Quaternary Interactions in Hemoglobin ,8 Subunit Tetramers KINETICS OF LIGAND BINDING AND SELF-ASSEMBLY*

We have investigated the rates of monomer tetra- mer self-association of oxygenated BSH subunits of hu-man hemoglobin A as well as the influence of self- association on the binding kinetics for O2 and CO. A 4B * 2B2 B4 assembly pathway can be used to describe the association equilibria and kinetics. We have determined all four elementary rate constants for this as- sembly pathway at 15 “C in 0.1 M Tris-HC1,O. 1 M NaCl, 1 mM Na2EDTA, pH 7.4. These data imply that a sig- nificant amount (-17%) of B2 can be present. Laser photolysis kinetic studies of O2 binding indi- cate that the O2 association rate constant is unaffected by the degree of self-association. In contrast, photolysis of BCO solutions shows an overall rate of CO binding that increases at higher protein concentrations. These data are consistent with a concentration-dependent equilibrium between two protein species with CO association rates differing by a factor of 2.5, but they do not appear to be compatible with a direct assignment of different CO binding rates to the different assembly states. Rather, we believe the data imply that CO binding to fi oligomers is heterogeneous, with both a fast binding and a slow binding form being present in single association states. The fast binding form predominates (-87%) in B4, while the /3 monomer has very little or none of the fast binding form. We propose that the slow binding component within B4 may be those subunits with rotationally disordered of 0.1 unit/ ml glucose oxidase (Behring Diagnostics 346385), 1 unit/ml catalase (Behring Diagnostics 219261), and 0.3% glucose was added. At the lowest P subunit concentrations this system caused some interference, and a trace of sodium dithionite (Virginia Smelting) was used instead (at higher (3 concentrations the enzyme system or dithionite gave essentially identical results). The concentration-dependent changes in the visible absorption spectrum of PC0 were measured using methods described previously (Philo et al., 1981), except that a Varian 2200 spectrophotometer was used. Protein concentrations were determined relative to the standard extinction for the cyanomet form at 540 nm of 11 m"' cm" (Tentori and Salvati, 1981). Oxygen and CO concentrations were based on Henry's law coefficients of 1.77 and 1.32 pM/mm Hg, respectively.

units associate to form deoxyhemoglobin A. The quaternary enhancement phenomenon was first found in isolated hemoglobin @ subunits when it was observed that the oxygenated subunits self-associate to tetramers more strongly than do the deoxy subunits Ackers, 1978a, 1978b). This data implies that the 0, affinity of p4 is about 4 times greater than that of @ monomers, and the overall linkage between oxygenation and subunit association for @ subunits is about half that in Hb A, but of opposite sign. Later hemoglobin A itself was also found to exhibit quaternary enhancement; the affinity of triply ligated tetramers was reported to be significantly higher than the mean affinity of the isolated a and @ subunits (Mills and Ackers, 1979;Chu et al., 1984), although recently this data has been questioned (Gibson and Edelstein, 1987). This 0, affinity difference between @ and p4 raises a number of interesting questions. First, given that there have been so many studies of Hb subunits, often aimed at using them to model the properties of the subunits in the oxy "R state" hemoglobin A tetramer, it is somewhat surprising that this difference went unnoticed for so long. More importantly, it is now unclear which form (if either) should be used as a model for an R state @ subunit. Second, while it is now clear that the ( 3 subunit changes ligand binding properties upon assembly of a tetramer, it remains unclear whether there is any hemeheme interaction within a @ tetramer. That is, the oxygen binding equilibria data are not sufficiently accurate to detect whether there are small (2-4-fold) differences in affinity between intermediate ligation states in p4. In an earlier study of the absorption spectra of oxy-Hb and its subunits, we discovered a tetramer-monomer difference spectrum for both oxy and carbonmonoxy @ subunits. The oxy @ tetramermonomer difference spectrum has features unlike other difference spectra for oxyhemoglobins (Philo et al., 1981). These spectral studies suggested (but certainly did not prove) that the PO, subunits in the HbO, tetramer have properties closer to the @ monomer than to the @ tetramer. Moreover, the unique nature of the oxy @ tetramer-monomer difference spectrum suggested that the mechanism leading to a change in the affinity of @ tetramers may differ from that producing the low affinity of deoxy "T state" Hb.
The association-dissociation kinetics of @ subunits also may be relevant to the assembly of Hb A tetramers in uiuo. Several investigations have shown that the rate-limiting step for in vitro assembly of Hb from a and /3 subunits is the dissociation of @ tetramers to monomers (Antonini et al., 1966;McGovern et al., 1976). It also appears that in some cases the proportion of mutant and variant hemoglobin tetramers in the red cell is governed by their assembly kinetics rather than by the rate of synthesis of the variant subunits (Shaeffer, 1980). Therefore, we decided to extend the earlier equilibrium studies and to explore the kinetic aspects of the @ tetramermonomer differences. In the present work we have measured the effect of the @ subunit association equilibrium on the Ligand Binding and Assembly Kinetics of Hb /3 Subunits 683 kinetics of CO and O2 binding. Further, we have used the tetramer-monomer difference spectrum as a marker to study the kinetics of the association reactions.

MATERIALS AND METHODS
The PSH subunits of Hb were prepared as described previously (Philo et al., 1981), stored under liquid nitrogen, and dialyzed against the "Ackers' buffer" (0.1 M Tris, 0.1 M NaC1, 1 mM EDTA, pH 7.4) prior to use. The kinetics of the oxy P association-dissociation reactions were measured using a temperature jump instrument producing 5 "C jumps in less than 10 /IS and by rapid dilutions with a Durrum stopped-flow instrument. The temperature jump instrument and data acquisition and analysis procedures have been described previously (Pennathur-Das and Schuster, 1982). The dissociation reaction was measured by hand mixing a small portion of concentrated material into a cuvette, using the optics of the temperature jump instrument.
As an aid to the interpretation of the association kinetics, a small perturbation relaxation computer simulation of a monomer-dimertetramer reaction scheme was performed. The program was derived from that by Ilgenfritz (1977). Kinetic data for CO and oxygen binding were obtained by laser photolysis. The optical monitoring was done using the optics and detection system of the temperature jump instrument. A 4 -/ I S photolysis pulse a t about 590 nm from a Candela SLL625 flashlamppumped dye laser using rhodamine 6G was brought into a sealed cuvette at right angles to the monitoring beam. For each sample, data were recorded for photolysis levels of -10, -30, and 100%. For all the kinetic data the changes in optical transmittance were digitally recorded. During subsequent analysis the data were converted to changes in absorbance and fit to sums of exponentials by a nonlinear least squares routine. For the CO binding kinetics typically 3000 data points were included in the least squares analysis in order to resolve multiple exponentials. For these CO binding kinetic studies, it was necessary to rigorously exclude oxygen, and two different methods were used. In some samples an enzyme system consisting of 0.1 unit/ ml glucose oxidase (Behring Diagnostics 346385), 1 unit/ml catalase (Behring Diagnostics 219261), and 0.3% glucose was added. At the lowest P subunit concentrations this system caused some interference, and a trace of sodium dithionite (Virginia Smelting) was used instead (at higher ( 3 concentrations the enzyme system or dithionite gave essentially identical results). The concentration-dependent changes in the visible absorption spectrum of PC0 were measured using methods described previously (Philo et al., 1981), except that a Varian 2200 spectrophotometer was used.
Protein concentrations were determined relative to the standard extinction for the cyanomet form a t 540 nm of 11 m"' cm" (Tentori and Salvati, 1981). Oxygen and CO concentrations were based on Henry's law coefficients of 1.77 and 1.32 pM/mm Hg, respectively.

Association-Dissociation Kinetics of Oxy
Subunits-In principle, there are two distinct experimental approaches to study the self-association kinetics of p subunits. Given the coupling between association and ligand binding, it is possible to use ligand binding as the indicator for association. An example of this is shown in Fig. 1, a temperature jump experiment on a partially 02-saturated p sample. A qualitative interpretation of the two reaction phases is that the faster phase is due to oxygen release (driven by the large negative enthalpy for O2 binding), while the slower phase is due to association to tetramers (driven by the large positive enthalpy for association), which in turn raises the affinity and leads to a net binding of oxygen. (This slower phase might have been seen in earlier temperature jump studies of / 3 subunits, but at lower protein concentrations it would probably be so slow that it would be obscured by cooling within the temperature jump cell (Brunori and Schuster, 1969;Nakamura et al., 1974).) However, to quantitatively treat the coupled system one must consider both the association and ligand binding kinetics to oxy, deoxy, and partially ligated monomers and tetramers (and dimers also, if present). Even with simplifying

Time (msec)
FIG. 1. Temperature jump relaxation kinetics of fl subunit self-association coupled to oxygen binding. Following a rapid release of 0, (which can be more clearly seen in the inset), there is a smaller slow phase of opposite sign (net binding of 0,) due to coupling with the self-association reactions. Sample conditions: 67 PM heme, 96% 0, saturation, 0.2 M (4-(2-hydroxyethyl))-l-piperazineethanesulfonic acid, 1 mM Na,EDTA, pH 7.0, 5-9 "C jump (L. E. Vickery and T. M. Schuster, unpublished observations). assumptions, such a system is too complex to extract elementary rate constants.
The second approach is to work only with fully deoxy or oxygenated samples and to rely on spectroscopic differences to monitor the association reaction. In our earlier study we found small association-dependent changes in absorption spectra of both oxy-and CO-p subunits but were unable to identify a spectral change for deoxy p ( P h i b et al., 1981).
Therefore, in this work we have concentrated on measuring the assembly kinetics of oxy fl using the peaks in the difference spectrum near 408, 422, or 582 nm to follow the reaction. We have chosen the buffer system used by Ackers (1978a, 1978b) in their equilibrium gel permeation association-dissociation studies so that we may make direct use of, and comparisons with, their data. Since we have found oxy p subunits to be significantly more stable below room temperature, these experiments were done at 15 "C (the final temperature for temperature jump experiments).
In treating the equilibrium association of fl subunits, it was found that a simple monomer-tetramer stoichiometry was adequate to describe the data for oxy, CO, and deoxy forms but that for cyanomet-P a substantial amount of dimer is also present Ackers, 1977, 1978b). From a kinetic point of view, a direct monomer to tetramer association would be extremely slow, since four-body collisions are rare. The most reasonable way to treat the self-association kinetics is to formulate a monomer-dimer-tetramer system. With corresponding equilibrium association constants where K4 is the overall monomer + tetramer association constant.
We have measured the self-association kinetics over a range of concentration from about 1 to 100 ~L M (heme); i.e. from  Table I. where the sample is predominantly monomer to where it is predominantly tetramer. For these studies we have used a combination of temperature jump and stopped-flow techniques. The former is generally preferable, but at the lower concentrations the reaction becomes too slow for conventional temperature jump because of cooling within the cell. Therefore, at the lower concentrations we used a stopped flow to mix samples at two different concentrations to perturb the assembly equilibrium. These experiments are rather difficult because the total change in absorbance is typically only 0.1%. In all these experiments what we observe in the time range from 100 p s to 20 s is a single exponential relaxation process.' The exponential decay time for this process drops from about 7 s at the lowest concentrations to 0.5 s at the highest. These rate data are summarized in Fig. 2. We also did large perturbation experiments where a high concentration stock was diluted 100-fold by hand mixing, giving essentially a complete dissociation from tetramers to monomers. Again a single exponential phase is observed with a rate of 0.1 k 0.01 s-I. In these latter experiments the observed rate will be essentially determined by k-] or k"2, whichever is slower.
In principle, one would expect to see two exponential relaxations for a monomer-dimer-tetramer system. In a small perturbation analysis, if the monomer-dimer and dimer-tetramer reactions were entirely uncoupled they would give relaxation rates.
For the coupled reactions it is expected that there would always be two relaxations, T , and T ,~, with rates While, in theory, two relaxations should always be present, experimentally only one relaxation may be seen if one of the In the temperature jump experiments there is an unresolved kinetic phase within the heating time due to the temperature dependence of the absorption spectra. Also, even though the oxygen saturation is >99.8%, a t some wavelengths a very fast (50 ps) phase can be seen due to an 0, binding relaxation. However, at this high 0 2 saturation the 0, binding and self-association reactions are essentially entirely uncoupled, and the self-association kinetics we report are not influenced by the kinetics of 0, binding and release. relaxation amplitudes is very small and/or if 7, and T , are too similar to be distinguished.
Since, in fact, we can distinguish only a single relaxation in our data, how can we tell whether this is T / or T,? One clue comes from the fact that at higher protein concentrations the rate we observe is approximately proportional to the square root of the total concentration. At high total [PI the dimer concentration should have this concentration dependence, which suggests (from Equation 2) that the phase we observe is dominated by the dimer + tetramer step. Furthermore, if we use the known assembly equilibrium constants for these conditions (Valdes and Ackers, 1978a;Philo et al., 1981) and assume that the association rate constants are within an order of magnitude of that for dimer-tetramer association of Hb, then our small perturbation relaxation simulation predicts that 1) the faster relaxation should be in the range of tens of milliseconds, should consist primarily of the monomer-dimer reaction, and should have a very small amplitude; and 2) the slower relaxation should be in the range of hundreds of milliseconds to seconds, should consist primarily of the dimertetramer reaction, and should have an amplitude about an order of magnitude larger than that of the fast relaxation.
We, therefore, believe the relaxation we observe and show in Fig. 2 is the slower one, T,, as given by Equation 3, and that the 0.1 s-' rate observed in the 100-fold dilution experiments can be assigned to k 2 . With these assignments, we have done a least squares fit of the relaxation rate data in Fig. 2 to Equation 3 with kl, k2, and I C l as adjustable parameters. We find that we can obtain good unique fits to this scheme. All attempts to interpret the observed relaxation as the fast phase and/or k-' as the slow dissociation step were incapable of fitting the experimental data.
A further constraint on the rate constants determined from the fitting procedure is that they should be consistent with the equilibrium association data and in particular with our earlier measurements of the concentration dependence of the spectral changes (Philo et al., 1981). As might be expected, the optimum fit to the kinetic data is not the optimum fit to the spectral data, but there is in fact a region of overlap between the confidence intervals of the independently determined parameters. From this overlap region we take a "consensus best fit," whose values are shown in Table I. The solid curve in Fig. 2 shows the theoretical T;' for these values.
As a further check on these rate constants, we can use them to calculate the expected amplitudes of the signals for comparison with the experimental data. To do this we must know the changes in absorption for each step of the monomer + dimer + tetramer pathway. While our earlier spectral studies determined the overall tetramer-monomer difference, there is  not enough dimer present at equilibrium to determine its spectral properties uniquely. Furthermore, to calculate amplitudes for the temperature jump data, we also must know the A H for each step, but only the overall monomer -+ tetramer A H is known. Therefore, we cannot make a rigorous comparison, but we can make some reasonable assumptions about the partitioning of the spectral changes and A H between the monomer -+ dimer and dimer --* tetramer steps, and then see whether the experimental and predicted amplitudes are in reasonable agreement and to what extent the predicted amplitudes are dependent on the assumptions we have made. (Note that it is entirely unnecessary to make any such assumptions in the above analysis of the relaxation rate data.) For the spectral changes, the simplest assumption is that there is only one spectral change and that it must occur at either the monomer -+ dimer or the dimer -+ tetramer step, and we have calculated the amplitudes for both of these extreme cases. It is unlikely that the A H would be zero or negative for either step, and it seems most reasonable to assume that the A H for the monomer + dimer step is the same as that for the monomer + dimer association of hemoglobin a subunits (Valdes and Ackers, 1978a). This choice also makes the A H values nearly equal for each intersubunit contact formed.
The experimental relaxation amplitudes are shown in  Fig. 2. Data monitored a t other wavelengths and/or pathlengths have been scaled to the equivalent amplitude a t 583 nm and a 1-cm pathlength using the known difference spectrum (Philo et ul., 1981). The stopped-flow amplitudes were scaled relative to the calculated data in the solid line (see text). The smooth curues were calculated from the relaxation simulation program using the values from Table I and AH values from Valdes and Ackers (1978a). The upper two curues are for the slower relaxation, the lower two are for the fast relaxation (which was not observed). The solid lines are calculated assuming the p dimer has the same absorption spectrum as the tetramer (i.e. the spectral change occurs at the monomer -+ dimer step). The broken lines m e calculated assuming the p dimer has the same spectrum as the monomer (i.e. the spectral change occurs at the dimer -+ tetramer step). The error burs on the experimental points represent the estimated uncertainty from the raw experimental amplitudes; they do not reflect the uncertainty introduced by the procedures to plot all the data on a uniform scale. Tjump, temperature jump. slow relaxations for two different assumptions about the spectral changes. In order to include the stopped-flow data with the temperature jump data, the observed stopped-flow amplitude was divided by the theoretical stopped-flow amplitude calculated from the pre-and postmixing species distributions predicted by the equilibrium constants from Table I. This ratio was then multiplied by the slow phase amplitude predicted by the temperature jump simulation at the postmixing concentration for the dimer = tetramer model. That is, the stopped-flow data points are positioned to show the correct ratio of experimental to theoretical amplitude, relative to the upper solid line in Fig. 3. If the spectrum of the dimer is, in fact, intermediate between that of monomer and tetramer, the theoretical curves for both temperature jump and stoppedflow data will lie between the two extreme cases shown.
Increasing the proportion of the overall AH which occurs at the monomer -+ dimer step will increase the amplitude of the fast phase and lower that of the slow phase while keeping the total constant.
The calculations show why we cannot detect the fast phase: its amplitude is always below A, which is too small for our instrument to detect at the higher bandwidth needed to follow it. The temperature jump data appear to agree better with the calculations for the dimer = tetramer model (spectral change at the monomer -+ dimer step), but given the uncertainty about the A H values, we would not say that these amplitude data rule out the dimer = monomer case or a dimer spectrum intermediate between that of monomer and tetramer. However, we note that good agreement with the dimer = monomer case would require the monomer -+ dimer A H to be less than half that for the monomer -+ dimer association of hemoglobin 01 subunits (i.e. only about 2 kcal/mol). Therefore, we favor the dimer = tetramer spectral model and consider the agreement of the experimental amplitudes with this model for both the temperature jump and stopped-flow data to be quite good considering the uncertainties involved in the calculations and in scaling the raw experimental data.
Ligand Binding Kinetics-The association equilibrium data of Ackers (1978a, 1978b) imply the oxygen affinity of p4 is 4 times greater than that of p monomers, while the corresponding CO affinities differ by a factor of 3. These differences in affinity imply that the association and/or dissociation kinetics for these ligands must also differ with association state. However, such differences were apparently not recognized in earlier kinetic studies of isolated p subunits (Antonini et al., 1965;Brunori et al., 1966;Brunori and Schuster, 1969;Geraci et al., 1969;Noble et al., 1969;Nakamura et al., 1974), although in some cases heterogeneous kinetics and/or inconsistencies with equilibrium data were noted (Antonini et al., 1965;Noble et al., 1969). We, therefore, wanted to look more carefully for such kinetic differences and wanted to determine whether the changes in affinity are due primarily to differences in ligand association or dissociation rates. We have measured the association rate constants for oxygen and CO using laser photolysis. Using the known affinity differences, we can also infer the difference in dissociation rates between species.
Oxygen Association Kinetics-Laser photolysis experiments on air-saturated PO2 solutions at 21.5 "C show only a single reaction phase. As shown in Fig. 4 FIG. 4. Oxygen binding kinetics. Data are shown for protein concentrations of 1.0 pM (solid truce, 100% photolysis) and 30 p~ (dotted truce, 48% photolysis). Oxygen rebinding after photolysis of air-saturated samples was monitored at 436 nm, and the data are plotted as a fraction of the initial absorbance change. For the sake of clarity, less than 20% of the data points are plotted. Sample conditions are the same as in Fig. 2 except that the temperature is 21.5 "C. ponential and will give a straight line in a semi-log plot such as Fig. 4. Under such pseudo-first order conditions, the maximum deviation of these kinetic data from a single exponential is no more than 0.8% of the total amplitude. We, therefore, conclude that the oxygen association rate constant, k', is essentially independent of association state, with a difference of at most 10% between monomers and tetramers. If the association rates are indeed independent of association state, these data give a value for k' of 6.8 (6.5, 7.1) X lo7 M" s".' Using the oxygen association constant for tetramers measured under these conditions, 2.24 (1.82, 2.75) x lo6 M" (Mills and Ackers, 1979), we infer a dissociation rate, k, for tetramers of 30 (24, 38) s-'. These association and dissociation rates are in reasonable agreement with earlier data for PSH (Brunori et al., 1966, Brunori andSchuster, 1969;Nobel et al., 1969;Nakamura et al., 1974). For monomers the 0, affinity is 4.2 (3.3, 5.5) times lower (Valdes and Ackers, 1978a), so we infer a dissociation rate of 130 (75, 220) s-'. This high dissociation rate would perhaps have been seen in earlier temperature jump studies except that these were done at concentrations where the protein samples were probably mostly tetramer (Brunori and Schuster, 1969;Nakamura et al., 1974).
CO Association Kinetics-In contrast to the behavior of PO,, photolysis of PC0 solutions gives rebinding kinetics which are slightly biphasic. Moreover, the overall rate of rebinding clearly increases with increasing protein concentration. Under conditions where the concentration of CO is greatly in excess over that of protein, CO rebinding after photolysis should be exponential and again should give a straight line on a semi-log plot. However, as can be seen in Fig. 5, the actual plots are clearly curved, and the overall rebinding is significantly faster at the higher protein concentration. The deviation of the data from a single exponential is never large (less than 4% of the total amplitude) but is well outside the noise level. In every case two exponentials are required to fit the kinetic data without systematic deviations, and in no case is a third exponential clearly required. When we use two exponentials, the fits consistently show two phases differing in rate by a factor of about 2.5, with the relative proportion of the amplitude of the faster phase increasing at higher protein concentrations.
There are several possible mechanisms which would pro-' Values in parentheses are 65% confidence limits. FIG. 5. CO binding kinetics. Data are shown for protein concentrations of 4.4 p M (solid truce, 36% photolysis) and 88 pM (dotted truce, 23% photolysis). The data were monitored at 436 nm and are plotted as a fraction of the initial absorbance change. For the sake of clarity, only 25% of the data points are plotted. Sample conditions are the same as in Fig. 4, except that these samples were equilibrated with a 50% CO, 50% N2 gas mixture, and they contained either a trace of dithionite or a (glucose oxidase, catalase, glucose) enzymatic system to scavenge 02. duce multiple phase kinetics in these experiments. The first would be that, as in the oxygen temperature jump data of Fig.  1, we are seeing extra phases due to the association-dissociation kinetics and the coupling between assembly and CO affinity. We do not believe this to be true. Because CO rebinding is rapid, there should be little change in association state driven by the photolysis, and both reaction phases are at least 2 orders of magnitude faster than the assembly kinetics observed for PO,. Also, assembly rates would be strongly dependent on protein concentration. We observe no significant concentration dependence of the rates but a strong variation of the proportions of the two phases. A second possibility might be that we are observing a conformational transition. However, as we vary the partial pressure of CO, the rates of both phases seem to be linear with [CO],,,. We observe no wavelength dependence of the proportions of the two phases and sharp kinetic isosbestic points, both of which imply that the absorption changes associated with both reaction phases are identical. We also see little, if any, change in the proportions of the two phases as we vary the photolysis level from 10 to 100%. Taken together, these facts strongly imply that we are observing two independent CO binding reactions, uncoupled from any association-dissociation process or conformational changes. Since the CO dissociation rate is negligible, we conclude that there must be at least two species present whose CO association rate constants, l', differ by about 2.5 and that the proportion of these species is dependent on protein concentration. We have measured the samples at both 21.5 and 10 "C, and the proportion of the faster phase is higher at the higher temperature, as expected if the formation of the faster binding species is driven by selfassociation. The rate constants we derive from these data are = 1.31 k .02 X lo7 M" s" and l'slow = 5.33 f .06 X lo6 very well with earlier flash photolysis studies a t high protein concentrations (Geraci et al., 1969), while the slower one is close to that from stopped-flow data at fairly low concentrations (and where the subunits are deoxy before mixing and, therefore, more dissociated) (Antonini et al., 1965;Brunori et al., 1966). Because the solvent and gas phase are probably not in equilibrium when we cool the cuvette to 10 "C, we are uncertain of the CO concentration at the lower temperature.
" 1 s-l at 21.5 "C. It is interesting that this faster rate agrees Therefore, we cannot give rate constants at 10 "C or derive activation energies. However, the data suggest that there is little, if any, difference in activation energy for CO binding between the fast and slow binding forms. Also, it is important to note that in order for us to observe these two distinct binding rates, interconversion between the fast and slow conformers must be slow compared to CO binding. This implies that the interconversion rate must be less than 100

S-1.
While the increase in the proportion of the faster binding phase with protein concentration implies a linkage to the association process, a direct assignment of the slower rate constant to the monomer and the faster to tetramer presents several difficulties. First, we noted that even at the lowest and highest protein concentrations we can use (about 0.5 and 100 PM), we still see both components, even though with such a large concentration range we should be going almost completely from monomer to tetramer. In view of our data for the association of PO,, we thought that perhaps the failure to see pure species at extremes of concentration was due to the presence of a significant amount of p dimer. The presence of dimers would tend to spread the overall monomer "-* tetramer association over a broader range of concentration and would also add a third component with an unknown rate constant. Since we saw no evidence for a third binding rate, we tried models where the dimer has the same rate constant as either monomer or tetramer. However, we found no set of association equilibrium constants that could reproduce the observed protein concentration dependence of the proportions of fast and slow CO binding material. That is, the quantitative modeling shows that even with dimer included, with nearly a 200-fold range in concentration, any reasonable values of association constants will give nearly pure fast or slow binding at one of the extremes of concentration. As shown in Fig. 6, we find the proportion of material binding at the slower rate is about 90% at the lowest concentrations and 20% at the highest.
Since we know that association of PC0 produces a change in the absorption spectrum (Philo et al., 1981), we decided to measure the concentration dependence of this spectral change in order to try to better define the association equilibrium. For the spectral data we have recorded the ratio of the absorbance at 579 nm (a peak in the difference spectrum) to that at 567 nm (an isosbestic point). The concentration dependence of this spectral ratio is also shown in Fig. 6. We then attempted to simultaneously least squares fit the concentration dependence of both data sets (weighted by their relative uncertainties) to a common set of association constants for monomer e dimer Ft tetramer assembly. As was the case for oxy P, there is uncertainty in assigning the spectral properties of the dimer. We tried models where the dimer has both spectral and CO binding properties either the same as monomer or the same as tetramer. With either assumption we can get only poor fits to the data, again because the strong concentration dependence of a monomer tetramer equilibrium tends to produce a larger variation in the proportion of fast and slow binding material than that which we observe. In attempting to match the data, the fits yield values for K4 either 1000 times greater or smaller than those from the gel permeation data. That is, these fits essentially reduce the system to a dimer e tetramer or a monomer dimer equilibrium (which have a lower concentration dependence), but this is clearly inconsistent with molecular weight data3 (Valdes and Ackers, 1978b).
The fitted association comtants can easily predict erroneous molecular weights because the properties we are measuring are only sensitive to the proportions of different assembly states and because the models treat the dimer properties as identical to either monomer or tetramer. The least squares fits can only attempt to determine the assembly stoichiometry from the shape of the association-dissociation curves. is determined from the relative amplitudes of a biexponential fit to the CO binding kinetics. The spectral change (closed circles, right-hand scale) is monitored by the ratio of the absorbance at 579 nm (a peak in the difference spectrum) to that at 569 nm (an isosbestic point). Sample conditions are the same as in Fig. 5. The solid line represents the simultaneous best fit of both sets of data to the model described in the text. This fit defines two curves, one for each type of data, but we have chosen the scales for plotting the spectral ratio data so that these two curves are superimposed.
We do not believe that the discrepancy between the data and this simple assembly model can be due to any impurities or degradation of the samples. These data include material derived from two different /3 chain preparations, and there is no significant difference between them. Oxidation to the ferric form can also be excluded by the spectra and by the inclusion of dithionite in some of the samples. Contamination with the p-mercuribenzoate form (PPMB) is ruled out both by the titration of free-SH groups and by control experiments which show that even the slower CO binding rate is much faster than that for pPMR. Instead, we believe the reason that these models fail to fit the data is that the assumption that each assembly state has only a single CO binding rate (either fast or slow) is incorrect. That is, we think the data indicate that (3 subunits have both a fast binding and a slow binding conformation and that both conformations might occur in all association states. However, the equilibrium between the two conformers is association-dependent, with the faster binding form more favored in tetramers. With this assumption, we again simultaneously fitted both the spectral data4 and the fast/slow proportion to a monomer-dimer-tetramer equilibrium. In this fitting process, the fast/slow proportions and spectral ratio for pure monomer and pure tetramer (i.e. the asymptotic values for each type of data at extremely low and high concentrations) are varied freely and independently, and the two types of data are linked only by the common association constants. We find that with this model we get much If we allow for two conformations within any association state, then the origin of the spectral change becomes ambiguous. The spectral change could be directly linked to association, or it could reflect a difference spectrum between the fast and slow binding conformers, which is indirectly linked to association. However, this distinction is unimportant for fitting the data, since the fitting function for the spectral ratio is the same in either case. ma Ligand Binding and Assembly Kinetics of H b 0 Subunits better fits of the data and that the best fit occurs when the dimer is assumed to have the same properties as the tetramer. This best fit is plotted as the solid line in Fig. 6. These parameters imply that monomers are 95% (88%, 100%) in the slow binding form while tetramers are 13% (9%, 17%) in the slow binding form. These data cannot distinguish whether the fast binding and slow binding conformers co-exist within the same tetramer or whether there are two classes of tetramers, with all four subunits in each type binding at the same rate.
This value for dimer association is very close to what we found from the PO, assembly kinetics, and the overall equilibrium is reasonably consistent with the gel permeation data (Valdes and Ackers, 1978b). These values imply a maximum dimer fraction of about 20%. On the other hand, when we use a model with dimer properties the same as monomer, the fits are significantly poorer, and the parameters imply that a very large fraction of dimer is present, which is inconsistent with the gel permeation data. Thus, it appears that the dimer has properties different from the monomer, but these data certainly cannot rule out the possibility that dimers have a proportion of fast and slow binding forms which is intermediate between monomers and tetramers. It may be important to note that while the optimum fit implies that the monomer contains about 5% of the fast binding conformation, it makes no significant difference if we restrict the monomers to be entirely slow binding. If this is true, the results are qualitatively different, since this would mean that only the associated states show binding heterogeneity. This point and a possible interpretation of the origin of the heterogeneity will be discussed below. With this assumption, the optimum fit implies that the tetramers are 12(f 4)% in the slow binding form and that slightly more dimer is present.

DISCUSSION
Presence and Properties of / 3 Dimers-There has been little agreement in the literature about the predominant stoichiometry for p self-assembly. The ultracentrifuge studies by Tainsky and Edelstein (1973) were interpreted as showing a dimer-tetramer equilibrium over a concentration range of 2-11 p~. In contrast, Valdes and Ackers (1977, 1978a, 1978b fitted their gel permeation data to a monomer-tetramer equilibrium over a 0.6-80 p~ concentration range. Based on attempts to fit their data with dimer included, Valdes and Ackers estimated that the maximum fraction of dimer is 10%. The association constants we obtained in this work imply up to 17 and 20% dimer for 00, and pC0, respectively, i.e. they imply that dimers are neither dominant nor negligible. Since the oxy ,8 assembly kinetics are dominated by the association of dimers, the assembly rates for 00, are very sensitive to the dimer concentration. Therefore, we feel there is little doubt that a significant amount of dimer is present. We do not feel there is really any discrepancy between our findings and the Valdes and Ackers data, since the inclusion of -20% dimer has a rather small effect on the shape of the associationdissociation curves. Rather, our ability to detect the presence of (PO,), simply reflects the fact that kinetic techniques are often more sensitive to low levels of reaction intermediates.
While our data have not given any direct evidence regarding the properties of p dimers, we note that the modeling suggests that both (PO,), and (PCO), have spectra and/or CO binding properties the same as the tetrameric forms. This seems to imply that the intersubunit interactions giving rise to quaternary enhancement in P subunits occur at the dimer level of assembly. This view is consistent with the observation that formation of ( 3 dimers is strongly enhanced in cyanomet p, which proves that the dimer intersubunit contacts are sensitive to heme ligation (Valdes and Ackers, 1978b). Kinetics and Energetics of Assembly for Oxy p-We find that the assembly rate constants kl and k, are equal and in fact are very close to values reported for the assembly of aP dimers to Hb tetramers (Ip et al., 1976) and for the association of cy and /3 into dimers (McGovern et al., 1976). Thus, it appears that Hb subunit association rates are rather independent of structural details, and the equilibrium constants are determined almost entirely by variation in the dissociation rates. These studies have established that the rate-limiting step in the dissociation of p tetramers is the tetramer + dimer step, and, therefore, it is this step which is rate-limiting for in uitro assembly of Hb A (McGovern et al., 1976).
These studies have shown that the energy of assembly of dimers is only slightly less than that for the dimer-tetramer step. This contradicts the suggestion by Valdes and Ackers (1978b) that the energy is approximately the same per intersubunit interface formed. Since there are six interfaces/tetramer, this argument predicts that the energy change at the dimer-tetramer step would be twice that at the monomerdimer step.
Quaternary Enhancement-The ligand binding kinetic data establish that the kinetic expression of the quaternary enhancement effect is opposite for 0, and CO. For 0, the higher affinity of tetramers is due exclusively to a lower dissociation rate, while for CO it is due almost exclusively to an increased association rate. This pattern is consistent with the manner in which the R uersus T affinity difference alters ligand binding kinetics in Hb A. For 0, binding to the p subunits in Hb A, the higher affinity of the R state is due to a -100-fold decrease in dissociation rate and only a =5-fold increase in association rate (Sawicki and Gibson, 1977). For CO binding, the Hb A data have not distinguished cy from p but suggest that the higher R state affinity is due primarily to a ~7 0 -f o l d increase in association rate, with only a -10-fold decrease in dissociation rate (Sawicki and Gibson, 1978). Therefore, these kinetic data do not suggest that the mechanisms altering the affinity of the heme between p tetramers and monomers is fundamentally different than the mechanisms altering heme affinities in Hb A.
Origin of the CO Binding Heterogeneity-The CO binding kinetic data clearly indicate that not all the subunits within P4 bind CO at the same rate and that the same is probably true of p,. A binding heterogeneity may also exist in the monomer, but the data are also consistent with homogeneous slow binding for the monomer. Can we saw anything about the origin and structural basis for this heterogeneity? In discussing the model used to treat these data, we postulated an equilibrium between a fast binding and a slow binding conformation for the subunits, and that association to dimers and tetramers tips this equilibrium strongly toward the faster binding conformer. It is not surprising that formation of intersubunit contacts might produce a conformation with different ligand binding properties, since hemoglobin subunits are clearly designed to make the heme sensitive to such contacts. We were surprised to find, however, that the formation of P4 does not seem to switch all of the hemes to the faster binding form.
We believe a possible explanation of the binding heterogeneity is that it is related to rotational disorder of the heme. It was recently shown that about 10% of the p subunits in native Hb A have the heme group rotated 180" about the a,y-meso axis from the "normal" orientation ( La Mar et al., 1985). For these "backward" hemes, the normal contacts between the heme vinyl and protein side chains are disrupted, including the one with Val-FG5 which has been suggested to be important for heme-heme interaction (Gelin and Karplus, 1977). There is evidence that the subunits with reversed hemes have different ligand affinity (Yamamoto and La Mar, 1986) and that heme reversal can affect the equilibrium between quaternary structure^.^ Since 10 f 3% of the isolated / 3 subunits also have reversed hemes: we would like to suggest that the subunits within p4 which bind CO at the slower rate are those which have the reversed heme orientation. That is, our preferred model for these data is that as monomers all the subunits bind CO at the same slow rate. Upon association, the intersubunit contacts cause a conformational change which raises the binding rate only in those subunits with the normal heme orientation.
We propose that the reversed heme prevents the subunit from "feeling" the association, so it retains monomer-like properties. This model provides both a structural basis for the heterogeneity and a quantitative explanation for its extent. We should also note that the true equilibrium distribution between the normal and reversed heme orientations in p subunits could itself vary with association state.
However, since it is known that heme reorientation in Hb A can require months (La Mar et aL, 1985;Yamamoto and La Mar, 1986), it is unlikely that any significant change in orientation occurs during the few hours these samples are at low concentrations (they are stored under conditions where they are tetrameric).
If this model is correct, it predicts that increasing the fraction of disordered heme would increase the fraction of slow binding material at high concentrations. Unfortunately, attempts to prepare heme-disordered p subunits by heme reconstitution have been unsuccessful. Surprisingly, NMR indicates that heme-reconstituted p subunits do not have significantly more reversed heme.5 Therefore, more definitive proof of this hypothesis will have to await the possible development of an alternative method for preparing heme-disordered p subunits.
Finally, we should also note that the association-dissociation reactions of oxy / 3 could also be similarly heterogeneous, but our methods are not sufficiently sensitive to have detected such heterogeneity.
p Subunits as Models for R State Hb A-As noted in the introduction, in many studies of Hb the isolated subunits are used as models for the properties of the Hb R state. Given the 3-4-fold differences between p and p4 in both kinetic (this work) and equilibrium Ackers, 1978a, 1978b) ligand binding properties, such modeling must be approached with great caution. Our earlier spectral studies (Philo et al., 1981) suggested that the p subunits in HbO, are more similar to PO, than to (PO,),. Is this supported by the ligand binding kinetics?
Unfortunately, a numerical comparison with literature Values for O2 and CO binding kinetics for R state Hb is difficult due to variations in sample conditions. Our new O2 association rate constants for p and p, are somewhat higher than the average a and p values reported for R state Hb A (Sawicki and Gibson, 1977;Gibson and Edelstein, 1987). The 0, dissociation rate for the p subunits in HbO, under these solution conditions was recently reported to be 28 s-' (Gibson and Edelstein, 1987). This is clearly much faster than our derived rate for p monomer but essentially identical to what we derive for P tetramers. G. N. La Mar, private communication.
For CO association, a comparison to Hb A is more difficult since the literature data do not distinguish a from p and because in many of the older studies it was not recognized that even low levels of photolysis lead to a partial switch to T. An average a and p R state rate of 1.2 X lo7 M" s-' under similar conditions has been reported (Campbell et al., 1984). Some preliminary studies of cu(cyanomet)p(CO) symmetric valence hybrids in our laboratory under these buffer conditions6 indicate a / 3 R state rate of about 1 X lo7 M" s-'.
Therefore, for CO binding it also appears that p4 is more similar to R state Hb A than is 8. Thus, both O2 and CO binding kinetics suggest that Pp is a better (but not perfect) model of the /3 subunit in R state Hb A, in contradiction to the spectral similarities. These results are further evidence that absorption spectra are a poor indicator of affinity differences.