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

We provide a short supplement to the paper MAPK networks and their capacity for multistationarity due to toric steady states by Perez Millan 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.