The interaction of 2,3-diphosphoglycerate with various human hemoglobins.

Oxygen equilibria were measured on a number of human hemoglobins, which had been "stripped" of organic phosphates and isolated by column chromatography. In the presence of 2 x 10(-4) M 2,3-diphosphoglycerate (2,3-DPG), the P(50) of hemoglobins A, A(2), S, and C increased about twofold, signifying a substantial and equal decrease in oxygen affinity. Furthermore, hemoglobins Chesapeake and M(Milwaukee-1) which have intrinsically high and low oxygen affinities, respectively, also showed a twofold increase in P(50) in the presence of 2 x 10(-4) M 2,3-DPG. In comparison to these, hemoglobins A(IC) and F were less reactive with 2,3-DPG while hemoglobin F(I) showed virtually no reactivity. The N-terminal amino of each beta-chain of hemoglobin A(IC) is linked to a hexose. In hemoglobin F(I) the N-terminal amino of each gamma-chain is acetylated. These results suggest that the N-terminal amino groups of the non-alpha-chains are involved in the binding of 2,3-DPG to hemoglobin.

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.

Nucleation, fiber growth and melting, and domain formation and structure in sickle cell hemoglobin gels.

Pathogenesis in sickle cell disease depends on polymerization and gelation of deoxyhemoglobin S. Under the double nucleation model, polymerization is initiated by homogeneous nucleation, followed by heterogeneous nucleation on pre-existing fibers. Fibers grow by non-cooperative addition of hemoglobin. The model derives from macroscopic results rather than direct observation of individual events. We observe individual events and structures by differential interference contrast (DIC) microscopy to show consistency with the model, to define structure and development of gel domains and their relation to kinetics, and to demonstrate the mechanism of fiber melting. Kinetics were controlled by producing deoxyhemoglobin by photolysis of CO hemoglobin under DIC observation. The first visible polymers appeared randomly and were usually linear aggregates less than 1 micron long, consistent with homogeneous nucleation and immediate post-nucleation aggregates. Aggregates then branched extensively, consistent with heterogeneous nucleation. This branching of new fibers was also induced at countable rates on isolated single fibers. Branching and fiber growth rapidly produced dense domains. Changes in photolytic intensity altered domain growth rates and domain structure. At low intensity and slow growth, fibers grew radially without branching. Domains lacked cross-links and polymer density was low. High intensity produced faster growth, much heterogeneous nucleation and highly cross-linked, dense, domains. At still higher intensity, homogeneous nucleation was very rapid, producing many small domains. These results show a hierarchy of processes: as deoxyhemoglobin concentration increases, growth occurs without observable nucleations, and then heterogeneous and finally homogeneous nucleation become dominant. This is consistent with the double nucleation model under which the concentration dependence of growth is low, and that of heterogeneous and homogeneous nucleation successively higher. Under decreased photolysis, fiber ends melted continuously without fiber breakage; increased photolysis reversed this, producing growth. Isolated fibers melted and grew at both ends. The results are consistent with a fiber melting mechanism that is the reverse of growth.

Length distributions of hemoglobin S fibers.

Electron microscopy of sickle cell hemoglobin fibers fixed at different times during gelation shows an exponential distribution of fiber lengths, with many short fibers and few long ones. The distribution does not change significantly with time as polymerization progresses. If this distribution of lengths reflects kinetic mechanism of fiber assembly, it complements information from studies of the progress of average properties of the polymers and, as has been done for other rod-like polymerizing systems, permits testing of models for the mechanism of fiber assembly. In this case, the results are consistent with the double nucleation model of Ferrone et al. or with a related alternative model based on fiber breakage. However, other possible causes of this microheterogeneity exist, including: breakage due to solution shearing of the long, rod-like, fibers; the presence of residual nuclei; equilibrium relations governing polymerization; and breakage of solid-like but weak gels that develop early and adhere to the grid. The arguments against the first three of these possibilities suggest that they are not responsible. However, breakage of entanglements or cross-links in a solid-like and adherent gel is consistent with the distributions.

Deforming biological membranes: how the cytoskeleton affects a polymerizing fiber.

We give a theoretical treatment of the force exerted by a fluctuating membrane on a polymer rod tip, taking into account the effects of an underlying biological cytoskeleton by way of a simple harmonic dependence on displacement. We also consider theoretically and experimentally the dynamics of a growing fiber tip under the influence of such a fluctuation-induced membrane force, including the effects of an underlying cytoskeletal network. We compare our model with new experimental data for the growth of hemoglobin fibers within red blood cells, revealing a good agreement. We are also able to estimate the force and membrane/cytoskeletal displacement required to stall growth of, or buckle, a growing fiber. We discuss the significance of our results in a biological context, including how the properties of the membrane and cytoskeleton relate to the thermodynamics of rod polymerization.

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.

Measuring forces between protein fibers by microscopy.

We propose a general scheme for measuring the attraction between mechanically frustrated semiflexible fibers by measuring their thermal fluctuations and shape. We apply this analysis to a system of sickle hemoglobin (HbS) fibers that laterally attract one another. These fibers appear to "zip" together before reaching mechanical equilibrium due to the existence of cross-links into a dilute fiber network. We are also able to estimate the rigidities of the fibers. These rigidities are found to be consistent with sickle hemoglobin "single" fibers 20 nm in diameter, despite recent experiments indicating that fiber bundling sometimes occurs. Our estimate of the magnitude of the interfiber attraction for HbS fibers is in the range 8 +/- 7 kBT/microm, or 4 +/- 3 k(B)T/microm if the fibers are assumed, a priori to be single fibers (such an assumption is fully consistent with the data). This value is sufficient to bind the fibers, overcoming entropic effects, although extremely chemically weak. Our results are compared to models for the interfiber attraction that include depletion and van der Waals forces. This technique should also facilitate a similar analysis of other filamentous protein assembles in the future, including beta-amyloid, actin, and tubulin.

Rheology of hemoglobin S gels: possible correlation with impaired microvascular circulation.

The sequence of pathogenic events in sickle cell disease begins with the genetic abnormality and proceeds through molecular and red cell abnormalities to clinical events of vascular obstruction, hemolysis, and crisis. The least studied event, central in the sequence, is altered viscosity and rheology of the gelled deoxyhemoglobin S. In this work, shear is shown not only to measure the formation of gels, but to alter the progress of gelation. Thus, intraerythrocytic shear may be an important factor in pathogenesis. Increasing shear decreases the delay time for gelation as measured directly and by experiments in which shear rate is altered during the delay period. After the delay time, during the growth stage, characterized by a large increase in viscosity, shearing increases the rate of viscosity increase. On the other hand, as previously shown, shearing breaks down solid-like gels. These two effects of shear, one detrimental and the other possibly beneficial, may contribute to the variations known to exist in the clinical picture of sickle cell disease. The growth stage progress curve of gelation is here shown to be exponential in shape. This suggests that fiber breakage occurs under shear and/or that new fibers nucleate on the surface of existing fibers (i.e. heterogeneous nucleation). Finally, the progress curve is shown to be composed of plastic (i.e. solid-like) as well as viscous components early in gel development.

The effects of shear on the delay time for gelation of hemoglobin S.

Intraerythrocytic gelation of deoxyhemoglobin S is the immediate cause of red cell deformation and rigidification in sickle cell disease. Hence the rheology of hemoglobin S gels, heretofore little studied, plays a central role in pathogenesis. I have shown previously [2] that unsheared gels are solid-like whereas sheared gels are viscous in character. In the present studies the delay (nucleation) time for gelation (after a temperature jump) was examined in the absence and presence of shear, employing a sharp rise in viscosity as the end point of the delay time. Shearing rates in the range from 1.92 to 384 s-1 progressively accelerate the delay time. In another series of experiments the temperature jump was carried out in the absence of shear, shear being instituted later in the delay period. There was a sparing effect of the time of incubation in the absence of shear on the subsequent time under shear needed to induce gelation. A linear relation was observed between these two parameters, indicating that shear operates with constant efficacy in accelerating gelation, independent of when in the delay time it is applied. This suggests that the mechanism of assembly is the same throughout the period. Since shear probably operates through the breakage of fibers with the creation of new growth centers, the results also suggest that long, breakable, fibers are present even relatively early in the delay period. Values were obtained for delay time under shear and (by extrapolation) delay time under no shear. The temperature dependence (and hence apparent activation energy) of the latter is larger than that of the former; i.e. the effect of shear in shortening delay time is more marked as temperatures are lowered. With respect to pathogenesis, these results suggest that intraerythrocytic shearing may be detrimental because it accelerates gelation. On the other hand shearing of gels converts them from solid-like to viscous, and hence more deformable, systems; this effect of shear might be expected to ameliorate pathogenesis. Consequently, the pathogenic effects of intraerythrocytic shearing may depend on when the shear is applied.

Solid-like behaviour of unsheared sickle haemoglobin gels and the effects of shear.

Pathogenesis in sickle cell disease depends on the polymerization of deoxyhaemoglobin S into long fibres followed by 'gel' formation. The 'gelation' renders the affected erythrocytes less deformable than normal so that they obstruct the microvasculature and the 'gelation' process has been one of the targets for the development of therapeutic treatments of sickle cell disease. 'Gelation', however, acts through the rheological properties it induces and rheological abnormalities are therefore the immediate bases of pathogenesis. Although there has been very little study of the rheology of haemoglobin S haemolysates, the 'gels' are generally considered to be highly viscous and thixotropic on the basis of gross observation. Limited and generally qualitative observations show that high viscosity depends on deoxygenation and exhibits hysteresis in gelling/ungelling cycles; also that shearing accelerates gelation and can induce formation of fibre aggregates and crystals in suspension which, in contrast to the 'gel', are fluid. Using transient and steady state methods it is shown here that unsheared sickle deoxyhaemoglobin preparations are solid-like, consistent with a gel-like nature, whereas shearing converts them to thixotropic viscous systems. These results underline the marked variation and thixotropy the system can undergo and may be relevant to the pathogenesis, clinical course and therapy of sickle cell disease.

Fragility and structure of hemoglobin S fibers and gels and their consequences for gelation kinetics and rheology.

Pathogenesis in sickle cell disease depends on whether red blood cells can pass the microvasculature during the delay time before hemoglobin S gelation and cell rigidification occur. Here we observe individual hemoglobin S fibers by differential interference contrast (DIC) microscopy and show that hemoglobin S gels and fibers are fragile and easily broken by mechanical perturbation, and that breakage results in vast acceleration of gelation kinetics due to the creation of new, growing fiber-ends. Hence, in vivo this may be an important factor, in addition to hemoglobin concentration and degree of deoxygenation, that governs delay time and pathogenesis. Pathogenesis also depends on gel rheology and cell rigidification, which depend on fiber cross-linking. We show different mechanisms by which X-shaped, Y-shaped, and "zippering" cross-links form. Finally, we estimate the "on" rate constant for fiber growth to be about 200 mmol/(L.s) and obtain a value for the heterogeneous nucleation rate at 13.5 mmol/L heme.

The absence of volume change in the gelation of hemoglobin-S.

The volume change for the gelation of deoxygenated sickle cell hemoglobin has been measured by dilatometry at 22.0 degrees C and found to be zero. The precision of the result is 0 +/- 1.4 ml/mol of protein present in the sample. When the solubility of the protein is taken into account, the precision is 0 +/- 5.1 ml/mol of gelled hemoglobin. The participation of "hydrophobic interactions" in sickle cell hemoglobin gelation and model compound studies of the volume change associated with transferring hydrophobic solutes from an aqueous to a hydrophobic milieu, as well as the volume changes of other globular protein polymerizations, led us, initially, to expect a large positive delta V. The results are discussed in the context of concentration effects in sickle cell hemoglobin solutions and of recent work on the pressure-induced denaturation of globular proteins, which also gives smaller volume effects than had been anticipated.

Tactoidal state and phase transitions in systems of linear polymers of variable length.

The tactoidal state in systems containing long, rod-like molecules consists of partially aligned solute molecules in equilibrium with and at a concentration not much higher than that in the conjugate isotropic phase. Under the liquid lattice model of Flory [Proc. R. Soc. London Ser. A, (1956) 234, 73-89], as well as under other models, tactoid formation by molecules of fixed axial ratio depends on nonideality induced by excluded volumes; the process is wholly entropy driven and requires no direct interactions between rods. Many rod-like biological polymers exhibit reversible polymerization, so that axial ratio and length are not fixed. Polymerization and rod length will then not only induce nonideality, alignment, and phase separation, but will be affected by these. In this work these interrelations are treated under the model of Flory, modified to include a free energy of polymerization and to permit reversible changes in rod length. The primary conclusion is that, in contrast to the situation for fixed lengths, excluded volume-dependent nonideality alone does not suffice to induce a tactoidal phase separation. In the absence of attractions or repulsions between rods the anisotropic phase is highly concentrated. This phase only becomes tactoidal when a minimal level of repulsive interaction between rods is reached. Under this model, tactoid formation in systems such as deoxygenated hemoglobin S and tobacco mosaic virus depends on repulsive interactions or metastability or both. As a secondary result it is shown that rod length in the anisotropic phase is much greater than in the conjugate isotropic phase.