Contents - Max-Planck-Institut für Physik komplexer Systeme

A
Ω/ω A
1.4
1.2
1.0
0.8
0.6
0
2π 4π 6π
τω A
B
τ = 0 min
C
τ = 7 min
D
τ = 21 min
T = 28 min T = 39 min T = 23.5 min
arrested
segments oscillating PSM
Arrest Front TA = 28 min
Figure 2: A time delay in the oscillator coupling affects the collective
frequency, the wavelength of gene expression patterns and segment
length. (A) Dimensionless collective frequency Ω as a function of
time delay τ of coupling for coupling strength ε/a 2 = 0.07 min −1 ,
and intrinsic frequency ωA = 0.224 min −1 . Solid lines: stable solutions
of Eq. (2). Dashed lines: unstable solutions of Eq. (2). Blue
dots correspond to the three cases shown in panels (B-D). (B-D) Snapshots
of numerical solutions of the model given by Eq. (1) in a twodimensional
geometry for different time delays as indicated. Color
intensity indicates the value sin θ of the phase θ.
Dynamic instabilities and the effects of fluctuations
Our theory reveals that regions of stable collective oscillations
are separated by unstable modes if the timedelay
of coupling is varied. This suggests that changes
of coupling delay can lead to a disruption of the wave
pattern at the limit between stable and unstable regions.
We have tested these predictions in experiments,
where coupling delays were reduced by overexpression
of the mindbomb (mib) gene. Above a critical level
of Mib overexpression, wave patterns are lost and embryos
die.
A B
period2π/Ω (min)
28
24
wt
des
aei
sat. DAPT
mib
wt +Mib
20
15 20 25
delay τ (min)
30
T/T(0)
1.2
1.1
1.0
0 20 40 60 80 100
DAPT concentration (μM)
Figure 3: (A) Collective period Ω as a function of time delay τ for
different coupling strengths (solid lines). The symbols indicate operating
points for wild-type and different mutants as indicated. At
saturated DAPT concentration no coupling exists (yellow). The blue
circle corresponds to Mib overexpression discussed in Fig. 4. (B) Experimentally
determined collective period T = Ω/2π (symbols) as
a function of DAPT concentration, which is a drug that influences
coupling strength. The theoretical prediction of the delayed coupling
theory is shown as a solid line.
If the instability is approached, starting from stable patterns,
we observed precursors of the dynamic instability
in the fluctuations of the pattern. In the vicinity of
the instability, correlation functions of oscillator phase
change their shape as the wave patterns becomes increasingly
noisy. The same signatures of the approach
to instability are also found in numerical simulations of
the system if we introduce noise terms. We can quantitatively
compare correlation functions obtained in Mib
overexpression experiments with noisy simulations of
our model, see Fig. 4.
A
C
autocorrelation (a.u.)
wild-type Mib overexpression
D
4.5
3.3
3.5
experiment
experiment
simulation
2.5
0 2 4 6 8
distance (cell diameters)
B
autocorrelation (a.u.)
E simulation F
2.3
experiment
τ = 21 min τ = 16 min
1.3
0 2 4 6 8
distance (cell diameters)
simulation
experiment
simulation
Figure 4: Coupling delays regulate the stability of the segmentation
clock. (A,B) Representative experimental patterns of the cyclic gene
dlc in wild-type (A), and Mib overexpression (B) conditions. Data
courtesy of A.C. Oates and L. Herrgen. (C,D) Average autocorrelation
function of spatial patterns in regions indicated by red boxes in
(A,E) and (B,F) respectively. (E,F) Snapshots of numerical simulations
of the model given by Eq. (1) with noise for wild-type parameters
(E) and shorter coupling delay (F).
Discussion The wave-like patterns of gene expression
that occur during vertebrate segmentation represent
an important example of the collective organization
of cells during development. We have shown that
generic features of this process can be understood by
using a simple coarse grained description of phase oscillators.
This theory provides several interesting predictions
that we could confirm in quantitative experiments.
In addition, the process of segmentation provides
an important example for the collective behaviors
of nonlinear oscillators for which coupling involves a
time delay. This example also involves moving boundary
conditions and inhomogeneous frequency profiles.
These features, which are absent in most earlier theoretical
studies of collective oscillator behaviors, introduce
new regimes of rich dynamic behaviors.
[1] O. Pourquié, Science 301 (2003) 328–330.
[2] H.P. Schuster, G. Wagner, Prog. Theor. Phys. 81 (1989) 939–945.
[3] L.G. Morelli, S. Ares, L. Herrgen, C. Schröter, F. Jülicher,
A.C. Oates, HFSP J. 3 (2009) 55–66.
[4] L. Herrgen, S. Ares, L.G. Morelli, C. Schröter, F. Jülicher,
A.C. Oates, Curr. Biol. 20 (2010) 1244–1253.
2.6. Delayed Coupling Theory of Vertebrate Segmentation 53