Compartment model with gamma-distributed elimination flow rate

Until now, the compartmental model was considered as consisting of compartments associated with several anatomical locations in the living system. The general definition of the compartment allows us to associate in the same location a different chemical form of the original molecule administered in the process. In other words, the compartmental analysis can include not only diffusion phenomena but also chemical reaction kinetics. 

One source of nonlinear compartmental models is processes of enzyme-catalyzed reactions that occur in living cells. In such reactions, the reactant combines with an enzyme to form an enzyme-substrate complex, which can then break down to release the product of the reaction and free enzyme or can release the substrate unchanged as well as free enzyme. Traditional compartmental analysis cannot be applied to model enzymatic reactions, but the law of mass-balance allows us to obtain a set of differential equations describing mechanisms implied in such reactions. An important feature of such reactions is that the enzyme 

The mathematical basis for enzymatic reactions stems from work done by Micha-elis and Menten in 1913 [315]. They proposed a situation in which a substrate reacts with an enzyme to form a complex, one molecule of the enzyme combining with one molecule of the substrate to form one molecule of complex. The complex can dissociate into one molecule of each of the enzyme and substrate, or it can produce a product and a recycled enzyme. Schematically, this can be represented by 

In this formulation k+ is the rate parameter for the forward substrate-enzyme reaction, k is the rate parameter for the backward reaction, and k 2 is the rate parameter for the creation of the product. 

Relying on a suggestion of Segel [316], we make the variables of the above equations dimensionless 

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Figure 8.2 One-compartment model with gamma-distributed elimination flow rate k Gam(2, 2). The solid line represents the expected profile E[q(t)], and dashed lines, the confidence intervals E [q (t) JVar [<7 ( )].

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