Figure 2.

Oscillations in Hes/Her protein numbers for the simplified two-cell gene regulatory system (depicted in figure 1) with no Delta-Notch coupling. (a) The results of individual-based simulations where noisy cycles persist. The simulations of the nonlinear stochastic model are performed using a modified version [53] of the Gillespie algorithm [54], which takes into account delayed reactions. This is in contrast to the deterministic trajectory in (b), where the oscillations are transient and eventually decay to a fixed point. A small difference between the delays in either cell gives rise to differing frequencies of oscillation—the noisy cycles are not synchronized. This is further illustrated by the power spectra of the cycles in (a), which are shown in figure 4a2. Referring to the model specified in electronic supplementary material, S1, the rate parameters used here are $a^{\alpha }=4.5$, $b^{\alpha }=0.23$, $c^{\alpha }=0.23$ and $k_{\alpha }=3.3$ for all α, the system size is N = 10, the reference protein levels are $n_0^{(p)h}=4N$ and $n_0^{(p)d}=100N$, the delay times are τ^(p)h = 2, τ^(p)d = 5 and τ^(m)d = 50 in both cells but τ^(m)h = 18 in cell 1 and τ^(m)h = 22 in cell 2. The rates associated with the Hill functions are $r_0^{\mathrm{h}}=r_d^{\mathrm{h}}=r_{hd}^{\mathrm{h}}=0$, $r_h^{\mathrm{h}}=1$, $r_0^{\mathrm{d}}=r_d^{\mathrm{d}}=r_{hd}^{\mathrm{d}}=0$ and $r_h^{\mathrm{d}}=1$. These values are taken from estimates provided in [19], which are justified therein. Times are in units of minutes, and rates have units of ${\text{min}}^{-1}$. (Online version in colour.)