Systems Biomechanics of the Cell - Systems Biomechanics of the Cell - page 38

Fig. 21 Equilibrium position of the centrosome as a function of microtubule length in a flat cell.
Diamonds show positions reached after a small perturbation of a fully symmetric cytoskeleton.
Lines show changes in the course of continuous elongation and shortening of microtubules.
Reproduced from Maly and Maly ( 2010 ) with permission from Elsevier
Taken together with the irreversible loss of symmetry in response to infinitesimal
perturbations, this permits a generalization: The equilibrium form of a microtubule
cytoskeleton confined in a flat cell exhibits memory of past perturbations.
The diamonds in Fig. 21 plot the positions to which the centrosome moves spon-
taneously in response to small perturbations of the fully symmetric cytoskeleton
with the given ratio of the microtubule length to the cell size. The plot demonstrates
that although the relationship Δ e = L − R is not exact, it is a useful approximation.
The sample points comprising this plot are not connected by a curve. This feature of
Fig. 21 is meant to reflect the fact that each equilibrium is a result of a separate
process of relaxation from the fully symmetric state. Because of the already noted
strong nonlinearity, the system will not move from one of these points to the other,
when the microtubule length is varied continuously. For comparison, the result of a
continuous tuning of the equilibrium by varying the length is shown in Fig. 21 by
continuous curves. The solid curve corresponds to microtubule elongation. Two
sample reverse branches of this function are also shown, which correspond to short-
ening from the two sample reversal points. It can be observed that the equilibrium
centrosome position as a function of the microtubule length exhibits hysteresis.
Figure 22 illustrates the underlying cause: Elongation of microtubules expands
the range of unstressed microtubule directions for which metastable forms no lon-
ger exist. The metastable forms lost during elongation are not restored during short-
ening, and this affects the equilibrium position of the centrosome. It does not return
to the same position that it had when the length had the same value during elonga-
tion. As shown in Fig. 23 , the equilibrium cytoskeleton as a whole exhibits memory
of past variations of the microtubule length.
Considering these results in toto, one may observe that although the centrosome
position can be controlled approximately linearly and approximately reversibly by
varying the microtubule length, the evolution of the overall cytoskeleton form dis-
plays irreversibility. The conclusion that the evolution of the microtubule aster is in