A note on "MAPK networks and their capacity for multistationarity due to toric steady states"

TL;DR

This paper demonstrates that the capacity for toric steady states in certain MAPK networks can be established through network translation, linking original systems to generalized mass action systems with guaranteed steady state properties.

Contribution

It shows how network translation can be used to verify toric steady states in MAPK networks, extending previous methods to specific network cases.

Findings

Network translation yields proper, weakly reversible networks with zero deficiency.

Toric steady states are guaranteed by structural and kinetic properties.

Steady state ideals can be derived from linkage class analysis.

Abstract

We provide a short supplement to the paper "MAPK networks and their capacity for multistationarity due to toric steady states" by P\'erez Mill\'an and Turjanski. We show that the capacity for toric steady states in the three networks analyzed in that paper can be derived using the process of network translation, which corresponds the original mass action system to a generalized mass action system with the same steady states. In all three cases, the translated chemical reaction network is proper, weakly reversible, and has both a structural and kinetic deficiency of zero. This is sufficient to guarantee toric steady states by previously established work on network translations. A basis of the steady state ideal is then derived by consideration of the linkage classes of the translated chemical reaction network.