Simulating calcium influx and free calcium concentrations in yeast

Abstract

Yeast can proliferate in environments containing very high Ca²⁺ primarily due to the activity of vacuolar Ca²⁺ transporters Pmc1 and Vcx1. Yeast mutants lacking these transporters fail to grow in high Ca²⁺ environments, but growth can be restored by small increases in environmental Mg²⁺. Low extracellular Mg²⁺ appeared to competitively inhibit novel Ca²⁺ influx pathways and to diminish the concentration of free Ca²⁺ in the cytoplasm, as judged from the luminescence of the photoprotein aequorin. These Mg²⁺-sensitive Ca²⁺ 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 Ca²⁺ transporters that undergo rapid feedback inhibition in response to elevated cytosolic free Ca²⁺ concentration. Finally, we show that Vcx1 helps return cytosolic Ca²⁺ toward resting levels after shock with high extracellular Ca²⁺ 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 Ca²⁺ transport and signaling in yeast can provide important insights into the dynamics of this complex system.

1. Introduction

In eukaryotic cells, Ca²⁺ 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 Ca²⁺ concentration at very low levels in growing yeast cells (50–200 nM) [31] through the actions of Ca²⁺ pumps and exchangers (see Fig. 1A) [7,12]. A vacuolar calcium ATPase Pmc1 [13,15] and a vacuolar Ca²⁺/H⁺ exchanger Vcx1 [14,32] independently transport cytosolic Ca²⁺ into large vacuoles, the major Ca²⁺ store and sink. Cytosolic Ca²⁺ is also pumped into ER and Golgi through a secretory pathway calcium ATPase termed Pmr1 [44,47], which may also promote Ca²⁺ extrusion from yeast cells [52]. Unlike in the case of plant and animal cells, Ca²⁺ 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 Ca²⁺ can be released into the cytosol through the mechanosensitive Ca²⁺ channel Yvc1, a distant homolog of mammalian TRPC-type Ca²⁺ channels [36], which results in transient elevation of cytosolic Ca²⁺ concentration. A plasma membrane voltage-gated Ca²⁺ channel (VGCC) composed of Cch1 and Mid1 becomes activated in response to depolarization [10], depletion of secretory Ca²⁺ [27], pheromone stimulation [34], hypotonic shock [4], In unstimulated yeast cells growing in standard laboratory conditions, small amounts of extracellular Ca²⁺ enter the cell through unidentified transporters on the plasma membrane [27].

Fig. 1.

Schematic graph of the system and control block diagram. Panel A, A schematic graph of Ca²⁺ 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 under extremely high extracellular Ca²⁺ concentration. Panel B, Control block diagram of our model. In yvc1 cch1 yeast cells, the cytosolic Ca²⁺ influx is through Transporter X and an assumed Transporter M, the cytosolic Ca²⁺ efflux is through Pmc1, Pmr1 and Vcx1. For yeast cells with fixed volume, the cytosolic Ca²⁺ concentration (i.e., x(t)) is the integral of the flux rate difference (i.e., influx rate subtracting efflux rate) divided by the cytosolic volume. The green lines describe the feedback loop: cytosolic Ca²⁺ concentration is sensed by the calmodulin and Ca²⁺-bound calmodulin is assumed to inhibit the activity of both transporters M and X. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

In response to sudden increases in extracellular Ca²⁺, cytosolic free Ca²⁺ 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 Ca²⁺ 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 Ca²⁺ shock. Therefore, the initially complex dynamics of cytosolic Ca²⁺ 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 Ca²⁺-dependent feedback inhibition of Ca²⁺ influx. Though the feedback mechanism and the Ca²⁺ influx transporters all remain to be identified, experimental evidence suggests that the primary Ca²⁺ influx transporters are competitively inhibited by extracellular Mg²⁺. Indeed, the impaired growth of yeast cells in high calcium environments was ameliorated by inclusion of Mg²⁺ salts in the medium and the best fitting mathematical model incorporates two Mg²⁺-sensitive Ca²⁺ influx pathways with different cation affinities. These findings extend a recent report of Ca²⁺ influx pathways in yeast that become (re)activated upon withdrawal of extracellular Mg²⁺ [49] and demonstrate the power of combining experimental and computational methodologies.

2. Methods

2.1. Experimental methods

The ability of extracellular Mg²⁺ to prevent toxicity of extra-cellular Ca²⁺ 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 CaCl₂ and MgCl₂, 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 CaCl₂ causing a 50% inhibition of growth (the IC₅₀) was obtained by fitting the data to the simple sigmoid equation.

Cytosolic free Ca²⁺ 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 MgCl₂, 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 CaCl₂ (300 μL). Output is plotted as relative luminescence units per second (RLU) over time using similar numbers of cells per sample.

2.2. Mathematical modeling

2.2.1. Control block diagram

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 Ca²⁺ (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 Ca²⁺ influx pathways [51], here for simplicity, we just assume that the activity of Transporter M and Transporter X are both directly inhibited by Ca²⁺-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.

2.2.2. Feedback part modeling

Sensing cytosolic Ca²⁺

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 Ca²⁺ [16]. By law of mass action, we can use the following equation to describe the cytosolic Ca²⁺ sensing process under the assumption of strong cooperativity existing among the three active sites [12,23]:

$$m^{\prime }(t)=k_{\text{M}}^{+}([{\text{CaM}}_{\text{total}}]-m(t))x{(t)}^3-k_{\text{M}}^{-}m(t)$$ (1)

where m(t) denotes the concentration of Ca²⁺-bound calmodulin and m′ (t) denotes its change rate, $k_{\text{M}}^{+}$ denotes the forward rate contant, $k_{\text{M}}^{-}$ denotes the backward rate constant, x(t) denotes cytosolic calcium ion concentration and [CaM_total] denotes the total concentration of calmodulin (including Ca²⁺-free and Ca²⁺-bound form).

Once we know the concentration of Ca²⁺-bound calmodulin, we use a putative algebraic expression to model its assumed inhibitory regulation on Transporter M and Transporter X as follows:

$$J_{\text{M}}=\frac{J_{\text{MO}}}{1+k_a\times m(t)}$$ (2)

$$J_{\text{X}}=\frac{J_{\text{XO}}}{1+k_b\times m(t)}$$ (3)

where k_a, k_b denote the feedback control constants. J_M and J_X denote the calcium ion flux through Transporter M and Transporter X, respectively. J_MO and J_XO denote the calcium ion flux through Transporter M and Transporter X in the case of without calmodulin inhibition. Obviously with such simple algebraic expressions the inhibitory relation (i.e., when m(t) rises, the activity of Transporter M and Transporter X will both drop) can be represented.

2.2.3. Volume evolution modeling (under hypertonic shock)

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 CaCl₂ 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]:

$$V^{\prime }(t)=-A(t)L_{\text{p}}\sigma \left({\mathrm{\Pi }}_{\text{ex}}-\frac{{\mathit{RTn}}_{\text{s}}}{V(t)-b}\right)$$ (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 = Π₀ + 3([Ca_ex] + [Mg_ex])RT where Π₀ denotes the initial osmotic pressure of the external medium, [Ca_ex] and [Mg_ex] denote extracellular Ca²⁺ concentration and extracellular Mg²⁺ concentration, respectively (please note that we can calculate A(t) = 4π (3V(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, Π_i(t) = RT(n₀ + x(t))/(V(t) − b) where x(t) denotes cytosolic calcium ion concentration. This means that the initial apparent number of osmotically active molecules n₀ in yeast cell is around 0.636 × 0.4 mol/L × V₀ = 3.82 × 10⁻¹⁴ mol (please note that b is 0.4V₀ and V₀ = 100 μm³ is the initial volume of the yeast cell [20]). As a common knowledge, x(t) is always <1 mM (even under extremely high hypertonic shock) which is small compared with n₀. So we can assume n_s = n₀ + x(t) as a constant.

2.2.4. Protein modeling

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 Mg²⁺ 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:

$$J_{\text{MO}}=\frac{V_{max}\times [{\text{Ca}}_{\text{ex}}]}{K_{\text{m}}(1+[{\text{Mg}}_{\text{ex}}]/K_{\text{IM}})+[{\text{Ca}}_{\text{ex}}]}$$ (5)

where J_MO denotes the uptake rate of Transporter M (without calmodulin inhibition), [Ca_ex] and [Mg_ex] denote extracellular Ca²⁺ concentration and extracellular Mg²⁺ 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 Mg²⁺-sensitive [27].

2.2.5. A concise model

Since now we have built the uptake models for all the five proteins involved in Ca²⁺ 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]:

$$x^{\prime }(t)=f\times \frac{(J_{\text{M}}+J_{\text{X}}-J_{\text{Pmc}1}-J_{\text{Vcx}1}-J_{\text{Pmr}1})-x(t)\text{Vo}{\text{l}}^{\prime }(t)}{\text{Vol}(t)}$$ (6)

where x′(t) denotes the change rate of cytosolic free Ca²⁺ 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., f) can be interpreted as the fraction of ${[{\text{Ca}}^{2+}]}_i^{\text{tot}}$ (i.e., total cytosolic calcium) which is free. Typical measured values for f_i are 0.01–0.05.” In yeast cells, the calcium buffer effect is especially severe (we believe that >99% of total cytosolic calcium is bound).

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

$$x^{\prime }(t)=f\times (\frac{1}{1+k_{a}m(t)}\frac{V_{\text{m}}\times [{\text{Ca}}_{\text{ex}}]}{K_{\text{m}}(1+[{\text{Mg}}_{\text{ex}}]/K_{\text{IM}})+[{\text{Ca}}_{\text{ex}}]}+\frac{1}{1+k_{b}m(t)}\frac{V_{\text{x}}\times [{\text{Ca}}_{\text{ex}}]}{K_{\text{x}}(1+[{\text{Mg}}_{\text{ex}}]/K_{\text{IX}})+[{\text{Ca}}_{\text{ex}}]}-\frac{V_1\times x(t)}{K_1+x(t)}-\frac{V_2\times x(t)}{K_2+x(t)}-\frac{V_3\times x(t)}{K_3+x(t)}-V^{\prime }(t)x(t))/V(t)$$ (7)

where x(t) denotes cytosolic calcium ion concentration, m(t) denotes the concentration of Ca²⁺-bound calmoculin, K_m, K_x, K₁, K₂ and K₃ are the Michaelis binding constants of Transporter M, Transporter X, Pmc1, Vcx1 and Pmr1, respectively, V_m, V_x, V₁, V₂ and V₃ are the corresponding rate parameters.

Previously published data [27] show that after 2.5 h of incubation, the maximum Ca²⁺ accumulation due to Transporter X is around 390 nmol/10⁹ 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]):

$$\begin{array}{l}{V}_{\text{x}}=V_{max}/10\%=2\times 390\text{nmol}/{10}^9\text{cells}/{\int }_0^{2.5\text{h}}{\text{e}}^{\alpha t}\text{d}t/10\%\\ =3.2\times {10}^{-17}\text{mol}{min}^{-1}\end{array}$$ (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.

Table 1.

Model parameters for which all results (except Fig. 4C) are calculated unless otherwise stated

Parameter	Value	Description
Km	505.43 mM	The binding constant of Transporter M
Kx	500 μM	The binding constant of Transporter X [27]
K1	4.3 μM	The binding constant of Pmc1 [48]
K2	0.1 μM	The binding constant of Pmr1 [44]
K3	25 μM	The binding constant of Vcx1 [35]
KIM	149.18 mM	The Mg2+ inhibition constant of Transporter M
KIX	3.51 × 10−6 μM	The Mg2+ inhibition constant of Transporter X
Vm	1.239 × 10−16 mol min−1	The rate parameter of Transporter M
Vx	3.2 × 10−17 mol min−1	The rate parameter of Transporter X (calculated from [27])
V1	4.459 × 10−17 mol min−1	The rate parameter of Pmc1
V2	4.394 × 10−17 mol min−1	The rate parameter of Pmr1
V3	1.682 × 10−15 mol min−1	The rate parameter of Vcx1
ka	0.0448	The feedback control constant of Transporter M
kb	4.319	The feedback control constant of Transporter X
$k_{\text{M}}^{+}$	3.751 (μM)−3 min−1	The forward rate constant of Eq. (1)
$k_{\text{M}}^{-}$	0.02445 min−1	The backward rate constant of Eq. (1)
[CaMtotal ]	25 μM	The total concentration of calmodulin [12]
Lmax	1.9996 × 109 RLUs	The maximal aequorin luminescence
f	0.007626	The calcium buffer effect constant
Lp	6 × 10−12 m s−1 Pa−1	The hydraulic membrane permeability [29]
σ	0.035	The reflection coefficient
b	0.4V0	The non-osmotic volume of yeast cell [20]
ns	3.82 × 10−14 mol	The apparent number of osmotically active molecules in the cell (calculated from [20])
Π0	0.24 Osm	The initial osmotic pressure of the external medium [20]
V0	100 μm3	The initial volume of the yeast cell [20]

2.2.6. Conversion to aequorin luminescence unit (RLUs)

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):

$$y(x)={\left(\frac{1+7x}{1+118+7x}\right)}^3\times L_{max}$$ (9)

where x denotes Ca²⁺ concentration value in unit μM and L_max denotes the maximal aequorin luminescence.

2.3. Parameter estimation method

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, V₁, V₂, V₃, k_a, k_b, $k_{\text{M}}^{+},k_{\text{M}}^{-}$, Lmax and f) are free parameters which need to be estimated. Here we use a hybrid optimization algorithm to estimate the model parameters by minimizing the difference between the experimental data and the model prediction. The hybrid optimization algorithm consists of a combination of a stochastic evolutionary algorithm [6,50] for the global search to find a good initial guess followed by a local search performed using Levenberg–Marquardt algorithm to refine the quality of the fit.

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:

$$F(\theta )=\sum _{a=1}^2\sum _{m=1}^5\sum _{n=0}^{n=\text{Tn}}{w}_{a,m,n}{(Y_{a,m}^n-F(a,m,n))}^2$$ (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 MgCI₂ is the mth 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 $Y_{a,m}^n$ denotes the concentration for the data type a with extracellular Mg²⁺ = 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.

Fig. 2.

Experimental results. Panel A, the concentrations of CaCl₂ that caused a 50% inhibition of growth (i.e., the IC₅₀ for CaCl₂ ) were shown for pmc1 (filled circles), pmc1 vcx1 (open circles), pmc1 vcx1 cnb1 (filled triangles) and vcx1 cnb1 (open triangles) mutants after 24 h of growth standard YPD culture medium supplemented with 0–32 mM MgCl₂ . Panel B, A 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 (0 mM, 0.3 mM, 3 mM, 30 mM, 90 mM) of MgCl₂ , placed into a tube luminometer, and monitored for luminescence before and after injection of 800 mM CaCl₂ . Panel C, the corresponding aequorin luminescence curves for vcx1 yvc1 cch1 mutant.

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, V₁, V₂, V₃, k_b, $k_{\text{M}}^{+},k_{\text{M}}^{-}$, L_max and f) are free parameters which need to be estimated. Because of the stochastic nature of the optimization strategy used here, lots of fitting were made for both the one transporter and two transporters model. In order to be able to compare both models, the same parameter boundary was given for each parameter and the optimization was run under the same condition (500 iterations of global search and 10,000 iterations of local search). The estimated parameters values together with the remaining parameter values for the best fit of two transporters model are shown in Table 1. The simulation results of the best fit for two transporters model and for one transporter model are shown in Fig. 4A–C.

Fig. 4.

Simulated response curves and volume evolution curves. Panel A, the simulated response curves of the best fit using two transporters model for yvc1 cch1 mutant. The black, blue, green, yellow and red curves depict the simulated 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, respectively (please note that at the same time t = 0 parameter [Ca_ex ] suddenly increases from 0 mM to 800 mM in these simulations). Panel B, the corresponding simulated response curves using two transporters model for vcx1 yvc1 cch1 mutant. Panel C, the simulated response curves of the best fit using one Mg²⁺-sensitive transporter model (i.e., in this case we assume that Transporter X is the sole influx pathway) for yvc1 cch1 mutant. The black, blue, green, yellow and red curves depict the simulated 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, respectively (please note that at the same time t = 0 parameter [Ca_ex ] suddenly increases from 0 mM to 800 mM in these simulations). Panel D, the simulated volume evolution curves using the two 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 ] to various concentrations). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

3. Results

3.1. Mg²⁺ blocks Ca²⁺ toxicity and Ca²⁺ influx in yeast

Yeast mutants lacking the vacuolar Ca²⁺ ATPase Pmc1p grow as well as wild-type yeast strains in standard culture media but they grow much poorer than wild-type when environmental Ca²⁺ is elevated [13], suggesting that elevated cytosolic free Ca²⁺ can be toxic to yeast. However, we noticed that Ca²⁺ toxicity was blocked by a contaminant present in certain batches of agar (data not shown). The contaminant that blocked Ca²⁺ toxicity was traced to Mg²⁺ because (1) the toxicity blocking activity was abolished by addition of the Mg²⁺ chelator EDTA, (2) crude preparations of agar are known to contain millimolar Mg²⁺, and (3) pure MgCl₂ but not NaCl could block Ca²⁺ toxicity in pmc1 mutants in standard culture media.

To investigate the mechanism of Mg²⁺ suppression of Ca²⁺ toxicity, the concentration of CaCl₂ that caused a 50% inhibition of growth (i.e., the IC₅₀ for CaCl₂) was determined for pmc1 mutants after 24 h of growth standard YPD culture medium supplemented with 0–32 mM MgCl₂. Remarkably, for pmc1 knockout mutants the IC₅₀ for CaCl₂ increased with increasing MgCl₂ up to ~8 mM after which the effectiveness of MgCl₂ began to decrease (Fig. 2A, filled circles). Double mutants lacking both Pmc1 and the vacuolar Ca²⁺/H⁺ exchanger Vcx1 behaved similarly except the IC₅₀ values were shifted downward by ~1.6-fold (Fig. 2A, open circles). The sole remaining Ca²⁺ transporter Pmr1, a Ca²⁺/Mn²⁺ 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 CaCl₂ than the pmc1 vcx1 double mutant and MgCl₂ suppressed CaCl₂ 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 IC₅₀ for CaCl₂ as expected and also exhibited MgCl₂ suppression of CaCl₂ toxicity (Fig. 2A, open triangles). These findings demonstrate that MgCl₂ suppresses CaCl₂ toxicity independent of all known Ca²⁺ transporters, which is consistent with the possibility that Mg²⁺ competitively inhibits one or more Ca²⁺ influx pathways.

Two Ca²⁺ channels have been characterized in yeast, the vacuolar Ca²⁺ release channel Yvc1 and the plasma membrane Ca²⁺ influx channel Cch1-Mid1. The yvc1 cch1 pmc1 vcx1 quadruple mutant exhibited similar levels of MgCl₂ suppression of CaCl₂ toxicity as the pmc1 vcx1 double mutants (data not shown), suggesting that Mg²⁺ blocks some other Ca²⁺ influx pathways. To test this possibility directly, cytosolic free Ca²⁺ concentrations were monitored directly after a sudden increase in extracellular CaCl₂ by following luminescence of the Ca²⁺-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 MgCl₂, placed into a tube luminometer, and monitored for luminescence before and after injection of 800 mM CaCl₂. In the absence of added MgCl₂, aequorin luminescence rose quickly after CaCl₂ injection, peaked after ~13.3 s, and declined to a level that was well above the starting level (Fig. 2B). Increasing concentrations of MgCl₂ 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 MgCl₂ 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 Ca²⁺-sequestering transporter in short-term responses to high Ca²⁺ environments and confirm the hypothesis that Mg²⁺ interferes with one or more novel Ca²⁺ influx pathways. A similar hypothesis was proposed recently to explain the increased Ca²⁺ influx and elevated cytosolic Ca²⁺ concentration observed in yeast upon Mg²⁺ sudden withdrawal [49]. The proteins responsible for Mg²⁺-sensitive Ca²⁺ influx have not been identified but the Alr, Alr2, and Mnr2 proteins have been identified as hetero-oligomeric proteins required for Mg²⁺ uptake that also promote sensitivity to high environmental Ca²⁺ [28].

3.2. Computational modeling of Ca²⁺ influx and sequestration

The multiphasic nature and calcineurin-independence of the aequorin luminescence curves suggested complex dynamics of the novel Ca²⁺ influx pathway(s). To help understand these dynamics, a mathematical model was constructed in which the Ca²⁺ 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 Mg²⁺-sensitive Ca²⁺ 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 Mg²⁺-sensitive Ca²⁺ influx transporter poorly fit the experimental data (see Fig. 4C and compare it with Fig. 2B).

3.2.1. Steady state properties

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.

Fig. 3.

Steady state analysis and flux analysis of the system. Panel A, the steady state value of x(t) as a function of parameter [Mg_ex ] for simulated yvc1 cch1 mutant in extracellular medium with high calcium concentration (parameter [Ca_ex ] = 800 mM). The simulated cytosolic calcium level of our model yvc1 cch1 mutant rests within 0.173–0.212 μM (regardless of the initial conditions) as the media Mg²⁺ level (parameter [Mg_ex ]) ranges from 0 mM to 90 mM. Panel B, the simulated flux proportion of Transporter M in the total cytosolic Ca²⁺ influx as a function of t for yvc1 cch1 mutant under hypertonic shock (at t = 0, parameter [Ca_ex ] suddenly increases from 0 mM to 800 mM and at the same time parameter [Mg_ex ] suddenly increases from 0 mM to various concentrations (0 mM, 0.3 mM, 3 mM, 30 mM, 90 mM)). Please note that the blue, green and yellow curves coincide with the red curve in this graph. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

3.2.2. Transients and mutant behavior

By setting reasonable initial conditions (x(0) = 100 nM, m(0) = 0 M and V(0) = 100 μM³) 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 V₃ 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 Mg²⁺-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.

3.2.3. Flux analysis and cell volume evolution

To discriminate the different influx contributions from two Ca²⁺ influx pathways, in Fig. 3B we depict the simulated flux proportion of Transporter M in the total cytosolic Ca²⁺ 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 Ca²⁺ 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.

3.2.4. Extracellular Mg²⁺ depletion and Ca²⁺ challenge

To check the behavior of cytosolic Ca²⁺ level (i.e., x(t)) of our model yvc1 cch1 mutant cell upon extracellular Mg²⁺ 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 Ca²⁺ 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.

Fig. 5.

Simulated cytosolic Ca²⁺ level (i.e., x(t)) for yvc1 cch1 mutant increases upon extracellular Mg²⁺ depletion or extracellular Ca²⁺ challenge. Panel A, the simulated x(t) response curve (the black curve) for parameter [Mg_ex ] (the red curve) suddenly decreasing from 1 mM to 0 mM at t = 0 and being reset to 1 mM after 90 s (please note that in this simulation, parameter [Ca_ex ] = 150 mM). Panel B, the simulated x(t) response curve (the black curve) for parameter [Ca_ex ] (the red curve) suddenly increasing from 150 mM to 200 mM at t = 0 and being reset to 150 mM after 90 s (please note that in this simulation, parameter [Mg_ex ] = 0 mM). The basal level of [Ca_ex ] in these two simulations is set to be 150 mM instead of 2 mM as in Wiesenberger’s paper [49] because our model is only valid for hypertonic shock (extracellular Ca²⁺ >132 mM). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

To further check the behavior of cytosolic Ca²⁺ level of our model yvc1 cch1 mutant cell upon extracellular Ca²⁺ 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 Ca²⁺ 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.

4. Discussion

We present experimental data that are consistent with the existence of Mg²⁺-sensitive Ca²⁺ influx pathways in yeast. Increasing concentrations of extracellular Mg²⁺ increased the concentration of extracellular Ca²⁺ necessary for inhibition of yeast cell growth over a 24 h period. By extrapolation of log-transformed plots of Mg²⁺ concentration versus IC₅₀, we estimate that standard YPD medium effectively contains 0.5 mM Mg²⁺, in good agreement with the assayed value of 0.7 mM [1]. Increasing concentrations of extra-cellular Mg²⁺ also inhibited Ca²⁺ influx as judged by decreasing rates of aequorin luminescence following a sudden increase of extracellular Ca²⁺. While this work was in progress, a complementary study also showed that sudden withdrawal of all extracellular Mg²⁺ from synthetic growth medium caused a rapid and reversible influx of Ca²⁺ into yeast cells [49]. An earlier study of 45_Ca²⁺ influx demonstrated Mg²⁺ inhibition of a constitutive Ca²⁺ influx pathway [27]. Finally, overexpression of the yeast Alr1 or Alr2 Mg²⁺ transporters – homologs of the bacterial CorA Mg²⁺ transporters –resulted in increased sensitivity to high environmental Ca²⁺ [28]. All these results are consistent with a model where Mg²⁺ competitively inhibits one or more Ca²⁺ influx pathways.

As just mentioned, previously published data demonstrate the existence of a constitutive Mg²⁺-sensitive Ca²⁺ influx pathway, which we named as Transporter X [27]. From Fig. 2B, we can see that a subtle difference of extracellular Mg²⁺ 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 Mg²⁺ 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

$$\frac{1}{1+k_{b}m(t)}\frac{V_{\text{x}}\times [{\text{Ca}}_{\text{ex}}]}{K_{\text{x}}(1+[{\text{Mg}}_{\text{ex}}]/K_{\text{IX}})+[{\text{Ca}}_{\text{ex}}]}.$$

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 Mg²⁺, there is another transporter with a high magnesium inhibition constant whose influx is much less sensitive to the change of extracellular Mg²⁺ than that of Transporter X.

As shown in Fig. 4C, simulations using just one Mg²⁺-sensitive Ca²⁺ influx transporter poorly fit the experimental data because the best fit of one Mg²⁺-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 Mg²⁺-sensitive Ca²⁺ 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 Ca²⁺ 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 Mg²⁺ can slow Ca²⁺ influx and diminish cytosolic free Ca²⁺ elevation. The model assumed that Transporter M has very low affinity for Ca²⁺ (K_m = 505.43 mM), competitive inhibition by extracellular Mg²⁺ (K_IM = 149.18 mM), and rapid feedback inhibition by intracellular Ca²⁺. The mechanism of feedback was found to be independent of calcineurin but otherwise its components remain wholly unknown. The relatively low affinity for Ca²⁺ relative to Mg²⁺ also suggests that Transporter M may function primarily as a Mg²⁺ transporter in physiological conditions. If so, the cellular response to high extracellular Ca²⁺ may include Mg²⁺ starvation in addition to Ca²⁺ 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 Mg²⁺ 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⁺/Ca²⁺ exchanger Vcx1 on cytosolic free Ca²⁺ 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 Ca²⁺ 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 Ca²⁺ 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 Mg²⁺ in extracellular medium (with the presence of extracellular Ca²⁺) resulted in an immediate increase in free cytoplasmic Ca²⁺ 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 Ca²⁺ level (i.e., x(t)) and after parameter [Mg_ex] is reset to 1 mM at t = 1.5 min, simulated cytosolic Ca²⁺ 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 Ca²⁺ level under Mg²⁺ depletion is quite similar to that under Ca²⁺ 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 Ca²⁺ sequestration through Vcx1 and Pmc1. Intuitively there should exist a Ca²⁺ 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 Ca²⁺ efflux from yeast vacuoles both in vitro and in vivo. Their experiments indicated that in vivo vacuolar Ca²⁺ 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 Ca²⁺ 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 Ca²⁺ and Mg²⁺ through the transporters is fast enough that it could be considered as steady-state at all times. Ion channels enable rapid (~10⁷ ion s⁻¹) movement of selected ions through pores in biological membranes [5]. Ca²⁺ 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 Ca²⁺ 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 Ca²⁺ 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 Ca²⁺ shock. This model needs to be modified for low extracellular Ca²⁺ (<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 Ca²⁺. Moreover, for simplicity, this model assumes direct inhibition of Ca²⁺-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 Mg²⁺-sensitive Ca²⁺ transporter named as Transporter M on the yeast plasma membrane of yvc1 cch1 mutant cell working together with previously found Transporter X under hypertonic Ca²⁺ shock. The eventual accomplishment of a complete and accurate mathematical model of dynamic Ca²⁺ 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.