  The canonical Wnt signaling pathway, mediated by $\beta$-catenin, is
crucially involved in development, adult stem cell tissue maintenance and a
host of diseases including cancer. We undertake analysis of different
mathematical models of Wnt from the literature, and compare them to a new
mechanistic model of Wnt signaling that targets spatial localization of key
molecules. Using Bayesian methods we infer parameters for each of the models to
mammalian Wnt signaling data and find that all models can fit this time course.
We are able to overcome this lack of data by appealing to algebraic methods
(concepts from chemical reaction network theory and matroid theory) to analyze
the models without recourse to specific parameter values. These approaches
provide insight into Wnt signaling: The new model (unlike any other
investigated) permits a bistable switch in the system via control of shuttling
and degradation parameters, corresponding to stem-like vs committed cell states
in the differentiation hierarchy. Our analysis also identifies groups of
variables that must be measured to fully characterize and discriminate between
competing models, and thus serves as a guide for performing minimal experiments
for model comparison.
