Complex oscillatory phenomena in a three-variable model for cAMP signalling

2 Complex oscillatory phenomena in a three-variable model for cAMP signalling 

The reduction of system (6.2) to only four variables can be performed as in the slightly different model analysed in the preceding chapter (details of this reduction are given in the Appendix to chapter 5 see also Martiel Goldbeter, 1987a). As previously, the four variables retained in the reduced system are the total fraction of receptor in active state (/ r), the concentration of the substrate ATP (a), as well as the concentrations of intracellular (jS) and extracellular (y) cAMP. 

As in the case of the four-variable system (5.9) obtained when the source of cooperativity lies in the activation of adenylate cyclase rather than in the binding of cAMP to the receptor, the level of ATP varies only sUghtly in the course of oscillations. This observation allows us to reduce the number of variables by considering that the level of ATP remains fixed in the course of time. For system (5.9), this simplification leads to the three-variable system (5.12). Such an assumption is not retained here indeed, as soon as the ATP concentration is held constant, all manifestations of complex oscillatory behaviour disappear in the model. The reasons for such a phenomenon are elucidated below. 

In order to reduce the number of variables down to three, it is thus desirable to eliminate some variable other than ATP, if we wish to retain the possibility of complex oscillations. As indicated in chapter 5, a quasi-steady-state hypothesis for variable /3, justified by the large value of parameter q, allows the transformation of the kinetic equation for p into an algebraic relation. It is precisely such a reduction that led 

To refine the bifurcation structure of the model, it is useful to make a vertical section through the diagram of fig. 6.2 in the v-k parameter space by fixing the substrate input rate v at a particular value (vertical arrow at the top of fig. 6.2). 

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