Complex oscillations in a seven-variable model for cAMP signalling

1 Complex oscillations in a seven-variable model for cAMP signalling 

The first indication of complex oscillations was that of the complex periodic behaviour of the bursting type shown in fig. 6.1 (Martiel Goldbeter, 1985). These oscillations resemble those obtained in the model for the multiply regulated enzyme system analysed in chapter 4. 

These complex periodic oscillations were observed in the model for cAMP signalling in D. discoideum by numerical integration of the differential equations, before reduction of the number of variables. As indicated in chapter 5, excitable and oscillatory behaviour occm in this model as a result of self-amplification in cAMP synthesis. The nonlin- 

In the model analysed in section 5.5, positive cooperativity originated from the activation of adenylate cyclase by two molecules of the cAMP-receptor complex. Similar results are obtained when this cooperativity is replaced by that associated with the binding of two molecules of cAMP to an allosteric receptor, while the activation of adenylate cyclase occurs through the coupling of the enzyme with a single molecule of the cAMP-receptor complex. The reaction scheme (5.5) is then replaced by the sequence of reaction steps  

The time evolution of the concentrations of the substrate ATP (a), of intracellular (j8) and extracellular (y) cAMP, and of the different complexes formed by adenylate cyclase and by the cAMP receptor is then governed by a system of nine differential equations, as in the slightly different model studied in chapter 5. When a quasi-steady-state hypothesis is adopted for the enzyme-substrate complexes formed by adenylate cyclase in its free (C) and activated (E) states, the dynamics is described by the system of seven differential equations (6.2). In these equations, variables and parameters are defined as in eqns (5.6) (see table 5.3), but for dimensional reasons, /3 and y represent the concentrations of intracellular and extracellular cAMP divided by moreover, c - ( Cr/A d) and - (1 + a) (Martiel Goldbeter, 1984 Goldbeter, Decroly Martiel, 1984). 

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