Linear noise approximation The LNA approximates the CTMC as a continuous-state Gaussian process, given in the form of a set of ODEs that describe the time evolution of expectation and variance of the species. The error of approximation is dependent upon the volumetric factor N, the structure and the rates of the CRN. Given a CRN \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {C}}=(\varLambda ,{\mathcal {R}})$$\end{document}C=(Λ,R) with initial configuration \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0 \in {\mathbb {N}}^{|\varLambda |}$$\end{document}x0∈N|Λ| and in a system of volume size N, we define the stochastic process \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y=N\cdot \varPhi + \sqrt{N}\cdot Z$$\end{document}Y=N·Φ+N·Z, where \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 is the deterministic process given in Eq. 1, and Z is a zero-mean Gaussian process (since we assume the initial condition is a fixed value), and with covariance C[Z(t)] described by the solution of the following ODEs with initial condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C[Z(0)]=0$$\end{document}C[Z(0)]=0:3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{{\rm d} C[Z(t)] }{{\rm d} t}= & {} F'(\varPhi (t), C[Z(t)]) \nonumber \\= & {} J_F(\varPhi (t))C[Z(t)] + C[Z(t)]J^T_F(\varPhi (t))+W(\varPhi (t)) \end{aligned}$$\end{document}dC[Z(t)]dt=F′(Φ(t),C[Z(t)])=JF(Φ(t))C[Z(t)]+C[Z(t)]JFT(Φ(t))+W(Φ(t))where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${J}_F(\varPhi (t))$$\end{document}JF(Φ(t)) is the Jacobian of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(\varPhi (t))$$\end{document}F(Φ(t)), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J^T_F(\varPhi (t))$$\end{document}JFT(Φ(t)) its transpose, and\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} W(\varPhi (t))= \sum _{\tau \in {\mathcal {R}}} \upsilon _{\tau } {\upsilon _{\tau }}^T k_\tau \prod _{S \in \varLambda }\varPhi _{S}^{r_{S,\tau }}(t). \end{aligned}$$\end{document}W(Φ(t))=∑τ∈RυτυτTkτ∏S∈ΛΦSrS,τ(t).Expected value and covariance matrix of Y(t) are completely characterized by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPhi (t)$$\end{document}Φ(t) and C[Z(t)] since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E[Y(t)]=N\varPhi (t)$$\end{document}E[Y(t)]=NΦ(t) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C[Y(t)]=\sqrt{N} C[Z(t)] \sqrt{N}=N C[Z(t)]$$\end{document}C[Y(t)]=NC[Z(t)]N=NC[Z(t)].