ABSTRACT
We have seen in section 3.1 that for first order linear chemical reaction networks, the mean abundance vector https://www.w3.org/1998/Math/MathML"> E ( t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429086663/acb7e1d1-9bf9-43aa-8a26-560ac513e49b/content/eq1515.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and the vector containing the second moments https://www.w3.org/1998/Math/MathML"> V ( t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429086663/acb7e1d1-9bf9-43aa-8a26-560ac513e49b/content/eq1516.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> satisfy ordinary differential equations (o.d.e.) that can be deduced from probabilistic arguments; see (3.1) and (3.2). Sections 12.1 and 12.4 will provide a law of large numbers and Gaussian approximations that lead to gaussian processes drifted by nonlinear o.d.e., which are the main actual mathematical tool for handling complex gene regulatory networks. We hence provide a short course on dynamical systems in chapters 9,10 and 11 for scientists having some knowledge of differential calculus; we will use parts of this material in chapters 12 and 13 when dealing with the linear noise approximation and mass action kinetics.
