Systems Biomechanics of the Cell - Systems Biomechanics of the Cell - page 15

theoretical modeling as such will be reviewed in the last section; here we will be
concerned with the general methodology of modeling the deformation of confined
microtubules, which was advanced by these models. In the new model by Grill,
bending of astral microtubules against the cell boundary was included in addition to
the pulling. The microtubule deformations were not explicitly computed. The force
associated with them was computed using a linear Hookean dependence on the dis-
tance between the pole and the boundary. The resulting oscillations of the pole in this
one-dimensional model were about the middle, symmetric position. A different
model for the oscillations (Kozlowski et al. 2007 ) computed the microtubule defor-
mations explicitly in three-dimensions. The deformations were caused by the viscous
drag in the cytoplasm and instantaneous pulling by the postulated pulling elements
on the ends of the microtubules that were coming in contact with the boundary. Upon
contact with the boundary, the microtubules in this model disassembled, preventing
development of a durable deformation of bending against the boundary.
It should be noted that in contrast with the reviewed simplifications of the treat-
ment of the astral microtubules in spindle models, the deformations of the microtu-
bules in the spindle proper (those that connect the two poles) have been treated with
the precision of the standard bending elasticity theory (Nedelec 2002 ; Rubinstein
et al. 2009 ).
The model by Maly ( 2012 ) extended and complemented these approaches, per-
mitting a general analysis of the effects of the microtubule elasticity on spindle
positioning. It focused on the question of symmetric vs. asymmetric positioning of
the spindle, as posed above. Deformations of astral microtubules were computed
explicitly, and the stability of the equilibria to perturbations was assessed. In view
of insufficient experimental data pertaining to quantitative descriptors of mitotic
spindles, a theoretical context had to be established in which the experimentally
observed structures might subsequently be placed. The model allowed treatment of
the different theoretical regimes as well as calculation of some sample structures
that was deemed representative.
Instead of one microtubule aster around the single centrosome in the interphase
model (Maly and Maly 2010 ), in the mitotic model (Maly 2012 ) there are two asters
around the two rigidly coupled centrosomes at the spindle poles. These asters may be
partial in the sense that the microtubules may not radiate in all directions. Material
spindle poles are small compared with the cell size, and the spindle proper which con-
nects them contains a large mass of crosslinked microtubules (Bray 2001 ). In view of
this, in the model the spindle proper is an absolutely rigid segment that connects two
points in space that represent the poles (centrosomes). In this respect, the model is
similar to the models by Bjerknes ( 1986 ), Kozlowski et al. ( 2007 ), and Théry et al.
( 2005 ). The length of the spindle proper (the interpolar distance) is denoted S .
To address the problem as posed above, axially symmetric situations are consid-
ered, in which the axis of symmetry coincides with the spindle axis (i.e., passes
through the poles). In the Cartesian coordinate system of the model, the x axis is col-
linear with the spindle axis. When isolated spindle poles are considered, and unless
otherwise noted, the pole on the right is meant. The pole coordinate is denoted x p ; the
coordinate of the middle of the spindle proper is x s . F p is the projection on the x axis