Two-way catenary compartment model

A good review of the master equation approach to chemical kinetics has been given by McQuarrie [383]. Jacquez [335] presents the master equation for the general ra-compartment, the catenary, and the mammillary models. That author further develops the equation for the one- and two-compartment models to obtain the expectation and variance of the number of particles in the model. Many others consider the m-compartment case [342,345,384], and Matis [385] gives a complete methodological rule to solve the Kolmogorov equations. 

We can, however, analyze these problems within the framework of the stochastic formulation by looking for an exact solution, or by using the probability generating functions, or the stochastic simulation algorithm. 

For simple cases with populations of small sizes, one can express the Kolmogorov equations in a matrix form. The elements of the grand probability function Pn 1.nm (t) can be considered in a vector form  

A very useful tool for finding analytically the distribution of Nft) is to obtain and solve partial differential equations for the associated cumulant generating functions. The moment generating function, denoted by Ai (0, t), is defined for a multivariate integer-valued variable N (t) as 

Statistical characteristics of the random vector N (t) can be directly obtained from cumulants nSl.Sm (t) with all Sj = 0 except  

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As an example application, we will develop the master equation for a fragment of a two-way catenary compartment model around three compartments spaced by Az, as illustrated in Figure 9.24. By assuming only one particle in movement, the master equation gives... 

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