Biomedical Engineering Reference
In-Depth Information
b is a parameter and m is the mass of the rod. The active velocities are given by
1
2
v
a
D
v
r
C
V
m
;
ˇ
2
.
u
2
˛
2l
;
v
r
D
u
/
C
V
m
D
A.
u
C
u
2
/
C
B.
u
2
u
/;
!
a
D
2Œ
P
C
NP
.
u
u
2
/.
u
u
2
/;
(15)
u
2
/; ˇ
D
ˇ.1
C
˛
D
˛.1
C
where
P
and
NP
are the rotational rates,
u
2
/;
A
D
.ˇ
˛.s
1
s
2
/=2/=12,andB
D
˛.s
1
s
2
/=24 for long thin rods [
4
].
˛ and ˇ are model parameters. All four parameters can be related to the stiffness of
the crosslinkers and to the rate
u
.s/ at which a motor cluster attached at position s
steps along a filament toward the polar end [
72
]. The concentration has a generalized
Fourier expansion
u
u
c.
x
;t/
2
c.
x
;
u
;t/
D
Œ1
C
2
p
u
C
4
Q
W
uu
C
:;
(16)
where the zeroth, first, and second moments are defined by:
Z
c.
x
;
u
;t/d
u
;
c.
x
;t/
D
Z
u
c.
x
;
u
;t/d
u
;
c
p
.
x
;t/
D
Z
uu
I
c.
x
;
u
;t/d
u
:
2
k
uu
k
c
Q
.
x
;t/
D
(17)
2
The force due to the stress for the system is given by
ZZ
c.
x
;
u
;t/
˝
ı.
s
u
/
F
h
.s/
˛
s
dd
u
;
r
D
(18)
h
.s/ is the hydrodynamic force per unit length exerted by the suspension at
position s along the rod. The force can be approximated by the first two terms in the
Taylor expansion
where
F
Z
s
l
2
u
c
h
Z
c.
x
;
u
;t/
F
h
d
u
l
r
D
d
u
:
(19)
s
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