Biology Reference
In-Depth Information
The specific assumptions in the Maly and Maly model about the mechanical
system of the microtubule aster confined within the cell boundary merit some dis-
cussion. The simple beam equation is the classical model for microtubule bending
that has found both theoretical and experimental applications (see Howard
1998
).
The adequacy of the beam equation, which predicts the simple Euler forms of buck-
ling, in application to intracellular microtubules is not universal in the light of
observations of high-frequency buckling (Brangwynne et al.
2006
). An example of
more advanced models for bending of single microtubules can be found in the work
of Gu et al. (
2009
). It may be expected that the qualitative conclusions pertaining to
the properties of symmetry, stability, and reversibility that the simple model has
illuminated do not depend on how simple the buckled form is. At the same time, the
simple buckling forms compatible with this model are also abundant in cells, as the
microphotographs in the cited papers attest.
That the microtubules are clamped rather than hinged on the centrosome may be
deduced from images in the cited experimental papers. The images invariably show
that even strongly bent microtubules radiate from the centrosome in all directions
before bending, instead of converging on the centrosome at sharp angles in a fan-
like arrangement. The situation is opposite in cell-free and centrosome-free experi-
mental models and their numerical representation (Pinot et al.
2009
) which are
discussed in more detail in the section devoted to boundary dynamics. The model
discussed here is concerned with the mechanical system responsible for the posi-
tioning of actual centrosomes and depends on the faithful description of the mode
of microtubule clamping on them.
Maly and Maly observed that when the centrosome is in the center of the cell, the
forms of a microtubule that differ in the direction of its buckling are symmetric
about the axis which is the direction at which the microtubule is clamped at the
centrosome, and they have the same energy. In three dimensions (in a spherical
cell), an infinite number of such forms are connected by continuous rotation about
this axis. In two dimensions (in a flat round cell), there are only two such forms of
a microtubule, and they correspond to diametrically opposed forms of the three-
dimensional case. With the displacement of the centrosome from the cell center, the
described equivalency of the buckling forms of a microtubule is lost. Now the form
which is convex in the direction of the centrosome displacement is bent less, and the
opposite form is bent more. In three dimensions, these two diametrically opposed
forms are the only remaining equilibrium forms. They lie in the plane defined by the
displacement of the centrosome and by the unstrained direction of the microtubule.
They are connected by a continuity of nonequilibrium forms, and there is no energy
barrier between them. The microtubule therefore will adopt the lowest-energy form
(the one which is convex in the direction of the centrosome displacement from the
center). In two dimensions, the two equilibrium forms are connected only by higher-
energy nonequilibrium forms in the plane of the cell. Therefore both of them are
locally stable, and both can be occupied even after the displacement of the centro-
some from the center. The higher-energy form can be called metastable, and the
lower-energy form, stable.
Each equilibrium form of a microtubule is characterized by the force and torque
that it exerts on the centrosome. If the total force and torque exerted on the
