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* Materials Department, Physics Department, and Molecular, Cellular, and Developmental Biology Department, University of California, Santa Barbara, California 93106
Correspondence: Address reprint requests to Cyrus R. Safinya, E-mail: safinya{at}mrl.ucsb.edu.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSUPPLEMENTARY MATERIALACKNOWLEDGEMENTSREFERENCES |
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ß-tubulin heterodimers that align head-to-tail in the MT wall, forming linear protofilaments that interact laterally. We introduce a probe of the interprotofilament interactions within MTs and show that this technique gives insight into the mechanisms by which MT-associated proteins (MAPs) and taxol stabilize MTs. In addition, we present further measurements of the mechanical properties of MT walls, MT-MT interactions, and the entry of polymers into the MT lumen. These results are obtained from a synchrotron small angle x-ray diffraction (SAXRD) study of MTs under osmotic stress. Above a critical osmotic pressure, Pcr, we observe rectangular bundles of MTs whose cross sections have buckled to a noncircular shape; further increases in pressure continue to distort MTs elastically. The Pcr of 
600 Pa provides, for the first time, a measure of the bending modulus of the interprotofilament bond within an MT. The presence of neuronal MAPs greatly increases Pcr, whereas surprisingly, the cancer chemotherapeutic drug taxol, which suppresses MT dynamics and inhibits MT depolymerization, does not affect the interprotofilament interactions. This SAXRD-osmotic stress technique, which has enabled measurements of the mechanical properties of MTs, should find broad application for studying interactions between MTs and of MTs with MAPs and MT-associated drugs. |
INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSUPPLEMENTARY MATERIALACKNOWLEDGEMENTSREFERENCES |
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ß-tubulin protein heterodimers. The structure of the dimeric tubulin subunit is known (1
) and high-resolution models of MTs are available (2). Tubulin dimers stack longitudinally into protofilaments that form the MT wall (Fig. 1). Protofilaments form sheets at the end of growing MTs (3). At the ends of shortening MTs, protofilaments curl and peel apart from each other, eventually detaching from the MT as highly curved oligomers (4). However, despite this detailed structural knowledge, the forces and interactions that control MT dynamics are not well understood. There is no comprehensive model for the unique and complex manner of MT polymerization, called dynamic instability, in which MTs stochastically switch between growing and shortening phases, but it is thought that the curvature of protofilaments and the interaction between protofilaments are two of the key microscopic determinants (5).
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) and the presence of drugs and MT-associated proteins (MAPs). Many anticancer drugs currently in use in clinical chemotherapy function by interfering with MT dynamics (7). For example, taxol, which is widely used to treat breast and ovarian cancer, prevents cell division by binding to and stabilizing MTs (8). However, the microscopic mechanism by which taxol affects MTs is unclear. It has been proposed that taxol stabilizes MTs by either increasing the lateral interactions between protofilaments (9) or by locking the protofilaments in a straight conformation (10). MAPs regulate dynamic instability in vitro (11) and in cells (12). A number of neurodegenerative diseases are associated with improper MAP-MT interactions (13). The microscopic mechanism by which distinct MAPs affect MT dynamics is not well understood. It is unclear if MT-stabilizing MAPs function by cross-linking tubulin dimers within a protofilament or between protofilaments, or if they change the protofilaments curvature. They may also cause a conformational change in tubulin or work by different mechanisms (14).
MTs are often organized into bundles in vivo. These bundles show a wide variety of internal organizations. In many instances the biological significance of MT bundle formation is unclear; for example, the role of MT bundles observed at the hillock region of nerve cells and also protruding along the axon are unknown (15). MT organization is directed by MAPs (16) in vivo. However, even though MAPs can bundle MTs in vitro (17) and in in vivo overexpression experiments (18), the mechanism of bundle formation remains controversial (19). It is unclear if MAPs directly cross-link MTs, act as spacers between MTs, or facilitate a preexisting attraction between MTs (19–21). Furthermore it is not known how these microscopic interactions between MTs, mediated by MAPs, could give rise to the MT bundles with different morphologies observed in vivo. The study of model systems should help to illuminate these issues.
The presence of inert macromolecules can lead to an effective attraction between particles in a manner that is well understood (22). The centers of the macromolecules are excluded from a region around each particle because they cannot interpenetrate. If these ‘excluded volumes’ around the particles overlap, then the volume available to the macromolecules increases and the total entropy increases, which results in an effective attraction between the particles. This ‘depletion attraction’ can lead rod-like particles to form a variety of phases (23,24). DNA (25), actin (26), and MTs (27) have all been observed to form bundles in the presence of inert macromolecules. It has been proposed that these excluded volume effects help organize the cytoplasm (28).
The depletion attraction can be used as a powerful probe of intermolecular interactions. If the induced attraction causes the macromolecules to phase separate from the rod-like particles, then the interaction between rod-like particles can be measured using the "osmotic stress" technique (29). In this method a known osmotic pressure is applied by controlling the concentration of inert macromolecule, and the resulting spacing between particles is determined by x-ray scattering. The osmotic stress technique has been used to study a wide variety of systems; for example, the forces between viruses (30) have been measured and the interactions between DNA molecules have been extensively studied (31).
Unlike actin, DNA, and many other rod-like biological polyelectrolytes, MTs are hollow, which leads to additional possible effects of applied osmotic pressure. If the added inert macromolecule is large enough to be excluded from the MT lumen, then a pressure imbalance is created between the inside and outside of the MT. For a large enough pressure difference, the MTs buckle in a manner analogous to the buckling of a hollow tube with closed ends subjected to enormous hydrostatic pressure, such as a submarine in deep sea (27). The deformation of hollow tubes under pressure has been extensively studied but, to the best of our knowledge, this work is the first unambiguous determination of the structure of bundles of hollow tubes under radial pressure.
For deformable objects such as MTs, there is a connection between the osmotic pressure-induced buckling and the bundling caused by excluded volume effects. The strength of the depletion attraction is highly dependent on the shape of the attracting particles (32,33). The induced attraction between flattened particles, such as distorted MTs, can be orders of magnitude greater than between rounded particles, such as undistorted MTs. Thus increasing the osmotic pressure causes the MTs to buckle, and upon buckling the depletion attraction between MTs greatly increases so the buckled MTs form bundles (27).
The deformation of tubes under pressure can be used to measure the mechanical properties of the tubes. Previous mechanical measurements of MTs have been performed by a number of groups using various methods. The axial bending rigidity of MTs has been measured by observing the thermal fluctuations of filaments (34–37) and by manipulation with optical tweezers or hydrodynamic flow (34,37–39). Atomic force microscopy (AFM) has been used to probe the radial mechanical properties of MTs (40,41). Image analysis of cryoelectron micrographs has been used to analyze how MTs respond to internal stresses (42–45). Knowledge of the mechanics of MTs is important for understanding their behaviors and functions in vivo. In addition, MT mechanics can be used to gain insight into the molecular and mesoscopic interactions that underlie MT dynamic instability (36,42,44). Our work expands on these earlier measurements by describing new features of the radial mechanical properties of MTs that can be used to probe the interprotofilament interactions. The osmotic stress-small angle x-ray diffraction (SAXRD) method described here allows the microscopic mode of action of MT stabilizing agents to be directly investigated using mechanical perturbations.
We have subjected MTs to osmotic stress by mixing them with poly-(ethylene-oxide) (PEO), dextran, and bovine serum albumin (BSA). PEO and polymeric sugars induce attractive interactions between proteins by excluded volume effects (46,47). Indeed, it has previously been demonstrated that the interaction between PEO and tubulin is purely unfavorable (48). PEO and dextran do enhance MT polymerization (49), presumably through a depletion attraction mechanism, in a manner similar to the stabilizing effects of osmolytes on actin (50).
We have characterized MTs under osmotic stress on length scales from millimeters to nanometers using polarized microscopy, video-enhanced differential interference contrast (DIC) microscopy, fluorescence microscopy, whole mount and plastic-embedded thin section electron microscopy, and synchrotron and rotating-anode SAXRD. Our results are summarized in Fig. 1. With no or low concentration of added osmolytes, the added polymers have no effect on the MTs, but the MTs align due to steric interactions with each other. When the concentration of stressing polymer is increased, the MTs form bundles. If the polymers do not enter the MT lumen, then at high enough concentration, the MTs buckle to a noncircular cross section and pack into a lattice with rectangular symmetry. The MTs continue to distort as the osmotic pressure increases. This buckling and distortion is reversible. If the polymer can enter the MT, because of its small size or at high concentrations of medium size polymer, then hexagonal bundles of undistorted MTs are formed. We have used the osmotic stress technique in the hexagonal bundle phase to measure the interactions between MTs. The transition to the rectangular bundle phase and the associated MT deformations provide information on the mechanical properties of MTs, which in turn give insight into how taxol and MAPs stabilize MTs. The work presented in this manuscript is an elaboration and continuation of a previous short article (27).
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSUPPLEMENTARY MATERIALACKNOWLEDGEMENTSREFERENCES |
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). MTP was obtained from three cycles of polymerization/depolymerization, and tubulin was purified from MTP by phosphocellulose chromatography. MTs were polymerized from tubulin at 4 mg/ml in 50 mM PIPES (pH 6.8, adjusted with 80 mM NaOH), 1 mM MgCl2, 1 mM EGTA, 1 mM guanine triphosphate (GTP), and 5% glycerol by incubating in a 37°C water bath for 20 min. Unless otherwise noted, MTs were stabilized by the addition of 20 µM taxol. Samples were made by diluting MTs 1:1 with PEO (Fluka, Seelze, Germany) solutions, so final buffer and taxol concentrations were half those listed above. Fluorescently labeled PEO was synthesized from mPEG20000-NH2 (Shearwater, Huntsville, AL) and Lissamine rhodamine B sulfonylchloride (Fluka) in dichloromethane with triethylamine (Fluka) and purified by flash chromatography on a silica gel. The osmotic pressure exerted by solutions of 20 k PEO was obtained from a publicly available database (http://www.brocku.ca/researchers/peter_rand/osmotic/osfile.html#data, see Supplementary Methods for further information).
X-ray scattering
Synchrotron SAXRD experiments were performed at beamline 4-2 of the Stanford Synchrotron Radiation Laboratory during four runs, over the course of a year, using tubulin purified from four different preps. X-ray scattering patterns were reproducible throughout, including over the volume of a sample and between different samples from different preps on different runs. The scattering was done at 8.98 keV with a beam size of 0.2 x 0.2 mm and sample-to-detector distances of 2.2015 m, 2.1579 m, 2.2022 m, and 2.1660 m, determined using silver behenate as a standard. A charge-coupled device-based area detector (MarCCD165, Mar USA, Evanston, IL) was used. Scans were performed for an average of 12 min, over which time no sample damage occurred, which was explicitly checked for by performing multiple short scans. In-house experiments were performed on a custom-built rotating anode SAXRD set-up. Rotating anode scans were typically performed for 12 h and have lower resolution and lower signal/noise than synchrotron scans but display the same features.
The samples showed powder scattering, so images were averaged over 360° to obtain plots of scattering intensity versus momentum transfer, 
Where 
is the x-ray's wavelength and 2
is the angle between the transmitted and scattered beams. All x-ray scans are raw data with no processing of any kind.
X-ray samples were centrifuged at 16,000 g for 1 h, transferred to 1.5 mm quartz capillaries, and left to sit at least 3 days before measurements were taken to ensure that the equilibrium osmotic stress was reached. Qualitatively similar data were obtained when SAXRD scans were taken immediately after samples were prepared. MTs are stable for days with sufficient taxol (10). To prevent MT depolymerization, measurements were taken immediately on samples with low concentrations of taxol.
Optical microscopy
A Nikon Diaphot 300 microscope with a Sutter Instrument (Novato, CA) Lambda LS xenon arc; an oil condenser; an oil, 1.4 numerical aperture, 60x objective with an additional 4x eye piece; and a Dage-MTI (Michigan City, IN) VE 1000 camera were used for high resolution video-enhanced DIC. Background subtraction, image enhancement, and two-frame averaging were performed with a Dage-MTI DSP-2000. Fluorescence images, and the corresponding DIC images, were taken with a Cooke (Romulus, MI) Sensicam QE monochrome digital camera, on the microscope described above, without the 4x eyepiece or image processing. Fluorescence images were high-pass filtered, Gaussian blurred, and contrast-adjusted with Adobe Photoshop. Coverslips and slides were cleaned with soap, ethanol, and water and sealed with wax.
Polarized microscopy images were taken with a Nikon Optiphot2-Pol microscope with a 5x objective and a Nikon Coolpix 990 camera; 3 mm x 0.3 mm rectangular capillaries were used.
Electron microscopy
Transmission electron microscopy experiments were performed at 80 kV. Whole mount samples were transferred to carbon-coated grids and stained with uranyl acetate. Samples for thin sections were centrifuged at 16,000 g for 1 h to form a pellet. Pellets were fixed with 2% glutaraldehyde (GA) and 0.2% tannic acid overnight, postfixed with 0.75% (w/v) OsO4 for 1 h, stained en bloc with 1% uranyl acetate for 1 h, dehydrated with acetone, embedded in spur plastic, and cut to 70 nm.
Modeling
For all calculations that treated MTs as homogeneous, isotropic, hollow cylinders, the MTs were taken to have an outer radius of 12.7 nm and an inner radius of 8.7 nm (2). Changing these values within physically reasonable limits does not qualitatively change the results. The surface charge of MTs at pH 6.8 was taken to be –0.87 e/nm2 (calculated from the primary structure of tubulin with 2 bound Mg2+ ions, one molecule GTP, and one molecule GDP per dimer), and the salt concentration was 40 mM Na+ and 0.5 mM MgCl2 (as in our buffer).
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RESULTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSUPPLEMENTARY MATERIALACKNOWLEDGEMENTSREFERENCES |
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), which is supported by a simple estimate of the volume fraction of rods necessary to form a nematic. Long rods of length L and radius R interacting through hardcore repulsion are predicted to form a nematic at a volume fraction of 
(52). Taking R = 12.7 nm and L = 10 µm, yields 
comparable to the MT volume fraction used in this study. Therefore, we refer to the oriented MTs as being in the nematic phase.
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250 nm, so other techniques are needed to probe the internal organization of the MT bundles.
Small angle x-ray scattering reveals the bundles' internal structure
We have performed a series of SAXRD experiments to study the nm scale structure of the MT bundles. The small angle scattering from solutions of MTs drastically changes as additional 20 k PEO is added (Fig. 3 A). For small concentrations of 20 k PEO the scattering is smoothly varying, but at higher concentrations multiple peaks are evident. These peaks shift, new peaks appear, and old peaks vanish as more 20 k PEO is added. All of these scattering data can be explained by the existence of three distinct structures: unbundled MTs, MT bundles with rectangular symmetry, and MT bundles with hexagonal symmetry.
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6–60 nm and therefore does not contain information on the structure of the tubulin subunit, which is available at wider scattering angles (53). At small q, the MT's form factor is dominated by its diameter and wall thickness, and, for comparison, Fig. 3 C shows a model calculation of the scattering from a hollow cylinder of appropriate dimensions (Fig. 3 C, unbuckled). The minimum in the scattering data (Fig. 3 B, NMT) are slightly filled in due to the presence of a small amount of tubulin which has not polymerized into MTs (54). If this background is properly accounted for, the hollow cylinder model (Fig. 3 C, unbuckled) can quantitatively describe the scattering from MTs in this q range (55). SAXRD is a sensitive measure of the average number of protofilaments in MTs (53). The measured form factors indicate that these MTs have an average of 13 protofilaments, determined by comparing these SAXRD data with results from Andreu (53). An average of 13 protofilaments is expected for MTs stabilized by taxol after polymerization (10), as MTs in this study were.
SAXRD experiments measure the Fourier transform of the electron density of objects in the sample (56). The electron density of an MT bundle is a convolution of the electron density of a single MT with the two-dimensional lattice of delta-functions that define the arrangement of MTs in the bundle. Therefore the scattering of an MT bundle is the multiplication of the MT form factor, the Fourier transform of the electron density of a single MT, with the bundle structure factor, the Fourier transform of the two-dimensional lattice (56). Thus the scattering consists of Bragg peaks, determined by the structure factor, and the peak heights are modified by the smoothly varying form factor. The height and shape of peaks in the structure factor are determined by lattice vibrations, defects, finite-size effects, and other distortions, whereas the peak positions determine the symmetry and dimensions of the lattice. There are only five lattice types in two dimensions (57), and indexing the Bragg peaks seen in the bundle phases allows the lattice type and the associated lattice constants to be uniquely determined. With high concentration of 20 k PEO (Fig. 3, A and B, 
), the four peaks index to a hexagonal lattice with lattice constant 
nm. Thus the MTs form hexagonally packed bundles with high amounts of added 20 k PEO (Fig. 1, 400,000 Pa). Fig. 3 C, 
shows the calculated scattering from a hexagonal array of hollow cylinders with aH = 27 nm. The structure factor peaks were taken to be Lorentzians with widths of 0.04 nm–1, and separate amplitudes for each peak were chosen to mimic the data (Fig. 3 B, ).
With moderate concentrations of 20 k PEO, the Bragg peaks index to a rectangular lattice (Fig. 3 B, 
). Surprisingly, these nine orders of diffraction peaks indicate that one of the lattice parameters, aR, is smaller than the unperturbed MT diameter of 25.4 nm, whereas the other lattice parameter, bR, is larger (Fig. 3 B, 
and 
; see Fig. 1 for illustrations of aR and bR). The only way that MTs can fit into this lattice structure is if the high pressure causes their shape to become distorted (Fig. 1). The bundles themselves are highly asymmetric. The bundle size, Li, along a direction i, can be measured with Warren's approximation, in which the scattering close to a reciprocal lattice vector, G, results in a structure factor 
This yields a domain size of L(1 0) = 166 nm along the (1 0) plane and a L(0 1) = 204 nm along the (1 0) plane, so these bundles are 5 MTs by 11 MTs, with more MTs along the axis that is more compressed. A model scattering curve for a rectangular lattice of ellipses with aR = 18.47 and bR = 33.40 nm is shown in Fig. 3 C (). In this model Lorentzian-shaped peaks were used with separate amplitudes for each peak, a width of 0.013 nm–1 in the (1 0) direction, and a width of 0.009 nm–1 in the (0 1) direction.
At intermediate concentrations of 20 k PEO, a coexistence of rectangular bundles and hexagonal bundles are observed (Fig. 3 C, 
). All of the peaks in the scattering data (Fig. 3 A) can be indexed to rectangular or hexagonal lattice structures, with lattice parameters varying as the PEO concentration varies.
Although the MTs in the rectangular bundle phase are distorted, SAXRD measurements show that the MTs in the nematic phase remain undistorted for all PEO concentrations and under all conditions investigated. Fig. 3 C compares the expected scattering from an MT with a circular cross section (unbuckled) with the calculated scattering from an MT with an elliptical cross section (buckled). The dimensions of the model-deformed MTs, with a semiminor axis of 7.2 nm and a semimajor axis of 13.8 nm to the center of the MT wall, are those of the least deformed MTs in the rectangular bundle phase. The model curves of the buckled and unbuckled MTs are quite different. Most notably, the second and third maxima of the buckled MT form factor occur at q values similar to minima in the unbuckled MT form factor. Thus, SAXRD can easily distinguish between buckled and unbuckled MTs in the nematic phase, and only unbuckled MTs have been observed. For example, the form factor of MTs with no added 20 k PEO (Fig. 3 A, 0%) is nearly identical to the form factor of MTs directly before the transition to rectangular bundles (Fig. 3 A, 0.2%). There may be some conditions, which we have not yet examined, in which MTs are buckled in the nematic phase.
The transition to the rectangular bundle phase is reversible
The transition from unbundled, nematic MTs to bundled MTs in the rectangular phase with increasing 20 k PEO can be reversed by decreasing the concentration of 20 k PEO. Fig. 4 A (1%) shows SAXRD results from MTs with 1% 20 k PEO in the rectangular bundle phase. When these MTs are centrifuged to a pellet and resuspended in buffer such that the final 20 k PEO concentration has been reduced to 0.2%, SAXRD scans show that the MTs return to unbundled, undamaged MTs (Fig. 4 A, 20 k PEO reduced to 0.2%). Furthermore, this SAXRD scan demonstrates that the number of protofilaments per MT remains unchanged before and after being in the rectangular bundle phase. Whole mount electron microscopy confirms that MTs previously in the rectangular bundle phase are intact and undamaged (Fig. 4 A, inset).
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The rectangular bundle phase forms in the presence of polymers that are excluded from the MT lumen
The nature of the formed bundle phase depends on the method or substance used to exert osmotic pressure, as summarized in Table 1. The rectangular bundle phase forms when MTs are mixed with BSA. For low concentrations of added BSA the MTs form a nematic, whereas at higher concentrations the rectangular bundle phase is observed. The hexagonal bundle phase is never seen even with up to 20% BSA in solution. BSA is a negatively charged globular protein 9 nm x 4 nm x 8 nm (58), with a radius of gyration (Rg) of 3.2 nm (59). Thus BSA is expected to be too large to appreciably enter the MT lumen. If BSA is modeled as a capsule of appropriate dimensions, then only 2% is expected to partition into the MTs (60).
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18 nm (61,62), essentially all of the added polymer should be excluded from the MT interior (63). Only unbundled MTs in the nematic phase and bundles in the rectangular phase are seen with 500 kd dextran, up to 8%, and 100 k PEO, up to 5%, the hexagonal bundle phase is never observed. In contrast, only unbundled MTs and hexagonal bundles are observed with added 600 PEO, bundles in the rectangular phase are never seen. 600 PEO is a small polymer, Rg 1 nm (61), and at least 75% of the added polymer is predicted to enter the MT lumen (63).
Furthermore, the polymer does not need to be in physical contact with the MTs to convert them to the rectangular bundle phase. If the MTs are incubated against a solution of 20 k PEO, with the two solutions separated by a semipermeable membrane, the rectangular bundle phase still forms; however, the hexagonal bundle phase is not observed. In these experiments, buffer is drawn across the semipermeable membrane, increasing the concentration of MTs until the mechanical pressure exerted by the MTs equals the osmotic pressure of the 20 k PEO. The transition from nematic, unbundled MTs to rectangular bundles occurs at a similar concentration of added 20 k PEO independent of whether the polymer is separated by a semipermeable membrane or directly mixed with the MTs. Less than 2% of added 20 k PEO, with Rg 7 nm (61), should partition into the MT interior for low concentrations of added polymer (63) even when the polymer is in contact with the MTs.
The connection between formation of the rectangular bundle phase and exclusion of stressing polymer from the MT lumen is further supported by optical microscopy measurements. Rectangular bundles created with fluorescent 500 kd dextran (Fig. 5, 8% 500 k dextran) or small amounts of fluorescently labeled 20 k PEO (Fig. 5, 1% 20 k PEO) appear dark in fluorescence. This proves that the dextran and 20 k PEO tend to be excluded from the rectangular bundle phase. In contrast, if the concentration of unlabeled 20 k PEO is increased, keeping the concentration of fluorescently labeled 20 k PEO constant so the bundles are in the hexagonal phase, then these bundles are not different from background in fluorescence images (Fig. 5, 20% 20 k PEO). This is consistent with 20 k PEO entering MTs when they convert to hexagonal bundles. The entry of polymer into MTs at high concentration is expected from theory (64), which predicts that the ease with which polymers enter a confining tube, such as an MT, greatly increases as the polymers begin to overlap and exert forces on each other. Consistent with this, the hexagonal bundles appear at a concentration of added 20 k PEO near their overlap concentration of c* 7.5% (w/w). Thus, if the stressing polymers are excluded from the MT lumen, then the pressure difference across the MT wall causes the MTs to buckle to a noncircular cross section. These buckled MTs form rectangular bundles due to depletion attraction. If the polymer can enter the MTs, either because the polymer is small or is forced in at high concentration, then the MTs are not distorted. Depletion attraction causes undistorted MTs to aggregate into hexagonal bundles at high enough concentrations of added polymer.
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30% MAP and 70% tubulin by weight (65). The presences of very small amounts of MAPs greatly suppress the nematic to rectangular bundle transition. MTs polymerized with 5% MTP and 95% tubulin contain only 1.5% MAPs by weight, but this substantially increases the concentrations of stressing 20 k PEO needed to induce the transition to rectangular bundles (compare Fig. 6 A and Fig. 3 A). MTs with 15% MAPs (50% MTP) do not display the rectangular bundle phase at any osmotic pressure, though the hexagonal bundle phase is still present with high concentrations of added 20 k PEO (Fig. 6 B). No rectangular bundles are observed with 100% MTP, with no taxol, at physiological ionic strength (Fig. 6 C). The osmotic pressure-%MTP phase diagram with 20 k PEO is shown in Fig. 7. As noted above, the most significant change with increasing MAPs is a large increase in the osmotic pressure required to induce rectangular bundles, and with enough MAPs no rectangular bundles are present.
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) and the stoichiometry of bound taxol to tubulin dimers in MTs varies from 0.2 to 0.8 (calculated using the known binding constant of taxol (66)). It is surprising that taxol and MAPs have such different effects on the transition to the rectangular bundle phase, as both stabilize MTs against depolymerization.
In the rectangular bundle phase, MTs continue to distort as the osmotic pressure increases
The MTs become increasingly distorted with increasing osmotic pressure. aR decreases linearly with the logarithm of the pressure, with the same slope for MTs with MAPs (Fig. 10 A), MTs with added KCl (Fig. 10 B), and MTs with various concentrations of added taxol (Fig. 10 C). The continued deformation of MTs beyond the initial buckling most likely provides additional information on the mechanics of the MT walls. We cannot rigorously rule out the possibility that the dependence of aR on osmotic pressure is controlled by the interactions between MT walls, but the independence of the postbuckling behavior on solution conditions makes this interpretation unlikely.
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In the hexagonal bundle phase, MT spacing decreases with increasing osmotic pressure
The center-to-center distance between MT in the hexagonal phase, aH, illustrated in Fig. 1, decreases with increasing osmotic pressure. This trend is clear from following the location of the (1 0) hexagonal peak in the raw x-ray data shown in Fig. 3 A (this peak is indexed in Fig. 3 B). The MT wall-to-wall distance is 
where R is the MT radius, and, since the MTs are in a hexagonal lattice, the force per unit length between them is 
where 
osm is the applied osmotic pressure (31). aH (and D) versus osmotic pressure (and F) is plotted in Fig. 11. These data measure the inter-MT interactions and are well fit by an exponential, 
(Fig. 11, dashed line). This measurement is in close agreement with the predicted force per unit length from Poisson-Boltzmann theory (67) for two charged cylinders with the appropriate radius, assuming constant surface potential: 
The surface potential was numerically calculated using the Graham equation from the MT surface charge. The slight discrepancy between the predicted and observed coefficient for the exponential may be caused by an increased repulsion from the localization of charge on the disordered, protruding C-terminus. MTs are highly negatively charged (68), so it is not surprising that the interaction between MTs is dominated by electrostatic double layer forces, as has also been measured for viruses (30) and DNA under appropriate conditions (69).
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSUPPLEMENTARY MATERIALACKNOWLEDGEMENTSREFERENCES |
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Pa by measuring unbuckled nematic MTs at 510 Pa and buckled MTs in rectangular bundles at 690 Pa.
If MTs are treated as homogeneous, isotropic, hollow cylinders, then MT bending measurements indicate a Young's modulus for tubulin of 2 GPa, which is similar to other proteins (70). Using the same model of MTs as elastic hollow cylinders that explains the bending measurements, a buckling pressure of 
MPa is predicted (71), where E is the Young's modulus of the MT wall, h is the MT wall thickness, and 
is the Poisson ratio for the material that makes up the MT wall (taken to be 0.4, similar to nylon). This calculated value of Pcr is over four orders of magnitude greater than our measured value. However, it is not surprising that the simple model described above fails because MTs are not homogeneous, isotropic objects. Large scale molecular level electrostatics calculations (72), simulations of course-grained mechanical properties (73), and simple models of dynamic instability (74) predict that the intermolecular interactions that connect protofilaments are much weaker than longitudinal tubulin bonds and the intramolecular interactions within a tubulin dimer. The derived value of the Young's modulus, along with cryoelectron microscopy measurements (43), argues that the resistance to axial bending is due to deformation of the tubulin subunit. This strongly suggests that low Pcr we measure is due to the weak compliance of the lateral bond between protofilaments.
An estimate of the lateral interaction between protofilaments may be obtained from Pcr by a standard linear elastic stability analysis (75) of a simple model of the MT. We treat the MT cross section as consisting of protofilaments of diameter l 5.0 nm, connected by linear springs of spring constant k, which enforce the desired angle, 
between the subunits. The protofilaments are considered to be incompressible at these low pressures because, assuming the protofilament can be treated as homogeneous, isotropic elastic solids with a Young's modulus of 2 GPa, which is typical for proteins (70), the expected relative change in dimensions is <1% at the buckling transition. Such a structure buckles at a critical pressure 
(see Supplementary Material). For our measured buckling pressure of Pa this leads to a very small spring constant of 
which implies that 28 tubulin dimers, which is 224 nm long, must stack to resist thermal fluctuations.
The continued deformation of MTs with increasing osmotic pressure provides additional information on the mechanical properties of MTs. The postbuckling compliance seems to have a different origin than the weak interprotofilament bonds that determine the initial buckling instability. The postbuckling behavior, unlike Pcr, is largely independent of solution conditions, and the initial deformation due to buckling requires less pressure than subsequent deformations of the same magnitude. For example, increasing the osmotic pressure from 300 Pa to 600 Pa causes the MTs to buckle from a circular cross section with a diameter of 25 nm to a noncircular shape with a short dimension of 19 nm, but further increasing the osmotic pressure to 200,000 Pa only decreases the short dimension to 16 nm. We currently have no model with which to interpret the postbuckling behavior, but a very rough understanding of the source of compliance can be obtained by calculating an effective Young's modulus of the entire MT, 
where 
is the fractional change in length of the short dimension caused by the change in pressure, 
This value of EMT suggests that the resistance to postbuckling deformation, like the resistance to bending, is due to the distortion of the tubulin subunits, as opposed to the weak interprotofilament bond that is responsible for the initial buckling.
Our work is complementary to previous imaging and mechanical radial indentation studies of MTs with AFM (40,41,76), which enables the properties of individual MTs to be examined. The mechanical properties measured in the AFM experiments may be consistent with the stiff postbuckling behavior we observe with the osmotic stress-SAXRD technique. The AFM experiments did not observe an initial soft deformation, which may be due to the large minimum pressure exerted by the AFM tip or the MTs being deformed due to absorption to a surface, and measured MT heights by AFM are slightly smaller than expected, which is consistent with both these explanations (40,76). However, a quantitative comparison of results from AFM and osmotic stress-SAXRD is difficult because in the AFM experiments MTs are absorbed to a surface, the interaction between the surface and MTs and the AFM tip and MTs is poorly categorized, and the absolute height of the MTs is uncertain. It is currently unclear if MT-MT interactions are important for interpreting the results of the osmotic stress-SAXRD technique. More detailed modeling of the mechanical properties of MTs is needed to synthesis the results from different experimental techniques.
Considerations on the structure of MT bundles
Whereas the MT buckling transition can be understood using simple models and the bundle formation occurs due to the depletion attraction, the reason that buckled MTs form bundles with rectangular internal symmetry is less clear. Some models of carbon nanotube bundles under pressure (77,78), and more abstract theories of stiff polymers with angular frustration (79), indicate that body-centered rectangular symmetry (perhaps with a herringbone structure) should be preferred. More recently, molecular dynamics simulations have shown that a face-centered oblique lattice of parallel carbon nanotubes buckled under pressure is more stable than the herringbone structure proposed in previous studies (80). Those authors did not compare the relative stability of those oblique lattices of parallel buckled tubes with the rectangular lattice of parallel buckled tubes found in our study. In addition, it is unclear why the overall shape of these MT bundles is asymmetric with more MTs along the more compressed axis, but it is interesting to note that depletion attraction between colloidal disks also leads to similarly asymmetric aggregates (33).
We have shown that osmotic stress can lead to the radial buckling of MTs. In theory, other types of elastic failures of MTs under osmotic stress are possible. At very high osmotic pressure, MTs might additionally fail via an axial local surface buckling (71) or an axial Euler buckling (71,75). It has previously been observed that MTs polymerizing against a barrier undergo an Euler buckling instability (81).
The hexagonal bundles form when the stressing polymer can enter the MT lumen. The entry of medium size polymer into MTs only at high concentration is consistent with theory (64), and similar effects have been seen with the partitioning of PEO in ion channels (82). Our measurements do not address the pathway by which the polymers enter MTs; they may enter through the open MT ends or through holes in the MT wall (2).
MT stabilization and interprotofilament interactions
We have provided strong evidence that Pcr provides a measure of the interprotofilament interactions and the osmotic stress-SAXRD technique is the first method that allows this to be probed. The interprotofilament bond is crucial for determining the dynamics of MT polymerization and depolymerization (5,74). The flexibility of the interprotofilament bond may also be important in MT nucleation and it may be responsible for variations in the number of protofilaments per MT.
MTs become destabilized in the presence of monovalent salt in excess of 100 mM (6), which corresponds to 60 mM added KCl in our experiments. Pcr decreases with added KCl in excess of this amount (Fig. 9 A), suggesting that the destabilization is due to weakening of the interprotofilament interactions. This is not surprising as observations from high resolution MT models (2), along with detailed computer simulations (72), suggest that the interprotofilament bonds are largely electrostatic in nature.
The addition of GA causes MTs in the rectangular bundle phase to become less distorted. If enough GA is added, these MTs unbuckle and revert to an unbundled state (Fig. 5 B). The unbundling of MTs is connected to their unbuckling, because when MTs revert to a rounded, undistorted shape the depletion attraction between then greatly decreases. These experiments demonstrate that GA increases the mechanical stiffness of MTs, including that due to compliance of the weak interprotofilament bound. The mechanical strengthening of MTs by GA has been previously measured by AFM (40).
MTs in neurons are much more stable than MTs in other cell types due to the action of MAPs (11,12). The presence of neuronal MAPs causes a dramatic increase in Pcr. MTs polymerized with 10% MTP are 3% MAP and 97% tubulin by weight, but the Pcr of such MTs is over 20 times greater than the Pcr of MTs polymerized from pure tubulin (Fig. 10). Thus the osmotic stress-SAXRD technique provides the first demonstration that MAPs increase interprotofilament interactions, presumably by directly cross-linking protofilaments, but this may also be due to a MAP-induced conformational change in tubulin.
Substoichiometric binding of the anticancer drug taxol enhances MT polymerization and inhibits depolymerization (8). Increasing taxol concentration from 2.5 µM to 20 µM has a large effect on MT polymerization, with the polymer mass at steady-state increasing by approximately a factor of two over this range of taxol concentrations (8). However, Pcr does not change, within error (approximately a factor of two), over this concentration range of taxol. In sharp contrast to the effects of taxol, increasing the concentration of MAPs from 0.2% to 3% (0.5% MTP to 10% MTP) has a more modest effect on MT stability, with polymer mass in steady state increasing by 40%, as estimated from results with 
(11), but a much larger effect on Pcr, which increases nearly 20-fold. Thus, over the tested concentration ranges, taxol has a large effect on MT polymerization and a small effect on Pcr, whereas MAPs have a more modest effect on MT polymerization but a much larger effect on Pcr. Thus taxol does not stabilize MTs by increasing the interaction between protofilaments. These results support an earlier suggestion that taxol functions by preventing the straight-to-curved conformational change normally associated with GTP hydrolysis (10). Our work shows that taxol and the MAPs tested here stabilize MTs through different microscopic mechanisms.
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Our work raises questions about the state of MTs inside the crowded environment found in cells. Cytoplasm is typically 20% protein by weight, and macromolecular crowding is believed to be significant at such high concentrations with important ramifications for cellular architecture and macromolecular interactions (28). The internal pressure of cells can be MPas (83), which is orders of magnitude greater than the osmotic pressure required to buckle MTs composed of pure tubulin. There seem to be three possibilities: MTs are sometimes buckled in vivo, one of the physiological roles of MAPs is to prevent MTs from buckling, or some of the osmotic stressing agents enter the MT lumen. Even if the rectangular phase of buckled MTs is suppressed inside cells, the osmotic pressure is still high enough to form hexagonal bundles through depletion attraction, and close packed hexagonal bundles of MTs have been observed in stressed cells (84).
More broadly, understanding dynamic instability of MTs, and the manner in which it is modified by distinct MAPs and drugs, is of fundamental biological and medical significance. The osmotic stress-SAXRD technique described in this work provides the first method, to our knowledge, for probing how various agents affect interprotofilament interactions. We hope that work on interactions, such as presented in this study, will be combined with structural models and MT polymerization measurements to develop a quantitative microscopic understanding of dynamic instability.
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ACKNOWLEDGEMENTS |
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U. Raviv acknowledges the support of the International Human Frontier Science Program Organization. This work was supported by National Science Foundation grants DMR 0503347, CTS 0404444, and CTS 0103516, National Institutes of Health grants GM-59288 and NS-13560, and Department of Energy contract No. W-7405-ENG-36 with the University of California. The Materials Research Science and Engineering Center at the University of California, Santa Barbara, is supported by National Science Foundation DMR-0080034. The Stanford Synchrotron Radiation Laboratory is supported by the Department of Energy.
Submitted on December 7, 2004; accepted for publication July 26, 2005.
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arrow indicates hexagonal (1 0) peak. All other peaks are from rectangular bundles, indices not shown). (C) Model calculations of the scattering from isolated MTs (unbuckled), unbundled, distorted MTs (buckled), bundles of distorted MTs with rectangular symmetry (
