We introduce an analytical framework that can be used to understand the alignment mechanism of ellipse-shaped cells interacting via self-propulsion and overlap avoidance. To do this, we consider two interacting particles with certain symmetries imposed to make the problem analytically tractable. This results in a dynamical system we can mathematically analyse in detail. We find that there is a half-stable steady state corresponding to a cell configuration that depicts perfect alignment, and that the presence of a separatrix splits the domain into two regions. These two regions characterise the outcome of a trajectory as moving towards this state or not. Understanding the asymptotic case corresponding to a small amount of self-propulsion offers an insight into the timescales at play when a trajectory is moving towards the point of perfect alignment. We find that the two cells initially move apart to avoid overlap over a fast timescale, and then the presence of self-propulsion causes them to move towards a configuration of perfect alignment over a much slower timescale. Overall, our analysis highlights how the interaction between self-propulsion and overlap avoidance is sufficient to generate alignment.

A dynamical analysis of the alignment mechanism between two interacting cells