Systems Biomechanics of the Cell - Systems Biomechanics of the Cell - page 37

Fig. 20 , point B). Beyond this point, a reversal of the outward movement of the cen-
trosome places the system on a different branch of the force function. There is a sepa-
rate reverse branch for each reversal point in the continuous model of Maly and Maly
( 2010 ), which makes the force function infinite-valued. A discrete model would have
the number of reverse branches on the order of the number of microtubules, which is
hundreds in a typical flat cultured cell. In general the force, according to the Maly and
Maly model, is a functional of the centrosome path.
The nondimensional form adopted in Fig. 20 reveals the parameter-independent
behavior of the model; refer to Table 1 in the last section for nomenclature. The
plotted quantity FR 2 /( N EI) is the average force per microtubule (among the N
microtubules in the cell), expressed in the natural units of force in the model.
The natural unit of force is the flexural rigidity of a microtubule EI divided by the
square of the cell radius R . Assuming, for example, a rigidity of 25 pN μm 2 , which
would be near the midpoint of the range of the measured values (see a compendium
table in Kikumoto et al. 2006 ), the force unit will be 1 pN for a cell that is 10 μm
in diameter. In this way, the force acting on the centrosome can be estimated in
common units when the parameters ( N , R , and EI) characterizing the individual cell
are known.
Continuing the concrete experimental example from the previous section, assume
now that the containing artificial chamber was prepared in such a way that it is shal-
low (Holy et al. 1997 ), and the microtubules are 12.5 μm long. In this case, using
Fig. 2 and the calculation strategy presented in the last section, we derive that the
equilibrium distance of the artificial centrosome from the center is predicted to be
very close to 2.5 μm. If an optical trap is then used to displace the bead serving as
the artificial centrosome from this position to 2 μm from the center, then the model
predicts that the force exerted by the trap on the bead will be very close to 1 N EI/ R 2 ,
as can be read directly from the plot in Fig. 20 . To compare this prediction with the
force as measured in common units by the optical trap technique, one should substi-
tute the values of N , EI, and R that characterize the specific experiment. Using the
above experimental estimate for the microtubule rigidity, we obtain 1
N EI/ R 2 = 20 × 25 pN μm 2 /(10 μm) 2 = 5 pN.
As examples in Fig. 20 illustrate, the reverse branches are non-zero at the zero
centrosome displacement. Thus, although the central position of the centrosome can
be restored by forces external to the microtubule cytoskeleton, the new central posi-
tion will not be an equilibrium (Fig. 19c ). The symmetry loss in response to a small
perturbation proceeds spontaneously beyond the range of reversibility. One may
also ask what happens if the centrosome is forced beyond the spontaneously
achieved equilibrium (Fig. 19d ). Calculations show that the reverse branches from
beyond the equilibrium point pass very close, within 1 %, of the original equilib-
rium (see the example in Fig. 20 ). Therefore, the irreversible effect of the perturba-
tion will not be detected by recording the equilibrium centrosome position, and the
eccentric equilibrium position of the centrosome in flat cells may, for practical pur-
poses, be called stable. Even through the centrosome position is almost precisely
restored after removal of the external force, forcing the centrosome farther away
from the center leaves an irreversible trace in the cytoskeleton structure (Fig. 19e ).