Parameter-free methods distinguish Wnt pathway models and guide design of experiments

Abstract

The canonical Wnt signaling pathway, mediated by β-catenin, is crucially involved in development, adult stem cell tissue maintenance, and a host of diseases including cancer. We analyze existing mathematical models of Wnt and compare them to a new Wnt signaling model that targets spatial localization; our aim is to distinguish between the models and distill biological insight from them. Using Bayesian methods we infer parameters for each model from mammalian Wnt signaling data and find that all models can fit this time course. We appeal 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 aspects of Wnt regulation: the new model, via control of shuttling and degradation parameters, permits multiple stable steady states corresponding to stem-like vs. committed cell states in the differentiation hierarchy. Our analysis also identifies groups of variables that should be measured to fully characterize and discriminate between competing models, and thus serves as a guide for performing minimal experiments for model comparison.

Keywords: Bayesian inference; bistability; chemical reaction network theory; experimental design; matroids.

Fig. 1.

Comparison of models of Wnt signaling. (A) Schematic of the new shuttle model in the absence (Left) and presence (Right) of a Wnt stimulus. DC and Dsh exist in inactive (i) and active (a) forms. P: phosphatase. For a full description of the reactions specifying this model, see SI Appendix. (B) Comparison of models: the shuttle model is compared with four others from literature (–8) based on the features that are present–absent in each model.

Fig. 2.

Bistability of shuttle model. (A) Bifurcations diagrams as a proof-of-concept for feasible shuttle model behavior. Reversible and irreversible behaviors are observed; however, this could change if parameter values are known. In each case the high state of target gene transcription corresponds to a stem-like state and the low state corresponds to a differentiated cell state. (B) Two of the eight necessary conditions for multistationarity of the shuttle model. (C) Model exhibits different behaviors as degradation parameter $k_{14}$ is varied: for low values, bistability; for moderate values, switch-like (ultrasensitive) response; and for high values, graded response. (D) Simulated trajectories for target gene transcription from five different initial conditions. In the bistable region we see two steady states reached; in the monostable region only the high (stem-like) state can be reached. Note that initial behavior in each region is similar: it is important to simulate for long enough to recover these differences in behavior.

Fig. 3.

Bayesian parameter inference of Wnt signaling models. (A) Description of the parameters that are inferred for each of the models used to fit the data describing β-catenin dynamics following a Wnt stimulus. (B) Depiction of the posterior probability distribution used for approximate Bayesian computation. The sequentially decreasing region of probability (blue ovals) defines the joint space of two parameters. Here we show synthesis and degradation rates that are a subset of the full parameter space. This is applicable to any of the Wnt models considered. (C) Fits to the data simulated from the posterior distribution for each model (1,000 particles simulated). Bars represent the 5% and 95% intervals.

Fig. 4.

Matroids allow for parameter-free model discrimination and prediction. (A) Depiction of linear dependence and independence in a vector space (see main text for additional information). (B) Schematic representation of the matroids for Wnt models. Each species represented by a loop is determined from the parameters alone; groups of species represented on a point can be determined by measurement of one of the species; groups of species represented on a line can be determined by measurement of two species. For notation used see SI Appendix, Table S1. (C) Schematic of rank, which corresponds to “what-to-measure.” So, for rank 1, measure 1 species to determine all of the others, for rank 2, measure 2, etc. (D) Deletion is a matroid-theoretic operation which removes a species x from the ground set of the matroid and only considers dependencies of the original model that excluded x. Deleting T and $C_{XT}$ gives a rank 4 matroid which can be visualized by planes in 3D space, as shown here.