We will use SciPy to simulate a one-dimensional model of frog oocyte maturation, based on the work of Ferrell and colleagues.
The MAP kinase kinase kinase Mos is synthesized in response to activation of MAP kinase which in turn is activated by Mos through the MAP kinase kinase MEK. Activated Mos therefore leads to further activation of Mos, and the system has positive feedback.
Ferrell and colleagues modelled levels of Mos through a rate equation with three terms: one describing basal synthesis of Mos, which is proportional to levels of extracellular progesterone; another describing the positive feedback generated by the MAP kinase cascade and modelled with a Hill function; and the third describing degradation.
Let $y$ denote levels of Mos, then we have: $$ {dy}/{dt}= k [P] + f y^n/{K^n + y^n} - y $$ where $k= 0.2$, $[P]$ is the concentration of progesterone, $f= 40$ is the strength of the feedback, $n$ is a Hill number ($n=5$), and $K= 20$ nM. Units of time are measured in units of the life-time of Mos.
Write code to simulate this equation using odeint. Show that there is a concentration of progesterone where you can find two different steady-state levels of Mos and a concentration of progesterone where you can find only one.
What happens to the bistability when $n=1$? What changes in the biochemistry could account for a Hill number of 1? What then are the key features that enable the bistability?