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First published online February 22, 2008
doi: 10.1242/10.1242/dev.012062
1 Max-Planck-Institute for the Physics of Complex Systems, Nöthnitzer
Strasse 38, 01187 Dresden, Germany.
2 Max-Planck-Institute of Molecular Cell Biology and Genetics, Pfotenhauer
Strasse 108, 01307 Dresden, Germany.
3 Department of Biochemistry, Sciences II, 30 Quai Ernest-Ansermet, 1211 Geneva
4, Switzerland.
¶ Authors for correspondence (e-mails: marcos.gonzalez{at}biochem.unige.ch; julicher{at}pks.mpg.de)
Accepted 24 December 2007
 |
SUMMARY |
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 TOP SUMMARY INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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Key words: Drosophila, Morphogen, Precision, Wing imaginal disk
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INTRODUCTION |
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TOPSUMMARYINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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). It
has been suggested that morphogen gradients pattern developing tissues by
providing positional information to the target cells by means of the
concentration-dependent expression of target genes
(Wolpert, 1969). Prominent
examples of morphogens are Bicoid (Bcd), which acts in the Drosophila
embryo (Driever and Nüsslein-Volhard,
1988a; Driever and
Nüsslein-Volhard, 1988b), Decapentaplegic (Dpp), which also
acts in the Drosophila wing disk
(Entchev et al., 2000;
Teleman and Cohen, 2000), and
Activin in the Xenopus embryo
(Gurdon et al., 1994).
The question of whether morphogen gradients can precisely guide the
patterning of growing tissues has deserved attention in recent years. In
particular, a number of studies have addressed the precision of the Bcd
gradient, which specifies the expression domain positions of gap genes, such
as hunchback (hb), Krüppel and giant,
in the early Drosophila embryo
(Bergmann et al., 2007;
Crauk and Dostatni, 2005;
Gregor et al., 2005;
Gregor et al., 2007;
Houchmandzadeh et al., 2002;
Houchmandzadeh et al., 2005;
Jaeger et al., 2004). These
domains are established while the embryo is still a syncytium, i.e. the nuclei
are not separated by cell membranes. In the context of the positional
information paradigm, the Bcd gradient has been scrutinized for its ability to
define by itself the precise domain of hb and its variability from
embryo to embryo. A recent report shows that the shape of the Bcd gradient
leads to relative concentration differences between neighboring cells of about
10%. This is also the concentration variability at corresponding positions in
multiple embryos and the relative magnitude of the noise in readout of Bcd by
the activation of hb (Gregor et
al., 2007). Together, these observations suggest that the system
exerts precise control over the absolute Bcd concentrations and responds
reliably to small concentration differences.
The issue of precision also arises for morphogens that form graded profiles
in fields of cells rather than a syncytium. In these fields, individual cells
have slightly different sizes, shapes, number of receptors and other
properties. As a consequence of this cell-to-cell variability, morphogens form
concentration profiles that are overall graded, but which exhibit fluctuations
on shorter length scales. Here, we study the Dpp morphogen gradient in the
Drosophila wing disk. Dpp signaling is initiated by Dpp binding to
its heterodimeric receptors, after which the type I receptor, Thickveins,
phosphorylates the R-Smad, Mothers against Dpp (Mad)
(Kim et al., 1997;
Newfeld et al., 1997).
Phosphorylated Mad (PMad), after binding to the co-Smad Medea, translocates to
the nucleus. There it represses directly the transcription of the
transcriptional repressor Brinker (Campbell
and Tomlinson, 1999; Jazwinska
et al., 1999; Moser and
Campbell, 2005; Müller et
al., 2003) (reviewed by
Affolter and Basler, 2007),
which therefore forms a gradient inverse to that of Dpp, and ultimately leads
to the determination of the boundaries of Dpp target genes, such as
spalt (sal; also known as spalt major)
(Kühnlein et al., 1994;
Barrio and de Celis, 2004).
A number of features distinguish the Bcd from the Dpp gradient: Bcd
operates in a syncytium and Dpp in a field of cells; Bcd spreads within a
field of fixed size and Dpp in a growing tissue; the Bcd gradient is likely to
be read out before the steady state is reached
(Bergmann et al., 2007),
whereas the Dpp readout might occur at all stages of development.
Combining the fluorescence profiles of GFP-Dpp
(Entchev et al., 2000) in a
large dataset, we analyze the Dpp concentration fluctuations as a function of
the distance from the source. We interpret the behavior of these fluctuations
using a theoretical description of morphogen production, transport and
degradation, which represents cell-to-cell variability in the target and
source regions by random variations of the effective diffusion coefficient,
the effective degradation rate and the Dpp production rate. We compare the
precision of the Dpp gradient to that of the nuclear concentration of PMad, as
an indicator of Dpp signaling activity. Finally, we investigate the
relationship between the Dpp concentration fluctuations, the precision of the
expression domain of the target gene spalt and that of the final
morphological vein pattern.
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MATERIALS AND METHODS |
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TOPSUMMARYINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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).
Third-instar larvae that developed for the same time under the same conditions
at 25°C were used. Wing disks were immunostained and prepared as
previously described (Entchev et al.,
2000; Kicheva et al.,
2007). Antibodies: mouse anti-Patched
(Capdevila et al., 1994) 1:200
dilution; rabbit anti-Spalt (Kühnlein
et al., 1994) 1:50; rat anti-Spalt
(de Celis et al., 1999) 1:600;
rabbit anti-PMad (Tanimoto et al.,
2000) 1:1000; secondary Alexa-546 and -633 conjugated (Molecular
Probes) 1:500.
Vein patterns in the adult wing
We quantified 88 wild-type male and female adult wings. We first drew a
line connecting the two ends of vein L4. Through this line, we drew a
perpendicular line closest to the position where vein L4 crosses the posterior
crossvein. This line is delimited by the wing margin and defines the
anterior-posterior length of the wing. We defined the distance between veins
L2 and L3 as the segment of the anterior-posterior length that lies between
veins L2 and L3.
Data analysis
We analyzed 30 GFP-Dpp gradients in 15 third-instar wing disks. The average
fluorescence intensity (FI) in a 3.1 µm square was measured in the middle
of the dorsal and ventral compartments at continuously varying distances to
the source from an additive projection of 15 z-sections (1 µm
apart) using ImageJ
(http://rsb.info.nih.gov/ij/)
(Fig. 4A). The source boundary
was determined using the abrupt decrease of PMad fluorescence in the source
cells (Fig. 4B). Background FI
(the lowest FI value far away from the source) was subtracted. The FI profiles
were normalized to the total FI IDpp defined as the
integral over each profile (Fig.
4C). 
and
Dpp(x) were averaged in x-bins of 3.1 µm
for Fig. 4E,F, and 2.5 µm
for Fig. 5F. Error bars in
Fig. 4E,F are twice the
standard deviation of 
and
Dpp in each bin, respectively.
For the PMad gradients (Fig.
5B), the maximum intensity
in one z-slice (width
1.8 µm) was measured for each individual nucleus i at a respective
distance from the source xi in the central third of the
dorsal and ventral parts of the posterior compartment using Metamorph
(Universal Imaging Corporation). The profiles were normalized to the total FI
IPMad, i.e. the integral over the curve that linearly
interpolates between the in the
anterior to posterior direction. Background fluorescence (measured outside the
range of the Dpp signal) was subtracted before normalization.
was calculated in
x bins of 2.5 µm via
where
is the exponential fit to the PMad concentration profile and the sum extends
over all N nuclei located in each bin.
The sal-expression domain was analyzed the same way as the GFP-Dpp profiles (Fig. 6A). The sal range was determined by fitting the sigmoidal function s(x)=s0(tanh((d -x)/w)+1)/2 to each profile using s0, d and w as fit parameters. The distance x*=d+w corresponds to the sal range.
All fits were performed using the nonlinear least-squares Marquardt-Levenberg algorithm (Mathematica 5.2, Wolfram Research). We estimated the disk sizes L as the linear extension of the GFP-Dpp source (Fig. 4A). The average cell diameter was estimated from the average area of a cell (5.46±0.82 µm, n=1200 cells), assuming that the cell is a circle.
Theoretical description of morphogen gradient precision
We discuss a simple model of morphogen gradient formation based on
non-directional transport, characterized by an effective diffusion coefficient
D, and degradation with an effective degradation rate k.
Morphogens secreted in the source (x<0) enter the tissue with a
current j at the boundary line x=0 of the two-dimensional
target area (Fig. 1A,B) (see
also Kruse et al., 2004;
Lander et al., 2002).
Cell-to-cell variability implies that the efficiency of transport and
degradation vary at different positions. We consider this by assuming that
D and k are functions of the position
x=(x,y), with
D(x)=D0+
(x) where
D0 is the average diffusion coefficient and
(x) is a random function with zero average that describes the
fluctuations. Analogously, we define
k(x)=k0+
(x).
Fig. 1C illustrates this
model in one dimension. In a continuum limit, it can be described by
 | (1) |
=(
/x,/y). Production
rate fluctuations in the source are captured by a boundary condition
j(x)=j0+
(x)|x=0
on the current at x=0, where j0 is the average
current across this boundary and
(x)|x=0 is a random
function with zero average.
Equation (1) is closely related to the well-known diffusion equation with
degradation term, but includes cell-to-cell variability. In steady state,
equation (1) leads to a graded concentration profile c(x)
for each realization of the random variables (x),
(x) and
(x)|x=0. Owing to the
presence of multiplicative noise in equation (1), the analytical calculation
of the steady-state solution is challenging. To calculate the first-order
correction to the noiseless case, we used a field theoretic perturbation
expansion in the noise strength. This allows us to express the concentration
fluctuations in terms of Green's functions and the fluctuation amplitudes of
D, k and j. A detailed theoretical study is included in the
supplementary material.
We can estimate the precision of the positional information conveyed by the
gradient. If the target position x* is defined by a
threshold concentration c* where
c(x*)=c*, the fluctuations of
c(x) generate an uncertainty
 | (2) |
is the average value of the target position,
is the average concentration at x*, and
is the concentration uncertainty at x*
(Fig. 4D). The average
steady-state gradient that results from (1) is given by an exponential decay,
(x)=c0exp(-x/
)
with 
.
One can thus express the uncertainty defined in (2) as
 | (3) |
(x*)=
c(x*)/(x*).
This reveals a direct relationship between the uncertainty of the threshold
position x* and (x*). |
RESULTS |
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TOPSUMMARYINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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; Colman-Lerner et al.,
2005; Elowitz et al.,
2002; Raser and O'Shea,
2005). In the following, we discuss the impact of such variability
on morphogen gradients.
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;
Eldar et al., 2002;
Eldar et al., 2003;
Howard and ten Wolde, 2005;
Kruse et al., 2004;
Lander et al., 2002). Here, we
use an effective description of non-directional morphogen transport,
independent of the specific underlying mechanisms [restricted extracellular
diffusion or planar transcytosis
(Bollenbach et al., 2007;
González-Gaitán,
2003; Strigini,
2005)]. Morphogen transport can be characterized by the influx
j of morphogens into the target tissue, an effective diffusion
coefficient D, and a degradation rate k. Using a
fluorescence recovery after photobleaching (FRAP) assay, we have recently
measured these parameters (j, D and k) in a manner
independent of the microscopic transport mechanism
(Kicheva et al., 2007). The
effective diffusion coefficient could correspond to passive extracellular
diffusion or transcytosis (Bollenbach et
al., 2005). Similarly, the effective degradation rate can be
affected by extracellular proteolytic degradation or lysosomal degradation.
Finally, morphogen secretion from the source determines the influx
j.
In our description of morphogen transport, we consider cell-to-cell
variability by introducing spatial fluctuations of the parameters D,
k and j (Fig. 1).
In an ensemble of gradients, this leads to an average steady-state
concentration profile of the form
(x)=C0exp(-x/)
where x denotes the distance from the source
(Fig. 1B),
C0 is the average concentration directly adjacent to the
source, and is the gradient decay length, i.e. the length over which
the concentration decreases by a factor of e. At each position
x, concentration fluctuations about this average profile are
characterized by the standard deviation of the concentration
c(x). To characterize the fluctuations of
the gradient, we study the `relative concentration uncertainty'
(x)=c(x)/(x),
which quantifies concentration fluctuations irrespective of the absolute value
of the local average concentration.
It has been proposed that morphogen gradients provide positional
information by activating target genes above distinct concentration thresholds
that occur at precise distances from the source
(Wolpert, 1969). In this
scenario, the relevant quantity is the precision of the distance from the
source at which a particular morphogen concentration occurs. This is captured
by the `positional uncertainty'
,
i.e. the standard deviation of the position x, carrying a given
concentration c. The positional uncertainty
x at a position x can be determined from
the relative concentration uncertainty at this position via
x(x)
(x) (see
Materials and methods). The precision of the gradient is inversely related to
x. We obtained analytical expressions for the
gradient precision, which were verified numerically (see the theoretical study
in the supplementary material).
Source versus target tissue variability: effect on precision
The effects of fluctuations can be discussed systematically by considering
two simplified situations: (1) cell-to-cell variations exist only in the
source and (2) they exist only in the target tissue. The real situation in the
wing disk is the combination of both. We first consider the hypothetical case
in which there is cell-to-cell variability in the source, whereas the target
cells are identical (Fig. 2A).
In this case, the influx j fluctuates along the source boundary,
while D and k in the target tissue are constant. Our
analysis shows that the fluctuations of j along the source boundary
lead to an uncertainty of the morphogen concentration in the target tissue
that decreases away from the source (Fig.
2B). This decrease results from the fact that morphogens that
arrive far from the source originate from many different source cells, and
from fewer source cells near the source. At large distances, the concentration
has low uncertainty because it results from an average over the production
rates of many source cells. In a two-dimensional epithelium, (x)
resulting from variations of j decreases following the power-law
(x)
x-1/4 for large distances x
(see continuum limit in the supplementary material).
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(x) in the region abutting the source, followed by a monotonic
increase at larger distances (Fig.
2D). This increasing uncertainty results from an accumulation of
fluctuations as the concentration profile forms at increasing distances from
the source. In a two-dimensional tissue, the concentration uncertainty
increases as the power-law (x)x1/4 for
large distances x (see continuum limit and Fig. S3 in the
supplementary material).
In general, cell-to-cell variability exists both in the source and the
target tissue. In this general scenario, (x) first decreases for
small distances from the source, reaches a minimum, and then increases for
large x (Fig. 2E).
In this analysis, fluctuations of the three trafficking parameters (D,
k and j) are uncorrelated. If both diffusion and degradation
rely on the same cellular machineries - e.g. if morphogen transport and
degradation are mediated by intracellular trafficking
(Entchev et al., 2000) - the
effective diffusion coefficient and the degradation rate could be correlated.
We found that positively correlated fluctuations in D and k
lead to reduced concentration uncertainty (see Figs S1, S2 in the
supplementary material). In addition to cell-to-cell variability within one
disk, certain parameters may vary from disk-to-disk. In particular, this
appears to be so for the morphogen secretion rate (see below). We consequently
studied the effects of such disk-to-disk variations of secretion in our
theoretical description. These variations increase the relative concentration
uncertainty and render its minimum less pronounced (see Fig. S4 in the
supplementary material).
|
) or three-dimensional tissues such as the ventral neural tube
in which the sonic hedgehog gradient forms during vertebrate nervous system
development (Ericson et al.,
1996) (reviewed by Briscoe and
Ericson, 1999) (Fig.
3A-C). In one dimension, a given fluctuation - e.g. a transport
deficiency in some cell - affects the steady-state concentration profile at
all positions farther away because all molecules moving from the source have
to pass the deficient cell (Fig.
3A). In two dimensions, this effect is weaker because the
morphogens can reach any position on many different paths through the tissue
(Fig. 3B). Analogously, the
impact of cell-to-cell variability is even weaker in three dimensions
(Fig. 3C). Our calculations
confirm these ideas: higher tissue dimensionality leads to a strong reduction
of the concentration uncertainty (x)
(Fig. 3D).
At large distances from the source, the concentration uncertainty obeys
general power-laws (x)x
as
described above for the two-dimensional scenario. The exponent v
depends on the dimensionality d of the tissue and is given by
=(3-d)/4 if d<3. For d=3 the exponent
=0, which implies an uncertainty that does not increase with increasing
distance from the source. Further analysis in this case reveals that there
remains a weak logarithmic increase (x)ln(x) for
large x. These behaviors also hold if fluctuations only occur in the
target, whereas the source does not fluctuate. However, if fluctuations occur
only in the source, the concentration uncertainty in the target obeys the
power-law
(x)x(1-d)/4
for large x.
Precision of the Dpp gradient in the Drosophila wing disk
Our theoretical framework to describe morphogen gradient precision enables
us to determine experimentally the precision of the Dpp gradient in the
Drosophila wing disk and to test our theoretical prediction that
(x) decreases for small values of the distance x from
the source and increases for larger x
(Fig. 2E). To analyze the Dpp
gradient precision, we determined the shapes of 30 Dpp profiles from
third-instar larvae. Using a GFP-Dpp fusion protein in a dpp mutant
background (see Materials and methods)
(Entchev et al., 2000), we
measured the GFP fluorescence intensity (FI) as a function of the distance to
the Dpp source in the posterior compartments
(Fig. 4A-C, Materials and
methods). Recently, we have calibrated FI to GFP-Dpp concentration
(Kicheva et al., 2007) using
GFP-tagged rotavirus-like particles
(Charpilienne et al., 2001).
Our analysis showed that we can reliably detect differences in concentration
of less than 2% of the GFP-Dpp concentration close to the source. This allows
us to undertake a quantitative analysis of the gradient and its
variability.
Indeed, the gradient shape varies from disk to disk
(Fig. 4C), as does the overall
level of fluorescence [variation coefficient 0.26 (standard deviation relative
to mean)]. We determined the average FI profile
and its standard deviation
(Fig. 4C-E, details in
Materials and methods). The shape of individual gradients is well described by
an exponential decay
cDpp(x)C0exp(-x/Dpp)
(red line in Fig. 4D) with an
average decay length Dpp=17.0±4.3
µm=6.5±1.7 cells (see Table S1 in the supplementary material) and
variable fluorescence levels at the source boundary, C0
(Kicheva et al., 2007). This
value of Dpp implies that neighboring cells (with diameter
2.6 µm) experience Dpp concentrations that differ on average by 15%,
similar to the 10% difference in the case of Bcd in the syncytial blastoderm
(Gregor et al., 2007). We
determined C0 and Dpp for each GFP-Dpp
profile by a fit to the function
C0exp(-x/Dpp). We find that
Dpp is not correlated with the wing disk size L,
defined as the dorsal-ventral extension of the source in the wing pouch
(Fig. 4A; see Table S2 in the
supplementary material). The standard deviation
decreases with increasing
distance to the source (Fig.
4E). This is partially because the average concentration decays
away from the source. To measure the gradient fluctuations, we therefore
investigated the relative concentration uncertainty
.
Close to the source, Dpp(x) decreases for a few cell
diameters until it reaches a minimal value at about x=4 µm, and
increases monotonously distally for larger distances from the source
(Fig. 4F).
As discussed above, we can determine the positional uncertainty
of the Dpp gradient at a
position x from Dpp(x) using
(Materials and methods). We find that, at its minimum 4 µm from the source,
the positional uncertainty of the Dpp gradient is smaller than two cell
diameters [
(4 µm4.4
µm=1.7 cells]. Next to the source,
(1 µm6.0 µm=2.3
cells, whereas at 40 µm from the source, a distance at around which the Dpp
gradient has been proposed to position target gene boundaries [e.g.
sal and omb (also known as bifid)],
(40 µm9.7 µm=3.7
cells.
We have so far presented our analysis based directly on the GFP-Dpp FI
profiles that we measured. To separate the variations of the gradient shape
from overall FI variations between different disks, we normalized the
fluorescence to the total intensity IDpp in the posterior
compartment of each disk (see Materials and methods,
Fig. 4C-F insets). The analysis
of the normalized data captures the effects of local cellular fluctuations on
the gradient precision, by ignoring global variations between different disks.
Next to the source, we find
(1 µm)4.1 µm =1.6
cells, and 40 µm away (40
µm)6.1 µm=2.3 cells. As in our theory, the minimum of
Dpp(x), corresponding to
(9 µm)2.4 µm=0.9
cells, is more pronounced and the gradient precision is higher if we normalize
and study only fluctuations of the gradient shape
(Fig. 4F, see Fig. S4 in the
supplementary material).
|
100 cells wide. These data also show that Dpp
precision is not maximal in the cells adjacent to the source, but at a certain
distance from the source. This is in qualitative agreement with our
theoretical prediction of the relative concentration uncertainty (compare
Fig. 2E with
Fig. 4F). The fact that Dpp has
maximal precision at a certain distance from the source indicates that both
fluctuations of the morphogen flux from the source and cell-to-cell
variability in the target tissue influence the gradient precision.
Precision of the Dpp activity gradient
In the previous sections, we analyzed the precision of the Dpp gradient and
the theoretical constraints that determine the spatial precision profile. How
is this precision conveyed downstream through the Dpp signaling pathway? Upon
Dpp binding to its receptor, the transcription factor Mad is phosphorylated
and imported into the nucleus, thereby initiating a signaling cascade that
leads to the transcriptional activation of target genes through the repression
of Brinker (Campbell and Tomlinson,
1999; Jazwinska et al.,
1999; Kim et al.,
1997; Newfeld et al.,
1997). Consequently, the nuclear PMad concentration is a direct,
early measure of the Dpp signaling activity.
We quantified the nuclear PMad concentration in the same disks in which we
quantified the GFP-Dpp profiles using an anti-PMad antibody
(Fig. 5A, Materials and
methods). The PMad concentration decreases away from the Dpp source in an
activity gradient cPMad(x), described by an
exponential decay
PMad(x)=P0exp(-x/PMad)
where P0 is the PMad level at x=0
(Fig. 5B). Analogous to
GFP-Dpp, we determined the decay length of the PMad gradient
(PMad=25.2±4.5 µm=9.7±1.7 cells) and the
total PMad level IPMad (variation coefficient 0.24). The
average PMad is comparable to the average decay length of
the Dpp gradient (Dpp=17.0 µm=6.5 cells;
Fig. 4C), supporting that the
Dpp gradient is transduced downstream into a parallel activity gradient. Here
we discuss normalized PMad profiles (Materials and methods), because the
determination of PMad levels by immunostaining captures relative PMad levels
within one disk, but does not permit one to reliably compare the overall
levels from different disks. The nuclear PMad and the Dpp concentrations at
the same distance from the source are correlated (correlation index
R=0.69 for the non-normalized and R=0.76 and normalized Dpp
data; Fig. 5C, see also Fig. S5
in the supplementary material). This is consistent with the notion that the
PMad gradient is controlled by a fraction of the total Dpp level, implying
that analysis of the total Dpp concentration can be used to estimate the Dpp
pool actively engaged in signaling.
As for Dpp, we estimated the precision of the PMad gradient by determining
the relative concentration uncertainty of PMad,
,
where 
is the uncertainty
of cPMad at a distance x
(Fig. 5B inset).
PMad(x) is similar to the relative uncertainty of the
Dpp concentration (Dpp(x); see
Fig. 5D versus
Fig. 4F): both have similar
values and exhibit a local minimum near the Dpp source. At the minimum of
PMad(x), the positional uncertainty of PMad
PMadPMad(x)=3.5
µm=1.4 cells, i.e. it is about one cell diameter and on the order of the
positional uncertainty of the Dpp gradient at this position. The positional
uncertainty at different x of the Dpp gradient
is correlated to that of the
PMad gradient
(R=0.74 for the
non-normalized and R=0.63 for the normalized Dpp data). The PMad
gradient has a similar precision as the Dpp gradient, as indicated by a
constant of proportionality for the two imprecisions that is close to one
(Fig. 5E). These observations
suggest that the precision of the Dpp gradient is maintained at the level of
the downstream PMad activity gradient.
|
).
This way, the extracellular Dpp morphogen establishes a finely tuned, graded
readout of transcriptional repression (reviewed by
Affolter and Basler, 2007). The
target gene sal is indeed expressed in response to the repression of
Brinker occurring upon Dpp signaling, up to a distance x*
from the Dpp source (Fig. 6A)
(Barrio and de Celis, 2004;
Campbell and Tomlinson, 1999;
Lecuit et al., 1996;
Nellen et al., 1996). The
spatial transition from sal-expressing to non-expressing cells is not
perfectly step-like, but graded in a region of finite width (see below).
Important questions are (1) whether the precision of the position
x* corresponds to the precision of the final wing vein
pattern and (2) whether the precision of the position x*
is related to the precision of the Dpp and PMad gradients at
x*. It is noteworthy that Dpp, PMad, Sal and the final
vein positioning represent only four steps in a complex cascade of events that
guides wing patterning. In principle, precision of the final pattern could be
affected in any step of this chain, such as by the distribution of the
receptor Thickveins, the conversion of PMad-Medea to an inverse Brinker
gradient, the control of knirps and knirps-related
transcription by Sal, etc. (Jazwinska et
al., 1999; Lecuit and Cohen,
1998; Lunde et al.,
1998). To address the patterning precision at the level of Dpp target genes, we determined the precision of the sal domain boundary in the same set of disks in which we analyzed the Dpp and PMad gradients. We determined the Sal FI profile as a function of the distance x to the Dpp source (Fig. 6B,C) and fitted a sigmoidal function s(x)=s0(tanh((d-x)/w)+1)/2 to each profile. At the distance d from the source, the Sal level does not decay abruptly, but transitions from a high to a low value within a region of width 2w (Fig. 6B). To describe the distance at which Sal is activated in response to Dpp, we operationally defined a Sal domain boundary position x* as x*=d+w. At x=x*, s(x) has dropped below 12% of its maximum value s0 (Fig. 6B, Materials and methods).
The average range of sal expression in the posterior compartment
is
*=
x*=39.1±6.1
µm=15.0±2.3 cells. Unlike the decay lengths of the Dpp and PMad
gradients, which only weakly correlate with the disk size L
(R=0.14 and R=0.03, respectively), x* is
correlated to L (R=0.56). This indicates that whereas the
sal domain scales with the size of the tissue, the decay lengths of
the Dpp concentration and signaling activity gradients do not at the end of
development.
In third-instar disks, the vein primordia form as narrow stripes of cells
at the boundaries between different sectors of gene expression along the
anterior-posterior axis (Biehs et al.,
1998; Sturtevant et al.,
1997; Sturtevant and Bier,
1995). Previous work has shown that the L2 vein primordium appears
at the anterior edge of the sal domain, whereas the vein L3
differentiates at the anterior boundary of the Dpp source
(Sturtevant et al., 1997). Sal
then initiates a cascade of events that continues in the prepupal and pupal
stages and leads to the final positioning and refinement of vein L2 in the
adult wing (de Celis and Barrio,
2000; Lunde et al.,
1998). The adult morphological vein pattern scales with wing size
and is extremely precise. To compare the precision of the sal range
in the disk to the precision of the vein pattern in the adult wing, we
measured the variability of the distance xV between veins
L2 and L3 in a set of 88 wings (see Materials and methods). This range
corresponds approximately to the sal expression range in the wing
primordium. We found that the variation coefficient of xV
in the adult wing (0.08) is smaller than that of x* in the
wing disk (0.16), indicating that the adult vein positioning is more
invariable, regardless of organ size. This increased precision is even more
obvious considering the sal range scaled to disk size and the
distance between L2 and L3 scaled to the anterior-posterior length of the wing
(variation coefficient 0.15 for the disk and 0.04 in the wing). Our data show
that although x* correlates to some extent with the size
of the third-instar disk, later events in wing development (e.g. cell
rearrangements, apoptosis) must contribute to the high precision of the final
morphological pattern.
|
(*)=(39.1
µm)9.7 µm=3.7 cells for the non-normalized Dpp concentrations and
even lower
((*)6.1
µm =2.3 cells) for the normalized data
(Fig. 4F). Thus the imprecision
of the Dpp gradient is similar to the variability of the Sal boundary
position. This is indeed consistent with a morphogen paradigm in which the
ligand concentration defines the target gene expression domains in a
concentration-dependent manner (Wolpert,
1969). It is in principle possible that the local Dpp level in a given disk determines the position of x* with higher precision than three to four cells. In this scenario, the fluctuations in position of the Sal domain would result directly from the fluctuations of the Dpp level, and therefore correlate with them. If this is the case, we expect to find similar Dpp concentrations at the Sal boundary x* regardless of the actual value of x* in a particular disk. Fig. 6D shows that the dispersion of the concentrations of Dpp at distance x* from the source corresponds to a positional uncertainty, which is not smaller than the three to four cells described above. This lack of clear correlation between the Dpp concentration and x* is also related to the fact that the position of the sal boundary correlates with disk size, whereas the Dpp gradient does not. In summary, these observations imply that Dpp is able to convey positional information only with a precision of three to four cells in each disk, and that additional mechanisms must contribute to the accurate vein patterning.
|
DISCUSSION |
|---|
|
TOPSUMMARYINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
|---|
Precision and dimensionality
We have shown that the precision of morphogen gradients rapidly increases
with increasing tissue dimensionality (Fig.
3). We speculate that this might influence the `choice' of
dimensionality in different developmental contexts: mesenchyme versus
epithelium versus linear arrays. For example, upon proximodistal patterning of
the limb, digits first develop in two-dimensional primordia. Later,
interdigital cells are eliminated by apoptosis leaving behind one-dimensional
structures (Wolpert, 2002).
According to our results, initiating the patterning in two-dimensional
structures would help to circumvent the potential uncertainty that is
characteristic to linear arrays.
Gradient precision and positional uncertainty
Our theory can explain the observed qualitative behavior of the gradient
precision. It also implies that the magnitude of the observed concentration
uncertainty could be caused by relative fluctuations of about 10% for the
values of j, D and k in the source and the target tissue
(j/j00.1,
D/D00.1,
k/k00.1; see Fig. S2D in
the supplementary material). However, two effects that are not considered here
might also contribute to the large observed uncertainty: additional sources of
cell-to-cell variability and experimental noise.
First, theoretical analysis shows that the uncertainty of the total
concentration is increased by a constant factor if a fraction of the observed
GFP-Dpp is immobilized in endosomes. This is due to cell-to-cell variations in
the rate of transfer to the immobile pool. Note, however, that correlations
between the fluctuations of D and k can increase precision.
Such correlations could occur in a scenario of active Dpp transport
(Entchev et al., 2000), in
which endocytosis mediates the movement of Dpp (i.e. its diffusion), but also
controls its intracellular lysosomal degradation.
A second source of the large observed concentration uncertainty is
background noise and measurement errors. Our inaccuracy of measurement is less
than 2% of the Dpp fluorescence next to the source
(Kicheva et al., 2007).
However, far from the source, the signal to noise ratio is low, and therefore
the experimentally determined concentration uncertainty exceeds the effects
stemming from cell-to-cell variability. We consequently underestimate the
precision of the Dpp gradient at greater distances from the source.
We can however use our theoretical results to obtain an estimate of the
precision of the Dpp gradient at large distances. In a tissue such as the wing
disk, our theory shows that the concentration uncertainty increases rather
moderately as (x)x1/4, which implies
that the precision does not deteriorate dramatically within the range over
which the morphogen signals. Assuming that Dpp signals up to
4Dpp=68 µm=26 cells away from its source, we estimate
from the concentration uncertainty at x=Dpp that
the positional uncertainty remains smaller than 3.5 cell diameters over the
range of Dpp signaling (smaller than two cell diameters if we base this
estimate on the normalized Dpp data).
Target precision and the morphogen model
In the case of the morphogen Bcd, it has been found that the gradient,
which shows an embryo-to-embryo variability of the concentration at a certain
position of 10%, is transduced into hb activation with a similar
imprecision of 10% (Gregor et al.,
2007). This implies a positional precision of about a single cell
diameter. For Dpp, we found that the uncertainty of the Sal range
x* is two to three cell diameters, which is similar to the
positional uncertainty of the Dpp gradient at this distance to the source
(Fig. 4F). This suggests that
Dpp can convey the positional information required to position the Sal
boundary with an accuracy of 3 cells. Considering the Dpp gradient decay
length (Dpp=6.5 cells), two neighboring cells would have to
discriminate a relative concentration difference of about 15% to respond with
an accuracy of a single cell, which is comparable to the 10% reported for the
Bcd gradient. This difference corresponds to about 65 GFP-Dpp molecules at the
sal boundary [estimated using fluorescence/concentration calibration
(Kicheva et al., 2007)]. Since
the actual precision at the sal boundary is 3 cells, this would
require the ability of cells to discriminate a difference of at least 200
molecules or 37% at the sal border. It is interesting to compare the
implications of these numbers for Dpp and Bcd. It has been suggested that an
averaging time of 2 hours might be required for a single cell to detect
variations of the Bcd level of 10% (Gregor
et al., 2007). This time results from the estimated slow kinetics
of binding of the transcription factor to the DNA target site. Because this
would be slow compared with the time scale for gene expression boundary
formation in the embryo, which is in the order of minutes, spatial averaging
has been invoked as a possible means to achieve precise readout
(Gregor et al., 2007). In
contrast to Bcd, several hundred receptors are available to detect Dpp ligands
on the cell surface (Gurdon et al.,
1994), which reduces the required time for averaging
significantly. Furthermore, in the wing disk, establishment of target gene
boundaries happens on longer time scales than in the embryo (hours or days as
opposed to several minutes). Finally, we have shown that the Dpp readout
precision is about 37%. Therefore, the observed kinetics and precision of Dpp
are fully consistent with the positioning of target gene boundaries in the
wing disk, and mechanisms other than readout of the Dpp concentration gradient
over time would not need to be invoked.
The final vein pattern in the wing is more precise than three cells,
implying that Dpp acts as a morphogen to coarsely provide positional
information, which is refined by downstream events. Such events are known to
occur later in development, during which dorsal to ventral signals and lateral
inhibition contribute to the fine positioning and the final alignment of the
veins on the two surfaces of the wing
(Biehs et al., 1998;
de Celis and Barrio, 2000;
Lunde et al., 1998;
Sturtevant and Bier,
1995).
Here it is important to mention that we analyzed both non-normalized and normalized FI profiles. The normalization (see Materials and methods) enabled us to separate variations of the gradient shape due to cell-to-cell variability from disk-to-disk variations of the overall gradient amplitude, reflecting the physiological variability which different animals experience in the culture. Similarly, our theory describes both fluctuations of the gradient shape and of the total Dpp level. The total FI (and therefore the Dpp concentration) in different disks indeed varies (variation coefficient 0.26). If the Dpp signal-transduction system in different disks responds identically to the same absolute concentrations of Dpp, the positional uncertainty conveyed by the Dpp gradient at the sal boundary would be 3.7, instead of 2.3 cell diameters.
Several scenarios are consistent with the idea that Dpp functions as a morphogen that provides information more accurate than three to four cell diameters. First, it could be that Dpp functions as an accurate morphogen that positions the boundaries of target genes such as sal precisely in earlier stages. At later stages, sal might respond to other cues or be maintained to some extent by lineage, which could explain the observed correlation of sal range with disk size. Second, Dpp could determine target gene expression precisely at and around the distance at which precision is highest and not at greater distances. In this scenario, Dpp would be a precise short-range, rather than long-range, morphogen. Although this is still unclear, we favor the former scenario. Alternatively, we also consider the possibility that Dpp is a morphogen with low precision, and other developmental mechanisms refine the coarse positional information it provides. In the future, it would be interesting to see whether this is a specific feature of sal or whether the same behavior can be seen for other target genes.
Supplementary material
Supplementary material for this article is available at
http://dev.biologists.org/cgi/content/full/135/6/1137/DC1
|
ACKNOWLEDGMENTS |
|---|
|
Footnotes |
|---|
Present address: Department of Systems Biology, Harvard Medical School, 200
Longwood Avenue, Boston, MA 02115, USA
Present address: California Institute of Technology, 1200 E. California
Blvd., Pasadena, CA 91125, USA
Present address: DFG Center for Regenerative Therapies, Tatzberg 47-49,
01307 Dresden, Germany
|
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and
describe transport between cells n
and n+1. They fluctuate about their mean value p0
because of cell-to-cell variability, as indicated by arrows of different
widths. The degradation rate kn also fluctuates.
of the PMad gradient as
a function of the positional uncertainty of the non-normalized GFP-Dpp
gradient, 
,
non-normalized GFP-Dpp data, and
,
normalized. Green horizontal bars in B,D, 5 cell diameters (13 µm).