Biology Reference
In-Depth Information
the micrometer scale, while the free-fluctuation measurements can be challenging
due to diffusion out of the plane of focus of the microscope used.
In this chapter, we describe a complementary, passive technique to measure
microtubule persistence length. The technique involves kinesin-driven gliding
assays, which ensure that the microtubule always remains in the microscope focal
plane. Moreover, it involves tracking single fluorophores attached permanently to
the polymer of interest so that specific locations along the polymer are well charac-
terized. By varying the density of kinesin used in the gliding assay, the effective
length of the microtubule can be changed. The primary restrictions on the addition
of other substances which may modify microtubule persistence length, such as
paclitaxel or MAPs, are that they must support kinesin motility.
2.1
THEORY
One way to characterize flexibility (or stiffness) of microtubules, and polymers in
general, is the persistence length, the length of polymer which bends by approxi-
mately 1 rad under thermal fluctuations at ambient temperature: a rigid or stiff poly-
mer has a long persistence length; a flexible polymer has a short persistence length.
Persistence length
l
p
is heuristically defined as the length over which polymers bend
significantly (
Fig. 2.1
); mathematically, it is defined as
e
s
=
l
p
h
cos
i ¼
(2.1)
y
s
where
s
is the distance between two points on the polymer,
y
s
is the angle between
tangent vectors to the polymer, and
indicates an average over all positions a dis-
tance
s
apart (
Phillips, Kondev, & Theriot, 2008
). For very stiff polymers,
s
/
l
p
hi
1
and
y
s
¼
d
/
s
are small, so a first-order Taylor expansion gives
y
s
2
D
E
2
ðÞ
d
=
s
¼
s
=
l
p
(2.2)
with
d
the lateral displacement of the polymer with respect to a straight line.
FIGURE 2.1
Cartoon of curved microtubule, showing tangent vectors and angles. The gray tube is a
cartoon microtubule (25 nm diameter). Straight arrows represent tangent vectors at two
locations along the microtubule axis;
is the path length along the microtubule between the
two tangent vectors. Each tangent vector makes an angle
s
y
with respect to the horizontal; the
difference between
y
0
and
y
s
is used to calculate the persistence length following Eq.
(2.1)
.
