Use of models in understanding the mechanism of action of haemoglobin

Use of models in understanding the mechanism of action of haemoglobin F. Borras-Cuesta European Molecular Biology Laboratory, Grenoble Outstation, c/o ILL, 156 X, 38042 Grenoble Cedex, France (Received 5 July 1982; revised 29 October 1982) Models with three,four and eight salt-bridges have been used to study the mechanism of action of haemoglobin. Both side chains forming a salt-bridge, i.e. the proton acceptor and the proton donor, are postulated to change pK on ligation of oxygen. The eight salt-bridge model is able to predict, as a unified theory, both the degree of oxygenation and the Bohr effect at any pH and Po2 value; this has not been done by any other published model. The predicted pK values for the Bohr groups correspond well with those measured experimentally. This model predicts the pK values of those side chains responsible for the acid Bohr effect, suggesting that these correspond to the proton acceptors of the salt-bridges. The model also fulfils the condition of linearity between the fractional degree of oxygenation and fractional number of protons released. It is postulated that there is a gradual change in structure on going from deoxy to oxyhaemoglobin, due to the rupture of salt-bridges. The path followed during this process will be both pH and po z dependent. A formula describing the number of intact or broken salt-bridges as a function of pH and po 2 was developed. This formula shows that the fractional number of broken salt-bridges reaches a minimum value of O.2 at around pH 6.3 in the absence of oxygen. However, if oxygen is added, this .fractional number approaches 1.0 soon after the partial pressure of oxygen goes above 40 mm Hg.

Keywords: Haemoglobin; Bohr effect; conformational changes; oxygenation; proton acceptor; proton donor; salt bridges

Introduction X-ray crystallographic studies ~-8 have demonstrated that there are structural differences between the ligated and unligated forms of haemoglobin. It is clear from these studies that unligated haemoglobin is stabilized by a number of salt-bridge interactions between certain amino acid side chains. These interactions would play a significant role in the cooperative binding of oxygen to haemoglobin 6. This theory prompted a number of experiments 9-24 which were designed to measure the relative contribution to the Bohr effect of those side chains that are involved in salt-bridge formation according to the X-ray model 6. The Bohr effect is the number of protons per mole of haemoglobin that are taken up or released when going from the deoxy to the oxy form of haemoglobin. The pH region in which there is an uptake of protons is called the acid Bohr effect, while the pH region in which there is release of protons is called the basic Bohr effect. The most recent studies 21,23 favour the existence of at least eight salt-bridges (due to symmetry, four groups of two) which are responsible for the alkaline Bohr effect. The exact nature and number of those groups responsible for the acid Bohr effect is not yet clear 23. Many theories 25-42 have been put forward to account for the cooperative binding of oxygen to haemoglob'~n. However, none of these theories has been able to reproduce the experimental observations of both the fractional degree of oxygenation and the Bohr effect as a function of both pH and Po,. Furthermore, they have not been able to predict to a good approximation the pK values of identified Bohr groups. It is the purpose of this work to show that the use of an eight salt-bridge model 0141 8130/83/020066 17503.00 O 1983 Bunerworth & Co. (Publishers) Ltd

66

Int. J. Biol. Macromol., 1983, Vol 5, April

can explain, on a unified basis, both the oxygenation curves and the Bohr effect at any pH and Po2 value; this has not been done by any other published model. Besides being compatible with the published experimental evidence, this model is successful in predicting to a good approximation the pK values of well identified Bohr groups. It also predicts the pK values of those side chains responsible for the acid Bohr effect, suggesting that they correspond to the proton acceptors of the salt-bridges.

Theory General considerations

Models were sought which would explain the following experimental data: (i) the Bohr effect; (ii) the degree of oxygenation at any pH and Po, value; (iii) the linear relationship between the degree of oxygenation and the fractional number of protons released; (iv) the difference in conformation between oxy and deoxyhaemoglobin; (v) the reported pK values for the identified Bohr groups. Although recent work 23 suggests that eight salt-bridges would account for the alkaline Bohr effect, models with three, four and eight salt-bridges were evaluated. Preliminary calculations showed that a minimum number of three salt-bridges would be necessary to account for the Bohr effect and the oxygenation curves of haemoglobin. It was thought that models with fewer than eight saltbridges should be tried in order to rule out simpler interpretations of the mechanism of haemoglobin action. If only the model with eight salt-bridges could account for points (i) to (v) above, the ideas from the X-ray model 6 would then be reinforced by an independent approach.

Models for understanding haemoglobin: F. Borras-Cuesta pH D

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Figure 1 Postulated equilibria during oxygenation of haemoglobin (Model 1; four salt-bridges). Ligation of one molecule of oxygen brings about the rupture of one salt-bridge. The (+) and ( - ) signs represent, respectively, the proton donor and the proton acceptor that participate in the formation of the salt-bridges. (o) Indicates that the group is in the neutral form. When a salt-bridge is closed, a straight line joins the (+) and ( ~ ) signs: If the salt-bridge is open either by the effect ofpH or oxygenation, no tine joins these signs. The different haemoglobin species are represented by a large circle (A to LL) and oxygen by a small filled circle. Although not explicitly shown, there is a release of one proton at every step, on going from left to right of the diagram. Acid-base constants are represented by (K) and oxygenation constants by (k). See text for a more detailed explanation

The basic ideas used in the formulation of these models are as follows. (a) The change in oxygen affinity is due to conformational changes of the haemoglobin molecule induced either by p H or by Po~ or by both. (b) These conformational changes are associated with the formation or rupture of salt-bridges. (c) When a salt-bridge is broken by oxygenation, the values of p K of the groups involved in the salt-bridge change. Thus the value of p K of the proton acceptor increases while that of the proton donor decreases. (d) A change in p H of the media can induce either saltbridge formation or rupture. If the proton acceptor is a carboxyl group and the proton donor an alpha amino group, the salt-bridge will only exist when the carboxyl and the alpha amino are in the C O O - and N H 3 forms, respectively. On going from low p H to high pH, saltbridges will be formed with the appearance of C O O - and be broken with the disappearance of NH~'. (e) The uptake of one mole of O z per m o l e of haemoglobin can induce the rupture of one or more ;altbridges.

Model I: four salt-bridges ]Let us consider a model of haemoglobin with four saltbridges. In the absence of oxygen, there will be a number of species A' to E in acid-base equilibria (see the top line of Figure 1). The relative proportion of these species will depend on the values of p K of the groups and the p H of the media. F r o m left to right, there is a release' of one proton at every step. Each species can take up oxygen

with greater or lesser difficulty depending on its structure, i.e. whether the salt-bridges are closed or open. F r o m top to bottom, one molecule of oxygen is bound at each step. The haemoglobin molecule is represented by the large circle and the salt-bridges by two lines protru.ding from the circle. When the lines are joined, they represent an intact salt-bridge; the lines are left open when the saltbridge is not formed as a result of p H or oxygenation state. The charges of both partners of every salt-bridge are indicated; note that the charge of the proton acceptor changes from (0) to ( - ) and the" of the proton donor from ( + ) to (0) with increasing pH, from left to right in Figure 1. A smal r filled circle is used to indicate the presence of a molecule of oxygen. Acid base constants are indicated by (K) and oxygenation constants by (k). The rupture of a salt-bridge by oxygen brings about a discreet structural change. This in turn induces pK changes in both the proton acceptor and the proton donor forming the salt-bridge, due to the different environment in which the groups are exposed in the new structural state. In this way a proton is released or taken up in a manner dependent on the relative p K values of the group in the two conformationai states. F r o m this model it is clear that the oxygenation path will be p H dependent since the concentration of some o f the species will approach zero at certain p H values. For instance at high pH, the five species in the last column on t h e right in Figure 1 will be involved in oxygenation, and the.Bohr effect will be nil due to the absence of salt-bridges. The same is true at acidic p H for the five species in the first column. On the basis of the Law of Mass Action, using the

Int. J. Biol. Macromol., 1983, Vol 5, April

67

Models for understanding haemoglobin: F. Borras-Cuesta constants of Figure 1, expressions for the Bohr effect for the degree of oxygenation and for the number of broken or intact salt-bridges can be deduced. Because of its relative simplicity, Model I will be used to illustrate the deduction of the formulae. The detailed equations for both Model I and Model II are given in the Appendix.

Bohr effect The Bohr effect has been defined 44 as the number of protons per mole of haemoglobin that are taken up or released in the deoxy to oxyhaemoglobin transition. This corresponds to subtracting the titration curve of oxyhaemoglobin from that of deoxyhaemoglobin, and thus obtaining the charge difference between the two states of haemoglobin. At acid pH values up to pH 6.0 this difference is negative and is known as the acid Bohr effect. Above pH 6.0 the difference is positive and is known as the alkaline Bohr effect. For reasons that will become apparent below, this definition of the Bohr effect is referred to as the extreme Bohr effect in the present study. A more general treatment of this phenomenon is now given. Let us define the function Z (pH, Po2) which describes the fractional number of protons released by the given population of haemoglobin species during oxygenation.

[9] + 2[h] + 3[i] + 4[j] + 5[k] + 6[11 + 7Ira] + 8[n] 8(If] + [9] + [hi + [i] + [j] + [k] + Eli + [m] + [n])

(1)

where If]; [g]; [h]; [i]; [j]; [k]; [/]; Era]; and [n] represent the concentration of haemoglobin species which have lost 0, 1,2, 3, 4, 5, 6, 7 and 8 protons, respectively. For instance, [f] represents the sum of concentrations of the species of the first column of Figure 1 (A'; F'; M'; R' and X'). It can be shown that: [/] = [A'](1 + klox + kloklzx 2 + klokl2kl4X 3

[h]

If]' U 3

.

.

Fractional degree of oxygenation The fractional degree of oxygenation (the average number of molecules of oxygen bound per mole of haem) can be described by a function Y (pH; x), where x is the partial pressure of oxygen. [H,] +2[H2] + 3[H3] +4[H4] 4([H o] + [H 1] + [H2] + [H s] + [H4])

where [Ho], [H1], [H2], [H3] and [H4] represent the concentration of those haemoglobin species having taken up 0, 1, 2, 3 and 4 molecules of oxygen, respectively. For instance [H0] represents the sum of the concentrations of those species of the top line of Figure 1 (A', B', C', D;, A, B, C, D and E). It can be shown that: [Ho] = [A'](10- 8pn+ K1210-TpH + K11K 1210-6pH -{-

K1K9KloKllK1210-3pH+K1K2K9KloKllK12 lO-2pn+

where x represents the partial pressure of oxygen. In a similar manner, expressions for [9], [h] . . . . , In] can be obtained. Since all these sums have [A'] as a common factor, it is possible to cancel [A'] by taking the ratios:

K1KeK3K9KloKlIKlelO-Pn+ K1K2K3K4K9KIoK11K12) and in general:

[n]

bq

[ H o ] = [A']fo(PH )

Then Z (pH; x) becomes:

[H,] = [A']fltpH)x

Z (pH; x)=

[H2] = [A']f2(pH)x 2

-[hi

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[j] _ [k] , [/] +[m]. [n]~

[H4] = [A'].f4(pH)x4

if] I - 2 ~ - + 3 ~~ ] + 4 ~ + 3 ~ + 6 ~ + / ~ ] + 8 + E?-l-* V E3*

The expressions for [9]/[f]... [n]/[f] as a function ofpH and x can be found in the Appendix. A new general expression for the Bohr effect can be defined by the function AH + (pH; x~; x2) where xa and x 2 represent partial pressures of oxygen:

68

.

KloKllKx210-sPn+K9KloKllK1210-4pn+

+ k lOk12k14k16X4)

[9].

(2)

This equation accounts for the number of protons taken up or released by the haemoglobin molecule when going from one partial pressure of oxygen to another. This equation could be defined as the function representing a generalized Bohr effect. In the special case when x 1~ and x 2 = 0 the extreme Bohr effect (the number of protons taken up or released when going from deoxy to oxyhaemoglobin) is obtained. It can be shown that, when x 1~ , the ratios [9]/[f] [n]/[f] will depend on the pH and only those acid-base constants which describe the equilibria between the species that are fully saturated with oxygen (those of the bottom row of Figure 1). Also, when x 2 =0, these ratios will depend on the pH and on only those acid-base constants that describe the equilibria between the species that do not contain oxygen (the top row of Figure 1: see Appendix).

Y(pH; x)=

Z (pH; P o ) =

D]

AH+(pH; x,; xz)=8[Z(pH; x~)-Z(pH; x2)]

Int. J. Biol. Macromol., 1983, Vol 5, April

Since all the sums have [A'] as a common factor, it is possible to cancel [A'] by taking the ratios: [Hi] ' [Ho]-'

[H2] " [H3] " [H,] [Ho]' [Ho]' [Ho]

Models for understanding haemoglobin: F. Borras-Cuesta This leads tO: Y(pH; x)= fl(PH)

. 9f2(PH)x2,-f3(P H) 3, .f4(P H) 4 *

*

4(1 +f~(pH)

So(-Tffn3

(3)

. f2(pH) 2 •f3(P H) 3 . f4(P H) 4"~

\

+So-rffm

)

The ratios fx(pH)/fo(PH) to f4(PH)/fo(PH) can be interpreted as functions describing the pH dependence of the Adair coefficients. The expressions forfo(PH) tof4(pH ) are given in the Appendix.

Number of broken salt-bridges Let us define the functions W(pH; x) and R(pH; x) =I-W which represent, respectively, the average fraction of broken or intact salt-bridges during oxygenation. W(pH; x)=

[Bx]+2[B2]+3[B3]+4[B'] 4([B0] + [B,] + [B2] + [B3] + [B#])

(4)

where x is the partial pressure of oxygen and [B0], [B1], [B2], [Bs] and [B4] represent the concentration of haemoglobin species having 0, 1, 2, 3 and 4 salt-bridges broken, respectively. From Figure 1:

[Bo]

= [A]

[BI] = [D'] + [F] + [B] + [G]

[BE] = [C'] + [ K'] + [ M ] + [C] + [J] + [ N ] + [iq] [ B 3 ] = [B'] -F [J'] + [P'] + [R] + [S] + [T] + [ D ] +

[K] + [P] + [U] [B4] = [A'] + [F'] + [M'] + [R'] + [X'] + [G'] + [N'] + IS'] + [Y'] + IN'] + IT'] + [Z'] + [U'] [W'] + [X] + EY] + [z] + [w] + ELL] + [E] + [L] + [Q] + [V] Since [B0] to [B,] can be expressed in terms of [A'], [A'] can be cancelled out without affecting the value of the function W(pH; x). The expressions for [Bo], [B1] , [B2] , [B3] and FB4] are given in the Appendix.

Model II: eight salt-bridges A model based on the formation and rupture of eight salt-bridges is shown in Figure 2, using the same nomenclature as for Figure 1. Because of the symmetry of the haemoglobin molecule, the salt-bridges can be subdivided into four sets of two equivalent salt-bridges. In this model, the uptake of one mole of 0 2 brings about the rupture of two salt-bridges at every step of oxygenation. Equations for the Bohr effect, the fractional degree of oxygenation and the number of broken salt-bridges can be obtained in a manner similar to that for the model with four salt-bridges. In Model II statistical factors taking into account the number of equivalent species are

considered. The nomenclature is exemplified as follows. 1A° means that there is only one species (prefix 1) containing no oxygen (upper subscript zero) and from which no protons have ionized (lower subscript zero). 2A~ means there are two equivalent haemoglobin species containing one molecule of oxygen and from which eight protons have ionized. The species A ° represents deoxyhaemoglobin containing all the eight salt-bridges responsible for the Bohr effect. In going from A ° to A °, eight protons belonging to the proton acceptors have ionized. As a result eight saltbridges are formed due to the interaction between the negatively charged proton acceptors and the positively charged proton donors. In going from A ° to A76 another eight protons dissociate, but this time belonging to the proton donors. This brings abour the rupture of the eight salt-bridges in question, this being due to the absence of the positive charges of the proton donors needed for the interactions with the proton acceptors. When a molecule of oxygen is taken up by A °, A~ is formed. This brings about the simultaneous rupture of two different bridges, illustrated in A~ of Figure 2 by the absence of two full li:_esjoining the plus and minus signs. A proton ionizes from A~ to A~ at a lower pH than from A ° to A °. This is because the breaking of the salt-bridges by oxygen in going from A ° to A~ changes the chemical environment of the proton acceptor and the proton donor. Thus pKll is lower than pK 5. There are two possible species of A ° because of symmetry considerations, even though only one is shown in Figure 2, i.e. a proton can ionize from two equivalent sites. Ionization of both protons from these two equivalent sites leads to only one species of A°o . All statistical factors together with the detailed equations for Model II are shown in the Appendix.

Results and discussion

Degree of oxygenation Two main sets of experimental data for the degree of oxygenation were used for the calculations, namely that of Ferry and Green 43 for horse haemoglobin and that of Imai and Yonetani 34 for human haemJglobin. For the extreme Bohr effect of horse haemoglobin, the data of German and W3. -~an44 were used. Eight pairs of pK values (four pairs corresponding to proton acceptors and four pairs to proton donors) were chosen in the following way. It was considered that the pK value of a proton acceptor is reduced when a salt-bridge is formed and increased when the salt-bridge is broken. By contrast, the pK value of a proton donor is reduced when the salt-bridge is broken and increased when the saltbridge is formed. The pK values of donors and acceptors increase from left to right in both Figure 1 and Figure 2. Preliminary calculations soon showed that in order to predict with reasonable approximation the extreme Bohr effect published by German and Wym~in44, the pK values for the proton acceptors had to be taken around the range 4.9-6.5, and for the proton donors around 6.5-8.5. The actual values depending of course on the model used to fit the experimental points. Once a set of eight pairs of pK values for Model I and another set of eight pairs for Model II had been found, their values were used to calculate the oxygenation

Int. J. Biol. Macromol., 1983, Vol 5, April

69

Models for understanding haemoglobin: F. Borras-Cuesta II

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70

In this way klo, k12, k~+ and k16 of Model I and kl, k2, k 3 and k 4 of Model II could be estimated. Once the oxygenation constants had been estimated, the degree of oxygenation at any p H and Po~ value (for either human or horse haemoglobin) could be calculated. This procedure was repeated for both models till sets of values for K~ to g]6 could be found that gave the best fit (as judged by graphical comparison) of the experimental oxygenation curves of human or horse haemoglobin. Differences of the order of 0.05 pK units between the pK values of the groups in theligated and unligated forms, had an effect on the quality of the fit. It was also found that for a given group of oxygenation curves within a certain pH range, the best simultaneous fit was obtained by only one combination of p K values. High p K values (those of the proton donors) mainly affected the oxygenation curves above pH 6.5 while lower pK values (those of the proton acceptors) mainly affected those curves below pH 6.5. In Figures 3a and 3b, a comparison is made between experimental (dots) and calculated (full lines) degree of oxygenation for horse and human haemoglobin, respectively. The expressions used for the calculations were those of Model II. An equally satisfactory fit can be obtained using those of Model I. This shows that, although the fit for the oxygenation curves is almost

4- 0

+ / \1

++~) -t-

constants (k 1o, k12, kl 4 and k 16 of Model I or kt, k2, k 3 and k 4 of Model II) by the following procedure. It was found that the set of Adair coefficients calculated by Imai and Yonetani 34 to fit the oxygenation curve of human haemoglobin at pH 7.2 can also predict very accurately the pH 7.65 oxygenation curve of horse haemoglobin reported by Ferry and Green 43. The numerical values of these coefficients calculated by Imai and Yonetani 34, together with the values of K~ to Kt6 calculated above (Bohr effect fit) were substituted into the expressions of Model I and Model II which describe the p H dependence of the Adair coefficients, that is:

+

+ ' 1 " + ~ o *'+

Int. J. Biol. Macromol., 1983, Vol 5, April

°W

Figure 2 Postulated equilibria during oxygenation of haemoglobin (Model II; 8 salt-bridges). Ligation of one molecule of oxygen brings about the simultaneous rupture of two saltbridges. The (+) and ( - ) signs represent, respectively, the proton donor and the proton acceptor that participate in the formation of the salt-bridges. (o) indicates that the group is in the neutral form. When a salt-bridge is closed, a straight line joins the (+) and ( - ) signs. If the salt-bridge is open either by the effect of pH or oxygenation, no line joins these signs. The different haemoglobin species are represented by a large circle (Ao° to A)6 ) and oxygen by a small filled circle. Although not explicitly shown, there is release of one proton at every step, on going from left to right of the diagram. Since this model takes into account the symmetry of the haemoglobin molecule, a given species from Ao° to A46 might occur more than once. For instance, there are two equivalent species for A ° because a proton can ionize from two equivalent sites but only one for A°o because there is only one possible species that has lost two protons each coming from an equivalent site with an ionization constant of K s. In practical terms this means that statistical factors have to be used when developing the formulae for Model II in order to weigh the relative contribution of each species. These factors are given in the Appendix. Acid-base constants are represented by (K) and oxygenation constants by (k). See text for a more detailed explanation

Models for understanding haemoglobin: F. Borras-Cuesta ÷ pH= • pH= ÷ pH= D pH= . pH= a pH= pH= 9 pH=

0.~

0.6

!

0.4

I

8.88 8.40 7.95 7.65 7.38 7.00 6.77 6.50

0.2 20

40 60 poz(mm Hg)

80

a

pH= 8,80 opH= 8.40 ~pH= 8.00 =pH= 7.65 • pH= 7.40 • pH= 7.20 • pH= 7.00 ~pH= 6.80 pH= 6,70 ~pH= 6.50

0.8

0,(~

0.4

0.2

If

20

40

60

80

po 2 (ram H9)

Figure 3 Fractional degree of oxygenation (Y) as a function of the partial pressure of oxygen (Po2) at different pH values. Comparison between experimental (dots) and calculated values (full lines) using equation (3') of Model II. (a) Horse haemoglobin, experimental data taken from Ref 43. At pH 7.65; fl(pH)/fo(PH)=O.1152; f2(PH)/fo(PH)=O.O08126; f3(pH)/fo(PH)=O.O0037359; fa(PH)/fo(PH)=O.O0082938 and PK1=4.80; pK2=5.025; PK3=5.525; PK4=6.00; pK5=6.8; PK6=7.50; PK7=8.05; PK8=8.30; PK9=4.90; PK1o=5.275; pKll =6.60; PK12=6.80; PK13= 5.775; PK14= 6.10; pK 15= 7.35; pK 16 8.10. (b) Human haemoglobin, experimental data taken from Ref 34. At pH 7.20; fl(pH)/fo(PH) = 0.1152; f2(PH)/fo(PH)= 0.008126; f3(pH)/Jo(PH)= 0.00037359; f4(PH)/Jo(PH)=O.O0082938 and pK] =4.60; pK 2=4.825; pK3=5.325; pK4=5.70; pK5=6.75; PK6=7.45; pKT=8.00; PK8 = 8.05; PK9 =4.70; pK10 = 5.075; pKl~ =6.375; PK~2=6.775; PK13=5.575; pK14=5.90; PK15=7.125; PK16 = 7.90. See text for a more detailed explanation of the method of calculation =

insensitive to the model, it is nevertheless dependent on the relative pK values of the Bohr groups within a given model. Since experimental pK values of well identified Bohr groups are available, a model can be selected or rejected on the basis of how well these experimental values correspond to those predicted by the model.

In Table I a comparison is made between the pK values of certain identified Bohr groups, and those predicted by Model I, Model II and another model where only three salt-bridges have been considered..The predicted pK values are arbitrarily assigned to a given side chain on the basis of the similarity of the pK. It can be seen that the pK values predicted by Model II are quite compatible with those reported in the literature ~°-14'1s'2°'22 The correspondence is even better when the ApK values between ligated and unligated haemoglobin are compared. It is interesting to note that these pK and ApK values were obtained as a result of the fit of the oxygenation curves and thus can be taken as a test of the predictive power of Model II. If the assignment of pK values made in Table 1 is accepted, Model II then predicts that the uptake of the first molecule of oxygen would bring about the simultaneous rupture of those bridges where His 122at and Val 1~ are involved. This is followed by those of His 146fl and another residue that has pK values of 8.05 and 7.9 in the unligated and ligated forms, respectively.

General Bohr effect Since in the present models, the oxygenation constants are linked to the acid-base constants, the values of k 1, k 2, k3, k4 and K 1 to K16 will suffice to calculate all the remaining oxygenation constants of Model II. The numerical values of all these constants allow the calculation of the Bohr effect at any pH and Po, value. Using the expression of Model II for the general Bohr effect shown in the Appendix, calculations for horse and human haemoglobin were carried out. The results are shown in Figures 4a and 4b, respectively. These two figures clearly illustrate the meaning of the extreme Bohr effect. Indeed, as soon as the partial pressure of oxygen is increased, the number of protons taken up or released on going from deoxy to oxyhaemoglobin tends to a limit. Most authors 45'46 use the formula of Wyman 25 for calculation of the extreme Bohr effect. This formula is based on a model which is a limiting case where only two 'oxygen-linked' groups per haem are considered. It is now clear 2a that at least eight salt-bridges (four sets of two) are required to account for the alkaline Bohr effect alone, and so Wyman's hypothesis is no longer valid. It could still be said that the extreme Bohr effect could be calculated by taking the sum of the individual contributions of the different Bohr groups, as it has been done by others 13. It can be readily shown that this reasoning is incorrect and that the formula to be used will be model dependent. A formula analogous to the one proposed by Wyman can be obtained, using similar considerations to the ones used to develop either Model I or Model II, but in which only one salt-bridge is involved. However, if there is more than one salt-bridge, the final formula is no longer the algebraic sum of the individual contributions. Let us take for instance, the equilibria between species A, B, F, G of Model I (see Figure 1) where K 1 and K 5 are the acid base constants of a side chain (forming a salt-bridge) in the unligated and ligated form, respectively. From first principles it follows that the function Z(pH; Po2) describing the average number of protons released during ligation is:

[B] + [G] Z(pH, P o ) = [A] + [B] + [F] + [G]

Int. J. Biol. Macromol., 1983, Vol 5, April

71

Models for understandin9 haemoglobin: F. Borras-Cuesta Table I Comparison between the reported pK values of identified Bohr groups in haemoglobin and those predicted by model calculations Predicted values, proton acceptors (acid Bohr effect) Model 3 salt-bridges Horse

Model I 4 salt-bridges Horse Human

Model II 8 salt-bridges Horse Human

U L ApK

4.90 5.10 0.20

4.70 (K12) 4.90 (K13) 0.20

4.80 4.90 0.10

4.60 (K 0 4.70 (K9) 0.10

U 4.90(K9) L 5.30 (K12) ApK 0.40

4.975 5.425 0.45

4.70(K 11) 5.20(K14) 0.50

5.025 5.275 0.25

4.825 (K2) 5.075 (K10) 0.25

U 5.10(Ks) L 5.90(Kxl) ApK 0.80

5.375 5.825 0.45

5.30(Kx0) 5.80(K15) 0.50

5.525 5.775 0.25

5.325 (K3) 5.575 (K13) 0.25

U 5.70(K7) L 6.10 (Kl0) ApK 0.40

5.70 5.90 0.20

5.65 (K9) 5.95 (KI6) 0.30

6.00 6.10 0.10

5.70 (K4) 5.90 (K14) 0.20

Experimental values, proton acceptors (acid Bohr effect)

No experimental values available

Predicted values, proton donors (alkaline Bohr effect) Model 3 salt-bridges

Model I 4 salt-bridges

Horse

Model II 8 salt-bridges

Horse

Human

Horse

Human

U 7.50(K 1) L 6.50 (K 4) ApK 1.00

7.15 6.55 0.60

7.55(K1) 6.60 (K 5) 0.95

6.80 6.60 0.20

6.75 (Ks) 6.375 (Kll) 0.375

Experimental values, proton donors (alkaline Bohr effect) (a) 6.60 (a) 6.10 His 122~ 0.50

U 8.40(K2) L 6.50 (K 5) ApK 1.90

7.90 6.60 1.30

7.85(K2) 6.65 (K6) 1.20

7.50 6.80 0.70

7.45 (g6) 6.775 (K 12) 0.675

(b) 7.79_+0.10 (c) 7.83_+0.19 (d) 8.00_+0.20 (e) 7.60 (b) 6.95_+0.13 (c) 7.16___0.36 (d) 7.25_+0.05 (e) 7.10 Val 1~ 0.84 0.67 0.75 0.50

U 8.40(K3) L 7.00 (K6) ApK 1.40

8.30 7.00 1.30

7.95(K3) 6.75 (K7) 1.20

8.05 7.35 0.70

8.00 (K.7) 7.125 (K15) 0.875

(0 8.10 (f) 7.00 1.10

8.40 7.70 0.70

8.05 (K4) 7.35 (Ks) 0.70

8.30 8.10 0.20

8.05 (Ks) 7.90 (KI6) 0.15

No experimental values available

U L ApK

(g) 8.20 (g) 7.00 1.20

(e) 8.00 (e) 7.10 0.90

(h) 8.08 (h) 7.14 His146fl 0.94

The terms K 1to K 16which are specified in each of the models, refer to the appropriate acid-base constants of the Bohr groups that are used in the model in question. L and U stand for ligated and unligated, respectively. ApK is the pK difference (in absolute value) between ligated and unligated forms. (a), (b), (c), (d), (e), (f), (g) and (h) correspond to references 15, 11, 14, 18, 10, 22, 12 and 20 of the present publication

[A] . _ [A][O2] K'[--H~+g'K5 Z(pH, P o ) =

[A]+ K

[A] [ 0 2]] ,~+k,[A][O2]+k,Ks[A][~]

K x 10pH+klK5 10PHpo~ Z(pH, Po2) = 1 + K 1 10 pn + k 1Pot + k i K 5 10PHPo~ If Po2 = 0 then Z

=

K1 10pH

po~=O I + K 1 10P" and if Po2 tends to infinity then: Z = K5 lOPtl po~-*~ 1 + K 5 lOpn

72

Int. J. Biol. Macromol., 1983, Vol 5, April

and the extreme Bohr effect will be: AH+=(

Z po2-'* ~6

-

Z) po~=0

10PH(10-pK, - 10-PK,) (1 + 10 -pK, 10Pn)(1 + 10-or,10 pH) which is analogous to the expression deduced by W y m a n 2 s. However, if this formula is applied for every set o f p K groups (K1, Ks; K2, K6; Ks, KT; K4, Ks; K9, K16; Klo, K15; KI~, K14 and K12 , K13 ) and the algebraic sum of these contributions is taken, the result will not correspond to the expression for the extreme Bohr effect deduced for Model I. The same is of course true if Model II is considered. In Figure 5, a comparison is made between the extreme Bohr effect predicted by Model II

Models for understanding haemoglobin: F. Borras-Cuesta

Li6 -14

99

- I0

N

99

-2.4

- 1.6

99 -.8

I~

-0

99

...... 4.5

-1 . . . . . . . . . . 5.0

I" . . . . . . . . . . 5.5

T. . . . . . . 6.0

T ....... 6.5

7.5

pH

| ........

~ ..........

| .............

T ..... 7.0

8.0

8.5 -16

-14

99

.12

- I0

-8

N

f

12"4 -6

_4

"61>

L. =o

-2

~-0

.8

1

I

4.5

|

5.0

I

5.5

I

6.0

!

I

6.5

7,0

I

I

7.5

8.0

I

I

8.5

pH

Figure 4 General Bohr effect and average number of protons released by haemoglobin as a function of pH at different partial pressures of oxygen. (a) Data for horse haemoglobin, Values o f p o ~(mm Hg): 0; 2; 4; 6; 8; 10; 12; 15; 20; 30; 99. (b) Data for human haemoglobin. Values ofpo (mm Hg): 0; 1; 2; 3; 4; 5; 6; 8; 10; 12; 15; 20; 9q. A, The average number of protons released (Z) by haemoglobin as a function of pH at di~erent partial pressures of oxygen, as predicted by Model II. B, The general Bohr effect. The number of protons (AH ÷) released or taken up as a function of pH as predicted by model II. These curves were obtained by subtracting the values of Z at a given Po_, from the value of Z when Po~ is zero, i.e. when x 1 = x 1 and x 2 = 0

Int. J. Biol• M a c r o m o l . , 1983, Vol 5, A p r i l

73

Models for understanding haemoglobin: F. Borras-Cuesta

2.0

1.6

l

1.2

I I

I

iI

I

ii

0.8 + -r

<] 0.4

0.0 L

I

I

iiIII1~/ -0.4

-0.8

4.8

5.6

6.4

7.2

8.0

8.8

pH

Figure 5 The extreme Bohr effect for horse haemoglobin predicted by Model II (full line). This is compared with the experimental values (/x) reported by German and Wyman 44 and with the calculated values using the method of adding the individual contributions of every salt-bridge (---). In both methods of calculation, the acid-base constants of Figure 3a were used. See text for a detailed explanation

and the one calculated using the algebraic sum of the individual contributions of every set of salt-bridges. The constants used were the same for both calculations. It is clear that both methods give quite different results. This should be taken into account when predicting the pK values of an unidentified salt-bridge. For instance, in Ref. 15 it is postulated that the pK values of the 'missing' alkaline Bohr group should be around pK 7, while Model II predicts values around pK 8 instead, i.e. pK values of 8.05 and 7.9 for unligated and ligated human haemoglobin, respectively, or pK 8.30 and 8.10 for horse haemoglobin (Table 1). A good fit to the extreme Bohr effect can be obtained using either method of calculation, provided that the acid-base constants are chosen adequately. However, the Bohr effect calculated using the formula of Model II predicts pK values that are closer to the experimentally measured pK values of well identified Bohr groups. It has been suggested 23 that about 50% of the acid Bohr effect is accounted for by His 143/3. In Model II it is postulated that the acid Bohr effect is due to pK differences between eight side chains (four sets of two) in ligated and unligated haemoglobin that act as the proton acceptor, that is the more negatively charged component of the salt-bridge. The hypothesis that a structural change is likely to affect the pK values of both partners in the saltbridge can be substantiated as follows. (i) In the case of the alkaline Bohr effect, the pK value of the proton donor decreases upon ligation because this

74

Int. J. Biol. Macromol., 1983, Vol 5, April

is further away from the proton acceptor. Since charge neutralization by a more negatively charged group (the proton acceptor) is removed, the proton will ionize more easily, i.e. at a lower pH value. (ii) In the case of the proton acceptor, ligation will have the opposite effect because the group will be further away from the positively charged group. It will be easier for the first to pick up a proton, i.e. the pK value will increase. Because of the reasons explained in (i) and (ii) the pK values of the proton acceptor and the proton donor change in opposite directions upon ligation (this giving rise to the so called acid and basic Bohr effects, respectively) due to their mutual influence. Once a bridge is broken, the pK values of both partners change, due to the disappearance of their mutual interaction and the exposure to new environments. If an analogy is made between Model II and the finding that His 143fl is responsible for about 50% of the acid Bohr effect13, then in terms of Model II His 143fl would have to be taken as a proton acceptor in a salt-bridge. This raises the question as to which side chain would be the proton donor of that salt-bridge. It has been reported 23 that the replacement of Lys 82//or Lys 144//by a neutral residue is responsible for a diminished alkaline Bohr effect and also for a diminished acid Bohr effect. Since these residues are close to His 143/3, it appears possible that one or both Lys residues might act as the proton donor of His 143/3. Because both the proton acceptor and the proton donor of every salt-bridge change pK on ligation, Model II postulates a total number of 16 protons involved during the transition deoxy-oxyhaemoglobin; this figure is close to the 18 'allosterically responsive hydrogens' measured by difference hydrogen exchange 41'42. It has been known for some time 45 that the Bohr effect is ionic strength dependent. More recently48'49 it has been suggested that chloride ions bind differently to oxy and deoxyhaemoglobin and are thus responsible for some proportion of the Bohr effect. It has also been said that salt-bridges would not be the dominant energetic factor in stabilizing the deoxy quaternary structure of haemoglobin 24. This was suggested on the basis of the relative effect of chloride ion on the dimer-tetramer equilibrium constants of human oxy and deoxyhaemoglobins. Model II does not consider the effect of dissociation of haemoglobin itno ~-/3 dimers on the oxygen equilibrium parametersS°'51; this was neglected in view of the experimental evidence 51 that between 60 #M haemoglobin (as haem) and 600/~M haemoglobin there is no significant difference in the equilibrium parameters. Since the data 34'43'44 used to develop Model II were obtained at constant or similar ionic strength, no allowance was made for the effect of this parameter on the Bohr effect. In future work, this parameter could be incorporated in the model. However, Model II as such makes a number of predictions (in particular the pK values of well identified Bohr groups) that no other model has yet been able to make.

Number of broken or intact salt-bridges Although calculations using equation (4') predict that no measurable amount of salt-bridges (responsible for the Bohr effect) are present in oxyhaemoglobin, the case for deoxyhaemoglobin is different. Here the number of intact bridges reaches a maximum at around pH 6.3 and is

Models for understanding haemoglobin: F. Borras-Cuesta fractional saturation and fractional number of protons released, as predicted by Models I and II. A function T (pH =constant; x) can be defined which predicts the fractional number of protons released during oxygenation at constant pH.

0.8

T(pH = constant; x) =

0.6

Z(pH = constant; x 1) -- Z(pH = constant; x 2 = 0) Z(pH =constant; xl) (5)

0.4

If the values of T, calculated using equation (5) for Model I (or the equivalent one for Model II) are plotted against the fractional degree of oxygenation Y[equation (3) for Model I and (3') for Model.II] it can be seen that the condition of linearity is fulfilled to a good approximation by Model II (see Figure 7a and 7b) and also by Model I (not shown).

0.2

A

a

D 0.8

0,6

0.4

0.2 A

b I 4

I

I 5

I

I 6

I

I 7

I

I 8

I

I 9

oH

Figure 6 The fractional number of broken salt-bridges(W) as a

function of pH at different partial pressures of oxygen (A, 0; B, 15; C, 20 and D, 40 mm Hg). (a) Horse haemoglobin(b) Human haemoglobin strongly pH dependent (see Figures 6a and 6b). It is interesting to note that this pH, predicted by the theory as the one where there is maximum preservation of the deoxy structure, is close to pH 6.5 where human deoxyhaemoglobin is normally crystallized. Further, above pH 6.5, other crystal forms 2 tend to predominate.

Linearity between degree of oxygenation and number of protons released Antonini et al.47 observed that at pH values of 6.97, 7.04, 7.14, 7.88 and 7.93 there is a linear relationship between the fractional saturation of the haemoglobin molecule with carbon monoxide and the number of protons released. These authors quote that this proportionality also exists in the case of oxygen, and attributed it to the invariance in shape with pH of the oxygen equilibrium curve of haemoglobin. The careful measurements of Imai and Yonetani 34 showed that the shape is pH dependent over a wide range of pH values, and thus indicated that the linear relationship between fractional saturation and number of protons released can no longer be attributed to a shape invariance with pH. This linear relationship is derived from experimental fact, and is therefore a necessary constraint to be satisfied by any model. Let us study the relationship between

Other models In addition to the models shown in Figures 1 and 2, another model was postulated which had only three saltbridges. Although the oxygenation curves could be predicted accurately using this model, a substantial deviation from the experimental extreme Bohr effect and the linear relationship between degree of oxygenation and fractional number of protons released was observed. This shows that the predictions are model dependent, which reinforces the value of Model II. It could be argued that models with numbers of salt-bridges other than eight might satisfy the experimental data. This is unlikely. If a lower number of bridges are considered, it is clear that the lower the number of bridges, the bigger the ApK values would become to account for the oxygenation curves (see Table 1). The inverse is also true, that is, the higher the number of salt-bridges the smaller the ApK values. This shows that a minimum of eight bridges is necessary. The possibility of a number higher than eight can only be ruled out on the basis that the extra bridges would be associated with very low ApK values between the ligated and unligated forms, and so, in practical terms they could be considered to be averaged out by Model II. Conclusions

A model based on the existence of four salt-bridges can quantiatively account for the Bohr effect; the degree of oxygenation and the. linear relationship between fractional saturation with oxygen and fractional number of protons released. However, the predicted pK and ApK values for the alkaline Bohr groups do not correspond as well with those reported in the literature (Table 1). The eight salt-bridges model (four sets of two equivalent bridges) not only predicts the Bohr effect, the fractional degree of oxygenation, and the linear relationship between degree of oxygenation and the fractional number of protons released, but also the predicted pK and ApK values for the alkaline Bohr groups agree well with those determined experimentally. This last model predicts a gradual change in structure between deoxy and oxyhaemoglobin due to the sequential rupture of the saltbridges that stabilize the deoxy structure. Calculations show that no measurable amount of saltbridges (responsible for the Bohr effect) is present in oxyhaemoglobin. However, in the absence of oxygen, the

Int. J. Bi01. Macromol., 1983, Vol 5, April

75

Models for understanding haemoglobin: F. Borras-Cuesta

References 1 2 0.8-

• •

3

!

0.6-

VV~&

•

0.4-

•

4 5 6 7 8 9 10

•

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11 12 13 14

/

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i

15 16

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17

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19 20

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21 22 o•

23

0.4 &•

0.2

~

v

24 25 26

o

27

~ 1

•

v

I

I

0.2

I

J

0.4

I 0.6

I

1

b

t b

0.8

Y

Figure 7 The relationship between the fractional degree of oxygenation (Y) and the fractional number of protons released (T). (a) data for hor3e haemoglobin: A, pH 6.5; v , pH 6.77; .., pH 7.0; o, pH 7.38; o, pH 7.65; ~, pH 7.95; + , pH 8.4 and ~7, pH 8.88. (b) data for human haemoglobin: A, pH 6.5; o, pH 6.7; A, pH 6.8; .., pH 7.0; o, pH 7.20; xT, pH 7.4; + , pH 7.65; o, pH 8.0 and v , pH 8.4. The lines correspond to a theoretical relationship of l:l between (Y) and (T)

28 29 30 31 32 33 34 35 36 37 38 39 40 41

number of intact salt-bridge reaches a maximum of about 80~o at around pH 6.3 and is strongly pH dependent. This shows that deoxyhaemoglobin always exists as a mixture of species having different numbers of intact salt-bridges, and not as one species with a fixed total of intact saltbridges.

42 43 44 45 46 47 48

Acknowledgements The author is very grateful to Dr S. Cusack, Dr D. Tochetti and F. Borras-Herrera for computer programs, and also to Dr S. J. Perkins, Dr J. C. Jesior and J. M. Bois for help with the final draft.

76

Int. J. Biol. M a c r o m o l . , 1983, V o l 5, A p r i l

49 50 51 52

Perutz, M. F. J. Mol. Biol. 1965, 13, 646 Muirhead, H., Cox, J. M., Mazzarella, L., and Perutz, M. F. J. Mol. Biol. 1967, 28, 117 Bolton, W., Cox, J. M., and Perutz, M. F. J. Mol. Biol. 1968, 33, 283 Perutz, M. F., Muirhead, H., Cox, J. M., and Goaman, L. C. G. Nature 1968, 219, 131 Perutz, M. F., Muirhead, H., Mazzarella, L., Crowther, R. A., Greer, J., and Kilmartin, J. V. Nature 1969, 222, 1240 Perutz, M. F. Nature 1970, 228, 726 Perutz, M. F. Br. Med. Bull 1976, 32, 195 Perutz, M. F. Proc. R. Soc. Lord. (B) 1980, 208, 135 Kilmartin, J. V. and Rossi-Bernardi Nature 1969, 222, 1243 Kilmartin, J. V., Breen, J. J., Roberts, G. C. K., and Ho, C. Proc. Natl. Acad. Sci. USA 1973, 70, 1246 Garner, M.H.,Bogardt, R . A . , G u r d , F . R . N . J . Biol. Chem. 1975

250, 4398 Ohe, M., and Kajita, A. J. Biochem. Jpn 1977, 81, 431 Kilmartin, J. V. T I B S 1977, November 2, 247 Matthew, J. B., Morrow, J. S., Wittebort, R. J., and Gurd, F. R. N. J. Biol. Chem. 1977, 252, 2234 Nishikura, K. Biochem. J. 1978, 173, 651 Matthew, J. B., Hanania, G. I. H., and Gurd, F. R. N. Biochemistry 1979, 18, 1919 Matthew, J. B., Hanania, G. I. H., and Gurd, F. R. N. Biochemistry 1979, 18, 1928 Van Beek, G. G. M., Zuiderweg, E. R. P., and De Bruin, S. H. Eur. J. Biochem 1978, 92, 309 VanBeek, G . G . M . , Z u i d e r w e g , E . R . P . , a n d D e B r u i n , S.H. Eur. J. Biochem. 1979, 99, 379 Russu, I. M., Tseng Ho, N., and Chien Ho Biochemistry 1980, 19,

1043 Kilmartin, J. V., Fogg, J. H., and Perutz, M. F. Biochemistry 1980, 19, 3189 Ohe, M., and Kajita, A. Biochemistry 1980, 19, 4443 Perutz, M. F., Kilmartin, J. V., Nishikura, K., Fogg, J. H., Butler, P. J. G., and Rollema, H. S. J. Mol. Biol. 1980, 138, 649 Chu, A. H., and Ackers, G. K. J. Biol. Chem. 1981, 256, 1199 Wyman, J. Adv. Prof. Chem. 1964, 19, 223 Monod, J., Wyman, J., and Changeux, J. P. J. Mol. Biol. 1965, 12, 88 Koshland, D. E., Nemethy, G., and Filmer, D. Biochemistry 1966, 5, 365 Sarof, H. A., and Yap, W. T. Biopolymers 1972, 11, 957 Ogata, R. T., and McConnell, H. M. Proe. Natl. Acad. Sci. USA 1972, 69, 335 Szabo, A., and Karplus, M. J. Mol. Biol. 1972, 72, 163 Hopfield, J. J. J. Mol. Biol. 1973, 77, 207 Imai, K. Biochemistry 1973, 12, 798 Herzfeld, J., and Stanley, E. J. Mol. Biol. 1974, 82, 231 Imai, K., and Yonetani, T. J. Biol. Chem. 1975, 250, 2227 Baldwin, J. M. Prog. Biophys. Mol. Biol. 1975, 29, 225 Baldwin, J. M. British Med. Bull. 1976, 32, 213 Baldwin, J. M., and Chothia, C. J. Mol. Biol. 1979, 129, 175 Imaizumi, K., and Tyuma, I. J. Biochem. 1979, 86, 1829 Imai, K. J. Mol. Biol. 1979, 133, 233 Imai, K., Ikeda-Saito, M., Yamamoto, H., and Yonetani, T. J. Mol. Biol. 1980, 138, 635 Liem, R. K. H., Calhoum, D. B., Englander, J. J., and Englander, S. W. J. Biol. Chem. 1980, 255, 10687 Malin, E. L., and Englander, S. W. J. Biol. Chem. 1980, 255, 10695 Ferry, M. R., and Green, A. A. J. Biol. Chem. 1929, 81, 175 German, B., and Wyman, Jr. J. J. Biol. Chem. 1937, 117, 533 Antonini, E., Wyman, Jr., J., Rossi-Fanelli, A., and Caputo, A. J. Biol. Chem. 1962, 237, 2773 Rossi-Bernardi, L., and Roughton, F. J. W. J. Biol. Chem. 1967, 242, 784 Antonini, E., Wyman, Jr. J., Brunori, M., Bucci, E., Fronticelli, C., Rossi-Fanelli, A. J. Biol. Chem. 1963, 238, 2950 Rollema, H. S. H., De Bruin, S., Jansen, H. M. L., and Van Os, A. J. G. J. Biol. Chem. 1975, 250, 1333 Van Beek, G. M. G., Zuiderweg, R. P. E., and De Bruin, H. S. Eur. J. Biochem. 1979, 99, 379 Ackers, G. K., Johnson, M. L., Mills, F. C., Halvorson, H. R., and Shapiro, S. Biochemistry 1975, 14, 5128 Johnson, M. L. and Ackers, G. K. Biophys. Chem. 1977, 7, 77 Imai, K. and Yonetani, T. Biochim. Biophys. Acta 1977, 490, 164

.....I

1::/.

'O

<

~D oo

O

l

~

~

~

1

6

l + k2x + k2kax2 + k2kaksx3 + k2k3k5k'Tx4 \l +k~o-o~k,okt2x2 +k,ok,2kl,sX3 +k,ok,2k,4k,6x'* f

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+k

;

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1

6

x

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Z

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Bohr effect

KI3

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kI - k

k2=klo K5 K13 K,K,2 K14 k3=k12~

J

10PH

Kl5 ks=k'4~Olo

k6= ,4K~3 Kxo

k K7 K5

(2)

(1)

KI6 k7 = k ' 6 ~ - 9

9

k =k KaK16 16~

if x--,

if x = 0

[m]/[f] =KsK6K7KIsK14K15K~6 107pH

[/]/[f] = KsK6KI3KlaK15K16 l06pH

[k]/[.f] = KsK 13K1,,K1 sK16 105prt

[j]/[f]=K13KIaK15K16 104pH

[i]/[f]=K13K14KI5 103pH

[n]/f] = KsK6KTKaK 13KI4KI 5K16 108pH

8

= K1 a

[h]/[f]=Kx3K14 102pH

[gJ/[f]

[n]/[f] =KIK2K3KaK9KloKIIKI2 108pH

[rail[f] = K 1K2KsK9KloK 11KI 2 107pH

[I]/U]=KIK2K9KIoKIIK12 106vH

[k]/[f]=KIK9K1oKlIK12 105pH

Z - Z )= general Bohr effect any value x=O of x

k4=k12~2 K14 KH

AH ÷=8(

[9]/[f] + 2[h]/[f] + 3[ill[f] + 4[j]/[f] + 5[k]/[f] + 6[/]/[f] + 7[m]/[f] + 8[n]/[f] 8(1 + [g]/[f] + [h]/[f] + [i]/[f] + [ j ] / [ f ] + [k]/[f] + I / l / I f ] + [m]/[f] + [n]/[f])

The relationship between oxygenation and acid base constants

Z(pH; x)

4

l+k2x+k2kax2+k2k4k6xa+kzk'*k6k8x4

The average charge (Z) of the haemoglobin molecule as a function of pH and Po:

[n]/[.f]=glg2g3g4g9glogllgl2

107pH(

l+k2x+k2k4x2+k2k~k6xa+k2kak6k7 x'* ~l +k~ox +k~ok~2x2 +k~ok~2kxax3 ~klok~2ktak~6xdJ

. ../ l+k~x+k2kaxZ+k2k4ksx3+k2k4kskvx ~" ./ \l +k~ox +klokx2x2 +k~okx2kt4x3 +ktokl2k~4k~6x~']

105pH(

[m]/[f]=K,K2 KaK9K'oK''K'2

[I]/[f]=K~K2KgK~oK~2

[k]/[f]=K,K9K'oK, xK'2

k

[j]/[f]= K9KIoKlIKI2 104pH

X4)

+

l+ktx+klk3x2+k~ kakSxa+k~kaksk'Tx'*

4 H ff ~j]/[f]=K9KloKlIK,2 lO p ~

l

Ill~Ill = K1oK 1IK12 103pH

j

( l+klx+klk3x2+klkaksxa+klk3kskl6 x4 ~ ~ 1 6 x 4 [i]/[f] = K~oK ~1KI2 103pH\ ] + k~o~ ~

[9]/[f]=K12 10pH [hi~If] = K 11K 12 102pH

l+klx+klk12x2+kxka2k14x3+klk12k14k16 x4 ~ l ~ - - ~ 7 . - - - 41 \l+gtox+klokl2X +klok12k14x +kaok12k14k16x /

pH /

[h]/[fJ=KllK12 102pn(l + l +klx +klk3x2 +klk3klgx3 +k~kak~4k~6x~ "~ k~ox -}-klok12X2 +klokl2kl4X 3+k~ok~2k~4kt6x4/I

[9]/[f]=K,210

•

Model I

Appendix

t~

i

~...°

o

.<

oo

?-

¢')

?Y(pH; x)

*f2(P H) 2 ~f~(pH) 3 J~(pH) ,

l+fa(pH) ,fz(PH) 2 f3(pH) 3 f4 (pH) 4 fo~PH))X+f o ~ x +fo(~)) x + f o ( p ~ x

f~(pH) (3)

W(pH; x)

[B,] + 2[B2] + 3[B3] + 4[B4] 4([Bo] + [B,] + [B2] + [B3] + [B,])

R(pH; x ) = 1 - W

xakaok12kl,,kl~,[1-',-K13 10PH+KI3K14 I02pH+ KIsKI4K15 |03pH + KI3KI4K15KIe(IO4pH + K 5 105pH+KsK6 106pH+ KsK~ K 7 107pH+ KsK6KvK8 108pH)]

K3K~KsK6K9KIoKt3Ktalo8pH)+x3klok I zkl4[1 + K~ 3 10PH + K I3KI4.(102pH + K~ ~ 103pH + Ka.KsK6K7KgK I 5 108pH)] +

[B4] = 1 + K~K2K3K4KoK~oK ~1K12 108pH + xklo(1 + K1310PH + K2K3K4KsKgKIoK11Kl 3 ]08pH) ÷ x2k~oklz(l + K1310 pH + K13K14102pH ÷

xaklkI2kI4K9KI2KI4K15(lo4pH-bK5 105pH+KsK 6 106pH+KsK6Kv 10?PH)]

[B3]=KI2[lOpH + K1K2K3KgKIoKlI I07pH+ xklKI I(IO2pH+ K2K3KsK9KIo 107pH)+x2klk3KIoKlIIlO3pH + KaKsK6K9 107pH)+

[B2]=K11K12[lO2pH + K1K2K9KloIO6pH + NklKIo(IO3pH + K2KsKglO6pH) + x2klkaK9KIo(IO4pH q-K 5 I05pH+KsK6 106pH)]

[B1]= KxoKI IKlz[ IO3pH+ KIK9 I05pH+ xkIKo(IO4pH + K5 105pH)]

[Bo] = KgKIoK1 i Kl 2 104pH

(equation 4)

The fractional number of broken (W) or intact (R) sak-bridges

[Hg]=[A']f4=[A']k~ok~2k14k~6(lO-SpH + K~lO-TpH + K~aK~410 6pH + Ki3Ki~K~sIO 5pH + K~aK~#K~K~6IO 4pH + KsKI3KI4K~sKi610 3pH + KsK6Ka ~K~K~sK~610-2pH + KsK6KTK~3K~4K~sK:610-P q" KsK6K7KsK~3KagK~5Ki~)

[H3]=[A~]f~ =[A~]k~k~2k~4(l~-8pH + K~31~-7pH + K~3K~*~-6pH + Kx~K~K151~-5pH + K9K~3K~K~5l~ 4pH + KsKgK~3Ki~KasIO-apH + KsK6K9K~aK~4K~sIO-2pH + KsK6KTKgK~3K~gKisIO-PH + KgKsK6K7K9K~aK~4K~5)

[H2] = [ A ' ] f 2 = [A']k~ok~ 2(10-8pH + K1310-7pH + K ~-~g~.,10-6pH + K~oK ~3K1 ~10-5pH+ KgK~ oK ~3K~,,I 0-4pH + KsKgK~oK ~3K ~410-3pH + KsK6K9K~oK ~3K1410- 2pH + K3KsK6K9KIoK~3K~410 -pH + KaKgKsK6K9K~oKi3K N

[H~]=[A']f~ =[A']k~o(lO 8pH+ K~3~-7pH + K1~K~3l~-6pH + K ~ K ~ K ~ 3 ~ - 5 p H + K 9 K ~ K ~ K ~ 3 ~ - 4 p H + K~K~K~K~K~3l~-3pH + K2K5K9K~K~K~3l~-2pH + KzK3KsK9K~oK1 IK~310 -pH + KzKaK4KsK9K~oK l ~gi 3)

[Ho] = [ A ' ] f 0 = [A'](10 -SpH + K~ 210 -TpH + K~ ~K 1210-6pH + KioK~ l K~ 210- 5pH + KgKioK~ iK~ 210-4oH + K~K9KioK~ iK~ 210 -3pH + K~K2KgK ~oK~Ki 210 -2pH + K~K2K3KgK~oK~iK~210 Pra+ K~K2K3KgKgKioKx~Ki2)

The degree of oxygenation

Model I (continued)

E

g:a.,

t~

p-i

,¢ o

oo

o_

0

x.

[]~Ir~=(K1)2(K2)2(K3) 2 1o6pH(j "l-2k6xq-k6kloX2-t-2k6ko10k13x3-I-k6kioki3klTX4)~

rJ]Ir~=(K1)~(K2) 2 104,H(l-t-2k6x~-k6kloX2~-2k~k6kloX3~-k3k4k6kloX4)~

kaki 2kt4kiax4 );

[v]/~r] = (Ki)2(K2)2(K3)e(K4)2(Ks)2(K6)2(KT)Z(K8)21016pH(1 + 2ksx + ksk , 2x2 + 2ksk 12kl 6X3~-ksk 12k' 6k20x4) D

[t]/[~(KI)2(K2)2(K3)2(K4)2(K5)2(K6)2(KT)2 1014pHt.1-~2k8x~-k8kl2x2~-2k8kD2kl5x3~-k8kl2kl5kl9x4). ~

[s]/[J] = 2(KI)2(K2)2(K3)2(K4)2(KS)2(K6)2K7 1013pH(1q-kax + kaki 2x2+ kSklo2k' 5x3 + kSk' 2k' 5kisx4)

[r3/[.f] = (Ki)2(K2)2(K3)2(K,i)2(Ks)2(K6)21012pHC + 2kax+ k8ki2x2 + 2kskl2ki'iX3

rq]lr i] : 2(KI)3(K2)2(K3)2(K4)2(K5)2g6101lpH(.1+ k8x + k8k' ix 2+ 2k8kDiki,x3 + ksk I ,ki4kiax" );

[h]/[f] = 2(KI)2(K2)2(K3)2(K4)2K5 109PH(_1-t-kTx 4- kTk I0X2q-2k~k Iokl,,x3+ k.lk 1okl 4klsx4) D

Fm]/rf~=2(K1)I(K2)2(K3)3K4 ,07pH(1-t-k6x-t-k6kloX2-I-k6kl;k14x3-i-k6klok14ki7x4_)~ r#l]/~u~=(gl)2(K2)2(g3)2(Ka)2 108pti(l+2k6x-t-k6kloX2-t-2k6kolOkl4X3-t-k6kiokl4kl8X4)."

rk]lrf~=2(Kl)2(K2)2g3 |05pH(|-t-k6x-t-~6klox2-I-k6k~ok13x3-t-k4k6k10k13x41.~

I03PH('~-k6x-t-k6k9x2~-2kDk6k9x3q-k3k4k6k9x4).~

+ 2kix + klk2X 2 + 2k,k2k3x 3 + k l k 2 k ~ , l '

10PH(( 1~ l + ksx +k2ksx2 + 2k2kaksxa + k2k3kgksx4

[i]/[~=2(K,)2K2

[O]/[f] = 2Kl

The averagecharge (Z) of the haemoglobinmoleculeas a function of pH and po2

1g~ 2A~ 1A~ 2A~ 1g~ 2A~ ig~ 2A~ 1A~ 2A~ 1g~o 2g~l 1A~2 2A~3 IA~4 2A~s IA~6

2Ag 4A~ 2A] 4A] 2A~ 2A~ 2A~ 2A~ 2A~ 4A~ 2A~o 4g~l 2A~2 2A~3 2A~ 2A~s 2A~6

lAg 2A~ 1A] 2A] 1A~ 2A~ 1g~ 2A~ 1A~ 2A~ 1A~o EArl 1A~2 2A~3 1A~ 2A~s 1A~6

2A~ 2A~ 2A~ 2A~ 2A~ 2A~ 2A~ 2A~ 2A~ 2A~ 2Alo EArl 2A12 2A~3 2A~4 2g~s 2A~6

1A~ 2A7 1A~ 2A~ 1A~ 2A~ 1A~ 2A~ Ig~ 2A~ Ig~o 2g~l IA~2 2A~3 1A~4 2A75 1A~6

The statistical hctors a~ounting ~r the differentnumber of equivalent species ~und during oxygenation

Model II

.s-

t~

4

-[~3 -[q

[hi

[i]

[j]

[k]

[/3

~-~+

[m]

In]

~]+9~c]+

ItS]

Is]

It]

[u]

Iv]\

[n]/[f]=(K,))2(K,o)2(Kx3)Z(K,,~) 2 108P H [h]/[J] ~2(Kg)2(Klo)2(KI I)(K13)Z(KI4) 2 l09pH

[p]/[f ] =( K9)2( K j o)~(K I ,)2( K13)2( K14) 2 101°p H [q]/[f] =2(Ko)2(Klo)2(K, 1}2(K12)(K13)2(K14) 2 10 llpH [r]/[./] =(Kg)Z(Klo)2(Kl 1)2(K12)2(K13}2(gla) 2 1012Pla

[s]/[J] = 2(K9) z (K 1o)z(K ~1)z(K ~2)2(K 13)2(K14 )2 K I 5 1013pH [t]/[J] =(K9)Z(Klo)2(K11)2(KI z)2(KI3)Z(KI~)2(KI 5)2 10J4PH

[p]/[J] = (K a)Z(K z)2(K 3)Z(K ,)2(K ~)~ 10~Op~

[r]/[f]=(KI)~(K2)Z(K3)Z[Ka)Z(Ks)Z(K6) 2 ]012p tt

~ ~O~4p~

[fi]/[[-J=2(K~)2(Kz)2(K3)2(Ka)ZK~ ]09P H

[q]/[f]=2(KO2(KE)2(K3)2(KJ2(Ks)2K6 10lip/q

1013pH

[n]/[f]-~(KO2(K2)2(K3}2(K4) 2 108pH

[S]/[f] = 2(K,~)Z(K2)2(K3}2(K4)2(Ksj2(K6)ZK7

[t]/U]=tKOZ(Kz)~(K~)~(K~):(K~tg~K~I~IK~)

(2')

if x ~

if x = 0

extreme Bohr effect = AH + = 16( Z - Z ) x - ~ x=O

[v],/[ff] = (Kg)2(K 10)2(K l I)2(KI 2)2(K 13)2(K 14)21K 15)2(K16) 2 1016pH

[v]/[f]=(Kt)~{Kz)2fK3)Z(K4)Z(Ks)Z(K6)Z(KT)2(Ks) z 1016pH

10 jSpH

[m]/[f]=2(Kg)~'(Klo)2(K13)2Kl,, 107p r/

[m]/[f]=2(KlJ2(K2J2(K3J2K4 107pH

[u]/[f]=2(Ko)2(Kxo)2(Kla)2(KI2)2(K13)2(KI4)2(K15)ZK16

[I~/[f]=(Kg)2(Klo)Z(K13) 2 106pH

[r3/u]=(KI)2(K2)2(K3) 2 106P~

lO15pH

[k]/[J3=2(Kg)2(Klo)ZK1a 105PH

[k]/[.]]=2{K1)2(K2)2K3 105pH

[u]/[[]= 2(K~)2(K2)~(K3)2(K4)Z(Ks)Z(K~,I2(K~):Ks

[j]/[_f] =(K9)2(Kl o)2 10 4pH

[j]/[fJ=(K1)2(K2) 2 104p H

(2')

[i]/[f]= 2(Ko)eKlo 103P//

[g]/[f] = 2 K 9 10pH

Z - Z)=generalBohreffect any value x=f) of x

[r]

[i]/[J]=2(KoZK2 103oH

AH +=16(

AH + = 16( Z - Z ) = e x t r e m e Bohr effect x~zc x=o

[q]

[u]

U] U] U]. . _ +U] U]] - - + - - U] + - - + -U] .

[p]

• ^[P] ..[s] .4[t] 7Ira] 8[',3 ~[~] ~ u ~ + ~..[q] t ~ j + 1 2 [r] ~-3+ ~]+ ~ ] + ~

U] U] + -U] (~3+~+~+- + - - +U] - - + - - U] + - - + -U] - + - - +If] ~

16/[g]

3[F]+4[f]+)~]+0~+

[h]/DC]=(Ko) 2 102PH

10Pa

If

[.u] _[hi _[i3 .U]

Z(pH; ~c) [ f ]

[h]/[J]=(K,) 2 102pH

[O]/O3 = 2 G

The extreme Bohr effect

(v)

Model II (continued) ..[v]

O0

~D oo Lo .< O L~

O ...,

O

O

=..

KgKso k~ =K~K~ k~

KI3K14 ks,=~-s
k,5-

K13KI4KI5 K3K4K7 k3

K9K~oK~ k 7 = K~K2K ~ k s •

K9K~oK~K~2 k~

"3 k~6- K~aK~4KssKS6bKaK#KTK8

ks-- K ~ K ~ 6 6

K9 2 k19

kl 2 =

K9KloKIsK12 K~K2KsK6 k2

Ks3K~4KsSk4K3K4K7 k20- K~aK~Ks~K~6~ k~

K9K~oK ~

kl s = K~K2Ks s k2

k~8 =K~k4KI3KI4

K9Klo k~° . . . .K~K2 k2

ks7 = KI3K3k4

ko=~k

+

10 3pbI+ K3K,,KT(Ko)2(KIo)2(KII)2(Ks2)2KIsKs4Ks

~ 10-2pH+

(Ko)2(K~o)2(K ~s12(K 12)2(K ~312(KI4)2(K 15}210- 2pH + (Kg)2(KI o)2(Ks 112(K s 2)2(g s 3)2(KI 4)2(K s 5)2K 1610 - pH + (Kg)2(K s0)2(K s s)2(K ~2)2(K s312(K 14)2(K t 5)2(K 16)2

2(Kg):(Kso)2(KsO2K~2(KI3)2(Ks~,)210-SpH+(K9)2(Kso)2(Ks~)2(Ks2)2(K~3)2(Ks4)210 4pH+ 2(Kg)2(Kso)2(Kss)2(Ks2)2(KIs)2(K~4)2KI~IO-3pH +

2(Kg)2(Kso)2(Kss)2K~410-gpH +(K9)2(K~o)2(Ks3)2(Ks,,)210-SPH+ 2(Kg)2(KIo)2K~ ~(K~3)2(K s,,)210- 7pH +(Kg)2(K~o)2(Ks ~)2(K~3)2(Ks,t)210-6pH +

[H4] =[A']k~kzkak,,{lO- )6pn +2K9 10-)spn +(K9)2 10- 14pH+2(K9)2 K~ ° 10-13pH +(Kg)2(K~o)2 10-)2pH +2(Kg)2(Kto)2K~a 10- ~ pH +(KO2(Kto)2(K~ s)2 10-topn +

K3K4KT(Kg)2(KIo)2(Ksl)2(KI2)2Ks3Ks,,K~sKs~, 10 pH+ K3K4KTKs(K9)2(K~o)2(K~s)2(K~2)2(K~3Ks4KssK~61

K3K4(Kg)2(Kso)2(Kss)2(Ks2)2Ks3KI,, lO-4pH + K3K,t(Kg)2(KIo)2(Ksl)2(KI2)2Ks3Ks4Ks5

K3K4(Kg)2(Kso)2KI3Ks4 IO-spH + 2K3K4(Ko)2(Kso)2KIsKs3Ks4. IO-7pH + K3K,~.(K9)2(Klo)2(Kll)2Ks3Ks,~. 10-6p H + 2K3Ka(Ko)2(KIo)2(Ksl)2Ks2KI3KI4 10-5pH+

[H33 = [A']2ksk2ks{1 O-16pH + 2K910-15pH+ (K9)210-14pH+ 2(Kg)2Ks o l 0 - 13pH+ (K9)2(KI 0)210-12pH+ (K9)2(KIo)2KI 310-llpH+ Ks(Ko)2(Ks 0)2K1310-10pH+ Ks(Kg)2(KIo)2K~ 3Kl.*10-9pH +

2(K3)2(K4)2(K7)2K8(Kg)2(K10)2(K 1i) 2(K s2)2 10- pH + (K3)2(K~.)2(Kv)2(K8)2(Kg)2(K i o)2(K 1s)2(K s2)2}

(K3)2(K4)2(K9)2(KIo)2(KII)2(Ks2)2 IO-4pH + 2(K3)2(K4.)2K7(K9)2(Kso)2(KII)2(Ks2)2 IO-3pH +(K3)2(K4)2(KT)2(Kg)2(Klo)2(Kll)2(K12) 2 10-2pH+

(K 3)2(Ka)2(K9)2(KI0)2 10-- 8pH + 2(K 3)2(Ko,)2(Kg)Z(Kso)2K 11 l 0 - 7pH + (K3)2(K,~.)2(K9)2(Kso)2(K 11)2 10-6pH + 2(K 3)2(K4)2(K9)2(KIo)2(K10 2K 12 10- 5pH +

[H2] = [A']klk2{ 10 16pH+ 2K ° I0-15oH + (K9)2 10 - 14pH+ 2(Kg)2KI o 10-lSpH +(Kg)2(Ks 0)2 10-12pH + 2K3(K9)2(Klo)2 10 - l lpH + (Kaj2(g9)2(glo}2 10 ]0pH+ 2(K3)2K4(K9)2(Klo)2 10 -9pH +

K1K2(K3)2(K't)2K-sK6(KO2K8K°Ks°KI 1K12 10-PH + KIK2(K3)2(K't)2KsK6(KT)2(Ks)2KgKsoK1 sKs 2}

K1K2(Ks)2(K4)2KsK6KaK10K1 sK12 I0--4pH + K1K2(K3)2(K4)2KsK6KTKqKIoK1 sKi 2 10-3pH +K1K2(Ks)2(K,t)2KsK6(KT)2K9KsoKI 1K12 10-2pH +

K sKz(Ks)2(K4)2KgKIo l0 8pH + K1K2(K3)2(K4)2KoKz oK I s 10- 7pH + K1K2(K3)2(K4)2KsKgKIoK~I 10_6p H + KI K2(K3)2(K4)2K~K9KioK 1l K ~2 10-5pH +

[ H 1 ] = [A']2k s { 10-16pH + K910- 15pH+ K1K910-14pH + K 1K9Ksol0-13pH + Ks K2K9Klol O- 12pH+ K1K2K3K9K1010 - l lpH + Ks K2(KB)2K9K1010-10pH + K1K2(Ka)2K4K9K1010_9pH

2(K I)2(K2)2(K3)2(K4)2(Ks)2(K6)2(KT)2K8| 0 -pH + (K1)2(K2)2(K3)2(K4)2(Ks)2(K6)2(KT)2(K8)2}

(K1)2(K2)2(K3)2(Ka)2(Ks)2(K6)2 10-4p H +2(KI)2(K2)2(KB)2(K4)2(Ks)2(K6)2K7 10-3p H +(K1)2(K2)2(KB)2(K4)2(Ks)2(K6)2(KT)2 10-2p H +

(KI)2(K2)2(K3)2(K,,)2 10-8p H +2(KI)2(K2)2(Ka)2(K4)2K5 10-7p H +(K1)2(K2)2(Ka)2(K4)2(Ks)2 10-6p H +2(K1)E(K2)2(K3)2(K4)2(Ks)2K6 10-5p H +

[Ho]=[A']{10-16pH+2K1 10-15pH+(K1) 2 10-14pH+2(K1)2K2 IO-13pH+(Ks)2(K2) 2 lO-12pH +2(K1)2(K2)2K3 10 IlpH+(KI)2(Kz)2(K3) 2 lo-lOpH+2(KI)2(K2)2(K3)2K4 10-9pH+

The degree of oxygenation

KI3 k , 3 = ~ 3 k3

K9 k5 = K~ k~

The relationship between oxygenation and acid base constants

r,

q~ t~

.¢ o.

o~

0 0

o

W(pH; x) =

3

f~(pH) ,,~

2[B/] + 4[B~] + 6[B6] + 8[Bs] 8([Bo] + [B2] + [B4] + [B6] + [Bs] )

~<+ i o ~ :< +so57mx )

fz(PH) 2 f3 (pH)

~o~<+lo~

4(1 - L ( p H )

f~(pH) - - x + 2 - - x -fz (pH) 2, .fs(P H) 3 .f~(pH) ,, Y(pH: x)= f°(PH) f°(PH) ±'~f°(pH)X + 4 f ° ( ~ x

(4')

(Y)

~K~ 2109PHI +

2(Kt I)2(K 12)2K~ s 105r'n +(Kt 1)2(K 12)2(K 15)2106ptt +2(Ki 1)2(KI2)2(K15)ZK16107pH +(KI ~)2(K12)2(Kts)Z(Kx6)210sptll]

(K°)2(KI°)2(KI3)2106pH +2(K'~)2(KIo)2(Kt 3)2K~4107pH +(Kg)Z(K~o)Z(Ka3)2(K~,t)2108PHIaI+2KlllO pH+(K~ I)2102P~ +2(K~ 02K~ 210~pH+(K1 02(K 12)2104pH+

(Kt t)2(K~2)2KI3Ka4K~51OI4PHI I + Kte lOPH+ KsKt6102pH I + k~k2ksk,,x4[l + 2KglOPtt +(Kg)2102pt~ + 2(Ko)2 KiolOSpH +(Kg)2(K~o)elOaPt~+ 2(Kg)2(K~o)ZK~slOSPH+

(Kg) 2102pH_~c2(K9)2K l o I03pH -'F(Kq)2(K lo) 2104pIt + (Kg)Z(K I o)2K 1310~PH + K 3(Kg)2(K i o)2K 13106pH + K3(Kg)2(K 1o)2K 13I'( 14107pH ] + 2kl k2k3 X3K 3K4K ?(K9)Z(K 10)2

2KB(Ko)2(KIo)2105pH +(K3)2(Ko)2(K~o)2106pHI +klk2x2(Ks)2(K4)2(Kv)2(K,~)2(K,o)2(K~,)2(K,2)21OHpH{1 +2KsI0PH+(Ks)2102pH~ + 2k,k2ksx3{1 +2Kgl0PH+

2klxK1K2(K s)2(K4JZKsK6(KTJ2K9KIoKIIKI21Ot'tPHCtl + KslOPH +(Ks)2102pH I + klk2x21~l + 2KolOpH +(Kg)2102pH + 2(K9)2KIolO3pH +(K9)2(Klo)2104pH +

[883 = 1 +2K, IOpH+(Kt)2102pH+(KI)2~K2)2(K3)2(K4.)2(Ks)2(K6)2(Kv)21OI4ptI{I + 2K810pH +(Ks)2102pH} +2kix{ 1 +K91OpH+KIK9102pH+KIK9KtolO3pH } +

2klk2x2(K3)2K4"(K9)2(KI°)ZlOTpHI~I+ K4KT(K~ 02(K12)2106pHI2k~k2k3x3K3K4(K9)2(KIo)2KI 3KI4108pH{ 1 + 2Kl 110PH +(K 11)2102pH +2(K 1I)2K 12103pH+ (K11)2(KI2)2104pH+{KII)2(KI2)2KIsIO5pHI

[B 6] =2(K 02K210SPH{1 + Kz(K3)2(K4)=IKs)2(K6J2KTIOtOpHI +2k~xK,K2KgK,olO4pHII +K31OPH+(K3)2102pH+(Ks)2(K4)2KsK6KTK,

k~k2x2(K3)2(K4)Z(K9)2(Kio) 2 108pH{I+2K~ 10PH+(KI~) 2 102pH+2(K,~)2KI2 103pH+(K~)2(KI2) 2 104PH}

2k~xK~K2(K3)2K,tKgK~o 107pH[I + K4KsK~ 103pH+ K4KsK~K m 104pH+ K,tKsK6K~K~2 105pHI +

[B4] = (Kx}2(K2j 2 104P~{ 1 + 2K 3 10pH + (K3) 2 102oH+ (Ky(K4)2(Ks) 2 106pH + 2(K3}2(K,)2(Ks)2K~ 107pH + (K3)2(K,,)2(Ks)2(K6) 2 108pH} +

[B2] =2(KO2(K2)2(K3)2K4 107pH{1 +K~K~ 102pH) +2k~xK,K2(K3)Z(K~,)2KoKIo 108pH{I +K~I 10pn}

[B o] =(K~)g(K2)2(K3)2(K,t) ~ 108pH

The fractional number of broken salt-bridges (W)

Model II (continued)

¢b ~t

-.

5"

e.,