphere model of Chlamydomonas with a phase-dependent driving force that can produce run-and-tumble behaviour when intrinsic noise is added, due to the nonlinear mechanics of the system. Here, we consider the noiseless case and explore numerically the parameter space in the driving force profiles, which determine whether or not the synchronized state evolves from a given initial condition, as well as the stability of the synchronized state. We find that phase-dependent forcing, or a beat pattern, is necessary for stable synchronization in the geometry we work with. The phase-dependent forcing allows this simple model of Chlamydomonas to produce a rich variety of behaviours.