Positive and negative regulations in the Ras/cAMP/PKA pathway. The diagram shows the logical relationships among the principal components of the Ras/cAMP/PKA pathway. The switch cycle of Ras2 protein between its inactive state (Ras2-GDP) and active state (Ras2-GTP) is regulated by the activity of Cdc25 (a) and Ira2 (b). The intracellular ratio of GTP and GDP also contributes to the regulation of the activity of Ras proteins (a), since Cdc25 stimulates the exchange of these nucleotides on Ras according to their relative concentration. Ras2-GTP controls the activity of the adenylate cyclase Cyr1 (c), which mediates the synthesis of the second messenger cAMP (d). cAMP activates PKA (e) by binding to its regulatory subunits and releasing its catalytic subunits. The degradation of cAMP is governed by Pde1 (i) and Pde2 (j), which constitute a major negative feedback in this pathway, as they both contribute to decrease the intracellular level of the second messenger. The active form of PKA exerts three main regulations in this pathway through the phosphorylation of different components: a positive regulation of Pde1 (h) and of Ira2 (g), and a negative regulation of Cdc25 (f). Since the increased activity of the phosphorylated forms of both Pde1 and Ira2 result in switching off the signal—that is, they both contribute to reducing the intracellular level of cAMP—due either to a faster degradation of cAMP by Pde1 (i) or to a diminished fraction of active Ras2 by Ira2 proteins (b), these two positive regulations actually have the effect of a negative feedback control on the whole pathway. The negative regulation of Cdc25 by PKA results in a partial inactivation of the GEF activity (a), and thus a reduced activation of the Ras2 protein, which results in a decreased activity of the adenylate cyclase and therefore contributes to lowering the cAMP level.

Dynamics of cAMP, Ras2-GTP and active PKA with and without the feedback on Ira2. When the feedback control on Ira2 proteins is not activated, the dynamics of cAMP shows an initial transient increase and the successive establishment of a stable steady state (top left). In this condition, neither Ras2-GTP nor active PKA show oscillatory behavior (top right). On the contrary, when the feedback on Ira2 proteins is activated, the initial peak on cAMP amount is followed by the establishment of a stable oscillatory state (bottom left), as also reflected in the dynamics of both Ras2-GTP and of the active fraction of PKA (bottom right). In S. cerevisiae, cAMP was experimentally determined to be around 2×10⁵ molecules after stimulation, while basal levels vary from 2×10⁴ to 5×10⁴ molecules/cell; as a consequence of stimulation, a cAMP peak was observed after 45–60 s, and then a new steady-state was reached in 3–5 min (see, e.g., Figure 2 in [28]). The expected number of cAMP molecules derived from stochastic simulations was calculated here by considering our own measurements [16] and data presented in [28].

Activation of the feedback on Ira2: cAMP oscillations starting from a steady state condition. The occurrence of oscillatory regimes in the cAMP dynamics is only affected by the activation of negative feedback mechanisms and is independent of the initial condition of the system. The plots show the establishment of stable oscillations in cAMP amount (left), as well as in Ras2-GTP and active PKA amounts (right), starting the simulation from an initial condition where the system is at a stable steady state. In this simulation, the initial amounts of cAMP, Ras2-GTP and active PKA are around 47,000, 50, and 400 molecules, respectively.

Effects of the modulation of the negative feedback on Cdc25 (reaction constantc₃₄). The figure shows the simulated dynamics of cAMP (left) and Ras2-GTP (right) resulting from a PSA-1D on the value of the reaction constant that modulates the phosphorylation of Cdc25 by means of active PKA. The varied parameter is constant c₃₄ (see Table 1), within the interval [1.0×10⁻³, 1.0×10³], being 1.0 the reference value (represented with the black thick line). Stable oscillations occur in both cAMP and Ras2-GTP dynamics, for any value of the reaction constant, regardless of the magnitude of the feedback exerted by PKA on Cdc25. For larger values of the constant, namely c₃₄ greater than 1.0, the only effect is a reduction in the amount of cAMP and Ras2-GTP (whose average values are reduced from around 40,000 to 35,000 molecules, and from around 270 to 130 molecules, respectively) and an increase of about fifty per cent in the frequency of oscillations with respect to the reference value.

Effects of the modulation of the negative feedback on Ira2 (reaction constant c₃₆). The figure shows the simulated dynamics of cAMP (left) and Ras2-GTP (right) resulting from a PSA-1D on the value of the reaction constant that modulates the phosphorylation of Ira2 by means of active PKA. The varied parameter is constant c₃₆(see Table 1), within the interval [1.0×10⁻⁶, 1.0], being 1.0×10⁻³ the reference value (represented with the black thick line). The reaction that describes the feedback on Ira2 is able to control the establishment of oscillatory regimes: for values of the reaction constant lower than the reference value, only stable steady states can be reached. In particular, cAMP and Ras2-GTP reach a noisy steady state around 60,000 and 10,000 molecules, respectively. Conversely, by increasing the value of the constant, that is, enhancing the GTPase activity of Ras2 proteins by means of the phosphorylation of Ira2, the amplitude of oscillations of cAMP and Ras2-GTP is reduced from around 40,000 to 25,000 molecules and from 600 to 140 molecules, respectively, as well as their average amounts, that decrease from around 40,000 to 20,000 and from around 270 to 50 molecules, respectively. For values of constant c₃₆greater than 1.0×10⁻² the oscillatory regime is lost and the average values of cAMP and Ras2-GTP drop to around 5,000 and 20 molecules, respectively.

Comparison of stochastic and deterministic simulations of cAMP dynamics with different initial amounts of Cdc25. The figure shows the dynamics of cAMP with different initial amounts of Cdc25 (150, 200, 300, and 400 molecules) obtained by means of stochastic (left) and deterministic (right) simulations. The deterministic and stochastic behaviors are comparable for Cdc25 equal to 200 and 300 molecules (sustained oscillations occur in both cases) and for Cdc25 equal to 400 (damped oscillations). On the contrary, when the initial amount of Cdc25 is low (i.e., 150 molecules), with the stochastic approach we obtain stable oscillations of cAMP, while in the deterministic case the dynamics shows damped oscillations.

Effects of the modulation of the negative feedback on Pde1 (reaction constant c₂₆). The figure shows the simulated dynamics of cAMP (top left), Ras2-GTP (top right), phosphorylated Ira2 (bottom left) and phosphorylated Pde1 (bottom right) resulting from a PSA-1D on the value of the reaction constant that modulates the phosphorylation of Pde1 by means of active PKA. The varied parameter is constant c₂₆ (see Table 1), within the interval [1.0×10⁻⁹, 1.0×10⁻³], being 1.0×10⁻⁶ the reference value (represented with the black thick line). For values of the constant lower than the reference value, stable oscillations still occur. On the contrary, by increasing the value of the reaction constant, that is, if we simulate a marked promotion of the activity of Pde1, after an initial transient increase the amount of cAMP gets almost completely degraded and no oscillations occur anymore (cAMP reaches a noisy steady state around 5,000 molecules, top left). Under the same condition, the amount of Ras2-GTP tends to a high steady state level (around 10,000 molecules, top right), which would intuitively induce a promotion of the cyclase activity and thus an expectable increase in the cAMP amount, which is instead counterbalanced by two concurrent processes: (i) the reduced activity of Ira2 proteins, whereby only a few copies of phosphorylated Ira2 are present inside the system (around 5 molecules, bottom left), and (ii) the strong phosphodiesterase activity taking place in this condition, that is also confirmed by the high level of phosphorylated Pde1 (around 1,200 molecules, bottom right).

Influence of the amount of Pde1 and Pde2 on the establishment of cAMP oscillatory regimes. The figure shows the oscillations amplitude of cAMP dynamics resulting from a PSA-2D on the values of the initial amounts of Pde1 and Pde2, varied in the intervals [0, 2, 800] and [0, 13, 000], respectively (being Pde1 = 1,400 and Pde2 = 6,500 molecules the reference values—see Table 2). In the plot, the values on the x- and y-axis were normalized in the interval [0, 1]; a total of 200 initial conditions were sampled from the specified bidimensional parameters space. This analysis shows that the deletion of the high-affinity phosphodiesterase Pde2 fosters the establishment of oscillations of cAMP, whose amplitude increases with the increase of Pde1 amount (from point A to point C); on the contrary, the deletion of the low-affinity cAMP phosphodiesterase Pde1 has the effect of diminishing or even abolishing the oscillations irrespective of the amount of Pde2 (from point A to B). The overexpression of both phosphodiesterases (point D) does not allow the establishment of oscillatory regimes.

cAMP dynamics obtained with different initial amounts of Pde1 and Pde2. Dynamics of cAMP with different initial amounts of the phosphodiesterases, as given in Figure 8, where A: Pde1 = 0, Pde2 = 0; B: Pde1 = 0, Pde2 = 13,000; C: Pde1 = 2,800, Pde2 = 0; D: Pde1 = 2,800, Pde2 = 13,000 molecules. The plots show that in the absence of both phosphodiesterases (A) we achieve, as expected, an unlimited accumulation of cAMP, since our model does not include any other mechanism to reduce the intracellular level of cAMP; with a very high initial amount of Pde2 (B and D), cAMP reaches a noisy steady state but no oscillations are observed; in the absence of Pde2, when only Pde1 is active (C), oscillations of cAMP can then be established, showing a very large amplitude and a mean cAMP amount that is slightly higher than standard conditions. The molecular amounts of cAMP reached in conditions A and C are higher than the physiological levels measured in corresponding experimental settings [12,14] though, from a computational point of view, they are indicative of the role played by the two phosphodiesterases, since they highlight the different dynamical behaviors of the pathway in extreme conditions.

Diagram of cAMP oscillations amplitude with respect to Cdc25 amount when deleting Pde1. The figure shows the oscillations amplitude of cAMP dynamics resulting from a PSA-1D where the amount of Cdc25 varies in the interval [0, 900], in the condition of deletion of Pde1. The mean and standard deviation of the average (squares), maximum (circles) and minimum (triangles) amount of cAMP during oscillations are plotted. The left and right shady areas correspond to noisy stochastic fluctuations and stable steady states, respectively, while the white area corresponds to oscillatory regimes. This analysis highlights that, when deleting Pde1, oscillatory regimes in cAMP can be established even when the amount of Cdc25 is at a two-fold overexpression with respect to the physiological amount of 300 molecules/cell (Table 2). Therefore, the deletion of Pde1 has the effect of widening the range of Cdc25 molecules under which sustained oscillations of cAMP occur, being approximately [150, 400] the interval whereby oscillatory regimes in cAMP are found when Pde1 is present in the system (see [17] for more details).

Diagram of cAMP oscillations amplitude with respect to Cdc25 amount when deleting Pde2. The figure shows the oscillations amplitude of cAMP dynamics resulting from a PSA-1D where the amount of Cdc25 varies in the interval [0, 900], in the condition of deletion of Pde2. The mean and standard deviation of the average (squares), maximum (circles) and minimum (triangles) amount of cAMP during oscillations are plotted. The left and right shady areas correspond to noisy stochastic fluctuations and stable steady states, respectively, while the white area corresponds to oscillatory regimes. Similarly to Figure 10, the analysis highlights that, under the deletion of Pde2, oscillatory regimes in cAMP can be established even when the amount of Cdc25 is at a two-fold overexpression with respect to its physiological amount. Therefore, also the deletion of Pde2 has the effect of changing the range of Cdc25 molecules under which sustained oscillations of cAMP occur.

Deterministic simulations of cAMP dynamics under deletion of Pde1 or Pde2 with different initial amounts of Cdc25. The figure shows the dynamics of cAMP with different initial amounts of Cdc25, obtained by means of deterministic simulations when deleting Pde1 (left) and Pde2 (right). In both conditions, no sustained oscillations are present in the system for any value of the initial amount of Cdc25.

Comparison of stochastic and deterministic simulations of cAMP dynamics under deletion of Pde1 or Pde2, with Cdc25 = 300 molecules. The figure shows the comparison of the dynamics of cAMP obtained by means of stochastic and deterministic simulations when deleting Pde1 (left) or Pde2 (right), with an initial amount of Cdc25 equal to 300 molecules. We note that, while in the deterministic case oscillations are damped or even not occurring, the stochastic case shows sustained oscillations of cAMP in both cases.

Frequency modulation of cAMP oscillations in the wild type condition and under the deletion of Pde1 and Pde2. The figure shows the comparison of the oscillations frequency of cAMP in the wild type condition (red square dots) and under the deletion of Pde1 (green circle dots) or Pde2 (blue diamond dots), with respect to the ratio Cdc25/Ira2 (with the initial amount of Cdc25 ranging from 100 to 600 molecules, and Ira2 = 200 molecules). The plot shows that the deletion of Pde1 has a minor effect with respect to the wild type condition on the modulation of frequency, while the deletion of Pde2 is able to diminish the oscillations frequency of cAMP. For values of the ratio Cdc25/Ira2 greater than 1.5, the points related to the wild type condition are not plotted, since no stable oscillations are established in these cases [17].

Influence of the amount of GTP and Cdc25 on the establishment of cAMP oscillatory regimes. The figure shows the oscillations amplitude of cAMP dynamics resulting from a PSA-2D on the values of the initial amounts of GTP and Cdc25, varied in the intervals [1.9×10⁴, 5.0×10⁶] and [0,600], respectively (being GTP = 5.0×10⁶ and Cdc25 = 300 molecules the reference values—see Table 2). In the plot, the values on the x- and y-axis have been normalized in the interval [0, 1]; a total of 200 initial conditions were sampled from the specified bidimensional parameters space. This analysis shows that when the amount of Cdc25 is approximately at normal condition or slightly lower, oscillatory regimes are established for basically any value of GTP (from point A to point B), being the amplitude of oscillations smaller in lower nutrient availability conditions. On the contrary, when the amount of Cdc25 increases, no oscillations of cAMP occur when GTP is high (point D), but oscillatory regimes are still present if GTP is low (point C).

cAMP dynamics obtained with different initial amounts of Cdc25 and GTP. The figure shows the dynamics of cAMP with different initial amounts of Cdc25 and GTP, as given in Figure 15, where A: Cdc25 = 10, GTP = 1.9×10⁴; B: Cdc25 = 10, GTP = 5.0×10⁶; C: Cdc25 = 600, GTP = 1.9×10⁴; D: Cdc25 = 600, GTP = 5.0×10⁶ molecules. The plots show that low amounts of Cdc25 (A and B), or high amounts of Cdc25 coupled with low amounts of GTP (C), lead to the production of very low amounts of cAMP, showing a noisy dynamics. On the other hand, when the amounts of both Cdc25 and GTP are high (D), we observe a dynamics of cAMP similar to that obtained without the feedback on Ira2 proteins (see Figure 2, top left), and no oscillations are established.