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Correspondence to David A. Weitz: weitz{at}deas.harvard.edu
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100 times that of actin filaments and a persistence length on the order of millimeters (Gittes et al., 1993). Cytoplasmic microtubules are particularly critical for stabilization of long cell extensions, such as nerve cell processes (Zheng et al., 1993), and their disruption results in process retraction and production of regular polygonal cell forms (Omelchenko et al., 2002). Microtubules also orient vertically in cells when they become columnar, as observed during neurulation in the embryo (Burnside, 1971), and they can physically interfere with cardiac myocyte contractility in certain heart conditions (Tsutsui et al., 1993, 1994; Tagawa et al., 1997). Despite their long persistence length, microtubules are also often highly curved in cultured cells, suggesting that they experience large forces within the cytoplasm. Indeed, both slow retrograde flow of the actin cytoskeleton at the cell periphery and stimulation of cell contraction appear to cause compressive buckling of microtubules (Wang et al., 2001; Gupton et al., 2002; Schaefer et al., 2002), whereas disruption of microtubules results in increased transfer of cytoskeletal contractile stress to extracellular matrix adhesions (Stamenovic et al., 2002). These observations suggest that microtubules can bear compressive loads, which is consistent with models for cellular mechanics in which microtubule compression helps stabilize cell shape by balancing tensional forces within a prestressed cytoskeleton (Wang et al., 1993; Ingber et al., 2000, 2003; Wang et al., 2001; Stamenovic et al., 2002). However, other biophysical studies suggest that individual microtubules cannot bear the large-scale compressive forces generated by the surrounding cytoskeleton in a whole living cell. Moreover, under compressive loading, isolated microtubules exhibit a classic Euler buckling instability, resulting in the formation of a single long-wavelength arc that is completely unlike the highly curved appearance of microtubules observed in living cells (Gittes et al., 1996; Dogterom and Yurke, 1997). These results suggest that microtubules may not support compressive loads because they should buckle at much larger wavelengths and should be unstable to very small forces (Dogterom et al., 2005; Deguchi et al., 2006). The structural role of microtubules in whole living cells thus remains highly controversial.
Despite the importance of understanding the forces associated with the microtubule cytoskeleton for control of cell shape and mechanics, there have been very few studies that quantitatively measure these forces, and the precise physical basis of the microtubule bending seen throughout the cytoplasm remains unknown (Heidemann et al., 1999; Ingber et al., 2000). In this paper, we address the question of whether or not microtubules bear large-scale compressive forces in living cells. Our results reveal that individual microtubules can and do bear levels of compressive force that are one hundred times greater in whole cells than in vitro. This is possible because of lateral mechanical reinforcement by the surrounding elastic cytoskeleton. To illustrate this principle, we present a macroscale model composed of a plastic rod embedded in an elastic gel, which mimics the short-wavelength curvature observed in microtubules in living cells. We show that a reinforced buckling theory accounting for the surrounding elastic network can quantitatively predict the wavelengths of buckling induced by compression at both the macro- and microscales. This simple reinforcement principle, which appears to be widely used by nature to enhance the structural stability of cells, provides an explanation for how microtubules can bear large compressive forces within the cytoskeleton of living cells.
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is the bending rigidity and L is the length of the microtubule (Landau and Lifshitz, 1986; Dogterom and Yurke, 1997). Within a cell, microtubules are typically even longer than those studied in vitro, suggesting that they should buckle easily under small loads of order 1 pN; forces larger than this can be generated by even a single kinesin or myosin motor protein (Gittes et al., 1996).
To investigate the response of microtubules to endogenous polymerization forces, Cos7 epithelial cells and bovine capillary endothelial cells were either transfected with tubulin labeled with EGFP or microinjected with rhodaminated tubulin and then analyzed using real-time fluorescence microscopy. The dynamic ends of growing microtubules that polymerized toward the edge of the cell consistently buckled when they hit the cell cortex (Fig. 1
and Videos 1 and 2, available at http://www.jcb.org/cgi/content/full/jcb.200601060/DC1), as observed in a previous study (Wang et al., 2001). Given the well defined (end-on) loading conditions that were visualized with this real-time imaging technique, these results strongly suggest that these particular microtubules are compressively loaded. These microtubules did not, however, exhibit the expected long-wavelength Euler buckling, but, instead, consistently formed multiple short-wavelength (
3 µm) arcs near the site of contact (Fig. 1 and Videos 1 and 2), which were similar to the microtubule shapes that were previously observed in various different cell types (Kaech et al., 1996; Wang et al., 2001; Gupton et al., 2002; Schaefer et al., 2002).
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2.9 ± 0.6 µm (Fig. 2 C). Thus, in addition to polymerization forces and external loads, internally generated actomyosin contractile forces can also compressively load microtubules under physiological conditions. The microtubules buckle at identical short wavelengths in each case, suggesting that the same physics governs their behavior.
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To test the biophysical hypothesis that lateral mechanical constraints lead to short-wavelength buckling in microtubules within living cytoplasm, we first performed studies using a macroscopic model system in which a thin plastic rod (0.1-mm-diam) was used to mimic a single microtubule. When this rod was compressed in aqueous solution, it yielded the expected long-wavelength Euler buckling mode (Fig. 5 A
). To mimic the effect of a surrounding elastic network, we embedded the rod in gelatin: when the rod was compressed, the long-wavelength mode was suppressed. Instead, shorter-wavelength buckling resulted (Fig. 5 B), which was strikingly similar to that displayed by microtubules in living cells (Figs. 1–4
). This finding suggests that similar lateral constraints may lead to the short-wavelength buckling behavior observed in living cells.
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is proportional to the elastic modulus G (see supplemental discussion). The first term represents the bending energy of the rod, the second term expresses the axial compression energy released by the buckling, f
L, where L is the change in axial length of the rod, and the third term is the elastic deformation energy associated with pushing laterally into the surrounding medium. The Euler buckling theory comprises the first two terms; in contrast to the long wavelength Euler buckling result, the additional deformation energy represented by the third term leads to buckling on a short wavelength. The wavelength is set by the ratio of the bending rigidity of the rod and the elastic modulus of the surrounding network in the following equation:
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This constrained buckling theory provides a quantitative description of our macroscopic observation. Using measured values of the elastic modulus of the gelatin (G = 1.5 kPa), and the bending rigidity of the plastic rod ( 10–7 Nm2), the theory predicts a buckling wavelength of 1 cm, which is in excellent agreement with the experimental result ( 1.1 cm). Moreover, by using an array of plastic rods with bending rigidities varying by over an order of magnitude, we confirmed that the theoretical predictions closely matched the measured wavelengths in all cases (Fig. 4 D).
The biological relevance of this theory was also examined quantitatively by testing its ability to predict the buckling wavelength of microtubules in living cells. Using the reported microtubule bending rigidity (Gittes et al., 1993), 2 x 10–23 Nm2, and the elastic modulus of the surrounding cytoskeletal network (Mahaffy et al., 2000; Fabry et al., 2001), G 1 kPa, the theory predicts that compressively loaded microtubules should buckle with a wavelength of 2 µm; this is in good agreement with the measured wavelength 3 µm (Fig. 2 C). We note that the effect of intracellular microtubule-associated proteins on the microtubule bending rigidity is unclear (Felgner et al., 1997), and the rigidity also appears to depend on the speed of their polymerization (Janson and Dogterom, 2004); indeed, measurements of microtubule bending rigidity have varied by an order of magnitude. The elastic modulus of the cytoskeleton is also locally heterogeneous within the same cell, and measurements of this quantity have varied. However, the one-quarter power dependence of the wavelength on both the bending rigidity and the elastic modulus makes the predicted wavelength relatively insensitive to uncertainty in these values. The close agreement between the predicted and observed wavelengths shows that this constrained buckling theory can quantitatively explain both the macroscale model and the microscale buckling of microtubules in living cells (Fig. 5 D). Despite the fact that these two systems differ by more than four orders of magnitude in spatial scale, with bending rigidity differing by over 16 orders of magnitude, the same physics governs their behavior.
A central component of the elastic cytoskeleton of the cell is the actin filament network (Wang et al., 1993; Fabry et al., 2001; Gardel et al., 2006), which surrounds and is connected to intracellular microtubules (Svitkina et al., 1995; Waterman-Storer and Salmon, 1997); therefore, we hypothesize that this actin network plays an important role in microtubule reinforcement. To test this hypothesis, we pretreated Cos7 cells with 2 µM cytochalasin D for 30 min to disrupt the surrounding actin filament network, and used a microneedle to compress microtubules. We found that the wavelength increased to 4.3 ± 1.0 µm in cytochalasin-treated cells, compared with 2.8 ± 0.5 µm in untreated cells (P < 0.0005; paired t test; Fig. 2 C). The wavelength of naturally buckled microtubules also increased to a similar degree in these cells. This increase, albeit small, supports the hypothesis that the actin network plays a role in reinforcing microtubules. The relatively small change is consistent with the weak dependence of the buckling wavelength on the elasticity of the surrounding cytoskeleton ( G–1/4). Indeed, the larger buckling wavelength corresponds to a decrease in the elastic modulus of the surrounding cytoskeleton by a factor of 5, which is consistent with previous measurements of cytochalasin-treated cells (Wang et al., 1993; Fabry et al., 2001). Thus, the lateral structural reinforcement of microtubules responsible for their enhanced compressive load-bearing capacity appears to be at least partly attributable to the surrounding actin cytoskeleton in living cells.
Effect of local mechanical properties on location and extent of buckling
In studies with both muscle and nonmuscle cells, we found that localized regions of single microtubules repeatedly underwent short-wavelength buckling at the same sites when analyzed over many minutes, whereas intervening regions of the same microtubule, and also neighboring microtubules, remained straight (Figs. 3 and 4; and Videos 6 and 8, available at http://www.jcb.org/cgi/content/full/jcb.200601060/DC1). Nonbuckled microtubules in cells may experience a large, but subcritical compressive force, even when other regions of the same microtubule exhibit short-wavelength buckling, perhaps because of local weak spots in the surrounding elastic network. We observed similar behavior in the macroscopic experiment when a localized region of the surrounding gelatin network was disrupted; the rod preferentially buckled in this same localized region when it was compressed a second time, even though adjacent segments of the same rod remained straight (Fig. 5 C). These observations suggest that the location of high curvature microtubule buckling may be linked to local variations in the stiffness of the surrounding cytoskeletal network.
Interestingly, when microtubules were compressed by forces acting at their tips, the buckling typically did not extend along the entire microtubule; instead, the buckling was localized to a region close to the site of force application (Figs. 1, 2, and 6 ). We found similar behavior when we performed the macroscopic buckling experiment in tall sample chambers; the amplitude of the short-wavelength buckling decayed with distance from the site of force application (Fig. 6 B). This localization of the buckling results from longitudinal mechanical coupling (sticking) of the rod to the network; this attenuates the compressive force along the axis of the rod (see supplemental discussion). Thus, there is likely a similar coupling of the microtubule to the surrounding network, causing the attenuation of its buckling. This is consistent with studies showing that microtubules are physically linked to the surrounding cytoskeleton through protein cross-linkers (Svitkina et al., 1995; Gupton et al., 2002). Thus, the strength and kinetics of these cross-linkers determine the distance over which forces are mechanically transmitted through the cell.
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1 pN of compressive force, and thus, they should not contribute significantly to the mechanical stability of the whole cell. This apparent discrepancy reflects the lack of information about the mechanical behavior of microtubules within the normal physical context of the living cytoplasm. In this study, we directly addressed the question of whether individual microtubules can bear the levels of compressive forces necessary to influence overall cell mechanical behavior by studying and modeling microtubule buckling behavior in the living cytoplasm. Our results show that microtubules exhibit similar buckling responses, with nearly identical short wavelengths and correspondingly high curvature, whether compressed by endogenous polymerization or contractile forces or by direct application of end-on compression using a micropipette. This buckling wavelength could be increased by weakening the reinforcement provided by the cytoskeletal actin network. Moreover, similar buckling behavior can be mimicked using a macroscale model of a plastic rod embedded in an elastic gelatin network. A constrained buckling theory provides a quantitative description of this behavior at all size scales.
The finding that microtubules buckle in the living cytoplasm implies that they are under a minimum level of compressive loading because buckling is a threshold phenomenon; it only occurs once the compressive force reaches a critical value. However, lateral reinforcement ensures that a microtubule can remain structurally stable and continue to support a compressive load even after it buckles. Within this picture, we can calculate the critical force using
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, which is the shorter wavelength of buckling (see supplemental discussion). This critical force depends linearly on the bending rigidity and, therefore, is sensitive to the large uncertainties in the microtubule bending rigidity. Nevertheless, using the measured wavelength, which is 3 µm, and the bending rigidity of microtubules, the minimum compressive force experienced by microtubules that exhibit short-wavelength buckling can be estimated, and we obtain fc 100 pN. Interestingly, this is about 10 times larger than the microtubule polymerization forces measured in vitro (Dogterom and Yurke, 1997), which could reflect larger forces in the cell caused by the complex molecular environment at the microtubule tip (Schuyler and Pellman, 2001; Dogterom et al., 2005). Short-wavelength shapes similar to those we describe have also been seen in microtubules that were buckled by retrograde flow of the actin network (Gupton et al., 2002; Schaefer et al., 2002), and can be seen in microtubules in various other cell types and species (Kaech et al., 1996; Heidemann et al., 1999; Wang et al., 2001). Some of this high curvature microtubule bending may result from transverse shear stresses (Heidemann et al., 1999). For example, the active viscoelastic flow of the cytoplasm generates a slowly evolving stress field (Lau et al., 2003) that can cause both longitudinal compression and transverse shear stresses, depending on the details of the local stress field. Indeed, microtubules also display bending on longer length scales that appears to result from this complex stress field (Figs. 1 and 2). However, it is highly unlikely that the observed multiple short-wavelength bending could be caused by effects of transverse stresses alone. Instead, our results suggest that this ubiquitous highly curved form of microtubule deformation reflects the generic nature of reinforced microtubule compression in living cytoplasm.
Our results suggest that microtubules can be used to probe the local mechanical environment within cells. Although we have focused on the cytoskeleton of interphase cells, the mitotic spindle is another important microtubule-based structure. This may provide additional insight into the poorly understood mechanical behavior of mitotic spindles (Pickett-Heaps et al., 1984; Maniotis et al., 1997; Kapoor and Mitchison, 2001; Scholey et al., 2001). Microtubules within spindles have been observed to buckle at somewhat longer wavelengths under natural conditions (Aist and Bayles, 1991), or after mechanical or pharmacological perturbations (Pickett-Heaps et al., 1997; Mitchison et al., 2005), which suggests that spindle microtubules also experience compressive forces. This long-wavelength buckling may reflect an increased effective stiffness of microtubules caused by reinforcement by intermicrotubule bundling connections within the complex structure of the spindle. However, in the absence of bundling, these results suggest that the elasticity of any surrounding matrix cannot be very large. Another cell in which longer-wavelength buckling is observed is the fission yeast, where nuclear positioning is thought to occur by compressive loading of microtubules (Tran et al., 2001). This again suggests that the elasticity of any surrounding network must be considerably less than that of the interphase animal cells we studied. Thus, in these particular microtubule arrays, structural reinforcement may be either unnecessary, or mediated by other mechanisms, such as microtubule bundling.
An important implication of this work is the demonstration that cytoplasmic microtubules are effectively stiffened when embedded in even a relatively soft (elastic modulus 1 kPa) cytoskeletal network; e.g., a reinforced 20-µm-long cytoplasmic microtubule can withstand a compressive force (>100 pN) >100 times larger than a free microtubule before buckling. Consequently, individual microtubules can withstand much larger compressive forces in a living cell than previously considered possible (Fig. 7
). Moreover, as demonstrated by our results with cytochalasin-treated cells, the lateral reinforcement is robust; even disruption of the surrounding actin network only slightly increases the buckling wavelength, with a corresponding decrease in the critical force by a factor of 2. This is likely caused by the presence of other sources of elasticity, such as intermediate filaments, which have been previously shown to both connect laterally to microtubules (Bloom et al., 1985), and to contribute to whole cytoskeletal mechanics (Wang et al., 1993). As illustrated with the macroscopic model, this reinforcement is a robust phenomenon that is insensitive to the specific molecular details; the only requirement is that the surrounding matrix must be elastic.
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1 mg/ml rhodamine-labeled tubulin (Cytoskeleton, Inc.) using a Femtojet microinjection system (Eppendorf) and allowed to incorporate fluorescent tubulin for at least 2 h. For some experiments, cells were incubated with 2 µM cytochalasin D (Sigma-Aldrich) for 30 min before imaging. Microtubules were buckled using Femtotip needles controlled with a micromanipulator (both Eppendorf).
Cardiac myocytes
Cardiac myocytes were isolated from 2-d-old Sprague-Dawley rats (Charles River Laboratories). In brief, whole hearts were removed from killed animals and subsequently homogenized and washed in HBSS, followed by trypsin and collagenase digestion for 14 h at 4°C with agitation. Once trypsinized, cells were resuspended in M199 culture medium supplemented with 10% heat-inactivated FBS, 10 mmol/liter Hepes, 3.5 g/liter glucose, 2 mmol/liter L-glutamine, 2 mg/liter vitamin B-12, and 50 U/ml penicillin at 37°C and agitated. Immediately after purification, cells were plated on glass-bottomed Petri dishes (MatTek Corp.). For some studies, the dishes were spin coated with polydimethylsiloxane silicone elastomer (Sylgard; Dow Corning) that was treated with 25 µg/ml human fibronectin in ddH20 for 1 h. Immediately before incubation with the protein solution, culture substrates were treated in a UVO cleaner for 8 min (Jelight Company, Inc.). Cells were kept in culture at 37°C with a 5% CO2 atmosphere. Medium was changed 24 h after plating to remove unattached and dead cells, followed by changes with supplemented M199 medium containing 2% FBS 48 and 96 h after plating. Cells were transfected with EGFP-tubulin or microinjected with fluorescent tubulin, as with nonmuscle cells. For some studies, beating was stimulated with 1 µmol/liter epinephrine immediately before imaging.
Microscopy and image analysis
Fluorescent images were acquired on an inverted microscope (DM-IRB; Leica) equipped with an intensified charge-coupled device camera (model C7190-21 EB-CCD; Hamamatsu) and automated image acquisition software (MetaMorph; Universal Imaging Corp.). Images were analyzed using custom-built filament tracking software to extract the contours of single microtubules as a function of time. The shape at each time point was then analyzed using a Fourier decomposition (Gittes et al., 1993).
Macroscopic buckling
The macroscopic rods were either hollow capillary-loading pipette tips (Eppendorf) or solid fishing line (Stren). The rods were placed in a 1-cm cuvette cell that was then filled with liquid gelatin (Sigma-Aldrich) and allowed to gel before applying a compressive force from the top. For the capillary-loading pipette tips, we used tabulated values for the elastic modulus of the polypropylene plastic (E = 1.9 GPa) and measured the outer (ro 116 µm) and inner radius (ri 93 µm) of the rod. For the fishing line, we determined the elastic modulus using a tensile test (E = 0.9 GPa) and measured the radius. We tested samples in the range r 103–227 µm. We estimated the bending rigidity (Landau and Lifshitz, 1986) using the following equation:
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Online supplemental material
Video 1 shows EGFP microtubules buckling into sinusoidal shapes when they polymerize into the edge of a Cos7 cell. Video 2 shows an EGFP microtubule buckling into a sinusoidal shape as it polymerizes against the edge of a capillary endothelial cell. Video 3 shows an initially straight EGFP microtubule at the edge of a Cos7 cell that is compressively loaded with a glass microneedle and undergoes buckling. Video 4 shows another example of an EGFP microtubule compressively loaded with a microneedle at the edge of a Cos7 cell. Video 5 shows a rhodamine-labeled microtubule repeatedly buckling into a short-wavelength shape with each cycle of contraction in a beating cardiac myocyte. Video 6 shows an example of a rhodamine-labeled microtubule in a beating cardiac myocyte that repeatedly buckles in the same spot, even while adjacent microtubules remain straight. Video 7 shows two microtubules within an EGFP-tubulin–transfected cardiac myocyte that can be seen to cyclically buckle in a coordinated manner with each contractile beat. Video 8 shows a microtubule locally buckling within the cytoplasm of an EGFP-tubulin–transfected capillary cell, whereas adjacent regions of the same microtubule remain straight. Online supplemental materials are available at http://www.jcb.org/cgi/content/full/jcb.200601060/DC1.
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This work was supported by the National Science Foundation (NSF) through the Harvard Materials Research Science and Engineering Center (DMR-0213805). C.P. Brangwynne was supported by the NSF through the Integrated Training Program in Biomechanics (DGE-0221682). D.A. Weitz was supported by the NSF (DMR-0243715). D.E. Ingber and S. Kumar were supported by grants from the National Aeronautics and Space Administration (NN-A04CC96G) and a National Institutes of Health postdoctoral fellowship (F32-NS048669). F.C. MacKintosh was supported by the Foundation for Fundamental Research on Matter. J. Talbot was supported by the Research Experience for Undergraduates/Research Experience for Teachers in Materials Research programs from the NSF (DMR-0353937).
Submitted: 12 January 2006
Accepted: 1 May 2006
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