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that is, those at angles greater than 5
to the original trajectory, are rejected. After
running through all trajectories and combining those which meet the above criteria,
unique microtubule trajectories are numbered, albeit with many overlapping fluor-
ophores. These are thinned into one, averaged, ordered microtubule trajectory fol-
lowing
Lee (2000)
. Example trajectories are shown in
Fig. 2.3
B.
We then calculate the tangent angle
as a function of path length,
s
, along the
trajectory. We compute the angle difference between all points separated by a given
distance
s
and then compute the average of the cosines of these angle differences,
again using custom software written in IDL. A fit of Eq.
(2.1)
to these cosine data
returns the persistence length, as shown in
Fig. 2.3
C.
y
2.3
DISCUSSION
The protocol described in this chapter provides a fast and efficient method of mea-
suring microtubule persistence lengths. We have found that we can measure the per-
sistence lengths of hundreds of microtubules per experiment and finish all analysis
within a few hours using a desktop computer. In our lab, the protocol has been used to
measure the persistence length of microtubules as a function of length, by varying the
density of kinesin on the surface. Simply changing the average concentration of kine-
sin added in step 7 of the microtubule gliding assay permits length-dependent exper-
iments. Calibration of the density of kinesin is straightforward with low-density
kinesins or very short microtubules (
Duke et al., 1995
): for very dilute surface coat-
ings, the average trajectory length scales with microtubule length until the microtu-
bule length is longer than the average spacing between kinesins.
In addition, our lab has also extended this technique to measure microtubule per-
sistence length as a function of microtubule diameter. Microtubules with different di-
ameters have different numbers of protofilaments (13 is typical for
in vitro
assembled
microtubules, and in many
in vivo
contexts). Thirteen protofilament microtubules
have protofilaments parallel to the long axis of the microtubule (
Wade, Meurer-
Grob, Metoz, &Arnal, 1998
), whilemicrotubules with different numbers of protofila-
ments have protofilaments skewed relative to themicrotubule axis into a super-helical
pitch, with a pitch length on the scale of micrometers. Because kinesin follows single
protofilaments, microtubules with skewed protofilaments will rotate in a gliding assay
(
Marchuk, Guo, Sun, Vela, & Fang, 2012; Nitzsche, Ruhnow, &Diez, 2008
). We use
this analysis to correlate the rotation rate of microtubules (and, hence, the number of
protofilaments and diameter) with the persistence length of individual microtubules.
These two examples indicate one advantage of the technique described above: it
permits the connection of microtubule mechanical and structural properties. We an-
ticipate, as well, that the use of a motility (gliding) assay at the same time as a me-
chanical (persistence length) assay will permit other experiments examining
connections between motility and mechanical properties.
On the other hand, there are limitations to this technique. Microtubules must be
stable on a time-scale of hundreds of seconds, meaning that they must be either
