Oxygen binding and its allosteric control in hemoglobin of the primitive branchiopod crustacean Triops cancriformis Ralph Pirow 1 , Nadja Hellmann 2 and Roy E. Weber 3 1 Institute of Zoophysiology, University of Mu ¨ nster, Germany 2 Institute of Molecular Biophysics, Johannes Gutenberg University of Mainz, Germany 3 Zoophysiology, Institute of Biological Sciences, University of Aarhus, Denmark The Branchiopoda are an ancient, primitive and, except for the Cladocera, conservative group of crusta- ceans [1]. The earliest known representatives were mar- ine and occurred % 500 million years ago in the Upper Cambrian [2]. Present-day branchiopods are predomin- antly freshwater animals, and the fossil records indi- cate that marine branchiopods invaded freshwater habitats early in evolution. Two of the four extant Keywords allosteric control; Crustacea; hemoglobin; oxygen binding Correspondence R. Pirow, Institute of Zoophysiology, Hindenburgplatz 55, University of Mu ¨ nster, D-48143 Mu ¨ nster, Germany Fax: +49 251 8323876 Tel: +49 251 8323858 E-mail: pirow@uni-muenster.de (Received 19 January 2007, revised 13 April 2007, accepted 8 May 2007) doi:10.1111/j.1742-4658.2007.05871.x Branchiopod crustaceans are endowed with extracellular, high-molecular- mass hemoglobins (Hbs), the functional and allosteric properties of which have largely remained obscure. The Hb of the phylogenetically ancient Tri- ops cancriformis (Notostraca) revealed moderate oxygen affinity, coopera- tivity and pH dependence (Bohr effect) coefficients: P 50 ¼ 13.3 mmHg, n 50 ¼ 2.3, and u ¼ )0.18, at 20 °C and pH 7.44 in Tris buffer. The in vivo hemolymph pH was 7.52. Bivalent cations increased oxygen affinity, Mg 2+ exerting a greater effect than Ca 2+ . Analysis of cooperative oxygen binding in terms of the nested Monod–Wyman–Changeux (MWC) model revealed an allosteric unit of four oxygen-binding sites and functional coupling of two to three allosteric units. The predicted 2 · 4 and 3 · 4 nested struc- tures are in accord with stoichiometric models of the quarternary structure. The allosteric control mechanism of protons comprises a left shift of the upper asymptote of extended Hill plots which is ascribable to the displace- ment of the equilibrium between (at least) two high-affinity (relaxed) states, similar to that found in extracellular annelid and pulmonate molluscan Hbs. Remarkably, Mg 2+ ions increased oxygen affinity solely by displacing the equilibrium between the tense and relaxed conformations towards the relaxed states, which accords with the original MWC concept, but appears to be unique among Hbs. This effect is distinctly different from those of ionic effectors (bivalent cations, protons and organic phosphates) on anne- lid, pulmonate and vertebrate Hbs, which involve changes in the oxygen affinity of the tense and ⁄ or relaxed conformations. Abbreviations Hb, hemoglobin; i r , i t , interaction parameters of the cooperon model; K i , Adair constants of the i th oxygenation step; K r , K s , K t , oxygen- binding constant for a particular conformation; K ab , oxygen-binding constant for a particular conformation; m, number of Mg 2+ -binding sites per oxygen-binding site; MWC, Monod–Wyman–Changeux; n 50 , Hill’s cooperativity coefficient at half-saturation; P 50 , half-saturation oxygen partial pressure; P m , median oxygen partial pressure; PO 2 , oxygen partial pressure; pK ab ,pK value of an oxygenation-linked acid group for a particular conformation; Q ab , q ab , magnesium and proton binding polynomials; q, size of the allosteric unit; rmse, root mean squared error; SSR, sum of the squared residuals; s, number of functionally coupled basic allosteric units; s Y , standard error of Y; w, size of the basic allosteric unit; x, oxygen partial pressure; Y, ^ Y , measured and predicted oxygen saturation; z, number of cooperons in a functional constellation; z ab ,Mg 2+ -binding constant of a particular conformation; ab (¼ tT, rT, tR, rR), particular conformation of the nested allosteric model; u, Bohr factor. 3374 FEBS Journal 274 (2007) 3374–3391 ª 2007 The Authors Journal compilation ª 2007 FEBS groups, the Conchostraca (clam shrimps) and the Not- ostraca (tadpole shrimps), can be traced back to the Devonian and the Late Carboniferous [3], respectively. The transition from the marine to the physicochemically more extreme inland water environments represented a great challenge for the physiological systems involved in regulating the internal milieu. Given the primitive and conservative morphological characteristics of many extant branchiopods, which seem to have chan- ged little over long periods of time, it may be pre- sumed that prehistoric adaptations to a highly variable environment remained preserved and essentially unsup- plemented by physiological ‘innovations’, allowing us to gain insight into homeostatic mechanisms operative in early crustaceans [4,5]. The tadpole shrimp Triops cancriformis (Notostraca) is one of the ‘oldest’ extant branchiopods; it was found to be inseparable from Triassic (250–205 million years ago) fossils on the basis of morphological criteria [6]. Notostracans comprising the two genera Triops and Lepidurus inhabit temporary water bodies that com- monly exhibit extreme physicochemical conditions. For desert ephemeral pools in south-western North America, a typical branchiopod habitat, Scholnick [7] reported large diurnal variations in oxygen tension (40–200 mmHg), carbon dioxide tension (0.07– 3 mmHg), pH (7.5–9.0), and temperature (17–35 °C) during summer months. Horne [8] similarly observed large diurnal fluctuations in oxygen concentration (1–6.5 mgÆL )1 ) and temperature (17–30 °C) in North American ephemeral ponds typically inhabited by Tri- ops longicaudatus. Notostracans seem to be well adapted to varying oxygen conditions, as reflected by their ability to maintain constant rates of oxygen consump- tion even when the ambient oxygen concentration decreases to critical levels of % 17% air saturation (1.2–1.3 mgÆL )1 )forLepidurus lemmoni at 12 °C [9] and % 25% air saturation for T. cancriformis at 20 °C (R. Pirow, unpublished data). Their oxyregulatory capa- city is impressive given that notostracans appear to lack extensive systemic (circulatory) regulatory capacities in response to variations in ambient oxygen availability. The open circulation of Triops lacks an arterial distri- bution system [10] and tissue capillaries, so that regula- tion of tissue oxygen supply by regional adjustments in perfusion rate is scarcely possible. In addition, no ana- tomical evidence has been found for a neuronal control of cardiac output by the central nervous system [10,11]. Although the heart is able to respond to neurohormones [12], it does not seem to be involved in the regulation of circulatory oxygen transport, as no compensatory adjustments in cardiac output were found in animals challenged by progressive hypoxia [13]. As argued previously [14,15], the greater differenti- ation and complexity of respiratory proteins in inverte- brates than in vertebrates may compensate for the lower morpho-functional organization at the organ level in the former and represent a shift of the homeo- static regulatory burden from the organ to the mole- cular level compared with vertebrates. This view is corroborated by the fact that exposure to hypoxic conditions increases hemolymph hemoglobin (Hb) concentrations in Triops spp. [16–18]. This homeostatic response may be complemented by changes in the functional properties of the protein. The extracellular Hbs of invertebrates are commonly high-molecular-mass complexes which exhibit high variability in oxygen-binding properties and their sensi- tivities to pH and ionic effectors [15]. So far, nothing appears to be known about the allosteric control of Hb–oxygen binding and its significance for the regula- tion of internal oxygen conditions in branchiopod crustaceans. This lack of knowledge contrasts with the detailed information available on the structure of sev- eral branchiopod Hbs [19,20]. To probe Hb function, its molecular correlates and organismic regulation in the phylogenetically ancient crustaceans, we investi- gated the oxygen-binding characteristics, their sensitivi- ties to pH, temperature and bivalent cations, and the allosteric mechanisms controlling oxygen binding in Hb from T. cancriformis. Results and Discussion Physicochemical characteristics of Triops hemolymph The in vivo pH of the hemolymph in the dorsal sinus of T. cancriformis was 7.52 ± 0.02 (n ¼ 3 animals) at 20 °C. A markedly lower pH value of 7.1 (presumably measured at 23 °C) has been given for T. longicaudatus [21]. The hemolymph had an osmolality of 150 ± 19 mosmolÆkg )1 (n ¼ 3). The chloride concentration was 58 meqÆL )1 (mean of a triple determination of one pooled hemolymph sample), which is comparable to the concentrations reported for nonmolting T. cancri- formis (40–57 meqÆL )1 ) [22] kept in distilled water and for T. longicaudatus (56 mm) [23]. Oxygen affinity, cooperativity and pH dependence The Hb of dialyzed hemolymph showed a moderate oxygen affinity (P 50 ¼ 13.3 mmHg), cooperativity (n 50 ¼ 2.3), and pH sensitivity (Bohr factor u ¼ dLogP 50 ⁄ dpH ¼ )0.18) at pH 7.44 (Tris buffer) and R. Pirow et al. Allosteric control of O 2 binding in crustacean Hb FEBS Journal 274 (2007) 3374–3391 ª 2007 The Authors Journal compilation ª 2007 FEBS 3375 20 °C (Fig. 1A,C). Raising the pH from 6.7 to 8.1 increased the n 50 from 1.9 to 2.9 and decreased the P 50 from 16.0 to 9.4 mmHg, respectively. The oxygen-binding characteristics of purified Hb (Fig. 1B,D) were comparable to those of dialyzed hemolymph. The P 50 of purified Hb, for example, was only 2–3% lower than that of dialyzed hemolymph under the same buffer and temperature conditions (Tris ⁄ Bis-Tris, pH 6.7–8.1, 20 °C). Experiments using Hepes as an alternative buffer revealed a somewhat higher P 50 than with Tris ⁄ Bis-Tris (Fig. 1D). At pH 7.5, for example, which represents the in vivo pH condition, the Hepes-buffered Hb showed a P 50 of 14.0 mmHg. The oxygen-binding properties of whole hemolymph were examined at three CO 2 levels at 20 °C (Fig. 2). Strong alkalinization of hemolymph induced by CO 2 - free conditions yielded an extreme affinity (P 50 ¼ 7.1 mmHg) and cooperativity (n 50 ¼ 3.80) which excee- ded the range of values obtained from Hepes-buffered Hb at pH 6.7–8.3 (Fig. 1B,D). The exposure of whole hemolymph to 1% and 2% carbon dioxide gave P 50 values of 13.4 and 16.5 mmHg, respectively, and n 50 values of 2.41 and 2.02, respectively. Essentially the same P 50 ⁄ n 50 combinations were observed for Hepes- buffered Hb at pH 7.6 and pH 7.0 (Fig. 1B,D), respectively. Analyses of buffer characteristics of T. cancriformis hemolymph at 1% and 2% CO 2 revealed pH values of 7.63 and 7.36, respectively (R. Pirow, unpublished data). These findings suggests that the replacement of the native hemolymph environ- ment by Hepes buffer does not significantly influence the pH dependence of the oxygen-binding properties of T. cancriformis Hb, at least in the physiological pH range. A comparison of oxygenation characteristics among branchiopod Hbs (Table 1) reveals the lowest affinities in the largest species, i.e. the notostracans (body length 10–100 mm [24]). This negative correlation extends to the smallest branchiopods such as the cladocerans (0.2–6 mm), which at high ambient oxygen tension rely predominantly on simple diffusion. Several lines of evidence [25–27], including the reduction in oxygen uptake when Hb–oxygen binding is blocked by carbon monoxide [28] and the striking induction of Hb under hypoxia in euryoxic species such as Daphnia magna (1–16 g HbÆL )1 ) [29], indicate that the high-affinity Hbs of cladocerans (P 50 ¼ 1.2–8.3 mmHg) function as oxygen carriers mainly at low ambient oxygen tension. Large branchiopods, in contrast, invariably require convective transport of oxygen. The moderate oxygen affinity (P 50 ¼ 6.8–14 mmHg) and the high concentra- tion of Hb (8–25 g HbÆL )1 ) [17,21] (this study) in Tri- ops spp. suggest that the respiratory protein mediates circulatory oxygen transport over a wide range of ambi- ent oxygen tensions i n Notostraca. The remarkably C pH 6.5 7.0 7.5 8.0 log P 50 0.7 0.8 0.9 1.0 1.1 1.2 D p H 6.5 7.0 7.5 8.0 A n 50 1.5 2.0 2.5 3.0 3.5 B dialyzed hemolymph purified Hb 20 °C Tris 10 °C Tris 20 °C Hepes 20 °C Tris Fig. 1. pH-dependence of oxygen-binding properties of T. cancrifor- mis Hb. Effects of pH on (A) Hill’s half-saturation cooperativity coef- ficients (n 50 ) and (C) half-saturation oxygen tensions (P 50 )at20°C (circles) and 10 °C (squares) in Tris ⁄ Bis-Tris-buffered (dialyzed) hemolymph. (B, D) Effects of pH on n 50 and P 50 of purified Hb in Hepes buffer (diamonds) and Tris ⁄ Bis-Tris buffer (triangles) at 20 °C. Oxygen partial pressure P O2 (mmHg) 0 1020304 0 Fractional oxygen saturation Y 0.0 0.5 1.0 0 % CO 2 1 % CO 2 2 % CO 2 whole hemolymph Fig. 2. Oxygen-binding curves of T. cancriformis Hb in whole hemo- lymph at three different CO 2 concentrations and 20 °C. Allosteric control of O 2 binding in crustacean Hb R. Pirow et al. 3376 FEBS Journal 274 (2007) 3374–3391 ª 2007 The Authors Journal compilation ª 2007 FEBS large variability in P 50 observed in notostracan Hbs (6.8–20 mmHg, pH 7.1–7.5, 20–23 °C) may reflect genus ⁄ species-specific variation in oxygen tolerance and temperature preference. In comparison with Lepidurus spp., species of the genus Triops are generally more warmth-demanding [6,24,30] and possess hemo- globins of higher oxygen affinity. The moderate coop- erativity (n 50 ¼ 1.8–3.1, pH 6.7–8.3, 20 °C) and the small Bohr effect (u ¼ )0.05 to )0.24, pH 7.0–8.0, 20 °C) found in T. cancriformis Hb conform with the homotropic and heterotropic interactions reported for other branchiopod Hbs. Prominent exceptions appear to be cladoceran (D. magna and Moina macrocopa) Hbs that seem to lack Bohr effects. Effect of temperature on oxygen binding The oxygenation of Hb is exothermic, and increasing temperature lowers oxygen affinity directly by weakening the bond between Hb and oxygen and indirectly via the Bohr effect because of the associated pH decrease. In T. cancriformis Hb, the increase in temperature from 10 to 20 °C increased the P 50 from 6.5 to 13.3 mmHg at pH 7.44 (Fig. 1C). The pH-dependence of n 50 was virtually unaffected by temperature (Fig. 1A). The temperature-dependence of the P 50 val- ues at pH 7.0 and pH 8.0 corresponded to the overall heats of oxygenation of )51.3 and )45.6 kJÆmol )1 , respectively, which include the heat of oxygen dissolu- tion and the heat of proton dissociation from oxygen- ation-linked acid groups. The reduction in the overall heat of oxygenation with increasing pH correlates with an intensification of the Bohr effect at higher pH and the endothermic nature of Bohr proton release [31]. In the physiological context, the reduction in oxygen affinity with increasing temperature may favor oxygen delivery to the tissues in synchrony with temperature- induced increases in oxygen consumption rates, but Table 1. Oxygenation characteristics of branchiopod Hbs. Data from whole hemolymph (WH) were obtained under normocapnic conditions (0.03% CO 2 ). In cases where information on the experimental buffer conditions was lacking, the extraction buffer type is given together with a question mark. Group ⁄ species P 50 (mmHg) n 50 T (°C) pH Buffer Bohr factor u b DH (kJÆmol )1 ) Reference Anostraca Artemia salina (Hb-I) 5.3 1.6 25 8.5 Borate )0.09 )45 [65] (Hb-II) 3.7 1.9 25 8.5 Borate )0.21 % )50 (Hb-III) 1.8 1.6 25 8.5 Borate )0.03 )23 Notostraca T. longicaudatus 6.8 1.5 23 7.1 Phosphate )0.23 )31 [21] T. cancriformis 13 2.3 20 7.4 Tris )0.18 )49 Present study 14 2.3 20 7.5 Hepes Lepidurus bilobatus 20 2 20 7.2 Tris )0.13 )31 [66] Lepidurus lynchi 21 3 20 8.0 Tris )0.19 [67] Lepidurus couesi 22 3 20 8.0 Tris [67] Conchostraca Caenestheria inopinata 7.8 1.2 25 6.8 Phosphate )0.19 )37 [68] Caenestheriella setosa 5.9 2.5 20 8.0 Tris )0.15 )21 [67] Cyzicus hierosolymitanus 0.035 2.3 28 7.3 Tris ⁄ maleate [69] Cladocera Daphnia magna 3.5 20 7.2 Phosphate (?) $0 [70] Daphnia magna Pale 3.8 1.3 20 7.2 Phosphate [71] Pale 8.3 20 WH [72] Pale 7.7 1.6 a 25 7.2 Bis-Tris ⁄ propane [57] Red 1.6 1.5 20 7.2 Phosphate [71] Red 2.5 20 WH [72] Red 2.6 1.8 a 25 7.2 Bis-Tris ⁄ propane [57] Daphnia pulex Hb-1 2.6 2.2 17 7.45 Tris (?) [73] Hb-3 1.2 1.4 17 7.45 Tris (?) [73] Moina macrocopa 2.1 20 7.2 Phosphate (?) $0 [70] a Calculated from the Adair constants; b dLogP 50 ⁄ dpH. R. Pirow et al. Allosteric control of O 2 binding in crustacean Hb FEBS Journal 274 (2007) 3374–3391 ª 2007 The Authors Journal compilation ª 2007 FEBS 3377 may also compromise oxygen loading in warm hypoxic water. Effect of bivalent cations on oxygen binding The addition of Mg 2+ and Ca 2+ increased the oxygen affinity of T. cancriformis Hb at cation concentrations higher than 5 mm (Fig. 3C,D). For example, increasing the Mg 2+ concentration from 0 to 20 mm decreased the P 50 from 14.6 to 11.4 mmHg at pH 7.1, and from 11.2 to 9.0 mmHg at pH 7.8. The effect of Ca 2+ was smaller than that of Mg 2+ (Fig. 3C). The cations had no strong effect on cooperativity (Fig. 3A,B); the indi- vidual values of n 50 were % 2.2 (pH 7.1, BisTris), % 2.3 (pH 7.6, Hepes), and % 2.6 (pH 7.8, Tris ⁄ HCl). Although the cation sensitivities were investigated by adding chloride salts, the measured effects are probably not attributable to the chloride counterions as Mg 2+ exerted a greater effect than Ca 2+ . The lack of a Cl – effect and the greater sensitivity of oxygen affinity to Mg 2+ than Ca 2+ are moreover consistent with a previ- ous study [21], which showed that univalent cations such as Na + and K + (added as chloride salts) had no significant effect on T. longicaudatus Hb. The differen- tial effects of Mg 2+ and Ca 2+ show that specific prop- erties of cationic effectors other than net ionic charge, such as size and the stereochemical orientation of their charges, influence oxygen affinity of Triops Hb. Allosteric control mechanisms and their physiological significance To reveal the allosteric control mechanisms, high-reso- lution oxygen-equilibrium curves of dialyzed hemo- lymph (in Tris ⁄ Bis-Tris buffer) and purified Hb (in Hepes buffer) were measured at different pH values (pH 6.7–8.3) and Mg 2+ concentrations (0–100 mm). As illustrated in the Hill-plot representation (Fig. 4), all oxygen-binding curves virtually approach the same asymptote of unity slope at low saturation (< 5%). This convergence shows that the Adair constant for the first oxygen-binding step (K 1 ¼ 0.027– 0.030 mmHg )1 ) is independent of proton and Mg 2+ concentration. Increasing the Mg 2+ concentration (0–100 mm) induced a left shift of the Hill plot in the half-saturation range (Fig. 4C,D) without affecting the affinity in the high-saturation range (> 95%), as reflected by the almost constant Adair parameter for the last oxygenation step (K N ). K N assumed values of 0.86–0.95 mmHg )1 at pH 7.50–7.60 (Hepes buffer) and 1.12–1.26 mmHg )1 at pH 7.77–7.80 (Tris buffer). This invariance of K 1 and K N reveals an apparently unique heterotropic control mechanism for bivalent cations. In contrast with the heterotropic interactions so far described for annelid (Arenicola marina [32]), pulmo- nate molluscan (Biomphalaria glabrata [33]) and ver- tebrate [34] Hbs, where modulation of oxygen affinity by ionic effectors invariably involves changes in K 1 and K N , increasing Mg 2+ concentrations raised the oxygen affinity of T. cancriformis Hb without affecting K 1 and K N . The physiological significance, if material, of the effects of bivalent c ations on Triops Hb is not clear. T he cation concentrations in T. cancriformis hemolymph (Ca 2+ , 1.6–3.1 mmolÆkg )1 ;Mg 2+ , 0.6–0.8 mmolÆkg )1 ; R. Pirow, unpublished data) and T. longicaudatus hemolymph (Ca 2+ , 0.8–1.7 mm;Mg 2+ , 0.6–0.9 mm) [21,23] are more than one order of magnitude below the values where the cations significantly increase oxy- gen affinity. Moreover, raised ambient Mg 2+ concen- trations are lethal to T. longicaudatus [35], and the regulation of internal Mg 2+ concentrations in this spe- cies breaks down when external levels exceed 14 mm, although Na + ,K + ,Ca 2+ ,Cl – and SO 4 2+ concentra- tions continue to be regulated [35]. C [M g 2+ ], [Ca 2+ ] (m M )(m M ) 12 5102050100 log P 50 0.7 0.8 0.9 1.0 1.1 1.2 D [M g 2+ ] 12 5102050100 A n 50 1.5 2.0 2.5 3.0 3.5 B dialyzed hemolymph purified Hb Mg 2+ pH 7.8 Ca 2+ pH 7.8 Mg 2+ pH 7.1 Ca 2+ pH 7.1 Mg 2+ pH 7.6 Fig. 3. Effects of bivalent cations on oxygen-binding properties of T. cancriformis Hb at 20 °C. Dependence of (A) n 50 and (C) P 50 on Mg 2+ (circles) and Ca 2+ (squares) in Tris ⁄ Bis-Tris-buffered (dialyzed) hemolymph at pH 7.1 (grey, filled symbols) and pH 7.8 (open sym- bols), where the dotted lines extrapolate to the values in the absence of bivalent cations at the same pH. (B, D) Effects of Mg 2+ on n 50 and P 50 of purified Hb in Hepes buffer (diamonds) at pH 7.6. Allosteric control of O 2 binding in crustacean Hb R. Pirow et al. 3378 FEBS Journal 274 (2007) 3374–3391 ª 2007 The Authors Journal compilation ª 2007 FEBS The proton and cation insensitivities at low satura- tions contrast with the pH-dependent divergence of oxygen-binding curves at high saturation (> 95%) (Fig. 4A,B). Increasing pH enhanced the affinity for binding the last oxygen molecule. Accordingly, the Adair constant for the last oxygenation step (K N ) increased from 0.49 mmHg )1 at pH 6.71 to 1.26 mmHg )1 at pH 7.77 in Tris buffer in the absence of Mg 2+ . In Hepes buffer, K N increased from 0.40 to 1.34 mmHg )1 when the pH changed from 6.69 to 8.30. This control mechanism, i.e. the left shift of the upper asymptote of extended Hill plots, is similar to that found in the extracellular annelid Hbs (Arenicola mar- ina [32] and Lumbricus terrestris [36]) and pulmonate molluscan Hb (Biomphalaria glabrata [33]). In water breathers such as Arenicola that exploit the upper part of the oxygenation curve maintaining a high ‘venous reserve’ [37], a pronounced Bohr effect at high satura- tion may be adaptive in favoring oxygen loading to blood perfusing the respiratory structures [32]. This characteristic contrasts with the tetrameric vertebrate Hbs, where increases in the concentrations of protons and anionic organic phosphates decrease Hb oxygen affinity by lowering the binding constant for the low- affinity (tense) state [38], which may favor oxygen unloading in the tissues under varying oxygen demand. The physiological significance of Hb in Triops has been questioned on the basis of reports that some indi- viduals lack Hb and because experiments with carbon monoxide poisoning of Hb resulted in no depression of the rate of oxygen consumption [21]. However, these observations may merely indicate that the oxygen requirements of the tissues may be satisfied by the oxy- gen carried in physical solution in the hemolymph, at least at rest and under normoxic conditions. Against the background of the limited systemic (circulatory) regulatory capacities in Triops, Hb becomes the key control component of the oxygen-transport cascade from environment to cell. Hb enables the animal to maintain aerobic respiration under environmental hypoxia by increasing the convective conductance for oxygen in the circulatory system [39]. When oxygen loading and unloading spans the steep part of the oxygen-equilibrium curve, Hb also exerts a stabilizing effect on the hemolymph oxygen tension (‘oxygen buf- fering’) [40], thereby reducing the risk of oxidative stress to the tissues. The oxyregulatory function of Hb may be enhanced by the Bohr effect, which enables the animal to optimize oxygen loading to the hemolymph at the respiratory surfaces via hyperventilation and res- piratory alkalosis under conditions of environmental oxygen deficiency. B % Ox y Hb 99 98 95 90 70 50 30 10 5 2 –log K N D log P O2 0.0 0.5 1.0 1.5 2.0 % Ox y Hb 99 98 95 90 70 50 30 10 5 2 A log [Y/(1–Y)] -1 0 1 2 C log P O2 0.0 0.5 1.0 1.5 2.0 log [Y/(1–Y)] -1 0 1 2 dialyzed hemolymph in Tris buffer purified Hb in Hepes buffer –log K 1 pH 8.30 pH 7.50 pH 7.12 pH 6.69 pH 7.77 pH 7.44 pH 6.71 [Mg 2+ ] pH 100 7.78 15 7.80 0 7.77 [Mg 2+ ] pH 100 7.60 50 7.58 0 7.50 Fig. 4. Extended Hill plots of T. cancriformis Hb at different pH values and different Mg 2+ concentrations at 20 °C. Effects of pH on Hb oxygen binding in Tris ⁄ Bis-Tris-buf- fered hemolymph (A) and in Hepes-buffered purified Hb (B) in the absence of Mg 2+ . The lower row shows the influence of Mg 2+ concentration (values in mmolÆL )1 )onHb oxygen binding in Tris ⁄ Bis-Tris-buffered hemolymph at pH 7.8 (C) and in Hepes- buffered purified Hb at pH 7.5–7.6 (D). The solid lines were fitted to the data by using the 2 · 4 nested MWC model with one oxy- genation-linked acid group and half a Mg 2+ - binding site per oxygen-binding site. Dashed lines with slopes of unity represent asymp- totes approached by the curve furthest to the left at very low and very high saturation. The intercepts of these dashed lines with the horizontal (dotted) line at log [Y ⁄ (1–Y)] ¼ 0 correspond to the negative logarithms of the Adair constants of the first and last oxygenation step (K 1 and K N ). R. Pirow et al. Allosteric control of O 2 binding in crustacean Hb FEBS Journal 274 (2007) 3374–3391 ª 2007 The Authors Journal compilation ª 2007 FEBS 3379 Structure–function relationships The multimeric Hb of T. cancriformis is composed of two subunit types, TcHbA (60–70%) and TcHbB (30–40%), which have polypeptide masses of 35775 and 36055 Da, respectively; each carry two heme groups and assemble into disulfide-bridged dimers that comprise a homodimer of TcHbA and a heterodimer [20]. These dimers assemble into three native isoforms, one 16-mer and two 18-mer species. Only the larger 18-meric species seems to possess the heterodimer con- taining subunit type TcHbB. Thus, several structural levels are present: the first level is the di-domain sub- unit, which carries two oxygen-binding sites (see Fig. 6A). The next level is the disulfide-bridged dimer D. As it seems unlikely that eight or nine copies of these dimers oligomerize into a big lump, additional substructures such as D 2 ,D 3 or D 4 have to be taken into account (Fig. 6A). These substructures carry 8, 12 and 16 oxygen-binding sites. In order to determine whether the hierarchical structure plays a role in the functional properties, and to obtain some indication of which of the possible structural organizations might occur, oxygen-binding data comprising six curves (Fig. 4B,D) were analyzed in terms of different models of cooperativity. As the largest Hill coefficient determined empirically from the oxygen-binding curves is 3.8 (whole hemo- lymph at 0% CO 2 , Fig. 2), it does not seem necessary to assume interactions beyond the dimer D, bearing in mind, that each dimer D carries four oxygen-binding sites. If this can be assumed, an analysis based on four Adair constants should describe the data well. The result of this analysis is shown in Table 2 together with the results of the other models. The residual plots of all models are shown in Fig. 5. In all cases, the sum of squared residuals (SSR) is given for the simultaneous analysis of all six binding curves, as in some cases parameters are shared between individual binding curves in order to reduce the number of free parameters as well as the uncertainty in the remaining parameters. In the Adair formalism none of the parameters are shared between the curves, thus, the total set contains 24 free parameters. Despite this very large number of parameters, the root mean squared error (rmse) is sig- nificantly larger than that of most other models tested (Table 2, model b). Furthermore, the residuals are not randomly distributed (Fig. 5B), indicating systematic deviations between the data and the fit. The obvious next level to take into account is a functional coupling of two dimers (Fig. 6A, substructure D 2 ), involving an interaction between eight oxygen-binding sites. This Table 2. Comparison of the goodness of fit of different oxygen-binding models. Each model was applied to the set of six oxygen-equilibrium curves of purified Hb shown in Fig. 4B,D. Shown are the total number of (shared and curve-specific) parameters, SSR, the degrees of free- dom (DF), the root mean squared error (rmse ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SSR=DF p ), and the best-fit parameter values of each model. Shared parameters and curve- specific parameters are given as single values and range of values, respectively. K values are in mmHg )1 . Model Number of parameters SSR DF rmse Best-fit values of model parameters (a) Two-state MWC 9 (¼ 6 · 1 +3) 300 105 1.69 K T ¼ 0.034 log L ¼ 5.2–6.6 with shared q K R ¼ 1.820 q ¼ 4.6 (b) Tetramerous Adair equation 24 (¼ 6 · 4) 130 90 1.20 (c) Two-state MWC 14 (¼ 6 · 2 +2) 88 100 0.94 K T ¼ 0.033 log L ¼ 4.6–6.5 with curve-specific q K R ¼ 1.516 q ¼ 4.1–5.4 (d) Two-state MWC 19 [¼ 2 · (6 +2 + 1) +1] 80 95 0.92 species 1: species 2: with two species and K T ¼ 0.022 K T ¼ 0.035 with species-specific q K R ¼ 0.219 q ¼ 11.6 log L ¼ 2.4–6.9 species ratio ¼ 15 : 85 K R ¼ 8.818 q ¼ 4.7 log L ¼ 8.4–10.0 (e) Three-state MWC 21 (¼ 6 · 3 +3) 42 93 0.67 K S ¼ 0.026 log L ¼ 5.3–9.0 with curve-specific q K T ¼ 0.074 K R ¼ 1.546 log M ¼ 5.9–10.2 q ¼ 5.3–8.0 (f) 4 · 2 Cooperon model 14 (¼ 6 · 2 +2) 32 100 0.56 K T ¼ 0.025 log L ¼ 8.3–10.8 z ¼ 4 dimeric cooperons K R ¼ 1.448 i T ¼ 4.6–11.3 with i R fixed to unity i R ¼ 1 (fix) (g) 4 · 2 Cooperon model 20 (¼ 6 · 3 +2) 20 94 0.46 K T ¼ 0.026 log L ¼ 5.6–6.7 z ¼ 4 dimeric cooperons K R ¼ 0.274 i T ¼ 3.6–9.6 with curve-specific i R i R ¼ 2.0–5.9 (h) 3 · 4 nested MWC (h ¼ 1, m ¼ 0.5) 15 20 99 0.45 (i) Octomerous Adair equation 48 (¼ 6 · 8) 11 66 0.40 Allosteric control of O 2 binding in crustacean Hb R. Pirow et al. 3380 FEBS Journal 274 (2007) 3374–3391 ª 2007 The Authors Journal compilation ª 2007 FEBS leads to the octomerous Adair equation with 48 free parameters. The agreement between the fit and the data is very good (Table 2, model i) and the residuals appear randomly distributed (Fig. 5I). The analysis based on the Adair formalism gives an idea about the minimal number of interacting binding sites, but does not give any information about the number of confor- mations (or substates) involved in the cooperative mechanism or insights into any more complex interaction pattern such as a hierarchical grouping of functional units. In order to obtain this kind of information, more specific models that take the structure of this Hb into account are needed. The simplest model to consider is the Monod– Wyman–Changeux (MWC) model, which predicts two conformations that are simultaneously adopted by a molecule-specific number of binding sites, the allosteric unit (Eqn 7, see Experimental procedures). We applied this model in an approach where the binding constants K t and K r are shared among the six binding curves, and the allosteric equilibrium constant (L) is specific for each curve. If one allows curve-specific values for the size of the allosteric unit (q), a reasonably good fit is obtained, but the values for q vary between 4.1 ± 0.2 and 5.4 ± 0.3 (means ± 95% confidence interval) (Table 2, model c). If one fixes q to the same value for all curves, a poor value for the rmse is obtained (Table 2, model a) and the residuals are non- randomly distributed (Fig. 5A), indicating that the MWC model does not reflect the complexity of the oxygen-binding process for Triops Hb. At the next level of complexity, we allowed one more conformation for the allosteric unit within the framework of the MWC model, and employed a three- state model (Eqn 8). In this case, the value for the size of the allosteric unit (q) is still highly variable, ranging from 5.3 to 8.0 (Table 2, model e), disfavoring this model too. Alternatively, we extended the simple MWC model to take a possible heterogeneity in the cooperative interactions because of the different sizes of the oligomers (16-mers and 18-mers; Fig. 6A) into account. This extension is based on superimposition of two binding curves corresponding to two types of mole- cules, each having an oxygen-binding characteristic that obeys the MWC formalism (Eqn 9). The size of the allosteric unit (q) is allowed to differ between these two types, but for each molecule species, q is shared among the binding curves obtained at different effector concentrations (Table 3, model d). The agreement between the fit and the data is much better than that obtained for a single molecule species with shared q. However, in total, the agreement is still not as good as I Fractional ox yg en saturation Y 0.0 0.2 0.4 0.6 0. .0 A Residuals -0.05 0.00 0.05 2-state MWC (shared q) B C Residuals -0.02 0.00 0.02 D E Residuals -0.02 0.00 0.02 F Residuals -0.02 0.00 0.02 G H Fractional ox yg en saturation Y 0.0 0.2 0.4 0.6 0.8 1.0 Residuals -0.02 0.00 0.02 0.0 pH 8.30 (0) pH 7.12 (0) pH 6.69 (0) pH 7.60 (100) pH 7.58 (50) pH 7.50 (0) tetramerous Adair 2-state MWC (curve-specific q -state MWC (2 species) 3-state MWC (curve-specific q) 4×2 cooperon (i R = 1) 4×2 cooperon (curve-specific i R ) 3×4 nested MW ctomerous Adair 1.0 Fig. 5. Comparison of the (unweighted) resi- duals of the fit provided by the different oxygen-binding models (see Table 2). (A) Two-state MWC with shared q. (B) Tetra- merous Adair equation. (C) Two-state MWC with curve-specific q. (D) Two-state MWC with two species and species-specific q. (E) Three-state MWC with curve-specific q. (F) 4 · 2 cooperon model consisting of four dimeric cooperons with i R fixed to unity. (G) 4 · 2 cooperon model consisting of four dimeric cooperons with curve-specific i R . (H) 3 · 4 nested MWC with one oxygenation- linked acid group and half a Mg 2+ -binding site per oxygen-binding site. (I) Octomerous Adair equation. R. Pirow et al. Allosteric control of O 2 binding in crustacean Hb FEBS Journal 274 (2007) 3374–3391 ª 2007 The Authors Journal compilation ª 2007 FEBS 3381 in the case of the octomerous Adair model, and the distribution of residuals is clearly not random (Fig. 5D). The two species as identified by this approach are present in a ratio of 15 : 85, indicating that the main part of the oxygen-binding curve is dom- inated by one species. Altogether, these results indicate that models allowing hierarchical functional properties need to be applied. The cooperon model includes both KNF-type and MWC-type interactions [41,42]. It describes a basic dimeric cooperon (the ab dimer in the case of verteb- rate Hb), in which cooperative interactions are allowed according to the induced-fit mechanism. The change in the binding affinity for the second oxygenation step compared with the first is quantified via a parameter i. A value for i larger than unity indicates positive coop- erativity, and a value smaller than unity indicates neg- ative cooperativity. The dimeric cooperon is nested into a higher-level oligomer formed by a number of z cooperons, which are regulated according to the MWC mechanism. Thus, for each conformation r and t, a specific interaction parameter (i r and i t ) is considered. We applied this model by equating the dimeric cooper- on with the di-domain subunit of the T. cancriformis Hb (Fig. 6A). The resulting fit describes the data somewhat better than the three-state model, with the additional plus that the values for the parameters do not violate model-inherent assumptions. The best fit for this model was achieved for a variant where four dimeric cooperons form an oligomeric structure, which functions according to the MWC model. Such a functional constellation is accommodated by the substructure D 2 (Fig. 6A). Similar to the results for vertebrate Hb, there is no need to include KNF-type interactions in the R-state: the agreement between fit and data is quite good for a fixed value of i r ¼ 1 (Fig. 5F; Table 2, model f). The value for the inter- action parameter i t is effector-dependent and ranges between 4.6 and 11.3. The rmse can be further reduced by allowing curve-specific values for i r (Fig. 5G; Table 2, model g). When this is done, the value for i r ranges between 2.0 and 5.9, with i t assuming values between 3.6 and 9.6. Thus, the KNF-type interaction predicts a positive cooperativity for the T and the R state at the level of the di-domain subunit of T. cancri- formis Hb. An alternative description of hierarchical interac- tions provides the nested MWC model [43,44]. Here, two levels of allosteric units, which both function according to the MWC model, are embedded into each other. This model has been successfully applied to describe the allosteric interactions in hierarchically structured, multimeric proteins such as arthropod hemocyanins [45,46], annelid Hbs [47], and chaperonin GroEL [48]. The nested MWC model was fitted to the data using different combinations of the size (w) of the basic allosteric unit and the number (s)ofw-sized basic allosteric units. In order to keep the number of free parameters as small as possible, the influence of effectors such as protons and Mg 2+ was directly inclu- ded in the model (Eqns 14–16, 18–20). The results obtained for different combinations of s and w are shown as contour maps (Fig. 7), which also visualize the influence of variations in the numbers of Mg 2+ - binding sites (m) and proton-binding sites (h) per oxygen-binding site. di-domain subunit disulfide-bridged dimer D possible sub-structures possible stoichiometries of native Hb isoforms D 2 D 3 D 4 (D ) 42 (D ) 24 (D ) 33 (D ) D 24 (D)D 42 16-mer: 18-mer: l R rR rR tR tR tT tT rT rT l T L A B Fig. 6. Possible stoichiometries of Hb quaternary structure and scheme of the nested MWC model. ( A) T. cancriformis Hb consists of di-domain subunits, which carry two heme groups and form disulfide-bridged dimers. Three possible assemblies of dimers (D 2 , D 3 and D 4 ) have been suggested as building blocks of the native 16-meric and 18-meric Hb isoforms [20]. (B) Nested 2 · 4 MWC model showing the conformational states (tR, rR, rT and tT) and transitions for a nested, basic allosteric unit containing w ¼ 4 oxy- gen-binding sites. A number of s ¼ 2 copies of the basic allosteric unit assemble into a larger structure. This s · w assemblage can adopt two overall conformations, R and T, which impose con- straints on the conformations of the constituent basic allosteric units. The conformational equilibria are described by the allosteric constants l R , l T , and L. Allosteric control of O 2 binding in crustacean Hb R. Pirow et al. 3382 FEBS Journal 274 (2007) 3374–3391 ª 2007 The Authors Journal compilation ª 2007 FEBS These contour maps suggest that the size of the basis allosteric unit is a functional tetramer (i.e. w ¼ 4), which is accommodated by the disulfide-bridged, dimeric structure D of the T. cancriformis Hb (Fig. 6A). The higher-level allosteric unit seems to con- sist of a number of s ¼ 2–3 functional tetramers. A number of s ¼ 2 would correspond to the substructure D 2 , whereas s ¼ 3 would refer to the D 3 (Fig. 6A). D 4 as functionally operative substructure can be ruled out on the basis of the SSR (Fig. 7). The details of the contour maps vary somewhat with changing number of binding sites for protons and Mg 2+ , but the principal behavior is maintained. The lowest values for SSR were obtained for (s · w) combinations of 2 · 4 and 3 · 4 with one (h ¼ 1) oxygenation-linked acid group and half (m ¼ 0.5) a Mg 2+ -binding site per oxygen-binding site. For both combinations, 2 · 4 and 3 · 4, the oxygen-binding and allosteric-equilibrium constants (Table 3, Fig. 6B) showed the typical pattern (K tT <K tR <K rR <K rT and L<<l R <l T ) found in anne- lid Hbs (Macrobdella decora [47]) and arthropod hemocyanins [45,46]. On the basis of the parameters of the 2 · 4 nested MWC model, conformational distributions were calcu- lated for three different situations. Under the condi- tions used for the measurement of oxygen-binding curves, the conformation tT is not strongly populated (Fig. 8A–C). Thus, neither protons nor Mg 2+ displace the conformational distribution sufficiently towards the tT state to visibly shift the lower asymptote of the Hill plot. This also explains the relatively large errors in the parameter describing the effector binding in the tT Size of the basic allosteric unit w 3456 Number of coupled allosteric units s 4 3 2 1 1/4 1/2 1 Number (m) of Mg 2+ -binding sites per heme Number (h) of proton-binding sites per heme SSR 20 30 40 2 1 1 2 – Fig. 7. Gallery of error contour maps showing the dependence of SSR on different parameter combinations. The s · w nested MWC model was globally fitted to a set of six oxygen-equilibrium curves of purified Hb in Hepes buffer (Fig. 4B,D). Error contour maps were calculated for nine different combinations of the number of proton-binding sites (h) and magnesium-binding sites (m) per oxygen- binding site. Each error contour map shows (in a grey-scale repre- sentation) the SSR in relation to the size of the basic allosteric unit (w) and the number of coupled allosteric units (s). The combination of h ¼ 0.5 and m ¼ 0.25 yielded the error contour map with the lowest SSR (¼ 17.8) which occurred at s ¼ 2.1 and w ¼ 4.5. The best fit using integer-sized (s · w) combinations gave the 3 · 4 model (SSR ¼ 20.3) in the presence of presence of one (h ¼ 1) oxygenation-linked acid group and half (m ¼ 0.5) a Mg 2+ -binding site per oxygen-binding site. The 68.3% (i.e. one standard devi- ation) confidence region of this best-fit integer-sized parameter combination lies within the SSR contour of 22.3. This confidence region excludes the 4 · 4 combination (SSR ¼ 34.6) but includes the 2 · 4 combination (SSR ¼ 22.1). The confidence region also includes all combinations of m and n (SSR < 21.9), for which the 3 · 4 nested model was tested. Note that the error contour maps are truncated at SSR levels higher than 40. Table 3. Best-fit parameter combinations of the nested MWC model. The best fit of oxygen-binding data from T. cancriformis Hb gave a model that assumed a basic allosteric unit with w ¼ four oxygen-binding sites and a functional coupling of s ¼ two to three allosteric units. The parameters refer to the presence of one oxy- genation-linked acid group and half a Mg 2+ -binding site per oxygen- binding site. Given are the oxygen-binding constants (K ab ), the pK ab values, and the Mg 2+ -binding constants (z ab ) for the four conforma- tions (ab ¼ tT, rT, tR, rR). The allosteric equilibrium constants (l T °, l R °, L°) refer to the reference condition at pH 6.5 and zero Mg 2+ concentration. The SSR was taken as a measure of the goodness of fit. The degrees of freedom (DF) represent the number of data points minus the number of fitted parameters. Parameter values are given as mean ± 95% confidence interval, either in absolute terms or as a percentage. Parameter s · w 2 · 43· 4 K tT (mmHg )1 ) 0 ± 0.0100 0.0073 ± 0.0070 K rT (mmHg )1 ) 1.41 ± 6% 1.38 ± 7% K tR (mmHg )1 ) 0.028 ± 6% 0.029 ± 2% K rR (mmHg )1 ) 0.181 ± 15% 0.337 ± 10% pK tT 7.97 ± 2% 7.40 ± 36% pK rT 7.65 ± 1% 7.63 ± 1% pK tR 8.16 ± 1% 8.15 ± 1% pK rR 7.95 ± 1% 7.84 ± 1% z tT (mM )1 ) 0.016 ± 50% 0 ± 0.147 z rT (mM )1 ) 0.023 ± 20% 0.024 ± 18% z tR (mM )1 ) 0.009 ± 33% 0.011 ± 25% z rR (mM )1 ) 0.025 ± 21% 0.028 ± 18% log l R ° 2.08 ± 12% 2.99 ± 6% log l T ° 4.66 ± 9% 3.12 ± 8.18 log L° )6.70 ± 6% )7.30 ± 6% SSR 22.1 20.3 DF 99 99 R. Pirow et al. Allosteric control of O 2 binding in crustacean Hb FEBS Journal 274 (2007) 3374–3391 ª 2007 The Authors Journal compilation ª 2007 FEBS 3383 [...]... implementing the corresponding effectorbinding functions This approach successfully describes proton binding [59] and urate binding [63] in hemocyanins Here, we assume the simplest effector -binding model with h identical proton -binding sites and m identical Mg2+ -binding sites per oxygen- binding site It is assumed that the binding of protons and Mg2+ occurs independently Given a nested s · w structure, the. .. data points Adair equation To determine the minimum number of interacting binding sites necessary to describe the data, the binding curves were analyzed by a tetramerous and octomerous Adair equation [54,56] The native Hb isoforms of T cancriformis contain 32 and 36 oxygen- binding sites [20], respectively, but often it is not mandatory to assume that all binding sites of large respiratory proteins are... within the quaternary structure of multimeric proteins into account In this model it is assumed that a number of w binding sites are functionally coupled to form an allosteric unit which, in accordance with the standard MWC model [60], can adopt two basic conformations (r and t) A number of s copies of these basic allosteric units assemble into a larger allosteric unit containing q ¼ s · w oxygenbinding... represents the oxygen partial pressure and N is the number of interacting binding sites For some models, the six binding curves were analyzed simultaneously, with some of the parameters shared between the curves This means, that the values of these particular parameters were forced to be the same for all six binding curves This strategy allows the number of free parameters to be reduced A comparison of the. .. On the nature of allosteric transitions: a plausible model J Mol Biol 12, 88–118 61 Minton AP & Imai K (1974) The three-state model: a minimal allosteric description of homotropic and heterotropic effects in the binding of ligands to hemoglobin Proc Natl Acad Sci USA 71, 1418–1424 ´ 62 Koshland DE, Nemethy G & Filmer D (1966) Comparison of experimental binding data and theoretical models in proteins... mass of macromolecules and subunits and the quarternary Allosteric control of O2 binding in crustacean Hb 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 structure of hemoglobins from the microcrustacean Daphnia magna FEBS J 273, 3393–3410 Rousselot M, Jaenicke E, Lamkemeyer T, Harris JR & Pirow R (2006) Native and subunit molecular mass and quarternary structure of the haemoglobin from the primitive branchiopod. .. for the least-squares curve-fitting analysis of oxygen- equilibrium curves The octomerous Adair equation proved to be appropriate to (phenomenologically) describe the oxygen binding of the related D magna Hb [57,58], which also contains 32 oxygen- binding sites [59] The binding polynomial for the Adair equation is given by [54]: BAdair ¼ 1 þ N i X N  Y xi Kj i i¼1 j¼1 ð5Þ where Kj denotes the intrinsic... upper asymptote of the Hill plot is shifted by changes in pH, and the preferential binding of protons to rR rather than to rT is reflected by the pK values (pKrT < pKrR) The presence of Mg2+ does not lead to shifts in the Hill-plot asymptotes, a phenomenon that is explained by the very similar values for the Mg2+ -binding constants zrR and zrT Thus, the effector-induced phenomena visualized by the Hill plot... Brunori M (1971) Hemoglobin and Myoglobin in Their Reactions with Ligands North-Holland Publishing Co, Amsterdam and London Weber RE (1981) Cationic control of O2 affinity in lugworm erythrocruorin Nature 292, 386–387 Bugge J & Weber RE (1999) Oxygen binding and its allosteric control in hemoglobin of the pulmonate snail, Biomphalaria glabrata Am J Physiol Regul Integr Comp Physiol 276, R347–R356 Tyuma... to include the binding properties of effectors explicitly into a saturation function (see below) The latter equation shows that the s-th root of L describes the allosteric equilibrium between the two relaxed conformations available to the basic allosteric unit in the unliganded state The in uence of effectors such as protons and Mg2+ on the allosteric equilibrium constants (lR, lT, L) was taken into . Oxygen binding and its allosteric control in hemoglobin of the primitive branchiopod crustacean Triops cancriformis Ralph Pirow 1 , Nadja Hellmann 2 and. group and half a Mg 2+ -binding site per oxygen- binding site. Given are the oxygen- binding constants (K ab ), the pK ab values, and the Mg 2+ -binding constants