use of mathematical models for predicting the metabolic effect of large-scale enzyme activity...

Eur. J. Biochem. 229,403-418 (1995) 0 FEBS 1995

Use of mathematical models for predicting the metabolic effect of large-scale enzyme activity alterations Application to enzyme deficiencies of red blood cells

Ronny SCHUSTER and Hermann-Georg HOLZHWER Institut fur Biochemie, Medizinische Fakultat (Charit&), Humboldt-Universitat zu Berlin, Germany

(Received 1 December 1994/26 January 1995) - EJB 94 1848/1

There are numerous examples showing that the metabolism of cells can be severely impaired if the activity of only one of the participating enzymes undergoes large-scale alterations, resulting, for example, from spontaneous mutations (inherited or aquired enzymopathies), the administration of toxic drugs or self-inactivation of enzymes during cell aging. However, a quantitative relationship between the degree of enzyme deficiency and the extent of metabolic dysfunction is very difficult to establish by experimental means. An alternative is to tackle this problem by mathematical modelling. Our approach is based on a comprehensive mathematical model of the energy and redox metabolism for human erythrocytes. We calculate stationary states of the cell metabolism, varying the activity of each of the participating enzymes by several orders of magnitude. The metabolic states are then evaluated in terms of a performance function which relates the metabolic variables to the overall functional fitness of the cell. The performance function for the erythrocyte takes into account the homeostasis of three essential metabolic variables : the energetic state (ATP), the reductive capacity (reduced glutathione), and the osmotic state. Based on the behaviour of the performance function at varying enzyme activities, we estimate those ranges of enzyme activities, in which the metabolic alterations should be either tolerable, associated with non-chronic or chronic diseases, or letal. For most enzymopathies, the experimental and clinical observations can be satisfactorily rationalized by the computational results. Moreover, a surprisingly high correlation is found between the range of the activity range where disease is predicted by the model and the observed number of diseased probands.

Another objective of our study was to contribute to the theory of metabolic control. The well-elaborated concept of the metabolic control theory is restricted to (infinitely) small activity alterations. In order to quantify the metabolic effect of finite (large-scale) changes in the activity of an enzyme, we propose, as a control measure, the effective activity Em, defined as the relative activity of an enzyme (with respect to the activity in a reference state) required to bring about a change in the stationary value of a metabolic variable by the (finite) factor a. We demonstrate that none of the existing extrapolation methods using the conventional control coefficient is capable to provide reliable predictions of the effective activities for all enzymes of erythrocyte metabolism.

Keywords. Mathematical modelling ; metabolic system ; enzyme deficiency ; control theory ; erythrocyte.

Correspondence to R. Schuster, Institut fur Biochemie, Medizinische Fakultat (Charit&), Humboldt-Universitat zu Berlin, Hessische Strasse 3-4, D-10115 Berlin, Germany

Fax: +49 30 28468600. Abbreviations. Glc6P, glucose 6-phosphate; Fru6P, fructose 6-phos-

phate; Frul,6P,, fructose 1,6-bisphosphate ; GraP, glyceraldehyde 3- phosphate ; DHAP, dihydroxyacetone phosphate; 1,3P2Gri, 1,3-bisphosphoglycerate; 2,3P,Gri, 2,3-bisphosphoglycerate; 3PGri, 3-phosphoglycerate ; 2PGri, 2-phosphoglycerate ; P-pyruvate; phosphoenolpy- ruvate; 6PGlcA, 6-phosphogluconate ; RulSP, ribulose 5-phosphate ; XulSP, xylulose 5-phosphate; RibSP, ribose 5-phosphate; S7P, sedohep- tulose 7-phosphate; E4P, erythrose 4-phosphate; GSH (GSSG), reduced (oxidized) glutathione; Ec, effective activity; MCT, metabolic control theory; Q, share of intermediates in osmotic pressure of the cell; HK, hexokinase ; GPI, glucose-6-phosphate isomerase ; PFK, phosphofructokinase ; TPI, triosephosphate isomerase; GAPD, glyceraldehyde phosphate dehydrogenase; PGK, phosphoglycerate kinase; DPGM, 2,3-bisphosphoglycerate mutase ; DPG, 2,3-bisphosphoglycerate phosphatase ;

PGM, 3-phosphoglycerate mutase ; PK, pyruvate kinase ; LDH, lactate dehydrogenase ; AK, adenylate kinase; G6PD, glucose-6-phosphate dehydrogenase ; 6PGD, 6-phosphogluconate dehydrogenase ; GSSGR, glutathione reductase; EP, ribose phosphate epimerase; KI, ribose phosphate isomerase; TK, transketolase; TA, transaldolase; PRPPS, phos- phoribosylpyrophosphate synthetase.

Enzymes. Hexokinase (EC 2.7.1 1) ; glucose-6-phosphate isomerase (EC 5.3.1.18); phosphofructokinase (EC 2.7.1.11); aldolase (EC 4.1.2.13); triosephosphate isomerase (EC5.3.1.1); glyceraldehyde phosphate dehydrogenase (EC 1.2.1.12) ; phosphoglycerate kinase (EC 2.7.2.3); 2,3-bisphosphoglycerate mutase (EC 5.4.2.4); 2,3-bisphosphoglycerate phosphatase (EC 3.1.3.13); 3-phosphoglycerate mutase (EC 5.4.2.1); pyruvate kinase (EC 2.7.1.40); lactate dehydrogenase (EC 1.1.1.28) ; adenylate kinase (EC 2.7.4.3) ; glucose-6-phosphate dehydrogenase (EC 1.1.1.49); 6-phosphogluconate dehydrogenase (EC 1.1.1.44); glutathione reductase (EC 1.6.4.2); ribose phosphate isomerase (EC 5.3.1.6); transketolase (E 2.2.1.1); transaldolase (EC 2.2.1.2).

404 Schuster and Holzhutter (Eul: J. Biochern. 229)

Ru5P

GraP Fru6P TK2 Energy consuming

PRPPS ATP- processes

GSHox [ GSSGR I GBPD] / I ADP ATP

P G S H jb NADP 4 Fleeduction

of ceff components DHAP ATP ADP x

AMP ADP

2PG IEN

t LAC

NADP NADPH

Fig. 1. Reaction scheme of energy and redox metabolism of erythrocytes as used for the mathematical model. X5P, xylulose 5-phosphate; S7P, sedulose 7-phosphate; E4P, erythrose 4-phosphate; R5P, ribose 5-phosphate; PEP, phosphoenolpymvate; 3PG, 3-phosphoglycerate; 2PG, 2-phosphoglycerate; 1 ,3P2G,1,3-bisphosphoglycerate ; 2,3P2G,2,3-bisphosphoglycerate ; 6PG, 6-phosphoglycerate.

The metabolism of living systems consists of many thou- sands of interconnected chemical reactions being, with a few exceptions, catalysed by enzymes which have a well-defined molecular structure and thus well-defined kinetic parameters as, for example, maximum activities, binding constants for sub- strates or allosteric effectors or half-life time. Changing the kinetic properties of only one single enzyme causes more or less pronounced changes in the metabolic states of the system under consideration and, consequently, of its viability and functional performance. Such changes of the kinetic properties of enzymes occur permanently during the time-course of biological evolution, due to spontaneous mutations affecting the amino acid se- quence and thus the spatial arrangement of enzyme molecules. If the resulting new steady state allows for a significantly im- proved functional performance of the system (e.g. owing to a higher flux rate through a pathway which is essential for self- reproduction), the mutant enzyme may become dominant in the process of natural selection. Thus, permanent evolutionary optimization of enzyme function may occur. In some recent theoretical papers, we have dealt with the problem of how to design the kinetic parameters of enzymes in order to achieve an optimal performance of the corresponding metabolic pathway (Heinrich and Holzhiitter, 1985; Heinrich et al., 1987, 1991; Schuster et al., 1991). If, alternatively, changes in the kinetic properties of an enzyme result in a reduced viability of the affected species, the kinetically modified enzyme is usually termed ‘deficient’. More and more diseases with unknown origin have been eluci- dated as being caused by inherited or aquired enzyme deficiencies (Belfiore, 1980) and a deeper understanding of the underly- ing molecular and cellular alterations represent a rapidly grow- ing field of medical and biochemical research (Fujii and Miwa, 1990).

Beside spontaneously occuring natural changes of enzyme- kinetic properties; there is substantial interest in medicine and biotechnology to modify the kinetic properties of enzymes in order to manipulate the metabolism and functional performance of specific cells. Computer-aided molecular design of drugs, directed towards distinct enzymes, has become an important branch of modem pharmacology (Earlick et al., 1991). Enzyme mutants generated by site-directed mutagenesis, as well as activity alterations due to intervention into gene expression, may increase the flux through given metabolic pathways, which paves the way for improving the efficiency of microorganisms employed as bioreactors in biotechnology (Small and Kacser, 1993; Kacser and Acerenza, 1993).

In all areas mentioned above, one important subject of math- ematically oriented theoretical research is to predict the metabolic changes caused by changing the activity of a given enzyme in a defined manner. Such analysis may contribute to understand the possible strategies governing the evolution of metabolic systems and to the elucidation of those enzymes which are a sui- table target for the manipulation of metabolic processes to fit medical or biotechnological purposes. This, however, requires a reliable mathematical model for the metabolic system under consideration. During the past two decades, we have reached at an advanced level in the mathematical modelling of the main metabolic pathways of the human erythrocyte (Rapoport et al., 1976; Holzhutter et al., 1985; Schuster et al., 1988). These models have been successfully employed to describe stationary and time-dependent metabolic states of the cell under normal physio- logical conditions as well as in the presence of enzyme deficiencies of pyruvate kinase (Holzhutter et al., 1985, 1990) and glucose 6-phosphate dehydrogenase (Schuster et al., 1989).

Schuster and Holzhiitter (Eul: J. Biochern. 229) 405

i;; 0 10 20 30 40 50 60 70 80 90 100

Vmax HK %normal achvity

I

C

600 I I

0.0 1 0.1 1 10

Vmax PGK %normal activity

D 600 I

2 200 8 2 4 100

s 2,3PzG

ATP 4 100 -- I s 2,3PzG

100

0 20 40 60 80 100 0.01 0. I 1 10 100 Vmax TPI %normal activity Vmax PK %normal actlvity

Fig. 2. The dependence of stationary metabolite concentrations on the activity of various enzymes. Dependences calculated using the model. Straight lines, stable steady states ; dotted lines, unstable steady states. (0) 2,3-bisphosphoglycerate concentration; (m) ATP concentration; (A) 1,3- bisphosphoglycerate. (A) Hexokinase deficiency. Experimental data from Ouwerkerk et al., 1989 (Vmax = 31.5%); Newman et al., 1980 (Vma = 27%); Rijksen and Staal, 1978 (VmaX = 26%). (B) Phosphoglycerate kinase deficiency. Experimental data from Krietsch et al., 1977 (V,,,,, = 19- 83%); Fujii et al., 1992 (V,- = 0.7%); Rosa et al., 1982 (V,,,,, = 2.7%). (C) Pymvate kinase deficiency. Experimental data from Staal et al., 1982 (Vma = 5 5 % ) ; Ouwerkerk et al., 1989 (Vnax = 22%); Zeres and Tanaka, 1987 (V,,, = 10.3%, 10.8%, 27.5%); Tani et al., 1988 (VmaX = 67%, 11.5 %); Jacobasch (private communication; V,, = lo%, 3.5 a). (D) Triosephosphate isomerase deficiency. The strong increase in the concentration of dihydroxyacetone phosphate as well as the relatively unchanged ATP and 2,3-bisphosphoglycerate (2,3P,G) levels are in accordance with experimental data (Eber et al., 1991; Zanella et al., 1985; Rosa et al., 1985; Vives-Corrons et al., 1978).

Based on a comprehensive mathematical model of the energy and redox metabolism of human erythrocytes, we present a systematic computational study on the metabolic changes caused by changing the activity of one of the participating enzymes by several orders of magnitude. The validity of our model is checked by comparing the theoretical predictions with experimental metabolic data for various enzyme deficiencies. For the human erythrocytes, deficiencies of about 20 enzymes, associated with widely different degrees of severity and complexity, have been identified so far (Valentine et al., 1983; Fujii and Miwa, 1990).

The main objective of this study is to assess the severity of cellular dysfunction associated with a given enzyme defect. Since, up to now, no reliable model exists which describes the relation between the viability and function of a cell and its metabolic state, we propose an empirical performance function. This function is used to predict the range of enzyme activities which should give rise to a chronic or non-chronic disease. The theore- tically predicted ‘ranges of disease’ are compared with data and clinical observations reported for various enzymopathies.

A second objective of the present study is to contribute to the field of ‘metabolic control’ at large-scale changes of enzyme-kinetic parameters. For small changes in an enzyme parameter, predictions of the new steady state can be made apply-

ing the well-elaborated concept of the metabolic control theory (MCT) (Kacser and Bums, 1973; Heinrich and Rapoport, 1974; Reder, 1988). In order to assess, in a quantitative manner, the metabolic consequences of large-scale changes in the activity of a particular enzyme, we introduce the ‘effective activity’ E, of an enzyme, which is defined as its relative activity (with respect to the activity in a reference state, which usually is chosen as the normal in vivo state of the system) required to produce a change in the stationary value of a metabolic variable (flux rate, metabolite concentration, etc.) by factor a. We calculate the effective activities of all enzymes for two representative cases, a = 0.5 and a = 1.5. Recently, control coefficients of the MCT have been used to predict the metabolic effect of large-scale changes in enzyme parameters by linear, logarithmic and hyperbolic extrapolations methods (Savageau, 1976; Small and Kacser; 1993). We judge the capability of these methods to predict the effective activities of red blood cell enzymes.

MATERIALS AND METHODS

Mathematical model. The mathematical model comprises the central pathways of the energy and redox metabolism of mature red blood cells, glycolysis, the pentose phosphate pathway

406 Schuster and Holzhiitter ( E m J. Biochem. 229)

and glutathione reduction (Fig. 1). ATP- and reduced glutathione (GSH)-consuming processes of the cell have been lumped to- gether into the generalized reactions ATPase and GSHox, respectively. In contrast to the 'internal' reactions constituting the metabolic pathways, the 'external' reactions of ATPase and GSHox represent a measure for the energetic and oxidative load, brought about by various processes needed to maintain cell viability (e.g. membrane pumps, reduction of lipid peroxides).

The mathematical model is based on ordinary differential equations which describe the temporal behaviour of metabolite concentrations. The total concentrations of adenine nucleotides, pyridine nucleotides and glutathione, as well as of phosphoribo- sylpyrophosphate, pyruvate and lactate are considered to be constant, implying that their concentrations are balanced by fluxes not included in the model. The equations are as follows:

[Glc6Pl = UHK - U G ~ P D - UGPI;

IFm6f'l = uGPI - 0pFK + UTA + UTIU;

Fru6f'J = UPFK - Ualdolase;

[GrW = Ualdolase UTPI U T K ~ - UTA + UTKZ - UGAPD;

All enzyme reactions are described by rate equations having the general form:

Here, c+ and c- are positive and negative elements of the stoi- chiometric matrix (for definition, see Schuster et al., 1992), 4 is the equilibrium constant, and the regulatory function r(X) contains all non-linearities due to saturation, allostery, etc. In previ- ous mathematical models, the non-equilibrium reactions cata- lyzed by HK, PFK, PK, DPGM, DPG, G6PD, 6PGD, GSSGR have been described as completely irreversible. Such a simplification is not justified when modelling enzyme defects, since high accumulation of intermediates may increase backward fluxes by several orders of magnitude. Therefore, the rate equations for these enzymes (listed in Schuster et al., 1988, 1989) were extended according to Eqn (2) ; the equilibrium constants are from Lueck and F r o m (1974), Barman (1969), McIntyre et al. (1989) and Boyer (1963). The kinetic equations for GPI, aldolase, TPI, EP, KI, TK1, TK2 and TA have been taken from McIntyre et al. (1989), the corresponding equilibrium constants are as listed in Schuster et al. (1988). For GAPD, we have taken the kinetic model of Verlick and Furfine (1963) and the kinetic constants from Jacobasch et al. (1974). We set up rate equations

i normal in vivo state i

,

I unstable states

- bifurcation point

0 0 20 40 60 80 100 120 140 160 180 200

Vmax Hexokmase I, %normal amvlty

B

0 20 40 60 80 100 120 140 160 180 200

Vmax DPGase %normal activity

Fig. 3. The dependence of the stationary flux through hexokinase on the maximum activities of hexokinase (A) and 2,3bis-phosphoglycerate phosphatase (B). (-) exact solution; (- - -) linear approxima-

+ 1 - CJ; (----I logarithmic approximation, ~ = C'- p 0 d E""" J

c' tion, ~ J = (s) ; (-) hyperbolic approximation, ~ J - -

.P-d Po- . The enzyme activities (VmaJ and the flux rates

1

are represented relative to the normal value. The control coefficients have been taken from Table 2. The bifurcation point in Fig. 3A represents the lower value of hexokinase activity where stationary states exist. At this point, the stationary states become unstable.

for AK, PGK, PGM and enolase, based on literature data. AK and PGK were assumed to have a random BiBi mechanism, with kinetic constants taken from Tsuboi and Chervenka (1975), Bar- man (1969) and Rapoport (1974). The kinetic expressions for PGM and enolase are based on an UniUni mechanism and the kinetic constants are from Barman (1969) and Rapoport (1974). All rate equations are listed in the Supplementary material.

The rate equations contain a variety of (macroscopic) parameters (V,,,, K,, Ki, etc.) which all can be changed by mutations, binding of effectors, lack of cofactors, etc. In this paper, we restrict our considerations to alterations in the maximum activity V,,, which is affected by changes in the concentration of the enzymes as well as in all structural modifications of the enzyme.

Moreover, we only consider alterations in the stationary metabolic states of red cells associated with permanent changes in the activity of a particular enzyme. In stationary states, the time- dependent variations of metabolite concentrations in Eqn (1) are zero. This results in a non-linear algebraic equation system for metabolite concentrations. The stability of the calculated steady states is checked by the eigen values of the Jacobian matrix.

Schuster and Holzhiitter (Eul: J. Biochern. 229) 407

Table 1. The effect of large-scale enzyme activity changes on stationary metabolic concentrations.

Enzyme Glc6P Fml,6P2 GraF‘ with altered activity

C E Eo 5 Ei s C;: Eo 5 El 5 C E Eo 5 El s

HK GPI PFK Aldolase TPI GAPD PGK DPGM DPG PGM Enolase PK AK G6PD 6PGP GSSGR EP KI TK1 TA TK2 ATPase GSH,,

1.403 -0.001 -0.034 -0.019

0 0 0.002

-0.239 0.001 0.01 1 0.02 0.068 0 0 0 0 0 0 0.01 1 0 0

- 1.253 -0.015

66

0.84

0

0.92 415

13.2, 760 3.75 6.7

12.3 0.01

167 5917

140 0.448 0.003 0.215

-0.077 -0.001 -0.002 -0.017

0.98 0.028 -0.014 -0.073 -0.135 -0.452

0 0 0 0 0 0 0.001 0 0

65.5 0.093 -0.002

36 210 0.73 6.8 502

0.26 0.76 5

1600 2.5, 3080

18.5 30.7

600 47.4 0.01

3 62

0.241 0.002 0.116 0.072 0.001

-0.001 -0.009

0.024 -0.034 -0.041 -0.076 -0.253

0 0 0 0 0 0 0.001 0 0

-0.039 -0.001

30 0.58 3.7

0.03

3000 3119

3100

374

326

1292

0.44 3

14 20 32.4 0.01

2,3P,Gri Enzyme Ppyruvate ATP with

activity

HK -0.233 1140 31 0.072 27 0.23 28.5 360 GPI -0.002 0.73 0 0.43 0.002 0.57 PFK -0.115 6.2 0.035 2.8 0.111 3.6 1570 Aldolase 0.101 11 -0.003 0.069 TPI 0.001 0.1 0 0.01 0.001 0.03 GAPD 0.006 0.44 0 0.002 PGK 0.022 2.5 0 0.18 -0.01 2.5 DPGM -0.018 1900 -0.037 2860 0.924 46 160 DPG 0.073 2250 0.017 0, 3119 -0.933 195 64 PGM 0.095 10 0.002 0.5 -0.039 10 Enolase 0.175 2 0.003 0.83 -0.072 19.7 PK -0.356 700 42.1 0.01 2.4 -0.242 2800 30 AK 0 0 0 0 0.01 G6PD 0 0 0 6PGD 0 0 0 GSSGR 0 0 0 EP 0 0 0 KI 0 0 0 TKI -0.002 0.001 0.001 TA 0 0 0 TK2 0 0 0 ATPase 0.265 2.4 307 -0.104 399 -0.04 381 GSH,, 0.003 -0.001 -0.001

altered c“, EO 5 Ei 5 C;: Eo s Ei s CX, EO 5 El 5

RESULTS Metabolic effects of large-scale activity changes. For each of the participating enzymes, the activity was varied between 0% and maximally 5000% of the normal value. Typical representa- tives of ‘activity characteristics’, i.e. the dependence of stationary metabolite concentrations and flux rates on enzyme activities, are depicted in Figs 2 and 3. As seen in Fig. 2, the calculated activity characteristics are in good accordance with experimental data depicted from the literature.

In most of the examples shown (Figs 2, 3A), the normal in vivo point is situated in a plateau. At a characteristic point on the activity scale, the slope of the curves increases abruptly. This phenomenon has recently been described as the ‘threshold effect’ (Letellier et al., 1994). It can be frequently observed in biological systems, e.g. when titrating the activity of an enzyme by means of a specific inhibitor (Mazat et al., 1993). A further peculiarity of the activity characteristics, shown in Figs 2 and 3A, is the occurence of a bifurcation point at low enzyme activi-

408 Schuster and Holzhiitter ( E m J. Biochern. 229)

Table 1. (Continued).

Enzyme 6PGlcA GSSG Rub5P S7P with

activity altered CE Eo5 Ei s G Eos G Eo 5 El 5 G Eo, El 5

HK GPI PFK Aldolase TPI GAPD PGK DPGM DPG PGM Enolase PK AK G6PD 6PGP GSSGR EP KI TKI TA TK2 ATPase GSH,

1.596 68 131 0.001 0.26

-0.004 1.5 0.003 0 0 0 0 0.16

-0.021 640 -0.239 7.5, 478 -0.002 0.47 -0.004 4 -0.012 9.8

0 0 0.926 50.6 162

-0.732 200 60 0 0

-0.002 0.38 -0.038 -0.001 -0.001 -1.362 159 65.5 -0.078 4100

-0.02 0 0.001 0.001 0 0 0 0.007

-0.003 0

-0.001 -0.002

-0.014 -0.001 -0.996

0

0 0 0 0 0 0.019 1.004

0.24 1540

3.6 0.8 1.6 4 0 4.9 0.14

200 66.7

50 143

0.619 0.004 0.029 0.019 0

-0.001 -0.002 -0.082 -0.012 -0.011 -0.02 -0.065

0 0.002 0 0 0

-0.005 -0.189 -0.004 -0.006 -0.492

0.131

30 180 0.51 1.9

0.02 0.28

2626 0.9, 3119

0.05 1

23.4 0.55 1

500 399 38.7

2.153 76 120 0.014 0.91

-0.099 3910 4.1

-0.001 0.03 -0.058

0 0.008 1.84

0.01 19.8, 572 0.033 7.5 0.061 14 0.203 24.6 1200 0 0.01

0 0 0 0.05 0.005 0.52 0.218 13.4

-0.39 275 17.1

-0.002

-0.016 3.3 -0.025 4.7 -2.002 136 74.4 -0.132 917

ties near the ‘threshold’ value. At the bifurcation point, the stability of the stationary state of the system changes and, beyond this point, steady states do not exist. Another type of activity characteristics is depicted in Fig. 3B. Lowering the DPG activity leads to a hyperbolic decrease in the flux through hexokinase down to a residual value of approximately 30% at complete lack of DPG. Here, a ‘threshold’ behaviour does not exist. Corre- spondingly, the flux control coefficient at the normal in vivo point is higher than in the former examples (see Fig. 3 B).

Effective activities. In order to condense the information deriv- able from such plots, we determined for each activity characteristics the effective activities E0., and E,,, defined as the maximum activities of the corresponding enzyme at which the stationary value of the considered metabolic variable attains 50% or 150% of its normal value, respectively. This definition is similar to that of the effective dose Z,., used in toxicology to indicate the dose of a chemical at which the response of the biological system attains 50% of its normal value.

Tables 1 and 2 depict the effective activities E,.,and E,., for selected metabolite concentrations and fluxe rates under steady- state conditions, as well as critical enzyme activities at which bifurcation points may occur. Blank cells indicate that the effective activities do not exist, i.e. the desired alteration cannot be obtained by any change in the enzyme activity.

From Tables 1 and 2, the following conclusions can be drawn :

(a) The effect of large-scale changes in the activity of a given enzyme on the stationary flux rates is almost exclusively restricted to the pathway to which the enzyme belongs. An excep- tion is the flux through AK which is influenced by most enzymes of the energy and redox metabolisms. With respect to metabolites, energy and redox metabolisms are strongly cou- pled; some metabolites of the redox metabolism are sensitive to activity alterations of enzymes belonging to glycolysis and the 2,3P,Gri bypass. (b) The effective activities E,, and E,, differ

by several orders of magnitude, ranging from 76% in the case of HK activity, to 0.0009% for AK. Most of effective activities are less than 10%. (c) Metabolite concentrations are generally more sensitive to activity alterations than fluxes, with the excep- tion of ATP and GSH. (d) Owing to the plateau in the activity characteristics for fluxes, the effective activity does exist only for external processes (ATPase and GSHox) and the enzymes of the 2,3P2Gri bypass.

Comparison of effective activities with approximation methods based on the metabolic control theory. Several approximation methods have appeared in the literature aimed at predicting the consequences of finite changes in enzyme activities, based on control coefficients of the MCT. In the case considered throughout this paper that only enzyme activities E (corresponding to V, in Eqn 2 ) are changed, the normalized control coefficient is defined as follows (Kacser and Burns, 1973; Heinrich and Rapoport, 1974; Reder, 1988):

ax E c;=--x--, aE X

(3)

where X denotes the stationary value of either a metabolite concentration or flux rate to be effected. E is the maximum enzyme activity. The control coefficients are calculated according to Eqn (3) for metabolite concentrations, and flux rates are depicted in Tables 1 and 2. It has to be emphasized that the control coefficient measures the metabolic effect of very small activity changes. In order to use this entity also in the case of large- scale activity changes, certain assumptions as to the shape of the activity characteristics in a sufficiently wide region around the in vivo point have to be made (Fig. 3A and B). The most simple type of approximation is the linear one, i.e. a straight line having a slope that equals the control coefficient. Integration of Eqn (3) yields a power law representation of X as a function of the enzyme activity (Fig. 3). In the finite-difference method of Small

Schuster and Holzhutter ( E m J. Biochem. 229) 409

Table 2. Steady-state fluxes for enzymes. ~

Enzyme HK PGK DPGM PK

CS Eo 5 Ei 5 G Eo 5 Ei 5 CS Eo 5 El 5 CS Eo s Ei s

DPGM -0.074 DPG 0.22 PGM 0.003 Enolase 0.005 PK 0.016 AK 0 G6PD 0 6PDG 0 HK 0.07 GPI 0 PFK 0.024 Aldolase -0.005 TPI 0 GAPD 0 PGK 0.001 GSSGR 0 EP 0 KI 0 TKI -0.007 TA 0 TK2 0 ATPase 0.763 GSH,, 0.01

3574 5.2 0.35 0.63 1.7 0

25.5 0.35 2.3

0.01

0.14

28.6

-0.102

0.004 0.007 0.024 0 0 0 0.073 0 0.028

-0.007 0 0 0.001 0 0 0

0 0

477 0.08

-0.005

176 0.951 6433 0.003

1968 3.8 0.6 1.1 3.5 0

26.4 0.43 2.7

0.02

0.16

44.6

0.058 0.941

-0.002 -0.005

0.015 0 0 0 0.015 0 0.007 0.004 0 0

0 0 0 0 0 0

0

-0.001

160 -0.003

3.9 49

0

1.5 1.4

0.01

-0.075 156 0.225

0.003 0.005 0.01 8 0 0 0 0.063 0 0.024

0 0 0.001 0 0 0

0 0 0.79 0.003

-0.005

-0.004

3495 5.6 458 0.39 0.7 1.75 0

25.5 0.36 2.3

0.01

0.15

31 171

Enzyme ATPase AK G6PD

c; E0.5 E1.5 P E Eo.5 El .5 G E0.5 e1.5

DPGM DPG PGM Enolase PK AK GGPD GPDG HK GPI PFK Aldolase TPI GAPD PGK GSSGR EP KI TK1 TA TK2 ATPase GSH,,

-0.093 0.076 0.004 0.007 0.025 0 0 0 0.051 0 0.025

0 0 0.001 0 0 0 0.001 0 0 0.907

-0.001

-0.007

1968 2.6 0.63 1.25 3.5 0

25.5 0.43 2.7

0.01

0.24

47.6 160

-0.083 -0.01 1 -0.01 -0.019 -0.064

0 0.002 0 0.615 0.004 0.029 0.018 0

-0.001 -0.002

0 0 0.006

-0.187 -0.004 -0.006 -0.49

0.127

2350 1.9 3111 0.1 1 0.2 0.55 0

29 193 0.53 1.98

0.02

0.05 0.68

23.4 0.73 2.1

399 433

-0.004 -0.002

0 0 0.001 0 0.008 0.3 0 0.02 0.011 0

-0.001 0 0 0 0 0 0 0 0 0 0

-0.01 0.495 0 200

and Kacser (1993), a hyperbolic shape of the activity characteristics is assumed. In order to check the applicability of the three approximation methods mentioned above, we applied them for

(B) Logarithmic approximation,

E;g = 100% X exp [Cc X In (OS)];

E'g = 100% X exp [Cc X In (1 .91 . estimating effective activities according to the following for- mulas.

(A) Linear approximation, 6'"- 0.5 - 100% - 50%1Cg; B'" - 1.5 - 100% + 50%1C$.

(C) Hyperbolic approximation,

w,p = 100% x CY(C2 + 1);

E;yp = 100% X CY(C$ - 113). (4)

410 Schuster and Holzhutter (Eur J. Biochern. 229)

Table 2. (Continued).

Enzyme EP GSH,, Maximum activity at birurcation point

C E EO.5 Ei 5 G Ell 5 E1.5

DPGM DPG PGM Enolase PK AK G6PD GPDG HK GPJ PFK Aldolase TPI GAPOD PGK GSSGR EP KI TK1 TA TK2 ATPase GSH,

0.024 0.006 0.004 0.007 0.023 0 0.01 0.001

-0.198 -0.001 -0.011 -0.007

0 0 0.001 0 0

-0.002 0.065 0.001 0.002 0.156 0.624

0.5 0.03

370

0.18

0.03 0.01

6.4 0.18 0.6 4.8 0 200

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

0.12 0.01

0.07

% 3119

0.18 0.293 0.55 0.001

24.48

0.82 2.3 0.0014 0.167 0.05

0.004

593 50 150

The comparison of true effective activities with their estimates (Eqn 4-Eqn 6 ) are illustrated Figs 3 and 4 and Table 3. In general, all three methods provide reasonably good results for those few enzymes possessing relatively high control coefficients (>0.1). Provided that the effective activity of a given enzyme does exist, the hyperbolic approach generally yields the most satisfactory estimates (Fig. 4, Table 3). The problem is, however, that this approach as well as the power law approach may predict effective activities although they do not exist. When not existing, the effective activities may not be predicted by the approximation methods which more often occurs accidentially rather than due to a correct description of the system behaviour.

Modelling of enzyme deficiencies. In what follows, we will predict for each enzyme a range of disease, encompassing those maximum activities leading to an observable impairment of the cell without cell death (Fig. 5).

Peflormuncefunction. Up to now, it has not been possible to describe by a mechanistic mathematical model the chain of events starting with metabolic disorder and leading to the loss of cell integrity. We, therefore, relate the metabolic states to the overall functional fitness of the cell in terms of an empirical performance function. Obviously, one objective of erythrocyte metabolism is the ensurance of red cell survival over 120 days in the blood circulation. Hence, the performance function takes into account the homeostasis of essential metabolic variables. Clinical and experimental studies have evidenced that the loss of cell integrity can be attributed to depletion of ATP or GSH, or to dramatic accumulation of metabolites proximal to the point of metabolic block (Valentine et al., 1983 ; Tanaka and Zerez, 1990). Moreover, the importance of ATP and GSH as cardinal metabolites is underlined by the above calculations, which re- vealed them to respond more inertly to activity changes than other metabolites (see Table 2) . Accumulation of intermediates occurs in almost all enzyme deficiencies. It may become delete- rious if the osmotic pressure of the cell exhausts the capacity of

Table 3. Prediction of the effective activities by various extrapolation methods using control coefficients. Effective activities were correctly predicted when the approximated values were of the same order of magnitude.

Approach Number of correctly predicted effective activities for

metabolites fluxes

%

Linear 27.3 24.3 Logarithmic 30.3 30.1 Hyperbolic 55.8 55.3

membrane pumps (see Fig. 2D). The problem, however, is to specify critical values for ATP, GSH and Q (share of intermediates in osmotic pressure of the cell), indicating the breakdown of homeostasis. For GSH, experimental findings suggest that its concentration may not fall below a critical level GSH""' = 0.9 GSHtotal in order to prevent hemolysis (Ataullakhanov et al., 1981). An upper limit for SZ can be calculated from possible cell volume changes at a constant surface. We obtained @'"' = 0.5 osmolarity of the cell (Georgiewa et al., 1989). For ATP, such a threshold value is not available in the literature. We have chosen the value ATF"' = 0.5 ATPnomal since concentrations lower than that critical limit have never been observed in enzyme deficiencies. From the activity characteristics calculated above, we can predict for each enzyme which of the three key metabolic variables is the most sensitive, i.e. which first attains its critical value when lowering the enzyme activity (Table 4). Note that, for some enzymes, none of these variables increases above its critical value during deficiency.

For defining a performance function, is has to be taken into account that red cells are exposed to varying external conditions

Schuster and Holzhutter ( E m J. Biochern. 229)

0.00000001

1E-10

41 1

0 0 0

--

-9 = -

A

1OOOOOOOO

I loowoo--

I ." X

--

8"

B I .- B G o .a

8 loo00

100 M '3 1 8

'J .e .* +

0

% 0.01 B - 2 o.oO01 2

0.000001

0.00000001

1E-10

t X

/---- a

,

D

o.Ooo1 0.001 0.01 0.1 1 10 100 lo00 loo00

Effective activities %

Fig. 4. Correlation between the effective activities E,, and E l , calculated using the model and those estimated by different approximation methods (x, linear; U, logarithmic; A, hyperbolic approximation) for metabolite concentrations (A) and flux rates (B). The points represent the effective activities listed in Table 1 or Table 2.

which, in our model, are represented by the load parameters kATPase and kGSHox. For example, passage through thin capillaries or free-radical generation by oxidative drugs will increase these parameters. Therefore, one has to demand that ATP, GSH and D do not exceed their critical values at small changes kATp,, and kGSHox. In the definition (Eqn 7) of the performance function, we have considered 10% variation in the load parameters, i.e.

We combined the values of the three key variables at different load conditions into a single performance function, such that the performance function equals unity at the normal in vivo point and becomes zero when at least one variable reaches its critical value.

P = l-I Pi, if all Pi > 0 or 0, if at least one Pi 6 0 ,

AkAm,, = 0.1 k m P a s e and AkGs,,, = 0.1 kGsHox.

x,, the stationary value of

[ATP], [GSH], 52 = intermediate concentrations all

at ~ A T P ~ C ~ A T P ~ ~

AkAwme = 0.1 kATpase and AkGsHox = 0.1 kGsHox. A k A w a S e t kGsHox, kGsHox * AkGsH,,, where

x"'. the critical value

ATp"' = 0.5 ATpnomd; GSHCr" = 0.9 GSH'o'd; 52'"' == 0.5 osmolarity of the cell;

x:oma', value at normal steady state (without enzyme defi-

In order to determine the activities of a deficient enzyme associated with metabolic disease, one has to distinguish between chronic and non-chronic diseases. Chronic diseases are characterized by an impairment of cell functions at normal values of the external load parameters. In contrast, non-chronic diseases occur only at drastically enhanced values of the load parameters, which can be due to a variety of stress factors, e.g. drugs (mostly oxidants), fava beans (Beutler, 1972), infections, fever or physi-

ciency) . (7)

412

cell death

Schuster and Holzhiitter ( E m J. Biochern. 229)

alteration without consequences RANGE OF DISEASE

0% MAXIMUMACITWIYOFAN

ENZYME

CHRONIC DISEASE NON-CHRONIC DISEASE

t I

I MINIMUM SURVIVAL ACTIVITY

Fig. 5. Definition of the 'range of disease'.

t

Table 4. Enzyme deficiencies where critical values of important metabolic variables are reached. Deficiencies were predicted by mathematical modelling.

\ MINIMUM NORMAL ACTIVITY

Most sensitive Deficiency metabolic variable

ATP GSH G6PD, GSSGR Osmotic pressure None

HK, GPI, PFK, PGK, DPG, PGM, enolase, PK, AK

aldolase, TPI, GAPD, 6PGD, KI DPGM; EP, TK1 , TK2, TA

cal exercise. Bearing this in mind, we propose the following definitions. (a) Minimum survival activity. This is the activity of a given enzyme at which the performance function becomes zero. It represents the minimum activity of an enzyme at which the metabolism of the cell is just sufficient for maintaining cell integrity. (b) Range of chronic disease. The range of enzyme activities giving rise to chronic diseases is defined by values of the performance function smaller than one but not yet zero. We fix the upper boundary of that range as the activity at which the performance function assumes a value of 0.8. (c) Range of non- chronic disease. For defining this range, we calculate particular activity Characteristics denoted as load characteristics, where the stationary state is varied with respect to the two load parameters k,,,, and kcsHox at fixed enzyme activities. Two typical examples are shown in Fig. 6. With respect to the energetic load, these curves exhibit a bifurcation point defining the upper limit kKiSe of kATPare (Fig. 6A). In the case of the oxidative load, load characteristics usually do not show bifurcations. Since, however, concentrations of GSH lower than GSWn" (90 % of the total glutathione) are not compatible with cell integrity, we may use GSH" for the definition of &$cox. As seen in Fig. 6, the upper critical boundaries for &vise and k:q& usually decrease when lowering the activity of a given enzyme. The enzyme activity at which the critical upper boundaries of kAmase or kGSHox are reduced to 50% of the value for normal cells defines the upper limit for the range of non-chronic disease. (d) Minimum normal activity. The minimum normal activity separates the range of disease (chronic or non-chronic) from the non-disease state. Ac- cording to the above definitions, the minimum normal activity is given by the upper limit of either the range of chronic or of non-chronic disease.

The calculated ranges of disease are summarized in Table 5 and illustrated in Fig. 7. Furthermore, Table 5 contains enzyme activities measured in patients with enzyme deficiency as well

A 2 ,

1.8

2

$ 1

1.4 C .s 1.2

B 0.8

5 0.6

0.4

0.2

0

0 1 2 3 4 5 6 7 8 9

kATPase '' B

normal in VIVO state

/ 100

2 50

40

- 2

-

30 ' I

0 1 2 3 4 5 6 7 8 9 10

kGSHox lh

Fig. 6. Energetic load characteristics (A) and oxidative load characteristics (B). (A) The dependence of stationary ATP concentration on the rate constant kATPase of ATPase for normal cells and those with hexokinase deficiency (55 % normal activity). The upper critical boundaries kpg&. for kAmasr correspond to the bifurcation points. For the deficient enzyme, the critical boundary for k,,,. is 4.5h which is 50% of the critical value in non-deficient cells. Thus, according to the definition of the range of non-chronic disease, below 55 % hexokinase activity, non- chronic diseases may appear. (B) The dependence of the stationary GSH concentration on the rate constant kcsHox of GSHox for normal cells and those with glucose 6-phosphate dehydrogenase deficiency (25 % normal activity). Since, for oxidative load characteristics, no bifurcation points exist, the upper critical boundaries k"G",, for kCSHai is given by the value at which [GSH] assumes the critical concentration of 0.9 GSH'"'.

as some typical characteristics of the deficient enzymes. We note that the minimum survival activities vary over several orders of magnitude. Enzymes such as HK, PK, PFK and G6PD have

Table 5. Calculated results and experimental data in red cell enzyme deficiencies. (X), rarely; X, frequently ; X X, predominantly.

Enzyme Calculated range of disease Properties of deficient enzymes Number of References diseased

minimum range of minimum enzyme unstable kinetic change in persons survival chronic normal activity variants change isoenzyme activity disease activity pattern

~-

HK

GPI

PFK

Aldolase

TPI

GAPD

PGK

DPGM

DPG

PGM

Enolase

PK

AK

G6PD

6PGD

GSSGR

EP

KI

TKI

TK2

TA

34

0.865

4.3

1.95

0.0317

0.22 0.15

0

2.8 0.533 0.82 2.4 0.0032

0.336 0.105

0.24

0

0.0067 0 0

0

34.0-52.0

0.865-1.82

4.3- 16.2

0

0.0317-0.038

0 0.15-2.5

0

2.8-43.33 0.533-8.5 0.82-14.5 2.4-22.3 0.003 -0.006

0.336-0.38 0.105 -0.825 0.24-1.27

0 0.0067-0.0096 0

0

0

%

58

1.82

16.2

3.7

0.055

0.42 2.5

0

43.33

8.5 14.5 22.3 0.025

25 1.6

15.5

0.043 0.13

5.1 6.4

6.6

15.0-50.0

5.0-40.0

40.0-60.0

4.0-6.0, 16.0

1.6-20.0

30 0.0- 10.0

0.0, 12.0-50.0

50 1

6.5, 8.5, 50.0 5 .O - 40.0 0.05. 0.5-44.0

0.0-30.0 2.4, 4.5, 30.0, 70.0 c9.0, 10.0

X

( X )

X

X

( X )

?

X

?

?

?

?

X

(X )

X

?

?

?

?

?

?

?

X

no

x x

no

no

?

no

no

no

?

no X

?

no

?

?

?

?

?

?

?

16

45

30

3

28

1 12

?

?

?

4

> 300 1

>4X10h 2

>1000*

?

?

?

?

?

- Tanaka, 1990; Valentine, 1983; Board, 1978; Rijksen, 1983; Miwa, 1985; Paglia, 1981; Necheles, 1970

Tanka, 1990; Paglia, 1974; Zanella, 1980; Neubauer, 1990; Arnold, 1973 Tanka, 1990; Vora, 1983; Valentine, 1984 ; Fogelfeld, 1990 Kishi, 1987; Miwa, 1981 ; Valentine, 1984 Eber, 1979, 1991 ; Valentine, 1984; Rosa, 1985 ; Zanella, 1985 Waller, 1974 Tanaka, 1990; Maeda, 1991 ; Fujii, 1980, 1992; Valentine, 1984 Schroter, 1965; Rosa, 1973, 1978; Buc, 1974; Rosa, 1989; Tanaka, 1990

Syllm-Rapoport, 1965

Stefanini, 1972; Tanaka, 1990 Valentine, 1983; Miwa, 1993 Beutler, 1983; Waller, 1974; Szeinberg, 1969; Matuura, 1989 Beutler, 1993; Valentine, 1984 Waller, 1974 ; Beutler, 1979

Loos 1976 ; Beutler, 1979 ; Frischer, 1987

a Mostly acquired enzyme deficiencies are due to lack of cofactor (Beutler, 1979) or drugs (Frischer, 1987; for only three patients were the deficiencies due to inherited glutathione reductase deficiency (Loos, 1976).

414 Schuster and Holzhutter ( E m J. Biochem. 229)

- 100 x

2 3 10 a

8 1

P

B

B g 0.1

g 0.01

0.001

cell death B chromc disease 0 non-chroruc Bsease u non-disease state

Fig. 7. Calculated ranges of disease. The range of disease corresponds to the two shadowed areas (chronic and non-chronic disease).

large ranges of disease (12-20%), whereas the range for other enzymes is much smaller (< 1 % for e.g. TPI, GAPD, aldolase, AK). Only for one enzyme, DPGM, does a range of disease not exist at all. For most enzymes, the range of disease practically corresponds to that of chronic disease (PK, enolase, PGM); that means that most deficiencies manifest as chronic diseases. In the case of G6PD and GSSGR, the range of disease is almost iden- tical with the range of non-chronic disease, i.e. deficiencies of these enzymes should become noticeable under stress conditions. Except DPGM, all enzymes which do not possess a minimum survival activity (EP, TK1, TK2, TA) only have a range of non-chronic disease.

Frequency of enzymopathies. As depicted in Table 5, the number of diagnosed enzyme defects varies considerably from one enzyme to the other. It is tempting to rationalize this phenomenon on the basis of the calculated results. Under the assumptions that (a) by a mutation, each activity of 0-100% of an enzyme can be generated with the same probability and (b) the mutation rate is similar for all enzymes, the frequency of enzyme defects associated with diseases should be proportional

to the ‘range of disease’ (minimum normal activity minus minimum survival activity). A check of this hypothesis is shown in Fig. 8, where the number of diseased persons is plotted versus the calculated ‘ranges of disease’ (difference between columns 4 and 2 in Table 6). The relationship can be well described by a hyperbolic curve, the parameters of which have been obtained by curve fitting. The existence of a plateau is due to the fact that the range of disease is restricted to 100%. Considerable discrepancies to the found relationship can only be seen in the case of GPI and TPI, which will be discussed below.

DISCUSSION

We used a comprehensive mathematical model to estimate the consequences of large-scale enzyme-parameter changes in erythrocyte metabolism. The mathematical model is based on detailed knowledge of the kinetic properties of red cell enzymes and has been proven to appropriately describe experiments car- ried out on erythrocyte metabolism under a variety of stationary and time-dependent conditions. Therefore, the computational results presented in this paper should provide a reliable description of metabolic alterations which are difficult to access by experimental means. The problems in obtaining experimental data for mature erythrocytes in the case of severe enzyme deficiencies is due to the fact that the population of red cells contains a very large amount of reticulocytes which generally have higher enzyme activities as well as higher intermediate concentrations than erythrocytes. Moreover, those erythrocytes, being near or beyond the critical threshold of loosing cell integrity, are almost not detectable within the population. The consequence is that the activity of the deficient enzyme determined in the red cell population overestimates the true value (Piomelli and Seaman, 1993). Comparing our theoretical results with the experimental data in Fig. 2, we carefully selected those studies where the above mentioned problems are overcomed, e.g. by reticulocyte controls.

According to the computational results, most stationary metabolite concentrations and flux rates depend on the maximum activities of ‘internal’ enzymes in a highly non-linear manner,

GPI

I

1 10 100 looa loo00 1 m loaM00 IWOOOOO

Number of diseased persons

Fig. 8. Correlation between the ‘range of disease’ (minimum normal activity minus minimum survival activity relative to normal activity) and the number of diseased persons due to enzyme deficiency (inborn or aquired e.g. by drugs or cofactor lack) (see Table 5). The straight line has been obtained by fitting the parameters of a saturation function to the data.

Schuster and Holzhiitter (Eur J. Biochem. 229) 415

characterized by sharp thresholds at low enzyme activities and plateaus at normal activities (see also Mazat et al., 1993). The normal in vivo point of the activity characteristics with respect to these enzymes mostly lies in the plateau, so that stationary fluxes cannot be significantly enlarged by increasing the activity of only a single enzyme. This finding supports the view of Small and Kacser (1993), that flux stimulation in a metabolic network necessarily involves the concerted activation of several enzymes. The activity characteristics reveal a marked homeostasis of fluxes against variations in enzyme activities. Apparently, this stability of metabolism is not simply due to the stoichiometry, but has been achieved by regulatory mechanisms which evolved during evolution. Omitting, for example, the allosteric AMP activation of PFK, the width of the plateau in the activity characteristics is significantly reduced (not shown). Unlike internal reactions, all metabolites and fluxes are more sensitive to finite changes in the activity of external load processes, even in the vicinity of the normal in vivo point. These theoretical findings indicate that evolutionary optimization of erythrocyte metabolism has been governed by two basic requirements (a) metabolic stability, i.e. to maintain substantial metabolic functions even at larger alterations in enzyme activities, which may occur during cell aging or as result of mutations, and (b) to respond sensi- tively to changes in the environemental conditions, such as energetic demand or oxidative load, sensed by these cells during blood circulation.

An important aspect of our theoretical analysis was to introduce the effective activity of an enzyme as a reasonable measure for the control of metabolism excerted by finite changes of enzyme kinetic properties. It represents the activity alteration required in order to produce a certain metabolic change. There are two problems associated with this entity. First, the concrete value of the effective activity depends on the extent of metabolic change under consideration. In analogy to toxicology, in this paper we have calculated the effective activities Eo.5 and El 5 . It should be noted that these effective activities cannot be determined in all cases, indicating that the metabolic change cannot be achieved at all. A second problem is that the quantification of the effective activities requires knowledge of all of the activity characteristics which can be either calculated by a mathematical model as presented in this paper or have to be measured. In any case, the computational or experimental effort is considerably higher than that for determining control coefficients of MCT which only require information on the metabolic behaviour in very close vicinity around the in vivo point. Unfortunately, approximation methods employing control coefficients of MCT do not yield reliable predictions for enzymes possessing small control coefficients.

In order to describe metabolic diseases in mathematical terms, we introduced the performance function (Eqn 7) relating the value of key metabolic variables to cell integrity. The selection of metabolic variables as well as their critical values enter- ing the performance function are based on experience of red cell metabolism and the physiology of cell hemolysis. It cannot be excluded that further metabolites may play a dominant role as well, so that in the progress of experimental research, modifications of the proposed performance function may become neces- sary. There is a certain arbitrariness in the choice of the characteristic values of the performance function and of the load characteristics, on which the estimation of the range of disease is based. These two values [ P = 0.8 as the upper boundary of the range of chronic disease; P P p ‘ = 0.5 k”Pp‘(norma1) for the range of non-chronic disease] have been optimized so that a maximum accordance between computational results and experimental data is achieved.

The range of disease for an individual enzyme was compared with residual activities observed in vivo (Table 5). For a group of enzymes (HK, PGK, PK, G6PD, GSSGR), most of the experimental and clinical observations can be satisfactorily accounted for by the model. There is a second group of enzymes (DPGM, aldolase, enolase, AK, 6PGD) for which contradictory data have been reported in the literature, i.e. where the correlation between the severity of the disease and the residual activity of the affected enzyme is missing. Generally, the results of the models for these enzymes fit better with observations found in those probationers equipped with the lowest activities reported. For example, Beutler et al. (1983) describes one case of AK deficiency, where two siblings had activities less than 0.05%. The girl had periodical hemolytic episodes whereas her brother was free of symptomes. The minimum normal activity of AK predicted by the model is approximately 0.025% and the range of chronic disease is 0.0032-0.006%. Thus, only small differences in AK activity might explain the differences between the siblings. For DPGM deficiency, no range of disease is predicted by the model. This supports the clinical finding in a family with DPGM activity of approximately 0.28 % and complete absence of nonspherocytic hemolytic anemia (Rosa et al., 1989; Tanaka and Zerez, 1990). In 6PGD deficiency, two patients are known with 30% and 70% activity and nonspherocytic hemolytic anemia (Waller and Benohr, 1974), while two others with 2.4% and 4.5% activity had no symptoms at all (Beutler, 1979). The corresponding minimum survival activity has been predicted to be 1.6% of the normal enzyme activity. In this light, it has to be questioned whether in deficiencies of enzymes belonging to this group, where residual activities higher than lo%, have been determined, the observed nonspherocytic hemolytic anemia was solely due to a defect of that enzyme.

For a third group of enzymes (PFK, GPI; TPI), theoretical and experimental data are conflicting. Diseases caused by phosphofructokinase defects are predominantly related to the lack or instability of one of the subunit types (M, L) forming the enzyme tetramer (Valentine and Paglia, 1984). Since the five iso- enzymes normally found in red blood cells substantially differ in their kinetic constants for ATP and DPG inhibition, lack of one subunit leads to a drastic change in PFK kinetics but a moderate activity decrease to 40-60% (Vora et al., 1983), which cannot be simulated by varying V,,, alone. The discrepancies for GPI and TPI deficiencies are probably due to the remarkable instability of almost all abnormal enzyme variants (Tanaka and Zerez, 1990). Thus, it is particularly doubtful whether the activities measured in the surviving cell fraction are representative for those critical activities causing cell damage. Bearing the instability in mind, the low values of maximum activities, constituting the ranges of disease in GPI and TPI deficiencies, are probably attained during normal cell life. Unfortunately, only experimental data from the thermostability test are available, which does not provide information on the dynamics of enzyme decay in vivo.

For some enzymes, the experimental data are not substanti- ated sufficiently (GAPD, only one case ; 2,3-bisphosphate phosphatase, not directly measured; EP, ET, TK, TA). For the enzymes of the non-oxidative pentose phosphate pathway, the model predicts a minimum survival activity only for the isomerase (KI). Other enzyme defects of this pathway should only cause non-chronic hemolytic crises under oxidative load conditions. Possibly, their importance is underestimated, since lowering the activities of these enzymes could limit the supply of ribose phosphates for the salvage pathway of adenine nucleotides, which was neglected in the model.

Besides the range of disease, we calculated activity ranges for chronic and non-chronic diseases. In good agreement with

,

416 Schuster and Holzhutter (Eul: J. Biochem. 229)

experimental data (Valentine et al., 1983), these results show that most enzymopathies of glycolysis lead to chronic hemolytic anemia, whereas enzymes responsible for maintenance of glutathione in its reduced state (G6PD, GSSGR etc.) have a wide range of activity where hemolytic crises are expected only under stress conditions. From the good correlation shown in Fig. 8, it can be concluded that the ‘range of disease’ is an important factor determining the frequency of the corresponding enzymo- pathy. Of course, other factors can be important as well; such as the frequency of mutations, which obviously is not equal for all the enzymes (Satoh et al., 1983), possible instabilities of enzyme variants, or the activity pattern generated by random mutations (Loeb et al., 1989).

The methods developed in this paper for the prediction of consequences of enzyme deficiencies in erythrocyte metabolism can also be applied to other cellular systems. A condition for this is, however, the establishment of a performance function, adapted to the particular function of the tissue under investiga- tion, on the basis of which the severity and frequencies of enzyme defects can be assessed.

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Supplementary material. Use of mathematical models for predicting the metabolic effect of large-scale enzyme activity alterations. Application to enzyme deficiencies of red blood cells. Enzymatic rate equations used in the mathematical model. This information is available, on request, from the Editorial Office. Nine pages are available.

use of mathematical models for predicting the metabolic effect of large-scale enzyme activity...

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