Model based on desensitization of the cAMP receptor

The model based on desensitization of the cAMP receptor (Martiel Goldbeter, 1984, 1987a) is represented schematically in fig. 5.28. This model retains a certain number of elements of the preceding model (fig. 5.16). Thus, extracellular cAMP binds to the receptor, but the latter now exists in two states, one of which is active (R) and the other desensitized (D) only the complex formed by the receptor in the R state with cAMP is capable of activating adenylate cyclase (C). In contrast with the preceding model, the latter enzyme is not linked here to the recep- 

The model corresponds to the detailed sequence (5.5) formed by the following reaction steps a-y. 

Step a Transitions between the active and desensitized states of the free receptor. The kinetic constants ki and k i relate to the reactions catalysed, respectively, by the kinase and phosphatase responsible for the phosphorylation and dephosphorylation of the receptor. 

Steps b and c Binding of extracellular cAMP (P) to the two states of the receptor, with affinities that may or may not differ, depending on the relative values of the dissociation constants = dja and 

Step d Transitions between the R and D states of the receptor bound to cAMP. The kinetic constants 2 and k 2 relate, respectively, to the kinetic activity of the kinase and phosphatase. 

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Models must be amended when they cannot account for the results of new experiments. Thus, several observations indicate the necessity of modifying certain assumptions of the allosteric model for cAMP signalling. This is done in the second half of chapter 5, which is devoted to the analysis of a model based on desensitization of the cAMP receptor (Martiel Goldbeter, 1987a). In D. discoideum, the receptor is phos-phorylated upon incubation with cAMP. This phosphorylation is associated with desensitization of the receptor and with the progressive decline in its capacity to activate adenylate cyclase (Devreotes ... 

As the model based on desensitization of the cAMP receptor accounts well for excitable and oscillatory behaviour in the synthesis of cAMP in D. discoideum, it is necessary to test its predictive power for the types of response observed in other experimental conditions. In particular, it remains to be shown whether the model can account for the response of cells subjected to constant stimulation by cAMP. 

Thus, the successive transitions between a nonexcitable state, relay and cAMP oscillations also occur in the stability diagram of fig. 5.29 obtained for the model based on desensitization of the cAMP receptor. This diagram is established as a function of parameters Lj and L2 linked to the activities of the protein kinase and phosphatase that control the level of receptor phosphorylation. A variation in the kinase or phosphatase activity in the course of time could give rise to the observed transitions. In each parameter space, the path predicted theoretically to account for the behavioural transitions should be compared with the variations observed experimentally for these parameters, as done above for adenylate cyclase and phosphodiesterase. 

Fig. 5.28. Model for the synthesis of cAMP in D. discoideum based on desensitization of the cAMP receptor. Extracellular cAMP binds to the active (R) and desensitized (D) forms of the receptor. Binding of cAMP to the R state elicits the activation of adenylate cyclase (C), via the action of a G-protein (not represented), as well as the reversible transition of the receptor into the D state. The intracellular cAMP thus synthesized is transported across the membrane into the extracellular medium. Arrows denote synthesis of ATP and hydrolysis of cAMP by the intra- and extracellular forms of phosphodiesterase (Martiel Goldbeter, 1984,1987a).

The qualitative and, in a a large measure, quantitative agreement of this model with experimental observations should not hide the fact that it is based on a certain number of simplifying assumptions or as yet unverified conjectures as to the precise mechanism of some steps of reaction scheme (5.5). Thus, one or more G-proteins play a role in the activation or inhibition of adenylate cyclase after binding of extracellular cAMP to the active and desensitized receptor states (Van Haastert, 1984 Janssens Van Haastert, 1987 Snaar-Jagalska Van Haastert, 1990) this issue is discussed further below in seetion 5.9. Ca could play a role in the control of adenylate cyclase and could contribute to the termination of a cAMP pulse by inhibiting the enzyme this as yet unverified conjecture is at the basis of an alternative model proposed for cAMP relay and oscillations in D. discoideum (Othmer, Monk Rapp, 1985 Rapp, Monk Othmer, 1985 Monk Othmer, 1989). The latter model, however, does not take into account the phenomenon of desensitization of the cAMP receptor, which plays an essential role here. 

The numerical study of the four-variable system (5.9a-d) reveals that it is capable of sustained oscillatory behaviour. These results, developed in further detail in the following section, also indicate that ATP remains practically constant in the course of cAMP oscillations (fig. 5.30, below). Thus, in contrast with the allosteric model considered above, the model based on receptor desensitization can account for the experimental observation (fig. 5.22) on the limited variation of ATP in the course of cAMP oscillations. Once we have established that the model predicts this characteristic feature of the experimental system, we may consider, as a first approximation, that ATP remains constant in time at the value given by eqn (5.11), in view of the relative smallness of the maximum rate of adenylate cyclase compared to the accumulative rate of ATP utilization in other pathways ... 

Fig. 5.30. Sustained oscillations of cAMP in the model based on receptor desensitization. (a) The evolution of intracellular cAMP (/3), ATP (a), the total fraction of receptor in active state (pr), and extracellular cAMP (-y). The latter is represented, on an enlarged scale, in (b) together with the total fraction of receptor in the desensitized state (Sp) and the saturation function (Y) measuring binding of cAMP to the two receptor states. The curves are obtained by numerical integration of the four-variable system (5.9) for the parameter values indicated in table 5.3 most of these values are those determined experimentally and available in the literature (see table 5.2). Similar curves are obtained by integration of the three-variable system (5.12) when ATP is maintained constant at the value a = 3 (Martiel Goldbeter, 1987a).

Table 5.3. Parameter values considered in numerical simulations of oscillations and relay of cAMP in the model based on receptor desensitization...

Like the allosteric model analysed in section 5.2 above, the model based on receptor desensitization is capable of describing the amplification of cAMP pulses by the signalling system when the latter is initially in a stable steady state, provided that the amplitude of the stimuli exceeds a threshold. 

Fig. 5.42. Response to four successive increments in extracellular cAMP. The level of extracellular cAMP is brought successively from zero to 10" M, lO" M, 10" M, and 10" M in four steps lasting 225 s each, (a) Experimental results (first part of Fig. 8 of Devreotes Steck, 1979) (b) Theoretical predictions from the model based on receptor desensitization the curve is obtained by integration of eqns (5.16) for the parameter values of fig. 5.38.

Incorporation of the role of G-proteins in transduction of the cAMP signal therefore allows us to account both for the results of experiments on oscillations and on adaptation to constant stimuli. With the exception mentioned above (see fig. 5.42), most experimental results can, however, be accounted for in a satisfactory manner when the role of G-proteins is not explicitly incorporated into the model for cAMP signalling. The model based on receptor desensitization thus permits us to unify the different types of dynamic behaviour observed under various... 

Fig. 6.11. Aperiodic oscillations (chaos) in cAMP synthesis predicted by the model based on receptor desensitization. The chaotic behaviour is obtained by numerical integration of the seven-variable system (6.2), for v = 7.545 x 10 s, other parameter values are those of fig. 6.1, divided by 10 for constants expressed in s (Martiel Goldbeter, 1985a).

Experimentally, the mechanism of intercellular communication by cAMP pulses in the course of D. discoideum aggregation is characterized by its periodicity. The latter is reflected by the wavelike movement of amoebae towards the aggregation centres, as a result of the periodic pulses of cAMP that the latter emit at regular intervals (Durston, 1974a). The periodic behaviour of the model based on receptor desensitization accounts for the periodic secretion of cAMP by aggregation centres, whereas excitable behaviour accounts for the relay of cAMP pulses by cells that amplify the suprathreshold signals emitted by the centres. 

Although the model for cAMP signalling based on receptor desensitization contains but a single feedback loop, in fact it hides two distinct mechanisms, each of which can, on its own, produce sustained oscillations. These two mechanisms, coupled in parallel, share the same process of self-amplification, namely the activation of adenylate cyclase that follows binding of cAMP to the receptor, but the two mechanisms differ by the process limiting this autocatalysis (fig. 6.25). In the first oscillatory mechanism, the limitation arises from the passage of the active receptor into the desensitized state at high concentrations of extracellular cAMP. In the second mechanism, it is the limitation by... 

Fig. 7.5. Incorporating the variation of biochemical parameters into the description of the evolution of the mechanism of intercellular communication in D. dis-coideum. The sigmoidal increase observed during the 6h that follow the beginning of starvation for adenylate cyclase (

In D. discoideum, receptor desensitization results from reversible phosphorylation (Devreotes Sherring, 1985 Klein, C. et al, 1985). The study of a model based on this process (chapter 5) allows us to account for the oscillations and relay of cAMP signals observed in the course of aggregation. In conditions where the level of extracellular cAMP is controlled and varied periodically, simulations of the model indicate (fig. 8.1) that cAMP synthesis is more important when the interval between successive stimuli is 5 min rather than 1 min. Before examining... 

Fig. 8.16. Synthesis of intracellular cAMP (/S, solid line) elicited by a periodic signal of extracellular cAMP (y, dotted line) in the model for the cAMP signalling system of Dictyostelium based on receptor desensitization. Also indicated is the variation of the fraction of active (unphosphorylated) receptor, pj (dashed line). During the on-phase of stimulation (t, = 3 min) the normalized level of y equals 10 for a dissociation constant /Cr of 10 M for the cAMP receptor, this level of y corresponds to 1 p.M extracellular cAMP. During the off-phase (t, = 5 min), the level of extracellular cAMP is nil. The curves are generated by integration of eqns (5.16) for the parameter values of table 5.3. The shaded area represents the integrated synthesis of cAMP, above the basal level obtained in the absence of stimulation. Actual levels of cAMP or of total cAMP synthesized over a period are obtained by multiplying Pot Pjby /Cr (Li Goldbeter, 1990).

Theoretical models shed light on additional aspects of pulsatile cAMP signaling in Dictyostelium. First, like Ca + spikes, cAMP pulses are frequency encoded. Only pulses delivered at 5-min intervals are capable of accelerating slime mold development after starvation. Simulations indicate that frequency encoding is based on reversible receptor desensitization [76]. The kinetics of receptor resensitization dictates the interval between successive pulses required for a maximum relay response [78]. Second, cAMP oscillations in... 

Table 5.2. Experimental values of parameters in the model for cAMP synthesis in D. discoideum based on receptor desensitization...

Despite its necessarily incomplete nature, it appears that the model analysed above for the synthesis of cAMP in D. discoideum amoebae is based on the most conspicuous properties of the signalling system, namely self-amplification due to the activation of adenylate cyclase that follows the binding of cAMP to the receptor, and desensitization of this receptor through cAMP-induced phosphorylation. 

In line with this explanation, complex periodic oscillations, birhyth-micity and chaos disappear in the model when the concentration of the substrate is held constant in the course of time. The system then admits a unique oscillatory mechanism based on the coupling between selfamplification in cAMP synthesis and its sole limitation by receptor desensitization. 

Fig. 8.1. Response to pulsatile stimulation by extracellular cAMP in the model for cAMP synthesis based on receptor desensitization. Equations (5.16) of the two-variable model are integrated in the case where periodic stimulation by extracellular cAMP (y) takes the form of a square wave. In (a), the stimulus consists in raising y from 0 to 10 for 5 min at 5 min intervals. In (b), the same pulse is applied at 1 min intervals. In each case the variation of intracellular cAMP (j8) is represented, as well as the variation of the total fraction of active receptor (pp). Parameter values are those of fig. 5.38 (Martiel Goldbeter, 1987a).

The results obtained by Wurster (1982) in Dictyostelium and by Knobil (1980) in the rhesus monkey show, nevertheless, that the frequency of stimulation is as important as the periodic nature of the signal in eliciting an adequate response. The results shown in fig. 8.1 obtained from the model for cAMP synthesis based on receptor desensitization in D. discoideum provide a theoretical basis for these observations. Confirming that the efficiency of the signal is a function of its frequency, these results suggest that the optimum frequency will be dictated by the relative values of the interval between successive stimuli and the time required for receptor resensitization. 

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