A biophysically based mathematical model for the kinetics of mitochondrial calcium uniporter

Theoretical approaches used in the modelling of reversible and irreversible mitochondrial swelling in vitro

Existing theoretical approaches were considered that allow modelling of mitochondrial swelling (MS) dynamics. Simple phenomenological kinetic models were reviewed. Simple and extended biophysical and bioenergetic models that ignore mechanical properties of inner mitochondrial membrane (IMM), and similar models that include these mechanical properties were also reviewed. Limitations of these models we considered, as regards correct modelling of MS dynamics. It was found that simple phenomenological kinetic models have significant limitations, due to dependence of the kinetic parameter values estimated by fitting of the experimental data on the experimental conditions. Additionally, such simple models provide no understanding of the detailed mechanisms behind the MS dynamics, nor of the dynamics of various system parameters during MS. Thus, biophysical and bioenergetic models ignoring IMM mechanical properties can't be used to model the transition between reversible and irreversible MS. However, simple and extended biophysical models that include IMM mechanical properties allow modelling the transition to irreversible swelling. These latter models are still limited due to significantly simplified description of biochemistry, compared to those of bioenergetic models. Finally, a strategy of model development is proposed, towards correct interpretation of the mitochondrial life cycle, including the effects of MS dynamics.

Substrate- and Calcium-Dependent Differential Regulation of Mitochondrial Oxidative Phosphorylation and Energy Production in the Heart and Kidney

Mitochondrial dehydrogenases are differentially stimulated by Ca2+. Ca2+ has also diverse regulatory effects on mitochondrial transporters and other enzymes. However, the consequences of these regulatory effects on mitochondrial oxidative phosphorylation (OxPhos) and ATP production, and the dependencies of these consequences on respiratory substrates, have not been investigated between the kidney and heart despite the fact that kidney energy requirements are second only to those of the heart. Our objective was, therefore, to elucidate these relationships in isolated mitochondria from the kidney outer medulla (OM) and heart. ADP-induced mitochondrial respiration was measured at different CaCl2 concentrations in the presence of various respiratory substrates, including pyruvate + malate (PM), glutamate + malate (GM), alpha-ketoglutarate + malate (AM), palmitoyl-carnitine + malate (PCM), and succinate + rotenone (SUC + ROT). The results showed that, in both heart and OM mitochondria, and for most complex I substrates, Ca2+ effects are biphasic: small increases in Ca2+ concentration stimulated, while large increases inhibited mitochondrial respiration. Furthermore, significant differences in substrate- and Ca2+-dependent O2 utilization towards ATP production between heart and OM mitochondria were observed. With PM and PCM substrates, Ca2+ showed more prominent stimulatory effects in OM than in heart mitochondria, while with GM and AM substrates, Ca2+ had similar biphasic regulatory effects in both OM and heart mitochondria. In contrast, with complex II substrate SUC + ROT, only inhibitory effects on mitochondrial respiration was observed in both the heart and the OM. We conclude that the regulatory effects of Ca2+ on mitochondrial OxPhos and ATP synthesis are biphasic, substrate-dependent, and tissue-specific.

Reversible and irreversible mitochondrial swelling: Effects of variable mitochondrial activity

An extended biophysical model was obtained by upgrading the previously reported one (Khmelinskii and Makarov, 2021). The upgraded model accommodates variations of solute transport rates through the inner mitochondrial membrane (IMM) within the mitochondrial population, described by a Gaussian distribution. However, the model may be used for any functional form of the distribution. The dynamics of system parameters as predicted by the current model differed from that predicted by the previous model in the same initial conditions (Khmelinskii and Makarov, 2021). The amount of change varied from one parameter to the other, remaining in the 1-38% range. The upgraded model fitted the available experimental data with a better accuracy (R = 0.993) compared to the previous model (R = 0.978) using the same experimental data (Khmelinskii and Makarov, 2021). The fitting procedure also estimated the Gaussian distribution parameters. The new model requires much larger computational resources, but given its higher accuracy, it may be used for better analysis of experimental data and for better prediction of MS dynamics in different initial conditions. Note that activities of individual mitochondria in mitochondrial populations should vary within biological tissues. Thus, the currently upgraded model is a better tool for biological and bio-medical applications. We believe that this model is much better adapted to the analysis of MS dynamics in vivo.

Reversible and irreversible mitochondrial swelling in vitro

Mitochondrial activity as regards ATP production strongly depends on mitochondrial swelling (MS) mode. Therefore, this work analyzes reversible and irreversible MS using a detailed biophysical model. The reported model includes mechanical properties of the inner mitochondrial membrane (IMM). The model describes MS dynamics for spherically symmetric, axisymmetric ellipsoidal and general ellipsoidal mitochondria. Mechanical stretching properties of the IMM were described by a second-rank rigidity tensor. The tensor components were estimated by fitting to the earlier reported results of in vitro experiments. The IMM rigidity constant of ca. 0.008 dyn/nm was obtained for linear deformations. The model also included membrane bending effects, which were small compared to those of membrane stretching. The model was also tested by simulation of the earlier reported experimental data and of the system dynamics at different initial conditions, predicting the system behavior. The transition criteria from reversible to irreversible swelling were determined and tested. The presently developed model is applicable directly to the analysis of in vitro experimental data, while additional improvements are necessary before it could be used to describe mitochondrial swelling in vivo. The reported theoretical model also provides an idea of physically consistent mechanism for the permeability transport pore (PTP) opening, which depends on the IMM stretching stress. In the current study, this idea is discussed briefly, but a detailed theoretical analysis of these ideas will be performed later. The currently developed model provides new understanding of the detailed MS mechanism and of the conditions for the transition between reversible and irreversible MS modes. On the other hand, the current model provides useful mathematical tools, that may be successfully used in mitochondrial biophysics research, and also in other applications, predicting the behavior of mitochondria in different conditions of the surrounding media in vitro or cellular cyto(sarco)plasm in vivo. These mathematical tools are based on real biophysical processes occurring in mitochondria. Thus, we note a significant progress in the theoretical approach, which may be used in real biological systems, compared to the earlier reported models. Significance of this study derives from inclusion of IMM mechanical properties, which directly impact the reversible and irreversible mitochondrial swelling dynamics. Reversible swelling corresponds to reversible IMM deformations, while irreversible swelling corresponds to irreversible deformations, with eventual membrane disruption. The IMM mechanical properties are directly dependent on the membrane biochemical composition and structure. The IMM deformationas are induced by osmotic pressure created by the ionic/neutral solute imbalance between the mitochondrial matrix media and the bulk solution in vitro, or cyto(sarco)plasm in vivo. The novelty of the reported model is in the biophysical mechanism detailing ionic and neutral solute transport for a large number of solutes, which were not taken into account in the earlier reported biophysical models of MS. Therefore, the reported model allows understanding response of mitochondria to the changes of initial concentration(s) of any of the solute(s) included in the model. Note that the values of all of the model parameters and kinetic constants have been estimated and the resulting complete model may be used for quantitative analysis of mitochondrial swelling dynamics in conditions of real in vitro experiments.

A Spatiotemporal Ventricular Myocyte Model Incorporating Mitochondrial Calcium Cycling

Intracellular calcium (Ca2+) cycling dynamics in cardiac myocytes are spatiotemporally generated by stochastic events arising from a spatially distributed network of coupled Ca2+ release units that interact with an intertwined mitochondrial network. In this study, we developed a spatiotemporal ventricular myocyte model that integrates mitochondria-related Ca2+ cycling components into our previously developed ventricular myocyte model consisting of a three-dimensional Ca2+ release unit network. Mathematical formulations of mitochondrial membrane potential, mitochondrial Ca2+ cycling, mitochondrial permeability transition pore stochastic opening and closing, intracellular reactive oxygen species signaling, and oxidized Ca2+/calmodulin-dependent protein kinase II signaling were incorporated into the model. We then used the model to simulate the effects of mitochondrial depolarization on mitochondrial Ca2+ cycling, Ca2+ spark frequency, and Ca2+ amplitude, which agree well with experimental data. We also simulated the effects of the strength of mitochondrial Ca2+ uniporters and their spatial localization on intracellular Ca2+ cycling properties, which substantially affected diastolic and systolic Ca2+ levels in the mitochondria but exhibited only a small effect on sarcoplasmic reticulum and cytosolic Ca2+ levels under normal conditions. We show that mitochondrial depolarization can cause Ca2+ waves and Ca2+ alternans, which agrees with previous experimental observations. We propose that this new, to our knowledge, spatiotemporal ventricular myocyte model, incorporating properties of mitochondrial Ca2+ cycling and reactive-oxygen-species-dependent signaling, will be useful for investigating the effects of mitochondria on intracellular Ca2+ cycling and action potential dynamics in ventricular myocytes.

Slow Ca2+ Efflux by Ca2+/H+ Exchange in Cardiac Mitochondria Is Modulated by Ca2+ Re-uptake via MCU, Extra-Mitochondrial pH, and H+ Pumping by FOF1-ATPase

Mitochondrial (m) Ca²⁺ influx is largely dependent on membrane potential (ΔΨ_m), whereas mCa²⁺ efflux occurs primarily via Ca²⁺ ion exchangers. We probed the kinetics of Ca²⁺/H⁺ exchange (CHE_m) in guinea pig cardiac muscle mitochondria. We tested if net mCa²⁺ flux is altered during a matrix inward H⁺ leak that is dependent on matrix H⁺ pumping by ATP_m hydrolysis at complex V (F_OF₁-ATPase). We measured [Ca²⁺]_m, extra-mitochondrial (e) [Ca²⁺]_e, ΔΨ_m, pH_m, pH_e, NADH, respiration, ADP/ATP ratios, and total [ATP]_m in the presence or absence of protonophore dinitrophenol (DNP), mitochondrial uniporter (MCU) blocker Ru360, and complex V blocker oligomycin (OMN). We proposed that net slow influx/efflux of Ca²⁺ after adding DNP and CaCl₂ is dependent on whether the ΔpH_m gradient is/is not maintained by reciprocal outward H⁺ pumping by complex V. We found that adding CaCl₂ enhanced DNP-induced increases in respiration and decreases in ΔΨ_m while [ATP]_m decreased, ΔpH_m gradient was maintained, and [Ca²⁺]_m continued to increase slowly, indicating net mCa²⁺ influx via MCU. In contrast, with complex V blocked by OMN, adding DNP and CaCl₂ caused larger declines in ΔΨ_m as well as a slow fall in pH_m to near pH_e while [Ca²⁺]_m continued to decrease slowly, indicating net mCa²⁺ efflux in exchange for H⁺ influx (CHE_m) until the ΔpH_m gradient was abolished. The kinetics of slow mCa²⁺ efflux with slow H⁺ influx via CHE_m was also observed at pH_e 6.9 vs. 7.6 by the slow fall in pH_m until ΔpH_m was abolished; if Ca²⁺ reuptake via the MCU was also blocked, mCa²⁺ efflux via CHE_m became more evident. Of the two components of the proton electrochemical gradient, our results indicate that CHE_m activity is driven largely by the ΔpH_m chemical gradient with H⁺ leak, while mCa²⁺ entry via MCU depends largely on the charge gradient ΔΨ_m. A fall in ΔΨ_m with excess mCa²⁺ loading can occur during cardiac cell stress. Cardiac cell injury due to mCa²⁺ overload may be reduced by temporarily inhibiting F_OF₁-ATPase from pumping H⁺ due to ΔΨ_m depolarization. This action would prevent additional slow mCa²⁺ loading via MCU and permit activation of CHE_m to mediate efflux of mCa²⁺.

HIGHLIGHTS-

We examined how slow mitochondrial (m) Ca²⁺ efflux via Ca²⁺/H⁺ exchange (CHE_m) is triggered by matrix acidity after a rapid increase in [Ca²⁺]_m by adding CaCl₂ in the presence of dinitrophenol (DNP) to permit H⁺ influx, and oligomycin (OMN) to block H⁺ pumping via F_OF₁-ATP synthase/ase (complex V).

-

Declines in ΔΨ_m and pH_m after DNP and added CaCl₂ were larger when complex V was blocked.

-

[Ca²⁺]_m slowly increased despite a fall in ΔΨ_m but maintained pH_m when H⁺ pumping by complex V was permitted.

-

[Ca²⁺]_m slowly decreased and external [Ca²⁺]_e increased with declines in both ΔΨ_m and pH_m when complex V was blocked.

-

ATP_m hydrolysis supports a falling pH_m and redox state and promotes a slow increase in [Ca²⁺]_m.

-

After rapid Ca²⁺ influx due to a bolus of CaCl₂, slow mCa²⁺ efflux by CHE_m occurs directly if pH_e is low.

Computational Modeling of In Vitro Swelling of Mitochondria: A Biophysical Approach

Swelling of mitochondria plays an important role in the pathogenesis of human diseases by stimulating mitochondria-mediated cell death through apoptosis, necrosis, and autophagy. Changes in the permeability of the inner mitochondrial membrane (IMM) of ions and other substances induce an increase in the colloid osmotic pressure, leading to matrix swelling. Modeling of mitochondrial swelling is important for simulation and prediction of in vivo events in the cell during oxidative and energy stress. In the present study, we developed a computational model that describes the mechanism of mitochondrial swelling based on osmosis, the rigidity of the IMM, and dynamics of ionic/neutral species. The model describes a new biophysical approach to swelling dynamics, where osmotic pressure created in the matrix is compensated for by the rigidity of the IMM, i.e., osmotic pressure induces membrane deformation, which compensates for the osmotic pressure effect. Thus, the effect is linear and reversible at small membrane deformations, allowing the membrane to restore its normal form. On the other hand, the membrane rigidity drops to zero at large deformations, and the swelling becomes irreversible. As a result, an increased number of dysfunctional mitochondria can activate mitophagy and initiate cell death. Numerical modeling analysis produced results that reasonably describe the experimental data reported earlier.

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.

The Involvement of Mg2+ in Regulation of Cellular and Mitochondrial Functions

Mg²⁺ is an essential mineral with pleotropic impacts on cellular physiology and functions. It acts as a cofactor of several important enzymes, as a regulator of ion channels such as voltage-dependent Ca²⁺ channels and K⁺ channels and on Ca²⁺-binding proteins. In general, Mg²⁺ is considered as the main intracellular antagonist of Ca²⁺, which is an essential secondary messenger initiating or regulating a great number of cellular functions. This review examines the effects of Mg²⁺ on mitochondrial functions with a particular focus on energy metabolism, mitochondrial Ca²⁺ handling, and apoptosis.

Inositol 1,4,5-trisphosphate-mediated sarcoplasmic reticulum-mitochondrial crosstalk influences adenosine triphosphate production via mitochondrial Ca2+ uptake through the mitochondrial ryanodine receptor in cardiac myocytes

AIMS

Elevated levels of inositol 1,4,5-trisphosphate (IP3) in adult cardiac myocytes are typically associated with the development of cardiac hypertrophy, arrhythmias, and heart failure. IP3 enhances intracellular Ca(2+ )release via IP3 receptors (IP3Rs) located at the sarcoplasmic reticulum (SR). We aimed to determine whether IP3-induced Ca(2+ )release affects mitochondrial function and determine the underlying mechanisms.

METHODS AND RESULTS

We compared the effects of IP3Rs- and ryanodine receptors (RyRs)-mediated cytosolic Ca(2+ )elevation achieved by endothelin-1 (ET-1) and isoproterenol (ISO) stimulation, respectively, on mitochondrial Ca(2+ )uptake and adenosine triphosphate (ATP) generation. Both ET-1 and isoproterenol induced an increase in mitochondrial Ca(2+ )(Ca(2 +) m) but only ET-1 led to an increase in ATP concentration. ET-1-induced effects were prevented by cell treatment with the IP3 antagonist 2-aminoethoxydiphenyl borate and absent in myocytes from transgenic mice expressing an IP3 chelating protein (IP3 sponge). Furthermore, ET-1-induced mitochondrial Ca(2+) uptake was insensitive to the mitochondrial Ca(2+ )uniporter inhibitor Ru360, however was attenuated by RyRs type 1 inhibitor dantrolene. Using real-time polymerase chain reaction, we detected the presence of all three isoforms of IP3Rs and RyRs in murine ventricular myocytes with a dominant presence of type 2 isoform for both receptors.

CONCLUSIONS

Stimulation of IP3Rs with ET-1 induces Ca(2+ )release from the SR which is tunnelled to mitochondria via mitochondrial RyR leading to stimulation of mitochondrial ATP production.

Cytoskeleton rotation relocates mitochondria to the immunological synapse and increases calcium signals

Ca²⁺ microdomains and spatially resolved Ca²⁺ signals are highly relevant for cell function. In T cells, local Ca²⁺ signaling at the immunological synapse (IS) is required for downstream effector functions. We present experimental evidence that the relocation of the MTOC towards the IS during polarization drags mitochondria along with the microtubule network. From time-lapse fluorescence microscopy we conclude that mitochondria rotate together with the cytoskeleton towards the IS. We hypothesize that this movement of mitochondria towards the IS together with their functionality of absorption and spatial redistribution of Ca²⁺ is sufficient to significantly increase the cytosolic Ca²⁺ concentration. To test this hypothesis we developed a whole cell model for Ca²⁺ homoeostasis involving specific geometries for mitochondria and use the model to calculate the spatial distribution of Ca²⁺ concentrations within the cell body as a function of the rotation angle and the distance from the IS. We find that an inhomogeneous distribution of PMCA pumps on the cell membrane, in particular an accumulation of PMCA at the IS, increases the global Ca²⁺ concentration and decreases the local Ca²⁺ concentration at the IS with decreasing distance of the MTOC from the IS. Unexpectedly, a change of CRAC/Orai activity is not required to explain the observed Ca²⁺ changes. We conclude that rotation-driven relocation of the MTOC towards the IS together with an accumulation of PMCA pumps at the IS are sufficient to control the observed Ca²⁺ dynamics in T-cells during polarization.

Determination of the catalytic mechanism for mitochondrial malate dehydrogenase

The kinetics of malate dehydrogenase (MDH) catalyzed oxidation/reduction of L-malate/oxaloacetate is pH-dependent due to the proton generated/taken up during the reaction. Previous kinetic studies on the mitochondrial MDH did not yield a consensus kinetic model that explains both substrate and pH dependency of the initial velocity. In this study, we propose, to our knowledge, a new kinetic mechanism to explain kinetic data acquired over a range of pH and substrate concentrations. Progress curves in the forward and reverse reaction directions were obtained under a variety of reactant concentrations to identify associated kinetic parameters. Experiments were conducted at physiologically relevant ionic strength of 0.17 M, pH ranging between 6.5 and 9.0, and at 25 °C. The developed model was built on the prior observation of proton uptake upon binding of NADH to MDH, and that the MDH-catalyzed oxidation of NADH may follow an ordered bi-bi mechanism with NADH/NAD binding to the enzyme first, followed by the binding of oxaloacetate/L-malate. This basic mechanism was expanded to account for additional ionic states to explain the pH dependency of the kinetic behavior, resulting in what we believe to be the first kinetic model explaining both substrate and pH dependency of the reaction velocity.

The calcium uniporter regulates the permeability transition pore in isolated cortical mitochondria

To investigate the influence of the mitochondrial calcium uniporter on the mitochondrial permeability transition pore, the present study observed mitochondrial morphology in cortical neurons isolated from adult rats using transmission electron microscopy, and confirmed the morphology and activity of isolated mitochondria by detecting succinic dehydrogenase and monoamine oxidase, two mitochondrial enzymes. Isolated mitochondria were treated with either ruthenium red, an inhibitor of the uniporter, spermine, an activator of the uniporter, or in combination with cyclosporin A, an inhibitor of the mitochondrial permeability transition pore. Results showed that ruthenium red inhibited CaCl2-induced mitochondrial permeability transition pore opening, spermine enhanced opening, and cyclosporin A attenuated the effects of spermine. Results demonstrated that the mitochondrial calcium uniporter plays a role in regulating the mitochondrial permeability transition pore in mitochondria isolated from the rat brain cortex.

Optimal microdomain crosstalk between endoplasmic reticulum and mitochondria for Ca2+ oscillations

A Ca²⁺ signaling model is proposed to consider the crosstalk of Ca²⁺ ions between endoplasmic reticulum (ER) and mitochondria within microdomains around inositol 1, 4, 5-trisphosphate receptors (IP₃R) and the mitochondrial Ca²⁺ uniporter (MCU). Our model predicts that there is a critical IP₃R-MCU distance at which 50% of the ER-released Ca²⁺ is taken up by mitochondria and that mitochondria modulate Ca²⁺ signals differently when outside of this critical distance. This study highlights the importance of the IP₃R-MCU distance on Ca²⁺ signaling dynamics. The model predicts that when MCU are too closely associated with IP₃Rs, the enhanced mitochondrial Ca²⁺ uptake will produce an increase of cytosolic Ca²⁺ spike amplitude. Notably, the model demonstrates the existence of an optimal IP₃R-MCU distance (30–85 nm) for effective Ca²⁺ transfer and the successful generation of Ca²⁺ signals in healthy cells. We suggest that the space between the inner and outer mitochondria membranes provides a defense mechanism against occurrences of high [Ca²⁺]_Cyt. Our results also hint at a possible pathological mechanism in which abnormally high [Ca²⁺]_Cyt arises when the IP₃R-MCU distance is in excess of the optimal range.

Computational analysis of Ca2+ dynamics in isolated cardiac mitochondria predicts two distinct modes of Ca2+ uptake

Cardiac mitochondria can act as a significant Ca(2+) sink and shape cytosolic Ca(2+) signals affecting various cellular processes, such as energy metabolism and excitation-contraction coupling. However, different mitochondrial Ca(2+) uptake mechanisms are still not well understood. In this study, we analysed recently published Ca(2+) uptake experiments performed on isolated guinea pig cardiac mitochondria using a computer model of mitochondrial bioenergetics and cation handling. The model analyses of the data suggest that the majority of mitochondrial Ca(2+) uptake, at physiological levels of cytosolic Ca(2+) and Mg(2+), occurs through a fast Ca(2+) uptake pathway, which is neither the Ca(2+) uniporter nor the rapid mode of Ca(2+) uptake. This fast Ca(2+) uptake component was explained by including a biophysical model of the ryanodine receptor (RyR) in the computer model. However, the Mg(2+)-dependent enhancement of the RyR adaptation was not evident in this RyR-type channel, in contrast to that of cardiac sarcoplasmic reticulum RyR. The extended computer model is corroborated by simulating an independent experimental dataset, featuring mitochondrial Ca(2+) uptake, egress and sequestration. The model analyses of the two datasets validate the existence of two classes of Ca(2+) buffers that comprise the mitochondrial Ca(2+) sequestration system. The modelling study further indicates that the Ca(2+) buffers respond differentially depending on the source of Ca(2+) uptake. In particular, it suggests that the Class 1 Ca(2+) buffering capacity is auto-regulated by the rate at which Ca(2+) is taken up by mitochondria.

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.

The connection between inner membrane topology and mitochondrial function

The mitochondrial inner membrane has a complex and dynamic structure that plays an important role in the function of this organelle. The internal compartments called cristae are created by processes that are just beginning to be understood. Crista size and morphology influence the internal diffusion of solutes and the surface area of the inner membrane, which is home to critical membrane proteins including ATP synthase and electron transport chain complexes; metabolite and ion transporters including the adenine nucleotide translocase, the calcium uniporter (MCU), and the sodium/calcium exchanger (NCLX); and many more. Here we provide a brief overview of what is known about crista structure and formation, and discuss mitochondrial function in the context of that structure. We also suggest that mathematical modeling of mitochondria that incorporates accurate information about the organelle's internal architecture can lead to a better understanding of its diverse functions. This article is part of a Special Issue entitled 'Calcium Signalling in Heart'.

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 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.

How do control-based approaches enter into biology?

Control is intrinsic to biological organisms, whose cells are in a constant state of sensing and response to numerous external and self-generated stimuli. Diverse means are used to study the complexity through control-based approaches in these cellular systems, including through chemical and genetic manipulations, input-output methodologies, feedback approaches, and feed-forward approaches. We first discuss what happens in control-based approaches when we are not actively examining or manipulating cells. We then present potential methods to determine what the cell is doing during these times and to reverse-engineer the cellular system. Finally, we discuss how we can control the cell's extracellular and intracellular environments, both to probe the response of the cells using defined experimental engineering-based technologies and to anticipate what might be achieved by applying control-based approaches to affect cellular processes. Much work remains to apply simplified control models and develop new technologies to aid researchers in studying and utilizing cellular and molecular processes.

Enhanced charge-independent mitochondrial free Ca(2+) and attenuated ADP-induced NADH oxidation by isoflurane: Implications for cardioprotection

Modulation of mitochondrial free Ca(2+) ([Ca(2+)](m)) is implicated as one of the possible upstream factors that initiates anesthetic-mediated cardioprotection against ischemia-reperfusion (IR) injury. To unravel possible mechanisms by which volatile anesthetics modulate [Ca(2+)](m) and mitochondrial bioenergetics, with implications for cardioprotection, experiments were conducted to spectrofluorometrically measure concentration-dependent effects of isoflurane (0.5, 1, 1.5, 2mM) on the magnitudes and time-courses of [Ca(2+)](m) and mitochondrial redox state (NADH), membrane potential (ΔΨ(m)), respiration, and matrix volume. Isolated mitochondria from rat hearts were energized with 10mM Na(+)- or K(+)-pyruvate/malate (NaPM or KPM) or Na(+)-succinate (NaSuc) followed by additions of isoflurane, 0.5mM CaCl(2) (≈200nM free Ca(2+) with 1mM EGTA buffer), and 250μM ADP. Isoflurane stepwise: (a) increased [Ca(2+)](m) in state 2 with NaPM, but not with KPM substrate, despite an isoflurane-induced slight fall in ΔΨ(m) and a mild matrix expansion, and (b) decreased NADH oxidation, respiration, ΔΨ(m), and matrix volume in state 3, while prolonging the duration of state 3 NADH oxidation, respiration, ΔΨ(m), and matrix contraction with PM substrates. These findings suggest that isoflurane's effects are mediated in part at the mitochondrial level: (1) to enhance the net rate of state 2 Ca(2+) uptake by inhibiting the Na(+)/Ca(2+) exchanger (NCE), independent of changes in ΔΨ(m) and matrix volume, and (2) to decrease the rates of state 3 electron transfer and ADP phosphorylation by inhibiting complex I. These direct effects of isoflurane to increase [Ca(2+)](m), while depressing NCE activity and oxidative phosphorylation, could underlie the mechanisms by which isoflurane provides cardioprotection against IR injury at the mitochondrial level.

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.

Cardiac myocytes and local signaling in nano-domains

It is well known that calcium-induced calcium-release in cardiac myocytes takes place in spatially restricted regions known as dyads, where discrete patches of junctional sarcoplasmic reticulum tightly associate with the t-tubule membrane. The dimensions of a dyad are so small that it contains only a few Ca²⁺ ions at any given time. Ca²⁺ signaling in the dyad is therefore noisy, and dominated by the Brownian motion of Ca²⁺ ions in a potential field. Remarkably, from this complexity emerges the integrated behavior of the myocyte in which, under normal conditions, precise control of Ca²⁺ release and muscle contraction is maintained over the life of the cell. This is but one example of how signal processing within the cardiac myocyte and other cells often occurs in small "nano-domains" where proteins and protein complexes interact at spatial dimensions on the order of ∼1-10 nm and at time-scales on the order of nanoseconds to perform the functions of the cell. In this article, we will review several examples of local signaling in nano-domains, how it contributes to the integrative behavior of the cardiac myocyte, and present computational methods for modeling signal processing within these domains across differing spatio-temporal scales.

The low conductance mitochondrial permeability transition pore confers excitability and CICR wave propagation in a computational model

Mitochondria have long been known to sequester cytosolic Ca(2+) and even to shape intracellular patterns of endoplasmic reticulum-based Ca(2+) signaling. Evidence suggests that the mitochondrial network is an excitable medium which can demonstrate independent Ca(2+) induced Ca(2+) release via the mitochondrial permeability transition. The role of this excitability remains unclear, but mitochondrial Ca(2+) handling appears to be a crucial element in diverse diseases as diabetes, neurodegeneration and cardiac dysfunction that also have bioenergetic components. In this paper, we extend the modular Magnus-Keizer computational model for respiration-driven Ca(2+) handling to include a permeability transition based on a channel-like pore mechanism. We demonstrate both excitability and Ca(2+) wave propagation accompanied by depolarizations qualitatively similar to those reported in cell and isolated mitochondria preparations. These waves depend on the energy state of the mitochondria, as well as other elements of mitochondrial physiology. Our results support the concept that mitochondria can transmit state dependent signals about their function across the mitochondrial network. Our model provides the tools for predictions about the internal physiology that leads to this qualitatively different Ca(2+) excitability seen in mitochondria.

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.

Mitochondrial free [Ca2+] increases during ATP/ADP antiport and ADP phosphorylation: exploration of mechanisms

ADP influx and ADP phosphorylation may alter mitochondrial free [Ca2+] ([Ca2+](m)) and consequently mitochondrial bioenergetics by several postulated mechanisms. We tested how [Ca2+](m) is affected by H2PO4(-) (P(i)), Mg2+, calcium uniporter activity, matrix volume changes, and the bioenergetic state. We measured [Ca2+](m), membrane potential, redox state, matrix volume, pH(m), and O2 consumption in guinea pig heart mitochondria with or without ruthenium red, carboxyatractyloside, or oligomycin, and at several levels of Mg2+ and P(i). Energized mitochondria showed a dose-dependent increase in [Ca2+](m) after adding CaCl2 equivalent to 20, 114, and 485 nM extramatrix free [Ca2+] ([Ca2+](e)); this uptake was attenuated at higher buffer Mg2+. Adding ADP transiently increased [Ca2+](m) up to twofold. The ADP effect on increasing [Ca2+](m) could be partially attributed to matrix contraction, but was little affected by ruthenium red or changes in Mg2+ or P(i). Oligomycin largely reduced the increase in [Ca2+](m) by ADP compared to control, and [Ca2+](m) did not return to baseline. Carboxyatractyloside prevented the ADP-induced [Ca2+](m) increase. Adding CaCl2 had no effect on bioenergetics, except for a small increase in state 2 and state 4 respiration at 485 nM [Ca2+](e). These data suggest that matrix ADP influx and subsequent phosphorylation increase [Ca2+](m) largely due to the interaction of matrix Ca2+ with ATP, ADP, P(i), and cation buffering proteins in the matrix.

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.

Mitochondrial Ca2+ sequestration and precipitation revisited

The ability of mitochondria to sequester and retain divalent cations in the form of precipitates consisting of organic and inorganic moieties has been known for decades. Of these cations, Ca(2+) has emerged as a major player in both signal transduction and cell death mechanisms, and, as a consequence, the importance of mitochondria in these processes was soon recognized. Early studies showed considerable effort in identifying the mechanisms of Ca(2+) sequestration, precipitation and release by uncouplers of oxidative phosphorylation; however, relatively little information was obtained, and these processes were eventually taken for granted. Here, we re-examine: (a) the thermodynamic aspects of mitochondrial Ca(2+) uptake and release, (b) the insufficiently explained effect of uncouplers in inducing mitochondrial Ca(2+) release, (c) the thermodynamic effects of exogenously added adenine nucleotides on mitochondrial Ca(2+) uptake capacity and precipitate formation, and (d) the elusive nature of the Ca(2+) -phosphate precipitates formed in the mitochondrial matrix.

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

Sodium-calcium antiporter is the primary efflux pathway for Ca(2+) in respiring mitochondria, and hence plays an important role in mitochondrial Ca(2+) homeostasis. Although experimental data on the kinetics of Na(+)-Ca(2+) antiporter are available, the structure and composition of its functional unit and kinetic mechanisms associated with the Na(+)-Ca(2+) exchange (including the stoichiometry) remains unclear. To gain a quantitative understanding of mitochondrial Ca(2+) homeostasis, a biophysical model of Na(+)-Ca(2+) antiporter is introduced that is thermodynamically balanced and satisfactorily describes a number of independent data sets under a variety of experimental conditions. The model is based on a multistate catalytic binding mechanism for carrier-mediated facilitated transport and Eyring's free energy barrier theory for interconversion and electrodiffusion. The model predicts the activating effect of membrane potential on the antiporter function for a 3Na(+):1Ca(2+) electrogenic exchange as well as the inhibitory effects of both high and low pH seen experimentally. The model is useful for further development of mechanistic integrated models of mitochondrial Ca(2+) handling and bioenergetics to understand the mechanisms by which Ca(2+) plays a role in mitochondrial signaling pathways and energy metabolism.

Cardiac cell modelling: observations from the heart of the cardiac physiome project

In this manuscript we review the state of cardiac cell modelling in the context of international initiatives such as the IUPS Physiome and Virtual Physiological Human Projects, which aim to integrate computational models across scales and physics. In particular we focus on the relationship between experimental data and model parameterisation across a range of model types and cellular physiological systems. Finally, in the context of parameter identification and model reuse within the Cardiac Physiome, we suggest some future priority areas for this field.