Pattern Formation in Polymerizing Actin Flocks: Spirals, Spots, and Waves without Nonlinear Chemistry

We propose a model solely based on actin treadmilling and polymerization which describes many characteristic states of actin-wave formation: spots, spirals, and traveling waves. In our model, as in experiments on cells recovering motility following actin depolymerization, we choose an isotropic low-density initial condition; polymerization of actin filaments then raises the density towards the Onsager threshold where they align. We show that this alignment, in turn, destabilizes the isotropic phase and generically induces transient actin spots or spirals as part of the dynamical pathway towards a polarized phase which can either be uniform or consist of a series of actin-wave trains (flocks). Our results uncover a universal route to actin-wave formation in the absence of any system-specific nonlinear biochemistry, and it may help to understand the mechanism underlying the observation of actin spots and waves in vivo. They also suggest a minimal setup to design similar patterns in vitro.

Figure 1

Representative snapshots for actin pattern formation as described in the main text. (a) $\mathrm{\Gamma }=1$; (b) $\mathrm{\Gamma }=4.3$; (c) $\mathrm{\Gamma }=10$. Other parameters: $r=1.1$, $\mathcal{D}=5$, and ${\mathrm{\Gamma }}_2=0.075$. The scale bar is 50.

Figure 2

Snapshots of the evolution of the $\mathbf{P}$ field during the formation of spots for $\mathrm{\Gamma }=4.3$, $r=1.1$, $\mathcal{D}=5$, and ${\mathrm{\Gamma }}_2=0.075$, shown in Fig. 1.

Figure 3

Real and imaginary parts of the dispersion relation of the uniform polarized phase (iii), for $\mathrm{\Gamma }=0$, 2, 4, 6, 8, 10 (following the sense of the arrows), and with $r=1.1$, $\mathcal{D}=5$, and ${\mathrm{\Gamma }}_2=0.075$.

Figure 4

Phase diagram with curves resulting from the linear stability analysis. They are consistent with numerical simulations (see [29]) and correspond to $\mathcal{D}=5$ and ${\mathrm{\Gamma }}_2$ equals 0.00075, 0.075, 0.1775, 0.3704, $3/8$, 0.3754, and 0.4688 from bottom to top.