Homeostasis-Bifurcation Singularities and Identifiability of Feedforward Networks

This dissertation addresses two aspects of dynamical systems arising from biological networks: homeostasis-bifurcation and identifiability.

Homeostasis occurs when a biological quantity does not change very much as a parameter is varied over a wide interval. Local bifurcation occurs when the multiplicity or stability of equilibria changes at a point. Both phenomena can occur simultaneously and as the result of a single mechanism. We show that this is the case in the feedback inhibition network motif. In addition we prove that longer feedback inhibition networks are less stable. Towards understanding interactions between homeostasis and bifurcations, we define a new type of singularity, the homeostasis-bifurcation point. Using singularity theory, the behavior of dynamical systems with homeostasis-bifurcation points is characterized. In particular, we show that multiple homeostatic plateaus separated by hysteretic switches and homeostatic limit cycle periods and amplitudes are common when these singularities occur.

Identifiability asks whether it is possible to infer model parameters from measurements. We characterize the structural identifiability properties for feedforward networks with linear reaction rate kinetics. Interestingly, the set of reaction rates corresponding to the edges of the graph are identifiable, but the assignment of rates to edges is not; Permutations of the reaction rates leads to the same measurements. We show how the identifiability results for linear kinetics can be extended to Michaelis-Menten kinetics using asymptotics.