Dpp gradient formation by dynamin-dependent endocytosis: receptor trafficking and the diffusion model

Morphogens are signaling molecules which are produced and secreted from a restricted region of a developing tissue and spread to form a graded profile of concentration. This concentration profile, also called gradient by developmental biologists, endows receiving cells with positional information. Indeed, cells in the target field interpret the gradient by responding with expression of different target genes above distinct concentration thresholds and thereby at different distances from the source. Secreted ligands of the TGF-beta, Wnt and Hh families among others have been shown to act as morphogens (reviewed by Tabata,2001).

Although the concept of morphogenetic signaling, first formulated by Turing(Turing, 1952) and modified by Wolpert (Wolpert, 1969), has pervaded the field of developmental biology, the cell biological basis of the spreading phenomenon itself for each particular morphogen molecule is still a matter of controversy (reviewed by González-Gaitán,2003; Vincent and Dubois,2002). Two scenarios for the dominant mechanisms of transport have been mainly discussed: extracellular diffusion and planar transcytosis, i.e. endocytosis and resecretion of the ligand that is thereby transported through the cells (reviewed by González-Gaitán,2003; Vincent and Dubois,2002).

In the case of the Drosophila TGF-beta-superfamily homolog Dpp,both diffusion and planar transcytosis have been proposed as transport mechanisms (Entchev et al.,2000; Lander et al.,2002) (reviewed by González-Gaitán,2003). Dpp is expressed within a narrow stripe of cells in the center of the developing wing epithelium(Basler and Struhl, 1994), from where it is secreted and forms a long-range gradient of concentration across 40 cell diameters (Entchev et al.,2000; Teleman and Cohen,2000). In experiments in which Dpp is pulsed from the source, the graded profile of Dpp concentration expands rapidly and reaches its steady state range of 40 cell diameters in less than 8 hours(Entchev et al., 2000; Teleman and Cohen, 2000). In addition, Dpp spreads equally in all directions.

The proposal that intracellular Dpp trafficking is implicated in its long-range dispersal stemmed from mosaic experiments in which endocytosis was impaired in mutant patches of cells: the `shibire rescue assay', the `shibire shadow assay' and the `Rab mutant assays'(Entchev et al., 2000). In the`shibire rescue assay', the source is wild-type (WT), whereas the target tissue is endocytosis-defective because of a Dynamin thermosensitive mutation(shibire^ts1, shi^ts1)(Chen et al., 1991). In this condition, Dpp is not internalized in the target cells and its range is restricted to the cells adjacent to the source. In the `shibire shadow assay',Dpp spreading from the source is confronted with an endocytosis-defective mutant patch of cell. In this situation, Dpp is unable to spread across the clone and forms a shadow distal to the clone. The shadow is transient and is finally filled by Dpp moving rapidly and in all directions from the sides of the clone. Finally, in the `Rab mutant assay', mutants for key Rab GTPases involved in the endocytosis/early endosomal dynamics (Rab5) or degradation(Rab7) are expressed in the receiving cells. When endocytosis is impaired or degradation is enhanced, the signaling range is reduced, whereas, conversely,an enhanced endocytosis/endosomal dynamics leads to an expansion of the signaling range. These data support the idea that Dpp dispersal is mediated by endocytosis and resecretion of the ligand in the receiving cells. In the absence of endocytosis, extracellular diffusion contributes only to spreading over a short-range (across 3-5 cells)(Entchev et al., 2000). However, the Dpp re-secretion event itself has not yet been directly monitored.

In general, ligand transport depends on complex non-linear kinetics, such as the kinetics of receptor binding/release and the kinetics of trafficking of ligands and receptors. Therefore, a quantitative analysis based on mathematical models is essential in order to establish that the observed ligand dynamics indeed emerge from a particular mechanism. Ligand dispersal has been early described theoretically using reaction-diffusion equations(Gierer, 1981; Gierer and Meinhardt, 1972; Koch and Meinhardt, 1994; Turing, 1952). Such theoretical approaches are useful to study robustness and precision in morphogen gradient formation (Eldar et al.,2002; Eldar et al.,2003; Houchmandzadeh et al.,2002) and have suggested that simple diffusion may not suffice to generate graded profiles of receptor occupation(Kerszberg and Wolpert, 1998). Furthermore, ligand trafficking in cells has been studied theoretically and a possible role of transcytosis to enhance transport efficiency has been proposed (Chu et al., 1996; Lauffenburger and Linderman,1993).

A recent theoretical analysis of Dpp spreading indicates, though, that transcytosis does not play an important part in this process(Lander et al., 2002). The properties of transport based on extracellular diffusion were studied using a model that takes into account diffusion and receptor binding. It was suggested that this `diffusion, binding and trafficking' (DBT) model can generate ligand profiles which are consistent with WT gradients and the results observed in the `shibire shadow assay' (Lander et al.,2002). A block of endocytosis could induce a higher level of surface receptors and thereby titrate out the pool of spreading free ligand,obstructing the ligand transport (Lecuit and Cohen, 1998). Lander et al. argued that this scenario generates a transient shadow. They solved reaction-diffusion equations in a one-dimensional geometry, suggesting that this description suffices to capture key features of these experiments.

Here, we perform a theoretical analysis of the DBT model in one and two dimensions. We discuss the role of the geometry, the appropriate boundary conditions, and the initial conditions in the `shibire shadow assay' and the`shibire rescue assay'. We then determine experimentally the levels of receptors and extracellular Dpp to compare them with the ligand and receptor profiles obtained in the DBT model. We show that although the DBT model cannot account for the observed transient shadows, a modified version of the model,introducing surface receptor saturation, is consistent with such shadows. However, the receptor and ligand profiles under this modified DBT model are inconsistent with the observed levels in the `shibire shadow' and `shibire rescue' experiments. We therefore conclude that current models in which transport occurs exclusively in the form of extracellular diffusion cannot explain the experimental data, suggesting that endocytosis plays an active role in the ligand transport beyond the regulation of receptors at the surface.

Equations 1, 2, 3, 4, 5, 6, 7 together with boundary conditions have been solved numerically using a forward Euler differencing algorithm (see supplementary material). In the one-dimensional calculations,the area of interest (AOI) was discretized by 200 sites, and in our two-dimensional calculations, we used a triangular lattice with 200×174 sites. The time-step chosen was Δt=0.025 seconds for the one-dimensional and Δt=0.033 seconds for the two-dimensional calculations.

shi^ts1 and tkv⁸ are described in Flybase(http://flybase.bio.indiana.edu). tkv⁸ is a Tkv receptor truncated at amino acid 144 before the transmembrane domain that presumably represents a null mutation of tkv (Nellen et al.,1994). UAS-Dynamin, UAS-Tkv and UAS-GFP-Dpp were previously described (Entchev et al.,2000; Nellen et al.,1996). UAS-GFP was graciously provided by Barry Dickson(Institute of Molecular Pathology, Vienna, Austria). The tub-DsRedconstruct was inserted into a P-element plasmid containing the promoter of the tubulinα1 gene and flanked at its 3′ end by the 3′ UTR of the tubulinα1 gene (Basler and Struhl, 1994). Tub-DsRed was recombined onto FRT18 chromosome to allow the generation of somatic clones by Flp-mediated mitotic recombination (Xu and Harrison,1994).

Rabbit anti-Tkv antibody was generated against two peptides corresponding to parts of the intracellular kinase domain (H2N-SQQLDPKQFEEFKRAC-CONH2 and H2N-GFRPPIPSRWQEDDVC-CONH2). Rabbit luminal anti-Tkv antibody was generated against two peptides corresponding to the luminal side of the Tkv peptide sequence outside the ligand binding cleft (H2N-YEEERTYGCMPPEDNG-CONH2 and H2N-KEDFCNRDLYPTYTP-CONH2). The immune sera were affinity chromatography purified using the corresponding Tkv peptides coupled to CNBr-activated Sepharose 4B (Amersham Biosciences). The specificity of the antibodies was tested by preincubating the purified antibody with 100 μg/ml Tkv peptide(or 500 μg/ml when performing the `extracellular immunostaining protocol with luminal anti-Tkv antibody') for 30 minutes at room temperature and performing subsequently an antibody staining on Tkv-overexpressing discs. No fluorescent signal was detected under these conditions, whereas preincubation with a control peptide did not affect the staining. Immunostainings were performed as previously described (Entchev et al., 2000) using Mouse anti-Myc, 1:25 dilution; Rabbit anti-Tkv(intracellular), 1:125; Goat anti-GFP, 1:100. Extracellular GFP-Dpp and cell surface-exposed Tkv were detected by incubating prior to fixation(Strigini and Cohen, 2000)with Goat anti-GFP antibody, 1:10 dilution, and Rabbit anti-Tkv (raised against the luminal domain of Tkv), 1:10 dilution, respectively. Dimmer GFP-Dpp signal was found upon extracellular immunostaining compared with the normal staining because of the different washing procedures. To estimate GFP-Dpp range in number of cells, a fluorescent phalloidin (Molecular Probes)counterstaining was performed to monitor the cell profiles. Cryostat z-sections at Cryo-Star HM 560 (Microm) were performed with PFA-fixed developing wing discs incubated for at least 12 hours at 4°C in 30%sucrose solution in PBS and mounted with Tissue-Tek (Sakura). Images were acquired in a Zeiss LSM510 confocal microscope and processed using Adobe Photoshop 7.0 (Adobe Systems). Quantifications were done with Image J(NIH).

tkv⁸ mutant Minute⁺/FRT clones(Xu and Harrison, 1994) were generated by heat shock (30 minutes, 36°C) in 3-day-old larvae(HS-Flp/+; M(2)z PMyc FRT40A/tkv⁸ FRT40A) and raised at 25°C to mid-third-instar larvae. To induce PMyc transcription larvae were heat-shocked at 38°C for 1 hour followed by at least 1 hour at 25°C to allow the translation of the PMyc transcript prior to fixation. shi^ts1 FRT mutant clones were generated in larvae of the genotype shi^ts1 FRT18A/HS-NM8A FRT18A; HS-Flp/+ and shi^ts1 FRT18A/tub-DsRed FRT18A; HS-Flp/+, respectively. Embryos were collected during one day at 18°C, larvae were raised for one day at 18°C and heat-shocked (90 minutes, 38.3°C). Larvae were subsequently kept at 25°C until third-instar larval stage. Afterwards,endocytosis was blocked either for 3 hours at 34°C followed by 1 hour at 38.3°C to induce both NMyc transcription and shibire block and 1 hour at 34°C to allow the translation of the NMyc transcript, or for 5 hours at 34°C in the case of larvae of the genotype shi^ts1FRT18A/tub-DsRed FRT18A; HS-Flp/+. Dissection of wing discs was performed at 34°C.

shi^ts1; UAS-Dynamin⁺/+; dpp-gal4/UAS-GFP-Dpplarvae were kept at the shi^ts1-permissive temperature(25°C or 18°C) to allow normal wing development until third-instar larval stage, when endocytosis was blocked for 6 hours at 34°C. Wing discs were dissected and fixed.

The wing disc forms a flat pouch consisting of two cell layers, which are connected at the edges (Fig. 1D). One cell layer is formed by columnar epithelial cells and includes the wing primordium (Fig. 1A,D; arrowhead); the other layer forms a squamous epithelium, the peripodial epithelium (Fig. 1A,D; asterisk). Dpp-producing cells are located within a narrow band, 5 cells wide, along the anterior-posterior compartment boundary(Fig. 1) in the center of both the columnar and the peripodial cell layer(Fig. 1D).

We are interested in the formation of a Dpp gradient in a particular area of the primordium. Fig. 1C and Fig. 2C define this AOI. The AOI corresponds to a rectangular piece of tissue in the primordial cell layer located in the posterior compartment and adjacent to the Dpp source(Fig. 1C, Fig. 2C). It extends L_x=200 μm (50 cells) in the x-direction and L_y=200 μm (50 cells) in the y-direction(Fig. 2C). In principle, we need to take into account the complete geometry of the wing disc in order to have a full mathematical description of the ligand kinetics. However, the ligand kinetics within the AOI depends only weakly on the kinetics outside,provided the size of the AOI is sufficiently large (see below).

Let us consider the boundary at x=L_x on the opposite side of the AOI with respect to the source (`distal boundary'; Fig. 2C). An outflux of ligand across this boundary is present which becomes small if the ligand concentration nearby is small. We expect the current across the boundary sufficiently far from the source to be small enough to be neglected and impose the current to be zero at the boundary. We choose the width L<sub>x</sub> of the AOI, such that this choice of boundary condition does not affect the ligand distribution in the region where the gradient develops. Indeed, one can show that for L<sub>x</sub>≥200 μm the choice of the `distal boundary' condition becomes irrelevant (see Fig. S3 in the supplementary material).

At the remaining boundary lines y=–L_y/2 and y=+L_y/2 of the AOI (`side boundaries'; Fig. 2C), we also impose `zero current' conditions across the boundary line, J_y=0. In the simplest case, in which the whole system is homogeneous in the y-direction, this condition is satisfied automatically. An example for the formation of a graded ligand profile using these boundary conditions(`current boundary' conditions) is displayed in Fig. 2D. A more interesting case arises if a patch of genetically modified cells (a clone) is present in the system. Then, ligand and receptor concentrations will vary along the y-coordinate and a current can cross the boundaries at y=–L_y/2 and y=+L_y/2. However, if the boundaries are located sufficiently far away from the clone, the ligand concentration in the vicinity of the clone is not affected by our choice of boundary conditions. For a clone of size 50 μm, we observed that the choice L_y=200 μm results in concentration profiles that are independent of the specific boundary conditions imposed (see supplementary material).

The role of endocytosis during Dpp gradient formation has been studied by inducing a patch of thermosensitive Dynamin mutant cells, a shi^ts1 clone, into a wing disc(Entchev et al., 2000). In this experiment, GFP-Dpp expression in the source was triggered using the thermosensitivity of the driver system. The experiment is performed under the following initial conditions: (1) Non-tagged endogenous Dpp is also expressed in the disc and is presumably in a steady-state distribution, because normal-looking adults emerge when applying these temperature conditions in WT animals; and (2) the thermosensitive Dynamin mutant cells can perform endocytosis at the permissive temperature (25°C) at t=0. The experiment starts by elevating the temperature (34°C), which causes an immediate block of endocytosis in the clone, and an immediate onset of GFP-Dpp production in the source.

Under these conditions, GFP-Dpp is entering the target tissue. When confronted with the endocytosis-defective Dynamin mutant clone, a `shadow', a region with reduced GFP-Dpp concentration distal to the clone, is generated. This shadow can only be observed for a limited time: after several hours, the shadow region is indistinguishable from the adjacent regions. This result provides evidence that endocytosis is essential for the long-range movement of Dpp. Based on this and the `shibire rescue assay' (see below), a working model was proposed in which free diffusion of Dpp only accounts for short-range spreading of Dpp, and long-range movement of Dpp is mediated by repeated rounds of internalization and resecretion through the receiving cells(Entchev et al., 2000).

We now address the question of whether the DBT model can account for the formation of a transient shadow behind the clone. For this purpose, we solve the DBT dynamic Eqns 3, 4, 5, 6, 7 using the `current boundary conditions' discussed above. We perform a two-dimensional calculation representing the clone by a rectangular region. In this region the internalization rates for the free- and the bound-receptor k_in and k_p are abruptly reduced at t=0 in order to model an impaired endocytosis.

A rapid reduction of the receptor internalization rates is consistent with the observation that endocytosis is blocked within seconds in the thermosensitive dynamin mutant (Entchev et al., 2000; Ramaswami et al.,1994). In the DBT model, completely blocking endocytosis would correspond to setting the internalization rates k_in and k_p to zero. However, this leads to an unrealistic unlimited increase of the cell surface receptor concentration. There are two obvious ways to limit the surface receptor concentration: introducing surface receptor saturation by defining the externalization rates of the receptor as a function of surface receptor concentration, and reducing internalization rates to a non-zero value. The effects of surface receptor saturation are discussed in the next section. In this section, the internalization rates are reduced by a factor of 10.

In our calculations, the initial receptor concentrations in the whole AOI are set to the WT steady-state values in the absence of ligands. The resulting concentration profiles in the presence of a clone are displayed in Fig. 3A-E. Behind the clone,the concentration exhibits a clear minimum along the y-axis in the ligand concentration (Fig. 3A,E). This minimum reflects a depletion or `shadow' that is reminiscent of the Dpp depletion observed by Entchev et al. However, in contrast to the experiments,in which the shadow disappeared with time, the DBT model does not generate transient shadows, but instead shadows become more pronounced with time and persist in the steady state (Fig. 3E). The contrast of the shadow can be quantified by comparing the total ligand concentrations at the points indicated by the arrows in Fig. 3E. As the Dpp gradient is built up, the contrast increases monotonously and attains a steady-state value with maximal contrast (Fig. 3F).

We have thus shown that, in the DBT model, shadows are generated that become more contrasted with time and remain persistent. Transient shadows as observed in the experiments are not obtained in the DBT model.

The DBT model becomes biologically meaningless if internalization rates become zero, because in this case the level of surface receptors tends to infinity. It is therefore not possible to describe the extreme case in which the endocytotic block is complete. However, at the restrictive temperature the internalization in shibire mutants is negligible(Entchev et al., 2000; Verstreken et al., 2002). Therefore, we modify the DBT model to include the saturation of surface receptor levels. This allows us to freely vary the internalization rates and even set them to zero.

Here, the parameters

The profiles of total (Fig. 4A-E) and internal bound (inset in Fig. 4C; see also Fig. 7E) Dpp have been obtained by a calculation of the DBTS model in two dimensions and in the presence of a clone. Inside the clone, the internalization rates k_p and k_in have been set to zero at t=0. The profile in the y-direction behind the clone displays a pronounced transient shadow similar to the experimental observation(Fig. 4A,C-E), followed by a weak persistent accumulation of ligand behind the clone (`anti-shadow') after long time periods (Fig. 4B-E). The corresponding contrast of this shadow attains a maximum after a few hours(Fig. 4F). The emergence of a shadow is a consequence of a rapid 20-fold increase of the surface receptor concentration inside the clone (Fig. 7G). In order to obtain such a rapid increase in the surface receptor concentration, the externalization and internalization rates had to be increased (by at least a factor of 10) compared with the values used in the DBT model (Table 1).

The models discussed so far assume that receptors produced de novo traffic through the secretory pathway via endocytic compartments(Fig. 2A). Although this biosynthetic trafficking pathway seems to exist in yeast cells(Harsay and Schekman, 2002),the conventional biosynthetic pathway does not necessarily involve endosomal structures (Alberts et al.,1994) (Fig. 2B). We therefore modified the DBTS model to consider the standard biosynthetic and endocytic pathways (Alberts et al.,1994) (see supplementary material). Numerical calculations show that this modification does not lead to major changes with respect to the DBTS model.

In summary, the DBTS model can generate Dpp profiles outside of the clone that qualitatively resemble the ones observed, although anti-shadows have not been observed experimentally. Note that the DBT and the DBTS model require a drastically increased surface receptor concentration within the clone in order to create significant shadows. This generates a large accumulation of surface-bound ligand inside the clone in both models.

The DBT model can generate morphogen gradients similar to those seen experimentally in the absence of a clone(Fig. 2D). In particular, the steady-state profiles of total Dpp and extracellular Dpp, monitored with a specific extracellular GFP-Dpp immunostaining(Strigini and Cohen, 1999)(Fig. 5), resemble the profiles obtained by the DBT model for total (Fig. 2D) and extracellular Dpp (not shown). In addition, the time needed to form the gradient upon a pulse of Dpp from the source(Fig. 2D) is consistent with experimental observations in which the gradient expands until it reaches a steady state 6 to 8 hours after the initiation of the pulse(Entchev et al., 2000).

In the presence of a shi^ts1 clone, there are, however,important differences between the results of the DBT model and the experimental data. If we choose parameters in such a way that the ligand profiles define a shadow behind the clone, the levels of (1) surface receptors, (2) total ligands inside the endocytosis-defective shibire clone and (3) extracellular ligand observed experimentally are qualitatively different from those obtained in the DBT model. The same conclusion holds true for the DBTS model, in which the levels of total(Fig. 4C) and internalized ligand (Fig. 4C, inset) in the clone are also elevated, unlike our experimental observations.

We first consider the cell surface receptor concentration. The essential prerequisite for forming a shadow in the DBTS model is a rapid accumulation by a factor of 10-20 of surface receptors in the clone(Fig. 7G). In order to compare this with the actual surface receptor levels in the clone, we raised an antibody that specifically recognizes the Dpp receptor, Tkv. Confirming the results published in Teleman and Cohen(Teleman and Cohen, 2000), we find that the receptor accumulates predominantly at the cell surface, although some intracellular vesicular structures can also be observed (see Fig. S5A in the supplementary material). The level of the Tkv protein follows the accumulation of the Tkv transcript, which is distributed in a graded fashion complementary to the Dpp gradient (Lecuit and Cohen, 1998) (Fig. S5A). The antibody specifically recognizes Tkv because it: (1) detects a corresponding band of 63 kDa in western blot experiments from developing larvae (not shown); (2) detects overexpression levels of Tkv, induced by the Gal4 system using a ptc-gal4 driver(Fig. S5B); (3) is titrated out by incubating it, prior to immunostaining,with the peptide used to raise the antibody (Fig. S5C); and (4) does not stain cells lacking Tkv in mutant mosaics (Fig. S5D). In addition, quantitative RT-PCR experiments show that our antibody-staining conditions can robustly detect overexpression levels above 5-fold (Fig. S5B and not shown).

We then addressed whether the levels of surface Tkv are altered in shibire mutant clones when endocytosis is blocked. Fig. 5E shows that in the shibire mutant cells after 5 hours at the restrictive temperature (the experimental conditions that generated the Dpp shadows in the shibire clones),the levels of receptors associated to the cell membranes are not changed. This result indicates that even though endocytosis is blocked during 5 hours,surface receptor levels do not change tenfold or more. To confirm that the Tkv pool associated to the cell profiles correspond to Tkv on the cell surface, we generated an antibody directed against the luminal domain of Tkv (see Materials and methods) and performed the `extracellular immunostaining'protocol (Strigini and Cohen,1999). We determined the specificity of this antibody following the same criteria discussed above (Fig. S5E-G). As with the other Tkv antibody, our antibody staining in this condition can robustly detect overexpression levels above 5-fold (Fig. S5), as monitored by RT-PCR (not shown). Figs 5F,G show that the levels of surface Tkv are not affected upon 5 hours of endocytic block in the shibire mutant clones. The observed shadow can therefore not result from a mechanism based on a high surface Tkv receptor concentration as in the DBT and DBTS models.

According to the DBT model, the levels of internalized Dpp are significantly increased inside the clone after long time periods(Fig. 7C,D). This may seem surprising when internalization is blocked. The effect is because decreased internalization by a factor of 10 leads to an accumulation of the surface receptors, which in turn increases the levels of receptor-mediated endocytosis of Dpp. Such an effect was not observed inside the shibire clone in which the levels of intracellular Dpp were reduced after 5 hours (M.G.-G., unpublished)(Entchev et al., 2000). A reduced internal Dpp concentration is achieved in the DBTS model if the internalization rates inside the clone are set to zero(Fig. 7E, Fig. 4C, inset). Furthermore,in the DBT model, the total concentration of ligand inside the clone is significantly higher as compared with outside the clone(Fig. 7B).

Finally, note that both for the DBT(Fig. 3C, inset) and the DBTS model (not shown), the extracellular level of ligand is significantly increased in the clone. Such an extracellular accumulation of Dpp was not observed (Entchev et al.,2000).

The `shibire rescue assay' allows us to monitor how blocking endocytosis in the receiving cells has an effect on the formation of the Dpp gradient and on the levels of intracellular and extracellular ligand and receptor(Entchev et al., 2000). In these experiments, the receiving cells cannot perform endocytosis at the restrictive temperature in a shibire mutant animal, whereas the secreting cells are rescued with a Dynamin⁺ transgene and can thereby perform endocytosis normally (see Materials and methods). At the permissive temperature, a GFP-Dpp gradient forms in the target tissue. After a temperature shift to the restrictive temperature, endocytosis is blocked in the receiving cells. Upon 6 hours of endocytic block, internalized Dpp has disappeared and no gradient can be observed(Entchev et al., 2000).

We use the DBTS model to calculate ligand profiles under conditions that correspond to this shibire rescue experiment. We modify the AOI and include a region with –10 μm<×<0 representing half of the stripe of producing cells (Fig. 2C). This region is described by the DBTS model with the same parameters as before, but in addition each cell in this region also secretes ligand with rate ν. Because of the symmetry of the source, we now impose zero ligand current as a boundary condition at x=–10 μm (see supplementary material). In WT, endocytosis is active and the ligand-concentration profiles(Fig. 6A, broken line) closely resemble those obtained without explicitly describing the source(Fig. 2D).

At the permissive temperature, the gradients of extracellular ligand generated by the DBTS model (Fig. 6B, broken line) are consistent with the observed levels of extracellular Dpp in WT (Fig. 5D) and in the control discs of the `shibire rescue' experiment(Fig. 6E,F). In the source, we observed a large accumulation of GFP-Dpp, probably corresponding to ligand in the secretory pathway (Fig. 6D,E,G,H,J,K). This pool of Dpp is not included in our model.

After 6 hours of endocytic block in the `shibire rescue' discs (modeled by setting the internalization rates in the target tissue to zero), the calculated total (Fig. 6A, red line) and extracellular ligand (Fig. 6B, red line) distributions in the DBTS model changes by a factor of up to four (total) and up to ten (extracellular) in the receiving tissue and generate a long-range gradient of high levels of Dpp. In contrast, when we monitor total and extracellular Dpp after 6 hours of the endocytotic block at the restrictive temperature in the experiments, the Dpp concentration decreases and the range of the extracellular Dpp gradient is reduced(Fig. 6D-F)(Entchev et al., 2000). Notably, whereas the DBTS model exhibits a discontinuous behavior of the external ligand concentration between the WT source and the receiving tissue with blocked endocytosis (Fig. 6B), no such discontinuity is observed in the experiment(Fig. 6D-F).

We also investigated the surface receptor levels in the `shibire rescue assay'. The DBTS model generates a discontinuity of the levels of surface receptors by a factor of 20 in the receiving cells when compared with the source after 6 hours of the endocytic block(Fig. 6C, red line). To monitor the surface receptor levels, we performed Tkv immunostainings in the `shibire rescue' discs after endocytic block. No significant increase in the receptor levels between the WT source and the receiving cells could be found(Fig. 6G-I). Furthermore, the levels of surface Tkv were not increased in the mutant cells as determined by`extracellular immunostaining' (Strigini and Cohen, 1999) using the antibody directed against the Tkv luminal domain (Fig. 6J-L).

The `shibire rescue assay' was compared with results of the DBTS model. The distributions of both the extracellular ligand and the surface receptor densities remain qualitatively the same in the DBT model as in the DBTS model(not shown). Our comparison of experiment and theory therefore leads to the conclusion that high surface receptor levels cannot be the origin of the shadows in the `shibire shadow assay'.

In this report, we have studied whether ligand transport by extracellular diffusion can explain the spreading of Dpp through the target tissue. We compared the theoretical results of two models, in which transport is exclusively because of extracellular diffusion of the ligand, with a direct measurement of the extracellular and intracellular pool of ligand as well as the levels of total and cell surface receptor in WT and in endocytosis-defective mosaic tissues. The current models in which ligand transport is solely because of free extracellular diffusion are inconsistent with the experimental results obtained by the `shibire shadow assay' and the`shibire rescue assay'.

Three points lead to this conclusion. First, a DBT model of the `shibire shadow assay' generates permanent shadows (Figs 3, 7), whereas the experimental shadows are transient (Entchev et al.,2000). Second, the DBTS model can generate transient shadows, but only if the surface receptor levels in the clone increase dramatically(Fig. 4). This leads to a strong increase in the levels of extracellular ligand in the clone(Fig. 4C). Using receptor antibodies in the `shibire shadow assay', we did not observe these higher levels of surface receptors in the clone(Fig. 5E-G). Similarly, the levels of extracellular ligand were not increased in the clone(Entchev et al., 2000). Third,in the DBTS model for the `shibire rescue assay', the levels of both the extracellular Dpp and the surface receptors are dramatically increased in the endocytosis-defective target cells as compared with the WT source(Fig. 6B,C). Such an increase is not seen experimentally (Fig. 6D,G). Instead, extracellular Dpp enters the receiving tissue over a distance of only 4-5 cells in steady-state(Fig. 6D,F). This is in contrast to both DBT and DBTS models of the `shibire rescue assay' in which ligand can enter the tissue over large distances. Therefore, in addition to downregulating surface receptors, endocytosis is likely to play additional roles in the transport of ligands during gradient formation.

These three caveats of the DBT/DBTS models are actually not caused by the choice of a particular set of parameters. The parameter values used in our calculations (Table 1) were chosen in such a way, that the typical distance over which the ligand gradient extends as well as the characteristic time to reach steady state are consistent with the experimentally observed profiles. Furthermore, if possible, parameters were chosen similar to values measured for the EGF receptor in a cell culture system (Table 1). In the case of the DBT model they are the same parameters used in Lander et al. (Lander et al.,2002) when they studied the diffusion model. Note that our results showing that a high surface receptor concentration inside the clone is required for shadows to appear is independent of any choice of parameters. Furthermore, convincing shadows appear in the DBT and DBTS models only for values of k_off, which are small compared with those typically measured in related systems(Table 1). It will be necessary to estimate the actual parameter values for Dpp during wing morphogenesis in order to ultimately understand its mechanism of spreading (see below).

The geometry and boundary conditions discussed here differ from those introduced in Lander et al. (Lander et al., 2002). There, the one-dimensional case is considered exclusively, i.e. concentrations of ligand and receptor that are independent of y, even in the presence of a clone. At the boundaries x=0 and x=L_x, Lander et al. imposed the ligand concentrations. In particular, at x=L_x the concentration was fixed to A=0, which implies that all ligands that reach x=L_x are instantaneously degraded. Such a Dpp sink does not exist in the wing disc. This sink has a significant influence on the shape of the gradient obtained in the calculations of Lander et al., with L_x=100 μm, whereas the difference becomes insignificant for L_x=200 μm (see Fig. S3C,D in the supplementary material).

At x=0, the boundary conditions imposed by Lander et al. are also problematic. These boundary conditions imply that at x=0 the ligand concentration is imposed by the secreting cells but is unaffected by the exchange of ligands between secreting and non-secreting cells via diffusion(for details, see supplementary material). We refer to the boundary conditions imposing the ligand concentration and the ligand current as `concentration boundary conditions' (Lander et al.,2002) and `current boundary conditions' (this work),respectively.

We also performed one-dimensional calculations in the presence of a clone(Fig. 7A-C). In these calculations, the clone region is represented by an interval on the x-axis. We find that a one-dimensional description can generate ligand profiles that qualitatively correspond to the profiles in the x-direction of a two-dimensional calculation(Fig. 3C), if the extension of the clone in the y-direction is larger than the distance over which the gradient forms (compare Fig. 7C and 7D). For the present choice of parameters and an extension of the clone of 50 μm in y-direction, this criterion is satisfied. The contrast c of shadows in the two-dimensional geometry can be determined approximately in a one-dimensional calculation by taking the difference of the concentration behind the clone and the concentration at the same position in a calculation without a clone. In general, however, a two-dimensional description is required to describe the effects of the clone.

Our results show that the DBT model generates permanent shadows behind the clone, whereas a DBTS model is able to generate transient shadows similar to those observed in the experiments by Entchev et al.(Entchev et al., 2000). Note,that this finding differs from the results of Lander et al.(Lander et al., 2002), who concluded that the DBT model can generate transient shadows.

In their one-dimensional calculations of the DBT model, like in ours,endocytosis block is modeled by a tenfold reduction of the internalization rates at t=0. However, in their study, the receptor concentrations[R_out] and [R_in] in the clone are simultaneously and abruptly set to the steady-state values corresponding to the reduced internalization rates. This assumption does not correspond to the experimental situation interpreted in the framework of the DBT model, because it would imply an instantaneous tenfold increase of the surface receptor concentration within the clone at the time of the temperature shift (see broken line in Fig. 7F). This is different from what is expected to happen in the experiment according to the DBT model: as the internalization rates in the clone are reduced in an abrupt fashion at t=0, the concentration of surface receptors only gradually increases (Fig. 7F,unbroken line). We have performed the same calculations as described in Lander et al., but changing the initial conditions for the receptor concentration(Fig. 7C). This one-dimensional calculation qualitatively leads to the same result as already discussed in two dimensions: a shadow develops that at t=5 hours is weak and becomes more pronounced after long time periods (compare Fig. 7C and 7D).

We have repeated the calculations of Lander et al., using the parameter values and system size, the boundary conditions, and the initial conditions chosen in their article. Note that in these calculations, we have set the initial surface receptor concentration in the clone to a tenfold larger value as compared with the remaining tissue as discussed above. As a result, we obtain ligand profiles that after long time periods exhibit a persistent shadow in the steady state identical to the situation in which the surface receptor level increases gradually (compare Fig. 7B and 7C). We found no transient shadows in these calculations. This result is different from the one published in their work (Fig. 7B).

The fact that these calculations lead to ligand profiles that differ from those published in Lander et al. (Lander et al., 2002) indicates a possible technical problem in their calculations. Repeating the calculations of Lander et al., we noticed that their results could be reproduced. However, this was possible only if the receptor production rate inside the clone was reduced by a factor of 10 at t=0 as compared with the one outside the clone. The results of our calculations with this additional change in the clone are displayed in Fig. 7A. In this case, the surface receptor concentration in the clone after undergoing an initial step-wise increase, relaxes to a steady-state value that is similar to the steady state in the tissue outside the clone(Fig. 7F, dotted line). The corresponding ligand profiles of Fig. 7A indeed coincide with the results published in Lander et al.(Lander et al., 2002), see Fig. 7 therein. It is possible that in these calculations the receptor production rate in the clone was reduced by a factor of 10. In summary, in the case of the calculations discussed in Lander et al. (Lander et al.,2002), the shadow most likely appears because of the sudden step-wise increase of the surface receptor level; the shadow disappears at 24 hours because the receptor production rate is reduced and the surface receptor level therefore relaxes to approximately the same steady state as outside the clone.

Our results emphasize the facts that the number of dimensions considered(in particular in the presence of mutant clone) (Figs 3, 4), the size of the AOI (see Fig. S3 in the supplementary material), the boundary conditions (in particular the `source boundary' and the `distal boundary')(Fig. 2), and the initial conditions (most notably the levels of surface receptor in the clone at the beginning of the experiment) (Fig. 7F) are of key importance.

We have shown in this work that neither the DBT nor the DBTS model can explain the observed ligand and receptor profiles during Dpp spreading in the wing disc. Why should these models fail even though they incorporate many essential phenomena such as ligand diffusion, internalization and resurfacing via receptor recycling?

The essential point of both the DBT and the DBTS model is that ligand transport, which is described by the ligand current given in Eqn 1, is solely because of diffusion. In other words, this means that ligand bound to the surface receptors when internalized can only resurface at the same position on the cell surface where it was internalized. Only in this case is Eqn 1 justified and the intracellular transport of the ligand would not contribute to the current of the ligand in the tissue. This implies that simple reaction diffusion models ignore that in principle, ligand could also be transported by traveling through cells and resurface at other positions on the cell surface when receptors are recycled.

The fact that the DBT and DBTS models, which ignore these effects, cannot account for observed Dpp spreading suggests that contributions of receptor trafficking to transport and ligand current may indeed play an important role. We are currently generalizing the DBT/DBTS models to incorporate all relevant transport phenomena (diffusion and transcytosis) in the ligand current as well as the possibility of extracellular degradation of the ligand.

Our working hypothesis is that two phenomena contribute to the Dpp current in the developing wing epithelium: extracellular diffusion and intracellular trafficking (i.e. endocytosis plus resecretion). What is the relative importance of these two phenomena to the spreading of the morphogen? Both might be important. Limited by binding to the extracellular matrix and/or degradation, extracellular transport of the morphogen may only account for the spreading of the ligand across a few cell diameters. Intracellular trafficking in turn accounts for the movement of the morphogen across one cell diameter. Both phenomena together then lead to the long-range spreading of the morphogen.

Although it is expected that extracellular diffusion plays a role during morphogen spreading (Crick,1970), it has been argued that extracellular diffusion alone is insufficient to understand the reliability and precision of the formed gradient (Kerszberg and Wolpert,1998). The important role of intracellular trafficking has been uncovered in experiments in which endocytosis is blocked during morphogenetic signaling (González-Gaitán,2003). When endocytosis is blocked in the receiving tissue, Dpp spreading does occur, but generates a short-range gradient and thereby signaling responses only within 3 to 5 cells(Entchev et al., 2000; González-Gaitán and Jäckle, 1999). In particular, in a thermosensitive alpha-adaptin mutant, Dpp activates transcription of its target gene spalt only within 4-5 cells from the source(González-Gaitán and Jäckle, 1999), instead of within 15 cells in WT. Similar results where obtained by expressing a dominant-negative Rab5 mutant, which impairs endocytosis and endosomal dynamics(Entchev et al., 2000). These results do not exclude a role of endocytosis in the transduction, rather than on the spreading of Dpp. However, in the `shibire rescue assay', the reduced range of the extracellular Dpp gradient(Fig. 6D,E) indicates that impaired endocytosis restricts the spreading of Dpp.

This report is a theoretical and experimental study to address whether diffusion as the sole transport mechanism can explain the spreading of Dpp. We are currently studying the role of different transport mechanisms for Dpp spreading. Based on the values given in Table 1, it has been argued that the rates of endocytosis and recycling known for the EGF receptor in cultured cells are too small to allow for a sufficiently rapid transport by transcytosis (Lander et al.,2002). Indeed, our first results based on generalized models(including diffusion and transport by planar transcytosis) show that the parameter values used in this work (Table 1) do not produce consistent gradients during reasonable times. In particular, these models require a faster rate of endocytosis and recycling than those known for the EGF receptor in cultured cells. Therefore, it is essential to measure directly the different dynamic parameters, including the extracellular diffusion coefficient as well as the rates of endocytosis,degradation and recycling of Dpp in the developing wing. To estimate these parameters in situ we are currently monitoring photoactivatable fusion proteins in different cellular locations (extracellular versus endosomal) in the very context of the developing wing epithelium.

We thank A. Schwabedissen and D. Backasch for their excellent technical assistance. T. Lecuit, K. Simons, D. Lubensky, K. Sekimoto and people in the González laboratory for comments on the manuscript. This work was supported by the Max Planck Society and Deutsche Forschungsgemeinschaft.