Biology Reference
In-Depth Information
Fig. 3
Equilibrium
conformation of the
microtubule cytoskeleton in a
spherical cell. Sample
microtubule forms that lie in
the plane passing through the
centrosome and the cell
center are shown.
Reproduced from Maly and
Maly (
2010
) with permission
from Elsevier
microtubule cytoskeleton in these cells adopts the minimum-energy conforma-
tion, while the antigen-mediated conjugation with another cell provides the exter-
nal reference frame with respect to which the other forces may orient the
constitutively asymmetric microtubule aster (Arkhipov and Maly
2006a
; Baratt
et al.
2008
).
As an example with real numbers, consider an experiment in which an aster of
N
= 20 microtubules, each
L
= 12 μm long, is assembled inside an approximately
spherical chamber of radius
R
= 10 μm, with a bead replacing the centrosome (Holy
1997
). The model predicts (Fig.
2
) that when
L
/
R
= 12 μm/10 μm = 1.2, the normal-
ized equilibrium distance of the centrosome from the center of the chamber will be
Δ
e
/
R
≈ 0.4. In the chamber of the assumed size, therefore, the distance of the centro-
some from the center will be Δ
e
= 0.4
R
= 4 μm. This example demonstrates how the
unit-invariant form in which the model results are presented can be applied to any
specific situation in a quantitative experiment.
It was assumed in the above that all microtubules in the cell have the same length.
A generalization of the model to a distribution of lengths is straightforward. To
preserve the intrinsic symmetry of the cytoskeleton, the distribution characterized
by a density function
q
(
L
) should be the same for each orientation of unstressed
emanation from the centrosome. The only modification to the model will be to inte-
grate with respect to
L
in addition to integrating with respect to the emanation angle
when finding the total force
F
. The formula for the total force in the three-
dimensional case becomes
=
∫
Ff
(,)()
q
0
LpqL
dΩ
L
Let for example
q
(
L
) be the density function of a uniform distribution between
1.05
R
and 1.15
R
. Following the same computational strategy, Maly and Maly
(
2010
) find in this case that Δ
eq
= 0.220
R
. In the model with the constant length,
when its value was equal to the mean of this distribution (
L
= 1.1
R
), the equilibrium