Systems Biomechanics of the Cell - Systems Biomechanics of the Cell - page 16

Fig. 5 Bipolar confined
mitotic spindle model.
Reproduced from Maly
( 2012 ) under the Creative
Commons Attribution
License
of the total force acting on an isolated pole. F s is the projection on the x axis of the
total force acting on the spindle (i.e., on both poles). Both forces are collinear with
the x axis due to the axial symmetry. Figure 5 illustrates the model and notation.
Microphotographs in the papers cited in the introduction confirm that microtu-
bules in mitotic cells do not converge on spindle poles at sharp angles in a fan-like
arrangement. This rules out a hinged (free pivoting) boundary condition on the cen-
trosome. The smooth bending forms seen in microphotographs argue in favor of
equilibrium flexure and against additional constraints. Accordingly, like the micro-
tubules in the interphase model, the astral microtubules are assumed to be rigidly
clamped at the centrosomes, and their contact with the spherical cell surface, fric-
tionless. The lowest-energy equilibrium solution is calculated, because, as discussed
in the last section, the alternative equilibria are unstable in three dimensions.
Unlike in the fully isotropic centrosome in the interphase model, the spindle
model must take into account the fact that inward-pointing microtubules at the spin-
dle poles may not be astral microtubules but may instead be part of the spindle proper.
Accordingly, the angle q between the clamped direction and the outward direction of
the spindle axis in the model does not exceed a certain angle q max . Whereas in the
interphase model N is the number of microtubules in the single microtubule aster, in
the mitotic model N denotes the number of microtubules that radiate from one of the
two poles. Intrinsically, the model cytoskeleton structure is centrally symmetric: EI,
L , N , and q max are equal on both poles. No assumptions need to be made a priori about
parameter values. The model behavior, insofar as it concerns the problem posed here,
is controlled entirely by three compound parameters: L / R , S / R , and q max . Analysis in
their space can be for the practical purposes exhaustive.
Limiting cases set the theoretical context for the mechanics of the more biologi-
cally relevant regimes. The simplest behavior is exhibited by the structural case of
astral microtubules that emanate from the pole exclusively along the axis of the
spindle ( q max = 0). As the pole moves collinearly with the spindle axis, the micro-
tubules will abut on the cell boundary, buckle, and bend, exerting force on the pole.