Wnt signaling is a vital biological mechanism that regulates crucial
development processes and maintenance of tissue homeostasis. Here, we extended
the parameter-free analysis of four mathematical models of the
beta-catenin-dependent Wnt signaling pathway performed by MacLean et al. (PNAS
USA 2015) using chemical reaction network theory. We showed that the reaction
networks of the four models considered (Lee, Schmitz, MacLean, and Feinberg)
coincide in basic structural and kinetic properties except in their
mono-stationarity/multi-stationarity, and their capacity for admitting a
degenerate equilibrium. Moreover, we showed that the embedded networks of the
Lee and Feinberg models are very similar, and the discordance of the Lee
network limits its mono-stationarity to mass action kinetics, which challenge
the absoluteness of model discrimination into mono-stationarity versus
multi-stationarity alone. Focusing, henceforth, on the three multi-stationary
networks, we showed that their finest independent decompositions are very
different and can be used to study further similarities and differences among
them. We also determined equilibria parametrizations of the networks and
inferred the presence of species with absolute concentration robustness.
Finally, direct comparison of the Schmitz and Feinberg networks with the
MacLean network yielded new results in three aspects: structural/kinetic
relationships between embedded networks relative to their set of common
species, connections between the positive equilibria of the subnetwork of
common reactions and the positive equilibria of the whole networks, and
construction of maximal concordant subnetwork containing the common reactions
of the networks under comparison. Thus, this work can provide general insights
in comparing mathematical models of the same or closely-related systems