An interesting motif in biological circuits is a switch, in which the system can change from (effectively) one binary state to another. An analysis of a continuous reaction network reveals a rise to a switch like response (also referred to as ultra sensitivity). Consider inter conversion of a protein from its native state P to an activated form P^ˆ—, catalyzed by the enzymes E₁and E₂: 

P + E₁ †” PE₁ †” P^ˆ— + E₁
P^ˆ— + E₂ †’ P^ˆ—E₂ †’ P + E₂

(a) Assume that all reaction steps obey mass-action kinetics. What is the steady-state dependence of P^ˆ— as a function of the concentration of E₁? (Assume that total amount of E₁ E₂, and P are all constant and that P is in excess compared to E₁ and E₂).
(b) Alternate starting point for problem: you should be able to rearrange the solution as follows

where V₁ is proportional to the total E₁ in the system, E₂ is proportional to the total V₂ in the system, P_T is the total protein concentration (in all forms), and K₁ and K₂ are suitable combination of the rate constants for the reactions previously described.

For K₁ = 1.0, K₂ = 1.0, plot the steady- state locus of solutions for P* /P_T versus V₁ / V₂.

(c) Assume that the two enzymes operate in a saturated regime, that is m the reactions follow zero-order kinetics with respect to the enzymes. Use the expression from part (b) to plot the steady-state locus for this extreme situation(i.e., K₁ = 0, K₂ = 0).

(d) Comment on the difference in shape of the gain functions in parts (b) and (c). Based on the initial problem description, explain how biology can produce switchlike behavior in this system.

V- P*/P7(1 P*/Pr + K,) V2 (1 P*/P7)(P*/Pr + K,) T.