Systems Biomechanics of the Cell - Systems Biomechanics of the Cell - page 20

Fig. 8 Limiting case of q max = π, long microtubules. ( a ) Pole force function. ( b ) Spindle force
function. ( c , d ) Equilibrium conformations. L = 1.1 R , S = 0.3 R . Plotting conventions as in Fig. 7 .
Reproduced from Maly ( 2012 ) under the Creative Commons Attribution License
function changes sign (Fig. 8a ), it is monotonically decreasing for x p > 0 like it did in
the case of short microtubules. Thus, the symmetry of the spindle will be stable.
Overall, however, the pole force function with long microtubules is not mono-
tonic (Fig. 8a ). This opens the possibility for existence of additional equilibria,
when both poles are on the same side of the cell. The system remains attracted to the
symmetric equilibrium until the spindle proper is moved entirely into one half of the
cell. Then the conformational transition that was described in the last section in con-
nection with the single three-dimensional aster occurs in the more central aster of
the spindle. It places the mitotic system on the other branch of the spindle force
function (Fig. 8b ). The monotonicity of each half of the single pole force function
ensures that following this transition, the spindle continues moving spontaneously
in the direction that was previously forced. Indeed, even if the more distal pole is
already in the region where it is repelled by the boundary, the more central pole
upon crossing the center always experiences a greater force repelling it from the
center. The new equilibrium will be reached when the two forces become equal in
magnitude. This strongly asymmetric equilibrium will be stable. This is due to the
piecewise monotonicity: Whether the poles are on different sides or on the same
side of the cell, the force function (Fig. 8a ) is locally decreasing for each pole.