interrelations between glycolysis and the hexose monophosphate shunt in erythrocytes as studied on...

BioSystems, 22 (1988) 1'9-36 19 Elsevier Scientific Publishers Ireland Ltd.

Interrelations be tween glycolysis and the hexose monophosphate shunt in erythrocytes as studied on the basis of a mathematical model*

Ronny Schuste]', Hermann-Georg Holzhfitter and Gisela Jacobasch Institut f~r Biochemie des Bereichs Medizi~z der Humboldt-Universitdt zu Berlin, 1040 Berlin, Hessische Str. 34 (G.D.R.)

(Received October 22n~[, 1987) (Revision received March 9th, 1988)

A mathematical model is presented which comprises the reactions of glycolysis, the hexose monophosphate shunt (HMS) and the glutathione system in erythrocytes. The model is used to calculate stationary and time~iependent metabolic states of the cell in vitro and in vivo. The model properly accounts for the following metabolic features observed in vitro: (a) stimulation of the o:ddative pentoee pathway after addition of pyruvate due to a NADP~iependent lactate dehydrogen- ace as coupling enzyme between glycolysls and the oxidative pentose pathway, Co) relative share of the oxidative pentose pathway in the total consumption of glucose amounting to approximately 10% in the normal case and to approximately 90% under conditions ,of oxidative stress excreted by methylene blue. From the application of the model to in vivo conditions it is predicted that (e) under normal conditions glycolysis and the HMS are independently regulated by the energetic and oxidative load, respectively, (d) under conditions of enhanced energetic or oxidative load both glycolysis and the HMS are mainly controlled by the hexokinase; in this situation the highest possible values of the energetic and oxidative load which are compatible with cell integrity are strongly coupled and considerably restricted in comparison with the normal case, (e) the stationary states possess bifurcation points at high and low values of the energetic load.

Keywords: Metabolism; Mathematical model; Erythrocyte; Quasi-steady state approximation.

I. Introduction

Over decades the metabolism of the erythrocyte has bee:a the subject of intensive experimental and theoretical investigations. The mature red blood cell exhibits a remark- able metabolic simplicity. Two major pathways are operative in this cell: glycolysis and the hexose monophosphate shunt. In erythrocytes, glycolysis is the only source for ATP. The main function of the pentose phosphate shunt is to form NADPH which is an essential fuel for several redox systems which protect the cell against oxidative damage.

Mathematical models of different complex-

Correspondence to: Dr. Hermann-Georg Holzhiitter. *Dedicated to Prof. S.M. Rapoport on the occasion of his 75th birthday.

ity have been developed to gain deeper insight into the regulation of both pathways. On the basis of very simple "skeleton-models" (Reich and Selkov, 1981) the possible existence of unstable steady states of glycolysis could be demonstrated. In a series of papers (Heinrich et al., 1977; Heinrich and Rapoport, 1974a; Rapoport et al., 1977b) more complex mathematical models of glycolysis have been developed which are suited for a quantitative description of the available experimental data. On applying the control theory (Kacser and Burns, 1973; Heinrich and Rapoport, 1974b) it has been pointed out that the hexokinase-phosphofructokinase system plays a key role in the regulation of glycolysis. Fur- ther steps in the mathematical modelling of glycolysis were to include the osmotic properties of the cell (Brumen and Heinrich, 1984),

0303-2647/88/$03.50 ~ 1988 Elsevier Scientific Publishers Ireland Ltd. Published and Printed in Ireland

20

the degradation of the adenine nucleotides (Schauer et al., 1981) and genetic defects of glycolytic enzymes (Holzhtitter et al., 1985), respectively.

Compared with the large number of glycolytic models only a few mathematical models of the pentose phosphate metabolism of erythrocytes have been developed until now, most of them taking into consideration only the oxidative part of the shunt. Based on a simple "skeleton-model" (Kothe et al., 1975), unstable steady states of the oxidative pentose phosphate pathway have been predicted. Simplified models taking into account only a few regulatory enzymes have been proposed (Ataullakhanov et al., 1981a) to describe experimental data on interrelations between the flux rates through glycolysis and the oxidative pentose phosphate pathway. More recently a realistic model of the HMS has been developed (Thorburn and Kuchel, 1985) and successfully applied to study the regulation of this pathway.

In this paper we present a mathematical model which comprises glycolysis, the HMS and the glutathione system in a realistic man- ner. The model takes into account the time- hierarchy among the various reaction rates in terms of a quasi-steady-state approximation (Romanovsky et al., 1984) which has been performed as described elsewhere (Schauer and Heinrich, 1983).

The model provides a satisfactory quantitative description of in vitro experiments carried out to study interrelations between glycolysis and the hexose monophosphate shunt. Application of the model to in vivo states of the cell predicts -- a strong coupling between the two path-

ways under conditions of an enhanced energetic or oxidative load

-- the occurrence of bifurcation regions in the stationary solutions of the model at high and low values of the energetic load.

2. Description of the model

The scheme of all reactions involved in the

model is shown in Fig. 1. As in a previous model (Holzhiitter et al., 1985), all reaction steps of glycolysis have been taken into account. The ATP~onsuming processes are represented by one overall reaction UAzp~.

Furthermore, we have included all reactions of the oxidative pentose phosphate pathway except the lactonase since little is known about the proportion between the sponta- neous and enzymatic hydrolysis of 6-phos- phogluconolacton. In our model the NADPH produced by the two dehydrogenases of the oxidative pentose pathway is exclusively uti- lized by the glutathione reductase which is by far the most important NADPH-consuming process (Grimes, 1980). The oxidative load u

OX

consists of the overall consumption of reduced glutathione [GSH].

The model does not take into consideration the fact that the glutathione system may protect enzymes and membrane lipids from an oxidative attack. There is a lot of experimental evidence that the cellular GSH/GSSG ratio may be taken as a rough measure for the oxidative stress imposed on the cell and that a lowered rate of glutathione reduction is accompanied by a higher portion of inacti- vated or kinetically-modified enzymes and by alterations in the content and composition of phospholipids due to peroxide formations (Stern, 1985). But the quantitative description of the mechanisms which underly the deterio- ration and recovery of enzyme function are not sufficient for a kinetic modelling.

As an important coupling reaction between glycolysis and the oxidative pentose pathway we have introduced a NADPH<lependent lactate dehydrogenase (Rapoport et al., 1979; Rapoport and Ababei, 1964).

The branching point between the HMS and the nucleotide metabolism of the cell is the synthesis ~of phosphoribosylpyrophosphate which has also been included in the model.

Besides enzyme-catalyzed reactions the model comprises a number of non-enzymatic complex formations: magnesium complex formation of the adenine nucleotides and 2,3-bisphosphoglycerate and binding of NADP and

21

Xy l5P =

~ GSSG \/NADPH ~'---~.I ~'~ ~ J'~"~ \ OSfGR 6P'~GD R5P GAP F6P

GSH " " NADP j j A T P ~ s TK

6PG A~p'~p Rpp / # ~'~ NADP'FI%IpD ~ ADP ATP

NAD'~--J'I ~ ~,o _ ~L ) G u~-~-~-G6P = PG,_ F6P ~_~___F_X~Fp2 ~--~GAP "-~--9-"1,3P.G ~ 3 P G

/ ( L

AMP ADP

U Lac L~"H '~ Py r

NADP NADPH

t N

. , PK P E P

ATP ADP

Fig. 1. Scheme of glycolysis, the hexose monophosphate shunt and the glutathione system in human erythrocytes which are involved in the m~tel. Arrows in both directions refer to reactions near equilibrium, arrows in one direction indicate irreversible reactions. The formation of magnesium complexes and the binding of NADP(H) to proteins are not shown in the scheme.

NADPH to proteins (Kirkman et al., 1986b; Kirkman and Gaetani, 1986a).

The following conservation equations have been used:

A = AMP + ADP + ATP (total sum of adenine nucleotides)

N = NAD + NADH; NP - NADP + NADPH (sum of pyridine nucleotides)

G = GSH/2 + GSSG (reduced and oxidized glutathione)

All calculations have been carried out for pH 7.2 and the (constant) intraceilular phosphate concentration of ~'] = 0.94 raM. The reaction scheme shown in ]Pig. 1 corresponds to a sys-

tern of 20 non-linear differential equations (mathematical model) which are listed in the Appendix. A considerable reduction of this equation system can be achieved if the time- hierarchy among the various reaction rates is taken into account.

According to their activities the reactions can be classified as either slow reactions (represented by single arrows in reaction scheme 1 and designated by velocity expressions u) or as fast quasi-equilibrium reactions (represented by double arrows in Fig. 1 and designated by velocity expressions w).

For the glycolytic enzymes such a subdivision has been performed on the basis of the f ree~nergy profile measured in vivo (Minak- ami and Yoshikawa, 1966). According to these measurements, only a few glycolytic reactions

22

catalysed by hexokinase, phosphofructokinase, pyruvate kinase, 2,3-bisphosphoglycerate mutase, 2,&bisphosphoglycerate phosphatase and ATPase are slow reactions far from the thermodynamic equilibrium.

With respect to the enzymes of the HMS and of the glutathione system the experimental basis for such a subdivision into slow and fast reactions is not so well settled. According to previous models (Thorburn and Kuehel, 1985; Kothe et al., 1975; Ataullak- hanov et al., 1981a), the glucose~phosphate dehydrogenase and the 6-phosphogluconate dehydrogenase are treated as slow non-equilibrium reactions whereas the reactions of the non~xidative part of the shunt catalyzed by the epimerase, isomerase, transaldolase and transketolase are considered as fast equilibrium reactions. The glutathione reductase has been regarded as an equilibrium enzyme (Kothe et al., 1975; Ataullakhanov et al., 1981a) as well as a non-equilibrium enzyme (Thorburn and Kuchel, 1985). Since the mass- action ratio between substrates and products differs considerably from the thermodynamic equilibrium constant we have regarded the glutathione reductase as a slow non-equilibrium enzyme.

The classifications: (i)fast equilibrium reac- t/ons PGI, TPI, ALD, GAPD, PGK, PGM, En, LDH, KI, EP, TK1, TK2, TA, AK, complex formations; (if) slow non-equilibrium reac- t/ons HK, PFK, P~GM, P2Gase, G6PD, 6PGD, Ox, ATPase, GSSGR, PRPPS, PK, NADP- LDH, have been chosen to reduce the size of the original set of differential equations by application of the quasi-steady-state approximation (see Appendix).

The rate equations for those non-equilibrium enzymes and mass-action ratios which have not been already published elsewhere are given in Tables 1 and 2, respectively. The numerical values of some of the kinetic parameters have been altered in comparison with the value reported in the cited reference. This was necessary in order to involve additional experimental data which have not been taken into consideration in the cited paper. The

TABLE 1

Rate equat ions for non~qui l ibr ium enzymes . Maximal activi t ies ( V ) are given in raM/h, affinit ies (K) are given in raM.

A TPase uA~v, = uAr~. [MgATP] kA~v, . = 1.68/h (normal in vivo s ta te)

Phosphoribosylpyrophosphate syntitetase (Becket et al., 1975)

V. [ M e ATP] [ RSP] um~ = (g^~+ [ Mg ATP] ) (/iRa+ [R5P] )

V, = 1.1 KAr~ = 0.03 KR~, = 0.57

Phospltogluconate dehydrogenase (kinetic parameters ca~ culated after Yoshida and Lin, 1976]

V,([6PG]/gemJ ([ NADPI]/KNAve) u~a = Deaominator

Deaominator = / , + [6PG] + ~ ) ~. g, eo, 2..v.o )

0 [NADP~]~ [ATP] [N~_.~_~__~]E [6PG)~ + j+ . -=+ I,'+F J

V = 1575 K ~ i = 0.01 Kep~l = 0.0588 KNADp = 0.018 KNADptl ---- 0.0045 KI~ Q = 0.12 A~AT v ~" 0.154

GSH-consumption %~ = ko= 0.5 [GSH] ko~ = 0.06/h (normal in vivo state)

NADP-dependent lactate dehydrogenase

UN e -- k [NADPHI ] [Pyr] "~ " ~ " - [ l :~') + r .

k = 162/ !1K = 0 .414mM

maximum activities of the non-equilibrium enzymes and the equilibrium constants of some equilibrium enzymes have been slightly modified to adjust the calculated flux rates and metabolite concentrations to the experimental data, they are all close to the values reported for the isolated enzymes.

3. Results

Simulation of in vitro ezperiments

We have simulated the following experi-

TABLE 2

Mass action relations for equilibrium reactions.

23

Enzyme Mass-action relation Equilibrium constant

Experimental Model value value

[Xy15P] 1.5--3 [2] Phosphoribulose q~- [RuSP] epimerase 2.65 [3]

[P.~] Ribose isomerase ¢~= [Ru5P] 3.13 [3]

3.3 [2] 2.6 [4]

2.7

3.0

[ScdTP] [GAP] Transketolase I q~' = [P.SP] [XytSP] 1.02 [2] 1.05

[F6P] [GAP] 1.2 [2] 1.2 Transketolase II q~ffi2= [Ery4P] [XylSP] 10 [1]

[ Ery4P] [ F6P] Transaldolase qrA = [GAP] [SedTP] 1.05 [2] 1.05

[ADP/] [ MgADP] Adenylate kinase q~ = [AMP l] [ MgATP] 0.25 [5] 0.25

[1] Reich and Sel'kov, I981; [2] Barman, 1969; [3] Dische and Shigeura, 1957; [4] Urivetzky and Tsuboi, 1963; [5] Blair, 1970.

ments (Rapoport et al., 1979) demonstrating a coupling between glycolysis and the oxidative pentose pathway.

Ezperiment A. Erythrocytes were given in a glucose-free incubation medium. It was observed that the amount of 2.3P2G degraded within 2 h was not completely accumulated in pyruvate.

Experiment B. Addition of pyruvate to the incubation medium was accompanied by a stimulation of the oxidative pentose pathway.

The interconversion of pyruvate into lactate in the absence of glucose (result of experiment A) is only possible ff NADH or NADPH are continuously delivered by the degradation of the metabolite pools "above" the GAPD reaction. The pyruvate deficit, i.e. the difference between the, degraded 2.3P2G and the

accumulated pyruvate, is proportional (not identical, since there is a recycling of carbon atoms in the HMS) to the total size of these pools.

As a plausible explanation for the result of experiment B the existence of a NADPH- dependent lactate dehydrogenase has been postulated (Rapoport et al., 1979). Similar experimental findings (Omachi et al., 1969; Jablonska and Bishop, 1975) have been inter- preted as indication for either the existence of such an enzyme or a t ranshydrogenase reaction transferring hydrogen directly between NAD and NADP. Therefore, we performed simulations of experiments A and B by taking into account (a) a t ranshydrogenase reaction (NAD + NADPH ~ NADH + NADP) linking the two hydrogen-transferring

24

TABLE 3

Accumulation and degradation of pyruvate in erythrocytes.

Ezperiment A: Erythrocytes were given to a giucose-free incubation medium [1]. The concentration of 2,3-bisphosphoglycerate and of pyruvate was measured at the beginning and after 120 rain of incubation.

Experiment [1] Model

Pyruvate deficit 350 40 AP~G + bPyr ( /~ ) during 120 rain

Ezperiment B: Pyruvate was added to an erythrocyte suspension and the degradation of pyruvate and the C02-formation were measured over 120 min [1].

Pyruvate added Pyruvate degraded C02-formation (+ Pyr) (mmol/ml cell (1~MI120 min) X 100% suspension)

Experiment [1] Model CO=-formation ( - Pyr)

Experiment [I] Model

250 520 505 253 265 1000 919 1150 375 378

[1] Rapoport et al., 1979.

redox couples, (b) a slow NADPH~iependent lactate dehydrogenase. Details of the mathematical computations are given in the Appendix. Only in case (b) could a qualitative agreement with the experimental observations be achieved (cf. Table 3).

The simulated time<lependency of the pyruvate concentration and of the flux through the oxidative pentose pathway for experiment B are shown in Fig. 2. A good quantitative agreement between observed and calculated pyruvate degradation and CO 2- formation could be obtained for experiment B whereas quantitative discrepancies between the observed and calculated pyruvate deficit in experiment A remain. This discrepancy might be due to (i) the to .s imple Michaelis- Menten model applied for the NADPH<lependent lactate dehydrogenase or (ii) an underestimation of the size of the metabolite pools above the GAPD reaction.

Further simulations have been performed to study the regulation of the oxidative pen-

A ~E E

>, Q.

0 .5

!:

0 .5

E O.25 - -

o 1 2

T i m e ( h )

Fig. 2. Simulation of the pyruvate effect on the oxidative pentose pathway. The time-dependent variations of the intraceliular pyruvate (solid line) and of the rate through the oxidative pentose pathway (broken line) are shown. At time t = 0 the concentration of external pyruvate was abruptly increased to 250 pmol/l cell suspension. There is a steep increase in the rate through the oxidative pentoee pathway within a few minutes after addition of pyruvate.

TABLE 4

Stimulation of the oxidtative pentose pathway by methylene blue.

25

Methylene blue Total [1-xq3]glueose utilization Percentage of [1-'C]glucese utilization by

mM/h % G6PD PFK

- - + - - + - - + - - +

Model 1.5 2.75 100 184 7 97.7 93 2.3

Experiment [1] 2 4.86 100 243 11.4 92.4 88.6 7.6 [2] 1.62 4.58 100 283 12.5 87 87.5 13 [3] / / / / 7.5 88 92.5 12

[1] Reigas et al., 1970; [2] Rose and O'Connell, 1964; [3] Brand et al., 1970.

tose pathway. Table 4 comprises experimental data from various sources on the stimulation of the oxidative pentose pathway obtained by methylene blue-induced oxidation of NADPH and theoretical flux rates computed on the basis of the in viLro model. The glucose consumption predicted by the model is slightly smaller than observed in the experiments. The computed relative share of glycolysis and of the oxidative pentose pathway in the glucose consumption (expressed as percentage of total glucose consumption) is in a good agreement with the experimental data. Under conditions of methylene blue stimulation the computed flux through the oxidative pentose pathway is soraewhat higher than the reported experimental values but this difference can b e abolished if the NADH oxidation by methylene blue is taken into account in the computations (not shown).

Ataullakhanov et al. (1981b) have investi- gated the influence of oxidative stress (t-butyl hydroperoxide) on the flux through the oxidative pentose pathway and on the GSSG concentration of human erythrocytes in vitro. Figure 3 shows the relation between the flux rate through the oxidative pentose pathway and the GSH concentration as calculated on the basis of the model (solidqine) and as measured (Ataullakhanov et al:, 1981b) (broken line). There is a guod quantitative agreement between experimental and theoretical data.

In vivo simulations

In Table 5 metabolite concentrations and flux rates for the normal in vivo steady state of the system are presented. There is a satisfactory quantitative agreement between experimental and calculated values. Simula- tions have been performed to study the

100

5O

x

\

, I

50 100

GSH I [GSH • 2 x GSSG) x 100 ' / *

Fig. 3. Relation between the rate through the oxidative pentose pathway (expressed through the rate of CO 2 formation) and the relative content of reduced glutathione. ( . . . . . . ) experimental data (Ataullakhanov et al., 1981b) obtained by adding increasing amounts of t-butyl hydroperoxide (the dotted part indicates the range of uncertainty due to enhanced hemolysis of cells); ( ) theoretical curve calculated on the basis of the in vitro model.

20

TABLE 5

Metabolic concentrations and flux rates in the normal in vivo steady state.

MetaboHte Metabolite concentrations

Experimental Calculated value (/~D value (/LM)

FTP 15.7 [1] 15.7 FP z 7.0 [1] 7.0 GAP 5.7 [1] 5.7 I~P~G 0.5 [1] 0.5 3PG 68.5 [1] 69 2PG 10.0 [1] 10 PEP 17.0 [1] 17 Pyr 85.0 [1] 84 Lac 1430 [1] 1680 RuSP 16.0 (in the 3.6 Xy15P presence of 9.8 RSP inosine [3]) 10.9

18.0 [2] SedTP Ery4P G6P 38.5 [1] DHAP 140 [4] 2,3P2G 5700 [1]

0700 [5]

0PG 0.7 [6] 4.9 [7]

ATP 1830 [1] 1580 [5] 1890 [8]

ADP 180 [I] 325 275 [8]

AMP 37 [8] 75 A (sum of adenine 2000 [9] 2000

nucleotides) Mg I 500 [I0] 531

670 [9] Mg~ 3500 [9] 2800

2800 [11] 25oo [lO]

Mg2,3P2G 1240 [9] 724 600 [10]

MgATP 1440 [9] 1409 1300 [10]

MgADP 150 [10] 134 MgAMP 2.3 NAD 56 [12] 65.4

85 [13] NADH 2~ [13] 0.14 NADP 1.4 [14] 2 NADPH 53 [14] 50 NP 45 [12] 52 NADPI/{NADP / + NADPj + NADPHI} = 0.005 NADPH f [15] 4.9

19.9 7.6

40 140

3000 (assuming that 50% are bound [28])

20

1600

0.020

TABLE 5

Metabelite Metabolite concentrations

Experimental Calculated value (~/I) value (~uM)

NP~ GSH

GSSG

Enzyme

Hexoldnase

4 [15] 4.923 3100 [8] 3319.6 2800 [16] 3150 [17]

7 [18] 0.2 5.7 [8]

Flux rates

Experimental Calculated value (~M/h} value (1~/h)

Phosphoh-uctokinase 2~-bisphospho-

glycerate mutase 2~%bisphospho-

glycerate phosphatase

Pyruvate kinase ATPase Glucose-6-phosphate dehydrogenase

Phosphogluconate dehydrogenase

Glutathione reductase

Overall GSH- consumption

Phosphoribosylpyro- phosphate synthetase

NADP-lactate dehydrogenase

1785 [u] 15oo 2000 [16] 1550 [201

1453 500 [111 500

3800 [10]

6-- 11% of the overall glucose consumption

[191

20 [21]

500

2933 2373 I00

100

100

100

20

100

[1] Jacobasch et al., 1974; [2] Scientific Tables Geigy, 1979; [3] Bartlett and Bucolo, 1968; [4] Schauer e t a l . , 1981; [5] Marshall and Omachi, 1974; [6] Morelli et al., 1979; [7] Kirkman and Gaetani, 1986a; [8] Beutler, 1971; [9] Gerber et al., 1975; [10] Rapoport et al., 1972; [11] HolzhStter et al., 1985; [12] Hasart et al., 1972; [13] Sanders e ta l . , 1976; [14] Omachi e t a l , 1989; [15] Kirkman e t a l . , 1986b; [10] Ataullakhanov e t a l , 1981b; [17] Giintberberg and Rost, 1966; [18] Srivastava and Beutler, 1967; [19] Thorburn and Kuebel, 1985; [20] Heinrieb et al., 1977; [21] Per re t and Dean, 1977.

-6

dependency of fluxes and metabolite concentrations upon the energetic load (represented by the parameter ~ A ~ . which corresponds to the overall ATP~onsumption) and the oxidative load (represented by the parameter kox which correspond~s to the overall GSH~onsumption). In these calculations we apply the

concentrations of lactate and pyruvate to constant values d i r ec t ed by the outer conditions of the cell (in vivo model). Details of the mathematical procedure are outlined in the Appendix.

Three~limensional load characteristics which show the steady-state concentrations of

4 2 A ~" C

i

0 5 10 0 5 10

kATPase(h "t ) kATPase ( h -1 )

7 - - / - - / - / / / '~-~ / ~o+

0 5 10 0 5 10

kATpase (h -1) kATpo ~ (h -l )

Fig. 4. Steady-state load characteristics of the in vivo model. The concentrations of ATP (Fig. 4A) and GSSG (Fig. 4B) and the glycolytic flux rate ~ (Fig. 4C) and the flux rate u ~ through the oxidative pentose pathway Pig. 4D) are plotted as functions of the two load parameters kA~, " and k . The heavily drawn solid lines in the figure indicate those values of the load parameters for which GSSG equals the critical value GSSG* = 0.167 mM above with a drastic hemolysis of cells is observed in experimenl~ (see main text). The dotted curve indicates critical values of the energetic load parameter kay, , at which the steady state become unstable. An enlarged section of Fig. 4B indicated by the circle is shown in Fig. 5.

27

28

ATP and GSSG and the flux rates through glycolysis (umz) and the oxidative pentose pathway (U~p D) as functions of the two load parameters kATPm are presented in Fig. 4. The solid lines and dotted lines refer to stable and unstable steady states, respectively. The normal in vivo state (indicated by the heavily drawn point in Fig. 4) refers to the parameter values kATp~ = 1.68/h and ko~ = 0.06/h.

With respect to the energetic load, stable steady states exist only up to critical values of kATp~ above which the level of ATP becomes too low to maintain the "sparking reactions" (HK, PFK) at the beginning of glycolysis (Reich and Selkov, 1981; Heinrich and Rapoport, 1974a). This critical value of the energetic load parameter kATVu e depends on the value of the oxidative load parameter kox (dashed line). An increase of the oxidative load kox from 0.06/h to 2 h is accompanied with a decline of the critical energetic load kATpm from 10/h to 8.2/h. Thus the capability of glycolysis to maintain a stable steady state under conditions of increasing energetic load depends on the oxidative load.

Moreover, it is seen from the profiles in Fig. 4 that an increasing energetic load may also give rise to an elevation of the GSSG concentration provided that the oxidative load exceeds the normal value. This means that under conditions of an enhanced oxidative load the energy metabolism and redox metabolism are strongly coupled.

Whereas the model predicts unstable steady states in cases where the energetic load exceeds a critical threshold, no instabilities may occur if the oxidative load parameter kox is increased. This theoretical finding obviously conflicts with the well~stablished phenomenon that oxidative stress exerted on red cells may result in severe damage to the cell due to the exhaustion of those enzyme systems which protect the cellular proteins and lipids from the attack of H20 ~, lipid perox- ides and other secondary reaction products of the radical metabolism.

Ataullakhanov et al. (1981b) observed that isolated cells were able to tolerate an oxida-

tive stress without noticeable change in structure and function only up to flux rates through the oxidative pentose pathway of 60% of the maximal rate. According to the experimental curve shown in Fig. 3 the GSSG concentration GSSG* = 0.167 raM, which corresponds to a rate of the oxidative pentose pathway amounting to 60% of its maximum rate, can be considered as an empirical measure for the highest possible tolerable oxidative stress.

If it is assumed that only steady states of the system are realistic (i.e. compatible with cell integrity) for which the concentration of GSSG does not exceed this critical value GSSG* (indicated by the heavily drawn solid line in Fig. 4) the allowed parameter range for kATPa ~ and kox becomes considerably restricted.

As shown in Fig. 4 there is a range of low ~ATPase values where the steady state of the system is not unambiguous: for one and the same value of ~ATPue' tWO stable and one unstable steady state do exist (an enlarged section of the steady-state curve is shown in Fig. 5). In the three-dimensional representation of steady states this bifurcation region has the shape of a "fold" which becomes less pro- nounced with increasing values of kox. If the energetic load parameter ~ATPase is diminished to values approaching this bifurcation region a switch~ver in the metabolite concentration occurs (cf. Fig. 5).

From Fig. 4 it is seen that for higher values of kox the concentration of GSSG increases steeply at low as well as at high values of ~iTPase • The flux control coefficient shown clearly demonstrates that in both cases the flux through the oxidative pentose pathway is controlled by the hexokinase (see also Table 6) being the branching point between giycolysis and the oxidative pentose pathway. The limited flux through the hexokinase is caused by the increased concentration of 2,3-bisphosphoglycerate at low values of kATP~ and the diminished ATP concentration at high values of kATp . It has to be men- tioned that under conditions of an enhanced oxidative load the concentration of G6P is

29

approximately constant and therefore this metabolite does not play any role in the regulation of the hexokinase (el. Fig. 6).

Hitherto our considerations have been devoted to the steady-state characteristics of the system. We have performed "simulation experiments" by changing the load parameters abruptly at time t -- 0. It turns out that

- - the average transient time of the system between two distinct s teady states amounts to 4 h

- 2,3-bisphosphoglycerate is by far the slow- est component of the system.

From these findings it can be concluded that a real s teady state of the system can only be achieved iJf the external conditions of the cell are kept constant over several hours. Such a situation may arise in vivo, for exam- ple, after administration of oxidative drugs.

For deriving conclusions as to the short- term regulation of the system we have considered "quasi-steady states" which are reached after perturbation of the original s teady state

t /

(1 , ; I i I

o.2s o.375 0.5 0 5 "LO

kkTPose ( h-1 ) kATPose( h )

at unrelaxed (constant) 2,3-bisphosphoglycerate (Heinrich and Rapoport, 1974a).

In Fig. 7 the quasi-steady-state load characteristics of the system are plotted as function of the two load parameters kATPu e and ko. It is interesting to compare these quasi-steady- state profiles with the steady-state profiles given in Fig. 4. We note that the quasi-steady states do not reveal instabilities: there is nei- ther a critical threshold for the energetic load separating stable and unstable regions nor a bifurcation fold in the region of low kATPase values. Therefore, one should expect that a short-term perturbation (of about 1 h or less) can be bet ter tolerated by the system than a long-term perturbation. A further difference between the steady states and the quasi- s teady states consists of the behaviour of GSSG. For quasi-steady states GSSG is a monotonic function of both kATPase and kox, i.e. there is no steep increase in the concentration of GSSG at low values of kATP. e.

The regulatory properties of the system

-. 3

N

i ' o

c

¢ o

Fig. 5. Bifurcation region in the steady-state load characteristics of the in vivo model. The shown functional relationship between GSSG and the load parameter kATp, - (at fixed value k = 3.8/h) is an enlarged section from Fig. 4B indicated there by a circle. For small values of k , ~ , between 0.3/h and 0.35/h there exist two stable states (indicated by the full points) and one unstable state (indicated by the empty point).

Fig. 6. Quasi-steady-state flux control coefficient 8In uc~v/ain Z.x (control of the flux through the oxidative pentose pathway in steady state by hexokinase) as function of the two load parameters kATp. - and kox.

80

T A B L E 6

F lux control coeff icients C~ = in u / ln z. in t he no rma l in vivo s t eady-s ta te .

HK P F K P2GM P2Gase P K GSSGR L D H A T P a s e G6PD 6PGD Ox P R P P S

H K + 0.075 + 0.023 - 0.074 + 0.221 + 0.015 0 + 0.005 + 0.700 0 0 + 0.005 + 0.028 P F K + 0.069 + 0.024 - 0.075 + 0.228 + 0.016 0 - 0.006 + 0.729 0 0 - 0.006 + 0.020 PzGM + 0.011 + 0.005 + 0.061 + 0.939 - 0.014 0 0 - 0.002 0 0 0 0 P2Gase +0.011 +0 .005 +0 .061 +0 .939 - 0 . 0 1 4 0 0 - 0 . 0 0 2 0 0 0 0 P K + 0.066 + 0.024 - 0.074 + 0.226 + 0.016 0 0 + 0.724 0 0 0 + 0.018 GSSGR 0 0 0 0 0 0 0 0 0 0 + 1.000 0 L D H + 0.022 - 0.001 - 0.007 + 0.004 + 0.002 0 + 0.992 - 0.018 + 0.016 0 - 0.008 - 0.001 A T P a s e +0 .055 +0 .028 - 0 . I 0 0 +0.081 +0 .025 0 0 +0 .915 0 0 0 - 0 . 0 0 3 G6PD + 0.011 0 - 0.004 + 0.002 + 0.001 0 + 0.496 - 0.009 + 0.008 0 + 0.496 0 6PGD + 0.011 0 - 0.004 + 0.002 + 0.001 0 + 0.496 - 0.009 + 0.008 0 + 0.496 0 Ox 0 0 0 0 0 0 0 0 0 0 + 1.000 0 P R P P S +0.986 +0 .015 - 0 . 1 6 5 +0 .004 +0 .036 0 - 0 . 0 0 5 - 0 . 7 6 3 0 0 - 0 . 0 0 5 +0.970

have been studied by applying the concept of metabolic control (Heinrieh and Rapoport, 1974b; Kacser and Burns, 1973). The control coefficients were calculated with respect to (infinitely small) changes in the maximum activities of the enzymes. Calculations have been carried out for the normal steady state and quasi-steady state of the system and for

kAT~s e ( h~ 1)

i

0 5 10

E

t~

the case of a high oxidative load (kox = 15/h). Table 6 shows the flux control coefficients for the "normal" in vivo state. For the sake of a better overview the most important regulatory enzymes are depicted in Table 7, i.e. those enzymes having flux control coefficients remarkably different from zero. In general, the control coefficients calculated for the steady state and for the quasi-steady state are similar except those concerning the enzymes of the 2,3P~G bypass (which is neg- lected in the calculation of the quasi-steady states).

From Table 7 the following conclusions can be drawn:

(1) In the normal in vivo state of the cell, glycolysis, the oxidative pentose pathway and the 2,3P2G bypass are independently controlled by the energetic load, the oxidative load and the bypass enzymes, respectively. In other words, the activity of the various pathways is controlled by the need of those fuel metabolites which they produce.

(2) Under conditions of high oxidative load the independence in the control of glycolysis and the oxidative pentose phosphate pathway

Fig. 7. Quas i - s teady-s ta te load cha rac te r i s t i c s of t h e in vivo model. The concen t ra t ion of GSSG is p lo t ted as funct ion of t he two load p a r a m e t e r s kxrp. - and kox.

31

TABLE 7

Important control enzymes of fluxes. The flux control coefficients of the non-equilibrium enzymes were calculated in the normal in-vivo steady state and quasi-steady state of the system and in the presence of a strong oxidative load (k~ = 15/h). Details of the numerioal computations are given in the supplemental material. The Table contains only those controlling enzymes for which the flux control coefficient was remarkably different from zero. If there is more than one single controlling enzyme the most important one (with the highest flux control coefficient) is underlined.

Flux Controlling enzyme(s) through

Norm.~ conditions High oxidative load

Steady Quasi-steady Steady Quasi-steady s~Late state state state

HK A,TPase, ATPase, P2GM, HK HK P2Gase HK, P~Gase

PFK ATPase, ATPase, P2GM, ATPase, HK ATPase, HK P2Gase HK, P~Gase

P2GM P2Gase, P2GM P~Gase P~GM P~Gase P2Gase, P2Gase P~Gase P2Gase PK ATPaze, ATPase ATPase ATPase GSSGR Ox Ox H._KK, G6PD, H__K, G6PD,

ATPase ATPase NADP-LDH NADP-LDH NADP-LDH HK HK ATPase ATPase, ATPase ATPase ATPase G6PD Ox, NADP-LDH Ox, NADP-LDH HK._.._, G6PD, H._KK, G6PD,

ATPase ATPase 6PGD Ox, NADP-LDH Ox, NADP-LDH H.~K, G6PD, HK, G6PD,

ATPase ATPase Ox Ox Ox H.__K, G6PD, H_KK, G6PD,

ATPase ATPase PRPPS BK, PRPPS, PRPPS, HK, HK, PRPPS, PRPPS, H__KK,

ATPase ATPase ATPase ATPase

disappears. Both pathways are mainly controlled by the hexokinase whereas the controlling influence of UATp~ and Uox is negligible. In other words, the activity of the two pathways is controlled by the supply of the common substr~Lte glucose 6-phosphate.

(3) The metabolites (except GSSG and 2.3P2G) are mainly controlled by the hexokinase and the ATPase.

4. Discussion

In this paper we have developed a mathematical model which includes a lot of experimental information available on the kinetic properties of the relevant regulatory enzymes of glycoly:sis, the oxidative pentose

phosphate shunt and the glutathione system in erythrocytes. Our model unifies previous comprehensive models of giycolysis (Holzhtit- ter et al., 1985; Schauer et al., 1981) and of the oxidative pentose pathway (Thorburn and Kuchel 1985; Kothe et al. 1975; Ataullakhanov et al., 1981a) supplemented by the reactions of the non-oxidative pentose pathway. Two "external" parameters enter the model: the rate constant for ATP-utilization kA~ ~ and the rate constant ko~ of GSH-consumption. No further specifications have been made as to these overall processes.

A considerable reduction of the original set of differential equations could be achieved by means of the quasi-steady-state approximation. In general, this approximation is a pow-

32

erful tool in handling large biochemical reaction systems because it allows a considerable reduction in the number of parameters and differential equations (7 instead of ini- tially 20 in our model). The quasi-steady-state approximation is based on a subdivision of all reactions into "slow" non~quflibrium and "fast" quasi-equilibrium reactions. However, the experimental verification of such a classification of the reactions catalyzed by the enzymes of the pentose phosphate pathway and of the glutathione metabolism has not been clearly demonstrated and deserves further investigations.

The model-based calculations of the stationary and time-dependent in vitro states are in a good accordance with experimental observations. On applying the model to in vivo states of the cell we predict a weak coupling between the two pathways under normal conditions, i.e. the fluxes through glycolysis and through the oxidative pentose pathway are independently controlled by the need of ATP and GSH, respectively. On the other hand, a strong coupling of both pathways is predicted if an extensive oxidative load is present. In particular, in such a situation the flux through both pathways is controlled by the hexokinase. This theoretical finding is in line with recent experimental and theoretical investigations (Thorburn and Kuchel, 1985).

The simulation of in vitro experiments (Ra- poport et al. 1979) provides strong evidence for the existence of a NADP-dependent lactate dehydrogenase. Since the kinetic properties of this enzyme have not been elucidated yet, the two kinetic parameters of the hypo- thetical Michaelis-Menten model (cf. Table 1) have been chosen intuitively in order to obtain a satisfactory description of both experiments. Owing to the small number of available experimental data no systematic attempts have been made to optimize the two parameters, i.e. to find parameter values for which the difference between experimental and theoretical data attains a minimum. This may account for the quantitative differences remaining in the simulation of the pyruvate-

accumulation in vitro in experiment A. On the other hand, this discrepancy might also be caused by an underestimation of the total size of the HMS pool since experimental data on the concentrations of the involved metabolites vary considerably in the literature.

In conclusion, the elucidation of the regulatory importance of the NADP-dependent lactate dehydrogenase requires further experimental studies in order to set up a more realistic kinetic model.

On elucidating the regulatory features of the system under consideration we have clearly distinguished between the steady state (defined by vanishing time-derivatives of all metabolites) and the quasi-steady state (defined by vanishing time-derivatives of all metabolites at fixed value of 2,3-bisphosphoglycerate).

According to "simulation experiments" the realization of a steady state is only possible if the outer conditions of the cell remain unchanged for at least 4 h. Under such conditions the model predicts metabolic instabilities at high and low values of the energetic load parameter kATP~. In particular, a multi- stationary behaviour of the metabolism is expected at low values of the energetic load. The experimental evidence for the existence of such bifurcations is still lacking.

On the other hand there is experimental evidence for the loss of cell structure and function caused by oxidative stress. Following experimental observations (Ataullakhanov et al., 1981b) we have defined the upper bound for the load parameters kATPu e and kox up to which the glutathione system is capable of protecting macromolecules against oxidative attack, in terms of GSSG* which denotes the concentration of GSSG at which the flux through the oxidative pentose pathway corresponds to 60% of its maximum capacity. Introduction of this crude empirical limit value demonstrates that the tolerable oxidative load is severely restricted by an increasing energetic load and vice versa. This finding is of great importance as far as disorders of the metabolism of erythrocytes caused by

enzyme deficiencies are concerned. Defects of giycolytic enzyme.,, (e.g. of the pyruvate kinase, Holzhiitter et al., 1985) primarily connected with a disorder of the energy metabolism are supposed to influence the flux rate through the pentose phosphate pathway and thus the flux through the glutathione reductase. Defects of the giucose-6-phosphate dehydrogenase, a widespread metabolic dis- ease manifested in a reduced capacity of the oxidative pentose pathway, are expected to reduce also the energetic capacity of the c~ll. In a forthcoming paper we will apply the proposed model to s tudy the influence of various enzyme defects on the metabolism of the red cell and to compare the theoretical findings with experimental and clinical data of selected patients.

Appendix

Kinetic equations ,~modeO

The reaction scheme shown in Fig. 1 corresponds to the following set of ordinary, non- linear differential equations:

d[G6P]/dt = ~ t H K - - UG6PD - - "WpG I (1)

d[F6P]/dt = - - UpF K "4" ~/)PGI

+ 7"OTA + ~/'~TIC2 (2)

d~'P~]/dt = UpF K - - 'WAL v (3)

d[GAP]/dt = WAL D + WTp I + WTK 1

"~" "/~)TK2 - - ~'/)TA - - ~'/)GAPD

( 4 )

d[DHAP]/dt = 'U-)AL v - - "tOTp I (5)

d[1,3P2G]/dt = - - UP2GM "~" ~OGAPD

- wpo K (6)

d[3PG]/dt = Up2G~ + Wpo K

- WpG M (7)

3 3

d[2,3P2G]/dt = UP2G M -- lgP2Ga N (8)

d[2PG]/dt = WpQ M - w~ n (9)

d[PEP]/dt = - - UpK "1" ~OEn (10 )

d[6PG]/dt = 'gG6PD - - 1/'6PGD (11)

d[Ru5P]/dt = U6PGD - - WEp - - ~OKI ( 1 2 )

dlXylSP]/dt = "t/)Ep - - ~OTK 1 - - 'LOTK 2 (13)

d[RSP~dt = I~/)KI - - ~OTK 1 - - UpRpp S ( 1 4 )

d[Sed7P]/dt = WTK 1 - WTA (15)

d[Ery4P]/dt = ~OTA -- WTK 2 (16)

d[NADP]/dt = - - •G6PD - - U6PGD

"~" UGSSG R "~" UNADP.LDH (17)

d[ATP]/dt = - - UHK - - UpF K

- - '/~ATPase - - UPRPPS

"{" UPK "[" ~J)PGK - - ~OAK(18)

d[AMP]/dt = UpRpp S - - ~,OAK (19)

d[GSSG]/dt = - UGSSG R + UOx (20)

At the right-hand sides of Eqn. 1--20 the flux rates catalyzed by (slow) non-equilibrium enzymes and (fast) equilibrium enzymes (cf. classification on page 23) are denoted by u and w, respectively. The explicit rate equations of the non~quilibrium reactions (u) are given in Table 1 and in the papers of Holzhtitter et al. (1985) (giycolytic enzymes), Thorburn and Kuchel (1985) (giutathione reductase) and Buckwitz et al. (1984) (giucose-6-phosphate dehydrogenase). Except for K~i6P and V of hexokinase which are estimated from experimental data (Rose and 0'Conneil, 1964), all parameters are taken from the references as well. The complete mathematical model consists of the differential Eqns. 1 - 2 0 and the algebraic mass-action relations for the formation of magnesium complexes and the

34

binding of NADP(H) to proteins (cL Holzhiit- ter et al., 1985; Kirkman et al., 1986b).

The system of differential Eqn. 1--20 can be written in a more compact form using a vector notation:

dX(nYdt = C(mm). Vim) (21)

The symbols in Eqn. 21 have the following meaning:

X(n) n~iimensional vector of metabolite concentrations (= state variables)

Vim) m-dimensional vector of fluxes

C(n,m) stoichiometric matrix

n number of metabolites

m number of fluxes

The modelling in terms of ordinary differential equations is based on the usual assump- tion concerning spatially homogeneous concentrations and time-invariant cell volume. Changes of the cell volume seem to play no significant role provided that extreme osmotic conditions in the incubation medium are excluded (Werner and Heinrich, 1985).

Quasi-steady-state approximation

In order to reduce the size of the differential equation system (21) by application of the quasi-steady-state approximation (Schauer and Heinrich, 1983; Romanovsky et al., 1984) it is convenient to split the right-hand terms into two parts:

dX(n)/dt = R(n,q)" U(q) + S(n,p)" W(p) (22)

Here U{q) and W(p) represent the vectors of the q slow and p fast reactions whereby q + p = m. R(n,q) and S(n,p) are the corresponding stoichiometric matrices.

In the lowest order of singular perturbation theory (Schauer and Heinrich, 1983) the

quasi-steady-state approximation of system (22) means to put

W(p) = 0 (23)

Putting all components of W(p) equal to zero we arrive at the mass-action relations of the fast equilibrium reactions listed in Table 2 and in the paper of Holzhiitter et al. (1985). According to the equilibrium condition (23) the vector X(n) of all variables can be split into a vector Xi(r) of r independent variables and Xd(n - r) of dependent variables which are related to the independent variables by the algebraic mass-action relations. The choice of the r independent variables is not unambiguous. We have chosen

Xi(r) = {[GAP], [Ru5P], [ATP], [2,3P2G ], [6PG], [GSSG], [NADP]} f (24)

for the in vivo model and

Xi(r)- {[GAP], [Ru5P], [ATP], [2,3P2G ], [6PG], [GSSG], [NADP], [NAD], [Pyr]} T (25)

for the in vitro model which includes the concentrations of pyruvate and lactate as variables. Using the mass-action relations for the fast equilibrium reactions the n - - r dependent variables can be expressed through the r independent variables. To derive the (reduced) system of differential equations which govern the independent variables a matrix A(r,n) has to be constructed which obeys the condition

A(r,n). S(n,p) = 0 (26)

It has to be noted that the construction of matrix A(r,n) is not unambiguous.

Having determined A(r,n), so called "pool- variables" Y(r) =~ A(r,n) • X(n) are introduced which obey the differential equations

dY{r)/dt = A(r,n)" dX(n)/dt (27)

Based on the decompos i t i on of X(n) into Xi(r) and Xd(n--r) w e m a y w r i t e

dX(n)/dt = G(n,r) cLYi(r)/dt (28)

where the elements of matrix G(n,r) are given by

(o),~ = a(z) , la (zi); (29)

Inserting Eqn. 28 into Eqn. 27 one obtains

dXi(r)/dt = { A(r,~) • G(n,r)} -1 • A(r,n). R(n,q). U(q) (30)

We have chosen a transformation matrix A(r,.) which generates the following pool variables:

Yl " 3 [F6P] + 2 [Ru5P] + 2 [XylSP] + 2 [RSP] + 4 [Sed7P] + [Ery4P] + 3 [GeP] (31)

Y2 -~- [FSP] + 2 [PP~] + [GAP] + [1,3P2G ] + [3PG] + [2PG] + [PEP] + [Ru5P] + [Xyl5V] + [R5P] + [SedTP] + [Ery4P] + [G6P] + [DHAP] (32)

Y8 = [3PG] + [2PG] + [PEP] + [AMP] - [ATP] (33)

Y4 = [2,3P~G] (34)

Y5 = [6PG] (35)

Ys = [NADP] (36)

Y7 = [GSSG] (37)

Y8 = [Pyr] + [Lac] (38)

Y9 = [1,3P~G] + [3PG] + [2PG] + [PEP] + [Pyr] + [NAD] (39)

where the pool-variables Ys and Y9 are taken into account only in the in vitro model.

R e f e r e n c e s

35

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