Biomedical Engineering Reference
In-Depth Information
v
0
is the translational diffusivity, D
S
,
v
0
is the speed of the moving rod, D
jj
and
D
?
are the diffusivity in the parallel and perpendicular direction of the rod, m is the
mass of the rod, and the additional fluxes due to collisions are given below.
D
ZZ
sin
2
.
1
2
/Œ‚.sin.
1
2
//
‚.
I
SP
D
sin.
1
2
//
u
1
c
x
1
C
s
u
1
2
u
2
;
u
2
;t
u
2
c
x
1
C
s
u
2
2
u
1
;
u
2
;t
dsd
u
2
;
l
l
C
z
ZZ
sin
2
.
1
2
/Œ‚.sin.
1
2
//
‚.
I
SP
r
D
sin.
1
2
//
sc
x
1
C
s
u
1
u
2
;
u
2
;t
l
2
l
2
C
cos.
1
2
/
c
x
1
C
s
u
2
u
1
;
u
2
;t
dsd
u
2
;
l
2
(26)
D
where
z
u
2
,
1
and are the initial angles of
u
and
u
2
, respectively, before
collision. ‚.x/ is the Heaviside function.
Taking the zeroth moment, the first moment, and the second moment of the
Smoluchowski equation, the transport equation for the rod density, polarity vector,
and the nematic order tensor can be derived [
9
].
These models are developed for dilute to semidilute suspensions of active
filaments and rods in viscous solvents. Inside a cell, the cytoplasm is comprised of
various cytoskeletal filaments, microtubules, and intermediate filaments immersed
in the cytosol. The resulting network structures and buffer solution behave like a
gel. We briefly review new models for active biogels next.
u
3
Models for Active Gels
In active gels, networks of active filaments can form either temporarily or on a
longer timescale. The solvent permeation into the network must be accounted for in
the gel. We next describe several relevant models for active gels briefly.
3.1
Isotropic Active Gel Model
Banerjee and Marchetti proposed a phenomenological model for isotropic active
gels based on a continuum model for physical gels [
7
]. The governing system of
equations are summarized. We denote by
u
the position vector of the network,
v
the
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