On the Origin of Hemoglobin Cooperativity under Non-equilibrium Conditions

Abnormal hemoglobins can have major consequences for tissue delivery of oxygen. Correct diagnosis of hemoglobinopathies with altered oxygen affinity requires a determination of hemoglobin oxygen dissociation curve, which relates the hemoglobin oxygen saturation to the partial pressure of oxygen in the blood. Determination of the oxygen dissociation curve of human hemoglobin is typically carried out under conditions in which hemoglobin is in equilibrium with O2 at each partial pressure. However, in the human body due to the fast transit of red blood cells through tissues hemoglobin oxygen exchanges occur under non-equilibrium conditions. We describe the determination of non-equilibrium oxygen dissociation curve and show that under these conditions the true nature of hemoglobin cooperativity is revealed as emerging solely from the consecutive binding of oxygen to each one of the four subunits of hemoglobin until the entire tetramer is saturated. We call this form of cooperativity the sequential cooperativity of hemoglobin and define the simplest model that includes it as the minimalist model of hemoglobin. A single instantiation of this model accounts for ~70% of hemoglobin cooperativity under non-equilibrium conditions. The total cooperativity of hemoglobin can be viewed more correctly as the summation of two instantiations of the minimalist model (each one corresponding to a tetramer of low and high affinity for O2, respectively) in equilibrium with each other, as in the Monod-Wyman-Changeux model of hemoglobin. In addition to offering new insights on the nature of hemoglobin reaction with oxygen, the methodology described here for the determination of hemoglobin non-equilibrium oxygen dissociation curve provides a simple, fast, low-cost alternative to complex spectrophotometric methods, which is expected to be particularly valuable in regions where hemoglobinopathies are a significant public health problem, but where highly specialized laboratories capable of determining a traditional oxygen dissociation curve are not easily accessible.


INTRODUCTION
Hemoglobin (Hb) oxygen dissociation curve (ODC), which relates oxygen saturation (SO2) and partial pressure of oxygen in the blood (PO2), is an important tool for understanding how blood carries and releases oxygen 1 .
Classically, factors recognized to influence the ODC include the local CO2 partial pressure (PCO2), pH, temperature, as well as allosteric metabolites like 2,3 diphosphoglycerate (2,3-DPG). The curve is shifted to the right (i.e. lower saturation for a given PO2) by higher PCO2, greater acidity (lower pH), higher temperature, and higher concentration of 2,3-DPG [1][2][3][4][5][6][7] . The factors that shift the ODC to the right are directly relevant to the conditions that prevail in metabolizing tissues, as they facilitate the unloading of oxygen from hemoglobin. The converse occurs during passage through the pulmonary capillaries, with the greater affinity accompanying a shift of the ODC to the left aiding the uptake of oxygen 8 .
The partial pressure of oxygen in the blood at which hemoglobin is 50% saturated is known as the P50. The P50 of normal hemoglobin is approximately 26 mmHg at a partial CO2 pressure of 40 mmHg 7 . In the presence of disease or other conditions that change hemoglobin oxygen affinity and, consequently, shift the curve to the right or left, the P50 changes accordingly 9 . Low affinity hemoglobins are characterized by higher P50, and high-affinity hemoglobins by a lower P50. Such abnormal hemoglobins can have major consequences for tissue delivery of oxygen, but their effects are mitigated by various compensatory mechanisms, one of which is the hemoglobin concentration. High-affinity molecules, by definition, release oxygen less readily than normal and, because tissue hypoxia is a stimulus to hemoglobin production, affected individuals often have polycythemia; approximately 100 hemoglobin variants with high oxygen affinity have been described in the literature 10 . By contrast individuals with low affinity hemoglobins are usually anemic. For example, the abnormal hemoglobin (HbS) of sickle-cell disease (SCD) has reduced oxygen affinity. SCD is a monogenic disorder where the patient is a heterozygous or homozygous carrier for the  S allele. This mutation causes a single amino acid substitution from glutamic acid to valine in the -globin chain of Hb, which results in hydrophobic interactions with adjacent HbS molecules leading to HbS polymers. This polymerization process is triggered at low PO2 and is also dependent on pH. Early recognition in life of the presence of HbS in the blood erythrocytes can better prepare the patient for an SCD 'event' in which rigid, sickled erythrocytes adhere to the endothelium and cause painful vaso-occlusion crises (VOC).
Neonatal cyanosis can also be due to Hb variants with reduced oxygen affinity and a pronounced shift of the ODC to the right. With the introduction of universal screening for congenital heart disease, the finding of abnormal ODC's will likely uncover more neonates with hemoglobinopathies with low oxygen affinity.
Finally, measurement of a complete ODC is important for detecting hemoglobin variants that may escape conventional tests measuring only P50 10,-12 . For example, some abnormal ODC's are characterized by bi-phasism due to the coexistence of both normal and high affinity hemoglobins, and to an exchange of subunits between them 12,13 . In these cases, measurement of the complete ODC is important to infer the functional properties of the individual hemoglobin components that cannot be easily separated.
Hemoglobin ODC was originally determined by manual methods in which the oxygen saturation of hemoglobin was measured by spectrophotometry at every stepwise change of PO2 by the addition of aliquots of air to the pre-deoxygenated sample in a tonometer, after equilibrium between Hb and O2 at each PO2 was reached. Although very accurate, this static method was very laborious and time consuming. To overcome its limitations, dynamic methods were developed in which PO2 was progressively changed in a close vessel, allowing for equilibration in between changes 1,14,-18 . In the most current implementation of these methods, hemoglobin oxygenation and deoxygenation are achieved with highly purified supplies of oxygen and nitrogen or argon gases from cylinders, a Clark oxygen electrode detects the change in oxygen tension, while the resulting increase in oxyhemoglobin fraction is simultaneously monitored by dual-wavelength spectrophotometry at 560 nm and 576 nm 19 .
However, equilibrium ODC does not reflect correctly physiological gas exchanges by Hb in the lungs (oxygenation) and in peripheral tissues (deoxygenation) as, due to the fast transit of red blood cells (RBCs) through microcirculation 20 , hemoglobin O2 exchanges occur under nonequilibrium conditions. Here, we describe a simple method for the determination of non-equilibrium ODC and analyze the contribution of different components of Hb cooperativity to the curve. Besides mimicking more closely physiological gas exchanges, the methodology offers a low-cost, lowtech alternative to complex and expensive commercial instruments for the determination of equilibrium ODC, which will undoubtedly be valuable in areas of the world where hemoglobinopathies represent a significant public health problem, and highly specialized laboratories capable of determining a traditional ODC are not easily or timely accessible.

MATERIALS AND METHODS
Rat liver mitochondria were purified by differential centrifugation of tissue homogenate as described in 21 , portioned in aliquots of 30-40 mg protein/ml in 0.25 mM sucrose, and stored at -80 °C till used. Stripped Hb devoid of allosteric heterotropic factors was prepared from whole human blood of healthy donors 22 and its concentration, as HbO2, was estimated spectrophotometrically from its heme content, with an εmM at 577 nm of 15.4 23 . Alternatively, Vacutainer (Becton, Dickinson and company) collected whole blood was used; in this case the HbO2 concentration was estimated, after dilution of the sample in double distilled water, from the air-equilibrated minus Na2S2O4-supplemented differential spectra, using a ΔεmM at 577-568 nm of 4.8 according to 24 ; the Hb content (as heme centers) was typically 9-11 nmol/(ml blood).
Determination of oxygen consumption was carried out by respirometry using Oxygraph-2k (O2k, OROBOROS Instruments, Innsbruck, Austria). The instrument has two measuring chambers (2 ml each) both equipped with a Clarktype electrode; calibration of the instrument was performed according to the manufacturer instructions and all the measurements were carried out at 37 °C. The assay medium constituted by 250 mM sucrose, 1 mg/ml bovine serum albumin, 10 mM KH2(PO4), 27 mM KCl, 1 mM MgCl2, 40 mM Hepes, 0.5 mM EGTA (pH 7.4) was supplemented with 20-50 g/ml mitochondrial proteins in the absence or presence of either purified Hb or whole blood samples. Oxygen consumption was initiated by the addition of 10 mM succinate as respiratory substrate in the presence of 2 µM rotenone, an inhibitor of the respiratory chain NADH-ubiquinone oxidoreductase/Complex I. Alternatively, the measurements were carried out with a singlechamber (0.5-1 ml) oxymeter equipped with a Clark type electrode disc (Hansatech Instruments Ltd, King's Lynn, UK).
Fitting of the deoxygenation/reoxygenation curves of Hb using four different kinetic models (minimalist, Adair, Perutz, MWC) was carried out using in-house code written for Matlab ® (deposited at https://github.com/dgattiwsu/HB_ODC). Using these models, a single set of rate constants was derived by global analysis of both the deoxygenation and reoxygenation traces.

Non-equilibrium ODC of human hemoglobin.
In the following we describe a typical experimental determination of human hemoglobin nonequilibrium ODC.
2 ml of the assay buffer solution (pH 7.4) were placed inside each of the two glass chambers of the Oroboros oxymeter and supplemented with a small amount of purified rat liver mitochondria to a final concentration of 30 g prot/ml. With the cell open to the environment, the solutions were allowed to equilibrate under stirring with atmospheric O2 until the observed O2 concentration (based on an earlier calibration of the electrode [25][26][27] ) remained stable (~183 µM) for a few minutes.
Upon insulating the glass cells from air with glass stoppers (containing a small port for microsyringe additions) progressive reduction of the PO2 in the cell was achieved by adding 10 mM succinate according to the reaction catalyzed by the succinate oxidase segment of the respiratory chain (complexes II+III+IV: succinate dehydrogenase, ubiquinol:cytochrome c reductase, cytochrome c oxidase, respectively): The final concentration of cytochrome c oxidase (the mitochondrial enzyme that uses up O2 reducing it to water) in the cell was estimated to be ~0.05 µ M by visible spectroscopy of the mitochondrial suspension in the 500-650 nanometer range. Upon starting respiration, O2 concentration decreased linearly until anaerobiosis was reached ( Figure 1A).
The linear respiratory activity is due to the affinity of cytochrome c oxidase for O2 whose KM is estimated in the sub-micromolar range 28,29 . Considering that the instrument limit of detection of O2 concentration is also in the submicromolar range 30 , the O2 concentration is practically never limiting the respiratory flux under the prevailing conditions reported here. After adding 5 M Antimycin A, an inhibitor of respiration at the level of ubiquinol:cytochrome c reductase, the glass stopper was removed, and the cell content was allowed to equilibrate again with air. During this phase oxygen diffuses back in the cell in a nonlinear fashion.
When the cycle of deoxygenation/reoxygenation is repeated in the presence of 50 M (as heme centers) human hemoglobin isolated from blood hydrolysate ( Figure 1B) the final part of the deoxygenation curve is no longer linear due to the release of oxygen from Hb (green arrow). The initial part of the reoxygenation curve is also slower due to the uptake of oxygen by Hb (yellow arrow).
Since the alterations of the deox-/reoxygenation curves are due to the release/uptake of O2 by Hb, it is possible to recover the non-equilibrium ODC using a kinetic model of the cell ensemble as a set of reversible reactions involving the species of Hb in different oxygenation state, cytochrome c oxidase as the terminal O2 acceptor, O2 in the cell, and external air in diffusive equilibrium with the O2 in the cell, when the latter is open. We have evaluated three different kinetic models (Figure 2): A. a sequential Adair style model with 4 refined parameters (one O2 koff for each Hb(O2)n), B. a two state sequential Perutz style model with 2 refined parameters (one O2 koff for Hb(O2)1,2 and one O2 koff for Hb(O2)3,4), and C. a two state concerted Monod-Wyman-Changeux (MWC) style model with 3 refined parameters (one Kequil between a tense (T) and a relaxed (R) state of Hb, one O2 koff for all T states and one O2 koff for all R states) (reviewed in 31,32 ).
The three models fit equally well the oxymetric trace with essentially identical sum of square errors (~150 µM 2 ) and R-square values (>0.99). Each model provides the contribution of all Hb species (Figure 3A,D,G) at all time points (O2 concentrations). Non-equilibrium ODC's are derived for each PO2 value as the ratio between the sum of all the oxygenated species and the total amount of Hb (Figure 3B,E,H). The corresponding Hill plots are shown in Figure 3C,F,I.
The observed variations in the ODC and Hill plots are due to the fact that for each time point the contributions of individual Hb species are different in the three models. Details of these contributions are shown in a blow-up of the terminal part of Hb deoxygenation phase (Figure 4A,C,E), and of the initial part of Hb reoxygenation phase ( Figure   4B,D,F). Regardless of the model used, concentration peaks are reached in the order Hb(O2)4 → Hb(O2)3 → Hb(O2)2 → Hb(O2)1 → Hb during deoxygenation, and Hb → Hb(O2)1 → Hb(O2)2 → Hb(O2)3 → Hb(O2)4 during reoxygenation.
In the Perutz and MWC models, intercepts with the X axis of the extrapolated lines from the asymptotic ends of the Hill plot are usually interpreted as the concentration of O2 at which the concentrations of the unliganded and liganded forms of the T and R states are equal (KD). Better values for these magnitudes are calculated directly from the cross-over points (Figure 4) of the T and R forms derived from the kinetic simulations (Table 1). Nonlinear least-squares fit of Adair equation 16, 33, 34 , to the combined experimental points from the deoxygenation and reoxygenation non-equilibrium ODC's can be used to derive values for the equilibrium Adair constants in the three models ( Table 1).

Sources of cooperativity in the models
A common feature of Adair, Perutz, and MWC models is the presence of sequential reactions. A minimalist model containing four sequential binding reactions, Hb  Hb(O2)1  Hb(O2)2  Hb(O2)3  Hb(O2)4, with a single O2 affinity for all states, and no conformational changes (Figure 5A), fits experimental observations surprisingly well (sse = 515.5, R 2 = 0.998) (Figure 5B,C,D), giving origin to a sigmoidal ODC ( Figure 5E) and a Hill plot with asymptotic components that suggest the presence of both low and high affinity sites (Figure 5C), despite none such exist in the model. Accordingly, a fit of Adair equation to the sigmoidal ODC shows that all four Adair constants collapse into a single one corresponding to the refined kon/koff ratio ( Table 1). This observation suggests that the most basic source of cooperativity in the Adair, Perutz, and MWC We define as the cooperativity gain (cgain) of a model the rms (root mean square) deviation between the model derived ODC and the ODC with the same P50 derived from a model containing 4 independent identical O2 sites. The cooperative gain of the minimalist, Perutz and MWC models is shown in Figure 5E,G,H.

CONCLUSION
We have presented a typical experiment showing the determination of hemoglobin non-equilibrium ODC, and from it the determination of key parameters such as the P50 and the Hill's coefficient. The experimental component of the method requires minimally the acquisition of a deoxygenation curve of Hb and, optionally, also that of a reoxygenation curve. The computational component is based on the minimization of the sum of square errors (sse) between an experimentally observed O2 polarographic trace and a simulated O2 trace based on a kinetic model of choice (Figures 2-4). Since it does not require an optical determination of the Hb saturation, this method can be directly used with a red cell suspension or whole blood without the added complications of the dual-wavelength or full sphere spectrometry that are necessary to eliminate light scattering noise 14 . Dedicated instruments capable of carrying out the determination of hemoglobin equilibrium ODC are commercially available at a significant cost, as they require supplies of highly purified gases (i.e., O2, N2, Ar) from cylinders, a Clark-type electrode to measure Table 1

. KD's (µM) from model refinement, and Adair equilibrium (association) constants (µM -1 ) for the four steps of Hb oxygenation from a fit of Adair equation to the non-equilibrium ODC's
For the Adair and Perutz model, these values are identical to those refined during the optimization of the two models. Notice the collapse of 4 Adair constants into just 2 in the Perutz model, and just 1 in the Minimalist model (see below, Sources of cooperativity in the models). In the MWC model, KT and KR are the refined association constants for the T and R states oxygenation, c = KT/KR, and L0 = [T0]/[R0] is the refined allosteric constant expressing the equilibrium in the absence of any oxygenation between the T and R state of Hb. Allosteric constants Ln for the n different oxygenation levels are not refined, but calculated according to the relationship Ln = L0c n . Imai  oxygen concentration, and multi-wavelength spectroscopy to detect Hb oxygen saturation inside a glass cuvette of optical quality. These instruments are typically only accessible in the specialized hematology laboratories of large medical centers. On the contrary, the protocol described here for the determination of non-equilibrium ODC requires only a widely available glass cell of non-optical quality equipped with a Clark-type electrode, and no flushing of the solutions with purified gases. In our experiment, we have used an in-house preparation of rat liver mitochondria as the source of the respiratory enzymes that drive anaerobiosis. However, a preparation of bacterial membrane particles with comparable properties is commercially available as EC-Oxyrase ® . Finally, while in our study we have used the proprietary computational platform Matlab ® for model generation and for the analysis of experimental traces, our code will run also with few modifications in the open-source platform GNU-Octave, and it can be easily converted into fully open source Python code. Thus, altogether, the methodology described here can be easily set up in a small laboratory of a neighborhood clinic as a lowcost, low-tech alternative to the expensive commercial instrumentation available only in large medical centers. A simple determination of Hb ODC can be very useful as part of an initial population screening for hemoglobinopathies before further molecular validation is sought and is expected to be particularly valuable in regions where hemoglobinopathies are a significant public health problem, but highly specialized hematology laboratories are not easily accessible.
It has not escaped our notice that the chemical constants derived from non-equilibrium ODC are significantly different from those reported by other authors (i.e. 1 , Table 1), based on equilibrium ODC. This is not surprising, since non-equilibrium ODC mimics the physiological condition of red blood cells moving rapidly in the blood stream across regions of different PO2, while this condition is not reproduced in equilibrium ODC.
Three kinetic models (Adair sequential, Perutz two state sequential, MWC two state concerted) were all equally effective in fitting the experimental polarographic traces. Thus, as such, the determination of non-equilibrium ODC does not offer any new ways to discriminate between these models. However, the predictions made by the three models with respect to the concentrations of oxygenation and conformational intermediates are quite different (Figure 4) and may offer inspiration for future experiments.
The origin of Hb sigmoidal ODC has been a point of intense debate for over a century (reviewed in 31,32,35 ), and this curve is perhaps the most frequently used 'book perfect' example of positive cooperativity. Our experiments were carried out with stripped Hb devoid of allosteric heterotropic factors, and thus under these conditions we observed the intrinsic cooperative behavior of tetrameric Hb. A minimalist model, containing no induced (by O2 binding) or intrinsic conformational equilibria between a low and a high affinity state, is sufficient to provide, by virtue of the constraint of sequential binding reactions of identical affinity, a large fraction (~69% in cgain scale) of the cooperative behavior of Hb, as judged by the magnitude of its cooperative gain (= 0.1052) with respect to a model containing 4 independent identical O2 binding sites. We call this basic level of cooperativity observed in the minimalist model, the sequential cooperativity of Hb. The constraints of sequential binding of oxygen molecules to Hb (rather than, for example, the simultaneous binding of two O2 molecules to distinct subunits of a single Hb molecule with formation of a single transition state) does not require the invocation of any special structural features of Hb, because according to collision theory the probability of three chemical species reacting simultaneously with each other in a dilute solution in a termolecular elementary reaction is negligible 36,37 . Therefore, apparent termolecular reactions are typically viewed as non-elementary reactions that can be broken down into a more fundamental set of bimolecular reactions in agreement with the law of mass action.
The total cooperativity of Hb under nonequilibrium conditions is accounted for by inclusion in the model of a single conformational equilibrium between a low and a high affinity state, as shown by the values of the cooperative gain in the Perutz (cgain = 0.1505) and MWC models (cgain = 0.1526). One particularly interesting way of explaining the conceptual relationship between the sequential and the total cooperativity of hemoglobin is to view total cooperativity as the summation of two or more realizations of sequential cooperativity. From this point of view, the constraint of sequential reactions is the only true requirement for the emergence of a cooperative behavior. For example, the MWC model ( Figure 2C) can be viewed as the summation of two minimalist models (Figure 5A), each one with a different single O2 affinity for all states, in equilibrium with each other. In this respect, the Perutz model does not offer any additional insights on the origin of Hb cooperativity, and historically has been perhaps just a source of confusion on this issue.
The observation that Hb cooperativity might originate simply as the consequence of consecutive reactions of a single Hb molecule with four oxygen molecules, paves the way for the exploration of other enzymatic systems, in which phenomena of cooperativity have been attributed solely to conformational changes. At this point it is not clear whether the sequential nature of Hb cooperativity was not discovered previously simply because all earlier determinations of Hb ODC were carried out under equilibrium conditions. When a reversible reaction is observed under equilibrium conditions, the forward and backward rates are necessarily identical. However, earlier computational work has revealed that the path followed by the forward and reverse reactions can be different in enzyme conformational space 38,39 . The constraint of equality between forward and backward rates at equilibrium is more easily fulfilled if two distinct conformations of Hb can affect both rates. Instead, if Hb oxygenation occurs under non-equilibrium conditions, in the absence of a constraint of equality between forward and backward rates at each oxygenation step, hemoglobin may exist almost entirely in one conformational state.
A major contribution of this study to the understanding of the cooperative behavior of allosteric proteins is the indication that the kinetic parameters associated with conformational transitions might be affected by the measuring conditions. The amount of conformational change required to account for the microscopic forward and backward reaction rates of each oxygenation step may depend on the rate at which PO2 changes. On this basis, it is tempting to speculate that in vivo the relative contribution of the two minimalist components that make up the MWC model of Hb, may be modulated by the rate at which erythrocyte move through the capillary bed. In general, one can anticipate that factors that increase this rate (e.g., lower plasma viscosity, higher flexibility of the erythrocytes membrane and cytoskeleton) will magnify the non-equilibrium character of Hb oxygenation, and thus will increase the contribution of only one conformational state of Hb. Conversely, factors that decrease this rate (e.g., higher plasma viscosity, as it occurs in extreme exercise 40 , infection and/or inflammation 41 , or decreased erythrocytes flexibility, as it occurs in an SCD 'crisis') will magnify the equilibrium character of Hb oxygenation, and thus will increase the contribution of both conformational states of Hb.