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The organisation of biological tissues during development is accompanied by the formation of sharp borders between distinct cell populations. During the morphogenesis of numerous tissues/organs, cells of the same type regroup into regions, creating niches with specific identities that drive the differentiation of particular cell types. This spatial organization is ensured via cell-cell signalling leading specific cells/tissues to form at the appropriate location. The maintenance of this cell segregation is key in adult tissue homeostatis, and its disruption can lead tumor cells to spread and form metastasis.
Motivated by experiments on cell segregation, we present a two-species model of interacting particles, aiming at a quantitative description of this phenomenon. Under precise scaling hypothesis, we derive from the microscopic model a macroscopic one and we analyze it. In particular, we determine the range of parameters for which segregation is expected. We compare our analytical results and numerical simulations of the macroscopic model to direct simulations of the particles, and comment on possible links with experiments.
We mainly consider repulsive springs and aim to quantify the influence of heterotypic/homotypic repulsion on cell segregation and border sharpening. By addressing the stability of a homogeneous distribution of particles for Hookean repulsive potentials, we obtain a precise condition for the phase transition, which links the system segregation ability to the model parameters and give further insight into the cell segregation processes. Our study shows that in a system composed of two-species repelling each other, the interspecies forces must be large enough to compensate both for the diffusion and for the intra-species repulsion, which both tend to homogeneize the system. Aggregation will therefore be ensured if and only if interspecies repulsion wins over diffusion and intraspecies repulsion. In the case where attractive interactions are considered, we have noted that a necessary condition for the aggregation of the species (or equivalently instability of the homogeneous steady-state) is that the interspecies forces are of the same sign. To observe aggregates, the two families must therefore either repulse or attract each other, but must have the same effect on each other. On the contrary, if one family is attracted by the other and the other repulses it, we will always observe a homogeneous distribution at equilibrium (intermingling of the two families). A third remark concerns the size of the clusters when aggregation occurs. As the interspecies repulsion force increases, the particles of a given family aggregate more together, leading to a decrease of the size of the local aggregates of the compressed family.
Publications
J. Barré, P. Degond, D. Peurichard, E. Zatorska, Modelling pattern formation through differential repulsion, Networks and Heterogeneous media, (2020), 15:307-352. Manuscript on arXiv
Simulations of the microscopic (agent-based) model in different regimes of parameters and for different initial conditions (initial mixing or initial segregation)
Simulations of the macroscopic (PDE) system in different regimes of parameters and for different initial conditions (initial mixing or initial segregation)
By playing on the parameters controling the inter- and intra- repulsion forces, our model is able to reproduce the experiments on cell segregation and border sharpening, suggesting that cell segregation between different cell populations could be mainly due to differential repulsion, more efficient than decreased heterotypic adhesion.
source experimental images :Taylor HB, Khuong A, Wu Z,Xu Q, Morley R, Gregory L, Poliakov A, TaylorWR, Wilkinson DG. 2017 Cell segregation andborder sharpening by Eph receptor–ephrin-mediated heterotypic repulsion.J. R. Soc.Interface14: 20170338.
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