Linkage classes

Feinberg also keeps track of how the arrows connecting any two complexes in a linkage class are oriented for instance, the case of the three xylene isomers could be regarded as A B C A, or as A <-> B C A, or any other in-... 

For A = 4, / is obviously 1 for A = 5, / = 2 and for N = 6, /f = 4, because the reaction 2A3 = Ag cannot be written as a linear combination of reactions of the type of Eq. (165). However, after that adding a new olefin only increases the number of independent reactions by one, so for N> 5 one has R = N — 2. For every component A/ / > 3, there is one linkage class all pairs which, as a complex, have the same carbon number (this still leaves out a large number of compatible complexes, because compatible triplets, etc., are excluded). However, the number of complexes and the deficiency now grow very rapidly with N, since they are delivered by a modified Fibonacci series ... 

One may ask why any compatible complexes should be excluded from any given linkage class. First of all, the Feinberg theory results are strong if the kinetics are... 

These are all independent, so R = N each one represents a linkage class, so N" = N and each one yields two complexes, so N = 2N, and 6 = 0. The deficiency tends to become very large in systems where most of the components have brute chemical formulas that are multiples of each other, which give rise to strongly interconnected networks (very many compatible complexes exist, even if complexes with more than two members are excluded) it stays at 0 or 1 in most systems of the hydrodesulfurization type, where parallel reactions with (almost) no interconnection take place. 

Feinberg s network theory [3-7]. The deficiency of a network is calculated from its number of complexes, n, linkage classes, Z, and rank, s. Here, a "complex" is defined as the reactant(s) or product(s) of a step. For example, the complexes of a step A + B K are A+B and K. A network consists of one or more "linkage classes," defined as portions consisting of complexes connected with one another by arrows, but not with complexes of another class For example, a network... 

The CSTR network 14.1 has four complexes (A, K, P, and 0), only one linkage class, and rank 3. Accordingly, it has a deficiency of zero and will sustain just one stable and no unstable steady states. 

Show that all the reactions with three short complexes and with a single linkage class show regular behaviour, either by proving the equality of the deficiency to zero, or by directly verifying the properties needed for quasithermodynamic behaviour. 

Another corollary of the first statement is if there exists precisely one critical species, and if within each linkage class there exists at most one complex in which that species appears, then multiple positive steady states cannot occur. 

Consider a weakly reversible reaction with L linkage classes. Let 8 denote the deficiency of the reaction and let 5, (/ = 1, 2,..., L) denote the deficiency of the /th linkage class and suppose that... 

Show that a weakly reversible reaction which is of deficiency zero or one and which consists of a single linkage class admits precisely one positive equilibrium point. 

Show that condition (4.16) is equivalent to the requirement that the stoichiometric space be the direct sum of the stoichiometric spaces generated by the linkage classes separately. 

For the sake of easy manipulation it would be useful to find an inducing reaction with the minimal number of complexes, elementary reactions, linkage classes etc. What kind of reasonable assumptions assure the... 

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