## Abstract

A system analysis of ATP free energy metabolism in skeletal muscle was made using the principles of metabolic control theory. We developed a network model of ATP free energy metabolism in muscle consisting of actomyosin ATPase, sarcoplasmic reticulum (SR) Ca^{2+}-ATPase, and mitochondria. These components were sufficient to capture the major aspects of the regulation of the cytosolic ATP-to-ADP concentration ratio (ATP/ADP) in muscle contraction and had inherent homeostatic properties regulating this free energy potential. As input for the analysis, we used ATP metabolic flux and the cytosolic ATP/ADP at steady state at six contraction frequencies between 0 and 2 Hz measured in human forearm flexor muscle by ^{31}P-NMR spectroscopy. We used the mathematical formalism of metabolic control theory to analyze the distribution of fractional kinetic control of ATPase flux and the ATP/ADP in the network at steady state among the components over this experimental range and an extrapolated range of stimulation frequencies (up to 10 Hz). The control analysis showed that the contractile actomyosin ATPase has dominant kinetic control of ATP flux in forearm flexor muscle over the 0- to 1.6-Hz range of contraction frequencies that resulted in steady states, as determined by ^{31}P-NMR. However, flux control begins to shift toward mitochondria at >1 Hz. This inversion of flux control from ATP demand to ATP supply control hierarchy progressed as the contraction frequency increased past 2 Hz and was nearly complete at 10 Hz. The functional significance of this result is that, at steady state, ATP free energy consumption cannot outstrip the ATP free energy supply. Therefore, this reduced, three-component muscle ATPase system is inherently homeostatic.

- cellular energetics
- skeletal muscle
- metabolic control analysis

a number of approaches have been developed to describe muscle energetics, starting with Hill's analysis of heat and mechanics. Present analyses of muscle energetics obtained by noninvasive ^{31}P-NMR measurements can be expressed as specific biochemical mechanisms. Chance (15, 16) worked out control of mitochondrial oxidative ADP phosphorylation by the cytosolic ADP concentration ([ADP]) with a transfer function connecting muscle work output to phosphocreatine (PCr) content ([PCr]). Meyer (36) analyzed chemical changes in contraction-recovery cycles by analogy with simple electrical circuits. This model connected cytosolic ATPase activity during contraction with [ADP]-controlled net mitochondrial ATP synthase activity by feedback control. In this energy balance system, the creatine kinase reaction functioned as a capacitance. The creatine kinase reaction was constrained to maintain local equilibrium with cellular content of ATP, ADP, PCr, and creatine (Cr) in the models of Chance and Meyer. Later work showed that relaxation of this constraint did not affect the analysis (34, 37). Meyer's model correctly matches experimental observations in rat and feline (36, 37) and human (3, 9, 28, 33, 45) muscle over typical physiological ranges. It also gives a conceptual approach to understand muscle energetics as an interdependent network with feedback for achieving energy balance. Kushmerick (34) recently published a set of equations similar in concept to Meyer's model, which simulated information from human forearm contraction. The features of this model were the inclusion of complete terms for the creatine kinase enzyme kinetics and new information of second-order ADP dependence of mitochondrial ATP synthesis (29). By including specific functions derived from mechanistic studies of components, this model provides a generic way to add additional mechanisms as their properties are defined. Thus the study of muscle energetics and cellular respiration as primary determinants of ATP levels has a rich experimental and analytic history.

It might be concluded from this discussion that we understand muscle energetics at a satisfactory conceptual and mechanistic level, despite continued debate on the mechanisms for controlling cellular respiration (4, 12, 29, 34, 37). However, there remain other, and we believe equally fundamental, aspects of the physiology of muscle cell energetics that are not explained by these models and concepts. These have been ignored so far. One such aspect is the conservation of cytosolic ATP free energy (2) {Δ*G*
_{ATP} = Δ*G*
_{ATP}
^{o′} +*RT*ln([ADP][P_{i}]/[ATP]), where Δ*G*
_{ATP}
^{o′} = −32.8 kJ/mol (41),*R* is the gas constant, *T* is the absolute temperature, and [P_{i}] and [ATP] are P_{i} and ATP concentrations} during contractile activity. ^{31}P-NMR measurements revealed that Δ*G*
_{ATP} in skeletal muscle ranges from approximately −64 kJ/mol at rest to −55 kJ/mol at maximal sustainable contractile activity (31). Thus Δ*G*
_{ATP} in repetitively stimulated muscle maximally falls only on the order of 9 kJ/mol, i.e., less than one-third of the available free energy. Sustained contractile activity means that force of contraction is maintained approximately constant, a condition that is associated with an intracellular pH (pH_{i}) between 7.1 and 6.8 and [PCr] values that are steady at levels lower than at rest (1). It is of course possible to activate contractile activity transiently at higher rates to even lower [PCr] and more acidic pH values, but this induces the non-steady state of fatigue (23, 46). There are several explanations in the literature for this narrow range of sustained muscle performance and narrow range of ATP free energy before fatigue begins (22, 31,32, 40). Each explanation invokes a single but different rate-limiting step in ATP free energy dissipation in contracting muscle: a thermodynamic constraint on sarcoplasmic reticulum (SR) Ca^{2+} pumping (22, 31, 32) and a pH constraint on the rise of [ADP] in contracting muscle (40).

We asked whether the structure, organization, and sensitivity to control of this energetic system was itself sufficient to account for the narrow range of ATP free energies observed. This question requires that the muscle be considered as a network of interdependent components. An analysis of individual component mechanisms cannot answer the question (21). The mathematical theorems of metabolic control analysis (MCA) (21, 26, 30, 39, 47) provide tools for just such an analysis. This formalism considers all enzymes in a metabolic network together and attributes to each component a fractional control strength over the value of each of the variables (fluxes and concentrations) in the network at a particular steady state. MCA has previously not been used to analyze muscle contractile activity. Here we present the results of a control analysis of ATP free energy metabolism in contracting muscle. The analysis was applied to a set of ^{31}P-NMR spectroscopy data on steady-state energetics obtained from human forearm flexor muscle. By this analysis, we could test whether a minimal network model of ATP metabolism in contracting muscle would be homeostatic with respect to the ATP free energy content of the cell, and we could learn how control by one or more of the components achieves regulation of the cytosolic ATP-to-ADP concentration ratio (ATP/ADP).

The results of this analysis show that three components [actomyosin (AM) ATPase, Ca^{2+}-ATPase in the SR (SR ATPase), and mitochondrial ATPase working as a net synthase] are necessary and sufficient to account for the steady-state behavior and narrow range of change in ATP free energy. This simplification means that additional mechanisms cited above embellish the richness of control and complexity but are not necessary for regulation of the system. Furthermore, the results expand on the concepts resulting from previous models by showing a strategy by which it can be determined how and over what range of function muscle energetics can correctly be viewed as an ATP demand providing feedback signals to the mitochondrial ATP supply. As contractile activity progresses toward its maximal sustainable activity, this system analysis shows that regulation of ATP free energy is maintained by redistributing control among the components. Thus quantitative consideration of the degree of control of each component over the performance of the system is crucial to understand energetics in muscle physiology and to show that the details of energetic regulation depend on the particular steady state being analyzed. This metabolic control analysis of muscle physiology also provides an integrative strategy to understand how alterations of muscle properties change system properties. For example, we include in the discussion examples of how alteration of the properties of only one component changes the energetic system as a whole; one of these alterations is a mitochondrial defect. This approach should also be useful to account, in a more systematic and integrative manner, for observations on the physiology of muscle in intentionally altered phenotypes, e.g., transgenically altered animals, and in specific training- and drug-induced changes in muscle. Finally, there is no reason to suppose that the approach developed here is limited to muscle, because most cells have similar metabolic pathways and are subject to various steady-state energy demands.

## METHODS

### Model Development

We found that three components dominate metabolism of ATP free energy in contracting skeletal muscle and were needed for the analysis that follows: *1*) AM ATPase, which is responsible for mechanical output; *2*) SR ATPase, which is responsible for relaxation; and *3*) the mitochondria, which produce ATP free energy. The first two components consume ATP free energy (Fig.1
*A*). ATP free energy consumption by the sarcolemmal Na^{+}-K^{+}-ATPase pump and other ion pumps (37) and ATP free energy production by glyco(geno)lytic ATP synthesis flux (37) are not considered here for reasons that will be explained.

ATP free energy metabolic flux in the in vivo skeletal muscle cell above the basal level is under neural, external control and is regulated by the cytosolic Ca^{2+} concentration ([Ca^{2+}]), because the activity of AM and SR ATPase is Ca^{2+} dependent (44) and, under conditions of saturating ATP free energy, is described by the Hill function
Equation 1where the Hill coefficient (*n*
_{H}) is 2 for SR ATPase and 3 for AM ATPase (44). [Ca^{2+}] required for half-maximal stimulation ([Ca^{2+}]_{50}) is ∼0.2 μM for SR ATPase and ∼0.8 μM for AM ATPase (44). In the unstimulated muscle cell, cytosolic [Ca^{2+}] is well below 0.2 μM (14), and thus AM and SR ATPase fluxes are minimal.

When a skeletal muscle cell is activated by an action potential, the SR releases a Ca^{2+} pulse, causing cytosolic [Ca^{2+}] to rapidly increase two orders of magnitude above the resting level (14). This concentration is sufficient to activate AM and SR ATPase and, thereby, muscle contraction and SR Ca^{2+} pumping, respectively (44). ATP free energy drives both processes in the forward direction (Fig.1
*B*). When cytosolic [Ca^{2+}] has returned to resting level, AM and SR ATPase are switched off again. Thus ATP hydrolysis flux in skeletal muscle is pulsatile and periodic. Energy balance is achieved by mitochondrial ATP synthesis flux via a closed-loop regulatory mechanism involving [ADP] (15,29) but with much slower kinetics than AM and SR ATPase fluxes (hundreds of seconds vs. subseconds) (14, 34, 44) because of temporal dampening of ATP/ADP transients by the activity of creatine kinase (15, 34, 36, 37). Fluctuations of mitochondrial ATP synthesis flux can be shown in simulations, as indicated in Fig.1
*B,* but have not been observed experimentally (19), likely because of the large damping effect of the creatine kinase reaction buffering ATP/ADP.

The minimal model that captures these main features of ATP free energy metabolism in contracting skeletal muscle given in Fig.2 is a modular (43) branched pathway consisting of three ATPase modules: E_{1} (cellular pool of mitochondria), E_{2} (total AM ATPase), and E_{3} (total SR ATPase). These modules consume or produce a common intermediate (S_{1}) that is related to the cytosolic ATP free energy at rates *v*
_{1},*v*
_{2}, and *v*
_{3}, respectively. We chose the cytosolic ATP/ADP for S_{1}. We were unable to use the full expression of the ATP free energy for reasons given in thediscussion. The analysis is simpler for ATP/ADP without loss of interpretation or significance. Each module is treated as homogeneous, and there is no diffusion limitation of S_{1}among the components at steady state. Metabolism in this branched network is characterized by four system variables: three fluxes (*J*
_{1}, *J*
_{2}, and*J*
_{3}, in moles per volume per time) and one concentration ratio (S_{1}). These system variables are distinguished in the MCA formalism from system parameters, such as temperature and enzyme concentrations, that on the time scale of metabolic events can be treated as constants (21, 26, 30,47).

At steady state, the relation between the three fluxes in the network is as follows: *J*
_{1} =*J*
_{2} + *J*
_{3}; i.e., only two of the three fluxes in this branched pathway are independent. The model (Fig. 2) is defined without considerations of the magnitude or direction of the fluxes; i.e., until the properties of the model are specified by analysis of the available kinetic data on the three components, the model is totally general. For this reason, throughout this study, three components (modules) in the network are termed ATPases, even though it will be clear that mitochondria in the muscle function as a net synthase [despite its reversibility (35)] and the AM and SR components function as net ATPases. Neural control of ATPase flux in the network occurs by Ca^{2+} regulation of the activity of AM and SR ATPase. Therefore, *J*
_{2} and *J*
_{3} are designated independent fluxes in the network. We define the variable α, denoting the magnitude ratio of these branch fluxes,*J*
_{2}/*J*
_{3}. In contracting muscle, AM ATPase flux accounts for ∼70% of total ATP utilization flux, and the remaining 30% is mostly due to SR ATPase activity (37, 44). Therefore, α = 2.3 for the network. It is a constant in our analysis but may depend on the type of contraction (isometric vs. working contractions) and even on frequency of stimulation, on which there is no information in human muscle.

Thus there is interdependence and connectivity of the rates of SR and AM ATPase via the ATP energy potential ATP/ADP. It is important to see that a second connectivity between these rates exists via cytosolic [Ca^{2+}] because of the periodic nature of the ATPases (Fig. 1
*A*). The cytosolic [Ca^{2+}] attained after neural stimulation may be a system parameter (i.e., constant and saturated) or a variable, depending among other factors on the activity of SR ATPase. This second connectivity between AM and SR ATPase activity is an important factor in the control analysis.

### Control Analysis

#### Calculation of control coefficients for flux and concentration.

There are nine flux control coefficients (*C*
_{i}
^{Jm}), three ATP/ADP control coefficients (*C*
_{i}
^{ATP/ADP}), and three ATP/ADP elasticity coefficients (ε_{ATP/ADP}
^{i}) for ATP free energy metabolism in the three-component network model of ATP metabolism in contracting muscle (Fig. 2). Briefly, in the nomenclature of MCA (21, 47), a flux control coefficient refers to the relative magnitude of change in a flux in the network due to a small change in the activity of a particular modular component *i*. Similarly, a concentration control coefficient refers to the relative magnitude of change in the concentration of the shared metabolite due to a small change in the activity of a particular modular component*i*. These definitions apply well to control by the mitochondria and the AM ATPase. With respect to the SR ATPase, the situation is more complicated, and so the control coefficients calculated here will only apply to control by the enzyme directly as effected through the ATP/ADP regulation in the system; indirect effects are not included, and these can be important (seediscussion). Elasticity coefficients toward ATP/ADP quantify the relative sensitivities of each modular component to a small change in ATP/ADP and are, as such, determined by the particular ATP and ADP kinetics of the reaction catalyzed by each component. For a standard saturable process, the elasticity coefficient decreases from its initial value at low substrate concentrations to zero at saturating substrate concentration. The set of values of these coefficients is specific for each steady state. (See the
for more complete definitions and derivations and Refs. 21 and 39 for an introductory account of metabolic control theory.)

Mathematical expressions for flux control and ATP/ADP control coefficients in terms of the ATPase elasticities toward ATP/ADP (ε_{ATP/ADP}
^{i}) and the ratio of fluxes in the branches (α) are given in *Eqs. A6*,*A10*, *A14,* and *A20*. These solutions were developed on the basis of the summation and connectivity theorems and the branch theorems for flux control and concentration control of MCA. The calculation of the set of 12 control coefficients at a particular steady state involved three steps. *1*) The relevant physiological variable chosen for graphical presentation of the results was the stimulation frequency; we also performed our analysis in terms of the normalized network flux (fraction of maximal) with the same overall results and conclusions, but stimulation frequency relates in a direct and simple way to experiments. For each stimulation rate in the range of sustainable steady states (up to ∼2 Hz in experiments described below), the values of the response variables (ATP/ADP and ATP synthesis flux, *J*
_{1}, in forearm flexor muscle) were determined experimentally by^{31}P-NMR spectroscopy (see *Experimental Methods*).*2*) The values of the ATP/ADP elasticities of each of the ATPase modules in the network at each experimentally determined steady state were calculated using the measured ATP/ADP, as described below. We extrapolated our analysis beyond the range of measured stimulation frequencies that gave steady states. *3*) With this set of elasticity values and α = 2.3, the corresponding set of control coefficients was calculated using *Eqs. A6, A10, A14*, and*A20*.

#### Calculation of ATPase elasticities toward ATP/ADP.

The elasticity of each module *i* in the network toward the cytosolic ATP/ADP at a particular steady state (*k*) of ATP free energy metabolism in muscle [(ε_{ATP/ADP}
^{i})_{k}] was calculated on the basis of the steady-state kinetics of the enzyme module according to (21, 39, 47)
Equation 2The term [(ATP/ADP)/*v _{i}
*]

_{k}is a normalization term for absolute reaction velocity, and the term [d

*v*/d(ATP/ADP)]

_{i}_{k}defines the particular ATP/ADP sensitivity at this point on the velocity curve. This second term is the partial derivative of rate

*v*with respect to ATP/ADP, i.e., ∂

_{i}*v*/∂(ATP/ADP). Therefore, this method to calculate (ε

_{i}_{ATP/ADP}

^{i})

_{k}requires that a function is used that was determined under conditions where only ATP/ADP effects on

*v*were measured; i.e., concentrations of any other affectors of

_{i}*v*were saturating or constant during the course of the experiment. We were able to obtain appropriate functions in the literature.

_{i}##### MITOCHONDRIA.

The elasticity of mitochondria toward ATP/ADP at steady-state*k*, (ε_{ATP/ADP}
^{1})_{k}, was calculated using *Eq. 2
* on the basis of the kinetic function that describes the dependence of mitochondrial ATP synthesis flux (*J*
_{1}) on ATP/ADP in human forearm flexor muscle (29)
Equation 3where *K*
_{0.5}
^{ATP/ADP} is ATP/ADP at half-maximal ATP synthesis flux (∼186 for human forearm flexor muscle in vivo). This relation was derived from the kinetic function that described the covariation (ADP, *J*
_{p}) in this skeletal muscle under conditions of saturating [P_{i}] with respect to the reaction (29).

##### AM ATPase.

The elasticity toward ATP/ADP of AM ATPase at steady state*k*, (ε_{ATP/ADP}
^{2})_{k}, was calculated using *Eq. 2
* on the basis of the reported kinetic function that describes the AM ATPase rate dependence on ATP and ADP (20)
Equation 4where *K*
_{m}
^{MgATP} is the Michaelis constant for the substrate (10–20 μM for ATPase activity) and*K*
_{i} is the inhibition constant for MgADP, which is on the order of 200–300 μM (20). We used 15 and 250 μM for *K*
_{m}
^{MgATP} and*K*
_{i}, respectively, in the calculations.

##### SR ATPase.

The elasticity toward ATP/ADP of SR ATPase at steady state*k*, (ε_{ATP/ADP}
^{3})_{k}, was calculated using *Eq. 2
* on the basis of the kinetic function describing the SR ATPase forward rate dependence on ATP and ADP
Equation 5where *K*
_{m}
^{MgATP} is the affinity for the substrate and *K*
_{i} and*K*′_{i} are ADP inhibition constants. This function was derived on the basis of a study of SR ATPase kinetics in solubilized fragmented SR from rabbit skeletal muscle from which it was concluded that ADP inhibition was of mixed type under conditions of low Ca^{2+} and high Mg^{2+} and the inverse conditions (42). *K*
_{m}
^{MgATP} of SR ATPase is at least two orders of magnitude lower than [ATP] in human skeletal muscle [10 μM (42) vs. 8 mM (25)], and so the term*K*
_{m}
^{MgATP}/[MgATP] (where [MgATP] is MgATP concentration) is <0.01, and *Eq. 5
* reduces to*v*
_{3} =*V*
_{3 max}/(1 + [ADP]/*K*′_{1}). We determined*K*′_{1} from data reported previously (Fig. 2 at high [ATP] in Ref. 42) and obtained an estimate of 0.52 ± 0.20 mM.

#### Calculation of effective elasticities of AM and SR ATPase toward ATP/ADP.

So far, periodicity of AM and SR ATPase fluxes in intermittently stimulated muscle and possible consequences for the analysis of the steady state have been ignored. However, because twitch contractions, not fused tetani, are normal physiological modes of contraction in mammalian muscle, they must be explicitly considered in the analysis. Periodicity introduces one more variable not previously considered in models of energetics into the set that determines the effective elasticity of AM and SR ATPase toward ATP/ADP. This variable is the amount of time (Δ*t*) between subsequent stimulations for reactions and processes to take place. Consider, for example, the case of a series of infrequent twitches. Then there is sufficient time between stimulations for the SR ATPase function (i.e., restoration of SR [Ca^{2+}] to resting level) to go to completion within a single contraction-relaxation cycle. This means that the amount of ATP hydrolyzed by SR ATPase in this cycle is determined solely by the Ca^{2+}-ATP stoichiometry of the pump and the amount of Ca^{2+} cleared, instead of the sensitivity of the SR ATPase to ATP/ADP. Consequently, at sufficiently low contraction frequencies (i.e., Δ*t* between contractions is >3 kinetic time constants of the Ca^{2+} uptake reaction), the effective elasticity of SR ATPase toward ATP/ADP is zero. Conversely, in a series of high-frequency twitches, where time between stimulations is short relative to the kinetics of the SR ATPase reaction, SR Ca^{2+}-ATPase recovery will not go to completion within a single contraction-relaxation cycle. Depending on Δ*t* and ATP/ADP, as well as the capacity for and on and off rates of Ca^{2+} binding by cytosolic Ca^{2+} buffers [e.g., parvalbumin (29) and mitochondria (30)], SR Ca^{2+} release and subsequent peak cytosolic [Ca^{2+}] per stimulation may decline in time to levels that are insufficient to maximally activate AM ATPase, causing twitch force to fall. Although this is a simplification of the complex physiology under these conditions, experimental evidence exists that this scenario at least contributes to the causes of muscle fatigue (18, 23, 46). In muscle cells with a large noncontractile cytosolic binding capacity of Ca^{2+}, such as fast-twitch fibers (11, 37), this scenario will be more prominent. The magnitude of the AM ATPase flux (*J*
_{2}) will, in this case, depend on ATP/ADP as well as cytosolic [Ca^{2+}]. In this way, the activity of AM ATPase will indirectly depend on the SR ATPase elasticity toward ATP/ADP. Consequently, the calculation of the elasticity of AM ATPase toward ATP/ADP must, in this case, take into account the SR ATPase elasticity toward ATP/ADP.

On the basis of these considerations, three frequency ranges of muscle twitch contraction are distinguished in the control analysis

##### CASE I: LOW-FREQUENCY CONTRACTIONS.

For the forearm flexor muscle studied here, *case I* is defined as stimulation frequencies <0.6 Hz, corresponding to >1.3 s between stimulations. Over this range of contraction frequencies, the effective elasticity of SR ATPase toward ATP/ADP (ε_{ATP/ADP}
^{3′}), is zero. The elasticities toward ATP/ADP of AM ATPase and mitochondria under these conditions were determined as defined above.

##### CASE II: INTERMEDIATE CONTRACTION FREQUENCIES.

Over this range of twitch frequencies, we consider the possibility that the time interval between subsequent stimulations is no longer sufficient for SR ATPase to recover all Ca^{2+} released on stimulation irrespective of the ATP/ADP sensed by the enzyme, but Ca^{2+} release is still sufficient for near-maximal activation of AM ATPase after each stimulation. In this case, all ATPase fluxes and, therefore, elasticities are determined only by the kinetics toward ATP and ADP. The frequency range where this condition applies for forearm flexor muscle was defined to be between 1 and 2 Hz, corresponding to the high end of steady states of energy balance that can be sustained (see results).

##### CASE III: HIGH-FREQUENCY CONTRACTIONS.

In this case (stimulation frequencies >5 Hz), SR Ca^{2+}recovery by SR ATPase with stimulations at ≤200-ms intervals is insufficient to ensure maximal Ca^{2+} activation of AM ATPase on subsequent stimulation, so twitch force over time declines. As a simplification, any buildup of cytosolic [Ca^{2+}] at these stimulation frequencies affecting force is not considered. The dependence of AM ATPase rate (and, therefore, force) on [Ca^{2+}] is very steep (*Eq. 1
*). If we assume for*case II* that enough SR Ca^{2+} is recovered to ensure that cytosolic [Ca^{2+}] will be at least twice [Ca^{2+}]_{50} after stimulation, so that AM ATPase will be stimulated to ≥89% of the maximal activity (*Eq.1
*), twitch force should not fall >10% at the intermediate contraction frequencies. If, in *case III,* cytosolic [Ca^{2+}] after stimulation is less than one-half that defined for *case II*, then AM ATPase rate (and, therefore, force) will fall to <50% of maximal. The effective elasticity of AM ATPase toward ATP/ADP (ε_{ATP/ADP}
^{2′}) for *case III* is in part determined by the SR ATPase elasticity toward ATP/ADP, and quantification required derivation of the relation between SR ATPase, cytosolic [Ca^{2+}], and AM ATPase (see below).

#### Calculation of effective elasticity of AM ATPase toward ATP/ADP at high frequencies of activation.

[Ca^{2+}] in the cytosol and in the SR lumen are each at their respective baseline value in the resting state. At time*t*
_{1}, the muscle cell is excited by an action potential from the motor nerve and the SR releases its Ca^{2+}into the cytosol; cytosolic [Ca^{2+}] reaches a maximum at time *t*
_{1} + τ, when Ca^{2+} in the lumen of the SR is lower and significant binding to cytoplasmic proteins occurs. The relation between the free [Ca^{2+}] in the two compartments after stimulation can thus be described as follows
Equation 6where β is a time-dependent proportionality factor determined by the permeability of the SR Ca^{2+} release channels and the duration Δ*t* of the pulse, V_{SR}/V_{cyto} is the ratio of SR volume to cytosol volume, and [Ca^{2+}]_{cyto∣t0} is cytosolic [Ca^{2+}] before stimulation. At the onset of the next stimulation of the cell at time *t*
_{2}, SR [Ca^{2+}] recovered by SR ATPase over the time interval Δ*t* (= *t*
_{2} −*t*
_{1}) between stimulations is determined by the activity of SR ATPase as follows
Equation 7where *n*
_{P}
^{Ca} is the Ca^{2+}-per-ATP pumping stoichiometry of the SR ATPase,*v*
_{3} is the SR ATPase rate, and [Ca^{2+}]_{SR∣t1} is the residual SR [Ca^{2+}] after Ca^{2+} release at time*t*
_{1}. An expression for cytosolic [Ca^{2+}] that is attained after stimulation at time*t*
_{2} is obtained by substituting *Eq. 7
*into *Eq. 6
* and neglecting the last term as a simplification in analogy to others (5), yielding
Equation 8where [Ca^{2+}]_{cyto∣t2} is cytosolic [Ca^{2+}] at the onset of the next stimulation. This expression can be generalized to describe the dependence of cytosolic [Ca^{2+}] after stimulation *n* in a series of *N* stimulations at a constant frequency on the SR ATPase activity *v*
_{3}
Equation 9Next, we defined a scaled rate equation for AM ATPase in analogy to modeling of SR ATPase by Baylor and Hollingworth (5) that contains ATP/ADP- and cytosolic [Ca^{2+}]-dependent terms that reduce to the ATP/ADP-dependent rate equation for AM ATPase (*Eq. 4
*) under conditions of saturating cytosolic [Ca^{2+}] (*v*
_{2} is a function of [ATP], [ADP], and cytosolic [Ca^{2+}])
Equation 10We then substituted the expression for cytosolic [Ca^{2+}] of *Eq. 9
* into the scaled rate equation for AM ATPase (*Eq. 10
*) and made two further simplifications. First, we neglected cytosolic [Ca^{2+}] at rest relative to cytosolic [Ca^{2+}] after stimulation. Second, we used the time-averaged SR ATPase rate *v*
_{3} so that ∫*v*
_{3}d*t* ≈ *v*
_{3}Δ*t*(Δ*t* is the time interval between muscle stimulations) and defined a constant *K*′_{50} = (*K*
_{50}/β*n*
_{P}
^{Ca}). We thus obtained the following expression for the scaled AM ATPase rate that is attained after the *n*th stimulation in a series of*N* stimulations at a particular stimulation frequency
Equation 11With the use of *Eq. 11
* and a method to quantify the Ca^{2+}-dependent term, the effective elasticity of AM ATPase toward ATP/ADP under conditions of submaximal cytosolic [Ca^{2+}] activation of the enzyme can now be calculated from *Eq. 2
*. We developed two different methods to quantify the Ca^{2+}-dependent term in *Eq. 11
* that are presented below. The first method used a strict analytic approach. The second method used a composite analytic-numerical approach that incorporated reported kinetics of SR ATPase Ca^{2+} pumping.

##### ANALYTIC SOLUTION FOR THE CASE OF HIGH CONTRACTION FREQUENCIES.

An alternative mathematical formulation of the elasticity of a module toward ATP/ADP at a particular steady state (*Eq. 2
*) is the log-to-log ratio of the velocity and ATP/ADP (26, 30, 47). Applying this to *Eq. 11
*, we obtain
Equation 12
Under the limit condition Δ*t*→0, i.e., at high contraction frequencies, the term [*K*′_{50}/(*v*
_{3}Δ*t*)]^{3}> 1. Also, *K*
_{50}, β, and*n*
_{P}
^{Ca} are mute with respect to ATP/ADP sensitivity, but the SR ATPase rate *v*
_{3} is not. So, under the limit condition of high contraction frequencies in*case III*, the elasticity of AM ATPase toward ATP/ADP equals by approximation
Equation 13The ATP/ADP elasticity of SR ATPase predicted from the ATP/ADP-dependent term of the respective rate equations is orders of magnitude higher than that of AM ATPase, especially at the low ATP/ADP values that apply here. Therefore, the effective elasticity of AM ATPase toward ATP/ADP for the limit condition of high contraction frequencies, ε_{ATP/ADP}
^{2′}, is mostly defined by the SR Ca^{2+}-ATPase elasticity toward ATP/ADP and is, by approximation, equal to three times this elasticity.

##### COMPOSITE ANALYTIC-NUMERICAL SOLUTION FOR THE CASE OF HIGH CONTRACTION FREQUENCIES.

In the first of three steps involved in this second approach, we used a reported analysis of the kinetics of SR ATPase-mediated Ca^{2+} removal from the cytosol after a Ca^{2+}release pulse (13). This enabled us to obtain a quantitative relation between SR [Ca^{2+}] and the SR ATPase rate *v*
_{3}. The reported biexponential kinetics of SR Ca^{2+} accumulation in rat extensor digitorum longus muscle at 15°C (Fig. 9*A* in Ref. 13) were digitized and analyzed by fitting double-exponential functions to obtain the relation
Equation 14where the time constant τ_{1} equals 0.035 s. At*time 0*, the amount of SR Ca^{2+} (relative to maximum) is zero. Rat extensor digitorum longus muscle is composed of predominantly fast-twitch muscle cells but also contains slow-twitch cells (7) and is, in this respect, not unlike forearm muscle (38).

Because of the 20°C higher temperature in forearm muscle and with a Q_{10} for SR ATPase of 2 at 15–35°C (13,44), we corrected τ_{1} to 0.009 s. The normalized SR [Ca^{2+}] reestablished over the time after stimulation was computed as a function of the time between subsequent stimulations for the eight stimulation frequencies of Tables1-3. This led to a set of paired values {Δ*t* and [Ca^{2+}]_{SR}/([Ca^{2+}]_{SR})_{max}}. To correct for different degrees of ADP inhibition of SR ATPase between different stimulation frequencies, the array of [Ca^{2+}]_{SR}/([Ca^{2+}]_{SR})_{max}(Δ*t*) values was multiplied by the corresponding array of*v*
_{3}/*V*
_{max}(Δ*t*) values that was calculated using *Eq. 5
* and [ADP](Δ*t*) for each stimulation frequency. For time intervals >500 ms (i.e., contraction frequencies <2 Hz), SR [Ca^{2+}] recovery was ≥85% of the maximum (data not shown). For contraction frequencies >5 Hz, recovery dropped sharply, to as low as 60% at 10 Hz.

In the next step, the computed [Ca^{2+}]_{SR}/([Ca^{2+}]_{SR})_{max}recovery immediately before subsequent stimulation was correlated for each stimulation frequency with the corresponding steady-state ATP/ADP determined by ^{31}P-NMR measurements for frequencies <2 Hz (see *Experimental Methods*) and with extrapolated ATP/ADP for frequencies >5 Hz. A biexponential function describing the covariation of [Ca^{2+}]_{SR}/([Ca^{2+}]_{SR})_{max}and ATP/ADP was obtained by curve fitting as follows
Equation 15
*Equation 15
* was used to compute*v*
_{2} as a function of ATP/ADP, cytosolic [Ca^{2+}] for each contraction frequency (and corresponding ATP/ADP) on the basis of *Eq. 11
* with β = 1,*K*
_{50} = 0.5, and*n*
_{P}
^{Ca}
*v*
_{3}Δ*t*= [Ca^{2+}]_{SR}. The resulting covariation of ATP/ADP and *v*
_{2} was biexponential and was determined by nonlinear curve fitting to give
Equation 16This relation was then used to compute the elasticity of AM ATPase toward ATP/ADP for a steady-state *k,* as described above using *Eq. 2
*.

The analytic solution (*Eq. 13
*) applies only to the limit condition of high contraction frequencies of *case III*. The composite analytic-numerical solution to calculate ε_{ATP/ADP}
^{2′} on the basis of *Eq. 16
* applies to conditions in which cytosolic [Ca^{2+}] is saturating and nonsaturating with respect to the AM ATPase. The analytic-numerical method allowed computation of the continuum of ε_{ATP/ADP}
^{2′} in contracting muscle on the basis of a continuous array of ATP/ADP values for stimulation frequencies >2 Hz. The latter could not be applied to human subjects, because such conditions are intolerable. Thus, in combination with *Eqs. 3
* and *
5
*, we could compute flux and concentration control coefficients over a 10-Hz range of contraction frequencies as a continuous function.

### Experimental Methods

^{31}P-NMR spectroscopy.

Human forearm flexor muscle (5 men and 1 woman, age 28–55 yr) was studied at rest and during twitch contractions evoked by external electrical stimulation of the ulnar and medial nerves at frequencies between 0.3 and 2.0 Hz. These frequencies were high enough to allow us to find the stimulation rate above which non-steady-state acidification occurred. ^{31}P-NMR spectroscopic data were acquired at 2.0 T according to methods described in detail elsewhere (9,28). This range of stimulation frequencies was sufficiently broad to ensure that the maximal steady state of oxidative ATP synthesis in forearm flexor muscle was attained in each subject studied. The end point of the sustainable steady states was determined by the achievement of a steady reduction in PCr without acidification to pH <6.9. ^{31}P-NMR signals were acquired from forearm flexor muscle during rest-stimulation-recovery experiments (3:6:3-min duration, respectively) in blocks of 7-s serial acquisitions [4 summed free induction decays (FIDs), 1.76-s delay, 2-kHz sweep width, and 1,024 data points]. Twitch contractions of the entire muscle mass were elicited by supramaximal percutaneous stimulation of ulnar and medial nerves (electric pulse duration 0.2 ms, amplitude 250–300 V) (6).

#### NMR data analysis.

Raw data were transferred to a Sparc II workstation (Sun Microsystems) and analyzed in three steps, as described in detail elsewhere (28). Briefly, data were batch processed using NMR1 software (New Methods Research), involving apodizing of FIDs using a matched Lorentzian filter, zero filling to 2,048 data points, Fourier transformation, and phase correction, and then analyzed in the frequency domain with respect to PCr, P_{i}, and ATP peak integrals and frequencies. Second, the time course of the PCr content of the muscle during contraction was analyzed using Fig.P software (Elsevier Biosoft) for each twitch frequency. A monoexponential function was fitted to the PCr time course to determine the time constant τ_{PCr} (in s) (28). The value of τ_{PCr} was used as a basis for the calculation of ATP metabolic fluxes in the contracting muscle (28) and to determine the time at which a new steady state of energy balance was attained during stimulation; this occurred at *t* > 3τ_{PCr} s at which d[PCr]/d*t* ∼ 0. Typically, the steady state occurred after ∼3 min of stimulation, and the subsequent 3 min of data were used to characterize the steady-state metabolite concentrations.

For each 3 min of steady state, the corresponding FIDs were summed and analyzed in the time domain for PCr, P_{i}, and ATP integrals and resonance frequencies (6) with Fitmasters software (Philips Medical Systems). Finally, [PCr], [P_{i}], and [ADP] at each steady state were calculated assuming [ATP] of 8.2 mM, total Cr concentration of 42.7 mM (25), and creatine kinase equilibration. The pH_{i} was estimated from the chemical shift difference between the PCr and P_{i} resonance (45). The cytosolic free energy of ATP hydrolysis (Δ*G*
_{ATP}) was calculated as Δ*G*
_{ATP}
^{o′} +*RT*ln([ADP][P_{i}]/[ATP]), where −32.8 kJ/mol was used for Δ*G*
_{ATP}
^{o′} (41).

#### Curve fitting and statistical analyses.

Correlations of variables were analyzed by nonlinear curve fitting with defined functions in Fig.P software (version 6.0, Elsevier Biosoft, Cambridge, UK).

## RESULTS

### General Solution of Control in the Network

The general solution for kinetic control of the particular value of the four system variables *J*
_{1},*J*
_{2}, *J*
_{3}, and S_{1} at a particular steady state of metabolism in the three-component branched network of Fig. 2 in terms of elasticities and the branch flux ratio α is given in the
(*Eqs. A6, A10, A14*, and *A20*). The general solution for flux control was reported previously but in terms of different variables (39). The general solution for concentration control in a branched network was not previously described.

### Steady-State ATP Metabolic Flux in Contracting Forearm Flexor Muscle

Steady states of ATP free energy metabolism were defined after the decrease in [PCr] and increase in [P_{i}] when those concentrations and pH became constant. This occurred after ∼3 min of continuous stimulation. In the steady state the summed ATPases equal the ATP synthesis. These steady states were measured as a function of twitch frequencies until the maximal sustained decrease in PCr was found. In one subject, this maximal steady state of oxidative ATP metabolism was attained at a twitch frequency of 1.3 Hz. The maximum was attained at 1.6 Hz in three subjects and at 1.8 Hz in the remaining two subjects. The average ATP hydrolysis rate in forearm flexor muscle during twitch contractions at a frequency of 1.6 Hz, estimated from the initial slope of the PCr time course during stimulation at *time 0* as described previously (28), was 0.15 ± 0.01 (SE) μmol ATP · s^{−1} · g muscle^{−1}[*n* = 6 muscles, with assumption of 0.67 liter cell water/kg muscle (25)]. Mitochondrial ATP synthesis flux accounted for 90 ± 2% (mean ± SE, *n* = 6 muscles) of the total matching cellular ATP synthesis flux at this ATPase rate and approached 84 ± 4% (*n* = 5) of maximal synthesis flux estimated for each individual muscle, as described elsewhere (28). The rate of 0.15 μmol ATP · s^{−1} · g muscle^{−1}constituted the apparent maximal ATPase flux that could be sustained in this muscle and, therefore, represents the maximal flux in the three-component ATPase network model of ATP free energy metabolism of Fig. 2 in this muscle. At higher contraction frequencies (and associated ATP hydrolysis rates), nonoxidative ATP synthesis increased as estimated from concomitant proton production in four of six subjects studied. These conditions resulted in a decline of pH_{i}below 6.9 to values as low as 6.7 (data not shown). These conditions were not steady states and were not analyzed further.

### Dynamic range of ATP/ADP in Contracting Forearm Flexor Muscle at Steady State

The steady-state [PCr] in contracting human forearm flexor muscle decreased from 28.7 ± 0.7 (mean ± SE) mM (*n* = 6) at 0.3-Hz contractions to 15.8 ± 0.2 (SE) mM (*n* = 2) at 1.8-Hz contractions (Fig.3
*A*). The steady-state pH_{i} decreased from 7.02 ± 0.03 (mean ± SE, *n* = 6) at 0.3-Hz contractions to 6.97 ± 0.02 (*n* = 2) at 1.8-Hz contractions. The steady-state ATP/ADP in the cytosol decreased from 314 ± 8 (mean ± SE, *n* = 6) at 0.3-Hz contractions to 98 ± 1 (*n* = 2) at 1.8-Hz contractions (Fig.3
*C*). Steady-state [PCr], pH_{i}, and ATP/ADP in resting human forearm flexor muscle were 31.9 ± 0.2 mM, 7.05 ± 0.01, and 445 ± 14 (mean ± SE, *n* = 6), respectively. The ATP free energy dynamic range in forearm flexor muscle was thus 8 kJ/mol, from an average of −62.5 kJ/mol in unstimulated muscle to −54.6 kJ/mol in muscle stimulated at 1.8 Hz.

### Dynamic Range of Elasticities Toward ATP/ADP in the Network

#### Case I: low-frequency contractions.

The elasticity toward ATP/ADP of mitochondria (ε_{ATP/ADP}
^{1}) and AM ATPase (ε_{ATP/ADP}
^{2}) calculated as described inmethods from the steady-state ATP/ADP measured in forearm flexor muscle during twitch contractions at 0.3 and 0.6 Hz is given in Table 1. The effective elasticity of SR ATPase toward ATP/ADP over this range of contraction frequencies (ε_{ATP/ADP}
^{3′}) was zero. Steady-state ATP/ADP and corresponding normalized ATP free energy metabolic flux [(*J*
_{1}/*J*
_{1−})_{max}] calculated using *Eq. 3
* are given for each contraction frequency (Table 1). Over this range of contraction frequencies, the steady-state ATP/ADP in the muscle fell to approximately one-half of the resting potential, whereas ATP metabolic flux (*J*
_{1}) in the network increased to ∼30% of maximal sustainable flux (Table 1). For this range of contraction frequencies, ε_{ATP/ADP}
^{1} was at least four orders of magnitude greater than ε_{ATP/ADP}
^{2}, even when the former decreased twofold as the contraction frequency doubled from 0.3 to 0.6 Hz (Table 1). These results mean that the functions of the AM and SR module are immune from changes in ATP/ADP for this contraction frequency range.

#### Case II: intermediate-frequency contractions.

The elasticities toward ATP/ADP of the three ATPase modules, calculated as described in methods on the basis of measured steady-state ATP/ADP in forearm flexor muscle during twitch contractions at 1.0, 1.3, 1.6, and 1.8 Hz, are given in Table 2. Steady-state ATP/ADP and corresponding normalized ATP free energy metabolic flux [(*J*
_{1}/*J*
_{1−})_{max}] calculated using *Eq. 3
* are given for each contraction frequency (Table 2). Over this range of contraction frequencies, the steady-state ATP/ADP in the muscle fell another twofold from ∼50% of the resting potential at 0.6-Hz contractions (Table 1) to ∼25% of the resting potential at 1.8-Hz contractions, whereas ATP metabolic flux (*J*
_{1}) in the network increased to ∼80% of maximal flux (Table 2). The absolute value of ε_{ATP/ADP}
^{1} decreased a further 2.5-fold over this range of contraction frequencies (Table 2). The value of ε_{ATP/ADP}
^{3} approached the same order of magnitude as ε_{ATP/ADP}
^{1} over this range of contraction frequencies and increased 1.5-fold between 1.0 and 1.8 Hz (Table 2). The value of ε_{ATP/ADP}
^{2} remained at least two orders of magnitude smaller than ε_{ATP/ADP}
^{1} and ε_{ATP/ADP}
^{3}, despite a 10-fold increase in its value over this frequency range (Table 2). These results mean that function of the AM module is still immune from changes in ATP/ADP also over this contraction frequency range.

#### Case III: high-frequency contractions.

Here the analysis is extrapolated into regions for which there are no experimental data and where it may be difficult, if not physiologically impossible, to explore human forearm muscle in situ. The elasticities toward ATP/ADP of mitochondria and AM and SR ATPase were calculated as described in methods by extrapolation of steady-state ATP/ADP to contraction frequencies of 5 and 10 Hz (Table 3). These extrapolated ATP/ADP values represent the conditions that should be attained in the muscle had anaerobic glyco(geno)lysis remained insignificant. The effective elasticity of AM ATPase toward ATP/ADP under these conditions, ε_{ATP/ADP}
^{2′}, was calculated using *Eq. 15
*. At 10 Hz, ATP/ADP in the muscle was extrapolated to 41, and normalized ATP flux (*J*
_{1}) in the network [(*J*
_{1}/*J*
_{1−})_{max}] would consequently have increased to 96% of maximal sustainable flux (Table 3). The absolute value of ε_{ATP/ADP}
^{1} decreased a further twofold over this range of contraction frequencies to a value 30-fold lower than at low contraction frequencies (Tables 1 and 3). The absolute value of ε_{ATP/ADP}
^{3} (and therefore of ε_{ATP/ADP}
^{2′}) over this range of contraction frequencies now exceeded that of ε_{ATP/ADP}
^{1} by as much as threefold (Table 3). In this case, the effective elasticity of AM ATPase, ε_{ATP/ADP}
^{2′}, was the highest in the network (Table 3). Only at these high stimulation rates does the function of the AM module therefore become significantly influenced by ATP/ADP.

### Kinetic Control of ATP Metabolic Flux and ATP/ADP in Contracting Forearm Flexor Muscle

Here the distributions of kinetic control of ATP free energy-consuming and energy-producing fluxes and cytosolic ATP/ADP in the network model of ATP free energy metabolism in contracting skeletal muscle are calculated as a function of stimulation frequency. We used*Eqs. A6, A10, A14*, and *A20*, the elasticities toward ATP/ADP (Tables 1-3), and α = 2.3. The following distributions of kinetic control in the three-component ATPase network model of ATP free energy metabolism in contracting skeletal muscle were obtained as a function of stimulation frequency.

#### Case I: low-frequency contractions.

Over this range of contraction frequencies and associated steady states of ATP free energy metabolism, the activities of the ATPase modules that consume ATP free energy during contraction, AM and SR ATPase, control the magnitude of all three fluxes in the network, i.e., ATP hydrolysis as well as ATP synthesis flux (Table4). With respect to the two ATP hydrolysis fluxes in the network (*J*
_{2} and*J*
_{3}), each ATPase module fully controls its respective ATPase flux by its activity (*C*
_{2}
^{J2} = 1 and *C*
_{3}
^{J3} = 1 at 0.3 and 0.6 Hz; Table 4). The activity of the mitochondria has no active control over the magnitude of any of the fluxes in the network at steady state during twitch contractions at 0.3 or 0.6 Hz, including its own ATP synthesis flux (*C*
_{1}
^{Ji
} = 0). This result means that mitochondrial ATP synthesis flux (*J*
_{1}) passively follows the ATP demand fluxes (*J*
_{2} and *J*
_{3}) set by the summed activity of AM and SR ATPase (*C*
_{2}
^{J1} +*C*
_{3}
^{J1} = 1 at 0.3 and 0.6 Hz; Table 4). In contrast, control of ATP/ADP at steady state over this range of contraction frequencies is shared by all three ATPase modules in the network, with the activity of the mitochondria having the highest control over the ratio attained at steady state. In absolute magnitude, *C*
_{1}
^{ATP/ADP} was 1.4-fold higher than *C*
_{2}
^{ATP/ADP} and 3.1-fold higher than *C*
_{1}
^{ATP/ADP} at 0.3 and 0.6 Hz (Table 4).

#### Case II: intermediate-frequency contractions.

Kinetic control of the ATPase fluxes *J*
_{1} and*J*
_{3} now resides in the specific activities of all three ATPase modules instead of only in AM and SR ATPase, as was the case for low frequencies (Table 5). At the higher end of the range of contraction frequencies considered in this case, control by the activity of mitochondria of*J*
_{1} and *J*
_{3} is substantial (*C*
_{1}
^{J1} = 0.05 and 0.07 respectively, and*C*
_{1}
^{J3} = 0.16 and 0.23, respectively, at 1.6- and 1.8-Hz contractions; Table 5). Accordingly, a change in the control hierarchy among the three modules in the network occurs in this range of contraction frequencies. Control of*J*
_{2} in the network remains exclusively in the activity of AM ATPase itself and is not shared with the activities of the three modules in the pathway (*C*
_{2}
^{J2} = 1 and*C*
_{1}
^{J2} =*C*
_{3}
^{J2} = 0 for all 4 contraction frequencies studied; Table 5).

With respect to ATP/ADP attained in the network at steady state at intermediate contraction frequencies, the control distribution was the same as that found for the case of low contraction frequencies: control remained shared by all three modules in the pathway, with the highest absolute kinetic control residing in the activity of the mitochondria (Table 5). However, absolute homeostasis of ATP/ADP in forearm muscle contracting at these intermediate frequencies deteriorated as the contraction frequency approached 2 Hz: the absolute change in ATP/ADP accompanying a unit increase in ATPase activity caused by an increase in contraction frequency increased 2.5-fold over this frequency range.

#### Case III: high-frequency contractions.

We continue this analysis beyond the range of experimentally measurable steady states of ATP free energy metabolism in contracting forearm flexor muscle. We assumed a continuous downward trend in ATP/ADP with increasing contraction frequency for as long as glyco(geno)lytic ATP synthesis remained insignificant. This assumption allowed us to assess how the change in the control hierarchy with respect to flux control in the three-component network and ATP/ADP control would develop with an increasing duty cycle of ATPase activity.

The most significant finding was the inversion of flux control from an ATP demand control hierarchy to an ATP supply control hierarchy. This began at contraction frequencies >1 Hz (Table 5), progressed as the contraction frequency increased past 2 Hz, and was near complete at 10 Hz (Table 6). Specifically, the distribution of ATPase flux and ATP/ADP control in the network, as calculated on the basis of elasticities of mitochondria and SR ATPase and the effective elasticity of AM ATPase for these conditions (Table3), indicated that the activity of mitochondria should largely determine the magnitude of ATP synthesis flux (*J*
_{1}), the ATP hydrolysis flux (*J*
_{2}) at high contraction frequencies, and ATP/ADP (Table 6). Only the magnitude of SR ATPase hydrolysis flux (*J*
_{3}) should not be dominantly determined by the activity of mitochondria (Table 6). At low ATP/ADP, the activity of mitochondria was stimulated to near-maximal velocity (Table 3). This result predicts that the ATPase flux at high contraction frequencies is determined by the ATP synthesis capacity of the cellular mitochondrial pool for a major, but not exclusive, part. The kinetic properties of the contractile proteins and the SR Ca^{2+} pumps are not so important here for that flux. The absolute values of ATP/ADP control coefficients of all three ATPase modules over this range of contraction frequencies are lower than the values at 1.8 Hz and decreased with increasing frequency (Tables 5 and 6). This result predicts that the change in ATP/ADP in the ATPase network accompanying a unit change in ATPase activity caused by an increase in stimulation frequency reaches a maximum between 1.6 and 5 Hz. This means that the ability of the network to regulate the cytosolic ATP/ADP ratio is minimal at or above the high end of the physiologically attainable range of steady states.

The distribution of kinetic control of ATP metabolic flux and ATP/ADP among the three ATPase modules was also calculated as a continuous function of time interval between contractions on the basis of continuous values of the ATPase elasticities toward ATP/ADP. A second connectivity between the activities of AM and SR ATPase via cytosolic [Ca^{2+}], in addition to the connectivity via ATP/ADP, was quantitatively considered in these computations over the 10-Hz range of stimulation frequencies (see methods). The results are shown in Fig. 4by solid lines. The open symbols in Fig. 4correspond to the discrete control coefficient values calculated for the six steady states studied in forearm flexor muscle without consideration of a second connectivity between the activities of AM and SR ATPase via cytosolic [Ca^{2+}] (Tables 4 and 5). The filled symbols correspond to the discrete control coefficients for extrapolated steady states at 5 and 10 Hz calculated using the analytic solution for additional connectivity between AM and SR ATPase via cytosolic [Ca^{2+}] (Table 6). The stippled area in Fig. 4indicates the stimulation frequency range in which steady states were not experimentally determined but defined on the basis of extrapolation of the experimental data.

The results shown in Fig. 4 make three important points that were not revealed by the results in Tables 4-6.

First, over the physiological range of steady states (Fig. 4, nonstippled area), the results of the discrete vs. continuous approaches taken in the calculation of flux control coefficients were almost identical. Both predicted onset of the inversion of kinetic control of respiration flux (*J*
_{1}) and SR ATPase flux (*J*
_{3}) at frequencies >1 Hz. Over this physiological range of steady states, only a single connectivity between the modules via ATP/ADP was considered in the discrete approach (*cases I* and *II*). This agreement of both approaches over this particular stimulation frequency range makes the case that the main conclusion of the control analysis (i.e., that flux control in the network inverses from ATP demand to ATP supply control hierarchy as the stimulation frequency increases at >1 Hz) results from the particular ATP/ADP sensitivities of the ATPase modules and their integration in a network. In the frequency range of 1–2 Hz (*case II*), all elasticities of the modules were calculated in a straightforward manner, with use of only the MCA definition (*Eq. 2
*) and the steady-state kinetics of each ATPase (*Eqs. 3-5
*), as described in methods; no “effective” elasticity was introduced for any module over this particular frequency range in which the flux control hierarchy inversion begins. The particular assumptions and simplifications made in the derivation of the additional connectivity between the activities of SR and AM ATPase via cytosolic [Ca^{2+}] (*Eqs.6-11
* and *
14-16
*) and its translation into an effective ATP/ADP elasticity of AM ATPase for steady states in this frequency range only affected the progression of the flux control inversion in the frequency range >2 Hz.

The principal ambiguity in the discrete solutions for the relation between flux control coefficient and stimulation frequency (Tables4-6) was the choice of the contraction frequencies discriminating*cases I-III*. The particular choices that were made (Tables1-3) used experimental observations of mechanical performance of isolated mouse muscles at 25°C obtained in our laboratory. We have found that fast-twitch muscle can maintain constant force for contraction frequencies up to 0.75 Hz (unpublished data). Extrapolating this to a mixed-fiber muscle at a 10°C higher temperature [and thus 2-fold faster SR ATPase kinetics (13, 44)], we assumed that the condition of constant force (*cases I* and*II*) applied to contraction frequencies up to ∼2 Hz. The continuous solution in Fig. 4, on the other hand, used an algorithm that incorporated quantitative information on the actual kinetics of SR Ca^{2+}-ATPase pumping (*Eq. 14
*). There are uncertainties in each approach. The finding of fair agreement between the results of both approaches suggests that mechanical performance is an acceptable criterion for determining the conditions that apply to*case III*.

Third, the inversion of kinetic control of flux in the ATPase network going from intermediate to high contraction frequencies is smooth, not abrupt, and occurs over an approximately fivefold increase of stimulation frequency (Fig. 4
*A*; 5-fold range was calculated for the increase of*C*
_{1}
^{J1} from 0.1 to 0.9).

Finally, the result in Fig. 4
*D* shows that the ATP/ADP control coefficient was maximal at a stimulation frequency of 5 Hz. The absolute value of *C*
_{1}
^{ATP/ADP} in this maximum was 20% higher than at 1.8 Hz. This indicated that the ability of the network to regulate ATP/ADP during contraction further deteriorates as the contraction frequency of the muscle increases above 2 Hz, causing even larger changes in ATP/ADP per unit increase in AM and SR ATPase activity associated with increased duty cycle of contraction.

## DISCUSSION

Muscle activity above the basal resting state is controlled by external signals: experimental electrical stimulation or neural input. Muscle contraction can be sustained in a steady state with certain frequencies of stimulation, but, with more intense activity levels, muscle fatigues. The characteristic feature of the sustainable steady state in a variety of muscles is a limited decrease in the ATP free energy potential (22, 31, 32, 37) from approximately −64 to −55 kJ/mol in human muscle. We investigated this homeostatic regulation in this study.

We used a model of an externally driven metabolic network consisting of AM ATPase, SR ATPase, and mitochondria interacting via ATP/ADP in the cytosol. The main finding of our study was that this network with additional interaction between AM and SR ATPase via cytosolic [Ca^{2+}] at high stimulation frequencies is, in and by itself, homeostatic. Over most of the range of steady states, kinetic control of the network resides in the major ATPase demand in the network, the AM. However, kinetic control of the network by the AM ATPase shifts toward the mitochondria at stimulation frequencies >1 Hz for a mixed-fiber skeletal muscle (Tables 5 and 6, Fig. 4,*A–C*). This result is the major finding, and it demonstrates the inherent homeostatic property of muscle by which ATP free energy consumption in this system cannot outstrip the capacity for ATP free energy supply. The second finding was that kinetic control of the ATPase fluxes and ATP/ADP in the network was distributed over the three ATPase modules rather than retained by a single “rate-limiting” module. Our third finding was that this kinetic control is dynamic rather than static; i.e., the distribution of control was different for different metabolic steady states (Tables4-6, Fig. 4). Nonetheless, for a large fraction of the homeostatic range (low and intermediate stimulation frequencies), the kinetic control of AM ATPase flux resided exclusively in its own ATPase (Tables4-6, Fig. 4
*B*). These three characteristics of control have important functional consequences discussed below.

These homeostatic features of the network are not inherent in branched metabolic networks themselves; i.e., regulation over this normal range of free energies is not a structural or mathematical property of the system. The particulars of kinetic control of ATP free energy metabolism in the network and the physiological implications of that control are purely consequences of the kinetic properties of the individual enzymes as a function of [ATP], [ADP], and [Ca^{2+}]. This control analysis leads to a novel insight into the regulation of muscle energetics. For low and intermediate frequencies of stimulation, kinetic control by AM and SR ATPase by and large accounts totally for the control of flux in the network; the mitochondria properties are essentially irrelevant in this regulation. At higher rates of stimulation, the control of flux in the network becomes inverted; i.e., AM ATPase loses its dominance, and control increasingly is found in the mitochondria. Thus there is no single answer to the following question: Do mitochondria properties or AM properties control muscle energetics? The answer depends on where the muscle operates within its normal physiological range. The functional consequence of this property of the regulation is that muscle cannot exceed the capacity of mitochondria to generate ATP on a sustained basis. Although this behavior of muscle is known (37), our work shows that the reason lies entirely in the properties of the simple network; other mechanisms are not needed, even though they may be present and functionally active.

### Requirements for Homeostatic Regulation of ATP Free Energy

Initially, we considered kinetic control of ATPase fluxes and ATP free energy in a linear model in a network composed of only AM ATPase and mitochondria (27). However, this network was not homeostatic. That analysis showed that the AM ATPase maintained dominant control of its ATPase flux, even in the range of very high stimulation frequencies. In contrast to the properties of the three-component system, the two-component model has a consequence that the AM ATPase could outstrip mitochondrial supply capacity. Thus the salient limitation of the two-component system was that it lacks intrinsic homeostasis. The same lack of homeostasis could be obtained with the three-component network studied if the Ca^{2+}interactions were not included; i.e., the kinetic effects in *Eqs.10
* and *
11
* were omitted (results not shown). The lack of homeostasis in both cases (the 2-component system and the 3-component system without Ca^{2+} interactions) is the same: the initial primacy of the AM ATPase on the free energy remains dominant throughout the entire range of stimulation frequencies. Thus it is clear that a certain degree of complexity among a few components is needed to achieve physiological regulation in muscle energetics. We conclude that the branched network of AM ATPase, SR ATPase, and mitochondria ATP synthesis with interactions between the components through ATP/ADP and cytosolic [Ca^{2+}] constitutes the minimal model of ATP free energy metabolism in contracting muscle that is sufficient to account for homeostasis of ATP free energy.

Of course, more complex models could be constructed that would also achieve free energy homeostasis. One example is inclusion of Ca^{2+} effects in mitochondria (24). Such effects would likely alter the details of distribution of control as a function of stimulation frequency; it would also increase the complexity of the equations considerably. Another example is the inclusion of a glyco(geno)lytic ATPase flux at high stimulation frequencies. This additional ATP synthesis flux and its associated proton load would negatively affect AM ATPase flux via pH alteration of the Ca^{2+} sensitivity of troponin (8), would negatively affect the mitochondria synthesis flux indirectly by decreasing [ADP] as a consequence of altered creatine kinase equilibrium (34, 37, 44), and would positively affect ATP free energy by the additional ATP synthesis flux (28). The existence of various mechanisms not included here and their clear functional consequences do not alter the significance of the main point of this work, which is that the minimal three-component network defined has intrinsic homeostatic properties. The possibility of additional components and the resultant extra modes of regulation show that the ATP free energy is redundantly controlled in normal muscle.

### Sensitivity of the Homeostatic Properties of the System to Altered Kinetic Properties of Single Components

The control analysis of our basic model revealed a broad and dynamic spectrum of ATP/ADP sensitivities in the network. On one side of the spectrum was the case of AM ATPase that is essentially insensitive to [ADP] over a concentration range far exceeding the physiological range (17). On the other side of the spectrum was the 1,000-fold higher ADP sensitivity of mitochondria under unstimulated conditions, which progressively decreased as much as 30-fold as the stimulation frequency approached 10 Hz (Tables1-3). The ADP sensitivity of SR ATPase in unstimulated muscle was intermediate between these two extremes and increased threefold over the 10-Hz stimulation frequency range (Tables 2 and 3). These biological constraints within the network are a consequence of the values of the kinetic constants of each ATPase such as*K*
_{m}, *K*
_{i}, and kinetic order*n* of the reaction (see *Eqs. 1
* and *
3-5
*). It is important to recognize that these particular properties are not fixed in nature. In mammalian muscle, these properties are subject to the particular genotype of the individual and resultant isoform expression as well as to the history of type and intensity of muscle activity, i.e., the adaptive phenotype (11). Furthermore, we assume that the particular characteristics for conservation of ATP free energy and neural control of muscle function are the result of evolutionary pressure and have survival value. When the kinetic properties of the components change, as they did during evolution and as they might in disease, the specifics of the network and its regulation as a system must also change.

To test how dependent the system homeostatic properties were on the particular values of the kinetic constants of each ATPase, we performed a sensitivity analysis. The results are shown in Table7. We tested for 10 kinetic constants the effect of a twofold change in value in either direction (doubling and halving) on the stimulation frequency dependence of the flux and concentration control distribution in the network. For none of the kinetic constants tested was it found that the control distribution and its frequency dependence changed fundamentally. The rate of inversion of the flux control hierarchy and the ATP/ADP homeostatic capacity of the network were affected, however (Table 7). The former was quantified by two parameters: *1*) the frequency coordinate of the intersection of the frequency dependences of*C*
_{1}
^{J1} and*C*
_{2}
^{J1}, respectively, and *2*) the slope (d*C*
_{1}
^{J1}/dfreq) of the frequency dependence of*C*
_{1}
^{J1} in the intersection point. The ATP/ADP homeostatic capacity of the network was quantified by the maximum value of the ATP/ADP control coefficient. The changes in these properties resulting from each kinetic constant doubling and halving are listed in Table 7 and are illustrated in Fig.5.

The sensitivity analysis showed that the flux and concentration control properties of the network are insensitive to the precise values (within the same order of magnitude) of the ATP and ADP affinity of the modules AM and SR ATPase (Table 7). However, these properties of the network were quite sensitive to the precise value of the time constant of Ca^{2+} clearance and to the uncompetitive inhibition constant for ADP of SR ATPase, *K*
_{i′}
^{MgADP}. The latter was especially important and relevant, because no value for this constant had been reported in the literature, let alone for the two different isoforms of the enzyme (I and IIA) in mammalian skeletal muscle (48). We obtained only a rough estimate of this value (0.52 ± 0.20 mM, see methods) on the basis of a report of the SR ATPase kinetics studied at a nonphysiological temperature in vesicles prepared from a mixed-fiber-type muscle that, therefore, may have contained both isoforms of the enzyme (42). For a proper understanding of functional differences between slow- and fast-twitch skeletal muscles, it is important to determine the precise value of *K*
_{i′}
^{MgADP} for both isoforms of SR ATPase.

As to the kinetic constants of mitochondria, the analysis showed that the network control properties were not very sensitive to the maximal ATP synthesis rate of the mitochondria module. The properties were much more sensitive to the precise “operational” point of oxidative phosphorylation, i.e., to the ATP/ADP at which respiration is half-maximally stimulated (29) (*K*
_{0.5}
^{ATP/ADP}) and to the precise apparent kinetic order of the ADP sensitivity of mitochondria (*n*
_{H}
^{ADP}) (29). This result is important and relevant, because both of these kinetic “constants” are not necessarily fixed numbers. On the contrary,*K*
_{0.5}
^{ATP/ADP} is a variable depending on*V*
_{max} and, as such, is subject to conditions affecting protonmotive force generation (47), such as oxidative substrate selection. As for *n*
_{H}
^{ADP}, a first-order reaction had been generally assumed (15) until we recently showed that it is at least second order (29). Therefore, precise determination of these values for specific conditions and for skeletal muscle phenotypes will be necessary for proper understanding of the physiology.

The results of the sensitivity analysis may also be read as a guide to what type of mutations in proteins should be expected to affect contractile and ATP free energy homeostatic function of skeletal muscle. Together with Fig. 4, they also define in which frequency domain these effects should be tested experimentally. For example, effects on contractile function of mutations in SR ATPase affecting uncompetitive binding of ADP should be tested in the high stimulation frequency domain, where SR ATPase is predicted to have substantial control of force production associated with contractile AM ATPase flux (Fig. 4
*B*). In contrast, the effects of such mutations on mitochondrial ATP synthesis flux should be tested in the low-to-intermediate stimulation frequency domain, where SR ATPase has substantial control of this flux.

The control analysis may also be used to predict the effects on contractile and homeostatic function of mutations affecting multiple kinetic constants of a network component. Figure 5 shows the results of a simulation for such a case: a genetic defect in a mitochondrial proton pump affecting *V*
_{max} and*K*
_{0.5}
^{ADP} of the mitochondria. For the simulation, we used results from ^{31}P-NMR spectroscopic measurements on forearm flexor muscle and oxygen polarography studies of mitochondria isolated from thigh muscle of patients with a mitochondrial myopathy caused by a partial defect of complex I of the respiratory chain (3). These patients have a pathologically constricted range of sustainable muscle function presenting clinically as exercise intolerance. The ATP/ADP in unstimulated forearm flexor muscle was twofold lower than in controls (3), indicating a compromised ability for ATP free energy homeostasis, even at rest. The maximal mitochondrial ATP synthesis capacity (*V*
_{max}) and the affinity for ADP were twofold lower than in controls (unpublished results; Ref. 33). The simulation results showed that, in this case, some ability to regulate ATP/ADP and mechanical performance of the muscle would be retained. However, the stimulation frequency range for this ability to regulate ATP/ADP was severely contracted compared with normal conditions, explaining the clinical presentation of a mitochondrial myopathy (Fig.5). Also, the simulation indicates that experimental design of clinical tests of mitochondrial function in skeletal muscle in this patient group should be tailored toward conducting measurements at the highest sustainable work loads of contractile work where mitochondrial properties dominantly control flux in the network. This is somewhat counterintuitive and at odds with common experimental designs of such studies (33).

### Simplicity vs. Complexity and Essentials vs. Details

In the development of the model of ATP free energy metabolism in contracting muscle, we strove for a balance between the level of complexity necessary to capture sufficient major aspects of the physiology of muscle contraction and the level of reduction needed to solve the control analysis. The result of this balance is a better understanding of the way in which components interact to obtain properties of the system. These system properties reside in the system, not the components. Of course, these system properties are defined by the kinetics of the components. Further developments of the analysis of muscle energetics will not be trivial. An increase of the algebraic complexity of the analysis will be required to add any other components, even when using the concepts given here. This complexity applies to incorporation of additional ATP free energy-consuming modules such as the Na^{+}-K^{+}-ATPase pump. More information will be required on the existing components, e.g., the kinetics of the ATPase components and their isoforms with respect to ADP, P_{i}, and pH. The insufficient accuracy of knowledge of*K*
_{i}
^{′} for SR ATPase has been discussed. Less is known about the effects of P_{i} and pH on AM and SR ATPase kinetics. This lack was one of the reasons we chose ATP/ADP as the common metabolic intermediate in the network instead of the full expression of Δ*G*
_{ATP}. Developments of MCA formalism and theory are also needed to make the analysis more complex and complete. For example, the use of the full expression of Δ*G*
_{ATP} in present MCA theory has been clarified only for the case in which P_{i} and the sum of ATP and ADP are constant (47), a condition that is violated in creatine kinase-containing cells such as muscle. Addition of a second ATP free energy synthesis component to the network, such as the glyco(geno)lytic ATPase system, would be a desirable development of the analysis. This constitutes a major challenge, because, in addition to increasing the algebraic complexity of the control analysis and requiring further MCA theory development just mentioned, the regulation of glycolysis is more complex and less well understood than the regulation of mitochondrial function, involving ATP, ADP, AMP, and P_{i} as well as Ca^{2+} (37).

Finally, this control analysis also provides a basis for further exploration of Ca^{2+} regulation of muscle function. Another MCA concept, the response coefficient (*R*
_{x}
^{y}), which is defined as *R*
_{x}
^{y} = Σ*C*
_{i}
^{y}ε_{x}
^{i}, where *y* is a system variable and *x* is a system parameter or external effector (47), may then be implemented. Such an extension of the analysis will address one particular and unique aspect of the SR ATPase module in muscle that has only partly been addressed. SR ATPase has the dual role of a modulator of the energetic state and a modulator of the externally controlled signal that controls AM ATPase. The ATPase aspect and its consequence for SR ATPase control were quantified in the present analysis (Fig. 4). However, to predict the overall effect of a change in activity of SR ATPase on the system steady state, the response coefficient*R*
_{SR ATPase}
^{y} must be used. For example, to assess the net effect of a change in SR ATPase activity on AM ATPase flux (*J*
_{2}), one would obtain [assuming Ca^{2+} stimulation of mitochondria in fast-twitch muscle is negligible (4)]
Equation 17where *C*
_{3}
^{J2}corresponds to the (negative) control over this flux exerted by SR ATPase via its effect on the cell energetic state (Fig. 4
*B*) and*C*
_{2}
^{J2}ε_{SR ATPase}
^{2}corresponds to the control aspect of a change in SR ATPase activity exerted on AM ATPase flux that is due to its effect on the [Ca^{2+}] attained after stimulation. The second term is composed of two positive values.*C*
_{2}
^{J2} is positive and generally >0.4 (Fig. 4
*B*). In our*case I*, ε_{SR ATPase}
^{2} will be zero. When the frequency of stimulation increases so as to enter *case II*, this apparent elasticity will increase to ≤3 in the extreme case (see *Eq. 1
*). The summed effect of an increase in SR ATPase activity on AM ATPase flux will thus be positive. This one example illustrates that the complete Ca^{2+} regulation analysis for all system fluxes and concentrations merits further development. However, such an extension of the analysis was beyond the scope of the present study.

These difficulties and complexities involved in further development of the MCA analysis of the energetics of contracting muscle are serious only if the problem is viewed as needing a solution to account for all the details of muscle physiology. We believe this study shows that a simpler, more synoptic view of the essentials makes a significant advance in understanding the system. It will perhaps be most important as a next step to design experiments that will test the predictions of the control analysis in the high-frequency domain. Slow-twitch skeletal muscle, such as cat soleus, in which anaerobic ATP free energy synthetic flux under those conditions will probably remain low (37), appears a suitable experimental preparation of skeletal muscle in which these tests could be successfully conducted.

## Acknowledgments

The authors are grateful to Bryant Chase, Robert Wiseman, Ron Meyer, and Rafael Moreno-Sanchez for valuable discussions.

## Appendix

### Flux Control in the System

#### Flux J_{1}.

*Equations EA1-EA3
* apply to the control of flux*J*
_{1} in the ATPase network at steady state
Equation A1This is the MCA principle that the flux control coefficients in a network like ours sum to 1 (21, 26, 30, 47). A flux control coefficient *C*
_{i}
^{J} is the control coefficient (also termed control strength) of module*i* over flux* J*, loosely defined as the percent increase in flux *J* under steady-state conditions resulting from a 1% increase in activity of module *i* (21, 26,30, 47)
Equation A2
*Equation EA2 *is a mathematical formulation of the property of a steady-state system that it is stable to its own fluctuations (e.g., a fluctuation in a system variable such as a metabolite concentration ratio) (47). ε_{S1}
^{1} is the elasticity coefficient (also termed sensitivity) of *module 1* toward S_{1}, loosely defined as the percent increase of the rate *v* of*module 1* resulting from a 1% increase in S_{1}under non-steady-state conditions; after the system returns to the steady state at which it was before perturbation, the change in S_{1} will be nullified (21, 26, 30, 47)
Equation A3
*Equation EA3
* is the branch theorem for control of flux*J*
_{1} in the system (39, 47).

By substitution of *Eq. EA3
* into the summation relation (*Eq. EA1
*), the flux *J*
_{1} control distribution is obtained in terms of the flux *J*
_{1}control strength of *module 2*,*C*
_{2}
^{J1}
Equation A4By combining *Eqs. EA1
* and *
EA4
* and substituting into the connectivity relation for flux*J*
_{1} (*Eq. EA2
*), the flux*J*
_{1} control strength of *module 2* can be expressed in terms of elasticities of the modules toward S_{1} and the ratio of fluxes *J*
_{2} and*J*
_{3}
Equation A5The flux *J*
_{1} control distribution within the system will thus depend on the relative elasticity of each of the three ATPases and the relative magnitude of the branch fluxes. Rearranging *Eq. EA5
* to contain only ratios of the elasticities of the modules toward S_{1} and the fluxes*J*
_{2} and *J*
_{3} and substituting this expression for*C*
_{2}
^{J1} into *Eq.EA4
*, we obtain the following expression for flux*J*
_{1} control in the system at steady state
Equation A6a
Equation A6b
Equation A6cwhere
For the particular metabolic pathway under consideration (Tables1-3), ε_{ATP/ADP}
^{1} is <0, whereas ε_{ATP/ADP}
^{2} and ε_{ATP/ADP}
^{3} are >0. With α > 0, the denominator *D* in *Eq. A6, a–c,* is a positive number, as are all nominator terms. Consequently, all flux *J*
_{1} control coefficients in the pathway of Fig. 2 are positive; i.e., activation of mitochondria, AM ATPase, or SR ATPase causes an increase of ATPase flux*J*
_{1} in the network.

#### Flux J_{2} control.

*Equations EA7-EA9
* apply to flux *J*
_{2}control in the pathway of our model at steady state
Equation A7
Equation A8
Equation A9Substituting into *Eq. EA7
* that*J*
_{2}/*J*
_{3} = α and, at steady state, *J*
_{1} =*J*
_{2} + *J*
_{3}, the flux*J*
_{2} control distribution is obtained in terms of the flux *J*
_{2} control strength of *module 1*, *C*
_{1}
^{J2} (not shown). Combining this relation with the connectivity theorem for flux*J*
_{2} (*Eq. EA8
*), one obtains the following flux *J*
_{2} control distribution in the branched pathway of Fig. 2
Equation A10a
Equation A10b
Equation A10c

#### Flux J_{3} control.

*Equations EA11-EA13
* apply to flux*J*
_{3} control in the pathway of our model at steady state
Equation A11
Equation A12
Equation A13Substituting into *Eq. EA11
* that*J*
_{2}/*J*
_{3} = α and at steady state *J*
_{1} =*J*
_{2} + *J*
_{3}, the flux*J*
_{3} control distribution is obtained in terms of the flux *J*
_{3} control strength of *module 1, C*
_{1}
^{J3} (not shown). Combining this relation with the connectivity theorem for flux*J*
_{3} (*Eq. EA12
*), one obtains the following flux *J*
_{3} control distribution in the branched pathway of Fig. 2
Equation A14a
Equation A14b
Equation A14c

### Concentration S_{1} Control in the System

*Equations EA15-EA17
* apply to concentration*S*
_{1} control in our model at steady state
Equation A15This is the principle that the concentration control coefficients in a network like ours sum to 0 (21, 47)
Equation A16
*Equation EA16
* is, like *Eq. EA2
*, a mathematical formulation of the property of a steady-state system that it is stable to its own fluctuations (47). This present formulation shows the particular counteractive nature of the system response to a fluctuation in metabolite S_{1}, restoring the initial steady state
Equation A17
*Equation EA17
* is the branch theorem for control of concentration S_{1} in the system (47).

One can now solve the S_{1} control distribution in the system at steady state in terms of the control coefficient for the control of S_{1} by *module 2, C*
_{2}
^{S1}. To this aim, one develops the summation relation for S_{1} control (*Eq. EA16
*) in analogy to the analysis for *J*
_{1} to obtain
Equation A18Developing the connectivity relation for S_{1} control (*Eq. EA17
*) using *Eq. EA18
*, we obtain the following expression for *C*
_{2}
^{S1}
Equation A19Substituting this expression for*C*
_{2}
^{S1} into *Eq.EA19
*, we obtain the following expression for the concentration S_{1} control distribution in our model at steady state *i*
Equation A20a
Equation A20b
Equation A20cwhere
The denominator *D* in *Eq. A20, a–c,* is a negative number for the particular pathway that we shall analyze. The numerator is negative in *Eq.EA20a
* but positive in *Eq. A20, b* and*c*. It follows that concentration S_{1} control by*module 1* is positive but is negative for *modules 2* and *3*; i.e., activation of mitochondria will increase ATP/ADP, but activation of AM or SR ATPase will decrease ATP/ADP.

## Footnotes

This work was supported in part by National Institute of Arthritis and Musculoskeletal and Skin Diseases Grant AR-36281 (to M. J. Kushmerick) and the Netherlands Organization for Scientific Research (H. V. Westerhoff).

Address for reprint requests and other correspondence: J. A. L. Jeneson, NMR Research Laboratory, Dept. of Radiology, Box 357115, University of Washington Medical Center, Seattle, WA 98195 (E-mail: utrecht{at}u.washington.edu).

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