Model Reduction on the Wnt Pathway Leads to Biological Adaptation

Complex systems are an unavoidable problem in the field of biology. One of the ways that scientists have tried to overcome this problem is by building mathematical models—manageable representations designed to look at specific physical phenomena. The Wnt Signaling Pathway is a complex system known to regulate cell-to-cell interactions, play a crucial role in Embryonic Development, and has been implicated in the study of cancer. Typically, the Wnt signal is observed through the behavior of a protein called beta-Catenin (β-Catenin). In 2003, Lee et al. built a model of the Wnt pathway which caused β-Catenin to increase over time. However, in 2010, Jensen et al. built a different model of the Wnt pathway which caused β-Catenin to oscillate over time. This project called for model reduction on the Jensen et al. model to identify the phenomenological parameter combinations that determined features of the Wnt oscillations. The method used to reduce the model is called the Manifold Boundary Approximation Method, which is a geometric, parameter-independent method of reducing the model one parameter at a time. Reduction of the model showed that there were 5 variables and 8 parameters which drove the oscillating behavior of the system. After comparing our results to the Lee et al. reduced model of the Wnt pathway done by student Dane Bjork, a minimal model was constructed which predicted a novel class behavior of the Wnt system: biological adaptation