Spiral and never-settling patterns in active systems

We present a combined numerical and analytical study of pattern formation in an active system where particles align, possess a density-dependent motility, and are subject to a logistic reaction. The model can describe suspensions of reproducing bacteria, as well as polymerizing actomyosin gels in vitro or in vivo. In the disordered phase, we find that motility suppression and growth compete to yield stable or blinking patterns, which, when dense enough, acquire internal orientational ordering to give asters or spirals. We predict these may be observed within chemotactic aggregates in bacterial fluids. In the ordered phase, the reaction term leads to previously unobserved never-settling patterns which can provide a simple framework to understand the formation of motile and spiral patterns in intracellular actin systems.

Figure 1

Color maps of the density obtained by numerical integration of Eqs. (1) in a box with periodic boundary conditions and an initial isotropic state ($\mathbf{w}=0$) of uniform mean density ${\rho }_0={\rho }_s$, with small random fluctuations. The color bars give the values of the local density $\tilde{\rho }=\rho /{\rho }_s$. All images are for $\tilde{\lambda }=1.4$, $\tilde{D}=0.01$, $\tilde{\alpha }=0.083$ and (from $A$ to $D$) $\tilde{\gamma }=0.50,0.58,0.75,0.95$. The high-density static bacterial dots in $A$ and $B$ have zero or very small local polar order, as in [18]. In $C$ and $D$ polar order builds up in each dot, as highlighted by the blowups of individual dots shown in the bottom row. Here the polarization is displayed as an arrow of length proportional to its magnitude. The color refers to the density, with the same color scheme as indicated in the side bars (see also Supplementary Movies 1 and 2) [34].

Figure 2

The boundary of linear stability of the homogeneous state ($\tilde{\gamma }<1$) in the $(\tilde{\lambda },\tilde{\gamma })$ plane for $\tilde{\alpha }=0$ (dashed curve, blue online), $\tilde{\alpha }=0.083$ (solid curve, red online), and $\tilde{\alpha }=0.30$ (dotted curve, black online). The homogeneous state is unstable in the region to the right of each curve, up to the vertical axis $\tilde{\gamma }=1$. The calculation is described in Appendix pp2. The dots labeled A, B, C, D show the location in parameter space of the images shown in Fig. 1 and refer to $\alpha =0.083$. For a fixed value of $\tilde{\alpha }$, the region of $\tilde{\lambda }$ where the system is unstable grows as alignment increases. Conversely, increasing the birth or death rate for fixed $\tilde{\gamma }$ stabilizes the homogeneous state. This is highlighted in the inset that shows the linear stability boundary in the $(\tilde{\alpha },\tilde{\gamma })$ plane for $\tilde{\lambda }=1.4$. Note that in the inset the horizontal axis is $\sqrt{1-\tilde{\gamma }}$, i.e., alignment increases to the left.

Figure 3

Snapshots of the spreading of a droplet inoculated at the center of the simulation sample for $\tilde{\gamma }=0.83$ (top row) and $\tilde{\gamma }=0.98$ (bottom row), with $\tilde{\alpha }=0.167$, $\tilde{\lambda }=1.40$, and $\tilde{D}=0.01$. The simulation times are 1, 4000, 22 000 and 1, 4000, 14 000 from left to right, respectively, in units of ${\epsilon }^{-1}$. The blowups to the right show the internal structure of the dots, as described in the caption of Fig. 1.

Figure 4

Patterns in the polar state obtained from a uniform initial state with small random fluctuations around $\tilde{\rho }=1$ and $\tilde{\mathbf{w}}=\mathbf{0}$ for $\tilde{\alpha }=0.08$, $\tilde{D}=0.01$, and $\tilde{\lambda }=1.10$. Rings and lanes emerge in (E), (F), (G) at $\tilde{\gamma }=1.11,1.25,2.00$, respectively. We stress that the net total polarization of the lanes shown in frame G is directed at an angle to the long direction of the lanes, resulting in a transverse drifting motion (See Supplementary Movie 4) [34].

Figure 5

The dispersion relation of the mode $s_{+}\left(q\right)$ is plotted as a function of the wave vector $q$ for $\tilde{\alpha }=0.08$, $\tilde{D}=0.01$, and $\tilde{\lambda }=1.2$. The curves from A to D correspond to $\tilde{\gamma }=0,0.55,0.9$, and $0.99$. The homogeneous isotropic state is unstable for $s_{+}\left(q\right)>0$ and stable for $s_{+}\left(q\right)<0$. The instability is enhanced upon increasing the alignment strength $\tilde{\gamma }$ due to the decrease of the decay rate ${\epsilon }_r$ of the polarization as the mean-field transition at $\tilde{\gamma }=1$ is approached from below. The instability exists in a band of wave vectors $q_1<q<q_2$. When $\alpha <{\epsilon }_r$, as in typical bacterial suspensions, the decay rate at zero wave vector is controlled by $\alpha$ (curves A to C), while when $\alpha >{\epsilon }_r$, it is controlled by ${\epsilon }_r$ (curve D).

Figure 6

Density heat maps for $\tilde{D}=0.01$, $\tilde{\lambda }=1.2$, and $\tilde{\alpha }=0$ showing macroscopic phase separation. The color scale is indicated to the right, and the values refer to $\tilde{\rho }=\rho /{\rho }_s$. From A to D, $\tilde{\gamma }=0.50,0.75,0.96$, and $1.11$. The time evolution shows the coarsening of the structures into a single large cluster. Upon increasing the alignment strength, the cluster starts to display polar order, as highlighted in the blowups in the bottom row showing maps of the polarization field.

Figure 7

Heat maps of density in the polar region obtained by numerical integration of Eqs. (D4) for $\tilde{\lambda }=1.4$, $\tilde{\alpha }=0.08$, and $\tilde{D}=0.01$ and (from left to right) $\tilde{\gamma }=1.11,1.25$, and $1.67$.

Figure 8

Heat maps of density in the polar region obtained by numerical integration of the simplified equations, Eqs. (1) of the paper for $\tilde{\lambda }=1.4$, $\tilde{\alpha }=0.08$, $\tilde{D}=0.01$ and (from left to right) $\tilde{\gamma }=1.11,1.25$, and $1.67$.