  The Wnt signaling pathway plays a critical role in various biochemical
processes, including embryonic development, tissue homeostasis, and cancer
progression. In this paper, we conduct a comparative analysis of
$\beta$-catenin-dependent Wnt signaling reaction networks, which we refer to as
the Feinberg, Schmitz, and MacLean models, based on the previous study by
MacLean et al. (PNAS USA 2015). Our analysis is based on the (unique) finest
independent decomposition (FID) of each reaction network and our comparative
techniques include equilibria parametrizations (EP) and the newly developed
methods of Common Reactions Equilibria (CORE) analysis and Concordance Profile
(CP) analysis. Our investigation yields three interesting results concerning
the equilibria sets of these models. Firstly, we explore the concept of
absolute concentration robustness (ACR), wherein a system exhibits ACR in a
specific species if the equilibrium value for that species is the same for any
positive equilibrium. Through ACR analysis employing FID and EP, we observe
that both the Schmitz and MacLean models lack ACR, whereas the Feinberg model
demonstrates ACR in a single species. Second, our analyses using FID and CORE
reveal important relationships within the equilibria sets of the augmented
Schmitz and MacLean models. Furthermore, FID and CORE identify the lack of a
substantial relationship between the equilibria sets of the Feinberg and
MacLean models. Hence, these methods detect subtle differences between the
Feinberg and MacLean models and also between the Schmitz and MacLean models,
which are not evident in the standard reaction network analysis. Finally, based
on the concordance levels, CP analysis indicates that the MacLean and Schmitz
models are more similar than the MacLean and Feinberg models.
