Biomedical Engineering Reference
In-Depth Information
phases and the rheology of the system. The filaments diffuse in the solution and can
be cross-linked by both active and stationary protein clusters. Active cross-linkers
are small clusters of motor proteins that use chemical fuel as an energy source to
generate forces and torques on the filaments, sliding and rotating filaments relative
to each other [2, 12]. In addition, other small proteins, such as
α
-actinins, act as
stationary cross-linkers and induce filament alignment [1].
As in passive solutions of rigid filaments, the large scale dynamics can be de-
scribed in terms of a set of hydrodynamic equations for continuum fields that relax
on time scales much longer than microscopic ones. These include the conserved vari-
ables of the systems, as well as any field associated with broken symmetries. Various
forms of these equations have been written down phenomenologically by other au-
thors. What distinguishes our work from these phenomenological approaches is that
we derive the hydrodynamic equations from a mesoscopic model of coupled motor-
filament dynamics. This allows us to estimate the various parameters in the hydro-
dynamic equations (that are undetermined in the phenomenological approach) and
relate them to quantities that can be controlled in experiments. To make contact
with the existing literature, we first present the equations and then discuss their
derivation via coarse-graining of a Smoluchowski equation for rigid rods in a viscous
solvent.
The conserved densities in a suspension of interacting filaments (rods) in a
solvent are the mass densities of filaments (rods)
ρ
r
(
r
,t
)=
m
c
(
r
,t
) and solvent
ρ
s
(
r
,t
), and the total momentum density
g
(
r
,t
)=
ρ
(
r
,t
)
v
(
r
,t
) of the solution
(rods+solvent), with
v
(
r
,t
) the flow velocity and
ρ
(
r
,t
)=
ρ
s
+
ρ
r
the total density.
Here,
c
(
r
,t
)isthe
number
density of rods and
m
the mass of a rod. The conserved
densities satisfy conservation laws, given by
∂
t
ρ
=
−
∇
·
g
,
(7.1)
∂
t
c
=
−
∇
·
J
,
(7.2)
∂
t
g
i
+
∂
j
(
g
i
g
j
/ρ
)=
∂
j
σ
ij
+
ρF
ext
i
,
(7.3)
where
J
(
r
,t
) is the current density of rods and
F
ext
the external force on the sus-
pension. The stress tensor
σ
ij
is the
i
-th component of the force exerted by the
surrounding fluid on a unit area perpendicular to the
j
-th direction of a volume
element of solution. It includes all forces on a volume of suspension exerted by the
surrounding fluid. It can be written as the sum of solvent and filament contributions
as
σ
ij
=
σ
ij
+
σ
ij
.
(7.4)
The solvent contribution has the usual form appropriate for a viscous fluid,
σ
ij
=2
η
0
u
ij
+(
η
b
− η
0
)
δ
ij
u
kk
− δ
ij
Π
s
(
ρ
)
,
(7.5)
where
η
0
and
η
b
are the shear and bulk viscosity of the solvent,
Π
s
(
ρ
) is the pressure
of the solvent, and
u
ij
is the symmetrized rate of strain tensor:
2
∂
i
v
j
+
∂
j
v
i
.
1
u
ij
=
(7.6)
In the low Reynolds number limit we can ignore the inertial terms on the left-hand
side of the solvent momentum equation, Equation (7.3).
