Regulation of mitochondrial respiration in heart cells
analyzed by reaction-diffusion model of energy transfer
Marko
Vendelin1,
Olav
Kongas1, and
Valdur
Saks2,3
1 Institute of Cybernetics and
2 Laboratory of Bioenergetics, Institute of
Chemical and Biological Physics, Tallinn, Estonia; and
3 Joseph Fourier University, BP 53X-38 041 Grenoble Cedex, France
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ABSTRACT |
The purpose of this study is to investigate theoretically
which intracellular factors may be important for regulation of
mitochondrial respiration in working heart cells in vivo. We
have developed a model that describes quantitatively the published
experimental data on dependence of the rate of oxygen consumption and
metabolic state of working isolated perfused rat heart on workload over its physiological range (Williamson JR, Ford G, Illingworth J, Safer B. Circ Res 38, Suppl I, I39-I51, 1976). Analysis of
this model shows that for phosphocreatine, creatine, and ATP the
equilibrium assumption is an acceptable approximation with respect to
their diffusion in the intracellular bulk water phase. However, the ADP
concentration changes in the contraction cycle in a nonequilibrium workload-dependent manner, showing the existence of the intracellular concentration gradients. The model shows that workload-dependent alteration of ADP concentration in the compartmentalized creatine kinase system may be taken, together with the changes in Pi
concentration, to be among the major components of the metabolic
feedback signal for regulation of respiration in muscle cells.
compartmentation; adenosine diphosphate; creatine kinase; metabolic
oscillations; mathematical modeling
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INTRODUCTION |
THE CLASSICAL PARADIGM of cellular energy metabolism,
i.e., its conceptual framework, is based on the assumption of the
equilibrium (or at least quasi-equilibrium) of the reactions involved.
Practically all experimental studies of muscle metabolism have been
performed during the last two decades within the framework of these
theoretical considerations, with creatine kinase (CK) equilibrium used
for calculation of cytoplasmic ADP concentration and all derived
thermodynamic parameters (23, 34). However, these theoretical concepts
are based on very few experimental works exclusively performed on resting muscles, where the equilibrium state is the easily expected one
(34). Recent reinvestigation of this problem with a mathematical model
of compartmentalized energy transfer has shown that this concept is not
totally valid: in the working heart cells the CK reaction is clearly
out of equilibrium during most of the contraction cycle (1, 32). This
observation raises the following question: For which compounds involved
in the energy metabolism of the cell are the calculations based on the
assumption of equilibrium reasonably correct? Also, another equally
important question relates to the existence and deepness of the
concentration gradients of ADP and other metabolites in the cells (20).
Answers to these two questions are important for understanding the
nature of the intracellular factors regulating the rate of
mitochondrial oxidative phosphorylation in the cells in vivo. We
investigated these problems theoretically by analysis of different
versions of mathematical models of energy transfer in muscle cells
developed on the basis of the previous model of Aliev and Saks (1).
This model was supplemented with the model of oxidative phosphorylation
developed by Korzeniewski and Froncisz (21, 22) to account for the
changes in mitochondrial membrane potential and to calculate the rates
of oxygen consumption (
O2)
for different workloads. The new model describes quantitatively the
experimental dependencies of metabolic parameters of isolated perfused
working rat hearts on the workload over the physiological range of
O2 and workloads published by
Williamson et al. (38). The results of analysis of intracellular
metabolic changes confirm that there are significant oscillations of
the cytoplasmic ADP concentration in the cells within the cardiac cycle
and significant concentration gradients of this metabolite in the
working muscle cells because of the nonequilibrium state of the CK
reaction. It is concluded that localized changes of ADP concentrations
(its oscillations) due to the nonequilibrium mode of functioning of the
CK system in the cells open new possibilities of the metabolic regulation of mitochondrial respiration, in comparison with the equilibrium state of the cellular metabolism.
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GLOSSARY |
General
AK |
Adenylate kinase
|
ANT |
Adenine nucleotide translocase
|
CK |
Creatine kinase
|
Cr |
Creatine
|
IM |
Intermembrane
|
PCr |
Phosphocreatine
|
Model variables
Met |
Concentration of metabolite (ATP, ADP, AMP, PCr, Cr, Pi) in
myofibril and cytoplasm
|
Meti |
Concentration of metabolite in mitochondrial IM space
|
ATPg, ADPg |
ATP and ADP concentrations in microcompartment
|
UQ |
Oxidized form of coenzyme Q
|
c3+ |
Oxidized form of cytochrome c
|
NAD+ |
NAD+ in mitochondrial matrix
|
Hx |
H+ in mitochondrial matrix
|
ATPx |
ATP in mitochondrial matrix
|
Pi,x |
Pi in mitochondrial matrix
|
Diffusion
2 |
Laplace operator
|
DMet |
Diffusion coefficient of metabolite
|
GCK |
Spatial distribution function of myofibrillar CK
|
GAK |
Spatial distribution function of myofibrillar AK
|
GH |
Spatial distribution function of myofibrillar ATPase
|
Myofibrillar ATPase
HATP |
Rate of ATP hydrolysis in myofibril
|
HATP(max) |
Maximal HATP
|
CK in myofibril, myoplasm, and IM space
vCK |
CK reaction rate
|
vMiCK |
vCK in IM space
|
vMiCK,G |
vCK, ATP and ADP from microcompartment
|
vMiCK,I |
vCK, ATP and ADP from IM space
|
V1, V 1 |
Maximal CK reaction rates in forward and reverse directions
|
Kia, Kb,
Kic, Kd,
Kid, Kib,
KIb |
Dissociation constants for MgATP, Cr, MgADP, PCr, PCr, Cr,
and Cr from CK-MgATP, CK-MgATP-Cr, CK-MgADP, CK-MgADP-PCr, CK-PCr, CK-Cr, and CK-MgADP-Cr complexes
|
KiaG, KicG |
MiCK dissociation constants for ATP and ADP in
microcompartment
|
AK in myofibrils and IM space
vAK |
AK reaction rate
|
vAKmit |
vAK in IM space
|
kfa, kba |
AK forward and backward reaction rate constants
|
Mg2+
buffering
mMet |
Mg2+-bound forms of metabolite (ATP or ADP)
|
fMet |
Mg2+-free forms of metabolite (ATP or ADP)
|
Mgext |
Mg2+ in cytoplasm and myofibril
|
Mgx |
Mg2+ in mitochondrial matrix
|
KDText, KDDext,
kDTx, kDDx |
Mg2+ dissociation constants
|
H+ buffering
Hext, Hx |
H+ concentration in cytoplasm and matrix
|
pHext, pHx |
pH in cytoplasm and matrix
|
p, pH,   |
Protonmotive force, pH difference, and membrane potential across inner
mitochondrial membrane
|
rbuff, rbuff,0 |
H+ buffering capacity coefficients
|
Diffusion through mitochondrial outer
membrane
 |
Unit vector, outward normal to the domain boundary
|
Li |
Width of the layer between inner and outer membranes
|
Lm |
Width of the layer between core of myofibril and mitochondrial outer
membrane
|
RMet |
Permeability of the mitochondiral outer membrane for the metabolite
|
ATP diffusion between microcompartment and IM space
vexchATP, vexchADP |
Rate of ATP or ADP flux between microcompartment and IM
space
|
RexchATP, RexchADP |
Rate constants
|
ATP/ADP translocation by ANT
vANT |
Net rate of ATP export by ANT from matrix
|
vX GANT |
Rate of ATP export by ANT from matrix to microcompartment
|
vX IANT |
Rate of ATP export by ANT from matrix to IM space
|
VANT |
Maximal net rate of ATP export by ANT from matrix
|
Kg, Ki |
ANT reaction dissociation constants
|
Fractional volumes
FVi |
Fractional volume of IM space with respect to cell volume
|
FVx |
Fractional volume of matrix with respect to cell volume
|
Substrate dehydrogenation
vdh, kdh |
Net and maximum rates of substrate dehydrogenation
|
KmN |
Michaelis-Menten constant of substrate dehydrogenation for the
NAD+/NADH ratio
|
pD |
Relative sensitivity coefficient of substrate
dehydrogenation to the NAD+/NADH ratio
|
Phosphate carrier
vPI, vf,PI,
vb,PI |
Net, forward (to matrix), and backward rates of phosphate carrier
|
pKa |
One-half dissociation constant for monovalent Pi
|
Proton leak
kL1, kL2 |
Phenomenological constants for proton leak
|
ATP synthesis
vsn, ksn |
Net and maximal rates of ATP synthase
|
G0, Gsn |
Gibbs free energy and thermodynamic span of ATP synthase
|
na |
H+/ATP stoichiometry for ATP synthase
|
Respiratory chain
vC1, vC3,
vO2 |
Rates of complexes I, III, and IV
|
kC1, kC3,
kC4 |
Maximum rates of complexes I, III, and IV
|
a2+, c2+, a3+, c3+ |
Concentrations of reduced and oxidized forms of cytochromes
a3 and c
|
kMO |
Michealis-Menten constant of complex IV for oxygen
|
En, Eu, Ec,
Ea |
Redox potentials of NAD, ubiquinone, and cytochromes c and
a3
|
En,0, Eu,0,
Ec,0, Ea,0 |
Standard redox potentials of NAD, ubiquinone, and cytochromes
c and a3
|
Ax, NADtot, O2 UQtot, ctot,
atot |
Total concentrations of adenine nucleotides, NAD, oxygen in
matrix, ubiquinone, and cytochromes c and
a3.
|
rbuff, rbuff,0 |
Buffering capacity coefficients
|
General constants
T, R, F |
Absolute temperature, gas constant, and Faraday number
|
Z, u,  |
Constants for unit conversion
|
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METHODS OF MODELING |
Principles.
The spatially inhomogeneous reaction-diffusion model of energy transfer
considers the reactions in three main compartments of cardiac cells:
the myofibril together with the myoplasm, the mitochondrial
intermembrane (IM) space, and the mitochondrial inner membrane-matrix
space (Fig. 1). The
metabolites described by the model in the myofibrils and IM space are
ATP, ADP, AMP, phosphocreatine (PCr), creatine (Cr), and
Pi. All these metabolites diffuse between the cytosolic and
IM compartments, where the metabolites are involved in the CK and
adenylate kinase (AK) reactions. In addition, the ATP is hydrolyzed in
the myofibrils. In the IM space the mitochondrial CK reaction is
coupled to the adenine nucleotide translocase (ANT) reaction via strong
coupling between these enzymes; the coupling is moderated by a
diffusional leak of the intermediates. The metabolites described by the
model in the matrix compartment and in the inner membrane are NADH,
coenzyme Q, cytochrome c, protons, ATP, ADP, and
Pi. Three coupled reactions representing the production of
protonmotive force by complexes I, III, and IV are included in the
model. Protonmotive force is consumed by ATP synthase and membrane
leak. The ANT rate is considered to depend on membrane potential.
Pi is transported by a phosphate carrier.

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Fig. 1.
Scheme of 2-dimensional compartmentalized energy transfer model of
cardiac cells. A: components of model. Total diffusion path in
transverse direction is 1.2 µm consisting of mitochondrial outer
membrane (heavy line below mitochondrion), 0.2 µm for myoplasm, and
1.0 µm for myofibril. Bottom edge of A corresponds to
core of myofibril. Total diffusion path in longitudinal direction is
1.0 µm. Creatine kinase (CK) and myosin magnesium ATPase activities
are distributed nonuniformly along myofibril (37). B:
activities of enzymes in dependence of longitudinal coordinate. CK
concentration in myoplasm is taken to be equal to that in I-band.
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Model description.
The mathematical modeling was started by reproducing the
one-dimensional model (1) of compartmentalized energy transfer from
mitochondria to the center of myofibrils in the cardiac cells. It was
then converted to a spatially inhomogeneous two-dimensional reaction-diffusion model (Fig. 1). ATP utilization by myofibrillar ATPases and regeneration from PCr and ADP by CK and AK in the myofibril
and the myoplasm are considered two-dimensional processes. Namely, the
active ATPases are not distributed uniformly along the myofibril but
are found only in the regions where actin and myosin overlap (Fig. 1).
According to Wegmann et al. (37), the myofibrillar CK isozyme is also
distributed inhomogeneously within the myofibril. The distribution
functions for CK and AK in myofibrillar space were constructed on the
basis of Fig. 4 in Ref. 37 and are shown in Fig. 1B. The
distributions of CK and AK in the myoplasm are uniform, and their
activities are assumed to be equal to those in the I-band. Then the
two-dimensional model was supplemented with oxidative phosphorylation
processes by combinination with the model by Korzeniewski (21).
The new model was used to calculate the concentration of ADP and its
gradients in the cell within the contraction cycle, concentration of
metabolites, and
O2. During
contraction, the myofibrils are supposed to hydrolyze ATP, with
kinetics predicted from change in the time derivative of pressure in
isovolumic rat hearts, i.e., linear increase in ATP hydrolysis rate up
to 30 ms followed by its linear decrease to zero at 60 ms at a heart
rate of 333 beats/min (180 ms for the cardiac cycle). The total amount
of ATP hydrolyzed by myofibrillar ATPases is used to regulate the
workload from zero to its maximum value of up to 960 µmol
ATP · g dry
wt
1 · min
1
corresponding to ~160 µmol O2 · g dry
wt
1 · min
1
for working heart (32, 37). We use also a spatially homogeneously distributed basal level of ATP consumption in the myofibrillar and
myoplasmic compartment with 18 µmol ATP · g dry
wt
1 · min
1
corresponding to ~3 µmol O2/ · g dry
wt
1 · min
1.
Addition of basal level consumption is of some importance at very low
workloads. The lengths of the full diffusion paths are taken to be 1.0 µm in the "longitudinal" direction and 1.2 µm in the
"transverse" direction to the myofibril (5). These are the
distances between the Z and M lines and between the myofibril core and
the mitochondrial outer membrane, respectively. The radius of the
myofibril is taken to be 1.0 µm. Thus in the model the myoplasm
covers the area 1.0 × 0.2 µm (Fig. 1). The fractional volumes
of the cell compartments for myofibrils, myoplasm, IM space, and matrix
are chosen to be 10:2:1:3. We assume that within the mitochondrial IM
space and matrix the concentrations of the metabolites depend only on
the longitudinal coordinate. In the model of Korzeniewski (21), we
modified slightly the constants in the membrane leak function to meet
the following two conditions: 1) at 40% of maximal
O2, 25% of oxygen is
consumed by proton leak (Fig. 3 in Ref. 14 and Fig. 1 in Ref. 10) and
2) for arrested rat heart, the minimal
O2 is 12-15 µmol
O2 · g dry wt
1 · min
1
(32).
To study the importance and significance of the use of the
two-dimensional model of compartmentalized energy transfer, we compare
it with one- and zero-dimensional models. The dimension of the model is
changed by variation of the diffusion coefficients. To ignore a
longitudinal or transverse dimension, the corresponding diffusion
constant is increased by 105 times. Thus the diffusion in
this direction is considered to be infinitely fast and equalizes the
concentration of the metabolites, thus resulting in zero partial
derivatives along this direction. In the one-dimensional case, the
longitudinal (along myofibrils) dimension is ignored and the
myofibrillar part of the model, except for AK, is reduced into the
original model by Aliev and Saks (1). In the zero-dimensional case,
both dimensions are ignored. To study the reduced diffusion in the
myofibrillar compartment (3), we reduced the diffusion coefficients by
a factor of 10. To investigate the equilibrium case for
metabolites, we increased the activity of CK
105-fold compared with its normal activity. It
is supposed that in this case the reaction rates establish the
equilibrium fast enough compared with the diffusion rate. In these
different models we also study the influence of the outer mitochondrial
membrane on the transport processes (see APPENDIX).
Functional coupling of mitochondrial CK (miCK) to ANT was assumed to
occur by means of high local ATP and ADP concentrations in a 10-nm (1,
13) narrow-space microcompartment between coupled enzymes. We have made
the following assumptions in our description of the ANT and miCK.
First, ANT is translocating the adenine nucleotides between matrix
space and microcompartment and, partly, IM space. Second, the miCK
adenine nucleotide binding center may be occupied by the adenine
nucleotides from the microcompartment or from the IM space.
Furthermore, it is assumed that when the reaction involves ATP from the
microcompartment, the miCK reaction product ADP will be released also
into the microcompartment. This kind of channeling of ADP from miCK to
ANT ("reverse coupling") was not considered in the previous model
(1). Third, diffusion between the microcompartment and the IM space is
restricted. Finally, the capacity of the microcompartment is considered
infinitely small; therefore, for each adenine nucleotide the influx is
taken to be equal to its efflux. With these assumptions, the local ADP and ATP concentrations are determined by the ANT reaction, the miCK
reaction, and the restricted diffusion. The values of the coefficients
of the diffusion restriction from the microcompartment and the
dissociation constants of adenine nucleotides from miCK were estimated
from 1) deviation of the mass action ratio of the miCK from the
equilibrium constant value observed experimentally (31, 32) in coupled
rat heart mitochondria under conditions of oxidative phosphorylation
and 2) change in the apparent dissociation constants of miCK
due to the oxidative phosphorylation (19).
All equations included in the model are shown in the
APPENDIX. All values of the experimentally determined
parameters used in the model were taken from literature and are shown
in Table 1.
Computation.
The model equations were numerically solved by a finite-element method
in conjunction with Galerkin's method. The resulting system of
ordinary differential equations was solved by the backward differentiation formula that is able to treat stiff equations. The
accuracy of the solution was tested by comparing different spatial
discretizations and varying the tolerance of the ordinary differential
equation solver. The finite-element discretization was performed using
the software package Diffpack (7), and the system was integrated using
the DVODE package (6). The calculation time varied from 1 s to a couple
of hours on an Alpha-Linux PC, depending on the complexity of the model.
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RESULTS |
Testing the new model for "in vitro" conditions.
We first tested the new model for its ability to describe the published
experimental data in vitro. Figure 2 shows
that the new model describes adequately the experimentally observed
increase in the rate of PCr production in the coupled miCK reaction
under the condition of oxidative phosphorylation in isolated rat heart mitochondria described by Jacobus and Saks (19) and Saks et al. (31).
The increase in the rate of PCr production by oxidative phosphorylation
is explained by increased turnover of adenine nucleotides in the miCK
and oxidative phosphorylation reactions coupled by ANT (31, 32).
Another important observation is that, under conditions of oxidative
phosphorylation, direct supply of ATP and removal of ADP by ANT exert
strong kinetic and thermodynamic control of the miCK, making the
reaction of PCr synthesis favorable (19, 31, 32). Under these
conditions, inhibition of the oxidative phosphorylation results in
rapid reversal of the miCK reaction, and production of ATP, rather than
PCr, becomes favorable because of kinetic and thermodynamic properties
of the isolated CK reaction (19, 31, 32). This alteration in the
behavior of the miCK in the presence and absence of oxidative
phosphorylation is quantitatively described by our new version of the
model (Fig. 3). Thus the inclusion of
Korzeniewski's model of mitochondrial reactions (21) and the detailed
description of coupling between miCK and ANT used in the new version of
the model give a satisfactory description of the mitochondrial
reactions of energy production and their control in vitro.

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Fig. 2.
Dependence of rate of phosphocreatine (PCr) production in mitochondrial
CK reaction on ATP concentration in medium with ( ) and without ( )
oxidative phosphorylation (ox phos). Experimental points for isolated
rat heart mitochondria are from Jacobus and Saks (19). Solid lines,
calculations by using model described in this work. Reaction medium
contained 5 mM Pi, 15 mM creatine (Cr), and 50 µM ADP.
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Fig. 3.
Analysis of CK reaction in isolated rat heart mitochondria.
Mitochondrial CK reaction was reversed by inhibition of oxidative
phosphorylation with oligomycin. Conditions of reaction correspond to
those used in experiments by Saks et al. (31): 2 mM PCr, 0.12 mM ATP,
0.05 mM ADP, 40 mM Cr, 5 mM Pi. Calculated curve for
relative rate [rate of reaction related to maximal velocity
(Vmax) of forward reaction of PCr synthesis in
mitochondria] is shown. At time 0, oxidative
phosphorylation was taken to be inhibited, and this quickly reversed
reaction from PCr synthesis [positive CK reaction rate
(vCK)] to its utilization (negative
vCK), as shown in experiments by Saks et
al.
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Theoretical description of experimental results on isolated perfused
hearts "in vivo."
It is important to know exactly how the model describes the
cardiac energy metabolism in vivo for the wide variety of conditions, notably for the workload changes. There is abundant information in the
literature on the metabolic changes in the perfused heart during
workload transitions, but in too many of these studies the workload is
changed in a rather narrow range (usually 1.5- to 3-fold), so one can
only guess what may really happen outside these limits (15, 17, 33).
Therefore, these works are not suitable for serious modeling.
Fortunately for us, there is one classical study by Williamson et al.
(38) in which Neely's working isolated rat heart perfusion protocol
was used to change the workload and, correspondingly,
O2 and energy fluxes
>10-fold, up to probably maximal possible values. Also, changes in
metabolite levels were very carefully analyzed in this work (38). For
this reason, we used their experimental data as a basis of application
of our model for analysis of the in vivo situation in the
heart cells.
Figure 4 shows the calculated (our results)
and experimental values [results of Williamson et al. (38)]
of
O2 as a function of the
workload that was altered by changing the rate of ventricular filling
of isolated rat heart perfused according to the "working heart"
protocol of Neely (38). The experimental and theoretical dependencies
are nearly linear functions perfectly fitting each other. The maximal
O2 recorded by Williamson et
al. (38) is close to 160 µmol O2 · g
dry
wt
1 · min
1
(38), which, according to the data of Mootha et al. (25), corresponds
closely to the maximal activity of mitochondria calculated from state 3 respiration rates in vitro and tissue contents of mitochondria. The
model describes equally satisfactorily the stable levels of main
metabolites: ATP level always stays at its initial value, and PCr level
starts to decline significantly only when
O2 exceeds
80-100 µmol O2 · g dry
wt
1 · min
1.
PCr-to-ATP and PCr-to-Cr ratios (very often the only parameters measured in an NMR experiment) change from their initial values of 1.5 very slowly at moderate workloads and more significantly at the highest
workload (Fig. 5), in good
accordance with the experimental data of Willamson et al. These results
are in accordance also with all published data of Balaban et al. (2).

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Fig. 4.
Computed (solid line) and experimental ( ) O2 consumption
rates ( O2) of working cardiac
muscle. Experimental points, obtained in experiments with isolated rat
hearts perfused according to working heart protocol of Neely, in which
workload was changed by increasing ventricular filling rate, are from
Fig. 5B of Ref. 38. Relative workload is fraction of maximal
workload applied [maximal filling rate (38)]; in
computations this is rate of ATP hydrolysis by actomyosin ATPase. Zero
workload simulates situation in arrested heart, where O2 is
consumed to sustain only basal level of ATP consumption and membrane
leak through mitochondrial inner membrane. CK+, normal rates in hearts
with intact CK activity. ( O2 = 41 µmol
O2 · g dry
wt 1 · min 1),
computed maximal O2 by
CK-deficient hearts (CK ). Computed relation for CK-deficient
hearts coincides with similar relation for normal heart at low
workload. Relation is not absolutely linear, i.e., slope of curve at
left edge is ~10% lower than that at right edge;
reason for this nonlinearity is exponential dependence of membrane leak
on protonmotive force. Calculations were made using 0-dimensional
model.
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Fig. 5.
Average PCR-to-Cr and PCr-to-ATP ratios over cardiac cycles as
functions of O2 calculated
with 0-dimensional model. Model solutions are compared with
experimental values of ratios of PCr to ATP ( ) and PCr to Cr ( )
from literature (32).
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Under conditions of this remarkable metabolic stability, however, the
model shows a very significant change in ADP during the contraction
cycle in the myofibrillar space, and the peak values of ADP increase
with elevation of workload and, correspondingly, of
O2 (Fig.
6). Minimal ADP levels are characteristic
for the diastolic phase of the contraction cycle and close to the
equilibrium values of ADP concentration. In accordance with many
earlier observations, they do not change with elevation of workload and
stay close to 50 µM (dotted line in Fig.
7). The average values of ADP concentration per cycle change more significantly (solid line in Fig. 7). At the same
time, ADP peak value increases >40-fold when
O2 changes 10-fold (Fig. 7).
To accommodate the traditional way of thinking in physiological and
biochemical audiences, which are more accustomed to
Michaelis-Menten-type analyses of reactions, in Fig.
8 these relationships are presented as
respiration rate vs. ADP concentration. The relationships between
O2 and ADP concentration
resemble the classical hyperbolic function, with half-maximal
respiration rate observed at peak ADP concentrations close to
160-200 µM. At least two conclusions can be made from these
results: 1) the relationship between cardiac tissue respiration
rate and intracellular ADP concentration depends on how we calculate
the latter (i.e., it is concept dependent) and, 2) clearly, ADP
may participate in the feedback regulation of respiration in working
cardiac cells, if its average or maximal peak values within the
contraction cycle are taken into account. The second metabolic
parameter that changes significantly is Pi (Fig.
9).

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Fig. 6.
ADP profiles over cardiac cycle at different workloads. Profiles
correspond to O2 = 25 (low),
50 (medium), and 100 (high) µmol · g dry
wt 1 · min 1.
Profiles were obtained with 0-dimensional model.
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Fig. 7.
Relationship between maximal (max), minimal (min), and average (aver)
levels of cytoplasmic ADP over a cardiac cycle and
O2 by working cardiac muscle.
Considerable difference between maximal and average ADP levels is
caused by significant change in ADP level during systole (see Fig.
11).
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Fig. 8.
Relationship from Fig. 12 presented in more usual and conventional
coordinates as relationship between
O2 by working cardiac muscle
and maximal, minimal, and average levels of cytoplasmic ADP over a
cardiac cycle. Considerable difference between maximal and average ADP
levels is caused by significant change in ADP level during systole (see
Fig. 11). Minimal ADP levels are close to those computed for CK
equilibrium (see Fig. 6).
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Fig. 9.
Average computed ATP, Cr, PCr, and Pi levels over cardiac
cycle as functions of O2 by
contracting heart muscle with increasing workload (see Fig. 7).
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CK reaction is important for the function of normoxic heart at
increased workloads.
Two reactions of the intracellular energy transfer network (11, 32) are
included in our model: CK and AK reactions. To evaluate their relative
importance, we analyzed the situation when the activities of these
enzymes were taken to be zero (modeling the experiments with inhibition
of these enzymes or with their "knockout" due to genetic
manipulations). With fully active CK, the blocking of AK activity did
not alter the model behavior. This result confirms the conclusion of
Dzeja et al. (11). Figure 4 shows the results of calculations with use
of the model but the zero activity of CK and AK. This limited very much
the workload and corresponding
O2 values that could be
achieved by the heart: both stayed at only 20% of their maximally
possible values. These values can be compared with those reported in
three types of experiments: CK inhibition by iodoacetate (15), PCr
replacement by feeding rats guanidinopropionate (39), or total knockout
of CK (CK-deficient mice) (33). In all cases, the recorded maximal
O2 was 35-50 µmol · g dry
wt
1 · min
1
and was significantly lower than the control values in normal hearts
(if those were correctly studied at sufficiently high workloads). These
experimental results are close to our calculated
O2 of 41 µmol · g dry
wt
1 · min
1
(Fig. 4, open circle). Decreased work capacities of the skeletal muscles of CK-deficient mice have also been reported (36). Thus the
model predicts that although the animals can easily survive without the
MM isoform of CK or miCK activity, their work capacities are
significantly compromised (15, 33, 36, 39).
Nonequilibrium state of CK and ADP concentration changes: importance
of the intracellular diffusion rates.
Figure 10 shows the numerical results of
two- and zero-dimensional model calculations of the ADP concentration
in intact cardiomyocytes, in myofibril core at 0.2 µm of the M line,
where the activity of actomyosin ATPase is high (Fig. 1), and ADP
levels at the boundary between the cytoplasm and the mitochondria
during one cardiac contraction cycle. Calculations of these
concentrations shown in Fig. 10 were made with a two-dimensional model
for fixed energy fluxes equal to 100 µmol
O2 · g dry
wt
1 · min
1
by using the values of the diffusion coefficients characteristic of
water. The dotted line in Fig. 10A was obtained for the case of
infinitely fast diffusion of ADP (zero-dimensional model, see METHODS OF MODELING). In all cases, the ADP concentration
is changed very significantly within the contraction cycle, its peak
value exceeding about five times the resting (equilibrium) value of the
ADP (see above). Remarkably, the two- and zero-dimensional models give
close results, the maximal difference being only 10% of the peak value
(solid and dotted lines in Fig. 10A). This means that
significant variations of the ADP concentration in the cell are not
related to problems of its diffusion but are intrinsic to the
compartmentalized energy metabolism in the cell because of the rather
low total CK activity and the specific intracellular distribution of
CK. For the case of rapid diffusion of ADP (as in water), there is a
small gradient of ADP between the myofibrillar core and the boundary of
the myofibrillar space at the moment of its peak value (Fig.
10A).


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Fig. 10.
Time course of changes in ADP concentration in contracting cardiac
cells. A: ADP concentrations [maximal in core of
myofibrils (solid lines) and minimal at boundary between cytoplasm and
mitochondria (dashed lines)] were calculated for 1 contraction
cycle by use of normal diffusion coefficients (as in water), and
average ADP concentrations (dotted lines) were obtained for infinitely
fast diffusion in myofibrils [0-dimensional (0D) model].
B: same as A, but ADP levels were computed with 2-dimensional
(2D) model with diffusion coefficients reduced 10-fold compared with
those in water space.
|
|
Because there are some indications of a possible restriction of ADP
diffusion in muscle cells (3), we repeated these calculations with a
10-fold decrease in the ADP diffusion coefficient (Fig. 10B).
This increased the peak value of ADP concentration in the core of
myofibrils and decreased it significantly at the boundary of the
cytoplasm and the mitochondria; the ratio of ADP peak values was close
to 3. Thus restriction of ADP diffusion increases its concentration
gradients in the cells, as could be intuitively predicted.
The results in Fig. 10 were obtained for two points: for one point in
the core of myofibrils, at 0.2 µm from the M line, and for another
point at the mitochondrial surface. However, it was interesting to find
out what may happen in other parts of the cell at the moment when the
ADP concentration in the core of myofibrils is maximal. To answer this
question, we fixed the time when the ADP concentration in the core of
myofibrils reaches its maximal value, i.e., 40 ms, and analyzed the
concentration gradients of ADP in the cell by using the two-dimensional
model of compartmentalized energy transfer. These results are shown in
Fig. 11. Figure 11A shows the ADP
concentration profiles for normal values of its diffusion coefficient.
The ADP concentrations at this moment in time are not equal in the cell
and show concentration gradients. ADP concentration is maximal in the
core of the sarcomere (0.2 µm at x-axis), where its value is
0.4 mM (Figs. 1 and 10) and decreases in the direction of the
x- and y-axes, corresponding to directions to
mitochondria and to the Z line. Solid and dashed lines in Fig. 10A correspond to maximal and minimal ADP concentrations in
Fig. 11A. Figure 11B shows the contour plots of the ADP
concentration in the myofibrillar space of cardiomyocytes for the case
of restricted diffusion of ADP, when its diffusion coefficient was
decreased 10-fold compared with its value in water. In accordance with
the data from Fig. 10B, in this case, one observes very
remarkable ADP concentration gradients within the cell.


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Fig. 11.
ADP distribution in 2 dimensions over myofibril and myoplasm in
systole. Contour lines, concentrations (µM) at 40 ms after onset of
contraction; arrows, direction and velocity of ADP flux. ADP
concentrations are computed by 2-dimensional model with normal (water)
diffusion coefficients (A) and with reduced (10 times less than
normal) diffusion coefficients (B).
|
|
However, what happens with metabolites other than ADP that take part in
the energy metabolism of the cells? The results of calculations, by use
of the different reaction-diffusion models, of the ADP, ATP, Cr, PCr,
and Pi concentrations in the core of myofibrils during the
contraction cycle are shown in Fig. 12.
The concentration changes within the contraction cycle are very
significant only for ADP (~5 times), whereas at the same time, ATP
concentration decreases only by 4%, Cr concentration increases 4%,
and correspondingly PCr concentration decreases by 4% (Fig. 12).
However, significant changes, by up to 25%, are seen for
Pi concentration (Fig. 12) because of its low initial
value. Assumption of an infinite rate of diffusion (zero-dimensional
model) modifies only very slightly, within a range of 1%, these
metabolite concentrations. Change of the dimension of the model has
some influence only on the ADP concentration profile and has little
influence on the others (Fig. 12). This means that the rate of
intracellular diffusion of metabolites characteristic of the
intracellular bulk water phase is sufficiently rapid for equilibration
of the metabolites in the intracellular space, and further increase of
diffusion coefficients does not alter significantly the metabolite
profiles in the cells.

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Fig. 12.
Dynamics of ADP, ATP, Cr, PCr, and Pi concentrations within
cardiac cycle at myoplasmic side of mitochondrial outer membrane (at
x = 0.5, y = 1.2 in Figs. 1 and 9) obtained from models
with different spatial dimensions and diffusion coefficients. Solutions
corresponding to 2-, 1-, and 0-dimensional models with normal (water)
diffusion coefficients and 2- and 1-dimensional models with reduced (10 times less than normal) diffusion coefficients are presented.
|
|
Another question, however, relates to the state of the CK system.
Significant variations in the myoplasmic ADP concentration within the
contraction cycle shown in Figs. 10-13
demonstrate that the system is out of equilibrium. Figure 13 shows the
two-dimensional model calculations of the metabolite profiles within a
cardiac cycle for normal cellular activities of the compartmentalized CK in the heart and for the case when the CK activity was increased manyfold to ensure its equilibrium state. In the equilibrium, as
expected, the levels of metabolites, including ADP, are practically constant, but in the real situation in the cell, the ADP concentration is close to its equilibrium only in the diastolic phase, its peak value
exceeding the equilibrium five- to sixfold. There are no big
differences between the real and equilibrium concentrations of Cr, PCr,
and Pi; all these parameters are relatively stable, and the
difference between real and equilibrium values is <10%. The
concentration of ATP is constant in equilibrium and varies in the real
situation
4.2%.

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Fig. 13.
Dynamics of ADP, ATP, Cr, PCr, and Pi concentrations within
cardiac cycle at myoplasmic side of mitochondrial outer membrane (at
x = 0.5, y = 1.2 in Figs. 1 and 9) obtained by
2-dimensional models for increased (105 times compared with
normal) MM CK activity (CK in equilibrium) and increased permeability
of mitochondrial outer membrane (no outer membrane). For comparison,
solutions of unmodified (normal) 2-dimensional model are also given.
MM, MM isoform of CK.
|
|
There is, however, some influence of the permeability of the outer
mitochondrial membrane for adenine nucleotides on the profiles of
metabolites. The peak values of ADP are decreased by 20% after the
outer mitochondrial membrane is "opened" (Fig. 13), and,
correspondingly, the higher levels of PCr and lower levels of phosphate
and free Cr are established (Fig. 13) within a cytoplasmic space.
 |
DISCUSSION |
The results of this study show that the linear dependence of
O2 on workload of working
cardiac muscle can be quantitatively described by the
reaction-diffusion model of compartmentalized intracellular energy
transfer, which is based on experimentally measured enzyme activities
and accounts for intracellular localization of coupled CK. The model
reproduces and explains the constant PCr-to-ATP ratios in a rather wide
range of workloads. It shows that ATP, Cr, and PCr concentrations and
their changes within the contraction cycle, calculated by using
different modifications of the model, do not differ very much from
those obtained on the basis of the classical assumption of CK
equilibrium. However, the calculated myoplasmic ADP concentration shows
very significant, workload-dependent variations within the cardiac
cycle, and in the systolic phase its level may exceed its equilibrium
value by an order of magnitude. In the metabolic research, the values of ATP, PCr, and Cr are the parameters determined experimentally (2,
12, 15, 17, 35, 38, 39). The results of this study show that
interpretation of these data, i.e., calculation of ADP values,
crucially depends on assumptions and models used. The validity of our
new model is confirmed by quantitative description of experimental data
for the whole physiological range of
O2 and, thus, energy fluxes
in the heart cells in vivo. An analysis of the model shows that
increasing infinitely the value of the diffusion constants does not
alter significantly the character of changes of cytoplasmic
concentrations of the metabolites within the cardiac cycle (Fig. 4).
This means that in the bulk water phase the diffusion is already rapid
and all calculated metabolite profiles in the given steady state are
formed relative to the levels of the compartmentalized enzyme
activities characteristic of the given cell. Because similar levels of
ATP, Cr, and PCr are calculated by the equilibrium and
compartmentalized energy transfer models, one may say that these
metabolites have quasi-equilibrium steady-state levels (9). The basis
of this phenomenon, according to our simulations, is that the diffusion
of these compounds in the bulk water phase of the cytoplasm is
sufficiently rapid and that, in the diastolic phase of the contraction
cycle, the CK system in the myofibrillar space approaches an
equilibrium. This allows one to use the system of ordinary differential
equations (zero-dimensional model) in the place of the system of
partial differential equations (two-dimensional model) to describe
qualitatively these parameters of intracellular energy metabolism in
the steady state. However, this conclusion of the small role of
diffusion in the myoplasm in determining the metabolic profiles is
based on the assumption that diffusion in the myoplasm may be modeled as a diffusion in the bulk water phase (26). Increasing experimental evidence shows that this may not be totally true in the living cells,
where diffusion may be influenced by such factors as intracellular structures and the bound water structure (28; see below). However, the
conclusion of the quasi-equilibrium is clearly not true for ADP
concentration in the myoplasmic space. The level of ADP is lower than
that of other metabolites by factor of 100, and its value changes
manyfold in the contraction cycle with formation of the concentration
gradients in the myofibrillar space revealed by two-dimensional
modeling. Because the total content of adenine nucleotides is conserved
in the model, ADP increases at the expense of ATP, but because of the
100-fold difference in initial levels, a decrease in ATP of only
3-4% is enough to change ADP by an order of magnitude. Our
calculations of the diffusion profiles of ADP (Fig. 10) show that,
because of its low concentration, ADP diffusion is not sufficiently
rapid to equilibrate its concentration in the cells: there is a
difference of 13% in ADP peak concentration between the core of
myofibrils and the mitochondrial-myoplasmic border even for normal
(characteristic for water) diffusion coefficients. This idea was first
advocated by Kammermeier (20). Restriction of ADP diffusion results in
formation of its very significant concentration gradients (Figs. 10 and
11). In this case, local ADP concentrations in myofibrils may reach
very significant values (11).
Thus, in the nonresting state, where the energy fluxes are significant,
the ADP concentration is clearly dissociated from quasi-equilibrium
concentrations of other metabolites. It depends on the value of the
energy flux in the system and on many other parameters (1).
Such a dissociation of ADP levels from levels of other metabolites that
are close to the levels predicted by the CK equilibrium reopens the
question of the importance of ADP in regulation of the rate of
mitochondrial respiration. In the classical experiments of Balaban et
al. (2), the increased
O2
values in hearts in vivo were observed at a constant PCr-to-ATP ratio.
Very similar data were obtained in many other laboratories for heart
and skeletal muscle (18, 35). The common conclusion from all these
works was that there is no metabolic regulation of mitochondrial
respiration by ADP in vivo, and a search for other alternative
mechanisms, in particular by calcium ions, began. Our studies show that
this conclusion, based on the assumption of CK equilibrium, is probably not valid because of the compartmentalization of the energy transfer system in the cell. In the compartmentalized CK system of the heart,
the activities of different CK isozymes are very well fitted to
generate the oscillations of ADP as a part of the metabolic feedback
signal between contraction and energy production at the background of a
constant level of other metabolites. These oscillations may be strongly
amplified by the coupled reactions of aerobic PCr production in
mitochondria (1). Furthermore, these oscillations of ADP concentration
may be synchronized with localized changes of calcium in cytoplasm and
in mitochondria, the latter modifying the activity of the enzymes of
mitochondrial systems to increase the rate of ADP rephosphorylation
(16), which in mitochondrial coupled reactions results in enhanced PCr
production (31, 32). This explanation is in accord with the concept of
parallel activation of energy-producing and -consuming processes
proposed recently by Korzeniewski (21) also on the basis of the
quantitative analysis. However, Korzeniewski and Froncisz (21, 22)
completely omitted the CK reaction and, thus, the possible feedback
metabolic regulation from their considerations. On the contrary, our
calculations have been made for the constant maximal activity of
mitochondrial oxidative phosphorylation. In fact, the phenomenon of
parallel activation and any possible effects of calcium on respiration
are ignored in our model. Without any need to account for these
effects, our model explains quantitatively the workload dependencies of
O2 and remarkable metabolic
stability, i.e., the unchanged levels of PCr for a significant range of
work transitions (2, 38). However, the effects of calcium may be
related to a change in the maximal velocity of respiration (thus
activation of the respiration), but not to the problems of feedback
regulation of mitochondrial oxidative phosphorylation in
contracting cardiac cells. The real value of the rate of respiration
is, according to our calculations, determined by the metabolic signal
from cytoplasm. This is in accord with recent data of Brandes and Bers
(4). This signal, in the form of changing ADP and Pi
concentrations, amplified in the coupled CK reaction (32), determines
the fraction of maximal reaction velocity used in any metabolic
conditions. Thus, taken together, these two theories may explain the
feedback regulation of the respiration under conditions of metabolic
stability by synchronized oscillations of the calcium and ADP
concentrations. These concentration changes may well be localized in
the intracellular microcompartments. As shown recently by Rizzuto et
al. (29), the calcium concentration changes may be very significantly
localized in the mitochondrial and surrounding microcompartments
because of close contacts between mitochondria and sarcoplasmic
reticulum. Our results and conclusions are also in accord with the
recent discovery of subcellular metabolic transients and mitochondrial redox waves in cardiomyocytes by O'Rourke et al. (27) and Romashko et
al. (30). Using confocal imaging of flavoprotein redox potential and
mitochondrial membrane potential, they showed that substrate deprivation leads to subcellular heterogeneity of mitochondrial energization in the cell and propagation of a redox wave at ~2 µm/s, which is comparable to the average diffusion rate of small molecules such as ADP, if the diffusion coefficient is decreased by a
factor of 5 compared with that for water (30). The authors concluded
that intracellular control of mitochondrial function involves
diffusible cytoplasmic messengers. The role of calcium as a mediator
was excluded, since the redox oscillations were independent of calcium
concentration (30). However, rapid release of ADP by flash photolysis
of intracellular caged ADP induced metabolic oscillations (27). It has
been proposed that cytoplasmic structures may control mitochondria and
that there exists an organized network of energy transfer and feedback
signal transduction in the cells (11, 32) by a mechanism of
"vectorial ligand conduction through organized cytoplasmic
multienzyme systems" [a definition proposed by Mitchell
(24)], including CK and AK and probably glycolytic and other
systems. In this system, free diffusion of ADP is replaced by its
vectorial conduction, along with other ligands, by enzymes, probably
organized in association with the cytoskeleton. In fact, free diffusion
of ADP itself seems to be very limited in the muscle cells. Multiple
experiments carried out by Ventura-Clapier and by Bessman et al. (see
Ref. 32 for references) on the skinned muscle fibers showed that
relaxation from the rigor state was more effective by an order of
magnitude by PCr because of local rephosphorylation of ADP than by
externally added ATP. This shows that, in the absence of PCr, ADP is
accumulating in the vicinity of the myosin-active centers and slows the
cross-bridge detachment, and diffusion of ATP into fibers is not as
effective as diffusion of PCr. Bereiter-Hahn and Voth (3) described a very interesting phenomenon of delayed morphological response of
mitochondria to ATP or ADP microinjected into living cells, indicating
that diffusion of these molecules is ~10 times slower than that of
other small hydrophilic molecules. This kind of structural organization
of the cytoplasmic space in the cells in vivo is not yet included in
our model. Obviously, this kind of organization localizes the changes
of the ADP concentration in the cytoplasmic space (ADP "sparks"),
which may propagate as a metabolic wave, giving rise to oscillations of
other metabolites (ligands) (8, 29, 32). Clearly, the next very
interesting step is to include this kind of organized and
microcompartmentalized vectorial process in our model of energy
transfer and feedback regulation of respiration. This may help in
analysis of the recently observed metabolic transients and redox waves
in cardiac cells (27, 30).
In conclusion, the new version of the mathematical model of
compartmentalized energy transfer in cardiac cells describes
quantitatively the in vitro data on isolated mitochondria and the in
vivo metabolic data on working perfused rat heart. It describes
1) dependence of the rate of PCr production in the coupled CK
reaction on oxidative phosphorylation in vitro, 2) linear
dependence of
O2 by perfused heart on the workload, 3) reduction of the maximal workload of perfused heart after inhibition of CK, and 4) experimentally
observed metabolic stability (constant PCr-to-ATP ratio) of heart cells at different workloads. That means that a feedback mechanism can regulate the rate of oxidative phosphorylation in a wide range of
workloads and keep the levels of metabolites, i.e., Cr, PCr, and ATP,
at metastable steady-state levels and, thus, can explain the apparent
controversy widely discussed in the literature (2, 4, 15-23, 30,
32-35, 38). The model shows that the amplitude of ADP oscillations
due to the nonequilibrium state of the CK system increases in cytoplasm
with a decrease in the diffusion coefficient and that ADP gradients are
larger along than across the myofibrils. The latter conclusions point
to the potential importance of developing independent experimental
methods for determination of the ADP concentrations in different
cellular compartments, which is still a challenge in cellular bioenergetics.
However, to evaluate quantitatively the contribution of changes in
cytoplasmic ADP and Pi concentrations to regulation of the
rate of mitochondrial respiration in vivo, further analysis of the
model is required by using approaches similar to those used in
metabolic control analysis.
 |
APPENDIX |
Here we describe the mathematical models. Depending on dimensions, we
call these two- and zero-dimensional models. The two-dimensional model
is a continuous version of the discrete one-dimensional model (1)
obtained by adding the longitudinal dimension. This is a system of
partial differential equations involving two spatial coordinates and
time. With the assumption of spatial homogeneity of the metabolites
within the myocyte, i.e., infinitely fast diffusion, the
two-dimensional model is simplified to a system of ordinary differential equations involving only time dependence, hence, the name
zero-dimensional model. The values of the parameters used in the models
are listed in Table 1.
Two-dimensional model.
The dynamics of the metabolites in the myofibril and myoplasm are
described by the following system of reaction-diffusion equations
|
(1)
|
|
(2)
|
|
(3)
|
|
(4)
|
|
(5)
|
|
(6)
|
where
ATP, ADP, AMP, PCr, Cr, and Pi are the total concentrations
of the metabolites in the myofibril and myoplasm,
DMet is the diffusion coefficient of the metabolite
(Met, i.e., ATP, ADP, AMP, PCr, Cr, Pi),
vCK is the CK reaction rate,
vAK is the AK reaction rate, and
HATP is the rate of ATP hydrolysis in the myofibril. The factors GCK,
GAK, and GH describe the
spatial inhomogeneities of myofibrillar CK, AK, and ATPase,
respectively. The symbol
2 denotes the Laplace operator.
The vCK is described by Eq. 7
|
(7)
|
where
|
(8)
|
and
|
(9)
|
where
mATP and mADP are the concentrations of the Mg2+-bound
forms of ATP and ADP, respectively, and fATP and fADP are the Mg2+-free forms. The total concentrations of adenylates and
their Mg2+-bound forms are related as follows
|
(10)
|
|
(11)
|
|
(12)
|
|
(13)
|
where
Mg is the level of free Mg2+ and KDT
and KDD are Mg2+ dissociation
constants. The level of free Mg2+ and its dissociation
constant for the matrix differ from their corresponding values for the
rest of the cell (22). The vAK is described as
follows
|
(14)
|
where
kfa and kba are the forward and
backward reaction rate constants for the AK reaction.
We approximate the hydrolysis rate by a piecewise linear time-periodic
function HATP. The rate increases from 0 to
HATP(max) during the first 30 ms, decreases to 0 during the next 30 ms, and remains 0 until the end of the cardiac cycle
at 180 ms
|
(15)
|
where
t is elapsed time from the beginning of the cycle (in ms).
The reaction rates of CK, AK, and ATPase are proportional to the
concentrations of the corresponding enzymes. The spatial distribution
of CK (37) is described by
|
(16)
|
the distribution of AK by
|
(17)
|
and
the distribution of ATPase (5) by
|
(18)
|
where x and y are spatial coordinates (in
µm): x corresponds to the longitudinal direction (increases
from the M line to the Z line), and y corresponds to the
transverse direction (increases from the core of myofibrils to the
mitochondrion); the point with coordinates (0,0) is located at the core
of myofibrils and the M line. The mean values of
GCK and GAK over the myofibril
and myoplasm (0
x
1, 0
y
1.2) and
GH over the myoplasm (0
x
1, 0
y
1) are equal to unity.
Because of the symmetry of the problem, we use no flux condition at the
boundaries x = 0, x = 1, and y = 0. The
restricted diffusion through the mitochondrial outer membrane takes
place at the boundary y = 1.2 and is calculated by
|
(19)
|
|
(20)
|
|
(21)
|
|
(22)
|
|
(23)
|
|
(24)
|
where
Meti is the concentration of the metabolite in the
mitochondrial IM space and RMet is the permeability
of the mitochondrial outer membrane.
The concentrations of the metabolites in the IM space are changed
because of the diffusion through the outer membrane, CK and AK
reactions, ATP/ADP translocation, and Pi transportation through the inner membrane according to the following
equations
|
(25)
|
|
(26)
|
|
(27)
|
|
(28)
|
|
(29)
|
|
(30)
|
where
vMiCK is MiCK reaction rate,
vANT is net rate of ATP export by ANT from the
matrix, vAKmit is AK reaction rate (see Eq. 14), vPi is
rate of the phosphate carrier in the inner membrane, Li is width of the layer between the inner and
outer membranes, and FVi is fractional volume of the IM
space with respect to the cell volume.
The MiCK reaction rate (vMiCK) is divided into
vMiCK,G, which involves microcompartment ATP(ADP),
and vMiCK,I, which involves ATP(ADP), from the IM
space
|
(31)
|
The
reaction rates are as follows
|
(32)
|
|
(33)
|
where
|
(34)
|
Restricted diffusion between the microcompartment and the IM space is
governed by
|
(35)
|
for
ATP and by
|
(36)
|
for ADP.
For the oxidative phosphorylation, we use the model by Korzeniewski
(21)
|
(37)
|
|
(38)
|
|
(39)
|
|
(40)
|
|
(41)
|
|
(42)
|
where
UQ is the oxidized form of coenzyme Q, c3+ is the oxidized
form of cytochrome c, NAD+ is NAD+ in
the matrix, Hx, ATPx, and Pi,x are
H+, ATP, and Pi in the matrix, and
FVx is fractional volume of the mitochondrial matrix with
respect to the cell volume. The following reaction rates are used in
the above equations
(21)
|
(43)
|
for
substrate dehydrogenation (dh)
|
(44)
|
for
complex I (C1)
|
(45)
|
for
complex III (C3)
|
(46)
|
for
complex IV (O2)
|
(47)
|
for
ATP synthase (sn)
|
(48)
|
for
phosphate carrier (PI),
and
|
(49)
|
for
proton leak (leak). In Eqs. 43-49, we used the total (tot)
metabolite pools for substrate (NAD), coenzyme Q (UQ), cytochrome c
(c), cytochrome a3 (a), and adenine nucleotides
(in matrix, Ax) as follows
|
(50)
|
|
(51)
|
|
(52)
|
|
(53)
|
|
(54)
|
The
concentrations of Mg2+-bound and Mg2+-free
forms of adenine nucleotides in the matrix are obtained using Eqs.
10-13. For computing pH in the matrix (pHx), pH
difference across the inner membrane (
pH), protonmotive force
(
p), electrical potential (
), and buffering rate
(rbuff), the following equations were used
|
(55)
|
|
(56)
|
|
(57)
|
|
(58)
|
|
(59)
|
|
(60)
|
where
the coefficient u = 
/
p is kept constant.
Because of the high permeability of the mitochondrial outer membrane
for protons, pH in IM space (pHext) is considered to be
equal to that in the cytoplasm and is considered to be constant. The
coefficient Z is used to convert between molar concentrations and the
concentrations used in the model (µM). The coeffiient Z is
expressed as
|
(61)
|
where
R is the gas constant, T is the absolute temperature,
and F is Faraday's number. Throughout this study, the
notations ln and log represent logarithms on bases e = 2.718,... and 10, respectively.
Description of ANT kinetics is based on the phenomenological model by
Korzeniewski (21). The net rate of ATP export by ANT from the matrix to
the microcompartment and the IM space is
|
(62)
|
where
vX
GANT is the
rate of ATP export by ANT from the matrix to the microcompartment and
vX
IANT is the
rate of ATP export by ANT from the matrix directly to the IM space.
These rates are described by the following kinetic
equations
|
(63)
|
|
(64)
|
The capacity of the
microcompartment is considered to be infinitesimal. Therefore, for ATP
and ADP, the influx of the metabolite is balanced by its efflux
|
(65)
|
|
(66)
|
Equations
65 and 66 can be solved for ATPg and
ADPg by using, for example, the Newton-Raphson method.
The thermodynamic span of ATP synthase is calculated as follows
|
(67)
|
where
na and
G0 are the
H+/ATP stoichiometry and standard free energy for ATP
synthase, respectively.
In Eqs. 44 and 45, the NAD, coenzyme Q, and cytochrome
c redox potentials are used
|
(68)
|
|
(69)
|
|
(70)
|
respectively.
For computing the concentration of reduced cytochrome
a3 used in Eq. 46, the redox potential of
cytochrome a3
|
(71)
|
is
substituted into the relation
|
(72)
|
which
gives the result (together with Eq. 53).
With the assumption of infinitely fast diffusion, the two-dimensional
model can be simplified to the zero-dimensional model. In this model
the concentrations of the metabolites are spatially homogeneous and all
the partial differential equations can be reduced to ordinary
differential equations, hence, the name. The zero-dimensional model is
obtained by setting
|
(73)
|
in
Eqs. 1-6, integrating them over the region of myofibril
and myoplasm, and dividing by the area of this region. The integration of space-dependent functions GCK,
GAK, and GH leads us to
|
(74)
|
|
(75)
|
and
|
(76)
|
Zero-dimensional model.
The concentrations of the free metabolites in cardiac myocytes are
described by the following equations
|
(77)
|
|
(78)
|
|
(79)
|
|
(80)
|
|
(81)
|
|
(82)
|
where
Lm is the distance between the core of the
myofibril and the mitochondrial membrane. In addition to Eqs.
77-82, the complete zero-dimensional model includes Eqs.
25-30 and 37-42.
 |
ACKNOWLEDGEMENTS |
The authors thank Prof. Juri Engelbrecht (Tallinn, Estonia) for
continuous support of this work, Drs. Bernard Korzeniewski (Krakow, Poland) and Maiys Aliev (Moscow, Russia) for very
instructive discussions, and Dr. Bernard Korzeniewski for providing the
computer program of his model. We have always greatly appreciated our
most interesting and fruitful discussions with the late Prof. John R. Williamson (Johnson Foundation, Univ. of Pennsylvania, Philadelphia, PA), on the problems of modeling heart metabolism, for which he supplied the most reliable experimental data.
 |
FOOTNOTES |
This work was supported by Estonian Science Foundation Grant 3204.
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. §1734 solely to indicate this fact.
Address for reprint requests and other correspondence: V. Saks,
Laboratory of Bioenergetics, Joseph Fourier University, BP 53X-38 041 Grenoble Cedex, France (E-mail:
Valdur.Saks{at}ujf-grenoble.fr).
Received 21 July 1999; accepted in final form 10 November 1999.
 |
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