Cellular memory and model formulation
Several possible mechanisms enable integral control (Figure 2C-D, Figure 4). Our aim was to test which of these mechanisms operates during osmoadaptation. Consider a signaling molecule that exists in either a phosphorylated or unphosphorylated state (Additional file 1: Figure S2). Its kinase activity is modulated by a signal, while its phosphatase has a constant activity. Given a constant signal, if the kinase and phosphatase both operate in the saturation regimes (i.e. the concentrations of the kinase and the phosphatase far exceed their respective Km values), using a total quasi-steady-state approximation to describe the phosphorylation and dephosphorylation processes, it can be demonstrated that the activity of this signaling molecule is mainly determined by the activities of its kinase and phosphatase ([27, 28], for details see Figure 4). Thus, the activity of this signaling molecule effectively integrates the difference between its kinase and phosphatase activities. In addition, this mechanism involving saturated enzyme kinetics would result in a memory whose duration is determined by the rates at which the kinase and the phosphatase activities change. This is predicted to endow a short-term memory that lasts seconds because changes in the kinase and phosphatase activities typically are rapid [28]. However, it is unlikely that any component in the HOG pathway would operate under both saturating phosphorylation and dephosphorylation conditions because of the low concentrations of MAPK pathway components in S. cerevisiae (see supplementary material section 2.1 for details). An alternative mechanism of saturated enzyme kinetics can give rise to perfect adaptation is discussed in Figure 2D. For the same reason as the previous case, it would leave only a short-term memory.
An alternative mechanism by which integral control can arise is via differential time scales in biological processes (see Figure 2C for details). Phosphorylated Hog1 activates several parallel mechanisms to increase the intracellular levels of glycerol in S. cerevisiae. Active Hog1 up-regulates the expression of genes encoding glycerol biosynthetic enzymes (GPD1 GPD2 and GPP2). Hog1 also increases the activity of a key enzyme (Pfk26) that diverts metabolic flux towards glycerol production [29]. Furthermore, Hog1 stimulates the production of Stl1, a component of the glycerol/H+ symport system that assimilates glycerol from the growth medium [11, 30]. In principle, if the timescale for the inactivation any of these mechanisms is much longer than the timescale for achieving adaptation to higher osmolarity, then we can safely ignore the inactivation of the adaptive processes during the adaptation process. Since the glycerol channel activities change rapidly in response to external osmotic pressure [27], the adaptive processes we consider here are only related to glycerol production. Hence, if inactivation of glycerol production is a much slower process than adaptation itself, then the change in glycerol production rate (i.e. the output of the HOG signaling network) is determined by the time integral of phosphorylated Hog1. Indeed, in S. cerevisiae the rate of increase in glycerol biosynthetic enzyme levels is proportional to the time integral of Hog1 phosphorylation [31]. This change in physiological status would be likely to generate a molecular memory whose duration is determined by the rate at which the glycerol production rate reduces to basal levels following a return to hypo-osmotic conditions. Therefore, a significant memory was predicted because the degradation of stable enzymes is relatively slow. A descriptive model detailing Hog1 integral control was experimentally validated [31]. However, the idea that this integral control arises in S. cerevisiae through a separation of time scales between signaling and glycerol production rate changes was not fully recognized. The connection between Hog1 adaptation and integral control was not reported till later [27].
Whilst upstream components of HOG signaling have diverged significantly in the evolutionarily divergent yeast, C. albicans[19, 20], Hog1 integral control of osmotic stress adaptation might be conserved. Therefore our first aim was to build a simple mathematical model that highlights Hog1 integral control for both S. cerevisiae and C. albicans, and then use it to test alternative mechanisms by which a long-term memory can arise.
Model construction
In our model, a pathway or module describing a certain biological function is represented by a single reaction, encapsulating the main dynamical features of several molecular events. Our model encompasses the HOG signaling network that transmits osmotic stress signal and modulates Hog1 activity (v1 to v4), Hog1-dependent intracellular glycerol accumulation (v5 and v7) and an osmolarity-regulated A quaG lyceroP orin activity (v6) (AGP, i.e. Fps1 in S. cerevisiae; unidentified in C. albicans) (Figure 5). An acute environmental hyperosmotic shift is proposed to cause rapid closure of this aquaglyceroporin to reduce glycerol efflux [27]. It is generally believed that this is a fast change mediated by a rapid conformational change of the aquaglyceroporin under hyperosmotic conditions, and that the aquaglyceroporin will reopen once the cell has adapted [11, 32].
In our model we also assume that glycerol productivity does not decrease under hyperosmotic conditions (v7 = 0). Thus, the increase in the activity of G lycerol P roduction M achinery (GPM) will be proportional to the time integral of phosphorylation levels of Hog1. We also note that in the long term in S. cerevisiae, active Hog1 is reported to phosphorylate the aquaglyceroporin to target it for subsequent degradation [11, 32]. However, this regulation has yet to be quantified in S. cerevisiae, and it is not known whether the aquaglyceroporin in C. albicans is regulated in the same way. Hence, the degradation of aquaglyceroporin, despite its potential interest, is not included in our model.
To construct our model for both S. cerevisiae and C. albicans, we investigated the dynamics of different physiological variables for various doses of NaCl. These included Hog1 activity, and intracellular and extracellular glycerol concentrations. S. cerevisiae data were taken from several publications [31, 33, 34], and we systematically measured these values in C. albicans (Figure 6). As discussed in supplemental section 2 in details, most kinetic parameters were set to reproduce the dose–response in Hog1 activation. The rest were manually tuned with respect to the single shock experiment.
The simple model explains published data
[27] investigated the dynamics of Hog1 activation by monitoring the nuclear enrichment of fluorescently-tagged Hog1 (Hog1-YFP) in individual S. cerevisiae cells following hyperosmotic shock. They defined the nuclear enrichment of Hog1 as the relative change in the proportion of nuclear localised Hog1-YFP compared with pre-stimulus levels [27]:
(1)
where F
nuc
is nuclear fluorescence signal, F
cell
is whole cell fluorescence.
They found that the time integral of nuclear Hog1 enrichment increases linearly with extracellular osmolarity [27]. It was subsequently shown in a computational study that the nuclear enrichment of Hog1 is highly correlated with the Hog1 phosphorylation level [35]. This implies that the time integral of Hog1 phosphorylation is linear with extracellular osmolarity:
(2)
Our model is consistent with this. First, following adaptation to the higher osmotic pressure, the rate of production of intracellular glycerol is then balanced by glycerol efflux so as to maintain constant intracellular glycerol levels. In principle, three factors contribute to glycerol efflux rate: [i] intracellular glycerol levels, [ii] extracellular glycerol levels, and [iii] aquaglyceroporin activity.
(3)
We argue that upon adaptation, intracellular glycerol concentration is the main factor that determines glycerol efflux. Firstly, in our experimental conditions for C. albicans and those reported in the literature for S. cerevisiae, intracellular glycerol concentration G
i
is at least 3-order of magnitudes higher than its extracellular concentration , mainly because extracellular space V
e
is much larger than cellular volume [33]. Thus, the concentration difference that drives the passive efflux of glycerol is mainly determined by intracellular glycerol concentration, and is not significantly affected by extracellular glycerol concentration. Secondly, we assume that the activity of aquaglyceroporin c
G
is regulated by the hyperosmotic signal:
(4)
This signal reduces to a small value following cellular adaptation. This suggests that the activity of aquaglyceroporin will return to its resting value b
AGP
after adaptation. Taken together, the rate of glycerol efflux is determined by and is proportional to the intracellular glycerol concentration G
i
alone. Because glycerol production P
G
balances glycerol efflux v
6
, the former is also proportional to intracellular glycerol concentration G
i
.
The increase in intracellular osmolarity caused by the rise in the intracellular glycerol concentration G
i
eventually matches the new extracellular osmolarity ∏
NaCl
to reduce the signal upon adaptation. Thus, increase in the extracellular osmolarity that triggered the response ∏
NaCl
is linearly proportional to the increase in glycerol production rate P
G
, which is determined by ∫ 0tHog 1P(t)dt due to the integral control. Hence, the time integral of the nuclear enrichment of Hog1, which is highly correlated with the phosphorylation of Hog1, has a linear relationship with ∏
NaCl
.
Hyperosmotic responses in
C. albicans
In response to hyperosmotic conditions, S. cerevisiae closes the aquaglyceroporin channel, Fps1, to prevent glycerol leakage and help to maintain a high intracellular glycerol concentration [27, 31]. Similarly, C. albicans appears to have a high restriction over glycerol efflux by an aquaglyceroporin. Our experimental data show that, in the presence of 1 M NaCl, the increase in total glycerol (i.e. the sum of intra- and extra-cellular glycerol) is mainly due to the induction of intracellular glycerol, and extracellular glycerol (i.e. the difference between total and intracellular glycerol) does not change significantly (Figure 6D). This is consistent with the idea that the aquaglyceroporin closes quickly in response to hyperosmotic stress thereby inhibiting glycerol leakage. This would maximize the efficiency with which cells could accumulate glycerol. However, we note that the experimental data suggest that intracellular glycerol does not plateau upon Hog1 adaptation (Figure 6B6D).
Our measurements of intracellular and total glycerol concentrations revealed relatively high initial intracellular glycerol concentrations in unstressed cells, which were induced about four-fold following a hyperosmotic shock. Our experimental procedures, which examined exponential cells grown in rich media at 30°C, were designed to minimize environmental stress. We would point out that the glycerol assay does not have the sensitivity to measure low concentrations of glycerol accurately [33]. Nevertheless, our data indicate that under our experimental conditions unstressed C. albicans cells contain intracellular glycerol (Figure 6D). In our model we assume that the intracellular glycerol concentration is in equilibrium with the extracellular glycerol concentration pre-stress.
Our experimental data revealed two significant differences between the responses of C. albicans and S. cerevisiae. Firstly, in S. cerevisiae the proportion of phosphorylated Hog1 reaches its maximum value when concentrations of NaCl are greater than or equal to about 0.2 M (Figure 6A), whereas in C. albicans the percentage of phosphorylated Hog1 continues to increase, even in the presence of 0.5 M NaCl (data not shown). The NaCl concentration at which Hog1 phosphorylation amplitude becomes saturated was used to constrain parameters of the HOG signaling pathway (for details see supplemental information, section 2.1). Secondly, in S. cerevisiae GPD1 expression was strongly induced at the protein level and this regulation was mediated by Hog1 [31]. However, in C. albicans GPD1 is expressed at high levels even in the absence of hyperosmotic stress, and Gpd1 is not highly induced under hyperosmotic conditions [36]. However, expression of GPD2 was found to be induced at the mRNA level [36]. This indicates differential regulation of the Gpd isoenzymes by Hog1 in C. albicans.
Model validation
As mentioned previously, the existence of a long term memory is the key to distinguishing the mode of control that Hog1 exerts upon glycerol production (i.e. proportional, derivative or integral) and to explain its underpinning mechanism (i.e. saturated enzymatic kinetics or the separation of time scales). The existence of a long-term cellular memory would imply a mechanism of Hog1 integral control that arises through the separation of time scales between Hog1 signaling events and glycerol production rate changes.
To test this, we developed an experimental assay whereby C. albicans cells were exposed to an initial hyperosmotic shock (1 M NaCl) until they fully adapted. These cells were then removed from the hyperosmotic condition for about 10 min, and were then exposed to an identical, second hyperosmotic shock. We found that the duration of Hog1 phosphorylation in the second response was much shorter than that for the first stress (Figure 7A). This provided direct experimental evidence that adaptation to 1 M NaCl results in a memory that lasts for at least 10 min. Our model, manually turned to reproduce single-shock data, reproduces these results well quantitatively. We also assayed the time course of Hog1 phosphorylation during sequential hyperosmotic shocks of 0.5 M NaCl. Once again the model predictions were in good agreement with experimental results (Figure 7B).
We then tested whether this molecular memory was due to the retention of a relatively high glycerol synthetic capacity after adaptation to the first hyperosmotic stress. This was achieved by monitoring the intracellular glycerol levels generated by C. albicans cells after repeated exposure to hyperosmotic conditions (Figure 7C). The data showed that intracellular glycerol levels plateau within the first 45 min, which was coincident with the disappearance of Hog1 phosphorylation during the first hyperosmotic shock. Intracellular glycerol levels were then restored within 20 min of the second shock, also coincident with the disappearance of Hog1 phosphorylation. This provided further evidence that C. albicans cells retain high glycerol productivity for some time after the first adaptation. It also strongly supports the idea that the integral control in the HOG system is endowed by the separation of time scales. We note that the total glycerol was induced about 2-fold while it is simulated to increase 4-fold. However this minor discrepancy does not change the main conclusions drawn above.
Duration of long-term memory
Having shown that C. albicans displays an osmotic stress response memory and discussed its relevance to integral control, we used our model to make predictions about the behaviour of this memory. Our model predicted that during the hypoosmotic period between the first and second hyperosmotic shocks, cells would rapidly release the intracellular glycerol that was accumulated during the first hyperosmotic shock. We also predicted that the proteolytic degradation of the glycerol production enzymes synthesized during first hyperosmotic shock might take a relatively long time. These factors influence how long the memory lasts. By changing the duration of the interval between two consecutive hyperosmotic shocks, our model predicted that a short interval, such as 2 min, would not result in significant reduction in the glycerol production rate (Figure 8E). We therefore inferred that the rapid closure of the aquaglyceroporin would be sufficient to maintain the intracellular glycerol level when the second hyperosmotic shock is applied. Interestingly, the model also predicted that even though the intracellular glycerol concentration is fully restored during the second adaption, the glycerol production rate might not (dashed line in Figure 8E). During a longer interval between hyperosmotic shocks the glycerol production machinery is predicted to decrease significantly. In this case the second hyperosmotic adaptation would require a higher glycerol production rate and hence more sustained Hog1 activation. For example, after a gap of 30 min between shocks, the C. albicans cell is predicted to return to its pre-shock physiology and respond to the second shock in nearly the same way as it did to the first stress (dotted line in Figure 8A, E).
Long-term memory impacts on short-term memory
The HOG signaling networks in S. cerevisiae and C. albicans also possesses a “short-term memory” of hyperosmolarity. This results from the time it takes for the signaling components to be inactivated via dephosphorylation/phosphorylation. In response to a fluctuating osmotic signal, if the duration of the “off” phase of the signal is longer than the time required to inactivate the signaling components on HOG pathway, the phosphorylation levels of Hog1 would be predicted to follow the fluctuating signal. However, if the duration of the “off” phase is shorter than the inactivation time of the HOG pathway, the HOG network would remain active into the next cycle. The short-term memory of HOG pathway (i.e. the inactivation time for HOG pathway) typically lasts several minutes [37, 38].
To better understand this phenomenon, we developed a theoretical framework for the basic HOG signaling module, which is represented by a protein that is phosphorylated and dephosphorylated by its kinase and phosphatase, respectively. Employing Michaelis-Menten kinetics, we show that this module is essentially a low-pass filter (supplemental section 2.2). In other words, given a constant phosphatase level, the phosphorylation level of the protein faithfully follows a low-frequency activation signal while integrating a high-frequency signal, as reported previously [28, 37, 38]. We also derived analytically how the frequency of an activation signal would impact upon the phosphorylation level of an enzyme (see supplemental section 2.2 for details of the solution). In this way, our analytic solution of the frequency response has furthered our understanding of the behaviour of this simple module. In addition, we showed that in response to an alternating signal, the amplitude of the output of a network decreases with the number of cascades it contains (supplemental section 2.2). This ability to attenuate high frequency signals would allow cells to filter out random fluctuations (white noise) in environmental osmolarities, whilst maintaining its ability to sense significant environmental changes over longer time scales.
These studies were carried out using theoretical frameworks that only modeled the signal transduction pathway itself. We then extended our analyses to include downstream adaptive processes by simulating how the entire system responds to a fluctuating signal. As previously mentioned, the HOG signaling network faithfully monitors low frequency signals. As expected, our simulation results suggest that Hog1 phosphorylation levels follow the sinusoidal signal when its frequency is 0.1 rad/min (Figure 9A). However, the average of Hog1 phosphorylation initially decreases to a constant value. This is due to the sequential accumulation of intracellular glycerol during each period. Cells that have acquired relatively high intracellular glycerol concentrations are essentially pre-adapted to a hyperosmotic shock posed by 1 M NaCl.
Meanwhile, the HOG signaling network integrates high frequency signals. It is expected that a cell will respond to a high frequency sinusoidal signal in a similar manner as to a constant hyperosmotic stress with the time-average amplitude of the sine signal. In addition, due to the integral control, Hog1 phosphorylation levels might be expected to return to basal levels once the cell has adapted. However, our simulations predict that Hog1 phosphorylation does not return to the initial condition. Instead the model predicts that, due to the quick release of glycerol in the “off” phase of the signal, the cell does not accumulate intracellular glycerol to the same level as it does under a constant signal (Figure 9B), and the glycerol production rate does not reach the same level as that under a constant stress (Figure 9E). This would be similar to the molecular response of mutants that are unable to close the aquaglyceroporin channel, which must constantly activate Hog1 to maintain a high intracellular glycerol concentration. Based on this consideration, we designed an asymmetric square wave signal with the same frequency of 1 rad/min, but where each “on” phase is longer than each “off” phase. The simulations predicted that under these conditions, Hog1 phosphorylation levels will recover when the “off” phase is sufficiently short (see supplement section 2.6 for details). These results suggest that the frequency of the cellular response is dependent upon both the short-term memory in the signaling pathway (i.e. the low-pass filter) and the property of glycerol channels. The results highlight the need to place a model of the signal transduction network into an appropriate context when studying its properties.
We also investigated whether the long-term memory, a property of the adaptive process itself, affects the short-term memory of the signaling pathway. We performed simulations of a cell pre-stressed in 1 M NaCl for 60 min and observed how it then reacted to a 1 rad/min sinusoidal osmotic signal of a peak-to-peak amplitude of 1 M NaCl centred at 0.5 M NaCl (Figure 10). The model predicted that it takes much shorter for the Hog1 activity to adapt to similar levels compared with those cells excited by the same signal but without a preconditioning history (compare Figure 10 with the dashed lines in Figure 9). This indicates that, although HOG signaling behaviour is typically conceived to be determined by the short-term memory alone, long-term memory also contributes to signaling behaviour. Our computational study suggests that a pre-stressed cell possesses relatively high levels of intracellular glycerol that offset the environmental osmotic shock. Hence, as a result of this long-term memory, a stronger osmotic shock is required to achieve an equivalent signaling amplitude to non-pre-stressed cells. Also, this effect partially overrides the short-term memory, at least in over short time scales. This insight again emphasises the importance of a systems approach to study the properties of signal transduction pathways in vivo.