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Cell Calcium. Author manuscript; available in PMC 2011 July 5.

Published in final edited form as:

Published online 2008 September 10. doi: 10.1016/j.ceca.2008.07.005

PMCID: PMC3130064

NIHMSID: NIHMS301364

Jiangjun Cui,^{a,}^{1} Jaap A. Kaandorp,^{a,}^{*} Olufisayo O. Ositelu,^{b,}^{2} Veronica Beaudry,^{b,}^{2} Alicia Knight,^{b,}^{2} Yves F. Nanfack,^{a,}^{1} and Kyle W. Cunningham^{b,}^{2}

The publisher's final edited version of this article is available at Cell Calcium

See other articles in PMC that cite the published article.

Yeast can proliferate in environments containing very high Ca^{2+} primarily due to the activity of vacuolar Ca^{2+} transporters Pmc1 and Vcx1. Yeast mutants lacking these transporters fail to grow in high Ca^{2+} environments, but growth can be restored by small increases in environmental Mg^{2+}. Low extracellular Mg^{2+} appeared to competitively inhibit novel Ca^{2+} influx pathways and to diminish the concentration of free Ca^{2+} in the cytoplasm, as judged from the luminescence of the photoprotein aequorin. These Mg^{2+}-sensitive Ca^{2+} influx pathways persisted in *yvc1 cch1* double mutants. Based on mathematical models of the aequorin luminescence traces, we propose the existence in yeast of at least two Ca^{2+} transporters that undergo rapid feedback inhibition in response to elevated cytosolic free Ca^{2+} concentration. Finally, we show that Vcx1 helps return cytosolic Ca^{2+} toward resting levels after shock with high extracellular Ca^{2+} much more effectively than Pmc1 and that calcineurin, a protein phosphatase regulator of Vcx1 and Pmc1, had no detectable effects on these factors within the first few minutes of its activation. Therefore, computational modeling of Ca^{2+} transport and signaling in yeast can provide important insights into the dynamics of this complex system.

In eukaryotic cells, Ca^{2+} functions as a highly versatile intracellular signal molecule regulating many different biological processes such as proliferation, muscle contraction, neurotransmitter release, programmed cell death and gene expression, etc. [2,22,41,42]. In budding yeast (*Saccharomyces cerevisiae*), calcium signals are used to effect important adaptations in response to mating pheromones [24], membrane damaging compounds [8,9], and a variety of environmental stresses such as high salt, high pH, high osmolarity, and others [37].

Like mammalian cells, budding yeast maintain cytosolic Ca^{2+} concentration at very low levels in growing yeast cells (50–200 nM) [31] through the actions of Ca^{2+} pumps and exchangers (see Fig. 1A) [7,12]. A vacuolar calcium ATPase Pmc1 [13,15] and a vacuolar Ca^{2+}/H^{+} exchanger Vcx1 [14,32] independently transport cytosolic Ca^{2+} into large vacuoles, the major Ca^{2+} store and sink. Cytosolic Ca^{2+} is also pumped into ER and Golgi through a secretory pathway calcium ATPase termed Pmr1 [44,47], which may also promote Ca^{2+} extrusion from yeast cells [52]. Unlike in the case of plant and animal cells, Ca^{2+} pumps and exchangers have not been detected on the plasma membrane of yeast cells [47,52]. When yeast cells experience a severe hypertonic shock, vacuolar Ca^{2+} can be released into the cytosol through the mechanosensitive Ca^{2+} channel Yvc1, a distant homolog of mammalian TRPC-type Ca^{2+} channels [36], which results in transient elevation of cytosolic Ca^{2+} concentration. A plasma membrane voltage-gated Ca^{2+} channel (VGCC) composed of Cch1 and Mid1 becomes activated in response to depolarization [10], depletion of secretory Ca^{2+} [27], pheromone stimulation [34], hypotonic shock [4], In unstimulated yeast cells growing in standard laboratory conditions, small amounts of extracellular Ca^{2+} enter the cell through unidentified transporters on the plasma membrane [27].

Schematic graph of the system and control block diagram. *Panel A*, A schematic graph of Ca^{2+} homeostasis/signaling system in yeast cells (for details, please see the text in Section 1). Transporter M is newly detected in this work and is assumed to open **...**

In response to sudden increases in extracellular Ca^{2+}, cytosolic free Ca^{2+} levels rapidly rise and fall with complex dynamics, as judged by luminescence measurements of yeast cells expressing aequorin [31]. In response to these changes, the universal Ca^{2+} sensor protein calmodulin can bind and activate the protein phosphatase calcineurin, which inhibits the function of Vcx1 and induces the expression of Pmc1 and Pmr1 via activation of the Crz1 transcription factor [13,14,16,25,46]. These calcineurin-sensitive adaptations appear to be crucial for proliferation of yeast cells in high calcium environments. Mathematical modeling of these feedback networks successfully recapitulated the observed changes in aequorin luminescence [12]. However, that initial model relied on extremely rapid effects of calcineurin on Pmc1 and Pmr1 expression that were not physiologically realistic.

Here we show that the calcineurin-dependent feedback networks described above have little or no effect on aequorin luminescence traces within the first few minutes of Ca^{2+} shock. Therefore, the initially complex dynamics of cytosolic Ca^{2+} homeostasis must arise through other mechanisms. The observed dynamics are well described by a new mathematical model that omits calcineurin-dependent feedback and instead includes rapid Ca^{2+}-dependent feedback inhibition of Ca^{2+} influx. Though the feedback mechanism and the Ca^{2+} influx transporters all remain to be identified, experimental evidence suggests that the primary Ca^{2+} influx transporters are competitively inhibited by extracellular Mg^{2+}. Indeed, the impaired growth of yeast cells in high calcium environments was ameliorated by inclusion of Mg^{2+} salts in the medium and the best fitting mathematical model incorporates two Mg^{2+}-sensitive Ca^{2+} influx pathways with different cation affinities. These findings extend a recent report of Ca^{2+} influx pathways in yeast that become (re)activated upon withdrawal of extracellular Mg^{2+} [49] and demonstrate the power of combining experimental and computational methodologies.

The ability of extracellular Mg^{2+} to prevent toxicity of extra-cellular Ca^{2+} to yeast was measured as in [13]. Briefly, stationary phase cultures of each yeast strain were diluted 1000-fold into fresh YPD pH 5.5 medium containing varying concentrations of CaCl_{2} and MgCl_{2}, incubated at 30 °C for 24 h, and then optical density was measured at 650 nm using a 96-well platereader (Molecular Devices). The concentration of CaCl_{2} causing a 50% inhibition of growth (the IC_{50}) was obtained by fitting the data to the simple sigmoid equation.

Cytosolic free Ca^{2+} concentrations were monitored in populations of yeast cells expressing aequorin essentially as described [34]. Briefly, yeast cells were transformed with plasmid pEVP11 and grown to mid-log phase in synthetic medium lacking leucine to select for plasmid maintenance. Cells were harvested by centrifugation, washed, and resuspended at OD600 = 20 in medium containing 10% ethanol and 25 μg/mL coelenterazine (Molecular Probes, Inc.). After 45 min incubation at room temperature in the dark, cells were washed twice with YPD pH 5.5 medium and shaken at 30 °C for an additional 90 min. Aliquots of the cell suspension (450 μL) were pipetted into tubes containing appropriate volumes of 1 M MgCl_{2}, mixed, and placed into a Sirius tube luminometer (Bertholdfinc) Inc.). Luminescence was recorded at 0.2 s intervals for 0.5 min prior to and 3.0 min post injection of 2 M CaCl_{2} (300 μL). Output is plotted as relative luminescence units per second (RLU) over time using similar numbers of cells per sample.

In the control block diagram as shown in Fig. 1B, we can see that for yeast cells with fixed volume, we can calculate the concentration of cytosolic Ca^{2+} (i.e., *x*(*t*)) by taking an integral of the flux rate difference (i.e., the influx rate through Transporter M and Transporter X into the cytosol subtracting the sequestering rates of Pmc1, Pmr1 and Vcx1) divided by the cytosolic volume. Since the slow gene expression feedback pathway through calcineurin has now been shown not to be accounting for the observed response spikes, there should be some other quick feedback mechanisms the details of which we are currently quite ignorant to. Since calmodulin has been reported to have the ability of directly regulating the opening probability of Ca^{2+} influx pathways [51], here for simplicity, we just assume that the activity of Transporter M and Transporter X are both directly inhibited by Ca^{2+}-bound calmodulin (see the green lines in Fig. 1B). If the volume of yeast cells changes due to growth or hypertonic shock, etc., then we need to further consider the effects caused by the cellular volume change.

The mathematical modeling for simulating calcium response curves in *yvc1 cch1* yeast cells can now be divided into three parts: feedback part modeling, protein modeling (including Transporter M, Transporter X, Pmc1, Pmr1 and Vcx1) and volume evolution modeling.

Due to the usual defect of one of the two C-terminal EF-hands (site IV), yeast calmodulin can only bind to a maximum of three molecules of Ca^{2+} [16]. By law of mass action, we can use the following equation to describe the cytosolic Ca^{2+} sensing process under the assumption of strong cooperativity existing among the three active sites [12,23]:

(1)

where *m*(*t*) denotes the concentration of Ca^{2+}-bound calmodulin and *m′* (*t*) denotes its change rate,
denotes the forward rate contant,
denotes the backward rate constant, *x*(*t*) denotes cytosolic calcium ion concentration and [CaM_{total}] denotes the total concentration of calmodulin (including Ca^{2+}-free and Ca^{2+}-bound form).

Once we know the concentration of Ca^{2+}-bound calmodulin, we use a putative algebraic expression to model its assumed inhibitory regulation on Transporter M and Transporter X as follows:

(2)

(3)

where *k _{a}*,

As we will see later, our aequorin response curves are all obtained under condition of applying hypertonic shocks (e.g., by the sudden injection of 800 mM CaCl_{2} into the extracellular medium) to the yeast cells. Under such great osmotic perturbation, yeast cells will quickly shrink due to the existence of water channels in the plasma membrane [20,29,45]. The evolution of the cell volume is governed by the following equation [29]:

(4)

where *V*(*t*) and *A*(*t*) denote the volume and the surface area of yeast cells at time *t*, respectively, *V′* (*t*) denotes the change rate of yeast cell volume, *L*_{p} denotes the hydraulic membrane permeability, *σ* denotes the reflection coefficient, *R* denotes the ideal gas constant, *T* denotes the room temperature (294.15 K), *n*_{s} denotes the apparent number of osmotically active molecules in the cell, *b* denotes the non-osmotic volume and *Π*_{ex} denotes the osmotic pressure of the external medium which can be calculated as *Π*_{ex} = *Π*_{0} + 3([Ca_{ex}] + [Mg_{ex}])*RT* where *Π*_{0} denotes the initial osmotic pressure of the external medium, [Ca_{ex}] and [Mg_{ex}] denote extracellular Ca^{2+} concentration and extracellular Mg^{2+} concentration, respectively (please note that we can calculate *A*(*t*) = 4*π* (3*V*(*t*)/4/*π*)^{2/3} because yeast cells are somewhat spherical).

The initial intracellular osmotic pressure (*Π _{i}*(0)) for yeast cells in standard medium is around 0.636 Osm [20], according to Boyle van’t Hoff’s law,

Experimental results show that the uptake behaviors of the four involved proteins (Transporter X [27], Pmc1 [48], Pmr1 [44] and Vcx1 [35]) conform to the Michaelis–Menten equation, which is a general equation for describing the single site ion uptake behavior of many kinds of transport proteins [2,11,12]. If we assume that the uptake behavior of Transporter M can also be modeled using this equation and that Mg^{2+} is a competitive inhibitor of this transporter, according to the classical enzyme kinetics theory of competitive inhibition [17,38], the uptake rate of Transporter M can be expressed as:

(5)

where *J*_{MO} denotes the uptake rate of Transporter M (without calmodulin inhibition), [Ca_{ex}] and [Mg_{ex}] denote extracellular Ca^{2+} concentration and extracellular Mg^{2+} concentration, respectively, *V*_{max} is the maximum uptake rate of Transporter M, *K*_{m} is the binding constant and *K*_{IM} is the inhibition constant. And we can build a similar mathematical model for Transporter X which has been shown to be Mg^{2+}-sensitive [27].

Since now we have built the uptake models for all the five proteins involved in Ca^{2+} transport and models for the relevant feedback regulation, according to the control block diagram (Fig. 1B) and by further taking consideration of the effect of cell volume shrinkage, we can derive the main equation of our calcium homeostasis problem as follows [19]:

(6)

where *x′*(*t*) denotes the change rate of cytosolic free Ca^{2+} concentration, *J*_{M}, *J*_{X}, *J*_{Pmc1}, *J*_{Vcx1} and *J*_{Pmr1} denote the calcium ion flux through Transporter M, Transporter X, Pmc1, Vcx1 and Pmr1, respectively, Vol(*t*) denotes the volume of the cytosol at time *t* which can be roughly calculated as 10%*V*(*t*) and Vol′(*t*) denotes its change rate, *f* denotes the calcium buffer effect constant [19]. As stated in page 105 of Ref. [19], “*f _{i}* (i.e.,

We can further write the above main equation in fully detailed mathematical form:

(7)

where *x*(*t*) denotes cytosolic calcium ion concentration, *m*(*t*) denotes the concentration of Ca^{2+}-bound calmoculin, *K*_{m}, *K*_{x}, *K*_{1}, *K*_{2} and *K*_{3} are the Michaelis binding constants of Transporter M, Transporter X, Pmc1, Vcx1 and Pmr1, respectively, *V*_{m}, *V*_{x}, *V*_{1}, *V*_{2} and *V*_{3} are the corresponding rate parameters.

Previously published data [27] show that after 2.5 h of incubation, the maximum Ca^{2+} accumulation due to Transporter X is around 390 nmol/10^{9} cells. We can roughly calculate parameter *V*_{x} in the main equation (Eq. (7)) as follows (for details, please see Eq. (18) of Ref. [12]):

(8)

where *α* denotes the growth rate constant the value of which is 0.006 min^{−1} [18].

The main equation (Eq. (7)) together with the calcium sensing equation (Eq. (1)) and volume evolution equation (Eq. (4)) constitute a concise mathematical model for simulating response curves in *yvc1 cch1* yeast cells under hypertonic shocks. The whole set of parameters (except the control parameters [Ca_{ex}] and [Mg_{ex}]) in our model which will be used in the subsequent simulations are listed in Table 1.

Here we use the following experimentally reported function [3,21] to convert calcium ion concentration from μM to aequorin luminescence unit (i.e., RLU: relative luminescence unit) so that we can make intuitive comparison between our numerical solutions (in μM) with experimentally obtained aequorin response curves (in RLU):

(9)

where *x* denotes Ca^{2+} concentration value in unit μM and *L*_{max} denotes the maximal aequorin luminescence.

The above described two transporters model (Eqs. (1), (4) and (7). We name this model as two transporter model because in this model, it is assumed that there are two influx pathways: transporters M and X) has 25 parameters, 13 of which (*K*_{m}, *K*_{IM}, *K*_{IX}, *V*_{m}, *V*_{1}, *V*_{2}, *V*_{3}, *k _{a}*,

To describe a bit more specifically, the stochastic evolutionary algorithm is a stochastic process that modifies an original population of individuals from iteration to iteration with the aim of minimizing an objective function. In this study, we use a modified (μ, λ)-Evolutionary Strategy (ES), based on stochastic fitness ranking [6,50]. This method is simple and has proven to be more efficient than most other classical evolutionary algorithms for large parameter estimation problems [33,39,40]. The local search was performed using the Matlab (The Mathworks, Inc.) function “LSQCURVEFIT” which employs Levenberg–Marquardt algorithm to do the curve fitting for non-linear data.

We simultaneously fit the model to both the wild-type data (see Fig. 2B) and the mutant data (see Fig. 2C) by minimizing the least square error (LSE). As usual, we proportionally weight the wild-type data based on the maximum amplitude of each data set and take mutant data weight to be 1/3 of the wild-type data because the mutant data is less important than the wild-type data. This leads us to minimize the weighted least square error given as:

(10)

where *a* denotes the type of data (wild-type or mutant). Let *S* = {0, 0.3 mM, 3 mM, 30 mM, 90 mM}, then *m* denotes the case in which extracellular MgCI_{2} is the *m*th element of *S. n* denotes the time point and Tn denotes the number of data points in the dataset for a single experimental response curve. Thus
denotes the concentration for the data type *a* with extracellular Mg^{2+} = *S*[*m*] at time *t* = *n* × td where td denotes the time difference between the consecutive data points (in our case, td = 0.2 s and Tn = 300), *F*(*a*, *m*, *n*) is the simulated data and *w _{a,m,n}* is the associated weight.

Experimental results. *Panel A*, the concentrations of CaCl_{2} that caused a 50% inhibition of growth (i.e., the IC_{50} for CaCl_{2} ) were shown for *pmc1* (filled circles), *pmc1 vcx1* (open circles), *pmc1 vcx1 cnb1* (filled triangles) and *vcx1 cnb1* (open triangles) **...**

In addition to fitting the two transporters model to the experimental data, we also performed fitting for the one transporter model. In this case, Transporter X is assumed to be the only influx pathway (i.e., under the condition of *V*_{m} = 0, Eqs. (1), (4) and (7) constitute the one transporter model) and only 9 parameters (*K*_{IX}, *V*_{1}, *V*_{2}, *V*_{3}, *k _{b}*,
,

Yeast mutants lacking the vacuolar Ca^{2+} ATPase Pmc1p grow as well as wild-type yeast strains in standard culture media but they grow much poorer than wild-type when environmental Ca^{2+} is elevated [13], suggesting that elevated cytosolic free Ca^{2+} can be toxic to yeast. However, we noticed that Ca^{2+} toxicity was blocked by a contaminant present in certain batches of agar (data not shown). The contaminant that blocked Ca^{2+} toxicity was traced to Mg^{2+} because (1) the toxicity blocking activity was abolished by addition of the Mg^{2+} chelator EDTA, (2) crude preparations of agar are known to contain millimolar Mg^{2+}, and (3) pure MgCl_{2} but not NaCl could block Ca^{2+} toxicity in *pmc1* mutants in standard culture media.

To investigate the mechanism of Mg^{2+} suppression of Ca^{2+} toxicity, the concentration of CaCl_{2} that caused a 50% inhibition of growth (i.e., the IC_{50} for CaCl_{2}) was determined for *pmc1* mutants after 24 h of growth standard YPD culture medium supplemented with 0–32 mM MgCl_{2}. Remarkably, for *pmc1* knockout mutants the IC_{50} for CaCl_{2} increased with increasing MgCl_{2} up to ~8 mM after which the effectiveness of MgCl_{2} began to decrease (Fig. 2A, filled circles). Double mutants lacking both Pmc1 and the vacuolar Ca^{2+}/H^{+} exchanger Vcx1 behaved similarly except the IC_{50} values were shifted downward by ~1.6-fold (Fig. 2A, open circles). The sole remaining Ca^{2+} transporter Pmr1, a Ca^{2+}/Mn^{2+} ATPase of the Golgi complex, is essential for growth of *pmc1 vcx1* double mutants and is strongly up-regulated by calcineurin [14]. A *pmc1 vcx1 cnb1* triple mutant that also lacks calcineurin exhibited ~2.3-fold lower tolerance to CaCl_{2} than the *pmc1 vcx1* double mutant and MgCl_{2} suppressed CaCl_{2} toxicity over a similar range of concentrations (Fig. 2A, filled triangles). A *vcx1 cnb1* double mutant in which Pmc1 and Pmr1 are expressed only at basal levels exhibited ~6.6-fold increase of IC_{50} for CaCl_{2} as expected and also exhibited MgCl_{2} suppression of CaCl_{2} toxicity (Fig. 2A, open triangles). These findings demonstrate that MgCl_{2} suppresses CaCl_{2} toxicity independent of all known Ca^{2+} transporters, which is consistent with the possibility that Mg^{2+} competitively inhibits one or more Ca^{2+} influx pathways.

Two Ca^{2+} channels have been characterized in yeast, the vacuolar Ca^{2+} release channel Yvc1 and the plasma membrane Ca^{2+} influx channel Cch1-Mid1. The *yvc1 cch1 pmc1 vcx1* quadruple mutant exhibited similar levels of MgCl_{2} suppression of CaCl_{2} toxicity as the *pmc1 vcx1* double mutants (data not shown), suggesting that Mg^{2+} blocks some other Ca^{2+} influx pathways. To test this possibility directly, cytosolic free Ca^{2+} concentrations were monitored directly after a sudden increase in extracellular CaCl_{2} by following luminescence of the Ca^{2+}-sensitive photoprotein aequorin. The *yvc1 cch1* double mutant expressing apo-aequorin from a plasmid was incubated with coelenterazine co-factor to reconstitute aequorin in situ. The cells bearing reconstituted aequorin were returned to growth medium for an additional 90 min, divided into equal aliquots, treated with varying amounts of MgCl_{2}, placed into a tube luminometer, and monitored for luminescence before and after injection of 800 mM CaCl_{2}. In the absence of added MgCl_{2}, aequorin luminescence rose quickly after CaCl_{2} injection, peaked after ~13.3 s, and declined to a level that was well above the starting level (Fig. 2B). Increasing concentrations of MgCl_{2} progressively lowered the rates of luminescence rise, the maximum achievable luminescence, and the new baseline levels following decline (Fig. 2B). Remarkably, the loss of Vcx1 resulted in a dramatic increase in the rate and peak height of aequorin luminescence (Fig. 2C). Additionally, the loss of Vcx1 resulted in ~60% slower rate of luminescence decline after the peak regardless of MgCl_{2} concentration. Though calcineurin inhibits Vcx1 function in long-term growth assays [14], the activity of Vcx1 in these short-term luminescence experiments was not detectably affected by addition of the calcineurin inhibitor FK506 (data not shown). The further loss of Pmc1 also had little effect on aequorin responses (data not shown). These findings identify Vcx1 as the major Ca^{2+}-sequestering transporter in short-term responses to high Ca^{2+} environments and confirm the hypothesis that Mg^{2+} interferes with one or more novel Ca^{2+} influx pathways. A similar hypothesis was proposed recently to explain the increased Ca^{2+} influx and elevated cytosolic Ca^{2+} concentration observed in yeast upon Mg^{2+} sudden withdrawal [49]. The proteins responsible for Mg^{2+}-sensitive Ca^{2+} influx have not been identified but the Alr, Alr2, and Mnr2 proteins have been identified as hetero-oligomeric proteins required for Mg^{2+} uptake that also promote sensitivity to high environmental Ca^{2+} [28].

The multiphasic nature and calcineurin-independence of the aequorin luminescence curves suggested complex dynamics of the novel Ca^{2+} influx pathway(s). To help understand these dynamics, a mathematical model was constructed in which the Ca^{2+} transporters Pmr1, Pmc1, and Vcx1 were assumed to function without calcineurin feedback *in vivo* according to standard Michaelis–Menten kinetics. The optimal fitting to the experimental data using a hybrid optimization algorithm (see Section 2.3) shows that simulations assuming two Mg^{2+}-sensitive Ca^{2+} influx transporters (termed transporters M and X) each with distinct properties (as discussed below in more detail) can closely fit the experimental data (see Fig. 4A and B and compare them with Fig. 2B and C) whereas simulations using just one Mg^{2+}-sensitive Ca^{2+} influx transporter poorly fit the experimental data (see Fig. 4C and compare it with Fig. 2B).

The model for *yvc1 cch1* mutant under hypertonic shock consists of three equations (Eqs. (1), (4) and (7), please see Section 2.2) with three unknowns: *x*(*t*), *m*(*t*) and *V*(*t*). By performing steady state analysis of our model with fixed parameter [Ca_{ex}] = 800 mM, we can first depict the steady state value of *x*(*t*) as a function of parameter [Mg_{ex}] as shown in Fig. 3A. In general, the resting level of *x*(*t*) decreases almost linearly as [Mg_{ex}] increases. Only in the case of very low [Mg_{ex}] (≤1 μM), the curves shows a strange bending.

By setting reasonable initial conditions (*x*(0) = 100 nM, *m*(0) = 0 M and *V*(0) = 100 μM^{3}) and then solving the three equations (Eqs. (1), (4) and (7), using the parameters listed in Table 1) numerically, we can depict the *x*(*t*) curves for parameter [Mg_{ex}] suddenly increasing from 0 mM to various concentrations (0 mM, 0.3 mM, 3 mM, 30 mM, 90 mM) at *t* = 0 as shown in Fig. 4A (please note that at the same time, parameter [Ca_{ex}] suddenly increases from 0 mM to 800 mM in these simulations). In general, the simulated *x*(*t*) rises due to the hypertonic shock, forms a peak and then declines to a resting value; for higher value of parameter [Mg_{ex}], the peak value of the curve is lower and appears later. For example, by observing the top black curve for parameter [Mg_{ex}] = 0 mM, we can see that due to the hypertonic shock, the simulated *x*(*t*) quickly rises (in around 9 s) from an initial value of 5700RLUs (~100 nM) to a high peak of around 116600RLUs (~0.54 μM), then gradually decreases to a value of 46110RLUs (~0.35 μM) when *t* = 1 min. Further investigation shows that it will further decreases to steady state value of 17500RLUs (~0.21 μM). By observing the blue curve for parameter [Mg_{ex}] rising from 0 mM to 0.3 mM, we can see that the simulated cytosolic *x*(*t*) rises slower than the black curve and its peak value (around 85200RLUs (~0.47 μM)) which appears when *t* = 14 s is much lower than that of the black curve. However, the green curve for parameter [Mg_{ex}] rising from 0 mM to 3 mM seems just a little bit lower and slower than the blue curve.

In Fig. 4B, we depict the simulated *x*(*t*) curves for *vcx1 yvc1 cch1* triple mutant under hypertonic shock (step increase of [Ca_{ex}] from 0 mM to 800 mM with simultaneous step increase of [Mg_{ex}] to various concentrations at *t* = 0, please note that *V*_{3} is set to be 0 in these simulations, for the rest parameters please see Table 1). By comparison of these curves with their corresponding curves for simulated *yvc1 cch1* double mutant in Fig. 4A, the peak values of these simulated response curves seem much higher. For example, the peak value of the top black curve for [Mg_{ex}] = 0 mM in Fig. 4B is around 609000RLUs (~1.07 μM) which is certainly much higher than the peak value 116600RLUs (~0.54 μM) of its corresponding black curve in Fig. 4A.

In Fig. 4C, the simulated *x*(*t*) curves of the best fit using one Mg^{2+}-sensitive transporter model (i.e., Transporter X is assumed to be the only influx pathway) for *yvc1 cch1* mutant under hyper-tonic shock (step increase of [Ca_{ex}] from 0 mM to 800 mM with simultaneous step increase of [Mg_{ex}] from 0 mM to various concentrations at *t* = 0) are shown. In this figure, the top two curves (i.e., the black curve and the blue curve) coincide with each other. The peak values of the blue, green, yellow, red curves are 129000RLUs (~0.413 μM), 127000RLUs (~0.41 μM), 109000RLUs (~0.382 μM), 82340RLUs (~0.334 μM), respectively. The peaks of both blue and green curves appear when *t* = 7.5 s whereas the peaks of the yellow and red curves appear at *t* = 7.9 s and *t* = 8.6 s, respectively.

To discriminate the different influx contributions from two Ca^{2+} influx pathways, in Fig. 3B we depict the simulated flux proportion of Transporter M in the total cytosolic Ca^{2+} influx as a function of *t* for *yvc1 cch1* mutant under hypertonic shock (step increase of [Ca_{ex}] from 0 mM to 800 mM with simultaneous step increase of [Mg_{ex}] from 0 mM to various concentrations at *t* = 0). By observing the bottom black curve for [Mg_{ex}] = 0 mM, we can see that at the beginning, 70.4% of the cytosolic Ca^{2+} influx is contributed by Transporter M and this proportion quickly rises to 98%. The other curves seem to almost coincide and are flat with a constant value of 100%.

In Fig. 4D, the simulated volume evolution curves using to transporters model for *yvc1 cch1* mutant under hypertonic shock (step increase of [Ca_{ex}] from 0 mM to 800 mM with simultaneous step increase of [Mg_{ex}] from 0 mM to various concentrations at *t* = 0) are shown. As we can see in this figure, in general, the cell volume quickly decreases (in about 1 s) to a steady state value of 53–55% of the initial volume and the difference among different color curves are quite subtle.

To check the behavior of cytosolic Ca^{2+} level (i.e., *x*(*t*)) of our model *yvc1 cch1* mutant cell upon extracellular Mg^{2+} depletion, in Fig. 5A we depict the simulated *x*(*t*) response curve for parameter [Mg_{ex}] suddenly decreasing from 1 mM to 0 mM at *t* = 0 and being reset to 1 mM after 90 s (parameter [Ca_{ex}] = 150 mM in this simulation). From this figure, we can see that when *t* = 0, the simulated cytosolic Ca^{2+} level rises quickly (in 15 s) from its original resting level of 0.081 μM to a value of 0.099 μM, almost keeps at that value and recovers quickly to its original resting level of 0.081 μM when *t* = 1.5 min.

Simulated cytosolic Ca^{2+} level (i.e., *x*(*t*)) for *yvc1 cch1* mutant increases upon extracellular Mg^{2+} depletion or extracellular Ca^{2+} challenge. *Panel A*, the simulated *x*(*t*) response curve (the black curve) for parameter [Mg_{ex} ] (the red curve) suddenly decreasing **...**

To further check the behavior of cytosolic Ca^{2+} level of our model *yvc1 cch1* mutant cell upon extracellular Ca^{2+} challenge, in Fig. 5B we depict the simulated *x*(*t*) response curve for parameter [Ca_{ex}] suddenly increasing from 150 mM to 200 mM at *t* = 0 and being reset to 150 mM after 90 s (parameter [Mg_{ex}] = 0 mM in this simulation). From this figure, we can see that the simulated cytosolic Ca^{2+} level rises quickly (in 15 s) from its original resting level of 0.091 μM to a value of 0.125 μM and decreases very gradually to a value of 0.123 μM (at *t* = 1.5 min), then it decreases quickly to its original resting value.

We present experimental data that are consistent with the existence of Mg^{2+}-sensitive Ca^{2+} influx pathways in yeast. Increasing concentrations of extracellular Mg^{2+} increased the concentration of extracellular Ca^{2+} necessary for inhibition of yeast cell growth over a 24 h period. By extrapolation of log-transformed plots of Mg^{2+} concentration versus IC_{50}, we estimate that standard YPD medium effectively contains 0.5 mM Mg^{2+}, in good agreement with the assayed value of 0.7 mM [1]. Increasing concentrations of extra-cellular Mg^{2+} also inhibited Ca^{2+} influx as judged by decreasing rates of aequorin luminescence following a sudden increase of extracellular Ca^{2+}. While this work was in progress, a complementary study also showed that sudden withdrawal of all extracellular Mg^{2+} from synthetic growth medium caused a rapid and reversible influx of Ca^{2+} into yeast cells [49]. An earlier study of 45_{Ca}^{2+} influx demonstrated Mg^{2+} inhibition of a constitutive Ca^{2+} influx pathway [27]. Finally, overexpression of the yeast Alr1 or Alr2 Mg^{2+} transporters – homologs of the bacterial CorA Mg^{2+} transporters –resulted in increased sensitivity to high environmental Ca^{2+} [28]. All these results are consistent with a model where Mg^{2+} competitively inhibits one or more Ca^{2+} influx pathways.

As just mentioned, previously published data demonstrate the existence of a constitutive Mg^{2+}-sensitive Ca^{2+} influx pathway, which we named as Transporter X [27]. From Fig. 2B, we can see that a subtle difference of extracellular Mg^{2+} concentration results in the great difference between the profiles of the black (for [Mg_{ex}] = 0) and blue curve (for [Mg_{ex}] = 0.3 mM). In the case of one transporter model, this can only happen when Transporter X has a very small inhibition constant *K*_{IX} so that a subtle increase of extracellular Mg^{2+} concentration will have great inhibitory effect on the calcium influx (see Eq. (7), please note that *V*_{m} = 0 for one transporter model. The only influx is through Transporter X which is expressed as

Given fixed *V*_{x}, *K*_{x} and [Ca_{ex}], smaller *K*_{IX} can make this influx term more sensitive to the change of [Mg_{ex}]). However, a very small *K*_{IX} will lead to even greater inhibitory effect when [Mg_{ex}] = 3 mM or higher concentration. Thus for one transporter model, it either cannot reproduce the great difference between the profiles of the black and the blue experimental curves in Fig. 2B or cannot reproduce the relatively subtle difference between the profiles of the blue and green curves in Fig. 2B. However, it is possible for two transporters model to reproduce both differences when in addition to Transporter X with an extremely low *K*_{IX} which functions only in the case of extremely low extracellular Mg^{2+}, there is another transporter with a high magnesium inhibition constant whose influx is much less sensitive to the change of extracellular Mg^{2+} than that of Transporter X.

As shown in Fig. 4C, simulations using just one Mg^{2+}-sensitive Ca^{2+} influx transporter poorly fit the experimental data because the best fit of one Mg^{2+}-sensitive transporter model (see Fig. 4C) using hybrid optimization algorithm cannot reproduce the great difference between the black and blue experimental curves shown in Fig. 2B whereas simulations assuming two Mg^{2+}-sensitive Ca^{2+} influx transporters (termed transporters X and M with very distinct inhibition constants: *K*_{IX} = 3.51 × 10^{−6} μM and *K*_{IM} = 149.18 mM) can closely fit the experimental data (see Fig. 4A and B). All these results confirm the theoretical analysis in the previous paragraph and our simulation results strongly suggest the existence of a new transporter named Channel M on the plasma membrane of *yvc1 cch1* mutant. As shown in Fig. 3B, for [Mg_{ex}] = 0 (see the black curve in Fig. 3B), both transporters M and X make considerable contribution to the total Ca^{2+} influx whereas for relatively high [Mg_{ex}] (see the rest curves in Fig. 3B) the flux contribution of Transporter X becomes negligible. The strange bending shown in Fig. 3A appears because of the opening of Transporter X in the case of extremely low [Mg_{ex}].

A mathematical model of Transporter M and other components of the system was constructed here. By comparison of simulated response curves in Fig. 4A with corresponding experimental curve in Fig. 2B for *yvc1 cch1* mutant, we can see that the numerical simulations of our model can quantitatively reproduce the main characteristics of the experimental results such as low levels of extracellular Mg^{2+} can slow Ca^{2+} influx and diminish cytosolic free Ca^{2+} elevation. The model assumed that Transporter M has very low affinity for Ca^{2+} (*K*_{m} = 505.43 mM), competitive inhibition by extracellular Mg^{2+} (*K*_{IM} = 149.18 mM), and rapid feedback inhibition by intracellular Ca^{2+}. The mechanism of feedback was found to be independent of calcineurin but otherwise its components remain wholly unknown. The relatively low affinity for Ca^{2+} relative to Mg^{2+} also suggests that Transporter M may function primarily as a Mg^{2+} transporter in physiological conditions. If so, the cellular response to high extracellular Ca^{2+} may include Mg^{2+} starvation in addition to Ca^{2+} influx, as suggested recently [49]. It is very likely that both transporters M and X are homo/hetero-oligomers of Alr1, Alr2 and Mnr2—uniporters primarily involved in Mg^{2+} uptake (note that uniporter is an integral membrane protein that is involved in facilitated diffusion whose uptake behavior can be well described with Michaelis–Menton kinetics [2]). A more realistic understanding of Transporter M and Transporter X should be possible after their genes and regulators are identified and characterized.

The computer simulations (see Fig. 4A and B) also reproduced the experimentally determined effects of the vacuolar H^{+}/Ca^{2+} exchanger Vcx1 on cytosolic free Ca^{2+} dynamics. The presence or absence of calcineurin inhibitor had no effect on the aequorin traces in the presence or absence of Vcx1 (data not shown), suggesting no significant inhibition of Vcx1 by calcineurin within 3 min following Ca^{2+} shock. In long-term growth experiments, calcineurin appears to strongly inhibit Vcx1 function in addition to strongly inducing Pmc1 function [14]. Perhaps these effects of calcineurin can be observed in aequorin experiments performed over longer time scales and used to computationally model the long-term effects of Ca^{2+} on yeast growth (Fig. 2A).

From Fig. 4D, we can see that the shrinkage of our simulated *yvc1 cch1* model cell is quickly accomplished (in less than 1 s). More simulations show that the volume shrinkage rate of our model cell is mainly determined by the reflection coefficient *σ*, the value of which used here (i.e., 0.035) is actually a value for sorbitol obtained by fitting the experimental curve [29]. Although the reflection coefficient *σ* for the current solute is not exactly known, further investigations show that the value of *σ* (in the investigations, we let this value range from 1000 to 0.001) seems to have insignificant influence on our main simulation results shown in Figs. 3 and 4A–C.

Wiesenberger et al. reported in their experimental paper [49] that removal of Mg^{2+} in extracellular medium (with the presence of extracellular Ca^{2+}) resulted in an immediate increase in free cytoplasmic Ca^{2+} and this signal was reversible. As we can see from Fig. 5A, the sudden step decrease of [Mg_{ex}] from 1 mM to 0 mM at *t* = 0 incurs an immediate quick rise of simulated cytosolic Ca^{2+} level (i.e., *x*(*t*)) and after parameter [Mg_{ex}] is reset to 1 mM at *t* = 1.5 min, simulated cytosolic Ca^{2+} drops and recovers to its original resting level. Moreover, by comparison of two simulated response curves shown in Fig. 5A and B, we can see that the manner of the behavior of the simulated cytosolic Ca^{2+} level under Mg^{2+} depletion is quite similar to that under Ca^{2+} challenge. All these simulation results show that our model can reproduce (although roughly) the relevant experimental results reported by Wiesenberger et al. [49].

Finally there are still two issues worthy of discussion here. The first issue is about the reversibility of Ca^{2+} sequestration through Vcx1 and Pmc1. Intuitively there should exist a Ca^{2+} efflux from the vacuole into the cytosol. However, this is not the case for yeast cells. Dunn et al. [18] ever measured the rate of Ca^{2+} efflux from yeast vacuoles both *in vitro* and *in vivo*. Their experiments indicated that *in vivo* vacuolar Ca^{2+} efflux is very low (essentially zero). We think that it is more appropriate to use standard Michaelis–Menten kinetics (as we did here) rather than reversible Michaelis–Menten kinetics to describe the uptake behavior of Vcx1 and Pmc1 because the later kinetics will introduce an unreal vacuolar Ca^{2+} efflux. The second issue is about the validity of using Michaelis–Menten kinetics for modeling the uptake behavior of the transporters M and X. It is well-known that Michaelis–Menten kinetics assumes a rapid equilibrium between the enzyme and substrate to form an intermediate complex [17]. So we need to check if the uptake of Ca^{2+} and Mg^{2+} through the transporters is fast enough that it could be considered as steady-state at all times. Ion channels enable rapid (~10^{7} ion s^{−1}) movement of selected ions through pores in biological membranes [5]. Ca^{2+} channel can recognize its substrate and let it permeate within 10–100 ns [30]. Carriers (i.e., transporters) are characterized by turnover numbers that are typically 1000-fold lower than ion channels [26]. This means the average time needed for a Ca^{2+} to pass through the plasma membrane via the help of an ion transporter is within 10–100 μs, which is several order smaller than the time frame (in seconds) of the Ca^{2+} peaks shown in Fig. 2B and C. Moreover, Michaelis–Menten kinetics has been used in a mathematical model for describing the iron uptake behavior of plasma membrane iron transporter in the iron homeostasis system of *Escherichia coli* (see the first term in Eq. (1) in Ref. [43]). So here we think that it is appropriate to use Michaelis–Menten kinetics for modeling the uptake behavior of the transporters M and X.

As mentioned before, the mathematical model presented here is only valid for short-term (about 3 min) following hypertonic Ca^{2+} shock. This model needs to be modified for low extracellular Ca^{2+} (<132 mM) because Eq. (4) is only valid for hypertonic shock and it does not include factors such as turgor pressure which will arise in the case of low extracellular Ca^{2+}. Moreover, for simplicity, this model assumes direct inhibition of Ca^{2+}-bound calmodulin on both two transporters M and X, which is not backed by any experimental data. In the real cells, the relevant feedback regulation pathways may be more complicated. And the relative coarseness of the model accounts for the differences (e.g., the noticeable bias in the downward phase of the peak) shown by the comparison of Fig. 2B with Fig. 4A (also by comparison of Fig. 2C with Fig. 4B). On the other hand, the novelty of the present work lies in that by combining computational and experimental methodology, we detect the existence of a new Mg^{2+}-sensitive Ca^{2+} transporter named as Transporter M on the yeast plasma membrane of *yvc1 cch1* mutant cell working together with previously found Transporter X under hypertonic Ca^{2+} shock. The eventual accomplishment of a complete and accurate mathematical model of dynamic Ca^{2+} signaling networks in yeast will facilitate similar endeavors in all cell types and organisms, but doing so still requires additional mechanistic insights obtained from the fusion of experimental data and mathematical models.

J. Cui sincerely thanks his group leader Prof. P.M.A. Sloot for sustaining support for his research. We would like to thank Dr. Gertien J. Smits (University of Amsterdam) for her critical comments on our work. We are indebted to Prof. Patrick Gervais (Laboratoire de Génie des Procédés Alimentaires et Biotechnologiques) for his advice on volume evolution modeling under hypertonic shock. We thank Dr. Ariel Caride and an anonymous referee for their detailed and valuable comments on the previous version of this paper. J. Cui was funded by the Dutch Science Foundation on his project “Mesoscale simulation paradigms in the silicon cell” and later funded by EU on MORPHEX project. This work was partially supported by grant GM053082 from the National Institute of Health, USA (to KWC).

**Conflict of interest statement**

None.

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