Recent experimental studies have shown that stress in yeast induces nucleocytoplasmic oscillations of the transcription factor Msn2. Corroborated by the study of a deterministic computational model, subsequent experiments suggested that the oscillations of Msn2, of a period of the order of minutes, originate from stress-induced oscillations of cAMP. The latter are accompanied by a periodic variation in PKA activity, which, in turn, controls the oscillatory changes in subcellular localization of Msn2. Fluorescence measurements of Msn2–GFP in individual yeast cells show that the oscillations in Msn2 shuttling are highly irregular. If the deterministic model can account for the oscillatory dynamics of Msn2 shuttling, it is however unable to reproduce the irregular oscillatory behaviour seen in the experiments. We therefore developed in this work a stochastic approach to determine the role of molecular noise in the origin of irregular Msn2 oscillations within individual yeast cells.

We first transformed the deterministic model described by equations

(2.1*a*)–(2.3*d*) into a non-developed stochastic version containing 22 reaction steps (). The stochastic oscillations predicted by this model () were compared with the experimental observations on oscillatory Msn2 shuttling (). We determined the effect of a decrease in the number of molecules on the robustness of stochastic oscillations in Msn2 and cAMP ( and ). Finally, we showed that the robustness of oscillations with respect to molecular noise depends on the distance from the bifurcation points () determined in the deterministic model (). The stochastic study indicates that large-amplitude oscillations of cAMP are reflected by large-amplitude oscillatory changes in Msn2 subcellular localization only in the middle of the range of stress intensity corresponding to limit cycle behaviour.

Owing to the uncertainty on parameter values, it is difficult at this stage to make specific predictions as to the number of molecules needed to ensure robust oscillations. However, the results of confirm the trend to be expected when lowering the number of molecules involved in the oscillatory mechanism. The results further show that coherent oscillations are progressively overcome by noise when the total number of Msn2 molecules decreases from 75 to 25. Moreover, the results of suggest that at least moderate zero-order ultrasensitivity in Msn2 reversible phosphorylation is likely to be present *in vivo* to account for the sizeable periodic variations observed for Msn2 shuttling in intact yeast.

In regard to the role of zero-order ultrasensitivity in the cAMP–PKA pathway, a relatively steep threshold in the modification cycle involving RGDP and RGTP appears to be needed for sustained oscillations. This is not the case for the other proteins such as GEF, GAP, and PDE. Steep ultrasensitivity in the RAS switch appears to be favoured by the large levels of RAS observed experimentally (

Garmendia-Torres *et al*. 2007). In determining the role of zero-order ultrasensitivity, we established the bifurcation diagram as a function of the Michaelis constants

*K*_{5} and

*K*_{6}, which are associated with the RAS interconversion cycle. We found that a stable steady state coexists with a stable limit cycle at the extremity of the oscillatory domain, a phenomenon known as hard excitation. Unexpectedly stochastic simulations showed in these conditions the occurrence of high-amplitude oscillations well outside the domain of sustained oscillations predicted by the deterministic model. This behaviour, which represents an instance of noise-induced oscillations (

Horsthemke & Lefever 1983), occurs close to a domain of hard excitation, and is not seen in the absence of such a domain (see figures S1 and S2 in the electronic supplementary material).

In coupling Msn2 shuttling to PKA oscillations, we considered for simplicity a single phosphorylation of the protein by PKA. We know that four PKA sites control the NLS, and the NES also contains a PKA phosphorylation site. It appears that more than two sites of the NLS need to be dephosphorylated to activate it (C. Garmendia-Torres and M. Jacquet 2007, unpublished observations). Multiple phosphorylations may lead to more complex regulation, as is well illustrated in the case of the yeast transcription factor Pho4 (

Springer *et al*. 2003). This transcriptional activator displays different degrees of phosphorylation as a function of the amount of inorganic phosphate in the medium. At a high phosphate level, Pho4 is fully phosphorylated and remains cytoplasmic. Under phosphate limitation there is only one specific phosphorylation; Pho4 then goes to the nucleus and activates a set of genes. In the absence of phosphate in the medium, no phosphorylation occurs; Pho4 then interacts with another effector and regulates another set of genes. It is likely that the multiple sites of phosphorylation of Msn2 by PKA could generate also a more complex behaviour than the one presented here.

The reason why we consider a single phosphorylation is that, as stated above in

§2.1, this suffices to couple Msn2 shuttling to cAMP oscillations and PKA. Moreover, the kinetics and role of multiple phosphorylation of Msn2 are far from being known in as much detail as in the case of Pho4 (

Jeffery *et al*. 2001). Incorporating multiple phosphorylation of Msn2 into the deterministic and stochastic models for oscillatory shuttling of this transcription factor represents a natural extension that awaits further experimental information.

Stochastic simulations of the oscillatory nucleocytoplasmic shuttling of Msn2 coupled to oscillations in the cAMP–PKA pathway yield good agreement with the experimentally observed oscillations of Msn2. This supports the view that the irregular nature of the oscillations in nucleocytoplasmic shuttling of Msn2 originates from the effect of molecular noise at the level of single yeast cells. Molecular noise is associated with the relatively small numbers of molecules involved in the mechanism of oscillations underlying the oscillatory shuttling of Msn2.

Here we considered the effect of noise in a model for a cellular oscillatory process based on metabolic rather than genetic regulation. The comparison between the results of deterministic and stochastic simulations of periodic Msn2 shuttling controlled by oscillations in the cAMP–PKA pathway indicates that the deterministic approach provides us with a qualitative picture that encompasses well the most conspicuous properties of this new oscillatory phenomenon, besides its irregular nature due to stochastic fluctuations. Beyond the difference in the nature of the underlying regulatory process, this conclusion holds with that reached (Gonze

*et al*.

2002*a*,

*b*,

2004;

Leloup *et al*. 2006) when comparing stochastic and deterministic models for circadian rhythms based on genetic regulation and post-transcriptional modification.