Localized States in Active Fluids

Biological active matter is typically tightly coupled to chemical reaction networks affecting its assembly-disassembly dynamics and stress generation. We show that localized states can emerge spontaneously if assembly of active matter is regulated by chemical species that are advected with flows resulting from gradients in the active stress. The mechanochemical localized patterns form via a subcritical bifurcation and for parameter values for which patterns do not exist in absence of the advective coupling. Our work identifies a generic mechanism underlying localized cellular patterns.

Figure 1

Localized patterns (LPs). (a) Density (black lines) and velocity (pink line) profiles of a LP without internal peaks, ${\mathrm{LP}}_0$. (b)–(g) Multipeaked ${\mathrm{LP}}_i$, where $i=2,\dots ,7$ is the number of internal peaks. Parameter values as in SM Table II [28], with ${\mathrm{\Omega }}_{d,0}=0$, ${\mathrm{\Omega }}_d=10$, $Z=15$, and $\mathrm{\Omega }=6$ (a), 10 (b),(c), 14 (d)–(f), 15.5 (g).

Figure 2

Slanted snaking. Bifurcation diagram of localized patterns (LPs). Unstable branches are not shown. In the gray-colored region, ${\mathrm{LP}}_0$ loses stability to oscillatory localized states. $\overline{{({\partial }_{X}C)}^2}=(1/L){\int }_{-L/2}^{L/2}{({\partial }_{X}C)}^{2}dX$. Parameters and labels as in Fig. 1.

Figure 3

Spikes vs localized patterns (LPs). (a) Profile of a spike. Parameter values as in SM Table II [28], with $\mathrm{\Omega }=0$, ${\mathrm{\Omega }}_{d,0}=3$, ${\mathrm{\Omega }}_d=0$, $Z=20$. (b) Spatial dynamics’ orbits corresponding to the spike in panel (a), solid line, and ${\mathrm{LP}}_0$ in Fig. 1, dashed line, projected onto the $(C,dC/dX)$ plane. (c) Eigenvalues of the spatial dynamics linearized around $H$, for parameter values corresponding to the spike in panel (a), $\times$, and ${\mathrm{LP}}_0$, $+$. In either case, the eigenvalue with real part equal to 1 has multiplicity 4. To facilitate the view, the horizontal axis is broken and its scale changes from left to right.

Figure 4

Localized states in 2D. (a),(b) Isotropically localized states, simple (a) and patterned (b). (c),(d) Anisotropically localized states, simple (c) and patterned (d). Arrows represent velocity fields (a),(c). Parameters as in SM Table II [28], with ${\mathrm{\Omega }}_{d,0}=0$, $Z=30$, and $\mathrm{\Omega }=0$ (a),(c), $\mathrm{\Omega }=9$ (b),(d). The code used to obtain these results is available online [31].