Determination of the Rate and Equilibrium Constants for Oxygen and Carbon Monoxide Binding to R-state Human Hemoglobin Cross-linked between the (X Subunits at Lysine 99”

The kinetics of O2 and CO binding to R-state human hemoglobin A. and human hemoglobin cross-linked between the a chains at Lyses residues were examined using ligand displacement and partial photolysis tech- niques. Oxygen equilibrium curves were measured by Imai’s continuous recording method (Imai, K. (1981) Methods Enxymol. 76,438-449). The rate of the R to T transition was determined after full laser photolysis of the carbon monoxide derivative by measuring the resultant absorbance changes at an isosbestic point for ligand binding. Chemical cross-linking caused the R-state 0 2 affinity of a subunits to decrease 6-fold compared with unmodified hemoglobin. This inhibition of O2 binding was the result of both a decrease in the rate constant for ligand association and an increase in the rate constant for dissociation. The O2 affinity of R-state B subunits was reduced 2-fold because of an increase in the O2 disso- ciation rate constant. These changes were attributed to proximal effects on the R-state hemes as the result of the covalent cross-link between a chain G helices. This proximal strain in cross-linked hemoglobin was also expressed as a &fold higher rate for the unli- ganded R to T allosteric transition. The fourth O2 equilibrium binding constant, K4, measured by could be used to ana- lyze

Recent advances in protein engineering combined with an increasing clinical demand for a blood substitute have prompted the design of modified hemoglobins for use in oxygen-carrying fluids. To be considered for clinical applications the cell-free form of the protein should not dissociate into dimers, which filter through the kidneys. In addition, the Ps0 and oxygen dissociation rate constants of the protein must be large enough to allow rapid O2 unloading in respiring tissues. Generally, ideal values for these latter properties are defined by those of native, intracellular human hemoglobin in the presence of allosteric effectors.
Human hemoglobin has been site specifically cross-linked with a four-carbon fumaryl group covalently bound between a Lysg9 residues (aaHb), and this derivative is being examined as a potential blood substitute. aa-Cross-linked hemoglobin has been studied functionally by measurements of O2 and C02 binding and structurally by x-ray crystallography and resonance Raman spectroscopy (Chatterjee et d., 1986;Vandegriff et al., 1989Vandegriff et al., , 1991Larsen et al., 1990). This protein is a stable tetramer with ligand binding properties similar to those of intracellular hemoglobin (ie. a high degree of cooperativity and reduced affinity for oxygen). The functional effects of aa-cross-linking hemoglobin appear to be manifested primarily in the later stages of ligand binding, when the protein is mostly in the R-state. Like native hemoglobin, aaHb switches from the T to the R quaternary conformation when two to three ligands are bound (Vandegriff et al., 1989); however, precise determinations of the R-and T-state ligand binding properties of this protein have not been made.
The purpose of this study was 2-fold. First, O2 and CO association and dissociation kinetics were measured to examine effects of the aa-cross-link on the ligand binding properties of R-state hemoglobin. The switchover between the T and R conformation occurs on average after 2.3 ligands have been bound to either human hemoglobin A. (HbAo) or aaHb.' Consequently, kinetic measurements of the binding of the fourth ligand ( i e . Hb4(02)B + O2 c-, Hb4(02)4) can be used to define R-state properties. The low R-state affinity of cYaHb has been interpreted by a structural model in which the cross-link attenuates movement of the G helices and the joint region between the FG corner of the a chains and the C helix of the p chains. This model predicts that movement of the proximal histidine toward the heme plane is still restricted after the quaternary transition and that R-state a hemes are ' The abbreviations used are: Hb, hemoglobin; bis-Tris, (2-[bis(2hydroxyethyl)amino]-2-(hydroxymethyl)-propane-l,3-diol); nBNC, n-butyl isocyanide; pMB, para-hydroxymercuribenzoate.
17049 unable to adopt a completely planar conformation (Vandegriff et al., 1989;Larsen et al., 1990). In this study, kinetic ligand binding reactions of R-state a a H b were evaluated in terms of this structural model and were consistent with its interpretation. The second goal was to determine the equilibrium constant for the binding of the fourth O2 molecule independently by kinetic measurements. Discrepancies between K4 values measured by kinetic and equilibrium techniques have been reported but not resolved (Gibson and Edelstein, 1987;Philo and Lary, 1990). The techniques for measuring oxygen binding at equilibrium are usually not precise enough to allow an accurate determination of K4, and an independent method for assigning values to this binding constant is necessary. In general, it is difficult to resolve all four binding constants from oxygen equilibrium curves using fitting procedures alone. The low levels of partially liganded intermediates which arise during oxygenation of HbAo make the intermediate binding constants difficult to evaluate Vandegriff et al., 1989), and the dissociation of hemoglobin in dilute solutions causes errors in analyses which assume that only tetramers are present (Mills et al., 1976;Johnson and Ackers, 1977). Often the upper asymptote of the Hill plot cannot be defined well experimentally, and only the product K3K4 can be obtained with certainty (Winslow et al., 1977). Slight variations in normalizing the upper ends of HbAo equilibrium curves cause significant variation in the fitted values of K3 and K4 (Marden et al., 1989). Tetramer dissociation does not occur for aaHb, and partially liganded intermediates build up to a greater extent during O2 binding for the cross-linked protein than for HbAo (Vandegriff et al., 1989). This eliminates two problems associated with the interpretation of oxygen binding data, and thus, aaHb provides a simpler system for testing the applicability of kinetically determined values of K4 to the analysis of equilibrium curves.

MATERIALS AND METHODS
Hemoglobin Samples-HbAo and cycvHb were prepared as described previously (Christensen et al., 1988;Vandegriff et al., 1989Vandegriff et al., , 1991. All equilibrium and kinetic ligand binding measurements were carried out in 50 mM his-Tris, 0.1 M NaC1, and 1 mM EDTA buffer at pH 7.4, and 25 "C. Ligand Displacement Reactions-The rates of ligand dissociation were measured by ligand displacement reactions (Olson, 1981a). For 0, and n-butyl isocyanide dissociation, the ligands were displaced by CO. For CO dissociation, CO was displaced by NO. The reactions were measured in a Gibson-Dionex stopped-flow, rapid mixing spectrophotometer, interfaced to an OLIS model 3820 data collection system (Athens, GA). All time courses, measured a t wavelengths giving the maximal absorbance change, were fitted to a two-exponential expression with equal spectral amplitudes for the two phases.
A A , is the absorbance change a t time t, Mo is the total change in absorbance, and kf and 12, are the observed fast and slow first-order rate constants.
The observed ligand replacement rates, r,,hr. are equal to where X is the displaced ligand and Y is the displacing ligand; k, and hi are rate constants for the dissociation and association of ligand X , respectively, and k;, is the association constant for ligand Y (Olson, 1981b For n-butyl isocyanide (nBNC) displacement by CO, knBNC was determined directly as robs since k L R~~ << l'.
R-state Ligand A S S O C~Q~~O~ Kinetics-0, association kinetics were measured using a flash photolysis apparatus equipped with a Phase-R 2100B flash lamp pumped dye laser (Durham, NH). The reactions were studied a t 100 and 20% photolytic light intensities to produce complete and partial 0 2 photolysis, respectively. Time courses were monitored a t 436 nm and fitted to a two-exponential expression in which the spectral amplitudes of the two phases were either fixed or allowed to vary in an attempt to account for differences in the photolytic quantum yield of the cy and @ subunits (Morris et al., 1984).
CO association kinetics were measured using a conventional thyristor flash apparatus with photographic strobes (Sunpak Auto 544). The light intensity was set to 2% of full photolytic intensity to measure R-state kinetics. The time courses were monitored a t 435 nm for 10 pM hemoglobin solutions equilibrated with either 0.1 or 0.2 mM CO. The time courses were fitted to Equation 1 to determine individual rates of CO binding to the (Y and @ subunits. Oxygen Equilibrium Curues-Oxygen equilibrium curves were measured by the continuous method of Imai (1981) as described by Vandegriff et al. (1989). The enzymatic methemoglobin reducing system of Hayashi et al. (1973) was included, and the amount of methemoglobin was < 4%. The equilibrium curves were analyzed by the Adair equation (Adair, 1925) where Y ( p ) is the fractional saturation of hemoglobin,p is the oxygen pressure in mm Hg, and al through a4 are the overall Adair constants that are the products of the stepwise equilibrium constants (Vandegriff et d . , 1989). In another version of Equation 3, a4 was constrained by the kinetic value for the fourth stepwise constant, K4, by setting a4 = a3K4/4, where 1/4 is the statistical binding factor for the fourth ligand.
The equilibrium curves were also fitted to the two-state Monod-Wyman-Changeux model (Monod et al., 1965) and the three-state cooperon model Vandegriff et al., 1989) for an estimation of the allosteric constant, L (i.e., L = [To]/[%]).   assigned the fast and slow components observed in O2 displacement reactions of hemoglobin A. to dissociation from the p and a subunits, respectively. Their assignments were based on two experimental results. 1) There was correspondence between the absorbance difference spectra of the fast and slow phases, respectively, with the p ( 0 2 ) minus p(C0) and (~( 0~) minus a ( C 0 ) difference spectra for isolated subunits. 2) The binding of pMB to p CysS3 markedly increased the O2 dissociation rate constant for the fast component and had only a small effect on the rate constant for the slow component. These assignments have been confirmed by kinetic studies with genetically engineered human hemoglobins (Mathews et al., 1989). In the latter studies, the altered Oe dissociation rate constants could be assigned to mutated subunits and the "normal" rate constants to native subunits.

Oxygen Dissociation Rates and Subunit
As in the earlier studies, native a and p subunits exhibited 0 2 dissociation rate constants equal to -12 S" and -20 s-', respectively, at 20 "C.
The two kinetic phases of O2 dissociation from a a H b were both twice as fast as the comparable rates for HbAo. TWO alternative interpretations were possible. Either the dissocia- [COl/[02l). doubled, or aa-cross-linking increased the rate of O2 release from the a chains 4-fold over the rate from HbAo, with no change in the p subunit rate. In the latter explanation, the slow phase observed for O2 dissociation from a a H b would represent ligand displacement in p subunits; in the former, the slow phase would still represent ligand displacement in a subunits.
To distinguish between these interpretations for aaHb, O2 replacement was measured near an isosbestic point (574.2 nm) for the reaction in the presence and absence of 2 eq of pMBltetramer. Fig. 3 shows that the time courses at the longer wavelength (574.6 nm) were predominantly slow, and the time courses at the shorter wavelength (574 nm) were predominantly fast. A similar pattern was observed for native HbAo, in which the fast phase has already been assigned to p subunits. The presence of pMB increased the rates of the 574 nm time courses for both HbAo and aaHb. Thus, the fast and slow phases for a a H b are also a result of ligand displacement from p and a subunits, respectively. This result allows us to infer that aa-cross-linking doubles the rate of O2 release from both subunits, accounting in part for the lower R-state O2 affinity of the molecule as a whole (Table I).
To confirm that aa-cross-linking affects ligand release from both subunits, n-butyl isocyanide dissociation from a a H b was measured by displacement with CO. For this larger ligand, the difference between the p and a dissociation rate constants is at least 10-fold, and the assignment of the faster rate to p subunits is unambiguous ). Time courses were measured a t 432 nm to provide equal amplitudes  The association (k') and dissociation ( k ) rate constants represent the kinetics for the last step in ligand binding to hemoglobin tetramers and are defined as R-state parameters for the 01 and @ subunits. Association rates were measured by 20% laser photolysis of hemoglobin (100 p M in heme) in 1 atmosphere of 0, (1.25 mM 0,). The values represent the average of four determinations for each rate constant. Dissociation rates were measured by 0, displacement by CO. Hemoglobin (10 PM in heme) was equilibrated with 0 2 a t concentrations from 1.25 to 0.0625 mM and rapidly mixed with 1 mM CO. Dissociation rates (k) were determined by fitting the data in were calculated from the ratio of rate constants, k'/k. Rate constants in parentheses were measured in the presence of pMB. The errors in kinetic measurements were estimated a t +20% based on the results of Mathews et al. (1989). For the equilibrium measurements of K,, the data in Fig. 7 were fitted to Equation 3. P,, is the O2 partial pressure at half saturation; n is the Hill coefficient. Kinetics mm Hg x10-6 " x10-6 " 5.3 2.9 4.7 + 0.03 3.6 + 1.0 14.1 2.5 0.9 k 0.01 0.9 k 0.4 for the fast ( p ) and slow ( a ) phases. The rates of the two phases were 2.5 s" ( p ) and 0.2 s" ( a ) for Hb& and 3.4 s-' and 0.4 s-' for aaHb. pMB binding increased the rates to 12.0 s" ( p ) and 0.3 s" ( a ) for HbAo and 12.4 s-l and 0.4 s-' for aaHb. As in the case of 02, the fast components for both HbAo and aaHb showed absorbance difference spectra identical to that of isolated p chains and the slow components, difference spectra identical to that of isolated a chains. These results allow unequivocal assignment of the fast and slow kinetic phases of aaHb to the 0 and a subunits, respectively.
Oxygen Association Rates-Partial laser photolysis time courses for oxy-HbAo and oxy-aaHb are shown in Fig. 4A.
These data were fitted to a form of Equation 1 in which the relative amplitudes of the two phases were varied to determine R-state O2 association rate constants ( k ; ) for the individual subunits ( Table I). Using site-directed mutants, Mathews et al. (1989) assigned the faster phase in partial photolysis experiments to O2 rebinding to native / 3 subunits and the slower phase to O2 rebinding to native a subunits. Philo and Lary (1990) made the same assignments simultaneously and independently. As a further check, we added pMB to HbAo and aaHb to test whether or not modifications of p Cysg3 would preferentially affect the fast, phase.
As shown in Table I, the mercurial had no effect on the slow rate of O2 rebinding and decreased the fast rate by 20-30%. In the absence of pMB, the rate of O2 association for a a H b p subunits was the same, within experimental error, as that for HbAo p subunits, but the rate of O2 rebinding to a subunits of a a H b was -2.5-fold lower than that to native a chains.
Rates of O2 rebinding to HbAo after complete photolysis (Fig. 4B) agree with rates determined when only 20% of the ligand molecules were photolyzed (Fig. 4A). For native oxyhemoglobin a t high O2 concentration (1.25 mM), ligand rebinding occurs much more rapidly than the relaxation from the R to T quaternary conformation (Sawicki and Gibson, 1977a). Thus, both types of experiments give the same result. For aaHb, however, O2 rebinding after 100% photolysis was measurably slower toward the end of the reaction than for rebinding after partial photolysis. This decrease in rate with increasing light intensity suggests that O2 is rebinding to a mixture of T-and R-state species and may be a result of a more rapid conformational transition to the T-state by a a H b compared with HbAo (see below).
Kinetics of Carbon Monoxide Binding-Rates of CO dissociation were measured by displacement with NO (Table 11). The effect of pMB was greater on the rates of the fast phase, and by analogy to the O2 and n-butyl isocyanide results, this phase was assigned to the p subunits. The p CO dissociation parameters listed in Table 111. The time courses were also fit to Equation 1 with fitted amplitudes for the two phases, which gave rates of 1.8 X lo5 s-l and 0.5 X lo5 s" for HbA, and 1.6 X lo5 s" and 0.16 X lo5 s" for aaHb. Those fitted curves are not shown. rate constant for a a H b was 2-fold greater than that for HbAo. The a CO dissociation rate constants were equivalent for the two proteins. CO association rate constants for the R-state hemoglobins were measured after 2% photolysis. The time courses were fitted to Equation 1, and the fast and slow phases were assigned to the /? and a subunits, respectively, based on the measurements of Mathews et al. (1989) using site-directed hemoglobin mutants. The CO equilibrium constants were calculated from the ratio of the kinetic constants (L4 = l ' / l ) . CO association rate constants for both subunits of a a H b were -2-fold lower compared with Hb& (Table 11), and the CO affinities of the R-state a a H b subunits were 2-4-fold lower than those for the corresponding subunits of HbAo.
The Rate of R to T Transition in aa-Cross-linked Hemoglobin-The unliganded heme groups in human deoxyhemoglobin exhibit a substantial absorbance change when the protein switches from the R-to the T-quaternary state. This change was first characterized as the Hb* to Hb transition by Gibson  1959) and later shown to occur when isolated subunits aggregate to form deoxyhemoglobin tetramers (Brunori et al., 1968). This R to T spectral transition can also be observed at 425 nm (the isosbestic point for ligand binding) immediately after complete photolysis of carbonmonoxyhemoglobin (Sawicki and Gibson, 1976;Hofrichter et al., 1983;Marden et al., 1986). At low concentrations of CO, the rate of ligand rebinding occurs on millisecond time scales whereas the R to T transitions occur in 100-500 ps. Time courses for the R to T transitions of completely photolyzed carbonmonoxy-HbAo and aaHb were measured at 425 nm and are shown in Fig. 5  (upper panel). Estimates of the rates of the R to T transitions (k,J were determined by fitting the absorbance changes at 425 nm as described by Sawicki and Gibson (1976). Under these conditions, h for unliganded a a H b was -5-fold higher than that for HbAo. The isosbestic point for the R,, to To transition is 436 nm. Measurements at this wavelength showed only a small degree of CO rebinding on the time scale used to and are defined as R-state parameters for the cy and p subunits. CO association rates were measured by conventional flash photolysis at 2% of maximum photolytic light intensity. Hemoglobin (10 p~ in heme) was equilibrated with either 0.1 mM or 0.2 mM CO, and the time courses were monitored a t 435 nm. CO dissociation rates were determined directly from the observed rates of CO displacement by NO. Hemoglobin a t either of two concentrations (10 or 100 p~ in heme) was equilibrated with 0.1 mM CO and mixed rapidly with 1 atmosphere of NO. The time courses were measured at 419 nm for the lower protein concentration and 580 nm for the higher protein concentration. The rates were determined by fitting the time courses to Equation 1. The errors were estimated at +20% based on the results of Mathews et al. (1989). The equilibrium constant, L4, was calculated as l'/l. Rate constants in parentheses were measured in the presence of pMB.  Table I1 (i.e. HbA, showed a slightly faster rate of CO rebinding compared with aaHb).
Since the R to T transition is faster and ligand rebinding is slower for aaHb, we calculated At (mM" cm") for the R to T transition of this protein from the fitted amplitudes of the absorbance changes measured 75 ps after full photolysis of 100 pM carbonmonoxy-aaHb a t wavelengths from 400 to 460 nm (Fig. 5 , lower panel). These results were compared with the difference spectrum between isolated deoxy a and p chains and native deoxyhemoglobin tetramers (Brunori et al., 1968). The results reported by Sawicki and Gibson (1976) for the difference spectrum between F&, (AA measured a t 2 ps after photolysis) and To showed similar values of At compared with those in Fig. 5. The difference spectrum for a a H b is consistent with both the earlier R uersw T difference spectra reported for HbAo but does not agree in absolute value of At, presumably because of the contribution of ligand binding to the spectral change (i.e. the middle panel in Fig. 5 shows a small degree of ligand rebinding to a a H b 75 ps after photolysis).
Simulations of O2 Rebinding after Full Photolysis-Analysis of the 100% photolysis time course for oxy-aaHb required consideration of the rates of binding to and formation of Tstate species since the R to T transition occurs at a rate comparable to ligand binding at high O2 concentration. Simulations in Fig. 4B were calculated by numerical integration of the reactions in Scheme 1, which were taken from Sawicki and Gibson (1977a). Time-dependent concentration distributions of Tand R-state hemoglobins were evaluated. These simulations take into account alp differences and, in the case of HbAo, the fraction of a/3 dimer present.
Initial values for & in Scheme 1 were set to the fitted rates for the 425 nm time courses (Fig. 5, upper panel). Initial estimates for L were obtained by allosteric modeling of equilibrium data. The value of the conformational switching factor ( d ) , which adjusts & for the number of bound ligands, was taken from Sawicki and Gibson (1977a). R-state O2 rate constants for the a and p subunits were taken from Table I. Initial estimates for T-state rate constants were taken from values for unmodified hemoglobin (Sawicki and Gibson, The total amplitudes were slightly different for the two hemoglobins so that AA for the aaHb data was normalized to the maximum AA for HbAo. The single and dotted lines were calculated by a single exponential function with average rates of I' listed in Table I1 for HbA, and cyaHb, respectively. Lowerpanel, difference spectrum of deoxy T-state (To) and deoxy R-state hemoglobin (b). The ordinates are Ac (mM" cm"). The solid line is taken from Brunori et al. (1968) and shows the kinetic difference spectrum between deoxyhemoglobin and an equimolar mixture of a-and pdeoxy subunits. The open circles are the fitted amplitudes of the spectral changes for a a H b 75 ps after 100% photolysis divided by the millimolar protein concentration and the path length of the cuvette. 1977b). The fraction of dimers for HbAo was set to 15% using 3 p~ as the dimer-tetramer dissociation constant (Chu and Ackers, 1981).
The final fitted parameters in Table  I11 allowed a good representation of the data in Fig. 4B. The R-state dissociation  k h , k d ( k l , , k k , , ) ; cij = k+,;kd(kT,;kL,,); d = conformational switch factor; Hb = tetramers; and D = dimers.  . D is the conformational switching factor that adjusts the rate of conformational change with the number of ligands bound (Sawicki and Gibson, 1977a). Initial estimates for R-state O2 association rate constants for the N (kk,,) and /3 (kk,,) subunits were taken from the rates of the partial photolysis time courses for oxyhemoglobin ( Fig. 4 . 4 and Table I). R-state 0, dissociation rate constants for the N ( k d and 0 (kR,d subunits were taken from the fitted dissociation rates from the 0, displacement reactions (Table I). Initial estimates of the T-state O2 association ( k h , and k+,;) and dissociation (kTc, and kTp) rate constants were taken from the literature (Sawicki and Gibson, 1977b Fig. 4B were calculated with T-state association rates that were slightly less for aaHb than for HbAo and simulated T-state dissociation rates that were equal for the two hemoglobins. However, these T-state rate constants are not well defined, especially for HbAo, because of the small percentage of T-state species in the reactions. As expected from the time courses in Fig. 5 (upperpanel), the distributions of Tand R-state species as a function of time were different for the two proteins. Fig. 6 shows that during 0 2 rebinding after 100% photolysis, the amount of T-state species stayed below 8% for HbAo but rose very quickly to over 30% for LucuHb. Simulations over longer time scales showed that HbAo was completely resaturated in 100 ps, with 92% of the oxyhemoglobin species in the R-state and 8% in the T-state. Levels of saturated Tand R-state species reached their final equilibrium values (i.e. T4 = 0.01% and R4 = 99.99%) in 2 ms. a a H b was completely resaturated in 240 ps, with 60% of the oxyhemoglobin species in the R-state and 40% in the T-state; equilibrium levels of T4 (0.05%) and R4 (99.95%) were achieved in 3 ms.
Oxygen Equilibrium Binding-Oxygen equilibrium curves for HbAo and aaHb are shown in Fig. 7. a a H b has a higher Pso and slightly reduced cooperativity (Table I). The upper ends of the curves were normalized by extrapolating to the final saturation at infinite Po2 from graphical analysis of percent saturation ( Y ) uersus 1/Po2. Equilibrium binding constants were evaluated by fitting the data to the Adair equation (Equation 3).
Fitted values of an and a4 for HbAo were sensitive to normalization, and K4 was difficult to determine precisely solely from equilibrium binding data. Variation in the upper end of the HbAo curve by as little as -0.3% changed the fitted value of a3 from 9 X to 1 x lo-" mm Hg-,". Precise values of a 2 are required to calculate K4, and with this variation in a:,, the fitted value of K4 changed from 7.65 to 10" mm Hg". Thus, it is clear that K,i and K4 are highly correlated for native HbAo. Analysis of the aaHb equilibrium curve was less sensitive to normalization. The same variation in the upper end of the curve changed as from 9 X to 4 X IO-" mm Hg-3, producing only a 2-fold variation in K4 from 1.55 to 3.2 mm Hg". K4 was assigned by using normalization factors that gave the minimal residual sum of squares. The fitted values of K4 were 7.65 mm Hg-l and 1.55 mm Hg" (4.7 X lo6 M" and 0.9 X lofi M-') for HbAo and aaHb, respectively, and the residuals are shown in Fig. 8.
Kinetic Determination O f K 4 " K 4 can also be computed from the ratio of association and dissociation rate constants obtained in the partial photolysis and ligand displacement experiments. In these kinetic experiments, differences between the 01 and p subunits were resolved, and the rate parameters were assigned to ligand binding to the R-state quaternary conformation. Equilibrium constants for O2 binding to the last unliganded N or p subunit were computed from the ratios KR-= kk,/kRcy and K R~ = kk,,/kR6. If the subunits are not equivalent, the intrinsic value of K4 for equilibrium binding is defined as K4 = 2KRcrKRIj/(KR(r + KRi?).' The kinetically determined values of K4 for HbAo and NaHb were (3.6 2 1.0) X lo6 M" (5.9 mm Hg") and (0.9 k 0.4) X lofi M" (1.5 mm Hg"), respectively, and are similar to those obtained from fitting the 0, binding curves.
The equilibrium data in Fig. 7 Table I11 and are plotted as percent of total hemoglobin. These populations correspond to the simulated time courses in Fig: 4B  (intrinsic)/4 = 1.5 mm Hg" and 0.4 mm Hg" for Hb& and aaHb, respectively). The constrained Adair equation fitted the data as well as the unconstrained function, and the residuals for both models have maximum deviations of less than 1%. Thus, the kinetic value of K4 is compatible with the equilibrium binding data and is not dependent on the normalization and fitting constants used to analyze the equilibrium curves. A significant advantage of constraining K4 in the Adair equation was in the determination of the third Adair constant. The standard errors for the fitted values of u3 were reduced 10-fold when K4 was constrained, from +9.0% to +0.4% for HbAo and from +4.9% to +0.3% for aaHb.

Structural Effects of a LysS9
Cross-linking-Resonance Raman and absorption spectroscopy of transient photoproducts suggest that the tertiary conformation of aaHb R-state subunits are more "T-state like" (Larsen et al., 1990). These data suggest that the cross-link partially inhibits in-plane movement of the a iron atom even in the R quaternary conformation and also exerts proximal constraints on the B heme which are not as readily interpreted. These structural changes are manifested as 2-6-fold decreases in K R~ and K R~ when comparing aaHb with HbAo (Tables I and 11).
An alternative interpretation is that the cross-link increases the value of L, the allosteric conformational equilibrium constant, inhibiting the quaternary transition until three or four ligands are bound. This would cause a decrease in the apparent equilibrium constant for the binding of the fourth ligand molecule. However, this explanation is ruled out by our previous equilibrium binding studies, which have shown that L for the cross-linked hemoglobin is smaller than that for HbAo and that the switch from Tto R-states occurs after 2.3 ligands O x y g e n E q u i l i b r i u m C u r v e s X 10" f 7.7 X mm Hg-', ad = 1.64 X f 6.7 X lo-' mm Hg"; for a a H b al = 0.0795 +-0.0011 mm Hg", u2 = 2.70 X f 9.4 X mm Hg-', a3 = 9.03 X 1-4.4 X mm Hg-", u4 = 3.49 X 2 1.2 X mm Hg-4. have been bound (Vandegriff et al., 1989). The aaHb O2 binding data were poorly fitted when L was set to values equal t o or greater than L for Hb&. L was better determined for both aaHb and HbAo when KR was constrained by the kinetic measurement of K4. Thus, it is clear that the cross-link lowers O2 affinity by affecting the intrinsic ligand binding properties of the R-state subunits rather than by increasing the allosteric constant.
Kinetic Differences between the Subunits in aa-Cross-linked Hemoglobin-The association rate constants for 0 2 and CO binding to the a subunits in aaHb are 2-3-fold smaller than those for a subunits in HbAo (Tables I and 11). The altered configuration of the proximal heme pocket in aaHb places the R-state iron atom further out of the plane of the heme ring (Larsen et al., 1990), which could account for the decreased reactivity with ligands. Since there is little difference in the position of the E helix in the R-and T-states of native a chains (Baldwin and Chothia, 1979;Shaanan, 1983), the decrease in k&, is not likely to be caused by distal steric interactions. The O2 dissociation rate constant from R-state a subunits is increased 2-fold by the cross-link, presumably because of the added proximal strain that inhibits association although distal hydrogen bonding effects cannot be ruled out. The net result of these rate changes is a 6-fold lowering of KRn for O2 binding.
In the case of R-state @ subunits, cross-linking causes no changes in k& for O2 binding, a %fold increase in kRU, and as a result, only a 50% decrease in O2 affinity. In contrast, the rate of CO rebinding to /3 subunits in aaHb is decreased compared with HbAo, and the rate of CO dissociation increases %fold. Thus, CO affinity decreases roughly 4-fold, and discrimination against CO in favor of O2 binding is increased in / 3 subunits by the chemical cross-link.
R to T Transition in aa-Cross-linked Hemoglobin-The slower rate of O2 rebinding after 100% photolysis of aaHb is the result of both a lower rate of O2 binding to R-state a chains and a faster transition from the R to the T quaternary conformation. Simulations of time courses in Fig. 4B by numerical integration of Scheme 1 revealed higher levels of T-state intermediates during O2 rebinding to LvcvHb (Fig. 6).
Rates of O2 association with R-state aaHb were underestimated by -20% when transition to the T-state was unac- The rates of the Rto T-state transition were also measured directly at 425 nm after complete photolysis of the CO complexes of Hb& and aaHb. The obsewed values of & were 1.3 X IO4 and 7.3 X IO4 s" for Hb& and aaHb, respectively. The rate for HbAa is similar to that reported by Sawicki andGibson (1976, 1977a), and that for aaHb is nearly identical to ko obtained by analyzing O2 rebinding as shown in Figs. 4B and 6. Thus, it is clear that the a Lys* cross-link significantly enhances the rate of switching from the Rto the T-state as might be expected if it stabilizes the subunits in intermediate tertiary conformations.
The Advantage of Determining K4 Kinetically-Finallyl the kinetic data for 0 2 binding and dissociation were used to compute the Adair equilibrium constants for the last step in ligand binding to HbAo and aaHb. The resultant values of K4 were very similar to those obtained by curve fitting of 0 2 equilibrium binding data. The kinetically determined value for K4 was used to reduce the number of fitted Adair constants from four to three, and the equilibrium curves were reanalyzed. Fixing K4 had no effect on the size or distribution of the residuals of the fit; however, the confidence limits of K3 were markedly improved. These data show unambiguously that K4 can be determined independently from kinetic measurements and then used to analyze equilibrium data. This procedure prevents a3 from becoming zero or negative when the equilibrium curve is analyzed, avoiding the necessity of obtaining parameter values that are chemically unacceptable.