Consider a cascade of three activators X→Y→ Z. Protein X is initially present in the cell in its inactive form. The input signal of X, S x, appears at time t = 0. As a result, X rapidly becomes active and binds the promoter of gene Y, so that protein Y starts to be produced at rate β. When Y levels exceed a threshold K, gene Z begins to be transcribed and translated at rate β. All proteins have the same degradation/dilution rate γ.
(i) What are the concentrations of proteins Y and Z as a function of time?
(ii) What is the minimum duration of the pulse S x such that Z will be produced?
(iii) What is the response time of protein Z with respect to the time of addition of S x?