Hypothesis of a quasi-steady state

This two-variable system (Goldbeter et al, 1978) presents the additional advantage of being formally identical with the system of eqns (2.7) studied in chapter 2 for glycolytic oscillations. This similarity stems from the basic structure common to the two models a substrate, injected at a constant rate, is transformed in a reaction catalysed by an allosteric enzyme activated by the reaction product. In the cAMP-synthesizing system in D. discoideum, activation is indirect as extracellular cAMP enhances the synthesis of intracellular cAMP, which is then transported into the extracellular medium. However, the hypothesis of a quasi-steady state for intracellular cAMP is tantamount to considering that the variation of )8 is so fast that the enzyme is, de facto, activated directly by its apparent product, extracellular cAMP. 

As indicated in table 5.2, if the value of parameter q is sufficiently high to justify the hypothesis of a quasi-steady state for variable /3, the situation is somewhat different for the experimental values of k, and ki-The latter are too low for the hypothesis to hold rigorously. It is for this reason that the preceding results on oscillations and relay were obtained by means of the three-variable system (5.12). It is nevertheless interesting to study the behaviour of the two-variable system (5.15), in order to provide an estimate for the error made upon reducing the number of variables to only two. This error could well turn out to be minute, and thus acceptable, in comparison with the advantage provided by the possibility of resorting to the phase plane analysis of a two-variable system. 

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