Directed Motility and Dictyostelium Aggregation - Cell Motility - page 87

To understand this new instability, we can again use the MG framework,
but this time augmented by a cell-density field acting as a new dynamical
variable. This was first done by Levine and Reynolds (LR) [59], who analyzed
an augmented MG kinetics model with cell density acting as a new dynamic
field. The density obeyed the advection equation

∂σ
dt

+ ∇ · ( σ v ) = 0 (3.10)

with a velocity determined by

d v
d t

= − Γ v + k ( γ ) ∇ ψ (3.11)

with Γ equal to a relaxation time. The rate k ( γ ) is chosen to be a (rapidly)
decreasing function of the cAMP concentration γ . Because of the nature of
the cAMP pulse, this motion rule corresponds to giving the cells a positive
kick every time a wavefront (but not a waveback) passes. LR showed that this
model exhibits a linear instability to density fluctuations, a sort of excitable
medium version of the Keller-Segel mechanism that leads to the density col-
lapse of bacterial colonies due to chemotactic signaling [12]. That is, regions
of high cell density are more excitable and the waves move more quickly
through them. This deforms the wavefront so that cells tend to detect waves
coming primarily from high-density locations and move accordingly. The mo-
tion leads to even higher density, closing the feedback loop. This instability is
strong enough that within a few wave cycles, noticeable inhomogeneities can
be seen in the density.
The basic mechanism relies on the dependence of wave speed on cell den-
sity. This dependence has been directly measured by Van Oss et al [62] and
the results were in agreement with predictions based on the MG framework
and utilized in LR; namely, speed increases with density. Of course, we have
already discussed the relationship between excitability and density in the con-
text of the wavefield pattern selection process, and the results there are con-
sistent as well with what we have used here to get streams.
Past onset, one can carry out numerical simulations of continuum models
to study the nonlinear streaming state - an example of one such computation is
shown in Figure 3.15. A variety of groups have done this with fairly consistent
findings. To take one specific example, Hofer et. al. [60, 61, 64] utilized the
equation

∂σ
∂t

= ∇ ( μ ( σ ) ∇ σ ) − ∇ ( χ ( r ) σ ∇ γ ) (3.12)

Here, μ ( σ ) is a non-linear diffusivity to take into account random cell mo-
tion and χ , the chemotaxis coecient, depends explicitly on the state of the
receptor variable r . The fact that χ decreases at large r will accomplish the
required rectification of the traveling wave. These authors showed in a series of
papers that this model has the streaming instability and that simulations can