Symmetry-Breaking Motility

Locomotion of bacteria by actin polymerization and in vitro motion of spherical beads coated with a protein catalyzing polymerization are examples of active motility. Starting from a simple model of forces locally normal to the surface of a bead, we construct a phenomenological equation for its motion. The singularities at a continuous transition between moving and stationary beads are shown to be related to the symmetries of its shape. Universal features of the phase behavior are calculated analytically and confirmed by simulations. Fluctuations in velocity are shown to be generically non-Maxwellian and correlated to the shape of the bead.

Figure 1

Phase diagram in the $a\mathrm{\text{−}}b$ plane for the motion of a spherical bead in the vicinity of continuous transitions for $a=a_c$ and $b_{c1}<b<b_{c2}$. For $a>a_c$, the bead moves with constant nonzero velocity $\mathbf{v}=\mathrm{const}$, the stationary $\mathbf{v}=\mathbf{0}$ state being unstable. For small $a$, the reverse is true and the bead is motionless. In the gray regions, both stationary and moving states are locally stable.

Figure 2

Rescaled bead speed versus rescaled time for $\mu =3$, $c=0.2$, $b=1.5$ ($b_{c1}\approx 0.83$ and $b_{c2}\approx 2.17$), and various values of $a>a_c=1$. The bead begins (very nearly) at rest at $t=0$ with a small random actin distribution. The curves are nearly perfectly superimposed.