Logical modelling of the role of the Hh pathway in the patterning of the Drosophila wing disc
1Institute for Virus Research, Kyoto University, Shogoin-Kawahara, Sakyo-ku, Kyoto, Japan, 2INSERM U928 - TAGC, Campus Scientifique de Luminy, Marseille and 3CONTRAINTES Project, INRIA-Rocquencourt, Le Chesnay, France
*To whom correspondence should be addressed.
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ABSTRACT |
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 TOP ABSTRACT 1 INTRODUCTION2 METHODS3 RESULTS4 DISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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Motivations: The development of most tissues and organs relies on a limited number of signal transduction pathways enabling the coordination of cellular differentiation. A proper understanding of the roles of signal transduction pathways requires the definition of formal models capturing the main qualitative features of these patterning processes. This is a challenging task because the underlying processes, diffusion, regulatory modifications, reception and sequestration of signalling molecules, transcriptional regulation of target genes, etc. are only partly characterized. In this context, qualitative models can be more readily proposed on the basis of available (molecular) genetic data. But this requires novel computational tools and proper qualitative representations of phenomena such as diffusion or sequestration.
To assess the power and limits of a logical formalism in this context, we propose a multi-level model of the multi-cellular network involved in the definition of the anterior–posterior boundary during the development of the wing disc of Drosophila melanogaster. The morphogen Hedgehog (Hh) is the inter-cellular signal coordinating this process. It diffuses from the posterior compartment of the disc to activate its pathway in cells immediately anterior to the boundary. In these boundary cells, the Hh gradient induces target genes in distinct domains as a function of the Hh concentration. One target of Hh signalling is the gene coding for the receptor Patched (Ptc), which sequesters Hh and impedes further diffusion, thereby refining the boundary.
Results: We have delineated a logical model of the patterning process defining the cellular anterior–posterior boundary in the developing imaginal disc of Drosophila melanogaster. This model qualitatively accounts for the formation of a gradient of Hh, as well as for the transduction of this signal through a balance between the activatory (CiA) and inhibitory (CiR) products of the gene cubitus interruptus (ci). Wild-type and mutant simulations have been carried out to assess the coherence of the model with experimental data. Interestingly, our computational analysis provides novel insights into poorly understood processes such as the regulation of Ptc by CiR, the formation of a functional gradient of CiA across boundary cells, or yet functional En differences between anterior and posterior cells.
In conclusion, our model analysis demonstrates the flexibility of the logical formalism, enabling consistent qualitative representation of diffusion, sequestration and post-transcriptional regulatory processes within and between neighbouring cells.
Availability: An XML file containing the proposed model together with annotations can be downloaded from our website (http://gin.univ-mrs.fr/GINsim/), along with GINsim, a logical modelling and simulation software freely available to academic groups.
Contact: thieffry{at}tagc.univ-mrs.fr
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1 INTRODUCTION |
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TOPABSTRACT1 INTRODUCTION2 METHODS3 RESULTS4 DISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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In the wing imaginal disc of Drosophila melanogaster, Hedgehog (Hh) is activated in a stripe of cells adjacent to the anterior–posterior (AP) compartment border. The definition of this cellular boundary is initiated by the asymmetric expression of engrailed (en) in the posterior compartment, which induces cell-autonomously hh expression (reviewed by Basler, 2000). Hh is secreted and diffuses to anterior cells (Figure 1(a)) (Capdevila et al., 1994; Ingham and Fietz, 1995; Sánchez-Herrero et al., 1996). On the other hand, En inhibits the transcription factor cubitus interruptus (ci) (de Celis and Ruiz-Gómez, 1995; Dominguez et al., 1996; Guillen et al., 1995; Schwartz et al., 1995), which is needed to transduce the Hh signal (Von Ohlen et al., 1997). Hh forms a complex with its receptor Patched (Ptc) and is endocytosed, thereby preventing Hh diffusion to more anterior cells (Chen and Struhl, 1996). Hence, the AP boundary is refined to a narrow cell stripe. Hh pathway activation results into decapentaplegic (dpp) expression in boundary cells (Basler and Struhl, 1994; Capdevila et al., 1994; Strigini and Cohen, 1997). Dpp then acts as a morphogen to organize the growth and patterning of surrounding tissue in agreement with the classical boundary model (Meinhardt, 1980).
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In the absence of Hh in anterior cells, the Ptc receptor accumulates in the plasma membrane and induces the endocytosis of the membrane protein Smoothened (Smo), leading to the inhibition of the targets of Hh signal (Figure 1(b), left panel) (Alcedo et al., 2000; Denef et al., 2000). In boundary cells, the Hh produced by the posterior cells binds to the Ptc receptor, leading to the accumulation and phosphorylation of Smo at the cell surface, and ultimately to the activation of the targets of Hh signal (Figure 1(b), middle panel) (Jia et al., 2004; Martín et al., 2001; Strutt et al., 2001; Zhu et al., 2003).
The transcriptional responses to Hh signal are mediated by two forms of Ci. In the absence of Hh input, Ci is phosphorylated (Chen et al., 1998; Price and Kalderon, 1999). This phosphorylation targets Ci to ubiquitin mediated proteolysis, which gives rise to an 
75 kDa N-terminal fragment that acts as a transcriptional repressor (CiR) (Aza-Blanc et al., 1997). Conversely, the presence of Hh inhibits the phosphorylation of Ci, thus increasing the concentration of full-length Ci, which acts as a transcriptional activator (CiA) (Chen et al., 1999; Ohlmeyer and Kalderon, 1998).
In boundary cells, Hh induces the activation of target genes in a concentration-dependent manner. The gene dpp requires low Hh levels, so that dpp is induced in a stripe of around 8–10 boundary cells. The gene ptc requires intermediate Hh levels. Consequently, ptc is induced in a stripe of around 4–5 boundary cells. Finally, en is induced by highest Hh levels, so that en expression is restricted to a stripe of around 3–4 boundary cells (Figure 1(b), middle panel) (Blair, 1992; Capdevila et al., 1994; Strigini and Cohen, 1997). Other target genes such as collier and the genes of the Iroquois complex are also regulated in a concentration-dependent manner (Gómez-Skarmeta and Modolell, 1996; Vervoort et al., 1999).
Target genes of the Hh pathway are regulated by distinct combinations of CiA and CiR levels. The expressions of en and ptc are regulated by CiA, dpp is regulated by both CiA and CiR, and hh expression is regulated only by CiR (Ho et al., 2005; Méthot and Basler, 1999; Muller and Basler, 2000). Interestingly, both molecules CiA and CiR bind to the same Gli consensus (Alexandre et al., 1996; Muller and Basler, 2000).
Defining a dynamical model accounting for the experimental data just summarized is daunting for two main reasons: (i) published data are mainly qualitative, while precise molecular mechanisms are often elusive; (ii) the diversity of the underlying mechanisms is difficult to subsume under a common formal approach. Several groups have developed quantitative models accounting for the roles of several signalling pathways in the control of patterning processes in Drosophila (see, e.g. Buceta et al., 2007; Le Garrec et al., 2006). However, due to the lack of quantitative data, the scope of these studies remains largely qualitative. Therefore, other groups have taken advantage of logical formalisms to analyse such signalling pathways (Albert and Othmer, 2003; González et al., 2006). Here, we propose a multi-level model of the coordinated development of cells across the AP boundary in the Drosophila imaginal disc. This model is composed by four iterations (or virtual cells) of the same cellular network and implements a logical representation of Hh diffusion between neighbouring cells, as well as of the sequestration of Hh by Ptc when this is highly expressed. Each of these four virtual cells corresponds to a compartment defined by a specific gene expression pattern. For proper defined logical rules (or parameters), this model enables the recapitulation of the main regulatory steps involved in AP boundary formation, including the differential activation of Hh targets depending on its concentration (the logical approach and the modelling assumptions are presented in Section 2). Using wild-type and mutant computational simulations, we demonstrate the coherence of this model with most if not all published experimental observations. Finally, as we shall see, this analysis leads to novel insights regarding partly characterized features of the regulatory system involved in the formation of the AP boundary.
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2 METHODS |
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TOPABSTRACT1 INTRODUCTION2 METHODS3 RESULTS4 DISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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2.1 Logical formalism
Briefly, a regulatory network is modelled in terms of a regulatory graph, where vertices represent genes or their regulatory products, whereas (signed) arcs represent regulatory interactions between these regulatory components (Chaouiya et al., 2003; González et al., 2006; Thomas and D‘Ari, 1990; Thomas et al., 1995). Different functional levels of regulatory products are modelled by discrete (or logical) variables with a limited number of discrete values: 0, 1, 2, .... The regulation of a given component is defined by logical functions involving the corresponding regulators (cf. Table 1). This formalism has been implemented in the GINsim software (González et al., 2006).
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2.2 Dynamical analysis and simulation of logical models
A logical simulation results in a state transition graph, where vertices and arcs represent expression states and state transitions, respectively. An expression state is defined as a vector encompassing integer values for all logical variables (i.e. the current levels of the regulatory components). Here, we are particularly interested in terminal, stable states, i.e. states without outgoing transitions, which correspond to persistent differentiated cellular states (Table 2). These stable states are found efficiently with an algorithm based on Ordered Multi-valued Decision Diagrams (Naldi et al., 2007).
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2.3 Representation of regulatory processes
The development of a logical multi-cellular model of the role of Hh pathway in the establishment of the AP patterning of Drosophila imaginal disc requires consistent representations of various kinds of regulatory processes:
- (Post)-transcriptional activation or inhibition processes are represented by activatory or inhibitory arcs.
- The diffusion of the Hh morphogen has been represented by symmetrical positive arcs between Hh nodes belonging to neighbouring virtual cells. This representation qualitatively models the transport of Hh from one cell to another.
- The sequestration of Hh by the receptor Ptc is represented by negative arcs from each Ptc node towards the Hh nodes of the neighbouring cells. Furthermore, in a given cell, the overriding of the negative interaction from Ptc on Smo is represented at the level of the logical function of Smo.
- Depending on Hh pathway activity, represented by the transducer protein Smo, the ci product is processed into repressor (CiR) or activator (CiA) forms. The kinases are omitted in our model, which implements the differential effect of Hh pathway on Ci variants in terms of direct positive and negative arrows from Smo onto CiA and CiR, respectively.
- Finally, the competition between the two Ci variants for the same cis-regulatory binding sites is represented by a positive arrow originating from CiA plus a negative arrow from CiR converging onto the regulated node (along with proper logical rule definition for this node). Indeed, such arrows from CiA and CiR converge onto Dpp within each cell as shown inFigure 1. However, inhibitory arrows are omitted in the case of the regulation of En, as no inhibitory effect has been reported for Ci mutation resulting in constitutive CiR expression. Regarding Ptc regulation by Ci product, a negative effect of constitutive CiR has been observed in somewhat artificial conditions, hence the consideration of a negative arc from CiR onto Ptc to cover this special case.
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3 RESULTS |
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TOPABSTRACT1 INTRODUCTION2 METHODS3 RESULTS4 DISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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3.1 Model definition
The building of a logical model first requires the abstraction of the crucial components and regulatory interactions, with the aim of covering a wide array of (molecular) genetic experiments, while avoiding useless complications. This requires the delineation of coherent representations for various kinds of regulatory processes, including the diffusion of Hh, its sequestration by Ptc, the balance between CiA and CiR products, etc. These logical representations are introduced in Section 2. For sake of brevity, we focus here on the description of deliberated simplifications.
Regarding the transduction of Hh signal, we only consider the crucial component Smo, placing it directly under the control of Ptc and Hh. The positive interaction from Hh targeting Smo thus represents the positive effect of Ptc inactivation by Hh onto Smo activity.
Several components and reactions involved in the emission, reception and transduction of the Hh signal are omitted for the sake of simplicity. For example, it is known that Hh signal is transduced by protein interactions involving Cos, Fu and Su(Fu), or yet that Ci proteolysis depends on PKA, GSK3β, CKI and Slimb (reviewed by Osterlund and Kogerman, 2006). However, in our model, the outcome of Hh signalling is mainly modelled through the balance between CiA and CiR products, under the control of Smo and En.
Different genes have been reported to be expressed in a concentration-dependent manner at the boundary, including dpp, collier, ptc, en, as well as the genes of the Iroquois complex (Jiang and Struhl, 1995; Lepage et al., 1995; Li et al., 1995; Méthot and Basler, 1999; Ohlmeyer and Kalderon, 1998; Wang et al., 1999). Here, we have selected dpp, ptc and en, because they are commonly used as readout of the activation of Hh signalling pathway.
Presented in Figure 2, our model encompasses four cells representing the anterior (cell 1 in Fig. 2), boundary (cells 2 and 3) and posterior compartments (cell 4). Table 1 lists the logical functions used to formally represent the regulation of the model elements, along with the experimental evidence supporting these logical functions. These parameters take identical values in the four cells, with the notable exception of the parameter corresponding to en basal expression, which takes its highest value 2 in posterior, but is set to 0 in anterior and boundary cells.
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3.2 Simulations of wild-type and mutant behaviors
Simulation of the wild-type situation leads to a unique stable state, where Hh and Smo activities form a (discrete) gradient towards the anterior cell (row 1 in Table 2). Smo activity favours CiA over CiR in the boundary cells (cells 2 and 3). By contrast, absence of Smo activity in the anterior cell (cell 1) favours CiR over CiA. Decreasing levels of CiA result in the activation of Dpp, Ptc and En in the domain of high Hh activity (cell 3), as well as of Dpp and Ptc in the domain of lower Hh activity (cell 2). In summary, our model qualitatively recovers the patterning of the AP boundary involving the initial asymmetric posterior En input, the formation of Hh gradient, the regulation of CiA and CiR, as well as the differential expressions of the target genes dpp, ptc and en.
The next step consists in assessing the coherence of this model with available gene perturbation experiments. To this aim, we have simulated various genetic perturbations, by blocking the levels of misexpressed genes (circled values in Table 2), e.g. at value zero for loss-of-function mutations.
Whereas absence of posterior Hh downregulates Hh pathway target genes such as dpp and ptc in boundary cells, loss of Ptc leads to up-regulation of these target genes in the most anterior cells (Basler and Struhl, 1994; Capdevila et al., 1994; Chen and Struhl, 1996). Our model reproduces these phenotypes in two simulations of mutants with non cell-autonomous phenotypes, where either the posterior Hh (row 2 in Table 2) or boundary Ptc (row 3) are fixed to value zero (0). Whereas the first simulation leads to the downregulation of the targets of Hh pathway, the second simulation leads to the activation of Hh targets in the anterior cell, in agreement with experimental observations.
Clones with a smo loss-of-function mutation located at the boundary fail to upregulate dpp, ptc and en expression (Chen and Struhl, 1996; Strigini and Cohen, 1997). By contrast, ectopic moderate Smo activity drives dpp and ptc but not en expression in the anterior compartment (Jia et al., 2003). Simulation of these perturbations agree with experimental observations (rows 4 and 5 in Table 2).
Ci is post-transcriptionally regulated to give rise to the CiA and CiR forms, which activate and inhibit target genes in boundary and anterior cells, respectively (Dominguez et al., 1996; Méthot and Basler, 1999; Sánchez-Herrero et al., 1996). Accordingly, simulation of an anterior ci- clone leads to the upregulation of anterior Dpp (row 6). This simulation results in a second stable state, because the regulatory cascade CiA 
Ptc 
Smo, which maintains Smo at inactive level 0, is disrupted. Then, two stable states are also obtained if we fix CiA and Ptc levels to 0, but not if we fix CiR at value 0 (data not shown). On the other hand, simulation of a boundary ci- clone downregulates boundary Ptc, as a result of the lack of CiA (row 7). This in silico mutant enables Hh diffusion, which results in non cell-autonomous activation of Ptc and En in the anterior cell, as observed experimentally (Dominguez et al., 1996).
We conclude that our model correctly represents the major features of the process of AP boundary formation, for the wild-type situation as well as for the published mutant phenotypes. We now turn to the use of this model to assess the behaviour of the system depending on alternative hypotheses about the regulation of crucial effectors of the Hh pathway.
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4 DISCUSSION |
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TOPABSTRACT1 INTRODUCTION2 METHODS3 RESULTS4 DISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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We have presented a model of the network controlling the patterning of the AP boundary in the wing imaginal disc of Drosophila melanogaster. Dispersed experimental data have been integrated into a model explicitly covering seven crucial regulatory components found in each of four distinct cells or regions (Fig. 2). Noteworthy, the diffusion of the morphogene Hh and its sequestering by Ptc have been represented by discrete functions. Likewise, the regulation of target genes by Hh in a concentration-dependent manner could be properly modelled with logical functions.
We have performed several rounds of model definition and simulation to obtain a compact model involving only seven components but still enabling the generation of expression patterns consistent with experimental observations for both wild-type and mutant situations. In this process, we have left aside some genes, like collier, which constitute simple outputs of our model. Several components of transduction signalling cascade are considered only implicitly, so far that they have been shown to play only an intermediate role (e.g. Cos is implicitly considered behind the interaction from Smo to CiA and CiR).
Our model further simplifies the AP boundary regions in terms of four regions or virtual cells (each of these virtual cells corresponds to several real cells). Such a simplification still allows the exploration of situations reminiscent of larger multicellular contexts. For instance, by fixing En level to the interval [0,1], we can simulate a cellular context in the most anterior region of the disc.
Finally, our logical model and its dynamical analysis can be used to test hypothetical regulatory mechanisms or to provide novel insights regarding issues debated in the recent experimental literature. We illustrate this point by addressing three of these questions in the following subsections.
4.1 Competition between CiA and CiR for the same Gli consensus sequence
The Ci protein is found in two forms, a full-length form CiA with activating effect, and a proteolysed shorter form CiR with inhibiting effect. Available evidence suggests that CiA and CiR bind to the same DNA (Gli) consensus sequence. In the wild-type situation, CiA and CiR appear to have similar affinity for their consensus sequence in the Dpp promoter. Consequently, Dpp expression state should depend on the relative concentration of CiA and CiR.
In the case of en, molecular genetic evidence shows that its promoter responds to CiA but not to CiR. Because no Gli consensus sites have been reported in the en promoter, it is difficult to assess how its transcription is regulated by CiA at the molecular level.
In the wild-type situation, CiA seems to have higher affinity for the Gli consensus sequence of the ptc promoter than CiR. This is based on genetic deletion experiments that indicate that Ptc is regulated by CiA but not by CiR (Méthot and Basler, 1999). However, ectopic expression of CiR apparently titrates CiA from Gli sequences in the ptc promoter (Muller and Basler, 2000), leading to ptc repression (Aza-Blanc et al., 1997). This competition is modelled through an additional level (value 2) for CiR and an inhibitory arrow from CiR to Ptc (in pale red in Fig. 2, to mark its non-physiological nature). The level 2 (in contrast to the physiological level 1) of CiR would enable the titration of CiA from the Gli consensus site. In the wild-type situation, where CiR takes values 0 or 1, the presence or absence of CiA determines the level of Ptc. When CiR is over-expressed, the presence of CiR at level 2 overrides the effect of CiA, and reduces Ptc to 1. To simulate this situation, we have fixed the level of CiR to 2 in all four cells and computed the stable state (row 8 of Table 2). This results in Ptc inactivation in all mutant cells as observed by (Aza-Blanc et al., 1997).
4.2 CiA is differentially activated by increasing Hh levels
It has been argued that CiA is activated by high Hh levels in the wing imaginal disc (Hooper and Scott, 2005; Smelkinson and Kalderon, 2006; Vervoort, 2000). In the course of the construction of our model, we noticed that significant CiA levels must be present in the domain of low Hh levels (cell 2) to fully reproduce experimental observations. For instance, in the absence of CiA in cell 2, then Dpp and Ptc cannot reach their highest levels in this cell (cf. row 9 in Table 2). By contrast, if CiA would be present at high levels in both cells 2 and 3, then the target genes dpp, ptc and en would not be differentially regulated in boundary cells 2 and 3. This is illustrated by the simulation result 10 (Table 2), where highest CiA levels are fixed in both boundary cells 2 and 3 (as in cell 1). In other words, there is thus no differential expressions of the target genes dpp and en between the boundary cells. Hence, our analysis supports the contention that CiA is distributed along a gradient matching that of Hh. Our model takes into account two levels of CiA that correlate with two levels of Hh. Under low CiA levels, En is not induced, so that Dpp can be induced at its highest levels in cell 2, in contrast with lower levels of Dpp in cell 3 (row 1 in Table 2). The consideration of this gradient of CiA in boundary cells is thus sufficient to reproduce all observations regarding the expression patterns of CiA targets.
4.3 Different levels of En might account for functional differences in boundary and posterior cells
Initially, En triggers the AP boundary network in the posterior compartment. Regarding this posterior en regulation, it is only known that it is independent of Hh signalling (references in Table 1). Later on, en is regulated by Hh signalling in boundary cells. Whereas early posterior En represses Ci and induces Hh, later boundary En does not repress Ci nor induce Hh (references in Table 1).
To account for these observations, we consider that an unknown mechanism brings En to higher levels (fixed value 2) in the posterior compartment. This is supported by ectopic en misexpression experiments, leading to ci and ptc cell-autonomous repression (de Celis and Ruiz-Gómez, 1995; Dominguez et al., 1996; Guillen et al., 1995). However, higher levels of En are able to activate Hh and repress Ci in neighbouring cells. This is illustrated by row 11 in Table 2, describing the unique stable state in the situation where En is fixed at value 2 in boundary cells (cells 2 and 3), which results in Hh up-regulation, versus CiA and CiR down-regulation in these cells, as if they would belong to the posterior compartment. In contrast, we consider that En can be induced only at lower levels (value 1) by CiA in the boundary cell (cell 3). Such lower levels affect the production of Dpp but not that of Ci or Hh. Indeed, blocking En at value 1 in posterior cells results in the loss of Hh and CiA gradients, because such low En levels are not able to induce Hh (row 12 in Table 2).
These simulations thus support the contention that different levels of En in anterior and posterior cells might be responsible for the different regulation of En target genes. However, the reality is probably subtler since, for instance, Dpp is also repressed in anterior cells of en misexpression experiments (Guillen et al., 1995). This observation cannot be explained through the repression of Ci as ci mutations partially upregulate Dpp (Dominguez et al., 1996; Méthot and Basler, 1999; Sánchez-Herrero et al., 1996). Moreover, the factor putatively responsible for high en expression in the posterior compartment still remains to be identified.
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ACKNOWLEDGEMENTS |
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TOPABSTRACT1 INTRODUCTION2 METHODS3 RESULTS4 DISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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Funding: A.G. has been supported by PhD grant JR/MLD/MDV-A05/4 from the Association pour la Recherche sur le Cancer (France), and postdoctoral grant P06237 [GenBank] from the Japan Society for the Promotion of Science. This project was further supported by a research grant from the French Ministry of Research (ANR project JC05-53969).
Conflict of Interest: none declared.
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