Estimation of parameters in a multi-affinity-state model for haemoglobin from oxygen binding data in whole blood and in concentrated haemoglobin solutions

J. Hoi.

Biol.

(1978)

121, 507-522

Estimation of Parameters in a Multi-affinity-state Model for Haemoglobin from Oxygen Binding Data in Whole Blood and in Concentrated Haemoglobin Solutions TORWY

GROTII, LARS

GARBY

AND

CAM,-HESRK

DE VEKDIER

Department of Physiology Odewe University, Odense, Denmark Croup for Biomedical Informatics, Uppsala 1 rniversity I)ata Centre and Department of Clinical Chemistry, 1’~ iversity Hospital Uppsalo , Sweden (Received 10 October 1977) Oxygen binding data on whole blood and concmt,rated haemoglobin solut,ions at different, pH, pCOz and 2,3-bis-phosphoglycerwte concentrations were analysed with the use of a t,wo-quaternary-stnt)e model regarding P,-Grit purely as a quaternary effector (Herzfeld & Stanley, 1972). The model was applied as a. multi-afflnit.y-state model and was extendetl on a molecular mechanistic level in t,erms of the so-called Per&z mechanisms. Binding of oxygen to a subunit, contained in th(b oxy qu&ernary state was found tjo be insensitive to changes in pH and pCOz (Koxy = 0.71 i 0.05 mm Hg-‘). At was t&mated to be lower than 37”C, pH 7.2 and pCOz = 22 mm Hg, K,,,,, of 130, the allosteric constant was (1.5 i 0.4)104 and the binding h’“,, by a factor constant, of P,-Gri for the deoxy quat.ernary conformation was estimatrd to bc 25 times higher than that for the oxy quaternary conformat,ion (130 + 50 1 mol- I). With regard to the influence of protons, carbon dioxide and P,-Gri, the results are consistent with t,he concept,s t.hat (1) P,-Cri acts primarily as a quaternary effector, tha,t (2) protons act as constraint effect.ors and second-order effecters. and that (3) carbon dioxide acts as a first and second-order quaternary effector. The calculations further indicate that, (1) quat)ernary transition most likely t.akes place after oxygenation of t.he second subunit, that (2) the probabilities of intermediary states of oxygenation are surprisingly low, and that (3) t,ht> difference in t,otal conformational energy between the two quaternary ligand-fret, states is almost exclusively confined to molecular constraints, and very little to the difference in quaternary conformationnl enctrpy.

1. Introduction Haemoglobin oxygen binding data have previously been analysed in terms of diEerent. mechanistic models for haemoglobin function (Tyuma et al., 1973a; Herzfeld & Stanley, 1974; Minton, 1974: Minton & Imai, 1974: cf. also Monod et al.. 1965: Koshland et al., 1966). In t,hese st,udies, however, the haemoglobin concentration has generally been very low. between 0.01 and 0.12 mM-tetramer. Ackers et al. (1975) and Johnson & Ackers (1977) have shown that under these circumstances t,he dissociation of tetramer molct Abbreviation 0022%2836/78/1214-0722 18

usetl:

P,-Gri, $02.00/O

2,3-bis-phosphoglyccrat~~. ii07 $3 1978 Academic

Press

Inc.

(London)

Ltc(.

50s

‘I’.

(:RO’I’H.

I,.

(:Xlil3\-

;\SI)

(‘.-H.

I)15

\‘I.;I
HI{

cults inbo dimcrs is not ncgligiblr. and l,hat the presences of’ eve11 small amounts (11 dimers may profoundly affect t hc shape of the binding c*urv(' and rnav hiivt. tlevastat~ing effects upon the reliability of? c.g. Adair constant estimations. Previous dat)a on oxygen binding in whole blood (Kernohan B Itoughton. I!E: Severinghaus et al., 19’72; Roughton et a2.. 1972) are incomplet,e \vith respect to. c.p. information on the concentration of 2,3-bis-phosphoglycerate and/or red cell hydrogetr ion act’ivity, and thus cannot be used for the purposes of tesbing detailed mod& f’or haemoglobin co-operativity and effect,or action. The recent very precise dat,a reported by Winslow et uE. (1977) are limit,ed to a single value of bot#h pH and concentration of P,-Grit.

TABLET Summary Data set no.

Experiment no.

of experimental PC02 (mm Hg)

PH

77

3

7.1

-

77 -

9

(Fig.

~~~

22

G

77

1.5 4.4 9.3

71(b))

22

7.2

~.-

15 16

~~~.. 1.5 4.4 9.3

(Fig.:(a))

12 13 14

1.5 9.3 1.5 9.3

22

4

Reference

1.5 9.3 1.5 9.3

7.0 2

10 11

[P,-Gri] (mmol l-I)

22

1 -______

conditions

.Data from Garby et al. (1972) and Arturson et al. (1974). Whole blood at 37.O”C. The pH refers to intracellular conditions and the Hb concentration is 5.0 mmol l-1 packed cells. The data cover the saturation range 0.10 to 0.95 and each experimental curve is repwsented by 10 points.

1.5 9.3

7.3 15

1.5 9.3

1 2 3 4 5 6

0.2 2.0 3.9 7.8 15.x 27.4

17

7 8 9 10 11 12 t Abbreviation

8

77

7.2

40

(Fig.Yl(c))

7.4

10

used:

P,-Gri,

40

2,3-bis-phosphoglycerate.

0.2 24 3.9 7.8 15.8 27.4

Data from Duhm (1974), OIL haernoglobin solutions at 37.OY’. Hb concentration is 3.8 to 4.0 mmol I -I. The data covc~ the saturation range 0.10 to 0.90 and each experimental curve is represented by 9 to 22 points, in total 70 and 88 points for data sets 9 and 10, rcspectively.

ANALYSIS

OF

OXYGEN

BINDING

DATA

.509

Several sets of oxygen binding data on whole blood and on concentrated haemoglobin solutions at different values of pH, pC0, and P,-Gri concentrations have been obtained by Garby et al. (1972), Arturson et al. (1974) and Duhm (1974).t These data fulfil, at least partly, such requirements with respect to experimental design that enable, in principle, reliable estimation of parameters describing haemoglobin function in terms of physical mechanisms. In the present work, these data have been analysed wit’h the use of the Herzfeld-Stanley (1972) two-quaternary-state model and further interpreted in terms of concepts suggested by Perutz (1970,1972,1976). Thus we have, contrary to Herzfeld & Stanley (1974) regarded the oxy quaternary conformation to be without quaternary-tertiary constraints. Furthermore, we have described the molecular constraints in the deoxy quat,ernary conformat8ion in a different way. The binding of oxygen to a subunit when the tetramer is in the quaternary oxy conformat,ion was found to be insensitive to moderate changes in pH, pCOz and P,-Gri concentration (cf. Tyuma et al., 1971,1973a,b; Imai, 1974; Imai & Yonet,ani,

20 0 O-80

l-00

l-20

1.40

1.60

2.00

I-00

. . Ot

l-4 4

o-so

I.00

1.6

1-S

l-20

1.40

l-60

2-O

P-02

FIG. 1. Experimental oxygen binding data compared with theoretical (the Herzfeld (model 1972)) as fitted with the NONLIN program. (a) Data set 5; expt no. 9 (-+-+-), 10 (-g-n-) and I1 (-/,-,;>--). (b) Data set 7; expt no. 15 (-+-+-) and I6 (-n-n-). (c) Data set 9 (cf. Table 1). expt no. 1 (-t); expt no. 2 (0); expt no. 3 (< ); expt no. 4 (Y); expt no. 6 (x); oxpt t We IS’

express

our

thanks

to Dr

J. Duhm

for

making

2.00

I.80

his

data

available

to us before

t

no.

Stanley

6 (Z).

publication.

610

T.

GROTH.

L.

GARHY

:INU

197%; see also Kilmartin et al., 1975). Wlizing to obtain, for the first time, unique estimates co-operativity and effector action.

(I.-H.

l)E

\‘EHI)IEII

this circumstance it has btben possibk of parameters relaktl t’o haemoglobin

2. Experimental

Data

The oxygen binding data used in the present study were taken from Garby rt al. (1972), Arturson et al. (1974) and Duhm (1974; and personal communication). The experimental conditions are shown in Table 1 (cf. also Fig. 1 (a) to (c)). These experiments were not primarily designed for t.he present, type of analysis. Thus, only 4 of’ the 10 data sets contain saturation curves for more than 2 P,-Gri concentrations. The experimental equipment used for oxygen binding assay in data sets 1 to 8, a Radiometer DCA-1, gave according to Arturson et al. (1974) precise measurements in the saturation range 0.10 to 0.95 (see Fig. 2). Outside this range the experimental error (of systcamatic and random type) was considcrcd to ho t)oo large t,o allow informative measurements. As found in a previous study (Qrot,h et al., manuscript in preparation) the lack of data from these extreme ends is not crit,ical for t)hc intended type of analysis.

I

Fm. 2. The standard deviation of y0, measurements as a function of fractional saturation, Y ,+. Filled circles represent experimental data (Arturson et al., 1974), the broken line represents the fitted approximating function and the unbroken line represents the corresponding calculated methods). relative weight function I/ var Y ,-+ (see computational

3. Theory (a) The oxygen saturation

function

The fractional saturation of oxygen binding sites, two-quaternary-state model by Herzfeld & Stanley purely as a quaternary effector, Y,, can be written

Y,,, was calculated from the (1972). If P,-Gri is regarded

(1)

ANALYSIS

OF

OXYGEN

BINDING

51 I

DATA

where K, is t’he equilibrium constant for the t’wo quaternary conformations (deoxyare the binding constants for P,-Gri 0x9) in the absence of oxygen, B,,,,, and R,,, to the two quaternary conformations, and Rdeoxy and hTOxYare the binding constants for oxygen to subunits when the quaternary conformation is given as stated. Each of the latter two constants may be different, for a- and /3-chains (cf. Results sect)ion (c) and Discussion section (c)). The Herzfeld-Stanley model has the advantage that the effector action of P,-Gri is eqdicitly described and thus allows the analysis of oxygen binding curves recorded at conditions with a varying free concentration of P,-Gri. No constraining assumption was made with regard to the behaviour of the parameters at different pH and pC0, values, thus allowing a multiplicity of affinity states. The free concentration of P,-Gri was here calculated from t,he modified equation

V’2-Grilf,,, = F’,-Gril,,, -- [Hbl* YPZmGrl *(1 - fMgz-t ),

(2)

where [P,-Gri],,, is the total concentration of P,-Gri, [Hb] is the haemoglobin molecules with P,-Gri bound, concentration, YP2-er, is t,he fraction of haemoglobin and fMpz+ is a correction factor for binding of P,-Gri tJo magnesium ions. This factor is a function of oxygen pressure and was calculated from the data of Berger et cd. (1973): fMgz+ = 0.17 - 0401 *PO,; PO, ( 60 mm Hg, Pa)

FMg2+ = 0.11; pOZ > 60 mm Hg. (b) Thermodynamic&

parameters

The free energy change related to the quaternary absence of oxygen, E,* is given by

and

(3b)

relations

transition

(oxy-deoxy)

in the

E,* = RTlnK,. Following the concepts of Peruta (1970,1972,1976), energy change as being made up of two parts: E,* = K,t

+ E,,

Pa) we

consider

t,he total

free WI

where E& is bhe energy related to molecular constraints imposed by bonds, e.g. saltbridges, which stabilize the deoxy quaternary conformation and which are broken during oxygenation of that structure; and E, is the difference in energy between the two quaternary conformations (oxy-deoxy) due t,o structural differences which are not changed during oxygenation. Binding of oxygen to a subunit induces a change in tertiary conformation, so termed induced-j2 binding. During oxygenation energy is released by the reaction of the haem groups with oxygen: RTln K,,,,, and RTln K,,, for each subunit in the two quaternary conformations, respectively. ln the unconstrained oxy quaternary conformation this energy is entirely related to induced-fit binding. In the constrained deoxy quaternary state, part of the energy released is expended (1) to break the stabilizing salt-bridges, and (2) t’o increase the constraint energy due to strains imposed on the oxy tertiary structure when confined in the “wrong” quaternary structure. Thus, we assume that the energy of t,he stabilizing salt-bridges goes from -E& t’o 0 during successive oxygenation of the four subunits. Furthermore, we assume that

512

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C:ROTH.

I,.

(:AH.liY

ASI)

(‘.-H.

I)C:

\~1~;1{1)11~:1;

the constraint energy due to wrong conformation increases concurrently from o to E& where Eyt is the maxima1 value of t’his constraint, energy at full oxygenation. when all four subunits are in t’he wrong conformat.ion. (!onsequently the follo\ving relation holds E, + E& + 4RTlnK,,,,, which

-- E,

+- Eh’,

- 4RTlnK,,,

-== 0,

(5a)

gives K

is the constant i.e. &my by the deoxy quaternary

deoxy

-

Km,

*exp[-

for induced-fit structure.

(E&,

oxygen

4. Computational

+

Eh',)/4RTl,

binding

(5b)

of a subunit

as constrained

Methods

and rf,,,) were estimated by fitting bhe The parameters (K,, Kdeoxy,K,,,, Bd8oxy Y,,-function in a weighted least-squares sense to the experimental data sets (see Table 1). The analysis was made with the assumption that P,-Gri acts purely as a quat,ernary effector, i.e. Pa-Gri is assumed not to affect the binding constants K,,, see Discussion). It was thus possible to include data from and Kdeoxy (w W); experiments at different Pa-Gri concentrations in the same fit using the five-parameter model described by equations (1) to (3). Grouping of the experimenhal data was therefore only necessary with regard to pH and pC0,. A computer program, NONLIN, for non-linear least-squares curve-fitting was used (Metzler et al., 1974). This program works iteratively according t)o a modified version of the Gauss-Newton method (Hartley, 1961; Hartley & Booker, 1965). Weights were given to the data points as calculated from the variance of p0, measurements (Fig. 2) : weight var Y,,

= l/v&r Yo,,

= 8 Y,,/6pO,

varp0,.

@a) (6b)

5. Results (a) Compatibility The five-parameter model described by equations (1) to (3) was able to describe the data quite well. Figure l(a) and (b) shows sets of experimental data (data sets 5 and 7) together with fitted curves. The largest deviations between theory and experiments for the whole blood data were seen in data set no. 5 (Fig. 1 (a)). For haemoglobin solutions (Fig. 1 (c)), the deviations were of t’he same magnitude at the two pH values. For data sets 1 to 8 (whole blood) the deviations were less than two standard deviations of the experimental error (see Fig. 2). The model was thus considered to be compatible with the experimental data sets. In some of the data sets (cf. Fig. 1) a systematic deviation was observed in the lower saturation range. Herzfeld & Stanley, observing the same type of deviation, took this as a reason for introducing an additional parameter for bhe constraining effect of P,-Gri (cf. Fig. 5 of Herzfeld & Stanley, 1974). In the present study, the errors in the experimental data in this part of the curve were nob considered a justification for such an extension of the model.

ANALYSIS

OF

(b) Parameter

OXTCEX

RINDTNC:

estimates and confidence

D.&T.\

.;I:!

limits

When the five-parameter model was fitted to the data with the NONLIN program, K, was found to correlate strongly with K,,, (T = O-99), as could be expected from previous studies (Groth et al., manuscript) in preparation). Despite the strong correlation between the parameters related to quaternary equilibrium and oxygen binding, it t)urned out that, “best” estimates of the oxygen binding const(ant Koxy were quite constant with a small scat,ter between values for different pH and pC0, values: truncated mean value equal t’o O-71 mm Hg - ‘, with a standard error of estimatjca equal to 0.05 mm Hg- l, tz = 9. The data sets were t’herefore reanalysed under t’htl assumption that, K,,, is insensitive to moderate changes in pH, pC0, and P,-Gri activity (see Discussion). With K,,, frozen at a constant value it. was possible tf) estimate confidently t)he other parameters K,, Kdeoxy,lfdeoxy and R,,,,. Best estimates of these parameters with standard errors are given in Table 2, for K,,, -= 0.71 mm as a, function of pH at the two different pC0, values. The influence of unHg-‘, certainty with regard to t,he exact value of K,,, is illustrakd by Figure 3, which gives also t,hc result,s from analyses based on K,,, values equal to WSI and W81. rcspectivel-v. P,-Gri binding could only be determined for data se& 5, 6, 9 and 10, where dat,a at three or more different P,-Gri concentrations were available. The effect of neglect,ing the contribution of P,-Gri binding to Mg2 + in equation (2) was st,udied for data set, 5 and was found to be quite small. a deoxy, and rf,,, are only presented with st)andard errors for t,he sets where they could be uniquely determined. Assumpt,ions concerning a difference in affinity of the a and /I-subunits in the deoxy quat,ernary conformation were studied for da6a set 5. The dat!a were reanalysed with

I

I 7.0

I 7.1

I 7.2

I 7.3

1 7.4

PH 3. The effect of pH and carbon dioxide pressure on the allosteric constant K,, assuming a constant value of K,,, = 0.71 i 0.05 mm Hg-‘. pCOz =m 22 mm Hg (-.-.-), pC0, =_= ‘77 mm Hg (---O-o-) and pCOz = 40 mm Hg ( A, concentrated hsemoglobin solutions). The estimates are given with +I 1 S.D. The thick solid line and the thick broken line correspond to the thin solid and broken lines represent K,,, 0.61 (lower ~wrws) Gw = 0.71 mm Hg- I, while 0.81 mm Hg- 1 (upper rwrws), respwtiurly. anti K,,, PIG.

A14

‘I’.

a/p rat.ios 7.5 X10-“,

GROTH.

I,.

(:ARB\’

of 2, 5 and 10. The 8.7 X1W3. 8.3 X10m3.

(c) &uaternary

con&mat&al

,\NL)

(‘.-H.

Ill3

V151~1)l~R

corresponding values for KS,,,, atitl Kj;;LOXs. \\<‘I’(. and 3.8 x 1W3. 1.8 X 1W3. 0% x IO-“. rcspc~ctircl~.. equ~~~br~u~~~ an,d transition

duritbg

oxygena.tiotr

The quaternary transition of haemoglobin, expressed as mole fract,ion globin per mole oxygen, was calculated from the estimates in Table 2. Figure 4 illuskates this so-termed transit’ion function for data set, 5 P,-Gri concentrations. A switch-over point, 7bS, may be expressed as the haems oxygenated when the probabilit’ies of finding t’he molecule in the and the oxy quaternary conformations are equal, i.e. the value of n that, integral of the transition-function in equal par&.

of hacmoat different, number of deoxy and divides th(a

[P,-Gr,]

I

0

0.25

0.50

0.75

FIG. 4. The quaternary deoxy-to-my transition function (mol influenced by Pz-Gri concentration; 0,2.5, 5 and 10 mm01 1-l.

I-0

fraction

of Hb/mol

0, bound)

a~

This parameter was significantly affected by pH, pCO,, and P,-Gri activity and ranged between 1.8 and 2.6, for the experimental conditions studied. At [P,-Gri] = 0 mmol l-l, the variation with pH( 7.0 to 7.3) was from 2.2 to 1.9, and at [P,-Gri] = 9.3 mmol l-l, it was from 2.6 to 2.5, all values at pC0, = 22 mm Hg. The effects of pC0, on rbS values and its variation with pH and P,-Gri concentration are shown in Figure 5.

6. Discussion (a) Parameter (i) Parameters

estimates and comparison

with estimates

from

other studies

related to oxygen binding

The final analyses were performed with the assumption that K,,, is not affected by changes in pH, pC0, and P,-Gri concentration. This assumption was considered plausible as (l), the “best” estimates of K,,, in

ANALYSIS

OF

OXYGEN

BINDING

DATA

515

1

-0,

I

710

I

7.1

mmI-

~---

7.2

L--I

7.3

7.4

PH

FIG. 5. The change in switch-over dioxide pressuw wrsus pH at two (---O-G-).

point, P,-Gri

d 91, for a change from concentrations: 0 (-a---)

22 to 77 mm Hg in carbon and 9.3 mmol I- 1

the first phase of the analysis clustered in a narrow range, 0.71 f 0.05 mm Hg-‘; and (2), results from other groups support the same assumption (Tyuma et al., 1971, 1973a,b; Imai, 1974; Imai & Yonetani, 1975a; Kilmartin et al., 1975). The oxygen binding constant for a subunit when the tetramer is in the deoxy quaternary state, Kdeoxy, was quite well-determined with best values between 4.4 x 10m3 and 6.6 x 10m3 mm Hg-l (Table 2) giving Koxu/Kdeoxy ratios between 110 and 160. These estimates are significantly different from previous estimates for dilute solutions. Compared to previous values for whole blood (at’ 37”C, pH 7.4 and pC0, = 40 mm Hg), the estimates of Kdeoxy agree very well with the recent estimate of (4-O & 0.4) x 10m3 mm Hg-’ by Winslow et aE. (1977), and an estimate of (5*4&O-3) x 10m3 mm Hg-l derived from the paper by Roughton et al. (1972). The K,,, values reported by Winslow et al. (k4 = 0.032 to O-045 mm Hg-l), on the other hand, differ significantly from our value of 0.71 mm Hg - l (and the value of 2.60 mm Hg - 1 from Roughton et al., 1972). However, a graphical analysis of the data by Winslow et al. (1977) (t’heir RW2 data in Table 2), plotted in a Hill plot, reveals that their estimate of 0.039 (-& 52%) is in obvious contradiction to t,he data, and that Ko,, should rather be estimated to fall in the range 0.50 to O-70 mm Hg-l. The very precise data presented by Winslow et al. (1977) thus suggest a Koxy/Kdeoxy rat.io of between 125 and 175. It can also be noted that our results from concentrat’ed haemoglobin solutions (data sets 9 and 10) are quite close to those from the data on whole blood. The oxygen binding constant Kdeoxy showed a significant increase with pH (Table 2). This is in agreement with the results published by Imai & Yonetani (1975a). Purthermore, according to the model concepts applied, protons acting on bhe saltbridges (i.e. changing E&J should affect not only the quaternary equilibrium but also t,he oxygen affinity of subunits confined in the deoxy quarternary structure (cf. Perutz, 1976; Kilmartin et al., 1975). Thus protons act as so-called constraint effecters (increasing E& $- Eh’, in eqn (5b)). (Effector mechanisms are classified according t,o the terminology introduced by Herzfeld & Stanley (1974).) Carbon dioxide on the other hand, does not influence K,,,,, values significantlp (Table 2), i.e. does not act as a constraint’ effector. As mentioned above, we did not find it necessary to describe P,-Gri as a constraint effector in addit,ion t’o its recognized role as a quarternary effector. T,yuma et a.1

516

‘I’.

UROTH,

I,.

(:AKHY

AEl,

C’:H.

I)E

X’I~;KI~II~:lI

(1971, 1973a) working on precisely determined oxygen binding curves. observed R right shift of the lower asymptote of the Hill-plotted curves \\-h(~~ adding P,-Gri. This was interpreted to indicate that’ Kdeoxy is sensitive to Pa-&i (cf. also Perutz, 1976). Keeping in mind t,hat their data refer to highly dihued haemoglobin sohnions (0.015 mM-tetraIner), the shift could possibly be just a manifestation of the stabilizing effect, of P,-Gri on the deoxy quaternary conformation, counteract,ing the dissociation into dimers (with higher oxygen affinity) at these condit,ions. To elucidat,e this problem further, additional measurements are required on whole blood in the low sat)uration range (below 0.05) at, different’ concent’rat)ions of I’,-Gri. (ii) Parameters

related

to the quuternary

conformational

equilibrium

The allosteric constant K, ranged from 1.2 x 104 t)o 5.5 x 104, with a pronounced pH and pC0, dependence (Fig. 3). These values agree well with our K, estimate of between 3.5 x lo4 and 1.5 x lo5 from the whole blood data by Winslow et al. (1977). For comparison the values of 2 3 x lo5 and 6.7 ~10~ reported by Edelstein (1971, 1975) may also be mentioned, as estimated by two independent, methods. The first was obtained by fitting a haemoglobin oxygen binding curve with a p50 of 11 mm Hg, i.e. a value indicating conditions far from physiological conditions. The values from Herzfeld & Stanley (1974) for several data sets from the literature were not given with confidence limits, nor documented as best esbimates. With regard to the strong inter-parameter correlation their point, estimates must, by necessity, be regarded as unreliable. This, in addition t’o the complication of differences in experimental conditions makes a further comparison meaningless. Both protons and carbon dioxide were found to stabilize the quaternary deoxy conformation (Fig. 3). Following the concepts of Perutz (eqn (4b)), this figure suggests that protons act as constraint effecters (changing E&) and/or quaternary effecters (changing E,), whereas carbon dioxide most’ likely just, acts as a quaternary effector (changing E, but not Eh,; cf. above). The effect’ of carbon dioxide is negligible at pH 7.0 and increases with pH, in accordance with the idea that t’he changes are linked to formation of carbonate (cf. Perutz, 1976; Kilmartin, 1976). The allosteric constant is by definition independent of t,he concentration of P,-Gri. One way of illustrating the effect of P,-Gri on t’he quat’ernary equilibrium of the molecule is to plot (Fig. 4) the mole fract,ion of haemoglobin that, undergoes quaternary transition when one mole of oxygen binds t,o the molecule. The stabilizing effect of P,-Gri on the deoxy quaternary conformation is clearly shown in hhis Figure. Figure 5 illustrates the effect of protons and carbon dioxide on quaternary transition in the absence and presence of P,-Gri. These data generated with the use of the model seem to be in accordance with the general view concerning t,he influence of t’hese effecters at a molecular level (e.g. see Kilmartin, 1976). (iii)

Parameters related

to

P,-G’ri

binding

Confident estimates for the binding constants of P,-Gri to the two quaternary conformations, Rdeoxv and Roxy,were obtained for data sets 5, 6, 9, and 10 (Table 2) which are the sets containing data from more than two different P,-Gri concentrations (Table 1). The values for data set,s 5 and 9 are in relatively good agreement, with those obtained previously for concentrated haemoglobin solutions at t’he same pH (7.2) and temperature (37”C), using different techniques: (1) data from direct binding

ANALYSIS

OF

OXYGEN

BINDING

51;

DATA

and ultrafiltration (Garby & de Verdier 1971; Berger et al., 1973); (2) dialysis (Hama1974), see Table 3. The saki & Rose, 1974); (3) estimation from p,, values (Duhm. values refer t,o different conditions with regard to pC0, and slightly different, ionic strengths. Confidence limits have not been reported. A strict comparison is therefore not possible. The Z?dBOXy/Z&, ratios confirm previous findings that the P,-Gri affinit’y of deoxyhaemoglobin is considerably higher t’han t,hat of oxyhaemoglobin. St may be noted t’hat the ratios are significant)ly lower t)han those generally cited (see e.g. TAULE 3 lzstimutes of Bdeoxy and lx,,, from previous investigations on concentrated haemoglobin solutions (at 37°C and pH 7.2) compared with the present results

PC02 (mmHg) 0 0 0 0 22 40 40 77

rl deoxv (1 mol-‘) 1000 5000 9710 9800 3300 IL 1300 2850 5100 & 1200 2000 * 800

fLi,

(1 mol-1) 107 250 210 164 130 + 50 60 180 * 40 230 5 80

!) 20 46 60 252 14 43 2x i 9 9 i- 5

Garby & de Verdier (197 I ) Berger et al. (1973) Hamasaki & Rose (1974) Uuhm (1974) This work Duhm (1954) This work This work

Herzfeld & Stanley, 1972; Perutz, 1976). It is also of interest to note the value value for reported by Hedlund & Lovrien (1974), I?,,, = 125 1 mol -l (extrapolated high haemoglobin concentration), and t,he value of I?,,, = 120 1 mole1 from the titration data given by de Bruin et al. (1974) at pH 6% and pC0, = 0 mm Hg. The corresponding Eaeoxy values were 360 1 mol- l. and 17,000 1 mol-l, respectively. For the dat,a sets 1 to 4 and 7 to 8 the confidence limits were too wide to permit meaningful estimates of Z?,,,,, and Z?,,,. Conclusions concerning the possible effects of pH and pC0, on P,-Gri binding can thus only be based on the former four data sets. The estimates given in Table 2 indicate that protons may act as second-order quaternary effecters both in the deoxygenat’ed and oxygenated st’ates. As could be expected, carbon dioxide acts as a second-order quat.ernary effect,or only in t,he deoxygenated state (cf. also Table 3). (b) Thermodynamic

aspects of the haemoglobin

oxygenation

process

In Figure 6, ten hypothesized states of the haemoglobin molecule are ordered on a relative energy scale in units of kJ mol-‘. The free energy changes on binding of oxygen and quaternary transition (at different, sbages of oxygenation) are given in this Figure as calculated from estimates of K,, K,,,,, and K,,, for pH 7.2 and pC0, = 22 mm Hg. The KOXY/KdBOXY ratio of 130 corresponds bo a free energy difference of 12.5 kJ/mol haem (the free energy of haem-haem interaction). Figure 6 gives a quantitative overall view of the thermodynamics of haemoglobin oxygenation. The relative probabilities of the ten stat.es were computed for the same conditions and are shown in Figure 7. Figures 6 and 7 indicate that (1) quaternary transition most likely takes place after oxygenation of t,he second subunit, and (2) t,hc probabilities of intermediary states are low.

Deoxy

state

Oxy

CT)

,f

state

CR)

[dddd]’

[oddd]’ -33.9

[ooddlD

I

;--

-0.2

!

[oooojD 1 -+.p \I

[oodd]’

[ oood]’

-33.9

' [oooo]

0

FIG. 6. Quantitative overview of the thermodynamics of haemoglobin oxygenation. The 10 hypothesized states of the molecule are ordered on a relative energy scale in units of kJ mol-’ for data set 5. The symbols d and o indicate the tertiary conformations of subunits (deoxy and oxy. respectively) while D and 0 indicate the deoxy and oxy quaternary conformations of the molecule, respectively.

Deoxy I.0

0.5

OXY 0.5 -,..,.,

0

I.0

o-

I 0

FIG. states O-75 data

7. The diagram of haemoglobin

and I.00 correspond

I 0.25

shows the probabilities, at 5 different levels

with an average to pH 7.2; pC0,

number of = 22 mm

I 0.50

expressed of haemoglobin

I 0.75

as mol fraction, oxygenation:

0, 1, 2, 3 and 4 subunits Hg (data set 5).

I l-00

for oxidized,

the L’,,

:

10 hypothesized 0, 0.25, 0.50,

respectively.

The

ANALYSIS

OF

OXYGEN

BINDING

DATA

519

The “switch-over” point, n,, was found to range between 1.8 and 2.6. Estimates reported in the literature are in the range 2.0 to 3.4 (Herzfeld & Stanley, 1972,1974; Bansil et al., 1974; Imai, 1973; Huestis and Raftery, 1972; Gibson & Parkhurst, 1968; Salhany et al., 1972; Hopfield et al., 1972; MacQuarrie & Gibson, 1972; Tyuma et al.. 1973a; Caldwell & Nagel, 1973). The variation may be ascribed to differences in experiment,al conditions and the numerical estimation procedures applied. Binding of P,-Gri preferentially to the deoxy state will decrease the deoxy energy levels relative t,o the energy levels of the corresponding oxy states by an amount, RTln(l? cmxylkw) (= 8.5 kJ mol-l for data set 5), moving the quat$ernary switchover t,o a later stage of the oxygenation process (Fig. 8; cf. Fig. 4). Deoxy

FIG. 8. The influence quaternary state will RT 8.5

In &wl~ox,. kJ molI,

date

(T)

Oxy

state

(R)

of Pz-Gri on the energy levels. Preferential binding of P,-Gri to the deoxy decrease the energy levels of the 5 different hypothesized substates by For data set 5, illustrated previously in Figs 6 and 7 the energy decrease is

The energy parameters E,, E& and Eh’, (for a definit’ion see Theory section (b)) cannot be estimated independently, but only as the sums E, + E& (eqn (4)) and h& + E$ (eqn (5b)), unless specific assumptions are made about the relation bet’ween two of the three parameters, or unless one is known from independent investigations. At pH 7.2 and pC0, = 22 mm Hg, the following estimates were obtained: E, -IL’& = 24.8 kJ mol-1 and E& + E& = 50,4 kJ mol- l. The bond energy per saltbridge has been estimated to be between 1 and 2 kcal mol- l (Perutz, 1970). Assuming that the deoxy quaternary conformation is stabilized by six salt-bridges (cf. Perutz, 1970), the corresponding constraint energy E& may reasonably be estimated to be between 25.2 and 50.4 kJ mol-l. The lower limit’ gives an estimate of (1) t#he maximal constraint energy due to “wrong” conformation, Ett;,, equal to 25.2 kJ mol-I, and of (2) the difference in energy between the two quaternary conformations exclusive of molecular const,raints, E,, equal t,o -0.4 kJ mol-l. The upper limit of Eb, gives Eh’, = 0, and E, = -25.6 kJ mol-l. These rough calculations show that our parameter estimates are consistent with Perutz’s est,imates on bond energy of stabilizing salt,-bridges. The lower value suggests that the difference in total conformational energy between the deoxy and oxy quat,ernary stat,es (in t#he absence of ligand) is

520

‘I’.

GHO’I’H.

1,.

GAlCl
:\?!I,

(‘.-H.

l)E

\~lCKI~II~:I~

almost exclusively confined to molecular conAraint,s and ow(‘s v(r). littlr to t hs difference in quaternary conformational energy. This relation between ,FJ,,and f$,,, seems rather plausible as the bond energy of salt-bridges could be expected to be larger than the types of bonds lumped in E,. Values of E& approaching bhc upper limit, on the other hand, seem less probable as Eh’, by definition should be greator than zero. (c) C0ncludin.g

remarks

The two-quaternary-state allosteric model for haemoglobin co-operat’ivity, as primarily formulated by Monod et al. (1965) and modified by others (e.g. Herzfeld & Stanley, 1972; Ogata & McConnel, 1972) has proved to be consistent with t’he major observations on haemoglobin (cf. Edelstein, 1975), and especially so when extended to allow the deoxy quaternary structure to exhibit) various oxygen affinities (lmai et al., 1975). Despite the fact that it is an over-simplification of the real system, this “twoquaternary-st)ate-multi-affinity model” may thus be regarded as a useful firsta pproximation for the description of the haemoglobin oxygenation process as reflected in oxygen binding data, With regard to the more detailed molecular processes, we applied the established concepts proposed by Perutz, namely that: the two quaternary conformations can accommodate different tertiary ones; subunits contained in the unconstrained oxy quaternary form have an oxygen affinity like that of isolated subunits, whereas subunits in the deoxy quaternary form have a lower oxygen affinity due to molecular e0nstraint.s; t,he changes in t,ertiary structure by so-called induced-fit oxygen binding (Koshland et al., 1966) alter the allosteric equilibrium in favour of the high affinity state. It is remarkable that the present analysis of oxygen dissociation curves generates results in such consistency with detailed mechanistic concepts within this framework regarding: the influence of protons, carbon dioxide and P,-Gri on quaternary equilibrium (Figs 3 to 5 and 8); the magnitude of energy changes related t’o rupture of stabilizing bonds during oxygenation in the absence and presence of I’,-Gri. These findings increase the credibility of the model used and the values of estimated parameters. The inherent redundancy of some of the parameters, K, and K,,,, was resolved by setting the K,,, p arameter to a constant value at, all pH, pC0, and P,-Gri concentration values. The values of 1 x 10’ to 6 x 104 for t)he allosteric constant, ratios of 110 to 160, and the XdeOXYIRDXY ratios between 25 and 30 at the K,XYI&eOXY physiological conditions, are significantly different from estimates on diluted haemoglobin solutions. In the present study it has not. seemed worthwhile to use more detailed models, e.g. to introduce parameters related to the known difference in oxygen binding of a and /3-subunits (cf. Ogata & McConnel, 1972; de Verdier et al., 1973; Herzfeld & Stanley, 1974). This type of overspecification will just add to the problem of estimat)ing unique parameter values, unless these new parameters can be established by independent investigations. Whether or not P,-Gri should be described as a consbraint effector (cf. Herzfeld & Stanley, 1974) in addition to its role as a quaternary effector, is a matter of judgement of model compatibility (cf. Results section (a)). Data indicating that the K deoxv parameter is sensitive to P,-Gri, made Perutz (1976) suggest a mechanism where “uptake of oxygen by the quaternary dooxy structure would entail t’he sequential loosening of all its constraining hydrogen bonds, including those made by

ANALYSIS

OF

OXYGEN

BINDING

DATA

52 1

DPGt, and would normally be accompanied by a sequential rise in oxygen affinity of all the haems even before any change in quaternary structure has occurred”. Although presently available data on this matter (Tyuma et al., 1971,1973a) may have an alternative explanation (see above), a sequential change of oxygen affinity seems plausible as such. However, our data on whole blood and concentrated haemoglobin solut)ions do not require further decomposition of the K,,,,, parameter. The technical assistance of Mr TorbjBrn Haglund in the computa.tional work is gratcfldly acknowledged. One of t,he aut,hors (T. G.) gratefully acknowledges the receipt of a visit)ing scientist, ftallowship (University of Odense) from the Danish Medical Research Council. A part of t,he work was supported by the Swedish Medical Research Council (projects 13X-152 and 19X-547). A preliminary report of this work was given at the VIIIth International Berlin Symposium on St~ructure and Function of Erythrocytcs, Berlin, August, 1976.

REFERENCES Ackers,

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as P,-Gri

in this

paper.

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C:HOTH,

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CARRY

ANJ)

C.-H.

I)E

VEKl)l15K

from

Kernohan, J. C. & Roughton, F. J. W. (1972). L’roceedinys the @red lknzorc Sywposium I 1. on Oxygen Afinity of Hemoglobin and Red Cell Acid Base Status (Rorth, RI. & Astrup, I’., eds), pp. 54-64, Munksgaard, Copenhagen and Acaclrmic t’rrss, N(tw, York. Kilmartin, J. V. (1976). &it. llfed. Bull. 32, 2099212. Kilmartin, J. V., Imai, K. & Jones, R. T. (1975). In Erythrocyte Stnlcture and /+‘ctrcctiol,, Prog. C&z. Biol. Res. (Brewer, G. J., cd.), vol. 1, p. 21, Liss, New York. Koshland, D. E., Nemethy, G. & Filmer, D. (1966). Biochemistry, 5, 365 ~38.5. MacQuarrie, R. & Gibson, Q. H. (1972). J. Biol. C/rem. 247, 5686-5694. Metzler, C., Elfring, G. & McEven, A. (1974). TTsers Manual, Research Biostatistics, ‘lk? Upjohn Co., Kalamazoo. Minton, A. P., (1974). Science, 184, 577-579. Minton, A. P. & Imai, K. (1974). Proc. Nat. Acad. Sci., U.S.A. 71, 1418-1421. Monod, J., Wyman, J. & Changeux, J.-P. (1965). J. Mol. Biol. 12, 88118. Ogata, R. T. & McConnel, H. M. (1972). Proc. Nat. Acad. Sci., U.S.A. 69, 335-339. Perutz, M. F. (1970). Nature (London), 228, 726-739. Perutz, M. F. (1972). Nature (London), 237, 495-499. Perutz, M. F. (1976). Brit. Med. Bull. 32, 195-208. Roughton, F. J. W., De Land, E. C., Kernohan, J. C. & Severinghaus, J. W. (1972). Proceedings from the Alfred Benson Symposium on Oxygen Afinity of Hemoglobin and Red Cell Acid Base Status (Rorth, M. & Astrup, P., eds), Munksgaard, Copenhagen and Academic Press, New York. Salhany, J. M., Mathers, D. H. & Eliot, R. S. (1972). J. Biol. Chem. 247, 6985-6990. Severinghaus, J. W., Roughton, F. J. W. & Bradley, A. F. (1972). Proceedings from the Alfred Benzon Symposium IV on Oxygen Afinity of Hemoglobin and Red Cell Acid Base Status (Rorth, M. & Astrup, P., eds), Munksgaard, Copenhagen and Academic Press, New York. Tyuma, I., Shimizu, K. & Imai, K. (1971). Biochem. Biophys. Res. Commun. 43, 423-428. Tyuma, I., Imai, K. & Shimizu, K. (1973a). Biochemistry, 12, 1491.-1498. Tyuma, J., Kamigawara, Y. dz Imai, K. (1973b). Biochim. Biophys. Acta, 310, 317-320. Winslow, R. M., Swenberg, M.-L , Berger, R. L., Shrager, R. I., Luzzana, M., Samaja, M. & Rossi-Bernardi, L. (1977). J. Biol. Chem. 252, 2331-2337.