Nicolas Bruot

Collective synchronization states in arrays of driven colloidal oscillators

The phenomenon of metachronal waves in cilia carpets has been well known for decades; these waves are widespread in biology, and have fundamental physiological importance. While it is accepted that in many cases cilia are mainly coupled together by the hydrodynamic velocity field, a clear understanding of which aspects determine the collective wave properties is lacking. It is a difficult problem, because both the behavior of the individual cilia and their coupling together are nonlinear. In this work, we coarse-grain the degrees of freedom of each cilium into a minimal description in terms of a configuration-based phase oscillator. Driving colloidal particles with optical tweezers, we then experimentally investigate the coupling through hydrodynamics in systems of many oscillators, showing that a collective dynamics emerges. This work generalizes to a wider class of systems our recent finding that the non-equilibrium steady state can be understood based on the equilibrium properties of the system, i.e. the positions and orientations of the active oscillators. In this model system, it is possible to design configurations of oscillators with the desired collective dynamics. The other face of this problem is to relate the collective patterns found in biology to the architecture and behavior of individual active elements.

Long-range interactions, wobbles, and phase defects in chains of model cilia

Eukaryotic cilia and flagella are chemo-mechanical oscillators capable of generating long-range coordinated motions known as metachronal waves. Pair synchronization is a fundamental requirement for these collective dynamics, but it is generally not sufficient for collective phase-locking, chiefly due to the effect of long-range interactions. Here we explore experimentally and numerically a minimal model for a ciliated surface: hydrodynamically coupled oscillators rotating above a no-slip plane. Increasing their distance from the wall profoundly affects the global dynamics, due to variations in hydrodynamic interaction range. The array undergoes a transition from a traveling wave to either a steady chevron pattern or one punctuated by periodic phase defects. Within the transition between these regimes the system displays behavior reminiscent of chimera states.

Emergence of polar order and cooperativity in hydrodynamically coupled model cilia

As a model of ciliary beat, we use two-state oscillators that have a defined direction of oscillation and have strong synchronization properties. By allowing the direction of oscillation to vary according to the interaction with the fluid, with a timescale longer than the timescale of synchronization, we show in simulations that several oscillators can align in a direction set by the geometrical configuration of the system. In this system, the alignment depends on the state of synchronization of the system, and is therefore linked to the beat pattern of the model cilia. By testing various configurations from two to 64 oscillators, we deduce empirically that, when the synchronization state of neighbouring oscillators is in phase, the angles of the oscillators align in a configuration of high hydrodynamic coupling. In arrays of oscillators that break the planar symmetry, a global direction of alignment emerges reflecting this polarity. In symmetric configurations, where several directions are geometrically equivalent, the array still displays strong internal cooperative behaviour. It also appears that the shape of the array is more important than the lattice type and orientation in determining the preferred direction.

Investigating hydrodynamic synchronisation using holographic optical tweezers

Coordinated motion at low Reynolds number is widely observed in biological micro-systems, but the underlying mechanisms are often unclear. A holographic optical tweezers system is used to experimentally study this phenomenon, by employing optical forces to drive a pair of coplanar microspheres in circular orbits with a constant tangential force. In this system synchronisation is caused by hydrodynamic forces arising from the motion of the two spheres. The timescales of their synchronisation from large initial phase differences are explored and found to be dependent on how stiffly the microspheres are confined to their circular orbits. These measured timescales show good agreement with numerical simulations.