Biomedical Engineering Reference
In-Depth Information
(
c
0
)
k
2
.
(7.65)
Density fluctuations become unstable when
z
c
(
k
)
<
0, corresponding to
D
(
c
0
)
<
0
or
c>c
B
,where
z
c
(
k
)=
D
3
D
4
mα −
3
Dv
0
∼
3
D
4
mα
c
B
=
(7.66)
is the concentration above which bundling overtakes diffusion. Using
α
(
b/l
)
u
0
,we
can express the density
c
B
in terms of the activity parameter
μ
a
defined in Equation
(7.40) as
c
B
=
∼
2
μ
a
(
lγ
P
/α
),wherewehaveused
D
r
=
D/
(6
l
2
). A possible location
of this instability line in the phase diagram is shown in Figure 7.4.
9
Ρ
2
Ρ
c
c/c
N
3
Inhomogeneous
B
P
2
P
N
Oscillatory
1
instability
c
I
Μ
NP
N
1
2
3
Inhomogeneous
1
States
c
c
IP
I
IN
Μ
Diffusive
instability
μ
a
1
2
3
μ
x
Figure 7.4.
(color online) The phase diagram of homogeneous states for
μ
s
=0
in the plane of filament density,
c
0
, and motor activity
μ
a
,asdefinedinEquation
(7.40), showing the location of the bundling instability at
c
0
=
c
B
. The horizontal
line at
c
0
=
c
N
for the isotropic-nematic transition crosses
c
IP
at
μ
a
μ
x
=1
/c
N
.
The
c
B
line may lie above the
c
NP
− c
IP
line or cross through the
N
and
I
states,
as shown (
lγ
P
/α
=0
.
1,
a
4
= 50), depending on the value of
lγ
P
/α
,anumerical
parameter to leading order independent of ATP consumption rate. The instability
of the
I
and
N
states is diffusive (dashed line), while the instability of the
P
state is
oscillatory (dotted line).
7.7.2 Nematic State
The continuum variables describing the large-scale dynamics of an active nematic
solution are the density and flow velocity of the solution and the concentration and
alignment tensor of the filaments. For simplicity, we consider only the case where
