Conceived and designed the experiments: JNB GTB AER. Performed the experiments: JNB GTB AER. Analyzed the data: JNB GTB AER. Contributed reagents/materials/analysis tools: JNB GTB AER. Wrote the paper: JNB GTB AER.
The authors have declared that no competing interests exist.
Mathematical models of mitochondrial bioenergetics provide powerful analytical tools to help interpret experimental data and facilitate experimental design for elucidating the supporting biochemical and physical processes. As a next step towards constructing a complete physiologically faithful mitochondrial bioenergetics model, a mathematical model was developed targeting the cardiac mitochondrial bioenergetic based upon previous efforts, and corroborated using both transient and steady state data. The model consists of several modified rate functions of mitochondrial bioenergetics, integrated calcium dynamics and a detailed description of the K^{+}-cycle and its effect on mitochondrial bioenergetics and matrix volume regulation. Model simulations were used to fit 42 adjustable parameters to four independent experimental data sets consisting of 32 data curves. During the model development, a certain network topology had to be in place and some assumptions about uncertain or unobserved experimental factors and conditions were explicitly constrained in order to faithfully reproduce all the data sets. These realizations are discussed, and their necessity helps contribute to the collective understanding of the mitochondrial bioenergetics.
Mathematically modeling biological systems challenges our current understanding of the physical and biochemical events contributing to the observed dynamics. It requires careful consideration of hypothesized mechanisms, model development assumptions and details regarding the experimental conditions. We have adopted a modeling approach to translate these factors that explicitly considers the thermodynamic constraints, biochemical states and reaction mechanisms during model development. Such models have numerous constant parameters that must be determined. Integrating thermodynamics and detailed mechanistic representation of the principal phenomena help constrain these parameter values; therefore, only a handful of the total number of model parameters (∼10%) must be adjusted during parameter estimation through model simulations. Additionally, all models must undergo some form of corroboration prior to application. In practice, this corroboration should challenge all possible dynamics of the model, but it is recognized that in this data rich world, we are surprisingly data poor. Eventually such developed and corroborated models are capable of supporting current hypotheses, guiding experimental designs and contributing to the overall knowledge base of biological processes.
The simulation of mathematical models of mitochondrial bioenergetics provides a powerful analytical alternative to performing numerous exhaustive experiments. Such models aid in the interpretation of experimental data and facilitate experimental design for elucidating the supporting biochemical and physical processes. Current experimental techniques limit the ability to resolve details of the mitochondrial bioenergetic processes
Several mathematical models have been developed
The model integrates mitochondrial bioenergetic processes as shown in
Abbreviations: Oxphos, oxidative phosphorylation elements; PYR, pyruvate; CoASH, coenzyme A; AcCoA, acetyl-coenzyme A; CIT, citrate; ISOC, isocitrate; αKG, α-ketoglutarate; SCoA, succinyl CoA; SUC, succinate; FUM, fumarate; MAL, malate; OAA, oxaloacetate; GLU, glutamate; ASP, aspartate; NADH, reduced nicotinamide adenine dinucleotide; NAD, GTP, guanidine triphosphate; GDP, guanidine diphosphate; oxidized nicotinamide adenine dinucleotide; Pi, inorganic phosphate; UQ, ubiquinone; UQH_{2}, ubiquinol; Cytc^{3+}, oxidized cytochrome c; Cytc^{2+}, reduced cytochrome c; PDH, pyruvate dehydrogenase; CS, citrate synthase; ACH, acontinase; IDH, isocitrate dehydrogenase; αKGDH, α-ketoglutarate dehydroganse; SCoAS; succinyl CoA synthetase; SDH, succinate dyhdrogenase; FH, fumarate hydratase; MDH, malate dehydrogenase; GOT, glutamate oxaloacetate transaminase; CI, Complex I; CIII, Complex III; CIV, Complex IV; mHleak, proton leak; F_{1}F_{O}, F_{1}F_{O} ATP synthase; ANT, adenine nucleotide transporter; PIC, inorganic phosphate carrier; GAE, glutamate/aspartate exchanger; OME, α-ketoglutarate/malate exchanger; DCC, dicarboxylate carrier; TCC, tricarboxylate carrier; PYRH, pyruvate-proton cotransporter; GLUH, glutamate-proton cotransporter; mKATP, ATP-dependent K^{+} channel; mKHE, K^{+}/H^{+} exchanger; mKleak, K^{+} leak; mNHE, Na^{+}/H^{+} exchanger; mNCE, Na^{+}/Ca^{2+} exchanger; CaUNI, Ca^{2+} uniporter; AK, adenylate kinase.
Reaction | Enzyme | Biochemical Reaction |
J_{PDH} | Pyruvate dehydrogenase | PYR + CoASH + NAD + H_{2}O ↔ CO_{2} + SCOA + NADH |
J_{CS} | Citrate synthase | OAA + AcCoA + H_{2}O ↔ CoASH + CIT |
J_{ACH} | Aconitase | CIT ↔ ISOC |
J_{IDH} | Isocitrate dehydrogenase | NAD + ISOC + H_{2}O ↔ αKG + NADH + CO_{2} |
J_{aKGDH} | α-Ketoglutarate dehydrogenase | αKG + CoASH + NAD + H_{2}O ↔ CO_{2} + SCoA + NADH |
J_{SCoAS} | Succinyl CoA synthase | SCoA + GDP + Pi ↔ SUC + GTP + CoASH |
J_{SDH} | Succinate dehydrogenase | SUC + UQ ↔ UQH_{2} + FUM |
J_{FH} | Fumarate hydratase | FUM + H_{2}O ↔ MAL |
J_{MDH} | Malate dehydrogenase | NAD + MAL ↔ OAA + NADH |
J_{NDK} | Nucleoside diphosphokinase | GTP + ADP ↔ GDP + ATP |
J_{GOT} | Glutamate oxaloacetate transaminase | ASP + αKG ↔ OAA + GLU |
J_{CI} | Complex I | NADH + UQ ↔ NAD + UQH_{2} |
J_{CIII} | Complex III | UQH_{2} + 2Cytc^{3+} ↔ UQ + 2Cytc^{2+} |
J_{CIV} | Complex IV | 2Cytc_{red} + ½O_{2} ↔ 2Cytc_{ox} + H_{2}O |
J_{F1FO} | F_{1}F_{O} ATP synthase | ADP + Pi ↔ ATP + H_{2}O |
J_{AK} | Adenylate Kinase | 2ADP ↔ ATP + AMP |
J_{GAE} | Glutamate-aspartate exchanger | GLU_{ims} + H^{+}_{ims} + ASP_{mtx} ↔ GLU_{mtx} + H^{+}_{mtx} + ASP_{ims} |
J_{OME} | α-Ketoglutarate-malate exchanger | αKG_{ims} + MAL_{mtx} ↔ αKG_{mtx} + MAL_{ims} |
J_{PYRH} | Pyruvate-proton cotransporter | PYR_{ims} + H^{+}_{ims} ↔ PYR_{mtx} + H^{+}_{mtx} |
J_{GLUH} | Glutamate-proton cotransporter | GLU_{ims} + H^{+}_{ims} ↔ GLU_{mtx} + H^{+}_{mtx} |
J_{CITMAL} | Tricarboxylate carrier | CIT_{ims} + MAL_{mtx} ↔ CIT_{mtx} + MAL_{ims} |
J_{ISCOMAL} | Tricarboxylate carrier | ISOC_{ims} + MAL_{mtx} ↔ ISOC_{mtx} + MAL_{ims} |
J_{SUCPi} | Dicarboxylate carrier | SUC_{mtx} + Pi_{ims} ↔ SU C_{ims} + Pi_{mtx} |
J_{MALPi} | Dicarboxylate carrier | MAL_{mtx} + Pi_{ims} ↔ MAL_{ims} + Pi_{mtx} |
J_{ANT} | Adenine nucleotide transporter | ATP_{mtx} + ADP_{ims} ↔ ATP_{ims} + ADP_{mtx} |
J_{PIC} | Inorganic phosphate carrier | Pi_{ims} + H^{+}_{ims} ↔ Pi_{mtx} + H^{+}_{mtx} |
J_{mHleak} | Proton leak | H^{+}_{ims} ↔ H^{+}_{mtx} |
J_{mKHE} | Potassium-hydrogen exchanger | H^{+}_{ims} + K^{+}_{mtx} ↔ K^{+}_{ims} + H^{+}_{mtx} |
J_{mKATP} | ATP-dependent potassium channel | K^{+}_{ims} ↔ K^{+}_{mtx} |
J_{mKleak} | Potassium leak | K^{+}_{ims} ↔ K^{+}_{mtx} |
J_{CaUNI} | Calcium uniporter | Ca^{2+}_{ims} ↔ Ca^{2+}_{mtx} |
J_{mNCE} | Sodium-calcium exchanger | Ca^{2+}_{mtx} + 3Na^{+}_{ims} ↔ Ca^{2+}_{ims} + 3Na^{+}_{mtx} |
J_{mNHE} | Sodium-hydrogen exchanger | Na^{+}_{mtx} + H^{+}_{ims} ↔ Na^{+}_{ims} + H^{+}_{mtx} |
J_{HK} | Hexokinase |
GLC + ATP ↔ G6P + ADP |
Reaction is not presented in
The model proposed in the manuscript is a 73 state system of differential-algebraic equations (DAEs) that consists of 65 non-linear ordinary differential equations (ODEs); five algebraic conservation expressions to compute matrix ATP, guanidine triphosphate (GTP), reduced nicotinamide adenine dinucleotide (NADH), ubiquinol (UQH_{2}) and reduced cytochrome c (c^{2+}); one algebraic expression to compute matrix water volume; one algebraic expression to compute inner membrane space (IMS) water volume and one algebraic expression to compute matrix chloride content (Cl^{−}). The majority of the experimental data used to parameterize the model proposed in this manuscript were derived from heart tissue of either bovine, porcine or rat with some data obtained from liver tissue. Part S1 of the Supplemental Material (
The TCA and related enzyme rate expressions are structurally identical to Wu et al. except for a few alterations. To include the calcium-dependence of matrix dehydrogenases, the rate expressions for pyruvate dehydrogenase (PDH), isocitrate dehydrogenase (IDH) and α-ketoglutarate dehydrogenase (αKGDH) were modified. PDH, an important regulatory enzyme involved with mitochondrial bioenergetics, is responsible for the oxidative decarboxylation of pyruvate, transacylation of an acetyl group to CoA and production of reducing equivalents for the ETS. A similar rate expression found in Wu et al. was used with a few notable modifications. The proton, divalent cation and adenine nucleotide regulatory mechanisms were inserted into the expression to reproduce the available data
Two additional rate expressions that deviated from Wu et al. are the glutamate-aspartate exchanger (GAE) and dicarboxylate carrier (DCC). The GAE is a key enzyme in the malate-aspartate shuttle and is particularly important maintaining state 3 NADH levels when mitochondria respire on glutamate and malate. This electrogenic exchanger is activated by calcium and swaps glutamate and a proton with aspartate taking advantage of the energized state of mitochondria established by the ETS. The enzyme reaction was modeled based on a rapid equilibrium bi-bi mechanism with a third substrate, protons, added to the rate expression; it was fit to data from bovine heart mitochondria
The ANT is the enzyme responsible for exchanging unchealated ATP and ADP across the mitochondrial inner membrane. Previous models used a ping-pong mechanism that employed a single adenine nucleotide binding site whereby Δψ affected only ATP binding
The model proposed in this manuscript incorporates mitochondrial calcium dynamics similar to Nguyen et al.
The ‘futile’ K^{+}-cycle plays a major role in mitochondrial volume homeostasis
To parameterize the model, four independent data sets consisting of 32 data curves were used from Bose et al.
Parameter | Definition | Value (LSCx100) | Units |
Pyruvate dehydrogenase max rate | 127 (4.88) | nmol/min/mg | |
Citrate synthase max rate | 584 (1.64) | nmol/min/mg | |
Acontinase max forward rate | 1.16×10^{5} (0.011) | nmol/min/mg | |
Isocitrate dehydrogenase max rate | 6.84×10^{4} (0.341) | nmol/min/mg | |
α-Ketoglutarate dehydrogenase max rate | 779 (2.18) | nmol/min/mg | |
Succinyl CoA synthase max forward rate | 3.93×10^{4} (0.216) | nmol/min/mg | |
Succinate dehydrogenase max forward rate | 7.00×10^{3} (2.26) | nmol/min/mg | |
Fumurate hydratase max forward rate | 7.67×10^{5} (0.010) | nmol/min/mg | |
Malate dehydrogenase max forward rate | 965 (0.432) | nmol/min/mg | |
Malate dehydrogenase Pi binding constant | 5.00×10^{−3} (0.156) | M | |
Malate dehydrogenase Pi activation constant | 57.7 (0.193) | unitless | |
Nucleoside diphosphokinase max forward rate | 6.95×10^{3} (0.016) | nmol/min/mg | |
Glutamate oxaloacetate transaminase max forward rate | 1.09×10^{6} (0.251) | nmol/min/mg | |
Pyruvate-hydrogen co-transporter activity | 3.12×10^{13} (0.651) | nmol/M^{2}/min/mg | |
Glutamate-hydrogen co-transporter activity | 9.44×10^{10} (0.763) | nmol/M^{2}/min/mg | |
Tricarboxylate activity | 7.13×10^{8} (0.471) | nmol/M^{2}/min/mg | |
α-Ketoglutarate-malate exchanger max forward rate | 8.83×10^{3} (0.275) | nmol/min/mg | |
Glutamate-aspartate exchanger un-stimulated max rate | 674 (0.865) | nmol/min/mg | |
Glutamate-aspartate calcium activation constant | 27.2 (0.299) | unitless | |
Dicarboxylate carrier max rate | 8.03×10^{3} (0.901) | nmol/min/mg | |
Complex I activity | 5.63×10^{6} (0.962) | nmol/M^{2}/min/mg | |
Complex III activity | 5.84×10^{5} (2.19) | nmol/M^{3/2}/ming/mg | |
Complex III Pi binding constant | 4.40×10^{−3} (0.976) | M | |
Complex III Pi activation constant | 148 (1.54) | unitless | |
Complex IV activity | 44.0 (2.19) | nmol/M/min/mg | |
F_{1}F_{O} ATP synthase activity | 1.91×10^{7} (0.027) | nmol/M/min/mg | |
Adenine nucleotide translocase max rate | 365 (7.52) | nmol/min/mg | |
Inorganic phosphate carrier max rate | 7.59×10^{7} (0.012) | nmol/min/mg | |
Proton permeability | 8.06×10^{7} (5.01) | nmol/M/min/mg | |
Potassium-hydrogen exchanger max rate | 6.25 (3.58) | nmol/nl/min/mg | |
ATP-dependent potassium channel conductance | 6.75 (0.181) | nmol/min/mg | |
ATP-dependent potassium channel MgATP inhibition constant | 4.10×10^{−9} (0.061) | M | |
ATP-dependent potassium channel MgADP inhibition constant | 10.0×10^{−6} (0.063) | M | |
Potassium permeability | 13.8 (4.87) | nmol/M/min/mg | |
Calcium uniporter calcium permeability | 157 (0.403) | nmol/min/mg | |
Sodium-calcium exchanger max rate | 0.731 (0.360) | nmol/min/mg | |
Sodium-calcium exchanger calcium binding constant | 4.17×10^{−6} (0.326) | M | |
Sodium-calcium exchanger sodium binding constant | 1.72×10^{−3} (0.018) | M | |
Sodium-calcium exchanger calcium activation binding constant | 1.66×10^{−6} (0.238) | M | |
Sodium-calcium exchanger calcium activation constant | 67.6 (0.296) | unitless | |
Sodium-hydrogen exchanger max rate | 9.45×10^{5} (0.061) | nmol/min/mg | |
Hexokinase max rate | 1.20×10^{5} (5.14) | nmol/min/mg |
The NADH-linked respiration components of the model were fitted against the Pi-titration experiments performed by Bose et al.
Mitochondria were incubated in the assay buffer identified in the
The TCA cycle intermediate dynamics of the model were fitted to the data set presented by LaNoue et al.
Mitochondria were incubated in the assay buffer identified in the
Mitochondria were incubated in the assay buffer identified in the
The Na^{+}/Ca^{2+} cycle was fitted to the steady state data from Wan et al.
Mitochondria were incubated in the assay buffer identified in the
The volume dynamics were fitted to the transient mitochondrial matrix swelling data published by Kowalowski et al.
Mitochondria were incubated in the assay buffer identified in the
Model corroboration was necessary to establish confidence in the model resulting from these efforts. Herein, the corroboration considered the robustness of the model to local parameter perturbations, the qualitative agreement of predicted trends with experimental observations, and the ability of the model to reproduce experimental data that was not used in fitting its parameters.
A local sensitivity analysis on the model was performed to determine how robust the model simulations were to local perturbations in the parameter values of all the experiments used in the parameter identification. The absolute-value normalized local sensitivity coefficients (
The model was also able to reproduce the well known mitochondrial shrinkage/swelling dynamics in the presence of Pi and ADP.
The model was directly corroborated by predicting the steady state accumulation of extra-mitochondrial α-ketoglutarate during state 2 respiration at various extra-mitochondrial malate concentrations as shown in
Mitochondria were incubated in the assay buffer identified in the
The model was again directly corroborated by predicting the experimentally observed mitochondrial volume dynamics after various bioenergetic and/or mKATP interventions.
Mitochondria were incubated in the assay buffer identified in the
Finally, the model's ability to reproduce the expected trends in bioenergetic variables under varying KCl buffer osmolarity conditions was explored. Devin et al.
Mitochondria were incubated in the assay buffer identified in the
The model presented in this manuscript is based on previous models
The mitochondrial volume dynamics and the associated K^{+}-cycle appear to play an important role in cellular and mitochondrial bioenergetics
The hypothesized volume-dependent mKHE by Garid
To faithfully reproduce all the data sets, a certain network topology had to be in place and some assumptions about uncertain or unobserved experimental factors and conditions were explicitly constrained. It was found that these network features and experimental assumptions described below were necessary to successfully recreate all the experimentally observed data and trends. The necessity of these realizations contributes to our collective understanding of the mitochondrial bioenergetics.
The intrinsic thermodynamic dissipation of a system can override or mitigate enzymatic regulation
Animal model species specific parameterization may be required for detailed mathematical models of the mitochondrial bioenergetics; however, at this time there is not sufficient data from a single species to fully characterize the dynamics. For example, all the kinetic parameters for the αKGDH expression were found using independent data sets obtained from porcine heart mitochondria as described in Part S3 of the Supplemental Material (
The tissue type used for the supporting experiments is also important for parameterization of a semi-mechanistic mathematical model. For example, in the model, several exchangers and cotransporters are reported to possess low activity in heart tissue compared with other tissues
Another exchanger reported to possess low activity in heart tissue is the dicarboxylate carrier
The chosen substrates for the DCC also affected the model simulation capabilities. In the model formulation, only Pi, malate and succinate were assigned as the DCC substrates. Fumarate is also reported to be a substrate for the DCC
An additional tissue source related phenomena uncovered during model development was the choice of the calcium dissociation constant for the CaUNI. During model development, a single calcium dissociation constant was chosen to model the CaUNI kinetics from rat heart and liver tissue. This was achieved by considering the competitive nature of magnesium inhibition with respect to calcium binding. It is plausible that the major difference between liver and heart CaUNI kinetics is due to different expression levels versus different calcium binding affinities (as may be plausible between different species); however, more research concerning this matter needs to be done.
Mitochondria from specific tissue types are phenotypically different and contain various amounts of electron transport proteins, matrix proteins and lipid types optimized to support their designated function. Specifically, heart mitochondria possess much higher electron transport activity relative to liver mitochondria
The translation of the biochemical processes to appropriate mathematically descriptive expressions plays a large role in the simulated dynamics. For example, the citrate and isocitrate overestimation by the model as shown in
During the model development, it is important to consider any artifacts in the experimental data that may have been inadvertently generated during the mitochondrial isolation. For example, the extraction medium must contain Pi to achieve stable, well-coupled mitochondria
To simulate the precise experimental conditions during model development, a few explicit assumptions were necessary. For example, the LaNoue data set reported using 3–4 mg of mitochondrial protein; however, as little as a 25% change in mitochondrial load can dramatically alter the total substrate consumption and product accumulation during high MVO_{2} rates. Hence for these simulations, the conditions needed to be known with more certainty. The reported malate concentration was used to estimate the mitochondrial load. The initial malate content in state 2 and state 3 experiments with pyruvate was reported to be 1430 nmol/mg. This required that the mitochondrial load be 3.5 mg/mL using the stated 5 mM malate concentration. To compute this estimate, it was necessary to also consider the pyruvate concentration. The pyruvate concentration was 2 mM; however after 8 minutes of state 3 respiration, the pyruvate utilization was 848 nmol/mg. Considering the 1 mL chamber volume, this implied that the total initial pyruvate concentration was 2.5–3.4 mM and not 2 mM. In an attempt to address these potential data inconsistencies, the model simulations were fit to the data using a mitochondrial load of 3.5 mg/mL and the reported state 3 pyruvate utilizations were subsequently adjusted to be consistent with an initial pyruvate concentration of 2 mM.
All mathematical models are abstractions of the underlying process; the level of detail included in the model is dependent upon the application. This is particularly true for the calcium dynamics associated with mitochondrial bioenergetics. There are known omissions in this and previous models of these calcium dynamics. The mitochondrial Na^{+}/Ca^{2+} dynamics were simulated using a simplified Na^{+}/Ca^{2+} cycling mechanisms with only the CaUNI, mNCE and mNHE processes represented. This simplification prohibited a mechanistic representation of the actual physiological event. For example, the omission of the rapid mode of calcium uptake (RAM)
In summary, the model presented in this manuscript proposes an extended mitochondrial bioenergetics model targeted at the cardiac myocyte with the parameters estimated using four independent data sets consisting of 32 data curves. It was capable of fitting the data with good fidelity, had relatively little parameter sensitivity relative to the experimental conditions modeled herein and was capable of adequately modeling metabolic trends during the various conditions simulated. The resulting model simulations reproduce observed mitochondrial volume dynamics lending additional support to the current prevailing theory of mitochondrial volume regulation through the mKHE volume-sensitive exchange rate. The model builds upon previous successes and helps refine and establish a global model framework relating to mitochondrial bioenergetics. During the model development, a certain network topology had to be in place and some assumptions about uncertain or unobserved experimental factors and conditions were explicitly constrained to reproduce all the data sets. Specifically, the effect of intrinsic thermodynamic dissipation of the system on enzymatic regulation, importance of animal species and tissue sources differences, mechanistic detail of the model and potential impact of the experimental environment all help constrain the model formulation contributing to the construction of a successful and physiologically faithful model.
The model can serve as a foundation for further extension and refinement efforts. Future work may consider more detailed and mechanistic mathematical abstractions for the ETS and TCC, Ca^{2+} dynamics including the RAM and Na^{+}-independent Ca^{2+} efflux pathways, catabolic (i.e., glutamate dehydrogenase) and anabolic (i.e., pyruvate carboxyalse) reactions and β-oxidation pathways enabling integration into whole cell models of cardiac myocytes. Each of these additions will require additional experimental data taken under well controlled and documented conditions in order to be properly reproduced by the model proposed in this manuscript. For example, changing the passive exchange mechanism of the TCC to a more mechanistic, saturable exchange process should enable better fits to the LaNoue data set. This change would keep matrix citrate and isocitrate at sufficient levels to maintain α-ketoglutarate and succinate concentrations experimentally observed allowing fumarate to be included in the list of DCC substrates. With fumarate being removed from the matrix by the DCC, SDH inhibition would be mitigated and the lower branch of the TCA cycle would accelerate and prevent net oxidation of malate observed in the LaNoue experimental data set. Additionally, reproducing the respiratory control ratios as done
The DAEs describing the model were numerically integrated using MATLAB® (2008b) and the stiff ode solver ode15s (10^{−3} relative tolerance and 10^{−9} absolute tolerance for matrix and IMS state variables and 10^{−6} for all others). To increase computational efficiency, vectorized functions were used during model development in the MATLAB® environment. Parameter optimizations and sensitivity analyses were done on a cluster of four 8-core Intel Xeon 3.4 GHz CPUs each with 16 GB of memory and running the Windows 2003 server platform using the Parallel Computing Toolbox. The results obtained were displayed using MATLAB®.
The objective function used for parameterization of the model is defined as
Fitting such large, non-linear models to data with many unknown parameters and initial conditions requires a robust model structure and many independent data sets to appropriately constrain the parameters. The model presented in this manuscript consists of a total of 359 parameters. These parameters were identified using three methods: i) 262 parameters were fixed according to previously published values in the literature (see Part S3 of the Supplemental Material (
The local sensitivity analysis was done using the absolute normalized local sensitivity coefficient (
To simulate the various data sets used to parameterize and corroborate the model, the appropriate experimental conditions were taken into consideration. The temperature at which the experimental data were obtained, the mitochondrial loads applied in each experiment, the initial state of the mitochondria in the experimental system and the precise nature of the experimental environment, specifically the buffer composition and osmolarity were considered. These points are discussed below.
The rates of biochemical reactions can be extremely sensitive to temperature. A temperature induced change in activity can easily result in doubling or even tripling of some enzymatic reactions for a temperature difference of only 10°C. Since the experiments used to parameterize the model were done at different temperatures, the rates were adjusted according to a standard Q_{10} value of 2.25 or based specifically on the enzyme's activation energy and the Arrhenius rate law. The Arrhenius rate law temperature correction is only truly valid for reactions whose substrates are in rapid equilibrium with their respective substrate-enzyme complexes, and the catalysis step is rate limiting
It is especially important to use the appropriate initial conditions by either including them in the fitting procedure as variables or conditioning the model by simulation to set common initial operating conditions. We chose the latter and standardized the initial conditions by simulating the model for sufficiently long times with the experimental conditions outlined in each paper. For example, the Bose data set was derived from de-energized, equilibrated mitochondria, the LaNoue data set was derived from de-energized, non-equilibrated mitochondria, the Wan data set was derived from energized, equilibrated mitochondria, the Kowaltowski data set was derived from de-energized, non-equilibrated mitochondria and the Devin data set was derived from energized, equilibrated mitochondria. Once the model was simulated to its fully oxidized state, appropriate modifications to state variables were performed to mimic the experimental conditions as described in each paper used for parameter estimation.
Each experiment used different mitochondrial loads; therefore, for each data set, the buffer water volume relative to the mitochondrial protein content was varied. This is important because the higher the mitochondrial load, the higher the absolute rate of consumption for carbon substrates and oxygen making time series data essential for parameterization purposes. The Bose data set experiments were performed at a mitochondrial load of 1 nmolCyta/mL, or approximately 1 mg mitochondria/mL
Before simulating the experimental conditions (T = 37°C) in the Bose data set, the model had to be initialized to recreate the Pi-depleted state of the mitochondria. This was done by simulating the model to its fully oxidative state in the presence of ATP followed by a quick Pi-depletion step. Unfortunately, the exact details concerning the Pi-depletion protocol could not be found. For parameter estimation, the basic buffer composition consisted of 125 mM KCl, 15 mM NaCl, 20 mM K-Hepes, 1 mM KEGTA, 1 mM K_{2}EDTA, 5 mM MgCl, 4 µM TPP^{+} at pH 7.1. The authors stated that the free [Ca^{2+}] was generally held between 500 and 600 nM using CaCl_{2}. For simulation purposes, the free [Ca^{2+}] was fixed at 550 nM. To initiate respiration, 5 mM glutamate/malate was added to the buffer preceding the Pi-titrations for the simulations. For state 3 respiration, 1.3 mM ADP was added to the medium after the glutamate/malate and Pi additions. For each Pi-titration, the free [K^{+}], [Na^{+}] and [Mg^{2+}] were calculated based on the dissociation constants defined in the Supplemental Material (
For the LaNoue data set simulations (T = 28°C), the model was first initialized to achieve the fully oxidized state. The basic buffer composition used for respiratory state initiation consisted of 150 mM KCl, 20 mM Tris-Cl, 20 mM KPi, 5 mM MgCl_{2} and 30 mM glucose at pH 7.2. The mitochondria were preincubated in the basic buffer composition for 30 seconds before addition of substrates. For state 2, either 2 mM pyruvate and 5 mM malate or 1 mM pyruvate was added to the basic buffer composition, and the model was simulated for the specified experimental time. (Note that in the original reference, LaNoue et al. define state 4 as state 2 is defined in this manuscript. They used the previous state nomenclature defined by Chance and Williams
The Wan Data set simulations (T = 28°C) also required model initializations from a fully oxidized state. The buffer composition consisted of 130 mM KCl, 20 mM HEPES, 5 mM MgCl_{2}, 5 mM ATP, 5 mM KH_{2}PO_{4}, 5 mM NaCl and 1 mM EGTA at pH 7.0. For the Na^{+}- and Mg^{2+}-titrations, the free [K^{+}], [Na^{+}] and [Mg^{2+}] were recalculated at each data point based on the dissociation constants defined in the Supplemental Material (
The Kowaltowski data set simulations (T = 28°C) were simulated from a fully oxidative and pre-osmotic equilibration state. The basic buffer composition used for respiratory state initiation consisted of 135 mM KCl, 5 mM succinate, 2.5 mM Pi, 100 µM EGTA, 0.5 mM MgCl_{2} at pH 7.2. They applied various bioenergetic pharmaceutical interventions and measured the matrix swelling dynamics. To block the F_{1}F_{O} ATP synthase, 0.5 µg/ml oligomycin was used. To simulate this condition, the parameter defining the enzyme activity,
For the Devin data set simulations (T = 26°C), the model was first initialized to achieve the fully oxidized state. The basic buffer composition used for respiratory state initiation consisted of 5 µM TPMP^{+}, 5 µM DMO, 5 µM manitol, 20 mM Tris-HCl, 1 mM EGTA, 6 mM Tris-glutamate, 6 mM Tris-malate, 5 mM Tris-Pi at pH 7.2 with varying amounts of KCl used to adjust the osmolarity. The buffer [K^{+}] was approximated by setting it equal to half the reported osmolarity. Since no divalent cation was present in the medium, the adenylate kinase reaction was turned off. The model was simulated under state 2 conditions until steady state was reached (a simulated time of 5 minutes) under the varying osmotic conditions. Then, at each osmotic condition, 1 mM of ADP was added to the buffer and the model was simulated to a pseudo-steady state reproducing the reported 10 minute experiments.
The supplemental material consists of three parts. Part S1 lists the state variables comprising the model, updated Gibbs free energy of formation values, additional and revised dissociation constants, temperature correction method, and general model parameters. Part S2 introduces of the set of 60 non-linear ODEs, five algebraic conservation expressions (for ATP, GTP, NADH, UQH_{2} and c^{2+}), five matrix cation ODEs (for H^{+}, K^{+}, Na^{+}, Mg^{2+} and Ca^{2+}) and the algebraic expressions for computing matrix and intermembrane space (IMS) water volumes and matrix Cl^{−} is presented. Part S3 discusses the model rate equation derivations and provides all the associated parameter definitions and values.
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We thank the reviewers for their comments that helped clarify and direct the revision of this model. We would like to thank Robert Balaban for his insightful communications regarding the model development. We would also like to thank Jim Jones for computer assistance with the cluster and Athurva Gore for preliminary work on the model presented in this manuscript.