A two-variable biochemical model for birhythmicity

Properties of the model in the absence of recycling excitability with a single threshold, oscillations and bistability 

The main properties of the two-variable allosteric model for glycolytic oscillations, in the absence of product recycling into substrate, are summarized in fig. 3.2. In the phase plane (a, y), the system governed by eqns (2.7) evolves toward a limit cycle (dashed line) when the steady state, located at the intersection of the nullclines (da/dr) = 0, (dy/dr) = 0, lies in a region of sufficiently negative slope (do/dy) on the latter nullcline. In fig. 3.2, this region extends, schematically, from A to B. 

Two other phenomena follow from the structure of the nullclines as represented in fig. 3.2. First, when the steady state lies in the region of stability located to the left of point A, for example in C, the system 

The median branch of the product nullcline thus defines an abrupt threshold for excitability when the initial condition corresponding to the addition of a pulse of product lies to the left of the threshold, the perturbation regresses and the system immediately returns to the steady state the perturbation, on the contrary, is amplified when it exceeds the threshold. The role of the nullcline in this regard results from the fact that this curve, which obeys the equation (dy/dt) = 0, separates the regions of the phase plane in which y will tend to grow or decay in the course of time. 

On the other hand, the S (or rather the N) shape of the product nullcline implies the existence of a phenomenon of bistability when the substrate concentration is held constant in the course of time. Three steady states can indeed be obtained when the horizontal corresponding to the fixed value of a intersects the product nullcline in the region where the latter possesses a region of negative slope. The numerical integration of eqns (2.7) in these conditions reveals the evolution to either one of two simultaneously stable steady states the choice of the final state depends on the initial product concentration, y (fig. 3.3). Each of the two stable states possesses its own basin of attraction, which is the set of all initial conditions in the phase plane from which the system evolves to reach eventually that particular state. 

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