Biomedical Engineering Reference
In-Depth Information
1.4.3 Steady State
To discuss the conditions for steady state, one still needs to express the
monomer concentration at the inner surface
c
i
as a function of its concentra-
tion at infinity
c
∞
. In a first approximation, it is reasonable to assume that
the actin monomer concentration obeys a standard diffusion law. Thus, in
steady state, and for
e
D
∂c
∂r
=const. in the gel and
c
=
c
∞
outside. Monomer conservation further imposes:
l
2
D
∂c
∂r
r
, the flux
j
=
−
d
n
i
d
t
d
n
e
d
t
. These
conditions specify entirely the problem. One finds two regimes connected by
a smooth crossover. For small radii, the inner concentration is essentially
c
∞
,
and the steady state thickness is solution of the equation
=
=
c
∞
k
b
+
k
p
0
−
k
b
0
−
k
p
0
−
exp
(
e
∗
)
2
e
−
2
0
+
e
−
1
+exp
e
∗
e
−
1
+(
e
∗
)
2
e
−
0
.
=
(1.20)
2
Because all the
a
i
scale like
r
, the steady state thickness
e
∗
also scales like
r
. Actually, with numbers relevant to experimental situations one expects
e
∗
e/
10
.
(1.21)
This is what is observed experimentally for radii smaller than 10 microns
[15]. Such numbers imply that the normal stress exerted by the gel on the
bead is of the order of one atmosphere.
Note that if the leading term is provided by the depolymerization at the
barbed end, one finds exactly the expression announced in Section 1.3.2 that
is (with transparent notations),
e
∗
=
e
2
log
v
p
v
dp
.
The other limit corresponds to what happens on a flat surface. Then no
stress develops, but the thickness is still limited by the monomer depletion
due to the need for the monomers to diffuse in from outside. Now, the steady
state condition reads simply
l
2
D
c
∞
−
=
c
i
k
b
+
−
c
i
=
k
p
−
k
b
−
.
(1.22)
e
∗
For all practical purposes,
k
b
−
can be neglected in this stress-free situation. It
is then easy to infer
e
∗
=
l
2
Dc
∞
k
p
−
.
(1.23)
For large enough beads, this regime is always obtained. The crossover
radius between the two regimes is given by
r
e
a
1
/
2
Dl
3
c
∞
E
1
/
2
(
kT
)
1
/
2
k
p
−
. Plugging
the value of the diffusion constant as measured in solution, we estimate the
crossover radius to be in the millimeter range. It turns out that one can clearly
observe the two regimes, which implies that the monomer diffusion constant
is about the same as in cells [36]. There may be many reasons for this large
