plosPLoS Comput BiolploscompPLoS Computational Biology1553-734X1553-7358Public Library of ScienceSan Francisco, USAPCOMPBIOL-D-10-0024410.1371/journal.pcbi.1002025Research ArticleBiologyComputational biologyBiochemical simulationsBiophysic al simulationsSignaling networksDevelopmental biologyMolecular developmentAdhesion moleculesMorphogensSignalingOrganism developmentPattern formationComputational BiologyDevelopmental BiologyCell-Sorting at the A/P Boundary in the Drosophila
Wing Primordium: A Computational Model to Consolidate Observed Non-Local Effects
of Hh SignalingHedgehog-Mediated Cell SortingSchillingSabine12WilleckeMaria1Aegerter-WilmsenTinri1CirpkaOlaf A.3BaslerKonrad1von MeringChristian12*Institute of Molecular Life Sciences, University of Zurich, Zurich,
SwitzerlandSwiss Institute of Bioinformatics, University of Zurich, Zurich,
SwitzerlandCenter for Applied Geoscience, University of Tuebingen, Tuebingen,
GermanyShvartsmanStanislavEditorPrinceton University, United States of America* E-mail: mering@imls.uzh.ch
Conceived and designed the experiments: SS MW KB CvM. Performed the
experiments: SS MW. Analyzed the data: SS TAW. Contributed
reagents/materials/analysis tools: MW TAW OAC. Wrote the paper: SS CvM.
The authors have declared that no competing interests exist.
4201174201174e1002025811201016220112011Schilling et alThis is an open-access article distributed under
the terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author and
source are credited.
Non-intermingling, adjacent populations of cells define compartment boundaries;
such boundaries are often essential for the positioning and the maintenance of
tissue-organizers during growth. In the developing wing primordium of
Drosophila melanogaster, signaling by the secreted protein
Hedgehog (Hh) is required for compartment boundary maintenance. However, the
precise mechanism of Hh input remains poorly understood. Here, we combine
experimental observations of perturbed Hh signaling with computer simulations of
cellular behavior, and connect physical properties of cells to their Hh
signaling status. We find that experimental disruption of Hh signaling has
observable effects on cell sorting surprisingly far from the compartment
boundary, which is in contrast to a previous model that confines Hh influence to
the compartment boundary itself. We have recapitulated our experimental
observations by simulations of Hh diffusion and transduction coupled to
mechanical tension along cell-to-cell contact surfaces. Intriguingly, the best
results were obtained under the assumption that Hh signaling cannot alter the
overall tension force of the cell, but will merely re-distribute it locally
inside the cell, relative to the signaling status of neighboring cells. Our
results suggest a scenario in which homotypic interactions of a putative Hh
target molecule at the cell surface are converted into a mechanical force. Such
a scenario could explain why the mechanical output of Hh signaling appears to be
confined to the compartment boundary, despite the longer range of the Hh
molecule itself. Our study is the first to couple a cellular vertex model
describing mechanical properties of cells in a growing tissue, to an explicit
model of an entire signaling pathway, including a freely diffusible component.
We discuss potential applications and challenges of such an approach.
Author Summary
In developing animal tissues, cells can often re-arrange locally and mix
relatively freely. However, in some stereotypic and crucially important
instances during body development, cells will strictly not intermingle, and
instead form sharp boundaries along which they will sort out from each other.
This mechanism helps organisms to establish signaling centers and to maintain
distinct cellular identities. Often, cells at such boundaries will remain in
close physical contact and are morphologically alike. Thus, the boundary itself
can be difficult to observe unless the expression status of specific marker
genes is monitored experimentally. How are these ‘compartment
boundaries’ established? Here we devise a computational model that aims to
describe one such boundary in a well-studied animal tissue: the developing wing
primordium of Drosophila melanogaster. We model the production,
diffusion and local sensing of an essential signaling molecule, the
Hedgehog protein. We reveal one possible mechanism by which
Hedgehog sensing can influence the mechanical properties of cells, and compare
the simulated outcome to observations in experimentally perturbed, actual wing
discs. Our relatively simple model suffices to establish a straight and stable
compartment boundary.
This work has been funded by the University of Zurich through its research
priority program “Systems Biology and Functional Genomics”
(http://www.sysbio.uzh.ch/), and by a FEBS Long-term Fellowship
(to MW; http://www.febs.org/index.php?id=83). TAW was supported by the
Roche Research Foundation (http://www.research-foundation.org/), by the SystemsX.ch
initiative (http://www.systemsx.ch/) within the framework of the WingX
Project and by the Forschungskredit of the UZH (http://www.researchers.uzh.ch/). The funders had no role in
study design, data collection and analysis, decision to publish, or preparation
of the manuscript.Introduction
During embryonic development of complex multicellular organisms, spatial reference
points need to be established within tissues. These are often formed by specialized
groups of cells that are capable of signaling to neighboring cells. Such signaling
centers define coordinate systems along which newly arising cells can orient
themselves and make crucial decisions regarding proliferation, differentiation or
migration [1],
[2], [3], [4], [5], [6]. Because of
their pervasive importance, tissue-organizing centers need to be precisely
controlled – both spatially and temporally, as well as with respect to their
signaling amplitude.
One possible mechanism for spatial control of tissue organizers is to restrict the
movement of cells at fixed boundary positions. This phenomenon is indeed observed,
and it involves the separation of groups of cells that have already been spatially
instructed to assume distinct identities, for example at segment- or
parasegment-boundaries. Akin to water in oil, the two cell populations are seen to
establish and maintain a relatively straight interface to each other, effectively
minimizing their contact area. The minimizing force is assumed to help stabilize the
interface against random perturbations that may arise from cell divisions or from
arbitrary cell movements; thus, any organizing activity that is associated with the
interface is likewise stabilized. How is this separation, or ‘sorting’,
of cells of distinct identities achieved? One line of work attributes this to
differential cell adhesion [7], [8]: cell populations might develop distinct adhesive
properties; these affinity differences would then allow them to sort out from one
another. Another line of reasoning is based on Differential Interfacial Tension
(DIT) [9], [10]: this
hypothesis suggests that cells might actively constrict surfaces that are in contact
with neighboring cells, depending on the cellular identity of neighbors and/or
depending on signaling events. Both mechanisms would ultimately lead to physical
forces that would help keep the cell populations apart.
The developing wing primordium of Drosophila (‘wing
disc’) is particularly well suited to study boundary formation (Figure 1). It is not required for
larval viability, can be manipulated experimentally through an advanced genetic
toolkit, and has been well characterized. The disc contains a compartment boundary
that separates anterior from posterior cells; this boundary is inherited from
specification events occurring early in the embryo. The initial embryonic events
that give rise to the boundary involve mutual signaling between stripes of cells,
mediated by an extensively studied network of genes (the ‘segment polarity
network’ [11], [12], [13], [14]). Once established, the cellular identities on both sides
of these boundaries are stable throughout larval development and well into adult
life. The compartment boundary in the disc is strictly respected by all cells, even
when cells on one side of the boundary are artificially provided with a competitive
growth advantage over cells on the other side of the boundary [15]. The wing disc itself is
a simple, flat, epithelial sheet, and the orientations of cell divisions appear
largely random [16]. Genetic analysis and computational modeling of this
tissue is simplified by the fact that daughter cells arising from cell divisions
usually remain in physical contact and do not migrate away from each other. This has
been shown experimentally by tracing descendents of single cells; in most cases such
a ‘clone’ of offspring cells will form a coherent patch of connected
cells. This behavior suggests that the complicated processes of cell intercalation
and migration can be neglected, to a first approximation, when studying boundary
maintenance in this tissue.
10.1371/journal.pcbi.1002025.g001
Two possible mechanisms for cell sorting in Drosophila wing
discs.
In the wing imaginal disc, cells of two distinct identities form the
posterior and anterior halves, respectively. The demarcation line between
them has been termed ‘compartment boundary’. The secreted
signaling protein Hedgehog is emanating from the posterior section, and
cells respond to this signal in the anterior section; this response is
needed to maintain a well-defined separation between the two compartments.
In the boxes to the right, two alternative scenarios for the molecular
events leading to boundary formation are outlined.
Working with such wing discs, a recent, seminal study has begun to shed light on
possible boundary formation mechanisms [16] (see also ref [17]). The authors
have directly demonstrated an increased mechanical tension at cell-to-cell
interfaces located immediately at the boundary, using laser ablation experiments.
Subsequent computer simulations then revealed that collectively such local forces
are sufficient to maintain a stable compartment boundary. These results are
intriguing, but they raise a number of new questions: Boundary formation in the wing
disc is known to depend on the secreted and diffusible signaling protein Hedgehog
(Hh), which is produced by posterior cells and specifically sensed and transduced by
anterior cells [18], [19] (Figure
2). If diffusible Hh indeed somehow influences mechanical tension, what
conditions must then be met to ensure a well-defined boundary? So far, all known
transcriptional responses of Hh signaling are occurring several cell-diameters wide
into the responding tissue. How is the response in this case restricted to the
immediate boundary region? Furthermore, experimental suppression of Hh signaling has
been shown to lead to ectopic boundary formation distant from the actual boundary
[20]. Does
this mean that the influence of Hh signaling does extend beyond the actual boundary,
and if so, why does this not have a noticeable consequence in the wild type
situation?
10.1371/journal.pcbi.1002025.g002
Summary of Hh signaling in the wing disc.
A) The diffusible signaling protein Hh exerts its function by binding to its
receptor Ptc. This relieves the transmembrane protein Smo from inhibition by
Ptc. Smo then signals to the cytoplasm, where the signal is eventually
relayed to the transcription factor Cubitus interruptus (Ci). Target genes
under the control of Ci thus respond to Hh signaling. B) Expression domains
of genes important in the process. Engrailed (en) and ci mark the two
compartments. Hh is produced under the direction of en, and then diffuses
freely. Smo is expressed throughout the disc. Ci is needed to activate or
repress target genes in response to Hh signaling. Ci is not present in the
posterior half, hence Hh target genes cannot be activated there. Immediately
at the boundary, target genes exhibit the largest difference in expression.
C) When small groups of cells are made mutant for the smo
gene, they can no longer activate target genes. When such clones are located
within the target gene stripe, they are known to round off and to tend to
minimize contact with neighboring cells.
Here, we propose a mechanistic model that can generate a localized outcome of Hh
signaling with respect to physical forces and mechanical properties, despite a
longer range of the molecular response in terms of target gene expression.
Furthermore, we estimate the distance from the boundary, up to which Hh signaling
may be able ‘prime’ cells for boundary formation; this distance is
inferred using both experimental results as well as modeling results, and we
estimate it to be at least 10 cell diameters. We approach the problem by first
formulating an explicit, two-dimensional model of Hh production, diffusion and
transduction, and by then coupling this setup to a physical model of the growing
tissue. In our modeling approach, cells and their contact surfaces are described as
a graph of connected vertexes. Our model essentially follows the Differential
Interface Tension hypothesis; it is a modified version of a model that has been
previously established for the very same tissue [21]. We observe good
compartment boundary formation over a range of simulation parameters, and the
modeling outcomes agree qualitatively with experimental perturbations specifically
performed for this study.
Results/Discussion
In principle, at least two distinct molecular scenarios could explain the local
generation of tensile forces at the boundary (Figure 1). In the first scenario (ref [16]), two
different cell-surface molecules would form a heterotypic interaction at the
boundary; their expression would essentially be under the control of the anterior or
posterior “identities” of cells on either side of the boundary. The
heterotypic interaction of these two molecules would be sensed locally at the
cell-interaction interfaces, which would then respond by generating increased
physical tension. This is a simple and attractive model, but it is not
straightforward to consolidate with the known requirement, on the anterior side, for
reception and transduction of the Hh signal. Loss of Hh transduction can generate
ectopic boundaries in the anterior compartment ([20], this study), but it is
generally not presumed that such loss of Hh signal will change the identity of
anterior cells into that of posterior cells: the expression status of the selector
genes engrailed and ci is not affected by Hh signaling. Thus, if the cell-identity
seems unchanged, then both of the putative cell-surface molecules required for this
model would have to be under Hh control: one as a direct molecular target, and one
as an “inverse” molecular target (i.e., de-repressed upon the loss of Hh
signal). Only in such a setup would loss of Hh transduction lead to ectopic boundary
formation within the anterior compartment. However, target gene de-repression upon
loss of Hh signal has to our knowledge never been reported for any known Hh target
gene, and it would likely require further, more complicated indirect signaling
mechanisms. Alternatively, one might imagine that one of the two molecules was
expressed ubiquitously throughout the tissue, and only the other molecule would be a
target of Hh signaling. However, in such a scenario heterotypic binding would occur
throughout the entire Hh target gene expression domain; increased tension would thus
not be restricted to the immediate compartment boundary only.
We therefore propose an alternative, somewhat more parsimonious scenario (Figure 1): The increased tensile
forces at the boundary would be the consequence of a single cell-surface molecule,
which would be a simple, direct molecular target gene of Hh signaling. This molecule
would be able to transmit a signal to the inside of the cell, but only upon its
activation by a homotypic interaction with molecules of the same type from a
neighboring cell. Crucially, as discussed in more detail below, this signal and its
conversion into mechanical tension would have to be rate-limited: relative tension
would be highest at the section of the cell where most of the molecule has been
activated, but the overall tension per cell would be constant (i.e. independent of
the absolute amount of activated cell-surface molecule).
To accurately take into account the role of Hh signaling in the disc (Figure 2), we first devised a
formal model for the production, diffusion and transduction of Hh in this
two-dimensional tissue. The model (Figure 3) includes the Hh receptor Patched (Ptc), as well as the
essential downstream signaling component Smoothened (Smo), together with an unknown,
putative co-factor of Smo; this co-factor is not further specified but has been
speculated to be a lipid [22], [23]. The Hh protein is allowed to freely diffuse throughout
the tissue, following its production in posterior cells. For each individual cell
within the tissue, we compute the concentrations of the modeled entities as they
develop in time by numerically solving a set of partial differential equations
(Figure 3B). Apart from
known or suspected players in Hh signaling, we implement an additional, putative
target gene of Hh signaling, which we term “TMx”. Unlike Ptc, this gene
is presumed to play no active role in the signaling pathway itself, instead it is a
downstream target of the pathway and does not feed back into the sending or
receiving of the Hh signal. We assume this gene to have the simplest possible
connection to Hh signaling, namely a production term proportional to the amount of
active Smo molecule in anterior cells. We further assume that the product of this
gene has a function in regulating cortical tension at the inner surface of cells. We
do not specify the molecular mechanism by which it regulates tension, but one could
for example envisage TMx being a transmembrane protein whose intracellular domain
recruits or otherwise influences cortical actin filaments [24]. Since TMx is modeled as a Hh
target gene, it provides a way to connect transcriptional Hh responses to physical
forces acting on cell shapes (Figure
3C). Our model is based on three central assumptions with regard to TMx:
first, that it would increase cortical tension only in response to homotypic
activation, i.e. upon binding another TMx molecule presented on the surface of a
neighboring cell. Second, that it cannot increase the overall
propensity of the cell for exacting cortical forces, but instead merely
re-distributes cortical tension factors among the various interfaces that a given
cell has with its neighbors. Again, we do not specify why this might be the case,
but one could envisage a dynamic equilibrium of cytoskeleton filament deposition,
and removal, at the cortex. In such a situation, each section of the cellular
surface competes with all other sections within the same cell for the build-up of
cytoskeleton material, and activated TMx might simply tip the balance towards
deposition, locally. Lastly, the TMx molecule itself (while initially expressed
isotropically) would enrich at cell surfaces at which it is activated by homotypic
binding, perhaps because it is stabilized or preferentially re-deposited there.
Thus, the overall effect of TMx would be that it changes the relative strength of
contractile forces at each individual cell/cell contact segment; we model this as
scaling factor within the line tension term of the physical energy function (Figure 3C).
10.1371/journal.pcbi.1002025.g003
The modeling strategy.
A) Mechanistic modeling of Hh signal transduction. Only events in the
anterior compartment are shown. Dotted arrows denote the decay of either
single molecules or complexes of molecules; the reversible formation of
complexes is denoted by groups of three grey arrows. B) Differential
equations describing the Hedgehog pathway. The mechanistic modeling depicted
in A) is translated into a set of differential equations. This equation
system is solved numerically for each cell after each growth step. C)
Mechanical energy function describing cell shapes. Similar to ref. [21], we
describe the tissue as a two-dimensional mesh of cells, whereby edges
between vertices denote interaction interfaces between neighboring cells.
Stable network configurations are defined by local minima of the energy
function describing the separate contributions from the line tension between
two vertices, each cell's area elasticity and the elasticity of each
cell's perimeter [21], [37]. However, the term
describing line tension (i.e. the energy ‘stored’ in a given
edge) has been modified to include an additional scaling factor
, which depends on the ratio of the concentrations of
“TMx” in the two cells sharing the edge
<ij>. A detailed derivation of the scaling factor
is given in Figure S3. D) Outcome of a simulation run. Simulations were
started from 220 cells, growing up to 6000 cells. Left: Concentration
gradient of the transmembrane protein TMx. The inset shows a magnified view
of cells at the compartment boundary. In the control, the TMx input into the
energy function has been disabled; note that in this case the boundary is
quite irregular, due to disturbances by random cell division events. Videos
of typical simulation runs for both control and experiment are provided in
the Supplemental Information of this article (Videos
S1 and S2).
For our implementation of the full model, a challenge was to accurately compute the
two-dimensional diffusion of the Hh protein on a geometry that is itself constantly
changing. We achieve this by alternating the mechanical relaxation/growth
computations with an explicit diffusion of Hh on finite volumes established by the
shapes of the cells (see Supplemental Material, Text S1). It
should be noted that our model does not address questions related to overall
regulation of tissue growth or to the determination of final organ size (nor does it
address issues of correct developmental timing). Detailed models for growth control
and mechanical forces affecting the tissue as a whole have been developed already
[25],
[26], [27], but
they do not need to be applied here because our readouts are local, and because we
stop the simulations well before the tissue would normally cease growing.
Having specified the model, we next set out to parameterize it. Experimentally
quantified data regarding the various kinetic parameters in Hh signaling are
difficult to obtain and are at present quite sparse. We therefore focused our
parameter exploration and validation on the modeled shapes of the
various concentration gradients in the tissue (rather than on the absolute molecular
concentrations); these shapes are already much better known, mainly from antibody
staining experiments. For simplification, we performed parameter exploration in one
dimension only, by projecting molecular concentration gradients along an
anterior-posterior transect of the tissue (Figure S1). The Ptc protein in particular served
as a guide for our manual parameter optimization – it is itself a target gene
of Hh, and its expression and activity gradients are understood comparatively well
[28]. As is
shown in Figure
S1, our model resulted in the characteristic up-regulation of Ptc in a
small stripe of cells anterior to the boundary. Remarkably, the Ptc protein
concentration gradient shows an approximately sigmoidal shape when projected along
the antero-posterior axis, with highest values close to the boundary; this is not
specified in the model as such, but instead follows naturally from the wiring of the
pathway, with Ptc being both the receptor and a direct target gene of Hh. Because
the parameter space of our model is fairly large, and each simulation run takes
several CPU hours, a fully systematic scan of the possible parameters is difficult.
Instead, we explored the parameter space manually. Thus, our parameter set should
and will be updated as experimental data on concentrations and kinetic constants
become available; any updates will again have to reproduce the known shapes of
concentration gradients in the tissue. Initial test runs of our model revealed that
several parameter sets resulted in the formation of a stable lineage boundary at the
anterior/posterior interface (see for example Figure 3D and Figure 4). The resulting overall tissue-shapes
often revealed a small constriction of the tissue margins at the position of the
boundary, suggesting that the boundary exerts long-range mechanical forces on the
tissue as a whole, as might be expected (Figure 3D).
10.1371/journal.pcbi.1002025.g004
Experimental and simulated disruptions of Hedgehog signaling.
The disruption of the crucial transduction protein Smo is an important assay
revealing the functional roles of Hh signaling. A) Confocal microscopy image
of a wing pouch with clones of cells mutant for smo.
Posterior cells are marked in blue (via a hh-lacZ enhancer
trap construct), cell outlines are marked in red (cadherin staining), and
loss of smo is indicated by loss of GFP staining (i.e.
absence of green color). Most clones are accompanied by a twin-spot (having
two functional smo genes; bright green). Notice how an
anterior clone close to the boundary has a ‘rounded’ appearance
and seems to minimize contact with neighboring cells (marked by a white
arrow). Another clone of anterior origin (i.e. absence of blue staining)
happened to originate in immediate contact with the boundary. It has
migrated into posterior territory, lost its roundness, and its leftward edge
now constitutes a new boundary interface. B) Simulated wing pouch with
clones of cells mutant for smo (marked in red) and their
corresponding twin spots (marked in bright green). Simulations reproduce the
experimentally observed ‘rounded’ appearance of clones in the
boundary region of the anterior compartment as well as the migration of
anterior clones close to the compartment boundary into the posterior
compartment (see also Video S3). C) – F) Roundness of
clones is used as an indirect measure of the strength of ectopic boundary
formation. In both experiments (C) and simulation (D),
smo− clones in the anterior
compartment show a highly significant trending for decreasing roundness away
from the boundary. In both cases, roundness is above background levels for
at least 20 µm (equivalent to at least 10 cells). In
controls – i.e. posterior compartment clones, or wild-type twin spots
– no significant trend is observed. For automatic image processing in
C) and D), clones were required to be located at a minimum distance of 5
µm from boundary, thus excluding
clones migrating from anterior to posterior compartment in the analysis of
roundness. C)–F) Straight lines: Moving average of the clonal (red)
and twin spot (blue) shape distributions; Blue dashed lines: Linear fit of
twin spot shapes vs. distance. E)–F): Red dashed lines: Linear fit of
clonal shapes vs. distance in the posterior compartment.
Next, we validated the overall distribution of cell shapes in the simulated tissue,
i.e. the distribution of cells over the various possible polygon classes (i.e.,
number of edges per cell), and the dependency between polygon class and cell surface
area. We based this on published experimental data (cell shape measurements) from
refs [16] and
[21]. This
test further constrained our model parameters (Figure 5). As shown previously, the relative
settings of the main parameters of the energy function (i.e., perimeter elasticity
factor vs. line tension factor ) can be varied over a
certain range, without resulting in much deviation between modeled and measured cell
shapes. In our case, the added requirement of a stable boundary, which should mimic
the actual boundary in the disc, constrained the parameters even further. For
example, we noticed that relaxing the relative strength of the ‘perimeter
elasticity’ parameter (third row in Figure 5) resulted in the best overall appearance
of the boundary; however this was accompanied with a reduced fit to the
polygon-distribution, and with somewhat unrealistic (elongated) cell shapes
immediately adjacent to the boundary. As the best subjective compromise, we
identified the parameter setting and
(first row in Figure 5). At this point in parameter space, we observed the best fit to
known cell sizes and shapes, while at the same time obtaining a fairly straight
boundary (see also the comparison to a negative control in Figure 3D). Immediately at the boundary, our
model posits an approximately twelve-fold difference between TMx expression levels
(anterior cells in row A1 having maximal TMx concentration of roughly 2.4 a.u. vs.
posterior cells in row P1 with a basal TMx concentration of 0.2 a.u.; see Figures 3D and S1D). For
average six-sided cells, this translates to a roughly two-fold increase in line
tension at the boundary (see Figure S3F) – in good agreement with
laser-ablation experiments [16], in which a 2.5 fold increase had been measured.
10.1371/journal.pcbi.1002025.g005
Parameter dependency.
Different parameter choices for the energy function are observed to lead to
compartment boundaries of varying stringency and straightness, but they will
also influence descriptive statistics for cell and clonal shape
distributions. The settings in the first row demonstrate the best overall
fit to experimental data generated here and in ref [16], [21].
For each parameter set, we ran the simulations 10 times without the
insertion of smo mutant cells (columns B–C), and 10
times with mutant cell insertions (column D). A) Zoom
() into the boundary region of a typical simulated
wing pouch. B) Average apical cross-section areas of
n-sided cells as a function of the polygon number
n. Areas are normalized to the average apical cross -
section area of each disc.
C) Distribution of average polygon numbers. B) and C): Mean, and standard
error of the mean (SEM), are shown for data and simulations. D) Straight
lines denote the moving average of the clonal (red) and twin spot (blue)
shape distributions.
Our model qualitatively recapitulated the configurations observed in actual wing
discs (including a localized boundary in spite of a longer-ranging response to Hh),
so we next tested whether it would also correctly recapitulate the effects of
genetic perturbations in the pathway. As described previously [20], [29], the transduction and response
to the Hh signal can be blocked, in cells anterior to the boundary, by the removal
of the essential Hh pathway protein Smo. This is achieved experimentally by inducing
mitotic recombination in a small subset of cells, early in development, in larvae
that are heterozygous for a mutant in the smo gene. The resulting
small patches of homozygous mutant cells (clones) have been demonstrated to display
two types of behavior [20], [29]: first, when situated close to the boundary, they tend to
round off, minimizing their contact with neighboring cells. Second, when situated
immediately adjacent to the boundary (specifically: at its anterior side), they tend
to sort out from anterior cells and migrate into posterior territory. Both effects
are interpreted as evidence for ectopic boundary formation – cells inside the
clone are not receiving the Hh signal, but are juxtaposed to cells that do (this
mimicks the situation at the boundary and leads to the rounding off, and/or to the
migration into the posterior compartment that also does not transduce Hh). For our
present study, we have repeated these experiments for a number of wing discs, and
used automated image processing to quantify the extent of the
“rounding-off” effect (Figure 4). We observed a highly significant distance-dependence of the
rounding-off behavior: clones farther away from the boundary are rounding off less
strongly than clones closer to the boundary
(p = 6·10−6). As expected, this
effect is not observed on the posterior side of the boundary, where Hh signaling has
no known effects. This suggests that, whatever the molecular response to Hh
signaling that is contributing to boundary formation, this response does extend
further into the anterior tissue than just the immediate first row of cells at the
boundary. In essence, cells seem to be “primed” for boundary formation,
by Hh, several cell-diameters wide into the tissue.
In our model, we can arbitrarily set the Smo production rate to zero for any cell
(and its descendents), thus mimicking the experimental situation. We find that we
can qualitatively recapitulate the behavior of
smo− clones in our simulations (Figure 4): clones situated close
to the anterior side of the boundary, but not on the posterior side, can be observed
to round off; in addition, we observe a tendency of clones that immediately straddle
the boundary to migrate from anterior towards posterior territory (but not in the
opposite direction). Importantly, similar to the experimental situation, we also
observed a highly significant distance-dependence for the extent of rounding-off
(with respect to the distance to the boundary, again only on the anterior side).
This confirms that our model can correctly recapitulate this important aspect of the
perturbation, and it supports our interpretation of the situation in the wing disc:
a hypothetical transcriptional target of Hh signaling could be sufficient to
generate a strictly local force that can establish a clearly delinated compartment
boundary, despite this target being expressed (like all known transcriptional
targets) over a certain distance away from the boundary. By assessing the shape of
experimental smo− clones, we can effectively chart
out the predicted expression level of this putative gene; it appears to be expressed
roughly similar to ptc or dpp (in a graded stripe
of expression along the boundary, at least 10 cell diameters wide).
When mutant cells are generated experimentally using mitotic recombination, a sister
cell is generated that is not homozygous mutant, but instead homozygous wild-type in
the smo gene. This so-called “twin-spot” provides
another relevant input for our modeling: it presumably contains a larger amount of
Smo protein (relative to the surrounding heterozygous tissue). We note that, both in
the experiment and in our simulation, this difference in Smo levels does not suffice
to generate a significant rounding-up of twin-spots (Figure 4). Indeed, the roundness of twin-spots is
identical in the anterior and posterior compartment and independent from the
distance towards the compartment boundary. Effectively, this observed behavior of
experimental twin spots served as another constraint for our model parameterization:
Differences in Hh pathway activity that are at most two-fold should not be
sufficient to generate an observable boundary; and, the actual change in pathway
activity at the endogenous boundary can thus be inferred to be much higher.
In an earlier version of the model, we had assumed that the amount of cortical
constriction would simply be directly proportional to the hypothetical Hh target TMx
(data not shown). However, under this assumption we were unable to find a parameter
set that would satisfy all constraints and that would result in realistic cell
shapes. Cells were either visibly too small or too large in the TMx expression
stripe, and/or were showing imbalances in the relative contributions of cortical
forces and area elasticity, leading to distorted cellular shapes (data not shown).
In our view, this indicates that the processes at the boundary are not simply based
on increasing or decreasing overall cortical constriction, but instead on a local
redistribution of a pre-existing, basal propensity for cortical constriction. As an
important consequence, it appears that it is not the absolute level of TMx that is
important, but the ratio of TMx expression between two neighboring
cells.
Our model is the first to couple tissue growth, driven by explicit cell divisions in
a force-balanced cellular vertex approach, to signal transduction processes
including diffusion, transcriptional responses and mechanical effects. This general
approach should be applicable to a number of crucial developmental mechanisms,
including growth control and body axis specification [24], [30], [31], [32]. In our case, we chose to model
the Hh pathway, despite lacking many of the kinetic parameters that are needed to
fully describe the pathway. This is probably the situation faced for most
developmental signal transduction pathways today. However, we do believe this
approach is justified, as long the as the outcome of the modeling is challenged
experimentally, and as long as the sought-after answers are not addressing merely
quantitative nuances in the pathway, but instead more fundamental mechanistic
choices. Here, we essentially aimed to clarify whether a homotypic boundary model
can work in principle (Figure
1), and whether a single, classical transcriptional target of Hh could be the
missing link between pathway activity and physical forces at the cellular level. We
find that this could indeed be the case, and that such a target gene might even be
expressed at a basal level outside the Hh signaling stripe (since only relative
differences at the boundary are needed). Our findings provide one possible
explanation why previous attempts to search for this gene were unsuccessful: often
it was assumed that the gene would be strongly expressed anteriorly, but not at all
posteriorly. Instead, in our model the gene can indeed be expressed posteriorly (in
fact, many configurations are possible, as long as they include a localized
difference in expression at the boundary). Overall, our study indicates that
mechanistic pathway modeling within whole tissues can help to choose among
hypothetical, conflicting scenarios, and that it can even constrain properties of
postulated missing players in a pathway.
Material and MethodsFly stocks and genetics
To generate smo mutant clones, the
smo3 allele was flipped against a CD2-marked FRT
chromosome. After mitotic recombination took place, non-CD2 expressing cells
were homozygous mutant for smo3. Cells of the
posterior compartment were marked by expression of a hh-lacZ
transgene. Flies had the following genotype: y w hsflp; FRT39
smo3/FRT39 hsCD2;
hhlacZ/+.
Antibody staining
Antibody stainings of imaginal discs were done as described previously [33]. The
following antibodies were used: rabbit α-E-Cadherin (Santa Cruz
Biotechnology, 1∶200), mouse α-CD2 (Serotec, 1∶500), chicken
α-βGal (Immunology Consultants Laboratory 1∶1000).
Shape measurements of mutant clones
The shapes of smo− clones were determined by
the absence of CD2 staining; correspondingly, the shapes of twin spots were
defined by increased CD2 staining. The ‘roundness’ of
smo− clones or twin spots was quantified
by the measure [34], where
A is the area of the clone (or of the twin spot) and
L its perimeter. Circular clones have
, all other clones have . Clonal position
was defined by the distance of the center of mass of the clone to the A/P
boundary as marked by hh-lacZ staining. All geometry
measurements in confocal microscopy images of wing discs, as well as in the
corresponding images from simulations, were fully automatized with the help of
the ImageProcessingToolbox™ of Matlab.
Modeling the Hedgehog pathway
We explicitly describe the Hedgehog pathway by a coupled system of ordinary and
partial differential equations. The Hh protein, produced in the posterior
compartment of the wing disc, diffuses into the anterior compartment, where it
binds reversibly to its receptor Patched (Ptc). Binding of Hh to Ptc relieves
the repression of the transmembrane protein Smoothened (Smo) by Ptc, but neither
the mechanism for Ptc repression of Smo nor the mechanism by which the complex
[Hh Ptch] relieves this repression has been fully understood. We
assume that the active form of Smo corresponds to a complex of Smo protein and
an unknown ligand Lx, [Lx Smo]. We further assume
that, in a membrane compartment inaccessible to Smo, there exists a reservoir of
Lx, from where it can flow towards Smo via a passive transport mechanism. We
assume that Lx gets pumped away from Smo (active transport) by unbound Ptc. Ptc
in turn is produced with a constant, low basal rate in the A-compartment, and is
additionally a transcriptional target downstream of the active form of Smo in
the A-compartment (via the transcription factor Ci, not modeled explicitly).
Finally, we assume that the putative transmembrane protein “TMx” is
likewise a transcriptional target downstream of the active form of Smo, with an
additional, basal expression throughout the tissue. In Figure 3A, the above players and their
interactions are summarized. This network of interactions is translated into a
system of coupled ordinary and partial differential equations, listed in Figure 3B. Since cell-to-cell
diffusion is restricted to the Hh molecule, only the first equation includes
spatial derivatives, whereas all other equations are ordinary differential
equations. We assume that protein kinetics can be described by a constant set of
parameters for each protein [35]. smo− clones were
mimicked by setting the corresponding production rate
to zero; the corresponding twin spots were modeled by
doubling this production rate as compared to wild type cells. The coefficients
appearing in the system of equations are provided and described in the
Supplemental Material (Text S1).
A change in line tension leads to a straight boundary
The apical side of Drosophila's wing disc is modeled as a two-dimensional
vertex model, where the junctions between cells are defined by straight lines
(edges) connecting vertices. The resulting tissue topology is obtained by
minimizing an energy function describing visco-elastic properties of the cells.
Our model is an extension of previously published models describing cells as
polygons [21], [26], [36]. In keeping with the framework of these previous
models, mechanical forces are not stated explicitly; instead, by minimizing the
‘work function’ that aims to reflect the potential energy of the
system, quasi-instantaneous relaxation of the system into local energy-minima is
achieved [37]. This is assumed to correspond to the outcome of balanced
forces acting in the elastic, dampened system of the tissue. It should be
stressed that the modeling takes place on three, well-separated time scales: at
the longest time scale (hours to days), cells divide and the tissue grows. At
the medium time scale (minutes to hours), signaling proteins diffuse and are
transduced into molecular responses inside the cell. The actual mechanics
(forces and movements) occur at the shortest time scale – on the order of
seconds – as has been demonstrated experimentally by tracking the
relaxation movements following laser ablations in the tissue [16].
In our extension of the published models, we assume that a putative transmembrane
protein downstream of the Hedgehog pathway (“TMx”) leads to a change
in the line tension term of the energy function. We assume that TMx molecules
are preferentially recruited to those edges that offer more binding partners
(i.e., other TMx molecules expressed on neighboring cells). At the inner side of
the cell membrane, TMx is assumed to signal to “effectors” that in
turn influence cortical tension. The total number of effectors in each cell is
not influenced by Hh signaling and is rate limiting. Both requirements are
reflected in the definition of an additional scaling
factor in the line tension contribution of the energy function
displayed in Figure 3. Note
that the sum in the definition of the scaling factor runs only over the edges of
cell α.
The size of the scaling factor only depends on the ratio of the
concentrations of two neighboring cells and is thus independent on the absolute
values of concentrations (Figure S3). For all cells outside the stripe
of increased TMx expression, the scaling factor computes to 1 and the energy
function thus corresponds to the original energy function published in ref [21]. The
scaling factor is strongly increased above the basal value of 1 on those edges
of posterior cells immediately straddling the boundary (and thus touching
anterior cells); and it is strongly decreased on all other edges of those cells.
The changes in the scaling factor for anterior cells are more subtle, as shown
in Figure
S3. As each edge belongs to two cells, and their scaling factors for
a given edge may not be the same, the energy function effectively takes into
account the average of the two factors.
With the dimensionless parameters and
, we obtain the following normalized energy function from
the energy function displayed in panel C of Figure 3:We
minimize the normalized energy function eq. (1) by a conjugate gradient method,
leading to a shortening of edges that have an increased scaling factor and a
lengthening of edges with a decreased scaling factor. This causes a
straightening of the boundary between anterior and posterior cells, and (as an
interesting side-effect) an increased average area of cells immediately
posterior to the boundary (i.e., “P1” cells in Figure S2A;
this has been experimentally observed as well [16]). Final simulations were
run with the following parameter set for the normalized energy function:
, (and
for edges of cells constituting the outer margin of the
tissue). Assuming an average cell edge length of , we applied the
same target area to all cells, based on a regular hexagon with edge length
l: .
Modeling growth and the Hedgehog signaling pathway
The simulation of tissue growth was implemented as described in ref. [21]. In
contrast to this previous work, we chose not to apply periodic boundary
conditions, but modeled the tissue margins explicitly. Diffusion of Hh was
discretized by the Finite Volume Method, using the cells as local control
volumes. Following each growth step, the diffusion step was executed, and then
the remaining of the differential equations displayed in Figure 3 (kinetic reactions) were solved
numerically within each cell (for further details see Supplementary Information,
Text
S1). Simulations were started with 220 cells placed in a regular grid
(always using the same starting formation). In simulations that included mutant
clones, 20 smo− cells (simulated by a zero Smo
production rate) adjacent to the corresponding twin spot cells (simulated by a
doubled production rate of Smo as compared to wild type cells) were distributed
uniformly in the starting configuration. We set the initial concentrations for
all proteins within each cell to zero. Between cells and the extracellular
medium we applied zero flux boundary conditions.
All simulations were run until the number of cells had increased to 6000; this
roughly corresponds to the total number of cells in the pouch of a third-instar
wing disc.
Supporting Information
Simulated concentrations of major players of Hedgehog pathway. A)–D):
Concentrations are displayed using Matlab's jet algorithm, where red
corresponds to high, and blue to low concentrations. All concentrations are
given in arbitrary units. B) PtcT is defined as the sum of free
Ptc and ligand bound Ptc. E) Concentrations projected onto the
anteroposterior axis. Note that the compartment boundary does not always
remain precisely at the zero position, hence the slight ‘spread’
of the concentration curves (each cell in the tissue corresponds to one dot
in the graphic).
(TIFF)
Average apical cross-section areas of n-sided cells as a function of n. Areas
are normalized to the average apical cross - section area
of each disc. Mean and standard error of the mean
(SEM) are shown for 10 wing discs in both experiment and simulations. Note
that the simulations reproduce the experimentally observed [16]
increase in area of P1 cells. A) Data vs. simulation for all anterior (A),
posterior (P), A1 and P1 cells. B) Data vs. simulation for all anterior (A),
posterior (P), A2 and P2 cells.
(TIFF)
Converting TMx expression differences to localized edge constriction.
Schematic representation of cells in the A2, A1, P1 and P2 rows of the wing
pouch. A) Concentration of transmembrane protein TMx, per cell. All
concentrations are expressed as multiples of the basal concentration
with . Only cells in
the boundary region of the anterior compartment are exposed to
concentrations of TMx higher than the basal concentration. The concentration
is highest for A cells directly adjacent to the P compartment
(“A1” cells) and decreases with the distance from boundary. B)
We associate to each cell edge a term proportional
to the TMx concentration of the cell it belongs to. For simplicity we have
chosen . C) For neighboring edges with different values of
, the higher value of both is given to both bonds
(named in the following ). Together
with the subsequence normalization, this mimics the fact that transmembrane
proteins are preferentially recruited to edges offering more binding
partners. D) Our assumption that the total line tension per cell is limited
is modeled by normalizing to the average
value of on all edges of a cell. Note that if all edges of a
cell have the same value of , the scaling
factor equals one for all edges of the cell. This is the
case for all cells outside the stripe of increased expression of the
transmembrane protein. E) Example for the calculation of the scaling factor
with and for regular
hexagons. F) The effective average scaling factor
of each edge.
(TIFF)
Dependency of boundary straightness on the ratio of TMx levels at the
boundary. In this figure, the parameters chosen for the first panel of Figure 5 have been fixed,
with the exception of kALxSmo,2 which
has been varied to achieve different ratios of TMx between cells on either
side of the boundary. TMx ratios of 6 and higher can be observed to result
in a boundary quality approaching the actual situation in the wing disc.
(TIFF)
Visualization of simulated dynamic boundary behavior. The movie shows a
simulation run from 220 cells to roughly 4000 cells. The concentration of
TMx is denoted by a white-to-purple color scale, where white denotes the
basal, and purple the highest TMx concentration. In our model, all
P-compartment cells express basal levels of TMx only. The TMx concentration
alters the energy function as described in Figure 3. Note the stable separation of A
and P compartment cells during growth.
(M4V)
Control: TMx input into energy function is needed for boundary formation. The
movie shows a simulation run from 220 cells to 6000 cells. Identical setup
as in movie M1, but the TMx input to the energy function has been disabled.
Note the mixing of cells at the compartment boundary, due to disturbances by
random cell division events.
(M4V)
Wing pouch with smo− clones and corresponding twin spots.
The movie shows a simulation run from 220 cells to 6000 cells. Wild type
A-compartment cells are depicted in dark green, wild type P-compartment
cells in blue. Cells mutant for the smo gene in the
A-compartment are denoted in bright red,
smo− cells in the P-compartment by a
darker red. Twin spot cells are labeled by two shades of bright green,
dependent on their compartment of origin. Note the migration of an anterior
clone, originating adjacent to the compartment boundary, into the posterior
compartment, as well as the rounding up of
smo− clones in the boundary region of
the anterior compartment.
(M4V)
This text describes the modeling procedure in more detail, and lists
parameters and initial conditions.
(PDF)
We are grateful to Sven Bergmann and Aitana Morton de LaChapelle for insightful
comments and discussions.
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