Biology Reference
In-Depth Information
Fig. 4
Total force exerted by
the microtubules on the
centrosome for small
deviations of the centrosome
from the center of a flat cell,
starting from the fully
symmetric cytoskeleton
conformation. Reproduced
from Maly and Maly (
2010
)
with permission from
Elsevier
distance was 0.214
R
. Thus, the equilibrium distance with the distribution of lengths
is not the same but larger, due to nonlinearity of microtubule bending.
The two-dimensional case corresponds to microtubules confined in an essen-
tially flat spatial domain, as in the thinly spread cells cultured on glass in most
experiments today. The lowest-energy conformation of the microtubule cytoskele-
ton, in which each microtubule is in the lower-energy (stable) equilibrium form,
can be computed as in the three-dimensional case. It will share the “combed”
appearance with the three-dimensional case (see Fig.
3
), but this appearance is
inconsistent with the images of flat cultured cells, in which neighboring microtu-
bules are typically buckled in opposite directions (Euteneuer and Schliwa
1992
). In
view of the claim that the centrosome in flat cells is maintained in the geometrical
center of the cell outline (Euteneuer and Schliwa
1992
; Burakov et al.
2003
; Gomes
et al.
2005
), the conformation of special interest in the flat-cell case is the fully
symmetric cytoskeleton. The full symmetry in the appropriate cell-biological sense
is a reflection symmetry with respect to any axis that can be drawn through the
center of the cell. This implies an infinite-fold rotational, or circular, symmetry, but
excludes the case of vortex polarization that the rotational symmetry by itself
would permit. The full symmetry thus requires that the centrosome be in the cell
center, that the unstrained directions of microtubules be uniformly distributed
around the centrosome, and that the two directions of buckling for each unstrained
direction be equally represented. It is clear that the fully symmetric conformation
is a static equilibrium.
Calculations show (Maly and Maly
2010
) that small deviations of the centro-
some from the center result in a third-power growth of the total force on the centro-
some (Fig.
4
). This force is directed outward. Consequently, the symmetry is
unstable. The new equilibrium is reached in which the centrosome is removed from
the center by a distance approximately equal to the difference of the microtubule
length and cell radius (Fig.
2
). Precise calculations show that it is slightly smaller
than this difference, which means that unlike in the three-dimensional case, all
microtubules in the predicted flat equilibrium structure are in contact with the cell
boundary and are bent.
