Some results about the structural properties of the Wnt pathway, its steady states and its non-associative commutative algebra

We consider a reaction network of the Wnt pathway endowed with mass-action kinetics. Using concepts in the theory of robustness and stability within Chemical Reaction Network Theory and advances in decomposing reaction networks, we perform a systematic analysis of the structural, structo-kinetic and kinetic properties of this pathway. We show that the network can be systematically decomposed into a set of subnetworks and we use elements matrix theory to study their stability properties. Considering the positive stoichiometric classes, we obtain the analytical expressions of the positive steady states and we identify three species with absolute concentration robustness within the core of the destruction complex of the pathway. We identify nonnegative stoichiometric classes that admit boundary steady states. We construct the non-associative commutative algebra associated with the system and combine algebraic and algorithmic approaches to characterize its structural properties, construct its subalgebras and show how they relate to the existence of boundary initial conditions that admit boundary steady state solutions. We also show the existence of a category of subalgebras that generate unbounded solutions for most of the nonnegative initial conditions.