Stability bounds of a delay visco-elastic rheological model with substrate friction

Author (s): Dawi, M.A., Muñoz, J.J.
Journal: Journal of Mathematical Biology

Volume: 83
Pages: 1 – 19
Date: 2021

Cells and tissues exhibit sustained oscillatory deformations during remodelling, migration or embryogenesis. Although it has been shown that these oscillations correlate with intracellular biochemical signalling, the role of these oscillations is as yet unclear, and whether they may trigger drastic cell reorganisation events or instabilities remains unknown.

Here, we present a rheological model that incorporates elastic, viscous and frictional components, and that is able to generate oscillatory response through a delay adaptive process of the rest-length. We analyse its stability as a function of the model parameters and deduce analytical bounds of the stable domain. While increasing values of the delay and remodelling rate render the model unstable, we also show that increasing friction with the substrate destabilises the oscillatory response. This fact was unexpected and still needs to be verified experimentally. Furthermore, we numerically verify that the extension of the model with non-linear deformation measures is able to generate sustained oscillations converging towards a limit cycle. We interpret this sustained regime in terms of non-linear time varying stiffness parameters that alternate between stable and unstable regions of the linear model. We also note that this limit cycle is not present in the linear model.



   author = “Malik A. Dawi, J.J. Mu{\~n}oz",
   title = "Stability bounds of a delay visco-elastic rheological model with substrate friction",
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   year = "2021",
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