https://doi.org/10.1140/epjp/s13360-023-04669-9
Regular Article
Onsager’s variational principle in proliferating biological tissues, in the presence of activity and anisotropy
1
Laboratoire Jean Perrin, Sorbonne Université, 4 Place Jussieu, 75005, Paris, France
2
Laboratoire de Physique de l’Ecole Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris Cité, 75005, Paris, France
3
Faculté de Médecine, Institut Universitaire de Cancérologie, Sorbonne Université, 91 Bd de l’Hôpital, 75013, Paris, France
a
joseph.ackermann@sorbonne-universite.fr
Received:
7
September
2023
Accepted:
2
November
2023
Published online:
12
December
2023
A 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 that describe their dynamics taking into account growth terms, chemical factors and the symmetry of the cells and of the other tissue components. 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 non-equilibrium systems. This 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 cellular activity, 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 on small scales and solve the resulting system of partial differential equations numerically 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.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
