Deterministic semantics Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C=(\varLambda ,R)$$\end{document}C=(Λ,R) be a CRN. The net change associated to a reaction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau \in R$$\end{document}τ∈R is defined by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upsilon _{\tau }=p_{\tau } - r_{\tau }$$\end{document}υτ=pτ-rτ. The deterministic semantics models the concentration of the species in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varLambda$$\end{document}Λ over time as a set of autonomous polynomial first order differential equations (ODEs):1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{{\rm d} \varPhi (t)}{{\rm d}t}= & {} \nonumber \\ F(\varPhi (t))= & {} \sum _{\tau =(r_{\tau },p_{\tau },k_{\tau }) \in R} \upsilon _{r} \cdot \left( k_{\tau }\prod _{i=1}^{|\varLambda |}{\varPhi _i(t)}^{r_{\tau ,i}}\right) . \end{aligned}$$\end{document}dΦ(t)dt=F(Φ(t))=∑τ=(rτ,pτ,kτ)∈Rυr·kτ∏i=1|Λ|Φi(t)rτ,i.where function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPhi : {\mathbb {R}}_{\ge 0} \rightarrow {\mathbb {R}}^{\varLambda }$$\end{document}Φ:R≥0→RΛ describes the behaviour of the system assuming a continuous state-space semantics, and therefore \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPhi (t) \in {\mathbb {R}}^{|\varLambda |}$$\end{document}Φ(t)∈R|Λ| is the vector of the species concentrations at time t and F is simply the derivative of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPhi$$\end{document}Φ with respect to time. Assuming \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_0=0$$\end{document}t0=0, the initial condition is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPhi (0)=[x_0]$$\end{document}Φ(0)=[x0], where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0$$\end{document}x0 is the initial configuration (vector of concentrations of molecules) of the CRN. It is well known that the deterministic semantics may be imprecise for low molecular counts, but is accurate in the limit for high populations Kampen (1992). However, the deterministic semantics produces the same proportion of molecules, regardless of total concentration.