With this study we determined, in a quantitative way, that the coupling between feedback mechanisms and the molecular levels of the Ras modulators can influence the oscillatory regimes of cAMP and PKA. In this context, our study highlights the role played by the feedback exerted by PKA on phosphodiesterases and on Ira2 proteins, that was never directly investigated so far. To this aim, stochastic and deterministic simulations were carried out to analyze the behavior of the Ras/cAMP/PKA pathway under different conditions. As also presented in , the comparison between the two approaches indicates that with deterministic simulations the interval of Cdc25 amount for obtaining stable oscillations of cAMP is reduced with respect to stochastic simulations. In particular, in  it was shown that stable oscillations occur when Cdc25 amount is approximately between 200 and 350 molecules in the first case, while in the second case noisy oscillations are still evident for lower and higher Cdc25 amounts (being the oscillatory regime interval around [150, 400] molecules). Within this oscillatory interval, the frequency and the amplitude of oscillations are well comparable in the stochastic and the deterministic simulations in standard conditions.
On the contrary, the comparison between stochastic and deterministic analysis performed in the perturbed conditions (that is, under the deletion of phosphodiesterases) shows qualitatively and quantitatively different results. Indeed, the dynamics of cAMP with different initial amounts of Cdc25, in both cases of deletion of Pde1 and Pde2, does not present sustained oscillations in the deterministic case, while stochastic simulations show stable oscillations.
Therefore, we can argue that molecular noise within the Ras/cAMP/PKA pathway can enhance the robustness of the system at least in response to the different perturbations we considered here, ensuring the presence of stable oscillatory regimes as also previously discussed for other biological systems (see  and references therein). Indeed, stochastic simulations show that the cell might be able to respond appropriately to an alteration of its pivotal components—such as the amount of protein Cdc25, which is related to the stress level [19, 36]—fostering the maintenance of stable oscillations during the signal propagation (i.e., the synthesis of the second messenger cAMP) and the activation of PKA. As such, this might suggest a stronger adaptation capability of yeast cells to various environmental stimuli or endogenous variations.
In  it was shown that in MIN6 beta cells PKA, cAMP and calcium are highly integrated in an oscillatory circuit that allows a fine spatiotemporal regulation of the kinase activity. Similarly, we think that the multi-level regulation carried out with different feedback mechanisms in the Ras/cAMP/PKA pathway in yeast might represent a way to extend the regulatory span of the system, therefore acting as a tuning mechanism for the numerous downstream targets of PKA. This assumption might be in line with the hypothesis of the “frequency-modulated” regulation that was recently proposed in yeast in relation to calcium oscillations , though further computational and experimental investigations should be carried out to ascertain the validity of this hypothesis also in relation to cAMP and PKA oscillations induced by the molecular interactions within the Ras/cAMP/PKA pathway.
Indeed, oscillations related to the Ras/cAMP/PKA pathway were experimentally observed, but only in indirect ways, e.g., through the analysis of the periodic nucleocytoplasmic shuttling of Msn2 [18, 19]. In this context, the observations presented in  show that, in single yeast cells subject to continuous light exposure, the oscillations frequency of Msn2 between the nucleus and the cytoplasm can be influenced by PKA as well as by Pde1 and Pde2, whereby phosphodiesterases indirectly affect the activity of PKA through the degradation of cAMP. In particular, in  it was shown that the oscillation frequency of Msn2 increases alongside the increase in the induced-stress condition, which can be also ascribed to a reduced activity of Cdc25 protein [19, 36], as tested in our study. Anyway, it is not clear whether the nucleocytoplasmic oscillations frequency of Msn2 can be interpreted solely in terms of the above mentioned frequency modulation control, since the response of the cell to cumulative light-induced stress might suggest a more complex scenario that is still to be unraveled .
The computational results presented in this study and in , in relation to the amplitude and the frequency of oscillations within the Ras/cAMP/PKA pathway, suggest that a frequency modulation can be achieved when perturbing the ratio between the amounts of Cdc25 and Ira2 proteins, that is, the Ras regulator proteins. In particular we showed here that, with respect to the wild type condition, the deletion of Pde2 is able to diminish the oscillations frequency of cAMP, while the deletion of Pde1 has a minor effect on its variation. These results represent a first step towards an in depth analysis of oscillatory regimes in the Ras/cAMP/PKA pathway, that we plan to carry out in different ways. On the one hand, we will investigate the correlation between the oscillations of cAMP, PKA and its downstream targets (such as Msn2) to analyze how this behavior propagates through the signal transduction pathway. On the other hand, we are currently developing a computational tool to quantitatively characterize the oscillations (whether stochastic, deterministic or noise-induced) by means of Fourier analysis, as already proposed for the study of the oscillatory shuttling of NF-κ B , whereby the power spectrum of simulated dynamics was analyzed to verify the occurrence of peaks at non-zero frequencies, as well as to calculate the signal-to-noise ratio, in order to inspect the presence of oscillations. In addition, qualitative analysis of the nature of bifurcation points, based on dynamical systems theory (see, e.g.,  for an application to the study of noise-induced stabilization in a genetic circuit, and  for a broad overview of the subject), could be exploited to better investigate how stochastic fluctuations are able to originate the stable oscillatory regimes in the Ras/cAMP/PKA model, which do not occur in the absence of noise.
Furthermore, as a future development of our study, we will investigate the response of the Ras/cAMP/PKA pathway to nutrients and to intracellular acidification (that likely causes an inhibition of GAP activity of the Ira proteins ), its crosstalk and integration with other pathways co-involved in glucose signaling and yeast metabolism, as well as the regulated expression of downstream target genes. In particular, we will define additional functional modules of reactions to describe the Gpr1/Gpa2 pathway, a signaling mechanism that responds only to high glucose concentration and operates in an addictive redundant way with Ras2-GTP to activate the adenylate cyclase [28, 42]. We also plan to define a multi-volume version of our mechanistic model, in order to characterize the intracellular localization of the central components of this pathway, since there exist experimental evidences that most of the Cdc25, Cyr1, Ira2 and Ras2 proteins localize at internal membranes, suggesting the presence of large signaling complexes inside yeast cells [43, 44]. Investigations about the topological distribution of the molecular species in distinct cellular regions will be performed by means of the tau-DPP framework . This will enable us, for instance, to study the dynamical movement of Cdc25 proteins to plasma membrane in response to nutrient starvation, and the hyper-activity of PKA to counteract the localization of Cdc25 and Ira2 proteins.
In conclusion, the computational model we developed allows to investigate in details the mechanisms that regulate the transition between stable and oscillatory regimes in the Ras/cAMP/PKA pathway, to make predictions on the conditions that lead to the insurgence of oscillations, and to eventually plan focused validation experiments. In particular, by directly operating on the modulation of specific components of the pathway, such as molecular amounts and reaction constants, we are able to study in details the influence of every molecular interaction on the pathway behavior. Nonetheless, although the Ras/cAMP/PKA pathway has gone through extensive investigations in S. cerevisiae, accurate wet data on the spatiotemporal dynamics of cAMP in single yeast cells are still lacking. To this aim, we are carrying out extensive laboratory work to develop a FRET-sensor based on Epac able to respond to cAMP levels in S. cerevisiae, in order to measure the changes in the level of cAMP in single cells and to directly test the presence of long term cAMP oscillations in vivo. This setup will allow us to conduct an in depth analysis of the response of the pathway to different nutritional and stress conditions, as well as to perform an accurate parameter estimation analysis , therefore working thoroughly on the experimental and computational validation of our model.