Eur. J. Biochem. 245, 71-83 (1997) 0 FEBS 1997
Model of the pH-dependence of the concentrations of complexes involving metabolites, haemoglobin and magnesium ions in the human erythrocyte Peter J. MULQUINEY and Philip W. KUCHEL Department of Biochemistry, University of Sydney, Australia (Received 6 Septembed23 December 1996) - EJB 96 1327/1
The rate of glucose consumption and the concentrations of glycolytic intermediates in human erythrocytes have long been known to be pH sensitive. Despite the extensive literature on modelling erythrocyte metabolism, no model developed so far can adequately describe all of these pH-dependent changes. None of these models have included all the significant association reactions between metabolites, Hb and Mg2+ that will influence metabolism. As part of a larger enterprise to develop a detailed model of erythrocyte glycolysis, we present a sub-model which predicts, as a function of pH and oxygenation state, the concentrations of free and Mg2+-bound metabolites that are substrates, co-factors and effectors of glycolysis. This model shows that pH changes around physiological values can cause large changes in the distribution of metabolites between free, bound and Mg2+-complexedforms, based on binding interactions alone ; in oxygenated cells, at pH 7.2 -7.6, many glycolytic intermediates undergo changes in concentration of 50100 %. The model also predicts intracellular concentrations of free Mg2+ in erythrocytes to be 0.4 mM and 0.64 mM in oxygenated and deoxygenated cells, respectively, assuming a total magnesium concentration of 3 mM ( ~ 8 % 8 of the total magnesium usually found in erythrocytes). This is in close agreement with the values found by Flatman [Flatman, P. W. (1980) J. Physiol. 300, 19-30] and the finding by Flatman and Lew [Flatman, P. & Lew, V. L. (1977) Nature 267, 360-3621 that the main Mg2+ buffer systems bind =90 % of Mg2+in the cell. Hexokinase has a high ‘flux control coefficient’ in human erythrocyte glycolysis, so the dependence of its rate on the pH and oxygenation state of haemoglobin is important. With a low oxygen tension and an intracellular pH of 7.34, the major inhibitor of its activity (2,3-bisphosphoglycerate) is 85 % bound to either haemoglobin or MgZf,and the maximum possible flux of substrate via it would be 2.05 mmol L erythrocytes-’ h-’. However, if the haemoglobin were saturated with oxygen, and the pH were 7.2, it was calculated that the maximum rate would be 1.48 mmol L erythrocytes-’ h-’; this is primarily due to a doubling of the free 2,3-bisphosphoglycerate concentration. However, the full extent of the inhibition is counteracted because the concentration of the Mg2+-2,3-bisphosphoglyceratewould be approximately doubled. Many other similar comparisons are possible with this new model, which highlights the complex network of interactions between haemoglobin, Mg”, H’ and the metabolites as substrates and effectors of the glycolytic reactions. Keywords: erythrocyte glycolysis ; regulation of metabolism ; pH effect; magnesium ions ; computer simulation.
Many metabolically significant ions, such as ATP, 2,3-bisphosphoglycerate [Gra(2,3)P2] and other phosphorylated metabolites in human erythrocytes, bind reversibly with haemoglobin (Hb) and Mg2+.These interactions significantly affect the free and bound concentrations of metabolites within the cells and thus significantly affect the activity of many enzymes (Gupta et al., 1978b; Hamasaki and Rose, 1974; Gerber et al., 1973; Bunn et al., 1971). For example, the erythrocyte glycolytic enzymes with the largest ‘flux control coefficient’, hexokinase and phosphofructokinase, require the Mg” complex of ATP as a subCorrespondence to P. W. Kuchel, Department of Biochemistry, NSW 2006, Australia Fux: +61 2 9351 4126. E-mail:
[email protected] Abbreviutions. Cra( 1,3)P2, 1,3-bisphosphoglycerate; Gra(2,3)P2, 2,3-bisphosphoglycerate;deoxy, deoxygenated; Fru(l,6)P2,fructose 1,6bisphosphate; Fru(2,6)P2, fructose 2,6-bisphosphate; Glc( 1,6)P,, glucose 1,6-bisphosphate; Glc6P, glucose 6-phosphate; oxy, oxygenated.
strate, while free ATP and Gra(2,3)P2 are negative effectors (Gerber et al., 1974). The interactions of many anions with Mg2+ and Hb are dependent on pH (O’Sullivan and Smithers, 1979; Garby et al., 1969; Janig et al., 1970; Rapoport et al., 1972). Changes in glycolytic rates and the concentration of glycolytic intermediates at various pH values have, in part, been explained by this fact (e.g. Gerber et al., 1974). In addition, the oxygenation state of Hb affects its affinity for different metabolites (Gupta et al., 1978a; Hamasaki and Rose, 1974; Gerber et al., 1973; Bunn et al., 1971). This fact has been used to explain the different rates of glycolysis in erythrocytes with different extents of oxygen staturation (Asakura et al., 1966), as well as to explain the increased concentration of Gra(2,3)P2 in cases of hypoxia (Duhm and Gerlach, 1971). These examples indicate that for a model of glycolysis to explain the many effects associated with pH or changes in oxygenation state, it must account for the binding of metabolites to
72
Mulquiney and Kuchel ( E m J. Biochem. 245)
Hb and Mgz+ as a function of pH and oxygenation state. As yet, to our knowledge, no quantitative model of erythrocyte metabolism has included all the significant association reactions between metabolites, Hb, and Mg’+, that will influence metabolism. This work is an essential part of a larger enterprise which is a detailed model of erythrocyte glycolysis. In particular, the aim has been to develop a model which can explain many aspects of Gra(2,3)P, metabolism. We are mindful that the former task has been addressed in a comprehensive way in previous publications (Rapoport et al., 1974, 1975, 1976; Heinrich et al., 1977; Brumen and Heinrich, 1984; Joshi and Palsson, 1989a, b, c, d ; Lee and Palsson, 1990; Lee et al., 1991), but all of the models have fallen short of providing a good comparison between their predictions and experimental data on carbon and phosphate fluxes into and out of the Rapoport-Luebering shunt, that is readily obtained by l3C-NMR and ?‘P-NMR spectroscopy (Oxley et al., 1984). However, the previous models did identify the key features of the regulation of erythrocyte metabolism and were instrumental in the development of the conceptually rich area of metabolic control theory (Cornish-Bowden, 1989; Burns et al., 1985). It is salutory to note, however, that a quantitative explanation of the control of Gra(2,3)P2 concentration as a function of oxygen partial pressure in erythrocytes does not yet exist. This control may be via an unidentified signalling compound which could act via enzyme phosphorylation or other forms of ligand association, but the answer could equally lie in piecing together the kinetic and binding data that currently exist in the literature. This approach of including the fine details of the kinetics of enzymes in computer models of metabolism, is very much that pioneered by Garfinkel (1966) and which we have employed to date in models of erythrocyte metabolism that have been used to interpret our NMR-derived data (Thorburn and Kuchel, 1985; McIntyre et al., 1989; Kuchel et al., 1990a, b; Berthon et al., 1993). Here we present a model which predicts, as a function of pH and oxygenation state, the concentrations of free and Mgz+-bound metabolites that are substrates, co-factors and effectors of glycolysis. The effects that pH and the haemoglobin-oxygenation state have on the intracellular distribution of metabolites are examined and the validity of the model is assessed by its ability to predict experimentally determinable concentrations of free Mg” in erythrocytes incubated under various conditions. It is important to realise that the simulated intracellular conditions presented pertain to a limited set of steady-states of the metabolic system. When the present sub-model is incorporated into a more complete model of glycolysis, the total concentrations of metabolites and selected ions, including protons, will be time-dependent. Specifically, they will be dependent variables in an array of differential flux equations that make up the model. Notwithstanding the ‘local’ nature of the present model, useful predictions relating to the effects on the rate of glycolysis of variations of total metabolite, Mg2+ and H’ concentrations, are possible. The profiles of free and bound, metabolite and ion concentrations calculated with the model are useful for predicting the rate of glycolysis under conditions of different pH and extent of oxygen saturation of haemoglobin, especially when viewed in conjunction with models of glycolysis that have identified those enzymes, such as hexokinase, which have the highest flux control coefficients (Rapoport et al., 1974, 1975, 1976; Heinrich et al., 1977; Brumen and Heinrich, 1984; Thorburn and Kuchel, 1985; McIntyre et al., 1989; Kuchel et al., 1990a, b; Berthon et al., 1993).
Table 1. Apparent associationconstants for haemoglobin complexes. Association constant
Association reaction
K, of Oxy-Hb
K, of deoxy-Hb
M-’ KmMFmP
Hb +MgATP
K,w.n
Hb ATP
KHhaoP KmCra(Z,3)PZ
Hb+ADP Hb+Gra(2,3)P2
+
K,,hPru(,.h)P2 Hb+Fru(l,6)P2 KHhC,c(,,h,PZ Hb+Glc(l,6)P2 KHhCrr(,,3)P2 Hb+Gra(l,3)P2
3.9 X 1 0 ” 1.9 X102‘ 3.6 X10’” 2.92 X 10” 2.5 X 10’’ 2.5 X 10’” 2.11 x lo2’ negligible binding‘ negligible bindingd 3.17 X loZd
1.4 X10’” 8.7 XIOZh 2.6 X lo3“ 7.87 X 1.20~103b 5.0 X10’” 9.71 X 1.01 X loJ‘ 1.01 X 10’” 1.55 X
3 7 T , pH 7.2, ,u = 0.15, [Hb] > 1.5 mM (Berger et al., 1973). 37”C, pH7.2, p = 0.14+0.01, [Hb] > 0.06 mM (Gupta et al., 1978b; Hamasaki and Rose, 1974). ‘ assumed to be as for Glc(l,6)P2 “ 25”C, pH 7.2, p = 0.12, [Hb] > 0.06 mM (Hamasaki and Rose, 1974). *
THEORY The following model considers interactions between Hb, Mg2+and the glycolytic metabolites. Interactions between Mg” and metabolites with an apparent association constant at pH 7.2 [K“pp(pH 7.2)] below 100 M-’ were considered insignificant (except for P,). Interactions between Hb and metabolites which have association constants lower that 10 M-’ were also assumed to be insignificant. Clearly, interactions which have been too small to be determined experimentally were assumed not to occur, and no speculations regarding likely interactions and their thermodynamic parameter values were made. The interactions that were considered are shown in Tables 1 and 2. The interaction of Mg2+with ions. MgATl? The interaction of Mg” with ATP involves many different association reactions. In solution, various protonated forms of ATP constitute a complex multiple-equilibrium mixture, with each species having different affinities for Mg2+. Mg2+ usually forms a 1 : l complex with ATP4-, however there is evidence that MgzATP and possibly MgATP, may also be formed (Noat et al., 1970; Bock et al., 1991). In addition, Mg2+ and ATP4- undergo association reactions with interfering ions not directly involved in metabolic processes such as Na+, K+ and CI- (Table 3). Calculation of the concentrations of all ionic species involved in the Mg:ATP association can be achieved in many ways. One method involves solving simultaneously the equations for the dissociation constants and those of the conservation of mass. Another method is to find iteratively the concentrations of the metabolites which satisfy these equilibrium equations (Perrin and Sayce, 1967; Storer and Cornish-Bowden, 1976). In light of the time consuming nature of such procedures and because knowledge of the concentration of all ATP species is often not needed, many studies on the interaction of Mg2+ with ATP have used ‘apparent constants’. Apparent constants are usually expressed in the form
where [MgATP], refers to the sum of all concentrations of the complexes containing ATP4- and Mg2+; [ATP], refers to complexes containing ATP4- but not Mg2+ and [Mg], refers to complexes containing Mg2+ but not ATP4-. Each apparent constant
Mulquiney and Kuchel ( E m J. Biochern. 245)
73
Table 2. Apparent association constants for magnesium complexes at 3 7 T , pH 7.2, ,u = 0.15 M, [K'] = 0.15 M, [CI-] = 0.15 M. Complex
fpPP
K"p" as a function of pH
M-I
MgATP
2.6 X lo4 where
MgADP
2.3 X 10'
MgGra(2,3)P2
6.7 X 10'
MgGra(l,3)P2
1.9X lo2
MgFru(l,6)P2
4.0X 10'
MgC1(1,6)P2
4.0X 10'
MgP,
3.4x 10
pH dependence assumed to be as for MgFru( 1,6)P2
will be a function of temperature, ionic strength and the concentrations of interacting ions such as Na', K', C1- and H' (Garfinkel and Garfinkel, 1984). Thus, each apparent constant is valid only for a specific set of experimental conditions. However, by specifically defining the apparent constant and by including all significant interactions, it is possible to evaluate a new apparent constant for different concentrations of interacting ions (Garfinkel and Garfinkel, 1984). By solving the algebraic equations which describe the equilibrium reactions shown (Table 3), under conditions close to physiological (37"C, ,u = 0.15 M, [K'] = 0.15 M and [CI-] = 0.15 M, [Mg''] = 1 mM, [AT€-'] = 1 mM, pH 7.2), a value of 8.1 X lo4M-' obtained for F$gATp where
m;,,
=
ations in the reported values for some dissociation constants; values reported for K, have varied over ==20000100000 M-' (O'Sullivan and Smithers, 1979; Garfinkel and Garfinkel, 1984). Second, other significant equilibria may play a part e.g., Gupta et al. (1983) have reported that MgATP2binds K' and Na' equally as well as does ATP4-. If this is the case, a value of 2.4X103 M-' would be obtained for F$iATp, which is in agreement with the value obtained by Gupta and coworkers (Gupta et al., 1978b). In light of these uncertainties, we have chosen the experimentally determined value for F$;ATp for our overall model. K&gATp varies with pH for four different models of Mg:ATP interaction (Fig. 1); the models are defined in the figure caption.
[MgATP*-] + [MgHATP-I ([Mg2*l + [MgCl']) ([ATP"]
This is =70 % lower than the experimentally obtained value for Ka&EATP determined under physiological conditions (2.7 2 0.3X lo3M-'; Gupta et al., 1978b). There are two major possibilities for the discrepancy. First, there have been wide vari-
+ [Mg2ATP] + 2 [MgATP,] + [HATP'-1 + [KATP"] + [H,ATP'-I)
(2)
The successive models exclude the weaker interactions of Table 3. Model C (Fig. 1) is in good agreement with model A (the model which includes all equilibrium reactions given in Table 3). It can also be seen that even when MgATP2- is assumed to
74
Mulquiuey and Kuchel (Eur J. Biochem. 245)
bind K' (as proposed by Gupta et al., 1983), there is little change in the pH dependence of P$EATp. It is possible to derive an algebraic equation which describes K:gATp as a function of pH for model C. Thus, the apparent association constant for model C is
From Eqn ( 3 ) ,
Eqn (4) was chosen to model P&gATP(pH) in all subsequent simulations shown here. MgADP. The reactions involved in the complexation of Mg" and ADP are shown in Table 4. By solving the algebraic equations describing the equilibria chown (Table 4) under conditions close to physiological ones, a value of 1.1X 10' M-' is obtained for P$;,,,, where =
+
([Mg"]
[MgHADP] [MgADP-] ([ADP'-] + [HADP*-]
+ [MgCI'])
+ [KADP'-])
The experimentally determined value is 2.34 2 0.17X lo3M-' (Gupta et al., 1978b). The algebraic equation which describes K g A u pas a function of pH can be derived from Eqn ( 5 ) and is shown in Table 2. MgGra(2,3)P2.Table 5 shows the equilibrium reactions that are important in the association of Gra(2,3)P2 with Mg?'. By solving the algebraic equations relevant to these equilibria for physiological conditions, a value of 2.10X102M-' is obtained for ~d~G,,~,2,3j,,,, where F P P
[MgGra(2,3)P:-]
-
MgGru(2,3)P> -
([Mg"]
+ [MgCl'])
([Gra(2,3)P:-]
+ [MgHGra(2,3)P:-l + [Mg2Gra(2,3)P;1
+ [HGra(2,3)e-] + [H2Gra(2,3)P:-] + [KGra(2,3)P-] + [KHGra(2,3)P:-]) (6)
The experimentally determined value for physiological conditions is 6.94-tO.14X1O2 M-' (Gupta et al., 1978b). If the formation of the MgZGra(2,3)P, complex is neglected, the pH dependence of P$&ru(Z,i)P2 is unchanged (Fig. 2). The alas a function of pH gebraic equation which describes P$gGru(Z,3)E2 when KMgZGra,2,3)Ez = 0 (Table 2) can be derived in the same manner as that for MgGra(I,3)P2,MgClc( 1,6)P2 and Fru(l,6)P2. The association constants for the MgGra( 1,3)P2and MgGlc( 1,6)P2com-
plexes have been reported as 1.9X102M-' (pH7.5, p = 0.1, 25°C; Hamasaki and Rose, 1974) and 4.0X102M-' (pH 8.0, 100 mM ethanolamine-HCl, temperature not stated; Rose et al., 1974), respectively. The association constant for MgFru( 1,6)P, under physiological conditions calculated using the above meth-
Table 3. Equilibrium reactions important in the association of Mg" with ATP4- at 37"C, p = 0.15 M. Association constant
Association reaction
K
Reference
M-' Hi + ATP4 H' +HATP'
H ' +MgATPLK++ATPNa'+ATP Mg*'+ATP MgZ'+HATP3Mg2' +MgATP2Mg2'+C1MgATP' +ATP4
PH Fig.1. The pH dependence of Kg& for five different models of Mg:ATP association (37"C, p = 0.15 M, [K'] = [Cl-] = 0.15 M, [Mg''] = [AT€'-] = 1 mM). Model A (-): all equilibrium reactions shown in Table 3. Model B (-'-.-): as for model A but assumes MgATPZ- binds K' with the same affinity as ATP-. Model C ( - - - - - ) : only significant asociation reactions are H' + ATP4 -, H' + MgATP2-, K' + ATP-, Mg" + A T P ~and , Mg" + HATP'-. Model D (---): as for model D but assumes that the H ' + MgATP2- and Mg" + HATP'- reactions are insignificant, i.e. KHMrATr and KMgHATP are both equal to 0. Model E (- - .): as for model D but assumes that the K' + ATP- reaction is insignificant, i.e. KKATP = 0.
9.07 X 10' Phillips et al. (1966) 1 X lo4" O'Sullivan and Smithers (1979) 1.57 X loSh Phillips et al. (1966) 1.4 X 10" O'Sullivan and Perrin (1964) 1.5 X 10" O'Sullivan and Perrin . (1964) 4.32 X lo4' Phillips et al. (1966) 7.48 X 10'' 4.0 X 10" Noat et al. (1970) 3.4' Grzybowski et al. ( 1970) 1.6 X l o r Bock et al. (1993)
Values measured at 3 0 T , p = 0.05 M. Value was corrected to 37°C using the van't Hoff equation with AH"= -5.10 kJ/mol since AH is independant of p for K,,,,, (Phillips et al., 1966). ' Value was corrected to 37°C using the van't Hoff equation with AH,,, 1) , = 13.81 kJ/mol (Phillips et al., 1966). ' Value calculated from K,,,, KHMgATP, K,,,,, according to the reaction cycle set out in Phillips et al. (1966). Value measured at 30"C, p = 0.3 M, pH 8.0. I Conditions at which this constant was measured were not indicated. Value measured at 24OC, p = 0.1 M, pH 7.2. "
=
75
Mulquiney and Kuchel ( E m J. Biachem. 245)
Table 4. Equilibrium reactions important in the association of Mg" with ADP'- at 37"C, p = 0.15 M at pH 5.0-9.0. [Note that since the pK,, of H,ADP is 3.9 at 3 0 T , ,u = 0.05 M (O'Sullivan and Smithers, 1979), it will not be present at a significant concentration at pH 5.09.0.1 Association constant
Association reaction
K
Reference
Ki,,rw KHMsanP
H++ADP' H++MgADPK++ADP'Na++ADP3Mg2++ADP3Mg2++HADPZ Mg" +Cl-
5.42 X 10' Phillips et al. (1966) 1.77 X lo5" Phillips et al. (1966) 4.8 Melchior (1954) 4.5 Melchior (1954) 3.29 X 10" Phillips et al. (1966) 1.07 X loLd 3.4' Grzybowski et al. (1970)
M-'
KK,
KNaADP K M P A DI,
KMgHAD, KMgci
~
~
5
=
Fig. 2. The pH dependence of K~&Gra(Zj)P2 for three different models of Mg:Gra(2,3)P2 association (37"C, p = 0.15 M, [K'] = [CI-] = 0.15 M, [Mg"] = [ATP4-] = 1 mM). Model A (-), all equilibrium reactions shown in Table 5. Model B (---), as for model A but assumes the Mg2+ + MgGra(2,3)P;- association reaction is insignificant, i.e. KMg2tir.,12,7)P2 = 0. Model C (-'-.-), as for model B but asumes that K~~tirai2.31~2 = 0.
Table 6. Equilibrium reactions important in the association of Mg" with Fru(l,6)P:- at 37"C, p = 0.15 M at pH 5.0-9.0. All values for K are from Achilles et al. (1975) except for K,,, which is from Grzybowski et al. (1970).
Table 5. Equilibrium reactions important in the association of Mg2+ Association constant with Gra(2,3)P:- at 3 7 T , p = 0.15 M at pH 5.0-9.0. [Note that the pK, of H,Gra(2,3)Pz (3.5; Achilles et al., 1972) indicates that this species will not be present at significant concentrations when the pH is at 5.0 9.0.1 All values for K are from Achilles et al. (1972) except KMgc,which is from Grzybowski et al. (1970). Association reaction
K M-'
H++Gra(2,3)E'H++HGra(2,3)e K++Gra(2,3)P:Na' +Gra(2,3)P:K++HGra(2,3)eNa'+HGra(2,3)eMg2++Gra(2,3)P:Mg" +HGra(2,3)eMg" +MgGra(2,3)P: Mg2++C1-
1.62X lo8 4.27 X 10' 8.51 X 10 8.51 X 10 8.9 9.5 7.41 x 103 5.13 X lo2 1.78 X 10' 3.4
8
PH
~~~
Value was corrected to 37°C using the van't Hoff equation with (Phillips AH"= -8.45 kJ/mol since AH is independant of ,u for KHMgADP et al., 1966). Values measured at 25 "C, ,u = 0.22 M. No adjustment was made for variation in K due to changes in temperature or ionic strength. Value was corrected to 37°C using the van't Hoff equation with = 15.06 kJ/mol (Phillips et al., 1966). AH,, Value calculated from the values for K from the reactions KHADP, K,,MoADl>r and KMsADP according to the reaction cycle set out in Phillips et al. (1966). ' Conditions at which this constant was measured were not indicated. *
Association constant
i
6
Association reaction
K M-'
H++Fru( 1,6)P:H++HFm(l,6)f'K++ FIX(1,6)P:Na++Fru(l,6)P:K++HFru( 1,6)l'Na++HFru(l,6)eMg'' +Fru( 1,6)P:Mg2++HFru(l,6)eMg" +MgFru(l,6)P:Mg2++C1-
7.56 X 10" 1.12x loh 1.07 X 10 1.23 X 10 3.3 3.7 3.63 X 10' 8.90 X 10 3.16X 10 3.4
algebraic equations describing the equilibria (Table 7), under physiological conditions, a value of 3.4X 10 M-' is obtained for K"JEP,, where
PdgP,= [MgHPO'lI ([Mg2+] + [MgCI']) ([HPO:-] [H2P0,]
+ [KHPO;]) ' (7) ods, and the association constants of Table 6 was 82 M-I. Given the structural similarity between Glc( l , 6 ) P 2 and Fru(l,6)P2 and No experimentally determined value was found for the same since calculations of the apparent constant have consistently un- conditions. Although this P p p is relatively low, it is included because P, can reach in vivo concentrations greater than 1 mM. derestimated the experimental value, the value of 4.0X lo2M - I was chosen for both Pd&u(,,6JP2 and KG5:(1,6JP2.If it is assumed At these concentrations, Mg2+:P,complexes can account for a significant proportion of bound MgZ+.The pH dependence of that KMg2Fru(1,6)Pz = 0, an algebraic equation which describes K"PP MgFm(,,6JP2 as a function of pH can be derived (Table 2 ) . Due to P$Ep is, readily derived from Eqn (7) and is given in Table 2. Other glycolytic intermediates. All other glycolytic intera lack of information on the other association reactions involved values of less than 100 M-' [calculated from and K&,,6)Pz were assumed to mediates have Ppp in these equilibria, K"d;Gra(,,3)f,2 vary with pH in an equivalent manner to P$;Gra(Z,3)Pz and the data of O'Sullivan and Smithers (1979) and Achilles et al. K""P MgFru(l,6)P2r respectively. (1973) using the methods described above] and concentrations MgP,. The association reactions involved in the complex- less than 0.2 mM under normal in vivo conditions (Beutler, ation of Mgz+ with P, are shown in Table 7. By solving the 1984), therefore, they were ignored for this model. Achilles et al.
+
Mulquiney and Kuchel (Eul: J. Biochern. 2451
76
Table 7. Equilibrium reactions important in the association of Mgz+ with Pi at 37"C, p = 0.15 M at pH 5.0-9.0. [Note that the pKa values of H,PO, and HP0:- (2.14 and 12.4, respectively; Mathews and van Holde, 1990) indicate that these species will not be present at significant concentrations when the pH is at 5.0-9.0.1
Association constant
Association reaction
KHZpo4
H++HPO:-
5.68X 10'"
KKHPOq
K++HPO:-
3.0h
KNsHPoJ
Na++HPO:-
3.9
KM8HP04
MgZ++HPO:-
9.31 X 10"
KM,,.,
Mg2++C1-
3.4'
K
Table 8. A model of Hb binding.
Association reaction
Association constant
Reference
M-'
Lawson and Veech (1979) Smith and Alberty (1956) Smith and Alberty (1956) Lawson and Veech (1979) Grzybowski et al. (1970)
Values measured at 38"C, p = 0.2 M. No adjustment was made for variation in K due to changes in temperature or ionic strength. Values measured at 25"C, p = 0.2 M. No adjustment was made for variation in K due to changes in temperature or ionic strength. ' Conditions at which this constant was measured were not indicated.
(1973) calculated that at the Mg" and metabolite concentrations normally found in erythrocytes in vivo, these other Mg2+-metabolite interactions accounted for only 4 % of the total Mg :metabolite complexes and that, at most, only 6 % of these other metabolites were complexed to Mg". The interaction of human haemoglobin A with ions. Gra(2,3)P2 has been shown to hind to deoxy-Hb A in a central cavity between the two p chains on the twofold symmetry axis of the Hb tetramer (Arnone, 1972). Four positively charged groups on each of the two p chains (the adjacent a-amino group of His2 and its imidazole side group and the side groups of Lys82 and His143) create a pocket of eight positive charges of which the negatively charged Gra(2,3)P2 is stereochemically complementary. On oxygenation, the central cavity narrows, becoming too small to accommodate Gra(2,3)P2. However, the site of Gra(2,3)P2binding to oxy-Hb is still uncertain, but two possibilities appear to exist; Gra(2,3)P2 may interact with positively charged residues at the entrance to the central cavity or another site or sites may exist. There is evidence that the binding site is close to that of deoxy-Hb (Gupta et al., 1979; Russu et al., 1990). Given the similar pH dependence of the binding of Gra(2,3)P2 and ATP as well as other anions (Garby et al., 1969; Janig et al., 1970; Rapoport et al., 1972) and the fact that Gra(2,3)P2 decreases the binding of other metabolites and vice versa (Garby et al., 1969; Berger et al., 1973; Hamasaki and Rose, 1974), it is assumed that ATP and other metabolites have the same binding site on both oxy-Hb and deoxy-Hb as does Gra(2,3)P2. There may be other binding sites as well. Using a Scatchard analysis, Garby et al. (1969) provided evidence of two Gra(2,3)P2-binding sites and found no evidence of saturation of ATP binding at ATP concentrations up to 6 mM, in a solution containing 3.1 mM Hb, 3.0 mM Mg", pH7.2. Janig et al. (1970) demonstrated that ATP binds to methaemoglobin at four sites at pH 7.2, and 10 sites at pH 6.2. In contrast to these findings, Hamasaki and Rose (1974) found that, at pH7.2, phosphorylated compounds bound to one sitehetramer for both oxyHb and deoxy-Hb. Thus, there is conflict over the number of binding sites, however, it seems clear that, as the pH decreases, the number of binding sites increases.
As indicated by the conflicting results from studies on the number of Hb-binding sites, the study of the interaction of haemoglobin with ions has been accompanied by many difficulties. These problems were outlined by Berger et al. (1973). Problems in accurately determining the amount of bound ligand at increasing concentrations of free ligand, as well as problems with keeping ionic strength constant with increasing free ligand concentrations, make accurate and detailed characterisation of binding parameters extremely difficult. Also, the binding of a number of polyvalent anions will reduce binding due to strong electrostatic repulsion between the bound anions. In addition, both the associating metabolite and the haemoglobin-binding site will be in an ionisation equilibrium around physiological PH. These difficulties led many investigators (e.g. Berger et al., 1973) to adopt a phenomenological model of the interaction of ions with Hb. For phospho compounds, this model assumes a single binding site with a single dissociation constant. Each dissociation constant is apparent, which is only valid for the conditions under which it is measured. The dissociation constants for all significant association reactions between Hb and metabolites are shown in Table 1. In a study on Hb affinity for Gra(2,3)P2 using pulsed-field gradient NMR, Lennon et al. (1994) found that the association constants reported by Berger et al. (1973) describe the relative affinities of deoxy-Hb and oxy-Hb for Gra(2,3)P2 more adequately than those of Hamasaki and Rose (1974) in intact erythrocytes. The use of the association constants of Berger et al. (1973) have also yielded closer matches between the value of the concentration of free Mgz+ calculated from "P-NMR data and the value determined by other experimental methods (Mulquiney, P. J. and Kuchel, P. W., unpublished results). In addition, it was found that the values of Berger et al. (1973) were in better agreement with the data of Garby et al. (1969) and Garby and de Verdier (1971). In this study, the values of Berger et al. (1973) were used in preference to the values of Hamasaki and Rose (1974) and Gupta et al. (1978b) when they were in conflict. A general feature of the binding of multivalent anions to Hb is a steep decline in binding from pH 6.0 to pH 8.0. This has led many researchers to suggest that histidine residues are involved in the binding (histidine pKa values are in the range 6-7; Janig, 1970). Given the difficulties in accurately characterising the binding of ions to Hb, a simple phenomenological description of the effect of pH on binding is given. It is assumed that, for a metabolite to bind to Hb, two histidine molecules (with equal pKLtvalues of 6.8) must he protonated. These histidines can he taken to be His143 from the p chains (pK, -6.8; Garby and de
77
Mulquiney and Kuchel (Eul: J. Biochem. 245)
h
5
0
i
0
I
I“
O
O
0
6
7
9
8
6
7
8 PH
PH Fig.3. The binding of Gra(2,3)Pzto oxyHb as a function of pH. ( 0 ) Garby and de Verdier (1971) 37°C; [Hb] = 5.3 mM; [Gra(2,3)P2] = 4.0mM; [Mg”] = 3.0mM. (0)Garby et al. (1969), 0°C; [Hb] = 3.1 mM; [Gra(2,3)P2] = 5.0 mM; [Mg2+] = 3.0 mM. (---). Theoretical value calculated using the equilibrium constants of Tables 1 and 2 with [Hb] = 3.1 mM, [Gra(2,3)P2] = 5.0mM, [Mg2+]= 3.0 mM. (-.-.-): Theoretical value calculated using the equilibrium constants of Tables 1 and 2 with [Hb] = 5.3 inM, [Gra(2,3)Pz] = 4.0 mM, [Mg”] = 3.0 mM.
Fig.4. The binding of Gra(2,3)Pz to deoxyHb as a function of pH. ( 0 ) Garby and de Verdier (1971), 37°C; [Hb] = 5.3mM; [Gra(2,3)P2] = 4.0 mM; [Mg2+]= 3.0 mM. (0) Garby et al. (1969), 0°C; [Hb] = 3.1 mM; [Gra(2,3)P2] = 5.0 mM; [Mg”] = 3.0 mM. (- - -) : Theoretical value calculated using the equilibrium constants of Tables 1 and 2 with [Hb] = 3.1 mM; [Gra(2,3)P2] = 5.0 mM; [Mg”] = 3.0 mM. (- .-. -): Theoretical value calculated using the equilibrium constants of Tables 1 and 2 with [Hb] = 5.3 mM; [Gra(2,3)P2] = 4.0 mM; [Mg”] = 3.0 mM.
Verdier, 1971), for example. This system can then be described by the equilibrium reactions in Table 8. From
KJ& =
[HbH, : L]
[L-I [Hb + HHb
+ HbH + HbH,]
n
’
(8)
is simple to derive
s-
:
;
v 1 CI-7 2
U
t
1 0
D
Data on the effect of pH on metabolite binding to Hb is limited. The only studies relevant to this work were performed by Garby et al. (1969) and Garby and de Verdier (1971). It is difficult to extract the effect of pH on association constants from these data because the incubations were performed with a constant total concentration of Mg2+, implying that different concentrations of free ligand and Mg*+-chelated ligand were present at different pH values. However, by solving the algebraic expressions for the association of the ligand with Mg”, as well as the expressions for the associations of Hb with the ligand and the Mg’+-chelated ligand, simultaneously at each pH value, it was possible to compare the experimental data with the theoretically calculated data (Figs 3 - 6). The theoretically derived curves match the data for binding to oxy-Hb at 37°C quite well and KHbATP within the (Figs 3 and 5). Adjustments to KHbMBATP range bounded by the values of Berger et al. (1973) and Gupta et al. (1978b) produced a good fit to the data for ATP binding to oxy-Hb (see Fig. 5). In addition, small changes in the value of KHbGra(2,3)Fz produced a better fit to the data for Gra(2,3)P2 binding to oxy-Hb (Fig. 3). The larger discrepancy between the theoretically derived curves and the experimental data obtained at 0°C is partly due to the increased binding at lower temperatures. The fact that these data give a mole ratio of greater than one for the binding of Gra(2,3)P2 to Hb shows the limitation of using an apparent constant assuming 1: 1 binding. For the binding of metabolites to deoxy-Hb, the model exaggerates the pH
E U 6
7 PH
8
Fig. 5. The binding of ATP to oxyHb as a function of pH. (0)Garby et al. (1969), 0°C; [Hb] = 3.1 mM; [ATP] = 1.OmM; [Mg”] = 3.0 mM. (-.-. -) : Theoretical curve calculated using the equilibrium constants of Gupta et al. (1978 b and Table 1) and Table 2. (- - -) : Theoretical curve calculated using the equilibrium constants of Berger et al. (1975 and Table 1) and Table 2.
dependence at 0°C and underestimates it at 37°C (Figs 4 and 6). However, overall, the model gives a satisfactory, at least semiquantitative description of the pH dependence of the associations, bearing in mind that, for this purpose, it is an overly simplified description.
Development of a dynamic model of metabolite interaction with Mg2+and Hb. In order to incorporate the above binding model into a kinetic model of metabolism, a kinetic model of metabolite association with Mg2+ and Hb is necessary. This allows Mg2’ and Hb interactions to be incorporated directly into the metabolic model as an extra set of differential equations. An NMR-based study of the exchange between free and Mgbound ATP showed that, under equilibrium-exchange condi-
Mulquiney and Kuchel ( E m J. Biochern. 245)
78
..
Table 10. Apparent rate constants for metabolically important haemoglobin complexes at 37"C, pH 7.2, p = 0.15 M, [K'] = 0.15 M, [ClF] = 0.15 M.
\
\
O
0
'\
Complex
6
0.20 -
\\
0
k,
\ \
M-1
ooo
\\ \
\
'
0.10-
0 \ \
0
\ \
\ \
-
I
k,
kd
k,
-~
\
0.00
DeoxyHb
OxyHb
...--. I
I
Hb :MgATP Hb : ATP Hb :ADP
Hb:Gra(2,3)P2 Hb:Fru(l,6)P2 Hb: Glc(1,6)P,
Hb: Gra( 1,.?)PI
PH
s-l
4.68 x 104
4.32 X 10' 3.0 X 10' 3.0X 10' negligible binding negligible binding 3.80 X 10'
s-'
M-I
1.2X1O3 1.2x 103 1.2~103 1.2x 103 negligible binding negligible binding 1.2x 10'
1.68 X 10' 3.12 x 106 1.44 X loh 6.00 x 106 1.21X10h
1.2 X 10' 1.2 x 103 1.2 X 10' 1.2 x 103 12x10'
1.21 x 106
1 . 2 103 ~
$-I
s-l
Fig.6. The binding of ATP to deoxyHb as a function of pH. (0) Garby et al. (1969),OOC; [Hb] = 3.1 mM; [ATP] = 1.0 mM; [Mg2+]= 3.0 mM. (---), theoretical curve calculated using the equilibrium con- in the expression. This approach assumes that k, is independent stants of Tables 1 and 2. of pH.
Table 9. Apparent rate constants for metabolically important magnesium complexes at 37"C, pH 7.2, p = 0.15 M, [K'] = 0.15 M, [CI-] = 0.15 M.
Complex
MgATP MgADP MgAMP MgGra(2,3)P2 MgGra( 1,3)P, MgFWl,6)P, MgGlc(1,6)P2 MgP,
k,
k,
M-' s-'
S-'
3.12X lo7 2.16 X 10' 3.84 x lo4 8.04 x 105 2.28 X 10' 4.80 x 105 4.80X 10' 4.08 x 104
1.2 x 103 1.2 x lo3 1.2X10' 1.2 x 10'
1.2 x lo3 1.2 x 10'
1.2X10' 1.2x 10'
tions, the rate-determining step is the unimolecular dissociation of the complex. This is true over a range of temperatures, concentrations and pH values (Bishop et al., 1990). The rate constant for dissociation ($) was 1200 s-' (25°C). Assuming that the apparent association constant under the NMR experimental conditions can be expressed as
where k, is the association rate constant and K " P P is the apparent association constant [and assuming that k, (25OC) is equal to kd (37"C)], then (Table 2) k,= 3.12X107M-'s-'. For all other complexes in the model (both Mg" and Hb), it was assumed that the exchange is rate determined by the unimolecular dissociation, and that the dissociation rate constant is 1200 s-.'. In this way, values of k, were evaluated for each complex (Table 7). Since the time scale of these associations and dissociations are extremely fast compared to the rate-limiting metabolic reactions, the exact value of the rate constants is unimportant; i.e. yuasiequilibrium concentrations of the complexes will be formed very rapidly and the equilibrium concentrations will depend on the ratio of the rate constants rather than their absolute values. The pH dependence of k, was modelled with the same eyuations as R P P by replacing Z P ' with k, whenever K"pp appeared
Calculations. All calculations were performed with Mathematica using the rate constants and pH dependencies defined in Tables 1, 2, 9, and 10. A copy of the model can be obtained from PWK (
[email protected]). RESULTS AND DISCUSSION The distribution of metabolites as a function of pH and its effect on enzyme kinetics. The concentrations of free and Mgz+-complexed metabolites were calculated over the pH range 6-8 (Fig. 7). For oxygenated cells, the effect of pH on metabolite distribution is greatest at pH 7.2-7.6, with many species undergoing changes in concentration of 50- 100 % within this range. Thus, pH changes around physiological values cause large changes in the distribution of metabolites between free, bound and Mg2+-complexedforms, based simply on binding interactions alone. It is a complex task to speculate on how these changes will affect the rates of various glycolytic enzymes because many of the changes that occur cause opposite effects on particular enzymes. For example, 2,3-bisphosphoglycerate synthase, the enzyme which catalyses the formation of Gra(2,3)P2 from Gra(l,3)P2, is competitively inhibited by free Gra(2,3)P2. An increase in pH causes a predicted (by the binding model) increase in both free Gra(2,3)P2 and free Gra(l,3)P2. Even though MgP, has a small association constant, the concentration of MgP, varies significantly with pH (Fig. 7). Therefore, it would be expected that other glycolytic metabolites, which form weak complexes with Mg2+,would also show large variations in the concentration of the Mgz+-chelated species with pH. We contend though that the neglect of these species in our model is valid, since it is only their free forms which appear to exert a direct effect on the enzymic reactions. Since the concentration of these Mg2+ complexes will be low compared to the total concentration of the metabolite, the concentration of the free metabolite will not vary greatly with pH. For example, the concentration of free P, varies by less than 1.3 % in solutions of both oxy-Hb and deoxy-Hb, over the entire pH range (Fig. 7). The distribution of metabolites in the oxygenated and deoxygenated erythrocyte. The distribution of metabolites and their various complexes, at pH 7.2, is shown in Table 11 for oxygenated and deoxygenated cells. This shows that upon oxygenation, an increase in free Mg" of 83% occurs. However, when the intracellular pH rise of ~ 0 . 1 that 4 occurs on deoxygenation (La-
Mulquiney and Kuchel (Eur: J. Biochem. 245)
79
3.5
3-
A
,
.Y
6.0
6.5
7 .O
7.5
6.0
8.0
6.5
7.O
7.5
8.0
7.5
8.0
PH
PH
6.0
6.5
7.O PH
PH
I
2.5
1
F
2 1.5 1
0.5
0 6 :O
615
710
7.5
8.0
PH
PH
Fig. 7. The distribution of metabolites between free and Mg*+-complexedforms as a function of pH and Hb oxygenation-state relative to their concentration at pH 7.2. The calculations were performed assuming the following total concentrations of metabolites to be Hb, 7 mM; Gra(2,3)P2, 6 mM; ATP, 2.1 mM; ADP, 0.31 mM; P,, 1 mM; Mgz+, 3 mM; Gra(l,3)PZ,0.7 pM; Fru(l,6)P2, 2.7 pM; Glc(l,6)P2, 122 pM. (A) (-) Mg2+ with oxy-Hb; (---) free Hb with oxy-Hb; ( - - - - - ) Mg” with deoxy-Hb; (-.-.-) free Hb with deoxy-Hb. (B) ATP. ( C ) ADP. (D) Gra(2,3)P2.(E) Gra(l,3)PZ.(F) Fru(l,6)P2. (G) P,. (B-G) (-) free metabolite with oxy-Hb; (---) Mg2’ bound metabolite with oxy-Hb (-----); free metabolite with deoxy-Hb; (- .-.-) Mg” bound metabolite with deoxy-Hb.
botka, 1984) is taken into account, an increase of 61 % is calculated. This relative change agrees well with that of 55 % determined by Flatman (1980). Other changes that occur on deoxygenation are also reduced if this pH effect is taken into account.
Another interesting point is that the free concentrations of the potent effectors of erythrocyte glycolysis, Glc(l,6)P2 and Fru(2,6)P2 [the latter having an obvious structural similarity to Fru(l,6P),] decrease by 50% upon deoxygenation.
Mulquiney and Kuchel (Eur. J. Biochem. 245)
80 1.5
IG ,
8
8
.---. .
8
. *.-. ---.-.-*. \
\
8
1-+-
: ,
\
Y
\’.\-. *.*.
,
3.
*.
Component f5 8.
-
.5
Table 11. The distribution of Hb, MgZ+,and phosophorylatedmetabolites in the human erythrocyte in oxygenated cells (pHi 7.2) and deoxygenated cells (pHi 7.2 and 7.34) with the following metabolite concentrations: Hb, 7 mM; Gra(2,3)P2,6 mM; ATP, 2.1 mM; ADP, 0.31 mM; P,, 1 mM; MgZ+,3 mM; Gra(l,3)Pz, 0.7 pM; Fru(l,6)P2, 2.7 pM; Glc(l,6)P2, 122 pM.
--
Intracellular concentration aerobic pH, 7.2
-*.
anaerobic pH, 7.2
pH, 7.34
0.733 (24) 1.472 (49) 0.279 (9) 0.121 (4) 0.358 (12) 0.037 (1) 0.077 (4) 1.472 (70) 0.272 (13) 0.279 (13) 0.072 (23) 0.121 (39) 0.1 17 (38) 0.727 (12) 0.358 (6) 4.915 (82) 2.16X (30) 0.30 X (4) 4.53 X 10 - 4 (65) 1.02 X 10 - 3 (38) 0.30X (11) 1.39 X (51) 0.046 (38) 0.014 (11) 0.063 (52) 0.976 (98) 0.243 (2) 1.352 (19) 0.272 (4) 0.279 (4) 0.117 (2) 4.915 (70) 0.065 (1 j
0.643 (21) 1.544 (51) 0.226 (8) 0.126 (4) 0.425 (14) 0.036 (1) 0.089 (4) 1.544 (74) 0.242 (13) 0.226 (11) 0.082 (26) 0.126 (41) 0.102 (33) 0.895 (IS) 0.425 (7) 4.680 (78) 2.54X (36) 0.34 X (5) 4.11 X (59) 1.16 X (43) 0.31 X lo-.’ (11) 1.23 X (46) 0.053 (43) 0.014 (11) 0.055 (45) 0.978 (78) 0.022 (2) 1.693 (24) 0.242 (3) 0.226 (3) 0.102 (1) 4.680 (67) 0.057 (1)
mM (%)”
0 6.0
6.5
7.0
7.5
8.0
PH Fig. 7. (Continued).
Validity of model. This is the first model of metabolite binding to Hb and Mg2+ which takes into account the effect of pH. Its ability to calculate the relative change in free Mg” is one test of its plausibility. The concentration of free Mg2+ inside the erythrocyte has been experimentally determined to be 0.4 mM and 0.62 mM for oxygenated and deoxygenated cells, respectively (Flatman, 1980). These values could only be approximated by the model by choosing a total concentration of Mg2+ of 3 mM. This is on the low end of the range of total Mgz+ found in erythrocytes (3.4 -+- 0.4; Millart et al., 1995) and is only 88% of the mean total cell magnesium. However, this is in agreement with the finding of Flatman and Lew (1977) that the main buffer systems bind ~ 9 0 % of the Mg” in the cell. It is speculated that the remaining Mg” may be bound to other intracellular components such as the cell membrane. Prior to the present attempt, the best model constructed to calculate the concentration of free Mg2+ in oxygenated and deoxygenated erythrocytes was that of Berger et al. (1973) and Gerber et al. (1973). These workers predicted intracellular Mg” concentrations of 0.67 mM and 1.12 mM for oxygenated and deoxygenated cells, respectively. The higher absolute values and relative change calculated by these workers results from their assumption that all Mg” bound by ATP or Gra(2,3)P2 only and from their neglect of the change in intracellular pH caused by deoxygenation.
Mg” 0.400 (13) MgATP 1.520 (51) HbMgATP 0.229 (8) MgADP 0.099 (3) MgGra(2,3)P2 0.721 (24) Mg other 0.031 (1) ATP 0.147 (7) MgATP 1.520 (72) HbATP 0.204 (10) HbMgATP 0.229 (11) ADP 0.107 (35) MgADP 0.099 (32) HbADP 0.104 (34) Gra(2,3)P2 2.684 (45) MgGra(2,3)P2 0.721 (12) HbGra(2,3)P2 2.595 (43) Gra(l,3)P2 3.04X lW4 (43) MgGra(l.3)P2 0.23 X (3) HbGra(l,3)P2 3.73 X (53) Fru(l,6)P, 2.33 X (86) MgFru(l,6)P2 0.37X (14) HbFru(l,6)P2 -
Glc(l,6)Pz MgC1(1,6)P2 HbGlc( 1,6)PI P, MgP, Hb HbATP HbMgATP HbADP HbGra(2,3)P2 Hb other
0.105 (86) 0.017 (14) -
0.987 (99) 0.013 (1) 3.868 ( 5 5 ) 0.204 (3) 0.229 (3) 0.104 (1) 2.595 (37) 0.000 (0)
Limitations of the model. Use of apparent constants. For sim-
The bracketed term after each concentration refers to the percentage of the total concentration of the common species within the group as grouped in the table.
plicity, we have chosen to use apparent binding constants to describe the binding of metabolites to Mg”. This was necessary since much of the available data on enzyme kinetics, as well as Hb :metabolite interactions, have been analysed in terms of simplified Mg2+:metabolite binding. The use of apparent constants, however, limits the scope of the model. For example, at high concentrations of Mg2+with respect to ATP, the formation of Mg2ATPbecomes significant in reducing the activity of hexokinase (Noat et al., 1970). By choosing to use apparent constants such phenomena cannot be explained. Another limitation of using apparent binding constants occurs when complexes with stoichiometries other than 1 : 1 form. In these cases, P P p will be a function of [Mg”] and the concentration of the associating metabolite. For example, KgATp will also be a function of [MgZ+]and [ATP-] due to the formation of Mg2ATP and Mg(ATP),. However, this does not become significant until the mole ratio, R = [MgZ+]/[ATP4-],becomes varies from 7.9X 10’ M-’ large or small. For R = 1, cJ;Arp
to 8.5X10’ M-’ between concentrations of 1.0XIO-’M and 5.0XlO-’M. At [ATP] = 5.0XIO-’M and [Mg’’] = 1X lo-” M, K“”’ increases to 11.3 X lo-’ M-’. However, at such a low concentration of Mg*+,the Concentration of Mg:ATP complexes will be extremely low, and the change in K“pp would be insignificant. Effect of haemoglobin concentrations on association constants. It is possible that in the interior of the erythrocyte, where the concentration of Hb is high (=7 mM), that excluded volume and other crowding effects (Ralston, 1990) cause the values of dissociation constants to be different from those obtained in dilute solutions that are often used in binding studies. There is contradictory evidence on the dependence of association constants on the concentration of Hb. Garby and de Verdier (1971) found that the affinity of oxy-Hb and deoxy-Hb for Gra(2,3)P2 decreased markedly as the concentration of Hb increased from
Mulquiney and Kuchel (Eur J. Biochern. 245)
0.4 mM to 5.5 mM; Rapoport et al. (1972) found a similarly strong dependence of association constants on Hb concentration. In contrast, Hamasaki and Rose (1974) found that the affinity of Hb for Gra(2,3)P, was independent of Hb concentration at 0.06-4 mM. In addition, Lennon et al. (1994) found no difference in the affinity of HbCO for Gra(2,3)P2 at Hb concentrations of 2.7 mM and 4.96 mM. For the present model, it was assumed that the values of association constants obtained at lower Hb concentrations remain valid for the concentrations of Hb found in the erythrocyte. Other significant interactions. Both Mgz+ (as discussed above) and Gra(2,3)P2 (Lennon et al., 1994) have been shown to bind to other sites in the erythrocyte. For Mg", this is accounted for in the model by choosing a lower value for the total concentration of Mg". No account is taken for other possible Gra(2,3)P2-binding sites. The fact that in the above analysis, the apparent equilibrium constants, calculated from the intrinsic constants, were lower than the experimental values, suggests that the descriptions of Mg*+-metabolite interactions as presented above may be incomplete. Specifically, the model does not take into account the fact that CO, may compete with the binding of phospho compounds to Hb, due to the formation of carbamino-Hb. Additionally, it has been suggested that HCO; complexes weakly with Mg2+ (Gerber et al., 1973) and that C1- may compete with phospho compounds in binding with Hb (Flatman, 1980). However, in a study of solutions containing physiological concentrations of ATP, Gra(2,3)P,, CI-, Mg", K+ and Hb, negligible amounts of C1- were found to be bound to Hb (Achilles et al., 1981).
Importance of including K' interactions in pH descriptions of magnesium binding constants. Figs 1 and 2 show the importance of including all significant association reactions in the calculation of the pH dependence of the various dissociation constants. In particular, they show that the competition between K'and H+ associations largely affect the pH dependence of these Mg"-binding constants. These interactions have often been ignored in previous models (e.g. Lawson and Veech, 1979; Bock et al., 1985) and, significantly, this is ignored in the most comprehensive erythrocyte metabolic model to date, namely that of Lee et al. (1991). The model of Lee et al. (1991) also neglects interactions between Hb and the glycolytic intermediates except for Gra(2,3)P,. Metabolite distribution and the activity of hexokinase. Using a minimal steady-state model of hexokinase (Kuchel et al., 1984) with the kinetic parameters c g A T p = 1.0 mM, = 1.0 mM, = 47 pM, elc= 47 pM, = 14 pM, ~ l ' ( l , b ) P 2 = 22 pM p ( 2 ZIP? = 2.7 mM and p g A D P = 1.0 mM with all other parameters of the model being as described by Kuchel et al. (1984), and assuming the metabolite concentrations shown in Table 11, the activity of hexokinase at [Glc] = 5 mM, under aerobic conditions, was calculated to be 1.48 mmol (L erythrocytes ' h-'. Under anaerobic conditions at pH, 7.34, the calculated activity increased to 2.05 mmol (L erythrocytes-' h-'. This 38% increase agrees well with the increase in glycolytic flux in deoxygenated erythrocytes found by Asakura et al. (1966). The similarity of these values is to be expected, given the high flux control coefficient of hexokinase in glycolysis. The increase arises primarily from the decrease in free Gra(2,3)P2 and the decrease in free Glc( 1,6)P, (both are hexokinase inhibitors) upon deoxygenation. Although this increase in hexokinase activity upon deoxygenation has been described before (e.g., Gerber et al., 1973, 1974) there is a fundamental difference in the explanations given. Gerber et al. (1973, 1974) believed that
e:c
c1c6p
>
I
81
along with the marked decrease in free Gra(2,3)P,, the cause of the hexokinase activation was an increase in free Mg". They reported that concentrations of free Mgz+ up to 4 mM were capable of activating hexokinase. They ignored the effect of Glc( 1,6)P, on hexokinase. In the present analysis, the apparent activation of hexokinase by Mg2' was ignored since it appears likely that this effect is an indirect one due to the relief of inhibition by free ATP (Rijksen and Staal, 1977). Using the present model of metabolite binding, and assuming the total metabolite concentrations from Table 11, at pH, 6.8 the following metabolite distribution is calculated : [Mg2+], 0.6 mM; [MgATP], 1.31 mM; [Glc(l,6)P2], 0.101 mM; [Gra(2,3)P2], 1.84 mM. Thus, a decrease in pH from pH, 7.2 to pH, 6.8 causes changes in the metabolite distribution that will have opposite effects on hexokinase activity ; the decrease in concentration of the inhibitors Gra(2,3)P2 and Glc(l,6)P2 activates hexokinase and the decrease in the substrate MgATP reduces the activity of hexokinase. Overall, however, it is calculated that the activity of hexokinase at pH, 6.8 would be 1.37 mmol (L erythrocytes-' h-' ; a decrease of =7 %. This decrease is much lower than the expected decrease of > 50 % (Minakami and Yoshikawa, 1966). This difference is primarily due to the previous failure to take into account the effect of pH on enzyme kinetic parameters. For example, the maximal velocity of hexokinase decreases by = l o % from pH 7.2 to pH 6.8 (Gerber et al., 1974). Additionally, phosphofructokinase is inhibited by H+ ions, causing a marked elevation in Glc6P concentration. This example illustrates that in order to model enzyme kinetics as a function of pH both the effect of pH on enzyme kinetic parameters, as well as interactions between metabolites, Hb, and Mg2+,have to be taken into account.
Concluding remarks. In conclusion, we have presented a model which predicts the distribution of glycolytic metabolites between free, Mgz+-bound,and Hb-bound forms under various experimental conditions. It is the first model of metabolite binding to Hb and Mg2+ which takes into account pH, and in doing so, it is the first model able to match experimental measurements of intracellular free Mg" in oxygenated and deoxygenated cells. An important finding to emerge from the simulations is that pH changes around physiological values can cause large changes in the distribution of metabolites. For example, in oxygenated cells at pH 7.2-7.6, many species undergo changes in concentration of 50- 100 %. Additionally, the potent effectors of erythrocyte glycolysis, Glc( 1,6)P2 and Fru(2,6)P2, decrease in concentration by ==50% upon deoxygenation. It seems likely that this decrease in Glc(l,6)P, is a major cause of the increased rate of glycolysis upon deoxygenation. This work was supported by the Australian National Health and Medical Research Council and P. J. M. received an Australian Commonwealth Postgraduate Research Award. Dr Julia Raftos is thanked for an extremely valuable exchange of ideas.
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