Biomedical Engineering Reference
In-Depth Information
ment, since the depolymerization mechanism is certainly stress dependent. In
fact, the dynamical equation for the gel density should read
∂ρ
∂t
+
v
·∇
ρ
=
S
−
k
d
ρ,
(1.38)
where
v
is the gel velocity relative to the bead or the bacterium,
S
is the gel
density source localized at or close to the surface, and
k
d
is the depolymeriza-
tion probability per monomer connected to the gel, per unit time. In general,
one expects stress dependence similar to the one already described, that is,
k
d
=
k
d
exp
σ
t
φ
2
a
3
kT
,inwhich
φ
2
=1
/
(
ρa
) is the average area spanned by
a filament, and
a
3
is a model-dependent length. Far from the surface, in an
essentially unstressed comet tail, one does obtain an exponential decrease of
density over a length
L
=
k
d
. Knowing
v
and the comet length, one deduces
k
d
easily. The physical mechanism behind this depolymerization is not obvious:
it could be that actin filaments can spontaneously break anywhere, or that
reticulation points stabilize the structure and provide the rate-limiting step
in the depolymerization, or that there is a one-to-one mapping of the pointed
end density on the connected monomer density. In all cases, a constant av-
erage number of monomers should leave the gel for each event. Note that a
depolymerization from the pointed ends of the filaments cannot generally be
represented by such a mathematical structure. For instance, if the filaments
were parallel on average, all starting from the surface at the barbed end and
with a length distribution, the term would read
v
k
p
−
ρ
,inwhich
p
is
the unit vector in the filaments direction. Indeed, under such circumstances,
a
p
−
a
p
·∇
ρ
is a measure of the pointed end density.
Because of the exponential dependence of the depolymerization coecient
on stress, the length over which the density significantly decreases may become
very short in the presence of such a stress and this mechanism could provide
an alternative interpretation of the steady state in spherical symmetry. Under
such circumstances, the density decrease occurs essentially over a length such
that
σ
t
=
·∇
kT
φ
2
a
3
, or with the scaling laws derived in spherical geometry,
e
kT ρ
Ea
3
r
. The sharpness of the density decrease is controlled by the
ρ
dependence
of
E
. In all reasonable cases, it is quite pronounced. For instance, if
E
is
proportional to
ρ
, which is the case whenever the cross-link angular elasticity
determines the elastic modulus, then
ρ
=
ρ
0
exp
r
0
αL
1
exp
α
(
r
.
−
r
0
)
−
(1.39)
r
0
In this equation,
r
0
,
ρ
0
denote the radius and density at the bead surface, and
α
is a dimensionless number of the order of ten. There is a sharp cutoff for
r
r
0
α
. Also considering that below a threshold density the gel integrity
is totally lost, it is clear that any thickness measurement will give a value
very close to
r
0
/α
, essentially equivalent to the one derived in Section 1.4.
The drawback of this type of presentation is the added complexity. Its merit
−
r
0
