(Received for publication, October 20, 1994; and in revised form, March 3, 1995)
From the
The hypothesis tested was whether creatine kinase (CK)
equilibrates with its substrates and products in the cytosol as if in
solution. We used the creatine analogs cyclocreatine (cCr) or
-guanidopropionate (
GPA) to test if mass action ratios
(
) for CK in muscle could be predicted from combined equilibrium
constants (K
) measured in solutions mimicking
the intracellular environment. Mice were fed cCr or
GPA and their
muscles assayed for substrates and products of the CK reaction by
P NMR spectroscopy and high performance liquid
chromatography. After three weeks of feeding,
was
indistinguishable from K
in cCr-treated muscles
demonstrating both PCr/Cr and phospho-analog/analog must have
equilibrated with a constant and uniform cellular ATP/ADP ratio. In
GPA-treated muscles,
was smaller than K
due to a
higher content of muscle
GPA. Feeding
GPA for
9-12 weeks resulted in a closer agreement between K
and
, suggesting ATP/ADP ratios are not
uniform within the muscle perhaps due to transient metabolic stress in
some cells. From this analysis it follows that calculation of free ADP
from the CK equilibrium for a heterogeneous population of cells with
respect to total Cr and ATP content is correct only if chemical
potentials of these cells are uniform.
One of the key tenets of bioenergetics is the near-equilibrium
property of creatine kinase (CK) ()coupled with cytosolic
substrates which are freely mixing and available to the
enzyme(1, 2, 3, 4) . Three criteria
have been used to demonstrate that a near-equilibrium condition exists
within the cytoplasm: (a) kinetic limitations are considered
minimal because concentrations of substrates are in the approximate
range of their K
values(5, 6) ; (b) CK activity is far
in excess of maximal ATPase activity within the cell(1) ; (c)
P NMR spin transfer methods have shown that
forward and reverse fluxes for CK are equal (7) and are far
greater than net ATPase activity(8) . Thermodynamic control of
PCr/Cr by ATP/ADP through CK equilibration implies that ATP/ADP is the
same in all loci within the cell(1, 9, 10) .
Alternatively, the cytoplasmic fraction of ATP and ADP must be so large
that other ``compartments'' (which may or may not be at the
same chemical potential) do not influence physicochemical properties of
bulk cytoplasm. For a simple system which is not diffusion limited, it
has been shown that this near-equilibrium formulation can be used for
calculation of ADP
and ATP chemical potential
(
G
)(11, 12, 13, 14) .
For complex populations of cells, comprising tissues or organs, a
crucial assumption for proper calculation of ADP
and
G
(which are derived values from measured
parameters (PCr, Cr, ATP, P
, and pH) on a population of
cells) is that not only is ATP/ADP within each cell uniform,
but it is uniform between cells as well.
By a variety of criteria, heterogeneity of cell types in individual muscles have been documented(15, 16, 17, 18, 19) , and chemical contents of these cell types have shown that their metabolite contents differ. Furthermore it has been shown that these differences are reflected in chemical potentials of predominantly fast versus slow muscles (-68 to -61 kJ/mol ATP)(20) . In addition to heterogeneity of metabolite distribution, there is also considerable information on the existence, properties, and intracellular distribution of CK isoforms (3, 21, 22) . Although the major isoform is MM-CK (which is thought to be in solution in the cytoplasm as well as bound to the M-line of myofibrils), there is also a substantial amount of ``mitochondrial'' isoforms existing in dynamic multimers in the intermembrane space of mitochondria(3) . Based on the existence of multiple isoforms of CK taken together with heterogeneity of metabolite contents(23, 24) , it has been argued that the simple concept of CK equilibration, with its substrates and products mixing in the bulk cytoplasm as if in solution, is not completely valid.
In view of the large number of physiological and
biochemical studies that have assumed a simple behavior of the cytosol
(with respect to CK function), we investigated this issue by
capitalizing upon the competitive inhibitor nature of two creatine
analogs (-guanidopropionic acid (
GPA) and cyclocreatine
(cCr)) to establish a criterion for testing CK function in
vivo. Application of creatine analogs to study cellular functions
of CK was pioneered by Fitch et al.(25, 26, 27) and
Walker(28, 29, 30, 31) . Analogs of
creatine deplete total Cr over a period of several weeks when fed to
rodents and chickens. Both phosphorylated and free forms of analog
accumulate in brain, heart, and skeletal
muscle(27, 28, 29, 32, 33, 34, 35, 36, 37) .
As competitive inhibitors, these compounds can be exploited to answer
questions in cellular biology concerning organization of enzymatic
activity in cells, because these analogs are also utilized by cells as
substrates for cellular energy
metabolism(34, 37, 38, 39, 40, 41) .
So it is possible to test whether competitive inhibitors and endogenous
substrates obey simple rules of enzyme kinetics and equilibration.
Feeding synthetic analogs of creatine (which as competitive
inhibitors are not expected to disturb equilibration) partially
displaces PCr and Cr from the cell. By choosing duration of feeding
such that there were approximately equal concentrations of total
creatine and total analog with minimal adaptive changes, ()we optimized the ability to measure mass action ratios for
CK and to test whether ratios in tissue equaled those predicted from
combined equilibrium constants measured in solutions mimicking the
intracellular ionic environment. There are a number of reasons why this
equality may not hold true: (a) analogs and their
phosphorylated forms may not mix and equilibrate with endogenous
substrates and products of CK (as would occur in a solution); (b) mixing occurs, but metabolites may be separated in
compartments with significantly different ATP/ADP or PCr/Cr ratios; or (c) ATP/ADP ratios between individual muscle cells may not be
uniform. Our results show that in steady state and during long-term
exposure to these analogs none of these possibilities is true, except
for an instructive case of feeding
GPA for short (3 week)
duration, where heterogeneity of ATP/ADP ratios must be considered.
In our experiments, combined reactions are
measured in solution at the temperature of interest with a similar set
of measurements then performed on muscles of animals treated with
analogs. If the reaction achieved equilibration with its substrates in
muscle as in solution, the value for K in
solution and the mass action ratio in muscle should agree within
experimental error. The following equations specify the approach taken.
consider the reaction for the analog (X) which is phosphorylated at equilibrium (PX),
At pH 7.0, equilibrium expression for and are written as follows.
With adenylates at equilibrium with both analog and creatine
species, then we can define a K as
follows.
In muscle, when the mixture of substrates and a competitive
inhibitor comes to equilibrium with the same adenylate pool, the
observed mass action ratio () for the combined reaction will equal K
. Unlike solution experiments in which there
is no net ATP utilization (ATPase flux) and ATP/ADP does not change,
equilibration may not be exact. In muscle, even at rest, there is a
small net ATPase activity. In this case, equilibration will only occur
if forward and reverse fluxes of CK are far in excess of steady state
fluxes of ATP through ATP synthesis and ATPase reactions.
Muscles were
superfused with phosphate-free saline equilibrated with 95%
O, 5% CO
and contained (mM) 116 NaCl,
4.6 KCl, 26.2 MOPS (titrated to pH 7.4 with NaOH), 2.5
CaCl
, 1.2 MgSO
, and gentamycin (10 mg/l) at pH
7.4. Since these preparations gave reproducible
P spectra
for up to 4 h, we concluded that they were in a metabolic steady state
during the 2-h duration of our experiment. Muscles were freeze-clamped
after completion of spectral acquisition. Contralateral muscles were
prepared similarly and frozen when its mate was put into the probe.
Muscles frozen about 30-45 min after dissection rather than after
3 h in the NMR probe showed no differences in metabolite contents.
Two methods of quantification were used for spectral analysis. In
solutions where signal-to-noise and spectral resolution were high and
base lines strictly flat, spectral peaks were integrated by summing
digitized data symmetrically around each peak. Each integral value was
expressed as fraction of the total phosphorus integral within the
spectrum and reported in absolute chemical content based upon ATP
concentration measured by optical spectroscopy as described above. In
isolated hindlimb muscles, time domain fitting of free induction decay
was used (45) in a commercially available package (FITMASTERS,
Philips Medical Systems). As with solution data, integrals were
expressed as fractional areas and normalized to chemical content by
HPLC-determined ATP content in µmolesgww
.
ATP content by
P NMR spectroscopy was the average of
,
, and
ATP resonances.
Figure 1:
P NMR
solution spectra for determination of combined equilibrium constants. A depicts a typical spectrum for
GPA experiments.
Chemical shift values were 2.5, -2.54, -2.98, -5.0,
-10.0, and -18.5 for P
, PCr,
GPAP, and
,
, and
resonances of ATP, respectively
. B depicts solutions used for cCr. Note that chemical shifts are
identical with the exception of PcCr at -2.38 and absence of
GPAP. Solutions contained (in mM): 100 KCl, 5.3 ATP, 15
PCr, 100 MOPS, 4 EGTA, 1 KH
PO
, 92
KC
H
O
, 70 Tris (pH 7.0), and 1 mg/ml
CK. Note chemical shift differences of PCr to
GPAP (0.44) and PcCr
(0.16) and that there is no ADP detected in either solution, indicating
that the source of phosphate for each analog is from PCr
hydrolysis.
Two such experiments (with least variance) were used to
construct HPLC standards for phospho-analog in HPLC assays. These
standards are essential for measuring PX concentrations in
freeze-clamped muscles and for cCr treated animals where P
NMR chemical shift differences are small.
The values in all
experiments for K with CK for both analogs are
given in Table 1. K
values for both
analogs with CK agree with previous
reports(34, 40, 48) .
Figure 2:
P NMR spectra of control (A) and
GPA (B)- and cCr (C)-treated
extensor digitorum longus muscles from mouse hindlimb. Data acquisition
parameters were as follows:
/2 pulse width (6.6 µs), 15-s
predelay, 5-KHz sweep width, and 400 acquisitions. Summed data were
filtered with a 15-Hz exponential prior to the Fourier transform. Inset of each panel contains an expanded region surrounding
the PCr resonance which was processed using 2-Hz line broadening to
show chemical shift differences of PCr to
GPAP (0.44, B)
and PCr and PcCr (0.16, C). Peak assignments are as follows: P
= inorganic phosphate, PCr = phosphocreatine,
GPAP =
-guanidophosphonic acid phosphate, PcCr =
phosphocyclocreatine, and
,
, and
reson
ances of
ATP.
Fractional peak areas of EDL and
SOL muscles of control and analog-fed mice are presented in Table 2. Control values for PCr, P, and ATP are
consistent with previous reports using Swiss-Webster (20) and
C-57 strains of mice (49, 50) from this laboratory.
Figure 3:
Endogenous substrates for CK (except ADP)
as determined by anion (PCr and ATP) and cation (free creatine)
exchange HPLC for control (open bars) and GPA (closed
bars)- and cCr (hatched)-treated EDL and soleus. Data are
presented as mean + S.E. with units of
µmoles
gww
.
The contents of phosphorylated and free analog from GPA (closed bars) and cCr (left-hatched bars) fed animals
are presented in Fig. 4. Accumulation of each phospho-analog
reached similar levels in both EDL and SOL during 21 days of feeding
while levels of free analog were 11-12-fold lower in cCr-treated
muscles than those from mice-administered
GPA.
Figure 4:
Synthetic substrates for CK in GPA (closed bars)- and cCr (hatched)-treated EDL and
soleus. PcCr and
GPAP were determined by anion exchange HPLC (left panel), whereas free cCr and
GPA
were determined
using a cation exchange column (right panel). In addition to
the measured value for free analog, B shows predicted values (cross-hatched) for each analog calculated from solution
equilibrium constants and therefore contain no error bars. Data are
presented as mean + S.E. with units of
µmoles
gww
.
Figure 5:
Comparison of HPLC (closed bars)
and NMR (open bars) ratios of PCr/ATP, PCr/GPAP, and
PCr/PcCr (presented as PCr/PX). ATP values for NMR were calculated as
mean of
+
+
ATP resonances. Value
s are
presented as mean + S.E.
Figure 6:
Combined equilibrium constants for cCr/Cr
and GPA/Cr via CK reaction compared for in vitro (solution) and in vivo. Values are presented as mean
+ S.E. (n = 6-8 for each group). The upper panel represents data from cCr experiments, whereas the lower panel contains data from
GPA experiments: left-
hatched bars = solutions, right-hatched bars = soleus, and cross-hatched bars = EDL. The lower panel contains data from experiments performed at 3
weeks as well as on a longer time course (9-12 week)
GPA
feeding (solid bars) to test for transient changes in tissue
. Asterisks were placed over 3-week data to indicate that
these values deviate significantly from solution K
.
The result in GPA-treated muscles suggests there is a
lack of equilibration between PCr/Cr and
GPAP/
GPA, since
for both tissues is considerably lower than solution K
values. Because
P NMR spectra
and HPLC results agree in both analog treatments, the apparent lack of
equilibration can only be explained by either low free Cr or high
GPA values (see ). Fig. 4shows depletion of
free Cr occurs to approximately the same extent in EDL and SOL of both
analog-treated groups; therefore lack of equilibration may result from
the presence of too much
GPA. This finding leads to the hypothesis
that there may be some nonspecific binding of
GPA in the intact
cell which is released during perchloric acid extraction and
contributes to HPLC measurement.
To test whether there is any
significant binding of GPA to subcellular fractions we performed
an equilibrium binding experiment using differential centrifugation
with a whole muscle homogenate. Rabbit tibialis anterior muscle was
homogenized (1:10 w/v) in 125 mM Tris buffer (pH 7.0). After
an initial incubation period of 30 min, the homogenate was divided into
4 equal volumes. Either
GPA or cCr (approximately 40-50
mM) was added to a pair of samples by diluting 1:1 with stock
solutions of each analog (also in 125 mM Tris), then incubated
for an additional period of 30 min at 25 °C. Differential
centrifugations were performed at 600, 5,000, and 13,000
g, and supernatants and pellets of each fraction were
subjected to perchloric acid extraction. The test for binding of analog
as compared with Cr was to measure the ratio of synthetic analog to
endogenous Cr in each fraction. Neutralized extracts were assayed by
cation exchange HPLC for free
GPA, cCr, and Cr. If some
GPA
binding were sufficient to account for the observed lack of
equilibration, then there should have been excess
GPA in at least
one of the fractions, and a different ratio of
GPA/Cr should have
been detected. Cyclocreatine was used as a negative control for this
experiment. The results presented in Table 3were decisive: the
ratio of free Cr to analog was constant in all fractions tested. On
this basis we exclude an unsuspected selective binding of
GPA as
an explanation for apparent disequilibration of
GPA experiments in
animals fed the diet for 3 weeks.
Rats fed GPA over longer time
periods did not exhibit this apparent lack of
equilibration(40) , suggesting that the present results showing
excess free
GPA might be transient, and if measured later in the
time course, would not be present. To test this hypothesis, we fed mice
over equivalent time periods as in rat studies (9-12 weeks) and
performed the same HPLC measurements as in the present study. The
results presented in Fig. 6show a marked deviation of mass
action ratios from K
in muscles from
GPA
animals with several weeks of feeding but with typical feeding regimens
used, 9-12 weeks, mass action ratios are statistically
indistinguishable from K
, indicating that
GPA effects on
are transient.
One of the key tenets of bioenergetics of excitable tissues
is the near-equilibrium property of CK coupled with the assumption that
cytosolic metabolites involved in the reaction are freely mixing and
hence available to the enzyme. The existence of multiple isozymes of CK
and demonstration of their localization into subcellular compartments,
perhaps with their associated substrate
pools(23, 24) , has suggested that the concept of
freely diffusing metabolites mixing cytoplasm may be too simplistic. In
light of the importance of this concept to interpretation of
biochemical data, in particular by P NMR spectroscopy, we
employed two competitive inhibitors of CK, cCr, and
GPA to
investigate these issues. Feeding synthetic analogs of creatine (which
as competitive inhibitors are not expected to disturb equilibration)
partially displaced the content of PCr and Cr from the cell. By
choosing the duration of feeding such that there were approximately
equal concentrations of total Cr and total analog, it was possible to
have an accurate measure of mass action ratios for CK in resting muscle
and test the hypothesis that these ratios equal those predicted from K
measured in solution as would be the case if
the reaction is equilibrating and the cytosol is freely mixing with
respect to substrates.
In cCr experiments, in muscle was
indistinguishable from those predicted by K
in
solution. Therefore results with this analog could not disprove the
central hypothesis of this work. In contrast, in
GPA experiments
(3 weeks of analog feeding),
in muscle was clearly different from
those predicted by K
in solution. However, with
longer periods of feeding
in muscle were indistinguishable from
that predicted by K
in solution, as has been
shown in other preparations(40) . Thus our results show that
muscles in the presence of
GPA undergo a significant bioenergetic
transformation from the onset of analog feeding to the time they
demonstrate large phenotype changes in contractile and metabolic
proteins(51, 52) . The primary observation of
near-equilibration of CK with its natural substrates and exogenous
competitive inhibitors is consistent with the idea that cytoplasmic
phosphorylation potential displays thermodynamic characteristics of
substrates and enzymes freely mixing. Thus, these data establish a
solid biochemical basis for the widely used concept of simple solution
dynamics for interpretation of
P NMR data.
The
remaining discussion considers three further issues that show cell to
cell differences in ATP/ADP are the only physiologically important
factors determining whether observed mass action ratio from tissue
equals K measured in solution. This conclusion
is derived from the principle that PCr/Cr (and also PX/X) ratios are
set by cell ATP/ADP ratios (which are determined by cytosolic chemical
potential) acting through CK thereby rendering calculation of
independent of total Cr or analog
content(1, 9, 10) . By extension, this
conclusion can be related to calculation of free ADP in whole tissue
where metabolite heterogeneity is present either as a characteristic of
phenotype (20) or induced via differential cellular responses
to metabolic stress (e.g. fiber recruitment during exercise)
or abnormal physiologic states (e.g. localized ischemia).
To test whether quantitative agreement
occurs in muscle between NMR and HPLC of their respective perchloric
acid extracts, a similar analysis was performed as in solutions for in vivo data from control and treated muscles. From Fig. 5, PCr/ATP for all muscles are in excellent agreement using P NMR and anion exchange HPLC. Quantitative agreement with
NMR and chemical methods illustrates that artifacts of freeze clamping
and extraction are absent using our technique irrespective of total Cr
content. Calculation of
might also be influenced by NMR
visibility of PCr and PX. NMR visibility of ATP and P
have
been examined in other tissues such as heart and liver, particularly
under different metabolic states (53, 54, 55, 56) . Our analysis
using metabolite ratios allowed unambiguous investigation of this issue
in muscle. To maintain agreement with stoichiometry of the chemical
results, any NMR ``invisible'' portion of these metabolites
would require an equivalent fraction of the other metabolites to be
``invisible'' as well. We illustrate this important point as
follows: consider PCr/ATP in control EDL (from Fig. 3) with
approximate metabolite contents of 20 µmole
gww
PCr and 5 µmol
gww
ATP. If a
``pool'' of ATP or PCr corresponding to 20% of total cellular
content was not NMR visible by some mechanism (e.g. rapid T
relaxation), this possibility is equivalent to 4
µmol of PCr being invisible but only 1 µmol of ATP. This
``NMR-invisible'' pool would have to be nonstoichiometric (4
PCr/1 ATP) to be consistent with observed ratios and these
``invisible fractions'' would have to be extracted and
quantified to the same extent for HPLC and NMR ratios to agree.
Additionally, in muscles from treated animals, PCr/PX ratios for both
methods also agree. Therefore, any NMR-invisible fraction which exists
must influence invisible metabolites in this pool via an identical
nonstoichiometric mechanism. We consider this scenario which so affects
equal fractions of PCr, PX, and ATP untenable and favor the simpler
explanation that phosphorus metabolites involved in the CK reaction,
namely PCr, ATP (and in analog-treated muscles PX) must be fully
NMR-visible.
The second possibility to explain lack of equilibration is
based upon enzyme kinetics and suggests that CK fluxes under conditions
of our muscle experiments may not be sufficient to maintain equilibrium
in resting muscle in vivo in the presence of competitive
inhibitors during 3 weeks of feeding or following metabolic
perturbations during tissue handling in the experiment. Arguments
against this hypothesis are derived from in vivo muscle
experiments from other investigators. P NMR spectroscopy
performed on intact hindlimb of
GPA-treated rats have shown that
GPAP levels decrease 20% (approximately 5 mM) during
acute exercise in rat hindlimb muscles and return to initial levels
within 13 min(40) . This translates into a phosphorylation rate
of 0.37 mM/min for rat fast-twitch muscles. During the course
of our NMR experiments (3 h), approximately 60 mM of
GPAP
could have been produced. This flux is far in excess of what is
required to meet equilibrium conditions assuming rat and mouse
fast-twitch muscles have similar CK activities. Thus a kinetic
limitation under resting conditions cannot explain the observations for
GPA-treated muscles.
What aspects of metabolite distribution could explain
the observed disequilibration with GPA at 3 weeks? From our
analysis based upon an expansion of from the
``Experimental Procedures'' (see in the
``Appendix''), lack of agreement for solution K
values with
from muscles of
GPA-treated animals cannot be explained by varying distributions
of Cr or its analogs. In fact, observed disequilibration can only be
explained if ATP/ADP ratios are no longer constrained to be equal (e.g.fPCr
fPCr
, Appendix, and ) between volume fractions. Although adenylates do not
directly enter into any of these calculations, under equilibrium
conditions, fractional PCr content is determined by ATP/ADP through CK (9, 10) and therefore ultimately sets PX/X ratio as
well. Thus, if ATP/ADP, or equivalently fPCr, in 1 volume
fraction is significantly different from the other then
no longer equals K
. This is possible
while physicochemical constraints for thermodynamic equilibrium are
still met in each volume fraction comprising the sample. This analysis
demonstrates that apparent lack of equilibration observed is a
physiologic phenomenon resulting from different fractional PCr contents
(hence different ATP/ADP ratios) in each volume fraction. Thus
measurements from whole muscle may appear to violate CK equilibration, i.e.
does not equal K
, despite the fact that each volume fraction
is in equilibrium as defined by and in the
``Appendix.'' We favor the hypothesis that
GPA is
selectively effecting a set of muscle fibers in a cell-specific manner,
and the nature of this perturbation is related to metabolism of Cr
ultimately altering cellular energetics. This fiber-specific effect may
in part explain adaptive changes in Cr analog administration in murine
muscle(51, 52, 57) .
Results with cCr and
previously established equilibration of CK with GPA in other
preparations exposed to long duration feeding (confirmed here in mouse
hindlimb muscles) are highly significant for describing and
understanding functional organization of CK in muscle cells. That we
observed muscle
to be indistinguishable from solution K
indicates that these major bioenergetic
reactions function in cells of resting muscles as they do in
homogeneous solutions in vitro, despite the presence of
mitochondrial and cytoplasmic isoforms, their partial binding to
macromolecular structures, and compartmentalization of the cell volume
(sarcoplasmic reticulum, mitochondria, etc.). Corollaries of this
conclusion, supported by other studies of CK function, are several: (a) calculations of metabolically active ADP concentration
from CK equilibration are valid; (b) concentrations of PCr,
phospho-analog, and ATP measured by
P NMR are equal to
those measured in perchloric acid extracts, thus there is no evidence
for NMR-invisible pools of these metabolites; (c) the net
quantitative contribution of metabolites in diffusion-limited
compartments to total measured quantities is small; (d)
effects of such compartmentalization, however important for certain
functions, are negligible with respect to total cellular bioenergetics
and metabolism. Thus our experiments unambiguously demonstrate that
thermodynamic characteristics of the cytosol can be predicted as if
these metabolites were freely mixing in solution. Finally, if there is
an apparent lack of equilibration due to two compartments in the muscle
with differing ATP/ADP ratios, bioenergetics can still be solved by
knowledge of either ATP/ADP (or fPCr) in one of the two
components.
The following equations illustrate the effects of analog, Cr,
and adenylate distribution within the sample on calculation of .
From expansion of in ``Experimental
Procedures,'' consider 2 volumes which comprise the entire sample.
This expression can be expanded to include mass action ratios for more
than 2 fractional volumes. Contribution of each volume fraction to the
total mass action ratio can be written as
follows,
where is the mass action ratio for the total
sample volume (V
),
is the first fractional
volume, and
is the second fractional volume and these are
constrained to be
+
= V
.
Mass action ratios for each fractional volume (
and
) can be written as
follows.
If we constrain each fractional volume to its thermodynamic
equilibrium by setting the solution combined equilibrium constant K =
=
, we see that for any analog concentration (TX),
the PX/X ratio is now entirely defined by the only independent
variables in and , namely PCr and Cr. Under
these circumstances the PX/X ratio reflects the concentrations of PCr
and Cr in each volume fraction independent of total analog content.
More importantly, one can also conclude that as long as concentrations
of PCr and Cr are equal in both fractional volumes then K
=
. These conclusions
are independent of total analog present or its distribution between
volume fractions.
As in Case 1, it follows from these equations that as long as fPCr for both volume fractions is identical then
=
, inde
pendent of
total Cr content of either volume fraction or distributions. Thus K
remains equal to
.
where K and K
are equilibrium constants for CK for each fractional volume
. Thus
only if ATP/ADP ratios for both frac-tional volumes are equal will
PCr/Cr and PX/X ratios agree and thus K
=
.