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PLoS One. 2010; 5(10): e13278.
Published online Oct 8, 2010. doi:  10.1371/journal.pone.0013278
PMCID: PMC2951907

Characterization of Membrane Potential Dependency of Mitochondrial Ca2+ Uptake by an Improved Biophysical Model of Mitochondrial Ca2+ Uniporter

Jörg Langowski, Editor

Abstract

Mitochondrial Ca2+ uniporter is the primary influx pathway for Ca2+ into respiring mitochondria, and hence plays a key role in mitochondrial Ca2+ homeostasis. Though the mechanism of extra-matrix Ca2+ dependency of mitochondrial Ca2+ uptake has been well characterized both experimentally and mathematically, the mechanism of membrane potential (ΔΨ) dependency of mitochondrial Ca2+ uptake has not been completely characterized. In this paper, we perform a quantitative reevaluation of a previous biophysical model of mitochondrial Ca2+ uniporter that characterized the possible mechanism of ΔΨ dependency of mitochondrial Ca2+ uptake. Based on a model simulation analysis, we show that model predictions with a variant assumption (Case 2: external and internal Ca2+ binding constants for the uniporter are distinct), that provides the best possible description of the ΔΨ dependency, are highly sensitive to variation in matrix [Ca2+], indicating limitations in the variant assumption (Case 2) in providing physiologically plausible description of the observed ΔΨ dependency. This sensitivity is attributed to negative estimate of a biophysical parameter that characterizes binding of internal Ca2+ to the uniporter. Reparameterization of the model with additional nonnengativity constraints on the biophysical parameters showed that the two variant assumptions (Case 1 and Case 2) are indistinguishable, indicating that the external and internal Ca2+ binding constants for the uniporter may be equal (Case 1). The model predictions in this case are insensitive to variation in matrix [Ca2+] but do not match the ΔΨ dependent data in the domain ΔΨ≤120 mV. To effectively characterize this ΔΨ dependency, we reformulate the ΔΨ dependencies of the rate constants of Ca2+ translocation via the uniporter by exclusively redefining the biophysical parameters associated with the free-energy barrier of Ca2+ translocation based on a generalized, non-linear Goldman-Hodgkin-Katz formulation. This alternate uniporter model has all the characteristics of the previous uniporter model and is also able to characterize the possible mechanisms of both the extra-matrix Ca2+ and ΔΨ dependencies of mitochondrial Ca2+ uptake. In addition, the model is insensitive to variation in matrix [Ca2+], predicting relatively stable physiological operation. The model is critical in developing mechanistic, integrated models of mitochondrial bioenergetics and Ca2+ handling.

Introduction

Mitochondrial Ca2+ uniporter is the primary influx pathway for Ca2+ into respiring mitochondria, and hence is a key regulator of mitochondrial Ca2+. Mitochondrial Ca2+ homeostasis is critical for metabolic regulation, mitochondrial function/dysfunction, and cell physiology/pathophysiology [1][9]. Therefore, a mechanistic characterization of mitochondrial Ca2+ uptake via the uniporter is essential for developing mechanistic, integrated models of mitochondrial bioenergetics and Ca2+ handling that can be helpful in understanding the mechanisms by which Ca2+ plays a role in mediating signaling pathways between cytosol and mitochondria and modulating mitochondrial energy metabolism in health and disease [10], [11].

The kinetics of mitochondrial Ca2+ uptake depends on the catalytic properties of the uniporter and also on the electrochemical gradient of Ca2+ across the inner mitochondrial membrane (IMM), which has been extensively studied both experimentally [12][18] and with the help of mathematical models [10], [11], [19][21]. Though the mechanism of extra-matrix Ca2+ dependency of mitochondrial Ca2+ uptake has been well characterized, the mechanism of membrane potential (ΔΨ) dependency of mitochondrial Ca2+ uptake has not been completely characterized.

In a recent paper [11], we introduced a mechanistic mathematical model of mitochondrial Ca2+ uniporter (presented briefly in Materials S1) that satisfactorily describes the available experimental data on the kinetics of mitochondrial Ca2+ uptake, measured in suspensions of respiring mitochondria isolated from rat hearts and rat livers under various experimental conditions [12], [13], [16]. This model is developed based on a multi-state catalytic binding and interconversion mechanism (Michaelis-Menten kinetics) for carrier-mediated facilitated transport [22], [23], and Eyring's free-energy barrier theory for interconversion and electrodiffusion [22], [24][26]. The model also accounts for possible allosteric, cooperative binding of Ca2+ to the uniporter, as seen experimentally [12], [13]. Therefore, the biophysical formulation, thermodynamic feasibility, and ability to explain a large number of independent experimental data sets are some of the remarkable features of the model [11], compared to the previous models of the uniporter [19][21]. The model was able to characterize the possible mechanisms of both the extra-matrix Ca2+ and ΔΨ dependencies of the uniporter-mediated mitochondrial Ca2+ uptake [12], [13], [16].

In the development of our recent model of the uniporter [11], two different kinetic models (Model 1 or Model 2: fully or partial cooperativity of Ca2+ binding to the uniporter) under two different kinetic assumptions (Case 1 or Case 2: external and internal Ca2+ binding constants for the uniporter are equal or distinct) were formulated to characterize the extra-matrix Ca2+ and ΔΨ dependencies of mitochondrial Ca2+ uptake via the uniporter [12], [13], [16] (see Materials S1). Both the models under both the cases were able to satisfactorily describe the extra-matrix Ca2+ dependent data [12], [13]. However, the models under two different cases provided two significantly different predictions of the ΔΨ dependent data [16], especially in the domain ΔΨ≤120 mV. While the models under Case 1 were not able to simulate the ΔΨ dependent data in the domain ΔΨ≤120 mV, the models under Case 2 were able to satisfactorily reproduce the ΔΨ dependent data in the entire ΔΨ domain for which data were available. Based on these kinetic analyses, Case 2 was determined to be the most plausible representation of the observed ΔΨ dependency of the uniporter-mediated mitochondrial Ca2+ uptake.

The four variant models of the uniporter [11] were parameterized exclusively based on the experimental data [12], [13], [16] in which matrix [Ca2+] was unknown from the measurements. For model parameterization, matrix [Ca2+] was fixed at 250 nM. Although the two variant models under Case 2 were able to adequately describe all the available experimental data with appropriate model perturbations as provided by the experimental protocols, it was unknown whether physiological variation of matrix [Ca2+], as seen in the intact myocyte, have significant impacts on the estimates of model parameters and model predicted trans-matrix Ca2+ fluxes via the uniporter. Therefore, it is important to test the robustness of the estimates of model parameters and model predictions subject to such physiological variation.

In the present paper, we attempt to provide a quantitative reevaluation of our previous model of the uniporter [11]. Based on a model simulation analysis, we show that the two variant model predictions under Case 2 are highly sensitive to variation in matrix [Ca2+] (ranging from 100 nM to 500 nM), suggesting that the model parameter estimates under Case 2 would vary significantly to variation in matrix [Ca2+], and hence can not be robust. This indeed indicates that the Case 2, in which the Ca2+ binding constants for the uniporter at the inside and outside of the IMM are distinct, is physiologically implausible, and hence can not be a feasible representation of the observed ΔΨ dependency of the uniporter-mediated mitochondrial Ca2+ uptake. Furthermore, the Case 2 is associated with negative estimates of the biophysical parameter αx (with αe = 0 fixed) (see Table S1), which is found to be contributing to the high sensitivities of the model predictions to variation in matrix [Ca2+]. To reconcile this issue, we reestimate model parameters subject to the constraint: αe = αx = α≥0, which implies that the Ca2+ binding sites on the uniporter are located at equal distances from the bulk phase on either side of the IMM. This reparameterization shows that the two variant assumptions on the Ca2+ binding to the uniporter (Case 1 and Case 2) are indistinguishable from each other, indicating that the external and internal Ca2+ binding constants for the uniporter may be equal (Case 1). The model predictions in this case are insensitive to variation in matrix [Ca2+], but do not match the ΔΨ dependent data [16] in the domain ΔΨ≤120 mV.

To accurately characterize the ΔΨ dependency of mitochondrial Ca2+ uptake via the uniporter in the entire ΔΨ domain for which data are available [16], we reformulate the ΔΨ dependencies of the rate constants kin and kout of Ca2+ translocation in our previous model of the uniporter [11] by exclusively redefining the biophysical parameters βe and βx associated with the free-energy barrier of Ca2+ translocation based on a generalized, non-linear GHK (Goldman-Hodgkin-Katz) formalism (see Materials and Methods). This alternative uniporter model has all the characteristics of our previous uniporter model [11], and is also able to satisfactorily characterize the possible mechanisms of both the extra-matrix Ca2+ and ΔΨ dependencies of the uniporter-mediated mitochondrial Ca2+ uptake [12], [13], [16]. Furthermore, the model is relatively insensitive to variation in matrix [Ca2+], making the model physiologically plausible.

Results

This section presents the detailed simulation analyses of our previous model of mitochondrial Ca2+ uniporter [11] that describe the sensitivity of the model predicted mitochondrial Ca2+ uptake in response to physiologically realistic variation in matrix [Ca2+] and is used to test the robustness of the estimates of model parameters and model predictions. This section also presents the reparameterization of our previous model of the uniporter [11] and parameterization of the present alternate model of the uniporter subject to the constraint: αe = αx = α≥0 based on the experimental data of Scarpa and coworkers [12], [13] and Gunter and coworkers [16] on the kinetics of Ca2+ fluxes via the uniporter. For the purpose of illustrations, only the fully cooperativity binding model (Model 1) under both the kinetic assumptions (Case 1 and Case 2) is chosen, because both the fully and partial cooperativity binding models (Model 1 and Model 2) are indistinguishable from the available experimental data [12], [13], [16] (see Dash et al. [11]).

The simulation analyses of mitochondrial Ca2+ uptake based on our previous model (Model 1) of mitochondrial Ca2+ uniporter [11] are shown in Figures 1 and and2.2. The upper and lower panels correspond to the simulation analyses for Case 1 and Case 2, while the left, middle, and right panels correspond to the simulation analyses based on the experimental protocols of Scarpa and Graziotti [12], Vinogradov and Scarpa [13], and Wingrove et al. [16], respectively. The model uses the same parameter values as estimated before (see Table S1).

Figure 1
Predicted sensitivities (color maps) of mitochondrial Ca2+ uptakes as functions of extra-matrix [Ca2+] and ΔΨ in response to variation in matrix [Ca2+] using our previous model of ...
Figure 2
Predictions of mitochondrial Ca2+ uptake as a function of extra-matrix [Ca2+] for a range of ΔΨ and as a function ΔΨ for a range of extra-matrix [Ca2+] based on our ...

In the experiments of Scarpa and Graziotti [12] and Vinogradov and Scarpa [13], the initial (or pseudo-steady state) rates of Ca2+ influx via the uniporter were measured in suspensions of energized mitochondria purified from rat hearts and rat livers following additions of varying levels of extra-matrix Ca2+ (with extra-matrix Mg2+ fixed at 5 mM and 2 mM, respectively) (Figure 1 (A,D) and 1 (B,E)). In the experiments of Wingrove et al. [16], the initial (or pseudo-steady state) rates of Ca2+ influx via the uniporter were measured as a function of ΔΨ in suspensions of energized mitochondria purified from rat livers with three different levels of extra-matrix Ca2+ ([Ca2+]e = 0.5 µM, 1.0 µM, and 1.5 µM; [Mg2+]e = 0 mM) (Figure 1 (C,F)); ΔΨ was varied by adding varying levels of malonate to the extra-matrix buffer medium. In these experiments, matrix [Ca2+] was fairly unknown. Our previous model of the uniporter [11] was parameterized based on these experimental data with a fixed matrix [Ca2+] of 250 nM. Figure 1 illustrates the effects of physiological variation of matrix [Ca2+] on the estimates of model parameters and model predicted Ca2+ fluxes via the uniporter.

Specifically, Figure 1 shows the model predicted sensitivities of Ca2+ fluxes via the uniporter (lines) as functions of extra-matrix [Ca2+] (ΔΨ = 190 mV) and ΔΨ ([Ca2+]e = 0.5 µM, 1.0 µM, and 1.5 µM) over a range of matrix [Ca2+] along with the experimental data [12], [13], [16] (points). In these simulations, matrix [Ca2+] was varied from 100 nM to 500 nM. The simulations corresponding to the model fits to the data are based on matrix [Ca2+] of 250 nM. Figure 2 shows the model predicted Ca2+ fluxes via the uniporter as a function of extra-matrix [Ca2+] for a range of ΔΨ (100 mV to 210 mV) and as a function of ΔΨ for a range of extra-matrix [Ca2+] (10 µM to 150 µM), with matrix [Ca2+] fixed at 250 nM.

It is apparent from the model simulation analyses in Figure 1 that though the model under Case 2 with matrix [Ca2+] fixed at 250 nM is able to fit well to all of the available experimental data [12], [13], [16] with suitable model perturbations as provided by the experimental protocols, the model predictions under this case are extremely sensitive to variation in matrix [Ca2+] (Figure 1 (D–F): lower panel). In contrast, the model predictions under Case 1, although do not fit well to the ΔΨ dependent data in the range ΔΨ≤120 mV, are insensitive to variation in matrix [Ca2+] (Figure 1 (A–C): upper panel). It is also observed from Figure 2 that the Ca2+ uptake profiles under Case 2 have stiff gradients with respect to ΔΨ and reach saturation for a lower level of extra-matrix [Ca2+], compared to that under Case 1. Note that the extra-matrix [Ca2+] is in µM range, while matrix [Ca2+] is in nM range. Therefore, with a positive ΔΨ (ΔΨ = Ψe−Ψx = outside potential – inside potential; Ψe is positive and Ψx is negative) (i.e., with a high electrochemical gradient of Ca2+ from the extra-matrix to matrix space), it is unlikely that physiological variation of matrix [Ca2+] would have any appreciable effects on the experimental measurements and model predictions on Ca2+ fluxes via the uniporter as well as on the estimates of the uniporter model parameters. Furthermore, it is unlikely that in the experiments of Scarpa and colleagues [12], [13] and Gunter and colleagues [16], matrix [Ca2+] would have been precisely maintained at 250 nM. Therefore, the present model simulation analyses suggest that the model parameter estimates under Case 2 would vary considerably with variation in matrix [Ca2+] as well as with different initial guesses for the parameters, compared to that under Case 1. In the other words, the model parameter estimates under Case 2 would be ambiguous and not unique (robust), and are expected to be different for different matrix [Ca2+] and different initial guesses for the parameters. Given matrix [Ca2+], the initial guesses for the parameters need to be close to the optimal parameter estimates for the optimization algorithm to converge to the optimal parameter estimates. In this case, the sensitivities of the model to variations in matrix [Ca2+] would also be different for different model parameter estimates with similar fittings of the model to the experimental data.

As shown in Table S1, Case 2 is associated with negative estimates of the biophysical parameter αx (with αe = 0 fixed), which is found to be contributing to the high sensitivities of the model predictions to variation in matrix [Ca2+] and stiff gradients of Ca2+ uptake profiles to variation in ΔΨ. The Ca2+ uptake profiles under Case 2 attaining saturation for a lower level of extra-matrix [Ca2+] is attributed to the lower estimates of An external file that holds a picture, illustration, etc.
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Object name is pone.0013278.e002.jpg parameters that characterize the binding of [Ca2+] to the uniporter (see Table S1). To reconcile this issue, we reestimate our previous uniporter model parameters with an additional constraint: αe = αx = α≥0, an assumption that implies that Ca2+ binding sites on the uniporter are located at equal distances from the bulk phase on either side of the IMM. With this constraint, four unknown parameters were estimated for Case 1, while five unknown parameters were estimated for Case 2, using the two kinetic and thermodynamic constraints of Eq. (S7), as in our previous paper on the uniporter [11] (see Materials S1). Here, we follow a three-step modular approach to reparameterize our previous uniporter model. In the first step, the binding constants An external file that holds a picture, illustration, etc.
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Based on this model reparameterization, Case 2 provides multiple estimates of the kinetic parameters An external file that holds a picture, illustration, etc.
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Object name is pone.0013278.e018.jpg (see Table 1), all giving exactly the same fittings of the model to the data as for Case 1 (not shown, see below). In this case, the sensitivities of the least-square error to these parameters are extremely low, compared to the other parameters. The biophysical parameters were uniquely estimated as αe = αx = α≈0, βe≈0.113, and βx≈0.887. The new model fittings to the data for both the cases (Case 1 and Case 2) are exactly same as those shown in Figure 1 (A–C; upper panel), and hence are not shown here again; the new kinetic parameter estimates are also of comparable order of magnitudes for both the cases (see Table 1). Therefore, Case 2 is unidentifiable as a distinct case and is indistinguishable from Case 1. These model simulation analyses merely suggest that Case 2 in which the external and internal Ca2+ binding constants for the uniporter were assumed to be distinct (An external file that holds a picture, illustration, etc.
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Table 1
Reestimated parameter values for our previous models of mitochondrial Ca2+ uniporter with additional constraint: αe = αx = α≥0.

The simulation analyses of mitochondrial Ca2+ uptake based on our present alternative model (Model 1) of mitochondrial Ca2+ uniporter, in which An external file that holds a picture, illustration, etc.
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Object name is pone.0013278.e050.jpg, αe = αx = α, and βe and βx are functions of ΔΨ and nH (see Eq. 7), are shown in Figures 3 and and4.4. For model parameterization, a similar modular approach is used as described above. Specifically, the extra-matrix Ca2+ dependent kinetic data of Scarpa and coworkers [12], [13] are first used to get an estimate of the binding constant An external file that holds a picture, illustration, etc.
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Object name is pone.0013278.e056.jpg for the data of Scarpa and coworkers [12], [13] is estimated. This approach enables us to obtain a unique and robust set of parameters for our alternate model of the uniporter (see Table 2).

Figure 3
Predicted sensitivities (color maps) of mitochondrial Ca2+ uptake as functions of extra-matrix [Ca2+] and ΔΨ in response to variation in matrix [Ca2+] using our present alternate ...
Figure 4
Predictions of mitochondrial Ca2+ uptake as a function of extra-matrix [Ca2+] for a range of ΔΨ and as a function ΔΨ for a range of extra-matrix [Ca2+] based on our ...
Table 2
Estimated parameter values for our present models of mitochondrial Ca2+ uniporter, in which An external file that holds a picture, illustration, etc.
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Figure 3 depicts the model predicted sensitivities of Ca2+ fluxes via the uniporter (lines) as functions of extra-matrix [Ca2+] (ΔΨ = 190 mV) and ΔΨ ([Ca2+]e = 0.5 µM, 1.0 µM, and 1.5 µM) with the variation in matrix [Ca2+] (100 nM to 500 nM) and their comparisons to the available experimental data [12], [13], [16] (points), obtained with matrix [Ca2+] = 250 nM. Figure 4 depicts the model predicted Ca2+ fluxes via the uniporter as a function of extra-matrix [Ca2+] for a range of ΔΨ (100 mV to 210 mV) and as a function of ΔΨ for a range of extra-matrix [Ca2+] (10 µM to 150 µM), obtained with matrix [Ca2+] = 250 nM. The left, middle, and right plots correspond to the model simulation analyses based on the experimental protocols of Scarpa and Graziotti [12], Vinogradov and Scarpa [13], and Wingrove et al. [16], respectively.

The model simulation analyses in Figure 3 demonstrate that our present alternative model of the uniporter is able to match all the available experimental data [12], [13], [16] on the kinetics of both the extra-matrix Ca2+ and ΔΨ dependencies of mitochondrial Ca2+ uptake via the uniporter in the entire ranges of extra-matrix [Ca2+] and ΔΨ for which data were available. In addition, this alternate uniporter model is insensitive to variation in matrix [Ca2+], making the model physiologically plausible. This characteristic of the model helps provide unique and accurate estimates of the model parameters with different matrix [Ca2+]. It is observed from Figure 4 that the Ca2+ uptake profiles, obtained from the present alternate uniporter model, do not have stiff gradients with respect to ΔΨ, and reach saturation for a higher level of extra-matrix [Ca2+], comparable to that obtained under Case 1 of our previous uniporter model (see Figure 2 (A–C, upper panel), but unlike to that obtained under Case 2 of our previous uniporter model (Figure 2 (D–F), lower panel). The estimates of the Michaelis-Menten kinetic parameters An external file that holds a picture, illustration, etc.
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The fittings of Model 1 (Case 1) to the extra-matrix Ca2+ dependent data of Scarpa and Graziotti [12] from cardiac mitochondria and Vinogradov and Scarpa [13] from liver mitochondria provides the estimates An external file that holds a picture, illustration, etc.
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Object name is pone.0013278.e074.jpg = 0.0142 nmol/mg/sec, respectively (see Table 2). These differences in the estimates of the kinetic parameters may be attributed towards the fact that the data are from two different mitochondrial preparations and two different experimental protocols (e.g., the differences in the amount of Mg2+ present in the two experimental buffer mediums, which is known to compete with Ca2+ for transport into mitochondria via the uniporter, and hence inhibits mitochondrial Ca2+ uptake [12][15], [17]). Analysis of these kinetic data with Model 2 (Case 1) showed that the model parameter values are readjusted to provide similar fits of the model to the three independent experimental data sets (see Table 2).

The ΔΨ dependencies of the biophysical parameters βe and βx that characterize the ΔΨ dependent factor E(ΔΨ) in the uniporter flux expressions (see Eqs. S8, 5, and 6), and hence the ΔΨ dependency of mitochondrial Ca2+ uptake via the uniporter [16], are demonstrated in Figure 5. The solid lines are based on our present alternate model of the uniporter in which βe and βx are analytical functions of ΔΨ (Eq. 7), while the dotted lines are based on our previous model of the uniporter [11] in which βe and βx are constants and are numerically estimated. The results show that βe and βx based on the previous model abruptly change their respective values at ΔΨ = 0, resulting in non-differentiability of the factor E(ΔΨ) at ΔΨ = 0. In contrast, βe, βx, and E(ΔΨ) based on the present model are smooth functions of ΔΨ. The ΔΨ dependent factor E(ΔΨ) in the present model has a larger spread, resulting in a better fit of the model to the ΔΨ dependent data [16]. Also the two ΔΨ dependent factors differ in the domain ΔΨ≤120 mV, describing the discrepancy of the fitting of the two models to the ΔΨ dependent data [16] in this domain.

Figure 5
Variation of biophysical parameters βe and βx as functions of ΔΨ and the corresponding ΔΨ dependent factors in the uniporter flux expressions as functions of βe and βx.

Discussion

The major contributions of the present paper lies in the improvements of our previous biophysical model of mitochondrial Ca2+ uniporter [11], which is the primary influx pathway for Ca2+ into energized (respiring) mitochondria, and hence plays an important role in mitochondrial Ca2+ homeostasis. Specifically, the present paper provides an alternate (improved) biophysical model of the uniporter that overcomes the limitations of our previous uniporter model by mechanistically recharacterizing the membrane potential (ΔΨ) dependency of mitochondrial Ca2+ uptake via the uniporter. This alternate uniporter model is able to accurately characterize the possible mechanisms of both the extra-matrix Ca2+ and ΔΨ dependencies of the uniporter-mediated mitochondrial Ca2+ uptake [12], [13], [16]. This model along with our recently developed biophysical model of mitochondrial Na+/Ca2+ exchanger [27] is important in developing mechanistic, integrated models of mitochondrial bioenergetics and Ca2+ handling that can be helpful in understanding the mechanisms by which Ca2+ plays a role in mediating signaling pathways between cytosol and mitochondria and modulating mitochondrial energy metabolism in health and disease.

Our previous model of the uniporter [11] was developed based on a multi-state catalytic binding and interconversion mechanism (Michaelis-Menten kinetics) for carrier-mediated facilitated transport [22], [23], combined with Eyring's free-energy barrier theory for interconversion and electrodiffusion (Ca2+ translocation) [22], [24][26]. The model also accounts for possible allosteric cooperative binding of Ca2+ to the uniporter, as depicted experimentally [12], [13]. The model was parameterized based on comparisons of model simulated outputs under various kinetic assumptions (Model 1 or Model 2: fully or partial cooperative binding of Ca2+ to the uniporter; Case 1 or Case 2: external and internal Ca2+ binding constants for the uniporter are equal or distinct) to several independent experimental data sets from the literature on the kinetics of Ca2+ fluxes via the uniporter [12], [13], [16], measured in suspensions of respiring mitochondria purified from rat hearts and rat livers under varying experimental conditions. The model was able to adequately describe the extra-matrix Ca2+ dependent data of Scarpa and coworkers [12], [13] as well as the ΔΨ dependent data of Gunter and coworkers [16] with the assumption that the dissociation constants associated with the binding of external and internal Ca2+ to the uniporter are distinct (Case 2). Therefore, the mechanistic formulation, thermodynamic feasibility, and ability to describe a large number of independent experimental data sets are some of the notable features of the model [11], compared to the previous models of the uniporter [19][21]. Since the present alternate uniporter model is developed from the previous uniporter model by exclusively reformulating the ΔΨ dependencies of the rate constants of Ca2+ translocation, the present model has all the characteristics of the previous model.

In the experimental studies of Scarpa and colleagues [12], [13] and Gunter and colleagues [16], matrix [Ca2+] was not measured. Our two variant models of the uniporter under two different cases [11] were parameterized entirely based on these experimental data [12], [13], [16] with a fixed matrix [Ca2+] of 250 nM. Although both the variant models under Case 2 were able to satisfactorily explain all the available experimental data with suitable model perturbations as provided by the experimental protocols, it was not realized whether physiological variation of matrix [Ca2+], as observed in the intact myocyte, will have any appreciable effect on the estimates of model parameters and model predicted trans-matrix Ca2+ fluxes via the uniporter. The present study provides a quantitative reevaluation of the previous uniporter model [11] to test the robustness of the estimates of model parameters and model predictions subject to physiologically reasonable variation in matrix [Ca2+] ranging from 100 nM to 500 nM. Based on this model simulation analysis, it is found that the two variant model predictions under Case 2 are highly sensitive to variation in matrix [Ca2+] (Figure 1 (D–F): lower panel), signifying that the model parameter estimates under Case 2 would vary considerably to variation in matrix [Ca2+], and hence can not be robust. In addition, Case 2 was associated negative estimates of the biophysical parameter αx (with αe = 0 fixed) (Table S1), attributing to the high sensitivities of the model predictions to variation in matrix [Ca2+] (Figure 1(D–F): lower panel) and stiff gradients of Ca2+ uptake profiles to variation in ΔΨ (Figure 2 (D–F): lower panel). The reparameterization of the model subject to the constraint: αe = αx = α≥0 showed that Case 2 is unidentifiable as a distinct case and is indistinguishable from Case 1.

In summary, the reevaluation of our previous model of mitochondrial Ca2+ uniporter [11] simply suggest that Case 2 in which the external and internal Ca2+ binding constants for the uniporter were assumed distinct (An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e075.jpgAn external file that holds a picture, illustration, etc.
Object name is pone.0013278.e076.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e077.jpgAn external file that holds a picture, illustration, etc.
Object name is pone.0013278.e078.jpg) is unacceptable as a possible explanation for the observed ΔΨ dependency of mitochondrial Ca2+ uptake via the uniporter [16]. On the other hand, the external and internal Ca2+ binding constants for the uniporter should be equal (An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e079.jpg = An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e080.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e081.jpg = An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e082.jpg) (Table 2), and the biophysical parameters βe and βx associated with the free-energy barrier of Ca2+ translocation via the uniporter should be dependent on ΔΨ (Eq. B7; Figure 5). The alternate uniporter model based on this revised ΔΨ dependent formulation for βe and βx is shown to satisfactorily reproduce all the available experimental data [12], [13], [16] on the kinetics of both the extra-matrix Ca2+ and ΔΨ dependencies of mitochondrial Ca2+ uptake via the uniporter in the entire ranges of extra-matrix [Ca2+] and ΔΨ for which data were available (Figure 3). In addition, the model is insensitive to variation in matrix [Ca2+], predicting relatively stable physiological operation of the uniporter.

Recently, mitochondria have been recognized as one of the key organelles that actively involves in physiological Ca2+ signaling [6], [8], [28]. Its ability to buffer Ca2+ in distinct region of the cells and maintain spatial Ca2+ concentration low even under strong global Ca2+ mobilization upon cell stimulation is critical for the Ca2+ sensitive signal transduction within the cell [29]. It is also evident that, mitochondria can potentially modulate the nature of intracellular Ca2+ oscillations and waves, generated by the Ca2+ release from the endoplasmic reticulum [30], [31]. While numerous efforts have been made to model intracellular Ca2+ oscillations, the mitochondrial Ca2+ uptake has also been shown to influence significantly the nature of Ca2+ oscillations [32]. Furthermore, in many cell types, the respiring mitochondria remain critical for the activity and maintenance of capacitive Ca2+ entry [33], [34]. For example, in a recent study it has been shown that, the mitochondrial Ca2+ uptake has considerable effect on the STIM1-Orai1-dependent store operated Ca2+ entry into endothelial cells [35]. In this context, the present model of mitochondrial Ca2+ uniporter will be crucial for developing an integrated model of intracellular and mitochondrial Ca2+ handling which can be helpful in understanding many aspects of signal transduction mechanisms.

Materials and Methods

Alternate Model of Mitochondrial Ca2+ Uniporter

This section provides the derivation of the present alternate (improved) biophysical model of mitochondrial Ca2+ uniporter that accurately characterizes the ΔΨ dependency of mitochondrial Ca2+ uptake via the uniporter [16] under Case 1 (An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e083.jpg = An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e084.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e085.jpg = An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e086.jpg) and overcomes the limitations of our previous biophysical model of the uniporter [11]. Specifically, to accurately characterize this ΔΨ dependency under Case 1, we reformulate the ΔΨ dependencies of the rate constants kin and kout of Ca2+ translocation (Eq. S5, Materials S1) in our previous uniporter model [11] by exclusively redefining the biophysical parameters βe and βx associated with the free-energy barrier of Ca2+ translocation based on a generalized, non-linear Goldman-Hodgkin-Katz (GHK) formulation. The expressions for the binding constants Ke and Kx and the equilibrium constants Keq remain the same as in Eqs. (S4) and (S6) of Materials S1. Therefore, the present alternative uniporter model has all the characteristics of our previous uniporter model [11]. We also illustrate that this uniporter model is able to characterize the possible mechanisms of both the extra-matrix Ca2+ and ΔΨ dependencies of the uniporter-mediated mitochondrial Ca2+ uptake [12], [13], [16] under Case 1. Furthermore, this uniporter model is insensitive to variation in matrix [Ca2+], making the model physiologically plausible.

Based on a generalized, non-linear GHK formulation, the rate constants kin and kout of Ca2+ translocation via the uniporter (see Eq. S5) can be expressed in the following form [1], [9]:

equation image
(1)

where An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e088.jpg is an unknown non-linear function to be determined. Substituting Eq. (1) for kin and kout and Eq. (S4) for Ke and Kx into Eqs. (S1–S2), the uniporter flux expression is reduced to

equation image
(2)

where D = D1 for Model 1 (Eq. S9a) and D = D2 for Model 2 (Eq. S9b). In order to derive the functional form of An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e090.jpg, we consider the equilibrium condition for trans-membrane Ca2+ transport via the uniporter An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e091.jpg, which in combination with Eqs. (S3) and (S6) gives:

equation image
(3)

Under Case 1 (An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e093.jpg = An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e094.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e095.jpg = An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e096.jpg), the kinetic constraint of Eq. (3) is automatically satisfied. However, the thermodynamic constraint of Eq. (3) provides multiple solutions for An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e097.jpg. The general solution that satisfies the equilibrium condition for passive Ca2+ transport via the uniporter in the absence of ΔΦ An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e098.jpg is given by

equation image
(4a)

where

equation image
(4b)

is an even function: An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e101.jpg; An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e102.jpg is an arbitrary number to be determined. Thus, the unknown function An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e103.jpg is fully characterized by only one unknown parameter nH, and hence the two rate constants kin and kout in Eq. (1) are fully characterized by only three unknown parameters An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e104.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e105.jpg and nH, in contrast to four unknown parameters An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e106.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e107.jpg, βe and βx in the previous formulation (see Materials S1). In standard GHK formulation (linear, constant field-type approximation) for interconversion and electrodiffusion of the uniporter-2Ca2+ complex, nH = 1, An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e108.jpg, and An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e109.jpg [1], [9]. Substituting Eq. (4) into Eq. (2), the uniporter flux expression can be expressed as

equation image
(5)

where D = D1 for Model 1 (Eq. S9a) and D = D2 for Model 2 (Eq. S9b). By comparing the present uniporter model (Eq. 5) with the previous uniporter model (Eq. S8), we obtain the following functional relationship between the biophysical parameters αe, αx, βe, βx, and nH:

equation image
(6)

Using Eq. (6) along with the thermodynamic constraint: An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e112.jpg, the biophysical parameters βe and βx can be expressed in terms of ΔΦ as

equation image
(7a)
equation image
(7b)

Therefore, the biophysical parameters βe and βx in the present model of the uniporter become functions of ΔΨ, compared to the previous model of the uniporter [11], in which βe and βx were constant with respect to ΔΨ. Furthermore, both the parameters are characterized by only one unknown parameter nH.

Model Parameterization

Both the kinetic models of the uniporter are characterized by seven unknown parameters: An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e115.jpg. These parameters are estimated based on the experimental data of Scarpa and coworkers [12], [13] and Gunter and coworkers [16] on the kinetics of Ca2+ fluxes via the uniporter subject to the constraints: An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e116.jpg = An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e117.jpg = An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e118.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e119.jpg = An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e120.jpg = An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e121.jpg, and αe = αx = α≥0 (Case 1). Thus, only four unknown parameters An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e122.jpg are estimated for both the uniporter models. A least-squares estimation technique is used in multiple steps to fit the model simulated outputs to the experimental data.

equation image
(8)

where Nexp is the number of experiments and Ndata is the number of data points in a particular experiment, An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e124.jpg are the experimental data on Ca2+ uptakes and An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e125.jpg are the corresponding model simulated outputs which depend on the model parameters An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e126.jpg, and An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e127.jpg is the maximum value of An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e128.jpg. A MATLAB function optimizer FMINCON is used to minimize the mean residual error E(An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e129.jpg) to estimate the model parameters An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e130.jpg. The accuracy and robustness of the model fitting to the data for a particular uniporter model is assessed based on the value of the mean residual error E(An external file that holds a picture, illustration, etc.
Object name is pone.0013278.e131.jpg) and its sensitivities to perturbations in the optimal parameter estimates.

Supporting Information

Materials S1

Supporting materials that briefly describe the previous models of mitochondrial Ca2+ uniporter.

(0.14 MB DOC)

Table S1

Estimated parameter values in the previous models of mitochondrial Ca2+ uniporter.

(0.09 MB DOC)

Footnotes

Competing Interests: The authors have declared that no competing interests exist.

Funding: This work was supported by National Institutes of Health grants R01-HL072011 (DAB) and R01-HL095122 (RKD) and American Heart Association grant SDG-0735093N (RKD). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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