Fluctuations in the polymerization of sickle hemoglobin. A simple analytic model

Using the stochastic theory of chemical reactions and the theory of first passage times, a simple analytic expression is derived for the distribution of delay times that has been observed in studies of the polymerization kinetics of sickle hemoglobin under conditions where the polymerization progress curves exhibit stochastic variation. The rate of homogeneous nucleation can be readily extracted from such experiments using this expression. This work constitutes a significant addition to the rather limited number of examples where contact can be successfully made between the stochastic theory of chemical kinetics and experiment.     

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.     

Metastable polymerization of sickle hemoglobin in droplets

Sickle cell disease arises from a genetic mutation of one amino acid in each of the two hemoglobin beta chains, leading to the polymerization of hemoglobin in the red cell upon deoxygenation, and is characterized by vascular crises and tissue damage due to the obstruction of small vessels by sickled cells. It has been an untested assumption that, in red cells that sickle, the growing polymer mass would consume monomers until the thermodynamically well-described monomer solubility was reached. By photolysing droplets of sickle hemoglobin suspended in oil we find that polymerization does not exhaust the available store of monomers, but stops prematurely, leaving the solutions in a supersaturated, metastable state typically 20% above solubility at 37 degrees C, though the particular values depend on the details of the experiment. We propose that polymer growth stops because the growing ends reach the droplet edge, whereas new polymer formation is thwarted by long nucleation times, since the concentration of hemoglobin is lowered by depletion of monomers into the polymers that have formed. This finding suggests a new aspect to the pathophysiology of sickle cell disease; namely, that cells deoxygenated in the microcirculation are not merely undeformable, but will actively wedge themselves tightly against the walls of the microvasculature by a ratchet-like mechanism driven by the supersaturated solution.     

Universal metastability of sickle hemoglobin polymerization

Sickle hemoglobin (HbS) polymerization occurs when the concentration of deoxyHbS exceeds a well-defined solubility. In experiments using sickle hemoglobin droplets suspended in oil, it has been shown that when polymerization ceases the monomer concentration is above equilibrium solubility. We find that the final concentration in uniform bulk solutions (i.e., with negligible boundaries) agrees with the droplet measurements, and both exceed the expected solubility. To measure hemoglobin in uniform solutions, we used modulated excitation of trace amounts of CO in gels of HbS. In this method, a small amount of CO is introduced to a spatially uniform deoxyHb sample, so that less than 2% of the sample is liganded. The liganded fraction is photolyzed repeatedly and the rate of recombination allows the concentration of deoxyHbS in the solution phase to be determined, even if polymers have formed. Both uniform and droplet samples exhibit the same quantitative behavior, exceeding solubility by an amount that depends on the initial concentration of the sample, as well as conditions under which the gel was formed. We hypothesize that the early termination of polymerization is due to the obstruction in polymer growth, which is consistent with the observation that pressing on slides lowers the final monomer concentration, making it closer to solubility. The thermodynamic solubility in free solution is thus achieved only in conditions with low polymer density or under external forces (such as found in sedimentation) that disrupt polymers. Since we find that only about 67% of the expected polymer mass forms, this result will impact any analysis predicated on predicting the polymer fraction in a given experiment.     

Gelation of sickle cell hemoglobin in mixtures with normal adult and fetal hemoglobins.

We report the results of thermodynamic and kinetic studies on the gelation of mixtures of sickle cell (S) deoxyhemoglobin with normal human adult (A) and fetal (F) deoxyhemoglobins. The delay time of thermally induced gelation was monitored by the increase in turbidity. At the completion of gelation the solubility was determined by sedimenting the polymers and measuring the supernatant concentration spectrophotometrically. Addition of hemoglobins A or F, at mole fractions from 0 to 0.6, resulted in large increases in both the solubility and the delay time. For a 50:50 mixture of deoxyhemoglobin F with deoxyhemoglobin S, the solubility increased by a factor of 1.8 and the delay time by a factor of 10⁷ relative to pure deoxyhemoglobin S at the same total concentration, while for a 50:50 mixture of deoxyhemoglobins A and S the solubility increased by a factor of 1.4 and the delay time by a factor of 10⁴. The relative delay times were independent of both temperature and total hemoglobin concentration. The data have been analyzed according to theoretical models which treat the effects of temperature, concentration, non-ideality and solution composition on the thermodynamics and kinetics of gelation. The increased solubility in mixtures with deoxyhemoglobin F is fully explained by a model in which only deoxyhemoglobin S molecules polymerize. The effect of fetal hemoglobin (α₂γ₂) and hybrid α₂γβ^S molecules is to increase the solution non-ideality through the contribution of their excluded volume. The smaller increase in the solubility observed in comparable mixtures with deoxyhemoglobin A requires that the hybrid α₂β^Aβ^S molecules copolymerize with the deoxyhemoglobin S. The kinetic results for the mixtures can be quantitatively accounted for using a nucleation model in which the equilibrium properties of the polymer are used to describe the critical nucleus. The very large increases in delay time observed for the $\text{S}\text{F}$ mixtures can be explained by assuming that only α₂β₂^S molecules participate in the formation of a nucleus containing about 25 monomers. As in the thermodynamic analysis, the smaller effect of adding deoxyhemoglobin A can be attributed to the contribution of the hybrid molecules in forming the critical nucleus. Thus the difference between the polymerization properties of mixtures of deoxyhemoglobin S with deoxyhemoglobins A and F can be attributed solely to the copolymerization of the α₂β^Aβ^S hybrid molecule and the absence of any significant copolymerization of the α₂γβ^S hybrid.     

Influence of sickle hemoglobin polymerization and membrane properties on deformability of sickle erythrocytes in the microcirculation.

The rheological properties of normal erythrocytes appear to be largely determined by those of the red cell membrane. In sickle cell disease, the intracellular polymerization of sickle hemoglobin upon deoxygenation leads to a marked increase in intracellular viscosity and elastic stiffness as well as having indirect effects on the cell membrane. To estimate the components of abnormal cell rheology due to the polymerization process and that due to the membrane abnormalities, we have developed a simple mathematical model of whole cell deformability in narrow vessels. This model uses hydrodynamic lubrication theory to describe the pulsatile flow in the gap between a cell and the vessel wall. The interior of the cell is modeled as a Voigt viscoelastic solid with parameters for the viscous and elastic moduli, while the membrane is assigned an elastic shear modulus. In response to an oscillatory fluid shear stress, the cell--modeled as a cylinder of constant volume and surface area--undergoes a conical deformation which may be calculated. We use published values of normal and sickle cell membrane elastic modulus and of sickle hemoglobin viscous and elastic moduli as a function of oxygen saturation, to estimate normalized tip displacement, d/ho, and relative hydrodynamic resistance, Rr, as a function of polymer fraction of hemoglobin for sickle erythrocytes. These results show the transition from membrane to internal polymer dominance of deformability as oxygen saturation is lowered. More detailed experimental data, including those at other oscillatory frequencies and for cells with higher concentrations of hemoglobin S, are needed to apply fully this approach to understanding the deformability of sickle erythrocytes in the microcirculation. The model should be useful for reconciling the vast and disparate sets of data available on the abnormal properties of sickle cell hemoglobin and sickle erythrocyte membranes, the two main factors that lead to pathology in patients with this disease.     

The rates of polymerization and depolymerization of sickle cell hemoglobin

The polymerization and depolymerization of concentrated solutions of sickle cell deoxyhemoglobin were initiated by raising and lowering the temperature, and the time courses of the reactions monitored by the change in apparent turbidity. The polymerization reaction exhibits a marked lag phase followed by a rapid increase in turbidity, and is dependent on a very high power of the hemoglobin concentration, roughly the fifteenth. The depolymerization reaction exhibits no such lag, and is much less dependent on concentration. The implications of these results for polymerization models are discussed.     

Kinetics of sickle hemoglobin polymerization. III. Nucleation rates determined from stochastic fluctuations in polymerization progress curves

The polymerization kinetics of sickle cell hemoglobin are found to exhibit stochastic variations when observed in very small volumes (approximately 10(-10) cm3). The distribution of progress curves has been measured at several temperatures for a 4.50 mM-hemoglobin S sample using a laser-photolysis, light-scattering technique. The progress curves at a given temperature are superimposable when translated along the time axis, showing that the variability of the kinetic progress curves results primarily from fluctuations in the time at which polymerization is initiated. The shapes of the initial part of the progress curves are well-fitted using the functional form I(t) = Io + As exp (Bt), derived from a dual nucleation model. When the distribution of the measured tenth times is broad, the rate of homogeneous nucleation can be obtained by fitting the exponential tail of the distribution. As the distribution sharpen, the rate of homogeneous nucleation can be estimated by modelling the width of the distribution function using a simple Monte-Carlo simulation of the polymerization kinetics. Using the rates of homogeneous nucleation obtained from the distributions, the rates of heterogeneous nucleation and polymer growth can be obtained from the experimental parameters As and B. The resulting nucleation rates are roughly 1000 times greater than those obtained from an analysis of bulk kinetic data. The results provide strong support for the dual-nucleation mechanism and show that the distribution of progress curves provides a powerful independent method for measuring the rate of homogeneous nucleation and thereby obtaining values for the other principal rates of the mechanism.     

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.     

Electron microscopic studies of the intracellular polymerization of sickle hemoglobin

Transmission electron microscopy has been used to study intracellular sickle hemoglobin polymer in unfractionated cells from the arterial and venous blood of patients and after external deoxygenation. We detect polymerized hemoglobin in up to 10% of the cells in the venous circulation, especially in cells that are "cigar-shaped" and appear to be irreversibly sickled. We could not see well-defined polymer in mixed arterial samples; nevertheless, we found electron opaque spots, which could be ferritin granules, hemosiderin, or small aggregates of hemoglobin S. However, upon sequential chemical deoxygenation using 1.0% sodium metabisulphite, polymer formation was seen at oxygen saturation values of 75%-85%. Cells that were physically deoxygenated using gas mixtures containing nitrogen-carbon dioxide-oxygen mixtures were found to contain distinct polymers of deoxyhemoglobin S at oxyhemoglobin saturation values of 50%-75%. As deoxygenation increases, we detect short, randomly arranged polymer in a loose network, with occasional long polymers. Upon further deoxygenation, the length and number of polymer forms increased. Between 0% and 50% saturation, most erythrocytes were full of long, parallel, closely packed polymers that tend to align and run parallel to the cell membrane. In both chemical and physically deoxygenated blood samples, cells were seen at 50%-75% oxyhemoglobin saturation that retained their normal biconcave disc shape, although they contained significant amounts of polymer. The structural changes in sickle erythrocytes seen in vitro due to physical or chemical deoxygenation of cells, may reflect in vivo intracellular changes in the sickle cell patient.     

Abnormal human hemoglobins X. A study of hemoglobin leporeBoston

An abnormal hemoglobin has been isolated from two related individuals of Italian extraction. This abnormal hemoglobin has been identified with hemoglobin Lepore_Boston on the basis of the fingerprinting pattern. Chemical studies have been performed on the “core” of this hemoglobin Lepore. Tryptic digestion after carboxy-methylation has been found to be the most suitable procedure for analyzing the “core” of the non-α chains. The fingerprinting analysis of the carboxymethylated hemoglobin has given sufficient information on the structure of this hemoglobin to pinpoint accurately the region along the amino acid sequence of the abnormal peptide chains, where the δ-like portion of these chains is joined to the β-like portion. Evidence has been obtained to show that the abnormal peptide chains of hemoglobin Lepore are equal in length to either β- or δ-chains (146 amino acids).     

Gelation of sickle hemoglobin. III. Nitrosyl hemoglobin.

Sickle cell nitrosyl hemoglobin was examined for gelation by an ultracentrifugal method previously described (Briehl &; Ewert, 1973) and by birefringence. In the presence of inositol hexaphosphate gelation which exhibited the endothermic temperature dependence seen in gels of deoxyhemoglobin S was observed by both techniques. In the absence of inositol hexaphosphate no gelation was observed, nor did nitrosyl hemoglobin A exhibit gelation. On the assumption that gelation is dependent on the deoxy or T (low ligand affinity) as opposed to the oxy or R (high ligand affinity) quaternary structure this supports the conclusion that nitrosyl hemoglobin S in inositol hexaphosphate assumes the T structure, in contrast to the other liganded ferrohemoglobin derivatives oxy and carbon monoxide hemoglobin. Assuming further that the quaternary structures and isomerizations are the same in hemoglobins A and S it can also be concluded that nitrosyl hemoglobin A in inositol hexaphosphate assumes the T state. Since no gelation was seen in stripped nitrosyl hemoglobin S, inositol hexaphosphate serves to effect an R to T switch in this derivative. Thus R-T isomerization in nitrosyl hemoglobin occurs without change in ligand binding at the sixth position of the heme group confirming the conclusion of Salhany (1974) and Salhany et al. (1974).Lowering of the pH toward 6 favors gelation of NO hemoglobin S as it does of deoxy and aquomethemoglobin S (Briehl &; Ewert, 1973,1974), consistent with a favoring of the T structure due to strengthening of the interchain salt bridges and the binding of inositol hexaphosphate and/or changes in site-to-site interactions on which gelation depends.     

Dielectric constant of sickle cell hemoglobin. Dielectric properties of sickle cell hemoglobin in solution and gel

The dielectric constants of sickle cell hemoglobin were determined before and after gelation. The dielectric properties of oxy and deoxy sickle cell hemoglobin in solution are nearly identical to those of oxy and deoxy hemoglobin A. Only in the gel state did deoxy sickle cell hemoglobin display dielectric behavior different from that in solution. Upon gelation of deoxy sickle cell hemoglobin, the dielectric constant showed a marked decrease, and the relaxation frequency shifted towards higher frequencies. This result suggests that dielectric constant measurement can be used for the investigation of the kinetics of polymerization of sickle cell hemoglobin molecules. Despite the marked decrease in the dielectric constant, deoxy sickle cell hemoglobin still showed a well-defined dielectric dispersion even in the gel state. This indicates that individual molecules have considerable freedom of rotation in gels. It was observed that the dielectric properties of gelled deoxy sickle cell hemoglobin were affected by electrical fields at the level of 10 to 20 V/cm. This observation suggests that electrical fields of moderate strengths are able to perturb the gel structure if the system is near the transition region. The non-linear electrical behavior of gelled sickle cell hemoglobin will be discussed further in subsequent papers.     

Interaction of serum albumin with normal and sickle hemoglobins.

The stability of oxyhemoglobin S during mechanical shaking was enhanced by the addition of human serum albumin. The stabilizing effect was maximum when the concentration of serum albumin approached that of oxyhemoglobin, suggesting a molecular level interaction between them. The effects of serum albumin on oxyhemoglobin A were essentially similar to those on oxyhemoglobin S. Deoxy- and methemoglobins were also stabilized by serum albumin. The addition of human serum albumin to a solution containing sickle cell oxyhemoglobin slowly formed a compound which had an absorbance peak at 620 nm. After purification by Sephadex G-200 column chromatography, this compound was identified as methemalbumin. Comparison of the rates of formation of methemalbumin from hemoglobin with various ligand states and human serum albumin showed that the rate of formation from hemichrome was much faster than from met-, oxy- and deoxyhemoglobin. About 60% of the heme was transferred from hemichrome to albumin when the mixture was kept standing at room temperature for 5 min, in contrast to only 5% from methemoglobin. This result suggests that hemichrome, rather than methemoglobin, is the intermediate in the formation of methemalbumin from oxyhemoglobin and human serum albumin. This hypothesis is supported by the finding that the rate of formation of methemalbumin was faster at alkaline pH values than at acid pH values. Serum albumin from various animal sources showed different stabilizing effects. The formation of methemalbumin from these animal albumins was far less than that from human albumin.     

Molecular dynamics of sickle and normal hemoglobins

Molecular dynamics (MD) simulations have been carried out for 62.5 ps on crystal structures of deoxy sickle cell hemoglobin (HbS) and normal deoxy hemoglobin (HbA) using the CHARMM MD algorithm, with a time step of 0.001 ps. In the trajectory analysis of the 12.5–62.5 (50 ps) simulation, oscillations of the radius of gyration and solvent-accessible surface area were calculated. HbS exhibited a general contraction during the simulation, while HbA exhibited a nearly constant size. The average deviations of simulated structures from the starting structures were found to be 1.8 Å for HbA and 2.3 Å for HbS. The average rms amplitudes of atomic motions (atomic flexibility) were about 0.7 Å for HbA and about 1.0 Å for HbS. The amplitudes of backbone motion correlate well with temperature factors derived from x-ray crystallography. A comparison of flexibility between the α- and β-chains in both HbA and HbS indicates that the β-chains generally exhibited greater flexibility than the α-chains, and that the HbS β-chains exhibit greater flexibility in the N-terminal and D- and F-helix regions than do those of HbA. The average amplitude of backbone torsional oscillations was about 9°, a value comparable with that of other simulations, with enhanced torsional oscillation occurring primarily at the ends of helices or in loop regions between helices. Comparison of atomic flexibility and torsional oscillation results suggests that the increased β-chain flexibility results from relatively concerted motions of secondary structure elements. The increased flexibility may play an important role in HbS polymerization. Time course analysis of conformational energy of association, hydrogen bonding and hydrophobic bonding (as calculated from solvent accessibility) shows that all three of these factors contribute to the stability of subunit association for both hemoglobins. © 1993 John Wiley & Sons, Inc.     

The hemoglobins of Artemia salina. IV. A model for genetic control of hemoglobin 1, hemoglobin 2, and hemoglobin X

Two loci account for all genetic variation resulting in difference in electrophoretic mobility in three hemoglobins (Hb1, Hb2, and HbX) in the hemolymph of the brine shrimp. Four alleles and nine alleles have been studied. In shrimps of all genotypes and in electrophoresis in media with varying degrees of molecular sieving, Hb2 is approximately equidistant from Hb1 and HbX. A shrimp heterozygous at both loci has a three-banded Hb1, a four-banded Hb2, and a three-banded HbX. We conclude that Hb2 contains n -polypeptides and n -polypeptides. Hb1 contains 2n -polypeptides. HbX contains 2n -polypeptides. During electrophoresis, the three native hemoglobins undergo reversible dissociation to n subunits. Subunits with the same charge reassemble to migrate as molecules of the same size as the native molecules. Although there is no evidence for an additional polypeptide in the three hemoglobins, we cannot exclude such a possibility. If it exists, it is under three constraints: (1) it must be present in equal amounts in each of the three hemoglobins; (2) it must have the same molecular weight as the - and -polypeptides; and (3) it must be free of genetic variation (detectable by electrophoresis).Supported by National Institutes of Health Grant HE-11445.     

The hemoglobins of Artemia salina. V. Genetic variation in hemoglobin 1, hemoglobin 2, and hemoglobin 3

Electrophoretic mobilities of three hemoglobins (Hb1, Hb2, and Hb3) were studied in 15 populations of brine shrimps. Genetic segregation data support the model that Hb2 contains n -polypeptides and n -polypeptides; Hb1 contains 2n -polypeptides. Hb3 contains neither - nor -polypeptides. There is no evidence of linkage of and loci with each other or with the locus (or loci) which governs Hb3 or with the nonhomologous portion of the sex chromosomes. Hemoglobins of different populations may be hybridized in vitro by incubation at high temperature. Reversible dissociation to subunits which contain only one ( or ) polypeptide occurs at 40 C (for Hb1) and at 50 C (for Hb2).Supported by Grant HD-11445 from the National Institutes of Health.