Generalised compartmental system

Analysing Exercise l(ii) try to characterise a large set of generalised compartmental systems for which the induced kinetic differential equation can be solved in closed form. 

Another answer to the same question is that all the compartmental systems so important in chemical and biological modelling (see, for example, Jacquez, 1972) are of deficiency zero (Horn, 1971). A generalisation of these mechanisms, the generalised compartmental system, also belongs to this class see Problem 2. 

Show that the deficiency of a (generalised) compartmental system is zero. How can this statement be generalised ... 

As is known from an earlier exercise (Problem 4(ii) of Section 3.5) a generalised compartmental system is a reaction consisting of elementary reactions of three types... 

Let us repeat that a generalised compartmental system is closed if it only contains elementary reactions of the type (4.22a), while it is strictly half open or strictly open according to whether it contains elementary reactions of the type (4.22b) or (4.22c) too. 

These definitions express that a generalised compartmental system is closed if and only if no matter enters it from and leaves it for the outside world. It is strictly half open if and only if no matter enters it but matter does leave it. Finally, it is strictly open if and only if matter does enter it and may leave it. 

As we know from Problem 2 of Subsection 4.2.5, the deficiency of a generalised compartmental system is zero. It is also known that there exists a directed route from one component into the another in the V-graph of a generalised compartmental system if and only if there is a directed route from the first given component into the second one in the FHJ-graph of the reaction. 

Earlier it was shown that if a reaction is acyclic then it cannot be weakly reversible (Exercise 1 of Subsection 4.2.4). This statement shows that in the case of generalised compartmental systems the connection between the two graphs is even stronger there exists not just a one-to-one correspondence between cycles and closed directed routes the two graphs are essentially identical. (Here the empty complex may not be excluded.)... 

The induced kinetic differential equations of generalised compartmental systems of the three types are as follows ... 

An easy-to-formulate (but still hard to work through, see Exercise l(ii) of Subsection 4.1.3 and Open Problem 1 of Subsection 4.1.5) generalisation of some of the results collected by Rodiguin Rodiguina (1964) is if the graph of a generalised compartmental system is a tree (i.e. it is acyclic) then the solutions of the induced kinetic differential equation may often happen to be explicitly determined (but not in all cases of tree graph, as one familiar with compartmental systems would expect). 

Now let us formulate a statement on the existence of a deficiency zero inducing mechanism. It will be shown that if the right-hand side of a kinetic differential equation is the sum of univariate monomials and if all the variables have the same exponent in all the rows (this assumption is necessary as well ) then — if an additional condition is met and only then — there exists an inducing generalised compartmental system to the system of differential equations. Let us formulate the statement more precisely. 

There exists an inducing generalised compartmental system of M compartments to the system of differential equations... 

Therefore there is no generalised compartmental system that induces this kinetic differential equation. The question is if there exists a deficiency zero mechanism that induces this kinetic differential equation ... 

A necessary and sufficient condition has been given for the existence of an inducing generalised compartmental system to the system of differential equations (4.23). However, certain generalised compartmental systems (and differential equations) are to be considered as identical, as they are essentially not different. This will be done below and thus a problem of the type (3) formulated in Subsection 4.7.1.2 will be solved here. 

The core of a generalised compartmental system is obtained through substituting all first-order endpoints by the zero complex. Thus the core of our example is ... 

The core of a system of differential equations of the form (4.23) is the differential equation where the variables having the same index as the zero column vectors of the matrix (a,y) have been deleted. The induced kinetic differential equation of our generalised compartmental system is ... 

The core of this differential equation consists of the first five equations. Clearly, these make up the induced kinetic differential equation of the core of the given generalised compartmental system. This is not by chance, because in general it is true that the induced kinetic differential equation of the core of a generalised compartmental system is the core of the induced kinetic differential equation of the generalised compartmental system. 

Let us call two differential equations of the form (4.23) equivalent, if their core is the same. Two generalised compartmental systems will also be called equivalent, if their core is the same. 

All the necessary preparations have been done to answer the question what is the number of generalised compartmental systems that essentially induce eqn (4.23) Part of the answer will be formulated here, without proof. The full answer can be found in the Problems, Section 4.7.4. 

Let us suppose that the coefficients of the right-hand side of the differential equation fulfil (4.24a). Then there exists a generalised compartmental system and 2 - 1 other strictly half-open generalised compartmental systems consisting of Af — A i (A i 1, 2,..., ) components inducing a differential... 

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