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 , 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 or cannot reproduce the relatively subtle difference between the profiles of the blue and green curves in . 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 , 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 ) using hybrid optimization algorithm cannot reproduce the great difference between the black and blue experimental curves shown in 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 ). 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 , for [Mg_{ex}] = 0 (see the black curve in ), 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 ) the flux contribution of Transporter X becomes negligible. The strange bending shown in 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 with corresponding experimental curve in 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 ) 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 ().

From , 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 and .

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 , 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 , 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 . 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 with (also by comparison of with ). 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.