Answer Happy

Simulate the Blood CO₂ problem: Cn+1-Cn+a3Cn-1 = m for relevant values of m, a and 3. HINT: Be sure to try 4aß> and 4aß < 1. Consider the reaction k₁ k_1 where S, E, C and P are substrate, enzyme, complex and product, respectively, and k₁, k1 and k₂ are positive rate constants. S+E SE=C P+E, (a) Use the Law of Mass Action, which you should state, to write down four equations for the concentrations s, e, c and p of S, E, C and P, respectively. (b) Initially s 80, e = co, c = 0 and p = 0, where so and eo are constant. Show that the total amount of enzyme is conserved. (c) Hence show that the system may be reduced to the following pair of equations ds dt de dt C=-. 80 k₂ = -kiegs + (kis + k_1)c, (d) With the non-dimensionalisation = kiegs- (kis + k_1+k₂)c.. A = k₂ k180 dv(a) do K= k_1+k₂ k180 E= ≈1-(1+K)v(o). eo 80 (5) (6) €0 use the rescaling in time o= k₁eot/e to show that if € < 1 then there is an initial fast transient solution given by u(o) 1 and (7) (8)
(e) Now use the rescaling in time 7 = k₁eot to show that the outer solution is given by du(T) dr ≈-u+(u+K-X)v and V≈ C= U u+K (f) Show that the null clines for Equations (5) and (6) are given by, respectively, Ds a+s Ds B+s¹ where a, ß and D are to be found in terms of k₁, k-1, k2 and eo. (g) Sketch the null clines and draw the phase trajectory which begins at s(0) = so, c(0) = 0. Indicate the fast transient and pseudo-steady-state portions on the trajectory. and c= (9)