Dissecting the Energies that Stabilize Sickle Hemoglobin Polymers

Abstract

Sickle hemoglobin forms long, multistranded polymers that account for the pathophysiology of the disease. The molecules in these polymers make significant contacts along the polymer axis (i.e., axial contacts) as well as making diagonally directed contacts (i.e., lateral contacts). The axial contacts do not engage the mutant β6 Val and its nonmutant receptor region on an adjacent molecule, in contrast to the lateral contacts which do involve the mutation site. We have studied the association process by elastic light scattering measurements as a function of temperature, concentration, and primary and quaternary structure, employing an instrument of our own construction. Even well below the solubility for polymer formation, we find a difference between the association behavior of deoxy sickle hemoglobin molecules (HbS), which can polymerize at higher concentration, in comparison to COHbS, COHbA, or deoxygenated Hemoglobin A (HbA), none of which have the capacity to form polymers. The nonpolymerizable species are all quite similar to one another, and show much less association than deoxy HbS. We conclude that axial contacts are significantly weaker than the lateral ones. All the associations are entropically favored, and enthalpically disfavored, typical of hydrophobic interactions. For nonpolymerizable Hemoglobin, ΔH_o was 35 ± 4 kcal/mol, and ΔS was 102.7 ± 0.5 cal/(mol−K). For deoxyHbS, ΔH_o was 19 ± 2 kcal/mol, and ΔS was 56.9 ± 0.5 cal/(mol−K). The results are quantitatively consistent with the thermodynamics of polymer assembly, suggesting that the dimer contacts and polymer contacts are very similar, and they explain a previously documented significant anisotropy between bending and torsional moduli. Unexpectedly, the results also imply that a substantial fraction of the hemoglobin has associated into dimeric species at physiological conditions.

Introduction

Sickle cell disease is a genetic disorder in which the substitution of a surface amino acid on the hemoglobin molecule (β6 Glu for β6Val) allows it to form large, helical, stiff polymers. This mass of polymers rigidifies the erythrocytes, thereby disrupting the microcirculation. Polymerization only occurs once oxygen has been delivered, and the hemoglobin has switched into the T (deoxygenated) quaternary structure. Hemoglobin A (HbA) does not form such polymers, nor does liganded Hemoglobin S (HbS). Although both the structure of the polymers and the thermodynamics of the reaction are known, their connection is relatively unexplored. This may account for the absence of any therapeutic drug that acts to interfere with the sickling process directly. (The only approved drug works by promoting the production of Hemoglobin F (HbF) rather than affecting the HbS assembly itself (1).)

The details of the polymer structure are based on deoxyHbS crystals, which form in a double-stranded half-staggered linear structure (2). The mutation site on one of the β-chains is docked in a hydrophobic receptor on another β in a contact region referred to as lateral, as illustrated as the indentations in Fig. 1. A second major contact region (axial) is found along the strand axis. The axial contact region does not contain the mutation site, and appears to be present without known perturbation in the unmutated HbA as well as HbS (3). The double strand was established to be a fundamental component of the polymer structure by a series of mutant/hybrid studies (4–6). The polymer structure itself was determined via electron microscopy and image reconstruction and consists of 14 strands in which the double strands of the crystal wind about the fiber axis in helical fashion (7,8), as also shown in Fig. 1. The exact location of all the contact sites is the subject of some controversy, centering on the various assumptions and corrections that must be employed in the reconstruction; however, the overall 14-stranded motif is well accepted (9,10).

Figure 1.

Structure of the double strand and polymer. (Top panel) Double strands, with the lateral contacts indicated by the highly exaggerated protrusion that signifies the β6 Val mutation, which fits in a receptor pocket. (Dashed diamonds) Location of the axial contacts. Below the double strand is shown a polymer structure composed of twisted double strands. Not visible is a core of two more pairs of double strands. (Lower right) A helical cross section is illustrated, showing the lateral contacts (solid red circles) between double strands, plus two additional contact sites (open blue circles) using the other β6 Val that was not already engaged in a contact. (Upper right) Effect of bending the lateral contact is shown as a possible means to generate a variation in pitch. (Dashed lines) Structure with a different intermolecular angle, which lengthens the distance between successive molecules. To see this figure in color, go online.

The relative strength of the polymer contacts is unknown. Model building of the polymer structures has proceeded under the assumption that the β6 mutation contact should be transformed intact between the crystalline double strand and the twisted double strands of the polymer. However, there has been no recorded attempt to construct fibers on any other set of assumptions (e.g., invariance of the axial contacts) and no prior experimental data mandates whether one or the other contact should be conserved in winding a double strand into a helical fiber. While it might be naively thought that the lateral region must have the greater impact because the β6 mutation is needed for assembly, simply eliminating the charged β6 Glu relieves a penalty for placing it in a hydrophobic pocket. The axial contacts therefore might equally well be the dominant interface. Crudely, the free energy of transfer for a Val from water to oil is 1.5 kcal/mol. Given two complimentary surfaces that mutually bury this hydrophobic surface, the energy of the lateral contact would be expected to be ∼3 kcal/mol. Given the much larger surfaces in contact in the axial regions, one might justifiably suspect the axial regions of being at least as significant. There has been an attempt at calculating some of these energies in a more systematic way, still based on burial of solvent-accessible areas (11). In those calculations, the lateral contact was computed to be 50% stronger than the axial, although, as the authors noted, the magnitude of the calculation was not expected to be accurate, and indeed the energies they calculated seem anomalously large relative to, e.g., the energetics of burial of the solvent-exposed β6 (12).

In the work presented here, the strengths of the lateral and axial contacts are finally resolved by light-scattering measurements. The central idea of the experiment was to measure light scattering under conditions in which polymers cannot form. Because HbA molecules have the same amino acids in the same configuration as those that appear in the HbS axial contact region, we reasoned that observing HbA could reveal information about the axial contacts. We could then compare these with HbS molecules that make transient associations below the solubility at which polymers will form, thus determining the strength of lateral contacts.

For all species studied, we find that light scattering (and thus association) increases exponentially with temperature, a result consistent with a reaction with positive enthalpy. Deoxy HbS has much greater scattering, implying more association, and thus establishes the predominance of the lateral contacts, and detailed analysis provides quantitative measures of the relative strengths.

Materials and Methods

Purified oxyhemoglobin was prepared by column chromatography according to standard procedures, in 0.15 M phosphate buffer, at pH 7.35 (13). DeoxyHb was prepared from oxyHb by the addition of 0.05 M sodium dithionite in a cold chamber flushed with humidified nitrogen. COHb was prepared by exposing oxyHb to CO. Cross-linked HbA (HbAxl) (14) was generously provided by Professor Robin Briehl and Staff Associate Suzanna Kwong of the Albert Einstein College of Medicine (Yeshiva University, Bronx, NY). All Hb concentrations were determined by dilution and subsequent absorbance measurement in the Soret band.

Light-scattering experiments were conducted on an apparatus of our own design. Samples were contained in a long, narrow, hollow, rectangular glass tube (W3520; Fiber Optic Center, New Bedford, MA) of cross-section 200 μm × 4 mm, which was filled with 24 μL of Hb solution. An optical fiber was inserted into one end. The fiber core had a 62.5-μm diameter, and including the cladding, had an outside diameter of 125 μm. Both ends of the rectangular tube (i.e., the free end and the end into which the fiber was inserted) were sealed with dental wax. Using a microscope objective, scattering was then collected at 90° to the incident fiber illumination. Because the tube had a rectangular cross section, the flatness of the tube permitted absorption spectra to be collected (at regular intervals) to insure the integrity of samples over long experiments. For such purposes, visible spectra were taken using a fiber-optic spectrometer (Ocean Optics, Dunedin, FL) and a 150 W Xenon lamp as a light source. Absorbance spectra of the samples were found to be unchanged for a minimum of 24 h.

The source for light scattering was a model No. DL7140-201S diode laser (Sanyo Laser Products, Richmond, IN) emitting at 785 nm. Laser-diode intensity was monitored by a model No. PDA100A Si switchable-gain detector (Thorlabs, Newton, NJ), and the scattered light was detected by a model No. R636 GaAs photomultiplier (Hamamatsu, Hamamatsu City, Japan), which was calibrated against the PDA100A. The scattered light was imaged at the phototube using a Leitz 32× LWD objective (Leica Microsystems, Wetzlar, Germany). The observed area was typically 140 × 10 μm. Heating due to the diode laser was determined by using Cresol Red dye, as described in Ferrone et al. (13). Intensity was then kept at ≤2 mW, corresponding to <0.2°C heating. On average, light-scattering intensity agreed with theoretical expectations for unaggregated solutions within 10%.

To analyze the temperature-dependence of the aggregate species, denoted c_n, the familiar mass-action equation may be written as

$$c_n=e^{\text{ln}K}{\left(\gamma c\right)}^n/{\gamma }_n,$$ (1)

in which γ-value denotes an activity coefficient (which are functions of concentration), c is the monomer concentration, and K is the equilibrium constant. In turn, ln K may be expanded in a Taylor series around a reference temperature T_o, i.e.,

$$\text{ln}K\left(T\right)=\text{ln}K\left(T_{\mathrm{o}}\right)+{\left(d\text{ln}K/dT\right)}_{\mathrm{o}}\delta T=\text{ln}K\left(T_{\mathrm{o}}\right)+\left(\Delta H/R{T}_{\text{o}}^2\right)\delta T,$$ (2)

where ΔH is the enthalpy. In the case where the enthalpy is temperature-dependent, then the ΔH value in the above expression is the value at the reference temperature T_o. The value T_o may conveniently be chosen as 0°C, so that δT is simply the temperature in Celsius. The excess scattering intensity ΔI, i.e., that above the background level due to density fluctuations of monomers, is taken as proportional to the aggregate concentration c_n from which an exponential form follows

$$\Delta I=a\text{exp}\left[\left(\Delta H_{\text{o}}/R{T}_{\text{o}}^2\right)\delta T\right].$$ (3)

The scattered intensity I at distance r is related to the incident intensity I_o by the Rayleigh ratio, R, defined as R = Ir²/I_o. For scattering from two species, monomers and dimers, the Rayleigh ratio R₁₂ is given by (15)

$$R_{12}=\frac{2{\pi }^2{n}^2}{N_A{\lambda }^4}{\left(\frac{dn}{dw}\right)}^2\left[M_1^2\langle \Delta c_1^2\rangle +M_2^2\langle \Delta c_2^2\rangle +2M_1{M}_2\langle c_2\Delta c_1\rangle \right],$$ (4)

in which the refractive index is n, the molecular weight is w, N_A is Avogadro’s number, M is the molar mass, and the wavelength is λ. The concentration fluctuations are given by

$$\langle \Delta c_i^2\rangle =\frac{c_i}{D}\left(1+{\zeta }_{jj}\right),$$ (5)

$$\langle c_j\Delta c_i\rangle =-\frac{c_i{\zeta }_{ij}}{D},$$ (6)

where

$$D=\left(1+{\zeta }_{11}\right)\left(1+{\zeta }_{22}\right)$$ (7)

and

$${\zeta }_{ij}=\frac{\partial \text{ln}{\gamma }_i}{\partial \text{ln}{c}_j},$$ (8)

in which γ_i is the activity coefficient for the ith species. By taking the ratio of R₁₂ (the Rayleigh ratio with monomers and dimers) to R₁ ( that of monomers alone), we eliminate the optical constants and obtain

$$\frac{R_{12}}{R_1}=\frac{\langle \Delta c_1^2\rangle +4\langle \Delta c_2^2\rangle +4\langle c_2\Delta c_1\rangle }{\langle \Delta c_1^2\rangle }=\frac{1+{\zeta }_{11}^{\text{o}}}{D}\left(1+{\zeta }_{22}-4{\zeta }_{12}+4\left(1+{\zeta }_{11}\right)\frac{c_2}{c_1^{\text{o}}}\right).$$ (9)

The monomer concentration is that taken before aggregation, i.e., at lower temperatures, and is thus denoted as c₁⁰. The activity coefficients generally depend on all species present in a solution. Thus, the various ζ_ij values depend on the particular concentration, and so should not be viewed as simple constants. For the same reason, the superscript zero indicates the value of this activity coefficient derivative before aggregation has begun. For monomers, a number of simple approaches are available to calculate activity coefficients (16). For dimers, we adopted scaled particle theory as described in detail by Minton (17).

The sample was mounted on a thermoelectric stage whose temperature was changed typically in 1°C intervals. A grid eyepiece was used to determine if the sample was displaced when temperature was changed, and allowed appropriate realignment to be made easily if that did occur. For deoxyHbS samples, which might polymerize, some experiments were done above the solubility, and subsequently below. Results on reentering the region below solubility were generally indistinguishable from the data taken before the solubility line was crossed.

See the Supporting Material for further information and schematics in Fig. S1 and Fig. S2 in the Supporting Material for the experimental setup.

Results

Light scattering was measured from samples of deoxyHbS, deoxyHbA, COHbS, COHbA, and deoxyHbAxl, where xl denotes a cross-linked molecule. Measurements spanned a range of concentrations, with 25 samples analyzed overall between 3 and 43°C; note that not all temperatures were collected on all samples. Although no polymerization is being observed in these experiments, it is nevertheless convenient to group the samples studied in two categories, i.e., either as nonpolymerizable (deoxyHbA, COHbS, COHbA, and deoxyHbAxl) or as polymerization-competent (deoxyHbS). Scattering intensities remained essentially constant as temperature was incremented until some temperature at which the intensity began to increase rapidly, as shown in Fig. 2. As can be seen in Fig. 2, the excess scattering could be well described by as an exponential function of temperature. This follows from simple free-energy descriptions for the equilibrium constant (compare to Eqs. 1–3).

Figure 2.

Light scattering observed as a function of temperature for different molecular species. The ordinate shows the intensity measured in nanoWatts. The wavelength of the scattered light was 785 nm. The scattering was fit with an exponential (drawn as continuous curves in the image). The exponent of the exponential yields the enthalpy of the reaction, ΔH_o. Each of the exponentials began from a flat baseline that is attributed to monomer scattering. For the deoxyHbS, the data were collected for temperatures below the solubility (vertical dashed line near 29°C) before proceeding to higher temperatures. (Data was collected subsequently that was above solubility, but was not used in the fit.)

The nonpolymerizable species showed scattering that began to increase noticeably only near 30°C, whereas the scattering from deoxy HbS began to rise at much lower temperatures. As shown in Fig. 3, the values for ΔH_o were consistent among all nonpolymerizable species (left) and were distinct from the polymerizable HbS (right) data. The values of ΔH_o were essentially concentration-independent, as one would expect, allowing an average to be determined (horizontal lines). For nonpolymerizable species, ΔH_o was 35 ± 4 kcal/mol, and for deoxyHbS, ΔH_o was 19 ± 2 kcal/mol. These are positive quantities, meaning association is enthalpically unfavorable.

Figure 3.

Enthalpy for the association reactions observed by light scattering. (Solid symbols) DeoxyHb species; (open symbols) COHb. (Triangles) HbA; (circles) HbS. (Solid diamonds) Cross-linked deoxyHbA. (Right data) DeoxyHbS; (left data) nonpolymerizing species. Data like that of Fig. 2 was analyzed to yield ΔH. Five different samples were studied: deoxygenated HbA and HbS, COHbA, COHbS, and deoxygenated cross-linked HbA, which exhibits greater thermal stability than HbA. (Left panel) The hemoglobins that are polymerization-incompetent, which includes all species except for deoxyHbS, showed a consistent ΔH_o; this averages 35 ± 4 kcal/mol in contrast to deoxyHbS (right panel), which has a lower ΔH_o of 19 ± 2 kcal/mol. (Long dashes) Average; (dotted lines) standard deviation of the average.

The overall free energy of association will also have an entropic contribution. To obtain that contribution requires accounting for the other constants, which include the various, well-known optical constants that are part of the Rayleigh ratio. Once this can be done, the concentration of the aggregate, and thus its thermodynamics, can be determined. A simple way to account for the various constants is to divide the excess intensity by the background intensity, thereby eliminating the common multiplicative factors. We divide the Rayleigh ratio of the aggregate signal to the Rayleigh ratio of the monomer signal (compare to Eq. 9).

We assume that the scattering comes mainly from association of Hb monomers into dimers. (Throughout this article, the term “dimer” will exclusively refer to a pair of hemoglobin molecules, which themselves are tetramers of their subunits. Dimers—meaning only two of the four subunits—are not considered here.) Such an assumption must be explicitly included in the analysis because of the presence of activity coefficients that vary with size. Under this assumption, and employing the average values of ΔH_o determined as described above (compare to Fig. 3), the values for ΔS can be ascertained. Once again there is a clear segregation of data between those species that are polymerization-incompetent, and those of deoxy HbS, as shown in Fig. 4. The species that do not polymerize have an average ΔS = 102.7 ± 0.5 cal/(mol−K) whereas the average for deoxyHbS is ΔS = 56.9 ± 0.5 cal/(mol−K).

Figure 4.

Entropy for the association reactions observed by light scattering. (Solid symbols) DeoxyHb; (open symbols) COHb. (Solid circles) DeoxyHbS; (open circles) deoxyHbA. (Solid diamond) Cross-linked HbA. The averaged ΔH_o values were used from Fig. 3 to permit calculation of ΔS. The species that cannot polymerize had an average ΔS = 102.7 ± 0.5 cal/(mol−K) while the average of the deoxyHbS was ΔS = 56.9 ± 0.5 cal/(mol−K). Note that all the deoxygenated HbS data were analyzed below solubility, where polymers cannot form.

Combining the enthalpy and entropy, we see quantitatively, as the raw data indicated qualitatively, that the concentration of deoxyHbS, which has assembled into dimers, is greater than the concentration of dimers formed from Hb, which does not polymerize (compare to Fig. 5).

Figure 5.

The fraction of dimers as a function of initial concentration and temperature. The free energy ΔG was used to compute an equilibrium constant, which with activity coefficients could determine the concentration of the dimers. The mole fractions are not the same as weight fractions because dimers are twice as massive as monomers. (Solid lines) 37°C; (dashed lines) 25°C. (Upper curves, red) HbS; (lower curves, black) nonpolymerizing species. Above the solubility, the HbS curves are shown (dotted). The prediction is for a strikingly large fraction of dimers in HbS, and even for nonpolymerizable Hb at high temperatures. To see this figure in color, go online.

At physiological temperatures (37°C), if we assign the results obtained for the polymerizable species to lateral contacts, and those of the nonpolymerizable species to axial contacts, the data presented here implies that the free energy for association of the axial contacts is ≥+3.1 kcal/mol and the free energy for lateral association is +1.3 kcal/mol. (Note that all free energies are taken with a standard state of 1 mM.) Because these are signed quantities, the lower value implies the greater population. Thus, a lateral contact is at least 1.8 kcal/mol more stable than an axial contact, which corresponds to a Boltzmann factor of 20.

Discussion

The identification of the lateral contacts as the cause of the dimer formation of deoxyHbS is unambiguous. Both liganded and unliganded molecules of HbS possess a surface Val at β6, but are otherwise identical to HbA. However, the receptor region that receives the β6 Val is not fully accessible in the COHbS. Thus what distinguishes deoxy HbS is that it alone has both a β6 Val donor and an exposed receptor region and the degree of its association into dimers is likewise unique among these species.

Assigning the axial contact strength is slightly more subtle. The nonpolymerizable molecules can all form axial contacts, but it is also possible that they form dimers using contact sites not found in the polymer because they would not be commensurate with the simultaneous formation of a lateral contact. Of course, if there were interactions that did not appear in polymers, yet were responsible for the dimers, this in turn would imply that the axial contacts that are present in the polymer are weaker still. Thus, strictly speaking, this work places an upper bound on the strength of the axial contacts.

It is interesting to compare these measurements of dimer contacts (between a pair of hemoglobin molecules, not within the molecule) with the measurements of the contacts within a polymer, because the docking process for a dimer may make a slightly different fit than the docking of the same contacts when several constraints must be satisfied simultaneously. The enthalpy is solely attributable to the interaction or bonds that form the assembly, and so is the more robust place to begin the comparison. ΔH_P, defined as the enthalpy measured for the polymer at 0°C, is +13 kcal/mol (6), whereas it is +19 kcal/mol for the lateral contacts in the dimer. These measurements must be properly normalized to permit their quantitative comparison because the polymer enthalpy is for an average molecule joining the polymer.

Adding a molecule to the fiber-end (compare to Fig. 1) always entails making at least one lateral and one axial contact. We shall denote the chemical potential due to the polymer contacts of each molecule as μ_PC. Given the finding here that the axial contact is much weaker, μ_PC is essentially that of lateral contacts. However, two molecules in the polymer involve both β6 sites, and this adds another two lateral contacts to the total in a slice through the polymer. Therefore, instead of seven lateral contacts in a given layer, one has nine lateral contacts. Assuming the lateral contact, of strength μ_2C^L, between monomers in the dimers we observe is identical to the lateral contact in the fiber, the average strength therefore is 9 μ_2C^L divided among 14 molecules. Therefore we would expect that μ_PC = 9/14 μ_2C^L. The enthalpic contributions to ΔG all come from the enthalpies of the contact chemical potential. Thus we expect that ΔH_P = 9/14 ΔH_o. The value of 9/14 ΔH_o =12.2 kcal/mol, which agrees very well with the 13 kcal/mol measured for the polymer. This therefore argues that the dimer contact is very similar if not identical to that found within the polymer.

The foregoing discussion can be extended to consider the full association free energy. We begin by equating the chemical potentials of the two associating monomers with that of the associated dimer. The value μ_TR is the chemical potential for monomer translations and rotations in solution, μ_2C is the dimer energy due to the presence of a bond or contact, and the vibrational chemical potential is given by μ_2V. The variables γ and γ₂ are activity coefficients for monomer and dimers, respectively, and c and c₂ are monomer and dimer concentrations. The 4RT ln2 comes from treating the dimer as a sphere of twice the volume of the monomer for computing its translational and rotation chemical potentials. Thus,

$$2\left({\mu }_{\mathrm{TR}}+RT\ln \gamma c\right)={\mu }_{\mathrm{TR}}\mathrm{-4}RT\ln 2+{\mu }_{2\mathrm{C}}+{\mu }_{2\mathrm{V}}+RT\ln {\gamma }_2{c}_{\mathrm{2}},$$ (10)

from which it follows that

$$\mathrm{\Delta }{G}_2\mathrm{\equiv -}RT\ln \left[{\gamma }_2{c}_{\mathrm{2}}/{\left(\gamma c\right)}^2\right]={\mu }_{2\mathrm{C}}+{\mu }_{2\mathrm{V}}\mathrm{-}{\mu }_{\mathrm{TR}}\mathrm{-4}RT\ln 2.$$ (11)

The above expression can be applied to lateral contacts, ΔG₂^L or axial contacts, ΔG₂^A. These energies entail the strength of their respective contacts as μ_2C^L and μ_2C^A. The vibrational terms do not differ for lateral and axial contacts, because the vibrational motion is a net result. We can write

$$\mathrm{\Delta }{G}_2^{\mathrm{L}}\mathrm{-\Delta }{G}_2^{\mathrm{A}}={\mu }_{2\mathrm{C}}^{\mathrm{L}}\mathrm{-}{\mu }_{2\mathrm{C}}^{\mathrm{A}}.$$ (12)

Likewise, ΔG_P, the free energy of adding a monomer to a polymer, is given by (18)

$$\mathrm{\Delta }{G}_{\mathrm{P}}\mathrm{\equiv -}RT\ln {\gamma }_{\mathrm{s}}{c}_{\mathrm{s}}={\mu }_{\mathrm{PC}}+{\mu }_{\mathrm{PV}}\mathrm{-}{\mu }_{\mathrm{TR}},$$ (13)

where μ_PC and μ_PV are polymer contact and vibrational chemical potentials, respectively, and the subscript s represents solubility. The value μ_PC represents the net contact energy per monomer in the polymer, so that having multiple bonds to adjacent molecules needs to be taken into account when comparing it to μ_2C, which represents the single bond between Hb monomers.

As described above, we concluded that μ_PC = 9/14 μ_2C^L. Because μ_RT refers to molecules in solution, it is the same regardless of which aggregate is formed. The question is whether μ_PV and μ_2V are the same. We shall test the adequacy of such an assumption. Taking the value of μ_PV previously determined from kinetic measurements (18), and equating it to μ_2V, we can proceed to compare (ΔG_P – 9/14 ΔG₂), which are measured quantities, with the values determined from their constituent equations (11,13) above. Multiplying by 9/14 allows μ_PC to be eliminated. We find that, at 25°C, the value of (ΔG_P −9/14 ΔG₂) is −2.2 kcal/mol whereas the expected value using Eqs. 11 and 13 is −2.3 kcal/mol. This excellent agreement thus supports the assumption that we can equate the vibrational chemical potentials, and thus that the lateral contact in the dimer and polymer are essentially the same.

The positive signs of the enthalpy indicate a repulsive interaction, which is typical of hydrophobic interactions (19). The larger value for ΔH for the axial contacts than the lateral ones (+35 vs. +19 kcal/mol) is likely the result of the larger contact area. The positive sign on ΔS is also typical of hydrophobic interactions, but this is not the only contribution to the entropy of the reaction because the motion of the molecules around their equilibrium points also contributes to the entropy (18).

The dominance of the lateral contact raises some interesting issues for the polymer structure. One question that naturally arises is how the seven double strands are held together in the 14-stranded structure using only lateral contacts. Of course, the same question would arise if lateral and axial contacts were comparable in strength, and only admits a trivial answer if significant contract strength existed between adjacent double strands (which mainly would be through the α-chains). Copolymerization studies argue persuasively that in a 14-stranded polymer, the average number of lateral contacts is not seven but nine, i.e., two HbS molecules have employed both β6 sites in the structure. We have elsewhere presented a model for the fiber, showing which strands use both β6 contact sites (20). That proposed model cross-links a four-strand structure—not a structure of 14 strands. An interesting alternative is that the polymer, much like a frustrated spin glass, might not possess a unique arrangement. In such structures, multiple minima compete with one another because one structure is made at the expense of another. This assumes that, while there are more than two added sites for intermolecular lateral contacts, only two can be made in a given arrangement. Thus, by varying the position at which that second lateral contact is made across the different strands, the structure would obtain overall stability. Such behavior would be obscured by the procedures used to reconstruct the images (21). An explicit assumption of such reconstruction algorithms is that all layers of the fiber are identical, except for rotation and variation in pitch, so that any structural variation due to the presence of different contacts would be treated as noise and averaged out.

It is also possible that the links between double strands are fixed at two extra sites within the polymer, and that other weak forces, including possibly depletion forces, may account for the interactions that hold the double-strands together (i.e., to other double strands). Fourteen-stranded polymers have been observed by differential-interference contrast microscopy as zippering together in solutions, and from analysis of such experiments, it was deduced that the attraction between fibers was ∼3 kT/μm or 1.8 kcal/mol at 25°C (22). If weak forces were what held the strands together, one should expect to observe polymers missing double-strand pairs. Such flawed polymers might not be plentiful, because, with the aforementioned energy, the probability of finding polymers with missing double strands in a collection of 1-μm-length polymers would only be ∼5%. Indeed, polymers lacking outer strands have been seen in both low- and high-phosphate buffers (7,23).

The dominance of the lateral contacts can also rationalize one of the more unusual mechanical properties of the polymers. The pitch of the fibers has been found to vary (8), and its variation was used to infer a torsional rigidity. This was found to be 2.5 kT−μm, which is 50 times lower than the 130 kT−μm found for the bending rigidity (24). In the absence of axial contacts of any significance, a change only in pitch would necessitate bond rotation; given the softness of such forces, this could lead to variation in apparent pitch as illustrated in Fig. 1 (upper right). On the other hand, bending of a polymer would still entail compression and extension of the lateral contacts, and so be a much more difficult deformation to induce.

One of the most interesting predictions to emerge from this study is that large fractions of the HbS molecules are present as dimers. The projected dimer fraction, shown in Fig. 5, assumes monomers and dimers (i.e., two Hb monomers together) are the only species, rather than going on to larger oligomers. It might seem that such a phenomenon should have been observed. We now turn to this question.

One might have expected a deviation in sedimentation studies, for example, that have deduced the known monomeric molecular weight of Hb, and have shown no difference between HbS and HbA or liganded HbS below the solubility (6). However, the sedimentation value for molecular weight is obtained in the limit of zero concentration, and the decrement in molecular weight as concentration increases is remarkably insensitive to dimerization, as Minton (15) has shown (compare to Minton’s Fig. 4A in the aforementioned reference).

Measurements using inelastic light scattering would also be expected to resolve such an effect. The net radius increase expected for a dimer is not great. A simple sphere of appropriate radius, to double the monomer volume, inflates the monomer radius by 25%. Two exponentials of decay constants differing by only 25%, as would be expected in correlation functions for light scattering, are generally impossible to resolve, so only a single exponential is observed, indicative of increasing mean particle size. There are several reports of inelastic scattering done in tandem with elastic scattering, and there an increasing hydrodynamic radius (25–27) was observed that correlated with the increase in elastic light scattering. There it was ascribed to critical fluctuations in a liquid-liquid demixing transition. While we did not elect to interpret our data within such a framework, the data presented here and the data previously obtained are fully consistent. (Inconsistencies between nucleation rate kinetics and the liquid-liquid demixing assumption, as discussed in Ferrone (28), have led us to employ the analysis reported here instead. Moreover, a dimer cannot be described per se as either a liquid or solid. Not until a significantly larger aggregate is formed can a phase, in principle, be properly defined.) Thus, we conclude that previous observations support the aggregation data and interpretation presented here.

As mentioned above, the analysis presented here is based on the assumption that monomers and dimers are the only significant species. The size of the dimer population mandates closer scrutiny of this assumption, because one might anticipate the formation of larger aggregates at these dimer fractions. For the deoxyHbS data, experiments were done at concentrations <∼20 g/dL (compare to Fig. 3), and those experiments were at low temperatures. Therefore, the conditions under which the data were actually collected will have dimer fractions lower than the high values seen in Fig. 5. The adequacy of the dimer assumption is further supported by the lack of significant concentration dependence of ΔH in Fig. 3, suggesting that the effect of aggregates of higher order is not a factor in this data. Thus, while limiting our analysis to dimer formation is indeed consistent with the data range employed, larger aggregates may well form in significant numbers beyond the range of this study. The detailed analysis of such larger aggregates becomes more speculative, however, because various added assumptions would be required about their structure and thermodynamics. A more precise set of measurements might also have observed a small effect of concentration on ΔH that could not be resolved in this data set. Likewise, the use of Hb mutants (either natural or site-directed) or other constructs could be useful in confirming some of these hypotheses. For example, cross-linked HbAS hybrids would be useful to ascertain whether higher-order structures formed, because the hybrid has only one mutation site as opposed to the usual two. Similarly, axial mutants (e.g., β95 Lys → Ile) could destabilize the axial contacts and confirm their assignment as the origin of the dimerization in nonpolymerizing species.

The dimer populations may also play a role in the highly concentration sensitive nucleation kinetics (13), but this has yet to be explored theoretically as well as experimentally. Although it is unsurprising that deoxyHbS is prone to aggregate as its solubility is approached, the behavior of the nonpolymerizing species is more unexpected and may even indicate an unknown assembly process (into crystals or fibers). Indeed, there have been reports of tubules formed in high phosphate buffer from these very species, which may be related to the processes seen here (29,30). Light scattering in high phosphate buffers has shown the presence of larger aggregates as well (31).