1
Department of Biological Sciences, Lehigh University, Bethlehem, PA, USA
2
Cell Biology Department, Duke University School of Medicine, Durham, NC,
USA
3
Biology Department, University of Utah, Salt Lake City, UT, USA
4
Biology Department, University of North Carolina, Chapel Hill, NC, USA
*
Present address: Department of Microbiology, Columbia University, New York,
NY, USA
Author for correspondence (e-mail:
lc07{at}lehigh.edu
)
Accepted May 17, 2001
![]() |
SUMMARY |
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Key words: Microtubules, MAPs, Electron microscopy, Flexural rigidity
![]() |
INTRODUCTION |
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These dynamic turnover mechanisms are probably regulated by microtubule
associated proteins (MAPs), as turnover in vivo can be faster than that
observed with purified tubulin in vitro (reviewed by Desai and Mitchison,
1997). One such regulator is
XMAP215, a protein initially purified from Xenopus egg extracts,
based on its ability to stimulate microtubule plus-end growth (Gard and
Kirschner, 1987
). Subsequent
studies have demonstrated that XMAP215 speeds microtubule plus-end assembly
seven- to tenfold, increases shortening rate threefold, and does not change
catastrophe frequency (Vasquez et al.,
1994
). By contrast, XMAP215
had negligible effects on minus-end assembly (Vasquez et al.,
1994
). Therefore, it is likely
that one function of XMAP215 is to speed microtubule plus-end growth, yet
allow rapid microtubule turnover in cells. This has been demonstrated in egg
extracts where partial depletion of XMAP215 results in approximately twofold
slower growth rate in interphase extracts (Tournebize et al.,
2000
), while 90% depletion
nearly eliminates microtubule assembly (Tournebize et al.,
2000
; Popov et al.,
2001
). XMAP215 also protects
microtubule plus ends from the catastrophe-promoting activity of the Kin I
kinesin, XKCM1 (Tournebize et al.,
2000
). XKCM1 and related
kinesins are thought to disrupt microtubule ends by peeling away
protofilaments and then releasing tubulin dimers by an ATP-dependent mechanism
(Desai et al., 1999
). How
XMAP215 is able to both speed plus-end assembly and protect microtubule ends
from XKCM1 is not understood.
XMAP215 probably functions during spindle assembly, as depletion of this
MAP abolishes spindle assembly in Xenopus egg extracts (Tournebize et
al., 2000; Wilde and Zheng,
1999
). The importance of
XMAP215 function during mitosis is also supported by studies of related
members of this MAP family: genetic mutations in Drosophila
mini-spindles protein (Msps; Cullen et al.,
1999
), C. elegans
Zyg-9 (Matthews et al., 1998
),
S. cerevisiae Stu2p (Wang and Huffacker, 1997) and S. pombe
p93dis1 (Nabeshima et al.,
1995
; Nabeshima et al.,
1998
) resulted in defective
spindle assembly, short microtubules or poor viability. Finally, depletion of
the human homolog, TOGp, from HeLa cell extracts prevented mitotic aster
assembly (Dionne et al., 2000
).
XMAP215 and the related MAP family members probably function in mitosis by
stimulating microtubule plus-end assembly and counteracting the destabilizing
activity of Kin I kinesins (Tournebize et al.,
2000
), but additionally
XMAP215 may serve as a scaffold to localize cyclin B/CDK1 to microtubules
during mitosis, as has been observed for TOGp (Charrasse et al.,
2000
).
Recent sequence analysis suggests that the XMAP215 family members contain a
large number of HEAT repeats (Neuwald and Hirano,
2000). While the primary amino
acid sequence is not well conserved, each HEAT repeat forms a common
structural motif consisting of two
helices that associate together in
an antiparallel orientation (reviewed by Kobe et al.,
1999
). The function of HEAT
repeats is not known, but may contribute significantly to the structure of
XMAP215.
To develop models of how XMAP215 functions as a regulator of microtubule assembly dynamics or as a scaffold, it is necessary to understand its structure. In the results presented here, we have used electron microscopy to examine the structures of XMAP215 and the partial tubulin rings assembled with this MAP. Additionally, we measured the flexural rigidity of microtubules assembled with XMAP215 to determine whether this MAP stiffens the microtubule lattice. Finally, we use the positions of identified HEAT repeats to align repeated sequences within the N-terminal half of XMAP215.
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MATERIALS AND METHODS |
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Tubulin was purified from porcine brains as described previously (Vasquez
et al., 1994). No MAPs were
detected in this preparation after silver staining of over-loaded SDS-PAGE
mini-gels (50 µg/lane; not shown).
Glycerol gradient centrifugation
Approximately 25 µg of purified XMAP215 was applied to a 4.5 ml linear
gradient of 15 to 40% glycerol in 0.2 M ammonium bicarbonate. Standards
consisting of catalase (11.3 S), albumin (4.6 S) and ovalbumin (3.5 S) were
loaded on a separate gradient. Samples were spun at 4°C for 16 hours at
130,000 g in an SW 55 rotor, fractionated and examined by
Coomassie Blue staining of 10% SDS-PAGE gels. The sedimentation coefficient
for XMAP215 was calculated by comparison with the standards by assuming linear
separation across the gradient. The maximum sedimentation value was calculated
(Ohashi and Erickson, 1997)
using 228,000 as the molecular weight of XMAP215 (Tournebize et al.,
2000
; Becker and Gard,
2000
).
Tubulin/XMAP215 partial rings
Tubulin (4 µM) with or without XMAP215 (0.4 µM) was incubated in
0.1xBRB80, 1 mM GTP and 15% glycerol for 1 hour on ice before spraying
onto mica and shadowing (described below). Additional tubulin samples (5
µM) lacking XMAP215 were incubated in 0.1xBRB80, 1 mM GTP and 30%
glycerol for several hours on ice before spraying onto mica.
C-terminal antibody to TOGp/XMAP215
A C-terminal fragment of TOGp (bp 5421-5961) was cloned into pGEX2T-L
(pGEX2T (Amersham) modified to include additional restriction sites; a
generous gift from C. Larroque), transformed into Escherichia coli
strain BL21 and expressed as a GST-fusion protein after induction by IPTG. The
fusion protein was purified on a glutathione-S-transferase column according to
the manufacturer's instructions. This GST-TOGp(amino acids 1807-1987) was used
to immunize rabbits (Pocono Rabbit Farms) and generated antiserum that
recognized a single band at 215 kDa in HeLa lysates (data not shown). The
antiserum was fractionated by ammonium sulfate precipitation (Harlowe and
Lane, 1988) and then further
purified on a 6xHis-TOGp (amino acids 1807-1987) affinity column. The
6xHis-tagged TOGp (amino acids 1807-1987) was generated by cloning TOGp
(bp 5421-5961) into the pQE30 vector (Qiagen) and expression of the fusion
protein in E. coli strain M15. 6xHis-TOGp (amino acids
1807-1987) was purified on a nickel column (Qiagen) and coupled to
CNBr-activated Sepharose (Pharmacia), according to the protocol provided by
the manufacturer. Antibodies were bound to the column in PBS and eluted with
0.1 M glycine, pH 2.5 and each 1 ml fraction was neutralized with 200 µl
Tris-HCl, pH 8.5 (Harlowe and Lane,
1988
). To demonstrate the
specificity of the antibody for XMAP215, purified XMAP215 and a crude
Xenopus egg extract were separated on 3-12.5% SDS PAGE gels,
transferred to Immobilon membrane (Millipore) and probed with the purifed
antibodies; antibody binding was detected by enhanced chemilumenescence
(Amersham Pharmacia Biotech, Arlington Heights, IL).
Shadowing and electron microscopy
The peak XMAP215 fraction from the glycerol gradient was diluted twofold
with 15% glycerol in ammonium bicarbonate and sprayed onto 1 cm square
pieces of freshly cleaved mica. The mica was then dried under vacuum in a
Balzers BAE 120 operated at room temperature. Specimens were rotary or
uni-directional shadowed by evaporation of platinum onto the mica at
6° and then coated with carbon at 90°. The platinum/carbon
replicas were removed from the mica by flotation on deionized H2O
and mounted on 400 mesh EM grids.
Replicas were examined using a Philips 301 electron microscope operated at
80 kV. Micrographs were taken at a magnification of 47,000-50,000x.
Magnification was calibrated using a negatively stained preparation of bovine
tropomyosin paracrystals (Erickson et al.,
1981
).
Image analysis
EM negatives were scanned at 600 dpi resolution using an Epson Perfection
1200 scanner in negative mode. Measurements were made from the digital images
using NIH image software. A distance of 2 nm was subtracted from each length
or width measurement to remove the contribution from the shell of metal. The
radius of curvature for partial rings was measured using Canvas 6.0 by drawing
and overlaying a circle on the partial ring. The radius of each circle was
then measured using the radius tool in Canvas. These latter measurements were
not corrected for the platinum/carbon coating, as the circle was drawn to
bisect the center of the arc of each partial ring. Statistical analysis was
performed using analysis of variance in Microsoft Excel.
Measurement of flexural rigidity by thermal flucuations
For an idealized beam (i.e. a linear rod of homogeneous material and no
structural variation), the curvature along its axis gives information about
the bending moment associated with the impact of external bending energy.
Bending moment and bending energy are coupled by Young's elastic modulus (E)
and the geometric moment of inertia (I) (Feynman et al.,
1964). For an idealized beam,
these two parameters can be separated. However, a microtubule, made of
subunits of
ß-tubulin, is not an idealized beam, and its
elasticity is often specified by the product EI, which is a
measurement of a the stiffness or flexural rigidity of the microtubule.
Several techniques have been used previously to measure the flexural rigidity
of microtubules by correlating bending energy with positional deflections of
the microtubule long axis. Bending has been induced by passive excitation
through thermal energy (Gittes et al., 1994; Venier et al.,
1994
; Mickey and Howard,
1995
), or by controlled forces
applied by optical trapping or hydrodynamic flow (Venier et al.,
1994
; Kurachi et al.,
1995
; Felgner et al.,
1996
; Felgner et al.,
1997
).
In the present study, microtubule flexural rigidity was derived from measurements of thermal fluctuations at the free ends of clamped microtubules. This type of flexural rigidity analysis relies the equation (see Appendix for derivation):
which suggests that the
flexural rigidity EI of a microtubule and the mean square deflection
<d2> of an axis point at a distance L from
the clamped end are inversely proportional (Venier et al.,
1994
). This equation assumed
that L is much smaller than the persistence length of the microtubule
(
6000 µm, Gittes et al., 1994). The remaining parameters are the
absolute temperature (T) and the Boltzman constant
(kB).
Microtubules were assembled from axoneme fragments and imaged using
video-enhanced DIC microscopy at video rates (30 frames/second; Vasquez et
al., 1994). Tubulin was
assembled at 10 µM with or without 0.2 µM XMAP215. The effects of
XMAP215 on dynamic instability have been previously published (Vasquez et al.,
1994
) and the video tapes were
analyzed here for rigidity by measuring microtubule thermal vibration.
Axonemes adhered to the glass surface and therefore acted as a clamped point
of the microtubule. Microtubule plus ends of less than 10 µm length and
free to vibrate, owing to thermal fluctuation, were selected for analysis.
S-VHS tape sequences were digitized and captured into a computer (Power
Macintosh) equipped with an image frame grabber board (Scion LG-3 PCI) and
running the NIH-Image software. The capture rate was 15 frames/second, which
translates to 67 msecond intervals between successive frames. For each series
of frames describing one microtubule, an arc line perpendicular to the
microtubule long axis was drawn near the vibrating microtubule tip. For each
advancing image frame, the cursor was overlaid on the image of the microtubule
lattice intersecting the arc line, and the x-y coordinates
of the position of interest were recorded. The point (x0,
y0) represents the position of the microtubule at the
axoneme attachment site. The 50 collected points (x1,
y1),..., (x50,
y50)) represent the positions of the vibrating tip of the
microtubule for each of the 50 successive frames. The 50 points represent a
spread of the different positions of the microtubule tip to the left and right
of a mean position (xmean, ymean),
where:
,
The mean-squared
deflection <d2> is calculated as:
where,
![]() | (5) |
![]() | (6) |
![]() |
RESULTS |
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The shape of XMAP215 was observed directly by electron microscopy after unidirectional shadowing and appeared as a long, thin molecule of uniform density along its length (Fig. 1A). A corresponding gel of a glycerol gradient fraction used for shadowing is shown in Fig. 1B. Based on electron microscopy, we found no evidence for globular domains within the protein. In general, most XMAP215 molecules had a straight conformation, but several molecules were bent, suggesting that XMAP215 had some flexibility. The average length of an XMAP215 molecule was 60.0±17.7 nm s.d. (n=86) (Fig. 1C).
|
Gard and Kirschner (Gard and Kirschner,
1987) have previously
concluded that XMAP215 is a monomer in solution, based on results from
size-exclusion chromatography and sucrose gradient sedimentation. To confirm
that XMAP215 is a monomer and ensure that the above length measurement
reflects the length of a single protein, we incubated XMAP215 with an affinity
purified antibody raised against the C-terminus of TOGp. This antibody also
recognizes XMAP215 (Fig. 2A).
After unidirectional shadowing, antibody labeling was detected as a knob at
one end of the protein (Fig.
2B) and the size of the knob (
13.5 nm) is consistent with the
size of single antibody molecules observed by others (Wille et al.,
1992
). No evidence for XMAP215
dimerization was observed, consistent with previous hydrodynamic experiments
(Gard and Kirschner, 1987
) and
it is therefore highly likely that the long rods shown in
Fig. 1A represent the
individual XMAP215 molecules.
|
Based on the molecular mass of XMAP215 (228,000; Tournebize et al.,
2000, Becker and Gard,
2000
) and the length (60 nm) of
a single molecule, XMAP215 has a mass density of
3800 Da/nm. This mass
density is approx. twofold higher than that determined for other MAPs (see
Table 1 for comparison).
|
XMAP215 binds partial tubulin rings
To examine whether XMAP215 can bind tubulin dimers or oligomers, XMAP215
and tubulin were incubated on ice to prevent microtubule assembly. Samples
were then rotary shadowed and observed by electron microscopy. Under these
conditions, addition of XMAP215 resulted in assembly of partial tubulin rings
(Fig. 3). These partial rings
were not observed in the absence of XMAP215
(Fig. 3). To compare the
XMAP215/tubulin partial rings with those assembled from tubulin alone, we
found that a higher glycerol concentration (30%) and a longer incubation time
(several hours) resulted in partial ring assembly from pure tubulin
(Fig. 3).
|
The partial rings assembled with XMAP215 appeared uniform in width and were wider than rings assembled from tubulin alone (Fig. 4A); XMAP215/tubulin partial rings had an average width of 8.8±1.8 nm (s.d.; n=59) compared with 5.6±1.1 (s.d.; n=64) for the tubulin partial rings. These mean widths are statistically different (P<0.01) and suggest that XMAP215 adds a width of approximately 3.2 nm to the curved tubulin protofilament. The lengths of the rings (Fig. 4B) also differed significantly (P<0.01); rings assembled from tubulin and XMAP215 measured 59.6±26.2 nm s.d. (n=86) compared with 48.6±24.8 nm (s.d.; n=133) for tubulin alone.
|
The curvature of the partial rings was also measured by determining the
radius of a circle overlapping the arc formed by the partial rings
(Fig. 4D). The partial rings
had an average radius of curvature of 26.9 nm for tubulin alone and 26.8 nm
for samples containing XMAP215 (Fig.
4C). These measured values are close to those reported previously
for a protofilament ring (Voter and Erickson,
1979; Mandelkow et al.,
1991
). Clearly, the presence
of XMAP215 had no effect on the curvature of the partial rings.
Microtubule bending stiffness is not altered by XMAP215
To determine whether XMAP215 affected the stiffness of microtubules, we
measured microtubule plus end tip deflections and used these displacements to
calculate microtubule flexural rigidity (see Materials and Methods). Vibration
measurements were made on microtubule plus ends elongating from axoneme
fragments in the absence or presence of XMAP215. Plus-end assembly was
stimulated eightfold using the maximium possible concentration of XMAP215.
Under these conditions, plus-end microtubules assembled without XMAP215 had a
flexural rigidity of 17.5±2.2 (10-24/Nm2;
n=27) compared with 18.5±2.0 (10-24/Nm2;
n=25) for microtubules assembled from tubulin alone. Thus, XMAP215
did not change microtubule stiffness.
Four N-terminal repeats in XMAP215 are composed of HEAT repeats
Previous alignments of the XMAP215 family protein sequences have recognized
that the segment corresponding to XMAP215 264-543 is highly conserved in all
family members (Graf et al.,
2000; Popov et al.,
2001
). Cullen et al. (Cullen
et al., 1999
) noted further
that there were actually four repeats of this subdomain in the N-terminal
region of TOGp and Drosophila Msps. They identified four separate
motifs that made a common signature for the subdomain, but the overall
sequence identity between the four subdomains is quite limited, making
alignment difficult. Recent sequence analysis suggests that the XMAP215 family
members contain a large number of HEAT repeats (Neuwald and Hirano,
2000
; see Discussion). We
found that use of these poorly conserved HEAT repeat sequences provided a
guide for alignment within the N terminus of XMAP215 and that this analysis
removed most ambiguities. As shown in Fig.
5A, there are five HEAT repeats in each of the four N terminal
subdomains (N1-N4), and possibly a sixth one following these not recognized by
sequence. The arrangement of HEAT repeats and the subdomains is shown in
Fig. 5B.
|
![]() |
DISCUSSION |
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The binding of XMAP215 to partial tubulin rings suggests that XMAP215 binds
along a single protofilament, since a tubulin ring consists of a curved
protofilament (Voter and Erickson,
1979). This conclusion is
consistent with results from experiments with TOGp, demonstrating binding to
microtubules, zinc sheets and tubulin rings (Spittle et al.,
2000
). The 60 nm length of
XMAP215 suggests that this MAP could span seven to eight dimers along a
protofilament. It is remarkable that the length of the partial tubulin rings
assembled with XMAP215 (59.6 nm) is identical to the average length of the MAP
(60 nm), suggesting that tubulin dimers may have bound all along the length of
a single XMAP215 molecule.
Our results are thus consistent with evidence that XMAP215/TOGp binds to
microtubules with a large surface of the MAP bound to the microtubule lattice.
Microtubule binding assays of TOGp and fragments demonstrated that one
microtubule-binding domain was located within a region of approximately 600
amino acids near the N terminus and a second region in the C-terminal half of
the protein was able to bind tubulin dimers or oligomers (Spittle et al.,
2000). Expression of truncated
segments of XMAP215 in vivo also suggests that the entire protein participates
in microtubule binding (Popov et al.,
2001
). Taken together, these
results suggest a widely distributed microtubule binding interface on XMAP215
and are consistent with recent insights into the structure of this protein, as
discussed below.
XMAP215 is composed of multiple HEAT repeats
Recent iterative sequence analysis identified 20 HEAT repeats in TOGp
(Neuwald and Hirano, 2000) and
we find that these are also present in XMAP215
(Fig. 5A; data not shown). To
compare our EM structure with the structures of well-defined HEAT repeat
proteins, we used RASMOL (Sayle and Milner-White,
1995
) to measure the spacings
of HEAT repeats in PR65/A (PDB # 1B3U), a subunit of PP2A that contains 15
tandem HEAT repeats (Groves et al.,
1999
). The spacing of 1.15 nm
corresponds to a mass density of approximately 3750 Da/nm. This value compares
well with the 3800 Da/nm estimated for XMAP215
(Table 1). Additionally, PR65/A
has a depth of 2 nm and a width of 3.5 nm (Groves et al.,
1999
), similar to the 3.2 nm
width we estimate for XMAP215, based on its binding to partial tubulin rings.
Finally, HEAT repeat proteins are thought to be flexible (Kobe et al.,
1999
), which is consistent
with the variable bending seen in our EM of XMAP215.
Based on the above considerations of sequence, mass density, shape and
flexibility, and the fact that XMAP215 appears in the EM to have a uniform
structure over its entire length, we propose that XMAP215 may be composed
entirely of HEAT repeats. Additional HEAT repeats are likely to be of such
divergent sequence that they have not yet been identified by the iterative
search and Gibbs sampling alignment procedures used by Neuwald and Hirano
(Neuwald and Hirano, 2000).
Therefore, TOGp and XMAP215 could be composed of up to 45 HEAT repeats. HEAT
repeat proteins typically act as scaffold proteins (Kobe et al.,
1999
) and a scaffold function
for TOGp has been proposed recently (Charrasse et al.,
2000
). Finally, it is
intriguing that another HEAT repeat protein, importin ß (Cingolani et
al., 1999
), has been identified
as an intermediate in Ran-dependent microtubule assembly (Wiese et al.,
2001
; Nachury et al.,
2001
), as has XMAP215 (Wilde
and Zheng, 1999
), suggesting
that these two HEAT repeat proteins may share binding partners.
Microtubule stiffness
Measurements of microtubule tip displacements were used to calculate
microtubule flexural rigidity and demonstrated that XMAP215 did not alter
microtubule stiffness. Thus, XMAP215 is able to speed plus-end assembly
without altering microtubule stiffness. Several studies have measured
microtubule flexural rigidity using several different methods; while the
measured values differ, neuronal MAPs consistently stiffen microtubules
(Kurachi et al., 1995; Mickey
and Howard, 1995
; Felgner et
al., 1996
; Felgner et al.,
1997
). The method used here
also detected stiffening of microtubules when they bound neuronal MAPs (P. T.
T. and E. D. Salmon, unpublished), suggesting that the lack of stiffening
observed with XMAP215 is not due simply to the thermal vibration method used
here. Thus, XMAP215 is the first MAP studied that does not stiffen
microtubules and demonstrates that MAPs need not stiffen the microtubule
lattice to stimulate assembly.
Several possible mechanisms could explain the lack of microtubule
stiffening by XMAP215. First, XMAP215 may not be present at sufficient
concentration to saturate the microtubule, although it was present in
sufficient concentration to increase microtubule growth rate eightfold.
Second, XMAP215 may be bound preferentially to microtubule ends where it would
not affect microtubule stiffness. Preferential binding to microtubule ends has
been observed with TOGp (Spittle et al.,
2000). Third, XMAP215 may bind
along the protofilament axis (discussed above); this has also been suggested
for TOGp (Spittle et al.,
2000
).
How does XMAP215 stimulate assembly and protect ends from
depolymerization by XKCM1?
Our results and those of Spittle et al. (Spittle et al.,
2000) show that XMAP215/TOGp
binds along the protofilament axis. In addition, XMAP215 does not appreciably
stiffen the microtubule lattice under conditions that promote rapid plusend
assembly. How, then, does XMAP215 stimulate plus end assembly? It seems
unlikely that XMAP215 stimulates assembly through an interaction with the
existing microtubule lattice. Rather, it is possible that XMAP215 enhances
dimer addition to microtubule plus ends, as suggested previously (Gard and
Kirschner, 1987
; Spittle et
al., 2000
).
XMAP215 also counteracts the destabilizing activity of XKCM1. As XMAP215
does not function to stiffen the microtubule lattice and therefore does not
increase longitudinal bond strength along a protofilament, XMAP215 cannot
block XKCM1 action by simply increasing the energy necessary to remove a
tubulin dimer from the microtubule end. XMAP215 may compete for binding sites
at microtubule ends and sterically hinder XKCM1 access to the microtubule.
Alternatively, Popov et al. (Popov et al.,
2001) have suggested that
XKCM1 and XMAP215 may interact when not bound to the microtubule. In this case
XMAP215 and microtubule ends would compete for binding to
XKCM1.
|
In summary, XMAP215 is a thin protein that has a size and shape consistent with a structure based on multiple HEAT repeats. This structure provides an extended interface for interaction with the microtubule lattice or other binding partners. The proposed domain structure may provide the basis for making smaller expression proteins for biochemical and structural studies; this may lead to an understanding of how XMAP215 functions to speed assembly and protect ends from XKCM1.
![]() |
APPENDIX |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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If the curvature is
large (i.e. the deflection d is small) then we can assume (dy/dx=0)
and Eqn 1 reduces to:
The bending moment
M (torque about the neutral axis of any cross section) of the
microtubule at a given point from the origin is:
where
F=perpendicular force vector at x, L=total length
of beam and EI=flexural rigidity. The flexural rigidity EI
is the product of E, the Young's elastic modulus, and I, the
moment area of inertia. Rearranging Eqns 2 and 3, we get:
We can integrate Eqn.
4 twice to yield,
and,
At x=L, Eqn 6 reduces
to,
The quantity
y(L) is the displacement, d, of the tip of the microtubule.
The condition at the tip of the clamped microtubule is also analogous to
Hooke's law for a spring. For a spring, the force and potential energy
associated with a deflection through a distance y are:
![]() | (A8) |
where
=the
spring constant, and y=d. Rearranging Eqns 7 and 8 we arrive at:
According to the
equipartition theorem (Reif,
1965
), the mean potential
energy is:
where
kB=Boltzman constant and T=absolute temperature.
So, from Eqn. 9,
Rearranging Eqns 11
and 12, we get:
Substituting Eqn 10
into 13, and rearranging, we get:
or,
where
<y2>=<d2> is the mean square
displacement of the microtubule tip.
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ACKNOWLEDGMENTS |
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![]() |
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