## S2 Ro R1 S2 R1 R2

Show that, under appropriate assumptions about the ratios k1/k-1 and k-2k+k3 the equations describing this reaction are of the form (1.68)-(1.69) with f (a1,a2) given by (1.85).

9. Use the law of mass action and the quasi-steady-state assumption for the enzymatic reactions to derive a system of equations of the form (1.68)-(1.69) for the Goldbeter-Lefever model. Verify (1.84).

10. When much of the ATP is depleted in a cell, a considerable amount of cAMP is formed as a product of ATP degradation. This cAMP activates an enzyme phophorylase that splits glycogen, releasing glucose that is rapidly metabolized, replenishing the ATP supply. Devise a model for this control loop and determine conditions under which the production of ATP is oscillatory.

11. By looking for solutions to (1.14) and (1.15) of the form a = ao + eo1 + e2CT2 +----, (1.115)

show that a0 and x0 satisfy the quasi-steady-state approximation. Thus, the quasi-steady-state approximation is the lowest-order term in an asymptotic expansion for the solution. Typical initial conditions are a = 1, x = 0. Does the lowest-order solution satisfy the initial conditions? Find a1 and x1 and plot the solution to first order in e. The variables a and x are called the outer solution, as they are valid for times outside some boundary layer around t = 0. Now rescale time by e, and use the same procedure to construct an asymptotic solution to (1.100) and (1.101), the so-called inner solution. Show that the inner solution satisfies the initial conditions. How can one construct a solution that satisfies the initial conditions and is valid for all times?

CHAPTER 2