Systems Biomechanics of the Cell - Systems Biomechanics of the Cell - page 17

Fig. 6 Limiting case of q max = 0. ( a ) Pole force function. ( b ) Spindle force function. ( c ) Symmetric
equilibrium. ( d ) Asymmetric equilibrium. ( b - d ) L = 0.8 R , S = 0.65 R . For clarity, only two microtu-
bule forms are plotted. These microtubules lie in the ( x , y ) plane that passes through the spindle
axis. The circumference is the section of the cell surface, and the thicker line segment depicts the
spindle proper. Reproduced from Maly ( 2012 ) under the Creative Commons Attribution License
A sample calculation is presented in Fig. 6 . When the axial distance of the pole from
the boundary is greater than the length of the astral microtubules, the force is zero.
When the distance is equal to the length, the magnitude of the force can take any
value between zero and the buckling force. For shorter distances, the force decreases,
as the increasingly bent microtubules become less efficient at resisting the displace-
ment of the pole. When the movement of two such poles is coupled through the
spindle proper, three regimes are possible. If the spindle proper is short, the astral
microtubules may not come in contact with the boundary. This happens when
L + S /2 < R . The symmetric position will then be a neutral equilibrium. The special
case of the entire structure just fitting in the cell without deformation is described by
L + S /2 = R . Naturally, this rather special condition is unlikely to be realized. It can
be noted parenthetically that models operating with microtubule buckling forces,
such as the pioneering Bjerknes model, presuppose this condition while also implic-
itly assuming that it can be maintained as the centrosomes shift. This condition is
difficult to cast in physical terms even hypothetically. Returning to the model being
reviewed here, the non-trivial situation arises when L + S /2 is greater than R . In this
case, microtubules emanating from one pole or from both must be bent.