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.

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.

Twisted protein aggregates and disease: the stability of sickle hemoglobin fibers.

We describe how twist could play an essential role in stabilizing 20 nm diameter sickle hemoglobin fibers. Our theory successfully reproduces the observed variation of helical pitch length with fiber diameter. With no remaining adjustable parameters it also yields a prediction for the torsional rigidity of sickle hemoglobin fibers that is in good agreement with experiment and hence retains the striking feature that such fibers can be highly mechanically anisotropic, even with a ratio of bending to torsional rigidity of about 50. We discuss how our study might be relevant to the development of treatment strategies.

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.

Nonideality and the nucleation of sickle hemoglobin.

The homogeneous and heterogeneous nucleation kinetics of sickle hemoglobin (HbS) have been studied for various degrees of solution crowding by substitution of cross-linked hemoglobin A, amounting to 50% of the total hemoglobin. By cross-linking hemoglobin A, hybrid formation between hemoglobin A and hemoglobin S was prevented, thus simplifying the analysis of the results. Polymerization was induced by laser photolysis, and homogeneous nucleation kinetics were determined by observation of the stochastic behavior of the onset of light scattering. Heterogeneous nucleation was determined by observing the exponential growth of the progress curves, monitored by light scattering. At concentrations between 4 and 5 mM tetramer (i.e., approximately 30 g/dl), the substitution of 50% HbA for HbS slows the reaction by a factor of 10(3) to 10(4). Using scaled particle theory to account for the crowding of HbA, the observed decrease in the homogeneous nucleation rate was accurately predicted, with no variation of parameters required. Heterogeneous nucleation, on the other hand, is not well described in the present formulation, and the theory for this process appears to require modification of the way in which nonideality is introduced. Nonetheless, the accuracy of the homogeneous nucleation description suggests that such an approach may be useful for other assembly processes that occur in a crowded intracellular milieu.

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.

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.

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.

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.

Solubility of sickle hemoglobin measured by a kinetic micromethod.

We have developed a photolytic method to determine the concentration of reactive hemes in a solution in the presence of a trace amount of CO. By measurement of the bimolecular rate of CO binding, and by calibration of the rate constant under equivalent conditions, the concentration of the reactive hemes can be determined. In a solution of sickle hemoglobin, the molecules in the gel contribute negligibly to the recombination rate, allowing the concentration of the molecules in the solution phase to be determined. To optimize signal to noise, modulated excitation methods were employed, although the method could also be used with pulse techniques and suitable signal averaging. Because the optical method employs a microspectrophotometer, only a few microliters of concentrated Hb solution is required to reproduce the entire temperature dependence of the solubility previously determined by centrifugation using milliliter quantities of solutions of the same concentration. This should be especially useful for studies of site-directed mutants, and we present results obtained on one such HbS in which Leu 88 beta has been replaced by Ala. The free energy difference in the polymerization of the Leu 88 beta double mutant is consistent with known differences in the amino acid hydrophobicities. The calibration required for these experiments also provides an excellent determination of the activation energy for binding the first CO to deoxy Hb.

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.

Modulated excitation of singly ligated carboxyhemoglobin.

We have extended the method of modulated excitation, a small perturbation kinetic method, to study ligand binding and conformational change of hemoglobin tetramers with a single ligand bound. To restrict the excitation to the first ligand, only 1% of the hemes have bound CO, and the remainder are kept unliganded. A detailed theory is presented which agrees well with the experimental observations. This method of observing ligand recombination also provides a novel and simple method for determination of hemoglobin concentration. Additional relaxation processes are also observed. By fitting independently determined spectra to the spectra associated with the relaxations, these processes are assigned as thermal excitation and thermally driven protonation/deprotonation reactions. These added relaxations arise from the deoxy-Hb portion of the samples, and demonstrate that modulated excitation can be used effectively for temperature perturbation in the absence of photodissociation. The spectra observed are not well described by the spectra of allosteric change, however, and we conclude that there is no significant mixing of quaternary states at the first ligation step. In an appendix we present a derivation of the particular features seen in thermally modulated protonation reactions.

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.

Allosteric kinetics and equilibria of triligated, cross-linked hemoglobin.

Using modulated excitation, we have measured the forward and reverse rates of the allosteric transition between relaxed (R) and tense (T) quaternary structures for triply ligated hemoglobin (Hb), cross-linked between the alpha chains at Lys 99. Oxygen, carbon monoxide, and water were used as ligands and were studied in phosphate and low Cl- bis-Tris buffers at neutral pH. Since the cross-link prohibits disproportionation, triply ligated aquomet Hb species with ferrous beta chains were specifically isolated by isoelectric focusing. Modulated excitation provides rate pairs and therefore gives equilibrium constants between quaternary structures. To coordinate with that information, oxygen binding curves of fully ferrous and tri-aquomet Hb were also measured. L3, the equilibrium constant between three liganded R and T structures, is determined by modulated excitation to be of order unity for O2 or CO (1.1 to 1.5 for 3O2 and 0.7 for 3CO bound), while with three aquomet subunits it is much greater (> or = 23). R-->T conversion rates are similar to those found for HbA, with weak sensitivity to changes in L3. The L3 values from HbXL O2 were used to obtain a unique allosteric decomposition of the ferrous O2 binding curve in terms of KT, KR, and L3. From these values and the O2 binding curve of tri-aquomet HbXL, L3 was calculated to be 2.7 for the tri-aquomet derivative. Consistency in L3 values between equilibrium and modulated excitation data for tri-aquomet-HbXL can be achieved if the equilibrium constant for O2 binding to the alpha chains is six times lower than that for binding to the beta chains in the R state, while the cooperative properties remain homogeneous. The results are in quantitative agreement with other studies, and suggest that the principal effect of the cross-link is to decrease the R state and T state affinity of the alpha subunits with almost no change in the affinity of the beta subunits, leaving the allosteric parameters L and c unchanged.

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.

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.

Monomer diffusion into polymer domains in sickle hemoglobin.

The gelation of sickle hemoglobin includes the formation of spherulitic arrays of polymers, known as polymer domains, which are an intrinsic result of the polymer formation mechanism. We have observed the diffusion of monomers into domains as they form, which substantially increases the total concentration of hemoglobin within the domain. The maximum total concentration attained is comparable with the pellet concentration of 0.5-0.55 g/cm3 obtained in sedimentation experiments. The half time for this process is approximately 50 s for domains of 25 microns radius, and is approximately independent of temperature. The shape of the diffusion progress curves as well as the deduced diffusion constants, and their weak temperature dependence are consistent with a simple model of hemoglobin monomer diffusion into the domain.

Theoretical description of the spatial dependence of sickle hemoglobin polymerization.

We have generalized the double nucleation mechanism of Ferrone et al. (Ferrone, F. A., J. Hofrichter, H. Sunshine, and W. A. Eaton. 1980. Biophys. J. 32:361-377; Ferrone, F. A., J. Hofrichter, and W. A. Eaton. 1985. J. Mol. Biol. 183:611-631) to describe the spatial dependence of the radial growth of polymer domains of sickle hemoglobin. Although this extended model requires the consideration of effects such as monomer diffusion, which are irrelevant to a spatially uniform description, no new adjustable parameters are required because diffusion constants are known independently. We find that monomer diffusion into the growing domain can keep the net unpolymerized monomer concentration approximately constant, and in that limit we present an analytic solution of the model. The model shows the features reported by Basak, S., F. A. Ferrone, and J. T. Wang (1988. Biophys J. 54:829-843) and provides a new means of determining the rate of polymer growth. When spatially integrated, the model exhibits the exponential growth seen in previous studies, although molecular parameters derived from analysis of the kinetics assuming uniformity must be modified in some cases to account for the spatially nonuniform growth. The model developed here can be easily adapted to any spatially dependent polymerization process.