Figure 2. Presence of oscillations in the model when miR-9 acts only to affect the stability of Hes1 mRNA.

(a) Different combinations of τ, p₀ and S(r) are used to test for the presence of oscillations. The different coloured lines denote Hopf bifurcations for different values of τ as indicated. A Hopf bifurcation denotes the transition of a system from a stable to an unstable, oscillatory state or vice versa, as a parameter of the system is varied. Fixed points exist to the right of the curves whereas oscillations are present to the left. The inset shows the curve for τ=29. The dashed red line indicates the value of p₀ for which two Hopf bifurcations exist for mRNA half-lives of ~35 and 20 min. (b) A window of oscillations emerges for changes in r, with p₀ fixed at 390 and τ=29 min. (c) Example time series when r is fixed to the values given by crosses in (b). These r values give rise to the half-lives (ln(2)/S(r)) as indicated. n₀=5, μ_p=22 min, b_l=ln(2)/20 min⁻¹, b_u=ln(2)/35 min⁻¹, r₀=100, m₀=5.