Precision of the Dpp gradient

Morphogens form graded concentration profiles in target tissues adjacent to a region of secreting cells (reviewed by Tabata and Takei, 2004). 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 Drosophilaembryo (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.

Genotypes: dpp^d8/dpp^d12; dppGal4/UAS-GFP-Dpp. In(2L)dpp^d8 and In(2L)dpp^d12 are described in FlyBase; UAS-GFP-Dpp has been described previously(Entchev et al., 2000). 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.

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.

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 I^Dpp 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

\({\sigma}_{c}^{\mathrm{Dpp}}\)

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 x_i 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 I^PMad, i.e. the integral over the curve that linearly interpolates between the

\(c_{i}^{\mathrm{Pmad}}\)

in the anterior to posterior direction. Background fluorescence (measured outside the range of the Dpp signal) was subtracted before normalization.

\({\sigma}_{c}^{\mathrm{Pmad}}\)

was calculated in x bins of 2.5 μm via

\({\sigma}_{c}^{\mathrm{Pmad}}=({\Sigma}_{i=1}^{N}(c_{i}^{\mathrm{PMad}}-{\bar{c}}_{i}^{\mathrm{Pmad}}(x_{i}))^{2}{/}N)^{1{/}2}\)

where

\({\bar{c}}_{i}^{\mathrm{PMad}}(x)=P_{0}\mathrm{exp}(-x{/}{\lambda}^{\mathrm{PMad}})\)

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 salrange.

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±0.82μm, n=1200 cells), assuming that the cell is a circle.

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+η(x) where D₀ is the average diffusion coefficient andη(x) is a random function with zero average that describes the fluctuations. Analogously, we define k(x)=k₀+ζ(x).

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 η(x),ζ(x) andχ(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 first asked what is at the origin of morphogen concentration fluctuations observed in an epithelium such as the wing disk. Individual cells can differ in size, shape, number of receptors and protein transport rates(Fig. 1A). This cell-to-cell variability affects morphogen transport, degradation, production and storage in endosomes and thus induces spatial variations in the resulting concentration profile (Fig. 1B). Cell-to-cell variability has been reported within various systems (Arias and Hayward,2006; 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.

Several theoretical descriptions of gradient formation have been developed(Bollenbach et al., 2005; 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 c̄(x)=C₀exp(-x/λ)where x denotes the distance from the source(Fig. 1B), C₀ is the average concentration directly adjacent to the source, and λ 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 concentrationσ _c(x). To characterize the fluctuations of the gradient, we study the `relative concentration uncertainty'∑(x)=σ_c(x)/c̄(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 uncertaintyσ _x at a position x can be determined from the relative concentration uncertainty ∑ at this position viaσ _x(x)≅λ∑(x) (see Materials and methods). The precision of the gradient is inversely related toσ _x. We obtained analytical expressions for the gradient precision, which were verified numerically (see the theoretical study in the supplementary material).

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).

In the second case, in which producing cells are identical, whereas there is variability in the target (Fig. 2C), the influx j is constant while D and k fluctuate. Our analysis reveals a sudden decrease in∑(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)∼x^1/4 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 klead 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).

We then generalized the theory of gradient precision for developmental contexts other than the two-dimensional epithelium. Indeed, morphogen gradients also form in one-dimensional chains of cells such as the mandibular arch in zebrafish (Kimmel et al.,2003) 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^ν as described above for the two-dimensional scenario. The exponent vdepends on the dimensionality d of the tissue and is given byν=(3-d)/4 if d<3. For d=3 the exponentν=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)/4for large x.

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

\({\sigma}_{c}^{\mathrm{Dpp}}(x)={\langle}(c^{\mathrm{Dpp}}(x)-{\bar{c}}^{\mathrm{Dpp}}(x))^{2}{\rangle}^{1{/}2}\)

(Fig. 4C-E, details in Materials and methods). The shape of individual gradients is well described by an exponential decay c^Dpp(x)≅C₀exp(-x/λ^Dpp)(red line in Fig. 4D) with an average decay length λ^Dpp=17.0±4.3μm=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 λ^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 λ^Dpp for each GFP-Dpp profile by a fit to the function C₀exp(-x/λ^Dpp). We find thatλ ^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

\({\sigma}_{c}^{\mathrm{Dpp}}(x)\)

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

\({\Sigma}^{\mathrm{Dpp}}(x)={\sigma}_{c}^{\mathrm{Dpp}}(x){/}{\bar{c}}^{\mathrm{Dpp}}(x)\)

⁠. 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

\({\sigma}_{x}^{\mathrm{Dpp}}(x){\cong}{\lambda}^{\mathrm{Dpp}}{\Sigma}^{\mathrm{Dpp}}(x)\)

(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 [

\({\sigma}_{x}^{\mathrm{Dpp}}\)

(4 μm≅4.4μm=1.7 cells]. Next to the source,

\({\sigma}_{x}^{\mathrm{Dpp}}\)

(1 μm≅6.0 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)],

\({\sigma}_{x}^{\mathrm{Dpp}}\)

(40 μm≅9.7 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 I^Dpp 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

\({\sigma}_{x}^{\mathrm{Dpp}}\)

(40μm)≅6.1 μm=2.3 cells. As in our theory, the minimum of∑ ^Dpp(x), corresponding to

\({\sigma}_{x}^{\mathrm{Dpp}}\)

(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).

In summary, although showing some variability, the Dpp gradient is precise enough to provide positional information with a resolution of about two cells in a target tissue which is ∼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.

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 c^PMad(x), described by an exponential decay c̄^PMad(x)=P₀exp(-x/λ^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(λ^PMad=25.2±4.5 μm=9.7±1.7 cells) and the total PMad level I^PMad (variation coefficient 0.24). The average λ^PMad is comparable to the average decay length of the Dpp gradient (λ^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

\({\sigma}_{c}^{\mathrm{PMad}}(x)\)

is the uncertainty of c^PMad 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

\({\sigma}_{x}^{\mathrm{PMad}}\)

≅λ^PMad∑^PMad(x)=3.5μm=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

\({\sigma}_{x}^{\mathrm{Dpp}}\)

is correlated to that of the PMad gradient

\({\sigma}_{x}^{\mathrm{PMad}}\)

(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.

Since a relatively precise Dpp concentration gradient is translated into a similarly precise Dpp activity gradient encoded in the phosphorylation state of Mad, we next asked how precise is the response of the Dpp target genes. The PMad-Medea protein complex binds directly to a silencer sequence in the brinker repressor gene(Müller et al., 2003). 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-relatedtranscription 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 x̄^*=〈x^*=39.1±6.1μm=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 x_V 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 x_Vin 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.

In a morphogen paradigm, the key question is whether the uncertainty of the sal domain position reflects the variability of the Dpp gradient, as would be expected for a Dpp target gene. The positional uncertainty of the Dpp gradient at the average sal boundary

(x̄^*)=

\({\sigma}_{x}^{\mathrm{Dpp}}\)

(39.1μm)≅9.7 μm=3.7 cells for the non-normalized Dpp concentrations and even lower(⁠

\({\sigma}_{x}^{\mathrm{Dpp}}\)

(x̄^*)≅6.1μm =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.

Morphogen gradients provide cells in developing tissues with positional information that is used to form patterns of distinct cell types. As a consequence, adult animals have organs with precisely defined shapes and stereotypically patterned tissues. It is therefore important to ask if such precise patterns could be laid down by morphogen gradients. Alternatively,these gradients could lay out a relatively coarse pattern, while mechanisms occurring later in development increase precision. In this report, we addressed how cell-to-cell variability affects the precision of morphogen gradients. The transport of morphogens through a tissue with irregularly shaped cells with individual properties leads to an uncertainty in the shape of the resulting graded concentration profile. We have shown experimentally and theoretically that this uncertainty depends on the distance from the source in a characteristic way. The uncertainty of the positional information encoded in the Dpp gradient is lowest a few cells away from the source, where it is about one cell diameter. Closer to the source, precision is reduced by fluctuations of the morphogen production rate in the source, whereas farther away from the source fluctuations of the morphogen concentration accumulate as molecules move through the tissue.

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.

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(σ_j/j₀≅0.1,σ _D/D₀≅0.1,σ _k/k₀≅0.1; 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)∼x^1/4, which implies that the precision does not deteriorate dramatically within the range over which the morphogen signals. Assuming that Dpp signals up to 4λ^Dpp=68 μm=26 cells away from its source, we estimate from the concentration uncertainty at x=λ^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).

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 (λ^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.

We thank all members of the M.G.-G. and F.J. groups for constructive criticism of the manuscript. This work was supported by the Max Planck Society, DFG, VW, EU, SNF and HFSP.