X-ray diffraction studies of 14-filament models of deoxygenated sickle cell hemoglobin fibers. Models based on electron micrograph reconstructions

The transforms of a large number of models of deoxygenated sickle hemoglobin fibers, related to that derived from image reconstruction of electron micrographs, have been calculated and compared with X-ray diffraction data of 15 A resolution. The model of the fiber, determined from the reconstructed image, is a helix consisting of 14 filaments that associate in a specific mode to form seven pairs, or protofilaments. Pairs were identified through the pattern of filament loss in partially disassembled fibers and by the separation between molecules, in adjacent filaments, of half a molecular diameter, along the fiber axis. An alternative mode of filament association can be derived also from the surface lattice of the reconstruction, which meets these criteria for the pairing of molecular filaments. Both pairing modes have been used in the search for structures whose transforms show the best agreement with the diffraction data. Models were generated by the systematic translation of six protofilaments, taken in symmetry related pairs, in steps of 3.5 A along the fiber axis relative to a fixed central protofilament. Each translation of a protofilament corresponds to a different fiber model, whose transform was compared with observed data. In all, over 11,000 transforms were calculated. Of all the models considered, three have been found whose residuals are minimal. At 30 A resolution, similar to that of electron micrographs, the model derived from image reconstruction and the three found through our search procedure are indistinguishable. At 15 A, however, the transforms of these models show better agreement with the observed data than the transform of the reconstructed image. Comparison of residuals shows that the model derived from the reconstructed image can be rejected with 99.5% probability relative to the model, with the same pairing scheme, found by our search procedures. The two other models, derived from the alternative pairing scheme, are also more credible than the reconstructed image, but at a lower confidence level. Each of our three models is equally acceptable. Their existence may reflect structural polymorphism of the fiber.     

Computer models of a new deoxy-sickle cell hemoglobin fiber based on x-ray diffraction data.

A new x-ray fiber diffraction pattern from deoxygenated sickle cell erythrocytes has been observed. It displays 14 layer lines with a 109 A periodicity compared with the 64 A periodicity of the "classic" sickle cell hemoglobin (HbS) fiber. These data and association energy calculations serve as a basis for computer model building. Systematic searches over four-dimensional parameter space yielded twelve protofilament models that satisfy the following constraints: (a) two HbS molecules be related by twofold screw symmetry with a translational repeat of 109 A; (b) at least one of the substituted residues in HbS, val beta 6, should participate in intermolecular contacts; and (c) the energy of intermolecular interaction be less than -24 kcal/mol. Each of the protofilament models is a zigzag mono-strand that stands in contrast to the double-stranded protofilament of the "classic" fiber. Fiber models were constructed with each of the 12 protofilament models, pseudo-hexagonally packed. Searches of variable packing parameters showed four fiber models with minimal protofilament association energies and minimal differences between calculated transforms and observed data. The R-factor was less than 0.24 for each of these four models. In three of the fiber models the protofilament association energy is between -(93 and 130) kcal, and in a fourth, the energy is -64 kcal. One protofilament model constituted three distinct fiber models of the lower energy class, and a second protofilament model packed with a higher association energy into a fourth fiber model. The selection of a unique fiber model from among these four cannot be made because of the limited available data.(ABSTRACT TRUNCATED AT 250 WORDS)     

Triclinic crystals associated with fibers of deoxygenated sickle hemoglobin.

Triclinic crystals have been found in capillaries that initially contained deoxygenated sickled erythrocytes, and in solutions of sickle hemoglobin that were stirred during deoxygenation. In both cases these crystals occur as a phase transition from fibers. They have been observed only as twins; the a-axis of one member is related to that of its twin by 180 degrees rotation about the b* direction. The volume of the triclinic crystal unit cell is half that of the monoclinic crystals that have also transformed from fibers. Analysis of X-ray diffraction data indicates that the two molecules in the triclinic unit cell that repeat at an interval of 64 A form double filaments similar to those found in the monoclinic crystals and in the fiber. The existence of the triclinic crystals which contain only one double filament per unit cell removes a postulated requirement that antipolar double filament pairs be the sole unit of the fiber architecture.     

Patterns in the quinary structures of proteins. Plasticity and inequivalence of individual molecules in helical arrays of sickle cell hemoglobin and tubulin.

The four recognized levels of organization of protein structure (primary through quaternary) are extended to add the designation quinary structure for the interactions within helical arrays, such as found for sickle cell hemoglobin fibers or tubulin units in microtubules. For sickle cell hemoglobin the main quinary structure is a 14-filament fiber, with a number of other minor forms also encountered. Degenerate forms of the 14-filament fibers can be characterized that lack specific pairs of filaments; evidence is presented which suggests an overall organization of the 14 filaments in pairs, with particular pairs aligned in an antiparallel orientation. For tubulin, a range of quinary structures can be detected depending on the number of protofilaments and whether adjacent protofilaments composed of alternating alpha- and beta-subunits are aligned with contacts between like or unlike subunits and with parallel or antiparallel polarity. Thus, in contrast to quarternary structure, which generally involves a fixed number of subunits, the quinary structures of proteins can exhibit marked plasticity and inequivalence in the juxtaposition of constituent molecules.     

Structural analysis of polymers of sickle cell hemoglobin. II. Sickle hemoglobin macrofibers

Sickle cell hemoglobin macrofibers are an important intermediate in the low pH crystallization pathway of deoxygenated hemoglobin S that link the fiber to the crystal. Macrofibers are a class of helical particles differing primarily in their diameters but are related by a common packing of their constituent subunits. We have performed three-dimensional reconstructions of three types of macrofibers. These reconstructions show that macrofibers are composed of rows of Wishner-Love double strands in an arrangement similar to that in the crystal. We have measured the orientation and co-ordinates of double strands in macrofibers using cross-correlation techniques. In this approach, the electron density projections of double strands calculated from the known high-resolution crystal structure are compared with regions along the length of the particles in which the distinct pattern of double strands in c-axis projection may be observed. Contrary to assertions by Makinen & Sigountos (1984), our results unambigously demonstrate that adjacent rows of double strands in macrofibers are oriented in an antiparallel manner, as in the Wishner-Love crystal. Adjacent rows of antiparallel double strands are displaced along the helical axis relative to their co-ordinates in the crystal. Electron density models of macrofibers based on the crystallographic structure of the sickle hemoglobin double strand are in good agreement with the projections of macrofibers observed in electron micrographs. We have studied the structure of a closely related crystallization intermediate, the sickle hemoglobin paracrystal. The arrangement of double strands in paracrystals is similar to that in Wishner-Love crystals, except that they are displaced along the a-axis of the crystal. Measurements of the double strand co-ordinates reveal that the distribution of strand positions is bimodal. These results further establish the close structural relationship between macrofibers and paracrystals as intermediates in the crystallization of deoxygenated sickle hemoglobin.     

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.     

Saturation-transfer electron paramagnetic resonance detection of anisotropic motion by sickle hemoglobin molecules in the polymer state

The motional behavior of spin-labeled deoxygenated sickle hemoglobin has been studied by using both 9- and 35-GHz saturation-transfer electron paramagnetic resonance (EPR). Using spectral subtraction techniques and saturation-transfer EPR parameter correlation plots, we find that the saturation-transfer EPR spectra for the sickle hemoglobin gel state at high temperature and high hemoglobin concentration cannot be described as a simple superposition of spectra from immobilized hemoglobin plus solution-state hemoglobin but instead suggest that the individual sickle hemoglobin molecules exhibit limited, anisotropic, rotational oscillation within the polymer fiber. The spectra also imply that the symmetry axis for sickle hemoglobin rotational oscillation is approximately coincident with the nitroxide z axis of the covalently attached spin-label. We suggest that this anisotropic rotational motion may be produced by one or two of the known intermolecular contact sites within the sickle hemoglobin fiber acting as strong intermolecular binding sites, and producing "motional alignment" within the fiber; determining the location of the strong binding site should be important in focusing the future development of antisickling agents.     

High-resolution proton nuclear magnetic resonance studies of sickle cell hemoglobin.

High-resoluiton proton nuclear magnetic resonance spectroscopy at 250 MHz has been used to investigate sickle cell hemoglobin. The hyperfine shifted, the ring-current shifted, and the exchangeable proton resonances suggest that the heme environment and the subunit interfaces of the sickle cell hemoglobin molecule are normal. These results suggest that the low oxygen affinity in sickle cell blood is not due to conformational alterations in the heme environment or the subunit interfaces. The C-2 proton resonances of certain histidyl residues can serve as structural probes for the surface conformation of the hemoglobin molecule. Several sharp resonances in sickle cell hemoglobin are shifted upfield from their positions in normal adult hemoglobin. These upfield shifts, which are observed in both oxy and deoxy forms of the molecule under various experimental conditions, suggest that some of the surface residues of sickle cell hemoglobin are altered and they may be in a more hydrophobic environment as compared with that of normal human adult hemoglobin. These differences in surface conformation are pH and ionic strength specific. In particular, upon the addition of organic phosphates to normal and sickle cell hemoglobin samples, the differences in their aromatic proton resonances diminish. These changes in the surface conformation may, in part, be responsible for the abnormal properties of sickle cell hemoglobin.     

Studies of the fiber to crystal transition of sickle cell hemoglobin in acidic polyethylene glycol

The crystallization of deoxygenated sickle cell hemoglobin in acidic (pH 5.2) polyethylene glycol (10%) has been studied in order to determine if the mechanism of crystal formation under such conditions has features in common with the mechanism of crystal formation at higher pH values in the absence of polyethylene glycol. The existence of a common mechanism of crystallization under different conditions is relevant in validating the use of the known high resolution crystal structure to interpret the fiber structure. Our findings indicate that deoxygenated sickle cell hemoglobin crystallization in acidic polyethylene glycol is initiated by fiber formation. Fibers, in turn, convert to larger structures called macrofibers within several hours (Wellems et al., 1981). Fibers and macrofibers (and their respective optical transforms) formed in acidic polyethylene glycol appear to have the same structure as their counterparts formed at higher pH values in the absence of polyethylene glycol. Early in the transition one can observe macrofibers in the process of alignment and fusion. The structural characterization of the intermediates leaves little doubt that crystallization in acidic polyethylene glycol is mediated by the same mechanism as that occurring under more physiological conditions, and that fibers are a metastable intermediate whose ultimate fate is to crystallize.     

Molecular packing in a second monoclinic crystal of deoxygenated sickle hemoglobin

A close correspondence has been demonstrated between double filaments of deoxygenated hemoglobin S molecules as found in monoclinic crystals, forms I and II, and in sickle fibers. We have carried out a low resolution study of monoclinic form II by X-ray diffraction analysis. Its structure differs from that of form I solely by a shift along the a-axis of the molecular centers of the asymmetric unit, which forms the double filament. The magnitude of the translation was determined from a minimum residual calculation. The x co-ordinates of the symmetry related molecular centers of antipolar double filaments are approximately the same. This means that the double filaments are nearly in register. A minor component associated with form II crystals proved to be form I. The possible existence of additional forms is discussed.The significance of the molecular arrangement in form II is related to its presence in sickle fibers. We have determined the contacts between antipolar double filaments in this form as well as a number in form I not tabulated previously. These new contacts represent additional stabilizing interactions that might provide targets for the design of stereospecific antisickling agents.     

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.     

Kinetics of increased deformability of deoxygenated sickle cells upon oxygenation

We have examined the kinetics of changes in the deformability of deoxygenated sickle red blood cells when they are exposed to oxygen (O(2)) or carbon monoxide. A flow-channel laser diffraction technique, similar to ektacytometry, was used to assess sickle cell deformability after mixing deoxygenated cells with buffer that was partially or fully saturated with either O(2) or carbon monoxide. We found that the deformability of deoxygenated sickle cells did not regain its optimal value for several seconds after mixing. Among density-fractionated cells, the deformability of the densest fraction was poor and didn't change as a function of O(2) pressure. The deformability of cells from the light and middle fraction increased when exposed to O(2) but only reached maximum deformability when equilibrated with supraphysiological O(2) concentrations. Cells from the middle and lightest fraction took several seconds to regain maximum deformability. These data imply that persistence of sickle cell hemoglobin polymers during circulation in vivo is likely, due to slow and incomplete polymer melting, contributing to the pathophysiology of sickle cell disease.     

Macrofiber structure and the dynamics of sickle cell hemoglobin crystallization

Fibers of deoxyhemoglobin S undergo spontaneous crystallization by a mechanism involving a variety of intermediate structures. These intermediate structures, in common with the fiber and crystal, consist of Wishner-Love double strands of hemoglobin S molecules arranged in different configurations. The structure of one of the key intermediates linking the fiber and crystal, called a macrofiber, has been studied by a variety of analytical procedures. The results of the analysis indicate that the intermediates involved in the fiber to crystal transition have many common structural features. Fourier analysis of electron micrographs of macrofibers confirms that they are composed of Wishner-Love double strands of hemoglobin molecules. Electron micrographs of macrofiber cross-sections reveal that the arrangement of the double strands in macrofibers resembles that seen in micrographs of the a axis projection of the crystal. This orientation provides an end-on view of the double strands which appear as paired dumb-bell-like masses. The structural detail becomes progressively less distinct towards the edge of the particle due to twisting of the double strands about the particle axis. Serial sections of macrofibers confirm that these particles do indeed rotate about their axes. The twist of the particle is right handed and its average pitch is 10,000 Å. The effect of rotation on the appearance of macrofiber cross-sections 300 to 400 Å thick can be simulated by a 15 ° rotation of an a axis crystal projection. The relative polarity of the double strands in macrofibers and crystals can be determined easily by direct inspection of the micrographs. In both macrofibers and crystals they are in an anti-parallel array.On the basis of these observations we conclude that crystallization of macrofibers involves untwisting and alignment of the double strands.     

The role of pH on instability and aggregation of sickle hemoglobin solutions

Understanding the physical basis of protein aggregation covers strong physical and biomedical interests. Sickle hemoglobin (HbS) is a point-mutant form of normal human adult hemoglobin (HbA). It is responsible for the first identified "molecular disease," as its propensity to aggregation is responsible for sickle cell disease. At moderately higher than physiological pH value, this propensity is inhibited: The rate of aggregate nucleation becomes exceedingly small and solubility after polymerization increases. These order-of-magnitude effects on polymer nucleation rates and concurrent relatively modest changes of solubility after polymerization are here shown to be related to both pH-induced changes of location and shape of the liquid-liquid demixing (LLD) region. This allows establishment of a self-consistent contact between the thermodynamics of the solution as such (i.e., the LLD region), the kinetics of fiber nucleation, the theory of percolation, and the thermodynamics of gelation. The observed pH-induced changes are largely attributable to strong perturbations of hydrophobic hydration configurations and related free energy by electric charges. Similar mechanisms of effective control of aggregate nucleation rates by means of agents such as cosolutes, pH, salts, and additives, shifting the LLD and associated regions of anomalous fluctuations, promise to be relevant to the whole field of protein aggregation pathologies.     

The neurotoxic effect of sickle cell hemoglobin

A growing body of experimental evidence suggests that the oxidative neurotoxicity of hemoglobin A may contribute to neuronal loss after CNS hemorrhage. Several hemoglobin variants, including hemoglobin S, are more potent oxidants in cell-free systems. However, despite the increased incidence of hemorrhagic stroke associated with sickle cell disease, little is known of the effect of hemoglobin S on cells of neural origin. In the present study, its toxicity was quantified and directly compared with that of hemoglobin A in murine cortical cell cultures. Reactive oxygen species production, as assessed by cellular fluorescence after treatment with dihydrorhodamine 123, was significantly increased by exposure to 10 μM hemoglobin S for 2-4 h. Neuronal death, as measured by propidium iodide staining and lactate dehydrogenase release, commenced at 4 h; for a 20-h exposure, the EC₅₀ was approximately 0.71 μm. Glial cells were not injured. Cell death was completely blocked by iron chelation with deferoxamine or phenanthroline. Direct comparison of sister cultures exposed to either hemoglobin A or hemoglobin S revealed a similar amount of cell injury in both groups. A significant difference was consistently observed only after treatment with 1 μM hemoglobin for 20 h, which resulted in death of approximately one third more neurons with hemoglobin S than with hemoglobin A. The results of this study suggest that sickle cell hemoglobin is neurotoxic at physiologically relevant concentrations. This toxicity is iron-dependent, oxidative, and quantitatively similar to that produced by hemoglobin A.     

Intermolecular interactions of oxygenated sickle hemoglobin molecules in cells and cell-free solutions.

We have measured the intermolecular interactions of oxygenated sickle hemoglobin molecules in cells and in cell-free solutions, and have compared the results with similar data for liganded normal adult hemoglobin. The experiments involve the measurement of the spin-lattice relaxation time T1 of protons of solvent water molecules, as a function of an externally applied static magnetic field. From such data, one can derive a correlation time tauc, for each sample, which is a measure of the time taken for a hemoglobin molecule to randomize its orientation due to Brownian motion. Thus tauc is a measure of the freedom of rotational motion, on a molecular or microscopic level, of hemoglobin molecules. Intermolecular interactions will reduce this freedom of motion and lengthen tauc. We find that oxygenated sickle hemoglobin molecules have an additional intermolecular interaction not found for normal hemoglobin. This extra interaction is increased by the presence of either inorganic phosphate or diphosphoglycerate, and is greater for sickle hemoglobin within cells than in cell-free solutions. By comparing the present results with published data on the viscosity of oxygenated sickle and normal hemoglobin, we conclude that, at concentrations comparable to intracellular values, oxygenated sickle hemoglobin molecules form aggregates several tetramers in size. The possibility exists that these aggregates are the earliest stage of fiber formation itself, the physical basis of the sickling phenomena.     

The binding of hemoglobin to membranes of normal and sickle erythrocytes.

The binding of hemoglobins A, S, and A2 to red cell membranes prepared by hypotonic lysis from normal blood and blood from persons with sickle cell anemia was quantified under a variety of conditions using hemoglobin labelled by alkylation with 14C-labelled Nitrogen Mustard. Membrane morphology was examined by electron microscopy. Normal membranes were found capable of binding native hemoglobin A and hemoglobin S in similar amounts when incubated at low hemoglobin: membrane ratios, but at high ratios hemoglobin saturation levels of the membranes increased progressively for hemoglobin A, hemoglobin S and hemoglobin A2, respectively, in order of increasing electropositivity. Binding was unaffected by variations in temperature (4-22 degrees C) and altered little by the presence of sulfhydryl reagents, but was inhibited at pH levels above 7.35; disrupted at high ionic strength; and dependent on the ionic composition of the media. These findings suggest that electrostatic, but not hydrophobic or sulfhydryl bonds are important in membrane binding of the hemoglobin under the conditions studied. An increased retention of hemoglobin in preparations of membranes from red cells of patients with sickle cell anemia (homozygote S) was attributable to the dense fraction of homozygote S red cells rich in irreversibly sickled cells, and the latter membranes had a smaller residual binding capacity for new hemoglobin. This suggests that in homozygote S cells which have become irreversibly sickled cells in vivo, there are membrane changes which involve alteration and/or blockade of hemoglobin binding sites. These findings support the notion that hemoglobin participates in the dynamic structure of the red cell membrane in a manner which differs in normal and pathological states.     

Characterization of microtubule protofilament numbers. How does the surface lattice accommodate?

Frozen-hydrated specimens of microtubules assembled in vitro were observed by cryoelectron microscopy. Specimens were of both pure tubulin, and of microtubule protein isolated by three cycles of assembly and disassembly. It is shown that the characteristic image contrast of individual microtubules allows the microtubule protofilament number to be determined unambiguously. Microtubules with 13, 14 and 15 protofilaments are observed to coexist in specimens prepared under various assembly conditions. Confirmation of these results is obtained by observations of thin sections of pelleted samples fixed and stained using the glutaraldehyde/tannic acid technique. Images of individual microtubules show both characteristic contrast profiles across their width and typical variations of these profiles along their length. The profiles across the images indicate the protofilament number of the microtubule. The lengthwise variations indicate how the protofilaments are aligned with respect to the microtubule axis giving what has previously been called a supertwist. In 13 protofilament microtubules the protofilaments are paraxial. In 14 and 15 protofilament microtubules, the protofilaments are skewed with respect to the microtubule axis. The skew is greater for the 15 protofilament case than for 14 protofilaments. The skew allows the extra protofilaments to be accommodated by the surface lattice. These results should also be relevant to situations in vivo.     

On the assembly of sickle hemoglobin fascicles

Deoxyhemoglobin S fibers associate into bundles, or fascicles, that subsequently crystallize by a process of alignment and fusion. We have used electron microscopy to study the formation of fascicles and the changes in fiber packing that occur during the conversion of fascicles to crystals. The first event in crystallization involves fibers forming fascicles that are initially small and poorly ordered but, with time, become progressively larger and more highly ordered. After six to eight hours, the fibers in a fascicle form a crystalline lattice. The three-dimensional unit cell parameters of this lattice are a = 1300 A, b = 365 A, and c = 210 A (the a axis is parallel to the fiber axis). Fibers have an elliptical cross-section whose major and minor axes are 250 A and 185 A, respectively. When projected on to the unit cell vectors, these dimensions are 210 A and 155 A, so the unit cell dimension of 365 A implies that there are two fibers per unit cell. Theoretically, fibers could pair so that each member of the unit cell is oriented in the same direction (parallel) or opposite directions (antiparallel). Fourier transforms of electron micrographs (or models) cannot distinguish between these alternatives, since the two arrangements produce very similar intensity distributions. The orientation of the fibers was determined from cross-sections of the fascicles in which the fibers are seen end-on. In this view the images of the fibers are rotationally blurred because the fibers twist 30 degrees to 40 degrees about their helical axis through the 300 A to 400 A thick section. We have been able to remove the rotational blur from each of the fibers in the unit cell using the procedures described by Carragher et al. The deblurred images of the two fibers in the unit cell are related by mirror symmetry. This relationship means that the fibers are antiparallel. These observations suggest that crystallization of fibers in fascicles is mediated by assembly of the fibers into antiparallel pairs that contain equal numbers of double strands running in each direction.     

On macrotubule structure.

The number of protofilament pairs in macrotubules was calculated as a function of macrotubule diameter, protofilament angle to the long axis, and pair width. A comparison of these calculations with published observations suggests utilizasion of all 13 microtubule protofilaments in the formation of macrotubules in vivo.