Systems Biomechanics of the Cell - Systems Biomechanics of the Cell - page 21

Fig. 9 Transition between
q max = 0 and q max = π in
the case of short astral
microtubules. L = 0.8 R .
( a ) Pole force function.
( b ) Spindle force function,
q max = π/5. Reproduced from
Maly ( 2012 ) under the
Creative Commons
Attribution License
A centripetal movement of the spindle from the asymmetric equilibrium increases
its repulsion from the center and decreases its repulsion from the nearest boundary,
and vice versa. The spindle will therefore return to the asymmetric equilibrium,
unless the forced movement brings one of the poles to the opposite half of the cell,
in which case it will spontaneously continue toward the symmetric equilibrium.
Transitions between the limiting cases further illuminate the systems-
biomechanical underpinnings of the spindle positioning. Let us first consider the
instance of short astral microtubules. We have seen that as q max increases from 0 to
π, stability of symmetry and nonexistence of asymmetric equilibria replace instabil-
ity of symmetry and stability of asymmetry. Computations demonstrate that increas-
ing q max from zero first makes the development of the extremum pole force smooth
(Fig. 9a, solid curve; cf. Fig. 6 ). The finite interval of x p in which the pole force
function is decreasing emerges immediately when q max exceeds zero. The range of
values of S that place symmetric poles in these intervals will correspond to stable
symmetry. Unlike in the extreme case of q max = π, however, the pole force function is
increasing beyond the extremum (Fig. 9a). Let us denote the pole position that cor-
responds to the extremum x p ′. For S exceeding 2 x p ′, the behavior seen with q max = 0
is retained, and symmetry is unstable (Fig. 9b). The corresponding stable asymmet-
ric equilibrium is derived from the one described in the case of q max = 0: One pole