Figure 3:

The leading order asymptotic solution (52) of the substrate concentration for the indicator reaction matches the numerical solution when the indicator reaction is faster than the non-observable reaction (δ_S ≪ 1). The solid black curves are numerical solutions to the mass action equations of the complete reaction (7) and the broken red curves are numerical solutions to the asymptotic differential equation (51). (a) The constants (without units) used in the numerical simulation are: $e_1^0=1$, $s_1^0=100$, k₁ = 1, k₂ = 1 and k₋₁ = 1. $s_2^0=10$, k₃ = 10, k₄ = 100 and k₋₃ = 10. (b) The constants (without units) used in the numerical simulation are: $e_1^0=1$, $s_1^0=100$, k₁ = 1, k₂ = 1 and k₋₁ = 1. $s_2^0=100$, k₃ = 10, k₄ = 100 and k₋₃ = 10. Time has been mapped to the t_∞ scale: t_∞(t) = 1 – 1/ ln(t + e).