Oxidative ATP synthesis in skeletal muscle is controlled by substrate feedback

Multiple ion binding equilibria, reaction kinetics, and thermodynamics in dynamic models of biochemical pathways

The operation of biochemical systems in vivo and in vitro is strongly influenced by complex interactions between biochemical reactants and ions such as H(+), Mg(2+), K(+), and Ca(2+). These are important second messengers in metabolic and signaling pathways that directly influence the kinetics and thermodynamics of biochemical systems. Herein we describe the biophysical theory and computational methods to account for multiple ion binding to biochemical reactants and demonstrate the crucial effects of ion binding on biochemical reaction kinetics and thermodynamics. In simulations of realistic systems, the concentrations of these ions change with time due to dynamic buffering and competitive binding. In turn, the effective thermodynamic properties vary as functions of cation concentrations and important environmental variables such as temperature and overall ionic strength. Physically realistic simulations of biochemical systems require incorporating all of these phenomena into a coherent mathematical description. Several applications to physiological systems are demonstrated based on this coherent simulation framework.

Models of muscle contraction and energetics

How does skeletal muscle manage to regulate the pathways of ATP synthesis during large-scale changes in work rate while maintaining metabolic homeostasis remains unknown. The classic model of metabolic regulation during muscle contraction states that accelerating ATP utilization leads to increasing concentrations of ADP and Pi, which serve as substrates for oxidative phosphorylation and thus accelerate ATP synthesis. An alternative model states that both the ATP demand and ATP supply pathways are simultaneously activated. Here, we review experimental and computational models of muscle contraction and energetics at various organizational levels and compare them with respect to their pros and cons in facilitating understanding of the regulation of energy metabolism during exercise in the intact organism.

Putting the pieces together in diabetes research: towards a hierarchical model of whole-body glucose homeostasis

Type 2 diabetes is one of the most widespread and rapidly spreading diseases world-wide and has been subject of extensive research efforts. However, understanding the molecular basis of the disease is increasing piecemeal and a consensus regarding the overall picture of normal metabolic regulation and malfunction in diabetes has not emerged. A systems biology approach, combining mathematical modelling with simultaneous high-throughput measurements, can be of considerable help. On the whole-body level, this has been done in pharmacokinetic and pharmacodynamic models, which recently have started to mature into more physiologically realistic organ-based models. At the other end of the spectrum, detailed models for crucial cellular processes are starting to mature into complete modules that potentially can be fitted into such whole-body organ-based models. The result of such a merge is a multi-level hierarchical model, which is a model type that has been common in technical systems. In this review, we report and exemplify some of the recent progress that has been made to achieve such a hierarchical model, and we argue why it is only through such a model that a complete picture of diabetes mellitus can be obtained.

Regulation of mitochondrial respiration by inorganic phosphate; comparing permeabilized muscle fibers and isolated mitochondria prepared from type-1 and type-2 rat skeletal muscle

ADP is generally accepted as a key regulator of oxygen consumption both in isolated mitochondria and in permeabilized fibers from skeletal muscle. The present study explored inorganic phosphate in a similar regulatory role. Saponin permeabilized fibers and isolated mitochondria from type-I and type-II muscle from male Wistar rats were prepared. Respiration was measured while the medium P(i) concentration was gradually increased. The apparent K(m) values for P(i) were 607 +/- 17 microM and 405 +/- 15 microM (P < 0.0001) for type-I and type-II fibers, respectively. For isolated mitochondria the values were significantly lower than type-1 permeabilized fibers, 338 +/- 130 microM and 235 +/- 30 microM (P < 0.05), but not different with respect to fiber type. The reason for this difference in K(m) values in the permeabilized muscle is unknown, but a similar pattern has been observed for K(m) of ADP. Our data indicate that phosphate may play a role in regulation of oxygen consumption in vitro and in vivo.

In vivo modular control analysis of energy metabolism in contracting skeletal muscle

We used (31)P MRS (magnetic resonance spectroscopy) measurements of energetic intermediates [ATP, P(i) and PCr (phosphocreatine)] in combination with the analytical tools of metabolic control analysis to study in vivo energy metabolism in the contracting skeletal muscle of anaesthetized rats over a broad range of workload. According to our recent MoCA (modular control analysis) used to describe regulatory mechanisms in beating heart, we defined the energetic system of muscle contraction as two modules (PCr-Producer and PCr-Consumer) connected by the energetic intermediates. Hypoxia and electrical stimulation were used in this in vivo study as reasonably selective modulations of Producer and Consumer respectively. As quantified by elasticity coefficients, the sensitivities of each module to PCr determine the control of steady-state contractile activity and metabolite concentrations. The magnitude of the elasticity of the producer was high (4.3+/-0.6) at low workloads and decreased 5-fold (to 0.9+/-0.2) at high workloads. By contrast, the elasticity of the consumer remained low (0.5-1.2) over the range of metabolic rates studied. The control exerted by each module over contraction was calculated from these elasticities. The control of contraction was found on the consumer at low workloads and then swung to the producer, due to the workload-dependent decrease in the elasticity of producer. The workload-dependent elasticity and control pattern of energy production in muscle is a major difference from heart. Since module rate and elasticity depend on the concentrations of substrates and products, the absence of homoeostasis of the energetic intermediates in muscle, by contrast with heart, is probably the origin of the workload-dependent elasticity of the producer module.

Role of NADH/NAD+ transport activity and glycogen store on skeletal muscle energy metabolism during exercise: in silico studies

Skeletal muscle can maintain ATP concentration constant during the transition from rest to exercise, whereas metabolic reaction rates may increase substantially. Among the key regulatory factors of skeletal muscle energy metabolism during exercise, the dynamics of cytosolic and mitochondrial NADH and NAD+ have not been characterized. To quantify these regulatory factors, we have developed a physiologically based computational model of skeletal muscle energy metabolism. This model integrates transport and reaction fluxes in distinct capillary, cytosolic, and mitochondrial domains and investigates the roles of mitochondrial NADH/NAD+ transport (shuttling) activity and muscle glycogen concentration (stores) during moderate intensity exercise (60% maximal O2 consumption). The underlying hypothesis is that the cytosolic redox state (NADH/NAD+) is much more sensitive to a metabolic disturbance in contracting skeletal muscle than the mitochondrial redox state. This hypothesis was tested by simulating the dynamic metabolic responses of skeletal muscle to exercise while altering the transport rate of reducing equivalents (NADH and NAD+) between cytosol and mitochondria and muscle glycogen stores. Simulations with optimal parameter estimates showed good agreement with the available experimental data from muscle biopsies in human subjects. Compared with these simulations, a 20% increase (or approximately 20% decrease) in mitochondrial NADH/NAD+ shuttling activity led to an approximately 70% decrease (or approximately 3-fold increase) in cytosolic redox state and an approximately 35% decrease (or approximately 25% increase) in muscle lactate level. Doubling (or halving) muscle glycogen concentration resulted in an approximately 50% increase (or approximately 35% decrease) in cytosolic redox state and an approximately 30% increase (or approximately 25% decrease) in muscle lactate concentration. In both cases, changes in mitochondrial redox state were minimal. In conclusion, the model simulations of exercise response are consistent with the hypothesis that mitochondrial NADH/NAD+ shuttling activity and muscle glycogen stores affect primarily the cytosolic redox state. Furthermore, muscle lactate production is regulated primarily by the cytosolic redox state.

Metabolic dynamics in skeletal muscle during acute reduction in blood flow and oxygen supply to mitochondria: in-silico studies using a multi-scale, top-down integrated model

Control mechanisms of cellular metabolism and energetics in skeletal muscle that may become evident in response to physiological stresses such as reduction in blood flow and oxygen supply to mitochondria can be quantitatively understood using a multi-scale computational model. The analysis of dynamic responses from such a model can provide insights into mechanisms of metabolic regulation that may not be evident from experimental studies. For the purpose, a physiologically-based, multi-scale computational model of skeletal muscle cellular metabolism and energetics was developed to describe dynamic responses of key chemical species and reaction fluxes to muscle ischemia. The model, which incorporates key transport and metabolic processes and subcellular compartmentalization, is based on dynamic mass balances of 30 chemical species in both capillary blood and tissue cells (cytosol and mitochondria) domains. The reaction fluxes in cytosol and mitochondria are expressed in terms of a general phenomenological Michaelis-Menten equation involving the compartmentalized energy controller ratios ATP/ADP and NADH/NAD(+). The unknown transport and reaction parameters in the model are estimated simultaneously by minimizing the differences between available in vivo experimental data on muscle ischemia and corresponding model outputs in coupled with the resting linear flux balance constraints using a robust, nonlinear, constrained-based, reduced gradient optimization algorithm. With the optimal parameter values, the model is able to simulate dynamic responses to reduced blood flow and oxygen supply to mitochondria associated with muscle ischemia of several key metabolite concentrations and metabolic fluxes in the subcellular cytosolic and mitochondrial compartments, some that can be measured and others that can not be measured with the current experimental techniques. The model can be applied to test complex hypotheses involving dynamic regulation of cellular metabolism and energetics in skeletal muscle during physiological stresses such as ischemia, hypoxia, and exercise.

Phosphate metabolite concentrations and ATP hydrolysis potential in normal and ischaemic hearts

To understand how cardiac ATP and CrP remain stable with changes in work rate - a phenomenon that has eluded mechanistic explanation for decades - data from (31)phosphate-magnetic resonance spectroscopy ((31)P-MRS) are analysed to estimate cytoplasmic and mitochondrial phosphate metabolite concentrations in the normal state, during high cardiac workstates, during acute ischaemia and reactive hyperaemic recovery. Analysis is based on simulating distributed heterogeneous oxygen transport in the myocardium integrated with a detailed model of cardiac energy metabolism. The model predicts that baseline myocardial free inorganic phosphate (P(i)) concentration in the canine myocyte cytoplasm - a variable not accessible to direct non-invasive measurement - is approximately 0.29 mm and increases to 2.3 mm near maximal cardiac oxygen consumption. During acute ischaemia (from ligation of the left anterior descending artery) P(i) increases to approximately 3.1 mm and ATP consumption in the ischaemic tissue is reduced quickly to less than half its baseline value before the creatine phosphate (CrP) pool is 18% depleted. It is determined from these experiments that the maximal rate of oxygen consumption of the heart is an emergent property and is limited not simply by the maximal rate of ATP synthesis, but by the maximal rate at which ATP can be synthesized at a potential at which it can be utilized. The critical free energy of ATP hydrolysis for cardiac contraction that is consistent with these findings is approximately -63.5 kJ mol(-1). Based on theoretical findings, we hypothesize that inorganic phosphate is both the primary feedback signal for stimulating oxidative phosphorylation in vivo and also the most significant product of ATP hydrolysis in limiting the capacity of the heart to hydrolyse ATP in vivo. Due to the lack of precise quantification of P(i) in vivo, these hypotheses and associated model predictions remain to be carefully tested experimentally.

Modeling cellular metabolism and energetics in skeletal muscle: large-scale parameter estimation and sensitivity analysis

Skeletal muscle plays a major role in the regulation of whole-body energy metabolism during physiological stresses such as ischemia, hypoxia, and exercise. Current experimental techniques provide relatively little in vivo data on dynamic responses of metabolite concentrations and metabolic fluxes in skeletal muscle to such physiological stimuli. As a complementary approach to experimental measurements and as a framework for quantitatively analyzing available in vivo data, a physiologically based model of skeletal muscle cellular metabolism and energetics is developed. This model, which incorporates key transport and reaction processes, is based on dynamic mass balances of 30 chemical species in capillary (blood) and tissue (cell) domains. The reaction fluxes in the cellular domain are expressed in terms of a generalized Michaelis?Menten equation involving energy controller ratios ATP/ADP and ATP/ADP and NADH/NAD+ . This formalism introduces a large number of unknown parameters ( approximately 90). Estimating these parameters from in vivo sparse data and evaluating dynamic sensitivities of the model outputs with respect to these parameters is a challenging problem. Parameter estimation is accomplished using an efficient, nonlinear, constraint-based, optimization algorithm that minimizes differences between available experimental data and corresponding model outputs by explicitly utilizing equality constraints on resting fluxes and concentrations. With the estimated parameter values, the model is able to simulate dynamic responses to reduced blood flow (ischemia) of key metabolite concentrations and metabolic fluxes, both measured and nonmeasured. A general parameter sensitivity analysis is carried out to determine and characterize the parameters having the most and least effects on the measured outputs.

Analysis of cardiac mitochondrial Na+-Ca2+ exchanger kinetics with a biophysical model of mitochondrial Ca2+ handling suggests a 3:1 stoichiometry

Calcium is a key ion and is known to mediate signalling pathways between cytosol and mitochondria and modulate mitochondrial energy metabolism. To gain a quantitative, biophysical understanding of mitochondrial Ca(2+) regulation, we developed a thermodynamically balanced model of mitochondrial Ca(2+) handling and bioenergetics by integrating kinetic models of mitochondrial Ca(2+) uniporter (CU), Na(+)-Ca(2+) exchanger (NCE), and Na(+)-H(+) exchanger (NHE) into an existing computational model of mitochondrial oxidative phosphorylation. Kinetic flux expressions for the CU, NCE and NHE were developed and individually parameterized based on independent data sets on flux rates measured in purified mitochondria. While available data support a wide range of possible values for the overall activity of the CU in cardiac and liver mitochondria, even at the highest estimated values, the Ca(2+) current through the CU does not have a significant effect on mitochondrial membrane potential. This integrated model was then used to analyse additional data on the dynamics and steady-states of mitochondrial Ca(2+) governed by mitochondrial CU and NCE. Our analysis of the data on the time course of matrix free [Ca(2+)] in respiring mitochondria purified from rabbit heart with addition of different levels of Na(+) to the external buffer medium (with the CU blocked) with two separate models--one with a 2:1 stoichiometry and the other with a 3:1 stoichiometry for the NCE--supports the hypothesis that the NCE is electrogenic with a stoichiometry of 3:1. This hypothesis was further tested by simulating an additional independent data set on the steady-state variations of matrix free [Ca(2+)] with respect to the variations in external free [Ca(2+)] in purified respiring mitochondria from rat heart to show that only the 3:1 stoichiometry model predictions are consistent with the data. Based on these analyses, it is concluded that the mitochondrial NCE is electrogenic with a stoichiometry of 3:1.

Computational biology of cardiac myocytes: proposed standards for the physiome

Predicting information about human physiology and pathophysiology from genomic data is a compelling, but unfulfilled goal of post-genomic biology. This is the aim of the so-called Physiome Project and is, undeniably, an ambitious goal. Yet if we can exploit even a small proportion of the rich and varied experimental data currently available, significant insights into clinically important aspects of human physiology will follow. To achieve this requires the integration of data from disparate sources into a common framework. Extrapolation of available data across species, laboratory techniques and conditions requires a quantitative approach. Mathematical models allow us to integrate molecular information into cellular, tissue and organ-level, and ultimately clinically relevant scales. In this paper we argue that biophysically detailed computational modelling provides the essential tool for this process and, furthermore, that an appropriate framework for annotating, databasing and critiquing these models will be essential for the development of integrative computational biology.

Model of oxygen transport and metabolism predicts effect of hyperoxia on canine muscle oxygen uptake dynamics

Previous studies have shown that increased oxygen delivery, via increased convection or arterial oxygen content, does not speed the dynamics of oxygen uptake, V̇o2m, in dog muscle electrically stimulated at a submaximal metabolic rate. However, the dynamics of transport and metabolic processes that occur within working muscle in situ is typically unavailable in this experimental setting. To investigate factors affecting V̇o2m dynamics at contraction onset, we combined dynamic experimental data across working muscle with a mechanistic model of oxygen transport and metabolism in muscle. The model is based on dynamic mass balances for O2, ATP, and PCr. Model equations account for changes in cellular ATPase, oxidative phosphorylation, and creatine kinase fluxes in skeletal muscle during exercise, and cellular respiration depends on [ADP] and [O2]. Model simulations were conducted at different levels of arterial oxygen content and blood flow to quantify the effects of convection and diffusion of oxygen on the regulation of cellular respiration during step transitions from rest to isometric contraction in dog gastrocnemius muscle. Simulations of arteriovenous O2 differences and V̇o2m dynamics were successfully compared with experimental data (Grassi B, Gladden LB, Samaja M, Stary CM, Hogan MC. J Appl Physiol 85: 1394–1403, 1998; and Grassi B, Gladden LB, Stary CM, Wagner PD, Hogan MC. J Appl Physiol 85: 1404–1412, 1998), thus demonstrating the validity of the model, as well as its predictive capability. The main findings of this study are: 1) the estimated dynamic response of oxygen utilization at contraction onset in muscle is faster than that of oxygen uptake; and 2) hyperoxia does not accelerate the dynamics of diffusion and consequently muscle oxygen uptake at contraction onset due to the hyperoxia-induced increase in oxygen stores. These in silico derived results cannot be obtained from experimental observations alone.

Computer modeling of mitochondrial tricarboxylic acid cycle, oxidative phosphorylation, metabolite transport, and electrophysiology

A computational model of mitochondrial metabolism and electrophysiology is introduced and applied to analysis of data from isolated cardiac mitochondria and data on phosphate metabolites in striated muscle in vivo. This model is constructed based on detailed kinetics and thermodynamically balanced reaction mechanisms and a strict accounting of rapidly equilibrating biochemical species. Since building such a model requires introducing a large number of adjustable kinetic parameters, a correspondingly large amount of independent data from isolated mitochondria respiring on different substrates and subject to a variety of protocols is used to parameterize the model and ensure that it is challenged by a wide range of data corresponding to diverse conditions. The developed model is further validated by both in vitro data on isolated cardiac mitochondria and in vivo experimental measurements on human skeletal muscle. The validated model is used to predict the roles of NAD and ADP in regulating the tricarboxylic acid cycle dehydrogenase fluxes, demonstrating that NAD is the more important regulator. Further model predictions reveal that a decrease of cytosolic pH value results in decreases in mitochondrial membrane potential and a corresponding drop in the ability of the mitochondria to synthesize ATP at the hydrolysis potential required for cellular function.

Mitochondrial inner membrane electrophysiology assessed by rhodamine-123 transport and fluorescence

Rhodamine-123 is widely used to make dynamic measurements of mitochondrial membrane potential both in vitro and in situ. Yet data interpretation is difficult due to a lack of quantitative understanding of how membrane potential and measured fluorescence are related. To develop such understanding, a model for dye transport across the mitochondrial inner membrane and partition into the membrane was developed. The model accounts for experimentally measured dye self-quenching and was integrated into a model of mitochondrial electrophysiology to estimate transients in mitochondrial membrane potential from kinetic fluorescence measurements. Our analysis indicates that (i) R123 fluorescence peaks at concentrations near 50 microM due to self-quenching; (ii) measured fluorescence intensity and membrane potential are related by a non-linear calibration curve sensitive to certain experimental details, including total concentration of dye and mitochondria in suspensions; and (iii) the time courses of membrane potential and electron transport fluxes following a perturbation (i.e. addition of ADP) significantly differ from observed transients in fluorescence intensity. These findings are consistent with the model predictions that mitochondria display a characteristic time of response to changes in substrate concentration of less than 0.1 s, corresponding to the time scale over which the rate of ATP synthesis changes to meet changes in ADP concentration.