Contents - Max-Planck-Institut für Physik komplexer Systeme

2.7 Reorientation of Large-Scale Polar Order in Two-Dimensional Tissues
DOUGLAS STAPLE, MATTHIAS MERKEL, FRANK JÜLICHER
Planar cell polarity. During the development of an
organism from a fertilized egg, cells multiply by cell
division and organize in space to form complex morphologies.
An important situation is the formation of
epithelia which are sheet-like, two-dimensional packings
of cells. Cells often exhibit a structural polarity in
the plane of the epithelium. Cell polarity implies that
a vectorial asymmetry exists and the cell morphology
defines a direction in the plane of the epithelium, see
Fig. 1. Typically, the polarity of individual cells is locally
aligned and large-scale patterns of cell polarity
emerge during the development of an epithelium [1].
The mechanisms by which cell polarity patterns are dynamically
reorganized are unknown. Furthermore, a
fundamental problem is to understand how patterns
of planar cell polarity (PCP) that are ordered on large
scales emerge at early stages of development.
A B
Figure 1: (A) Hair pattern on the adult wing of the fly exhibiting
large-scale order with wing hairs pointing towards the distal end of
the wing. Arrows indicate the hair orientation in different regions.
(B) Higher magnification of region in (A) reveals individual wing
hairs. (Experimental images courtesy of Suzanne Eaton, MPI-CBG.)
A
B
Figure 2: (A) Schematic representation of the distribution of planar
cell polarity (PCP) proteins along cell bonds. Distal and proximal
proteins are shown in red and blue, respectively. In a vertex model
cell shapes are described by a polygonal network of cell bonds. We
describe levels of polarity proteins on a bond i by variables σα i , where
α is a cell index. (B) The distribution of polarity proteins defines a polarity
vector in each cell, represented by an arrow. In simulations of
the vertex model, we find that large-scale polar order emerges during
growth.
Genetic and cell biological studies have revealed that a
family of proteins called PCP proteins play a key role
in the dynamic organization of polarity patterns. Mutants
in these proteins lead to the formation of polarity
defects such as swirls [1]. These PCP proteins are recruited
along the junctions between neighboring cells
where they form complexes and aggregates that span
from the membrane of one cell to its neighbor. Furthermore,
within a given cell, subsets of these proteins are
typically located at opposite sides of the cell, implying
a polar pattern of PCP proteins in the cell, see Fig. 2.
The wing of the fruit fly Drosophila provides an important
model system for the study of epithelia and
the development of planar polarity patterns. The precursor
of the wing in the larvae and the pupa undergoes
dynamic remodeling processes. The patterns of
cell polarity in the wing epithelium become apparent in
the orientation patterns of hairs that grow on the adult
wing, see Fig. 1. In close collaboration with experimentalists,
we have quantified the planar polarity patterns
in the fly wing at different stages of development, and
have identified principles underlying the reorientation
of planar polarity patterns. In particular, we find that
cell rearrangements and cell flows play a key role in
guiding polarity reorientation.
Polarity patterns and cell flows. Using flies expressing
fluorescently labelled PCP proteins, we measure
the polarity patterns on the single cell level. The signatures
of cell polarity can be characterized by a nematic,
which is a traceless symmetric tensor. Locally
averaged PCP patterns obtained by this method are
shown in Fig. 3. This analysis revealed for the first
time that large-scale polarity exists already at early pupal
stages of the fly wing, see Fig. 3A. We find that
this initial pattern is subsequently reoriented in a process
that takes about one day. It eventually leads to
the previously known polarity pattern of wing hairs
in the adult wing, see Fig. 3B. During this reorientation
process the tissue is remodeled by cell rearrangements.
We have quantified these movements by measuring
the time-dependent cell flow field v(r,t), see
Fig. 3C. Inhomogeneities of the cell flow give rise to local
rates of rotation ω = (∂xvy − ∂yvx)/2, compression
C = (∂xvx + ∂yvy)/2 and shear. Shear is characterized
by a traceless symmetric velocity gradient tensor:
S1 S2 cos 2θs sin2θs
= ks
, (1)
S2 −S1 sin 2θs −cos 2θs
which defines a shear axis at an angle θs and a shear
rate ks. In general, flow profiles are expected to reorient
polarity and we developed theoretical approaches
to study these phenomena.
54 Selection of Research Results