Coarsening dynamics of chemotactic aggregates

Abstract

Auto-chemotaxis, the directed movement of cells along gradients in chemicals they secrete, is central to the formation of complex spatiotemporal patterns in biological systems. Since the introduction of the Keller--Segel model, numerous variants have been analyzed, revealing phenomena such as coarsening of aggregates, stable aggregate sizes, and spatiotemporally chaotic dynamics. Here, we consider general mass-conserving Keller--Segel models, that is, models without cell growth and death, and analyze the generic long-time dynamics of the chemotactic aggregates. Building on and extending our previous work, which demonstrated that chemotactic aggregation can be understood through a generalized Maxwell construction balancing density fluxes and reactive turnover, we use singular perturbation theory to derive the rates of mass competition between well-separated aggregates. We analyze how this mass-competition process drives coarsening in both diffusion- and reaction-limited regimes, with the diffusion-limited rate aligning with our previous quasi-steady-state analyses. Our results generalize earlier mathematical findings, demonstrating that coarsening is driven by self-amplifying mass transport and aggregate coalescence. Additionally, we provide a linear stability analysis of the lateral instability, predicting it through a nullcline-slope criterion that parallels the curvature criterion in spinodal decomposition. Overall, our findings suggest that chemotactic aggregates behave similarly to phase-separating droplets, providing a robust framework for understanding the coarse-grained dynamics of auto-chemotactic cell populations and a quantitative basis for comparing chemotactic coarsening to canonical non-equilibrium phase separation.

Figures (6)

• Figure 1: The phenomenology of chemotactic aggregation. (a) If chemotaxis is sufficiently strong (see Sec. \ref{['sec:lsa']}), a uniform distribution of the cells is unstable against spatially varying perturbations (lateral instability). The result is the aggregation of cells into aggregates that are either peak- (b) or mesa-shaped (c). These aggregates are quasi-stationary but interact leading to a coarsening of the aggregates by competition for cells and coalescence (mass-competition instability, see Sec. \ref{['sec:mass-comp']}). The final stationary pattern contains a single aggregate that moves to the system boundary in the case of no-flux boundary conditions of the domain (illustrated here).
• Figure 2: Dispersion relation for the lateral instability of the hss. (a) If the nullcline slope is negative, all growth rates $\sigma_q^+$ are negative (blue), as also the QSS approximation of the dispersion relation for $q\to0$ (black). (b) A band of unstable modes that extends to $q=0$ emerges if the nullcline slope is negative.
• Figure 3: Construction of the stationary elementary mesa pattern. (a) The elementary mesa pattern (blue) consists of a single interface separating a high- and a low-density plateau on a domain of length $\Lambda/2$ with no-flux boundary conditions. The high- and low-density plateau lengths $L_\pm$ depend on the mesa mass $M$ (blue-shaded region) and, equivalently, the average cell density $\bar{\rho}$ in the system. The profile of a single interface on the infinite line (black profile) approaches the plateau densities $\rho_\pm$ exponentially far from the interface. The profile on the finite interval of length $\Lambda/2$ (blue) shows deviations in the plateaus close to the boundaries as it must fulfill the boundary conditions. As a result, the pattern maximum $\hat{\rho} = \rho_+-\delta\rho_+$ differs from the plateau density $\rho_+$ by an amount $\delta\rho$ exponentially small in the plateau lengths. Similarly, the pattern minimum is $\check{\rho}=\rho_-+\delta\rho_-$. (b) In the local $(\rho,\eta)$ phase space, the stationary pattern is restricted to the flux-balance subspace (FBS) $\eta_\mathrm{stat}=\mathrm{const.}$ The value of $\eta_\mathrm{stat}$ is fixed qualitatively by the balance of the red-shaded areas between the FBS and the nullcline $\eta^*(\rho)$ (see Ref. Weyer.etalinpreparation). The plateau densities $\rho_\pm$ are the intersections of the FBS $\eta_\mathrm{stat}^\infty$ of the pattern on the infinite line with the nullcline, which have a positive nullcline slope. The middle intersection with a negative nullcline slope (blue dot) indicates the inflection point of the pattern profile. The shift $\eta_\mathrm{stat}-\eta_\mathrm{stat}^\infty$ of the stationary mass-redistribution potential $\eta_\mathrm{stat}$ of the pattern on a finite domain compared to its value $\eta_\mathrm{stat}^\infty$ for the pattern on the infinite line is due to the changes of the red-shaded compared to the gray-shaded areas induced by the deviations $\delta\rho_\pm$ in the plateaus.
• Figure 4: Construction of the stationary elementary peak pattern. (a) The elementary peak pattern (blue) consists of half a peak with a low-density plateau on a domain of length $\Lambda/2$ with no-flux boundary conditions. The peak height $\hat{\rho}$ depends on the peak mass $M$ (blue-shaded region). As for mesa patterns (see Fig. \ref{['fig:stat-mesa']}), the pattern profile deviates in the plateau from the profile (black) on the half-infinite line $[0,\infty)$ due to the boundary condition. (b) Again, the pattern can be constructed in the local $(\rho,\eta)$ phase space (see Ref. Weyer.etalinpreparation). Peak patterns arise if the cell density does not saturate and does not form a high-density plateau. Therefore, peak patterns occur if the nullcline is $\mathsf{\Lambda}$- rather than $\mathsf{N}$-shaped and no (third) high-density intersection between the flux-balance subspace $\eta_\mathrm{stat}=\mathrm{const.}$ and the reactive nullcline $\eta^*(\rho)$ (FBS-NC intersection) exists (cf. Fig. \ref{['fig:stat-mesa']}). Alternatively, peak patterns also form if the third FBS-NC intersection lies at higher densities $\rho_+\gg\hat{\rho}$ that are not reached for the given peak mass $M$. The value of $\eta_\mathrm{stat}$ is fixed qualitatively by the balance of the red-shaded areas between the FBS and the nullcline $\eta^*(\rho)$. It differs by an amount exponentially small in the plateau width from the value of the pattern on the half-infinite line due to the change of the red- compared to gray-shaded areas induced by the deviation $\delta\rho_-=\check{\rho}-\rho_-(\eta_\mathrm{stat}^\infty)$. The mass-redistribution potential $\eta_\mathrm{stat}$ depends, via the area balance, also on the peak height $\hat{\rho}$.
• Figure 5: Coarsening scenarios of a peak pattern. (a) Peak competition: Peak patterns (dark-blue profile) can undergo coarsening by the transport of mass from smaller to larger peaks, resulting in the collapse of the smaller peaks (red arrows and light-blue profile). This competition for mass between peaks renders stationary periodic patterns with equally large peaks unstable to small disturbances in the peak masses. Due to the symmetry of the perturbation mode, peak competition can be analyzed by the interaction of two half peaks on a domain of length $\Lambda$ with no-flux boundary conditions (gray lines). (b) Peak coalescence: Peak patterns can also undergo coarsening due to the coalescence of peaks (red arrows and light-blue profile). This peak coalescence renders stationary periodic pattern unstable to small variations in the peak distances. It can be analyzed by examining the movement of a single peak on a domain of length $\Lambda$ with no-flux boundary conditions (gray lines). The coalescence with the neighboring peak of the periodic pattern then corresponds to the coalescence with the boundary.