Multistationarity questions in reduced versus extended biochemical networks.

We address several questions in reduced versus extended networks via the elimination or addition of intermediate complexes in the framework of chemical reaction networks with mass-action kinetics. We clarify and extend advances in the literature concerning multistationarity in this context, mainly from Feliu and Wiuf (J R Soc Interface 10:20130484, 2013), Sadeghimanesh and Feliu (Bull Math Biol 81:2428-2462, 2019), Pérez Millán and Dickenstein (SIAM J Appl Dyn Syst 17(2):1650-1682, 2018), Dickenstein et al. (Bull Math Biol 81:1527-1581, 2019). We establish general results about MESSI systems, which we use to compute the circuits of multistationarity for significant biochemical networks.

Parameter regions that give rise to 2[n/2] +1 positive steady states in the n-site phosphorylation system.

The distributive sequential n-site phosphorylation/dephosphorylation system is an important building block in networks of chemical reactions arising in molecular biology, which has been intensively studied. In the nice paper of Wang and Sontag (2008) it is shown that for certain choices of the reaction rate constants and total conservation constants, the system can have 2[n/2] +1 positive steady states (that is, n+1 positive steady states for n even and n positive steady states for n odd). In this paper we give open parameter regions in the space of reaction rate constants and total conservation constants that ensure these number of positive steady states, while assuming in the modeling that roughly only 1/4 of the intermediates occur in the reaction mechanism. This result is based on the general framework developed by Bihan, Dickenstein, and Giaroli (2018), which can be applied to other networks. We also describe how to implement these tools to search for multistationarity regions in a computer algebra system and present some computer aided results.

Regions of multistationarity in cascades of Goldbeter-Koshland loops.

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).