Goldbeter and Segel's Model

Goldbeter and Segel's model for the dynamic behavior of the cAMP signalling system for Dictyostelium discoideum is sketched in Figure 1. The steps are as follows:

• I: synthesis of ATP
• II: binding of extracellular cAMP to the receptor (labelled R)
• III: activation of adenylate cyclase (labelled C)
• IV: production of cAMP from ATP, catalyzed by adenylate cyclase
• V: transport of cAMP into the extracellular medium
• VI: cAMP hydrolysis by phosphodiesterase

Figure 1: Sketch of Goldbeter and Segel's model of the cAMP signalling system for Dictyostelium discoideum.

The variables in the model are dimensionless concentrations of the intracellular ATP (called $\alpha$), the intracellular cAMP (called $\beta$), and the extracellular cAMP (called $\gamma$). Assuming that the system remains spatially homogeneous (which corresponds to conditions in continuously stirred suspensions), the time evolution of the system is governed by a three-dimensional set of ODES:

$$\frac{d \alpha}{dt} = \nu - \sigma \phi(\alpha,\gamma),$$ (1)

$$\frac{d \beta}{dt} = q \sigma \phi(\alpha,\gamma) - k_t \beta,$$ (2)

$$\frac{d \gamma}{dt} = \frac{k_t \beta}{h} - k \gamma,$$ (3)

where the rate function $\phi$ for adenylate cyclase is given by

$$\phi(\alpha,\gamma) = \frac{\alpha (1+\alpha) (1+\gamma)^2}{L + (1+\alpha)^2 (1+\gamma)^2}.$$ (4)

The definitions for the constants $\nu, \sigma, q, k_t, k, h$, and $L$ are given in Goldbeter and Segel. For our purposes, the important things are:

• $\nu$ in (1) represents the (constant) synthesis of ATP
• $-\sigma \phi(\alpha,\gamma)$ in (1) and $q \sigma \phi(\alpha,\gamma)$ in (2) represent the production of intracellular cAMP from ATP
• $-k_t \beta$ in (2) and $k_t \beta/h$ in (3) represent the transport of cAMP into the extracellular medium
• $-k \gamma$ in (3) represents the hydrolysis of extracellular cAMP by phosphodiesterase.

Following Goldbeter and Segel, the constants will be taken to be

$$\sigma = 1.2 s^{-1}, \qquad k = k_t = 0.4 s^{-1}, \qquad L = 10^6,$$

$$q = 100, \qquad h = 10.$$

We will consider different values of $\nu$, the rate of synthesis of ATP.