Biology Reference
In-Depth Information
Consider the symmetric case, in which both sides are bent equally. This, obviously,
is an equilibrium. However, the magnitude of the force exerted on the pole in this situ-
ation is a locally decreasing function of the pole's distance from the boundary
(Fig.
6a
). Therefore movement of both poles to the right will decrease the magnitude
of the force exerted by the boundary on the right pole. By symmetry, the magnitude of
the force exerted on the left pole will be increased. The symmetric equilibrium proves
unstable. The coupled poles will continue moving spontaneously, until the new equi-
librium is reached. In this other equilibrium, the force of the bent microtubules acting
on one pole is balanced by the force of the straight microtubules acting on the other
pole. This is always possible, because the magnitude of the force of a bent microtu-
bule is always lower than the buckling force. The asymmetric equilibrium is stable.
Indeed, movement in the direction of the bent microtubules will leave the opposite
pole unsupported, as the straight microtubules lose contact with the boundary.
Movement in the opposite direction will place the system in a state it already passed
during its spontaneous movement from the unstable symmetric equilibrium.
Thus, in this special case (the limiting case of
q
max
= 0), we observe instability of
symmetry, stability of asymmetry, and the possibility to predict the stable confor-
mation from the structural parameters that include the length of the spindle
S
and
the length of the astral microtubules
L
in relation to the cell radius
R
. The forces are
proportional to the microtubule bending rigidity EI and to the number of the micro-
tubules
N
, but the positions and stability of the equilibria do not depend on these
parameters. The more general and biologically relevant cases exhibit more complex
behavior but retain these fundamental characteristics.
The opposite extreme case is also revealing—the special case of complete,
intrinsically spherical asters at each pole (
q
max
= π). Now the behavior depends on
whether the astral microtubules are longer or shorter than the cell radius. The case
of short microtubules is simple. After the aster comes in contact with the boundary,
the force exerted on the pole increases gradually with
x
p
. The graduality is due to the
number of the microtubules in contact with the boundary increasing gradually,
unlike in the case of
q
max
= 0. In addition, only the axial microtubules (
q
= 0), whose
contribution to an aster with
q
max
≠0 is infinitesimal, go through developing the
buckling force during the axial movement of the spindle; all others deflect on con-
tact. The bending leads to a decrease in stiffness, as can be seen in Fig.
7
. The
decrease in the stiffness of the aster (a progressively shallower slope of the force
curve) is different from the previous case, where the magnitude of the elastic force
was decreasing with the progressing deformation. The numerical results (Fig.
7
)
indicate that the softening effect of the deformation (Fig.
6
) in the case of the com-
plete aster is more than offset by the increasing numbers of microtubules that come
in contact with the boundary. Thus, even though the total force is a nonlinear func-
tion of the pole position, the force resisting the outward movement of the pole is
monotonic. Due to the monotonicity, the stability of symmetry, which would be
expected with simple Hookean (linear) elasticity, is exhibited by a spindle with two
complete asters of short microtubules. In addition, the monotonicity means that
there is only one equilibrium conformation of the mitotic microtubule cytoskeleton,
insofar as the latter is large enough to maintain contact with the cell boundary.
