Open Access

Research Article

• Download: XML | PDF | Citation
• E-mail this Article
• Order Reprints
• Print this Article
• Bookmark this page:
  

Introduction

Skeletal muscle plays a major role in the regulation of whole-body substrates and energy metabolism, especially under changing physiological conditions such as ischemia (reduced blood flow), hypoxia (reduced oxygen supply), and exercise (increased energy demand). Current experimental techniques provide relatively little in vivo data on dynamic responses of muscle metabolite concentrations and metabolic fluxes to such physiological stimuli, especially in subcellular domains, such as mitochondria. To quantitatively analyze available in vivo experimental data and predict nonmeasurable dynamic responses, we developed a physiologically-based, multi-scale computational model of skeletal muscle cellular metabolism and energetics. The model is developed here from our previous model of cellular metabolism and energetics in skeletal muscle [1] and incorporates inter-domain transport processes and compartmentalized metabolic reactions of many key chemical species in both cytosol and mitochondria.

Developing a mechanistic computational model of substrates and energy metabolism in complex, multi-scale metabolic systems, such as skeletal muscle, using a detailed, bottom-up systems approach with sparse in vivo experimental data—with an objective of achieving a quantitative understanding of the system to physiological perturbations—is a challenging task. Such a modeling approach requires information about the general structural features and catalytic mechanisms of the associated transporters and enzymes, subcellular metabolic pathways and fluxes and their control mechanisms, and tissue/organ specific metabolic characteristics. Such a modeling approach also requires mechanistic models for key functional components of the system (e.g., inter-domain transport processes, glycolysis, TCA cycle, oxidative phosphorylation, fatty acid β-oxidation) to be first individually developed and validated and then integrated to emulate the systems behavior at the molecular, subcellular, cellular, and tissue/organ levels. To avoid this complex approach and facilitate analysis of available sparse in vivo experimental data to understand dynamic responses of the system to physiological stresses, approximations are often made to obtain a simplified model of the system that includes key functional components regulating cellular metabolic processes at the desired level of complexity.

A top-down systems approach is an alternative approach which has been previously applied to determine and integrate a representative set of lumped biochemical reactions in vivo metabolic systems that incorporate primary substrates and key intermediate metabolites with coupled metabolic energy controllers ATP-ADP and NADH-NAD⁺ [1]–[8]. This approach is similar to the top-down systems approach in metabolic control analysis proposed by Brand and co-workers [9], [10] and is intended to provide an essential or minimal set of stoichiometrically balanced lumped biochemical reactions participating in ATP synthesis within mitochondria from the metabolism of nutrients (e.g., glucose, fatty acids, amino acids). Even with such simplifications, a large number of phenomenological kinetic parameters are introduced in the governing model equations, which must be estimated from available sparse in vivo experimental data. To estimate these unknown parameters, constraint-based, robust nonlinear optimization methods are needed, as established in our previous work [1].

To date, no physiologically-based, whole-organ level model of skeletal muscle cellular metabolism and energetics has been developed that can be applied to analyze available in vivo experimental data and predict dynamic metabolic responses to physiological stimuli in subcellular compartments, such as mitochondria. Previous models have incorporated some aspects of glycolysis, TCA cycle, oxidative phosphorylation, and fatty acid β-oxidation [3]–[5], [11]–[21]. None of them, however, include sufficient key substrates/metabolites and/or integrate metabolic pathways/reactions that are essential in the regulation of cellular metabolic processes in skeletal muscle in vivo at whole-organ level. While our recently developed models of skeletal muscle cellular metabolism and energetics [1] have incorporated key metabolic pathways and reactions, the intracellular cytosolic and mitochondrial compartments were not distinguished. The energy controller pairs ATP-ADP and NADH-NAD⁺ that modulate several key metabolic reactions in the cytosol and mitochondria have different concentrations in these two subcellular domains [6]–[8]. As a consequence, the model may not accurately predict the dynamics of several metabolite concentrations and metabolic fluxes that are critical in the regulation of fuel (carbohydrate, fat, and lactate) metabolism and cellular respiration during physiological stresses such as ischemia, hypoxia, and exercise.

In this paper, a physiologically-based, whole-organ level model of skeletal muscle cellular metabolism and energetics is developed and applied to study dynamic cellular metabolic responses to reduced blood flow and oxygen supply to mitochondria (muscle ischemia). The model, which is extended from our previous model [1], is based on a multi-scale, top-down systems approach [1]–[8], accounts for subcellular compartmentalization, and includes primary substrates (carbohydrates and fats) and key intermediate metabolites and metabolic reactions specific to skeletal muscle metabolic system. The model equations are based on dynamic mass balances of chemical species in capillary blood and tissue cells (cytosol and mitochondria) domains. The model also distinguishes the free and bound forms of O₂ and CO₂ transport in the blood and cells. The inter-domain species transport processes are considered either by passive diffusion or by carrier-mediated (facilitated) transport. The metabolic reaction fluxes in the cytosolic and mitochondrial domains are represented by a general phenomenological Michaelis-Menten equation involving the compartmentalized ATP/ADP and NADH/NAD⁺ energy controller ratios. The phenomenological kinetic parameters of the model are estimated by using our recently developed constraint-based, robust nonlinear optimization approach [1]. In this estimation process, we fit the model output to sparse in vivo dynamic data on glycolytic and energetic metabolite concentrations from experiments in humans with muscle ischemia previously published [22]. With the estimated optimal parameter values, the model is able to simulate dynamic responses of key chemical species and reaction fluxes to reduced blood flow and oxygen supply to mitochondria associated with the muscle ischemia.

Materials and Methods

Model Development

The first step in our development of a multi-scale, top-down computational model of cellular metabolism and energetics in skeletal muscle was to identify the key intermediate metabolites and regulatory enzymes in the cellular metabolic pathways of skeletal muscle. We then integrated available information on cellular metabolic pathways and fluxes, cellular metabolic control mechanisms, catalytic enzyme kinetic mechanisms, subcellular compartmentation and metabolites volumes of distributions, inter-domain transport mechanisms, and skeletal muscle tissue-specific metabolic characteristics. A simplified map of the compartmentalized cellular metabolic pathways in skeletal muscle is shown in Figure 1. The lumped biochemical reactions in the metabolic pathways are generated by stoichiometrically coupling several sequential elementary reactions. These reactions include the compartmentalized metabolic energy controller pairs ATP-ADP and NADH-NAD⁺ whose ratios are known to modulate (fine tune) the reaction fluxes [2], [23] in the subcellular compartments. Many of these lumped reactions are considered irreversible for which the resting Gibbs free energy (ΔG) is high and negative in favor of product formation [24]. As a part of a general formalism for modeling in vivo metabolic systems (nonequilibrium open systems), the reversible reactions like lactate dehydrogenase (LDH), creatine kinase (CK), and adenylate kinase (AK) were decomposed into two separate irreversible reactions with distinct kinetics [1].

Dynamic mass balance equations

The dynamic mass balance equations are based on a multi-domain model structure for skeletal muscle consisting of a spatially-lumped capillary blood domain which exchanges nutrients and metabolic waste products with a spatially-lumped domain of tissue cells (Figure 2). Although these two domains are separated by the interstitial fluid (ISF) space, we assume phase-equilibrium of chemical species between the blood and ISF domains, and consider them together as the “blood” domain. Furthermore, the tissue cells domain is compartmentalized into the cytosolic and mitochondrial domains. The chemical species are assumed to be distributed in these two subcellular domains as per their mass fractions and volumes of distributions (Table 1). Here, the mass fraction of a species in a particular domain is defined as the fractional amount of the species in that domain in comparison to the total amount of the species in the whole muscle tissue cells. The volume of distribution of a species in a particular domain is defined as the anatomical volume of the domain plus the binding space of the domain for the species. In addition, the species that are common to both of these domains are assumed to have the similar dynamics in these two domains, because the species transport processes between these two domains can be sufficiently fast [5]. Consequently, a change in the species concentration in one domain will be proportional to the change in the species concentration in the other domain: C_mit,j(t) = σ_j.C_cyt,j(t), where σ_j is the equilibrium concentration ratio (or partition coefficient) of species j between mitochondria and cytosol. In the present model, a total of 10 chemical species (pyruvate, fatty acyl-CoA, CoA, ATP, ADP, inorganic phosphate, NADH, NAD⁺, O₂, and CO₂) are considered to exist in both the cytosolic and mitochondrial compartments with negligible transport flux between the compartments.

The dynamic mass balance of a chemical species j in the spatially-lumped blood domain has the following general form:
(1)
where C_art,j is the arterial species concentration; C_bl,j is the capillary blood species concentration (equal to the venous species concentration C_ven,j); V_bl,j and V_isf,j are the volumes of distribution of species j in blood and ISF, and Q is the regional blood flow; J_bl↔cyt,j is the net transport flux (mass per unit time) across the blood-cytosol exchange barrier (consisting of capillary membrane, ISF, and tissue cell membrane).

The dynamic mass balance of the chemical species j in the spatially-lumped tissue cells domain (cytosol and/or mitochondria) has the following general forms:
(2a)

(2b)

(2c)
where
(2d)

Here C_x,j is the species concentration in domain x (cytosol/mitochondria); V_x,j is the volume of distribution of species j in domain x; P_x,j and U_x,j are the production and utilization of species j in domain x; φ_x,p and φ_x,u are the reaction fluxes of the reactions processes that produce and utilize species j in domain x; β_x,j,p and β_x,j,u are the corresponding stoichiometric coefficients. For chemical species which are in the tissue cells (cytosol and/or mitochondria) but not in the capillary blood, the transport flux J_bl↔cyt,j is zero.

The dynamic mass balance equations for all the chemical species in blood, cytosol and mitochondrial domains can be rewritten from our previous model of skeletal muscle metabolism [1] by accounting for the species compartmentalized volumes of distributions as laid out in Eqs. (1) and (2a–2c). The dynamic mass balance equations for O₂ and CO₂ in these domains are developed by considering their distinct transport and binding mechanisms [25], [26]. The detailed mass balance equations of the chemical species, including O₂ and CO₂, are provided in Materials S1.

Transport and reaction flux equations

The reversible transport flux J_bl↔cyt,j (mass per unit time) of the species j across the blood-cytosol exchange barrier is related to the concentrations C_bl,j and C_cyt,j by
(3)
where λ_bl↔cyt,j is the effective permeability surface area product for diffusive mass transport across the barrier (for passive transport); T_bl↔cyt,j is the maximal transport flux across the barrier (T_max) and M_bl↔cyt,j is the corresponding Michaelis-Menten (M-M) constant (M_m) (for facilitated transport). The flux expressions (3) satisfy the thermodynamic equilibrium conditions across the barrier.

The species involved in the blood-cytosol exchange are glucose, lactate, pyruvate, alanine, glycerol, free fatty acid, CO₂, and O₂ (Table 2). The transport processes may be passive or carrier-mediated (facilitated) (Eq. 3). The transport of glucose, pyruvate/lactate, and free fatty acid across the sarcolemma of skeletal muscle is facilitated via the GLUT4 [27], MCT1 or MCT4 [28], [29], and FABP or FAT/CD36 [30] proteins, respectively. The sarcolemmal transport of the remaining species (alanine, glycerol, CO₂, and O₂) is considered to be passive. For the 10 species that exist in both cytosol and mitochondria, the inter-domain transport flux J_cyt↔mit,j is zero.

The lumped metabolic reactions of skeletal muscle cellular metabolism and energetics can be considered as special cases of a general, irreversible, multi-reactant multi-product enzymatic reaction coupled with the metabolic energy controller pairs:

where P₁ and P₂ are ATP and ADP or vice-versa (PS±:
(4)
phosphorylation state); R₁ and R₂ are NADH and NAD⁺ and vice-versa (RS±: redox state). The corresponding coupled phosphorylation and redox reactions are ATP→ADP (PS+) and/or NADH→NAD⁺ (RS+) or vice-versa (PS- and RS-). Assuming a phenomenological, single-step enzyme kinetic mechanism [31], the flux expression for the lumped metabolic reaction can be written as (see Ref. [1] for detailed description):
(5)
where V_max and K_m are the phenomenological maximal/limiting velocity and M-M parameter of the reaction; are the phenomenological M-M parameters for the coupled phosphorylation and redox reactions; C_P1/C_P2 is the phosphorylation ratio: C_ATP/C_ADP (PS+) or C_ADP/C_ATP (PS-), and C_R1/C_R2 is the redox ratio: C_NADH/C_NAD+ (RS+) or C_NAD+/C_NADH (RS-). So the reaction fluxes are explicitly regulated by the coupled energy controller ratios C_P1/C_P2 and/or C_R1/C_R2 (if coupled).

The flux expressions for all the lumped metabolic reactions in the subcellular compartments of cytosol and mitochondria can be rewritten from our previous model of skeletal muscle metabolism [1] in terms of the compartmentalized metabolites concentrations and energy controller ratios ATP/ADP and NADH/NAD⁺. The details are provided in Materials S2.

In summary, in this mathematical model of skeletal muscle cellular metabolism and energetics, the number of chemical species (primary substrates, intermediate metabolites, and energy controllers) in the tissue cells domain (cytosol and mitochondria) is 30, which participate in 26 metabolic reactions. The number of chemical species that exist in both blood and cells is 8, and that in both cytosol and mitochondria is 10. Accordingly, the model includes 8 J_bl↔cyt,j transport fluxes characterized by 4 λ, 4 T_max and 4 M_m parameters, and 26 φ reaction fluxes characterized by 26 V_max, 27 K_m (1 K_m for G6P inhibition of the hexokinase reaction; see Reaction 1, Materials S2), parameters (as in our previous model [1]). Besides, there are 10 partition coefficients σ for the chemical species that exists in both cytosol and mitochondria (since the 10 J_cyt↔mit,j transport fluxes are considered negligible). Therefore, the present model is characterized by a total of 99 unknown parameters that need to be estimated to fit the model outputs to the available in vivo experimental data. For convenience, the transport and reaction fluxes are written here in vector form: , where the parameter vector for these transport and reaction fluxes is denoted by ; is the concentration vector. Note that the 12 transport parameters (4 λ, 4 T_max and 4 M_m) can be estimated here uniquely from the resting, steady-state transport flux-concentration relationships (Table 3) [1]. The 10 partition coefficients (10 σ) can be estimated uniquely from the resting, steady-state species concentration ratios between the mitochondria and cytosol (σ_j = C_mit,j/C_cyt,j) (Table 3).

Model simulation

With specified parameter values, the mathematical model is solved numerically to simulate dynamic responses of the system to ischemia produced by reducing muscle blood flow. Typically, the initial conditions: are assumed to be at a normal, resting steady-state. These are fixed based on average resting species concentrations gathered from various literature sources on skeletal muscle cellular metabolism that are consistent with the resting, steady-state flux-concentration relationships (Tables 1 and 2). The species mass fractions and volumes of distributions in the subcellular cytosolic and mitochondrial domains are set to have appropriate phosphorylation and redox potentials in the cytosol and mitochondria. For numerical solution of this stiff initial-value problem, a robust implicit integrator DLSODES (https://computation.llnl.gov/casc/odepack/odepack_home.html; http://www.netlib.org/odepack; [32]) is used. Specifically, the Gear's implicit integration method based on backward difference formula (BDF) is most suitable for this problem. An absolute and relative error of tolerance of 10⁻¹⁰ guarantees high accuracy and convergence of the iterative solutions of the ODEs. The DLSODES solver is usually very fast; a typical simulation of this problem using the DLSODES solver in a standard desktop computer (Intel Xeon or Core 2 Duo CPU 5160 @ 3 GHz) takes only about 5 seconds of the CPU time. As a check, the numerical solutions of the initial value problem were also obtained using the ODE15S solver in MATLAB (http://www.mathworks.com) with the similar tolerance levels. These solutions were of comparable accuracy with that obtained using the DLSODES solver.

Parameter Estimation

The large number of unknown parameters of this model are estimated by comparing model outputs to in vivo experimental data of Katz [22] with the optimization procedure described previously [1]. The data consist of key metabolites concentration dynamics measured during circulatory occlusion and recovery (reperfusion) in human skeletal muscle at the whole tissue-organ level [22]. Specifically, the data is based on the biopsy measurements of glucose and glycolytic/glycogenolytic intermediates (i.e., glucose 6-phosphate, pyruvate, and lactate) and creatine and high-energy phosphates (i.e., phosphocreatine, inorganic phosphate, ATP, ADP, and AMP) in quadriceps femoris muscle tissue at rest, after 30 minutes of ischemia, and after 15 minutes of reperfusion. The tissue metabolite contents or concentrations were measured in the units of mmol/kg dry weight. For our analysis, these concentrations are converted to the units of mmol/kg wet weight (mmol/L or mM) by multiplying a conversion factor of 0.25 kg dry weight/kg wet weight, corresponding to the muscle tissue [33]. Consistent with the experimental measurements of Katz [22], the following ischemia protocol is used for model simulations and parameter estimation: the muscle blood flow is reduced from Q = 0.9 L/min at rest (t<0 min) to Q = Q_isch (unknown) at the onset of ischemia (0≤t≤30 min), and then increased to the starting level at the onset of recovery (t>30 min). The active muscle volume (V_mus) and ischemic muscle blood flow (Q_isch) responsible for the predicted metabolic dynamics in skeletal muscle during ischemia and recovery were not known from the experimental study of Katz [22]. Therefore, V_mus and Q_isch are also included as the unknown parameters for estimation, making the total number of unknown parameters in the model to 101. This is an increase of 10 unknown parameters (10 σ) from our previous model [1].

With this relatively sparse data and large number of unknown parameters, the parameter estimation problem is ill-conditioned and under-determined. Nevertheless, an efficient estimation method is devised based on our earlier work [1] to obtain physiologically reasonable parameter values that minimize the sum of squared differences between the available experimental data and corresponding model outputs. The estimation procedure proceeds in two main stages. In the first stage, the published normal, resting species concentration data (Tables 1 and 2) are used to evaluate the resting transport and reaction fluxes from a steady-state flux balance analysis (Tables 2, 3 and 4). From the resting flux-concentration relationships: Eqs. (3) and (5), the preliminary estimates of the transport and reaction parameters are obtained. The preliminary estimates of the 10 partition coefficients are obtained from the resting, steady-state species concentration ratios between mitochondria and cytosol (σ_j = C_mit,j/C_cyt,j) (Table 3). In the second stage, the dynamic species concentration data [22] together with the resting, steady-state flux balance equations as equality constraints are used to obtain the optimal parameter estimates (Tables 3 and 4).

A detailed description of this robust estimation approach for obtaining optimal parameter estimates from species concentration dynamics during muscle ischemia and recovery is presented in Ref. [1]. This method has been established powerful in the analysis of large-scale in vivo metabolic systems. The efficiency and robustness of this parameter estimation approach was tested with parameter sensitivity analysis as well as with repeated parameter estimation with various initial parameter estimates (initial guesses). Various estimates of the more sensitive model parameters spanned in a small neighborhood of the optimal parameter estimates (see Ref. [1] for details). Since the preliminary estimates of the 12 transport parameters and 10 partition coefficients based on the resting species concentrations and transport fluxes were accurate enough, these 22 parameters were not re-estimated from the dynamic species concentration data. Therefore, as in our previous paper [1], a total of 101–22 = 79 parameters (77 reaction parameters+V_mus+Q_isch) are effectively estimated from the dynamic data.

Results

Comparison of model simulations with experimental data

The optimal estimates of the transport parameters (λ,T_max,M_m), partition coefficients (σ), active muscle volume and ischemic muscle blood flow (V_mus, Q_isch), and reaction parameters for the model are shown in Tables 3 and 4. These parameter estimates yield the best fit of the model outputs to the published experimental data [22] with minimal residual errors and minimal objective function. The optimal estimates of V_mus and Q_isch corresponding to the experimental data were ~4.0 L (~16% of normal two-legs muscle volume of ~25 L) and ~0.216 L/min (~76% reduction from normal, resting two-legs muscle blood flow of Q = 0.9 L/min). These optimal parameter values were used for model simulations during the resting, ischemia and recovery periods. The blood flow reduction levels of Q_isch = 0.18, 0.27 and 0.36 L/min (80%, 70% and 60%) were used for simulating severe to moderate to mild ischemic conditions.

The correspondence of model simulations and experimental data is demonstrated through Figures 3 and 4. Specifically, shown are the concentration dynamics of 4 glycolytic metabolites (GLC, G6P, LAC, PYR) and 6 energy metabolites (PCR, CR, PI, ATP, ADP, AMP) in the muscle tissue cells (i.e., weighted volume averages of concentrations in cytosol and mitochondria) for which experimental data were available [22] for four different blood flow reduction levels Q_isch = 0.18, 0.216, 0.27 and 0.36 L/min. Figure 3 also includes the concentration dynamics of NADH and NAD⁺ in the muscle tissue cells. The dynamics of compartmentalized cytosolic and mitochondrial phosphorylation and redox potentials (i.e., [ATP]/[ADP] and [NAD⁺]/[NADH] ratios) and cytosolic [LAC]/[PYR] and [PCR]/[CR] ratios are shown in Figure 5. The level Q_isch = 0.216 L/min corresponds to the experimental data of Katz [22] and the corresponding model simulations match to the data reasonably well within the experimental noise (Table 5). The experimental data are normalized here with respect to the resting metabolites concentrations (control). Furthermore, the individual metabolites responses are shown in separate plots in order to distinguish metabolite responses to different levels of blood flow reductions.

The model simulations of cellular glucose (GLC), glucose-6-phosphate (G6P), pyruvate (PYR), and lactate (LAC) concentrations at the end of 76% ischemia and reperfusion are in close agreement with the experimental data (Fig. 3(A–D), Table 5). Cellular [G6P] increased quickly (exponentially) by ~60% during ischemia and returned rapidly to its resting level at the onset of reperfusion. In contrast, the cellular [GLC] increased slowly (almost linearly) by ~35% during ischemia and remained at an elevated level even after 30 minutes of reperfusion. Blood [GLC], however, decreased rapidly, but only by ~10%, during ischemia and returned quickly to its baseline value during reperfusion (not shown). Furthermore, the model-predicted changes in the cellular [G6P] and [GLC] during the mild 60% and 70% blood flow reduction levels were not significant when compared to the changes during the high 76% and above (i.e., 80%) blood flow reduction levels.

Both cellular [LAC] and [PYR] increased by ~125% during 76% ischemia and almost returned to the resting levels after 30 minutes of reperfusion. However, the dynamic response of [PYR] was different from that of [LAC]. During ischemia, [PYR] first decreased and then increased, but during reperfusion, [PYR] first sharply increased and then rapidly decreased to the resting level. On the other hand, [LAC] slowly (almost linearly) increased during ischemia and slowly (almost linearly) decreased during reperfusion. Blood [LAC] had the similar dynamics as that of cellular [LAC] (not shown). These differential dynamics of cellular [LAC] and [PYR] characterize the dynamics of cellular or cytosolic [LAC]/[PYR] ratio. The dynamics of cytosolic [LAC]/[PYR] ratio and cytosolic and mitochondrial redox states ([NADH] and [NAD⁺]) and redox potentials ([NAD⁺]/[NADH] ratios) were all similar having biphasic behaviors during the ischemia and reperfusion periods (Figs. 3(E,F) and 5(A–C)). This is in contrast to our previous model (in which the cytosolic and mitochondrial compartments were lumped into a single tissue cells compartment [1]) predictions that the dynamic responses of cellular [LAC]/[PYR] and [NAD⁺]/[NADH] ratios to ischemia and reperfusion are distinct. Thus the present compartmentalized model is able to correctly simulate the dynamic responses of cytosolic and cellular [LAC]/[PYR] ratios as well as cytosolic and mitochondrial [NAD⁺]/[NADH] ratios. The cytosolic [LAC]/[PYR] ratio rapidly increased from a resting level of ~16.4 to ~26.8 at the onset of 76% ischemia, then quickly decreased almost to the baseline value (~17.22) as ischemia progressed, in agreement with the data (Table 5). In contrast, the cytosolic [NAD⁺]/[NADH] ratio decreased quickly from a resting level of ~540 to ~265 at the onset of 76% ischemia, then increased and reached a new steady state value of ~332 as ischemia progressed. The LDH mass action ratio ([LAC][NAD⁺])/([PYR][NADH]) decreased considerably from a resting level of ~8856 to ~5585 during 76% ischemia. At different levels of blood flow reductions, the model-predicted changes were not proportional. The changes were negligible below 70% blood flow reduction levels and significant above 76% blood flow reduction levels.

The model simulations of muscle phosphocreatine (PCR), creatine (CR), and inorganic phosphate (PI) concentrations at the end of 76% ischemia and recovery are in good agreement with the experimental data (Fig. 4(A–C)). After 30 minutes, 76% ischemia resulted in ~17.5% decrease in muscle PCR content, which was fully resynthesized after 30 minutes of recovery. Muscle CR and PI contents increased by ~35% and ~130%, respectively, at the end of 76% ischemia and returned to their resting levels at the end of recovery. The dynamics of [PCR] drop during ischemia and rise during recovery were exponential, but faster during the recovery period. The dynamics of both [CR] and [PI] show the opposite trends (both are mirror images of [PCR] having the similar time constants). The cytosolic [PCR]/[CR] ratio decreased substantially from a resting value of ~2 to ~1.2 during ischemia (Fig. 5D, Table 5). The model-predicted muscle [PCR], [CR], [PI] and [PCR]/[CR] ratio during the mild 60% and 70% blood flow reduction levels did not change appreciably from their baseline, resting levels.

The model-simulated muscle [ATP] decreased slightly during ischemia which resulted in appropriate increases in muscle [ADP] and [AMP] (Fig. 4(D–F)). A 76% blood flow reduction resulted in ~5% and ~25% increases in [ADP] and [AMP], respectively, in accordance with the data. The cytosolic and mitochondrial phosphorylation potentials ([ATP]/[ADP] ratios) had the similar dynamics as of the cytosolic [PCR]/[CR] ratio (Fig. 5(D–F)). However, the magnitude of [ATP]/[ADP] decrease (~6%) during ischemia was negligible in comparison to the magnitude of [PCR]/[CR] decrease (~40%). The CK mass action ratio ([CR][ATP])/([PCR][ADP]) increased from a resting value of ~166 to ~260 during the peak ischemia. The total adenylate nucleotide pool (TAN = [ATP]+[ADP]+[AMP]) was maintained at a constant level of ~7.0 mM in accordance with the data (Table 5). The model-predicted changes in muscle [ATP], [ADP], [AMP], and [ATP]/[ADP] ratio during the mild 60% and 70% blood flow reduction conditions were negligible. Thus, in contrast to our previous model [1], the present compartmentalized model is able to simulate the dynamic responses of cytosolic and cellular [PCR]/[CR] ratios as well as cytosolic and mitochondrial [ATP]/[ADP] ratios appropriately.

Simulated dynamics of mitochondrial metabolites

The model-simulated dynamic responses of mitochondrial FAC, ACoA, CIT, AKG, SCoA, SUC, MAL and OXA during the ischemia and recovery periods with blood flow reduction levels of 80%, 76%, 70% and 60% (Q_isch = 0.18, 0.216, 0.27 and 0.36 L/min) are shown in Figure 6. Experimental data were not available for these key mitochondrial and TCA cycle intermediate metabolites. In fact, due to the limitations in the available experimental techniques, most of these subcellular metabolites can not be measured conveniently in skeletal muscle tissue cells in vivo at the whole tissue-organ level. However, the present compartmentalized model is able to simulate the dynamic responses of these subcellular metabolites to physiological stresses such as muscle ischemia. Model simulations show that the changes in these metabolites concentrations during muscle ischemia are not simply proportional to the extent of blood flow reduction. The changes are negligible during the mild 60% and moderate 70% blood flow reduction levels, but significant above the severe 76% blood flow reduction level.