1. A ODE-based model for an auto-regulated gene may be expressed as: dXprot) dt = w[Xrnal - yprot[Xprot] d[Xrna) dt = [Xprot] 2 K2 1/2 +[Xprot] 2 - Xina Xrna) (a) Outline how you would implement this as a stochastic model using Gillespie's algorithm. Your answer should be given as a descriptive step-by-step procedure, not Matlab code. (b) Plot the null clines of the ODE-based model, assuming 2K1/2 u = Sprot w. Note: Think carefully about what this relationship in parameters means. (c) Sketch what you expect to observe from such a simulation in a plot of [Xrnal and Xprot) vs time. from an initial concentration of Xma] = [Xprot] = 0, assuming parameters consistent with (b). (d) Sketch what you expect to observe from such a simulation in a plot of [Xrna) and [Xprot] vs time from an initial concentration of [Xina) = p 2 and [Xprot] = k1/2, again assuming parameters consistent with (b). 2. A model of the regulation of lysis and lysogeny in an E. coli bacteria infected by phage-A involves eight reactions with the following rate laws: v1 = w1(CIRNA) v2 = 2 CrORNA) v3 = x3[CIRNAI 14 = x4[CroRNAJ V5 = 15 1 - (croprot] 2 (K5) 2+[croprot] 2 v6 = u6 1 - Eclprot) 2 (K6) 2+(ciprot] 2 v7 = x/[clerot] v8 = *8[croprot) For all the following questions, consider a model where w1 > w2, X3 6, K5 > K6 and X7 w2, X3 6, K5 > K6 and X7