Biology Reference
In-Depth Information
are more deformable, and the drop of the force magnitude past the buckling-force
maximum that they exhibit is deeper. Because of this, the increase in resistance at
large pole displacements in this case cannot erase the nonmonotonicity efficiently.
Instead, at around
q
max
= π/2 the pole force function develops two descending
branches (Fig.
10a
). The difference from the already considered limiting case of
complete asters (Fig.
8a
) is that the pole force function with the intermediate
q
max
may not change sign across the discontinuity. This reflects the comparatively
simple behavior of asters with
q
max
< π/2: The individual pole would find equilib-
rium when all microtubules that are emanating from it are straight, unlike in the
more complete asters, where bending is unavoidable. Despite this difference in
the individual behavior of separate poles, poles coupled by the spindle will behave
in the same way as with the more complete asters of long microtubules, because
of the fundamental similarity of the pole force function with the two descending
branches. The symmetric position is stable, as is the asymmetric equilibrium
(Fig.
10b
, dashed curve).
The intermediate case
L
≈
R
presents special interest. Theoretically, it connects
the qualitatively different types of behavior described for long and short microtu-
bules. Biologically, astral microtubules seem to be comparable in length with the
cell radius in the morphogenetically important instances of large cells of early
embryos (Lillie
1901
; Hyman and White
1987
; Symes and Weisblat
1992
; Grill
et al.
2001
). Evidently, the case of
L
=
R
is by itself unrealistic, because the two
quantities cannot be exactly equal. This special case, however, establishes a useful
reference in the space of the model parameters and regimes of behavior.
Calculations show that a complete (
q
max
= π) aster with
L
=
R
develops the peak
force just before its pole reaches the cell center (Fig.
11a
). This force is associated
with the buckling force of the microtubules that all straighten when the pole reaches
the center. When the pole is in the center, the elastic force can take any value
between the positive and negative extremum, and will be zero for an aster not sub-
jected to any external force. Thus, an individual separate aster exhibits a special
kind of stability of the central position, in which a finite restoring force develops
upon an infinitesimal perturbation.
However, the magnitude of the restoring force decreases with the magnitude of
the perturbation (Fig.
11a
). A pole that is closer to the center therefore experiences
a higher centripetal force than a pole farther away from the center. This is a condi-
tion for instability of symmetry of the spindle. One pole's centripetal movement will
be completed at the expense of the increasingly eccentric position of the coupled
pole. This is demonstrated by the spindle force function (Fig.
11b
). In the asymmet-
ric equilibrium, which is stable, the force on the eccentric pole is balanced by an
equal force supported by the straight microtubules of the centrally positioned pole.
The described behavior is observed for spindles with interpolar distances not
exceeding a certain value. Inspection of the pole force function (Fig.
11a
) shows
that the stiffening of the aster at large eccentricities of the pole creates a range of
extreme pole positions where the magnitude of the force is increasing with the dis-
tance from the center. This creates conditions for stability of symmetry of suffi-
ciently long spindles (Fig.
11c
).
