Biomedical Engineering Reference
In-Depth Information
we present here to give a more complete “biophysical” picture of motility in cells in
which the laws of physics provide important constraints on the possible “system”
dynamics. Finally, even within our limited frame of reference, we will also make a
number of simplifying assumptions in developing the models. Some of the important
physical phenomena ignored here, such as active polymerization, treadmilling [19, 20]
and filament flexibility [21, 22, 23, 24, 25, 26, 27], will be incorporated in future work.
We first review some recent theoretical approaches to describe active filament
suspensions. We then describe some of our current work and give perspectives for
the future.
7.2 Theoretical Modeling of Active Systems
There have been a number of recent theoretical studies of the collective dynamics of
mixtures of rigid filaments and motor clusters. First and most microscopic, numeri-
cal simulations with detailed modeling of the filament-motor coupling have yielded
patterns similar to those found in experiments [11, 13], including vortices and asters.
These simulations modeled the filaments as elastic rods with motor clusters being
parameterized by three binding parameters, the on and off rates, and the off-rate
at the plus end of the filament. It was found that the rate of motor unbinding at
the polar end of the filaments plays a crucial role in controlling the vortex to aster
transitions at high motor densities [12].
A second interesting development has been the proposal of “mesoscopic” mean-
field kinetic equations first studied in one dimension [28, 29], where the effect of
motors is incorporated via a motor-induced relative velocity of pairs of filaments,
with the form of such a velocity inferred from general symmetry considerations.
Kruse and collaborators [30, 31, 32] proposed a one-dimensional model of filament
dynamics and showed the existence of instabilities from the homogeneous state to
contractile states [30] and traveling-wave solutions [32]. We generalized the kinetic
model to higher dimensions [33, 34] and used it to classify the nature of the ho-
mogeneous states and their stability [35, 36]. Related kinetic models have also been
discussed by other authors [37, 38, 39, 40].
Finally, phenomenological continuum theories have been proposed where the
mixture is described in terms of a few coarse-grained fields whose dynamics are
inferred from symmetry considerations [41, 43, 42, 44, 45, 46, 47, 48, 49, 50, 51].
Lee and Kardar [43] proposed a simple hydrodynamic model for the coupled dy-
namics of a coarse-grained filament orientation and the motor concentration, ignor-
ing fluctuations in the filament density. These authors argued that filament growth
by polymerization provides a mechanism for an instability of the system from an
isotropic to an oriented state [43, 42], with large-scale aster and vortex structures.
They obtained a phase diagram for the system showing a transition from vortices
to asters. This model was subsequently generalized by Sankararaman et al. [44] to
include varying populations of bound and free motors, as well as an additional cou-
pling of filament orientation to motor gradients. The effects of boundary conditions
on the steady states of the system was also studied numerically.
A phenomenological hydrodynamic description for polar gels and suspensions
including momentum conservation has been discussed by several authors [45, 46,
47, 48]. These equations generally consider incompressible suspensions and incorpo-
rate momentum conservation in the Stokes approximation, by assuming the form
