# Regions of multistationarity in cascades of Goldbeter–Koshland loops

## Abstract

We consider cascades of enzymatic Goldbeter–Koshland loops (Goldbeter and Koshland in Proc Natl Acad Sci 78(11):6840–6844, 1981) with any number *n* of layers, for which there exist two layers involving the same phosphatase. Even if the number of variables and the number of conservation laws grow linearly with *n*, we find explicit regions in *reaction rate constant* and *total conservation constant* space for which the associated mass-action kinetics dynamical system is multistationary. Our computations are based on the theoretical results of our companion paper (Bihan, Dickenstein and Giaroli 2018, preprint: arXiv:1807.05157) which are inspired by results in real algebraic geometry by Bihan et al. (SIAM J Appl Algebra Geom, 2018).

## Keywords

Enzymatic cascades Goldbeter–Koshland loops Sparse polynomial systems Multistationarity## Mathematics Subject Classification

92C42 80A30 14P99 14L32## Notes

### Acknowledgements

The authors are grateful to the Kurt and Alice Wallenberg Foundation and to the Institut Mittag-Leffler, Sweden, for their support to work on this project. We are also grateful to the Mathematics Department of the Royal Institute of Technology, Sweden, for the wonderful hospitality we enjoyed, and to the French Program PREFALC and the University of Buenos Aires, which made possible the visit of F. Bihan.

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