Onsager’s variational principle in proliferating biological tissues, in presence of activity and anisotropy

AbstractA hallmark of biological cells is their ability to proliferate and of tissues their ability to grow. This is common in morphogenesis and embryogenesis but also in pathological conditions such as tumour growth. To consider these tissues from a physical point of view, it is necessary to derive fundamental relationships, in particular for velocities and density components, taking into account growth terms, chemical factors and the symmetry of cells and tissues. The aim is then to develop a consistent coarse-grained approach to these complex systems, which exhibit proliferation, disorder, anisotropy and activity at small scales. To this end, Onsager’s variational principle allows the systematic derivation of flux-force relations in systems out of equilibrium and the principle of the extremum of dissipation, first formulated by Rayleigh and revisited by Onsager, finally leads to a consistent formulation for a continuous approach in terms of a coupled set of partial differential equations. Considering the growth and death rates as fluxes, as well as the chemical reactions driving the cellular activities, we derive the momentum equations based on a leading order physical expansion. Furthermore, we illustrate the different interactions for systems with nematic or polar order at small scales, and numerically solve the resulting system of partial differential equations in relevant biophysical growth examples. To conclude, we show that Onsager’s variational principle is useful for systematically exploring the different scenarios in proliferating systems, and how morphogenesis depends on these interactions.