A metabolic control analysis of kinetic controls in ATP free
energy metabolism in contracting skeletal muscle
J. A. L.
Jeneson1,
H. V.
Westerhoff2,3, and
M.
J.
Kushmerick1,4,5
Departments of 1 Bioengineering, 4 Physiology and
Biophysics, and 5 Radiology, University of Washington School of
Medicine, Seattle, Washington 98195; 2 Department of Molecular
Cell Physiology, Faculty of Biology, Free University, Amsterdam; and
3 Department of Mathematical Biochemistry, Biocenter, University
of Amsterdam, Amsterdam, The Netherlands
 |
ABSTRACT |
A system analysis of
ATP free energy metabolism in skeletal muscle was made using the
principles of metabolic control theory. We developed a network model of
ATP free energy metabolism in muscle consisting of actomyosin ATPase,
sarcoplasmic reticulum (SR) 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.
cellular energetics; skeletal muscle; metabolic control analysis
 |
INTRODUCTION |
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 mechanisms. Chance (15, 16) worked out control of mitochondrial
oxidative ADP phosphorylation by the cytosolic ADP concentration
([ADP]) with a transfer function connecting muscle work output to
phosphocreatine (PCr) content ([PCr]). Meyer (36)
analyzed chemical changes in contraction-recovery cycles by analogy
with simple electrical circuits. This model connected cytosolic ATPase
activity during contraction with [ADP]-controlled net mitochondrial
ATP synthase activity by feedback control. In this energy balance
system, the creatine kinase reaction functioned as a capacitance.
The creatine kinase reaction was constrained to maintain local
equilibrium with cellular content of ATP, ADP, PCr, and creatine (Cr)
in the models of Chance and Meyer. Later work showed that relaxation of
this constraint did not affect the analysis (34, 37).
Meyer's model correctly matches experimental observations in rat and
feline (36, 37) and human (3, 9, 28, 33, 45)
muscle over typical physiological ranges. It also gives a conceptual
approach to understand muscle energetics as an interdependent network
with feedback for achieving energy balance. Kushmerick
(34) recently published a set of equations similar in
concept to Meyer's model, which simulated information from human
forearm contraction. The features of this model were the inclusion of
complete terms for the creatine kinase enzyme kinetics and new
information of second-order ADP dependence of mitochondrial ATP
synthesis (29). By including specific functions derived
from mechanistic studies of components, this model provides a generic
way to add additional mechanisms as their properties are defined. Thus
the study of muscle energetics and cellular respiration as primary
determinants of ATP levels has a rich experimental and analytic history.
It might be concluded from this discussion that we understand muscle
energetics at a satisfactory conceptual and mechanistic level, despite
continued debate on the mechanisms for controlling cellular respiration
(4, 12, 29, 34, 37). However, there remain other, and we
believe equally fundamental, aspects of the physiology of muscle cell
energetics that are not explained by these models and concepts. These
have been ignored so far. One such aspect is the conservation of
cytosolic ATP free energy (2)
{
GATP =
GATPo' + RTln([ADP][Pi]/[ATP]), where
GATPo' =
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 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.

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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 steady-state 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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|
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
|
(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 µM for SR ATPase and
~0.8 µM for AM ATPase (44). In the unstimulated muscle
cell, cytosolic [Ca2+] is well below 0.2 µM
(14), and thus AM and SR ATPase fluxes are minimal.
When a skeletal muscle cell is activated by an action potential, the SR
releases a 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 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).

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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+.
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|
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.
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
elasticity coefficients (
ATP/ADPi) 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 ATP/ADP (
ATP/ADPi) 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
31P-NMR spectroscopy (see Experimental Methods).
2) The values of the ATP/ADP elasticities of each of the
ATPase modules in the network at each experimentally determined steady
state were calculated using the measured ATP/ADP, as described below.
We extrapolated our analysis beyond the range of measured stimulation
frequencies that gave steady states. 3) With this set of
elasticity values and
= 2.3, the corresponding set of control
coefficients was calculated using Eqs. A6, A10, A14, and
A20.
Calculation of ATPase elasticities toward ATP/ADP.
The elasticity of each module i in the network toward the
cytosolic ATP/ADP at a particular steady state (k) of ATP
free energy metabolism in muscle
[(
ATP/ADPi)k]
was calculated on the basis of the steady-state kinetics of the enzyme
module according to (21, 39, 47)
|
(2)
|
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). Therefore, this method to
calculate
(
ATP/ADPi)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/ADP at steady-state
k, (
ATP/ADP1)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)
|
(3)
|
where K0.5ATP/ADP is ATP/ADP at
half-maximal ATP synthesis flux (~186 for human forearm flexor muscle
in vivo). This relation was derived from the kinetic function that
described the covariation (ADP, 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 at steady state
k, (
ATP/ADP2)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)
|
(4)
|
where KmMgATP is the Michaelis
constant for the substrate (10-20 µM for ATPase activity) and
Ki is the inhibition constant for MgADP, which
is on the order of 200-300 µM (20). We used 15 and
250 µM for KmMgATP and
Ki, respectively, in the calculations.
SR ATPase.
The elasticity toward ATP/ADP of SR ATPase at steady state
k, (
ATP/ADP3)k,
was calculated using Eq. 2 on the basis of the kinetic function describing the SR ATPase forward rate dependence on ATP and
ADP
|
(5)
|
where KmMgATP 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
conditions (42). KmMgATP of SR
ATPase is at least two orders of magnitude lower than [ATP] in human
skeletal muscle [10 µM (42) vs. 8 mM
(25)], and so the term
KmMgATP/[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 elasticity of SR ATPase toward ATP/ADP
(
ATP/ADP3'), is zero. The elasticities toward
ATP/ADP of AM ATPase and mitochondria under these conditions were
determined as defined above.
CASE II: INTERMEDIATE CONTRACTION FREQUENCIES.
Over this range of twitch frequencies, we consider the possibility that
the time interval between subsequent stimulations is no longer
sufficient for SR ATPase to recover all 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
200-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
89% 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/ADP (
ATP/ADP2') 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
|
(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+]cyto|t0 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
|
(7)
|
where nPCa is the
Ca2+-per-ATP pumping stoichiometry of the SR ATPase,
v3 is the SR ATPase rate, and
[Ca2+]SR|t1 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
|
(8)
|
where
[Ca2+]cyto|t2 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
|
(9)
|
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+])
|
(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
v3dt
v3
t
(
t is the time interval between muscle stimulations) and
defined a constant K'50 = (K50/
nPCa). 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
|
(11)
|
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
|
(12)
|
Under the limit condition
t
0, i.e., at high
contraction frequencies, the term
[K'50/(v3
t)]3 > 1. Also, K50,
, and
nPCa 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
|
(13)
|
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 the limit condition of high contraction frequencies,
ATP/ADP2', 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
|
(14)
|
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 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
85% of the maximum (data not
shown). For contraction frequencies >5 Hz, recovery dropped sharply,
to as low as 60% at 10 Hz.
In the next step, the computed
[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
|
(15)
|
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, K50 = 0.5, and
nPCav3
t = [Ca2+]SR. The resulting covariation of
ATP/ADP and v2 was biexponential and was
determined by nonlinear curve fitting to give
|
(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.
The analytic solution (Eq. 13) applies only to the limit
condition of high contraction frequencies of case III. The
composite analytic-numerical solution to calculate
ATP/ADP2' 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 computation of the continuum of
ATP/ADP2' 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
31P-NMR spectroscopy.
Human forearm flexor muscle (5 men and 1 woman, age 28-55 yr) was
studied at rest and during twitch contractions evoked by external
electrical stimulation of the ulnar and medial nerves at frequencies
between 0.3 and 2.0 Hz. These frequencies were high enough to allow us
to find the stimulation rate above which non-steady-state acidification
occurred. 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. 31P-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 (
GATP) was calculated as
GATPo' + RTln([ADP][Pi]/[ATP]), where
32.8 kJ/mol
was used for
GATPo' (41).
Curve fitting and statistical analyses.
Correlations of variables were analyzed by nonlinear curve fitting with
defined functions in Fig.P software (version 6.0, Elsevier Biosoft,
Cambridge, UK).
 |
RESULTS |
General Solution of Control in the Network
The general solution for kinetic control of the particular value
of the four system variables 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) µmol
ATP · s
1 · g muscle
1
[n = 6 muscles, with assumption of 0.67 liter cell
water/kg muscle (25)]. Mitochondrial ATP synthesis flux
accounted for 90 ± 2% (mean ± SE, n = 6 muscles) of the total matching cellular ATP synthesis flux at this
ATPase rate and approached 84 ± 4% (n = 5)
of maximal synthesis flux estimated for each individual muscle, as
described elsewhere (28). The rate of 0.15 µmol
ATP · s
1 · g muscle
1
constituted the apparent maximal ATPase flux that could be sustained in
this muscle and, therefore, represents the maximal flux in the
three-component ATPase network model of ATP free energy metabolism of
Fig. 2 in this muscle. At higher contraction frequencies (and associated ATP hydrolysis rates), nonoxidative ATP synthesis increased as estimated from concomitant proton production in four of six subjects
studied. These conditions resulted in a decline of 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.
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
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.

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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.
|
|
Dynamic Range of Elasticities Toward ATP/ADP in the Network
Case I: low-frequency contractions.
The elasticity toward ATP/ADP of mitochondria
(
ATP/ADP1) and AM ATPase
(
ATP/ADP2) 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 contraction frequencies (
ATP/ADP3') 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 frequencies,
ATP/ADP1 was at least four orders of
magnitude greater than
ATP/ADP2, 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/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 absolute value of
ATP/ADP1 decreased a further 2.5-fold over this
range of contraction frequencies (Table 2). The value of
ATP/ADP3 approached the same order of magnitude
as
ATP/ADP1 over this range of contraction
frequencies and increased 1.5-fold between 1.0 and 1.8 Hz (Table 2).
The value of
ATP/ADP2 remained at least two
orders of magnitude smaller than
ATP/ADP1 and
ATP/ADP3, despite a 10-fold increase in its value
over this frequency range (Table 2). These results mean that function
of the AM module is still immune from changes in ATP/ADP also over this
contraction frequency range.
Case III: high-frequency contractions.
Here the analysis is extrapolated into regions for which there are no
experimental data and where it may be difficult, if not physiologically
impossible, to explore human forearm muscle in situ. The elasticities
toward ATP/ADP of mitochondria and AM and SR ATPase were calculated as
described in METHODS by extrapolation of steady-state
ATP/ADP to contraction frequencies of 5 and 10 Hz (Table 3). These
extrapolated ATP/ADP values represent the conditions that should be
attained in the muscle had anaerobic glyco(geno)lysis remained
insignificant. The effective elasticity of AM ATPase toward ATP/ADP
under these conditions,
ATP/ADP2', 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). The absolute value of
ATP/ADP1 decreased
a further twofold over this range of contraction frequencies to a value
30-fold lower than at low contraction frequencies (Tables 1 and 3). The
absolute value of
ATP/ADP3 (and therefore of
ATP/ADP2') over this range of contraction
frequencies now exceeded that of
ATP/ADP1 by as much
as threefold (Table 3). In this case, the effective elasticity of AM
ATPase,
ATP/ADP2', was the highest in the network
(Table 3). Only at these high stimulation rates does the function of
the AM module therefore become significantly influenced by ATP/ADP.
Kinetic Control of ATP Metabolic Flux and ATP/ADP in Contracting
Forearm Flexor Muscle
Here the distributions of kinetic control of ATP free
energy-consuming and energy-producing fluxes and cytosolic ATP/ADP in the network model of ATP free energy metabolism in contracting skeletal
muscle are calculated as a function of stimulation frequency. We used
Eqs. A6, A10, A14, and A20, the elasticities
toward ATP/ADP (Tables 1-3), and
= 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).
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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
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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).
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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
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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 three-component network and ATP/ADP control would develop with an
increasing duty cycle of ATPase activity.
The most significant finding was the inversion of flux control from an
ATP demand control hierarchy to an ATP supply control hierarchy. This
began at contraction frequencies >1 Hz (Table 5), progressed as the
contraction frequency increased past 2 Hz, and was near complete at 10 Hz (Table 6). Specifically, the distribution of ATPase flux and ATP/ADP control in the network, as
calculated on the basis of elasticities of mitochondria and SR ATPase
and the effective elasticity of AM ATPase for these conditions (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.
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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
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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 between 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.

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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).
|
|
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 stimulation frequency increases at >1 Hz) results
from the particular ATP/ADP sensitivities of the ATPase modules and
their integration in a network. In the frequency range of 1-2 Hz
(case II), all elasticities of the modules were calculated
in a straightforward manner, with use of only the MCA definition
(Eq. 2) and the steady-state kinetics of each ATPase
(Eqs. 3-5), as described in METHODS; no
"effective" elasticity was introduced for any module over this
particular frequency range in which the flux control hierarchy
inversion begins. The particular assumptions and simplifications made
in the derivation of the additional connectivity between the activities of SR and AM ATPase via cytosolic [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 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 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 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.
Requirements for Homeostatic Regulation of ATP Free Energy
Initially, we considered kinetic control of ATPase fluxes and ATP
free energy in a linear model in a network composed of only AM ATPase
and mitochondria (27). However, this network was not homeostatic. That analysis showed that the AM ATPase maintained dominant control of its ATPase flux, even in the range of very high
stimulation frequencies. In contrast to the properties of the
three-component system, the two-component model has a consequence that
the AM ATPase could outstrip mitochondrial supply capacity. Thus
the salient limitation of the two-component system was that it lacks
intrinsic homeostasis. The same lack of homeostasis could be obtained
with the three-component network studied if the 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
spectrum was the 1,000-fold higher ADP sensitivity of mitochondria
under unstimulated conditions, which progressively decreased as much as
30-fold as the stimulation frequency approached 10 Hz (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 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.
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Table 7.
Dependence of the result of the control analysis on the precise
value of the kinetic constants of each module
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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.
|
|
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 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 which respiration is
half-maximally stimulated (29)
(K0.5ATP/ADP) and to the precise apparent
kinetic order of the ADP sensitivity of mitochondria
(nHADP) (29). This result is
important and relevant, because both of these kinetic "constants"
are not necessarily fixed numbers. On the contrary,
K0.5ATP/ADP 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 nHADP,
a first-order reaction had been generally assumed (15)
until we recently showed that it is at least second order
(29). Therefore, precise determination of these values for
specific conditions and for skeletal muscle phenotypes will be
necessary for proper understanding of the physiology.
The results of the sensitivity analysis may also be read as a guide to
what type of mutations in proteins should be expected to affect
contractile and ATP free energy homeostatic function of skeletal
muscle. Together with Fig. 4, they also define in which frequency
domain these effects should be tested experimentally. For example,
effects on contractile function of mutations in SR ATPase affecting
uncompetitive binding of ADP should be tested in the high stimulation
frequency domain, where SR ATPase is predicted to have substantial
control of force production associated with contractile AM ATPase flux
(Fig. 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 mitochondrial
proton pump affecting Vmax and
K0.5ADP of the mitochondria. For the
simulation, we used results 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).
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
(Rxy), which is
defined as Rxy =
Ciy
xi,
where y is a system variable and x is a system
parameter or external effector (47), may then be
implemented. Such an extension of the analysis will address one
particular and unique aspect of the SR ATPase module in muscle that has
only partly been addressed. SR ATPase has the dual role of a modulator
of the energetic state and a modulator of the externally controlled
signal that controls AM ATPase. The ATPase aspect and its consequence
for SR ATPase control were quantified in the present analysis (Fig. 4).
However, to predict the overall effect of a change in activity of SR
ATPase on the system steady state, the response coefficient
RSR ATPasey 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)]
|
|
|
(17)
|
where C3J2
corresponds to the (negative) control over this flux exerted by SR
ATPase via its effect on the cell energetic state (Fig. 4B)
and
C2J2
SR ATPase2
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 generally >0.4 (Fig. 4B). In our
case I,
SR ATPase2 will be zero. When
the frequency of stimulation increases so as to enter case
II, this apparent elasticity will increase to
3 in the extreme
case (see Eq. 1). The summed effect of an increase in SR
ATPase activity on AM ATPase flux will thus be positive. This one
example illustrates that the complete 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 essentials makes a significant
advance in understanding the system. It will perhaps be most important
as a next step to design experiments that will test the predictions of
the control analysis in the high-frequency domain. Slow-twitch skeletal
muscle, such as cat soleus, in which anaerobic ATP free energy
synthetic flux under those conditions will probably remain low
(37), appears a suitable experimental preparation of
skeletal muscle in which these tests could be successfully conducted.
 |
APPENDIX |
Flux Control in the System
Flux J1.
Equations A1-A3 apply to the control of flux
J1 in the ATPase network at steady state
|
(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)
|
(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).
S11 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)
|
(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
|
(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
|
(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
|
(A6a)
|
|
(A6b)
|
|
(A6c)
|
where
For the particular metabolic pathway under consideration (Tables
1-3),
ATP/ADP1 is <0, whereas
ATP/ADP2 and
ATP/ADP3 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
|
(A7)
|
|
(A8)
|
|
(A9)
|
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
|
(A10a)
|
|
(A10b)
|
|
(A10c)
|
Flux J3 control.
Equations A11-A13 apply to flux
J3 control in the pathway of our model at steady
state
|
(A11)
|
|
(A12)
|
|
(A13)
|
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
|
(A14a)
|
|
(A14b)
|
|
(A14c)
|
Concentration S1 Control in the System
Equations A15-A17 apply to concentration
S1 control in our model at steady state
|
(A15)
|
This is the principle that the concentration control
coefficients in a network like ours sum to 0 (21, 47)
|
(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
|
(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
|
(A18)
|
Developing the connectivity relation for S1 control
(Eq. A17) using Eq. A18, we obtain the
following expression for C2S1
|
(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
|
(A20a)
|
|
(A20b)
|
|
(A20c)
|
where
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.
 |
ACKNOWLEDGEMENTS |
The authors are grateful to Bryant Chase, Robert Wiseman, Ron
Meyer, and Rafael Moreno-Sanchez for valuable discussions.
 |
FOOTNOTES |
This work was supported in part by National Institute of Arthritis and
Musculoskeletal and Skin Diseases Grant AR-36281 (to M. J. Kushmerick) and the Netherlands Organization for Scientific Research
(H. V. Westerhoff).
Address for reprint requests and other correspondence: J. A. L. Jeneson, NMR Research Laboratory, Dept. of Radiology, Box 357115, University of Washington Medical Center, Seattle, WA 98195 (E-mail: utrecht{at}u.washington.edu).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
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in accordance with 18 U.S.C. §1734 solely to indicate this fact.
Received 7 July 1999; accepted in final form 30 March 2000.
 |
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