Recently, Gennemark and coworkers reported a simple mathematical model of hyperosmotic responses in S. cerevisiae
]. Their model concisely described the biophysical changes of a hyperosmotically stressed yeast cell, and related those changes to glycerol production. This model is able to reproduce some published data on the changes in intracellular and extracellular glycerol concentrations in response to different physiological perturbations. Instead of integral control, their model assumes a time-delayed proportional control of glycerol production, i.e. glycerol productivity at a particular time is assumed to be proportional to the change in turgor pressure (relative to a constant reference turgor pressure). A consequence of this assumption is that there should be no memory effects in the stress response dynamics. This motivated us to investigate whether a model including integral control would be more appropriate.
Also inspired by the recent finding of Hog1 integral control in S. cerevisiae
], we investigated osmotic stress responses in C. albicans
by constructing a low granularity mathematical model assuming Hog1 integral control. This model integrates the HOG signal transduction pathway with downstream adaptive processes, and encompasses both hyper- and hypo-osmotic conditions. In this model, a signaling or biochemical pathway was concisely represented by an overall reaction (Figure ). Hence, the property of the pathway was defined by the associated kinetic parameters. The model was calibrated using in-house generated time-course data of Hog1 phosphorylation plus intracellular and total glycerol levels following exposure to two doses of NaCl. Subsequently, Hog1 integral control was experimentally verified in C. albicans
using experiments in which cells were subjected to repeated NaCl insults. The data showed that pre-stressed cells adapted to a second hyperosmotic shock more quickly, as if they remembered the hyperosmotic history. This phenomenon is known as the “long-term stress memory”, because its duration was longer than the time required for signal transduction to take place. Using this model, we reasoned that the cellular memory observed under hyperosmotic conditions is a hallmark of the separation of time scales between HOG signaling and reduction of glycerol production rates. In this case this mechanism enables Hog1 integral control.
The separation of time scales in different biological processes is a universal phenomenon. Often, signaling networks that perceive environmental conditions such as nutrient availability, chemical insults, and mating pheromones are required to function at much faster rates (i.e. seconds) than the ensuing regulatory networks which encompass transcriptional and translational controls (i.e. tens of minutes). It is also common for signals to be converted, via modulation of transcriptional regulators, into the expression of a stable product (i.e. a stable mRNA or protein). Such a separation of time scales could result in an integral control mechanism that would generate an experimentally testable long-term memory. Therefore our experimental approach could be further applied to other signaling networks to examine such a property. In addition, the identification of such a phenomenon allows mathematical simplifications of the regulatory network under study, an important step towards a thorough understanding of signaling networks which may be obscured in exhaustive descriptive models.
The teleological question as to why the yeast cell employs an integral control mechanism remains to be answered. Among the various types of feedback mechanism, proportional control and derivative control mechanisms would allow a system to adapt to a new status that is different from the original condition. However, integral control ensures perfect adaptation of the system, whereby signaling returns to basal levels once adaptation is achieved. This would reduce the potential for cross-talk between signaling pathways. The ability of the cell to restore turgor pressure in the face of an osmotic challenge is crucial for the restoration of growth. Should Hog1 signaling regulate glycerol production via proportional control, a change in osmotic pressure would be reflected by a proportional change in Hog1 activity. In other words, the ability to sense diverse amplitudes of osmotic signal would require high concentrations of HOG signaling proteins, the production of which would demand considerable cellular resources. In contrast, the integral control mechanism converts the error signal into the temporal integral of HOG activity. This avoids the need to express HOG pathway components at high levels. In fact in S. cerevisiae
, Hog1 phosphorylation levels become saturated in response to relatively low hyperosmotic signals. In addition, derivative control offers gradual changes in a system, which might be too slow to generate the necessary adaptation to external osmotic challenges. Indeed, derivative control is often found to provide a buffering capacity in cellular systems. For instance, in energy metabolism, phosphorylated molecules, such as phosphocreatine, act as a buffer during high energy demand periods. Phosphocreatine implements a biological derivative control over ATP concentration [26
Our modeling further suggests that the frequency response of the HOG signaling pathway can be affected by the regulation over the glycerol channels. This prediction is significant because this frequency response is generally thought to be a property of the short-term memory mediated by the regulation of intermediate protein kinases alone. In addition, our simulations show that the long-term memory may also affect the way a cell responds to a high frequency signal. Our simulations suggest that a cell that has previously adapted to hyperosmolarity retains relatively high intracellular glycerol levels for a period. Consequently, the high intracellular osmolarity counterbalances the hyperosmotic signal that the cell perceives. These results highlight the importance of a systems approach to study the signaling networks.