Biomedical Engineering Reference
In-Depth Information
1+
a
4
R
1
−
b
1
1+
2
a
4
R
(1
−
R
)
b
1
1
μ
a
c
NP
=
(7.52)
where
R
=
c
N
/c
IP
. The polarized state at
c
0
>c
NP
has
P
i
=
P
0
p
i
,
(7.53)
Q
ij
=
S
P
(
p
i
p
j
− δ
ij
/
2)
,
(7.54)
with
p
a unit vector in the direction of broken symmetry and
c
0
b
1
b
2
1
−
2
a
1
b
1
c
0
S
0
2
,
2
a
1
a
2
P
0
=
(7.55)
S
P
=
S
0
1
−
2
a
1
a
2
p
0
=2
!
!
!
!
!
!
c
0
b
1
b
2
a
1
c
0
b
1
.
(7.56)
II) For
μ
a
>
1
/c
N
,
c
IP
>c
N
and the polarity of motor clusters renders the
nematic state unstable at all densities and the system goes directly from the I to the
P state at
c
IP
, without an intervening N state. The phase diagram has the topology
shown in Figure 7.3. At the onset of the polarized state the alignment tensor is again
slaved to the polarization field,
Q
ij
=
b
2
a
2
c
0
(
P
i
P
j
−
2
δ
ij
P
2
)
,
and
P
=
P
0
p
.The
value of
P
0
is determined by cubic terms in Equation (7.44) not included here.
1
Ρ
c/c
N
c
NP
P
1.5
N
1
I
c
c
IN
0.5
I
Μ
μ
a
1
2
3
μ
x
Figure 7.3.
(color online) The phase diagram for
μ
s
=0.For
μ
a
>
1
/c
N
,where
c
IN
and
c
IP
intersect, no N state exists and the system goes directly from the I to
the P state (
γ
P
/D
r
= 1 and
a
4
= 50).
For a fixed, but nonzero value of
μ
s
, the phase diagram has the same topology
as shown in Figure 7.3, but with
c
N
replaced by
c
IN
given in Equation (7.50). The
value of
μ
a
, where the three phases coexist, is shifted to a larger value, given by
μ
a
=(1+
μ
s
c
N
/
4)
/c
N
.
Estimates of the various parameters can be obtained using a microscopic model
of the motor-filament interaction of the type described in the Appendix. Us-
ing parameter values appropriate for kinesin (
κ
10
−
22
nm/rad [65]), we esti-
∼
10
−
1
sec
−
1
mate
γ
P
∼
γ
NP
∼
κ/ζ
r
=
κD
r
/
(
k
B
T
a
)
∼
where we used the value
