Fig. 3

The deterministic QSSA can identify key parameters determining the accuracy of the stochastic QSSA. a The diagrams of the full model (Eqs. 9) and the reduced model (Eq. 10). b–c) Whereas the deterministic QSSA is accurate (b), but the stochastic QSSA is inaccurate (c) with the same initial condition: \( S(0)=E_{S}(0)=E_{S_{2}}(0)=0\protect \phantom {\dot {i}\!}\). The deterministic QSSA becomes inaccurate with a different initial condition: S(0)=410 or 500, E _S (0)=0, \(E_{S_{2}}(0)=1\) (the inset of (b)). The colored ranges and histograms represent a standard deviation of S from its mean and the distribution of S at steady state, respectively (c). The parameters of the model is adopted from Thomas et al. [27]: k _in =0.5s ⁻¹, k ₋₁=k ₋₂=100s ⁻¹, k _p =1s ⁻¹, \(K_{m_{1}}=2 \cdot 10^{6}\), \(K_{m_{2}}=0.101\). d–e Both the errors of the stochastic and the deterministic QSSA depends on k ₋₁ (d), but not k ₋₂ (e). The errors were measured as in Fig. 2. In particular, the error of the deterministic QSSA is estimated with Eq. 8, where T=4000 and X(t)=S(t) were used. See Additional file 1: Figure S3 for the distribution of S from the stochastic simulations