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Fluctuations in Self-Assembled Sickle Hemoglobin Fibers† Matthew S. Turner,*,‡ Jiangcheng Wang,§ Christopher W. Jones,‡ Frank A. Ferrone,| Robert Josephs,⊥ and Robin W. Briehl§ Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom, Department of Physiology & Biophysics, Albert Einstein College of Medicine, Bronx, New York 10461, Department of Physics, Drexel University, Philadelphia, Pennsylvania 19104, and Department of Molecular Genetics & Cell Biology, University of Chicago, Chicago, Illinois 60637 Received January 15, 2002. In Final Form: April 15, 2002 We discuss microscopic measurements of thermal fluctuations in self-assembled sickle hemoglobin fibers. These permit the measurement of various mechanical moduli that control the rheology of the fiber gel responsible for the pathology of the disease. Differential interference contrast microscopy of isolated fibers undergoing equilibrium thermal bending fluctuations leads to estimates of thermal persistence lengths of between 0.23 and 10.4 mm for the most flexible and stiffest fibers, respectively. We argue that this large range may reflect the formation of fiber bundles. If the most flexible fibers are single fibers, then the stiffest measured object is consistent with a “closed shell” hexagonal bundle of 7 single fibers. We estimate the spectrum of persistence lengths associated with bundles consisting of different numbers of single fibers and compare this with the experimental data. The equivalent Young’s modulus for the material is 0.1 GPa, less than for structural proteins but much larger than for extensible proteins.
1. Introduction Sickle cell disease arises from the polymerization of sickle cell hemoglobin (HbS) at low oxygen concentrations. Molecules of HbS aggregate into long, stiff, rodlike fibers. The 20 nm diameter fibers that result form noncovalent cross-links1-3 and create a gel that deforms and rigidifies red blood cells in vivo. Such cells obstruct capillaries, resulting in a local drop in oxygen concentration and thus feedback positively on fiber stability. At the scale of the red blood cell, it is the rheology and phase equilibria of HbS gels that controls this pathology. However, the gel rheology will itself depend on the microscopic properties of individual fibers and the energetics of their interactions. Much is known about how fiber structure and intermolecular interactions within the fiber (intrafiber interactions) affect the gelation equilibria. Fibers are themselves made up of seven molecular double strands that come together in the form of a twisted helical fiber,4,5 with a variable pitch averaging 270 nm.6 Mutations in hemo† This article is part of the special issue of Langmuir devoted to the emerging field of self-assembled fibrillar networks. ‡ Department of Physics, University of Warwick, Coventry CV4 7AL, U.K. § Department of Physiology & Biophysics, Albert Einstein College of Medicine, Bronx, NY 10461. | Department of Physics, Drexel University, Philadelphia, PA 19104. ⊥ Department of Molecular Genetics & Cell Biology, University of Chicago, Chicago, IL 60637.
(1) Samuel, R. E.; Salmon, E. D.; Briehl, R. W. Nucleation and growth of fibers and gel formation in sickle hemoglobin. Nature 1990, 345, 833-835. (2) Briehl, R. W.; Guzman, A. E. Structure and fragility of hemoglobin S gels and their consequences for gelation kinetics and gel rheology. Blood 1994, 83, 573-579. (3) Briehl, R. W. Nucleation, fiber growth and melting and domain formation and structure in sickle hemoglobin gels. J. Mol. Biol. 1994, 245, 710-723. (4) Dykes, G.; Crepeau, R. H.; Edelstein, S. J. Three-dimensional reconstruction of the fibres of sickle cell haemoglobin. Nature 1978, 272, 506-510. (5) Dykes, G.; Crepeau, R. H.; Edelstein, S. J. Three-dimensional reconstruction of the 14-filament fibers of hemoglobin S. J. Mol. Biol. 1979, 130, 451-472.
globin that alter the gelation equilibria usually correspond closely to sites of intermolecular contacts within double strands as obtained from crystallographic data.7-10 The kinetics of the self-assembly process are very important in determining the pathogenesis of the disease.11 They strongly depend on hemoglobin concentration, deoxygenation, and temperature. We will not discuss them further here except to say that the so-called double nucleation model12,13 is the most successful model of the kinetics because it takes account of the nucleation of new fibers on the surfaces of existing ones. Although HbS fibers normally occur in dense networks, we are able to take measurements of single fibers by exploiting a procedure known as “fiber surgery” in which a gel is selectively depolymerized until only an isolated fiber remains.3 This method takes advantage of the high sensitivity of CO hemoglobin to photolytic dissociation of CO, producing deoxyhemoglobin. Under microscopic observation, we photolyze CO HbS to produce a gel and then allow CO to recombine in selected regions, thereby depolymerizing those regions until the desired fiber is isolated in free solution. (6) Carragher, B.; Blumeke, D. A.; Gabriel, B.; Potel, M. J.; Josephs, R. Structural analysis of polymers of sickle cell hemoglobin. J. Mol. Biol. 1988, 199, 315-331. (7) Eaton, W. A.; Hofrichter, J. Sickle cell hemoglobin polymerization. Adv. Protein Chem. 1990, 40, 63-279. (8) Wishner, B. C.; Ward, K. B.; Lattman, E. E.; Love, W. E. Crystal structure of sickle-cell deoxyhemoglobin at 5å resolution. J. Mol. Biol. 1975, 98, 179-198. (9) Padlan, E.; Love, W. E. Refined crystal structure of deoxyhemoglobin S. J. Biol. Chem. 1985, 260, 8272-8285. (10) Harrington, D. J.; Adachi, K.; Royer, W. E. The high resolution crystal structure of deoxyhemoglobin S. J. Mol. Biol. 1997, 272, 398407. (11) Mozzarelli, A.; Hofrichter, J.; Eaton, W. A. Delay time of hemoglobin s gelation prevents most cells from sickling in vivo. Science 1987, 237, 500-506. (12) Ferrone, F. A.; Hofrichter, J.; Eaton, W. A. Kinetics of sickle hemoglobin polymerization. i. studies using temperature-jump and laser photolysis techniques. J. Mol. Biol. 1985, 183, 591-610. (13) Ferrone, F. A.; Hofrichter, J.; Eaton, W. A. Kinetics of sickle hemoglobin polymerization. ii. a double nucleation mechanism. J. Mol. Biol. 1985, 183, 611-631.
10.1021/la025539k CCC: $22.00 © 2002 American Chemical Society Published on Web 06/06/2002
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Figure 1. Three video frames of a sickle hemoglobin fiber being observed by DIC are shown. The images are of the same fiber at different times. The thermal bending fluctuations are quite apparent. Measurements of the deviations allow us to extract the thermal persistence length and corresponding material properties. This fiber is found to have a persistence length of 0.2 mm. The video field is 25 µm across.
We have found evidence that HbS fibers form bundles spontaneously.14 Thus, an isolated fiber may consist of a bundle of two or more single fibers. (We use “fiber” to indicate either a single fiber consisting of seven double strands4,5 or a bundle of single fibers and employ “single fiber” and “fiber bundle” to designate those structures only.) There is evidence elsewhere for spontaneous bundle formation in unperturbed HbS fibers including, but not limited to, pairs of fibers being observed zippering together laterally.1,2 We have measured the spontaneous, thermally induced bending motions of isolated fibers. These are obtained noninvasively and are used to estimate the bending moduli of the fibers which are free in solution. The use of videoenhanced differential interference contrast (DIC) microscopy permits visualization of the fibers, which are too narrow to observe by conventional microscopic techniques; see Figure 1. In section 5, we discuss how the new information on the mechanical response of fibers reported here may make possible future studies of the interaction energy between two fibers and more complete understanding of the stability of helical hemoglobin aggregates. We also briefly mention that analysis of rapidly frozen fibers by electron microscopy provides an independent check of the accuracy of our estimates of the thermal persistence length obtained from DIC.
except for one fiber formed in a 40 µm spot. Fibers are necessarily restricted to regions under photolysis, and hence lengths are governed both by photolytic spot size and the location of the fiber within the spot. Fibers were observed with DIC optics using a 100× PlanNeofluar oil objective. We were careful to measure fibers that existed in free solution, adhering neither to glass or to other constraining objects. Measurement errors and corrections applied to obtain deviations and variances are described in detail elsewhere,14 although in the present article the fiber deviations are uncorrected. By this, we mean that the raw fiber deviations are used to obtain the fiber rigidity and persistence length (as described below). No attempt is here made to “correct” these deviations to take account of the small nonzero mean deviations which are sometimes present. All studies were done in 0.1 M potassium phosphate, pH 7.2, at room temperature. The hemoglobin concentration was 3.75 mM in hemoglobin tetramers (24.4 gm/dL). 2.2. Sample Preparation: Electron Microscopy. Sickle hemoglobin fibers were dispersed on a copper electron microscope grid before rapid freezing in liquid ethane (in order to immobilize them). After freezing, the grids were maintained at -170 °C in liquid nitrogen. Only fibers that were not touching any other fiber or object and whose full length was contained in the micrograph were selected for analysis. Since the grids were covered with a fenestrated (holey) carbon film, we chose only fibers that were suspended over the holes for measurement. These criteria ensure that only fibers suspended freely in solution were analyzed.
2. Materials and Methods
Thermally induced shape fluctuations of semiflexible fibers can be used to obtain fiber bending moduli. A method that has proved efficient in other contexts consists of measuring the mean squared amplitude of each Fourier mode, a procedure that provides an independent measure of the modulus from each mode.15,16 However, HbS fiber segments bend little within the microscope field, being relatively stiff on this length scale. Thus, the signal-tonoise ratio of higher Fourier modes is too small for them to be useful. In this study, we measure deviations of the center of a fiber from a straight line joining its ends. The mean squared amplitude of this deviation is directly related to κ, the bending modulus of the fiber, as we now demonstrate. This technique corresponds roughly to measuring the lowest Fourier mode and has the added benefit of improved computational convenience. The total bending energy, Ebend, of a semiflexible fiber of length L in the absence of external forces is17
2.1. Sample Preparation: DIC Microscopy. HbS was purified chromatographically and converted to CO hemoglobin as previously.3 Slides of CO HbS were sealed anaerobically and observed by video-enhanced DIC microscopy using a Zeiss Axioplan microscope. Path lengths through the slide were approximately 10 µm. Unlike bright field microscopy, this technique permits visualization of fibers, whose nucleation, growth, interactions, and movement can be observed unperturbed in real time. Because their diameter is below the diffraction limit, fibers appear approximately 20 times wider than their actual widths, which therefore cannot be ascertained from the images. This restriction does not affect the observations we report here. Deoxyhemoglobin S was produced by photolyzing CO HbS with mercury arc epiillumination at 436 nm in situ under microscopic observation at 546 nm with a separate, transmitted mercury source. Individual fibers were isolated from dense gels as previously3 by selective adjustment of photolytic intensity and the region being photolyzed. By this procedure, one may gradually depolymerize all except the desired fiber. Regions of photolytic illumination were circular and either 15 or 25 µm in diameter, (14) Wang, J. C.; Turner, M. S.; Agarwal, G.; Kwong, S.; Josephs, R.; Ferrone, F. A.; Briehl, R. W. Micromechanics of isolated sickle hemoglobin fibers: Bending moduli and persistence lengths. J. Mol. Biol. 2000, 315, 601-612.
3. Statistical Mechanics of Fiber Fluctuations
(15) Gittes, F.; Mickey, B.; Nettleton, J.; Howard, J. Flexural rigidity of microtubules and actin filaments measured from thermal fluctuations in shape. J. Cell Biol. 1993, 120, 2923-2934. (16) Jin, A. J.; Nossal, R. Rigidity of triskelion arms and clathrin nets. Biophys. J. 2000, 78, 1183-1194.
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Ebend )
κ 2
2
∫0L ∂∂zu2 ‚
Turner et al.
∂2u dz ∂z2
(1)
where the vector u represents normal deviations from the fiber (z) axis, defined so as to pass through the fiber ends. Since we only measure the fluctuations projected onto the x-z (focal) plane, we further decompose (1) into two contributions Ebend ) E(x) + E(y) with u the x-component of u and
E(x) )
( ) 2
∫0L dz ∂∂zu2
κ 2
2
(2)
similarly for E(y). The contributions E(x) and E(y) are completely independent at the level of (1), valid for small deviations |u| , L. We exploit the fact that we can determine the magnitude of the three-dimensional fluctuations by examining their projection onto the focal plane. Thus, the out-of-plane fluctuations, which are independent degrees of freedom and are not measured experimentally, may be neglected for this purpose. We Fourier decompose the fiber shape according to ∞
u(z) )
∑ an sin
(3)
L
Substituting (3) into (2), integrating, and exploiting orthogonality, we find the energy of each mode
E(x) )
κπ4 3
4L
∞
∞
n4an2 ) ∑ En(x) ∑ n)1 n)1
(4)
By the principle of equipartition of energy, 〈En(x)〉 ) kBT/2 where kB is the Boltzmann constant. Together with (4), this yields the mean squared Fourier amplitude
〈an 〉 ) 2
2kBTL3
(5)
4 4
κπ n
Relations similar to (5) have previously been used to obtain an independent estimate of κ from the mean squared amplitude of each mode (actually the angular deviation was used in at least one case15). However, for reasons already discussed, we here employ a new and computationally straightforward technique that involves measuring the spatial displacements of the projected fiber midpoint u(L/2) from the z-axis to obtain κ and the thermal persistence length l ) κ/kBT. To see how this method works, consider the mean squared amplitude at the midpoint that follows from (3):
〈u(L/2)2〉 )
〈
∞
∞
∑ ∑ an sin n)1 m)1
nπ 2
am sin
〉
mπ 2
(6)
Since the Fourier modes are uncorrelated, 〈anam〉 ∝ δnm. Exploiting (5), we can sum the resulting diagonal terms in (6) and rearrange to obtain
κ ) kBTl )
kBTL3 48〈u(L/2)2〉
between measurements of u(L/2). In this case, the rescaled deviation
∆ ) 4x3u(L/2)/L3/2
nπz
n)1
Figure 2. This figure shows the persistence lengths l (in mm) of all 22 fibers measured by DIC (solid circles). The data are arbitrarily sorted in order of descending persistence length (stiffness) and assigned an index from 1 to 22 accordingly. The values indicated by the dashed lines correspond to the persistence length of fiber bundles with, for increasing stiffness, from n ) 1 to 7 fibers per fiber bundle. The values shown correspond to the best one-parameter fit of our model to the data (see text).
(7)
To this point, we have treated the fiber length L as a constant. In general, L is not fixed but rather varies slightly (17) Landau, L. D.; Lifshitz, E. M. Theory of Elasticity; ButterworthHeinemann: Oxford, U.K., 1986.
(8)
is a Gaussian random variable with mean squared value
〈∆2〉 ) 1/l ) kBT/κ
(9)
For an isotropic, homogeneous elastic medium, the bending modulus κ ) EI depends on both Young’s modulus, E, and a cross-sectional moment I equal to the geometrical moment of inertia, that is, the moment calculated with unit density.17 For a solid rod of circular cross-section with radius b, this moment is I ) πb4/4 and the Young’s modulus is
E)
4kBTl 4κ ) 4 πb πb4
(10)
4. DIC Results Figure 2 shows the persistence lengths l (in mm) of all 22 fibers measured by DIC. These persistence lengths are obtained by numerous sequential measurements of the lengths and deviations at midpoint for each fiber.14 These measurements were separated by at least 3, and usually 10, times the longest hydrodynamic correlation time τ for the fibers. This time has been estimated to be of the order of τ ≈ 1 s for fibers of this length.14 The persistence length then follows from eq 9. The average persistence length for the two most flexible fibers is 0.23 mm, and for the two stiffest fibers the average is 10.4 mm, giving a stiff/flexible ratio of 45:1. We postulate that this variation in stiffness may be due to the lateral aggregation of single fibers into fiber bundles as shown in Figure 3. Our model of bundling treats fibers as having approximately circular cross sections and fiber bundles as consisting of a hexagonal array of fibers. This model is clearly an approximate one as the fibers are known to have slightly elliptical cross sections. In fact, the cross sections resemble ellipses with eccentricities that are only O(10%) different from circles. However, we expect that it can be used to obtain fairly accurate estimates of the persistence lengths of fiber bundles consistent with our hypothesis of lateral fiber aggregation into a hexagonal array. Other forms of bundling are also possible, including assembly of bundles directly from different numbers of double strands, rather than from an integer number of
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Figure 3. Cross sections of idealized fiber bundles consisting of, from left to right, from n ) 1 to 7 single fibers.
Figure 4. The variation of the average moment hI(n) with the number of fibers in the fiber bundle is shown in units of b4, with b the radius of each fiber (solid circles). The persistence length and rigidity are both proportional to this moment. For comparison, the solid curve is the quadratic function hI(1)n2 which is the natural continuum approximation to hI that slightly underestimates the fiber stiffness.
single fibers each containing seven double strands. We are not able to test such hypotheses here. Fiber bundles in the (focal) plane are free to rotate along their long axis. By considering first a fiber bundle with arbitrary rotational orientation and the resulting rigidity with respect to fluctuations in the focal plane, we will be able to average over all axial rotational angles ψ to obtain the effective average rigidity. The rigidity of a fiber rotated by ψ with respect to bending in the (focal) plane follows from the second moment of the cross section according to17
κ(ψ)/E ) I(ψ) )
∫s d2r(r cos(θ - ψ))2
(11)
where the integral is over part of the plane normal to the fiber bundle, S is the interior of all circular fiber cross sections in the bundle, and the coordinate system is chosen so that the origin is at the “center of mass”, that is, ∫s r d2r ) 0. Thus
I(ψ) ) Ix cos2 ψ + Iy sin2 ψ
(12)
where Ix ) ∫s x2d2r and Iy ) ∫s y2d2r are the moments measured along the two principal axes. The fluctuations have a mean squared amplitude that is proportional to the following average (integral) over all rotational orientations
1/Ih )
1 2π
∫02π dψ/I(ψ) w hI ) xIxIy
(13)
The variation of the average moment hI with the number n of fibers in the bundle is shown in Figure 4. The average moment hI is proportional to the measured persistence length according to
l ) EIh/kBT
(14)
We see from Figure 4 that the persistence length of fiber bundles increases approximately quadratically with the number of fibers in the bundle. We are now in a position to fit our model of the persistence lengths of fiber bundles to the experimental data of Figure 2 by adopting the hypothesis that the variation in persistence length is due only to fiber bundling. We determine the constant of
Figure 5. Histogram showing the distribution of normal deviations of the fiber midpoints u(L/2) (in microns) of the two softest fibers. The vertical axis shows the number N of observed normal deviations within each range. The superimposed Gaussian is a good fit, indicating that the deviations are approximately Gaussian distributed as would be expected for thermal bending fluctuations (see discussion in the text). A total of 127 measurements of u(L/2) were made.
Figure 6. Histogram showing the distribution of instantaneously observed fiber lengths L for the two softest fibers. The vertical axis shows the number N of observed lengths within each range. A total of 127 measurements of L were made.
proportionality between hI and l in eq 14 by optimizing the least-squares fit between the data for l and the closest discrete value of the bundle rigidity for 1 e n e 7. The best fit corresponds to a value of l ) 0.19 mm, a little lower than the value estimated from the softest two fibers alone. The stiffest objects measured, corresponding to the two longest values of l shown in Figure 2, seem to be broadly consistent with them being n ) 7 “closed shell” fiber bundles; see Figure 3. It seems physically reasonable that this would be a particularly stable bundle configuration. It is difficult to quantify the significance of the fit of the data to the spectrum of predicted persistence lengths derived from our bundling hypothesis. One crude approach involves a computational comparison of the data with artificial statistical data. To this end, we generated 5000 artificial data sets consisting of “persistence lengths” chosen at random from a probability distribution uniformly distributed between the softest and stiffest observed values of l. A least-squares fit was then conducted between each artificial data set and the closest value of the spectrum hI(n), in each case linearly scaled by a (different) single fit parameter. In only six cases did the artificial data set have a smaller least-squares fit residue than that obtained for the experimental data. This approach neglects the fact that the experimental data are not uniformly distributed over their entire range, a feature which tends to improve the fit of the experimental data compared with the artificial data. Thus, it would be dangerous to interpret this as a quantitative measure of a confidence bound on the bundling hypothesis but rather it should be taken as a qualitative indicator.
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Figure 7. Electron micrograph of a typical frozen hydrated sickle hemoglobin fiber. The scale bar is 200 nm. Variations in the bending of these fibers can serve as a check on the results of DIC.
To obtain Young’s modulus, we need to estimate the fiber radius, b. The average radius of a fiber can be obtained from electron micrographs. The helical fiber has slightly elliptical cross section, and the effective mean radius, for the purposes of calculating the moment of inertia, is the geometric mean of the semimajor and semiminor axes; see eq 13. These radii for HbS single fibers are, respectively, 11.5 and 9.0 nm for the fiber model of Cretegny & Edelstein18 and 12.5 and 9.25 nm for the less tightly packed model of Watowich et al.19 Using b ) 10.5 nm and eq 10 and assuming the two most flexible fibers represent single fibers, we obtain the estimate E ) 0.1 GPa. The distribution of deviations u(L/2) for the two softest fibers is shown in Figure 5 while the distribution of the instantaneously measured lengths L for the same two fibers is shown in Figure 6, demonstrating that the fibers actually have rather similar lengths that do not vary dramatically. A total of 127 snapshot measurements were made on these two fibers. As can be seen from Figure 5, the normal deviation u(L/2) is quite well approximated by a Gaussian distribution function with mean squared deviation 〈L〉3/(48l) as might be expected from eqs 8 and 9 although this is only precise in the limit that variations in L between measurements are negligibly small. The motivation for showing Figure 6 is to show that this is approximately the case. When variations in L are more substantial, it is only the rescaled deviation ∆ that will exhibit the statistics of a Gaussian random variable. The quality of the Gaussian fit in Figure 5 is significant in that it indicates the deviations are consistent with an origin in equilibrium thermal fluctuations, as assumed in section 3. 4.1. Measurements of Bending Fluctuations of Frozen Hydrated Fibers. Electron microscopy measurements of frozen hydrated fibers have been carried out and measurements made of bending fluctuations, characterized by fiber deviations at midpoint, analogous to those obtained by DIC measurements and discussed above. A typical electron micrograph is shown in Figure 7. This provides an independent check of the accuracy of our estimates of the thermal persistence length obtained from DIC. This procedure reveals slightly smaller mean squared amplitude fluctuations than we would have expected with a persistence length roughly half that obtained by DIC20 which represents rather good agreement.
4.2. Comparison of Sickle Hemoglobin Fibers with Other Linear Biopolymers. Gittes et al.15 found Young’s modulus for microtubules and actin to be 1.2 and 2.6 GPa, respectively. That fibers of actin, tubulin, and other structural proteins that need to be firm should be less compliant21 and proteins such as elastin (E ≈ 0.6 MPa) that are required to stretch, more compliant than hemoglobin S fibers is not surprising. That actin has much shorter persistence lengths than hemoglobin S fibers results from its small diameter rather than material properties.
(18) Cretegny, I.; Edelstein, S. J. Double strand packing in HbS fibers. J. Mol. Biol. 1993, 230, 638-733. (19) Watowich, S. J.; Gross, L. J.; Josephs, R. Analysis of the intermolecular contact sites within sickle hemoglobin fibers: Effect of site-specific substitutions, fiber pitch and double strand disorder. J. Struct. Biol. 1993, 111, 161-179.
(20) Turner, M. S.; Briehl, R. W.; Wang, J. C.; Ferrone, F. A.; Josephs, R. Anisotropy in sickle hemoglobin fibers from variations in bending and twist. In preparation. (21) Wainwright, S. A.; Biggs, W. D.; Currey, J. D.; Gosline, J. M. Mechanical Design in Organisms; Princeton University Press: Princeton, NJ, 1976.
5. Future Studies In this section, we briefly discuss the prospects for future studies based on both the techniques and measurements that are reported here. Sickle hemoglobin fibers have an approximately elliptical cross section and are helical with a pitch length of 270 nm. By studying variations in this pitch length, it may be possible to establish a torsional modulus in much the same way as bending fluctuations give us information on the bending rigidity. Preliminary results indicate that the fibers are anomalously soft with respect to torsional fluctuations.20 As mentioned in the Introduction, sickle hemoglobin fibers are sometimes observed to zipper together with neighboring, parallel fibers.2 This observation is entirely consistent with our bundling hypothesis. Sometimes the fibers cannot zipper completely but form a frustrated, partially zippered structure. In this situation, the results for the rigidity of fibers reported here may allow mechanical force balance arguments to be applied to these frustrated fiber structures in order to obtain the lateral interaction force. The thermodynamic stability, or metastability, of sickle hemoglobin fibers ultimately controls the pathology of the disease. We argue that a knowledge of properties of the hemoglobin material may facilitate the construction of models for the free energy of the fiber, and hence determination of the relative thermodynamic stability of the fiber state itself. 6. Conclusions HbS fibers may be isolated from the noncovalently crosslinked gels in which they normally exist allowing mea-
Self-Assembled Sickle Hemoglobin Fibers
surement of their spontaneous bending under thermal forces. A wide range of persistence lengths is observed, from 0.23 to 10.4 mm, and we argue that increasing persistence length is associated with increasing fiber thickness due to bundle formation. We have been able for the first time to predict the spectrum of rigidities associated with fiber bundles involving different numbers of single fibers. Thus we are able to demonstrate satisfactory agreement between this model and the experimental distribution of persistence lengths. The effective Young’s modulus of the fibers is approximately 0.10 GPa, somewhat less than for structural proteins evolutionarily “designed” for a mechanical role. HbS fibers are very stiff on the scale of red blood cells, and so fibers and fiber bundles can support large forces within the cell leading to deformation and rigidification of the cell. These results may provide a basis for understanding the rheology of the pathogenic cross-linked fiber gel network formed spontaneously. Our technique, in which individual HbS fibers are isolated from gels and their microrheological properties are then measured by observation of thermal shape fluctuations,
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can be applied to other polymerizing mutants and to mixtures and hybrids of HbS with hemoglobin A and other hemoglobins. We have discussed future possible extensions, including (i) measurements of torsional fluctuations within fibers, allowing us to estimate their torsional rigidity, (ii) how mechanical force balance arguments applied to frustrated fiber structures can give the lateral fiber-fiber attractive force, and (iii) theoretical studies of the stability of helical fibers in which the twist modulus may aid in understanding the free energy of the fiber, and hence its equilibrium state. Acknowledgment. This work was supported by National Institutes of Health (NHLBI) Program Project Grant HL 58512 (R.W.B. (PI), F.A.F., and R.J.) and Grant HL22654 (R.J.). One of us (M.S.T.) gratefully acknowledges support from The Royal Society (U.K.) in the form of a University Research Fellowship. LA025539K
