Cells adhere to one another through cell–cell junctions, mediated by adhesion proteins such as cadherins that bind across neighbouring cell membranes. This allows large groups of cells to migrate together as a cohesive collective. Cell–cell adhesion can also drive cells of different types to self-organise into distinct patterns. Much like oil and water separate because of the high interfacial tension between them, differences in adhesion between cell populations create effective tensions at the interfaces between aggregates of different cell types, leading to sorting phenomena. Depending on how different cell types interact, other kinds of patterns can also emerge, in line with what we now understand as the differential adhesion hypothesis.

Modelling adhesive interactions has attracted a great deal of interest over the last two decades, and in my research we’ve developed a continuum model that predicts cell sorting patterns from the underlying cell–cell interactions. Interestingly, this model resembles thin-film equations — the equations used to describe the evolution of thin layers of fluid on surfaces — which brings us back to the oil-and-water analogy! These models can be implemented efficiently in simulations, while still giving analytical and mechanistic insight. See below for some examples, though there is still plenty left to understand!

Cells in our bodies typically migrate through the extracellular matrix, a complex protein-rich network surrounding tissues. How cells sense this matrix, adhere to it, and use it to move is still not fully understood. We’ve studied cell–matrix interactions in multicellular spheroids and how these interactions help cells coordinate their migration. Current work focuses on understanding the dynamical (and nonreciprocal) interactions between cells and the matrix, deriving and analysing continuum models, and identifying the conditions that give rise to coordinated migration or cell invasion into the surrounding matrix.

The proper coordination of cell proliferation is fundamental in development. Think of the first divisions in an embryo: they need to be highly synchronised to ensure healthy growth — althogh some variability in the internal clock is also needed! Cell proliferation also plays a key role in tissue regeneration, while a failure to stop dividing can lead to uncontrolled growth pathologies such as cancer. I develop mathematical models of cell cycle dynamics, sometimes coupled to cell migration, in which cell cycle progression depends on external cues such as crowding. We combine cell cycle fluorescence markers with modern inference methods to learn cell cycle checkpoints directly from data. More recent work focuses on the use of so-called universal differential equations to learn how cell cycle transition rates depend on crowding.

Cell invasion is one of the clearest manifestations of self-organisation. In mathematical models, it often appears through travelling wave solutions. Because these waves arise naturally, it is easy to overlook how remarkable their emergence really is: simple mechanisms driving motility and proliferation can organise into a coherent invading structure with a well-defined speed and shape. Travelling wave analysis gives us tools to characterise these waves, including their speed, spatial structure, and, in heterogeneous populations, the organisation of the invasion front. Questions like these motivate new mathematics, using asymptotic and variational methods to understand how such invading waves emerge and persist. In my work, I have studied these types of solutions in: haptotaxis, chemotaxis, and go-or-grow systems.

References:

• C. F., R. M. Crossley, M. Conte, T. Lorenzi (2026). Speed and stability of interfaces between cell populations with different mobilities. in preparation.  

• M. Watts, C. F., G. L. Celora (2026). Patterns of collective migration in a model of heterogeneous, self-generated chemotaxis . in preparation.

• C. F., S. Johnson, M. Dalwadi, P. K. Maini (2026). Theory of adhesion-driven self-organisation in growing tissues. arxiv preprint

• R. M. Crossley, C. F., R. E. Baker (2026). An optimal control approach to nonlinear wave speed selection in reaction-diffusion equations. arxiv preprint

• C. F., D. J. Cohen, J. A. Carrillo, R. E. Baker (2025). Quantifying cell cycle regulation by tissue crowding. Biophys. J., 124, 923-932.

• C. F., R. M. Crossley, R. E. Baker (2024). Travelling waves in a minimal go-or-grow model of cell invasion. Appl. Math. Lett., 158, 109209.