Biomedical Engineering Reference
In-Depth Information
velocities have equilibrated on microscopic time scales to a local Maxwell-Boltzmann
distribution at a temperature
T
a
[59, 60]. The effective temperature
T
a
incorporates
nonthermal noise sources as may arise from fluctuations in motor concentration and
ATP consumption rate. The Smoluchowski equation is given by
∂
t
c
+
∇
·
J
c
+
R
·
J
c
=0
,
(7.19)
where
∂
u
is the rotation operator. The
translational
probability current,
J
c
,andthe
rotational
probability current,
R
=
u
×
J
c
, are given by
D
ij
k
B
T
a
c ∇
j
U
ex
+
J
ci
,
J
ci
=
cv
i
− D
ij
∇
j
c −
(7.20)
D
r
k
B
T
a
cR
i
U
ex
+
J
A
ci
,
(7.21)
where
ω
i
=
ijk
u
j
u
l
∂
l
v
k
.Also
D
ij
=
D
u
i
u
j
+
D
⊥
δ
ij
− u
i
u
j
is the translational
diffusion tensor and
D
r
is the rotational diffusion rate. For a low-density solution
of long, thin rods
D
⊥
=
D
/
2
≡ D/
2, where
D
=
k
B
T
a
ln(
l/b
)
/
(2
πη
0
l
), and
D
r
=
6
D/l
2
.Thepotential
U
ex
incorporates excluded volume effects that give rise to the
nematic transition in a solution of hard rods. It can be written by generalizing the
Onsager interaction to inhomogeneous systems as
k
B
T
a
times the probability of
finding another rod within the interaction area of a given rod. In two dimensions,
this gives
J
ci
=
cω
i
− D
r
R
i
c −
U
ex
(
r
1
,
u
1
)=
k
B
T
a
d
u
2
s
1
s
2
|
u
1
×
u
2
| c
(
r
1
+
ξ
,
u
2
,t
)
,
(7.22)
where
s
i
,with
−l/
2
≤ s
i
≤ l/
2, parameterizes the position along the length of
the
i
-th filament for
i
=1
,
2, and
s
i
... ≡
l/
2
−l/
2
ds
i
... ≡..
s
i
. The filaments are
constrained to be within each other's interaction volume, i.e., in the thin rod limit
b l
considered here, have a point of contact. The factor
|
u
1
×
u
2
|
represents
the excluded area of two thin filaments of orientation
u
1
and
u
2
touching at one
point [61]. Finally,
ξ
=
r
2
−
r
1
u
2
s
2
, is the separation of the centers of
mass of the two rods. The translational and rotational active current of filaments
with center of mass at
r
1
and orientation along
u
1
are written as
u
1
s
1
−
J
c
(
r
1
,
u
1
)=
c
(
r
1
,
u
1
,t
)
b
2
m
v
a
(1; 2)
c
(
r
1
+
ξ
,
u
2
,t
)
,
(7.23)
u
2
s
1
s
2
c
(
r
1
,
u
1
)=
c
(
r
1
,
u
1
,t
)
b
2
m
J
ω
a
(1; 2)
c
(
r
1
+
ξ
,
u
2
,t
)
,
(7.24)
u
2
s
1
s
2
where
m
is the density of bound cross-linkers and (1; 2) = (
s
1
,
u
1
;
s
2
,
u
2
). Finally,
v
a
(1; 2) and
ω
a
(1; 2) are the translational and rotational velocities, respectively, that
Filament 1 acquires due to the cross-linker-mediated interaction with Filament 2,
when the centers of mass of the two filaments are separated by
ξ
(see Figure 7.1).
The derivation of the form of the active velocities in terms of motor parameters
(the stepping rate
u
(
s
) and the torsional stiffness
κ
) has been discussed in detail
elsewhere [56, 36]. The angular velocity is
ω
a
=2[
γ
P
+
γ
NP
(
u
1
·
u
2
)] (
u
1
×
u
2
)
,
(7.25)
