Biomedical Engineering Reference
In-Depth Information
rigid links that can walk along the filaments towards the polar end at a prescribed
rate
u
(
s
) proportional to the rate of ATP consumption. Generally,
u
(
s
) varies with
the point
s
of attachment along the filament (0
≤ s ≤ l
). Both active and stationary
cross-links also mediate the exchange of torques between the filaments by acting as
torsional springs of prescribed stiffness,
κ
. Our goal is to obtain a coarse-grained de-
scription of the system, where all the parameters in the hydrodynamic equations are
characterized in terms of
u
(
s
),
κ
, and the density of cross-linkers. Collective effects
arising from multiple cross-linkers are neglected and the density of cross-linkers is
assumed constant for simplicity. We also neglect the dynamics of cross-linkers bind-
ing and unbinding, which occurs on faster time scales than those of interest here, so
that we can treat a constant fraction of them as
bound
. The dynamics of cross-linkers
binding and unbinding was considered, for instance, in [44] and it was found that
varying the rates of motor binding and unbinding did not affect the nonequilibrium
steady states of the active solution. The derivation of the active contributions to the
various fluxes has been presented elsewhere [36] and will be summarized here for
completeness. We also present novel results on the evaluation of the filament contri-
bution to the stress tensor up to terms of first order in gradients of the hydrodynamic
fields.
To proceed, we also make a series of simplifying assumptions on the dynamics
of the solution. First, we assume that the friction between filaments and solvent is
large and the filaments move at the flow velocity
v
=
g
/ρ
of the solution. In many
fluid mixtures, internal friction mechanisms are so strong that the flow velocities of
the two components relax on microscopic time scales to the common value
v
.There
are situations, however, where the relaxation time of the relative momenta of the
two species is slow enough to have a significant influence even on hydrodynamic
time scales. In this case, a two-fluid description is appropriate and useful. Such
a “two-fluid model” of the system (rods and fluid background) will be described
elsewhere, where we will show under which conditions one approaches the one-fluid
model (which is always the true hydrodynamic limit).
We also limit ourselves to the case of incompressible solutions, with
ρ
=
ρ
s
+
ρ
r
=
constant, which requires
∇
·
v
=0
.
(7.16)
Finally, we neglect fluid inertial effect compared to the frictional forces between the
colloidal rods and the solvent. In this limit, the momentum equation (7.3) reduces
to the Stokes equation
∂
j
σ
ij
=
−ρF
ext
i
,
(7.17)
or, in the absence of external forces,
2
v
i
− ∂
i
Π
s
=
−∂
j
σ
ij
.
η
0
∇
(7.18)
Equation (7.18) shows that the flow velocity of the solution is determined by the
stress introduced by the filaments. In turn, the forces that the filaments exert on
each other and on the solvent depend on the flow of the suspension in which they
are immersed, and the problem must be solved self-consistently.
The dynamics of a dilute suspension of rods in the presence of a macroscopic
flow field
v
(
r
) can de described by the Smoluchowski equation for the probability
distribution
c
(
r
,
u
,t
)ofrodswithcenterofmassat
r
and orientation
u
at time
t
.The
Smoluchowski equation describes the mean-field Brownian dynamics of extended
colloidal particles at low Reynolds number, under the assumption that the particles'
