Figure 6: The “catch time,” tc, for fast indicator reactions when β ≪ 1 in the reaction mechanism (3)–(4).
Panel (a): The solid black curve is the numerical solution to the mass action equations (36). The solid black dots correspond to the location of the phase-plane trajectory at times: t = 0.33 · tc, 0.67 · tc, tc, 1.33 · tc and 1.67 · tc The thick, dashed/dotted red curve is the c2-nullcline at time t = tc, and the thin, dashed/dotted red curves are the locations of the c2-nullcline at the additional time points: t = 0.33 · tc, 0.67 · tc, 1.33 · tc and 1.67 · tc. Notice the phae-plane trajectory lies just below the c2 nullcline for t < tc and just above it for t > tc. Panel (b): The evolution of c2 in the concentration/time plane. The black dashed line corresponds to = tc, and clearly indicates the time at which c2 reaches its threshold value and intercept the c2-nullcline. The constants and initial conditions (both without units) used in (a) and (b) are: , , k1 = 1, k2 = 1 and k−1 = 1. , k3 = 1, k4 = 100 and k−3 = 1. The substrate concentrations in (a) and (b) has been scaled by their maximum values, and time has been mapped to the t∞ scale: t∞(t) = 1 − 1/ln(t + e).