An extended dynamic model of oxidative phosphorylation

Kinetic Mathematical Modeling of Oxidative Phosphorylation in Cardiomyocyte Mitochondria

Oxidative phosphorylation (OXPHOS) is an oxygen-dependent process that consumes catabolized nutrients to produce adenosine triphosphate (ATP) to drive energy-dependent biological processes such as excitation-contraction coupling in cardiomyocytes. In addition to in vivo and in vitro experiments, in silico models are valuable for investigating the underlying mechanisms of OXPHOS and predicting its consequences in both physiological and pathological conditions. Here, we compare several prominent kinetic models of OXPHOS in cardiomyocytes. We examine how their mathematical expressions were derived, how their parameters were obtained, the conditions of their experimental counterparts, and the predictions they generated. We aim to explore the general landscape of energy production mechanisms in cardiomyocytes for future in silico models.

Computational Model of the Effect of Mitochondrial Dysfunction on Excitation–Contraction Coupling in Skeletal Muscle

It has become well established that mitochondria not only regulate myoplasmic calcium in skeletal muscle, but also use that calcium to stimulate oxidative phosphorylation (OXPHOS). While experimental approaches have allowed for imaging of mitochondrial calcium and membrane potentials in isolated fibers, capturing the role of mitochondria and the impact of mitochondrial impairments on excitation–contraction coupling (ECC) remains difficult to explore in intact muscle. Computational models have been widely used to examine the structure and function of skeletal muscle contraction; however, models of ECC to date lack communication between the myoplasm and mitochondria for regulating calcium and ATP during sustained contractions. To address this, a mathematical model of mitochondrial calcium handling and OXPHOS was integrated into a physiological model of ECC incorporating action potential propagation, calcium handling between the sarcoplasmic reticulum (SR) and the myoplasm, and crossbridge cycling. The model was used to examine the protective role of mitochondria during repeated stimulation and the impact of mitochondrial dysfunction on ECC resulting from progressive OXPHOS inhibition. Pathological myoplasmic calcium accumulation occurred through distinct mechanisms in the model in the case of either electron transport chain, F1F0 ATP synthase, or adenine nucleotide transporter impairments. To investigate the effect of each impairment on force, a model of calcium-stimulated apoptosis was utilized to capture dysfunction-induced reductions in muscle mass, driving whole muscle force loss. The model presented in this study can be used to examine the role of mitochondria in the regulation of calcium, ATP, and force generation during voluntary contraction.

Power Failure of Mitochondria and Oxidative Stress in Neurodegeneration and Its Computational Models

The brain needs more energy than other organs in the body. Mitochondria are the generator of vital power in the living organism. Not only do mitochondria sense signals from the outside of a cell, but they also orchestrate the cascade of subcellular events by supplying adenosine-5′-triphosphate (ATP), the biochemical energy. It is known that impaired mitochondrial function and oxidative stress contribute or lead to neuronal damage and degeneration of the brain. This mini-review focuses on addressing how mitochondrial dysfunction and oxidative stress are associated with the pathogenesis of neurodegenerative disorders including Alzheimer’s disease, amyotrophic lateral sclerosis, Huntington’s disease, and Parkinson’s disease. In addition, we discuss state-of-the-art computational models of mitochondrial functions in relation to oxidative stress and neurodegeneration. Together, a better understanding of brain disease-specific mitochondrial dysfunction and oxidative stress can pave the way to developing antioxidant therapeutic strategies to ameliorate neuronal activity and prevent neurodegeneration.

Transient heat release during induced mitochondrial proton uncoupling

Non-shivering thermogenesis through mitochondrial proton uncoupling is one of the dominant thermoregulatory mechanisms crucial for normal cellular functions. The metabolic pathway for intracellular temperature rise has widely been considered as steady-state substrate oxidation. Here, we show that a transient proton motive force (pmf) dissipation is more dominant than steady-state substrate oxidation in stimulated thermogenesis. Using transient intracellular thermometry during stimulated proton uncoupling in neurons of Aplysia californica, we observe temperature spikes of ~7.5 K that decay over two time scales: a rapid decay of ~4.8 K over ~1 s followed by a slower decay over ~17 s. The rapid decay correlates well in time with transient electrical heating from proton transport across the mitochondrial inner membrane. Beyond ~33 s, we do not observe any heating from intracellular sources, including substrate oxidation and pmf dissipation. Our measurements demonstrate the utility of transient thermometry in better understanding the thermochemistry of mitochondrial metabolism.

Modeling mitochondrial function and its role in disease

Many neurodegenerative diseases involve defects in mitochondrial function. These defects often arise from mutations carried in the genes that code mitochondrial proteins. Many of these defective proteins are involved in mitochondrial energy metabolism which is one of the primary roles of the mitochondria. However, others proteins have other roles that are related to signaling or mitochondrial structure. The interaction of the mitochondrial proteins is complex. Understanding these complex dynamics requires the use of computational models. Studies have started to exploit such models, but much further work is necessary to understand mitochondrial function and its role in neurodegenerative disease.

Mathematical models of mitochondrial aging and dynamics

Research on the role of mitochondria in aging and disease is rapidly growing. Furthermore, in recent years, it also became clear that mitochondria are dynamic structures undergoing constant and rapid cycles of fusion and fission. The involvement of mitochondria in multiple complex processes makes them a prime target for mathematical and computational modeling. This review consists of two parts. In the first (Section 2), we provide a detailed introduction to the underlying concepts of mathematical modeling to help the reader who is not so familiar with these techniques to judge the requirements and results that can be obtained through modeling. In the second part (Section 3), we review existing mathematical and computational models that investigate mitochondrial dynamics and the role of mitochondria for the aging process.

A kinetic-metabolic model based on cell energetic state: study of CHO cell behavior under Na-butyrate stimulation

A kinetic-metabolic model approach describing and simulating Chinese hamster ovary (CHO) cell behavior is presented. The model includes glycolysis, pentose phosphate pathway, TCA cycle, respiratory chain, redox state and energetic metabolism. Growth kinetic is defined as a function of the major precursors for the synthesis of cell building blocks. Michaelis-Menten type kinetic is used for metabolic intermediates as well as for regulatory functions from energy shuttles (ATP/ADP) and cofactors (NAD/H and NADP/H). Model structure and parameters were first calibrated using results from bioreactor cultures of CHO cells expressing recombinant t-PA. It is shown that the model can simulate experimental data for all available experimental data, such as extracellular glucose, glutamine, lactate and ammonium concentration time profiles, as well as cell energetic state. A sensitivity analysis allowed identifying the most sensitive parameters. The model was then shown to be readily adaptable for studying the effect of sodium butyrate on CHO cells metabolism, where it was applied to the cases with sodium butyrate addition either at mid-exponential growth phase (48 h) or at the early plateau phase (74 h). In both cases, a global optimization routine was used for the simultaneous estimation of the most sensitive parameters, while the insensitive parameters were considered as constants. Finally, confidence intervals for the estimated parameters were calculated. Results presented here further substantiate our previous findings that butyrate treatment at mid-exponential phase may cause a shift in cellular metabolism toward a sustained and increased efficiency of glucose utilization channeled through the TCA cycle.

In silico studies on the sensitivity of myocardial PCr/ATP to changes in mitochondrial enzyme activity and oxygen concentration

The ratio of myocardial phosphocreatine (PCr)/ATP reflects the balance of energy consumption and energy supply in the heart. It is reduced in a range of important physiological conditions including during and after acute hypoxia, after a prolonged visit to high-altitude, and in those suffering from both type 2 diabetes mellitus and various forms of heart failure. Yet despite its significance, the factors underlying the reduced PCr/ATP ratio seen in heart failure remain poorly understood. Given that oxidative phosphorylation is the only viable steady-state provider of ATP in the heart, the argument has been put forward that the observed reduction in myocardial PCr/ATP in all these conditions can be accounted for by some form of mitochondrial insufficiency. Thus we used a computer model of oxidative phosphorylation, coupled with creatine kinase, to study the effects of hypoxia and mitochondrial dysfunction on myocardial PCr/ATP. In physiological normoxia, all oxidative phosphorylation complexes, NADH supply and proton leak exerted comparable (of the same order of magnitude) control over PCr/ATP, as defined within Metabolic Control Analysis (MCA). Under hypoxia, the control increased considerably for all components of the system, especially for cytochrome oxidase and mitochondrial proton leak. Hypoxia alone, without any changes in other factors, exerted a pronounced effect on PCr/ATP. Our simulations support three important ideas: First, that mitochondrial abnormalities can contribute considerably to a blunted PCr/ATP; second, that hypoxia and mitochondrial dysfunction can interact in important ways to determine the energy status of the failing heart; and third, that hypoxia alone can account for significant decreases in cardiac PCr/ATP.

Magnitude and control of mitochondrial sensitivity to ADP

The transduction function for ADP stimulation of mitochondrial ATP synthesis in skeletal muscle was reconstructed in vivo and in silico to investigate the magnitude and origin of mitochondrial sensitivity to cytoplasmic ADP concentration changes. Dynamic in vivo measurements of human leg muscle phosphocreatine (PCr) content during metabolic recovery from contractions were performed by (31)P-NMR spectroscopy. The cytoplasmic ADP concentration ([ADP]) and rate of oxidative ATP synthesis (Jp) at each time point were calculated from creatine kinase equilibrium and the derivative of a monoexponential fit to the PCr recovery data, respectively. Reconstructed [ADP]-Jp relations for individual muscles containing more than 100 data points were kinetically characterized by nonlinear curve fitting yielding an apparent kinetic order and ADP affinity of 1.9 +/- 0.2 and 0.022 +/- 0.003 mM, respectively (means +/- SD; n = 6). Next, in silico [ADP]-Jp relations for skeletal muscle were generated using a computational model of muscle oxidative ATP metabolism whereby model parameters corresponding to mitochondrial enzymes were randomly changed by 50-150% to determine control of mitochondrial ADP sensitivity. The multiparametric sensitivity analysis showed that mitochondrial ADP ultrasensitivity is an emergent property of the integrated mitochondrial enzyme network controlled primarily by kinetic properties of the adenine nucleotide translocator.

Mitochondrial energetic metabolism: a simplified model of TCA cycle with ATP production

Mitochondria play a central role in cellular energetic metabolism. The essential parts of this metabolism are the tricarboxylic acid (TCA) cycle, the respiratory chain and the adenosine triphosphate (ATP) synthesis machinery. Here a simplified model of these three metabolic components with a limited set of differential equations is presented. The existence of a steady state is demonstrated and results of numerical simulations are presented. The relevance of a simple model to represent actual in vivo behavior is discussed.

Control over the contribution of the mitochondrial membrane potential (DeltaPsi) and proton gradient (DeltapH) to the protonmotive force (Deltap). In silico studies

The protonmotive force across the inner mitochondrial membrane (Deltap) has two components: membrane potential (DeltaPsi) and the gradient of proton concentration (DeltapH). The computer model of oxidative phosphorylation developed previously by Korzeniewski et al. (Korzeniewski, B., Noma, A., and Matsuoka, S. (2005) Biophys. Chem. 116, 145-157) was modified by including the K+ uniport, K+/H+ exchange across the inner mitochondrial membrane, and membrane capacitance to replace the fixed DeltaPsi/DeltapH ratio used previously with a variable one determined mechanistically. The extended model gave good agreement with experimental results. Computer simulations showed that the contribution of DeltaPsi and DeltapH to Deltap is determined by the ratio of the rate constants of the K+ uniport and K+/H+ exchange and not by the absolute values of these constants. The value of Deltap is mostly controlled by ATP usage. The metabolic control over the DeltaPsi/DeltapH ratio is exerted mostly by K+ uniport and K+/H+ exchange in the presence of these processes, and by the ATP usage, ATP/ADP carrier, and phosphate carrier in the absence of them. The K+ circulation across the inner mitochondrial membrane is controlled mainly by K+ uniport and K+/H+ exchange, whereas H+ circulation by ATP usage. It is demonstrated that the secondary K+ ion transport is not necessary for maintaining the physiological DeltaPsi/DeltapH ratio.

Adenine nucleotide-creatine-phosphate module in myocardial metabolic system explains fast phase of dynamic regulation of oxidative phosphorylation

Computational models of a large metabolic system can be assembled from modules that represent a biological function emerging from interaction of a small subset of molecules. A "skeleton model" is tested here for a module that regulates the first phase of dynamic adaptation of oxidative phosphorylation (OxPhos) to demand in heart muscle cells. The model contains only diffusion, mitochondrial outer membrane (MOM) permeation, and two isoforms of creatine kinase (CK), in cytosol and mitochondrial intermembrane space (IMS), respectively. The communication with two neighboring modules occurs via stimulation of mitochondrial ATP production by ADP and P(i) from the IMS and via time-varying cytosolic ATP hydrolysis during contraction. Assuming normal cytosolic diffusion and high MOM permeability for ADP, the response time of OxPhos (t(mito); generalized time constant) to steps in cardiac pacing rate is predicted to be 2.4 s. In contrast, with low MOM permeability, t(mito) is predicted to be 15 s. An optimized MOM permeability of 21 mum/s gives t(mito) = 3.7 s, in agreement with experiments on rabbit heart with blocked glycolytic ATP synthesis. The model correctly predicts a lower t(mito) if CK activity is reduced by 98%. Among others, the following predictions result from the model analysis: 1) CK activity buffers large ADP oscillations; 2) ATP production is pulsatile in beating heart, although it adapts slowly to demand with "time constant" approximately 14 heartbeats; 3) if the muscle isoform of CK is overexpressed, OxPhos reacts slower to changing workload; and 4) if mitochondrial CK is overexpressed, OxPhos reacts faster.

Modeling mitochondrial function

The mitochondrion represents a unique opportunity to apply mathematical modeling to a complex biological system. Understanding mitochondrial function and control is important since this organelle is critical in energy metabolism as well as playing key roles in biochemical synthesis, redox control/signaling, and apoptosis. A mathematical model, or hypothesis, provides several useful insights including a rigorous test of the consensus view of the operation of a biological process as well as providing methods of testing and creating new hypotheses. The advantages of the mitochondrial system for applying a mathematical model include the relative simplicity and understanding of the matrix reactions, the ability to study the mitochondria as a independent contained organelle, and, most importantly, one can dynamically measure many of the internal reaction intermediates, on line. The developing ability to internally monitor events within the metabolic network, rather than just the inflow and outflow, is extremely useful in creating critical bounds on complex mathematical models using the individual reaction mechanisms available. However, many serious problems remain in creating a working model of mitochondrial function including the incomplete definition of metabolic pathways, the uncertainty of using in vitro enzyme kinetics, as well as regulatory data in the intact system and the unknown chemical activities of relevant molecules in the matrix. Despite these formidable limitations, the advantages of the mitochondrial system make it one of the best defined mammalian metabolic networks that can be used as a model system for understanding the application and use of mathematical models to study biological systems.

CK flux or direct ATP transfer: versatility of energy transfer pathways evidenced by NMR in the perfused heart

How the myocardium is able to permanently coordinate its intracellular fluxes of ATP synthesis, transfer and utilization is difficult to investigate in the whole organ due to the cellular complexity. The adult myocardium represents a paradigm of an energetically compartmented cell since 50% of total CK activity is bound in the vicinity of other enzymes (myofibrillar sarcolemmal and sarcoplasmic reticulum ATPases as well as mitochondrial adenine nucleotide translocator, ANT). Such vicinity of enzymes is well known in vitro as well as in preparations of skinned fibers to influence the kinetic properties of these enzymes and thus the functioning of the subcellular organelles. Intracellular compartmentation has often been neglected in the NMR analysis of CK kinetics in the whole organ. It is indeed a methodological challenge to reveal subcellular kinetics in a working organ by a global approach such as NMR. To get insight in the energy transfer pathway in the perfused rat heart, we developed a combined analysis of several protocols of magnetization transfer associated with biochemical data and quantitatively evaluated which scheme of energetic exchange best describes the NMR data. This allows to show the kinetic compartmentation of subcellular CKs and to quantify their fluxes. Interestingly, we could show that the energy transfer pathway shifts from the phosphocreatine shuttle in the oxygenated perfused heart to a direct ATP diffusion from mitochondria to cytosol under moderate inhibition of ATP synthesis. Furthermore using NMR measured fluxes and the known kinetic properties of the enzymes, it is possible to model the system, estimate local ADP concentrations and propose hypothesis for the versatility of energy transfer pathway. In the normoxic heart, a 3-fold ADP gradient was found between mitochondrial intermembrane space, cytosol and ADP in the vicinity of ATPases. The shift from PCr to ATP transport observed when ATP synthesis decreases might result from a balance in the activity of two populations of ANT, either coupled or uncoupled to CK. We believe this NMR approach could be a valuable tool to reinvestigate the control of respiration by ADP in the whole heart reconciling the biochemical knowledge of mitochondrial obtained in vitro or in skinned fibers with data on the whole heart as well as to identify the implication of bioenergetics in the pathological heart.

On the mechanism and biology of cytochrome oxidase inhibition by nitric oxide

The detailed molecular mechanism for the reversible inhibition of mitochondrial respiration by NO has puzzled investigators: The rate constants for the binding of NO and O2 to the reduced binuclear center CuB/a3 of cytochrome oxidase (COX) are similar, and NO is able to dissociate slowly from this center whereas O2 is kinetically trapped, which altogether seems to favor the complex of COX with O2 over the complex of COX with NO. Paradoxically, the inhibition of COX by NO is observed at high ratios of O2 to NO (in the 40-500 range) and is very fast (seconds or faster). In this work, we used simple mathematical models to investigate this paradox and other important biological questions concerning the inhibition of COX by NO. The results showed that all known features of the inhibition of COX by NO can be accounted for by a direct competition between NO and O2 for the reduced binuclear center CuB/a3 of COX. Besides conciliating apparently contradictory data, this work provided an explanation for the so-called excess capacity of COX by showing that the COX activity found in tissues actually is optimized to avoid an excessive inhibition of mitochondrial respiration by NO, allowing a moderate, but not excessive, overlap between the roles of NO in COX inhibition and in cellular signaling. In pathological situations such as COX-deficiency diseases and chronic inflammation, an excessive inhibition of the mitochondrial respiration is predicted.

Modeling the energy transfer pathways. creatine kinase activities and heterogeneous distribution of ADP in the perfused heart

The exchange scheme of high energy phosphate transport in a whole heart relies on a system of CK functioning in different ways. This suggests that the CKs are able to act both like a shuttle and like a buffer for the energy transfer. The challenge is to understand how these two functions are balanced in the CK system. One key of this balance is the knowledge of the local concentrations of the ADP nucleotide. These concentrations cannot be directly measured, but they may be derived by computation. In the present report we introduce the known properties of the enzymes catalyzing the exchange of high energy phosphate into the model of flux pathways derived from NMR experiments to compute both the maximum activity of each enzyme and the local concentrations of all the substrates. We show that the ADP distribution must be heterogeneous for the system to work. Its concentration is 50% higher in the vicinity of ATPase sites and 50% lower in the intermembrane space of the mitochondria than in the cytosol. Another result of this analysis is that the apparent large unbalance of the CKmito pathway is imposed by the adenosine nucleotide transferase fluxes. This analysis proves that it is possible to deduce biochemistry the local concentrations of a substrate by combining data originating from NMR, and enzymology into a common model.

Virtual mitochondria: metabolic modelling and control

Inside the eukaryotic cell, mitochondria are internal organelles of prokaryotic origin, responsible for energy supply in the cell. The control of the mitochondrial ATP production is a complex problem with different patterns according to different tissues and organs. Our aim is to continue to develop the modelling of oxidative phosphorylation in different tissues, to model other parts of mitochondrial metabolism and to include this virtual mitochondria in a virtual cell. In constructing the complete metabolic map of mitochondria, we will take advantage of the sequenced genomes of eukaryotic organism (10-15% of the yeast genome concerns mitochondria).

Kinetics of electron transfer through the respiratory chain

We show that the rate at which electrons pass through the respiratory chain in mitochondria and respiring prokaryotic cells is described by the product of three terms, one describing electron donation, one acceptance, and a third, the thermodynamic drive. We apply the theory of nonequilibrium thermodynamics in the context of the chemiosmotic model of proton translocation and energy conservation. This approach leads to a closed-form expression that predicts steady-state electron flux as a function of chemical conditions and the proton motive force across the mitochondrial inner membrane or prokaryotic cytoplasmic membrane. The rate expression, derived considering reverse and forward electron flow, is the first to account for both thermodynamic and kinetic controls on the respiration rate. The expression can be simplified under specific conditions to give rate laws of various forms familiar in cellular physiology and microbial ecology. The expression explains the nonlinear dependence of flux on electrical potential gradient, its hyperbolic dependence on substrate concentration, and the inhibiting effects of reaction products. It provides a theoretical basis for investigating life under unusual conditions, such as microbial respiration in alkaline waters.

A model of oxidative phosphorylation in mammalian skeletal muscle

A dynamic computer model of oxidative phosphorylation in oxidative mammalian skeletal muscle was developed. The previously published model of oxidative phosphorylation in isolated skeletal muscle mitochondria was extended by incorporation of the creatine kinase system (creatine kinase plus phosphocreatine/creatine pair), cytosolic proton production/consumption system (proton production/consumption by the creatine kinase-catalysed reaction, efflux/influx of protons), physiological size of the adenine nucleotide pool and some additional minor changes. Theoretical studies performed by means of the extended model demonstrated that the CK system, which allows for large changes in P(i) in relation to isolated mitochondria system, has no significant influence on the kinetic properties of oxidative phosphorylation, as inorganic phosphate only slightly modifies the relationship between the respiration rate and [ADP]. Computer simulations also suggested that the second-order dependence of oxidative phosphorylation on [ADP] proposed in the literature refers only to the ATP synthesis flux, but not to the oxygen consumption flux (the difference between these two fluxes being due to the proton leak). Next, time courses of changes in fluxes and metabolite concentrations during transition between different steady-states were simulated. The model suggests, in accordance with previous theoretical predictions, that activation of oxidative phosphorylation by an increase in [ADP] can (roughly) explain the behaviour of the system only at low work intensities, while at higher work intensities parallel activation of different steps of oxidative phosphorylation is involved.

Mathematical analysis of a mechanism for autonomous metabolic oscillations in continuous culture of Saccharomyces cerevisiae

Autonomous metabolic oscillations were observed in aerobic continuous culture of Saccharomyces cerevisiae. Experimental investigation of the underlying mechanism revealed that several pathways and regulatory couplings are involved. Here a hypothetical mechanism including the sulfate assimilation pathway, ethanol degradation and respiration is transformed into a mathematical model. Simulations confirm the ability of the model to produce limit cycle oscillations which reproduce most of the characteristic features of the system.

Theoretical studies on the regulation of oxidative phosphorylation in intact tissues

The theoretical studies on the regulation of oxidative phosphorylation that were performed with the aid of kinetic models of this process are overviewed. A definition of the regulation of the flux through a metabolic pathway is proposed and opposed to the control exerted by particular enzymes over this flux. Different kinetic models of oxidative phosphorylation proposed in the literature are presented, of which only the model proposed by myself and co-workers was extensively used in theoretical studies on the regulation and compensation in the oxidative phosphorylation system. These theoretical studies have led to the following conclusions: (1) in isolated mitochondria, an increase in the activity of an artificial ATP-using system stimulates mitochondria mainly via changes in [ADP], while changes in [ATP] and [P(i)] play only a minor role; (2) in non-excitable tissues (e.g. liver), hormones (acting via some cytosolic factor(s)) activate directly both ATP usage and at least some enzymes of the ATP-producing block; (3) in excitable tissues (e.g. skeletal muscle), neural signals stimulate (via some cytosolic factor(s)) in parallel all the steps of oxidative phosphorylation together with ATP usage and substrate dehydrogenation; (4) the decrease in the flux through cytochrome oxidase caused by a decrease in oxygen concentration is, at least partially, compensated by a decrease in Delta p and increase in the reduction level of cytochrome c. A theoretical prediction is formulated that there should exist and be observable a universal cytosolic factor/regulatory mechanism which directly activates (at least in excitable tissues) all complexes of oxidative phosphorylation during an increased energy demand.

An elementary kinetic model of energy coupling in biological membranes

The purpose of this work is to contribute to the understanding of the fundamental kinetic properties of the processes of energy coupling in biological membranes. For this, we consider a model of a microorganism that, in its plasma membrane, expresses two electrogenic enzymes (E(1) and E(2)) transporting the same monovalent cation C and electrodiffusive paths for C and for a monovalent anion A. E(1) (E(2)) couples transport C to the reaction S(1)<-->P(1) (S(2)<-->P(2)). We developed a mathematical model that describes the rate of change of the electrical potential difference across the membrane, of the internal concentrations of C and A, and of the concentrations of S(2) and P(2). The enzymes are incorporated via two-state kinetic models; the passive ionic fluxes are represented by classical formulations of electrodiffusion. The microorganism volume is maintained constant by accessory regulatory devices. The model is utilized for stationary and dynamic studies for the case of bacteria employing the electrochemical gradient of Na(+) as energetic intermediate. Among other conclusions, the results show that the membrane potential represents the relevant kinetic intermediate for the overall coupling between the energy donor reaction S(1)<-->P(1) and the synthesis of S(2).

Metabolic control of contractile performance in isolated perfused rat heart. Analysis of experimental data by reaction:diffusion mathematical model

The intracellular mechanisms of regulation of energy fluxes and respiration in contracting heart cells were studied. For this, we investigated the workload dependencies of the rate of oxygen consumption and metabolic parameters in Langendorff-perfused isolated rat hearts.(31)P NMR spectroscopy was used to study the metabolic changes during transition from perfusion with glucose to that with pyruvate with and without active creatine kinase system. The experimental results showed that transition from perfusion with glucose to that with pyruvate increased the phosphocreatine content and stability of its level at increased workloads. Inhibition of creatine kinase reaction by 15-min infusion of iodoacetamide decreased the maximal developed tension and respiration rates by a factor of two.(31)P NMR data were analyzed by a mathematical model of compartmentalized energy transfer, which is independent from the restrictions of the classical concept of creatine kinase equilibrium. The analysis of experimental data by this model shows that metabolic stability-constant levels of phosphocreatine, ATP and inorganic phosphate-at increased energy fluxes is an inherent property of the compartmentalized system. This explains the observed substrate specificity by changes in mitochondrial membrane potential. The decreased maximal respiration rate and maximal work output of the heart with inhibited creatine kinase is well explained by the rise in myoplasmic ADP concentration. This activates the adenylate kinase reaction in the myofibrillar space and in the mitochondria to fulfil the energy transfer and signal transmission functions, usually performed by creatine kinase. The activity of this system, however, is not sufficient to maintain high enough energy fluxes. Therefore, there is a kinetic explanation for the decreased maximal respiration rate of the heart with inhibited creatine kinase: i.e. a kinetically induced switch from an efficient energy transfer pathway (PCr-CK system) to a non-efficient one (myokinase pathway) within the energy transfer network of the cell under conditions of low apparent affinity of mitochondria to ADP in vivo. This may result in a significant decrease in the thermodynamic affinity of compartmentalized ATPase systems and finally in heart failure.

Regulation of mitochondrial respiration in heart cells analyzed by reaction-diffusion model of energy transfer

The purpose of this study is to investigate theoretically which intracellular factors may be important for regulation of mitochondrial respiration in working heart cells in vivo. We have developed a model that describes quantitatively the published experimental data on dependence of the rate of oxygen consumption and metabolic state of working isolated perfused rat heart on workload over its physiological range (Williamson JR, Ford G, Illingworth J, Safer B. Circ Res 38, Suppl I, I39-I51, 1976). Analysis of this model shows that for phosphocreatine, creatine, and ATP the equilibrium assumption is an acceptable approximation with respect to their diffusion in the intracellular bulk water phase. However, the ADP concentration changes in the contraction cycle in a nonequilibrium workload-dependent manner, showing the existence of the intracellular concentration gradients. The model shows that workload-dependent alteration of ADP concentration in the compartmentalized creatine kinase system may be taken, together with the changes in P(i) concentration, to be among the major components of the metabolic feedback signal for regulation of respiration in muscle cells.

Theoretical studies on control of oxidative phosphorylation in muscle mitochondria at different energy demands and oxygen concentrations

The mathematical dynamic model of oxidative phosphorylation in muscle mitochondria developed previously was used to calculate the flux control coefficients of particular steps of this process in isolated mitochondria at different amounts of hexokinase and oxygen concentrations. The pattern of control was completely different under different conditions. For normoxic concentration, the main controlling steps in state 4, state 3.5 and state 3 were proton leak, ATP usage (hexokinase) and complex III, respectively. The pattern of control in state 4 was not changed at hypoxic oxygen concentration, while in state 3.5 and state 3 much of the control was shifted from other steps to cytochrome oxidase. The implications of the theoretical results obtained for the regulation of oxidative phosphorylation in intact muscle are discussed.

Theoretical studies on the control of oxidative phosphorylation in muscle mitochondria: application to mitochondrial deficiencies

1. The dynamic model of oxidative phosphorylation developed previously for rat liver mitochondria incubated with succinate was adapted for muscle mitochondria respiring on pyruvate. We introduced the following changes considering: (1) a higher external ATP/ADP ratio and an ATP/ADP carrier less displaced from equilibrium; (2) a substrate dehydrogenation more sensitive to the NADH/NAD+ ratio; and (3) the respiratory chain, ATP synthase and phosphate carrier being more displaced from equilibrium. The experimental flux control coefficients already determined in state 3 for respiratory rate and ATP synthesis were used to adjust some parameters. This new oxidative phosphorylation model enabled us to simulate the whole titration curves obtained experimentally in state 3. These curves, which mimic the effect of mitochondrial complex deficiencies on oxidative phosphorylation, show a threshold effect, which is reproduced by the model. 2. the model was also used to simulate other physiological conditions such as (i) state 3.5, conditions in-between state 4 and state 3; and (ii) hypoxic conditions. In both cases a profound change in the pattern of the control coefficients was shown. 3. This model was thus found useful in investigating a variety of new conditions, the most interesting of which can then be experimentally studied.

Regulation of cytochrome oxidase: theoretical studies

Using the dynamic model of oxidative phosphorylation developed previously and tested for its validity under a broad range of conditions some properties of cytochrome oxidase in the whole system considered were simulated. The regulation of this enzyme by oxygen concentration, delta p and reduction level of cytochrome c were studied. Assuming at least qualitative validity of the model, the following conclusions were drawn: (1) Regulation of cytochrome oxidase is different under the same conditions, when changes in the system (oxidative phosphorylation in isolated mitochondria) are imposed by a decrease in oxygen concentration (aerobiosis-->anaerobiosis transition) or by addition of hexokinase (state 4-->state 3 transition). In the former case, cytochrome c and delta p play a very similar role in the compensation for a decrease in the respiration rate caused by lowered oxygen concentration, while in the latter case changes in delta p activate cytochrome oxidase much stronger than changes in the reduction level of cytochrome c. (2) There is no unique thermodynamic flux-force relationship for cytochrome oxidase. This relationship depends on how the thermodynamic span of the reaction catalyzed by this enzyme is changed (aerobiosis-->anaerobiosis transition vs. state 4-->state 3 transition). (3) Under some conditions (aerobiosis-->anaerobiosis transition) the flux-force relationship can be inverse, i.e. increase in a thermodynamic force occurs simultaneously with decrease in a reaction rate.

Simulation of oxidative phosphorylation in hepatocytes

The dynamic mathematical model of oxidative phosphorylation proposed previously was modified, developed and further tested. The description of cytochrome oxidase kinetics was changed to involve dependence on Deltap. Simple, phenomenological descriptions of the kinetics of substrate dehydrogenation and ATP usage, able to reflect experimental data correctly, were found. The kinetic response of the oxidation subsystem (substrate dehydrogenation, respiratory chain), phosphorylation subsystem (ATP synthase, ATP/ADP carrier, phosphate carrier, ATP usage) and proton leak to the changes of Deltap in isolated hepatocytes incubated with different respiratory substrates was simulated. The simulations revealed a good agreement with the experimental results. Simple, intuitive assumptions were able, when introduced into the model, to explain differences in the properties of the oxidative phosphorylation system working with different respiratory substrates. It was proposed, therefore, that our explicit understanding of the oxidative phosphorylation system was good enough to explain many properties of this system correctly, at least in the range of physiological conditions tested.

Simulation of state 4 → state 3 transition in isolated mitochondria

The mathematical dynamic model of oxidative phosphorylation developed previously and in the accompanying paper was modified to involve isolated mitochondria conditions; it was also used to simulate state 4 --> state 3 transition in rat liver mitochondria incubated with succinate as respiratory substrate and glucose-hexokinase as an ADP-regenerating system. Changes in the respiration rate, protonmotive force and reduction level of ubiquinone and cytochrome c as well as the internal (i) and external (e) ATP/ADP ratio between state 4 and state 3 were calculated and compared with the experimental data. Flux control coefficients with respect to oxygen consumption flux for different reactions and processes of oxidative phosphorylation were simulated for different values of the respiration rate (state 4, state 3 and intermediate states). Flux control coefficients for the oxidation, phosphorylation and proton leak subsystems with respect to the oxidation, phosphorylation and proton leak fluxes for different values of the respiration rate were also calculated. These theoretical data were compared with the experimental results obtained in the frame of metabolic control analysis and the 'top-down' approach to this analysis. A good agreement was obtained. Simulated time courses of the respiration rate, the protonmotive force (Deltap) and other parameters after addition of a small amount of ADP to mitochondria in state 4 mimicked at least semiquantitatively the experimentally measured time courses of these parameters. It was concluded, therefore, that in the present stage, the model is able to reflect different properties of the oxidative phosphorylation system in a broad range of conditions fairly well, allows deeper insight into the mechanisms responsible for control and regulation of this process, and can be used for simulation of new experiments, thus inspiring experimental verification of the theoretical predictions.

Proportional activation coefficients during stimulation of oxidative phosphorylation by lactate and pyruvate or by vasopressin

A 'proportional activation' approach designed to deal with the influence of external effectors within biochemical systems is described. The proportional activation coefficient, which enables the quantitative determination of the relative stimulation of different parts of a system by a given effector, is defined. The proportional activation approach was used to calculate the relative activation of delta p-producing and delta p-consuming subsystems during stimulation of the respiration rate of cells by a variety of different effectors. Oxidative phosphorylation was stimulated by the addition of either lactate and pyruvate (10 mM and 1 mM) or vasopressin. The addition of lactate and pyruvate to suspensions of resting hepatocytes increased the respiration rate by about 50%. The delta p-consuming subsystem was stimulated about 60% as much as the delta p-producing subsystem. Quinolinic acid, commonly considered to be a specific inhibitor of gluconeogenesis, was found to block the delta p-producing oxidative subsystem as well as the delta p-consuming subsystem, indicating some nonspecific effects of this inhibitor. Addition of vasopressin to hepatocytes that were incubated in the presence of lactate and pyruvate resulted in an increase of the respiratory rate by up to 35%. The relative stimulation of the delta p-producing and delta p-consuming subsystems was essentially equal. Using the 'proportional activation approach' to analyse these and previously published data, it is shown that substrates (lactate/pyruvate and fatty acids), Ca(2+)-acting hormones (vasopressin and others) and calcium in muscles (heart muscle and skeletal muscle) activate both subsystems to a similar extent (it concerns especially Ca(2+)-dependent systems).

Theoretical studies on the control of the oxidative phosphorylation system

The dynamic model developed in our previous publications [1,2] was used to calculate the flux control coefficients of oxidation, phosphorylation and proton leak fluxes for isolated mitochondria and for three modes of work of intact cells (hepatocytes). The results obtained were compared with experimental data, especially those measured in the frame of the 'top-down approach' of the metabolic control theory. A good agreement for mitochondria and for intact cells was found. The control of the oxygen consumption flux is shared between the ATP utilization (main controlling factor), substrate dehydrogenation, proton leak and, in some conditions, the ATP/ADP carrier. The phosphorylation subsystem seemed to be controlled mainly by itself, while the proton leak was influenced by all three subsystems. It was also shown that the large relative change in the enzyme activity during inhibitor titration of mitochondria or cells could lead to the overestimation of some flux control coefficient values in experimental measurements. An influence of some hormones (glucagon, vasopressin, adrenaline and others) on the mitochondrial respiration was also simulated. Our results suggest that these hormones stimulate the substrate dehydrogenation as well as the phosphorylation system (ATP usage and, possibly, the ATP/ADP carrier).

Control of respiration and ATP synthesis in mammalian mitochondria and cells

We have seen that there is no simple answer to the question 'what controls respiration?' The answer varies with (a) the size of the system examined (mitochondria, cell or organ), (b) the conditions (rate of ATP use, level of hormonal stimulation), and (c) the particular organ examined. Of the various theories of control of respiration outlined in the introduction the ideas of Chance & Williams (1955, 1956) give the basic mechanism of how respiration is regulated. Increased ATP usage can cause increased respiration and ATP synthesis by mass action in all the main tissues. Superimposed on this basic mechanism is calcium control of matrix dehydrogenases (at least in heart and liver), and possibly also of the respiratory chain (at least in liver) and ATP synthase (at least in heart). In many tissues calcium also stimulates ATP usage directly; thus calcium may stimulate energy metabolism at (at least) four possible sites, the importance of each regulation varying with tissue. Regulation of multiple sites may occur (from a teleological point of view) because: (a) energy metabolism is branched and thus proportionate regulation of branches is required in order to maintain constant fluxes to branches (e.g. to proton leak or different ATP uses); and/or (b) control over fluxes is shared by a number of reactions, so that large increases in flux requires stimulation at multiple sites because each site has relatively little control. Control may be distributed throughout energy metabolism, possibly due to the necessity of minimizing cell protein levels (see Brown, 1991). The idea that energy metabolism is regulated by energy charge (as proposed by Atkinson, 1968, 1977) is misleading in mammals. Neither mitochondrial ATP synthesis nor cellular ATP usage is a unique function of energy charge as AMP is not a significant regulator (see for example Erecinska et al., 1977). The near-equilibrium hypothesis of Klingenberg (1961) and Erecinska & Wilson (1982) is partially correct in that oxidative phosphorylation is often close to equilibrium (apart from cytochrome oxidase) and as a consequence respiration and ATP synthesis are mainly regulated by (a) the phosphorylation potential, and (b) the NADH/NAD+ ratio. However, oxidative phosphorylation is not always close to equilibrium, at least in isolated mitochondria, and relative proximity to equilibrium does not prevent the respiratory chain, the proton leak, the ATP synthase and ANC having significant control over the fluxes. Thus in some conditions respiration rate correlates better with [ADP] than with phosphorylation potential, and may be relatively insensitive to mitochondrial NADH/NAD+ ratio.(ABSTRACT TRUNCATED AT 400 WORDS)