Parameter regions that give rise to $2 \lfloor \frac{n}{2}\rfloor +1$ positive steady states in the $n$-site phosphorylation system

Abstract

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.

Robustness and parameter geography in post-translational modification systems

“…For the two-site, distributive system studied here, in which we restricted attention to E tot = F tot , we found bistable points in H 4 at S tot /E tot = σ = 1.0078125, which suggests that the threshold lies at σ = 1, so that T(E tot , F tot ) = E tot when E tot = F tot . The conjecture above is consistent with recent work which showed that, when all enzymes follow the Michaelis-Menten reaction scheme, if S tot < E tot or S tot < F tot , parameter values can be found at which there are multiple steady-states but such multistationarity does not exist if S tot > E tot and S tot > F tot [58,89,90].…”

Algebraic Geometry Tools in Systems Biology

“…We refer to [11] for a simple example and basic definitions, together with an explicit application to cascades with any number of layers. These degeneration tools are also applied to describe open regions where there are (or + 1) scpss for any in [12], where we also provide a basic systematic implementation in a computer algebra system.…”

Dynamics of ERK regulation in the processive limit

“…[15,Lemma B.3]), as follows. We first found a vertex of the polytope whose corresponding coefficient is a positive polynomial (namely, the leading coefficient in (12)). Next, we chose a vector v in the interior of the corresponding cone in the polytope's outer normal fan.…”

“…Our results fit into related literature as follows. First, as other authors have done for their models of interest [5,12,18], we analyze simplified versions of the ERK network obtained by removing intermediate species and/or reactions (in some cases, bistability and oscillations can be "lifted" from smaller networks to larger ones [1,2,4,10,13]). Also, our proofs harness two results from previous work: a Hopf-bifurcation criterion for the reduced ERK network [15], and a criterion for multistationarity arising from degree theory [6,9].…”