Upon HTS, after the transient osmotic swelling, mammalian cells re-adjust their volume by a mechanism known as regulatory volume decrease (RVD) (Okada et al., 2001). In some cell types, RVD is accomplished by means of stretch-activated Ca²⁺ channels which mediate a rapid increase of the cytosolic Ca²⁺ concentration due to both Ca²⁺ influx and Ca²⁺ release from intracellular stores, which in turn is often the signal that triggers release of osmolytes, reducing osmotic gradients and helping volume regulation (Jakab et al., 2002). In addition, the HTS-induced activation of ion conducting pathways leads to profound changes in the plasma membrane potential, which determines the direction of the ion fluxes depending on the respective equilibrium potentials and the modulation of voltage-gated ionic channels.

Upon HTS, a large fraction of glycerol, an intracellular osmolyte, is released in yeast cells to the extracellular environment within 2–3 min through the activation of Fps1 channels (Tamas et al., 1999), which is inhibited neither by gadolinium, which instead completely blocks Ca²⁺ increase (Batiza et al., 1996), nor by membrane potential alterations (Kayingo et al., 2001). Fps1 was proposed as a mechanosensitive channel directly activated by the induced membrane stretch, whereas the known post-translational modifications probably fine-tune its activity under basal conditions (Ahmadpour et al., 2014).

Phosphoinositides are negatively charged membrane lipids that serve as versatile molecules involved in protein regulation, assembly of actin cytoskeleton, vesicle trafficking and Ca²⁺ signaling. Several phosphoinositides are substrates for phospholipases, thereby generating a number of products that serve as second messenger with biological functions on their own (Strahl and Thorner, 2007). For example, the plasma membrane lipid phosphatidylinositol 4,5-bisphosphate (PtdIns (4,5) P₂) can be depleted by activation of phospholipase C (PLC), producing diacylglycerol (DAG) and the diffusible molecule Ins (1,4,5) P₃ (IP₃), which in mammals is a fundamental signaling molecule. In budding yeast, as in all eukaryotic cells, different phosphoinositide species are generated in a compartment-specific manner and, hence, can be regarded as distinct markers for each organelle (Odorizzi et al., 2000; Strahl and Thorner, 2007; Balla, 2013) and contribute to specific regulation of protein activity, such as ion channels (Hille et al., 2015). Upon HTS, S. cerevisiae cells hydrolyze plasma membrane PtdIns (4,5) P₂ with a mechanism dependent on Plc1 but independent on the extracellular Ca²⁺ concentration. This process liberates IP₃, which is then rapidly phosphorylated in IP₆. In addition, another phosphoinositide, PtdIns4P, and is rapidly synthesized and then progressively consumed in the next minutes (Perera et al., 2004). Although a link between these dynamics and Ca²⁺ signaling has not been explored yet, it is tempting to suggest one. In fact, the rapid depletion of the plasma membrane signature lipid PtdIns-(4,5)-P₂ could influence the activity of some channels. Flc2 is the best candidate, since it resides on plasma membrane and contains a putative lipid-binding domain. Channels localized on Golgi or ER could also be influenced by changes in phosphoinositides abundances: PtdIns4P transient increase could regulate Golgi channels, since PtdIns4P is the signature lipid of this organelle. Some Ca²⁺ regulation seems to be at stake in HTS because PLC1 gene deletion, which abolishes PtdIns-(4,5)-P₂ depletion and IP₆ production, causes a greater increase in calcium influx after HTS compared to a wild type strain (Tisi et al., 2002). It is worth noting, however, that PLC1 deletion does not affect PtdIns4P dynamics upon HTS (Perera et al., 2004).

The immediate and transient (∼2 min) cytosolic Ca²⁺ pulse (Batiza et al., 1996; Rigamonti et al., 2015) triggered by HTS in S. cerevisiae cells is generated both by influx from the extracellular medium and efflux from intracellular stores. An early study on yeast cells grown in synthetic medium reported that this increase was mediated by an instantaneous release of Ca²⁺ from intracellular stores and then sustained by influx of extracellular calcium. In fact, addition of an extracellular Ca²⁺ chelator, BAPTA, affected later stages of the response without affecting the initial, and rapid cytosolic Ca²⁺ rise (Batiza et al., 1996). Moreover, the same study showed that the HTS-induced Ca²⁺ response was dependent on both intensity of the shock and type of growth medium, the latter affecting also the pre-stimulus baseline [Ca²⁺]_cyt. The increase of Ca²⁺ was inhibited in a dose-dependent manner by pre-treatment with gadolinium, a blocker of stretch-activated channels, and suggesting that the membrane stretching that occurs following HTS-induced cell swelling is directly sensed by Ca²⁺ channels (Batiza et al., 1996). In yeast cells grown in YPD—a complex, nutrient-rich medium—and challenged with HTS by diluting the medium with distilled water, an estimate of the initial rate of calcium increase at micromolar [Ca²⁺]_ext could be fitted by a Hill function, suggesting that the calcium increase in response to HTS was caused by the activation of a single channel or transporter located on the plasma membrane (Rigamonti et al., 2015).

The HTS response was also measured for mutants lacking proteins known to be involved in calcium signaling and homeostasis. cch1Δ mutants, lacking a functional HACS channel on the plasma membrane, responded to HTS with a higher calcium peak at all [Ca²⁺]_ext considered, suggesting a negative regulatory role for this protein during HTS. Mutants in other known influx pathways, on the other hand, and had a response similar to the wild-type. Calcium levels are affected in calcineurin mutants, suggesting that this Ca²⁺-dependent effector shapes the calcium signal during HTS through dephosphorylation of target transporters and/or by modulating their long-term expression. In addition to display altered resting calcium levels, mutants lacking calcineurin respond to HTS with a dramatically reduced peak. Flc2 was found to be involved in the HTS-induced calcium response, since deletion of FLC2 increases the initial rate of the Ca²⁺ increase compared with wild-type, suggesting an inhibitory role of this protein on the channel that is activated by the HTS. Based on the experimental evidences described above, a model is proposed that includes only the essential players in the HTS-induced calcium response. In non signaling conditions, Ca²⁺ enters the cell through HACS channel and other unidentified influx pathways. A still unidentified mechanosensitive calcium channel is located on the plasma membrane and activated by the increased membrane tension caused by HTS. In addition, this channel appears to be negatively regulated by Flc2. Since the elevation of [Ca²⁺]_cyt is transient, some intracellular transporters must restore the steady-state levels of cytosolic Ca²⁺. This signal attenuation is probably performed by the Golgi-localized Pmr1, together with vacuolar-localized Pmc1 and Vcx1.

The mathematical model of HTS response in S. cerevisiae was defined on the basis of available experimental evidences (Rigamonti et al., 2015), and can be conceptually divided in two modules:

1 the biophysical module describes the changes in volume and other cell parameters, such as turgor pressure and membrane tension. This module allows for quantitatively following all changes in the physical state of the cell depending on cytosolic and extracellular osmolarities. Such parameters, in turn, regulate the activity of some components of the biochemical module. In particular, stretch-activated channels open following a sudden increase of membrane tension, promoting Ca²⁺ diffusion through them;

The model was formalized as a system of coupled ODEs. Mass-action kinetics was used to model the physical interactions between Ca²⁺, calmodulin, and calcineurin. Most transport reactions were modeled by means of the Michaelis-Menten kinetics, as substantiated by previous studies (Ohsumi and Anraku, 1983; Wei et al., 1999; Takita et al., 2001; Teng et al., 2008).

Stretch-activated channels were modeled differently: they can be viewed as pores, whose opening probability depends on membrane tension. In particular, the opening probability of mechanosensitive channels was shown to follow a Boltzmann distribution (Gustin et al., 1988; Sukharev et al., 1999; Jiang and Sun, 2013). For the sake of simplicity, in this work the Boltzmann equation employs turgor pressure (note that membrane tensions can be calculated from turgor pressure using Laplace’s law for a thin-walled sphere (Gustin et al., 1988; Sackin, 1995)). The opening probability (P _open) of mechanosensitive channels was thus formally defined as: P open = 1 − 1 1 + e P − P MS g MS (1) where P is turgor pressure, P _MS is the value of turgor pressure at which P _open is equal to 0.5, and g _MS is a slope parameter.

Since ions pass through the channel pore down their electrochemical gradient, we assume that calcium ions flow according to their concentration gradient. The calcium flux j is then given by: j = P open ⋅ k ⋅ Δ c (2) where k is a rate parameter and Δc is the Ca²⁺ concentration gradient across the membrane.

The model was simulated with COPASI (version: 4.19) (Hoops et al., 2006), using the LSODA algorithm (Petzold, 1983) with default settings. Simulation outputs consist in time traces of species concentrations over a period of 160 s, in line with the stress response duration of yeast cells. The model is available as an SBML Level 2 Version 5 file (ID MODEL2112030001) in the BioModels repository (Malik-Sheriff et al., 2020): https://www.ebi.ac.uk/biomodels/MODEL2112030001.

Unknown parameters were estimated with COPASI (Hoops et al., 2006), using the available implementation of the Particle Swarm Optimization (PSO) algorithm (Kennedy and Eberhart, 1995). PSO is a global optimization algorithm based on the concept of “swarm intelligence”, which was shown to be effective in solving optimization problems characterized by multi-modal and noisy fitness landscapes, such as the ones related to the parameter estimation problem of biochemical systems (Nobile et al., 2018; Tangherloni et al., 2019; Besozzi et al., 2020).

To estimate the unknown parameters, simulations were fitted against available experimental time traces of [Ca_cyt], measured in different experimental conditions in Rigamonti et al. (2015), including deletion mutants in key proteins of the HTS response and a wide range of extracellular Ca²⁺ concentrations. It is worth mentioning that, in the parameter estimation process, we took into account the well known fact that the intracellular calcium levels are kept in a narrow range despite wide variations in external conditions, thanks to calcium buffering and sensing and feedback mechanisms (Cunningham K. W. and Fink G. R., 1994). Thus, while reaction constants must be the same across different experiments, the concentrations of proteins bound to Ca²⁺ can vary. According to this line of reasoning, these parameters were not forced to be the same for all experiments. Since the model describes a stimulus response, the steady-state pre-stress conditions were also included in the parameter estimation process. Search ranges for all unknown parameters were set to be within biologically plausible numeric intervals.

The biophysical module describes the variation of the physical parameters of the cell, such as the cell volume and turgor pressure. The following mathematical description of volume regulation under osmotic stress is a simplified version of a previously published model, which was carefully parameterized using hyperosmotic shock data (Schaber and Klipp, 2008).

The concentration of calcium ions in the cytosol changes due to fluxes across different channels and transporters. The cytosol is also provided with calmodulin—a protein involved in the binding and sensing of calcium ions (Cyert, 2001)—that can activate calcineurin, the main calmodulin effector. HTS is applied by diluting the medium in which cells grow with distilled water, which has also the effect of reducing the availability of calcium ions. Since the extracellular volume is way larger than the volume occupied by all cells, the reduction of extracellular calcium ions caused by the cell uptake can be neglected. Therefore, the extracellular Ca²⁺ concentration depends only on dilution factor and mixing time (see Eq. 14): Ca ex = Ca ex 0 for t < t o f f , Ca ex = Ca ex 0 − Ca ex 0 d e t o f f − t t m + Ca ex 0 d otherwise, (15) with all concentrations expressed in nM, and t, t _off and t _m expressed in seconds.

The rate of change of the Ca²⁺ concentration in the cytosol can be written as the sum of fluxes of the relevant channels and transporters (described below): d Ca cyt d t = j IN + j Cch 1 + j MS − j Pmr 1 − j Vcx 1 − j Pmc 1 , (16) where the js are Ca²⁺ fluxes (in nM⋅s⁻¹) across calcium channels and transporters.

Very often, channels and transporters show kinetics that can be described with the Michaelis-Menten equation (i.e. the transporter saturates at high substrate concentrations) (Christopher, 2002; Weijiu, 2012; Fridlyand et al., 2003). Indeed, this is the case for the intracellular transporters Pmr1, Vcx1, Pmc1 (see references in Table 2). Pmr1 indirectly replenishes the endoplasmic reticulum with Ca²⁺, while Vcx1 and Pmc1 are responsible for its sequestration into the vacuole. Ca²⁺ enters the cell through the plasma membrane-located HACS channel and other transporters whose molecular identities are yet unknown (Batiza et al., 1996; Locke et al., 2000; Tisi et al., 2002; Cui et al., 2009a). Here, the influx associated with this unknown transport is simply called j _IN—to recall its function—and is assumed to have a Michaelis-Menten kinetics.

Experimental evidences strongly suggest that the increase of cytosolic Ca²⁺ in cells challenged with HTS is caused by the opening of a MS channel on the plasma membrane (Batiza et al., 1996; Rigamonti et al., 2015). This calcium influx pathway has not been molecularly identified yet, and it is here denoted by j _MS. The equations describing the fluxes are: j IN = v IN Ca ex k IN + Ca ex , (17) j Pmr 1 = v Pmr 1 Ca cyt k Pmr 1 + Ca cyt , (18) j Vcx 1 0 = v Vcx 1 Ca cyt k Vcx 1 + Ca cyt , (19) j Pmc 1 = v Pmc 1 Ca cyt k Pmc 1 + Ca cyt , (20) j MS 0 = P open k MS Ca ex − Ca cyt , (21) j Cch 1 0 = k Cch 1 Ca ex − Ca cyt , (22) where P _open is the opening probability of the MS channel (see below), k _Cch1 and k _MS are rate parameters (in s⁻¹), vs. are rate constants (in nM⋅s⁻¹), and all other ks are Michaelis constants (in nM).

The opening probability of MS channels follows a Boltzmann distribution (Gustin et al., 1988; Sukharev et al., 1999; Jiang and Sun, 2013). MS channels are gated by membrane tension (Gustin et al., 1988; Sackin, 1995) but here, for the sake of simplicity, turgor pressure is used instead (see Section 3.1 and Eq. 1 for a justification). The opening probability of the MS channel is then: P open = 1 − 1 1 + e P − P MS g MS , (23) where P is the cell turgor pressure in MPa (see Eq. 13), P _MS is the turgor pressure (in MPa) at which P _open is equal to 0.5, and g _MS is a slope parameter (in MPa).

Unknown parameters of the model were fitted against time traces of cytosolic Ca²⁺ measurements from HTS experiments conducted by Rigamonti et al. (2015) and by R. Tisi, unpublished results (Figure 2), as described in Materials and Methods. In these experiments, wild-type S. cerevisiae cells were grown in YPD medium, which has an estimated osmolarity of 0.26 Osm/L (Schaber et al., 2010). Specifically, the parameters of the biophysical module (Table 1) were set to reflect those experimental conditions.

In addition to the wild-type strain, we simulated three mutant strains: cnb1Δ, lacking functional calcineurin; flc2Δ, lacking a putative TRP-like channel subunit; and cch1Δ, lacking a functional HACS channel. Within the model, a mutant can be simulated by setting to zero the parameter associated with the function carried out by the removed gene. Hence, the cnb1Δ strain was obtained by setting to zero the total amount of calcineurin in the cell (CaNt), and the cch1Δ by setting to zero the rate parameter k _Cch1. However, these parameter settings were not sufficient to reproduce the behavior displayed by the three mutants. Thus, we performed additional parameter estimations exploiting available experimental time traces of the mutant strains (from Rigamonti et al. (2015) and R. Tisi, unpublished results, Figure 2).

In particular, PMC1 expression strongly depends on calcineurin (Cunningham and Fink, 1996), but the model can not automatically adjust PMC1 expression in the cnb1Δ mutant, since it lacks any description of transcriptional processes. To take into account the reduction of PMC1 expression in this mutant, we performed a separate parameter estimation to infer the value of the v _Pmc1 parameter (i.e., the V _max parameter of a Michaelis-Menten equation (Eq. 20), which is proportional to the enzyme abundance) in the cnb1Δ model. By doing so, it was possible to assess the reduction of Pmc1 transporters in this mutant with respect to the wild-type. We argue that the estimated value of v _Pmc1 for the cnb1Δ mutant could be regarded as an estimate of the basal expression level of the Pmc1 protein, when Cnb1 is not stimulating the PMC1 gene transcription. Thus, we used the same value of v _Pmc1 to simulate the cch1Δ model. In fact, in order to maintain a physiological Ca²⁺ level when the Ca²⁺ income is lower, a novel steady state balance has to be achieved, where Pmc1 abundance has to be adjusted so that Ca²⁺ remains available to be provided to the secretory pathway as well. This is achieved by finely tuning the homeostasis regulatory circuit, through calcineurin transcriptional control on PMC1 gene, leading to a Pmc1 activity nearby the basal level estimated for the cnb1Δ mutant.

Adopting a similar approach to the one described above, we estimated the value of k _MS for the flc2Δ strain, in order to account for the observed increased activity of the MS channel in this strain (Rigamonti et al., 2015). It is not known whether the increased activity of this channel is due to an increased channel abundance or simply to an increased channel activity. In any case, the rate parameter k _MS is—like the V _max of a Michaelis-Menten equation—proportional to the number of channels.

Analysis of the parameter estimation results led to the conclusion that the model could be simplified without affecting the simulations. In particular, removal of (a) the feedback inhibition on the HACS channel, mediated by calcineurin (Eq. 27), and of (b) the influx pathway that was called “IN” (Eq. 17), did not cause any significant difference on the simulation outcomes. Therefore, the feedback on HACS and the j _IN influx were removed and all simulations and parameter values reported here are related to this simplified model. Table 2 lists all the parameters used for the wild-type model, while Table 3 contains only the values that changed depending on the strain.

The simulations of the full model of HTS response correctly reproduce the dynamics of the Ca²⁺ transients (Figures 3A–D), as well as the steady-state levels and peak values of cytosolic Ca²⁺ (Figures 3E,F) of both wild-type and all mutant strains.

The results of the simulations for the wild-type strain are shown in Figure 4. According to the model, after the HTS the HACS activity decreases due to the sudden shortage of the extracellular Ca²⁺ (Figure 4A), while the MS channels on the plasma membrane open rapidly (Figure 4B, inset). The cytosolic Ca²⁺-dependent inhibition on the MS channels provides a way to decrease the activity of this channel, thus helping to restore low intracellular calcium levels after the stimulus. In fact, the dynamics reported in Figure 4B shows that only in the presence of feedback inhibition the flux through this channel is still decreasing at time t > 280s. These outcomes also suggest that the main pump responsible for signal attenuation is Pmc1, the Ca²⁺-ATPase located on the vacuole, while Pmr1 and Vcx1 have a negligible role in this respect (Figures 4C,D). In particular, while the predicted rate of Ca²⁺ sequestration by Pmr1 is predicted to be consistently slow, the Vcx1 activity is kept down by the feedback inhibition mediated by calcineurin (Figure 4D).

As described above, the peak response of cch1Δ mutants—lacking functional HACS—is considerably higher than in the wild-type strain (Figure 3F). According to the model, the peak difference is not ascribable to an increased activity of the MS channels in this strain with respect to the wild-type. In fact, in this mutant, Ca²⁺ fluxes through both MS channels and Pmc1 are considerably lower than both wild-type and flc2Δ strains (Figures 5A,C). This behaviour is the result of the reduced PMC1 expression (Table 3):

Since Pmc1 is the main transporter responsible for Ca²⁺ sequestration from the cytosol, reducing its activity increases the level of Ca²⁺, which in turn inhibits the MS channels through the activated calmodulin (Figure 5B). This feedback process ensures that the steady-state calcium levels are comparable to those of the wild-type. However, the reduced activity of Pmc1 causes a higher peak when the MS channels open following HTS. Rigamonti et al. suggested that the increased Ca²⁺ peak in flc2Δ is caused by the removal of an inhibitory effect on the MS channels (Rigamonti et al., 2015). To test this hypothesis, during the parameter estimation process we let the k _MS parameter of flc2Δ mutant assume different values compared to other strains (Table 3). The simulations show that, in line with this, a difference in the MS channels rate compared to the wild-type is sufficient to explain the increased peak observed during HTS (Figure 5A).

The cnb1Δ mutants show an increased steady-state level of cytosolic Ca²⁺ and a lower peak compared to all other strains (Figure 3). As described in Section 4.3, for this mutant cells we estimated the reduction of PMC1 expression. Our model predicts that Pmc1 abundance is about 16 times lower compared to the wild-type, which is counterbalanced by the increased activity of Vcx1 that is no longer inhibited by calcineurin. This compensation mechanism, however, fails to fully replace the decreased activity of Pmc1, and thus the mutant displays higher steady-state calcium levels. The higher calcium levels result in more activated calmodulin (Figure 5B), which in turn inhibits the MS channels leading to a lower peak (Figure 5A).

Figure 6 shows a prediction of HTS response in not yet tested experimental conditions, since the technical difficulty in achieving a tight control of the speed of dilution could not be overcome. Variability in the speed and shape of the signal was observed in different experiments, particularly when an automatic injector was not applied, thus we decided to investigate the effect of this parameter variation with in silico experiments. The t _m parameter of the model defines the mixing time, i.e. the speed of dilution. The simulations of the wild-type strain with different mixing times produces an unexpected pattern (Figure 6A). In particular, while the speed of the response turns out to be linearly dependent on the mixing time (Figure 6B, red curve), the peak Ca²⁺ value is not. The peak values grow with the mixing time until a maximum at t _m = 15s, before decreasing again (Figure 6B, blue curve).

The model also predicts that the inactivation of the MS channels—simulated by setting to zero the parameter k _MS—would reduce the steady-state cytosolic Ca²⁺ concentration from 215 to 160 nM.