Fig. 1

The relationship between the accuracy of the deterministic and the stochastic QSSA. a The diagrams for the full model (Eqs. 1–3) and the reduced model (Eq. 5). b–c The deterministic QSSA is accurate when both k _f =10⁻¹ h ⁻¹ and k _f =10¹ h ⁻¹ (the insets). However, the corresponding stochastic QSSA is accurate only when k _f =10¹. The colored ranges and histograms represent a standard deviation of R from its mean and the distribution of R at steady state, respectively. Here, K _D =10, α _M =300h ⁻¹, β _M =β _R =1h ⁻¹. M(0)=R(0)=100 and D _A (0)=0. d–e The phase plots of deterministic trajectories with various initial conditions when k _f =10⁻¹ (d) and k _f =10¹ (e). The star and the square indicate initial conditions used in (b) and (c) and the insets of (d) and (e), respectively. The black circle indicates the fixed point. Here, we assumed M(0)=R(0)