- Email: [email protected]
Computer simulation of entry into glycolysis and lactate output in the ischemic rat heart
Journal
of Molecular
and Cellular Cardiology
Computer
10,779-796
Simulation of Entry into Output in the Ischemic MICHAEL
Department
(1978)
of Computer University (Received
C. KOHN
Glycolysis Rat Heart
AND DAVID
and Information Sciences, Moore of Pennsylvania, Philadelphia,
29 September
and Lactate
GARFINKEL School of Electrical Engineering Pa. 19104, U.S.A.
1977, accepted in revisedform
20 April
02,
1978)
M. C. KOHN AND D. GARFINKJIL. Computer Simulation of Entry into Glycolysis and Lactate Output in the Ischemic Rat Heart. 3oumal of Molecular andcellular CardioloQ (1978) 10,779-796. A computer model of glucose uptake, glycogenolysis, and export of pyruvate and lactate in the ischemic, isolated rat heart was constructed. This model predicts that, after an initial burst of glycogenolysis which consumes the outer glycogen branches, glycolysis is limited by glucose uptake. Reduced glucose uptake in this model then results from decreased rate of substrate delivery despite increased afiinity of the cell membrane carrier for glucose. Lactate accumulation in this model is due partially to its incomplete washout from the interstitium at low coronary flow rates and partially to a dramatic reduction in the activity of a cell membrane lactate permease. KEY WORDS: Glucose utilization; Glycogenolysis; Cardiac &hernia; Cardiac interstitium.
Lactate
production;
Lactate
export;
1. Introduction Utilization of exogenous glucose by the ischemic, isolated rat heart is suppressed in hearts receiving approximately 10% of normal coronary flow [12, 191 and accelerated in hearts receiving 20 to 40% of normal flow [13, 181. The reduced glycolytic rates in severely ischemic rat hearts occur despite changes in creatine phosphate and adenine nucleotide tissue contents similar to those observed for anoxic hearts where glycolysis is greatly increased [19]. On the other hand, tissue lactate accumulates to a much greater degree in ischemic hearts than anoxic hearts [ 191. A possible explanation for these consequences of reduced coronary flow was suggested by our simulation study of the depletion of adenine nucleotides in ischemic rat hearts [II]. Local equilibration between the perfusate and the inter-
stitium appeared to be diminishing the net rate of transfer of material between these two compartments. This effect might also reduce glucose delivery to, and lactate
washout
from,
the interstitial
fluid.
Such
behavior
was
indicated
by
preliminary experiments with a crude model and supported by experimental data on changesin cardiac lymph composition in ischemia [3]. 0022-2828/78/090779+
18 $02.00/O
0
1978 Academic
Press Inc. (London)
Limited
780
M. C. KOHN
AND
D.
GARFINKEL
The work reported here represents a fuller exploration of this subject and is intended to determine how much this behavior contributes to the known effects of ischemia. We have constructed a more realistic computer model of the entry into glycolysis (glucose uptake plus glycogenolysis) and the cell membrane transport of pyruvate and lactate in the ischemic, isolated rat heart based on the experimental studies of Neely et al. [12] and Rovetto et al. [19], and on the perfused rat heart computer model of Achs and Garfinkel [fl. The glycolytic enzyme portion of this model was abstracted for inclusion here, and representations of physiological processes such as membrane transport were added or readjusted for the new physiological conditions. The parameters specifying these physiological processes are not merely mathematical conveniences, but do have biological significance, although they will have different values for different experimental preparations. This model is a mechanistic representation of the molecular processes known to be involved in this portion of metabolism (with approximations as required for the complexity of glycogenolysis, as detailed in the appropriate section below). It will not properly represent severe long-term &hernia after there is structural damage to membranes or alteration of properties of enzymes. The simulation results describe processes occuring during global ischemia in the isolated rat heart perfused with oxygenated salt solution [12], and they may not reflect the quantitative behavior of other preparations. However, we have encountered similar behavior in the course of simulating other ischemic heart preparations. Therefore, we believe that similar effects are likely in severely ischemic cardiac tissue in physiologically more realistic situations.
2. Model-building
Methodology
Our model-building techniques have been reviewed in detail elsewhere [S, 71. Only a brief summary is appropriate here. Model construction begins by evaluating the flow of material through the appropriate metabolic pathways, applying stoichiometric mass balance techniques routinely used in chemical engineering. The results are expressed in flux maps (exemplified in Figure 1) showing material flow through enzymes and membranes at a given time (2 min after start of ischemia in Figure 1). Metabolites for which tissue totals were experimentally determined must also be distributed between compartments by methods described elsewhere [lo]. In the present model the only compartments considered are the cytoplasm, the interstitium, and the perfusate. The time derivative of each metabolite concentration is determined as the sum of the fluxes producing it minus the sum of the fluxes consuming it. The activity of each enzyme (or membrane transport mechanism) is specified by its rate law, an algebraic relation between the enzymatic velocity and the enzyme’s kinetic parameters and the levels of its substrates, products, and inhibitors, and whose form is determined by the enzyme mechanism. The formal
781
I
lnterstltium
1 I
-Perfusote
Glycogen inner .bmnches
lf
0.115 + Wycogen ’ outer branches 13.574 t
GIP
I
13.566
G6P
t Gtucosa
5.362
W. -5.352
5 18504 F6l’ 1
Glucose
t1803
Pyruvote
-
Glucose
‘I I
18532
J FDP
. . .
I
30.701
Pyruvofe
I
4
0.157
+
Pyruvote 0.153-w
30.535
Imtote
t Locfofe
-
30.593+
I I I
! 1 FIGURE 1. Diagram of the metabolism 2 min of ischemia. Numbera are in *01/g net flow through re-vcmiblc reactions.
Lactatr 26.1044
included in this model, dry wt min. The small
in the form of a flux map after arrows indicate the direction of
782
M.
C. KOHN
AND
D.
GARFINKJXL
techniques for determining these rate laws have been reviewed elsewhere [7]. When combined, the rate laws make up most of the set of differential equations which constitute the model. These equations are then solved by numerical methods, and the results are compared with experimental observation. A successful model must therefore fit both the properties of the constituent enzymes and the particular set of whole-tissue observations which is being simulated. In the course of model construction, a considerable body of numerical data and other information is abstracted from the literature. This is used to formulate hypotheses explaining the sequence of biochemical and physiological events leading to the observed behavior of the biological preparation. The modeling process checks that these hypotheses are internally consistent and actually explain (quantitatively) the experimental observations. Several metabolic processes which lie outside the scope of the simulation, but whose products affect the metabolism simulated, are represented by time-dependent functions. These functions are normally time profiles of metabolite concentrations. The time profiles of these functions and the related parameter values which will enable us to reproduce the observed behavior are determined by algebraically solving the appropriate rate law for the unknown metabolite concentration required to reproduce the observed flux through the corresponding enzyme.
3. Construction
and Behavior
of the Model
Since measurements of tissue glycogen content are not sufficiently reliable for the estimation of the net glycogenolytic flux, we had to construct this portion of the model by iterative procedures. We estimated the net rate of glycogenolysis from tissue glycogen measurements [19]. We computed the net glycolytic flux (through phosphofructokinase) from this value and the measured glucose uptake rate. The concentrations of several small molecules involved in determining this net flux, because they are variables in the appropriate enzymatic rate laws, were calculated as described below. These chemical concentrations were used to compute new glucose uptake and glycogenolysis rates, and the process was repeated until the flux changed by less than 0.1 o/obetween repetitions. Further details are given below.
Be&or
of related small nm!ecules
Because several glycolytic enzymes are sensitive to adenine nucleotide levels, pH, and cytosolic Mgs+ level, these were determined by a previously described procedure [IO]. Briefly, total cytosolic ATP and ADP are computed by subtracting
GLUCOSE
AND
LACTATE
TRANSPORT
IN
CARDIAC
ISCHEMIA
783
from the total tissue levels the mitochondrial content calculated from the respiration rate. Then cytosolic AMP was computed from the adenylate kinase equilibrium, and cytosolic pH from the creatine kinase rate law. Cytosolic Mga+ was adjusted to yield the mixture of chelated, protonated, and free forms of the adenine nucleotides which reproduces the current flux through phosphofructokinase. As all these cytoplasmic factors are interdependent, the procedure is iterated by computer until the adenine nucleotide distribution is constant to within 0.1 o/o between consecutive iterations. The time profile of the glucose-l-phosphate level was then computed [q as that necessary to reproduce the flux through phosphoglucomutase for the several time points. Similarly the time profile of fructose-6-phosphate was determined from the phosphohexose isomerase flux, and that of glucose from the total flux through hexokinase I and II. Measured values of glucose-6-phosphate and fructose-1,6diphosphate were used in these calculations. Debranching
enzyme
We found that the original model of the glycogen debranching enzyme in our perfused rat heart model [I] did not reproduce the observed time profiles of tissue glycogen [19]. This model was designed to reproduce glycogen breakdown over 2 min of anoxia, where only 27% of the initial tissuelevel [2lj is depleted and there is still a considerable amount of outer-branched glycogen remaining. In contrast, 84% of the tissueglycogen was depleted after 30 min of &hernia in the preparation simulated [19]. We interpret the observed glycolytic flux profiles to indicate that debranching enzyme is not limiting at 2 min (since the flux is greater than the tissuecapacity of the enzyme [14]), but that it doesbecome limiting by 4 min, when glycolytic flux is substantially lower. This enzyme hydrolytically cleaves and rearranges a polymeric substrate with a “binary tree” structure. The number of exposed branch points is a complex function of the number of glucose residues per branch and the total size of the “tree”. As the outer branches are degraded, branch points are exposed, inner branches are converted to outer branches and degraded, and the potential concentration of branch points falls exponentially. The fractional number of exposed branch points falls faster than the fractional number of glucose residues.We have introduced sigmoidal substrate activation of this enzyme into the model to represent the exponential decline in the number of branch points, as each successivetier of glycogen branches is degraded, together with sigmoidal inhibition of the enzyme by outer branches to represent their shielding of underlying branch points. As we are considering the inner branches asthe substrate of the enzyme, the above corrections are needed for the highly degraded glycogen we have here, but not for the less degraded glycogen represented in our previous model [I] (where hydrolysis is still accurately represented after the correction).
784
M. C. KOHN
AND
D. GARFINKEL
The resulting rate law that we used for the debranching
“=-(q[($)”
+ l{Tl}
[@)”
enzyme is
+ l]
where (in terms of glucose residues) Vmax = 4.0 ~mol/min-g dry wt, K, = 5.56 maa, XI = 16.7 mM, KO = 0.82 n-m, and I and 0 denote the concentration of glycogen inner and outer branches respectively. These values were empirically determined as those required to reproduce the tissue glycogen level upon solution of the final differential equations (Figure 2). As the initial ratio of inner to outer branch glycogen and the cyclic AMP time profile were not measured and may be preparation-dependent, these were treated as parameters, and their values were selected so as to reproduce the observed tissue glycogen profile upon integration. We estimated that glycogen was initially 56% inner branches and 44% outer branches. The cyclic AMP level was computed to rise from 0.5 nmol/g dry wt initially to 4.0 nmol/g dry wt at 1 min, after which it slowly fell, reaching its steady state value of 0.75 nmol/g dry wt at 4 min. Ovedglycogen
metabolism
The modified glycogen metabolism model consists of differential equations for glycogen synthetases and phosphorylases, branching and debranching enzymes, phosphoglucomutase, and equations for the interconversion of synthetase I and D
FIGURE
2. Comparison
of computed
Time (mid and olxerved
[18] tissue glycogen
content.
GLUCOSE
AND
LACTATE
TRANSPORT
IN CARDIAC
ISCHEMIA
785
and phosphorylase Q and b [I]. It was integrated using a restructured version [17] of our simulation program [4]. The resulting glycogen profile is given in Figure 2. The glycogen outer branches fall from 40 pmol/g dry wt initially to 24.63 pmol/g dry wt after 2 min, as required by our flux map (Figure 1) and then to 0.38 pmol/g dry wt, after 4 min. This strongly suggests that the remaining glycogen resembles a limit dextrin whose inner branches are starting to be converted to outer branches. If the initial distribution of glycogen between inner and outer branches were different, a different cyclic AMP profile might produce similar behavior, but the total amount of glycogen degraded would differ from that observed. Therefore we believe that the above represents a unique set of parameters for the preparation simulated here and corresponds with the biochemical events actually occuring in the tissue. Glucose uptake Passive diffusion of glucose between the perfusate and the interstitial fluid was represented as a reversible, first-order process. We had previously [II] deduced that the rate constants for this process in the perfused rat heart with high coronary flow rate should be 0,742 min-1 for nonionic small molecules. Coronary flow continually decreases with time in the preparation we are modeling [12]. The increased residence time of the perfusate in the tissue should then tend to reduce the local concentration gradient between the perfusate and interstitial fluid and result in lower net rates of diffusion for the whole heart. Accordingly, we allowed the diffusion rate constant to be a function of the coronary flow rate. The function selected after a series of trials was : k = 0.742 exp [(F-F,)/6.2973], where F and F, are the instantaneous and initial coronary flow rates, respectively. The denominator in the exponent yielded a smoothly decreasing diffusion rate, ultimately producing a 90% reduction for a comparable reduction in coronary flow. Other functions tried predicted such rapid depletion of interstitial glucose that the differential equations were insoluble. The differential equations constituting the submodel representing this process were integrated to determine the time profile of cytosolic glucose profile is for the interstitial glucose level; determination described above. The kinetics of glucose transport across the cell membrane are modified by anoxia [I] with changes in both K, and Pm=. Similar changes are expected here, as ischemic hearts are likely to be hypoxic. Therefore, we allowed the rate constants for inward glucose transport in our aerobic rat heart model [I] (identical MichaelisMenten kinetics in both directions) to vary sigmoidally with time : (t’&)“Ak k=k,+ s + (qt,)” t’ = t-to
786
M. C. KOHN
AND
D. QARFINKEL
where k. is the initial rate constant, Ak is the maximal change in k at final steady state for both glucose binding (kl) and turnover number (kg), to is the time when the transition begins, t+ is the half-time for the transition to the new kinetics, and S and n are parameters. A sigmoidal function was selected because it has a time derivative of zero at the beginning and end of the transition and allows the rate constant to change smoothly to its new value. Those values of the parameters which most closely reproduced the observed glucose uptake rates [I91 were determined by formal optimization techniques [7, 81 using the interstitial and cytosolic glucose profiles determined above. Optimal parameter values are : to = 0.0 min (i.e. this change actually begins immediately at the start of the ischemia) t+ = 25.7 min after the start of the transition Ab = 7.332 PM-~ min-1 (glucose binding) Akz = 1678 min-1 (turnover number)
n = 2.75 s = 3.77. This results in a change of kinetic parameters from the aerobic values, I& = 8.89 mu and V- = 11.52 ymol/min-g dry wt (which are the same as those previously determined [I] for the aerobic heart without insulin), to the following values after 30 min of ischemia: I& = 3.87 mM and Vmax = 12.39 pmol/min-g dry wt. These parameter values enabled us to reproduce the observed glucose uptake rate (Figure 3) and indicate that the glucose carrier increases both its affinity for glucose and its maximal permeation rate.
0.
I 0
I 6
I
I 12 Time
FIGURE
3. Comparison
of computed
I
I 18
I
I 24
I
glucose
uptake
30
(mini
and observed
[l8]
rate.
GLUCOSE 2.0
AND
LACTATE
TRANSPORT
IN
787
ISCHEMIA
4
I
I
1
I
I
I
I
I
I
1 6
I
I 12
1
I 18
I
I 24
I
OO
Time
FIGURE
CARDIAC
4. Comparison
of computed
and observed
30
(min)
[18] glucose&phosphate
tissue content.
The glucose uptake and net glycogenolysis fluxes at each of several timerpoints were computed and the results were used to construct more accurate flux maps, as described above. A new adenine nucleotide distribution was computed, and then this entire process was iterated until the flux maps did not change by more than 0.1 oh between successive cycles. The optimal parameter values cited above were obtained after five such iterations. The portion of our perfused rat heart model [lj including the entry into glycolysis as modified above through the glycolytic sequence to phosphofructokinase was integrated as a check on our procedure. The resulting glucose 6-phosphate profile was in good agreement with the available data (Figure 4), indicating internal consistency of the model. Pyruvate and la&ah tranqbrt Although metabolite levels as a function of time for the preparation we are modeling [12, 191 were determined for a recirculating perfusate, the lactate output rates were determined with a flow-through perfusion. Possibly as a result of this difference, more lactate was observed in the nonrecirculated perfirsate than could be accounted for by the sum of glucose units taken up from the per&sate or produced from glycogen [I91 in the recirculating perfusate experiment. Because strict material balance must be maintained in order to produce an internally consistent model, we elected to ignore the lactate output data while attempting to reproduce the observed tissue lactate levels. Since the initial lactate export rate is more likely to be accurate, we obtained the initial rate of pyruvate oxidation as the difference
788
M.
C. KOHN
AND
D.
GARPINKEL
between the rates of pyruvate and lactate production. The initial rate ofendogenous fatty acid oxidation was then computed as that required to account for the observed respiration rate. We assumed that the fatty acid oxidation profile had the same curve shape as the observed respiration rate [l.?] and extrapolated fatty acid oxidation accordingly. We computed the rate of pyruvate oxidation at each time point by difference. Throughout most of the experiment pyruvate oxidation is small, and the model is thus not sensitive to errors in its value. We estimated the rate of conversion of pyruvate to alanine from measured amino acid levels in aerobic and ischemic hearts [9]. With these two rates, tissue lactate data, and the glycolytic flux computed as above we were able to draw tentative flux maps for the pyruvate-lactate system. The diffusion of lactate between the interstitial fluid and the perfusate was treated like that of glucose, but an initial rate constant of 0.742 min-i required that interstitial lactate exceed the total tissue lactate at times of high glycolytic flux. Therefore, we treated this rate constant as a parameter. A value of 8.327 min-1 at zero time enabled us to reproduce the rate of lactate appearance in the perfusate given on the flux maps while leaving a significant amount of cytosolic lactate at times of rapid lactate production. This larger diffusion rate constant may be due to the smaller size or ionic nature of lactate. The compartmentation of tissue lactate between the cytosol and interstitial fluid which would reproduce the lactate output rate on the flux maps for the several time points was then calculated. It should be noted that the rate constants for diffusion between the interstitial fluid and the perfusate are parameters in our system, and their values should not be taken literally. Tissue pyruvate data were corrected for the mitochondrial content using the ratio of spaces (9 : 1, cytosol :mitochondria) in the full heart model, and assuming uniform concentration. The remainder was initially distributed between the cytosol and interstitium so as to produce equilibrium at the plasma membrane pyruvate carrier in our model [I]. Py ruvate diffusion between the interstitial fluid and perfusate was treated in an identical manner to lactate. The pyruvate transport portion of the full rate heart model was integrated separately to obtain a time profile of cytosolic pyruvate and the net pyruvate export rate. The flux through lactate dehydrogenase on the flux maps was corrected for the computed pyruvate output rate. The cytosolic NAD and NADH levels (whose sum was held constant at 2.20 pmol/g dry wt) which would reproduce the corrected flux through this enzyme were then calculated [6j. Our previous simulations [I, 21 suggested that the lactate transport rate was dependent on factors other than the cytosolic lactate level. Spencer and Lehninger [20] found evidence for a pHsensitive plasma membrane lactate carrier (in ascites cells), but construction of a reliable model for its regulation in heart is not possible without more detailed information. Therefore, the activity of the lactate carrier in our model is input as a
GLUCOSE
AND
LACTATE
TRANSPORT
IN
CARDIAC
ISCHEML4
789
time-varying first order rate constant which, when multiplied by the cytosolic lactate level, yields the lactate export rate. A section of the rat heart model comprising lactate and pyruvate transport and lactate dehydrogenase was integrated. The reasonably accurate reproduction of the pyruvate (Figure 5) and lactate (Figure 6) levels serves as a check on the modeling technique. The mathematical behavior of the propagation of errors in a model as complex as ours is too complicated to permit estimation of the confidence intervals for computed concentrations and fluxes. In fact, this problem is presently intractable except for the crudest (linear) approximation. After many years of model construction, we have the impression that uncertainties in our computed values are usually comparable to experimental error. Furthermore, predictions based on published models have generally been confirmed by subsequent experiments.
4. Discussion Glucose transport
The results of our simulation lead us to conclude that the reduced utilization of exogenous glucose by the ischemic rat heart is due to its reduced rate of delivery and not to any biochemical event peculiar to ischemia. Rovetto et al. [19] found that only 6% of the per&sate glucose was extracted on a single passthrough the heart, and they concluded that substrate delivery was not limiting. We compute an extraction ratio of 5.5% for a coronary flow rate of 1 ml/min and a glucose uptake rate of 3 ~mol/min-g dry wt from an insulin-free perfusate containing 11 mM glucose. However, for a coronary flow rate of 15 ml/min for aerobic rat hearts perfused with salt solution, and for a glucoseuptake rate of 6.3 pmol/min-g dry wt, we compute an extraction ratio of only 0.8%. The decreasedrate of delivery results in a reduced glucose uptake rate despite the greater degree of extraction. The increased extraction ratio may be due to the longer residence time of the perfusate in the coronary vasculature as well asthe decreasedK,,, for glucose uptake, and the reduced glucose uptake rate may be due to depletion of interstitial glucose (Figure 7). In the absence of any published information, we set the initial interstitial glucose concentration equal to the perfusate concentration as an upper limit. This approximation, aswell as those involved in obtaining the rate constant for diffusion of glucose between the perfusate and interstitial fluid, may result in the computed fall of interstitial glucose (Figure 7) being greater than that which actually occurs in the tissue. As Rovetto et al. [19] were not able to measure tissueglucose in the absenceof insulin, a direct comparison is impossible. This helps explain the increased glucose uptake rates observed for ischemic rat hearts with 20 to 40% of normal coronary flow. Here the increased affinity of the cell membrane carrier for glucose overcomes the reduced flow rate. In a computer
790
M. C. KOHN
AND
D. OARFINKEL
Interstitial I OO
I
I 6
I 12 Time
I
I
t 18
I
I 24
I
I 30
(min)
FIGURE 5. Comparison of computed pyruvate levels with observed 1eve.l.s: (e) J. R. Necly, pexsonal communication. Total pyruvate does not include mitochondrial data points are corrected for this.
J
from [18]; pyruvate,
Interstitial /
I
6
FIGURE
6. Comparison
ofcomputed
t
I
12 Time
and observed
I
I
I
18
I
I
I
24
30
(min)
lactate
levels:
(A)
from
[I-?];
(0)
from
[18].
GLUCOSE
AND
o-
FIGURE
LACTATE
’
’ 6
TRANSPORT
’
7. Computed
’ 12 Time
’
IN CARDIAC
’ 18
’
’ 24
ISCHEMIA
’
791
30
(mid
interstitial
glucose
level.
experiment with our model, we allowed the coronary flow rate to fall only to 6 ml/mm and used the same parameters for the increase in the glucose uptake rate constants as given above. The result (Figure 7) was that after an initial moderate decline, the glucose uptake rate increased. By suitable adjustment of the glucose carrier parameters, we could compute an increase in glucose uptake above the control aerobic value. The changes in the activity of the glucose transport mechanism are likely to reflect changes in the behavior of the protein molecules involved. It is unlikely that these changes are produced directb by changes in physical factors such as coronary flow rate or perfusate pressure. We suggest that they are instead under metabolic control, resulting from changes in concentration of small molecules which bind to the proteins involved. Gbcogen metabolism Neely et al. [12] observed that the final stage of ventricular failure begins alter 12 min of low coronary flow perfusion. Our simulation suggests that because of the reduced glucose uptake and respiration rates, severely ischemic cardiac tissue remains viable only as long as there is sufficient glycogen to maintain a minimal ATP level by glycolysis. (The ATP level is falling most rapidly at about 12 mm [22].) About 17% of the glycogen initially present in the tissue is depleted after
792
M.
C. KOHN
AND
D.
GARFINKEL
2 min of &hernia. Most of this is due to phosphorylase a activity. By 12 min 69% of the original glycogen is consumed. Nearly all of this additional depletion is due to phosphorylase b activity. By 16 min the glycogen consumption has reached nearly the same level as at 30 min (no significant statistical difference), so that glycogenolysis must become much slower at about this time. The curve in Figure 2 exhibits a marked change in slope at about 4 min. The net glycogenolysis rate drops from 14.6 pmol/min-g dry wt at 3 min to 4.86 pmol/min-g dry wt at 4 min. At this point glycogenolysis becomes limited by the activity of the debranching enzyme, even though the cytosolic AMP level (325 nmol/g dry wt) is sufficient to permit substantial phosphorylase b activity. This effect is not expected intuitively and is not altered by moderate changes in the kinetic parameters in the debranching enzyme rate equation. In a computer experiment with our model, we doubled the binding constant (Ko) for inhibition of this enzyme by undegraded glycogen outer branches. The result was that debranching activity began earlier, but the much greater tissue capacity of phosphorylase b still caused a depletion of outer branches and a dramatic decrease in the slope of the glycogen profile at about 4 min. The lag in conversion of glycogen inner branches to outer branches sufficient to meet the demand of phosphofructokinase for hexose phosphate results in the computed transient fall in glucose 6-phosphate at 4 min (Figure 4). Determination of whether any of this is an artefact of our crude model of the debranching enzyme, by building an accurate and biochemically realistic model of this enzyme, would be a substantial research project in itself.
Time
(min)
FIGURE 8. Computed glucose uptake rate if coronary 4 min, using kinetic parameters for glucose uptake identical
flow drops from 15 mI/min to those in Figure 3.
to 6 ml/min
in
GLUCOSE
AND
LACTATE
TRANSPORT
Lactate
IN
CARDIAC
793
ISCHEMIA
transport
Lactate export also seems to be affected by reduced coronary flow, but our simulation suggests that unknown factors are involved in this process. The results of a preliminary investigation of this process [5] are qualitatively similar to those of the more rigorous treatment reported here. Our previous simulations [I, 21 have indicated that the lactate export rate generally follows the flux through lactate dehydrogenase and is effective at preventing excessive accumulation of cytosolic lactate in normal-flow situations. Because the lactate carrier seems not to be controlled entirely by the lactate concentration, we describe it as a “permease” to indicate that its activity is regulated like that of enzymes. In the present simulation lactate permease is activated during the initial spurt of glycogenolysis (0 to 4 min) but not afterwards, even though lactate production exceeds that for the initial high flow conditions (Figure 9). Part of this effect may be explained by the computed rise in interstitial lactate from 0.20 ells initially to 2.44 mM after 30 min, which is largely due to the reduced rate of diffusion between the interstitial fluid and perfusate. Lactate may be exported by a cell membrane carrier with a lower Km for extracellular than intracellular lactate, analogous to the pyruvate carrier in our model [I]. The accumulation of lactate in the interstitium may result in appreciable back-uptake of lactate, thus reducing the net export rate and hence the computed lactate permease activity.
Lactate
permease
Time
FIGURE
9. Computed
lactate
production
IminI
rate and lactate
permease
activity.
794
M.
C. KOHN
AND
D. GARF’INKEL
The actual lactate output process must be more complex than described above. Interstitial lactate is computed to be 4.12 mu and cytosolic lactate to be 0.80 mu at 2 min, when lactate permease is maximally active. The permease is apparently transporting lactate against a concentration gradient. After 30 min however, interstitial lactate is computed to be only 4.22 mM while cytosolic lactate is 37.9 nq and the permease is nearly inactive. This suggests that the lactate carrier is under metabolic control as well. The cytosolic NADH level deduced for the ischemic rat heart preparation we are simulating is given in Figure 10. During the first few minutes, when lactate permease activity is high, the cytosol is relatively oxidized. Later, when the cytosol is much more reduced, lactate permease activity is very low. This result suggests that NADH and/or some metabolite which is closely correlated with highly reduced cytosol is an intracellular inhibitor of the lactate carrier. Spencer and Lehninger [2U] studied pyruvate and lactate transport in ascites cells and found it to be dependent on pH; low pH reduces the Km of the permease for lactate. Figures 6 and 9 indicate that increased lactate export correlates weakly with both the initial fall in pH and the second fall after 20 min (Figure 10). However, it is obvious that such an effect is superimposed on other, more important, biochemical controls. These may also be affected by the (pathological) low flow condition; we had previously concluded from models of normalflow experiments [I, 21 that lactate permease tends to be activated by rapidly rising H+ and lactate levels, a relationship that does not hold or is inapplicable here after the first few minutes.
- 6.4
I OO
FIGURE
t 6
I
10. Computed
t 12 Time
I
cytosolic
I 18
t
I 24
I 30
(min)
pH and cytcsolic
NADH
level.
GLUCOSE
AND
LACTATE
TRANSPORT
IN
CARDIAC
ISCHEMIA
795
If lactate permease is under metabolic control, it presents a logical site for pharmacological intervention in the treatment of ischemia. In view of the common belief that lactate accumulation in ischemia is harmful [16, 18, ,221, activation of lactate permease by a suitable drug may be an important potential contribution to the therapeutic management of ischemia. When lactate permease activity falls most greatly below the lactate production rate (at 10 min), the lactate export rate (permease activity times cytosolic lactate level) nevertheless equals 69% of the flux through lactate dehydrogenase. Thus, even a moderate increase in lactate permease activity might be sufficient to prevent deleterious accumulations of lactate. Rovetto et al. [19] considered the possibility that lactate actually crossesthe membrane by passive diffusion as unionized lactic acid. We believe that the complex nonlinear behavior we deduced for lactate export is more readily accommodated by active transport. Finally, these investigators assumedthat the interstitial lactate concentration was the same as that measured in the perfusate. Our simulation suggeststhat this (common) assumption is inaccurate. After 30 min of ischemia lactate accumulated in the perfusate to only 444 FM compared to 2.44 mM computed for the interstitium. The experimental results of Opie and Mansford [15] confirm that there is a concentration gradient between cytosol and interstitium for lactate and pyruvate. Comparison of Figures 5, 6, and 10 indicate that the commonly used method of estimating the cytosolic NADH/NAD ratio from the tissue lactate/pyruvate ratio is inaccurate here. The L/P ratio is often in error here by a factor of several times and sometimeschangesin the wrong direction with time. This method, ascommonly used, is basedon the assumptionsthat lactate dehydrogenase is sufficiently active to equilibrate tissue lactate, tissue pyruvate, NADH, and NAD, and that the intracellular pH is constant, usually 7.0. The early location of most of the lactate in the interstitial fluid rather than within the cells, and the fall in pH, deviate considerably from theseassumptions. We believe that our results indicate the need for a closer investigation of the nature of metabolic control of glucose and lactate transport acrossthe cell membrane, that alterations of these transport processesare very important, perhaps the primary, contributors to the harmful effects of ischemia, and that events at the biochemical level follow from thesephysiological changes. Acknowledgements
We thank J. R. Neely of the Milton S. Hershey Medical Center of Pennsylvania State University for his helpful criticisms during the preparation of this manuscript. This work has been supported by NIH Grant HL 15622. REFERENCES 1. ACHS, M. J. & GARFINKEL, D. Computersimulationof energy metabolismin anoxic perfusedrat heart. American Journal of Physiology 232, R164-R174 (1977).
796 2. 3. 4. 5. 6.
7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
17. 18. 19. 20. 21. 22.
Id.
C. KOHN
AND
D.
GARFINKEL
M. J. & GARFINKE L, D. Computer simulation of rat heart metabolism after adding glucose to the per&sate. American Journal of Physiology 232, R175-R184 (1977). FEOLA, M. & GLICK, G. Cardiac lymph flow and composition in acute myocardial ischemia in dogs. AmericanJournal of Physiology 229,44-48 (1975). GARFINKEL, D. A machine-independent language for the simulation of complex chemical and biochemical systems. Computers and Biomedical Research 2,3 1-44 (1968). GARFINICEL, D. Lactate permeation. Circulation Research 38, (Suppl. l), 1-13-I-14 (1976). GARFINKEL, D., ACHS, M. J., KOHN, M. C., PHIFER, J. & ROMAN, G.-C. Construction of more reliable metabolic models without repeated solution of the differential equations composing them. Proceedings of the Summer Computer Simulation Conference, 493499 (1976). GARFIIVKEL, L., KOHN, M. C. & GARFINKEL, D. Systems analysis in enzyme kinetics. CritiGal Reviews in Bioengineering 2,329-361 (1977). JOHNSON, L. E. Computer models and optimization in physiological kinetics. Criticat Reviews in Bioengineering 2, l-37 (1974). KAO, R., RANNELS, D. E. & MORGAN, H. E. Effects of anoxia and ischemia on protein synthesis in perfused rat hearts. CircutationResearch 38, (Suppl. l), 1-124-I-130 (1976). KOHN, M. C., ACHS, M. J. & GARFINKEL, D. Distribution of adenine nucleotides in the perfused rat heart. AmericanJournal of Physiology 232, R158-R163 (1977). KOHN, M. C. & GARFINKEL, D. Computer simulation of ischemic rat heart purine metabolism. I. Model construction. AmerzbnJournal ofPhysiology232,H386-H393 (1977). NEELY, J. R., ROVETTO, M. J., WHITMER, J. T. & MORGAN, H. E. Effects of ischemia on function and metabolism of the isolated working rat heart. American 3ournalof Physiology 225,65 l-658 ( 1973). NEELY, J. R., WHITMER, J. T. & Rovzrro, M. J. Effect of coronary blood flow on glycolytic flux and intracellular pH in isolated rat hearts. Circulation Research 37, 733741 (1975). NELSON, T. E., KOLB, E. & LARNER, J. Purification and properties of rabbit muscle amylo-1,6-glucosidase-oligo-1,4-t 1,4 transferase. Biochemistry 8, 1419-1428 (1969). OPIE, L. H. & MANSFORD, K. R. L. The value of lactate and pyruvate measurements in the assessment of the redox state of free nicotinamide-adenine dinucleotide in the cytoplasmofperfused rat heart. EuropeanJournaLofClinicalInvestigation 1,295-306 (1971). REGAN, T. J., EFFROS, R. M., HIAIDER, B., OLDEWURTEL, H. A., ETTINGER, P. 0. & AHMED, S. S. Myocardial ischemia and cell acidosis: modification by alkali and the effects on ventricular function and cation composition. American 3ouml of Cardiology 37, 501-507 (1976). ROMAN, G.-C. Ph.D. dissertation, University of Pennsylvania (1976). ROVETTO, M. J., LABERTON, W. F. & NEELY, J. R. Mechanisms ofglycolytic inhibition in ischemic rat hearts. Circulation Research 37,742-751 (1975). ROVETTO, M. J., WHITMER, J. T. St NEELY, J. R. Comparison of the effects of anoxia and whole heart ischemia on carbohydrate utilization in isolated working rat hearts. ACHS,
Circulation Research 32,699-711 (1973). SPENCER, T. L. & LEHNINOER, A. cells. BiochemicalJournal 154,#5--414 WILLIAMSON, J. R. Glycolytic control
L. L-Lactate transport in Ehrlich ascites-tumor (1976). mechanisms. II. Kinetics of intermediate changes in perfused rat heart. Journal of Biological Chemistry
during the aerobic-anoxic transition 241,5026-5036 (1966). WILLIAMSON, J. R., SCHAFFER, S. W., FORD, C. & SAFER, B. Contribution of tissue acidosis to ischemic injury in the perfused rat heart. Circulation 53 (Suppl. I), 1-3-I-14 (1976).
of Molecular
and Cellular Cardiology
Computer
10,779-796
Simulation of Entry into Output in the Ischemic MICHAEL
Department
(1978)
of Computer University (Received
C. KOHN
Glycolysis Rat Heart
AND DAVID
and Information Sciences, Moore of Pennsylvania, Philadelphia,
29 September
and Lactate
GARFINKEL School of Electrical Engineering Pa. 19104, U.S.A.
1977, accepted in revisedform
20 April
02,
1978)
M. C. KOHN AND D. GARFINKJIL. Computer Simulation of Entry into Glycolysis and Lactate Output in the Ischemic Rat Heart. 3oumal of Molecular andcellular CardioloQ (1978) 10,779-796. A computer model of glucose uptake, glycogenolysis, and export of pyruvate and lactate in the ischemic, isolated rat heart was constructed. This model predicts that, after an initial burst of glycogenolysis which consumes the outer glycogen branches, glycolysis is limited by glucose uptake. Reduced glucose uptake in this model then results from decreased rate of substrate delivery despite increased afiinity of the cell membrane carrier for glucose. Lactate accumulation in this model is due partially to its incomplete washout from the interstitium at low coronary flow rates and partially to a dramatic reduction in the activity of a cell membrane lactate permease. KEY WORDS: Glucose utilization; Glycogenolysis; Cardiac &hernia; Cardiac interstitium.
Lactate
production;
Lactate
export;
1. Introduction Utilization of exogenous glucose by the ischemic, isolated rat heart is suppressed in hearts receiving approximately 10% of normal coronary flow [12, 191 and accelerated in hearts receiving 20 to 40% of normal flow [13, 181. The reduced glycolytic rates in severely ischemic rat hearts occur despite changes in creatine phosphate and adenine nucleotide tissue contents similar to those observed for anoxic hearts where glycolysis is greatly increased [19]. On the other hand, tissue lactate accumulates to a much greater degree in ischemic hearts than anoxic hearts [ 191. A possible explanation for these consequences of reduced coronary flow was suggested by our simulation study of the depletion of adenine nucleotides in ischemic rat hearts [II]. Local equilibration between the perfusate and the inter-
stitium appeared to be diminishing the net rate of transfer of material between these two compartments. This effect might also reduce glucose delivery to, and lactate
washout
from,
the interstitial
fluid.
Such
behavior
was
indicated
by
preliminary experiments with a crude model and supported by experimental data on changesin cardiac lymph composition in ischemia [3]. 0022-2828/78/090779+
18 $02.00/O
0
1978 Academic
Press Inc. (London)
Limited
780
M. C. KOHN
AND
D.
GARFINKEL
The work reported here represents a fuller exploration of this subject and is intended to determine how much this behavior contributes to the known effects of ischemia. We have constructed a more realistic computer model of the entry into glycolysis (glucose uptake plus glycogenolysis) and the cell membrane transport of pyruvate and lactate in the ischemic, isolated rat heart based on the experimental studies of Neely et al. [12] and Rovetto et al. [19], and on the perfused rat heart computer model of Achs and Garfinkel [fl. The glycolytic enzyme portion of this model was abstracted for inclusion here, and representations of physiological processes such as membrane transport were added or readjusted for the new physiological conditions. The parameters specifying these physiological processes are not merely mathematical conveniences, but do have biological significance, although they will have different values for different experimental preparations. This model is a mechanistic representation of the molecular processes known to be involved in this portion of metabolism (with approximations as required for the complexity of glycogenolysis, as detailed in the appropriate section below). It will not properly represent severe long-term &hernia after there is structural damage to membranes or alteration of properties of enzymes. The simulation results describe processes occuring during global ischemia in the isolated rat heart perfused with oxygenated salt solution [12], and they may not reflect the quantitative behavior of other preparations. However, we have encountered similar behavior in the course of simulating other ischemic heart preparations. Therefore, we believe that similar effects are likely in severely ischemic cardiac tissue in physiologically more realistic situations.
2. Model-building
Methodology
Our model-building techniques have been reviewed in detail elsewhere [S, 71. Only a brief summary is appropriate here. Model construction begins by evaluating the flow of material through the appropriate metabolic pathways, applying stoichiometric mass balance techniques routinely used in chemical engineering. The results are expressed in flux maps (exemplified in Figure 1) showing material flow through enzymes and membranes at a given time (2 min after start of ischemia in Figure 1). Metabolites for which tissue totals were experimentally determined must also be distributed between compartments by methods described elsewhere [lo]. In the present model the only compartments considered are the cytoplasm, the interstitium, and the perfusate. The time derivative of each metabolite concentration is determined as the sum of the fluxes producing it minus the sum of the fluxes consuming it. The activity of each enzyme (or membrane transport mechanism) is specified by its rate law, an algebraic relation between the enzymatic velocity and the enzyme’s kinetic parameters and the levels of its substrates, products, and inhibitors, and whose form is determined by the enzyme mechanism. The formal
781
I
lnterstltium
1 I
-Perfusote
Glycogen inner .bmnches
lf
0.115 + Wycogen ’ outer branches 13.574 t
GIP
I
13.566
G6P
t Gtucosa
5.362
W. -5.352
5 18504 F6l’ 1
Glucose
t1803
Pyruvote
-
Glucose
‘I I
18532
J FDP
. . .
I
30.701
Pyruvofe
I
4
0.157
+
Pyruvote 0.153-w
30.535
Imtote
t Locfofe
-
30.593+
I I I
! 1 FIGURE 1. Diagram of the metabolism 2 min of ischemia. Numbera are in *01/g net flow through re-vcmiblc reactions.
Lactatr 26.1044
included in this model, dry wt min. The small
in the form of a flux map after arrows indicate the direction of
782
M.
C. KOHN
AND
D.
GARFINKJXL
techniques for determining these rate laws have been reviewed elsewhere [7]. When combined, the rate laws make up most of the set of differential equations which constitute the model. These equations are then solved by numerical methods, and the results are compared with experimental observation. A successful model must therefore fit both the properties of the constituent enzymes and the particular set of whole-tissue observations which is being simulated. In the course of model construction, a considerable body of numerical data and other information is abstracted from the literature. This is used to formulate hypotheses explaining the sequence of biochemical and physiological events leading to the observed behavior of the biological preparation. The modeling process checks that these hypotheses are internally consistent and actually explain (quantitatively) the experimental observations. Several metabolic processes which lie outside the scope of the simulation, but whose products affect the metabolism simulated, are represented by time-dependent functions. These functions are normally time profiles of metabolite concentrations. The time profiles of these functions and the related parameter values which will enable us to reproduce the observed behavior are determined by algebraically solving the appropriate rate law for the unknown metabolite concentration required to reproduce the observed flux through the corresponding enzyme.
3. Construction
and Behavior
of the Model
Since measurements of tissue glycogen content are not sufficiently reliable for the estimation of the net glycogenolytic flux, we had to construct this portion of the model by iterative procedures. We estimated the net rate of glycogenolysis from tissue glycogen measurements [19]. We computed the net glycolytic flux (through phosphofructokinase) from this value and the measured glucose uptake rate. The concentrations of several small molecules involved in determining this net flux, because they are variables in the appropriate enzymatic rate laws, were calculated as described below. These chemical concentrations were used to compute new glucose uptake and glycogenolysis rates, and the process was repeated until the flux changed by less than 0.1 o/obetween repetitions. Further details are given below.
Be&or
of related small nm!ecules
Because several glycolytic enzymes are sensitive to adenine nucleotide levels, pH, and cytosolic Mgs+ level, these were determined by a previously described procedure [IO]. Briefly, total cytosolic ATP and ADP are computed by subtracting
GLUCOSE
AND
LACTATE
TRANSPORT
IN
CARDIAC
ISCHEMIA
783
from the total tissue levels the mitochondrial content calculated from the respiration rate. Then cytosolic AMP was computed from the adenylate kinase equilibrium, and cytosolic pH from the creatine kinase rate law. Cytosolic Mga+ was adjusted to yield the mixture of chelated, protonated, and free forms of the adenine nucleotides which reproduces the current flux through phosphofructokinase. As all these cytoplasmic factors are interdependent, the procedure is iterated by computer until the adenine nucleotide distribution is constant to within 0.1 o/o between consecutive iterations. The time profile of the glucose-l-phosphate level was then computed [q as that necessary to reproduce the flux through phosphoglucomutase for the several time points. Similarly the time profile of fructose-6-phosphate was determined from the phosphohexose isomerase flux, and that of glucose from the total flux through hexokinase I and II. Measured values of glucose-6-phosphate and fructose-1,6diphosphate were used in these calculations. Debranching
enzyme
We found that the original model of the glycogen debranching enzyme in our perfused rat heart model [I] did not reproduce the observed time profiles of tissue glycogen [19]. This model was designed to reproduce glycogen breakdown over 2 min of anoxia, where only 27% of the initial tissuelevel [2lj is depleted and there is still a considerable amount of outer-branched glycogen remaining. In contrast, 84% of the tissueglycogen was depleted after 30 min of &hernia in the preparation simulated [19]. We interpret the observed glycolytic flux profiles to indicate that debranching enzyme is not limiting at 2 min (since the flux is greater than the tissuecapacity of the enzyme [14]), but that it doesbecome limiting by 4 min, when glycolytic flux is substantially lower. This enzyme hydrolytically cleaves and rearranges a polymeric substrate with a “binary tree” structure. The number of exposed branch points is a complex function of the number of glucose residues per branch and the total size of the “tree”. As the outer branches are degraded, branch points are exposed, inner branches are converted to outer branches and degraded, and the potential concentration of branch points falls exponentially. The fractional number of exposed branch points falls faster than the fractional number of glucose residues.We have introduced sigmoidal substrate activation of this enzyme into the model to represent the exponential decline in the number of branch points, as each successivetier of glycogen branches is degraded, together with sigmoidal inhibition of the enzyme by outer branches to represent their shielding of underlying branch points. As we are considering the inner branches asthe substrate of the enzyme, the above corrections are needed for the highly degraded glycogen we have here, but not for the less degraded glycogen represented in our previous model [I] (where hydrolysis is still accurately represented after the correction).
784
M. C. KOHN
AND
D. GARFINKEL
The resulting rate law that we used for the debranching
“=-(q[($)”
+ l{Tl}
[@)”
enzyme is
+ l]
where (in terms of glucose residues) Vmax = 4.0 ~mol/min-g dry wt, K, = 5.56 maa, XI = 16.7 mM, KO = 0.82 n-m, and I and 0 denote the concentration of glycogen inner and outer branches respectively. These values were empirically determined as those required to reproduce the tissue glycogen level upon solution of the final differential equations (Figure 2). As the initial ratio of inner to outer branch glycogen and the cyclic AMP time profile were not measured and may be preparation-dependent, these were treated as parameters, and their values were selected so as to reproduce the observed tissue glycogen profile upon integration. We estimated that glycogen was initially 56% inner branches and 44% outer branches. The cyclic AMP level was computed to rise from 0.5 nmol/g dry wt initially to 4.0 nmol/g dry wt at 1 min, after which it slowly fell, reaching its steady state value of 0.75 nmol/g dry wt at 4 min. Ovedglycogen
metabolism
The modified glycogen metabolism model consists of differential equations for glycogen synthetases and phosphorylases, branching and debranching enzymes, phosphoglucomutase, and equations for the interconversion of synthetase I and D
FIGURE
2. Comparison
of computed
Time (mid and olxerved
[18] tissue glycogen
content.
GLUCOSE
AND
LACTATE
TRANSPORT
IN CARDIAC
ISCHEMIA
785
and phosphorylase Q and b [I]. It was integrated using a restructured version [17] of our simulation program [4]. The resulting glycogen profile is given in Figure 2. The glycogen outer branches fall from 40 pmol/g dry wt initially to 24.63 pmol/g dry wt after 2 min, as required by our flux map (Figure 1) and then to 0.38 pmol/g dry wt, after 4 min. This strongly suggests that the remaining glycogen resembles a limit dextrin whose inner branches are starting to be converted to outer branches. If the initial distribution of glycogen between inner and outer branches were different, a different cyclic AMP profile might produce similar behavior, but the total amount of glycogen degraded would differ from that observed. Therefore we believe that the above represents a unique set of parameters for the preparation simulated here and corresponds with the biochemical events actually occuring in the tissue. Glucose uptake Passive diffusion of glucose between the perfusate and the interstitial fluid was represented as a reversible, first-order process. We had previously [II] deduced that the rate constants for this process in the perfused rat heart with high coronary flow rate should be 0,742 min-1 for nonionic small molecules. Coronary flow continually decreases with time in the preparation we are modeling [12]. The increased residence time of the perfusate in the tissue should then tend to reduce the local concentration gradient between the perfusate and interstitial fluid and result in lower net rates of diffusion for the whole heart. Accordingly, we allowed the diffusion rate constant to be a function of the coronary flow rate. The function selected after a series of trials was : k = 0.742 exp [(F-F,)/6.2973], where F and F, are the instantaneous and initial coronary flow rates, respectively. The denominator in the exponent yielded a smoothly decreasing diffusion rate, ultimately producing a 90% reduction for a comparable reduction in coronary flow. Other functions tried predicted such rapid depletion of interstitial glucose that the differential equations were insoluble. The differential equations constituting the submodel representing this process were integrated to determine the time profile of cytosolic glucose profile is for the interstitial glucose level; determination described above. The kinetics of glucose transport across the cell membrane are modified by anoxia [I] with changes in both K, and Pm=. Similar changes are expected here, as ischemic hearts are likely to be hypoxic. Therefore, we allowed the rate constants for inward glucose transport in our aerobic rat heart model [I] (identical MichaelisMenten kinetics in both directions) to vary sigmoidally with time : (t’&)“Ak k=k,+ s + (qt,)” t’ = t-to
786
M. C. KOHN
AND
D. QARFINKEL
where k. is the initial rate constant, Ak is the maximal change in k at final steady state for both glucose binding (kl) and turnover number (kg), to is the time when the transition begins, t+ is the half-time for the transition to the new kinetics, and S and n are parameters. A sigmoidal function was selected because it has a time derivative of zero at the beginning and end of the transition and allows the rate constant to change smoothly to its new value. Those values of the parameters which most closely reproduced the observed glucose uptake rates [I91 were determined by formal optimization techniques [7, 81 using the interstitial and cytosolic glucose profiles determined above. Optimal parameter values are : to = 0.0 min (i.e. this change actually begins immediately at the start of the ischemia) t+ = 25.7 min after the start of the transition Ab = 7.332 PM-~ min-1 (glucose binding) Akz = 1678 min-1 (turnover number)
n = 2.75 s = 3.77. This results in a change of kinetic parameters from the aerobic values, I& = 8.89 mu and V- = 11.52 ymol/min-g dry wt (which are the same as those previously determined [I] for the aerobic heart without insulin), to the following values after 30 min of ischemia: I& = 3.87 mM and Vmax = 12.39 pmol/min-g dry wt. These parameter values enabled us to reproduce the observed glucose uptake rate (Figure 3) and indicate that the glucose carrier increases both its affinity for glucose and its maximal permeation rate.
0.
I 0
I 6
I
I 12 Time
FIGURE
3. Comparison
of computed
I
I 18
I
I 24
I
glucose
uptake
30
(mini
and observed
[l8]
rate.
GLUCOSE 2.0
AND
LACTATE
TRANSPORT
IN
787
ISCHEMIA
4
I
I
1
I
I
I
I
I
I
1 6
I
I 12
1
I 18
I
I 24
I
OO
Time
FIGURE
CARDIAC
4. Comparison
of computed
and observed
30
(min)
[18] glucose&phosphate
tissue content.
The glucose uptake and net glycogenolysis fluxes at each of several timerpoints were computed and the results were used to construct more accurate flux maps, as described above. A new adenine nucleotide distribution was computed, and then this entire process was iterated until the flux maps did not change by more than 0.1 oh between successive cycles. The optimal parameter values cited above were obtained after five such iterations. The portion of our perfused rat heart model [lj including the entry into glycolysis as modified above through the glycolytic sequence to phosphofructokinase was integrated as a check on our procedure. The resulting glucose 6-phosphate profile was in good agreement with the available data (Figure 4), indicating internal consistency of the model. Pyruvate and la&ah tranqbrt Although metabolite levels as a function of time for the preparation we are modeling [12, 191 were determined for a recirculating perfusate, the lactate output rates were determined with a flow-through perfusion. Possibly as a result of this difference, more lactate was observed in the nonrecirculated perfirsate than could be accounted for by the sum of glucose units taken up from the per&sate or produced from glycogen [I91 in the recirculating perfusate experiment. Because strict material balance must be maintained in order to produce an internally consistent model, we elected to ignore the lactate output data while attempting to reproduce the observed tissue lactate levels. Since the initial lactate export rate is more likely to be accurate, we obtained the initial rate of pyruvate oxidation as the difference
788
M.
C. KOHN
AND
D.
GARPINKEL
between the rates of pyruvate and lactate production. The initial rate ofendogenous fatty acid oxidation was then computed as that required to account for the observed respiration rate. We assumed that the fatty acid oxidation profile had the same curve shape as the observed respiration rate [l.?] and extrapolated fatty acid oxidation accordingly. We computed the rate of pyruvate oxidation at each time point by difference. Throughout most of the experiment pyruvate oxidation is small, and the model is thus not sensitive to errors in its value. We estimated the rate of conversion of pyruvate to alanine from measured amino acid levels in aerobic and ischemic hearts [9]. With these two rates, tissue lactate data, and the glycolytic flux computed as above we were able to draw tentative flux maps for the pyruvate-lactate system. The diffusion of lactate between the interstitial fluid and the perfusate was treated like that of glucose, but an initial rate constant of 0.742 min-i required that interstitial lactate exceed the total tissue lactate at times of high glycolytic flux. Therefore, we treated this rate constant as a parameter. A value of 8.327 min-1 at zero time enabled us to reproduce the rate of lactate appearance in the perfusate given on the flux maps while leaving a significant amount of cytosolic lactate at times of rapid lactate production. This larger diffusion rate constant may be due to the smaller size or ionic nature of lactate. The compartmentation of tissue lactate between the cytosol and interstitial fluid which would reproduce the lactate output rate on the flux maps for the several time points was then calculated. It should be noted that the rate constants for diffusion between the interstitial fluid and the perfusate are parameters in our system, and their values should not be taken literally. Tissue pyruvate data were corrected for the mitochondrial content using the ratio of spaces (9 : 1, cytosol :mitochondria) in the full heart model, and assuming uniform concentration. The remainder was initially distributed between the cytosol and interstitium so as to produce equilibrium at the plasma membrane pyruvate carrier in our model [I]. Py ruvate diffusion between the interstitial fluid and perfusate was treated in an identical manner to lactate. The pyruvate transport portion of the full rate heart model was integrated separately to obtain a time profile of cytosolic pyruvate and the net pyruvate export rate. The flux through lactate dehydrogenase on the flux maps was corrected for the computed pyruvate output rate. The cytosolic NAD and NADH levels (whose sum was held constant at 2.20 pmol/g dry wt) which would reproduce the corrected flux through this enzyme were then calculated [6j. Our previous simulations [I, 21 suggested that the lactate transport rate was dependent on factors other than the cytosolic lactate level. Spencer and Lehninger [20] found evidence for a pHsensitive plasma membrane lactate carrier (in ascites cells), but construction of a reliable model for its regulation in heart is not possible without more detailed information. Therefore, the activity of the lactate carrier in our model is input as a
GLUCOSE
AND
LACTATE
TRANSPORT
IN
CARDIAC
ISCHEML4
789
time-varying first order rate constant which, when multiplied by the cytosolic lactate level, yields the lactate export rate. A section of the rat heart model comprising lactate and pyruvate transport and lactate dehydrogenase was integrated. The reasonably accurate reproduction of the pyruvate (Figure 5) and lactate (Figure 6) levels serves as a check on the modeling technique. The mathematical behavior of the propagation of errors in a model as complex as ours is too complicated to permit estimation of the confidence intervals for computed concentrations and fluxes. In fact, this problem is presently intractable except for the crudest (linear) approximation. After many years of model construction, we have the impression that uncertainties in our computed values are usually comparable to experimental error. Furthermore, predictions based on published models have generally been confirmed by subsequent experiments.
4. Discussion Glucose transport
The results of our simulation lead us to conclude that the reduced utilization of exogenous glucose by the ischemic rat heart is due to its reduced rate of delivery and not to any biochemical event peculiar to ischemia. Rovetto et al. [19] found that only 6% of the per&sate glucose was extracted on a single passthrough the heart, and they concluded that substrate delivery was not limiting. We compute an extraction ratio of 5.5% for a coronary flow rate of 1 ml/min and a glucose uptake rate of 3 ~mol/min-g dry wt from an insulin-free perfusate containing 11 mM glucose. However, for a coronary flow rate of 15 ml/min for aerobic rat hearts perfused with salt solution, and for a glucoseuptake rate of 6.3 pmol/min-g dry wt, we compute an extraction ratio of only 0.8%. The decreasedrate of delivery results in a reduced glucose uptake rate despite the greater degree of extraction. The increased extraction ratio may be due to the longer residence time of the perfusate in the coronary vasculature as well asthe decreasedK,,, for glucose uptake, and the reduced glucose uptake rate may be due to depletion of interstitial glucose (Figure 7). In the absence of any published information, we set the initial interstitial glucose concentration equal to the perfusate concentration as an upper limit. This approximation, aswell as those involved in obtaining the rate constant for diffusion of glucose between the perfusate and interstitial fluid, may result in the computed fall of interstitial glucose (Figure 7) being greater than that which actually occurs in the tissue. As Rovetto et al. [19] were not able to measure tissueglucose in the absenceof insulin, a direct comparison is impossible. This helps explain the increased glucose uptake rates observed for ischemic rat hearts with 20 to 40% of normal coronary flow. Here the increased affinity of the cell membrane carrier for glucose overcomes the reduced flow rate. In a computer
790
M. C. KOHN
AND
D. OARFINKEL
Interstitial I OO
I
I 6
I 12 Time
I
I
t 18
I
I 24
I
I 30
(min)
FIGURE 5. Comparison of computed pyruvate levels with observed 1eve.l.s: (e) J. R. Necly, pexsonal communication. Total pyruvate does not include mitochondrial data points are corrected for this.
J
from [18]; pyruvate,
Interstitial /
I
6
FIGURE
6. Comparison
ofcomputed
t
I
12 Time
and observed
I
I
I
18
I
I
I
24
30
(min)
lactate
levels:
(A)
from
[I-?];
(0)
from
[18].
GLUCOSE
AND
o-
FIGURE
LACTATE
’
’ 6
TRANSPORT
’
7. Computed
’ 12 Time
’
IN CARDIAC
’ 18
’
’ 24
ISCHEMIA
’
791
30
(mid
interstitial
glucose
level.
experiment with our model, we allowed the coronary flow rate to fall only to 6 ml/mm and used the same parameters for the increase in the glucose uptake rate constants as given above. The result (Figure 7) was that after an initial moderate decline, the glucose uptake rate increased. By suitable adjustment of the glucose carrier parameters, we could compute an increase in glucose uptake above the control aerobic value. The changes in the activity of the glucose transport mechanism are likely to reflect changes in the behavior of the protein molecules involved. It is unlikely that these changes are produced directb by changes in physical factors such as coronary flow rate or perfusate pressure. We suggest that they are instead under metabolic control, resulting from changes in concentration of small molecules which bind to the proteins involved. Gbcogen metabolism Neely et al. [12] observed that the final stage of ventricular failure begins alter 12 min of low coronary flow perfusion. Our simulation suggests that because of the reduced glucose uptake and respiration rates, severely ischemic cardiac tissue remains viable only as long as there is sufficient glycogen to maintain a minimal ATP level by glycolysis. (The ATP level is falling most rapidly at about 12 mm [22].) About 17% of the glycogen initially present in the tissue is depleted after
792
M.
C. KOHN
AND
D.
GARFINKEL
2 min of &hernia. Most of this is due to phosphorylase a activity. By 12 min 69% of the original glycogen is consumed. Nearly all of this additional depletion is due to phosphorylase b activity. By 16 min the glycogen consumption has reached nearly the same level as at 30 min (no significant statistical difference), so that glycogenolysis must become much slower at about this time. The curve in Figure 2 exhibits a marked change in slope at about 4 min. The net glycogenolysis rate drops from 14.6 pmol/min-g dry wt at 3 min to 4.86 pmol/min-g dry wt at 4 min. At this point glycogenolysis becomes limited by the activity of the debranching enzyme, even though the cytosolic AMP level (325 nmol/g dry wt) is sufficient to permit substantial phosphorylase b activity. This effect is not expected intuitively and is not altered by moderate changes in the kinetic parameters in the debranching enzyme rate equation. In a computer experiment with our model, we doubled the binding constant (Ko) for inhibition of this enzyme by undegraded glycogen outer branches. The result was that debranching activity began earlier, but the much greater tissue capacity of phosphorylase b still caused a depletion of outer branches and a dramatic decrease in the slope of the glycogen profile at about 4 min. The lag in conversion of glycogen inner branches to outer branches sufficient to meet the demand of phosphofructokinase for hexose phosphate results in the computed transient fall in glucose 6-phosphate at 4 min (Figure 4). Determination of whether any of this is an artefact of our crude model of the debranching enzyme, by building an accurate and biochemically realistic model of this enzyme, would be a substantial research project in itself.
Time
(min)
FIGURE 8. Computed glucose uptake rate if coronary 4 min, using kinetic parameters for glucose uptake identical
flow drops from 15 mI/min to those in Figure 3.
to 6 ml/min
in
GLUCOSE
AND
LACTATE
TRANSPORT
Lactate
IN
CARDIAC
793
ISCHEMIA
transport
Lactate export also seems to be affected by reduced coronary flow, but our simulation suggests that unknown factors are involved in this process. The results of a preliminary investigation of this process [5] are qualitatively similar to those of the more rigorous treatment reported here. Our previous simulations [I, 21 have indicated that the lactate export rate generally follows the flux through lactate dehydrogenase and is effective at preventing excessive accumulation of cytosolic lactate in normal-flow situations. Because the lactate carrier seems not to be controlled entirely by the lactate concentration, we describe it as a “permease” to indicate that its activity is regulated like that of enzymes. In the present simulation lactate permease is activated during the initial spurt of glycogenolysis (0 to 4 min) but not afterwards, even though lactate production exceeds that for the initial high flow conditions (Figure 9). Part of this effect may be explained by the computed rise in interstitial lactate from 0.20 ells initially to 2.44 mM after 30 min, which is largely due to the reduced rate of diffusion between the interstitial fluid and perfusate. Lactate may be exported by a cell membrane carrier with a lower Km for extracellular than intracellular lactate, analogous to the pyruvate carrier in our model [I]. The accumulation of lactate in the interstitium may result in appreciable back-uptake of lactate, thus reducing the net export rate and hence the computed lactate permease activity.
Lactate
permease
Time
FIGURE
9. Computed
lactate
production
IminI
rate and lactate
permease
activity.
794
M.
C. KOHN
AND
D. GARF’INKEL
The actual lactate output process must be more complex than described above. Interstitial lactate is computed to be 4.12 mu and cytosolic lactate to be 0.80 mu at 2 min, when lactate permease is maximally active. The permease is apparently transporting lactate against a concentration gradient. After 30 min however, interstitial lactate is computed to be only 4.22 mM while cytosolic lactate is 37.9 nq and the permease is nearly inactive. This suggests that the lactate carrier is under metabolic control as well. The cytosolic NADH level deduced for the ischemic rat heart preparation we are simulating is given in Figure 10. During the first few minutes, when lactate permease activity is high, the cytosol is relatively oxidized. Later, when the cytosol is much more reduced, lactate permease activity is very low. This result suggests that NADH and/or some metabolite which is closely correlated with highly reduced cytosol is an intracellular inhibitor of the lactate carrier. Spencer and Lehninger [2U] studied pyruvate and lactate transport in ascites cells and found it to be dependent on pH; low pH reduces the Km of the permease for lactate. Figures 6 and 9 indicate that increased lactate export correlates weakly with both the initial fall in pH and the second fall after 20 min (Figure 10). However, it is obvious that such an effect is superimposed on other, more important, biochemical controls. These may also be affected by the (pathological) low flow condition; we had previously concluded from models of normalflow experiments [I, 21 that lactate permease tends to be activated by rapidly rising H+ and lactate levels, a relationship that does not hold or is inapplicable here after the first few minutes.
- 6.4
I OO
FIGURE
t 6
I
10. Computed
t 12 Time
I
cytosolic
I 18
t
I 24
I 30
(min)
pH and cytcsolic
NADH
level.
GLUCOSE
AND
LACTATE
TRANSPORT
IN
CARDIAC
ISCHEMIA
795
If lactate permease is under metabolic control, it presents a logical site for pharmacological intervention in the treatment of ischemia. In view of the common belief that lactate accumulation in ischemia is harmful [16, 18, ,221, activation of lactate permease by a suitable drug may be an important potential contribution to the therapeutic management of ischemia. When lactate permease activity falls most greatly below the lactate production rate (at 10 min), the lactate export rate (permease activity times cytosolic lactate level) nevertheless equals 69% of the flux through lactate dehydrogenase. Thus, even a moderate increase in lactate permease activity might be sufficient to prevent deleterious accumulations of lactate. Rovetto et al. [19] considered the possibility that lactate actually crossesthe membrane by passive diffusion as unionized lactic acid. We believe that the complex nonlinear behavior we deduced for lactate export is more readily accommodated by active transport. Finally, these investigators assumedthat the interstitial lactate concentration was the same as that measured in the perfusate. Our simulation suggeststhat this (common) assumption is inaccurate. After 30 min of ischemia lactate accumulated in the perfusate to only 444 FM compared to 2.44 mM computed for the interstitium. The experimental results of Opie and Mansford [15] confirm that there is a concentration gradient between cytosol and interstitium for lactate and pyruvate. Comparison of Figures 5, 6, and 10 indicate that the commonly used method of estimating the cytosolic NADH/NAD ratio from the tissue lactate/pyruvate ratio is inaccurate here. The L/P ratio is often in error here by a factor of several times and sometimeschangesin the wrong direction with time. This method, ascommonly used, is basedon the assumptionsthat lactate dehydrogenase is sufficiently active to equilibrate tissue lactate, tissue pyruvate, NADH, and NAD, and that the intracellular pH is constant, usually 7.0. The early location of most of the lactate in the interstitial fluid rather than within the cells, and the fall in pH, deviate considerably from theseassumptions. We believe that our results indicate the need for a closer investigation of the nature of metabolic control of glucose and lactate transport acrossthe cell membrane, that alterations of these transport processesare very important, perhaps the primary, contributors to the harmful effects of ischemia, and that events at the biochemical level follow from thesephysiological changes. Acknowledgements
We thank J. R. Neely of the Milton S. Hershey Medical Center of Pennsylvania State University for his helpful criticisms during the preparation of this manuscript. This work has been supported by NIH Grant HL 15622. REFERENCES 1. ACHS, M. J. & GARFINKEL, D. Computersimulationof energy metabolismin anoxic perfusedrat heart. American Journal of Physiology 232, R164-R174 (1977).
796 2. 3. 4. 5. 6.
7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
17. 18. 19. 20. 21. 22.
Id.
C. KOHN
AND
D.
GARFINKEL
M. J. & GARFINKE L, D. Computer simulation of rat heart metabolism after adding glucose to the per&sate. American Journal of Physiology 232, R175-R184 (1977). FEOLA, M. & GLICK, G. Cardiac lymph flow and composition in acute myocardial ischemia in dogs. AmericanJournal of Physiology 229,44-48 (1975). GARFINKEL, D. A machine-independent language for the simulation of complex chemical and biochemical systems. Computers and Biomedical Research 2,3 1-44 (1968). GARFINICEL, D. Lactate permeation. Circulation Research 38, (Suppl. l), 1-13-I-14 (1976). GARFINKEL, D., ACHS, M. J., KOHN, M. C., PHIFER, J. & ROMAN, G.-C. Construction of more reliable metabolic models without repeated solution of the differential equations composing them. Proceedings of the Summer Computer Simulation Conference, 493499 (1976). GARFIIVKEL, L., KOHN, M. C. & GARFINKEL, D. Systems analysis in enzyme kinetics. CritiGal Reviews in Bioengineering 2,329-361 (1977). JOHNSON, L. E. Computer models and optimization in physiological kinetics. Criticat Reviews in Bioengineering 2, l-37 (1974). KAO, R., RANNELS, D. E. & MORGAN, H. E. Effects of anoxia and ischemia on protein synthesis in perfused rat hearts. CircutationResearch 38, (Suppl. l), 1-124-I-130 (1976). KOHN, M. C., ACHS, M. J. & GARFINKEL, D. Distribution of adenine nucleotides in the perfused rat heart. AmericanJournal of Physiology 232, R158-R163 (1977). KOHN, M. C. & GARFINKEL, D. Computer simulation of ischemic rat heart purine metabolism. I. Model construction. AmerzbnJournal ofPhysiology232,H386-H393 (1977). NEELY, J. R., ROVETTO, M. J., WHITMER, J. T. & MORGAN, H. E. Effects of ischemia on function and metabolism of the isolated working rat heart. American 3ournalof Physiology 225,65 l-658 ( 1973). NEELY, J. R., WHITMER, J. T. & Rovzrro, M. J. Effect of coronary blood flow on glycolytic flux and intracellular pH in isolated rat hearts. Circulation Research 37, 733741 (1975). NELSON, T. E., KOLB, E. & LARNER, J. Purification and properties of rabbit muscle amylo-1,6-glucosidase-oligo-1,4-t 1,4 transferase. Biochemistry 8, 1419-1428 (1969). OPIE, L. H. & MANSFORD, K. R. L. The value of lactate and pyruvate measurements in the assessment of the redox state of free nicotinamide-adenine dinucleotide in the cytoplasmofperfused rat heart. EuropeanJournaLofClinicalInvestigation 1,295-306 (1971). REGAN, T. J., EFFROS, R. M., HIAIDER, B., OLDEWURTEL, H. A., ETTINGER, P. 0. & AHMED, S. S. Myocardial ischemia and cell acidosis: modification by alkali and the effects on ventricular function and cation composition. American 3ouml of Cardiology 37, 501-507 (1976). ROMAN, G.-C. Ph.D. dissertation, University of Pennsylvania (1976). ROVETTO, M. J., LABERTON, W. F. & NEELY, J. R. Mechanisms ofglycolytic inhibition in ischemic rat hearts. Circulation Research 37,742-751 (1975). ROVETTO, M. J., WHITMER, J. T. St NEELY, J. R. Comparison of the effects of anoxia and whole heart ischemia on carbohydrate utilization in isolated working rat hearts. ACHS,
Circulation Research 32,699-711 (1973). SPENCER, T. L. & LEHNINOER, A. cells. BiochemicalJournal 154,#5--414 WILLIAMSON, J. R. Glycolytic control
L. L-Lactate transport in Ehrlich ascites-tumor (1976). mechanisms. II. Kinetics of intermediate changes in perfused rat heart. Journal of Biological Chemistry
during the aerobic-anoxic transition 241,5026-5036 (1966). WILLIAMSON, J. R., SCHAFFER, S. W., FORD, C. & SAFER, B. Contribution of tissue acidosis to ischemic injury in the perfused rat heart. Circulation 53 (Suppl. I), 1-3-I-14 (1976).