Polymerization Force Driven Buckling of Microtubule Bundles Determines the Wavelength of
Patterns Formed in Tubulin Solutions
Yongxing Guo, Yifeng Liu, Jay X. Tang, and James M. Valles, Jr.
Physics Department, Brown University, Providence, RI 02912
(Dated: June 8, 2021)
We present a model for the spontaneous formation of a striated pattern in polymerizing microtubule solutions.
It describes the buckling of a single microtubule (MT) bundle within an elastic network formed by other similarly
aligned and buckling bundles and unaligned MTs. Phase contrast and polarization microscopy studies of the
temporal evolution of the pattern imply that the polymerization of MTs within the bundles creates the driving
compressional force. Using the measured rate of buckling, the established MT force-velocity curve and the
pattern wavelength, we obtain reasonable estimates for the MT bundle bending rigidity and the elastic constant
of the network. The analysis implies that the bundles buckle as solid rods.
Microtubules (MTs), a major component of the eukary-
otic cytoskeleton [1], can form various structures and pat-
terns. For example, in vivo, MTs organize into the spindles
and asters essential for mitosis [2] and the parallel arrays and
stripes necessary for directing early processes in embryogene-
sis [3, 4]. Many in vitro studies of MT organization have been
performed in order to elucidate the mechanisms underlying
the formation of these structures [5, 6, 7, 8]. Of particular
relevance here are the striped birefringent patterns [Fig. 1(a)],
which spontaneously form from polymerizing a purified tubu-
lin solution without motor proteins or MT associated proteins.
Hitt et al. attributed these patterns to the formation of ne-
matic liquid crystalline domains [5]. Tabony et al., on the
other hand, proposed that a reaction-diffusion based mecha-
nism drives the formation of MT stripes [6]. Our recent in-
vestigations imply a starkly different scenario in which the
local MT alignment into wave-like structures occurs through
a collective process of MT bundling and buckling [9]. MTs
that are aligned by a static magnetic field [10, 11] or convec-
tive flow [9] during the initial stage of polymerization sponta-
neously form bundles in tubulin solutions with concentrations
of a few mg/ml. These bundles elongate and buckle in co-
ordination with neighboring bundles into a wave-like shape.
The nesting of the buckled bundles can quantitatively account
for the MT density and orientation variations leading to the
striped birefringent pattern [9]. We proposed that a compres-
sional force is generated by MT polymerization occurring uni-
formly along the bundle contours. The buckling wavelength
is controlled by the bending rigidity of the bundles and the
elasticity of the background network of MTs. This interesting
initial assessment calls for further investigation of the micro-
scopic picture of the bundle elongation, the MT buckling force
and the buckling mode selection mechanism.
Here we present a mechanical model for the process in
addition to new experimental data on the time evolution of
the bundle contour length and solution birefringence that pro-
vide direct support for the validity of the model. The model
considers the instability of a single MT bundle under a com-
pressional force, embedded in an elastic network formed by
both bundled and dispersed MTs. Time lapse phase contrast
and quantitative polarized light microscopy imply that MT
polymerization within the bundles provides the compressional
FIG. 1: Image of a MT birefringent pattern and a sketch of the me-
chanical buckling model. (a) Striped birefringent pattern [23]. The
image was taken between crossed polarizers with the polarization di-
rections at 45◦ with respect to the x axis. (b) Schematic drawing
of buckled MT bundles surrounded by an elastic MT network (gray
background). The white dashed line depicts the central bundle before
buckling. The white sinusoidal curve depicts the elongated bundle
after buckling and the gray sinusoidal curves represent the neighbor-
ing MT bundles. ξ (x) is the transverse displacement for the central
bundle.
force. Specifically, they reveal that the bundles elongate uni-
formly along their contours while maintaining constant radii
consistent with growth through the elongation of the individ-
ual MTs comprising them. We make predictions for the char-
acteristic buckling wavelength using the bundle bending rigid-
ity and the critical buckling force estimated from the measured
MT force-velocity curve. The measured wavelength of about
600 µm implies that the bundles bend as solid rods.
We envision initially the microtubule solution to consist of
an array of straight and parallel bundles aligned along the x
axis and embedded in a network composed of dispersed MTs
as in Fig. 1(b). All of the bundles experience a similar com-
pressional force that grows to a critical value, causing them to
buckle. To describe the buckling, we consider a single bun-
dle in the center of the sample and characterize its interaction
with the network using a single elastic constant, α , such that
αξ (x) is the elastic restoring force exerted by the network on
the bundle per unit length. Treating the bundle as a rod with a
bending rigidity, K, under a uniform compressional force, F ,
the force balance in the y direction at the onset of the buckling
is given by [12, 13, 14]
ξ (x)
∂ξ (x)
]+αξ (x) = 0 (1)
Performing a standard normal mode stability analysis of
Eq. (1) using ξ (x) ∝ eikx yields a relation between the angu-
lar wavenumber, k, and the compressional force, F = α/k2 +
Kk2, which suggests a minimum or critical compressional
force Fc for a buckling solution. The critical compressional
force is Fc = 2
Kα , and the characteristic wavelength is
λc = 2π/k = π
8K/Fc = 2π
K/α (2)
The resultant characteristic wavelength [Eq. (2)] agrees
with the prediction for λc based on energy minimization [9].
This model predicts buckling in a higher mode than the fun-
damental one as in classic Euler buckling.
In agreement with experiments, this model implies that the
orientation of MT bundles in a striped sample varies continu-
ously in space [9]. In contrast, previous models had suggested
that discrete and alternate angular orientations of the MTs
formed the striated patterns [15]. In addition, the weak depen-
dence of the buckling wavelength on the mechanical parame-
ters is consistent with the small variations in both the observed
buckling wavelength across a single macroscopic sample and
the patterns formed under different conditions (for example,
samples with different tubulin concentrations and samples in
containers with different size.).
Time lapse phase contrast microscopy reveals that the MT
bundles elongate uniformly along their contour during buck-
ling, which is consistent with polymerization occurring uni-
formly along the bundles. The elongation is illustrated in the
phase images Fig. 2(a) and Fig. 2(b), showing a fixed region
taken 12 and 100 minutes after polymerization initiation, re-
spectively. The three white curves in each image are computer
generated traces of bundle contours that extend between se-
lected fiducial marks. The fiducial marks are visible as dark
spots in the images. To generate the white curves, we pre-
sumed that the bundles followed the striations in the images
and traced the stripes between the fiducial marks, whose po-
sitions were tracked using the MetaMorph imaging software
(Universal Imaging, West Chester, PA). Specifically, we de-
termined the local striation orientation at each pixel by calcu-
lating a Fast Fourier Transform (FFT) of the area around the
pixel, shown, for example, in Fig. 2(c). The FFT appeared
as an elongated spot oriented perpendicular to the striation di-
rection [Fig. 2(d)]. The radially integrated FFT intensity has
a peak at a specific azimuthal angle [Fig. 2(d)] that is perpen-
dicular to the striation orientation. In this way, the lengths
of three segments along a MT bundle were recorded every 30
seconds and plotted in Fig. 2(f), (g) and (h). The normal-
ized lengths of these three segments grew at nearly the same,
constant rate, shown in Fig. 2(i), implying that the MT bun-
dles elongate uniformly along their contour instead of growing
FIG. 2: Illustration and measurements of the uniform elongation of
MT bundles [23]. (a,b) Phase contrast images of a sample region,
show progression of the pattern over one hour. MT bundles are dis-
cerned by the thin striations. The image contrast is enhanced for bet-
ter visualization. Segments 1 through 3 are adjacent pieces of a con-
tour followed by bundles. The segment ends are defined by fiducial
marks. (c) Magnified view of the region denoted by the white box in
(b), showing an encircled fiducial mark. (d) Fast Fourier Transform
(FFT) of (c). (e) The radially averaged FFT intensity plotted versus
the azimuthal angle θ and fit using a Gaussian function. The local
bundle orientation is orthogonal to the angle at which the Gaussian
fit peaks. (f-h) Length of segments 1 (f), 2 (g) and 3 (h) as a func-
tion of time. (i) Lengths of the three segments as functions of time,
normalized to their lengths at 46 minutes.
solely at their ends. It further suggests that the bundles elon-
gate through polymerization of their constituent MTs, which
start and end at random places along a bundle. The uni-
form growth of all MTs within the bundle justifies a uniform
elongation rate and the use of a uniform compressional force
throughout the bundle in the mechanical model, giving rise to
the sinusoidal ξ (x) over the entire pattern.
Additional quantitative information about the microscopic
picture of the buckling is gained through time-lapse bire-
fringence measurements. PolScope (CRI, Cambridge, MA)
images, taken sequentially at a fixed sample region [16],
yielded the time evolution at each pixel of both the retar-
dance (∆ ≡ bire f ringence× h, where h is the sample thick-
ness) and the slow axis direction (ϕ(x), orientation of MT
bundles) [17]. Two representative PolScope images of a sin-
gle region taken at different stages of self-organization are
shown in Fig. 3(a) and 3(b). The slow axis variation, ϕ(x),
along the white lines in Fig. 3(a) and 3(b) can be fit to
ϕ(x) = atan[A 2π
cos( 2π
(x + x0))], indicating that the bundle
follows ξ (x) = Asin( 2π
(x+ x0)) with a single wavelength λ ,
buckling amplitude A, and offset x0 [Fig. 3(c)]. The resultant
wavelength, λ ≈ 600 µm, is plotted in Fig. 3(d). The nor-
FIG. 3: Time evolution of a MT pattern obtained by measuring the
retardance and slow axis of the sample using a PolScope imaging
system [16]. (a,b) Retardance images of a sample region at 12 and
100 min of self-organization, respectively. The gray bar shows the
retardance magnitude scale and the white pins provide the slow axis
orientation. The straight white lines represent the slow axis line scan
position. (c) Slow axis line scan (black) and the fitted slow axis ori-
entation ϕ(x) = atan[A 2π
cos( 2π
(x+x0))] (gray) at 100 min. (d) The
dominant buckling wavelength λ , obtained from the fitted shapes of
the bundle at individual time points. (e) The length evolution of the
fitted bundle contour. L0 = 1544 µm is the initial unbuckled length of
the bundle. The segment before the arrow designates a latent period
prior to the onset of the buckling. (f) The magnitude of the retardance
averaged over the white lines as shown in (a,b) versus the normalized
length L/L0.
malized contour length calculated from the fits, L(t)/L0, grew
nearly linearly with time at a normalized rate of L̇(t)/L0≈ 1 %
per min [Fig. 3(e)]. Simultaneously, the retardance magnitude
averaged over the white line in Fig. 3(a) increased roughly
in proportion to L(t)/L0 [Fig. 3(f)]. Based on the nesting
model we proposed earlier and assuming that neighboring
MT bundles do not coalesce, the average retardance goes as
∆(t) ∼ δ × n(t)L(t)/L0 [9, 17], where n(t) is the number of
MTs in the cross section of a bundle and δ is the retardance of
a single MT. Therefore, the linear relation between ∆(t) and
L(t)/L0 implies that n(t) remains constant throughout buck-
ling. Thus, the elongation of MT bundles occurs through the
polymerization of MTs within the bundles and does not in-
volve the incorporation of new MTs to existing bundles.
With the above observations and model, we can quanti-
tatively characterize the elastic properties of the bundle (K)
and network (α). We begin with the implications of the mea-
sured wavelength λ . In order to predict λ from the mechani-
cal buckling model, we need to estimate K and F [Eq. (2)].
Two limits exist for K. If tight packing (solid model) of
the MTs inside the bundle is assumed, then Ksolid = n2KMT,
where KMT ≈ 3.4× 10−23 N ·m2 is the bending rigidity of
a single MT [18, 19]. If MTs slide freely inside the bun-
dle, then Kslip = nKMT. We employ the measured force-
velocity relation, f (v) = C1 ln[C2/(v + C3)] (C1 = 1.89 pN,
C2 = 1.13 µm/min and C3 = −0.08 µm/min [18]), for a sin-
gle MT and presume F = n f (v), where v is the average
elongation rate of individual MT inside the bundle. Writ-
ing the average length of MTs inside the bundle as lMT,
the elongation rate of a single MT is then approximately
v(lMT) = lMT × L̇(t)/L0. Using the models for K, F and
Eq. (2), we derive predictions of λ for both the solid model,
λsolid = π
8nKMT/ f (v(lMT)), and the slip model, λslip =
8KMT/ f (v(lMT)). Each depends on lMT and n. Using
n = 280 [9], we plot the wavelength over a reasonable range
of individual MT lengths ([1]) in Fig. 4. The solid model for
K appears much more reasonable than the slip model. The
fact that K depends quadratically on n in our system suggests
that MTs are fully coupled (acting like a solid material) inside
the bundle, similar to the behavior of F-actin bundles held to-
gether through depletion forces [20]. The bundling of initially
aligned MTs can be attributed to the depletion force induced
by unpolymerized tubulin dimers, oligomers and even short
MTs [9].
The conclusion that the bundles bend as solid rods appar-
ently conflicts with the picture of elongation, that involves the
growth and relative sliding of individual MTs within the bun-
dles. We speculate that the explanation involves two distinct
time scales: the time for a MT to come to mechanical equi-
librium with its neighbors following the insertion of a tubulin
dimer to its end, τmech, and the average interval between in-
sertions, τdimer. In the limit τmech < τdimer, strong coupling be-
tween the MTs in the bundle can occur leading to the solid rod
result. The opposite limit intuitively leads to weak coupling
between the MTs within a bundle. We estimate τdimer ≈ 0.1s
from our data, which seems quite long compared to the times
characterizing the relative motion of neighboring MTs on the
molecular length scales relevant to τmech. The exact molecu-
lar picture, which goes beyond the scope of our model, needs
further study.
Using the solid model for K, we can calculate the remain-
ing model parameter, α , from Eq. (2): α = Kslip(2π/λexpt)4 ≈
0.032Pa. This value is remarkably small compared to that
estimated for a single MT buckling inside a cell (α∗ ≈
2700Pa [12]). We identify two contributors to the difference
between α and α∗. In general, α ∼ G, where G is the elas-
tic shear modulus of the surrounding network. G ∼ 1Pa in
our system [21], while G∗ ∼ 1000Pa for the surrounding cy-
toskeleton network inside the cell [22]. The other contributor
is the coordination of the buckling of the MT bundles, which
reduces the distortion of the surrounding network, and thus
weakens the effective restoring force and α (analysis in prepa-
FIG. 4: Theoretically calculated wavelength (λ ) as a function of the
average length of MTs (lMT) inside the bundle at the onset of buck-
ling. In the solid model λsolid = π
8nKMT/ f (v(lMT)), and in the
slip model λslip = π
8KMT/ f (v(lMT))). λexpt is the experimentally
observed buckling wavelength (dashed line).
ration).
In summary, using microscopic studies of the temporal evo-
lution of the striated MT patterns, we show that the polymer-
ization of MTs within the bundles causes uniform elongation.
This in turn creates the driving compressional force which
ultimately causes the MT bundles to buckle. It is this coor-
dinated buckling that produces the striped birefringent pat-
tern. The proposed mechanical buckling model adequately
describes the buckling process. It predicts a critical buckling
force and a characteristic wavelength, which depend on the
elasticity of the surrounding network and the bending rigid-
ity of the MT bundles. Combing the bending rigidity of MT
bundles and the established MT force-velocity curve with the
mechanical model, we obtain a reasonable estimate for the
elastic constant of the network and find that MTs inside the
bundle are fully coupled.
We thank Allan Bower for help in understanding the elas-
tic constant α and thank L. Mahadevan and Thomas R.
Powers for valuable discussions. This work was supported
by NASA (NNA04CC57G, NAG3-2882) and NSF (DMR
0405156, DMR 0605797).
[1] A. Desai and T. J. Mitchison, Annu. Rev. Cell Dev. Biol. 13, 83
(1997).
[2] D. Bray, Cell Movement: From Molecules to Motility (Garland,
New York, 2001).
[3] R. P. Elinson and B. Rowning, Dev. Biol. 128, 185 (1988).
[4] G. Callaini, Development 107, 35 (1989).
[5] A. L. Hitt, A. R. Cross, and R. C. Williams, J. Biol. Chem. 265,
1639 (1990).
[6] J. Tabony, Science 264, 245 (1994).
[7] F. J. Nédélec, T. Surrey, A. C. Maggs, and S. Leibler, Nature
389, 305 (1997).
[8] C. E. Walczak, I. Vernos, T. J. Mitchison, E. Karsenti, and
R. Heald, Current Biology 8, 903 (1998).
[9] Y. Liu, Y. Guo, J. M. Valles, and J. Tang, Proc. Natl. Acad. Sci.
U.S.A. 103, 10654 (2006).
[10] W. Bras, G. P. Diakun, J. F. Dı́az, G. Maret, H. Kramer, J. Bor-
das, and F. J. Medrano, Biophys. J. 74 (1998).
[11] N. Glade and J. Tabony, Biophys. Chem. 115, 29 (2005).
[12] C. P. Brangwynne, F. C. MacKintosh, S. Kumar, N. A. Geisse,
J. Talbot, L. Mahadevan, K. K. Parker, D. E. Ingber, and D. A.
Weitz, J. Cell Biol. 173, 733 (2006).
[13] J. R. Gladden, N. Z. Handzy, A. Belmonte, and E. Villermaux,
Phys. Rev. Lett. 94, 035503 (2005).
[14] L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Oxford,
New York, 1986), 3rd ed.
[15] J. Tabony and N. Glade, Langmuir 18, 7196 (2002); C. Papa-
seit, L. Vuillard, and J. Tabony, Biophys. Chem. 79, 33 (1999);
J. Tuszynski, M. V. Sataric, and et al., Physics Letters A 340,
175 (2005).
[16] The sample was polymerized from a 5 mg/ml tubulin solution
(2 mM GTP, 3.5% in molar ratio of Oregon Green conjugated
taxol to tubulin dimers, 100 mM pipes, 1 mM EGTA, 2 mM
MgSO4, PH 6.9) in a 40×8×0.4mm3 glass cuvette which was
exposed to 9 T vertical static magnetic field for 5 minutes at
37◦C (the magnetic field direction is along the long axis of the
cuvette). The cuvette was then laid flat on the microscope stage
and a coherently buckled area was chosen for observation and
measurement at 30◦C.
[17] R. Oldenbourg, E. D. Salmon, and P. T. Tran, Biophys. J. 74,
645 (1998).
[18] M. Dogterom and B. Yurke, Science 278, 856 (1997).
[19] J. A. Tuszyński, T. Luchko, S. Portet, and J. M. Dixon, Eur.
Phys. J. E 17, 29 (2005).
[20] M. M. A. E. Claessens, M. Bathe, E. Frey, and A. R. Bausch,
Nature Materials 5, 748 (2006).
[21] M. Sato, W. H. Schwartz, S. C. Selden, and T. D. Pollard, J.
Cell Biol. 106, 1205 (1988).
[22] R. E. Mahaffy, C. K. Shih, F. C. MacKintosh, and J. Käs, Phys.
Rev. Lett. 85, 880 (2000).
[23] The sample was polymerized from 8 mg/ml tubulin solution
(same buffer condition as in [16]) in a 40×10×1mm3 quartz
cuvette and was subjected to convective flow (induced by asym-
metrical thermal contacts, with the left and bottom surfaces in
contact with a 37◦C waterbath-warmed aluminum holder and
other sides exposed to 30◦C ambient) for the first 9 minutes.
The cuvette was then laid flat on the microscope stage for ob-
servation and measurement at 30◦C.
	References

  We present a model for the spontaneous formation of a striated pattern in
polymerizing microtubule solutions. It describes the buckling of a single
microtubule (MT) bundle within an elastic network formed by other similarly
aligned and buckling bundles and unaligned MTs. Phase contrast and polarization
microscopy studies of the temporal evolution of the pattern imply that the
polymerization of MTs within the bundles creates the driving compressional
force. Using the measured rate of buckling, the established MT force-velocity
curve and the pattern wavelength, we obtain reasonable estimates for the MT
bundle bending rigidity and the elastic constant of the network. The analysis
implies that the bundles buckle as solid rods.

Polymerization Force Driven Buckling of Microtubule Bundles Determines the Wavelength of
Patterns Formed in Tubulin Solutions
Yongxing Guo, Yifeng Liu, Jay X. Tang, and James M. Valles, Jr.
Physics Department, Brown University, Providence, RI 02912
(Dated: June 8, 2021)
We present a model for the spontaneous formation of a striated pattern in polymerizing microtubule solutions.
It describes the buckling of a single microtubule (MT) bundle within an elastic network formed by other similarly
aligned and buckling bundles and unaligned MTs. Phase contrast and polarization microscopy studies of the
temporal evolution of the pattern imply that the polymerization of MTs within the bundles creates the driving
compressional force. Using the measured rate of buckling, the established MT force-velocity curve and the
pattern wavelength, we obtain reasonable estimates for the MT bundle bending rigidity and the elastic constant
of the network. The analysis implies that the bundles buckle as solid rods.
Microtubules (MTs), a major component of the eukary-
otic cytoskeleton [1], can form various structures and pat-
terns. For example, in vivo, MTs organize into the spindles
and asters essential for mitosis [2] and the parallel arrays and
stripes necessary for directing early processes in embryogene-
sis [3, 4]. Many in vitro studies of MT organization have been
performed in order to elucidate the mechanisms underlying
the formation of these structures [5, 6, 7, 8]. Of particular
relevance here are the striped birefringent patterns [Fig. 1(a)],
which spontaneously form from polymerizing a purified tubu-
lin solution without motor proteins or MT associated proteins.
Hitt et al. attributed these patterns to the formation of ne-
matic liquid crystalline domains [5]. Tabony et al., on the
other hand, proposed that a reaction-diffusion based mecha-
nism drives the formation of MT stripes [6]. Our recent in-
vestigations imply a starkly different scenario in which the
local MT alignment into wave-like structures occurs through
a collective process of MT bundling and buckling [9]. MTs
that are aligned by a static magnetic field [10, 11] or convec-
tive flow [9] during the initial stage of polymerization sponta-
neously form bundles in tubulin solutions with concentrations
of a few mg/ml. These bundles elongate and buckle in co-
ordination with neighboring bundles into a wave-like shape.
The nesting of the buckled bundles can quantitatively account
for the MT density and orientation variations leading to the
striped birefringent pattern [9]. We proposed that a compres-
sional force is generated by MT polymerization occurring uni-
formly along the bundle contours. The buckling wavelength
is controlled by the bending rigidity of the bundles and the
elasticity of the background network of MTs. This interesting
initial assessment calls for further investigation of the micro-
scopic picture of the bundle elongation, the MT buckling force
and the buckling mode selection mechanism.
Here we present a mechanical model for the process in
addition to new experimental data on the time evolution of
the bundle contour length and solution birefringence that pro-
vide direct support for the validity of the model. The model
considers the instability of a single MT bundle under a com-
pressional force, embedded in an elastic network formed by
both bundled and dispersed MTs. Time lapse phase contrast
and quantitative polarized light microscopy imply that MT
polymerization within the bundles provides the compressional
FIG. 1: Image of a MT birefringent pattern and a sketch of the me-
chanical buckling model. (a) Striped birefringent pattern [23]. The
image was taken between crossed polarizers with the polarization di-
rections at 45◦ with respect to the x axis. (b) Schematic drawing
of buckled MT bundles surrounded by an elastic MT network (gray
background). The white dashed line depicts the central bundle before
buckling. The white sinusoidal curve depicts the elongated bundle
after buckling and the gray sinusoidal curves represent the neighbor-
ing MT bundles. ξ (x) is the transverse displacement for the central
bundle.
force. Specifically, they reveal that the bundles elongate uni-
formly along their contours while maintaining constant radii
consistent with growth through the elongation of the individ-
ual MTs comprising them. We make predictions for the char-
acteristic buckling wavelength using the bundle bending rigid-
ity and the critical buckling force estimated from the measured
MT force-velocity curve. The measured wavelength of about
600 µm implies that the bundles bend as solid rods.
We envision initially the microtubule solution to consist of
an array of straight and parallel bundles aligned along the x
axis and embedded in a network composed of dispersed MTs
as in Fig. 1(b). All of the bundles experience a similar com-
pressional force that grows to a critical value, causing them to
buckle. To describe the buckling, we consider a single bun-
dle in the center of the sample and characterize its interaction
with the network using a single elastic constant, α , such that
αξ (x) is the elastic restoring force exerted by the network on
the bundle per unit length. Treating the bundle as a rod with a
bending rigidity, K, under a uniform compressional force, F ,
the force balance in the y direction at the onset of the buckling
is given by [12, 13, 14]
ξ (x)
∂ξ (x)
]+αξ (x) = 0 (1)
Performing a standard normal mode stability analysis of
Eq. (1) using ξ (x) ∝ eikx yields a relation between the angu-
lar wavenumber, k, and the compressional force, F = α/k2 +
Kk2, which suggests a minimum or critical compressional
force Fc for a buckling solution. The critical compressional
force is Fc = 2
Kα , and the characteristic wavelength is
λc = 2π/k = π
8K/Fc = 2π
K/α (2)
The resultant characteristic wavelength [Eq. (2)] agrees
with the prediction for λc based on energy minimization [9].
This model predicts buckling in a higher mode than the fun-
damental one as in classic Euler buckling.
In agreement with experiments, this model implies that the
orientation of MT bundles in a striped sample varies continu-
ously in space [9]. In contrast, previous models had suggested
that discrete and alternate angular orientations of the MTs
formed the striated patterns [15]. In addition, the weak depen-
dence of the buckling wavelength on the mechanical parame-
ters is consistent with the small variations in both the observed
buckling wavelength across a single macroscopic sample and
the patterns formed under different conditions (for example,
samples with different tubulin concentrations and samples in
containers with different size.).
Time lapse phase contrast microscopy reveals that the MT
bundles elongate uniformly along their contour during buck-
ling, which is consistent with polymerization occurring uni-
formly along the bundles. The elongation is illustrated in the
phase images Fig. 2(a) and Fig. 2(b), showing a fixed region
taken 12 and 100 minutes after polymerization initiation, re-
spectively. The three white curves in each image are computer
generated traces of bundle contours that extend between se-
lected fiducial marks. The fiducial marks are visible as dark
spots in the images. To generate the white curves, we pre-
sumed that the bundles followed the striations in the images
and traced the stripes between the fiducial marks, whose po-
sitions were tracked using the MetaMorph imaging software
(Universal Imaging, West Chester, PA). Specifically, we de-
termined the local striation orientation at each pixel by calcu-
lating a Fast Fourier Transform (FFT) of the area around the
pixel, shown, for example, in Fig. 2(c). The FFT appeared
as an elongated spot oriented perpendicular to the striation di-
rection [Fig. 2(d)]. The radially integrated FFT intensity has
a peak at a specific azimuthal angle [Fig. 2(d)] that is perpen-
dicular to the striation orientation. In this way, the lengths
of three segments along a MT bundle were recorded every 30
seconds and plotted in Fig. 2(f), (g) and (h). The normal-
ized lengths of these three segments grew at nearly the same,
constant rate, shown in Fig. 2(i), implying that the MT bun-
dles elongate uniformly along their contour instead of growing
FIG. 2: Illustration and measurements of the uniform elongation of
MT bundles [23]. (a,b) Phase contrast images of a sample region,
show progression of the pattern over one hour. MT bundles are dis-
cerned by the thin striations. The image contrast is enhanced for bet-
ter visualization. Segments 1 through 3 are adjacent pieces of a con-
tour followed by bundles. The segment ends are defined by fiducial
marks. (c) Magnified view of the region denoted by the white box in
(b), showing an encircled fiducial mark. (d) Fast Fourier Transform
(FFT) of (c). (e) The radially averaged FFT intensity plotted versus
the azimuthal angle θ and fit using a Gaussian function. The local
bundle orientation is orthogonal to the angle at which the Gaussian
fit peaks. (f-h) Length of segments 1 (f), 2 (g) and 3 (h) as a func-
tion of time. (i) Lengths of the three segments as functions of time,
normalized to their lengths at 46 minutes.
solely at their ends. It further suggests that the bundles elon-
gate through polymerization of their constituent MTs, which
start and end at random places along a bundle. The uni-
form growth of all MTs within the bundle justifies a uniform
elongation rate and the use of a uniform compressional force
throughout the bundle in the mechanical model, giving rise to
the sinusoidal ξ (x) over the entire pattern.
Additional quantitative information about the microscopic
picture of the buckling is gained through time-lapse bire-
fringence measurements. PolScope (CRI, Cambridge, MA)
images, taken sequentially at a fixed sample region [16],
yielded the time evolution at each pixel of both the retar-
dance (∆ ≡ bire f ringence× h, where h is the sample thick-
ness) and the slow axis direction (ϕ(x), orientation of MT
bundles) [17]. Two representative PolScope images of a sin-
gle region taken at different stages of self-organization are
shown in Fig. 3(a) and 3(b). The slow axis variation, ϕ(x),
along the white lines in Fig. 3(a) and 3(b) can be fit to
ϕ(x) = atan[A 2π
cos( 2π
(x + x0))], indicating that the bundle
follows ξ (x) = Asin( 2π
(x+ x0)) with a single wavelength λ ,
buckling amplitude A, and offset x0 [Fig. 3(c)]. The resultant
wavelength, λ ≈ 600 µm, is plotted in Fig. 3(d). The nor-
FIG. 3: Time evolution of a MT pattern obtained by measuring the
retardance and slow axis of the sample using a PolScope imaging
system [16]. (a,b) Retardance images of a sample region at 12 and
100 min of self-organization, respectively. The gray bar shows the
retardance magnitude scale and the white pins provide the slow axis
orientation. The straight white lines represent the slow axis line scan
position. (c) Slow axis line scan (black) and the fitted slow axis ori-
entation ϕ(x) = atan[A 2π
cos( 2π
(x+x0))] (gray) at 100 min. (d) The
dominant buckling wavelength λ , obtained from the fitted shapes of
the bundle at individual time points. (e) The length evolution of the
fitted bundle contour. L0 = 1544 µm is the initial unbuckled length of
the bundle. The segment before the arrow designates a latent period
prior to the onset of the buckling. (f) The magnitude of the retardance
averaged over the white lines as shown in (a,b) versus the normalized
length L/L0.
malized contour length calculated from the fits, L(t)/L0, grew
nearly linearly with time at a normalized rate of L̇(t)/L0≈ 1 %
per min [Fig. 3(e)]. Simultaneously, the retardance magnitude
averaged over the white line in Fig. 3(a) increased roughly
in proportion to L(t)/L0 [Fig. 3(f)]. Based on the nesting
model we proposed earlier and assuming that neighboring
MT bundles do not coalesce, the average retardance goes as
∆(t) ∼ δ × n(t)L(t)/L0 [9, 17], where n(t) is the number of
MTs in the cross section of a bundle and δ is the retardance of
a single MT. Therefore, the linear relation between ∆(t) and
L(t)/L0 implies that n(t) remains constant throughout buck-
ling. Thus, the elongation of MT bundles occurs through the
polymerization of MTs within the bundles and does not in-
volve the incorporation of new MTs to existing bundles.
With the above observations and model, we can quanti-
tatively characterize the elastic properties of the bundle (K)
and network (α). We begin with the implications of the mea-
sured wavelength λ . In order to predict λ from the mechani-
cal buckling model, we need to estimate K and F [Eq. (2)].
Two limits exist for K. If tight packing (solid model) of
the MTs inside the bundle is assumed, then Ksolid = n2KMT,
where KMT ≈ 3.4× 10−23 N ·m2 is the bending rigidity of
a single MT [18, 19]. If MTs slide freely inside the bun-
dle, then Kslip = nKMT. We employ the measured force-
velocity relation, f (v) = C1 ln[C2/(v + C3)] (C1 = 1.89 pN,
C2 = 1.13 µm/min and C3 = −0.08 µm/min [18]), for a sin-
gle MT and presume F = n f (v), where v is the average
elongation rate of individual MT inside the bundle. Writ-
ing the average length of MTs inside the bundle as lMT,
the elongation rate of a single MT is then approximately
v(lMT) = lMT × L̇(t)/L0. Using the models for K, F and
Eq. (2), we derive predictions of λ for both the solid model,
λsolid = π
8nKMT/ f (v(lMT)), and the slip model, λslip =
8KMT/ f (v(lMT)). Each depends on lMT and n. Using
n = 280 [9], we plot the wavelength over a reasonable range
of individual MT lengths ([1]) in Fig. 4. The solid model for
K appears much more reasonable than the slip model. The
fact that K depends quadratically on n in our system suggests
that MTs are fully coupled (acting like a solid material) inside
the bundle, similar to the behavior of F-actin bundles held to-
gether through depletion forces [20]. The bundling of initially
aligned MTs can be attributed to the depletion force induced
by unpolymerized tubulin dimers, oligomers and even short
MTs [9].
The conclusion that the bundles bend as solid rods appar-
ently conflicts with the picture of elongation, that involves the
growth and relative sliding of individual MTs within the bun-
dles. We speculate that the explanation involves two distinct
time scales: the time for a MT to come to mechanical equi-
librium with its neighbors following the insertion of a tubulin
dimer to its end, τmech, and the average interval between in-
sertions, τdimer. In the limit τmech < τdimer, strong coupling be-
tween the MTs in the bundle can occur leading to the solid rod
result. The opposite limit intuitively leads to weak coupling
between the MTs within a bundle. We estimate τdimer ≈ 0.1s
from our data, which seems quite long compared to the times
characterizing the relative motion of neighboring MTs on the
molecular length scales relevant to τmech. The exact molecu-
lar picture, which goes beyond the scope of our model, needs
further study.
Using the solid model for K, we can calculate the remain-
ing model parameter, α , from Eq. (2): α = Kslip(2π/λexpt)4 ≈
0.032Pa. This value is remarkably small compared to that
estimated for a single MT buckling inside a cell (α∗ ≈
2700Pa [12]). We identify two contributors to the difference
between α and α∗. In general, α ∼ G, where G is the elas-
tic shear modulus of the surrounding network. G ∼ 1Pa in
our system [21], while G∗ ∼ 1000Pa for the surrounding cy-
toskeleton network inside the cell [22]. The other contributor
is the coordination of the buckling of the MT bundles, which
reduces the distortion of the surrounding network, and thus
weakens the effective restoring force and α (analysis in prepa-
FIG. 4: Theoretically calculated wavelength (λ ) as a function of the
average length of MTs (lMT) inside the bundle at the onset of buck-
ling. In the solid model λsolid = π
8nKMT/ f (v(lMT)), and in the
slip model λslip = π
8KMT/ f (v(lMT))). λexpt is the experimentally
observed buckling wavelength (dashed line).
ration).
In summary, using microscopic studies of the temporal evo-
lution of the striated MT patterns, we show that the polymer-
ization of MTs within the bundles causes uniform elongation.
This in turn creates the driving compressional force which
ultimately causes the MT bundles to buckle. It is this coor-
dinated buckling that produces the striped birefringent pat-
tern. The proposed mechanical buckling model adequately
describes the buckling process. It predicts a critical buckling
force and a characteristic wavelength, which depend on the
elasticity of the surrounding network and the bending rigid-
ity of the MT bundles. Combing the bending rigidity of MT
bundles and the established MT force-velocity curve with the
mechanical model, we obtain a reasonable estimate for the
elastic constant of the network and find that MTs inside the
bundle are fully coupled.
We thank Allan Bower for help in understanding the elas-
tic constant α and thank L. Mahadevan and Thomas R.
Powers for valuable discussions. This work was supported
by NASA (NNA04CC57G, NAG3-2882) and NSF (DMR
0405156, DMR 0605797).
[1] A. Desai and T. J. Mitchison, Annu. Rev. Cell Dev. Biol. 13, 83
(1997).
[2] D. Bray, Cell Movement: From Molecules to Motility (Garland,
New York, 2001).
[3] R. P. Elinson and B. Rowning, Dev. Biol. 128, 185 (1988).
[4] G. Callaini, Development 107, 35 (1989).
[5] A. L. Hitt, A. R. Cross, and R. C. Williams, J. Biol. Chem. 265,
1639 (1990).
[6] J. Tabony, Science 264, 245 (1994).
[7] F. J. Nédélec, T. Surrey, A. C. Maggs, and S. Leibler, Nature
389, 305 (1997).
[8] C. E. Walczak, I. Vernos, T. J. Mitchison, E. Karsenti, and
R. Heald, Current Biology 8, 903 (1998).
[9] Y. Liu, Y. Guo, J. M. Valles, and J. Tang, Proc. Natl. Acad. Sci.
U.S.A. 103, 10654 (2006).
[10] W. Bras, G. P. Diakun, J. F. Dı́az, G. Maret, H. Kramer, J. Bor-
das, and F. J. Medrano, Biophys. J. 74 (1998).
[11] N. Glade and J. Tabony, Biophys. Chem. 115, 29 (2005).
[12] C. P. Brangwynne, F. C. MacKintosh, S. Kumar, N. A. Geisse,
J. Talbot, L. Mahadevan, K. K. Parker, D. E. Ingber, and D. A.
Weitz, J. Cell Biol. 173, 733 (2006).
[13] J. R. Gladden, N. Z. Handzy, A. Belmonte, and E. Villermaux,
Phys. Rev. Lett. 94, 035503 (2005).
[14] L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Oxford,
New York, 1986), 3rd ed.
[15] J. Tabony and N. Glade, Langmuir 18, 7196 (2002); C. Papa-
seit, L. Vuillard, and J. Tabony, Biophys. Chem. 79, 33 (1999);
J. Tuszynski, M. V. Sataric, and et al., Physics Letters A 340,
175 (2005).
[16] The sample was polymerized from a 5 mg/ml tubulin solution
(2 mM GTP, 3.5% in molar ratio of Oregon Green conjugated
taxol to tubulin dimers, 100 mM pipes, 1 mM EGTA, 2 mM
MgSO4, PH 6.9) in a 40×8×0.4mm3 glass cuvette which was
exposed to 9 T vertical static magnetic field for 5 minutes at
37◦C (the magnetic field direction is along the long axis of the
cuvette). The cuvette was then laid flat on the microscope stage
and a coherently buckled area was chosen for observation and
measurement at 30◦C.
[17] R. Oldenbourg, E. D. Salmon, and P. T. Tran, Biophys. J. 74,
645 (1998).
[18] M. Dogterom and B. Yurke, Science 278, 856 (1997).
[19] J. A. Tuszyński, T. Luchko, S. Portet, and J. M. Dixon, Eur.
Phys. J. E 17, 29 (2005).
[20] M. M. A. E. Claessens, M. Bathe, E. Frey, and A. R. Bausch,
Nature Materials 5, 748 (2006).
[21] M. Sato, W. H. Schwartz, S. C. Selden, and T. D. Pollard, J.
Cell Biol. 106, 1205 (1988).
[22] R. E. Mahaffy, C. K. Shih, F. C. MacKintosh, and J. Käs, Phys.
Rev. Lett. 85, 880 (2000).
[23] The sample was polymerized from 8 mg/ml tubulin solution
(same buffer condition as in [16]) in a 40×10×1mm3 quartz
cuvette and was subjected to convective flow (induced by asym-
metrical thermal contacts, with the left and bottom surfaces in
contact with a 37◦C waterbath-warmed aluminum holder and
other sides exposed to 30◦C ambient) for the first 9 minutes.
The cuvette was then laid flat on the microscope stage for ob-
servation and measurement at 30◦C.
	References
