Anisotropy in sickle hemoglobin fibers from variations in bending and twist.

We have studied the variations of twist and bend in sickle hemoglobin fibers. We find that these variations are consistent with an origin in equilibrium thermal fluctuations, which allows us to estimate the bending and torsional rigidities and effective corresponding material moduli. We measure bending by electron microscopy of frozen hydrated fibers and find that the bending persistence length, a measure of the length of fiber required before it starts to be significantly bent due to thermal fluctuations, is 130microm, somewhat shorter than that previously reported using light microscopy. The torsional persistence length, obtained by re-analysis of previously published experiments, is found to be only 2.5microm. Strikingly this means that the corresponding torsional rigidity of the fibers is only 6x10(-27)Jm, much less than their bending rigidity of 5x10(-25)Jm. For (normal) isotropic materials, one would instead expect these to be similar. Thus, we present the first quantitative evidence of a very significant material anisotropy in sickle hemoglobin fibers, as might arise from the difference between axial and lateral contacts within the fiber. We suggest that the relative softness of the fiber with respect to twist deformation contributes to the metastability of HbS fibers: HbS double strands are twisted in the fiber but not in the equilibrium crystalline state. Our measurements inform a theoretical model of the thermodynamic stability of fibers that takes account of both bending and extension/compression of hemoglobin (double) strands within the fiber.

Effect of T-R conformational change on sickle-cell hemoglobin interactions and aggregation.

We compare the role of a conformational switch and that of a point mutation in the thermodynamic stability of a protein solution and in the consequent propensity toward aggregation. We study sickle-cell hemoglobin (HbS), the beta6 Glu-Val point mutant of adult human hemoglobin (HbA), in its R (CO-liganded) conformation, and compare its aggregation properties to those of both HbS and HbA in their T (unliganded) conformation. Static and dynamic light scattering measurements performed for various hemoglobin concentrations showed critical divergences with mean field exponents as temperature was increased. This allowed determining spinodal data points T(S)(c) by extrapolation. These points were fitted to theoretical expressions of the T(S)(c) spinodal line, which delimits the region where the homogeneous solution becomes thermodynamically unstable against demixing in two sets of denser and dilute mesoscopic domains, while remaining still liquid. Fitting provided model-free numerical values of enthalpy and entropy parameters measuring the stability of solutions against demixing, namely, 93.2 kJ/mol and 314 J/ degrees K-mol, respectively. Aggregation was observed also for R-HbS, but in amorphous form and above physiological temperatures close to the spinodal, consistent with the role played in nucleation by anomalous fluctuations governed by the parameter epsilon = (T - T(S))/T(S). Fourier transform infrared (FTIR) and optical spectroscopy showed that aggregation is neither preceded nor followed by denaturation. Transient multiple interprotein contacts occur in the denser liquid domains for R-HbS, T-HbS, and T-HbA. The distinct effects of their specific nature and configurations, and those of desolvation on the demixing and aggregation thermodynamics, and on the aggregate structure are highlighted.

A 50th order reaction predicted and observed for sickle hemoglobin nucleation.

Sickle hemoglobin polymerization exhibits a striking sensitivity to initial concentration, with characteristic reaction times that exhibit 30th power dependence on concentration. This extraordinary reaction order is encompassed by a novel double nucleation mechanism that predicts 50th power dependence of the homogeneous nucleation rate. Using a technique that allows individual homogeneous nucleation events to be monitored, we have measured a concentration dependence of 47+/-, in excellent agreement with the predictions of the model. Absolute nucleation rates agree with predictions as well.

The structural link between polymerization and sickle cell disease.

Sickle hemoglobin molecules assemble into polymers composed of seven helically twisted double strands. Intermolecular contacts involving the mutation sites within the double strands are well established. We show that the same contact sites are present at the polymer surface on four of the ten exterior molecules in each layer, and demonstrate that the identical contact geometry can be achieved between polymers as found within the double strands. This provides a structural rationale for the exponential rate of polymer growth that characterizes the kinetics of gelation. This also gives a structural basis for the cross-linking which solidifies the polymer gel. In the absence of these surface contact regions sickle cell disease would be a much milder syndrome.

Kinetics of sickle hemoglobin polymerization. I. Studies using temperature-jump and laser photolysis techniques.

Using a combination of laser photolysis and temperature-jump techniques, the kinetics of hemoglobin S polymerization have been studied over a wide range of delay times (10(-3) to 10(5)s), concentrations (0.2 to 0.4 g/cm3) and temperatures (5 to 50 degrees C). A slow temperature-jump technique was used to induce polymerization in samples with delay times between 10(2) seconds and 10(5) seconds by heating a solution of completely deoxygenated hemoglobin S. For samples with shorter delay times, polymerization was induced by photodissociating the carbon monoxide complex in small volumes (10(-9) cm3) using a microspectrophotometer equipped with a cw argon ion laser. The photolysis technique is described in some detail because of its importance in studying hemoglobin S polymerization at physiological concentrations and temperatures. In order, to establish conditions for complete photodissociation with minimal laser heating, a series of control experiments on normal human hemoglobin was performed and theoretically modeled. The concentration dependence of the tenth time is found to decrease with increasing hemoglobin S concentration. In the range 0.2 to 0.3 g/cm3, the tenth time varies as the 36th power of the hemoglobin S concentration, while in the range 0.3 to 0.4 g/cm3 it decreases to 16th power. As the tenth times become shorter, the progress curves broaden, with the onset of polymerization becoming less abrupt. For tenth times greater than about 30 seconds, measurements with the laser photolysis technique on small volumes yield highly irreproducible tenth times, but superimposable progress curves, indicating stochastic behavior. The initial part of the progress curves from both temperature-jump and laser photolysis experiments is well fit with an equation for the concentration of polymerized monomer, delta (t) = A[cosh (Bt) -1], which results from integration of the linearized rate equations for the double nucleation mechanism described in the accompanying paper (Ferrone et al., 1985). The dependence of the parameters A and B on temperature and concentration is obtained from fitting over 300 progress curves. The rate B has a large concentration dependence, varying at 25 degrees C from about 10(-4) S-1 at 0.2 g/cm3 to about 100 s-1 at 0.4 g/cm3.

Kinetics of sickle hemoglobin polymerization. II. A double nucleation mechanism.

A double nucleation mechanism for the polymerization of sickle hemoglobin is described. The mechanism accounts for all of the major kinetic observations: the appearance of a delay, the high concentration dependence of the delay time, and the stochastic behavior of slowly polymerizing samples in small volumes. The mechanism postulates that there are two pathways for polymer formation: polymerization is initiated by homogeneous nucleation in the solution phase, followed by nucleation of additional polymers on the surface of existing ones. This second pathway is called heterogeneous nucleation. Since the surface of polymers is continuously increasing with time, heterogeneous nucleation provides a mechanism for the extreme autocatalysis that is manifested as an apparent delay in the kinetic progress curves. In this mechanism, each spherulitic domain of polymers is considered to be initiated by a single homogeneous nucleation event. The mechanism explains the irreproducibility of the delay time for single domain formation as arising from stochastic fluctuations in the time at which the homogeneous nucleus for the first polymer is formed. Integration of the linearized rate equations that describe this model results in a simple kinetic form: A[cosh(Bt)-1] (Bishop & Ferrone, 1984). In the accompanying paper (Ferrone et al., 1985) it was shown that the initial 10 to 15% of progress curves, with delay times varying from a few milliseconds to over 10(5) seconds, is well fit by this equation. In this paper, we present an approximate statistical thermodynamic treatment of the equilibrium nucleation processes that shows how the nucleus sizes and nucleation equilibrium constants depend on monomer concentration. The equilibrium model results in expressions for B and B2A as a function of monomer concentration in terms of five adjustable parameters: the bimolecular addition rate of a monomer to the growing aggregate, the fraction of polymerized monomers that serve as heterogeneous nucleation sites, the free energy of intermolecular bonding within the polymer, and two parameters that describe the free energy change as a function of size for the bonding of the heterogeneous nucleus to a polymer surface. This model provides an excellent fit to the data for B and B2A as a function of concentration using physically reasonable parameters. The model also correctly predicts the time regime in which stochastic behavior is observed for polymerization in small volumes.

Nucleation and polymerization of sickle hemoglobin with Leu beta 88 substituted by Ala.

We have measured the solubility, and the rates of homogeneous and heterogeneous nucleation on sickle hemoglobin (HbS beta 6 Glu-->Val) additionally modified by site-directed mutagenesis to possess Ala rather than Leu at beta 88, which forms part of the receptor site for beta 6 Val in the sickle polymer. The solubility of the hemoglobin is increased at all temperatures, and is about 29 g/dl at 25 degrees C. Polymerization kinetics, induced by laser photolysis and observed by light-scattering intensity, showed exponential growth with rates about 300 times slower than experiments done on similar concentrations of HbS. When polymerization is carried out in small volumes, the time of measurable light-scattering signal to reach one-tenth of its final value (denoted as the tenth time) showed stochastic fluctuations, as is seen in pure HbS. Homogeneous nucleation rates were measured by observing distributions of tenth times and these rates were slowed by the mutation by almost 1000-fold relative to pure HbS. The kinetics, including the exponential progress curves and shape of the tenth time distributions, are well described by the double nucleation mechanism for polymerization. Analysis of the homogeneous nucleation rates leads to the surprising conclusion that the mutation has scarcely changed the energy of the intermolecular contacts despite the increase in solubility of the double mutant. This conclusion is supported by the stereochemistry of the modified contact site, in which the amount of exposed hydrophobic surface appears to be unchanged by the mutation. The increased solubility must therefore result from decreased motional freedom of molecules within the polymer, which could arise from tighter packing into the enlarged receptor pocket. This points up the ability of kinetic analysis to reveal important thermodynamic properties of assembly, and underlines the importance of the vibrational degrees of freedom in setting the final equilibrium constant. Chemical modifications to restrict vibrations and enhance the cost of polymerization may prove useful in constructing compounds to act as inhibitors of sickle cell gelation.

Flexibility and nucleation in sickle hemoglobin.

We have studied the self-assembly of Hemoglobin C-Harlem (HbC-Harlem), a double mutant of hemoglobin that possesses the beta6 Glu-->Val mutation of sickle hemoglobin (HbS) plus beta73 Asp-->Asn. By electron microscopy we find it forms crystals, rather than the wrapped multistranded fibers seen in HbS. Fourier transforms of the crystals yield unit cell parameters indistinguishable from crystals of HbS. Differential interference contrast (DIC) microscopy and birefringence also show crystal formation rather than the polymers or domains seen for HbS, while the growth patterns showed radiating crystal structures rather than simple linear crystalline forms. The solubility of the assembly was measured using a photolytic micromethod over a temperature range of 17-31 degrees C in 0.15 M phosphate buffer and found to be essentially the same as that of fibers of HbS. The assembly kinetics were observed by photolysis of the carbon monoxide derivative, and the mass of assembled hemoglobin was found to grow exponentially, with onset times that were stochastically distributed for small volumes. The stochastic onset of assembly showed strong concentration dependence, similar to but slightly greater than that seen in sickle hemoglobin nucleation. These observations suggest that like HbS, HbC-Harlem assembly proceeds by a homogeneous nucleation process, followed by heterogeneous nucleation. However, relative to HbS, both homogeneous and heterogeneous nucleation are suppressed by almost 11 orders of magnitude. The slowness of nucleation can be reconciled with the similarity of the solubility to HbS by an increase in contact energy coupled with a decrease in vibrational entropy recovered on assembly. This also explains the linearity of the double-strands, and agrees with the chemical nature of the structural replacement.

A model for the sickle hemoglobin fiber using both mutation sites.

The standard molecular model of the fiber of the sickle hemoglobin (HbS: beta6 Glu-->Val) has been revised to allow both beta6 mutation sites to participate in intermolecular contacts, rather than only one beta6 site as previously thought, for four molecules per 14-molecule fiber cross section. This structure accurately predicts the copolymerization of hybridized mixtures of HbS with HbA or HbC (beta6 Glu-->Lys), which could not be reconciled with prior models in which only half the beta6 sites were required for assembly. This model suggests new contacts within the fiber and raises the question of whether these cross-linked double strands could possess added stability important in such processes as nucleation.

Fiber depolymerization.

Depolymerization is, by definition, a crucial process in the reversible assembly of various biopolymers. It may also be an important factor in the pathology of sickle cell disease. If sickle hemoglobin fibers fail to depolymerize fully during passage through the lungs then they will reintroduce aggregates into the systemic circulation and eliminate or shorten the protective delay (nucleation) time for the subsequent growth of fibers. We study how depolymerization depends on the rates of end- and side-depolymerization, k(end) and k(side), which are, respectively, the rates at which fiber length is lost at each end and the rate at which new breaks appear per unit fiber length. We present both an analytic mean field theory and supporting simulations showing that the characteristic fiber depolymerization time tau= square root 1/k(end)k(side) depends on both rates, but not on the fiber length L, in a large intermediate regime 1 << k(side)L(2)/k(end) << (L/d)(2), with d the fiber diameter. We present new experimental data which confirms that both mechanisms are important and shows how the rate of side depolymerization depends strongly on the concentration of CO, acting as a proxy for oxygen. Our theory remains rather general and could be applied to the depolymerization of an entire class of linear aggregates, not just sickle hemoglobin fibers.

Allosteric interpretation of the measurement of cooperative free energy in cyanomethemoglobin.

The experimentally resolved cooperative energies in partially ligated cyanomethemoglobin [F. R. Smith & G. K. Ackers (1985) Proc. Natl. Acad. Sci. USA 82, 5347-5351] have been compared with the predictions of an allosteric description of hemoglobin. A pattern of energetics similar to that observed (a "combinatorial switch") arises naturally from such an analysis using parameters in excellent agreement with other determinations. Although the energies for 2 of the 10 ligation states (namely, doubly ligated asymmetric tetramers) differ from the predictions, the remaining 8 of the 10 states exhibit excellent quantitative agreement with an allosteric description. This explains the discrepancy between previous analyses, which had found cyanomethemoglobin to be allosteric, and provides support for the basic allosteric concept that quaternary structure is the primary modifier for ligand affinity in hemoglobin.

The polymerization of sickle hemoglobin in solutions and cells.

The polymerization of sickle hemoglobin occurs by the same mechanisms in solutions and in cells, and involves the formation of 14 stranded fibers from hemoglobin molecules which have assumed a deoxy quaternary structure. The fibers form via two types of highly concentration-dependent nucleation processes: homogeneous nucleation in solutions with hemoglobin activity above a critical activity, and heterogeneous nucleation in similarly supersaturated solutions which also contain hemoglobin polymers. The latter pathway is dominant, and creates polymer arrays called domains. The individual polymers bend, but also cross-link, and the resulting mass behaves as a solid. The concentration of polymerized hemoglobin increases exponentially unless clamped by rate limiting effects such as oxygen delivery.

Simulated formation of polymer domains in sickle hemoglobin.

Using experimentally observed processes of linear growth, heterogeneous nucleation, and polymer bending, with no additional assumptions, we have been able to model the two-dimensional formation of polymer domains by sickle hemoglobin. The domains begin with twofold symmetry and proceed toward closure into spherulites at a constant rate. Relationships derived from the simulations presented and the requirements of scaling result in simple expressions for the sensitivity of the closure times to the model input parameters and allow the results to be extended to regions not actually simulated. For concentrations above approximately 25 g/dl, closure times are longer than the time required for the conclusion of the polymerization reaction, and thus incomplete spherulites will be the dominant geometry at high concentrations. Moreover, spherulites are not predicted to form in times less than a few seconds, implying that spherulites will not form during the transit of erythrocytes through the capillaries. Polymer-polymer exclusion, surface nucleation, and monomer exhaustion were also explored and found to have only weak effects on the results.

Rate of allosteric change in hemoglobin measured by modulated excitation using fluorescence detection.

We have measured the forward and reverse rates of the allosteric transition of hemoglobin A with three CO molecules bound by using modulated excitation coupled with fluorescence quenching of the DPG analogue, PTS (8-hydroxy-1,3,6 pyrene trisulfonic acid). This dye is observed to bind to the T state with significantly larger affinity than to the R state, and thus provides an unequivocal marker for the molecule's conformational change. The allosteric rates obtained with the fluorescent dye (pH 7.0, bis-Tris buffer) are (3.4 +/- 1.0) x 10(3)s-1 for the R to T transition and (2.1 +/- 0.5) x 10(4)s-1 for the T to R transition. This gives an equilibrium constant L3 of 0.16 +/- 0.06. These results provide good agreement with modulated difference spectra calibrated from model compounds, arguing that there is little if any difference in the kinetics observed by the heme spectra and the kinetics of the full subunit motion. The equilibrium constant between structures (L3) is smaller in the absence of phosphates than observed in phosphate buffer (0.33). However, the rates of the allosteric transition increase in the absence of phosphates as compared with the corresponding rates in phosphate buffer of 1.0 x 10(3)s-1 and 3.0 x 10(3)s-1. The effects of inorganic phosphates on the equilibrium can be separated from the effects on kinetics. We find that phosphates also affect the dynamic behavior of hemoglobin, and the presence of 0.15 M phosphate can be viewed as raising the transition state energy between R and T conformations by approximately 0.5 kcal/mol exclusive of the T state stabilization. Dissociation constants for the dye were measured to be 104 +/- 25 microM for unligated T state and 930 +/- 300 microM for the fully ligated R state. The best fit equilibrium constant (125 +/- 40 microM) for three ligands bound does not differ significantly from that measured without ligands bound. Incidental to the measurement technique is the determination of the rates of binding and release of the dye. The association rate for binding to the T state is large, (at least 4 x 10(9) M-1 s-1) and may be diffusion limited, while the association and dissociation rates for R state binding, while not determined with precision, are clearly much smaller, of the scale of 10(5) M-1 s-1 for association.

Monomer diffusion and polymer alignment in domains of sickle hemoglobin.

We have used polarized absorbance to observe the process of monomer accretion and polymer alignment which occurs in domains of sickle hemoglobin that are formed and maintained by laser photolysis. These diffusion and alignment processes have been studied as a function of initial concentration and temperature (initial and final), as well as beam size and domain number. Monomers are found to diffuse into growing polymer domains with a rate that is essentially temperature and concentration independent, but which depends on the size of the final domain boundaries, and the number of domains within a boundary. The final concentrations achieved are very close to those found in packed centrifugation experiments (50-55 g/dl) and are approximately independent of starting temperature and concentration. The influx of monomers is accompanied by polymer alignment, and the amount aligned is proportional to the amount diffused throughout the process. We propose that polymer alignment controls the influx of added monomers into the growing domain.

Homogeneous nucleation in sickle hemoglobin: stochastic measurements with a parallel method.

The homogeneous nucleation rate for sickle hemoglobin polymerization has been measured for concentrations from 3.9 to 4.9 mM and temperatures from 13 degrees C to 35 degrees C by observing the stochastic fluctuations of the time to complete 10% of the reaction after photolysis of the carboxy derivative. To allow efficient data collection, a mesh was used to divide the photolysis beam into an array of smaller beams, which allowed parallel observation of about 100 different regions. Nucleation rates measured here are consistent with more restricted previously published data and, when combined with directly measured monomer addition rates, are consistent with previous analysis of progress curves. By describing these rates with equilibrium nucleation theory, the concentration of nuclei and hence their stability can be ascertained. Consequently, the chemical potential by which a monomer is attached to the polymer is determined. This attachment energy ranges from -6.6 to -8.0 kcal/mol between 15 degrees C and 35 degrees C. The enthalpic part of that chemical potential is found to be equal to the enthalpy determined by solubility measurements, as expected from thermodynamic considerations. The entropic portion of the contact chemical potential contributes from -21.4 to -8.7 kcal/mol. The vibrational chemical potential of monomers in the polymer ranges from -25.7 to -27.4 kcal/mol over the same temperatures.

Rate of quaternary structure change in hemoglobin measured by modulated excitation.

Using a novel technique of modulated photo-dissociation of carbon monoxide from hemoglobin, we have obtained the rates for conversion between the two quaternary states, R, and T, at 3-fold ligation. Our measurements at pH 7 and 22 degrees give rates of 780 +/- 40 sec-1 for going from R to T, and 2500 +/- 200 sec-1 from T to R. This yields an equilibrium constant of 0.31 +/- 0.04, which is in good agreement with previous estimates. The degree of agreement between this equilibrium constant and that predicted from the allosteric model provides a new, quantitative test of the allosteric description. A sequential model for the change in structure was found incompatible with the data, even if kinetic subunit inequivalence was assumed. The technique described here is quite general and can be used as long as the system under investigation can be repetitively excited in a regime in which it responds linearly to the excitation.