Biomedical Engineering Reference
In-Depth Information
difference (temporary fixation sites on the gel, steric hindrance, other objects
like the Arp2/3 complex diffusing slowly and limiting the polymerization rate).
What is more important is that the measured value (
D
.
02
μ
m
2
s
−
1
)is
such that diffusion processes cannot be neglected in cells. The crossover length
corresponds precisely to typical cell lengths.
1.4.4 Growth with Spherical Symmetry
The arguments developed above allow us to write the gel thickness growth as
=
a
k
b
+
exp
exp
,
e
2
e
0
e
2
e
0
d
e
d
t
k
p
0
−
−
−
−
(1.24)
where we have omitted the
k
b
−
term for the sake of simplicity. The solution of
this equation, with initial condition
e
= 0, is a continuous monotonic function
with essentially two regimes
k
p
0
−
at short times, the growth is predicted to be linear:
d
d
t
a
(
k
b
+
−
•
), that
is with obvious notations,
e
=
v
p
t
. Such a relation is probably too naive
because it ignores the problems of the nucleation of filaments and of their
multiplication with Arp2/3, etc.
at long times, the gel thickness is close to its steady state value
e
∗
and the
dynamical equation can be linearized as a function of
δe
=
e
•
e
∗
.The
−
solution is then
δe
(
t
)=
δe
(
t
1
)exp
t
−
t
1
−
(1.25)
τ
τ
−
1
=
k
b
+
2
e
∗
a
exp
=
2
e
∗
e
0
v
p
.
(
e
∗
)
2
e
0
a
e
2
1
e
2
+
−
+
(1.26)
e
0
The thickness approaches its steady state value exponentially with a time
constant growing linearly with the sphere radius
r
.
1.4.5 Symmetry Breaking
Next, we consider the transition from a spherical gel to the symmetry-broken
state. In the following, we keep the arguments as simple as possible, and deal
only with the thin shell regime. A more elaborate version can be found in [37].
As understood first by K. Sekimoto, the important point is that mechanical
equilibrium requires that the total integrated tension across the gel thickness
must be constant everywhere. The tensile stress must be larger in regions of
smaller gel thickness because the integrated tension is the same. Now, because
the depolymerization rate at the exterior surface depends exponentially on the
tensile stress, this means that the depolymerization is faster where the gel is
