Contents - Max-Planck-Institut für Physik komplexer Systeme

Figure 3: (a) Linear stability diagram in the absence of flow (Pe = 0)
for ρa = 1, assuming L ≫ 1. Here, d ≡ Ds/Da. Region I: This
is the pattern-forming region. Region II: Stable homogeneous steady
state. Region III: Patterns may form in this region, depending on
initial conditions. Region IV: A homogeneous concentration profile
oscillates in time. Region V: Unphysical region. (b) Linear stability
diagram for Pe = 7, ρa = 1, and ζ∆µ = a/(1 + a).
Interestingly, Pe, L, and ζc are material properties of the
contractile fluid (e.g., an actomyosin cortex or a tissue),
while c0 may be regulated by gene expression. Thus,
increasing (or decreasing) c0 may shift a system from a
homogeneous state to one with steady flow.
The resulting nonhomogeneous steady concentration
profile consists of evenly-spaced symmetric extrema.
The velocity profile, given by v(x) = Pe −1 ∂x lnc,
crosses zero at the peaks and valleys such that fluid
flows into the peaks and out of the valleys. This provides
a mechanism for chemical pattern maintenance
in which a peak in concentration gives an active stress
gradient which drives fluid flow, resulting in advective
flux into a peak balancing diffusive flux out of it. In
this way, active stress-driven fluid flow provides both
the local activation and lateral inhibition that are the
general requirements for spontaneous chemical pattern
formation by at once delivering regulator into a peak
and depleting it from a valley.
Case study 2: ASDM with active stress regulation.
The activator substrate depletion model (ASDM) is a
classic model for biochemical pattern formation first
proposed by Gierer and Meinhardt in 1972 [5]. The
ASDM consists of an activator (with concentration a)
and a substrate (with concentration s) that undergo
chemical reactions with the following properties: 1)
The activator is auto catalytic, consuming substrate in
the process. 2) The activator has a constant degradation
rate. 3) The substrate has a constant production
rate. The simplest form of the chemical kinetics is
Ra = ρa(a 2 s−a) and Rs = ρs(1−a 2 s), where ρa and ρs
are chemical rate constants. The unique homogeneous
steady state is a0 = s0 = 1. When Pe = 0, or in the
absence of active stress, we recover the classic ASDM,
[1] T. Mammoto, D. Ingber, Development, 137 (2010) 1407.
[2] J. S. Bois, F. Jülicher, S. W. Grill, Phys. Rev. Lett., 106 (2011) 028103.
[3] G. Salbreux, J. Prost, J.-F. Joanny, Phys. Rev. Lett., 103 (2006) 061913.
[4] M. Mayer, M. Depken, J. S. Bois, F. Jülicher, S. W. Grill, Nature, 467 (2010) 617.
[5] A. Gierer and H. Meinhardt, Kybernetik, 12 (1972) 30.
whose linear stability diagram is depicted in Fig. 3a. In
region I of parameter space, the substrate acts as a fastdiffusing
“inhibitor” (since its depletion slows the production
of the activator), the hallmark of a Turing instability.
In the presence of advection due to active stress
up-regulation by the activator, the region of parameter
space in which patterns form grows, as depicted in
Fig. 3b. The steady concentration profile features peaks
in activator concentration that contract fluid into them,
providing an influx of activator, as shown in Fig. 4. This
stabilizes the peaks against diffusion in a manner similar
to the previous example, even in regimes where the
activator diffuses faster than the inhibitor. Thus, active
stress-driven flow serves to broaden the patternforming
region of parameter space, giving patterns not
only in concentration of chemical species but also in
mechanical stress and fluid motion.
Summary We have developed a general hydrodynamic
theory of chemically-regulated active stress in
viscous fluids and identified general principles of pattern
formation in such systems. This approach can be
extended to higher dimensions and to include polar
and nematic order, viscoelasticity, and other reaction
systems. In the two scenarios presented here, peaks in
concentration of stress activator are amplified by advective
influx due to active flows. This simultaneously
depletes activator from areas around a concentration
maximum. Thus, active stress-driven flow provides
both the local activation and lateral inhibition required
to form stable patterns. This mechanism leads to pattern
formation even in the absence of chemical reactions.
We conclude that integration of active hydrodynamics
with molecular signaling processes is an important
theme for future studies of biological processes.
Figure 4: Nonhomogeneous steady state for Pe = 7 and parameters
given by the “×” in Fig. 3b with no-flux boundary conditions.
The activator and substrate concentrations are given by the gray and
dashed lines, respectively. The velocity is the solid black curve.
2.24. Pattern Formation in Active Fluids 89