Contents - Max-Planck-Institut für Physik komplexer Systeme

A B C
50 µm 15 hAPF 30 hAPF 16 hAPF
Figure 3: (A and B) Patterns of nematic and vectorial order in planar cell polarity (PCP) protein distributions quantified from microscope
images of developing fly wings in the pupal stage. The magnitude and axis of nematic order averaged over groups of cells is represented by
yellow bars at early times (A) and later times (B). Times is given in hours after puparium formation (hAPF). (C) Pattern of cell flow computed
from time-lapse experimental images. Local averages of cell velocity are indicated by arrows.
Theory of planar cell polarity dynamics. We describe
the dynamics of cell polarity reorientation on
two different scales: the cellular scale and a hydrodynamic
continuum limit. On the cellular scale, we use
a vertex model for cell shapes and cell mechanics [2, 3]
in which we introduce bond variables σα i to describe
PCP distributions [4], see Fig. 2. Each bond i possesses
two variables σ α i
and σβ
i
that correspond to proteins on
bond i in adjacent cells α and β, respectively. The dynamics
of cell bond variables is governed by a potential
function
E = J1
i
σ α i σ β
i
− J2
{i, j}
σ α i σ α j − J3
ǫ α · Q α , (2)
as dσα i /dt = −γ∂E/∂σα i , where t is time and γ is a kinetic
coefficient. Here the parameter J1 describes interactions
that favor alignment of the polarities of adjacent
cells and the parameter J2 describes interactions
that stabilize polarity within each cell. The coupling
strength of cell polarity (represented by the PCP nematic
Qα ) to cell elongation (described by the tensor
ǫα ) is denoted J3.
Using this model, we identify a simple and general
mechanism to generate large-scale polar order, see Fig.
2B. This is achieved by starting from a small number
of cells with random PCP variables and then slowly
growing the network by repeated cell division. At early
times polarity orders in small networks and this order
is maintained during growth. Interestingly, depending
on the parameter regime, no topological defects in the
orientation pattern are generated by this process. Furthermore,
we have studied the influence of shear on
[1] J. A. Zallen, Cell 129 (2007) 1051.
[2] R. Farhadifar, J.-C. Röper, B. Aigouy, S. Eaton, and F. Jülicher, Curr. Biol. 17 (2007) 2095–2104.
α
PCP order, and have shown that shear generally reorients
planar polarity in the vertex model [4].
The reorientation of a polarity field in an inhomogeneous
flow is most easily understood in a continuum
description, valid on large scales [4,5]. Considering for
simplicity a homogeneous polarity pattern, the angle of
polarity θ changes dynamically as
∂θ
∂t = νks sin2(θ − θs) + ω. (3)
This implies that both local shear and rotation reorient
polarity. The effects of shear are captured by a dimensionless
phenomenological coefficient ν. Solving Eq.
(3) for the measured flow field in the fly wing, we show
that most of the observed reorientation of cell polarity
patters can be accounted for by the effects of local rotations
and shear. This comparison between theory and
experiment allowed us to estimate ν ≃ −3. A negative
value of ν implies that polarity preferentially aligns
with the shear axis.
Discussion. We have shown that the dynamic organization
of polarity patterns in tissues results from the
collective behaviors of many cells both in terms of cell
flow and polarity. The emergence of large-scale order
can be facilitated by growth processes that allow the
system to suppress topological defects in the orientation
field, which usually exist in large two-dimensional
polar systems. The flow-induced reorientation of polarity
patterns reported here show that general hydrodynamic
concepts developed originally to describe liquid
crystals also apply in living systems.
[3] D. B. Staple, R. Farhadifar, J.-C. Röper, B. Aigouy, S. Eaton, and F. Jülicher, Eur. Phys. J. E 33 (2010) 117–127.
[4] B. Aigouy, R. Farhadifar, D. B. Staple, A. Sagner, J.-C. Röper, and F. Jülicher, Cell 142 (2010) 773.
[5] P. G. de Gennes and J. Prost, “The Physics of Liquid Crystals”(Oxford University Press, 1995).
2.7. Reorientation of Large-Scale Polar Order in Two-Dimensional Tissues 55