Biomedical Engineering Reference
In-Depth Information
Another symmetry-breaking source is the lack of sphericity of the beads.
Arguments very similar to the one we have used for enzymes heterogeneity can
be made. One has essentially to replace
l
a
/l
i
by
r
a
/r
i
, in which the subscripts
refer to the beads' radius of curvature variation, with obvious meaning. It is
dicult to put numbers on this term, because it depends on the preparation
chemistry of the beads. It seems that beads smaller than a micron, have ex-
cellent sphericity, controlled by surface tension. In that range, one expects
enzyme heterogeneity to provide the main symmetry-breaking term. It seems
more dicult to obtain beads larger than a few microns, in which case spher-
ical aberrations might provide the main symmetry-breaking term. It would
be interesting to design carefully controlled experiments, for quantitatively
checking these predictions. Clever microscopic models have been imagined in
order to obtain symmetry-breaking conditions [23, 38]. However, they do not
take into account the very nature of the actin gel.
Finally, degradation of the gel could also occur through rupture (instead
of depolymerization). In both cases, degradation starts at the outer surface.
In one case, however, symmetry is broken by a smooth perturbation of the
thickness (depolymerization), and in the other by a sharp (localized) opening
of the gel (rupture) [37]. Indeed, symmetry-breaking of an elastic actin gel
driven by fracture was observed in Reference [39] (see Figure 1.8). Saltatory
motion can also occur in this case, if time scales are such that the ruptured
gel can regrow before the bead moves.
1.4.6 Limitations of the Approach and Possible Improvements
In the discussion developed above, we have kept only diagonal stresses. How-
ever, as soon as the gel thickness is inhomogeneous, some of the elastic energy
is released by shear, and a complete analysis should contain the corresponding
terms. One can show, however, that the exposed results are not changed in
any significant way [37]. Furthermore, whenever the anisotropy changes with
time, the gel redistribution causes friction at the bead/gel interface, a phe-
nomenon that is not included in the present analysis. It is possible in fact
to show that it does not modify the structure of the equations but simply
renormalizes the onset time
τ
0
of the anisotropy.
Indeed, under such circumstances, the tension has an angular dependent
part, which must be proportional to the friction coecient, the velocity of the
gel relative to the bead, and have the right dimensions
T
(
θ
)=
T
i
+
κξr
d
ε
d
t
cos
θ,
(1.36)
where
κ
is a dimensionless number and
ξ
is the bead/gel friction coecient
already discussed. The equations are formally unchanged and only the onset
time of the modulation is modified to a new value
τ
0
=
τ
0
+
r
2
v
i
e
2
e
0
.
1
e
∗
+
(1.37)
