Chemotaxis-induced phase separation

Abstract

Chemotaxis allows single cells to self-organize at the population level, as classically described by Keller-Segel models. We show that chemotactic aggregation can be understood using a generalized Maxwell construction based on the balance of density fluxes and reactive turnover. This formulation implies that aggregates generically undergo coarsening, which is interrupted and reversed by cell growth and death. Together, both stable and spatiotemporally dynamic aggregates emerge. Our theory mechanistically links chemotactic self-organization to phase separation and reaction-diffusion patterns.

Figures (3)

• Figure 1: The mass-redistribution potential (orange) induces redistribution (orange arrows) of the cell density (blue) that counteracts (a) or amplifies (b) initial perturbations (black arrows). (c) $\mathsf{\Lambda}$-shaped NCs $\eta^*(\rho)$ lead to stationary peak patterns. The low-density intersection between the NC (black) and the flux-balance subspace ${\eta = \eta_\mathrm{stat}}$ (orange) determines the low-density plateau $\rho_-$ (pattern minimum ${\check{\rho}\gtrsim \rho_-}$). The value $\eta_\mathrm{stat}$ is determined by the reactive area balance (balance of the red-shaded areas). (d) $N$-shaped NCs give an additional high-density NC-FBS intersection $\rho_+$ (pattern maximum ${\hat{\rho}\lesssim \rho_+}$). The resulting pattern is mesa-shaped.
• Figure 2: Illustration of the mass-competition instability for the competition of two peaks (a) and two plateaus (b; coalescence of the peak with its mirror image) destabilizing the symmetric pattern (dashed lines). In both cases (a) and (b), the changes in the stationary mass-redistribution potentials $\delta\eta$ of the two elementary patterns with the peak mass (blue-shaded areas) and plateau lengths lead to self-amplifying mass transport (orange). The generalized Maxwell area balance (shown for the right elementary patterns in the lower panels) shows that the (quasi-)stationary mass-redistribution potential decreases at the larger peak and shorter plateau (red- and green-shaded areas). (c) In a large system, these instabilities induce a coarsening process of the pattern by competition (examples of peak collapse marked by red arrows) and coalescence (blue arrows) as shown in two kymographs of simulations with $T\Bar{\rho}/D_\rho= 1.5, 3$. The faster mass-competition process [cf. inset in (d)] dominates the dynamics. (d) During the coarsening process, the average pattern wavelength $\langle\Lambda\rangle(t)$ continuously grows. Numerical simulations at $T\Bar{\rho}/D_\rho=1.25, 1.5, 2, 3$ (average over 3, 1, 8, and 2 replicates; light green to dark blue) are compared with the prediction ${\langle\Lambda\rangle(t)\sim \max(\gamma_\pm) \log(t)}$ with ${\gamma \propto 5/8,3/4,1,1}$ (black lines; see supplemental_material). Apart from the average density $\Bar{\rho}$ specified above, the simulation parameters are $T = 5$, $D_\rho = 0.1$, $D_c = 1$, with system length $L= 1000$.
• Figure 3: (a) Illustration of interrupted coarsening due to production and degradation [$s(\rho)$; purple] counteracting the mass-competition instability (orange) which destabilizes the symmetric peak position against shifts $\delta x$. (b) Plateau splitting occurs when the plateau density enters the laterally unstable region (green) due to production in the low-density plateau. In mesa patterns, high-density plateaus can split as well due to degradation when the plateau density decreases into the laterally unstable density regime. (c) The stability of patterns with wavelength $\Lambda$ in the mKS model with a logistic source term ${s(\rho)=\rho (1-\rho)}$ of strength $\varepsilon$. The threshold of plateau splitting (green) lies at low wavelengths $\Lambda$ and deforms the threshold of interrupted coarsening at large source strength $\varepsilon$ (solid compared to dashed blue line). The analytic thresholds (lines) well describe the numerical thresholds (circles and squares) obtained by numerical continuation and linear stability analysis (see supplemental_material). (d) The kymographs show $\log(1+\rho)$ (grayscale, $0$ to $1$) for $\varepsilon = 0.01, 0.1$ simulated on a domain with reflective boundaries and length $L=149, 74$ corresponding (approximately) to a multiple of the mean of the numerical thresholds $\Lambda_\mathrm{stop,split}$. The same simulation parameters are used as in Fig. \ref{['fig:2']}(d).