Biomedical Engineering Reference
In-Depth Information
In the second regime, the thickness saturates to a value
e
∗
controlled either
by the developed stresses or by the actin monomer diffusion process. Then,
v
=
v
i
(
e
∗
)
3
r
2
L
.
(1.11)
We will show that under appropriate circumstances
e
∗
=
e
2
log(
v
p
/v
dp
),
in which
e
2
is a length proportional to the bacterium radius
r
,and
v
p
,v
dp
are the polymerization and depolymerization rates in the absence of stress,
respectively. In this case, the dependence of the bacterium velocity on the
polymerization rate is extremely weak!
1.3.3 Gel/Bacterium Friction and Saltatory Behavior
In the above discussion, we have used the notion of surface friction without fur-
ther justification. The physical nature of this friction may be understood the
following way: during the polymerization process, the actin filaments spend
some time
τ
c
connected to the bacterium surface, and some other time
τ
d
disconnected to it. When the gel moves with respect to the bacterium, the
connected filaments gradually distort until they detach. A force results from
the distortion, as first understood by Tawada and Sekimoto [30]. The aver-
age force per unit area reads
F
f
=
n
c
φ
, where
n
c
is the average number
of connected filaments and
φ
is a typical force per filament. In steady state:
n
c
=
n
τ
c
τ
c
+
τ
d
, where
n
is the number of enzymes per unit area on the bacterium
surface. The force
φ
is simply given in terms of the product of a filament elastic
modulus
K
multiplied by a typical displacement
vτ
c
.
A first regime of small velocities is easy to discuss. Both
τ
c
and
τ
d
have
their intrinsic thermodynamic value,
τ
c
and
τ
d
, and the notion of a friction
coecient with a velocity independent value emerges as anticipated:
(
τ
c
)
2
τ
c
+
τ
d
ξ
=
nK.
(1.12)
If one estimates the gel elastic modulus on dimensional grounds by
E
kTλ
p
/λ
4
, and the filament surface modulus by
K
kTλ
p
/λ
3
,inwhich
λ
p
10
μ
m is the actin filament persistence length and
λ
the average dis-
tance between cross-links, then the intrinsic velocity takes the very simple
form:
v
i
(
τ
c
+
τ
d
)
λn
(
τ
c
)
2
, which further simplifies to
λ
(
τ
c
+
τ
d
)
(
τ
c
)
2
v
i
.
(1.13)
Although it is possible to have reasonable values of
λ
, nothing is known on
the connected and disconnected times.
The second regime is that of high velocities. The connections are broken
in times much shorter than the thermodynamic connection time
τ
c
, such
