FIGURE 4.

Positive feedback and constant activation can both lead to oscillations. (A) Per2/Cry mRNA concentration with varying transcription rate of Bmal1 (v_4b). For a low transcription rate of Bmal1, the Per2/Cry concentration reaches a steady state. In the Hopf bifurcation point at v_4b = 0.35 the steady state becomes unstable; for higher values of v_4b the Per2/Cry mRNA concentration oscillates. Thick lines represent the maximum and minimum of the oscillation. For instance, with the default parameter value v_4b = 3.6 the Per2/Cry mRNA concentration oscillates between 0.3 and 1.5 nM (see Fig. 3 A). The dashed line marks the unstable steady state. (C) Per2/Cry mRNA concentration with varying amount of constantly expressed activator c. With low levels of activator the Per2/Cry mRNA concentration reaches a steady state. An activator concentration above the Hopf bifurcation point at c = 0.02 results in Per2/Cry mRNA oscillations. (B and D) Oscillation periods as a function of Bmal1 transcription rate v_4b and constantly expressed activator c, respectively. In both cases the period is stable for a broad parameter range and shows a stronger parameter dependence close to the Hopf bifurcation point. (E, F) The positive feedback is gradually replaced by constant activation. The average amount of total activator (y7_peak + y7_trough)/2 + c is kept constant. Zero-percent positive feedback corresponds to v_4b = 0 and c = 1; 100% corresponds to v_4b = 3.6 and c = 0.01. Oscillations of Per2/Cry mRNA occur in all cases (E); the period remains stable (F).