Exercise 7.3: Dependence of signaling response on total histidine kinase and response regulator concentrations

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Consider again the simpler of the two models of a signaling circuit presented in Chapter 7.

nonparadoxical signaling

The chapter featured a figure, shown below, that gave the steady-state output (which is proportional to \(y_p\)) as a function of signal concentration, \(s\). The figure depicts dependence of the steady-state output on the level of signal for various dimensionless parameters depending on the total amount of histidine kinase X and response regulator Y in the system.

Code 
from IPython.display import HTML
HTML('../figs/monofunctional_kinase_response.html')
Response of circuit with monofunctional kinase

a) Derive an expression for the steady-state concentration of response regulator, \(y_p\).

b) Make a plot like the one above using the expression you derived in part (a).