A simple model of aerobic metabolism: applications to work transitions in muscle.

A simple model of aerobic metabolism: applications to work transitions in muscle CATHERINE

I. FUNK,

ALFRED

CLARK,

Departments of Mechanical Engineering Rochester, New York 14627

JR., AND

and Physiology,

RICHARD

University

J. CONNETT

of Rochester,

FUNK, CATHERINE I., ALFRED CLARK, JR., AND RICHARD J. CONNETT. A simple model of aerobic metabolism: applications to work transitions in muscle. Am. 3. Physiol. 258 (Cell Physiol.

energy-related species, namely the adenine nucleotides (ATP, ADP, AMP), phosphocreatine, creatine, and Pi; 2) the enzymes creatine kinase and adenylate kinase; 27): C995-C1005, 1990.-Adding kinetics to the model of the and 3) the interactions with pH, Mg+, and K+ (7, 23). phosphate energy system [Connett. Am. J. Physiol. 254 (RegIn our extension of this model, we describe the structure ulatory Integrative Comp. Physiol. 23): R949-R959, 19881, we of the phosphate energy system in terms of chemical provide a framework for analyzing metabolic transients in muscle tissue. We modify the formalism of the earlier model parameters, cell environment parameters (which include is and introduce a buffering factor, which measures buffering of pH), and state variables. Once the cell environment defined, the equilibrium relations determine the phosadenine nucleotides by phosphocreatine. The time course of phate energy state from a knowledge of any one of the the phosphate energy state can be calculated given the following: I) adenosinetriphosphatase (ATPase) rate, 2) pH, and 3) concentrations; in other words, the system has one degree a mitochondrial driving function, i.e., ATP production in terms of freedom. We choose to analyze the system in terms of of the phosphate energy state. We use mitochondrial driving the creatine charge, defined as the ratio of phosphocreafunctions derived from steady-state measurements to predict tine to total creatine concentration. We also define a the time courses for rest-work transitions. Predictions for tranbuffering factor, which measures the buffering of adenine sitions in the rat gastrocnemius muscle agree with published nucleotides by phosphocreatine and plays an important values. The model is used to test different existing hypotheses role in the transient analysis. of oxygen consumption (VO,) regulation. Each hypothesis genThe phosphate potential energy, or pool of available erates a specific mitochondrial driving function, which in turn generates a specific time course of phosphate energy state high-energy phosphate bonds, varies with time in reduring transitions. A mitochondrial driving function based on sponse to any change in ATP demand and/or supply. phosphate demand is determined by the enzyme kinetics with ADP as a substrate leads to time courses High-energy total adenosinetriphosphate (ATPase) rate. High-energy not matching the data. Mitochondrial driving functions that are linear with phosphocreatine, Pi, phosphorylation potential, phosphate supply is, under aerobic conditions, dominated or the pool of high-energy phosphate bonds (phosphate potenby oxidative phosphorylation and thus related to the rate tial energy) gave good agreement with the data. of 00~. We take the point of view that VOW is some function of the phosphate energy state of the cell, and phosphate energy system; metabolic regulation; creatine charge; we call this function the mitochondrial driving function. adenosine 53riphosphate; oxygen consumption; phosphocreaMuch of the present work is concerned with exploring tine; inorganic phosphate; skeletal muscle the consequences of different choices for this function. With this formulation, a single differential equation describes transients in the system. Given the chemical AFTER STEP CHANGES IN WORK INTENSITIES of skeletal parameters, the cell environment parameters, and the muscle, both phosphocreatine concentration and oxygen mitochondrial driving function scaled by steady-state consumption rate approach a new steady state along an data for skeletal muscle, the steady-state values of conapproximately exponential time course (e.g., Refs. 8, 20, centrations and time courses during any transition in 24, 30). We have extended an existing model of the energy turnover rate can be predicted. We use the model phosphate energy system (7), thereby obtaining a kinetic with a mitochondrial driving function derived by Meyer model of cell energetics to analyze this process. This (24) from steady-state data (16, 24) for rat gastrocnemodel is used to simulate several rest-work transitions, mius. Three in vivo rest-to-work transitions of the rat and exponential behavior is obtained. The model gives gastrocnemius are simulated. The results are tested an explicit expression for the exponential time scale in against data from Meyer (24), and the observed expoterms of the fundamental parameters of the muscle and nential time courses of VOW and phosphocreatine level steady-state data relating oxygen consumption (VO& and are obtained, validating our model. In these simulation the phosphate energy state. Moreover, our simulations cases, the model is quite similar to the electrical analog fit experimental measurements of rest-work transitions model used by Meyer (24). (using data from Refs. 8, 24). Different forms for the mitochondrial driving function Central to our model is the existing model of the correspond to different biochemical hypotheses describphosphate energy system (7, 23), which includes 1) the ing the regulation of v02. We show that for the same cell 0363-6143/90 $1.50 Copyright 0 1990 the American Physiological

Society

c995

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C996

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environment, different choices for this function give different time courses for Vo2 and the creatine charge during a rest-work transition. Thus measurements of Vo2 during a transition allow a discrimination among the choices. In this way, the VOW data give some insight into in vivo control of mitochondrial function. This is illustrated using simulations based on data for the dog gracilis muscle (8). FORMULATION

OF THE MODEL

Essential features of the phosphate energy system. We use the term phosphate energy state to designate the concentrations of the six energy state variables, namely the concentrations of ATP, ADP, AMP, phosphocreatine, creatine, and Pi, which we denote by [ATP], [ADP], [AMP], [PCr], [Cr], and [Pi], respectively. For the type of simulations considered here, specifically restwork transitions (with time constants on the order of lo-100 s), we can show that cytosolic concentrations are very homogeneous and that variations across the cell associated with diffusive transport are small compared with the mean values. Therefore, we may, with negligible error, analyze the cell energetics in terms of spatially averaged concentrations (see APPENDIX B for justification). In the cell in vivo, the interacting species constitute a system, which we call the phosphate energy system. For a detailed description, see Ref. 7. A summary of notation and equations is given in APPENDIX A. At this point, we summarize the main features of the system, since they are central to our model. The description of the phosphate energy system is based on the equilibrium of the creatine kinase reaction (2, 11, 22, 26) MgADP-

+ PCr2- + H+ + MgATP2-

+ Cr

and of the adenylate kinase reaction (23) ADP3- + MgADP-

s AMP2- + MgATP2-

Cation and hydrogen ion binding to adenine nucleotides are incorporated (23,32). Values for all equilibrium constants can be found in Table 1 of Ref. 7. They constitute the chemical parameters of the system. We now define cell environment parameters. There are three important parameters characteristic of the tissue: the free adenine nucleotide pool ([AdT]) = [ATP] + [ADP] + [AMP], the total creatine pool ([CrT]) = [Per] + [Cr], and the reactive phosphate pool ([PT]) = [PCr] + 3[ATP] + 2[ADP] + [AMP] + [Pi]. The creatine pool has been shown to remain very constant in skeletal muscle (31). All pools have been shown to be constant over the normal range of work rates in a red skeletal muscle (9). Under extreme conditions (e.g., ischemic work or maximal work), the free adenine nucleotide pool and the phosphate pool may change. We are not attempting to describe these extreme states here, and we therefore hold the three pools constant during our simulations. The numerical values depend on the species and the muscle being studied. Adenine nucleotides are found mainly in association with potassium, magnesium, and hydrogen ions in the cell. Therefore, the concentrations of these ions also

FIG. 1. Phosphate energy system. [ATP], [ADP], [AMP], [Pi], and phosphate potential energy [-PI as a function of the creatine charge (PCr) = [PCr]/[CrT]. [CrT] = 37.1 mM, [AdT] = 6.68 mM, and [PT] = 48.2 mM, pH 7.0. Inset is expanded scale. See text for definitions.

must be specified as part of the cell environment. We can assume that the potassium concentration is constant at 0.1 M in the cells (7). Although precise information is lacking, it is reasonable to take the magnesium concentration to be constant at 1 mM (5). Finally, we need to know the intracellular pH, which is usually in the range from 6.5 to 7.5 (5, 25, 26). In a more elaborate model, it might be possible to calculate pH as a function of the energy state of the cells, but this would require including all reactions which affect pH. In the present stage of the model, pH is treated as one of the inputs that define the cell environment. Once the chemical parameters and the cell environment are defined, the phosphate energy state is determined by knowing how the high-energy phosphate bonds are distributed within the different pools. For example, we need to know what fraction of the free adenine pool is ATP or what fraction of the total creatine pool is phosphorylated. The system, as it is presented here, has one degree of freedom. This means that the value of all concentrations can be determined by specifying any one of the phosphate energy state variables, or any particular combination of some variables. To characterize the phosphate energy state, we choose a convenient variable, namely the creatine charge (fPCr)), which is defined as the ratio of the phosphocreatine to the total creatine concentration, {PCr) = [PCr]/[CrT]. The expressions for all concentrations in terms of {PCr) are given in Eq. AI of APPENDIX A. Figure 1 illustrates the dependence of all concentrations on the creatine charge, where we have used data for the dog gracilis muscle to define the cell environment. One can see that the ATP, ADP, and AMP concentrations remain constant down to very small values of the creatine charge. In fact, an important effect of the presence of phosphocreatine in the cells is the regulation of adenine nucleotide concentrations (7,12). To describe this effect quantitatively, we define a buffering factor (fB) as follows d[PCr] + [ADP])

f B = d(2[ATP]


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OF AEROBIC

FIG. 2. Log of the buffering factor (f~) as a function of the creatine charge (PCr) = [PCr]/[CrT], for [CrT] = 37.1 mM, [AdT] = 6.68 mM, and [PT] = 48.2 mM, pH 7.0. See text.

This choice for the definition of fB proves to be useful in the formulation of the transient model (see Transient model below). The factor fB is one measure of how effectively phosphocreatine buffers adenine nucleotide concentrations. Because the phosphate energy system has one degree of freedom, we can express fB in terms of the creatine charge (see APPENDIX A, Eq. A2). In addition to the creatine charge, fB depends on equilibrium constants and on the ratio [CrT]/[AdT]. Because [CrT]/[AdT] is the same in all fiber types (7)) the buffering is an identical function of creatine charge in all muscle fibers. In the transient model, we will be looking at changes of the pool of available high-energy phosphate bonds (phosphate potential energy) in the tissue ([-PI = [PCr] + Z[ATP] + [ADP]). We can use fB to write d{PCrj

(1)

Figure 2 shows the value of fB as a function of the creatine charge for three different pH values in the physiological range. Buffering is extremely efficient (large fB) down to very low values of the creatine charge. Thus Eq. 1 can be approximated by d[-P] = [CrT]d{PCr). Transient model. The transient model is obtained by linking the phosphate energy system to two other subsystems of the cell: the ATPases, the sinks for ATP and the mitochondria, the sources for ATP. During transients in work rate, the pool of available high-energy phosphate bonds varies because of a temporary imbalance between ATP consumption and production, according to the equation

-d[-PI 1. I-IP - r dt

0

C

where Pp is the ATP production rate, and & is the ATP consumption rate. This equation can be derived by combining the relevant mass balances for the system. By using Eq. 1, we may write Eq. 2 as a differential equation for the transient variations of the creatine charge 9

=

rp - rc

(3)

METABOLISM

c997

This equation defines the basic structure of the model. We now examine each part of it in detail. The specific structure of the phosphate energy system enters the equation through the buffering factor and the total creatine pool. For all situations except the very highest work rates, fB is very large, and the term l/f~ can be neglected in the equation. rC represents ATP consumption by all the ATPases active in the cell. This includes the actomyosin splitting of ATP into ADP and Pi during muscle activity, as well as the Ca2+-ATPase at the sarcoplasmic reticulum. There are several other ATPases that are active even in the muscle at rest. All consumption of ATP is summed, and rC denotes the global ATPase rate per unit volume of the cell. Any process we wish to describe is specified by giving rc as a function of time. rp represents the production of ATP. During aerobic work, the mitochondria are the main sources for ATP, and the A.TP production rate rp is then equal to six times the VOW(assuming a normal P/O of 3). A number of hypotheses have been proposed for describing quantitatively the relationship between VOWand the energy state (e.g., 1,3,4,10,15,17-20,33,34). Some also include an effect of mitochondrial redox, independent of the phosphorylation state (15,18,34). All hypotheses predict a relation between VOWand some function of the phosphorylation state of the cytosol, whether expressed in terms of adenine nucleotides or creatine phosphate. Mitochondrial redox and oxygen are also substrates for the oxidative phosphorylation reaction and thus potentially important variables in the driving function. Under the submaximal conditions evaluated here, oxygen concentration is not limiting to VOW (13). It will therefore not be a variable in our analysis. If redox is stable, or related to the phosphorylation state, it also is not an independent variable in the system. For a first approximation, therefore, it is assumed in the model that Vo2 is a defined function of the phosphate energy state only i.e., the mitochondrial driving function. This function *can be determined from steady-state measurements of Vo2 and the phosphate energy state. The resulting expression for rp as a function of the phosphate energy state can then be used in Eq. 3 to predict transient behavior. The setup of Eq. 3 for each simulation requires three elements: 1) all parameters defining the phosphate energy system, including pH as a function of time; 2) a specification of PC an a function of time; and 3) an expression for rp as a function of the phosphate energy state, which can be obtained from steady-state measurements. Application of the transient model to rest-work transitions. During a transition in work rate, the ATP demand changes first, affecting the phosphate energy system, which then directs the change to the mitochondria. The VOW adjusts in response to the phosphate energy state until the ATP production rate matches the new ATP demand. The system is then at a new steady state. We simulate this process by solving Eq. 3 for the conditions specified below. The predictions of the model depend on the structure of the phosphate energy system, characterized by the


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chemical constants and the cell environment. In particular, for the resting cell, a pH of 7.0 is assumed. On the basis of available data (5, 14, 26), during a transition from rest to work, pH rises during the initial phase of the transition and then drops approximately exponentially. This is an important input of the model, and more data on the time course of pH during work transitions would be very useful. For a transition from rest to work, PC changes very rapidly from a very small number to a new larger value. The change in PC occurs on a much shorter time scale than changes in 00~. Therefore, in the model we can take this change in demand to be a step function connecting the given initial and final work rates. Assuming that Pp (or VO,) as a function of the phosphate energy state is known, we can use Eq. 3 to predict the time course of concentrations during a transition between different work rates. Thus this model provides a specific way to transform steady-state data (mitochondrial driving function) into transient data (time histories of concentrations). RESULTS

AND

DISCUSSION

METABOLISM 0.85 0.75

0.55

0

o*35

l

l

l

6a d( PCr) 7 + GP(PCr) = -

- rc

[Cfu

(4

The solution for (PCrJ is then PW

= (PCr)mp(-t/7)

+ {PC&J1

where (PCr),, = (6a! - PJ/(GP[CrT])

- exp(-t/7)]

(5)

is the steady-state

I

b

I l&O

1 240

. 360

. 480

Time

e&

I

Qs)

780

.

840

FIG. 3. Comparison of theory with data from Meyer (24). Time course of creatine charge (PCr) = [PCr]/[CrT] during 3 rest-work transitions to submaximal ATP consumption rates of rat gastrocnemius muscle followed by work-rest transitions (stimulation followed by recovery). For [CrT] = 34 pmol/g muscle and [AdT] = 7 gmol/g muscle, such that [AdT]/[CrT] = 0.2 (from Ref. 7). pH equals 7 during the transitions. Oxygen driving function: Voz = 3.432 - 0.1144 [PCr] (in pmol/g muscle-’ gmin-‘; Ref. 24). Resting [ PCrlo = 26.6 pmol/g muscle, ATP consumption rates were I) 0.088 prnol-g muscle-‘*rnin-’ (A); 2) 0.13 pm01 g muscle-l min-l (0); and 3) 0.18 pmol= g muscle-’ ,rnin-’ (*). See text. l

Testing the model against data (validation of model). The model was tested against data from Meyer (24) for the rat gastrocnemius muscle. The mitochondrial driving function 60, = 3.432- 0.1144 [PCr] (in pmo1.g muscle-l . min-l), derived by Meyer from steady-state in vivo data (16, 24) was used. Note that this function is linear with creatine charge. We used Meyer’s initial condition ([PCr],) = 26.6 pmol/g muscle. To generate the simulations, two additional parameters were specified, [AdT] and pH. Using the ratio [AdT]/[CrT] = 0.2 from Ref. 7 and Meyer’s (24) value of [CrT] = 34 pmol/g muscle, we obtained [AdT] = 7 pmol/g muscle. We assumed pH 7 during the transitions. Three different transitions from rest to submaximal work followed by recovery to resting state were then simulated, using Eq. 3. For each simulation, we assumed that PCwas constant and equal to the steady-state value. The values obtained from Meyer’s experiments were 1) 0.088 ~rnol g muscle-l .min+, 2) 0.13 ~rnol g muscle-l min-l, and 3) 0.18 pmol g muscle-1 min? We then solved Eq. 3 using a standard fourth-order Runge-Kutta routine. The three time courses for (PCr) obtained this way were compared with the time courses measured by Meyer (Fig. 3). The calculated and measured time courses for (PCr) agree. For submaximal work rates, the term 1lfB in Eq. 3 is small and can be neglected. Then, because the mitochondrial driving function measured by Meyer (24) is linear with (PCr), Eq. 3 is linear and can be solved analytically. More generally, if we can write voz = a - P[PCr] = a ,B[CrT] (PCr), we get from Eq. 3

1

l

creatine charge at the given turnover rate, and 7 = l/(6@). The time course for the phosphocreatine concentration 1s exponential with time constant 7. Usin .g the constant data, we calculated the P given in Meyer’s steady-state decay time as 1 ’ = ij =

1

= 1.46 min

6(0.1144)

Through a curve-fitting procedure applied to his 24 experiments, Meyer obtained an actual mean time constant of 1.44 min for the exponential variations of phosphocreatine concentration. Therefore, our model, which includes the phosphate energy system, ATP sinks, and ATP sources (regulation of Vop by creatine concentration), predicts time courses of concentrations in agreement with Meyer’s measurements and includes the known physiological chemistry. This is a validation of our model. It shows that the set of reactions included in the model is sufficiently complete for a description of rest-work transitions. If diffusion significantly delayed or reduced the transmission of the signals from myofibrils to mitochondria, or vice versa, our simulations could not have accounted for the data. In his study, Meyer (24) suggested a linear electrical circuit analog to describe the metabolic system. Our conclusions are similar, but our analysis is based on a nonlinear biochemical model, and the linear behavior of the model for the described rest-work transitions results from the large value of the buffering factor and from the linearity of Voz as a function of (PCr) exhibited by the steady-state data. The driving function derived from measurements by Meyer (24) is linear with phosphocreatine concentration. If this is the case for other muscles, our model provides a specific quantitative link between steady-state-data (@) and transient data (7). The time constant 7 has a specific


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OF

physiological meaning in the sense that it is directly related to the value of ,8 in the mitochondrial driving function obtained from steady-state data. Differences in decay time for the rest-work transitions of two different muscles correspond to differences in the values of ,& Therefore, fl is an important characteristic of a muscle, and an interesting question is whether we can relate ,8 to fundamental cell parameters. For example, p may be proportional to mitochondrial volume fraction. This is a topic for further investigation. Deviations from exponential time courses can occur, and they correspond to nonlinearities in the model. Such nonlinearities can occur only when 1) the buffering factor decreases significantly as the creatine charge drops to zero (very high work rates), 2) the muscles are artificially depleted of creatine, and 3) the mitochondrial driving function deviates from linearity in its dependence on the creatine charge. Different mitochondrial driving functions give different time courses. This can be used as a means to select between different mechanisms for the regulation of Tog, as discussed below. Using the model to study VOW regulation from measured transients. In the previous section, a linear function for vo2 in terms of phosphocreatine fit the steady-state data and was transformed, through the model, into time histories of concentrations that fit the transient data during a transition between work rates. In this section, we explore the sensitivity of predictions for transient behavior to the form of the mitochondrial driving function. We consider muscles and work rates for which oxygen is not limiting to $702 and for which redox changes are small or proportional to the phosphorylation state. This is true for example in dog gracilis muscle at submaximal work rates (29). Then any specific form of the mitochondrial driving function corresponds to some biochemical mechanism for the control of mitochondrial function. Many mechanisms have been suggested and discussed in studies performed on isolated mitochondria. Enzymatic models with ADP as a substrate (3,4, 17), or both ADP and Pi (33), have been suggested. Control mechanisms by [ATP]/[ADP] (10, 17, 19) or by the phosphorylation potential log ( [ATP4-]/[ADP3-] [HgPO:]) (15, 17) have been considered. Finally, a creatine shuttle mechanism, in which VO~ is controlled by phosphocreatine concentration, has also been suggested (20). Steady-state data have so far proven unable to completely identify the in vivo mechanism because such data can be fit, within experimental error, to several different hypotheses of biochemical mechanisms. The transient model developed here allows the use of transient measurements to investigate the different mechanisms and discriminate between them. It provides a theoretical basis for this discrimination. Because of the structure of the phosphate energy system, all quantities are functions of one variable (for example, the creatine charge). Therefore, any mechanism for VOQregulation by the phosphate energy system, whether it is driven by ADP, Pi, or other, can be translated into a specific relation between Voz and (PC& This permits testing of mitochondrial driving functions derived from biochemical studies. Each biochemical

AEROBIC

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CQQQ

mechanism for VOW regulation can be used in the model to predict a rest-work transition of the creatine charge and of vo2. The time courses can then be compared on the basis of 1) the shape of the transition, 2) the time scale for the transition, and 3) the initial and final steady-state values. If measurements of time courses are available, comparison with this data can be made. The model was used to predict the time courses for a rest-work transition in the dog gracilis muscle, using four different biochemical mechanisms for Voz regulation previously introduced in the literature (Table 1 shows the 4 forms of the mitochondrial driving function). To provide some consistency for comparison, the constants that appear in the expressions were determined using a single set of measurements of 002 and concentrations (6). The available data consisted of VOW and (PCr) as functions of time during a transition from rest (570~ = 0.05 pm01 g-l l rein-‘) to a submaximal work rate of 60, = 5 pm01 8-l. min-l, In choosing the constants, we imposed the same maximal oxygen consumption capacity &) for all mechanisms. This was obtained by fitting the data to function 3 in Table l(oo2 linear with creatine concentration) and then using a realistic minimum of {PCr) = 0.01.This gave V, = 7.4 prnola g-l. min-‘, which is in reasonable agreement with values measured in vivo (9). Also given were the three pools [CrT] = 37.1 mM, [AdT] = 6.68 mM, and [PT] = 48.2 mM (6). Figures 4-7 show the time courses of VOW and (PCr) predicted using each biochemical mechanism (obtained by solving &. 3 with a standard fourth-order RungeKutta routine). Also indicated in each Figs. 4-7 are the available data points. In the model simulations, we included pH changes, i.e., the early alkalinization and the subsequent acidification, that occur during a rest-work transition. The chosen input function for pH was a linear increase from 7.0 to 7.5 during the first 2 s, followed by an exponential decrease to 6.5 (time constant 20 s, Ref. 5) . Figure 4, A and B, shows the results obtained assuming Voz regulation by a simple enzymatic mechanism with only ADP as a substrate (mitochondrial driuing function 1 in Table 1). In generating the mitochondrial driving function, we were unable to find constants that fit the whole data set. It was still of interest to see what kind of rest-work transition could be predicted from the model if this mechanism were assumed. Therefore, to generate the simulations, all starting from the same resting VqZ, with the same maximal oxygen consumption capacity Vm = 7.4 pm01 .g-l. min-l, we used three different constants = 0.0043, KA2 = 0.043 (Q), and KA3 = 0.43 for KA [KAl mM]. We chose these constants to cover a large range (3 different orders of magnitude) in the parameter KA, around the value of published Michaelis constants (Km) for isolated mitochondria (17, 33). The calculations are then completely constrained. The initial and final values of the creatine charge, and the shape of the curves, are determined once these simulation conditions are chosen. This mechanism of 002 regulation predicts time courses for ~oZ during a rest-work transition which include a first phase of sharp rise, followed by a phase of much slower increase (plateau), preceeding another l

l


Cl000

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TABLE 1. Different

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forms of vo2 regulation (6)

Oxygen Driving

Function

Ref. No.

1

Equation

3, 4

KA iq)p

l

-

‘,’

2

33

3 4 See text for details and definitions.

’ + [ADP] >

K KA KP = Vm’ ’ + [ADP] + [HZPO,] + [ADP][&PO;I To, = A0 + A1IPCr) 60, = B. + &log ([ATP4-]/[ADP3-][H2P0,])

voz

20,24 15

---_

-- -- ---

Data points KA 0.0043 KA 0.043 KA 0.43 I / OY--

’

0

I

1

I

60

Tird?

I 180

1

(s)

.&ant

pH

s-l

1

00

\-

I 60

I

-120

l

-

Time

-

1

(s)

-

so-

7.

’

7.

Data points KA 0.0043 KA 0.043 KA 0.43

\

\

Constant

pH

\ --

OF0

r

1 60

I

Tird?

I

I

(s)

I 180

?

00

I

I

60

-m-e--

I

I

I

120

Time

I

180

(s)

FIG. 4. Sensitivity of model predictions to different mechanisms for vo2 regulation, mitochondrial driving function I (Table l), VO* and (PCr) during a rest-work transition, and comparison with data from Connett (6, 8) are shown. [CrT] = 37.1 mM, [AdT] = 6.68 mM, and [PT] = 48.2 mM. Transition from rest (vo2 = 0.05 pmol*g-‘emin-‘) to a submaximal work rate of VOW = 5 ~rnol-g-’ emin-‘; Tjm = 7.4 pmol-g-l .min-‘. fro2 (A) and (Per) (B) for 3 values for KA: 0.0043, 0.043, and 0.43 mM are shown. pH has a linear increase from 7.0 to 7.5 during the first 2 s followed by exponential decrease to 6.5 (time constant 20 s). To2 (C) and {PCr) (D) for the same 3 simulations as in A and B are shown except that pH is kept constant (pH 7.0) instead of varying as in A and B. See text.

phase of sharp rise toward steady state, and a final saturation at steady state. As & is increased, the initial sharp rise in Vo2 gets smaller, the second phase (“plateau”) gets longer and at a much lower value of VOW,and the transition to the third phase becomes sharper. Finally, the saturation behavior becomes more apparent with increasing values of KA. Also for high values of KA,

there is a significant delay before any significant increase in vo2 can be observed (60 s for KA3). All simulations show that steady state is reached with a significant delay with respect to the data points. If KA was to be further increased, the system would not be able to sustain ATP demand, and vo2 (along with ADP, see inset in Fig. 1) would start decreasing before reaching the working


. vo

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MODEL

OF

dbmputed time courses of phosphocreatine, seen in Fig. 4B, also deviate significantly from the data points for every value of &. All simulations include an initial phase at very high creatine charge. The structure of the phosphate energy system, parameterized for the gracilis muscle, is such that the corresponding values for Pi are negative (for our set of system parameters, [Pi] vanishes at (PCr) = 0.76). Therefore, the initial phases of all simulations (first 20 s or more, depending on KA) have no physiological relevance and need to be discarded. This problem can be overcome by increasing &. In Fig. 4B, the simulation corresponding to KA9 starts at lower creatine charge than the simulation corresponding to &. But, as & increases, the time course of VOW becomes less and less realistic (Fig. 4A). If K’ is increased to the point at which the resting creatine charge becomes physiologically realistic, then the system cannot sustain the ATP demand and runs out of phosphocreatine. These general predictions do not appear to be compatible with experimental observations. The saturation behavior observed for high values of KA is compatible with experimental observations, but the corresponding very low values of the creatine charge are not. The sensitivity of the predictions to the input time course of pH is important. Figure 4, C and D, shows the time courses of creatine charge and vo2 for the same values of & if constant pH is assumed during the simulations. The plateau phase essentially disappears. For low values of KA, computed time courses of vo2, show a much better fit to the data. For KA = 0.02 mM (not shown in Fig. 4C), the fit of 60, to the data is excellent. However, the corresponding time course of the creatine charge deviates from the data points. The initial and final values of the creatine charge are determined once the initial and final vo2 are chosen. In the case of large KA, there is still a delay of -40 s before observing a significant increase in VOW. The burst in glycolysis observed during the first 5 s of the transition in the original study (8) will, through ATP production, reduce the drive on Vo2. This could extend the delay before significant increases in VOW occur. Therefore, it is unlikely that this mechanism has a physiological relevance for the regulation of VOW in vivo. This means that if Vo2 is regulated in vivo by ADP concentration, it is not according to simple Michaelis-Menten kinetics. If the redox potential is the factor that compensates for the poor fit of this mitochondrial driving function, then redox as a function of the phosphorylation state has to be very complex. To our knowledge, no current mechanistic model of mitochondrial regulation includes this kind of redox compensation. Because of the biochemical structure of the system, the ADP concentration is well buffered, which makes ADP a very poor control variable. Figure 5, A and B, shows the transient results of the model simulations, assuming a mechanism obtained by adding a phosphate term to the regulation of Vo2 as suggested by Stoner and Sirak (33) (mitochondrial driuing function 2 in Table 1). We used their constants KA = 0.0044mM; KAp = 0.000523mM; and three different values for Kp, 0.33 from Ref. 33, 0.66, and 3.3 mM, to

AEROBIC

Cl001

METABOLISM

generate the simulations. Again, we chose $, = 7.4 prnol. g-l min-l, and once the initial and final VO, are chosen, the initial and final creatine a charge are determined. This mechanism for vo2 regulation predicts slightly different results, depending on the magnitude of the phosphate component. The same problems as with the previous mechanism appear at the resting state 1.For low values of Kp, the predicted Vo2 starts by rising sharply, then decreases, before rising again more slowly to reach its steady-state value after about 3 min. Again, time courses for the creatine charge are not exponential. These observations do not support a physiological relevance for this mechanism, but the predictions are very sensitive to pH changes. If we keep pH constant during the simulations (Fig. 5, C and D), the decay of the creatine charge and the increase in VOW are almost exponential, and the fit to the data is improved. More precise time courses of pH would therefore be essential to decide if this mechanism is to be rejected. For a high value of Kp, the time courses obtained by model simulation are in better agreement with the data, but by using a very large value of Kp, we assume a different mode of regulation, essentially based on [Pi]. This is mathematically analogous to having a mitochondrial driving function linear with creatine, [Pi] is a linear function of the creatine charge over the given range of creatine charge and of pH (Fig. l), and thus VOWregulation driven by [Pi] or [Cr] are equivalent. Therefore, by increasing E(p we do not provide a supportive a.rgument for the mechanism described by function 2. Figure 6, A and B, shows the results obtained using function 3, scaled by fitting the Voz-PCr data with a mitochondrial driving function linear with creatine charge. The values for the constants were A0 = 7.57 pmol . g-l min-’ and Al = -12.75 pmol . g-l gmin-‘. This mechanism for voz regulation predicts time * courses for Vo2 during a rest-work transition in excellent agreement with the transient data, except for a slight delay in reaching steady state. Predicted time courses are exponential, and measured time courses deviate slightly from a true exponential after 60 s. These simulations still yield the best agreement with the data. Predictions using this mechanism do not depend on pH changes. With this mitochondrial driving function, pH changes do not affect VO, at all, but only the values of the adenine nucleotide and Pi concentrations. Figure 7, A and B, shows the results obtained using function 4, in which vo2 is a linear function of the phosphorylation potential, with Bo = 18.18pmolag-l min-l and B1 = -4.21 pmol . g-’ min-’ mM-‘. Within most of the range of creatine charge considered for the simulations, the phosphorylation potential is a linear function of the creatine charge, for any given pH value (7), so we calculated Bo and B1 using the linear conversion of the creatine charge into the phosphorylation potential for values of the creatine charge between 0.1 and 0.7, at resting pH (7.0) and at “working” pH (6.5). The two sets of constants obtained this way provide a reasonable range for the study of rest-work transitions with this mechanism, and the corresponding rest-work transitions were simulated. The first set of constants l

l

l

l

l


KINETIC

Cl002

MODEL

OF AEROBIC

METABOLISM

4 B 0.

:I I

l

2 f I /.

:t 11 I

1

•a~..

Data points Kp 0.33 Kp 0.66 Kp 3.3 -- ---_ -- -- -- ----_

points . . . Data Kp 0.33 w-eKp 0.66 -Kp 3.3 l

0

Yl

60

1

1

I 60

1

1 60

, TirE

1

,

1

180

(s)

6b

I

I

Tirdr

(s)

varying constant

----

O! 0

--

.

Tird:

'

(s)

I 180

pH pH

7.

180

FIG. 5. Sensitivity of model predictions to different mechanisms for 602 regulation, mitochondrial driving function 2 (Table l), and vo2 and {PCr) during a rest-work transition are shown. Same pool values, initial and final 60, values, and data points are used as in Fig. 4. 0, = 7.4 PrnoL 8-l min-I; & = 0.0044 mM; &p = 0.000523mMI 60, (A) and (PCr) (B) for 3 values for Kp: 0.33, 0.66, and 3.3 mM are shown. pH has a linear increase from 7.0 to 7.5 during the first 2 s followed by exponential decrease to 6.5 (time constant 20 s). vo2 (C) and {PCr) (D) for the case of constant pH 7.0 are compared with the case of pH varying as in A and I?. Kp = 0.33 mM. See text. l

(linear conversion at pH 7) fit the data better, and this simulation only is shown in Fig. 7, A and 23. A driving function for which VOWis linearly related to the phosphorylation potential predicts time courses for vo2 during a rest-work transition within b-15% of the data. The major differences from the previous function are that there is a slight delay in reaching steady state and that the shape of the time course of VOW is not exactly exponential. In particular, one can note that the VOWat 60 s is 15% lower than the measured value. Transient predictions using this mechanism for VOW regulation differ from the previous ones because the phosphorylation potential as a function of the creatine charge deviates from linearity both at high creatine charge (“resting” state) and at very low creatine charge (“heavily working” states) (7, 9). Unfortunately for our discrimination between mechanisms, the resting state is very difficult to identify and to quantify (28), and the heavily working state cannot be modeled by our system

(without having to introduce some new mass balances to take into account AMP deaminase). Therefore, mechanisms 3 and 4 give very similar results in the range of energy turnover rates considered here. These results are in accordance with the results of Connett and Honig (9), based on steady-state measurements under different pH conditions in red skeletal muscle. They could differentiate easily between mechanisms 1 and 2 and mechanisms 3 and 4, and they coneluded that the first two were not relevant for physiological regulation of VOW in vivo. They were unable to differentiate between mechanisms 3 and 4. Because the mitochondria are state driven, the model allows one to explore many forms of VOWregulation by the phosphate energy state. Although apparently unnecessary for red skeletal muscle working under submaximal conditions, a more general mitochondrial driving fi.mction that includes redox and oxygen concentrations could be included in the model. Complete testing would require


KINETIC

MODEL

OF AEROBIC

METABOLISM

Cl003

I

I

60

60

-

120

Time

*

(s)

180

6. Sensitivity of model predictions to different mechanisms for regulation and mitochondrial driving function 3 (Table 1) are shown. fToz (A) and (PCr) (B) during a rest-work transition are shown. Same pool values, initial and final v02 values, and data points are used as in Fig. 4. A,-, = 7.57 ,umol*g-l. min”, and A1 = -12.75 pmol g-l min? pH has a linear increase from 7.0 to 7.5 during the first 2 s followed by exponential decrease to 6.5 (time constant 20 s). FIG,

~oZ

l

appropriate data on the time courses of all variables. Such data are to our knowledge not currently available. The results obtained for dog gracilis muscle show that in vivo VO~ is linearly related to the phosphate energy state. A mitochondrial driving function of the MichaelisMenten form with ADP as a substrate cannot account for the transient data in vivo. Moreover, we cannot differentiate at this point between mitochondrial functions linear with creatine charge, Pi concentration, phosphate potential energy [-PI, or the phosphorylation potential. All of these mechanisms can be grouped under the term “linear.” This is justified since all three variables are linear functions of the creatine charge (or of creatine) over the range of pH during a rest-work transition. Which of these four variables is most suitable for the formulation of the model? Because the creatine charge shows the largest relative variation over a transition, it is natural to express the model in its terms, as seen in Eq. 3. From a physiological point of view, the phosphate potential energy is conceptually more fundamental. The production of the phosphate potential energy stored in the high-energy phosphate bond pool must be ultimately

l

FIG. 7. Sensitivity of model predictions to different mechanisms for VOW regulation and mitochondrial driving function 4 (Table 1) are shown. voz (A) and (PCr) (B) during a rest-work transition. Same pool values, initial and final Vo2 values, and data points are used as in Fig. 4. B0 = 18.18 pmol gg-’ min-‘, and B1 = -4.21 pmol g-l min“ gmM-l. l

l

l

pH has a linear increase from 7.0 to 7.5 during the first 2 s followed by exponential decrease to 6.5 (time constant 20 s).

s/e/ /

FIG. 8. Spatial variations, in a slab model with consumption concentrated at fibril center and production at the extremities, of [ATP], NW, NW, PC r 1, and [Pi], expressed as a percent of the mean value, as a function of the fibril radius for [CrT] = 37.1 mM, [AdT] = 6.68 mM, and [PT] = 48.2 mM, pH 6.5. 60, = 5 prnol g-l gmin-? Spatial variation of [ATP] is negligible.


Cl004

KINETIC

MODEL

OF

AEROBIC

matched to the consumption required for muscle function. For a phosphate potential energy driven mitochondria, Eq. 3 can be further simplified, assuming that fB is large

-+w

dt

= r,([-P]) - r,(t)

(6)

where, as we have shown, the choice of a linear function for P,( [HP]) works well. This model fits the data remarkably well, is very simple to use, and is conceptually fundamental. Conclusions. The transient model presented here allows for a theoretical analysis of transient experiments. The model is obtained from the extension of the existing model of the phosphate energy system (7) by including ATP consumption and production. We emphasize that the resulting system has one degree of freedom. We also define a buffering factor (measuring buffering of the adenine nucleotides by phosphocreatine) that is very large for most physiological phosphate states. This buffering factor is an identical function of creatine charge in all muscle fibers. When the model was parameterized using steady-state data, it then reproduced the transient measurements in two species with very different time constants from two different laboratories. This shows that from a biochemical point of view, VOW in skeletal muscle in vivo is linearly related to the phosphate energy state, whether measured by creatine charge, Pi, phosphorylation potential, or the pool of high-energy phosphate bonds. The origin of this linearity deserves further exploration. APPENDIX

all equilibrium constants can be found in Table 1 of Ref. 7. Then

Example for dog gracilis muscle (data from Ref. 6): if [CrT] = 37.1 mM, [AdT] = 6.68 mM, and [PT] = 48.2 mM, at pH 7.0, then RAdk = 1.69 and RCpk = 5.34 X 10e3. All concentrations can be expressed in terms of the creatine 1 - {PCr) charge {PCr). If we let c = [ADP]/[ATP] = &pK (PCr) ‘we

get

PCr

Energy

The concentrations of ATP, ADP, AMP, PCr, Cr, and Pi are denoted by [ATP], [ADP], [AMP], [PCr], [Cr], and [Pi]. The constant total free adenine nucleotide pool is [AdT] = [ATP] + [ADP] + [AMP]; the constant total creatine pool is [CrT] = [PCr] + [Cr]; and the constant total reactive phosphate pool is [PT] = [PCr] + 3[ATP] + 2[ADP] + [AMP] + [Pi]. In the cell in vivo, the adenine nucleotides are bound to magnesium, potassium, and hydrogen ions. Once the concentrations of these cations and anions are known, the binding factors &, B,, B,, &, and B, (7,23,32) and the combined equilibrium terms RCpK and RAdK(7) can be calculated. The binding of any phosphorylated compound with the ions [H+], [M$+], and [K’] can be described by a binding constant KG = [XYn-m]/[Xm+][Yn-], where X refers to the cation binding, e.g., Mg2+, K+, and H+, Y refers to the anionic phosphate compound (ATP, ADP, Pi, and so forth), and n and m are arbitrary numbers. The total concentration of any compound can then be written as the sum of the free ionic form and the bound forms [Y] = [Y”-]eBi

where Bi = (1 + 2 K$*[X”+]) X

and subscript i refers to the particular Y; c = PCr; t = ATP; d = ADP; m = AMP; and p = Pi. This notation permits inclusion of both primary and secondary reactions, e.g., Bd = 1 + KrBp[Mg+] + K$&[H’](l + K$$JpH[Mg2+]) + a*. Values for l

WTI

[ATP

WTI

[ADP [AMP

= RAac2

(Al) [AdTl

1 + E + RAmc2

P = [PT] - {PCr)[CrT] i

- [AdT](3 + 2~ + RAac2) 1 + c + RAae2 The buffering factor fB is given by (1 + E + RA~E~)~

fB

Spatial

System (7)

= (PCr)[CrT]

[C r = (1 - (PCr))[CrT]

APPENDIX

A

The Phosphate

METABOLISM

B

Issues

In this paper we use a model in which concentrations are averaged over the cell, and the averaged values depend on time only. In this section, we provide an estimate of spatial variations to justify our approximation. Muscle fibers are made of long roughly cylindrical myofibrils separated by cytosolic tissue, with scattered mitochondria. The distance (d) separating consumption sites from production sites of high-energy phosphates varies between 1 and 10 pm (27). Following the reasoning of previous theoretical analyses (12, 21, 27), we assume that all species are free to diffuse between mitochondria and myofibrils. We also assume that both the creatine kinase and adenylate kinase reactions are at equilibrium. The relevant diffusion coefficients are DATP(=&nP) = 2 X IoN6 cm2/s, DPcr (=DcJ = 1.5 X 10M6cm2/s, and Dpi = 3 X low6 cm2/s (27). Therefore, the estimated time scale for diffusion of high-energy phosphates is TJJ= d2/&-rp = 5-500 ms. The time scale for transients in rest-work transitions in red muscle fibers ranges from several seconds to 100 s [for example, in dog muscle, half-times are of the order of 15 s (13,30)]. On such time scales, diffusion is so efficient that the transition can be viewed as a succession of diffusive steady states at various oxygen consumption rates. The gradients are nearly instantaneously established at every state. The largest gradients are obtained for the final (largest) oxygen consumption rate. Therefore, a study of spatial variations under steady-state working conditions is sufficient for estimating the largest gradients. To get an order of magnitude of the spatial variations between production and consumption sites in steady-state work, a one-dimensional slab model was considered. A mitochondrion


KINETIC

MODEL

OF AEROBIC

at one end delivers a certain phosphate potential energy flux (the phosphate potential energy being defined as [-PI = 2[ATP] + [ADP] + [PCr]) into a myofibril which consumes ATP at a constant rate. A worst-case situation was analyzed for which the consumption of the whole myofibril was concentrated at its center. Figure 8 shows the spatial range of each concentration, expressed in percent of the mean value, as a function of the myofibril’s radius. Parameters defining the cell environment are typical of the dog gracilis muscle (34 mM total creatine, 7 mM total adenine, and 48 mM total phosphate), and the work rate is moderately high, as for all simulations considered in this paper (VO, = 5 pmol *g-l. min-l). The largest percent variation is in [AMP], primarily because its absolute mean value is the smallest. We see that [ATP] is extremely well buffered. Also, all concentrations are well within a 10% range about their mean value. In most physiological cases, the distances to be considered do not exceed 1 or 2 pm, in which case this range of variation is ~0.2%. These calculations agree in order of magnitude with others done using more sophisticated geometries (21). These estimates show that diffusion is so efficient that the concentrations do not vary spatially by any significant amount. Thus we can use averaged concentrations to analyze rest-work transitions, and we do not need to resolve the small spatial variations. We thank Drs. P. A. A. Clark, C. R. Honig, and K. Groebe for encouragement and many useful discussions of this work. We are grateful for financial support from National Institutes of Health Grants HL-37205 and AR-36154. Address for reprint requests: C. I. Funk, Dept. of Mechanical Engineering, Univ. of Rochester, Rochester, NY 14627. Received 1 September 1989; accepted in final form 23 January 1990. REFERENCES 1. ALTSCHULD, R. A., AND G. P. BRIERLEY. Interaction between the creatine kinase of heart mitochondria and oxidative phosphorylation. J. Mol. Cell, Cardiol. 9: 875-896, 1977. 2. BALABAN, R. S., H. L. KANTOR, AND J. A. FERRETTI. In vivo flux between phosphocreatine and adenosine triphosphate determined by two dimensional phosphorus NMR. J. Biol. Chem. 258: 1278712789,1983. 3. CHANCE, B., J. S. LEIGH, J. KENT, K. MCCULLY, S. NIOKA, B. J. CLARK, J. M. MARIS, AND T. GRAHAM. Multiple controls of oxidative metabolism in living tissues as studied by phosphorus magnetic resonance. Proc. Natl. Acad. Sci. USA 83: 9458-9462, 1986. 4. CHANCE, B., AND G, R. WILLIAMS. Respiratory enzymes in oxidative phosphorylation. I. Kinetics of oxygen utilization. J. Biol. Chem, 217: 385-393,1955. 5. CONNETT, R. J, Cytosolic pH during a rest-to-work transition in red muscle: application of enzyme equilibria. J. Appl. Physiol. 63: 2360-2365,1987. 6. CONNETT, R. 7.

8.

9.

10. 11.

J. In vivo recruitment of vo2: test of current models using tissue data. Adu. Exp. Med. Biol. 215: 141-151, 1987. CONNETT, R. J. Analysis of metabolism control: new insights using scaled creatine kinase model. Am. J. Physiol. 254 (Regulatory Integrative Camp. Physiol. 23): R949-R959, 1988. CONNETT, R. J., T. GAYESKI, AND C. R, HONIG. Energy sources in fully aerobic rest-work transitions. Am. J. Physiol. 248 (Heart Circ. Physiol. 17): H922-H929, 1985. CONNETT, R. J., AND C. R. HONIG. Regulation of VOW in red muscle: do current biochemical hypotheses fit in vivo data? Am. J, Physiol. 256 (Regulatory Integrative Comp. Physiol. 25): FU398R906,1989. DAVIS, E. J,, AND W. I. A. DAVIS-VAN THIENEN. Control of mitochondrial metabolism by the ATP/ADP ratio. Biochem. Biophys. Res, Commun. 83: 1260-1266,1978. DAWSON. M. J.. D. J. GOODMAN. AND D. R. WILKIE. Contraction

METABOLISM

Cl005

and recovery of living muscles studied by 32P nuclear magnetic resonance. J. Physiol. Lond. 267: 703-735, 1977. 12. FUNK, C., A. CLARK, JR., AND R. J. CONNETT. HOW phosphocreatine buffers cyclic changes in ATP demand. Adv. Exp. Med. Biol. 258: 687-692,1989.

13. GAYESKI, T. E. J., R. J. CONNETT, AND C. R. HONIG. Oxygen transport in rest-work transition illustrates new functions for myoglobin. Am. J. Physiol. 248 (Heart Circ. Physiol. 17): H914H921,1985. 14. HILL, D. K. Hydrogen ion concentration changes in frog muscle following activity. J. Physiol. Lond. 98: 467-479, 1940. 15. HOLIAN, A., C. S. OWEN, AND D. F. WILSON. Control of respiration in isolated mitochondria: quantitative evaluation of the dependence of respiratory rates on [ATP], [ADP], and [Pi]. Arch. Biochem, Biophys. 181: 164-171,1977. 16. HOOD, D. A., J. GORSKI, AND R. L. TEFUUNG. Oxygen cost and tetanic isometric contractions in rat skeletal muscle. Am. J. Physiol. 250 (Endocrinot. Metub. 13): E449-E456, 1986, 17. JACOBUS, W. E., R. W. MOREADITH, AND K. M. VANDEGAER. Mitochondrial respiratory control. Evidence against the regulation of respiration by extramitochondrial phosphorylation potentials or by [ATP]/[ADP] ratios. J. Biol. Chem. 257: 2397-2402,1982. 18. KATZ, L. A., A. P. KORETSKY, AND R. S. BALABAN. Respiratory control in the glucose perfused heart. A 31P-NMR and NADH fluorescence study. FEBS L&t. 221: 270-276,1987. 19. LETKO, G., U. KUSTER, AND W. KUNZ. Investigation of the dependence of the intramitochondrial [ATP]/[ADP] ratio on the respiration rate. B&him. Biophys. Acta 593: 196-203,198O. 20. MAHLER, M. First-order kinetics of muscle oxygen consumption, and an equivalent proportionality between Qo~ and phosphorylcreatine level. J. Gen. Physiol. 86: 135-165, 1985. 21. MAINWOOD, G. W., AND K. RAKUSAN. A model of intracellular energy transport. Can. J. Pharmacol. 60: 98-102,1982. 22. MATTHEWS, P. M., J. L. BLAND, D. G. GADIAN, AND G. K. RADDA. A 31P-NMR saturation transfer study of the regulation of creatine kinase in the rat heart. Biochim. Biophys. Acta 721: 312-320,1982. 23. MCGILVERY, R. W., AND T. W. MURRAY. Calculated equilibria of phosphocreatine and adenosine phosphates during utilization of high energy phosphate by muscle. J. Biol. Chem. 249: 5845-5850, 1974. 24. MEYER, R. A. A linear model of muscle respiration explains monoexponential phosphocreatine changes. Am. J. Physiol. 254 (Cell Physiol. 23): C548-C553,1988. 25. MEYER, R. A., T. R. BROWN, B. L. KRILOWICZ, AND M. J, KUSHMERICK. Phosphagen and intracellular pH changes during contraction of creatine-depleted rat muscle. Am. J. Physiol. 250 (CeZl Physiol. 19): C264C274, 1986. 26. MEYER, R. A., T. R. BROWN, AND M. J. KUSHMERICK. Phosphorus nuclear magnetic resonance of fast- and slow-twitch muscle. Am. J. Physiol. 248 (Cell Physiol. 17): C279-C287, 1985. 27. MEYER, R. A., H. L. SWEENEY, AND M. J. KUSHMERICK. A simple analysis of the “phosphocreatine shuttle.” Am. J. Physiol. 246 (CeU Physiol. 15): C3654379, 1984. 28. MOMMAERTS, W. F. H. M. On the concept of resting muscle. News Physiol. Sci. 2: 30-32, 1987. 29. OLGIN, J., R. J. CONNETT, AND B. CHANCE. Mitochondrial redox changes during rest-work transition in dog gracilis muscle. Adu. Exp. Biol. 200: 545-554, 1986. 30. PIIPER, J., P. E. DI PRAMPERO, AND P. CERRETELLI. Oxygen debt and high-energy phosphates in gastrocnemius muscle of the dog. Am. J. Physiol. 215: 523-531, 1968. 31. SABINA, R. L., J. L. SWAIN, J. J. HINES, AND E. W. HOLMES. A comparison of methods for quantitation of metabolites in skeletal muscle. J. Appl. Physiol. 55: 624-627,1983. R. M., AND R. A. ALBERTY. The apparent stability con32. SMITH, stants of ionic complexes of various adenosine phosphates with monovalent ions. J. Phys. Chem. 60: 180-184,1956. 33. STONER, C. D., AND H. D. SIRAK. Steady-state kinetics of the overall oxidative phosphorylation reaction in heart mitochondria. J. Bioenerg. Biomembr. 11: 113-145,1979. 34. VAN DER MEER, R., T. P. M. AKERBOOM, A. K. GROEN, AND J. M. TAGER. Relationship between oxygen uptake of perifused ratliver cells and the cytosolic phosphorylation state calculated from indicator metabolites and a redetermined constant. Eur. J. Biochem. 84: 421-428,1978.


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