Am J Physiol Cell Physiol
279: C813–C832, 2000.
A metabolic control analysis of kinetic controls in ATP
free energy metabolism in contracting skeletal muscle
J. A. L. JENESON,1 H. V. WESTERHOFF,2,3 AND M. J. KUSHMERICK1,4,5
Departments of 1Bioengineering, 4Physiology and Biophysics, and 5Radiology, University of
Washington School of Medicine, Seattle, Washington 98195; 2Department of Molecular Cell
Physiology, Faculty of Biology, Free University, Amsterdam; and 3Department of Mathematical
Biochemistry, Biocenter, University of Amsterdam, Amsterdam, The Netherlands
Received 7 July 1999; accepted in final form 30 March 2000
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 31P-NMR measurements can be expressed as specific biochemical mech-
anisms. 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 secondorder 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
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: [email protected]).
The costs of publication of this article were defrayed in part by the
payment of page charges. The article must therefore be hereby
marked ‘‘advertisement’’ in accordance with 18 U.S.C. Section 1734
solely to indicate this fact.
cellular energetics; skeletal muscle; metabolic control analysis
http://www.ajpcell.org
0363-6143/00 $5.00 Copyright © 2000 the American Physiological Society
C813
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Jeneson, J. A. L., H. V. Westerhoff, and M. J. Kushmerick. A metabolic control analysis of kinetic controls in ATP
free energy metabolism in contracting skeletal muscle. Am J
Physiol Cell Physiol 279: C813–C832, 2000.—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) Ca2⫹-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 31P-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 31P-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.
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CONTROL ANALYSIS OF ATP FREE ENERGY METABOLISM IN MUSCLE
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. 1A). 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 Ca2⫹ concentration
([Ca2⫹]), because the activity of AM and SR ATPase is Ca2⫹
dependent (44) and, under conditions of saturating ATP free
energy, is described by the Hill function
v ATPase([Ca 2⫹]) ⫽ V max 䡠
([Ca 2⫹]) n H
([Ca 2⫹] 50 ) n H ⫹ ([Ca 2⫹]) n H
(1)
where the Hill coefficient (nH) is 2 for SR ATPase and 3 for
AM ATPase (44). [Ca2⫹] required for half-maximal stimulation ([Ca2⫹]50) is ⬃0.2 ␮M for SR ATPase and ⬃0.8 ␮M for
AM ATPase (44). In the unstimulated muscle cell, cytosolic
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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
o⬘
ATP free energy (2) {⌬GATP ⫽ ⌬GATP
⫹ RTln([ADP]
o⬘
[Pi]/[ATP]), where ⌬GATP ⫽ ⫺32.8 kJ/mol (41), R is the
gas constant, T is the absolute temperature, and [Pi]
and [ATP] are Pi and ATP concentrations} during contractile activity. 31P-NMR measurements revealed
that ⌬GATP in skeletal muscle ranges from approximately ⫺64 kJ/mol at rest to ⫺55 kJ/mol at maximal
sustainable contractile activity (31). Thus ⌬GATP 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 (pHi) 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) Ca2⫹ 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 31P-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, Ca2⫹-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
CONTROL ANALYSIS OF ATP FREE ENERGY METABOLISM IN MUSCLE
C815
[Ca2⫹] is well below 0.2 ␮M (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 Ca2⫹ pulse, causing cytosolic
[Ca2⫹] 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
Ca2⫹ pumping, respectively (44). ATP free energy drives both
processes in the forward direction (Fig. 1B). When cytosolic
[Ca2⫹] 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. 1B, 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
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Fig. 1. A: schematic diagram of ATP free energy metabolism in skeletal muscle. The 3 main ATPases in the
muscle cell [the contractile actomyosin (AM) ATPase,
the sarcoplasmic reticulum (SR) Ca2⫹ pump ATPase
(SR ATPase), and the mitochondrial ATP synthase] are
indicated by ellipsoids. The reversible activation of the
contractile AM ATPase is initiated by reversible Ca2⫹
binding to troponin. B: schematic diagram of the energetic events from onset of stimulation to attainment of
a steady state. Events in the top 2 curves diagram
measured variables, the phosphocreatine (PCr) content
([PCr]) and the rate of oxidative phosphorylation (J1, in
minutes); the bottom 3 curves display events at higher
time resolution (in seconds) that are not measured in
this study. [PCr] is initially high before the muscle is
stimulated. At the onset of twitch stimulation, [PCr]
declines to a steady state if the flux of oxidative phosphorylation is sufficiently high to achieve energy balance. J1 rises gradually to its maximum for the steadystate condition. In the steady state, total ATPase flux
equals total ATP synthesis flux. Dashed lines, higher
time resolution. With each twitch stimulation, AM
and SR ATPase are pulsatile. Force and AM ATPase
transient persist longer than the transient increase in
cytoplasmic Ca2⫹ concentration ([Ca2⫹]) and the SR
ATPase rate. The bottom curve displays a gradually
rising rate of oxidative phosphorylation with a transient pulse superimposed. This pulsatile flux of oxidative phosphorylation has not been observed (19).
The curve was obtained with a model (34) in which all
the ATPase activity was constrained to occur during
100 ms.
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CONTROL ANALYSIS OF ATP FREE ENERGY METABOLISM IN MUSCLE
Fig. 2. Network interactions of the model of muscle energy analyzed.
E1, E2, and E3, enzyme assemblies. E1, mitochondria pool in muscle;
E2, AM array in the filaments; E3, SR Ca2⫹ pump. S1, common
metabolite shared by E1, E2, and E3. E1 and E3 have a reversible flux
(v) into and out of S1; the ATPase of E2 is irreversible. The activities
of each component, the magnitude of the shared metabolite pool, and
the fluxes are considered in our analysis. Implicit (but explicitly
considered) in the control analysis is a second interdependence of
rates v2 and v3 via Ca2⫹.
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 [Ca2⫹] because of the
periodic nature of the ATPases (Fig. 1A). The cytosolic [Ca2⫹]
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 (CiJm), three
ATP/ADP control coefficients (CiATP/ADP), and three ATP/ADP
i
elasticity coefficients (⑀ATP/ADP
) 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 (see DISCUSSION). 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 APPENDIX 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
i
ATP/ADP (⑀ATP/ADP
) and the ratio of fluxes in the branches
(␣) 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, J1, 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
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given in Fig. 2 is a modular (43) branched pathway consisting
of three ATPase modules: E1 (cellular pool of mitochondria),
E2 (total AM ATPase), and E3 (total SR ATPase). These
modules consume or produce a common intermediate (S1)
that is related to the cytosolic ATP free energy at rates v1, v2,
and v3, respectively. We chose the cytosolic ATP/ADP for S1.
We were unable to use the full expression of the ATP free
energy for reasons given in the DISCUSSION. 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 S1 among the components at steady
state. Metabolism in this branched network is characterized
by four system variables: three fluxes (J1, J2, and J3, in moles
per volume per time) and one concentration ratio (S1). 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: J1 ⫽ J2 ⫹ J3; 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 Ca2⫹ regulation of the
activity of AM and SR ATPase. Therefore, J2 and J3 are
designated independent fluxes in the network. We define the
variable ␣, denoting the magnitude ratio of these branch
fluxes, J2/J3. 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, ␣ ⫽ 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.
CONTROL ANALYSIS OF ATP FREE ENERGY METABOLISM IN MUSCLE
states. 3) With this set of elasticity values and ␣ ⫽ 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
i
metabolism in muscle [(⑀ATP/ADP
)k] was calculated on the
basis of the steady-state kinetics of the enzyme module according to (21, 39, 47)
i
) k ⫽ [(ATP/ADP)/v i ] k 䡠 [dv i /d(ATP/ADP)] k
(⑀ ATP/ADP
(2)
J p ⫽ [(J p) max ⫺ (J p) min]
⫻
1
ATP/ADP 2 ⫺ (J p) min
1 ⫹ [(ATP/ADP)/K 0.5
]
(3)
ATP/ADP
where K0.5
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, Jp) in this skeletal muscle under
conditions of saturating [Pi] with respect to the reaction
(29).
AM ATPase. The elasticity toward ATP/ADP of AM ATPase
2
at steady state k, (⑀ATP/ADP
)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)
v ⫽ V max 䡠
1⫹K
MgATP
m
1
/MgATP 䡠 [1 ⫹ (MgADP/K i)]
(4)
MgATP
where Km
is the Michaelis constant for the substrate
(10–20 ␮M for ATPase activity) and Ki is the inhibition
constant for MgADP, which is on the order of 200–300 ␮M
MgATP
(20). We used 15 and 250 ␮M for Km
and Ki, respectively,
in the calculations.
SR ATPase. The elasticity toward ATP/ADP of SR ATPase at
3
steady state k, (⑀ATP/ADP
)k, was calculated using Eq. 2 on the
basis of the kinetic function describing the SR ATPase forward rate dependence on ATP and ADP
v ⫽ Vmax 䡠
1
(K ATP
i )]
m /MgATP) 䡠 [1 ⫹ (MgADP/Ki)] ⫹ [1 ⫹ (MgADP/K⬘
(5)
MgATP
where Km
is the affinity for the substrate and Ki 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 Ca2⫹ and high Mg2⫹ and the inverse conMgATP
ditions (42). Km
of SR ATPase is at least two orders of
magnitude lower than [ATP] in human skeletal muscle [10
MgATP
␮M (42) vs. 8 mM (25)], and so the term Km
/[MgATP]
(where [MgATP] is MgATP concentration) is ⬍0.01, and Eq.
5 reduces to v3 ⫽ V3 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 (⌬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 [Ca2⫹] 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 Ca2⫹-ATP stoichiometry of the pump and the
amount of Ca2⫹ cleared, instead of the sensitivity of the SR
ATPase to ATP/ADP. Consequently, at sufficiently low contraction frequencies (i.e., ⌬t between contractions is ⬎3 kinetic time constants of the Ca2⫹ 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 Ca2⫹-ATPase recovery will not
go to completion within a single contraction-relaxation cycle.
Depending on ⌬t and ATP/ADP, as well as the capacity for
and on and off rates of Ca2⫹ binding by cytosolic Ca2⫹ buffers
[e.g., parvalbumin (29) and mitochondria (30)], SR Ca2⫹
release and subsequent peak cytosolic [Ca2⫹] 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 Ca2⫹, such as fast-twitch
fibers (11, 37), this scenario will be more prominent. The
magnitude of the AM ATPase flux (J2) will, in this case,
depend on ATP/ADP as well as cytosolic [Ca2⫹]. 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 elas3⬘
ticity of SR ATPase toward ATP/ADP (⑀ATP/ADP
), 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
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The term [(ATP/ADP)/vi]k is a normalization term for absolute reaction velocity, and the term [dvi/d(ATP/ADP)]k defines the particular ATP/ADP sensitivity at this point on the
velocity curve. This second term is the partial derivative of
rate vi with respect to ATP/ADP, i.e., ⳵vi/⳵(ATP/ADP). Therei
fore, this method to calculate (⑀ATP/ADP
)k requires that a
function is used that was determined under conditions where
only ATP/ADP effects on vi were measured; i.e., concentrations of any other affectors of vi were saturating or constant
during the course of the experiment. We were able to obtain
appropriate functions in the literature.
MITOCHONDRIA. The elasticity of mitochondria toward ATP/
1
ADP at steady-state k, (⑀ATP/ADP
)k, was calculated using Eq.
2 on the basis of the kinetic function that describes the
dependence of mitochondrial ATP synthesis flux (J1) on ATP/
ADP in human forearm flexor muscle (29)
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CONTROL ANALYSIS OF ATP FREE ENERGY METABOLISM IN MUSCLE
V SR
[Ca 2⫹] cyto兩t 1⫹␶ ⫽ ␤ 䡠 [Ca 2⫹] SR兩t 0 䡠
⫹ [Ca 2⫹] cyto兩t 0
V cyto
(6)
where ␤ is a time-dependent proportionality factor determined by the permeability of the SR Ca2⫹ release channels
and the duration ⌬t of the pulse, VSR/Vcyto is the ratio of SR
volume to cytosol volume, and [Ca2⫹]cytoⱍt0 is cytosolic [Ca2⫹]
before stimulation. At the onset of the next stimulation of the
cell at time t2, SR [Ca2⫹] recovered by SR ATPase over the
time interval ⌬t (⫽ t2 ⫺ t1) between stimulations is determined by the activity of SR ATPase as follows
[Ca 2⫹] SR兩t 2 ⫽ n PCa 䡠
兰
t2
v 3 dt 䡠
t1
V cyto
⫹ [Ca 2⫹] SR兩t 1
V SR
(7)
Ca
where nP
is the Ca2⫹-per-ATP pumping stoichiometry of the
SR ATPase, v3 is the SR ATPase rate, and [Ca2⫹]SRⱍt1 is the
residual SR [Ca2⫹] after Ca2⫹ release at time t1. An expression for cytosolic [Ca2⫹] that is attained after stimulation at
time t2 is obtained by substituting Eq. 7 into Eq. 6 and
neglecting the last term as a simplification in analogy to
others (5), yielding
[Ca 2⫹] cyto兩t 2 ⫹ ␶ ⫽ ␤ 䡠 n PCa 䡠
兰
t2
t1
v 3 dt ⫹ [Ca 2⫹] cyto兩t 2
(8)
where [Ca2⫹]cytoⱍt2 is cytosolic [Ca2⫹] at the onset of the next
stimulation. This expression can be generalized to describe
the dependence of cytosolic [Ca2⫹] after stimulation n in a
series of N stimulations at a constant frequency on the SR
ATPase activity v3
[Ca 2⫹] cyto兩t n ⫹ ␶ ⫽ ␤ 䡠 n PCa 䡠
兰
tn
v 3 dt ⫹ [Ca 2⫹] cyto兩t n
(9)
tn ⫺ 1
Next, 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 [Ca2⫹]-dependent terms that reduce to the ATP/ADP-dependent rate
equation for AM ATPase (Eq. 4) under conditions of saturating cytosolic [Ca2⫹] (v2 is a function of [ATP], [ADP], and
cytosolic [Ca2⫹])
v 2 ⫽ V 2 max 䡠
1
1 ⫹ (K ATP /[ATP]) 䡠 兵1 ⫹ ([ADP]/K i)其
([Ca 2⫹] cyto) 3
⫻
(K 50 ) 3 ⫹ ([Ca 2⫹] cyto) 3
(10)
We then substituted the expression for cytosolic [Ca2⫹] of Eq.
9 into the scaled rate equation for AM ATPase (Eq. 10) and
made two further simplifications. First, we neglected cytosolic [Ca2⫹] at rest relative to cytosolic [Ca2⫹] after stimulation. Second, we used the time-averaged SR ATPase rate v3
so that 兰v3dt ⬇ v3⌬t (⌬t is the time interval between muscle
Ca
stimulations) and defined a constant K 50
⬘ ⫽ (K50/␤nP
). We
thus obtained the following expression for the scaled AM
ATPase rate that is attained after the nth stimulation in a
series of N stimulations at a particular stimulation frequency
v 2 ⫽ V 2 max 䡠
1
1 ⫹ (K ATP / [ATP]) 䡠 兵1 ⫹ ([ADP])/(K i)其
(11)
1
⫻
兵1 ⫹ [K⬘5 0 /(v 3 䡠 ⌬t)] 3 其
With the use of Eq. 11 and a method to quantify the Ca2⫹dependent term, the effective elasticity of AM ATPase toward
ATP/ADP under conditions of submaximal cytosolic [Ca2⫹]
activation of the enzyme can now be calculated from Eq. 2.
We developed two different methods to quantify the Ca2⫹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 Ca2⫹ 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
2
⑀ ATP/ADP
⫽
[(
d ln v 2
d
⫽
d ln(ATP/ADP) d ln(ATP/ADP)
⫻ ln V 2 max 䡠
1
1 ⫹ (K ATP /[ATP]) 䡠 兵1 ⫹ ([ADP]/K i)其
⫺ ln兵1 ⫹ [K⬘5 0 /(v 3 䡠 ⌬t)] 3 其
)
]
(12)
Under the limit condition ⌬t30, i.e., at high contraction
Ca
frequencies, the term [K⬘50/(v3⌬t)]3 ⬎ 1. Also, K50, ␤, and nP
Downloaded from http://ajpcell.physiology.org/ by 10.220.33.6 on June 15, 2017
longer sufficient for SR ATPase to recover all Ca2⫹ released
on stimulation irrespective of the ATP/ADP sensed by the
enzyme, but Ca2⫹ 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 Ca2⫹ recovery by SR ATPase
with stimulations at ⱕ200-ms intervals is insufficient to
ensure maximal Ca2⫹ activation of AM ATPase on subsequent stimulation, so twitch force over time declines. As a
simplification, any buildup of cytosolic [Ca2⫹] at these stimulation frequencies affecting force is not considered. The
dependence of AM ATPase rate (and, therefore, force) on
[Ca2⫹] is very steep (Eq. 1). If we assume for case II that
enough SR Ca2⫹ is recovered to ensure that cytosolic [Ca2⫹]
will be at least twice [Ca2⫹]50 after stimulation, so that AM
ATPase will be stimulated to ⱖ89% of the maximal activity
(Eq. 1), twitch force should not fall ⬎10% at the intermediate
contraction frequencies. If, in case III, cytosolic [Ca2⫹] 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/
2⬘
ADP (⑀ATP/ADP
) 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 [Ca2⫹], and AM ATPase (see below).
Calculation of effective elasticity of AM ATPase toward
ATP/ADP at high frequencies of activation. [Ca2⫹] in the
cytosol and in the SR lumen are each at their respective
baseline value in the resting state. At time t1, the muscle cell
is excited by an action potential from the motor nerve and the
SR releases its Ca2⫹ into the cytosol; cytosolic [Ca2⫹] reaches
a maximum at time t1 ⫹ ␶, when Ca2⫹ in the lumen of the SR
is lower and significant binding to cytoplasmic proteins occurs. The relation between the free [Ca2⫹] in the two compartments after stimulation can thus be described as follows
C819
CONTROL ANALYSIS OF ATP FREE ENERGY METABOLISM IN MUSCLE
are mute with respect to ATP/ADP sensitivity, but the SR
ATPase rate v3 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
2
⑀ ATP/ADP兩⌬t
30 ⬇
d ln v 2 ([ATP], [ADP])
d ln(ATP/ADP)
3d ln v 3 ([ATP], [ADP])
2
3
⫹ 3 䡠 ⑀ ATP/ADP
⫹
⬇ ⑀ ATP/ADP
d ln (ATP/ADP)
(13)
[Ca 2⫹] SR
⫽ 0.59 䡠 [1 ⫺ exp(t/␶ 1 )]
([Ca 2⫹] SR) max
(14)
vstim, Hz
ATP/ADP
J1/J1 max
1
⑀ATP/ADP
2
⑀ATP/ADP
3
⑀ATP/ADP
1.0
1.3
1.6
1.8
174
141
115
98
0.48
0.60
0.70
0.77
⫺1.21
⫺0.90
⫺0.65
⫺0.49
0.0009
0.0023
0.0046
0.0072
0.09
0.10
0.12
0.14
Elasticities were calculated for intermediate contraction frequencies (case II). See Table 1 footnote for definition of abbreviations and
further explanation.
different stimulation frequencies, the array of [Ca2⫹]SR/
([Ca2⫹]SR)max(⌬t) values was multiplied by the corresponding
array of v3 /Vmax(⌬t) values that was calculated using Eq. 5
and [ADP](⌬t) for each stimulation frequency. For time intervals ⬎500 ms (i.e., contraction frequencies ⬍2 Hz), SR
[Ca2⫹] recovery was ⱖ85% 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 [Ca2⫹]SR/([Ca2⫹]SR)max
recovery immediately before subsequent stimulation was correlated for each stimulation frequency with the corresponding steady-state ATP/ADP determined by 31P-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 [Ca2⫹]SR/
([Ca2⫹]SR)max and ATP/ADP was obtained by curve fitting as
follows
[
⫹ 0.41 䡠 [1 ⫺ exp(t/10␶ 1 )]
where the time constant ␶1 equals 0.035 s. At time 0, the
amount of SR Ca2⫹ (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 Q10 for SR ATPase of 2 at 15–35°C (13, 44), we
corrected ␶1 to 0.009 s. The normalized SR [Ca2⫹] reestablished over the time after stimulation was computed as a
function of the time between subsequent stimulations for the
eight stimulation frequencies of Tables 1–3. This led to a set
of paired values {⌬t and [Ca2⫹]SR/([Ca2⫹]SR)max}. To correct
for different degrees of ADP inhibition of SR ATPase between
1
2
3⬘
Table 1. ⑀ATP/ADP
, ⑀ATP/ADP
, and ⑀ATP/ADP
as a
function of stimulation frequency: case I
vstim, Hz
ATP/ADP
J1/J1 max
1
⑀ATP/ADP
0.3
0.6
315
234
0.16
0.31
⫺2.68
⫺1.77
2
⑀ATP/ADP
1.26E⫺05
1.54E⫺04
3⬘
⑀ATP/ADP
0
0
1
Elasticities of mitochondria (⑀ATP/ADP
) and actomyosin ATPase
(⑀
) and effective elasticity of sarcoplasmic reticulum ATPase
(⑀
) toward ATP-to-ADP concentration ratio (ATP/ADP) as a
function of stimulation frequency (vstim) were calculated for low
contraction frequencies (case I). Steady-state ATP/ADP and relative
ATP free energy metabolic flux ( J1/J1 max) in human forearm flexor
muscle at each stimulation frequency were calculated from 31P-NMR
spectroscopic measurements.
2
ATP/ADP
3⬘
ATP/ADP
(
[
[Ca 2⫹] SR
ATP/ADP
⫽ 0.83 䡠 1 ⫺ exp
2⫹
([Ca ] SR) max
31
)]
(15)
(
ATP/ADP
⫹ 0.14 䡠 1 ⫺ exp
159
)]
Equation 15 was used to compute v2 as a function of ATP/
ADP, cytosolic [Ca2⫹] for each contraction frequency (and
corresponding ATP/ADP) on the basis of Eq. 11 with ␤ ⫽ 1,
Ca
K50 ⫽ 0.5, and nP
v3⌬t ⫽ [Ca2⫹]SR. The resulting covariation
of ATP/ADP and v2 was biexponential and was determined by
nonlinear curve fitting to give
[ (
[ (
v2(ATP/ADP, [Ca2⫹]cyto) ⫽ 0.81 䡠 1 ⫺ exp
)]
ATP/ADP
24
ATP/ADP
⫹ 0.06 䡠 1 ⫺ exp
112
)]
(16)
This 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.
1
3
2⬘
Table 3. ⑀ATP/ADP
, ⑀ATP/ADP
, and ⑀ATP/ADP
as a
function of stimulation frequency: case III
vstim, Hz
ATP/ADP
J1/J1 max
1
⑀ATP/ADP
2⬘
⑀ATP/ADP
3
⑀ATP/ADP
5
10
55
41
0.92
0.96
⫺0.17
⫺0.09
0.63
0.90
0.21
0.30
Elasticities were calculated for high contraction frequencies (case
2⬘
III). ⑀ATP/ADP
was calculated using Eq. 13. Steady-state ATP/ADP at
these stimulation frequencies were extrapolated from the measured
relation between stimulation frequency and ATP/ADP (Fig. 3C).
J1/J1 max in human forearm flexor muscle at these extrapolated
ATP/ADP was calculated using Eqn. 3.
Downloaded from http://ajpcell.physiology.org/ by 10.220.33.6 on June 15, 2017
The 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
2⬘
the limit condition of high contraction frequencies, ⑀ATP/ADP
,
is mostly defined by the SR Ca2⫹-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 Ca2⫹ removal from the cytosol after a Ca2⫹ release pulse (13). This enabled us to
obtain a quantitative relation between SR [Ca2⫹] and the SR
ATPase rate v3. The reported biexponential kinetics of SR
Ca2⫹ accumulation in rat extensor digitorum longus muscle
at 15°C (Fig. 9A in Ref. 13) were digitized and analyzed by
fitting double-exponential functions to obtain the relation
1
2
3
Table 2. ⑀ATP/ADP
, ⑀ATP/ADP
, and ⑀ATP/ADP
as a
function of stimulation frequency: case II
C820
CONTROL ANALYSIS OF ATP FREE ENERGY METABOLISM IN MUSCLE
The analytic solution (Eq. 13) applies only to the limit
condition of high contraction frequencies of case III. The
2⬘
composite analytic-numerical solution to calculate ⑀ATP/ADP
on the basis of Eq. 16 applies to conditions in which cytosolic
[Ca2⫹] is saturating and nonsaturating with respect to the
AM ATPase. The analytic-numerical method allowed compu2⬘
tation of the continuum of ⑀ATP/ADP
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
RESULTS
General Solution of Control in the Network
The general solution for kinetic control of the particular value of the four system variables J1, J2, J3, and
S1 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 ␣ is given in the
APPENDIX (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
[Pi] 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)
␮mol ATP 䡠 s⫺1 䡠 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 ␮mol
ATP 䡠 s⫺1 䡠 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 pHi below 6.9 to values
as low as 6.7 (data not shown). These conditions were
not steady states and were not analyzed further.
Downloaded from http://ajpcell.physiology.org/ by 10.220.33.6 on June 15, 2017
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-steadystate acidification occurred. 31P-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, Pi, 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 ␶PCr (in s) (28). The value of
␶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 ⬎ 3␶PCr s at
which d[PCr]/dt ⬃ 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, Pi,
and ATP integrals and resonance frequencies (6) with Fitmasters software (Philips Medical Systems). Finally, [PCr],
[Pi], 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 pHi was estimated
from the chemical shift difference between the PCr and Pi
resonance (45). The cytosolic free energy of ATP hydrolysis
o⬘
(⌬GATP) was calculated as ⌬GATP
⫹ RTln([ADP][Pi]/[ATP]),
o⬘
where ⫺32.8 kJ/mol was used for ⌬GATP
(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).
CONTROL ANALYSIS OF ATP FREE ENERGY METABOLISM IN MUSCLE
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. 3A). The
C821
steady-state pHi 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. 3C). Steady-state [PCr], pHi, 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
Fig. 3. Dynamic range of the steady-state ATP free energy potential
in electrically stimulated human forearm flexor muscle. A: twitch
contraction frequency-[PCr] relation at steady state in forearm flexor
muscle measured by 31P-NMR spectroscopy. Values are means ⫾ SE
measured in 6 subjects, except for 1.6 and 1.8 Hz. B: relation between
the twitch contraction frequency and the difference between intracellular pH (pHi) at stimulated steady state and pH in unstimulated
forearm flexor muscle measured by 31P-NMR spectroscopy. Values
are means ⫾ SE measured in 6 subjects, except for 1.6 and 1.8 Hz. C:
relation between the twitch contraction frequency and the cytosolic
ATP-to-ADP concentration ratio ([ATP]/[ADP]) at steady state in
forearm flexor muscle measured by 31P-NMR spectroscopy. Values
are means ⫾ SE measured in 6 subjects, except for 1.6 and 1.8 Hz.
Downloaded from http://ajpcell.physiology.org/ by 10.220.33.6 on June 15, 2017
Case I: low-frequency contractions. The elasticity to1
ward ATP/ADP of mitochondria (⑀ATP/ADP
) and AM
2
ATPase (⑀ATP/ADP) calculated as described in METHODS
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 contrac3⬘
tion frequencies (⑀ATP/ADP
) was zero. Steady-state ATP/
ADP and corresponding normalized ATP free energy
metabolic flux [(J1/J1⫺)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
(J1) in the network increased to ⬃30% of maximal
sustainable flux (Table 1). For this range of contraction
1
frequencies, ⑀ATP/ADP
was at least four orders of mag2
nitude greater than ⑀ATP/ADP
, 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 [(J1/J1⫺)max] calculated using Eq. 3 are given
for each contraction frequency (Table 2). Over this
range of contraction frequencies, the steady-state ATP/
C822
CONTROL ANALYSIS OF ATP FREE ENERGY METABOLISM IN MUSCLE
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 ␣ ⫽ 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 (Table 4). With
respect to the two ATP hydrolysis fluxes in the network
(J2 and J3), each ATPase module fully controls its
respective ATPase flux by its activity (C2J2 ⫽ 1 and
C3J3 ⫽ 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 (C1Ji ⫽ 0). This result means
that mitochondrial ATP synthesis flux (J1) passively
follows the ATP demand fluxes (J2 and J3) set by the
summed activity of AM and SR ATPase (C2J1 ⫹ C3J1
⫽ 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, C1ATP/ADP was
1.4-fold higher than C2ATP/ADP and 3.1-fold higher than
C1ATP/ADP at 0.3 and 0.6 Hz (Table 4).
Case II: intermediate-frequency contractions. Kinetic
control of the ATPase fluxes J1 and J3 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 J1 and
J3 is substantial (C1J1 ⫽ 0.05 and 0.07 respectively,
and C1J3 ⫽ 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 J2 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 (C2J2 ⫽ 1
and C1J2 ⫽ C3J2 ⫽ 0 for all 4 contraction frequencies
studied; Table 5).
Table 4. Distribution of kinetic control of J1, J2, J3,
and ATP/ADP among the three ATPase modules of
the network model of ATP free energy metabolism in
contracting skeletal muscle as a function of
stimulation frequency for human forearm flexor
muscle stimulated at low frequencies: case I
0.3 Hz
J1
J2
J3
ATP/ADP
0.6 Hz
C1y
C2y
C3y
C1y
C2y
C3y
0.00
0.00
0.00
0.37
0.69
1.00
0.00
⫺0.26
0.31
0.00
1.00
⫺0.12
0.00
0.00
0.00
0.56
0.69
1.00
0.00
⫺0.39
0.31
0.00
1.00
⫺0.18
J1, J2, J3, and ATP/ADP represent steady-state variables. Flux
and ATP/ADP control coefficients were calculated for each steadystate in the network, as described in METHODS.
Downloaded from http://ajpcell.physiology.org/ by 10.220.33.6 on June 15, 2017
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 (J1) in the network increased to ⬃80% of maximal flux (Table 2). The abso1
lute value of ⑀ATP/ADP
decreased a further 2.5-fold over
this range of contraction frequencies (Table 2). The
3
value of ⑀ATP/ADP
approached the same order of mag1
nitude as ⑀ATP/ADP
over this range of contraction frequencies and increased 1.5-fold between 1.0 and 1.8 Hz
2
(Table 2). The value of ⑀ATP/ADP
remained at least
1
two orders of magnitude smaller than ⑀ATP/ADP
and
3
⑀ATP/ADP, 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 steadystate 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
2⬘
ATP/ADP under these conditions, ⑀ATP/ADP
, was calculated using Eq. 15. At 10 Hz, ATP/ADP in the muscle
was extrapolated to 41, and normalized ATP flux (J1)
in the network [(J1/J1⫺)max] would consequently have
increased to 96% of maximal sustainable flux (Table 3).
1
The absolute value of ⑀ATP/ADP
decreased a further
twofold over this range of contraction frequencies to a
value 30-fold lower than at low contraction frequencies
3
(Tables 1 and 3). The absolute value of ⑀ATP/ADP
(and
2⬘
therefore of ⑀ATP/ADP) over this range of contraction
1
frequencies now exceeded that of ⑀ATP/ADP
by as much
as threefold (Table 3). In this case, the effective elas2⬘
ticity of AM ATPase, ⑀ATP/ADP
, 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.
C823
CONTROL ANALYSIS OF ATP FREE ENERGY METABOLISM IN MUSCLE
Table 5. Distribution of kinetic control of J1, J2, J3, and ATP/ADP among the three ATPase modules of the
network model of ATP free energy metabolism in contracting skeletal muscle as a function of stimulation
frequency for human forearm flexor muscle stimulated at intermediate frequencies: case II
1.0 Hz
J1
J2
J3
ATP/ADP
1.3 Hz
1.6 Hz
1.8 Hz
C1y
C2y
C3y
C1y
C2y
C3y
C1y
C2y
C3y
C1y
C2y
C3y
0.02
0.00
0.07
0.81
0.67
1.00
⫺0.05
⫺0.56
0.31
0.00
0.98
⫺0.25
0.03
0.00
0.11
1.07
0.66
1.00
⫺0.07
⫺0.74
0.30
0.00
0.97
⫺0.34
0.05
0.00
0.16
1.46
0.65
1.00
⫺0.11
⫺1.00
0.30
0.00
0.95
⫺0.46
0.07
0.00
0.23
1.90
0.64
1.00
⫺0.16
⫺1.30
0.29
0.00
0.93
⫺0.59
Flux and ATP/ADP control coefficients were calculated for each steady state in the network, as described in
Table 6. Distribution of kinetic control of J1, J2, J3
and ATP/ADP among the three ATPase modules of
the network model of ATP free energy metabolism in
contracting skeletal muscle as a function of
stimulation frequency for human forearm flexor
muscle stimulated at high frequencies: case III
5 Hz
J1
J2
J3
ATP/ADP
10 Hz
C1y
C2y
C3y
C1y
C2y
C3y
0.75
0.94
0.31
1.50
0.17
0.35
⫺0.22
⫺1.03
0.08
⫺0.29
0.90
⫺0.47
0.89
1.12
0.37
1.25
0.08
0.23
⫺0.26
⫺0.86
0.04
⫺0.35
0.88
⫺0.39
Flux and ATP/ADP control coefficients were calculated for each
steady state in the network, as described in METHODS.
the network, as calculated on the basis of elasticities of
mitochondria and SR ATPase and the effective elasticity of AM ATPase for these conditions (Table 3), indicated that the activity of mitochondria should largely
determine the magnitude of ATP synthesis flux (J1),
the ATP hydrolysis flux (J2) at high contraction frequencies, and ATP/ADP (Table 6). Only the magnitude
of SR ATPase hydrolysis flux (J3) 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 Ca2⫹ 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 [Ca2⫹], 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. 4 by solid lines.
The open symbols in Fig. 4 correspond 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
[Ca2⫹] (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 be-
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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 threecomponent 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
METHODS.
C824
CONTROL ANALYSIS OF ATP FREE ENERGY METABOLISM IN MUSCLE
lation 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 [Ca2⫹] (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 (Tables 4–6) was the choice of the
contraction frequencies discriminating cases I-III. The
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tween AM and SR ATPase via cytosolic [Ca2⫹] (Table
6). The stippled area in Fig. 4 indicates 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 (J1) and SR ATPase flux (J3) 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 stimu-
CONTROL ANALYSIS OF ATP FREE ENERGY METABOLISM IN MUSCLE
frequency of 5 Hz. The absolute value of C1ATP/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
Fig. 4. Distributions of ATPase flux control and ATP/ADP control in the branched ATPase network as a function
of the contraction frequency of the muscle. A: distribution of kinetic control of the net synthase flux of mitochondria
(J1) at steady state over the 3 components in the branched ATPase network of Fig. 2 as a function of the stimulation
frequency of the muscle (log scale). The continuous stimulation frequency dependence of the control distribution
(solid and dashed lines) was calculated on the basis of a continuous array of (1/vstim, ATP/ADP) values inter- and
extrapolated from the measured relation (Fig. 3C). On the basis of this array, continuous elasticities toward
ATP/ADP were calculated using Eq. 2 and for mitochondria using Eq. 3, for SR ATPase using Eq. 5, and for AM
ATPase using Eqs. 14–16. With these values and a branch flux ratio (␣) of 2.3, Eq. A6 was used to calculate
fractional flux J1 control coefficients (CiJ1) for each stimulation frequency. The open and filled symbols correspond
to the flux J1 control coefficients for the 6 experimentally studied steady states in Tables 4 and 5 and the two
extrapolated steady states in Table 6, respectively. Stippled area, stimulation frequency range in which steady
states were not experimentally determined. The continuous stimulation frequency dependence of C3J1 is shown as
a dashed line to indicate that these values correspond only to that part of the control by SR ATPase that is exerted
through the ATP/ADP regulation in the muscle (cf. DISCUSSION). B: distribution of kinetic control of the ATPase flux
of AM ATPase (J2) at steady state over the 3 components in the branched ATPase network of Fig. 2 as a function
of the stimulation frequency of the muscle (log scale). The continuous stimulation frequency dependence (solid and
dashed lines) was calculated as described for A, except Eq. A10 was used to calculate the fractional flux J2 control
coefficients (CiJ2). The open and filled symbols correspond to the flux J2 control coefficients for the 6 experimentally
studied steady states in Tables 4 and 5 and the 2 extrapolated steady states in Table 6, respectively. Stippled area,
stimulation frequency range in which steady states were not experimentally determined. The continuous stimulation frequency dependence of C3J2 is shown as a dashed line to indicate that these values correspond only to that
part of the control by SR ATPase that is exerted through the ATP/ADP regulation in the muscle (cf. DISCUSSION).
C: distribution of kinetic control of the ATPase flux of the SR Ca2⫹-ATPase (J3) at steady state over the 3
components in the branched ATPase network of Fig. 2 as a function of the stimulation frequency of the muscle (log
scale). The continuous stimulation frequency dependence of the control distribution (solid and dashed lines) was
calculated as described in A, except Eq. A14 was used to calculate the fractional flux J3 control coefficients (CiJ3).
The open and filled symbols correspond to the flux J3 control coefficients for the 6 experimentally studied steady
states in Tables 4 and 5 and the 2 extrapolated steady states in Table 6, respectively. Stippled area, stimulation
frequency range in which steady states were not experimentally determined. The continuous stimulation frequency
dependence of C3J3 is shown as a dashed line to indicate that these values correspond only to that part of the control
by SR ATPase that is exerted through the ATP/ADP regulation in the muscle (cf. DISCUSSION). D: distribution of
kinetic control of cytosolic ATP/ADP at steady state over the 3 components in the branched ATPase network of Fig.
2 as a function of the stimulation frequency of the muscle (log scale). The continuous stimulation frequency
dependence of the control distribution (solid and dashed lines) was calculated as described in A, except Eq. A20 was
used to calculate the fractional ATP/ADP control coefficients (CiS1). The open and filled symbols correspond to the
ATP/ADP control coefficients for the 6 experimentally studied steady states in Tables 4 and 5 and the 2
extrapolated steady states in Table 6, respectively. Stippled area, stimulation frequency range in which steady
states were not experimentally determined. The continuous stimulation frequency dependence of C3S1 is shown as
a dashed line to indicate that these values correspond only to that part of the control by SR ATPase that is exerted
through the ATP/ADP regulation in the muscle (cf. DISCUSSION).
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particular choices that were made (Tables 1–3) 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 Ca2⫹-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. 4A; 5-fold range was calculated for the
increase of C1J1 from 0.1 to 0.9).
Finally, the result in Fig. 4D shows that the ATP/
ADP control coefficient was maximal at a stimulation
C825
C826
CONTROL ANALYSIS OF ATP FREE ENERGY METABOLISM IN MUSCLE
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 twocomponent 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 threecomponent network studied if the Ca2⫹ 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 Ca2⫹ 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 [Ca2⫹] 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 Ca2⫹ 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 Ca2⫹ 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 spec-
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additional interaction between AM and SR ATPase via
cytosolic [Ca2⫹] 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 (Tables 4–6, 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 (Tables 4–6, Fig. 4B). 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 [Ca2⫹]. 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.
C827
CONTROL ANALYSIS OF ATP FREE ENERGY METABOLISM IN MUSCLE
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 C1J1 and C2J1, respectively,
and 2) the slope (dC1J1/dfreq) of the frequency dependence of C1J1 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 Ca2⫹ clearance
and to the uncompetitive inhibition constant for ADP
of SR ATPase, Ki⬘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
Table 7. Dependence of the result of the control analysis on the precise value
of the kinetic constants of each module
Parameter Doubling
Parameter
Eq. No.
Frequency
shift, Hz
Change of
slope, %
⫹0.6
⫹3.3
⫺1.0
0
⫺40
⫹120
Parameter Halving
Change of
(C1S1)max, %
Frequency
shift, Hz
Change of
slope, %
Change of
(C1S1)max, %
⫺0.1
⫺1.6
⫹0.5
⫺13
⫹100
⫹50
⫺12
⫹76
⫺16
⫺2
0
⫹2
0.0
0.0
⫹0.1
0
0
⫺7
0
0
⫹4
Mitochondria
Vmax
ATP/ADP
K0.5
ADP
nH
3
3
3
0
⫺37
⫹56
AM ATPase
MgATP
m
MgADP
i
Ca2⫹
H
4
4
10
⫺0.1
0.0
0.0
0
0
0
MgATP
Km
5
5
5
0.0
0.0
⫹1.0
0
0
⫺13
0
0
⫹31
0.0
0.0
⫺0.8
0
0
⫹13
0
0
⫺24
14
⫺0.1
0
⫺4
⫹0.4
⫺29
⫹20
K
K
n
*
SR ATPase
KiMgADP
Ki⬘MgADP
␶1
For the 10 kinetic parameters listed, the influence of a 2-fold change in value (doubling and halving) was determined and quantified by 3
parameters: 1) frequency shift (in Hz) of the intersection of C1J1 and C2J1 (see also Fig. 5A), 2) change of slope (in %) of the C1J1 (freq) function
ATP/ADP
in the intersection, and 3) change of the maximal value of C1S1 (in %). Vmax, maximal velocity; K0.5
, ATP/ADP at half-maximal
ADP
MgATP
flux; nH
, apparent kinetic order of ADP sensitivity of mitochondria; Km
, Michaelis constant for MgATP; KiMgADP, inhibition constant
for MgADP; ␶i, time constant.
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trum 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 (Tables 1–3). 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
Km, Ki, 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 Table 7. 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
C828
CONTROL ANALYSIS OF ATP FREE ENERGY METABOLISM IN MUSCLE
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 Ki⬘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
Fig. 5. Simulation of the change in the distributions for net ATP
synthase flux (J1), AM ATPase flux (J2), and ATP/ADP control in the
network for the case of mitochondrial dysfunction. A: distribution of
J1 control in the network for the case of mitochondria with partial
complex I defect (CID; dashed lines), in comparison to control (solid
lines), as a function of the stimulation frequency of the muscle (log
scale). The elasticity of mitochondria toward ATP/ADP for the case of
CID was calculated using Eq. 3 and Jmin/Jmax ⫽ ⫺0.2 and K0.5 ⫽ 93.
ATP/ADP in unstimulated muscle in CID was 2-fold lower than in
controls (3, 33); this difference was extrapolated to stimulated conditions to obtain the (1/vstim, ATP/ADP) for CID and calculate the
continuum of the elasticities of mitochondria and AM and SR ATPase for this case. Control coefficients were calculated on the basis of
these elasticities as described for Fig. 4A. The relation for normal
mitochondria (solid lines) is the same as in Fig. 4A. B: distribution of
J2 control in the network for the case of mitochondria with partial
CID (dashed lines), in comparison to control (solid lines), as a
function of the stimulation frequency of the muscle (log scale).
Computation of the relation for the case of mitochondrial dysfunction
as described for A. The relation for normal mitochondria (solid lines)
is the same as in Fig. 4B. C: distribution of ATP/ADP control in the
network for the case of mitochondria with partial CID (dashed lines),
in comparison to control (solid lines), as a function of the stimulation
frequency of the muscle (log scale). Computation of the relation for
the case of mitochondrial dysfunction as described for A. The relation
for normal mitochondria (solid lines) is the same as in Fig. 4D.
Downloaded from http://ajpcell.physiology.org/ by 10.220.33.6 on June 15, 2017
which respiration is half-maximally stimulated (29)
ATP/ADP
(K0.5
) and to the precise apparent kinetic order of
the ADP sensitivity of mitochondria (nADP
) (29). This
H
result is important and relevant, because both of these
kinetic “constants” are not necessarily fixed numbers.
ATP/ADP
On the contrary, K0.5
is a variable depending on
Vmax and, as such, is subject to conditions affecting
protonmotive force generation (47), such as oxidative
substrate selection. As for nADP
, a first-order reaction
H
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. 4B). 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
ADP
mitochondrial proton pump affecting Vmax and K0.5
of
the mitochondria. For the simulation, we used results
CONTROL ANALYSIS OF ATP FREE ENERGY METABOLISM IN MUSCLE
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, Pi,
and pH. The insufficient accuracy of knowledge of Ki⬘
for SR ATPase has been discussed. Less is known
about the effects of Pi 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 ⌬GATP.
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
⌬GATP in present MCA theory has been clarified only
for the case in which Pi 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 Pi as well as Ca2⫹ (37).
Finally, this control analysis also provides a basis for
further exploration of Ca2⫹ regulation of muscle function. Another MCA concept, the response coefficient
(Ryx), which is defined as Ryx ⫽ ⌺Cyi ⑀ix, 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
y
system steady state, the response coefficient RSR
ATPase
must be used. For example, to assess the net effect of a
change in SR ATPase activity on AM ATPase flux (J2),
one would obtain [assuming Ca2⫹ stimulation of mitochondria in fast-twitch muscle is negligible (4)]
J2
J2
1
J2
2
R SR
ATPase ⫽ C 1 䡠 ⑀ SR ATPase ⫹ C 2 䡠 ⑀ SR ATPase
3
J2
J2
2
⫹ C 3J 2 䡠 ⑀ SR
ATPase ⫽ C 3 ⫹ C 2 䡠 ⑀ SR ATPase
(17)
where C3J2 corresponds to the (negative) control over
this flux exerted by SR ATPase via its effect on the cell
2
energetic state (Fig. 4B) and C2J2⑀SR
ATPase 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 [Ca2⫹] attained after stimulation. The second term
is composed of two positive values. C2J2 is positive and
2
generally ⬎0.4 (Fig. 4B). In our case I, ⑀SR
ATPase will be
zero. When the frequency of stimulation increases so as
to enter case II, this apparent elasticity will increase to
ⱕ3 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 Ca2⫹ 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 essen-
Downloaded from http://ajpcell.physiology.org/ by 10.220.33.6 on June 15, 2017
from 31P-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 (Vmax) 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).
C829
C830
CONTROL ANALYSIS OF ATP FREE ENERGY METABOLISM IN MUSCLE
tials 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.
C 1J 1 ⫽
C 3J 1 ⫽
(A1)
This 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 CiJ 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)
C 䡠⑀ ⫹C 䡠⑀ ⫹C 䡠⑀ ⫽0
J1
1
1
S1
J1
2
2
S1
J1
3
3
S1
(A2)
Equation A2 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). ⑀S1 1 is the elasticity coefficient (also
termed sensitivity) of module 1 toward S1, loosely defined
as the percent increase of the rate v of module 1 resulting
from a 1% increase in S1 under non-steady-state conditions;
after the system returns to the steady state at which it was
before perturbation, the change in S1 will be nullified (21, 26,
30, 47)
C /C ⫽ J 2 /J 3 ⫽ ␣
J1
2
J1
3
[ ][
C
C
C
J1
1
J1
2
J1
3
⫽
⫺(1 ⫹ ␣)/␣
1
1/␣
] []
䡠 C 2J 1 ⫹
1
0
0
C 2J 1 ⫽
⫺⑀ S1 1
⫺(1 ⫹ ␣)/␣ 䡠 ⑀ S1 1 ⫹ ⑀ S2 1 ⫹ 1/␣ 䡠 ⑀ S3 1
(A5)
The flux J1 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. A5 to contain only ratios of the elasticities of the modules
toward S1 and the fluxes J2 and J3 and substituting this
expression for C2J1 into Eq. A4, we obtain the following expression for flux J1 control in the system at steady state
1 1
䡠
␣ D
(A6c)
(
⑀ S2 1
1⫹␣
1 ⑀ S3 1
⫹
⫹
䡠
␣
⫺⑀ S1 1 ␣ ⫺⑀ S1 1
)
C 1J 2 ⫹ C 2J 2 ⫹ C 3J 2 ⫽ 1
(A7)
C 䡠⑀ ⫹C 䡠⑀ ⫹C 䡠⑀ ⫽0
(A8)
C 1J 2/C 3J 2 ⫽ ⫺J 1 /J 3
(A9)
J2
1
1
S1
J2
2
2
S1
J2
3
3
S1
Substituting into Eq. A7 that J2/J3 ⫽ ␣ and, at steady state,
J1 ⫽ J2 ⫹ J3, the flux J2 control distribution is obtained in
terms of the flux J2 control strength of module 1, C1J2 (not
shown). Combining this relation with the connectivity theorem for flux J2 (Eq. A8), one obtains the following flux J2
control distribution in the branched pathway of Fig. 2
C 1J 2 ⫽
1
␣/(1 ⫹ ␣) ⫹ ⫺⑀ S1 1/⑀ S2 1 ⫹ 1/(1 ⫹ ␣) 䡠 ⑀ S3 1/⑀ S2 1
(A10a)
C 2J 2 ⫽ 1 ⫺
␣/(1 ⫹ ␣)
␣/(1 ⫹ ␣) ⫹ ⫺⑀ S1 1/⑀ S2 1 ⫹ 1/(1 ⫹ ␣) 䡠 ⑀ S3 1/⑀ S2 1
(A10b)
C 3J 2 ⫽
⫺1/(1 ⫹ ␣)
␣/(1 ⫹ ␣) ⫹ ⫺⑀ S1 1/⑀ S2 1 ⫹ 1/(1 ⫹ ␣) 䡠 ⑀ S3 1/⑀ S2 1
(A10c)
Flux J3 control. Equations A11–A13 apply to flux J3 control in the pathway of our model at steady state
C 1J 3 ⫹ C 2J 3 ⫹ C 3J 3 ⫽ 1
(A11)
C 䡠⑀ ⫹C 䡠⑀ ⫹C 䡠⑀ ⫽0
(A12)
C /C ⫽ ⫺J 1 /J 2
(A13)
J3
1
(A4)
By combining Eqs. A1 and A4 and substituting into the
connectivity relation for flux J1 (Eq. A2), the flux J1 control
strength of module 2 can be expressed in terms of elasticities
of the modules toward S1 and the ratio of fluxes J2 and J3
(A6b)
For the particular metabolic pathway under consideration
1
2
3
(Tables 1–3), ⑀ATP/ADP
is ⬍0, whereas ⑀ATP/ADP
and ⑀ATP/ADP
are ⬎0. With ␣ ⬎ 0, the denominator D in Eq. A6, a–c, is a
positive number, as are all nominator terms. Consequently,
all flux J1 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 J1 in the network.
Flux J2 control. Equations A7–A9 apply to flux J2 control
in the pathway of our model at steady state
(A3)
Equation A3 is the branch theorem for control of flux J1 in
the system (39, 47).
By substitution of Eq. A3 into the summation relation (Eq.
A1), the flux J1 control distribution is obtained in terms of the
flux J1 control strength of module 2, C2J1
1
D
1
S1
J3
2
J3
1
2
S1
J3
3
3
S1
J3
2
Substituting into Eq. A11 that J2/J3 ⫽ ␣ and at steady state
J1 ⫽ J2 ⫹ J3, the flux J3 control distribution is obtained in
terms of the flux J3 control strength of module 1, C1J3 (not
shown). Combining this relation with the connectivity theorem for flux J3 (Eq. A12), one obtains the following flux J3
control distribution in the branched pathway of Fig. 2
C 1J 3 ⫽
1
1/(1 ⫹ ␣) ⫹ ⫺⑀ S1 1/⑀ S3 1 ⫹ ␣/(1 ⫹ ␣) 䡠 ⑀ S2 1/⑀ S3 1
(A14a)
C 2J 3 ⫽
⫺␣/(1 ⫹ ␣)
1/(1 ⫹ ␣) ⫹ ⫺⑀ S1 1/⑀ S3 1 ⫹ ␣/(1 ⫹ ␣) 䡠 ⑀ S2 1/⑀ S3 1
(A14b)
C 3J 3 ⫽ 1 ⫺
1/(1 ⫹ ␣)
1/(1 ⫹ ␣) ⫹ ⫺⑀ S1 1/⑀ S3 1 ⫹ ␣/(1 ⫹ ␣) 䡠 ⑀ S2 1/⑀ S3 1
(A14c)
Downloaded from http://ajpcell.physiology.org/ by 10.220.33.6 on June 15, 2017
C 1J 1 ⫹ C 2J 1 ⫹ C 3J 1 ⫽ 1
(A6a)
where
APPENDIX
Flux J1. Equations A1–A3 apply to the control of flux J1 in
the ATPase network at steady state
D
C 2J 1 ⫽
D⫽
Flux Control in the System
⑀ S2 1/⫺⑀ S1 1 ⫹ 1/␣ 䡠 ⑀ S3 1/⫺⑀ S1 1
CONTROL ANALYSIS OF ATP FREE ENERGY METABOLISM IN MUSCLE
C831
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Equations A15–A17 apply to concentration S1 control in
our model at steady state
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C 1S1 ⫹ C 2S1 ⫹ C 3S1 ⫽ 0
(A15)
This is the principle that the concentration control coefficients in a network like ours sum to 0 (21, 47)
C 1S1 䡠 ⑀ S1 1 ⫹ C 2S1 䡠 ⑀ S2 1 ⫹ C 3S1 䡠 ⑀ S3 1 ⫽ ⫺1
(A16)
Equation A16 is, like Eq. A2, 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 S1, restoring the initial steady state
C 2S1/C 3S1⫽J 2/J 3⫽␣
(A17)
Equation A17 is the branch theorem for control of concentration S1 in the system (47).
One can now solve the S1 control distribution in the system
at steady state in terms of the control coefficient for the
control of S1 by module 2, C2S1. To this aim, one develops the
summation relation for S1 control (Eq. A16) in analogy to the
analysis for J1 to obtain
[ ][
C 1S1
C 2S1
C 3S1
⫽
⫺(1 ⫹ ␣)/␣
1
1/␣
]
䡠 C 2S1
(A18)
Developing the connectivity relation for S1 control (Eq. A17)
using Eq. A18, we obtain the following expression for C2S1
C 2S1 ⫽
⫺1/⑀ S1 1
⫺(1 ⫹ ␣)/␣ ⫹ ⑀ S2 1/⑀ S1 1 ⫹ 1/␣ 䡠 ⑀ S3 1/⑀ S1 1
(A19)
Substituting this expression for C2S1 into Eq. A19, we obtain
the following expression for the concentration S1 control
distribution in our model at steady state i
C 1S1 ⫽
(1 ⫹ ␣)/␣ 䡠 ⫺1/⫺⑀ S1 1
(A20a)
D
C 2S1 ⫽
C 3S1 ⫽
1/⫺⑀ S1 1
(A20b)
D
1/␣ 䡠 1/⫺⑀ S1 1
(A20c)
D
where
(
D⫽ ⫺
⑀ S2 1
1⫹␣
1 ⑀ S3 1
⫺
⫺
䡠
␣
⫺ ⑀ S1 1 ␣ ⫺ ⑀ S1 1
)
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. A20a but positive in Eq. A20, b and c. It
follows that concentration S1 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.
The authors are grateful to Bryant Chase, Robert Wiseman, Ron
Meyer, and Rafael Moreno-Sanchez for valuable discussions.
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).
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Concentration S1 Control in the System
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