This work concerns the question of how two important dynamical properties, oscillations and bistability, emerge in an important biological signaling network. Specifically, we consider a model for dual-site phosphorylation and dephosphorylation of extracellular signal-regulated kinase (ERK). We prove that oscillations persist even as the model is greatly simplified (reactions are made irreversible and intermediates are removed). Bistability, however, is much less robust -this property is lost when intermediates are removed or even when all reactions are made irreversible. Moreover, bistability is characterized by the presence of two reversible, catalytic reactions: as other reactions are made irreversible, bistability persists as long as one or both of the specified reactions is preserved. Finally, we investigate the maximum number of steady states, aided by a network's "mixed volume" (a concept from convex geometry). Taken together, our results shed light on the question of how oscillations and bistability emerge from a limiting network of the ERK network -namely, the fully processive dual-site network -which is known to be globally stable and therefore lack both oscillations and bistability. Our proofs are enabled by a Hopf bifurcation criterion due to Yang, analyses of Newton polytopes arising from Hurwitz determinants, and recent characterizations of multistationarity for networks having a steady-state parametrization.
Neural codes are binary codes in {0, 1} n ; here we focus on the ones which represent the firing patterns of a type of neurons called place cells. There is much interest in determining which neural codes can be realized by a collection of convex sets. However, drawing representations of these convex sets, particularly as the number of neurons in a code increases, can be very difficult. Nevertheless, for a class of codes that are said to be k-inductively pierced for k = 0, 1, 2 there is an algorithm for drawing Euler diagrams. Here we use the toric ideal of a code to show sufficient conditions for a code to be 1-or 2-inductively pierced, so that we may use the existing algorithm to draw realizations of such codes.
We consider a model of extracellular signal-regulated kinase (ERK) regulation by dual-site phosphorylation and dephosphorylation, which exhibits bistability and oscillations, but loses these properties in the limit in which the mechanisms underlying phosphorylation and dephosphorylation become processive. Our results suggest that anywhere along the way to becoming processive, the model remains bistable and oscillatory. More precisely, in simplified versions of the model, precursors to bistability and oscillations (specifically, multistationarity and Hopf bifurcations, respectively) exist at all "processivity levels". Finally, we investigate whether bistability and oscillations can exist together.
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