A G protein-based model of adaptation in Dictyostelium discoideum

Abstract

A new model is proposed based on signal transduction via G proteins for adaptation of the signal relay process in the cellular slime mold Dictyostelium discoideum. The kinetic constants involved in the model are estimated from Dictyostelium discoideum and other systems. A qualitative analysis of the model shows how adaptation arises, and numerical computations show that the model agrees with observations in both perfusion and suspension experiments. Several experiments that can serve to test the model are suggested.

FIG. 1.

A schematic diagram of the activation of adenylate cyclase via G_s proteins. H_s denotes the stimulus signal, R_s the stimulus receptor, α_sGDPβγ unactivated G_s protein, and UC unactivated adenylate cyclase. It is believed that upon binding of H_s with R_s, G_s is activated by the H_sR_s complex. This involves the release of the βγ subunits and the addition of GTP to the α_s chain. α_sGTP, which is denoted by ${\alpha }_s^{\prime }$ in this figure, then activates adenylate cyclase, which catalyzes the conversion of ATP to cAMP.

FIG. 2.

The scheme of interactions for the regulation of AC proposed by van Haastert and coworkers [–41].

FIG. 3.

A schematic diagram of the interactions in the proposed model. An extracellular cAMP stimulus serves as both the stimulus and the inhibitory signal. Adaptation arises from the action of ${\text{G}}_{\text{s}}^{\prime }$ on the hormone–receptor cmplex. Viewed at the level of cAMP production, this interaction produces a feedforward mechanism for the control of cAMP production.

FIG. 4.

A qualitative description of latency and adaptation in the signal transduction process under a constant stimulus. The time for activation of G protein is the major component of the latency between stimulus arrival and activation of the adenylate cyclase. Adaptation at the receptor level results from the interaction of ${\text{G}}_{\text{i}}^{\prime }$ with HR_s, which reduces the amount of activated G_s and cyclase. (a) Response curves for HR_s and HR_i under a constant stimulus. T_s1 and T_i1, represent the half-maximal times for the respective components. The dashed line indicates the response for HR_s in the absence of an inhibitory effect. (b) Response curves for the concentrations of ${\text{G}}_{\text{s}}^{\prime }$ and ${\text{G}}_{\text{i}}^{\prime }$. T_s2 and T_i2 represent the half-maximal times for the respective components. The dashed line indicates the response if the inhibitory effect is not present. (c) In the presence of ${\text{G}}_{\text{i}}^{\prime }$, HR_s is diminished due to the formation of ${\text{HR}}_{\text{s}}{\text{G}}_{\text{i}}^{\prime }$. The half-maximal time, T_s3, for HR_s in the presence of ${\text{G}}_{\text{i}}^{\prime }$ is approximately the same as T_s1, whereas T_i3 > T_i2. (d) Response curves for the concentration of ${\text{G}}_{\text{s}}^{\prime }$ and ${\text{G}}_{\text{s}}^{\prime }\text{AC}$. The half-maximal time for ${\text{G}}_{\text{s}}^{\prime }$, T_s4, is approximately equal to T_s2, whereas the half-maximal time for ${\text{G}}_{\text{s}}^{\prime }\text{AC}$, T_s5, is greater than T_s4. The rate of cAMP_i production is proportional to the concentration of G_sAC.

FIG. 4.

A qualitative description of latency and adaptation in the signal transduction process under a constant stimulus. The time for activation of G protein is the major component of the latency between stimulus arrival and activation of the adenylate cyclase. Adaptation at the receptor level results from the interaction of ${\text{G}}_{\text{i}}^{\prime }$ with HR_s, which reduces the amount of activated G_s and cyclase. (a) Response curves for HR_s and HR_i under a constant stimulus. T_s1 and T_i1, represent the half-maximal times for the respective components. The dashed line indicates the response for HR_s in the absence of an inhibitory effect. (b) Response curves for the concentrations of ${\text{G}}_{\text{s}}^{\prime }$ and ${\text{G}}_{\text{i}}^{\prime }$. T_s2 and T_i2 represent the half-maximal times for the respective components. The dashed line indicates the response if the inhibitory effect is not present. (c) In the presence of ${\text{G}}_{\text{i}}^{\prime }$, HR_s is diminished due to the formation of ${\text{HR}}_{\text{s}}{\text{G}}_{\text{i}}^{\prime }$. The half-maximal time, T_s3, for HR_s in the presence of ${\text{G}}_{\text{i}}^{\prime }$ is approximately the same as T_s1, whereas T_i3 > T_i2. (d) Response curves for the concentration of ${\text{G}}_{\text{s}}^{\prime }$ and ${\text{G}}_{\text{s}}^{\prime }\text{AC}$. The half-maximal time for ${\text{G}}_{\text{s}}^{\prime }$, T_s4, is approximately equal to T_s2, whereas the half-maximal time for ${\text{G}}_{\text{s}}^{\prime }\text{AC}$, T_s5, is greater than T_s4. The rate of cAMP_i production is proportional to the concentration of G_sAC.

FIG. 5.

The response of the uncoupled system. The stimulus is [H] = 0.1 μM for t ∈ (0, 5). The numerical results can be compared with the analytical results obtained from the analysis in Section 4 and with Figure 6 to verify the steady-state additivity of the uncoupled system. (a) The dimensionless concentrations of HR_s (solid line) and HR_i (dashed line). (b) The dimensionless concentrations of ${\text{G}}_{\text{s}}^{\prime }$ (solid line) and ${\text{G}}_{\text{i}}^{\prime }$ (dashed line). Note that the half-time for the buildup of ${\text{G}}_{\text{i}}^{\prime }$ agrees well with the theoretical estimate but that the rise time for ${\text{G}}_{\text{s}}^{\prime }$ is longer than the theoretical estimate. This discrepancy indicates that the pseudo-steady-state hypothesis applied to [HR_sG_s] is probably not valid on this time scale.

FIG. 6.

The response of the uncoupled system to sequential increases of the stimulus. The sequence is [H] = 0.001 μM for t ∈ (0, 5); [H] = 0.001 μM for t ∈ (5, 10); [H] = 0.1 μM for t ∈ (10, 15); and [H] = 1.0 μM for t ∈ (15, 20). (a) The dimensionless concentrations of HR_s (solid line) and HR_i (dashed line). (b) The dimensionless concentrations of ${\text{G}}_{\text{s}}^{\prime }$ (solid line) and ${\text{G}}_{\text{i}}^{\prime }$ (dahsed line).

FIG. 7.

The experimental response of cAMP secretion to a single prolonged stimulus of 10⁻⁶ M extracellular cAMP (from [11]). The stimulus is a 20-min step function of 1 μM magnitude (dashed line). Two experimental results (circles and squares) are shown here for intracellular cAMP concentration (filled symbols) and cAMP secretion (open symbols).

FIG. 8.

The response of the model to a prolonged stimulus of 10⁻⁶ M extracellular cAMP. The stimulus is [H] = 1.0 μM from t = 0 to t = 8 min. (a) The dimensionless concentrations of HR_s (solid line) and cAMP_i/10 (dashed line). (b) The dimensional secretion rate of cAMP_i in molecules per cell per minute. Compare with Figure 7 for experimental data.

FIG. 9.

The experimentally observed cAMP secretion in response to a multistep stimulus. The first step is from 0 to 10⁻⁹ M, and this is followed by three 10-fold increases. (Adapted from [9].)

FIG. 10.

The numerical response of the system using the measured k₋₁. Starting at t = 0 min, extracellular cAMP was increased from 0 to 10⁻⁹ M and then to 10⁻⁶ M in three 10-fold steps of 4 min duration. Dimensional parameter values are given in Table 4. The dimensionless concentrations of HR_s (solid line) and cAMP_i/10 (dashed line).

FIG. 11.

The numerical response to sequential 10-fold increases of stimuli with k₋₁ in Table 4 changed from 0.45 to 0.075. Other parameter values are unchanged. The stimulus is the same as in Figure 10. (a) The dimensionless concentrations of HR_s (solid line) and cAMP_i/10 (dashed line). (b) The dimensional secretion rate of cAMP_i (molecules per cell per minute). Compare with Figure 9 for experimental data.

FIG. 12.

The experimental and model response to a two-step sequential stimulus. Note that after removal of the stimulus the unadapted portion decays very rapidly. The parameter value are the same as in Figure 11. The stimulus sequence is [H] = 10⁻⁹ M for t = 0 to t = 10, [H] = 10⁻⁶ M for t = 10 to t = 15, and [H] = 0 M for t = 15 to t = 20. (a) The experimentally observed response (adapted from [9]). (b) The dimensional secretion rate of cAMP_i in molecules per cell per minute.

FIG. 13.

The experimental and numerical response to short two-step stimuli. The time course of the extracellular cAMP stimulus is 10⁻⁸ M (t = 0–6), 0 M (t = 6–10), 10⁻⁸ M (t = 10–16), and 0 M (t = 16–20). This experiment probes the time course of deadaptation, because the magnitude of the response to the second stimulus depends upon the degree to which the cells have recovered from the first stimulus. Parameter values are the same in Figure 11. (a) Experimental results (adapted from [11]). (b) The dimensional secretion rate of cAMP_i (molecules per cell per minute). The response to the second signal of the same magnitude is much smaller because ${\text{G}}_{\text{i}}^{\prime }$ has not decayed completely. The magnitude of the second response increases with the elapsed time since the preceding signal.

FIG. 14.

The dimensionless [cAMP_i] as a function of time, illustrating decaying oscillations. Parameter values are taken from Tables 8 and 13. The initial stimulus is [H] = 10⁻⁸ M. In this figure we scale both γ₂ and γ₅ by a factor f. Here γ₂ = f × 0.048, and γ₅ = f × 0.224, with f = 5.0. (b) A stable oscillation for f = 10.0. Shown are the dimensional intracellular cAMP concentration [cAMP_i] (solid line) and 10 times the extracellular cAMP concentration [cAMP_o] (dashed line).

FIG. 15.

The experimental results for oscillations in suspensions and the corresponding numerical results. Parameter values are the same as in Figure 14 except f = 25.0. (a) Experimental measurements of intracellular (◯) and extracellular (Δ). Redrawn from Figure 2 of Gerisch and Wick [20]. (b) Dimensional intracellular cAMP concentration [cAMP_i] (solid line) and five times the extracellular cAMP concentration [cAMP_o] (dashed line).

FIG. 16.

The magnitude of oscillation is significantly influenced by the parameters reflecting the activities of AC, iPDE, and mPDE. Here γ₆ is decreased by a factor of 10 (γ₆ = 0.29) and other parameter values are the same as in Figure 15. Shown are the dimensional intracellular cAMP concentration [cAMP_i] (solid line) and 10 times the extracellular cAMP concentration [cAMP_o] (dashed line).

FIG. 17.

An illustration of how changes in β₂ and β₅ significantly change the oscillation frequency. Here β₂ and β₅ are divided by a factor of 2. Other parameter values are the same as in Figure 15. Compare with Figure 15. (a) The dimensionless concentration of ${\text{G}}_{\text{i}}^{\prime }$. (b) Five times the intracellular cAMP concentration [cAMP_i] (solid line) and the extracellular cAMP concentration [cAMP_o] (dashed line).

FIG. 18.

The steady-state additivity of cAMP production in the response to an extracellular cAMP signal, predicted for cholera toxin–treated cells. The steady-state concentration should depend only on the final stage of extracellular stimulus and not on the stimulus history. (a) A two-step stimulus of magnitude from S₀ to S₀ + S₁. (b) The corresponding response of activated adenylate cyclase to the stimulus in (a). The steady-state concentration should be from A₀ to A₀ + A₁. (c) A single-step stimulus of magnitude S₀ + S₁. (d) The corresponding response of activated adenylate cyclase to the stimulus in (c).

FIG. 19.

The effect of cholera toxin as predicted by the model. The result should be compared with Figure 18 to see the steady-state additivity. Parameter values are given in Table 4 with k₅ = 0 to mimic the effect of cholera toxin. The stimulus is 10⁻⁸ M for t ∈ (0, 5) and 10⁻⁷ M for t ∈ (5, 10). (a) Dimensionless concentrations of HR_s (solid line) and cAMP_i/10 (dashed line). (b) Dimensional secretion rate of cAMP_i (molecules per cell per minute).

FIG. 20.

Numerical output for the experiment when GTP is replaced by GTPγS. The parameter values are the same as in Figure 19 with h₅ = 0 to simulate the fact that α_i-GTPγS is also nonhydrolyzable. The stimulus is 10⁻⁸ M for t ∈ (0, 5) and 10⁻⁷ M for t ∈ (5, 10). (a) Dimensionless concentrations of HR_s (solid line) and cAMP_i/10 (dashed line). (b) Dimensional secretion rate of cAMP_i (molecules per cell per minute). (b) should be compared with (b) in Figure 19.