A biophysically based mathematical model for the kinetics of mitochondrial Na+-Ca2+ antiporter

Understanding Mechanisms of Secondary Active Transport by Analyzing the Effects of Mutations and Stoichiometry

Biological cells frequently exhibit a so-called secondary active transport by moving various species across their membranes. In this mode of transport, an energetically favorable transmembrane gradient of one type of molecule is used to drive another type of molecule in the energetically unfavorable direction against their gradient. Although it is well established that conformational transitions play a critical role in functioning of transporters, the molecular details of underlying mechanisms remain not well understood. Here, we utilize a recently developed theoretical method to understand better the microscopic picture of secondary active transport. Specifically, we evaluate how mutations in different parts of transporters affect their dynamic properties. In addition, we present a possible explanation on existence of different stoichiometries in the secondary active transport. Our theoretical analysis clarifies several important aspects of complex biological transport phenomena.

Influence of T-Bar on Calcium Concentration Impacting Release Probability

The relation of form and function, namely the impact of the synaptic anatomy on calcium dynamics in the presynaptic bouton, is a major challenge of present (computational) neuroscience at a cellular level. The Drosophila larval neuromuscular junction (NMJ) is a simple model system, which allows studying basic effects in a rather simple way. This synapse harbors several special structures. In particular, in opposite to standard vertebrate synapses, the presynaptic boutons are rather large, and they have several presynaptic zones. In these zones, different types of anatomical structures are present. Some of the zones bear a so-called T-bar, a particular anatomical structure. The geometric form of the T-bar resembles the shape of the letter “T” or a table with one leg. When an action potential arises, calcium influx is triggered. The probability of vesicle docking and neurotransmitter release is superlinearly proportional to the concentration of calcium close to the vesicular release site. It is tempting to assume that the T-bar causes some sort of calcium accumulation and hence triggers a higher release probability and thus enhances neurotransmitter exocytosis. In order to study this influence in a quantitative manner, we constructed a typical T-bar geometry and compared the calcium concentration close to the active zones (AZs). We compared the case of synapses with and without T-bars. Indeed, we found a substantial influence of the T-bar structure on the presynaptic calcium concentrations close to the AZs, indicating that this anatomical structure increases vesicle release probability. Therefore, our study reveals how the T-bar zone implies a strong relation between form and function. Our study answers the question of experimental studies (namely “Wichmann and Sigrist, Journal of neurogenetics 2010”) concerning the sense of the anatomical structure of the T-bar.

Theoretical study of active secondary transport: Unexpected differences in molecular mechanisms for antiporters and symporters

Successful functioning of biological cells relies on efficient translocation of different materials across cellular membranes. An important part of this transportation system is membrane channels that are known as antiporters and symporters. They exploit the energy stored as a trans-membrane gradient of one type of molecules to transport the other types of molecules against their gradients. For symporters, the directions of both fluxes for driving and driven species coincide, while for antiporters, the fluxes move in opposite directions. There are surprising experimental observations that despite differing only by the direction of transport fluxes, the molecular mechanisms of translocation adopted by antiporters and symporters seem to be drastically different. We present chemical-kinetic models to quantitatively investigate this phenomenon. Our theoretical approach allows us to explain why antiporters mostly utilize a single-site transportation when only one molecule of any type might be associated with the channel. At the same time, the transport in symporters requires two molecules of different types to be simultaneously associated with the channel. In addition, we investigate the kinetic constraints and efficiency of symporters and compare them with the same properties of antiporters. Our theoretical analysis clarifies some important physical–chemical features of cellular trans-membrane transport.

Molecular Mechanisms of Active Transport in Antiporters: Kinetic Constraints and Efficiency

A vital role in supporting successful functioning of biological cells is played by membrane channels called antiporters. These channel proteins utilize the concentration gradient of one type of species to move another type of species in the opposite direction and against their concentration gradient. It is believed that antiporters operate via alternating conformational transitions that expose these proteins to different sides of the membrane, and that only thermodynamics controls the activation of these channels. Here we explicitly investigate a chemical-kinetic model of antiporters to argue that there are additional kinetic constraints that need to be satisfied for these channels to be operational. This implies that kinetics and not thermodynamics governs the functioning of antiporters. In addition, the efficiency of antiporters is analyzed and the most optimal operating conditions are discussed. Our theoretical analysis clarifies some important aspects of the molecular mechanisms of biological membrane transport.

Calcium phosphate precipitation inhibits mitochondrial energy metabolism

Early studies have shown that moderate levels of calcium overload can cause lower oxidative phosphorylation rates. However, the mechanistic interpretations of these findings were inadequate. And while the effect of excessive calcium overload on mitochondrial function is well appreciated, there has been little to no reports on the consequences of low to moderate calcium overload. To resolve this inadequacy, mitochondrial function from guinea pig hearts was quantified using several well-established methods including high-resolution respirometry and spectrofluorimetry and analyzed using mathematical modeling. We measured key mitochondrial variables such as respiration, mitochondrial membrane potential, buffer calcium, and substrate effects for a range of mitochondrial calcium loads from near zero to levels approaching mitochondrial permeability transition. In addition, we developed a computer model closely mimicking the experimental conditions and used this model to design experiments capable of eliminating many hypotheses generated from the data analysis. We subsequently performed those experiments and determined why mitochondrial ADP-stimulated respiration is significantly lowered during calcium overload. We found that when calcium phosphate levels, not matrix free calcium, reached sufficient levels, complex I activity is inhibited, and the rate of ATP synthesis is reduced. Our findings suggest that calcium phosphate granules form physical barriers that isolate complex I from NADH, disrupt complex I activity, or destabilize cristae and inhibit NADH-dependent respiration.

Mitochondrial calcium handling has been studied for nearly half a century. As we understand it today, low concentrations (1–10 nmol/mg mitochondria) of calcium exert beneficial effects on energy transduction. And high concentrations (>500 nmol/mg mitochondria) lead to respiratory uncoupling and membrane damage. But relatively little is known about the effect of moderate calcium concentrations (10–500 nmol/mg mitochondria) on mitochondrial function. At these concentrations, mitochondrial membrane integrity remains intact and energized, while ATP synthesis becomes significantly impaired. Prior studies have postulated several possible mechanisms, but the precise consequence of calcium overload on mitochondrial ATP production remained obscure. In this study, we combine experimental and computational approaches to show that calcium phosphate precipitation, as opposed to matrix free calcium, inhibits respiratory function at complex I just enough to limit proton pumping during oxidative phosphorylation and decrease ATP synthesis rates.

Different approaches to modeling analysis of mitochondrial swelling

Mitochondria are critical players involved in both cell life and death through multiple pathways. Structural integrity, metabolism and function of mitochondria are regulated by matrix volume due to physiological changes of ion homeostasis in cellular cytoplasm and mitochondria. Ca²⁺ and K⁺ presumably play a critical role in physiological and pathological swelling of mitochondria when increased uptake (influx)/decreased release (efflux) of these ions enhances osmotic pressure accompanied by high water accumulation in the matrix. Changes in the matrix volume in the physiological range have a stimulatory effect on electron transfer chain and oxidative phosphorylation to satisfy metabolic requirements of the cell. However, excessive matrix swelling associated with the sustained opening of mitochondrial permeability transition pores (PTP) and other PTP-independent mechanisms compromises mitochondrial function and integrity leading to cell death. The mechanisms of transition from reversible (physiological) to irreversible (pathological) swelling of mitochondria remain unknown. Mitochondrial swelling is involved in the pathogenesis of many human diseases such as neurodegenerative and cardiovascular diseases. Therefore, modeling analysis of the swelling process is important for understanding the mechanisms of cell dysfunction. This review attempts to describe the role of mitochondrial swelling in cell life and death and the main mechanisms involved in the maintenance of ion homeostasis and swelling. The review also summarizes and discusses different kinetic models and approaches that can be useful for the development of new models for better simulation and prediction of in vivo mitochondrial swelling.

Interplay Between Intracellular Ca(2+) Oscillations and Ca(2+)-stimulated Mitochondrial Metabolism

Oscillations of cytosolic Ca(2+) concentration are a widespread mode of signalling. Oscillatory spikes rely on repetitive exchanges of Ca(2+) between the endoplasmic reticulum (ER) and the cytosol, due to the regulation of inositol 1,4,5-trisphosphate receptors. Mitochondria also sequester and release Ca(2+), thus affecting Ca(2+) signalling. Mitochondrial Ca(2+) activates key enzymes involved in ATP synthesis. We propose a new integrative model for Ca(2+) signalling and mitochondrial metabolism in electrically non-excitable cells. The model accounts for (1) the phase relationship of the Ca(2+) changes in the cytosol, the ER and mitochondria, (2) the dynamics of mitochondrial metabolites in response to cytosolic Ca(2+) changes, and (3) the impacts of cytosol/mitochondria Ca(2+) exchanges and of mitochondrial metabolism on Ca(2+) oscillations. Simulations predict that as expected, oscillations are slowed down by decreasing the rate of Ca(2+) efflux from mitochondria, but also by decreasing the rate of Ca(2+) influx through the mitochondrial Ca(2+) uniporter (MCU). These predictions were experimentally validated by inhibiting MCU expression. Despite the highly non-linear character of Ca(2+) dynamics and mitochondrial metabolism, bioenergetics were found to be robust with respect to changes in frequency and amplitude of Ca(2+) oscillations.

Sensitivity of rabbit ventricular action potential and Ca²⁺ dynamics to small variations in membrane currents and ion diffusion coefficients

Little is known about how small variations in ionic currents and Ca²⁺ and Na⁺ diffusion coefficients impact action potential and Ca²⁺ dynamics in rabbit ventricular myocytes. We applied sensitivity analysis to quantify the sensitivity of Shannon et al. model (Biophys. J., 2004) to 5%-10% changes in currents conductance, channels distribution, and ion diffusion in rabbit ventricular cells. We found that action potential duration and Ca²⁺ peaks are highly sensitive to 10% increase in L-type Ca²⁺ current; moderately influenced by 10% increase in Na⁺-Ca²⁺ exchanger, Na⁺-K⁺ pump, rapid delayed and slow transient outward K⁺ currents, and Cl⁻ background current; insensitive to 10% increases in all other ionic currents and sarcoplasmic reticulum Ca²⁺ fluxes. Cell electrical activity is strongly affected by 5% shift of L-type Ca²⁺ channels and Na⁺-Ca²⁺ exchanger in between junctional and submembrane spaces while Ca²⁺-activated Cl⁻-channel redistribution has the modest effect. Small changes in submembrane and cytosolic diffusion coefficients for Ca²⁺, but not in Na⁺ transfer, may alter notably myocyte contraction. Our studies highlight the need for more precise measurements and further extending and testing of the Shannon et al. model. Our results demonstrate usefulness of sensitivity analysis to identify specific knowledge gaps and controversies related to ventricular cell electrophysiology and Ca²⁺ signaling.

Mitochondrial calcium uptake

Calcium (Ca(2+)) uptake into the mitochondrial matrix is critically important to cellular function. As a regulator of matrix Ca(2+) levels, this flux influences energy production and can initiate cell death. If large, this flux could potentially alter intracellular Ca(2+) ([Ca(2+)]i) signals. Despite years of study, fundamental disagreements on the extent and speed of mitochondrial Ca(2+) uptake still exist. Here, we review and quantitatively analyze mitochondrial Ca(2+) uptake fluxes from different tissues and interpret the results with respect to the recently proposed mitochondrial Ca(2+) uniporter (MCU) candidate. This quantitative analysis yields four clear results: (i) under physiological conditions, Ca(2+) influx into the mitochondria via the MCU is small relative to other cytosolic Ca(2+) extrusion pathways; (ii) single MCU conductance is ∼6-7 pS (105 mM [Ca(2+)]), and MCU flux appears to be modulated by [Ca(2+)]i, suggesting Ca(2+) regulation of MCU open probability (P(O)); (iii) in the heart, two features are clear: the number of MCU channels per mitochondrion can be calculated, and MCU probability is low under normal conditions; and (iv) in skeletal muscle and liver cells, uptake per mitochondrion varies in magnitude but total uptake per cell still appears to be modest. Based on our analysis of available quantitative data, we conclude that although Ca(2+) critically regulates mitochondrial function, the mitochondria do not act as a significant dynamic buffer of cytosolic Ca(2+) under physiological conditions. Nevertheless, with prolonged (superphysiological) elevations of [Ca(2+)]i, mitochondrial Ca(2+) uptake can increase 10- to 1,000-fold and begin to shape [Ca(2+)]i dynamics.

Sodium accumulation in SERCA knockout-induced heart failure

In cardiomyocytes, a major decrease in the level of sarco/endoplasmic reticulum Ca(2+) ATPase (SERCA) can severely impair systolic and diastolic functions. In mice with cardiomyocyte-specific conditional excision of the Serca2 gene (SERCA2 KO), end-stage heart failure developed between four and seven weeks after gene deletion combined with [Na(+)](i) elevation and intracellular acidosis. In this study, to investigate the underpinning changes in Ca(2+) dynamics and metabolic homeostasis, we developed data-driven mathematical models of Ca(2+) dynamics in the ventricular myocytes of the control, four-week, and seven-week SERCA2 knockout (KO) mice. The seven-week KO model showed that elevated [Na(+)](i) was due to increased Na(+) influxes through the Na(+)/Ca(2+) exchanger (NCX) and the Na(+)/H(+) exchanger, with the latter exacerbated by intracellular acidosis. Furthermore, NCX upregulation in the seven-week KO model resulted in increased ATP consumption for ion transport. Na(+) accumulation in the SERCA KO due to NCX upregulation and intracellular acidosis potentially play a role in the development of heart failure, by initiating a reinforcing cycle involving: a mismatch between ATP demand and supply; an increasingly compromised metabolism; a decreased pH(i); and, finally, an even greater [Na(+)](i) elevation.

Dynamic buffering of mitochondrial Ca2+ during Ca2+ uptake and Na+-induced Ca2+ release

In cardiac mitochondria, matrix free Ca(2+) ([Ca(2+)]m) is primarily regulated by Ca(2+) uptake and release via the Ca(2+) uniporter (CU) and Na(+)/Ca(2+) exchanger (NCE) as well as by Ca(2+) buffering. Although experimental and computational studies on the CU and NCE dynamics exist, it is not well understood how matrix Ca(2+) buffering affects these dynamics under various Ca(2+) uptake and release conditions, and whether this influences the stoichiometry of the NCE. To elucidate the role of matrix Ca(2+) buffering on the uptake and release of Ca(2+), we monitored Ca(2+) dynamics in isolated mitochondria by measuring both the extra-matrix free [Ca(2+)] ([Ca(2+)]e) and [Ca(2+)]m. A detailed protocol was developed and freshly isolated mitochondria from guinea pig hearts were exposed to five different [CaCl2] followed by ruthenium red and six different [NaCl]. By using the fluorescent probe indo-1, [Ca(2+)]e and [Ca(2+)]m were spectrofluorometrically quantified, and the stoichiometry of the NCE was determined. In addition, we measured NADH, membrane potential, matrix volume and matrix pH to monitor Ca(2+)-induced changes in mitochondrial bioenergetics. Our [Ca(2+)]e and [Ca(2+)]m measurements demonstrate that Ca(2+) uptake and release do not show reciprocal Ca(2+) dynamics in the extra-matrix and matrix compartments. This salient finding is likely caused by a dynamic Ca(2+) buffering system in the matrix compartment. The Na(+)- induced Ca(2+) release demonstrates an electrogenic exchange via the NCE by excluding an electroneutral exchange. Mitochondrial bioenergetics were only transiently affected by Ca(2+) uptake in the presence of large amounts of CaCl2, but not by Na(+)- induced Ca(2+) release.

Modeling the calcium sequestration system in isolated guinea pig cardiac mitochondria

Under high Ca(2+) load conditions, Ca(2+) concentrations in the extra-mitochondrial and mitochondrial compartments do not display reciprocal dynamics. This is due to a paradoxical increase in the mitochondrial Ca(2+) buffering power as the Ca(2+) load increases. Here we develop and characterize a mechanism of the mitochondrial Ca(2+) sequestration system using an experimental data set from isolated guinea pig cardiac mitochondria. The proposed mechanism elucidates this phenomenon and others in a mathematical framework and is integrated into a previously corroborated model of oxidative phosphorylation including the Na(+)/Ca(2+) cycle. The integrated model reproduces the Ca(2+) dynamics observed in both compartments of the isolated mitochondria respiring on pyruvate after a bolus of CaCl2 followed by ruthenium red and a bolus of NaCl. The model reveals why changes in mitochondrial Ca(2+) concentration of Ca(2+) loaded mitochondria appear significantly mitigated relative to the corresponding extra-mitochondrial Ca(2+) concentration changes after Ca(2+) efflux is initiated. The integrated model was corroborated by simulating the set-point phenomenon. The computational results support the conclusion that the Ca(2+) sequestration system is composed of at least two classes of Ca(2+) buffers. The first class represents prototypical Ca(2+) buffering, and the second class encompasses the complex binding events associated with the formation of amorphous calcium phosphate. With the Ca(2+) sequestration system in mitochondria more precisely defined, computer simulations can aid in the development of innovative therapeutics aimed at addressing the myriad of complications that arise due to mitochondrial Ca(2+) overload.

Modeling effects of L-type ca(2+) current and na(+)-ca(2+) exchanger on ca(2+) trigger flux in rabbit myocytes with realistic T-tubule geometries

The transverse tubular system of rabbit ventricular myocytes consists of cell membrane invaginations (t-tubules) that are essential for efficient cardiac excitation-contraction coupling. In this study, we investigate how t-tubule micro-anatomy, L-type Ca²⁺ channel (LCC) clustering, and allosteric activation of Na⁺/Ca²⁺ exchanger by L-type Ca²⁺ current affects intracellular Ca²⁺ dynamics. Our model includes a realistic 3D geometry of a single t-tubule and its surrounding half-sarcomeres for rabbit ventricular myocytes. The effects of spatially distributed membrane ion-transporters (LCC, Na⁺/Ca²⁺ exchanger, sarcolemmal Ca²⁺ pump, and sarcolemmal Ca²⁺ leak), and stationary and mobile Ca²⁺ buffers (troponin C, ATP, calmodulin, and Fluo-3) are also considered. We used a coupled reaction-diffusion system to describe the spatio-temporal concentration profiles of free and buffered intracellular Ca²⁺. We obtained parameters from voltage-clamp protocols of L-type Ca²⁺ current and line-scan recordings of Ca²⁺ concentration profiles in rabbit cells, in which the sarcoplasmic reticulum is disabled. Our model results agree with experimental measurements of global Ca²⁺ transient in myocytes loaded with 50 μM Fluo-3. We found that local Ca²⁺ concentrations within the cytosol and sub-sarcolemma, as well as the local trigger fluxes of Ca²⁺ crossing the cell membrane, are sensitive to details of t-tubule micro-structure and membrane Ca²⁺ flux distribution. The model additionally predicts that local Ca²⁺ trigger fluxes are at least threefold to eightfold higher than the whole-cell Ca²⁺ trigger flux. We found also that the activation of allosteric Ca²⁺-binding sites on the Na⁺/Ca²⁺ exchanger could provide a mechanism for regulating global and local Ca²⁺ trigger fluxes in vivo. Our studies indicate that improved structural and functional models could improve our understanding of the contributions of L-type and Na⁺/Ca²⁺ exchanger fluxes to intracellular Ca²⁺ dynamics.

Characterization of Mg2+ inhibition of mitochondrial Ca2+ uptake by a mechanistic model of mitochondrial Ca2+ uniporter

Ca(2+) is an important regulatory ion and alteration of mitochondrial Ca(2+) homeostasis can lead to cellular dysfunction and apoptosis. Ca(2+) is transported into respiring mitochondria via the Ca(2+) uniporter, which is known to be inhibited by Mg(2+). This uniporter-mediated mitochondrial Ca(2+) transport is also shown to be influenced by inorganic phosphate (Pi). Despite a large number of experimental studies, the kinetic mechanisms associated with the Mg(2+) inhibition and Pi regulation of the uniporter function are not well established. To gain a quantitative understanding of the effects of Mg(2+) and Pi on the uniporter function, we developed here a mathematical model based on known kinetic properties of the uniporter and presumed Mg(2+) inhibition and Pi regulation mechanisms. The model is extended from our previous model of the uniporter that is based on a multistate catalytic binding and interconversion mechanism and Eyring's free energy barrier theory for interconversion. The model satisfactorily describes a wide variety of experimental data sets on the kinetics of mitochondrial Ca(2+) uptake. The model also appropriately depicts the inhibitory effect of Mg(2+) on the uniporter function, in which Ca(2+) uptake is hyperbolic in the absence of Mg(2+) and sigmoid in the presence of Mg(2+). The model suggests a mixed-type inhibition mechanism for Mg(2+) inhibition of the uniporter function. This model is critical for building mechanistic models of mitochondrial bioenergetics and Ca(2+) handling to understand the mechanisms by which Ca(2+) mediates signaling pathways and modulates energy metabolism.

Detailed kinetics and regulation of mammalian 2-oxoglutarate dehydrogenase

Background

Mitochondrial 2-oxoglutarate (α-ketoglutarate) dehydrogenase complex (OGDHC), a key regulatory point of tricarboxylic acid (TCA) cycle, plays vital roles in multiple pathways of energy metabolism and biosynthesis. The catalytic mechanism and allosteric regulation of this large enzyme complex are not fully understood. Here computer simulation is used to test possible catalytic mechanisms and mechanisms of allosteric regulation of the enzyme by nucleotides (ATP, ADP), pH, and metal ion cofactors (Ca²⁺ and Mg²⁺).

Results

A model was developed based on an ordered ter-ter enzyme kinetic mechanism combined with con-formational changes that involve rotation of one lipoic acid between three catalytic sites inside the enzyme complex. The model was parameterized using a large number of kinetic data sets on the activity of OGDHC, and validated by comparison of model predictions to independent data.

Conclusions

The developed model suggests a hybrid rapid-equilibrium ping-pong random mechanism for the kinetics of OGDHC, consistent with previously reported mechanisms, and accurately describes the experimentally observed regulatory effects of cofactors on the OGDHC activity. This analysis provides a single consistent theoretical explanation for a number of apparently contradictory results on the roles of phosphorylation potential, NAD (H) oxidation-reduction state ratio, as well as the regulatory effects of metal ions on ODGHC function.

Simulation of cellular biochemical system kinetics

The goal of realistically and reliably simulating the biochemical processes underlying cellular function is achievable through a systematic approach that makes use of the broadest possible amount of in vitro and in vivo data, and is consistent with all applicable physical chemical theories. Progress will be facilitated by establishing: (1) a concrete self-consistent theoretical foundation for systems simulation; (2) extensive and accurate databases of thermodynamic properties of biochemical reactions; (3) parameterized and validated models of enzyme and transporter catalytic mechanisms that are consistent with physical chemical theoretical foundation; and (4) software tools for integrating all these concepts, data, and models into a cohesive representation of cellular biochemical systems. Ongoing initiatives are laying the groundwork for the broad-based community cooperation that will be necessary to pursue these elements of a strategic infrastructure for systems simulation on a large scale.

Characterization of membrane potential dependency of mitochondrial Ca2+ uptake by an improved biophysical model of mitochondrial Ca2+ uniporter

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.