Polymer Structure and Solubility of Deoxyhemoglobin S in the Presence of High Concentrations of Volume-excluding 70-kDa Dextran
EFFECTS OF NON-S HEMOGLOBINS AND INHIBITORS*

Robert M. BookchinDagger and Tania Balazs

From the Department of Medicine, Albert Einstein College of Medicine, Bronx, New York 10461

Zhiping Wang, and Robert Josephs

From the Department of Molecular Genetics and Cell Biology, University of Chicago, Chicago, Illinois 60637

Virgilio L. Lew

From the Physiological Laboratory, University of Cambridge, Cambridge CB2 3EG, United Kingdom

    ABSTRACT
Top
Abstract
Introduction
Appendix
References

Earlier observations indicated that volume exclusion by admixed non-hemoglobin macromolecules lowered the polymer solubility ("Csat") of deoxyhemoglobin (Hb) S, presumably by increasing its activity. In view of the potential usefulness of these observations for in vitro studies of sickling-related polymerization, we examined the ultrastructure, solubility behavior, and phase distributions of deoxygenated mixtures of Hb S with 70-kDa dextran, a relatively inert, low ionic strength space-filling macromolecule. Increasing admixture of dextran progressively lowered the Csat of deoxyHb S. With 12 g/dl dextran, a 5-fold decrease in apparent Csat ("dextran-Csat") was obtained together with acceptable sensitivity and proportionality with the standard Csat when assessing the effects of non-S Hb admixtures (A, C, and F) or polymerization inhibitors (alkylureas or phenylalanine). The volume fraction of dextran excluding Hb was 70-75% of total deoxyHb-dextran (12 g/dl) volumes. Electron microscopy showed polymer fibers and fiber-to-crystal transitions indistinguishable from those formed without dextran. Thus when Hb quantities are limited, as with genetically engineered recombinant Hbs or transgenic sickle mice, the dextran-Csat provides convenient and reliable screening of effects of Hb S modifications on polymerization under near-physiological conditions, avoiding problems of high ionic strength.

    INTRODUCTION
Top
Abstract
Introduction
Appendix
References

The solubility of deoxyHb1 S polymers at equilibrium with the tetrameric "monomers" is a basic thermodynamic property of the polymerization process that underlies red cell sickling. Measurement of the polymer solubility, "Csat"(1) (also termed Cs (2) or Csol (3)), of Hb S in conditions similar to those within the cells has served as a critical estimate of the relative polymerization tendency when comparing the native or modified Hb S, alone or in mixtures with non-S Hbs or inhibitors of polymerization. A concentrated solution of Hb S is fully deoxygenated to form a gel, and after high speed centrifugation to sediment the polymerized solid phase, the concentration of soluble Hb in the supernatant is the measured Csat.

It is well documented that volume exclusion by macromolecules admixed with proteins results in their increased activity and corresponding interactions (4). Thus, when protein macromolecules capable of interacting, such as deoxyHb S, occupy a substantial fraction of the total volume of their medium, the addition of a second species of noninteracting macromolecules crowds the protein molecules closer together, increasing their interactions. More specifically, addition of macromolecules that do not copolymerize with deoxyHb S reduces its polymer solubility by this mechanism (5, 6). However, for the use of such observations in further investigations on the properties of deoxyHb S polymers, it is first necessary to characterize the morphology and properties of the polymer fibers generated in volume-excluded domains by comparison with those previously characterized for standard conditions.

In the course of our recent experiments designed to measure the Hb concentration in the deoxyHb S polymers, CP, we sought a macromolecular extrapolymer marker that would be excluded from the polymer water compartment and also would have minimal interaction with the polymer (7), and we found that dextran (70 kDa) appeared to behave as such an inert extrapolymer space filler.

In the present work, we demonstrate that volume exclusion by admixture of relatively high concentrations of 70-kDa dextran with Hb S, at close to physiological ionic strength, markedly reduces the solubility (Csat) of the deoxyHb while maintaining both its overall polymer architecture and general proportionality with the standard Csat in binary mixtures of Hb S with Hbs A, F, or C or in the presence of noncovalent inhibitors of polymerization.

    EXPERIMENTAL PROCEDURES

Preparation of Hemoglobin-- Heparinized venous blood was obtained, with informed consent, from SS patients, and hemolysates were prepared by the method of Drabkin (8). For experiments using highly purified Hb S, it was separated from the minor components Hb A2 and Hb F by chromatography on columns of microgranular diethylaminoethyl cellulose (DE52, Whatman) and developed with 0.05 mol/liter Tris-HCl, pH 8.3. The Hb S solutions were dialyzed overnight at 4 °C against 0.05 M potassium phosphate buffer, pH 7.50, concentrated by vacuum ultrafiltration to 30-35 g/dl (with continued simultaneous dialysis), and the concentrated Hb S either used at once or stored as frozen pellets in liquid nitrogen until use.

Standard Csat Assay-- This assay was performed as described previously (7), in the presence of 50 mM sodium dithionite, at 37 °C. When sufficient data were available from recent measurements in our laboratory using this method, they were used for comparison with the dextran-Csat, without repetition. Following addition of the dithionite, the pH of the Hb solution was 7.10 ± 0.05.

Dextran-Csat Assay-- A stock solution of dextran (average mass 70-kDa, Sigma), 32 g/dl, was prepared in 0.05 M potassium phosphate buffer, pH 7.50. Mixtures of dextran and Hb, either Hb S alone or in equimolar mixtures with non-S Hbs ("Hb S-X"), were initially prepared in 1.5-ml microcentrifuge tubes, to give final concentrations of 6-12 g Hb·dl-1 and 10-21 g dextran·dl-1 in 0.05 M potassium phosphate, pH 7.50. The solutions were overlaid with paraffin oil, chilled on ice, and deoxygenated by adding and stirring in the deoxy solution of sodium dithionite with a gas-tight syringe to give a final concentration of 0.05 M, and total volume of 500---700 µl. After incubation for 30 min at room temperature (22 °C) or in a 37 °C water bath, the resulting gel under the oil layer was carefully but vigorously disrupted with a narrow plunger or wire loop,2 and the tubes were spun at room temperature in an Eppendorf model 5415 centrifuge at 14,000 rpm (16,000 × g) for 20 min. The gel disruption was repeated twice, followed by a final 30-min spin. Pilot experiments indicated that no further decrease in supernatant Hb concentration occurred with additional or longer incubation or centrifugation. After determining the presence or absence of a pelleted solid Hb phase by viewing the tube before a bright light, the oil was aspirated, and replicate 10- or 20-µl samples of the supernatant were mixed with Drabkin's solution to measure the supernatant Hb concentration that represented the dextran-Csat.3

For the reasons noted below, the final total dextran concentrations chosen for the assay were 10 or 12 g/dl. It was found possible to scale down the total volume of the mixtures to 100 µl, reducing the total Hb needed per measurement to about 5-10 mg per tube, and still maintain good reproducibility of results. In order to have a sufficient volume of the soluble phase after centrifugation to permit replicate measurements of the Hb concentrations, it was important to limit the volume fraction of the polymer; this was found to be satisfactory when the initial total Hb concentrations were about 1-2 g/dl higher than the dextran-Csat, whereby the polymer pellet did not comprise most of the Hb. As with the standard Csat measurements, achieving this balance between soluble and pellet volume sometimes required trial and error, i.e. one or more pilot experiments to determine the appropriate total Hb concentrations with Hb mixtures or with inhibitors of polymerization.

Electron Microscopy of DeoxyHb S Fibers-- These studies employed a Philips CM120 electron microscope operating at 120 kV. Micrographs were recorded at a magnification of 45,000. Hb S fibers were prepared as described previously (9, 10) except that in the present studies, Hb S fibers were obtained in the presence of a variety of dextran concentrations up to 20 g/dl, and with Hb S concentrations of 7 g/dl. Higher concentrations of dextran were not used because the high viscosity of the solutions resulted in poor staining. It was necessary to ensure that nearly all of the dextran was removed during the negative staining, since residual dextran prevented penetration of the stain into the particles. Even with this precaution, individual fibers produced very weak diffraction patterns; bundles of fibers, however, which have a higher signal-to-noise ratio, did produce optical diffraction patterns whose spacings and intensity distributions were clearly visible.

    RESULTS

Effects of Varying Concentrations of Dextran-- We first explored the effect of dextran concentrations on the Csat of deoxyHb S. Fig. 1 shows that the dextran-Csat fell progressively with increasing concentrations of dextran. Increasing the initial Hb concentrations at constant dextran had a much smaller effect on the dextran-Csat, as shown in Figs. 1 and 2. As analyzed below, these results are consistent with the increased activity (or "effective" concentration) of the Hb with increasing volume exclusion by dextran, whereas most of an increase in the initial Hb concentration is incorporated into a larger polymer fraction, resulting in small or no changes in the Csat (Fig. 2).


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Fig. 1.   Effect of increasing concentration of 70-kDa dextran on the Csat of deoxyHb S. Each point represents the mean of at least two replicate measurements, none of which differed by more than 0.3 g/dl. Samples were incubated at room temperature (21 °C) (circles, with initial Hb concentration 4 g/dl, open circle , or 5 g/dl, ; or 37 °C, inverted triangles, with initial Hb concentration 4 g/dl, down-triangle, or 5 g/dl, black-down-triangle , prior to centrifugation (see "Experimental Procedures").


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Fig. 2.   Effect of varying initial total Hb concentrations on the dextran (12 g/dl)- Csat of deoxyHb S alone (open circle ) and of deoxygenated equimolar mixtures of Hbs S + A (), S + C (down-triangle), or S + F (black-down-triangle ).

Although the dextran-Csat could be reduced to 1 g of Hb/dl or less with dextran concentrations over 20 g/dl, the sensitivity of the assay for detecting polymer solubility differences induced by inhibitors or admixtures of non-S Hbs was correspondingly reduced in these conditions by larger proportionate errors (not shown). We therefore sought one or more dextran concentrations that would provide a satisfactory balance of a relatively low dextran-Csat together with reasonably good sensitivity in detecting inhibition of polymerization. As seen in Fig. 2, with 12 g/dl dextran, the dextran-Csat values were fairly stable over a range of initial Hb concentrations, and the expected directional increases in dextran-Csat occurred upon equimolar admixture of Hb S with the non-S Hbs A and F. The dextran-Csat values with mixtures of Hbs S and C were close to, but consistently slightly lower than, those of Hb S + A mixtures4 with both dextran concentrations, 10 or 12 g/dl (Tables I and II).

                              
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Table I
Comparison of standard and dextran (12 g/dl)-Csat; effects of admixture of non-S Hbs and of non-covalent inhibitors of polymerization
All Csat measurements were performed at 37 °C. Values for Standard Csat (in the absence of dextran) were taken from previous studies in this laboratory for Hb S alone and Hb mixtures; new measurements were made for Hb S mixed with the non-covalent inhibitors, where each value represents the mean of at least two replicate measurements; the two or more replicates were always within 2% of one another.

                              
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Table II
Comparison of standard and dextran (10 g/dl)-Csat; effects of admixture of non-S Hbs and of non-covalent inhibitors of polymerization
All Csat measurements were performed at 37 °C. Values for Standard Csat (in the absence of dextran) were taken from previous studies in this laboratory for Hb S alone and Hb mixtures; new measurements were made for Hb S mixed with the non-covalent inhibitors, where each value represents the mean of at least two replicate measurements; the two or more replicates were always within 2% of one another.

Comparison of Standard and Dextran-Csat, Effects of Hb Mixtures and Inhibitors of Polymerization-- Tables I and II compare the standard Csat and dextran-Csat values (in the presence and absence of 10 and 12 g/dl dextran, respectively) of Hb S alone, the above Hb mixtures, and of Hb S in the presence of some noncovalent inhibitors of deoxyHb S polymerization, phenylalanine, ethylurea, and butylurea. For each test condition, the directional increases of the dextran-Csat paralleled those of the standard Csat. But the ratio of dextran-Csat/Csat was not constant; it was lowest with Hb S alone, and generally tended to increase with increasing Csat (see explanation derived under "Appendix" and summarized under "Discussion"). Nevertheless, there was good overall proportionality between the relative increases in Csat and dextran-Csat, in that the relative order of differences was maintained.

Ultrastructure of DeoxyHb S Polymers Formed in the Presence of Dextran-- We wished to determine if dextran caused changes in the structure of deoxyHb S fibers, which could affect their appearance or their interactions. We examined electron micrographs to determine if the presence of dextran had an effect on the fiber appearance or the interactions associated with the fiber-to-crystal transition.

An electron micrograph of negatively stained deoxyHb S fibers formed in the presence of dextran, shown in Fig. 3, reveals that the overall appearance and dimensions of the fibers were the same as those observed previously in the absence of dextran (9, 10). With dextran, optical diffraction patterns from individual fibers were very faint, barely showing the 32-Å layer line, and the 64-Å and near equatorial diffraction were also not evident. We attribute this to incomplete removal of the dextran, which tends to adhere to the fibers. Such binding occurs with a variety of specimens suspended in viscous solutions. With fiber bundles, however, each of those characteristic features of the optical diffraction patterns were seen (Fig. 4).


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Fig. 3.   An electron micrograph of deoxyHb S fibers formed in the presence of dextran (12 g/dl). The fiber diameter is about 220 Å. Optical diffraction from individual fibers was very weak, probably because some dextran remained associated with the fiber and inhibited the penetration of the negative stain. However, the 32- and 64-Å layer lines and near equatorial diffraction previously observed in fibers formed in dilute phosphate were evident in micrographs of fiber bundles (see Fig. 4).


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Fig. 4.   a, an electron micrograph of a bundle of deoxyHb S fibers formed in the presence of dextran. b, an optical diffraction pattern of the bundle. The diffracted intensity was stronger than in individual fibers, since many fibers contribute to the pattern, increasing the signal-to-noise ratio. The pattern shows the 32- and 64-Å layer lines and the near equatorial region and is typical of those previously observed for fiber bundles formed in 0.05 M phosphate buffer (9).

It was previously shown that deoxyHb S fibers undergo a series of transitions over several hours, ultimately forming crystals (9, 10). There are two crystallization pathways. At pH levels above 6.5, fibers form bundles which then crystallize (10). Fig. 4a shows a micrograph of a bundle of fibers formed in dextran. Optical diffraction patterns from such bundles display the 32- and 64-Å layer lines seen with fibers formed in dilute phosphate buffers. In addition, there is diffracted intensity near the equator arising from the helical twist of the fibers. The 32-Å meridional reflection coupled with the 64-Å layer line confirm that the Hb molecules are arranged in the same half-staggered disposition as with fibers formed under physiological conditions. Since bundles contain many fibers, they generate a higher signal-to-noise ratio and reveal the diffraction pattern more clearly than do single fibers.

Below pH 6.5, fibers form macrofibers (10) in which the arrangement of the hemoglobin molecules is similar to that in crystals, except that they form a helical array, whereas in crystals the molecules are arranged in a linear array. Fig. 5a presents an electron micrograph of a macrofiber formed from fibers grown in the presence of dextran. Its appearance and the spacings in its optical diffraction pattern are indistinguishable from macrofibers obtained in the absence of dextran.


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Fig. 5.   a, an electron micrograph of a deoxyHb S macrofiber formed in the presence of dextran. The diameter of the macrofiber varies from a maximum of 620 Å to a minimum of 520 Å, as observed previously for macrofibers formed in 0.05 M phosphate buffer (31). b, an optical transform of the macrofiber, displaying the 32- and 64-Å layer lines and near equatorial intensity.

Both macrofibers and bundles form crystals. An example of a typical crystal formed from fibers grown in dextran is shown in Fig. 6a along with its optical diffraction pattern (Fig. 6b). The crystal spacings of 53 × 64 Å are the same as those of crystals grown without dextran. This view corresponds to a b axis projection of the crystal. Vassar et al. (12) found that deoxyHb S polymer fibers and crystals formed in polyethylene glycol (PEG) exhibit the same structure as those formed in dilute phosphate. Crystallization was preceded by the formation of short precursors, similar to those seen in Fig. 6, suggesting that dextran and PEG may have similar effects in promoting polymerization.


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Fig. 6.   a, an electron micrograph of a crystal of deoxyHb S formed in the presence of dextran. The unit cell dimensions are 53 × 64 Å, which corresponds to a b axis projection of the crystal. The background also shows short crystallization precursors (see arrow) similar to those seen when crystals form in the presence of PEG (see text). b, the optical diffraction displays unit cell dimensions of 64 Å in the axial direction and spacings of 53 Å in the radial direction. These are the same as the unit cell dimensions of the Wishner-Love crystal of deoxyHb S (32).


    DISCUSSION

The present results demonstrate that admixture with 70-kDa dextran markedly reduced the polymer solubility of deoxyHb-S, as reflected in the dextran-Csat, while maintaining general proportionality with the standard Csat when assessing the effects of non-S Hb admixtures and polymerization inhibitors. Furthermore, the polymer architecture, viewed by electron microscopy, appeared unchanged by the presence of dextran. The general appearance and the spacings in optical diffraction patterns of deoxyHb S fibers formed in the presence of dextran were indistinguishable from those of fibers formed without dextran. In addition, the fiber-to-crystal transitions exhibited by deoxyHb S, which must depend on the basic fiber structure, were also unaltered by the presence of dextran.

These findings suggest that the effects of 70-kDa dextran in this system can be attributed essentially to its excluded volume and are consistent with our previous studies using [14C]dextran (70 kDa) as a marker of the extrapolymer phase (7), which indicated that it showed no measurable interaction with Hb and might serve as an inert space-filling macromolecule. This behavior stands in contrast to that of some other non-polymerizing macromolecules such as albumin, which contribute volume to the system and lower the polymer solubility (6) but do not behave consistently as an inert macromolecule, probably because its extensive surface charges result in ionic interactions with Hb (7).

A quantitative characterization of the various phases or compartments generated by deoxygenation of dextran-Hb mixtures, as derived and analyzed under "Appendix," is presented in Table III. We consider first the condition with deoxyHb S alone, without admixture of other Hbs. The coefficients of variation (CV) for the different parameters show that P and fP, which are largely determined by the total Hb concentration (CT) used in each assay, show large variations, whereas the other parameters, which reflect intrinsic physical properties of the system and would thus be expected to vary little with the different experimental conditions, are relatively constant. The volume fraction of dextran that excludes Hb (fDex) constitutes about 75% of the total volume of the dextran-Hb system, and the remaining volume is distributed between that occupied by deoxyHb S polymer (fP, about 8%) and a soluble phase containing the soluble deoxyHb S molecules at equilibrium with polymer (fs, about 17%). The ratio between the volume of dextran that excludes soluble Hb S to that which excludes water (rex) is about 9.4, indicating that the molecular frame of the 70-kDa dextran molecules is nearly 10-fold less accessible to macromolecules than to water.

                              
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Table III
Statistical parameters of the dextran-deoxy-Hb system for SS, SA, SC, and SF mixtures, as described in the Appendix
The table presents the mean ± S.E. of the mean, standard deviation, and coefficient of variation for the dextran 12-Csat (Csatdex) for the fraction of Hb which is polymerized (P), the volume fractions of polymer (fp), of dextran which excludes soluble macromolecules (fDex), of solution available for soluble deoxy-Hb molecules (fS), and for the ratios of polymer volume which excludes macromolecules to that which excludes water (rex), and for fS/(1 - fP), which is equivalent to dextran-Csat/standard Csat reported in Table I, according to Equation 22. See "Appendix" for the derivation of each parameter.

Similar results were obtained with the different binary Hb mixtures (S-X) investigated. However, some characteristic differences were apparent. The volume fraction of dextran (fDex) was somewhat reduced, to about 70% of the total volume, and this was matched by a proportional reduction in dextran's macromolecule/water exclusion ratio (rex). In this system, the Hb S-X mixtures differed from Hb S alone primarily in their forming a proportionally larger residual (non-dextran) volume (fs + fP, about 30% compared with 25% for Hb S alone) which was partitioned between smaller polymer phases (fP, 3.5-5.5% relative to 8% for Hb S alone) and larger soluble Hb phases (fs, 23-30% relative to 17% for Hb S alone). Similar changes of a smaller degree were seen when the noncovalent polymerization inhibitors were added to Hb S. Possible explanations for these differences from Hb S alone are currently under investigation.

These relative changes in the total fP and fs and their partitioning are reflected in the observed variations in the ratio of dextran-Csat/standard Csat shown in Tables I and II. The net effect is a rise in this ratio with a rise in the dextran-Csat, which serves to magnify the effects of polymer inhibition on the dextran-Csat.

Refinements in our knowledge of the mechanisms of polymerization of deoxyHb S and the effects of molecular and cellular alterations that might prove useful in treating sickle cell disease have become possible with the advent of techniques such as site-directed mutagenesis for preparing selected mutant Hbs (13, 14), recombinant Hbs having both the Val-beta 6 (sickle) substitution and another chosen substitution (double mutants) (15), as well as the development of varieties of sickling transgenic mice (16, 17). In each case, it is important to assess accurately the polymerization and sickling tendency obtained with the new Hb or mixture, yet these Hbs are generally available in very limited amounts.

The standard Csat assay requires relatively large quantities of Hb (usually over 100-150 mg) at high concentrations (usually more than 25 g/dl). Two alternative assays that require smaller amounts of Hb than the standard Csat have been used to try to circumvent this problem. One assay, devised by Benesch et al. (18), is based on the sharp decrease in Hb O2 affinity associated with Hb S polymerization and gelation. In experienced hands, this can be a fairly accurate scanning assay of the minimum Hb concentration for gelation, albeit with some limitations (19, 20); but each measurement requires multiple determinations of Hb O2 affinity (p50) and for micro-scale determinations requires equipment (such as the Hem-O-Scan from Aminco, no longer manufactured) that is not readily available.

Another method commonly employed to assay deoxyHb S solubility with small amounts of Hb is based on its marked decrease in solubility in concentrated phosphate buffers (21), originally described by Itano (22) as a "salting out" of the protein in 2.24 M phosphate. This method was modified by Adachi and Asakura (23, 24) into a kinetic assay which measures the delay time of deoxyHb aggregation, generally in 1.8 M phosphate buffers. Adachi and co-workers (23, 24) have found many similarities in the high phosphate solubility behavior of some structurally modified Hb S or Hb S in mixtures with other Hbs, as compared with the more physiological Csat assay. However, the solubility behavior of modified deoxyHb S or Hb mixtures in high ionic strength media does not consistently reflect its properties in more physiological conditions (25, 26), in part because of the shielding of electrostatic (ionic) intermolecular interactions that appear to be involved in deoxyHb S polymerization (27, 28).

A new kinetic micromethod that determines the solubility of deoxyHb S in a gel by measuring the recombination rate of tracer amounts of CO after laser photolysis was recently described by Ferrone and co-workers (20). This elegant method requires only a few microliters of concentrated Hb and can estimate the temperature dependence of the solubility. The sophisticated equipment required for the method (29), however, limits its general application as a screening procedure on small samples.

Recent application of our dextran-Csat method to studies of a recombinant sickle Hb triple mutant showed similar inhibitory effects on polymerization when compared with the method based on the shift in oxygen affinity (18, 30). That study employed 10 g/dl 70-kDa dextran in mixtures with Hb S and the variants. Our present work indicates that a satisfactory balance between reduction in the Csat values and adequate sensitivity for screening polymer solubility can be obtained with 12 g/dl dextran in the mixtures with Hb, which reduces further the quantities of Hb needed for assay.

Furthermore, the increase in the ratio of dextran Csat/standard Csat observed in conditions that increase polymer solubility (Tables I and II) magnifies the effects of polymer inhibition on the dextran-Csat. This effect should help counterbalance the lesser sensitivity that accompanies the much smaller absolute values obtained with this method.

    FOOTNOTES

* This work was supported by Grants HL28018, HL58512, and HL22654 from the National Institutes of Health and the Wellcome Trust (UK).The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

Dagger To whom correspondence should be addressed: Albert Einstein College of Medicine, 1300 Morris Park Ave., Bronx, NY 10461. Tel.: 718-430-2123; Fax: 718-904-1164; E-mail: bookchin{at}aecom.yu.edu.

2 In the presence of high concentrations of dextran, centrifugation of the gels without prior disruption, either in the Eppendorf 5415 centrifuge at 14,000 rpm or, as with the standard Csat determination (7), in a Sorvall RC-80 ultracentrifuge with a TST 60.4 rotor at 45,000 rpm, did not sediment the polymer phase. After mechanical disruption of the gel, however, the solid phase was readily sedimented by centrifugation in the microcentrifuge.

3 In many instances, after centrifugation, a light yellow firm pellet layer was observed between the supernatant and the oil. It varied considerably in thickness, even between replicate tubes, and was presumed to be a dextran gel. Since the dextran-Csat values measured in such replicates were consistently in good agreement despite grossly different sized dextran pellets, we concluded that the concentration of dextran in the gel must have been about equal to that in the soluble phase and that its formation and size did not affect the results.

4 We found somewhat larger differences (with Csat of S+ C < S+ A) in measurements closer to physiological ionic strength (11). For the purposes of the present studies, however, delineation of those modest differences was considered secondary to the need to maintain full deoxygenation with 50 mM sodium dithionite, which raised the ionic strength above normal levels.

    ABBREVIATIONS

The abbreviations used are: Hb, hemoglobin; Csat, polymer solubility of deoxyhemoglobin S; CP, concentration of Hb in the polymer; PEG, polyethylene glycol; see "Appendix" for definitions used in equations.

    APPENDIX

The experimental protocol described under "Experimental Procedures" provides the quantities from which we can compute the relevant parameters of the dextran-Hb system, which are listed in Tables II and III and discussed above. We consider a solution containing dextran (70 kDa) and Hb; we measure the density of dextran (delta D), the total concentrations of dextran (CDT) and of hemoglobin (CT), and the concentration of soluble deoxyHb S in the supernatant after sedimenting the gel (Csat). By using the equations derived below with these measured quantities, we can compute the concentration of dextran in its water-exclusion volume (CD), the fraction of polymerized deoxyHb S (P), the ratio between dextran volume which excludes Hb molecules relative to that which excludes water molecules (fex), the volume fractions of Hb excluding dextran (fDex), of Hb polymer (fP), and of extra polymer solution containing soluble Hb (fs). We can also analyze observed differences in the empirically determined ratios between the standard Csat values, measured in the absence of dextran, and the dextran-Csat, in pure Hb S solutions in the presence and absence of polymerization inhibitors and in mixtures of Hb S with non-S Hbs.

Definitions, Conventions and Glossary-- Units are given only when relevant. Since most equations use and derive relative quantities, those measurements used for ratios must be expressed in identical units for comparability. The definitions used are as follows: CDT, total dextran concentration (in g/dl); CD, concentration of dextran in its water-exclusion volume (inverse of the specific volume of dextran, in g/dl); CH, concentration of soluble Hb in its water-exclusion volume (inverse of the specific volume of Hb, in g/dl); CP, concentration of deoxyHb S within the polymer, either in g/dl or mM, defined by CP = PQ/VP. For the purpose of this analysis, we assume that CP is constant under all the experimental conditions, but this has not been established. Csatdex, measured concentration of soluble deoxyHb in the supernatant (in g/dl) after centrifugation of the gel of deoxygenated Hb S (or Hbs S + Hb X, a non-S Hb) and dextran mixtures, identical to the "dextran Csat" in the text above; Csat, identical to Csat in the text above; concentration (in g/dl) of soluble deoxyHb S (or Hbs S + X) in the aqueous phase external to Hb S polymers and to dextran, when present, and at equilibrium with polymer. The standard Csat was measured in the absence of dextran and is assumed to remain unchanged in its presence. The value obtained before and used here was 16 g/dl. CT measured total concentration of Hb in tube (in g/dl). fDex, volume fraction of dextran that excludes Hb molecules; fDwex, volume fraction of dextran that excludes water; fP, volume fraction of polymerized Hb (S or S + X); fs, volume fraction of extra-polymer solution (sol) containing soluble Hb; delta D and delta x, densities of dextran solution and water, respectively (in g/dl); P, fraction of total Hb in polymer; Q, total amount of Hb in tube (in g); rex, ratio of dextran volume that excludes Hb molecules to that which excludes water molecules; rexH, ratio of deoxyHb polymer volume that excludes Hb or dextran molecules to that which excludes water molecules; VS, volume of solution available for soluble macromolecules. Vws, volume of water in extrapolymer solution (in dl). The volume of small osmoticants (salts) is neglected. VD volume of dextran inaccessible to water. VP volume of polymer, comprising the volume occupied by Hb tetramers in the polymer and by the polymer water compartment; it can also be defined by exclusion, as the volume of polymer which excludes dextran and soluble Hb. VT, total volume of Hb solution in tube. This volume has been shown to remain constant during polymerization (28). VDex, volume of dextran that excludes soluble macromolecules (Hb). wD, weight of dextran used to prepare solution of concentration CDT. ww, weight of water used to prepare dextran solution of concentration CDT.

Computation of CD, the Inverse of the Molar Volume of Dextran-- We weigh an amount wD of dextran and add a sufficient volume of water, Vw, to dissolve it in a total measured volume VT. By definition (see Equations 1-4),
C<SUB>D</SUB>=<FR><NU>w<SUB>D</SUB></NU><DE>V<SUB>T</SUB>−V<SUB>w</SUB></DE></FR> (Eq. 1)
V<SUB>D</SUB>=V<SUB>T</SUB>−V<SUB>w</SUB> (Eq. 2)
C<SUP>T</SUP><SUB>D</SUB>=<FR><NU>w<SUB>D</SUB></NU><DE>V<SUB>T</SUB></DE></FR> (Eq. 3)
and
&dgr;<SUB>D</SUB>=<FR><NU>w<SUB>D</SUB>+w<SUB>w</SUB></NU><DE>V<SUB>T</SUB></DE></FR> (Eq. 4)
CD is estimated from the measured density and total dextran concentration, by combining Equations 1, 3, and 4 and using delta w = ww/Vw = 100 g/dl, as a units conversion factor. We obtain Equation 5.
C<SUB>D</SUB>=<FR><NU>&dgr;<SUB>w</SUB></NU><DE>1−<FR><NU>(&dgr;<SUB>D</SUB>−100)</NU><DE>C<SUP>T</SUP><SUB>D</SUB></DE></FR></DE></FR> (Eq. 5)

Definition of Csatdex and Computation of P, the Polymer Fraction Formed on Deoxygenation of a Solution of Hb + Dextran with Known Total Hb Concentration (CT)-- We measure Csatdex in the supernatant after centrifugation (see Equations 6-8). By definition,
C<SUP><UP>dex</UP></SUP><SUB><UP>sat</UP></SUB>=<FR><NU>(1−P)Q</NU><DE>V<SUB>T</SUB>−V<SUB>P</SUB></DE></FR> (Eq. 6)
Q=C<SUB>T</SUB>V<SUB>T</SUB> (Eq. 7)
and
V<SUB>P</SUB>=P<FR><NU>C<SUB>T</SUB>V<SUB>T</SUB></NU><DE>C<SUB>P</SUB></DE></FR> (Eq. 8)
Solving for P, we obtain Equation 9.
P=<FR><NU>1−<FR><NU>C<SUP><UP>dex</UP></SUP><SUB><UP>sat</UP></SUB></NU><DE>C<SUB>T</SUB></DE></FR></NU><DE>1−<FR><NU>C<SUP><UP>dex</UP></SUP><SUB><UP>sat</UP></SUB></NU><DE>C<SUB>P</SUB></DE></FR></DE></FR> (Eq. 9)

Computation of the Different Volume Fractions Present in Solutions of Hb S and Dextran after Deoxygenation and Polymerization of DeoxyHb S, at Equilibrium-- The total volume (VT) includes the polymer volume (VP), the volume of dextran which excludes macromolecules (VDex), and the volume of solution available for soluble deoxyHb molecules (VS) (Equation 10). Thus
V<SUB>T</SUB>=V<SUB>P</SUB>+V<SUB>D</SUB><SUP><UP>ex</UP></SUP>+V<SUB>S</SUB> (Eq. 10)
Dividing by VT we rewrite Equation 1 using volume fractions (Equation 11).
f<SUB>P</SUB>+f<SUB>D</SUB><SUP><UP>ex</UP></SUP>+f<SUB>s</SUB>=1 (Eq. 11)
We can compute each of these volume fractions in different ways, depending of the choice of measured variables and parameters we wish displayed in the equations. We will compute fP, fS, and fDex independently from definitions of CP and Csat.

By definition (Equation 12),
C<SUB>P</SUB>=<FR><NU>C<SUB>T</SUB>V<SUB>T</SUB>−C<SUP><UP>dex</UP></SUP><SUB><UP>sat</UP></SUB>(V<SUB>T</SUB>−V<SUB>P</SUB>)</NU><DE>V<SUB>P</SUB></DE></FR> (Eq. 12)
Rearranging and solving for fP = VP/VT we obtain (Equation 13).
f<SUB>P</SUB>=<FR><NU>C<SUB>T</SUB>−C<SUP><UP>dex</UP></SUP><SUB><UP>sat</UP></SUB></NU><DE>C<SUB>P</SUB>−C<SUP><UP>dex</UP></SUP><SUB><UP>sat</UP></SUB></DE></FR> (Eq. 13)
Assuming that dextran behaves as an inert space filler that does not disturb the equilibrium between polymer and soluble deoxyHb S, the equilibrium concentration of soluble deoxyHb S in the aqueous phase, external to both the insoluble polymer and soluble dextran molecules, is Csat. Thus, by definition (Equation 14),
C<SUB><UP>sat</UP></SUB>=<FR><NU>(1−P)C<SUB>T</SUB>V<SUB>T</SUB></NU><DE>V<SUB>s</SUB></DE></FR> (Eq. 14)
from which we obtain Equation 15
f<SUB>s</SUB>=<FR><NU>(1−P)C<SUB>T</SUB></NU><DE>C<SUB><UP>sat</UP></SUB></DE></FR> (Eq. 15)
To derive fDex we re-write Equation 14 as Equation 16
C<SUB><UP>sat</UP></SUB>=<FR><NU>(1−P)C<SUB>T</SUB>V<SUB>T</SUB></NU><DE>V<SUB>T</SUB>−V<SUB>P</SUB>−V<SUB>D</SUB><SUP><UP>ex</UP></SUP></DE></FR> (Eq. 16)
Combining Equations 6 and 16 and solving for fDex we obtain Equation 17:
f<SUB>D</SUB><SUP><UP>ex</UP></SUP>=<FENCE>1−<FR><NU>PC<SUB>T</SUB></NU><DE>C<SUB>P</SUB></DE></FR></FENCE><FENCE>1−<FR><NU>C<SUP><UP>dex</UP></SUP><SUB><UP>sat</UP></SUB></NU><DE>C<SUB><UP>sat</UP></SUB></DE></FR></FENCE> (Eq. 17)
By using Equations 1-3 we can also define the volume fraction of dextran which excludes water (fDwex) at each dextran concentration. By definition,
f<SUB>D</SUB><SUP><UP>wex</UP></SUP>=<FR><NU>V<SUB>D</SUB></NU><DE>V<SUB>T</SUB></DE></FR> (Eq. 18)
From Equations 1-3, VD/VT in Equation 18 may be replaced by measured CDT and CD; thus Equation 19
f<SUB>D</SUB><SUP><UP>wex</UP></SUP>=<FR><NU>C<SUP>T</SUP><SUB>D</SUB></NU><DE>C<SUB>D</SUB></DE></FR> (Eq. 19)
We can now estimate the ratio rex between the volume of dextran that excludes soluble Hb, VDex, to that which excludes water, VD. From Equations 18 and 19 we obtain Equation 20
r<SUB><UP>ex</UP></SUB>=<FR><NU>f<SUB>D</SUB><SUP><UP>ex</UP></SUP></NU><DE>f<SUB>D</SUB><SUP><UP>wex</UP></SUP></DE></FR> (Eq. 20)
For reference, it is useful to compute the equivalent rex parameter for deoxyHb polymers, rexH. By definition, rexH is the ratio of polymer volume that excludes macromolecules to that which excludes water. This is given by Equation 21:
r<SUB><UP>ex</UP></SUB><SUP>H</SUP>=<FR><NU><FR><NU>PC<SUB>T</SUB>V<SUB>T</SUB></NU><DE>C<SUB>P</SUB></DE></FR></NU><DE><FR><NU>PC<SUB>T</SUB>V<SUB>T</SUB></NU><DE>C<SUB>H</SUB></DE></FR></DE></FR>=<FR><NU>C<SUB>H</SUB></NU><DE>C<SUB>P</SUB></DE></FR> (Eq. 21)
With CH and CP values of 136 and 55 g/dl, respectively {7}, rexH is about 2.5.

Comparison between CsatValues Obtained in the Presence and Absence of Dextran-- The ratio Csatdex/Csat which may be obtained by dividing Equation 6 by Equation 14. This renders Equation 22
<FR><NU>C<SUP><UP>dex</UP></SUP><SUB><UP>sat</UP></SUB></NU><DE>C<SUB><UP>sat</UP></SUB></DE></FR>=<FR><NU>f<SUB>s</SUB></NU><DE>1−f<SUB>P</SUB></DE></FR> (Eq. 22)
which expresses the comparison in terms of the volume fractions of polymer and extrapolymer solution.
    REFERENCES
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Abstract
Introduction
Appendix
References

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