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Collective effects of active particles and oscillators

Working Group 8.43

Active particles are defined by their ability for converting stored energy (partially) into directed motion. Inter-particle interactions and/or interactions between particles and the surrounding medium may give rise to collective effects and structures far from equilibrium. Examples are primarily found in living nature: Colonies of the bacterium Myxococcus xanthus, for instance, form wavelike patterns and the beating of cilia, hairlike projections of eukaryotic cells, appear highly coordinated. A single bacterium is a prototypic active particle which converts nutrition into directed motion. A cilium, on the other hand, can be characterized as an active oscillator: Similar to a mechanical clock, it converts stored energy into periodic motion.

In biological systems, the interactions of particles and therefore the origin of collective effects are typically unknown as the experimental discrimination between adhesive, steric, hydrodynamic, and chemotactic interactions is very challenging. In this context, modeling active particles and computing collective effects may help to verify hypothetical mechanisms and rule out others.

 

Synchronization of coupled oscillators

A basic collective phenomenon of several oscillators is synchronization where the phase difference between oscillators is constant in time. We investigate various scenarios in which synchronization of coupled oscillators is prominent. In a minimal model of beating cilia, hydrodynamic interactions are sufficient to explain synchronization and the observed metachronal waves [Niedermayer et.al., Chaos 18, 2008]. In some biological systems, the motion of individual units is also coupled to the synchronization of their internal degrees of freedom. In this context, an interesting theoretical problem is the influence of oscillator motion on synchronization. This issue has been studied by mean of a particle based model in conjunction with a field theory. It turns out that pattern formation relies on particle transport. In particular, active motion facilitates the formation of synchronized states [Großmann et.al., Phys.Rev.E., 2016].

 

Diffusion of active particles with directional reversal

Various bacterial species, such as Myxococcus xanthus and Pseudomonas putida, exhibit a so-called "run-and-reverse" motion: Persistent motion is interrupted by sudden directional reversal. Our analysis of the "run-and-reverse" motion pattern suggests the existence of finite parameter values which maximize locomotion efficiency [Großmann et.al., New J.Phys., 2016].

 

Self-propelled rods and myxobacteria

Motivated by the diverse dynamical structures which are observed during the life cycle of Myxococcus xanthus, various models of active ("self-propelled") rods have been developed, simulated, analyzed, and compared with measurements. Extended self-propelled rods in two dimensions with excluded volume interactions exhibit a strong tendency for cluster formation [Peruani et.al., Phys. Rev. E, 2006]. These model predictions have been confirmed in experiments with myxobacteria [Peruani et.al., Phys. Rev. Lett., 2012]. Simplified models allow for the simulation of millions of self-propelled rods and exhibit fascinating macroscopic patterns, such as nematic bands and turbulent fragments [Ginelli et.al., Phys. Rev. Lett., 2010]. The analysis of the nonlinear dynamics of these bands and their description via effective hydrodynamic equations are subject to current research.

 

Mesoscale turbulence and vortex arrays

Recently, experiments with dense suspensions of Bacillus subtilis revealed dynamical vortex patterns. These structures are reminiscent of turbulence, but display some characteristics which turbulence of classical fluids does not feature. In particular, effective length scales emerge which are larger than individual particles, but smaller than the system size. The phenomenon is termed mesoscale turbulence. The structures are described qualitatively by a simple model of self-propelled particles with competing interactions: Particles align due to steric interactions, but global order is prevented by an opposing, long-range interaction [Großmann et.al., Phys.Rev.Lett., 2014; Großmann et.al., Eur.Phy.J.Spec.Top. 224, 2015]. These rules give rise to an instability of the velocity field of particles which determines convective mass transport. In consequence, complex patterns such as mesoscale turbulence and vortex arrays emerge, see Fig. 3.

Fig. 3: Mesoscale turbulence (left) and vortex array (right) in a simulation of self-propelled particles with competing interactions.