3420581

7/29/2019 3420581

1/16

Biochem. J. (1999) 342, 581596 (Printed in Great Britain) 581

Model of 2,3-bisphosphoglycerate metabolism in the human

erythrocyte based on detailed enzyme kinetic equations1 :

equations and parameter refinement

Peter J. MULQUINEY and Philip W. KUCHEL2

Department of Biochemistry, University of Sydney, Sydney, NSW 2006, Australia

Over thelast 25 years, several mathematicalmodels of erythrocytemetabolism have been developed. Although these models haveidentified the key features in the regulation and control oferythrocyte metabolism, many important aspects remain un-explained. In particular, none of these models have satisfactorilyaccounted for 2,3-bisphosphoglycerate (2,3-BPG) metabolism.2,3-BPG is an important modulator of haemoglobin oxygenaffinity, and hence an understanding of the regulation of 2,3-BPG concentration is important for understanding blood oxygentransport. A detailed, comprehensive, and hence realistic math-ematical model of erythrocyte metabolism is presented that can

explain the regulation and control of 2,3-BPG concentration andturnover. The model is restricted to the core metabolic pathways,namely glycolysis, the 2,3-BPG shunt and the pentose phosphatepathway (PPP), and includes membrane transport of metabolites,the binding of metabolites to haemoglobin and Mg#+, as well aspH effects on key enzymic reactions and binding processes. The

INTRODUCTION

Of all cell types, the human erythrocyte is the most extensivelycharacterized in terms of its intermediary and secondary metab-olism [1]. This, along with its relative metabolic simplicity, has

made it the subject of several mathematical models [26].Although these models have identified many key features in theregulation and control of erythrocyte metabolism, many im-portant aspects have remained unexplained. In particular, noneof these models have satisfactorily accounted for 2,3-bisphosphoglycerate (2,3-BPG) metabolism. 2,3-BPG is an im-portant modulator of haemoglobin oxygen affinity and hence anunderstanding of the regulation of 2,3-BPG concentration andturnover is important for understanding blood oxygen transport.

A mathematical model of erythrocyte glycolysis was firstpresented in 1974 [2]. Since that time the main focus in theliterature has been to extend the mathematical description ofglycolysis to include all metabolically important events in an

Abbreviations used: 1,3-BPG, 1,3-bisphosphoglycerate; 2,3-BPG, 2,3-bisphosphoglycerate; BPGP, 2,3-BPG phosphatase; BPGS, 2,3-BPG synthase;

BPGS/P, 2,3-BPG synthase/phosphatase; 2-PGA, 2-phosphoglycerate; 3-PGA, 3-phosphoglycerate; 6-PG, 6-phosphogluconate; 6-PGDH, 6-

phosphogluconate dehydrogenase; 6-PGL, 6-phosphogluconolactone; AK, adenylate kinase; Ald, aldolase; Ery4P, erythrose 4-phosphate; Fru(1,6)P2,

fructose 1,6-bisphosphate; Fru6P, fructose 6-phosphate; G6PDH, glucose-6-phosphate dehydrogenase; GAPDH, glyceraldehyde-3-phosphate

dehydrogenase; Glc(1,6)P2, glucose 1,6-bisphosphate; Glc, glucose; Glc6P, glucose 6-phosphate; GPI, glucosephosphate isomerase; GraP,

glyceraldehyde 3-phosphate; GrnP, dihydroxyacetone phosphate; Hct, haematocrit; HK, hexokinase; kATPase, non-glycolytic energy consumption; kox,

reduction processes consuming GSH; koxNADH, reducing processes requiring NADH; Lac, lactate; Lace, extracellular lactate; lactonase, -gluconolactonase; LDH(P), NADPH-dependent lactate dehydrogenase; LDH, lactate dehydrogenase; PEP, phosphoenolpyruvate; PFK, phospho-

fructokinase; PGK, phosphoglycerate kinase; PGM, phosphoglycerate mutase; P ie, extracellular inorganic phosphate; PK, pyruvate kinase; PPP,

pentose phosphate pathway; Pyr, pyruvate; Pyre, extracellular pyruvate; R5P, ribose-5-phosphate isomerase; Rib5P, ribose 5-phosphate; Ru5E,

ribulose-5-phosphate epimerase; Ru5P, ribulose 5-phosphate; Sed7P, sedoheptulose 7-phosphate; TA, transaldolase; TK, transketolase; TPI,

triosephosphate isomerase; Xu5P, xylulose 5-phosphate.1 This is the second of a series of three papers on this topic; the first and third papers are [7] and [8] of the main paper respectively.2 To whom correspondence should be sent (e-mail p.kuchel!biochem.usyd.edu.au).

model is necessarily complex, since it is intended to describe theregulation and control of 2,3-BPG metabolism under a widevariety of physiological and experimental conditions. In addition,since H+ and blood oxygen tension are important externaleffectors of 2,3-BPG concentration, it was important that themodel take into account the large array of kinetic and bindingphenomena that result from changes in these effectors. Throughan iterative loop of experimental and simulation analysis manyvalues of enzyme-kinetic parameters of the model were refined toyield close conformity between model simulations and realexperimental data. This iterative process enabled a single set of

parameters to be found which described well the metabolicbehaviour of the erythrocyte under a wide variety of conditions.

Key words: computer model, erythrocyte enzymes, glycolysis,metabolic modelling, RapoportLuebering shunt.

attempt to develop a comprehensive model of erythrocytemetabolism. Brief histories of the development of erythro-cyte models are presented by Joshi and Palsson [6] and Heinrichand Schuster [9]. The most comprehensive model published todate is that of Palssons group [6]. This model includes glycolysis,

the 2,3-BPG shunt, the PPP, adenine nucleotide metabolism,various transmembrane processes, osmotic and electrostaticconditions, as well as pH effects on kinetic processes. Previouswork in our laboratory on the PPP [10,11] forms an importantpart of this model. Another notable model of the main metabolicpathways of the human erythrocyte has been presented bySchuster and Holzhu$ tter [5].

There are a number of processes which have been ignored bythese models that are significant in the description of 2,3-BPGmetabolism.

(1) No model presented so far has given an adequate accountof the binding of metabolites to Mg#+ and Hb. These interactionsare particularly important under conditions of changing pH

# 1999 Biochemical Society

7/29/2019 3420581

2/16

582 P. Mulquiney and P. W. Kuchel

and\or oxygen tension and need to be considered since oxygenand pH are important external effectors of 2,3-BPG concen-tration [8,12,13].

(2) Theinhibitory effects of 2,3-BPG on many enzyme activitieshave been ignored [1417]. While the inhibition of hexokinase(HK) and 2,3-BPG synthase by 2,3-BPG have been included inmost models (e.g. [18]), its inhibition of phosphofructokinase(PFK), aldolase (Ald) and glyceraldehyde-3-phosphate dehydro-genase (GAPDH) have been ignored. These inhibitions act as

negative feedbacks, controlling the production of 2,3-BPG [17].(3) The kinetic descriptions of both 2,3-BPG synthase (BPGS)

and phosphatase (BPGP) have been incomplete or inadequate(see [7]).

(4) Many important pH effects, as well as potentially importantglycolytic effectors, such as glucose 1,6-bisphosphate [Glc(1,6)P

#]

[19,20], have been neglected.(5) In most models, enzymes such as HK, PFK and pyruvate

kinase (PK) have been modelled as irreversible reactions withkinetic equations which ignore the presence of the products ofthese reactions (except [5]). This assumption is invalid undermany experimental and physiological conditions where accumu-lation of intermediates may lead to significant product inhibitionand\or significant increase in the backwards flux through these

reactions.(6) Finally, most models (except [5]) have modelled manyenzymic reactions using the rapid equilibrium assumption (e.g.see [9]). Again, this is not valid under all physiological conditions.

In the present paper these omissions are addressed and acomprehensive and more realistic model of the main metabolicpathways of the human erythrocyte is presented. The modelprovides a quantitative description of the regulation and controlof 2,3-BPG metabolism. By iterating experimental and simulationanalysis the parameter values of the model were refined to yieldclose conformity between the model simulation and realexperiments. This process enabled a single set of parametervalues to be found which described the metabolic behaviour oferythrocytes under a wide variety of physiological and ex-perimental conditions. Note that the focus of this paper is the

prediction of metabolism under normal in io steady-stateconditions. The response of the glycolytic portion of the modelto various external effectors is also considered. The predictionsof the model relative to real experimental data, and with respectto flux control coefficients for 2,3-BPG metabolism, are dealtwith in detail in the first [7] and third [8] papers of this seriesrespectively.

EXPERIMENTAL

Scope of model

The model encompasses the core metabolic pathways of thehuman erythrocyte: glycolysis, the 2,3-BPG shunt and the PPP(Scheme 1). Membrane transport of metabolites, the binding ofmetabolites to haemoglobin (Hb) and Mg#+ [21], as well as pHeffects on key enzyme reactions and binding processes, are alsoincluded. The model (at this stage) does not include osmotic andelectric (membrane potential) effects. The concentrations ofinorganic ions (except P

i) and cell volume were assumed to be

constants while transmembrane potential and pH were taken tobe external parameters. The neglect of many reactions ofnucleotide metabolism is justified in the Appendix. The concen-trations of GSH (turnover time $ 8 days; [22]), Glc(1,6)P

#(turnover time $ 20 h; [23]), and the sum of the NAD+ andNADP+ nucleotides were treated as constant external parameterson account of their very low turnover times.

Strategy of model development

Rate equations were derived for each of the enzyme-catalysedreactions using the method of King and Altman [24]; this processwas simplified with a computer program written in Mathematica(version 3.01, Wolfram Research Inc., Champaign, IL, U.S.A.)(available from P.W.K. by e-mail). In deriving rate equations,consideration was given to the reaction mechanism and tophysiologically and experimentally important inhibitors and

activators. From the rate equation for each enzyme, a non-linearalgebraic relationship between the steady-state kinetic parametersand the unitary rate constants was written [25]. Sets of unitaryrate constants, as consistent as possible with the steady-stateparameters, were then determined in order to check the adequacyof each model. This also assisted in parameter choice when facedwith a variety of literature values. A knowledge of the relation-ships between unitary rate constants and Michaelis constants isalso necessary for modelling enzyme deficiencies in a consistentway (e.g. [10,25]).

In determining unitary rate constants, constraints were placedon their possible values. Second-order rate constants were notallowed to exceed the diffusion limit of enzyme-catalysedreactions [26], namely 10* M":s". When applying the rapid-equilibrium assumption [27] to the derivation of rate equations,

all first-order rate constants (not part of a dead-end step, cf. [28])were made at least two orders of magnitude larger than the rateconstants for the interconversion of ternary complexes.

The kinetic behaviour of each enzyme was modelled in one oftwo ways. For most enzymes the steady-state rate equation wasused. For others [e.g. 2,3-BPG synthase\phosphatase (BPGS\P),adenylate kinase (AK) and transketolase (TK)] it was simpleroverall to use the elementary rate equations that described theformation and degradation of all metabolite and enzyme speciesinvolved in the reaction.

The individual kinetic equations for the reactions and transportprocesses used in the model are given in the Appendix. Thekinetic equations describing the formation of Mg#+ and Hbcomplexes are presented in [21]. A system of 108 differential

equations was constructed from these kinetic equations. Thissystem of differential equations was solved numerically usingMathematica on a desktop PC (Pentium Pro 200 processor, 64megabyte RAM). A typical simulation of 10 h of metabolismtime took $ 2 min.

pH-dependence of kinetic parameters

The pH-dependence of enzyme activity was included in modelsof enzymes only if this dependence was of possible physiologicalsignificance. Models of pH-dependence were chosen so that theydescribed the available (usually limited) data in as simple amanner as possible (i.e. Occams razor was invoked). Thusmodels were essentially phenomenological; only a few models ofenzymes were supported by detailed experimental evidence.

However, in all cases (except BPGS; see [7]) the model was basedon a simplekinetic reaction schemeand empirical pH-dependenceequations were avoided.

Refinement of parameter values

As discussed above, the parameter values used were chosen tomatch the available experimental kinetic and binding-parametervalues as closely as possible. Most of the kinetic data used weredetermined from isolated enzymes studied in itro and underconditions far removed from the typical intracellular environ-ment. Thus some of these values may not be applicable in situ


7/29/2019 3420581

3/16

583Model of erythrocyte metabolism based on enzyme kinetics

Scheme 1 Reaction scheme of the major metabolic pathways in the human erythrocyte: glycolysis, the 2,3-BPG (RapoportLuebering) shunt and the PPP

This reaction scheme is the basis of the model that is presented in this paper. Abbreviations not already defined: kATPase, non-glycolytic energy consumption; kox, reduction processes consuming

GSH; kox,NADH, reducing processes requiring NADH ; GSSGR, glutathione reductase, lactonase, -gluconolactonase; LDH, lactate dehydrogenase; note that the model also includes an NADPH-dependent LDH ; PFK, phosphofructokinase; PGK, phosphoglycerate kinase; PGM, phosphoglycerate mutase; R5PI, Rib5P isomerase; Ru5PE, ribulose 5-phosphate epimerase; Lace, extracellular

lactate; Pie, extracellular Pi ; PEP, phosphoenolpyruvate; 2- and 3-PGA, 2- and 3-phosphoglycerate; 6-PG, 6-phosphogluconate; 6-PGL, 6-phosphogluconolactone; Pyre, extracellular pyruvate;

Sed7P, sedoheptulose 7-phosphate.

[29,30]. Many of the parameter values used in the model had notbeen measured or were not known precisely. Therefore, in manyinstances, it was necessary to iteratively change parameter values

so that the model was able to consistently simulate the behaviourof redcells under a wide variety of experimental and physiologicalconditions (see the Results and discussion section and [7]).


7/29/2019 3420581

4/16


RESULTS AND DISCUSSION

Metabolite concentrations and fluxes in the normal in vivo steadystate

The steady-state metabolite concentrations and fluxes that werepredicted by the model in the normal in io state are presentedin Tables 1 and 2. The normal in io steady state is defined bythe values of the external parameters in Table 3. The steady statewas determined by solving the system of differential equations

that define the model, with all external parameters held at aconstant value, over a time period long enough to ensure that thevariables were constant to three significant figures.

In all cases there was a close match between the predicted andthe experimental values; however a few points need to be noted.

Values of equilibrium constants

In order to match the experimentally observed metaboliteconcentrations (Table 1), the equilibrium constants of triose-

Table 1 Steady-state metabolite concentrations predicted by the model in the normal in vivo steady-state

The simulation was made with the assumption that Hb was fully oxygenated.

Concentration (mM)

Metabolite* Predicted Observed Reference

ADPT 0.304 0.310 [48]

ADP 0.0937

MgADP 0.110

AMP 0.0292 0.030 [48]

ATPT 2.11 2.10 [48]

ATP 0.159

MgATP 1.52

1,3-BPGT 0.000822 0.000700 [49]

1,3-BPG 0.000369

2,3-BPGT 6.70 6.70 [7]

2,3-BPG 3.10Ery4P 0.000721

Fru(1,6)P2T 0.00265 0.00270 [48]

Fru(1,6)P2 0.00231

Fru6P 0.0122 0.0130 [48]

Glc(1,6)P2 0.106

Glc6P 0.0375 0.0390 [48]

GraP 0.00531 0.00570 [35]

GrnP 0.0221 0.0170 [35]

GSH 3.20 3.20 [48]

GSSG 0.0000864 0.0060 [48]

Lac 1.40 1.40 [35]

Mg2+ 0.369 0.40 [50]

NAD+ 0.0599 0.0400.090 [35]

NADH 0.000245NADP+ 0.065 0.0643 [51]

NADPH 0.000132 0.0014 [51]

2-PGA 0.0120 0.010 [35]3-PGA 0.0721 0.069 [35]

6-PG 0.0271 0.0049 [52]

6-PGL 0.0000113

PEP 0.0203 0.017 [48]

Pi 0.995 1.00 [35]

Pyr 0.0586 0.085 [35]

Rib5P 0.00473

Ru5P 0.00401

Sed7P 0.00545

Xu5P 0.00732

* Subscript T refers to total metabolite concentration and subscript e denotes extracellular concentration; all other concentrations are of the free metabolite in the cell water.

Experimental value is difficult to determine free of artifacts [2].

phosphate isomerase (TPI) and GAPDH were reduced by afactor of 5 and 3 respectively (see Tables A5 and A6 of theAppendix). The Mass Action ratio for the reaction catalysed byTPI in the normal in io state has been reported by manydifferent authors to be $ 3 (See Table 3.1 in [31]), while theequilibrium constant is around 20 [32,33]. The reason for thisdiscrepancy is unclear given the very high catalytic capacity ofTPI; TPI has been described as the perfect enzyme [34].

An important point to note from Table 1 is that many of the

reactions that have traditionally been treated as rapid-equi-librium reactions in the normal in io steady-state situation(e.g. [4,6]) have Mass Action ratios significantly smaller thantheir equilibrium constants. The ratios of the Mass Action ratioto the equilibrium constants for some are: Ald, 0.60; GAPDH,0.95 ; enolase, 0.56; LDH, 0.92; TK [xylulose 5-phosphatejribose 5-phosphate (Xu5PjRib5P), 0.40; TK [Xu5PjEry4P(erythrose 4-phosphate)], 0.41; transaldolase (TA), 0.84.


7/29/2019 3420581

5/16


Table 2 Important steady-state metabolite fluxes predicted by the erythrocyte metabolism model in the normal in vivo steady-state

It was assumed that Hb was fully oxygenated.

Numerical value (mmol:litre of erythrocytes1:h1)

Reaction Predicted Observed Reference

HK 1.41 1.44 [53]

1.35 [49]

G6PDH 0.147 611 % of HK rate [10]TK (Xu5PjRib5P) 0.0486

TK (Xu5PjEry5P) 0.0486

TA 0.0486

PFK 1.37

GAPDH 2.78*

BPGS 0.538 0.480 [43]PK 2.78

Glutathione reductase 0.274

NADPH-dependent LDH 0.0177

Non-glycolytic ATP consumption 2.24

GSH oxidation 0.548 0.560 [10]

Non-glycolytic NADH consumption 0.0101 0.0100 [31]

* Note that the rate through GAPDH is less than twice that through HK, due to the loss of CO2 in the reaction catalysed by 6PGDH.

See [7] for a full discussion of this value.

Table 3 External parameters of the model that describes the normal in vivo metabolic steady-state of the human erythrocyte

Numerical value

Parameter Model Literature Reference

Glc concentration 5 mM

Total Glc(1,6)P2 concentration 122 M 122 M [44]Extracellular Lac 1.82 mM

Extracellular Pyr 85 MPie 1.92 mM 1.20 mM [54]

CO2 concentration 1.2 mM 1.2 mM [55]

Total Mg2+ 3.0 mM 3.4 mM [56]

Total glutathione 3.21 mM 3.26 mM [48]

Total adenosine nucleotides 2.44 mM 2.44 mM [48]Total NAD+ nucleotides 60.14 M 4090 M [35]Total NADP+ nucleotides* 66 M 66 M [51]Total Hb concentration 7.0 mM 7.0 mM [57]

pHi 7.2 7.2 [58]

Donnan ratio 0.69 0.69 [59]

Hct 0.50 0.50 [60]

* The putative binding of NADP(H) to intracellular proteins was ignored ; see the Results and discussion section.

NAD+/NADH ratio

The model predicts an NAD+\NADH ratio of$240. There area number of experimental difficulties in accurately determining

this ratio; however, its value is probably between 40 and 1000[35].

NADP+/NADPH concentrations

A significant proportion of NADP(H) has been reported to bindto proteins in the erythrocyte [36]. However, when this putativebinding was incorporated into the present model, it was predictedthat 6-phosphogluconate (6-PG) would be present at aboutmillimolar concentrations in the normal in io steady state. Thisanomaly was traced to being primarily due to the inhibition of 6-phosphogluconate dehydrogenase (6-PGDH) by the reducedlevel of free NADP+. Hence, in order to retain the close fit

between experimental and predicted metabolite concentrations,this putative binding had to be ignored.

Response of the model to external effectors

Since the current model was developed in an attempt to describethe regulatory properties of erythrocytes under a variety ofexperimental and physiological conditions, it was important totest the response of the model to a number of external effectors.This process was also used to refine some of the parameters of themodel.

For the present paper the focus was primarily on the responseof the glycolytic component of the model. Note that theresponse of the 2,3-BPG shunt to external effectors is examined inthe accompanying papers [7,8]. In addition, the pentose phosphate


7/29/2019 3420581

6/16


Figure 1 pH-dependence of the glycolytic intermediates in humanerythrocytes at 37 mC

(A) pH 6.9: , [37];, simulated value. (B) pH 7.47.5:#, [35], pH 7.4;$, simulated

value, pH 7.4; , [37], pH 7.5; , simulated value, pH 7.5.

component of the model was previously subjected to detailedtesting (e.g. [10,11]) and is considered to be well verified.

In all following simulations it was assumed that the totalconcentration of 2,3-BPG was constant. This was done becausein most experimental work, flux and metabolite changes weredetermined before a new steady-state concentration of 2,3-BPGhad been achieved. Thus flux and intermediate concentrationsrefer to a quasi steady state. In addition, concentrations ofintracellular lactate (Lac), pyruvate (Pyr), and P

i, as well as the

parameters in Table 2, were all held constant in the simulations;these values are the same as those given in Tables 1 and 2 unlessotherwise specified.

pH

A plot of the changes that occur in the concentrations ofglycolytic intermediates with a change in pH is shown in Figure1. It is seen that, for both a decrease and an increase in pH, thepredictions of the model compared favourably with the exper-imental values. The large differences in the concentrations offructose 1,6-bisphosphate [Fru(1,6)P

#], glyceraldehyde 3-phos-

phate (GraP) and dihydroxyacetone phosphate (GrnP) at pH 7.4

and pH 7.5 were mainly due to the different ratios of Lac to Pyr,and hence NAD+ to NADH, that were present under each of theexperimental conditions.

The predicted changes in glycolytic rate, as a function of pH,matched the experimentally determined results well. At pH6.9 Minakami and Yoshikawa [37] measured a decrease inglycolytic rate of 57% relative to pH 7.2; a decrease of 55% waspredicted by the model. At pH 7.5, those authors measured anincrease in glycolytic rate of 46%, while the model predictedan increase of 53%.

The fit between the model predictions and the experimentaldata was shown to be particularly sensitive to the value of n ineqn. (A5) of the Appendix. In the PFK model originallydeveloped by Pettigrew and Frieden [38], these authors deducedthat nl8. This was based on calorimetric studies that indicated

that two protons per protomer were involved in the binding ofATP to the regulatory site [39,40]. However, it is not clearwhether all these protons are involved in the R-to-T transition;some may simply be involved in ATP binding. Pavelich andHammes [41] found that the inactivation of PFK with a fall inpH could be modelled by assuming that about two protons werebound per tetramer. Also, values of L, determined at differentvaluesof pH reported by Goldhammer andHammes [42], indicatethat n$ 2. We found, however, that a value of nl5 gave thebest matches between model predictions and experimental data.Thus a value of nl5 was settled upon for the present model.

One of the possible limitations with the simulations is that theyassumed that the first-order rate constant describing theATPase was independent of pH. However, the good match

between experimental and simulated values was justification forignoring a pH effect on the ATPase. In addition, using thisassumption it was predicted that the concentration of ATP atboth pH 6.9 and pH 7.4 would be equal. This is in agreementwith the finding of Rapoport et al. [43] that the concentration ofATP in erythrocytes incubated with glucose (Glc) at different pHvalues is constant; this finding was also reported by Jacobasch etal. [35]. Note, however that Minakami and Yoshikawa [37] havereported a pH-dependence of ATP concentration.

Glc(1,6)P2

Glc(1,6)P#

is present in the human erythrocyte at concentrationsof$ 120 M [44]. This makes it the third most concentratedphosphorylated metabolite, after 2,3-BPG and ATP, in theerythrocyte. Its major route of synthesis is via phospho-glucomutase, and at pH 7.4 it has a turnover time of$ 20 h [23].Glc(1,6)P

#is an inhibitor of HK and an activator of PFK and

PK in itro (see the Appendix). In addition, the work reported in[21] indicates that the free concentration of Glc(1,6)P

#is very

sensitive to pH and oxygenation state. Thus this compound maybe an important modulator of erythrocyte metabolism in io.

Piattiet al. [20] overloaded human erythroycteswith Glc(1,6)P#

to investigate possible regulatory roles of it in io ; they reportedthat, when erythrocytes are overloaded with about five timesthe normal concentration of Glc(1,6)P

#, the glycolytic rate falls

from 1.42p0.15 to 0.71p0.08 mmol:litre of erythrocytes": h",the concentrations of glucose 6-phosphate (Glc6P) and


7/29/2019 3420581

7/16


fructose 6-phosphate (Fru6P) fall by $ 50%, and the con-centration of ATP remains approximately constant. It was onlypossible to obtain a good match between these findings and theoutput of the model if it was assumed that PFK is weaklyactivated by Glc(1,6)P

#. This involved setting the value of

KR,Glc(",')P#

to 10 mM; its in itro value is 0.0151 mM [45]. Withthis assumption, and by setting total ATP concentrations to aconstant value (2.1 mM), the model predicted a glycolytic rate of0.70 mmol:litre of erythrocytes":h" and a 45% reduction in

Glc6Pand Fru6Pconcentrations. If the ATP concentration wasnot set to a constant value, a glycolytic rate of 0.59 mmol:litre oferythrocytes":h" was simulated. This reduced rate was largelydue to the predicted decrease in total ATP concentration. Thus,under these conditions the simple model of ATPase may havebeen inadequate. However, at a Glc(1,6)P

#concentration

approximately double the normal value, the model predicted aglycolytic rate of 1.13 mmol:litre of erythrocytes":h" with orwithout the assumption of constant ATP concentration, whilePiatti et al. [20] measured a value of 1.06p0.10 mmol:litre oferythrocytes":h"

Pi

At very high phosphate concentrations (up to$ 40 mM) the rate

of glycolysis is increased by a factor of more than 2 [31]. This ismost probably due to phosphate decreasing the inhibition ofPFK by ATP [35]. From eqn. (A5) of the Appendix it is seen thatthis effect could also be accounted for by P

ibinding preferentially

to the R state of PFK. At phosphate concentrations below10 mM, phosphate has only a small effect on the glycolytic rate ;at 10 mM P

ithe glycolytic rate increases by$ 1020% over the

normal value in io [31]. If the phosphate-binding constant forPFK was set to 0.431 mM [45], the glycolytic rate became muchhigher at 10 mM P

i. However, if the constant value was increased

to 30 mM, the model predicted an increase of only$ 1012% inthe glycolytic rate, in the presence of 10 mM P

i; this prediction

is consistent with the data of Grimes [31]. However this modelpredicted only a 2030% increase in glycolytic rate when [P

i] was

40 mM. Thus the model appears to account adequately for Piactivation if the Pi

concentrations are low ( 10 mM).

Total Mg2+

Laughlin and Thompson [46] recently reported that the con-centration of free Mg#+ needed for a half-maximal rate ofglycolysis in human erythrocytes is 0.03 mM. Given that the free[Mg#+] concentration in erythrocytes is usually 0.30.6 mM,these authors concluded that Mg#+ would have very little role inthe regulation of glycolysis in io. However, the model presentedin the present paper predicts a value of 0.3 mM for a half-maximal rate of glycolysis. Part of the reason for the abovediscrepancy could have been the use of inappropriate Hb-metabolite-binding constants for the $"P NMR determination ofintracellular free [Mg#+] by Laughlin and Thompson [46] (see[47]). This would have led to an underestimation of the free[Mg#+]. If the current model is correct, it implies that [Mg#+] mayplay a regulatory role in erythrocyte glycolysis. In support of thecurrent models prediction of [Mg#+]-sensitivity is the finding thatif the concentration of total Mg#+ was reduced by a factor of 2(which corresponds to a decrease in free Mg#+ by a factor of$3) the model predicts that the glycolytic rate would decrease by$30%. This is in agreement with the decline in the glycolyticrate measured in dog and rat erythrocytes which have had theirMg#+ concentrations reduced by a factor of$ 2 [35]. Anotherinteresting observation to emerge from the simulations was thatthe elasticity of the ATPase to free Mg#+ concentration is the

parameter most responsible for the response of glycolytic flux toMg#+ at low total concentrations of Mg#+. However HK, PFK,enolase, and PK were also found to be important in the Mg #+

response.

Oxygen tension

Since the effect of oxygen tension on glycolysis is closely linkedto 2,3-BPG metabolism, a discussion of this is deferred until the

last paper in the series of three [8].

CONCLUSIONS

A detailed, comprehensive (and hence realistic) model of themain metabolic pathways of the human erythrocyte is presented.Through an iterative loop of experimental and simulationanalysis some values of kinetic parameters of the model wererefined to yield close conformity of the model simulations andreal experiments. The model simulated a normal in io steady-state which matched well with the available experimental data. Inaddition the response of the model to various effectors such asH+, Glc(1,6)P

#, P

i, and total magnesium concentrations was also

seen to be in agreement with experimental reality.The refinement process revealed that the parameter values

used in the model of phosphofructokinase were uncertain. So far,no adequate in itro study has been performed on the humanerythrocyte enzyme; most of the data used in the model ofphosphofructokinase were from rat erythrocytes (see Table A3).The present study, along with that reported in [7], indicated thatthe allosteric effectors of PFK bind to it with dissociationconstants at least an order of magnitude higher than thosedetermined for the rat erythrocyte enzyme. Whether this is a truekinetic difference between the species, or whether it is due todifferent mechanisms of the enzyme in situ, is unclear.

A puzzling finding to emerge from the present work is that theMass Action ratio of metabolites involved in the TPI reactionunder steady-state conditions is significantly different from theequilibrium constant of this reaction. There are no simpleexplanations for this finding.

This work was supported by the Australian National Health and Medical ResearchCouncil, and P.J.M. received an Australian Commonwealth Postgraduate ResearchAward. Dr. Hilary Berthon, Dr. Serena Hyslop, Dr. Lisa McIntyre, Nicola Nygh, Dr.Julia Raftos, and Dr. David Thorburn are thanked for their contributions in the earlierstages of this work.

REFERENCES

1 Rapoport, S. M. (1988) in The Roots of Modern Biochemistry: Fritz Lipmanns

Squiggle and its Consequences (Kleinkauf, H., von Do$ hren, H. and Jaenicke, L.,

eds.), pp. 157164, Walter de Gruyter, Berlin

2 Rapoport, T. A., Heinrich, R., Jacobasch, G. and Rapoport, S. (1974) Eur. J. Biochem.

42, 107120

3 Holzhu$ tter, H.-G., Jacobasch, G. and Bisdorff, A. (1985) Eur. J. Biochem. 149,

101111

4 Schuster, R., Holzhu$ tter, H.-G. and Jacobasch, G. (1988) BioSystems22

, 1936

5 Schuster, R. and Holzhu$ tter, H.-G. (1995) Eur. J. Biochem. 229, 403418

6 Joshi, A. and Palsson, B. O. (1989) J. Theor. Biol. 141, 515528

7 Mulquiney, P. J., Bubb, W. A. and Kuchel, P. W. (1999) Biochem. J. 342, 565578

8 Mulquiney, P. J. and Kuchel, P. W. (1999) Biochem. J. 342, 595602

9 Heinrich, R. and Schuster, S. (1996) The Regulation of Cellular Systems. Chapman

and Hall, New York

10 Thorburn, D. R. and Kuchel, P. W. (1985) Eur. J. Biochem. 150, 371386

11 McIntyre, L. M., Thorburn, D. R., Bubb, W. A. and Kuchel, P. W. (1989) Eur. J.

Biochem. 180, 399420

12 Meldon, J. H. (1985) Adv. Exp. Med. Biol. 191, 6373

13 Harken, A. H. (1977) Surg. Gynecol. Obstet. 144, 935955

14 Beutler, E. (1971) Nature (London) 232, 2021

15 Ponce, J., Roth, S. and Harkness, D. R. (1971) Biochim. Biophys. Acta 250, 6374

16 Srivastava, S. K. and Beutler, E. (1972) Arch. Biochem. Biophys. 148, 249255


7/29/2019 3420581

8/16


17 Beutler, E., Matsumoto, F. and Guinito, E. (1974) Experientia 30, 190191

18 Kuchel, P. W., Chapman, B. E., Lovric, V. A., Raftos, J. E., Stewart, I. M. and

Thorburn, D. R. (1984) Biochim. Biophys. Acta 805, 191203

19 Fornaini, G., Magnani, M., Fazi, A., Accorsi, A., Stocchi, V. and Dacha, M. (1985)

Arch. Biochem. Biophys. 239, 352358

20 Piatti, E., Accorsi, A., Piacentini, M. P. and Fazi, A. (1992) Arch. Biochem. Biophys.

293, 117121

21 Mulquiney, P. J. and Kuchel, P. W. (1997) Eur. J. Biochem. 245, 7183

22 Dimant, E., Landsberg, E. and London, I. M. (1955) J. Biol. Chem. 213, 769776

23 Gerber, G., Winczuk, E. and Rapoport, S. (1973) Acta Biol. Med. Germ. 30, 759771

24 King, G. F. and Altman, C. (1956) J. Phys. Chem. 60, 1375137825 Kuchel, P. W., Roberts, D. V. and Nichol, L. W. (1977) Aust. J. Exp. Biol. Med. Sci.

55, 309326

26 Fersht, A. (1977) Enzyme Structure and Mechanism, Freeman, San Francisco

27 Cha, S. (1968) J. Biol. Chem. 243, 820825

28 Cornish-Bowden, A. (1995) Fundamentals of Enzyme Kinetics, Portland Press Ltd.,

London

29 Clegg, J. S. (1984) Am. J. Physiol. 246, R133R151

30 Srere, P., Jones, M. E. and Mathews, C. (1989) Structural and Organizational Aspects

of Metabolic Regulation, Liss, New York

31 Grimes, A. J. (1980) Human Red Cell Metabolism, Blackwell Scientific Publications,

Oxford

32 Meyerhof, O. and Junowicz-Kocholaty, R. (1943) J. Biol. Chem. 149, 7192

33 Fox, I. H. and Kelley, W. N. (1978) Annu. Rev. Biochem. 47, 655686

34 Knowles, J. R. and Alberty, W. J. (1977) Acc. Chem. Res. 10, 105111

35 Jacobasch, G., Minakami, S. and Rapoport, S. M. (1974) in Cellular and Molecular

Biology of Erythrocytes (Yoshikawa, H. and Rapoport, S. M., eds.), pp. 5592,University Park Press, Baltimore

36 Kirkman, H. N., Gaetani, G. F. and Clemons, E. H. (1986) J. Biol. Chem. 255,

40394045

37 Minakami, S. and Yoshikawa, H. (1966) J. Biochem. 59, 145150

38 Pettigrew, D. W. and Frieden, C. (1979) J. Biol. Chem. 254, 18961901

APPENDIX

Equations used in the mathematical model of erythrocyte metabolism

Glycolytic enzymesHK (EC 2.7.1.1). The steady-state kinetic expression for theenzyme reaction scheme shown in Scheme A1 is:

Scheme A1 A minimal model of human erythrocyte HK

This model is a modification of that presented in [35]. Symbols: E, free enzyme; A, MgATP;

B, Glc; P, Glc6P; Q, MgADP; I1, Pi ; I2, 2,3-BPG; I3, Glc(1,6)P2 ; I4, GSH; all other species are

the corresponding enzymemetabolite complexes. All reactions, except the interconversion of

EAB and EPQ, are assumed to be in rapid equilibrium. Expressions for the steady-state kinetic

parameters in terms of the rate constants are: kcat,fl k9, kcat,rl k10, Km,Al k8/k7, Km,Bl

k4/k3, Ki,Al k2/k1, Ki,Bl k6/k5, Km,Pl k11/k12, Km,Ql k15/k16, Ki,Pl k17/k18, Ki,Ql

k13/k14 and Khi,jl k18+2j/k17+2j,

jl 1, , 4.


40 Wolfman, N. M. and Hammes, G. G. (1979) J. Biol. Chem. 254, 1228912290

41 Pavelich, M. J. and Hammes, G. G. (1973) Biochemistry 12, 14081414

42 Goldhammer, A. R. and Hammes, G. G. (1978) Biochemistry 17, 18181822

43 Rapoport, I., Berger, H., Elsner, R. and Rapoport, S. M. (1977) Eur. J. Biochem. 73,

421427

44 Thorburn, D. R. and Kuchel, P. W. (1987) Clin. Chim. Acta 110, 7074

45 Otto, M., Heinrich, R., Jacobasch, G. and Rapoport, S. (1977) Eur. J. Biochem. 74,

413420

46 Laughlin, M. R. and Thompson, D. (1996) J. Biol. Chem. 271, 2897728983

47 Mulquiney, P. J. and Kuchel, P. W. (1997) NMR Biomed. 10, 12913748 Beutler, E. (1984) Red Cell Metabolism: A Manual of Biochemical Methods, 3rd edn.,

Grune and Stratton, New York

49 Momsen, G. and Vestergaard-Bogind, B. (1978) Arch. Biochem. Biophys. 10, 6784

50 Flatman, P. W. (1980) J. Physiol. (London) 300, 1930

51 Omachi, A., Scott, C. B. and Hegarty, H. (1969) Biochim. Biophys. Acta 184,

139147

52 Kirkman, H. N. and Gaetani, G. F. (1986) J. Biol. Chem. 261, 40334038

53 Gerlach, E., Duhm, J. and Deuticke, B. (1970) in Red Cell Metabolism and Function

(Brewer, G. J., ed.), pp. 155174, Academic Press, New York

54 Kemp, G. J., Bevington, A. and Russell, G. G. (1988) Mineral Electrolyte Metab. 14,

266270

55 McGilvery, R. W. (1979) Biochemistry, a Functional Approach, W. B. Saunders

Company, Philadelphia

56 Millart, H., Durlach, V. and Durlach, J. (1995) Magnesium Res. 8, 6576

57 Gerber, G., Berger, H., Ja$nig, G.-R. and Rapoport, S. M. (1973) Eur. J. Biochem. 38,

56357158 Stewart, I. M., Chapman, B. E., Kirk, K., Kuchel, P. W., Lovric, V. A. and Raftos, J. E.

(1986) Biochim. Biophys. Acta 885, 2333

59 Kirk, K., Kuchel, P. W. and Labotka, R. J. (1988) Biophys. J. 54, 241247

60 Dacie, J. V. and Lewis, S. M. (1975) Practical Haematology, Churchill Livingstone,

Edinburgh

dp

dtl

e!0 kcat,fabK

i,BK

m,A

kkcat,rpq

Ki,Q

Km,P

11j

a

Ki,A

jb

Ki,B

jab

Ki,B

Km,A

jp

Ki,P

jq

Ki,Q

jpq

Ki,Q

Km,P

j%

j="

ijb

Ki,Ij

Ki,B

(A1)

where e!

is the total enzyme concentration, and the lower-caseitalicized letters are the concentrations of the metabolite speciesthat are denoted by the corresponding uppercase letters. Theequation was derived using the method of Cha [1] and assumesrapid equilibrium of all steps in the mechanism except for the tworeactive-ternary complexes. A set of rate constants consistentwith the values of the steady-state kinetic parameters is given inTable A1.

The activity of HK shows an optimum at about pH 8.2 [24].The pH-dependence of mammalian erythrocyte HK activity hasbeen most extensively studied with the rabbit enzyme [5]. Thesedata can be modelled well with the bell function:

kcat,f

(pH)lk

01j10pH10pK"j10pK#

10pH 1(A2)

where k is a constant, pK"l7.02 and pK

#l9.0; these pK

values are independent of the buffer used [5]. kcat,r

was assumedto be dependent on pH in the same manner as k

cat,f. It was


7/29/2019 3420581

9/16


Table A1 A comparison of the reported kinetic parameters of human erythrocyte hexokinase with those predicted (at pH 7.2) by the scheme described inScheme A1

The ki values that yielded steady-state parameters consistent with literature values were: (M1:s1i108), k1l k7l k12l k18l 0.3, k3l k5l k14l k16l 6.4, k19l 7.1, k21l 0.075,

k23l 4.5, k25l 0.10; (s1 i104), k2l k4l k6l k8l k11l k13l k15l k17l k22l k26l 3.0, k20l k24l 1.0, k9l 1.80i10

2, and k10l 1.36i104. In this and Tables

A2A12 the references refer to the Appendix reference list, not that of the main paper. Notes: a[37], 37 mC, pH 7.2, human erythrocytes; b[38], 37 mC, pH 7.25, human erythrocytes, range of

estimates for different enzyme subtypes; c[39], 37 mC, pH 7.2, human erythrocytes, range of estimates for different enzyme subtypes; d[40], 37 mC, pH 7.7, human erythrocytes; the pH dependence

of KGSH has been cited [41] as the reason that these workers failed to observe inhibition of HK by GSH;e[42], average value for mammalian and yeast HK type I from 165 references; f[43],

30 mC, pH 6.0; gkcat,f for the most highly purified mammalian erythrocyte enzyme (rabbit; purification 300000i) was calculated from the data of [44] (specific activity 145 units/mg, pH 8.1, 37 mC ;

Mr 110000) to be 266 s1 ; therefore, from eqn. (A2), kcat,f (pHl 7.2) was calculated to be 180 s

1 ; hthe concentration of HK in human erythrocytes, eo, was estimated from the measured human

erythrocyte activity (182 units:litre of erythrocytes1, 37 mC, pH 7.2; [35]) and the value of kcat,f (pHl 7.2).

Reported value

Parameter (a) (b) (c) (d) (e) Predicted value

Km,MgATP 2.23.0 mM 735p45 M 600p20 M 400p230 M 1.0 mMKi,MgATP 1.02.1 mM 1.0 mM

Km,Glc 5154 M 6278 M 4648 M 49p18 M 47 MKi,Glc 3840 M 47 MKm,Glc6P 47 MKi,Glc6P 47 MKm,MgADP 1.0 mM

Ki,MgADP 1.0 mM

KiBPG 2.7 mM 4.0 mM 4.0p0.2 mM 4.0 mM

Ki,Glc(1,6)P2 69 M 3870 M 2122 M 30 MKi,Glc6P 69 M 9.020 M 1315 M 11p1 M 74p80 M 10 M

Ki,GSH 3.0 mM No inhibition 3.0 mMKeq 155

f 155

kcat,f 180 s1g

kcat,r 1.16 s1

eo 24 nMh

Scheme A2 A minimal model of human erythocyte GPI, TPI and enolase

Symbols: for GPI, A is Glc6P and P is Fru6P; for TPI, A is GraP and P is GrnP; for enolase,

A is 2-PGA and P is PEP. For all enzymes E represents the free enzyme and EA the corresonding

enzymemetabolite complex. Expressions for the steady-state kinetic parameters in terms of the

rate contants are: kcat,fl k3, kcat,rl k2, Km,Al (k2jk3)/k1, and Km,Pl (k2jk3)/k4, and

Keql k1k3/(k2k4).

assumed that all kinetic parameters (apart from kcat,f

and kcat,r

)were independent of pH.

GPI (EC 5.3.1.9). GPI was assumed to operate via a reversibleMichaelisMenten mechanism (Scheme A2) with the steady-state rate equation given by:

dp

dtl

e! 0kcat,f aK

m,A

kkcat,r

p

Km,P

11j

a

Km,A

jp

Km,P

(A3)

A unique set of rate constants was found from the steady-statekinetic parameters (Table A2).

PFK (EC 2.7.1.11). PFK was modelled with a concerted two-state symmetry scheme [6]; it is a modification of the two-

Table A2 Kinetic parameters used in the model of GPI at pH 7.2

The values of rate constants that yielded steady-state parameter values closest to the literature

values were: (M1:s1), k1l 1.78i107, k4l 4.55i10

7 ; (s1), k2l 1760, and k3l

1470. Notes: a[45]; b[46]; cKeql p/a; value measured at pH 8.0, 38 mC [47];dkcat,r was

calculated from the specific activity of the purified (51100i) human erythrocyte enzyme (843

units/mg at pH 8.3, 30 mC; [48]) adjusted to pH 7.2 by fitting eqn. (A2) to data for the pH

dependence of GPI activity [95] and adjusted to 37 mC with the temperature correction factor

v (30 mC)l 0.76iv(37 mC) [37]; GPI is reported to be a dimer of identical subunits with a

total Mr of 132000p2000 [48];ethe specific activity of the reverse reaction was found to be

1.2 times that of the forward reaction [45,47] ; fthe concentration of GPI in human erythrocytes,

eo, was estimated from the measured human erythrocyte activity (20064 units/litre of

erythrocytes in the reverse direction, pH 7.6, 37 mC; [37,62]) adjusted to pH 7.2 as in d, and

the value of kcat,r(pH 7.2).

Parameter Reported value Predicted value

Km,Glc6P 300500 Ma, 125p10 Mb 181 M

Km,Fru6P 5080 Ma, 71.3p9.1 Mb 71 M

Keq 0.327c 0.327

kcat,r 1760 s1d 1760 s1

kcat,f

1470 s1e 1470 s1

eo 218 nMf

substrate symmetry model of Pettigrew and Frieden [7]. Hencethe proportion of PFK in the R state is given by:

pl1

1jL*(A4)


7/29/2019 3420581

10/16


where

L*l0 hK

a

1n01j atpKT,ATP

1%01j mgKT,Mg#+

1%01j 2,3-bpgKT,#,$-BPG

1%

01j fru6pKm,R,Fru'P

jfru(1,6)p

#K

m,R,Fru(",')P#

1%01j ampKR,AMP

1%01j piKR,Pi

1%01j glc(1,6)p#KR,Glc(",')P#

1%(A5)

and so

dp

dtl

e!0 kcat,fabK

m,R,AK

m,R,B

kkcat,r

pq

Km,R,P

Km,R,Q

11j

a

Km,R,A

jb

Km,R,B

jab

Km,R,A

Km,R,B

jp

Km,R,P

jq

Km,R,Q

jpq

Km,R,P

Km,R,Q

ip (A6)

where al [MgATP], bl [Fru6P], cl [Fru(1,6)P#] and ql

[MgADP]. A comparison of the parameter values used in themodel and literature values is shown in Table A3. The values ofthe binding constants for allosteric effectors used in the modelare much larger than those reported in the literature; these valueswere necessary to gain adequate fits of model simulations toexperimental data (see the Results and discussion section of the

main paper and the accompanying paper [8]). Also, the pH-dependence of PFK is included by noting that the model assumesthat the equilibrium between the R and the T states is defined bythe infinitely co-operative protonation of n identical groups ofpK

asuch that:

LlT!

R!

l0 hKa

1n (A7)where h is [H+] and the subscript

!indicates that these are the free

enzyme species [9].Aldolase (EC 4.1.2.13). The steady-state rate equation for the

mechanism shown in Scheme A3 is:

dp

dtl

e!0kcat,f

a

Km,Ak

kcat,f

pq

Km,P

Ki,Q1

1ji

Ki,I

ja

Km,A

jK

m,Qp

Km,P

Ki,Q

j01j iKi,I

1j qKi,Q

jK

m,Qap

Ki,A

Km,P

Ki,Q

jpq

Km,P

Ki,Q

(A8)

Kinetic parametes are reported in Table A4.TPI (EC 5.3.1.1). TPI was assumed to operate via a reversible

MichaelisMenten mechanism (Scheme A2) with the steady-state rate equation given by eqn. (A3). Kinetic parameters arereported in Table A5.

GAPDH (EC 1.2.1.12). GAPDH was modelled by thesubstituted-enzyme mechanism of Scheme A4. The steady-staterate equation for this mechanism is:

dp

dtl

e! 0 kcat,f abcK

m,AK

i,BK

i,C

kkcat,r

pqh

Ki,P

Km,Q

1c

Ki,C

01j cKi,C

1j pKi,P

01j cKi,C

1j Km,P qhKi,P

Km,Q

jK

m,Cab

Km,A

Ki,B

Ki,C

jac

Ki,A

Ki,C

jbc

Ki,B

Ki,C

01j cKi,C

1j apKi,A

Ki,P

jK

m,Pbqh

Ki,B

Ki,P

Km,Q

jcqh

Ki,C

Ki,Q

jpqh

Ki,P

Km,Q

jabc

Km,A

Ki,B

Ki,C

jK

m,Cabp

Ki,C

Km,A

Ki,B

Ki,P

jbcqh

Ki,B

Ki,C

Ki,Q

jK

m,Pbpqh

Ki,P

Km,Q

Ki,B

Ki,P

(A9)

A graph of activity of GAPDH, in the forward direction, as afunction of pH is bell-shaped with a maximum value at pH $8.6and pK

"and pK

#values of 7.5 and 10 respectively [10]. In the

present work, in the absence of any relevant data, it was assumed

that the reverse reaction shows the same pH-dependence. Theeffects of changes in pH were incorporated into the model byassuming that all k

ivalues vary in accordance with eqn. (A2);

thus kcat,f

, kcat,r

, KiC

, and KiP

became functions of pH. Theparameters used in the model are summarized in Table A6.

PGK (EC 2.7.2.3). The steady-state rate equation for themodel shown in Scheme A5 is:

dp

dtl

e!0 kcat,f abK

i,BK

m,A

kkcat,r

pq

Ki,Q

Km,P

11j

a

Ki,A

jb

Ki,B

jab

Ki,B

Km,A

jp

Ki,P

jq

Ki,Q

jpq

Ki,Q

Km,P

(A10)

Kinetic parameters are reported in Table A7.PGM (EC 5.4.2.1). From the discussion in [8], to a good

approximation PGM could be modelled as a reversibleMichaelisMenten enzyme (eqn. A3). The parameters used in themodel are given in Table A8.

Enolase (EC 4.2.1.11). The steady-state rate equation for themodel shown in Scheme A5 is:

dp

dtl

e!0 kcat,f abK

i,BK

m,A

kkcat,r

pq

Ki,Q

Km,P

11j

a

Ki,A

jb

Ki,B

jab

Ki,B

Km,A

jq

Ki,Q

jbq

Ki,Q

Km,P

(A11)

Note that this is different from eqn. (A9), since Mg#+ is both asubstrate and a product. The parameters used in the model aresummarized in Table A9.


7/29/2019 3420581

11/16


Table A3 Comparison of the reported kinetic parameters of humanerythrocyte PFK with those used in the model

Notes:a[49], rat erythrocytes, 37 mC, pH 7.2; b[50], rat erythrocytes, 37 mC, pH 7.2; c[51],

rabbit muscle, 30 mC, pH 8.0; d[52], rabbit muscle, 30 mC, pH 8.0; eKeql [fru(1,6)p2mgadp[/(mgatp fru6p),[51], 30 mC, pH 8.0; fkcat,f was calculated from the specific activity of

the purified (22 500i) human erythrocyte enzyme (90 units/mg, pH 8.0, 25 mC) multiplied

by a temperature correction factor of 1.72 [62]; the mean Mr of the PFK tetramer was taken

to be 318000 [53]; gaccording to Hanson et al. [51] kcat,f is $ 23 times kcat,r ;hthe

concentration of PFK in human erythrocytes, eo, was estimated from the measured human

erythrocyte activity (3600 units/litre of erythrocytes, pH 7.6, 37 mC; [62]) and the value of kcat,f ;jsee the text for details.


Km,R,Fru6P 0.075 mMa, 0.150.3 mMb 0.075 mM

Km,R,MgATP 0.068 mMa 0.068 mM

Km,T,MgATP 0.068 mMa 0.068 mM

Km,R,Fru(1,6)P2 0.42 mMc, 0.310.4 mMd 0.50 mM

Km,R,MgADP 2069 mMc, 0.210.27 mMd 0.54 mM

Km,T,MgADP 0.54 mM

KT,ATP 9.8 Ma 100 M

KT,2,3-BPG 1.44 mMb 5.0 mM

KT,Mg2+ 0.44 mM 4.0 mM

KR,AMP 0.035 mMb 0.300 mM

KR,Pi 0.431 mMb 30 mM

KR,Glc(1,6)P2 0.0151 mMb 10 mM

Keq 1.2 i 103e 1.2i103

kcat,f 822 s1f 822 s1

kcat,r 36 s1g 36 s1

eo 1.1i107h 1.1i107 M

pKa 6.64j 7.05j

n 2, 8j 5.0j

Scheme A3 A minimal model of human erythrocyte Ald

The model is consistent with the reported ordered Uni Bi mechanism for Ald [61] and the

competitive inhibition of it by 2,3-BPG with respect to Fru(1,6)P2 [59,60]. Symbols: E, free

enzyme; A, Fru(1,6)P2 ; P, GraP; Q, GrnP; I, 2,3-BPGjMg:2,3-BPG; all other species are the

corresponding enzymemetabolite complexes. Expressions for the steady-state kinetic

parameters in terms of the rate constants are: kcat,fl k3k5/(k3jk5), kcat,rl k2, Km,Al

k5(k2jk3)/[k1(k3jk5)], Ki,Al k2/k1, Km,Ql k2/k6, Ki,Ql k5/k6, Km,Pl (k2jk3)/k4, Kil

k8/k7, and Keql k1k3k5/(k2k4k6).

PK (EC 2.7.1.40). Like PFK, PK was modelled with aconcerted two-state symmetry scheme [6]. The present model ofPK was based on that of Holzhu$ tter et al. [11] and the rateequation is given by eqns. (A4)(A6), where

L*l0 hK

a

1n01j atpKT,ATP

1%

01j pepKm,R,PEP

jpyr

Km,R,Pyr

1%01j fru(1,6)p#KR,Fru(",')P#

jglc(1,6)p

#K

R,Glc(",')P#

1%(A12)

Table A4 Kinetic parameters used in the aldolase model at pH 7.2

The rate constants that yielded steady-state parameters closest to the literature values were:

(M1:s1), k1l 1.18i107, k4l 6.5i10

6, k6l 6.62i106, k7l 1i10

9 ; (s1), k2l

234, k3l 995, k5l 73, and k8l 1.5i106. Notes:a[54], human erythrocyte; b[55], human

erythrocyte; c[56], rabbit muscle; d[57], rabbit muscle; e[58], rabbit muscle; f[59,60]; g[61],

pH 7.0, 35 mC; hkcat,f was calculated from the specific activity of the purified (8000i) human

erythrocyte enzyme (16.1 units/mg at pH 8.0, 30 mC; [54]) adjusted to 37 mC by multiplying

by the temperature correction factor 1.59 [62]; Ald is reported to be a tetramer of total Mr158 000 [54]; ithe specific activity of the reverse reaction in rabbit muscle was found to be 3.45

times that of the forward reaction, at 25 mC [57]; jthe concentration of GPI in human

erythrocytes, eo, was estimated from the measured human erythrocyte activity (1050 units/litre

of erythrocytes, pH 7.6, 37 mC; [62]) adjusted to pH 7.2 with eqn. (A2) and the value of kcat,f.


Km,Fru(1,6)P2 7.1Ma, 18 Mb 7.1 M

Ki,Fru(1,6)P2 19.8 MKm,GraP 190 M

c, 1 mMd 189 MKm,GrnP 2 m M

d 35 MKi,GrnP 1030 M

e 11 MKi,2,3-BPG 1.5 mM

f 1.5 mM

Keq 8.5i105 Mg 8.5i105 M

kcat,f 68 s1h 68 s1

kcat,r 234 s1j 234 s1

eo 37 Mj

Table A5 Kinetic parameters in the model of TPI

The rate constants that yielded steady-state parameters closest to literature values were:

(M1:s1), k1l 3.55i107, k4l 9.75i10

7 ; (s1), k2l 1 280, and k3l 14 560. Notes:a[63], kinetic parameters of the isoenzyme which makes up 70k75% of the total activity in

the human erythrocyte (three isoenzymes of TPI have been distinguished); b[64], human

erythrocyte; cKeql p/a, [65], 38 mC ;dkcat,f was calculated from the specific activity of the

purified (4265i) human erythrocyte enzyme (10236 units/mg at pH 7.6, 30 mC; [66])

adjusted to 37 mC by multiplication by the temperature correction factor of 1.52 [62]; TPI is

reported to be a dimer with a total Mr of 56000 [63];ekcat,f is 11.4 times kcat,r in rabbit muscle

[62]; fthe concentration of TPI in human erythrocytes, eo, was estimated from the measured

human erythrocyte activity (697000 units/litre of erythrocytes, pH 7.6, 37 mC; [62]) and the

value of kcat,f.


Km,GraP 434p56 Ma, 350 Mb 446 M

Km,GrnP 822p165 Ma 162.4 M

Keq 20.7c 4.14

kcat,f 14560 s1d 14560 s1

kcat,r 1280 s1e 1280 s1

eo 1.14 Mf

The parameters used in the model are given in Table A10.LDH (EC 1.1.1.27). LDH has been shown repeatedly, from a

wide variety of sources, to have a compulsory-order ternary-complex mechanism [12]. A model that is consistent with thismechanism as well as the reported pH behaviour [13,14] of theenzyme was proposed in which a single amino acid residue(probably His"*&) with a pK

aof 6.8 was necessarily protonated

for Pyr to bind and deprotonated for Lac to bind [15] (SchemeA6A). This scheme can be simplified by assuming that theprotonation steps are in equilibrium (Scheme A6B). The steady-state kinetic expression for this simplified reaction scheme isgiven by eqn. (A13).


7/29/2019 3420581

12/16


Scheme A4 A minimal model of human erythrocyte GAPDH based on thesubstituted-enzyme mechanism described by Segal and Boyer [94]

Symbols: E, thioacyl-enzyme; F, NAD+enzyme complex; A, NAD; B, Pi ; C, GraP; P, 1,3-BPG;

Q, NADH; H, H+. Expressions for the steady-state kinetic parameters in terms of rate constants

are: kcat,fl k5k9/(k5jk9), kcat,rl k2k4k8/(k2k4jk2k8jk4k8), Km,Al k5k9/[k1(k5jk9)], Ki,Al k2/k1, Km,Bl k9(k4jk5)/[k3(k5jk9)], Ki,Bl k2(k4jk5)/(k3k5), Km,Cl

k5(k8jk9)/[k7(k5jk9)], Ki,Cl 1/[k2k7k9(k4jk5)], Khi,Cl k12/k11, Km,Pl

k2k8(k4jk5)/[k6(k2k4jk2k8jk4k8)], Ki,Pl 1/[k2k4k6(k8jk9)], Khi,Pl k5/k6, Km,Ql

k2k4(k8jk9)/[k10(k2k4jk2k8jk4k8)], Ki,Ql k9/k10, and Keql k1k3k5k7k9/(k2k4k6k8k10).

Table A6 Comparison of the reported kinetic parameters of human erythrocyte GAPDH with those used (at pH 7.2) in Scheme A4

The ki values that yielded steady-state parameters consistent with the literature values were: (M1:s1), k1l 5.16i10

6, k3l k11 l 1.0i109, k6l k10l 2.55i10

8, k7l 3.06i106 ; (s1),

k2l 232, k4l 3.48i106, k5l 255, k8l 6.50i10

2, k9l 2.55i103, and k12l 3.1i10

4. Note that k10, Km,Q and Ki,Q are apparent values; the true value of these constants is found

by dividing by 107.2. Notes: a[68], human erythrocytes, 37 mC, pH 7.4; some constants calculated assuming sequential mechanism; b[69], human tissue, 25 mC, pH 7.0; c[70], human GAPDH,

37 mC, pH 7.2, 0.15 M KCl; d[71], rabbit muscle, 26 mC, pH 7.4; eKeql (1,3-bpg:h:nadh)/(grap:nad:pi) [72];fkcat,f was calculated from the specific activity of the purified (75i) human

erythrocyte enzyme (98 units/mg; pH 7.4, 37 mC; [68]). The Mr of the GAPDH tetramer was taken to be 142000 [69];gFurfine and Velick [71] measured the maximal velocity in the reverse

direction to be 11.92 times that in the forward direction; hthe concentration of GAPDH in human erythrocytes, eo, was estimated from the measured human erythrocyte activity (74600 units/litre

of erythrocytes, pH 7.6, 37 mC; [62]) and the value of kcat,f.


Km,NAD+ 45 Ma, 55210 Mb, 50 Mc, 90 Md 45 M

Ki,NAD+ 45 Ma, 100 Md 45 M

Km,Pi 78 Ma, 3.9 mMc 3.16 mM

Ki,Pi 3.16 mMKm,GraP 95 M

a, 1021 Mb, 5 Mc, 2.5 Md 95 MKi,GraP 0.06 M

d 1.59i1019 M

Ki,hGraP 0.031 mMa 0.031 mM

Km,NADH 8.3 Mc, 3.3 Md 3.3 M

Ki,NADH 218 Mb, 0.45 Mc, 3 Md 10 M

Km,1,3-BPG 0.8 Md, 3.5 Mc 0.671 M

Ki,1,3-BPG 1.52i1021 M

Kd,1,3-BPG 0.22 Mc, 1 Md 1 M

Keq 5.4i108e 1.9i108

kcat,f 232 s1f 232 s1

kcat,r 2765 s1g 171 s1

eo 7.66 Mh 7.66 M

Scheme A5 A minimal model of human erythrocyte PK, enolase and AK

This model is consistent with reports that the substrates of phosphoglycerate kinase bind in

a rapid-random mechanism in the forward direction [77]. Enolase is reported to have a

compulsory-order ternary-complex mechanism with Mg2+ acting as the second substrate in

both directions [95,96]. However, the MichaelisMenten parameters for such a mechanism

have not been determined in detail, and so to a first approximation enolase was modelled with

a rapid-random ternary-complex mechanism. AK has been shown have a random-order ternary-

complex mechanism [25]. Symbols: for PGK, E is free enzyme, A is 1,3-BPG, B is MgADP,

P is 3-PG and Q is MgATP; for enolase, A is 2-PGA, B is Mg2+, C is Mg2+, D is PEP; for

AK, A is ADP, B is MgADP, P is AMP and Q is MgATP; all other species are the corresponding

enzymemetabolite complexes. For PGK and enolase, all reactions except the interconversion

of EAB and EPQ were assumed to be in rapid equilibrium. Hence expressions for the steady-

state kinetic parameters in terms of the rate constants are: kcat,fl k9, kcat,rl k10, Km,Al

k8/k7, Km,Bl k4/k3, Ki,Al k2/k1, Ki,Bl k6/k5, Km,Pl k11/k12, Km,Ql k15/k16, Ki,Pl

k17

/k18

, and Ki,Q

l k13

/k14

. The rapid-equilibrium assumption is not valid for AK [25], hence

expressions for the steady-state kinetic parameters in terms of the rate constants are: Keql

(k1 k3 k9 k11 k13)/(k2 k4 k10 k12 k14)l (k1 k3 k9 k15 k17)/(k2 k4 k10 k16 k18)l (k5 k7 k9 k11k13)/(k6 k8 k10 k12 k14)l (k5 k7 k9 k15 k17)/(k6 k8 k10 k16 k18), kcat,fl (N1jN2ajN3b)/

(D7jD16ajD19b), kcat,rl (N18jN16pjN17q)/(D14jD30pjD31q), Km,Al (D3jD10b)/

(D7jD16ajD19b), Km,Bl (D2jD6a)/(D7jD16ajD19b), Km,Pl (D5jD15q)/(D14j

D30pjD31q), and Km,Ql (D4jD13p)/(D14jD30pjD31q).


7/29/2019 3420581

13/16


Table A7 Comparison of the reported kinetic parameters of humanerythrocyte PGK with those predicted from the scheme described in Scheme

A5

The ki values which yielded steady-state parameters consistent with literature values were:

(M1:s1i109), k1l k3l k5l k7lk12l k14l k16l k18l 1.0 ; (s1), k2l

1.6i103, k4l 1.0i105, k6l 8.0i10

4, k8l 2.0i103, k11l 1.0i10

6, k13l

2.05i105, k15l 1.1i106, k17l 1.86i10

5, k9l 2290, and k10l 917. Notes:a[73], pH

7.2, 37 mC; inhibitions are competitive with respect to the metabolite in parentheses; b[74],

rabbit and yeast; c[75], pH 7.5, 25 mC ; d[76] 37 mC ; e[77], 25 mC, pH 8.0; fkcat,r for the most

purified human erythrocyte enzyme (4467i) was calculated from the data of Yoshida and

Watanabe ([75]; specific activity 670 units/mg, pH 7.5, 25 mC) assuming that the Mr of the

enzyme is 49600; gcalculated from kcat,r assuming the activity in the direction of glycolysis is

2.5 times this value [74]; hthe concentration of PGK in human erythrocytes, eo, was estimated

from the measured human erythrocyte activity (105 600 units/litre of erythrocytes, 37 mC, pH

7.6; [62]).


Km,MgADP 100 Ma,d 100 M

Ki,MgADP 0.050.08 mMe 80 M

Km,1,3-BPG 2 Ma, 1.9 Md 2 M

Ki,1,3-BPG 1.6 mM

Km,MgATP 1 mMc 1 mM

Ki,MgATP 1.87 mM (MgADP)a, 0.45 mM (1,3-BPG)a 0.186 mM

Km,3-PGA 1.1 mMc 1.1 mM

Ki,3-PGA 0.205 mM

Keq 3.2-3.6i103b 3.2i103

kcat,f 2290 s1g 2290 s1

kcat,r 917 s1f 917 s1

eo 2.74 Mh 2.74 Mh

dp

dtl

e!0 kcat,f abK

i,AK

m,B

kkcat,r

pq

Km,P

Ki,Q

1

01j

Km,A

b

Ki,A Km,B

jK

m,Qp

Km,P Ki,Q1 01j

b

Ki,B1j

a

Ki,A

jq

Ki,Q

jab

Ki,A Km,B

jK

m,Qap

Ki,A Km,P Ki,Q

jK

m,Abq

Ki,A Km,B Ki,Q

jpq

Km,P

Ki,Q

jabp

Ki,A

Km,B

Ki,P

jbpq

Ki,B

Km,P

Ki,Q

(A13)

where e!

is the total enzyme concentration. See Table A11 forparameter values.

LDH(P). Following Schuster et al. [16] an NADPH-dependentLDH [LDH(P)] ; [17,18] was included in the overall schemebecause of its importance as a means of coupling between theredox carriers of glycolysis and the oxidative PPP. The kineticswere modelled assuming a random-order ternary-complex mech-anism, using the rapid-equilibrium assumption, that the K

mvalues for NADP+ and NADPH are significantly larger than

their in io concentrations, and that Km,LaclKm,Pyr. Thus:

dp

dtl

e! 0 kcat,f pyr nadphK

m,PyrK

m,NADPH

kkcat,r

lac nadp

Km,Lac

Km,NADP

11j

pyr

Km,Pyr

jlac

Km,Lac

(A14)

Owing to the equality of the NAD+ and NADP+ redox potentials,K

eqfor this reaction was taken to be the same as if NAD+

were a reactant. Thus from the data of [16] the parametervalues at pH 7.2 were as follows : e

!kcat,f

\Km,NADPH

,3.46i10$ s" ; K

m,Pyr, 4.14i10% mM; e

!kcat,f

\Km,NADP

,

Table A8 Kinetic parameters used in the simplified PGM reaction

See [8,78] for references of reported values.

Paramet er Report ed value Predicted val ue

Km,3-PGA 168 M 168 MKm,2-PGA 14 M 25.6 Mkcat,f 795 s

1 795 s1

kcat,r 714 s1 714 s1

Keq 0.08, 0.17 0.17eo 410 nM 410 nM

Table A9 Kinetic parameters used in the enolase model

Notes: a[79] rat liver enzyme, pH 7.5, 0 mC; areported value is an activation constant; ckcat,fwas calculated from the specific activity of the purified (270i) rat liver enzyme (87 units/mg

at pH 7.3, 30 mC; [79]) adjusted to 37 mC by multiplication with the temperature correction

factor of 1.43 [62]; enolase from rat liver is a dimer with a total Mr of 91000 [79];dthe maximal

rate of the forward reaction has been reported to be 3.78 times that of the reverse reaction [79];eKeql p/a [80,81];

fthe concentration of enolase in human erythrocytes, eo, was estimated

from the measured human erythrocyte activity (1800 units/litre of erythrocytes, pH 7.6, 37 mC ;

[62]) and the value of kcat,f.


Km,2-PGA, Ki,2-PGA 140p20 Ma 140 M

Km,PEP, Ki,PEP 310p45 Ma 110.5 M

Km,Mg, Ki,Mg 460p55 Ma,b 46 M

kcat,f 190 s1c 190 s1

kcat,r 50 s1d 50 s1

Keq 2.96.3e 3.0

eo 0.22 M 0.22 Mf

5.43i10( s" ; and Km,Lac

, 4.14i10% mM. The values ofe!kcat,f

\Km,NADPH

and eokcat,f

\Km,NADP

were reduced in comparisonwith those used in [16] on account of the higher values of freeNADP(+)(H) used in the present model and in order to keep PPPcarbon flux within the normal limits.

Enzymes of the 2,3-BPG shunt

The rate equations and parameter values used to model BPGS\P

are reported in [8].

Enzymes of the PPP

The kinetic equations used to model the oxidative part of thePPP [G6PDH, lactonase (including the spontaneous hydrolysisof 6-phosphogluconolactone, and 6-PGDH] were from [19].However, when the kinetic parameters reported for 6-PGDHwere incorporated into the complete metabolic model, simu-lations of the in io steady-state gave concentrations of 6-PG of$ 0.3 mM; 6-PG is usually present in about micromolar concen-trations [20]. In order to increase the activity of 6-PGDH, andhence lower the simulated in io steady-state concentration of 6-


7/29/2019 3420581

14/16


Table A10 Comparison of the reported kinetic parameters of humanerythrocyte PK with those used in the model

Notes: a[82], human erythrocytes, 37 mC, pH 7.6; bestimated from the findings of Koster et al.

[83] that half-maximal stimulation of PK by Glc(1,6)P2 occurs at a concentration of 60 M ;cKeq

l (pyr:mgatp)/(mgadp:pep); the value of 370 at pH 9.0 [84] was adjusted to pH 7.2assuming Keql (pyr:mgatp)/(mgadp:pep:h) ;

dkcat,f was calculated from the specific activity

of the purified (36 264i) human erythrocyte enzyme (330 units/mg; pH 7.6, 37 mC; [85]);

the Mr of the PK tetramer varies from 240000 to 252000 depending on cell age [85], so Mrwas taken to be the average of these values, 246000, for calculations; e[86], human

erythrocytes; fthe concentration of PK in human erythrocytes, eo, was estimated from the

measured human erythrocyte activity (4950 units/litre of erythrocytes, pH 7.6, 37 mC [62]) and

the value of kcat,f ;gRozengurt et al. [87] showed that the allosteric and co-operative properties

of PK are strongly influenced by pH and this suggested that the transition from R to T involves

a deprotonation; examination of the values that these workers obtained for L at different pH

values shows that they are most closely matched if it is assumed that nl 1 (to the nearest

integer); the pH-dependence of the inhibition of erythrocyte PK by iodoacetamide involves a

group that has a pKa of 6.8 [88]; it was assumed that this was the group that was involved

in the R-to-T transition.


Km,R,PEP 0.225 mMa 0.225 mM

Km,R,MgADP 0.474 mMa 0.474 mM

Km,T,MgADP 0.474 mMa 0.474 mM

Km,R,Pyr 2.0 mM

Km,R,MgATP 3.0 mM

Km,T,MgATP 3.0 mMKT,ATP 3.39 mM

a 3.39 mM

KR,Fru(1,6)P2 0.005 mMa 0.005 mM

KR,Glc(1,6)P2 0.1 mMb 0.1 mM

Keq 23 345c 23 345

kcat,f 1 386 s1d 1386 s1

kcat,r 3.26 s1

eo 69 nMe 87 nMf

pKa 6.8g 6.8

n 1g 1

Table A11 Kinetic parameters used in the LDH model, at pH7.2

The predicted values were calculated from unitary rate constants determined by Borgmann et

al. [89] and adjusted to 37 mC ; k1l k9l 5.42i107

, k2l 1.33i102

, k3l 4.34i106

, k4l 8.58i102, k5l 2.91i10

3, k6l 4.71i105, k7l 5.43i10

2, k8l 1.08i106, and

k10l 5.46i103. Notes: a[90], human eryrthrocytes, 37 mC, pH 7.4; b[91] rabbit muscle, pH

7.15, 28 mC ; ccalculated from the unitary rate constants reported in Borgmann et al. [89]

adjusted to 37 mC using the vant Hoff equation; dKeql Lac NAD/(NADH Pyr);ekcat,f was

calculated from the specific activity of the partially purified human erythrocyte enzyme (400

units/mg at pH 7.4, 37 mC; [90]); the Mr of the tetramer was taken to be 144000 (see [90]);

the specific activity of the forward reaction was found to be 4.7 times that of the reverse reaction

[91]; fthe concentration of LDH in human erythrocytes, eo, was estimated from the measured

human erythrocyte activity (66000 units/litre of erythrocytes, pH 7.6, 37 mC; [62]) and the

value of kcat,f.


Km,NADH 7.1 Ma, 7.43 Mb, 8.44 Mc 8.44 M

Ki,NADH 6.8 Ma, 2.45 Mc 2.45 M

Km,Pyr 71 Ma

, 209 Mb

, 137 Mc

137 MKi,Pyr 228 M

c 228 MKhi,Pyr 101 M 101 MKm,NAD+ 85 M

b, 106.6 Mc 107 MKi,NAD+ 502.8 M

c 503 MKm,Lac 916 M

b, 1070 Mc 1070 MKi,Lac 7.33 mM

c 7.33 mM

Keq 1.01i1011c,d 1.01i1011

kcat,f 960 s1e, 458 s1c 458 s1

kcat,r 204 s1e, 115 s1c 115 s1

eo 3.12 Mf 3.43 M

Scheme A6 Kinetic models of human erythrocyte LDH

(A) A minimal model of human erythrocyte lactate dehydrogenase which accounts for pH effects

on enzyme parameter values. Symbols: E, free enzyme; EH, protonated free enzyme; A, NADH;

B, Pyr; P, Lac; Q, NAD+ ; all other species are the corresponding enzymemetabolite complexes.

(B) The assumption that the protonation steps are in equilibrium allows Model (A) to be reduced

by using a composite enzyme species consisting of a protonated and an unprotonated enzyme

form. Expressions for the steady-state kinetic parameters in terms of the rate constants given

in Model (B) are, where the underbars indicate the relevant parameters in (A) : kcat,fl

k5k7/(k5jk7), kcat,rl k2k4/(k2jk4), Km,Al k5k7/[k1(k5jk7)], Ki,Al k2/k1, Km,Bl

(k4jk5)k7/[k3(k5 j k7)], Ki,Bl (k2jk4)/k3, Km,Pl k2(k4jk5)/[(k2jk4)k6], Ki,Pl

(k5jk7)/k6, Km,Ql k2k4/[(k2jk4)k8], Ki,Ql k7/k8, KhiBl k10/k9 and Keql pq/(abh)l

k1k3k5k7/(k2k4k6k8h). For all i, kil ki, except that k3l k3h/(hjKa) and k6l k6Ka/(hjKa).

PG, the Km

values of NADP+ and 6-PG were decreased by afactor of 2, while the K

mvalues for Ru5P and NADPH were

increased by a factor of 2. This involved increasing k"

and k$

bya factor of 2 and decreasing k

"!and k

"#by the same factor. Also,

in [19], k$

should have the value 10* M":s" and not the reported10( M":s".

The kinetic equations used to model the non-oxidative PPP(R5PI, Ru5PE, TA and TK) were from McIntyre et al. [21]. TKcatalyses two reactions at a common active site and hence itproved simpler to model its operation with elementary rateequations rather than a steady-state rate equation.

Enzymes of nucleotide metabolism

In the presence of glucose, the catabolism of the adenine moietyof adenosine phosphates is very slow (turnover time $ 18 days)[22]. Therefore, for all simulations in which glucose was thesubstrate, the sum of the adenine nucleotides was treated as aconstant; hence the neglect of nucleotide metabolism appearedto be justified.

In the absence of Glc (pHe

7.4), AMP catabolism occurs at arate that is between 40 and 120 mol:litre of erythrocytes":h"

[2224]. Under these conditions $75% of AMP catabolismoccurs via AMP phosphohydrolase [22]. For simulations ofmetabolism in the absence of Glc, or some other fuel sources, thecatabolism of AMP was modelled as a simple zero-order process.


7/29/2019 3420581

15/16


Table A12 Comparison of the reported kinetic parameters of humanerythrocyte AK with those predicted (at pH 7.2) from the reaction describedin Scheme A5

The ki values that yielded steady-state parameters consistent with literature values were:

(M1:s1i108), k1l k7l 3.2, k3l k5l 1.7, k12l k18l 0.27, k14l k16l 0.23;(s1i104), k2l k8l 0.39, k4l k6l 30, k9l 17.5, k10l 28.0, k11l k17l 0.23,

and k13l k15l 4.0. Note that, since the MichaelisMenten kinetic parameters are a function

of substrate concentration (Scheme A5), predicted values were calculated at saturating

concentrations of substrates. Notes: aRabbit muscle AK at 25 mC, pH 8.0 [25]; bcalculated

from:

KappAK l

mgatp.amp

mgadp.adpl

K(1j10pH KHADPjkKKADPjmgKMgADP)2

(1j10pH KHATPjkKKATPjmgKMgATP) (1j10pH KHATMPjkKKAMP)

KMgATP

KMgADP

where Kl (ATPT AMPT)/ADPT2, and ATPT, ADPT, and AMPT refer to all complexes containing

ATP, ADP and AMP respectively; K has been reported to be 0.744 [92]; ckcat,f for purified

(18800i) human erythrocyte AK was calculated from the data of Tsuboi ([93]; 3200 units/mg,

25 mC, pH 8.0) using the temperature correction factor of 1.81 [62]; the Mr of the major AK

isoenzyme in human erythrocytes was taken to be 21300 [93]; dthe concentration of AK in

human erythrocytes, eo, was estimated from the measured human erythrocyte activity (85100

units/litre of erythrocytes, [62]) and the value of kcat,f.


Km,ADP 0.01 mMa

0.0026 mMKm,MgADP 0.3 mM

a 0.11 mM

Km,AMP 0.11 mMa 0.11 mM

Km,MgATP 0.067 mMa 0.066 mM

Keq 4.3b 4.3

kcat,f 2.06i103 s1c 2.08i103 s1

kcat,r 3.8i103 s1

eo 0.97 Md 0.97 M

AK (EC 2.7.4.3)

The rate-limiting steps of the mechanism shown in Scheme A5have been reported to be the dissociation of substrates [25,26].The steady-state rate equation for such a mechanism is notreported here (although it was derived). The set of rate-constant

values that yielded steady-state kinetic parameter values thatwere consistent with the known experimental ones, and obeyedthe principle of detailed balance [27], are given in Table A12.

ATPase

Following others (e.g. [16]) ATP consumption was modelled witha first-order rate constant, k

ATPase. No attempt was made to

model the individual ATP-consuming processes. It is noteworthythat, by 1988 [28], not all of these processes had been identified,and since then no progress has been made in this area.

Non-glycolytic NADH consumption

Erythrocytes possess an NADH-dependent metHb reductasesystem which maintains the iron atoms in Hb in their Fe(II)state. In normal erythrocytes the rate of Fe(II) oxidation andsubsequent reduction is $ 1.5% of the total subunit concen-tration per day [29]; in other words $ 10 mol:litre oferythrocytes":h". This process was modelled with the first-order rate constant k

oxNADHto account for the effect of non-

glycolytic NADH demand on erythrocyte metabolism.

GSH metabolism

The loss and synthesis of GSH is slow, with a turnover time of$8 days [30]. Hence in the current model these processes wereignored. GSSG reductase was modelled using the enzyme-kineticexpression given by Thorburn and Kuchel [19]. Note that in [19]the value of k

%should be 7.2i10% s" and not 7.2i10$ s". A

number of spontaneous and enzyme-catalysed reactions oxidizeGSH. Following Schuster et al. [16], these were simulated withthe first-order rate constant, k

ox.

Metabolite transport

The overall model of metabolism was constructed with theassumption that the only metabolic species that cross the cellmembrane in the time courses that are simulated are Glc, Lac,Pyr and P

i. It was assumed that the transport of each metabolite

across the cell membrane is characterized by a first-order rateconstant. Thus :

dMe

dtlkk

iM

ejk

!M0 Hct1kHct1 (A15)

dM

dtlkk

oMjk

iM

e01kHctHct 1 (A16)where M

eand M are the extracellular and intracellular concen-

trations of metabolite m, ki

and ko

are first-order rate constantscharacterizing the transport into and out of the cell, Hct is thehaematocrit of the sample, and is the fraction of the total cellvolume accessible to the intracellular solute. From eqns. (A15)and (A16):

Keql

M

Me

lki

ko

01kHctHct 1 (A17)If k

iis given a fixed value, then:

kol

ki

Keq

01kHctHct 1 (A18)If there is no transmembrane electrical potential difference (l0) then K

eql1. However, when 0, the Nernst equation

implies that:

Mz

Mze

l exp0kzFRT 1l rz (A19)where F is the Faraday constant, z is the ionic charge onmetabolite m, R is the gas constant, T is absolute temperature,and r is the so-called Donnan ratio ; the latter is the con-centration ratio of the species inside and outside the cell. Notethat this relationship is true only if the intracellular and extra-cellular activity coefficients of the metabolite are equal. In otherwords, it is assumed that all metabolites, m, distribute passivelyin response to the transmembrane electrical potential difference.Thus for a metabolite m, which ionizes to m" with a dissociationconstant pK

a", the transmembrane equilibrium ratio is:

Keql

MjM"

MejM"

e

l1j10pHipKa,"

1j10pHipKa," r"(A20)

where pHi

is the intracellular pH. For a metabolite m" whichionizes to m# with a dissociation constant pK

a,#, the trans-

membrane equilibrium ratio is:

Keql

M"jM#

M"ejM#

e

l1j10pHipKa,#

r"j10pHipKa,# r#(A21)

In human erythrocytes, glucose transport across the plasmamembrane is 2501000 times the rate of glucose utilization[3133]. Hence k

iwas set to$1 s". k

iwas taken to be 0.018 s"

for pyruvate transport [34] and 5.06i10$ s" for lactate trans-port [35]. Since the pK

a"value for pyruvate is 2.39, for most

physiological pH values Keql r. The K

eqfor lactate transport

was modelled with eqn. (A20) with pKa"l3.73. k

ifor phosphate

transport was taken to be 5.6i10% s" [36] at 37 mC and pH 7.4.


7/29/2019 3420581

16/16


The Keq

for phosphate transport was modelled with eqn. (A21)with pK

a#l6.75.

REFERENCES

1 Cha, S. (1968) J. Biol. Chem. 243, 820825

2 Jacobasch, G., Minakami, S. and Rapoport, S. M. (1974) in Cellular and Molecular

Biology of Erythrocytes (Yoshikawa, H. and Rapoport, S. M., eds.), pp. 5592,

University Park Press, Baltimore

3 Hinterberger, U., Ockel, E., Gerkscher-Mothes, W. and Rapoport, S. (1961) Acta Biol.

Med. Germ. 7, 5056

4 Brewer, G. J. (1974) in The Red Blood Cell (Surgenor, D. M., ed.), pp. 387433,Academic Press, New York

5 Magnani, M., Stocchi, V., Serafini, G. and Bossu, M. (1983) Ital. J. Biochem. 32,

2835

6 Monod, J., Wyman, J. and Changeux, J. P. (1965) J. Mol. Biol. 12, 88118


8 Mulquiney, P. J., Bubb, W. A. and Kuchel, P. W. (1999) Biochem. J. 342, 565578


10 Oguchi, M. (1970) J. Biochem. (Tokyo) 8, 427439


101111

12 Holbrook, J. J., Liljas, A., Steindel, S. J. and Rossmann, M. G. (1975) Enzymes 3rd

Ed. 11, 191292

13 Schwert, G. W., Miller, B. R. and Peanasky, R. J. (1967) J. Biol. Chem. 242,

32453252

14 Schwert, G. W. (1969) J. Biol. Chem. 244, 12851290

15 Holbrook, J. J. and Stinson, R. A. (1973) Biochem. J. 131, 739748

16 Schuster, R., Holzhu$ tter, H.-G. and Jacobasch, G. (1988) BioSystems 22, 193617 Rapoport, S. and Ababei, L. (1964) Acta Biol. Med. Germ. 13, 852864

18 Rapoport, I., Elsner, R., Mu$ ller, M., Dumdey, R. and Rapoport, S. (1979) Acta Biol.

Med. Germ. 38, 901908

19 Thorburn, D. R. and Kuchel, P. W. (1985) Eur. J. Biochem. 150, 371386

20 Kirkman, H. N. and Gaetani, G. F. (1986) J. Biol. Chem. 261, 40334038

21 McIntyre, L. M., Thorburn, D. R., Bubb, W. A. and Kuchel, P. W. (1989) Eur. J.

Biochem. 180, 399420

22 Van den Berghe, G. and Bontemps, F. (1990) Biomed. Biochim. Acta 49, S117S122


421427


421427

25 Rhoads, D. G. and Lowenstein, J. M. (1968) J. Biol. Chem. 243, 39633972

26 Nageswara Rao, B. D. and Cohn, M. (1978) J. Biol. Chem. 253, 11491158

27 Lam, C. F. (1981) Techniques for the Analysis and Modelling of Enzyme Kinetic

Mechanisms, Research Studies Press, Chichester

28 Rapoport, S. M. (1988) in The Roots of Modern Biochemistry: Fritz LipmannsSquiggle and its Consequences (Kleinkauf, H., von Do$hren, H. and Jaenicke, L.,

eds.), pp. 157164, Walter de Gruyter, Berlin

29 Grimes, A. J. (1980) Human Red Cell Metabolism, Blackwell Scientific Publications,

Oxford

30 Dimant, E., Landsberg, E. and London, I. M. (1955) J. Biol. Chem. 213, 769776

31 Widdas, W. F. (1954) J. Physiol. (London) 125, 163180

32 Potts, J. R., Hounslow, A. M. and Kuchel, P. W. (1990) Biochem. J. 266, 925928

33 Potts, J. R. and Kuchel, P. W. (1992) Biochem. J. 281, 753759

34 Joshi, A. and Palsson, B. O. (1989) J. Theor. Biol. 141, 515528

35 Kuchel, P. W., Chapman, B. E., Lovric, V. A., Raftos, J. E., Stewart, I. M. and

Thorburn, D. R. (1984) Biochim. Biophys. Acta 805, 191203

36 Kemp, G. J., Bevington, A., Khodja, D. and Russell, G. G. (1988) Biochim. Biophys.

Acta 969, 148157

37 Gerber, G., Preissler, H., Heinrich, R. and Rapoport, S. M. (1974) Eur. J. Biochem.

45, 3952

38 Rijksen, G., Jansen, G., Kraaijenhagen, R. J., van der Vllist, M. J. M., Vlug, A. M. C.

and Staal, G. E. J. (1981) Biochim. Biophys. Acta 659, 29230139 Fornaini, G., Magnani, M., Fazi, A., Accorsi, A., Stocchi, V. and Dacha, M. (1985)

Arch. Biochem. Biophys. 239, 352358

40 Rijksen, G. and Staal, G. E. J. (1977) Biochim. Biophys. Acta 485, 7586

41 Magnani, M., Stocchi, V., Ninfali, P., Dacha, M. and Fornaini, G. (1980) Biochim.

Biophys. Acta 615, 113120

42 Garfinkel, L., Garfinkel, D., Matsiras, P. and Matschinsky, B. (1987) Biochem. J. 244,

351357

43 Robbins, E. A. and Boyer, P. D. (1957) J. Biol. Chem. 224, 121135

44 Magnani, M., Dacha, M., Stocchi, V., Ninfali, P. and Fornaini, G. (1980) J. Biol.

Chem. 255, 17521756

Received 15 March 1999/17 May 1999; accepted 22 June 1999

45 Arnold, H., Hoffmann, A., Engelhardt, R. and Lohr, G. W. (1973) in Erythrocytes,

Thrombocytes, Leukocytes: Recent Advances in Membrane and Metabolic Research

(Gerlach, E., Moses, K., Deutsch, E. and Wilmanns, W., eds.), pp. 177180, Georg

Thieme, Stuttgart

46 Tilley, B. E., Gracy, R. W. and Welch, S. G. (1974) J. Biol. Chem. 249, 45714579

47 Kahana, S.E., Lowry, O. H., Schulz, D. W., Passoneau, J. V. and Crawford, E. J.

(1960) J. Biol. Chem. 235, 21782184

48 Gracy, R. W. and Tilley, B. E. (1975) Methods Enzymol. 41, 392400

49 Otto, M., Heinrich, R., Ku$ hn, B. and Jacobasch, G. (1974) Eur. J. Biochem. 49,

169178

50 Otto, M., Heinrich, R., Jacobasch, G. and Rapoport, S. (1977) Eur. J. Biochem. 74,413420

51 Hanson, R. L., Rudolph, F. B. and Lardy, H. A. (1973) J. Biol. Chem. 248,

78527859

52 Merry, S. and Britton, H. G. (1985) Biochem. J. 226, 1328

53 Dunaway, G. A., Kasten, T. P., Sebo, T. and Trapp, R. (1988) Biochem. J. 251,

677683

54 Yeltman, D. R. and Harris, B. G. (1977) Biochim. Biophys. Acta 484, 188198

55 Strapazon, E. and Steck, T. L. (1977) Biochemistry 16, 29662971

56 Mehler, A. H. (1963) J. Biol. Chem. 238, 100104

57 Penhoet, E. E., Kochman, M. and Rutter, W. J. (1969) Biochemistry 8, 43964402

58 Mehler, A. H. and Bloom, B. (1963) J. Biol. Chem. 238, 105107

59 Beutler, E. (1971) Nature (London) 232, 2021

60 Srivastava, S. K. and Beutler, E. (1972) Arch. Biochem. Biophys. 148, 249255

61 Rose, I. A., OConnell, E. L. and Mehler, A. H. (1965) J. Biol. Chem. 240,

17581765

62 Beutler, E. (1984) Red Cell Metabolism: A Manual of Biochemical Methods, 3rd edn.,Grune and Stratton, New York

63 Sawyer, T. H., Tilley, B. E. and Gracy, R. W. (1972) J. Biol. Chem. 247, 64996505

64 Schneider, A. S., Valentine, W. N., Hattori, M. and Heins, H. L. (1965) New Engl. J.

Med. 272, 229235

65 Meyerhof, O. and Junowicz-Kocholaty, R. (1943) J. Biol. Chem. 149, 7192

66 Gracy, R. W. (1975) Methods Enzymol. 41, 442447

67 Burton, P. M. and Waley, S. G. (1968) Biochim. Biophys. Acta 151, 714715

68 Wang, C.-S. and Alaupovic, P. (1980) Arch. Biochem. Biophys. 205, 136145

69 Heinz, F. and Freimu$ ller, B. (1982) Methods Enzymol. 89, 301309

70 Maretzki, D., Tsamalukas, A. G., Ku$ ttner, G., Kru$ ger, S., Groth, J. and Rapoport, S.

(1974) in Abhandlugen der Akademie der Wissenschaften der DDR Jahrgang 1973:

VII Internationales Symposium u$ber Struckter und Funktion der Erythrozyten

(Rapoport, S. and Jung, F., eds.), pp. 4751, Akademie-Verlag, Berlin

71 Furfine, C. S. and Velick, S. F. (1965) J. Biol. Chem. 240, 844855

72 Cori, C. F., Verlick, S. F. and Cori, G. T. (1950) Biochim. Biophys. Acta 4, 160169

73 Jacobasch, G., Minakami, S. and Rapoport, S. M. (1974) in Cellular and MolecularBiology of Erythrocytes (Yoshikawa, H. and Rapoport, S. M., eds.), pp. 5592,

University Park Press, Baltimore

74 Krietsch, W. K. G. and Bu$ cher, T. (1970) Eur. J. Biochem. 17, 568580

75 Yoshida, T. and Watanabe, S. (1972) J. Biol. Chem. 247, 440445

76 Ali, M. and Brownstone, Y. S. (1976) Biochim. Biophys. Acta 445, 89103

77 Lee, C. S. and OSullivan, W. J. (1975) J. Biol. Chem. 250, 12751281

78 Mulquiney, P. J. and Kuchel, P. W. (1999) Biochem. J. 342, 595602

79 Rider, C. C. and Taylor, C. B. (1974) Biochim. Biophys. Acta 365, 285300

80 Wold, F. and Ballou, C. E. (1957) J. Biol. Chem. 227, 301312

81 Garfinkel, L. and Garfinkel, D. (1985) Magnesium 4, 6072


101111

83 Koster, J. F., Slee, R. G., Staal, G. E. J. and van Berkel, T. J. C. (1972) Biochim.

Biophys. Acta 258, 763768

84 McQuate, J. T. and Utter, M. F. (1959) J. Biol. Chem. 234, 21512157

85 Kahn, A. and Marie, J. (1982) Methods Enzymol. 90, 13114086 Albe, K. R., Butler, M. H. and Wright, B. E. (1990) J. Theor. Biol. 143, 163195

87 Rozengurt, E., de Asua, L. J. and Carminatti, H. (1969) J. Biol. Chem. 244,

31423147

88 Jacobson, K. W. and Black, J. A. (1971) J. Biol. Chem. 246, 55

Documents

3420581