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. T he Wnt signaling pathway plays a key role in essential cellular processes ranging from proliferation and cell specification during development to adult stem cell maintenance and wound repair (1). Dysfunction of Wnt signaling is implicated in many pathological conditions, including degenerative diseases and cancer (2-4). Despite many molecular advances, the pathway dynamics are still not well understood. Theoretical investigations of the Wnt/β-catenin pathway serve as testbeds for working hypotheses (5-12).We focus on models of canonical Wnt pathway processes with the aim of elucidating mechanisms, predicting function, and identifying key pathway components in adult tissues, such as colonic crypts. We compare four preexisting ordinary differential equation models (5-8) and find, using injectivity theory, that for any given conditions and parameter values, none of the models is capable of multiple cellular responses.In many tissues Wnt plays a crucial role in cell fate specification (3). At the base of colonic crypts, cells exist in a stem-like, proliferative phenotype in the presence of Wnt. As these cells' progeny move up the crypt axis they enter a Wnt-low environment and change fate (perhaps reversibly), becoming differentiated, specialized gut cells (13). In neuronal and endocrinal tissues, Wnt/β-catenin data suggest cell fate plasticity under different environmental conditions (14, 15). Here, we introduce a new model motivated by experimental findings not described in previous models (16-18) to investigate bistable switching in the Wnt pathway. We find the new model to be capable of multiple cellular responses; furthermore, our parameter-free techniques identify that molecular shuttling (between cytoplasm and nucleus) and degradation together may serve as a possible mechanism for governing bistability in the pathway, corresponding to, for example, ...

Joining and decomposing reaction networks

“…Steady-state invariants are polynomial equations satisfied by the species concentrations at steady state (Gunawardena 2007, Manrai & Gunawardena 2008). These polynomials are used for model comparison and are particularly useful when only incomplete data are available (Harrington et al 2012, 2016, MacLean et al 2015. Specifically, when only some of the species concentrations are measurable, an ideal obtained by eliminating non-measurable species variables from the steady-state equations is computed, and then the generators of this ideal are used to test goodness-of-fit.…”

Solving polynomial systems via homotopy continuation and monodromy

“…A family of interesting examples arises from chemical reaction network theory. A chemical reaction network considered under the laws of mass-action kinetics leads to a dynamical polynomial system, the solutions of which represent all the equilibria for the given reaction network [14,25]. These polynomial systems are not generically sparse and we cannot easily compute their root count.…”

Gröbner bases of reaction networks with intermediate species

“…Specifically, the evolution of the concentrations of the species of a chemical reaction network in time is described under mass-action by a system of ordinary differential equations in R n The question of restricting to non-negative steady states remains challenging and no straightforward solutions have been proposed. Despite of this, Gröbner bases have been for example used for model discrimination [13,14,[18][19][20]. They are also used to decide whether the steady state ideal is binomial, that is, whether any reduced Gröbner basis consists of polynomials with at most two terms.…”