Reasons for dependence and the impossibility of determining parameters

A well-known dependence of the equilibrium constants appears in the case in which some step is a linear combination of the others. For example, if we have three steps (1) A B, (2) B C, and (3) A F C, where step (3) is a linear combination of the other two, its equilibrium constant satisfies 

These dependences must always be taken into account in the solution of inverse kinetic problems. For example, when finding constants for eqns. (16) and (81) we must take into account that 

A different dependence of the parameters in kinetic equations was reported by Horiuti [11] who suggested a method for determining the number of independent parameters. The method consists of the numerical estimation of a rank for some Jacobian matrix. (It is known that this procedure can result in a considerable error.) Later, these problems were analyzed in detail by Spivak and Gorskii [52, 53] but they did not aim at the elucidation of the physico-chemical reasons for the appearance of dependent and undeterminable parameters. It is this aspect that we will discuss below. 

We have already noted that a denominator of the steady-state kinetic equation is the concentration polynomial Z. Each summand of this polynomial is the spanning tree weight and corresponds to some path for the formation of a chosen intermediate from the rest. Among the reaction paths there can be dependent paths due to the reversibility of the sufficiently large number of steps. 

The dimension of the basis for the weights of spanning trees is 2n all the rest of n2 - 2n spanning trees will be dependent. For the spanning tree weights Bk i these dependences can be expressed as 

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