Biomedical Engineering Reference
In-Depth Information
treated with a suitable actin nucleator, the other side is not. As we will see in
Section 1.5.3, approaching conditions have been experimentally obtained [25].
Actin polymerizes on the treated side only, a comet develops, and the disk
moves, see Figure 1.3. The argument that we used to describe the bacterium
motion is exactly the same provided we replace
v
b
by
v
d
the disk velocity. The
conservation of the polymerized actin flux tells us that
v
is the polymerization
rate. There are slight corrections due to a possible strain relaxation of the gel,
discussed in [10]. With this proviso, one can write:
v
v
p
=
a
(
k
b
+
c
i
−
k
b
−
).
Figure 1.3.
Intheabsenceofexternalforce,aflatdiskmoveswiththepolymer-
ization velocity
v
=
v
d
− v
c
v
p
. Note, that motion also occurs if the disk is larger
than the comet tail.
We will describe the stress dependence of the polymerization process in
more detail, but we know experimentally that forces have to reach values of the
order of piconewtons per filament to significantly influence the polymerization
process. All we need to know here is that polymerizing filaments exert a force
of
f
=
σl
2
on the nucleator, where
σ
is the stress normal to the surface
and
l
2
is the average area per polymerizing filament. Where does the stress
come from in this example? In view of the small velocities we are considering
here, the actin gel is to a very good approximation in mechanical equilibrium.
Hence the divergence of the stress vanishes and the sum of all forces exerted
by the gel on its external surrounding add up to zero. This implies that the
force exerted by the gel on the disk is exactly equal and opposite to the force
exerted by the fluid on the gel. The condition of a constant polymerization
rate requires a constant stress, and in the absence of external force we obtain
ζ
c
v
c
πR
2
ζ
d
ζ
c
v
(
ζ
c
+
ζ
d
)
πR
2
ζ
d
v
πR
2
,
σ
=
=
(1.3)
and
ηvl
2
/R.
f
(1.4)
