Synchronized rotation in swarms of magnetotactic bacteria

Self-organizing behavior has been widely reported in both natural and artificial systems, typically distinguishing between temporal organization (synchronization) and spatial organization (swarming). Swarming has been experimentally observed in systems of magnetotactic bacteria under the action of external magnetic fields. Here we present a model of ensembles of magnetotactic bacteria in which hydrodynamic interactions lead to temporal synchronization in addition to the swarming. After a period of stabilization during which the bacteria form a quasiregular hexagonal lattice structure, the entire swarm begins to rotate in a direction opposite to the direction of the rotation of the magnetic field. We thus illustrate an emergent mechanism of macroscopic motion arising from the synchronized microscopic rotations of hydrodynamically interacting bacteria, reminiscent of the recently proposed concept of swarmalators.

Figure 1

Stationary rotating structures at $N=3,4,7,19, \lambda =-0.1$. Positions of particles are averaged over 10 periods of the rotating magnetic field.

Figure 2

Formation of the quasiregular hexagonal structure shown by the trajectories of the centers of the bacterial rotations ${\vec{R}}_i$. Initial random positions are constrained to the square $[-5,5;-5,5], N=7, \lambda =-0.1$.

Figure 3

Formation of the quasiregular hexagonal structure shown by the trajectories of the centers of the bacterial rotations ${\vec{R}}_i$. Initial random positions are constrained to the square $[-5,5;-5,5], N=19, \lambda =-0.1$.

Figure 4

Trajectories of the bacteria at established mean positions of the centers of the bacterial rotations on the vertices of the quasiregular hexagonal structure (see Fig. 1) for 10 periods of a rotating field. Initial random positions are constrained to the square $[-5,5;-5,5], N=7, \lambda =-0.1$.

Figure 5

Trajectories of the bacteria at established mean positions of the centers of the bacterial rotations on the vertices of the quasiregular hexagonal structure (see Fig. 1) for 10 periods of a rotating field. Initial random positions are constrained to the square $[-5,5;-5,5], N=19, \lambda =-0.1$.

Figure 6

Particle phases in a single swarm, $N=3, \lambda =-0.1$. (a) Time period from a random initial condition up to the establishment of a quasiregular lattice. (b) Synchronized rotations of the bacteria in the swarm after the establishment of the ordered lattice state.

Figure 7

Particle phases in the boundary of a hexagonal swarm with $N=7, \lambda =-0.1$. (a) Time period from a random initial condition up to the establishment of a quasiregular lattice. (b) Synchronized rotations of the bacteria in the swarm after the establishment of the ordered lattice state.

Figure 8

Mean angular velocity of the swarm as a function of the gyration radius, $N=7, \lambda =-0.1$.

Figure 9

Mean angular velocity as a function of the bacteria count $N, \lambda =-0.1$.

Figure 10

Mean angular velocity as a function of $\lambda$ at $N=7$ (filled circles), 19 (open circles), 40 (squares). Lines are fits by second order polynomials.

Figure 11

Mean angular velocity of synchronized rotation in a swarm, $N=3$. The circles denote numerical data, the line shows a quadratic fit, satisfactory for small $\lambda$.

Figure 12

Hexatic order parameter as a function of time for $N=7$ (top) and $N=19$ (bottom). The random initial condition is different from the one used for Figs. 2 and 3, $\lambda =-0.1$.

Figure 13

Establishment of the order as measured by deviations in the angles obtained from the Delaunay triangulation from the angles of a regular triangle ($\pi /3$), $\lambda =-0.1$. (a) $N=7$; (b) $N=19$.

Figure 14

Establishment of the order as measured by deviations in the angles obtained from the Delaunay triangulation from the angles of a regular triangle ($\pi /3$) at different gyration radius values for the swarm $(0.39;0.66;0.88)$. Disorder of the synchronized swarm increases as the gyration radius is decreased, $N=7, \lambda =-0.1$.