First-order approximation of Michaelis-Menten kinetics

Consider the four-step reaction chain shown below. Here the label v_i is the reaction rate; it is not a mass action rate constant. So, for example d/dt s₂(t) = v₁ v₂. (This dt is the conventional notation when labeling the rates of enzyme-catalyzed reactions. It is unfortunately the same convention that's used for mass-action rate constants. We use the context and the variable name (v vs. k) to distinguish which case we're considering.)

Suppose the rate v₀ is fixed and presume the other rates are given by Michaelis-Menten kinetics, with

(a) Simulate the system from initial conditions (in mM) (s₁ , s₂ , s₃ ) = (0.3, 0.2, 0.1) and (6, 4, 4).

(b) Construct an approximate model in which the rates of reactions 1, 2, and 3 are replaced by mass action rate laws: v_i = k_is_i. For i = 1, 2, 3, choose rate constants ki so that the mass action rate is the linearization of the corresponding Michaelis- Menten rate law, centered at s_i = 0. This is called the first-order approximation for the Michaelis-Menten rate.

(c) Simulate your simpler (mass-action based) model from the two sets of initial conditions in part (a). Explain why the approximation is better in one case than the other.