Chemotaxis of cell aggregates: morphology and dynamics of migrating active droplets

TL;DR

A minimal model of a growing thin active droplet driven by a self-generated chemical gradient is developed and reveals that chemotacting droplets exhibit proliferation-driven morphological transitions, which can occur continuously or through a discontinuous bifurcation.

Abstract

Biological tissues have been observed to display emergent fluid-like properties, owing to physical interactions between cells. However, it remains unclear in general how these fluid-like properties affect tissue structure and function. Here, we are motivated by recent experiments in which cell aggregates were observed to behave as active droplets during collective migration along chemical gradients, or chemotaxis. To understand this process, we develop a minimal model of a growing thin active droplet driven by a self-generated chemical gradient. In broad agreement with the experiments, dynamic simulations reveal that chemotacting droplets exhibit proliferation-driven morphological transitions. To fully characterise these transitions, we perform a multiple scales analysis to show that the droplet dynamics follow a sequence of travelling wave solutions defined by a nonlinear eigenvalue problem parametrised by the slowly increasing droplet volume. Our analysis reveals that morphological transitions can occur continuously or through a discontinuous bifurcation. Further asymptotic analysis of the travelling wave problem reveals that these morphological transitions arise from exponentially small ("beyond-all-orders") asymptotic terms that originate from the rear and front contact lines. Moreover, we show that the nature of the transitions is fully determined by two key dimensionless parameters, which quantify the internal stress balance within the droplet and the strength of the coupling between the droplet migration dynamics and the external chemical field. Overall, our results provide a complete characterisation of the morphodynamics of a class of migrating active thin droplets, with implications in a range of biological systems where cell aggregates exhibit fluid-like behaviour.

Figures (6)

• Figure 1: Schematic of the thin active droplet model of collective self-generated chemotaxis of cell groups. The cell group is defined by the evolution of the free-surface $y=h(x,t)$, which touches the substrate at the contact lines $x=x_\pm(t)$. The migration of the active droplet is driven by active pressure gradients coupled to an external chemotactic field, $c=c(x,t)$, which is shaped by the advancing droplet itself via depletion of the chemoattractant source. Since we consider a thin droplet, the chemoattractant is taken to be homogeneous in the vertical direction.
• Figure 2: Representative simulations of morphological transitions in a chemotactic growing droplet \ref{['general governing equations']} (see \ref{['app:numerics']} for details on the numerical implementation). The shaded colors highlight results from two distinct parameter sets corresponding to (blue) a smooth (set 1 in \ref{['tab:parameters']}) and (yellow) a sharp (set 2 in \ref{['tab:parameters']}) morphological transition. (a) and (f) Example of the droplet height profile $h(x,t)$ at distinct times. Time increases from left to right (see corresponding colored dots in other panels). Note that the solution is shifted so that $x=0$ corresponds to the location of the left boundary $x_-(t)$. (b) and (g) Time evolution of the droplet volume, $v=\int_{x_-}^{x_+} \! h(x,t)\, \mathrm{d}x$. (c) and (h) Plots of the droplet length, $\ell=x_+(t)-x_-(t)$ vs droplet volume $v$. (d) and (i) Plots of the speed of the droplet centre of mass, $u_{\bar{x}}$ vs droplet volume $v$. (e) and (j) Plots of the shape parameter $s$ vs droplet volume $v$.
• Figure 3: (a)-(b) Numerical solutions of the full travelling wave problem \ref{['eq:general_TW_problem']} corresponding to the parameter values used in \ref{['fig:SG_travelling_homogeneous_example']}. (i)-(ii) Numerical bifurcation diagram (black curve) showing the estimated (i) volume-speed and (ii) volume-length relations. Results are compared with the prediction of the asymptotic approximations (dashed lines) discussed in \ref{['sec analysis:summary']}. (iii) Examples of droplet profile of travelling wave solutions for increasing droplet length ($\ell$).
• Figure 4: (a) Schematic of the structure of the droplet morphology in the large droplet limit. Two boundary layers form near the rear and front contact lines, in which the droplet profile rapidly changes until settling to a uniform solution in the middle region connecting the two. (b) A flow chart displaying the information flow for the large droplet asymptotic limit. The black arrows show the direction of the information flow, and the grey arrows indicate secondary connections. The grey and red boxes indicate results associated with the approximation of the travelling droplet profile and speed, respectively. (c)-(f) Characterisation of the leading-order behaviour as a function of the non-dimensional parameter $\mathcal{N}_\theta$ (see \ref{['eq:def_ntheta']}). Mapping between $\mathcal{N}_\theta$ and (c) the leading-order velocity $U_0$ (see \ref{['eq constant: U_0']}), and (d) $\tilde{\sigma}_0$ and (e) $\tilde{\psi}_0$, which characterise the far-field behaviour of the solution in the front boundary layer (see \ref{['eq:farfield front']}). The grey shaded area indicates the range of values taken by each variable, with dotted yellow curves indicating asymptotes. (f) Examples of drop profiles in the front boundary layer for increasing values of $\mathcal{N}_\theta$.
• Figure 5: Elongated-droplet theory of morphological transitions. Theoretical predictions for how (a) the critical length $L_{cr}(\gamma_\theta)$ at which $u$\ref{['eq leading-order function of L']} attains its maximal value, (b) the drop in migration speed $\Delta u(\gamma_\theta,b)$\ref{['eq:Delta u asymptotic']}, and (c) the volume variation $\Delta v(\gamma_\theta,b)$\ref{['eq:Delta v']} depend on the non-dimensional parameter $\gamma_\theta$\ref{['eq:gammatheta']}. In (b)-(c), colors indicate different values of the non-dimensional parameter $b$\ref{['eq:definition b']}, which ranges from $b=0$ (dark blue) to $b=1$ (light blue). (d) Phase diagram of elongated droplets undergoing self-generated chemotaxis predicted by the asymptotic theory. The two parameters $\gamma_\theta$ and $b$ characterise the balance between the different physical mechanisms governing the migration of elongated droplets (more details in the main text). In the left panel, the colormap indicates the relative reduction in the migration speed $\Delta u$ (defined by Eq. \ref{['eq:Delta u asymptotic']}) following morphological transitions. The white curve indicates the value of the parameter for which $\Delta v=0$, partitioning the region of parameter space where the theory predicts proliferation to drive a continuous (blue shaded area) or a discontinuous (red shaded area) morphological transition.