ABSTRACT
Section 12.1 focuses on laws of large numbers for reaction networks involving transitions of the form https://www.w3.org/1998/Math/MathML"> x → x + l https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429086663/acb7e1d1-9bf9-43aa-8a26-560ac513e49b/content/eq2431.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , where https://www.w3.org/1998/Math/MathML"> x ∈ N M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429086663/acb7e1d1-9bf9-43aa-8a26-560ac513e49b/content/eq2432.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and https://www.w3.org/1998/Math/MathML"> l ∈ Z M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429086663/acb7e1d1-9bf9-43aa-8a26-560ac513e49b/content/eq2433.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , of rates https://www.w3.org/1998/Math/MathML"> q xx + l = N β l ( x / N ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429086663/acb7e1d1-9bf9-43aa-8a26-560ac513e49b/content/eq2434.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , for a large parameter N. We here consider the special case where the rate functions are defined using the mass action principle, assuming well-mixed reactors of macroscopic proportions. Biological cells are neither well-mixed nor macroscopic; see, e.g., the discussion in [75]. Models based on mass action principles are, however, useful as first approximations where precise mathematical results can be obtained, which help understanding complex systems. For example, [5] showed that the steady state distributions associated with stochastic mass action dynamics are often of product form and are strongly related mathematically to the equilibrium points of the associated deterministic versions. They also extend these results to Markov processes based on Michaelis Menten transition rates, shedding light, e.g., on the results described in section 4.1 for metabolic pathways. A beautiful mathematical theory has been developed for deterministic mass action kinetics by [90], [89], [48], [49], [50] and [51]
