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Data‐driven modelling of a gene regulatory network for cell fate decisions in the growing limb bud

2015; Springer Nature; Volume: 11; Issue: 7 Linguagem: Inglês

10.15252/msb.20145882

1744-4292

Manu Uzkudun, Luciano Marcon, James Sharpe,

Genomics and Chromatin Dynamics

Article14 July 2015Open Access Data-driven modelling of a gene regulatory network for cell fate decisions in the growing limb bud Manu Uzkudun Manu Uzkudun EMBL-CRG Systems Biology Program, Centre for Genomic Regulation (CRG), Universitat Pompeu Fabra (UPF), Barcelona, Spain Search for more papers by this author Luciano Marcon Luciano Marcon EMBL-CRG Systems Biology Program, Centre for Genomic Regulation (CRG), Universitat Pompeu Fabra (UPF), Barcelona, Spain Search for more papers by this author James Sharpe Corresponding Author James Sharpe EMBL-CRG Systems Biology Program, Centre for Genomic Regulation (CRG), Universitat Pompeu Fabra (UPF), Barcelona, Spain Institucio Catalana de Recerca i Estudis Avancats (ICREA), Barcelona, Spain Search for more papers by this author Manu Uzkudun Manu Uzkudun EMBL-CRG Systems Biology Program, Centre for Genomic Regulation (CRG), Universitat Pompeu Fabra (UPF), Barcelona, Spain Search for more papers by this author Luciano Marcon Luciano Marcon EMBL-CRG Systems Biology Program, Centre for Genomic Regulation (CRG), Universitat Pompeu Fabra (UPF), Barcelona, Spain Search for more papers by this author James Sharpe Corresponding Author James Sharpe EMBL-CRG Systems Biology Program, Centre for Genomic Regulation (CRG), Universitat Pompeu Fabra (UPF), Barcelona, Spain Institucio Catalana de Recerca i Estudis Avancats (ICREA), Barcelona, Spain Search for more papers by this author Author Information Manu Uzkudun1, Luciano Marcon1 and James Sharpe 1,2 1EMBL-CRG Systems Biology Program, Centre for Genomic Regulation (CRG), Universitat Pompeu Fabra (UPF), Barcelona, Spain 2Institucio Catalana de Recerca i Estudis Avancats (ICREA), Barcelona, Spain *Corresponding author. Tel: +34 93 316 0098; E-mail: [email protected] Molecular Systems Biology (2015)11:815https://doi.org/10.15252/msb.20145882 PDFDownload PDF of article text and main figures. Peer ReviewDownload a summary of the editorial decision process including editorial decision letters, reviewer comments and author responses to feedback. ToolsAdd to favoritesDownload CitationsTrack CitationsPermissions Figures & Info Abstract Parameter optimization coupled with model selection is a convenient approach to infer gene regulatory networks from experimental gene expression data, but so far it has been limited to single cells or static tissues where growth is not significant. Here, we present a computational study in which we determine an optimal gene regulatory network from the spatiotemporal dynamics of gene expression patterns in a complex 2D growing tissue (non-isotropic and heterogeneous growth rates). We use this method to predict the regulatory mechanisms that underlie proximodistal (PD) patterning of the developing limb bud. First, we map the expression patterns of the PD markers Meis1, Hoxa11 and Hoxa13 into a dynamic description of the tissue movements that drive limb morphogenesis. Secondly, we use reverse-engineering to test how different gene regulatory networks can interpret the opposing gradients of fibroblast growth factors (FGF) and retinoic acid (RA) to pattern the PD markers. Finally, we validate and extend the best model against various previously published manipulative experiments, including exogenous application of RA, surgical removal of the FGF source and genetic ectopic expression of Meis1. Our approach identifies the most parsimonious gene regulatory network that can correctly pattern the PD markers downstream of FGF and RA. This network reveals a new model of PD regulation which we call the "crossover model", because the proximal morphogen (RA) controls the distal boundary of Hoxa11, while conversely the distal morphogens (FGFs) control the proximal boundary. Synopsis A dynamical 2D computer model of limb development, combining tissue movements and spatially controlled gene regulatory interactions, allows reverse-engineering the regulatory network controlling cell fate decisions along the main proximodistal (PD) axis. The expression patterns of the PD markers Meis1, Hoxa11 and Hoxa13 are mapped into a dynamic description of the tissue movements that drive limb morphogenesis. Reverse-engineering is used to test how different gene regulatory networks can interpret the opposing gradient of FGF and RA to pattern the PD markers. Experimental validations reveal a new "crossover model" explaining how Hoxa11 and Hoxa13 are spatially regulated. Introduction The vertebrate limb is a classical model system to study the specification of a growing organ during development. In the mouse, limb development starts with the outgrowth of the limb bud from the lateral plate mesoderm, which in just two days is able to form the main skeletal elements present in the adult limb. The limb skeleton is divided into three main proximodistal (PD) segments: the stylopod (upper arm/leg), the zeugopod (lower arm/leg) and the autopod (hand/foot) (Fig 1A). The three segments are specified in a proximal-to-distal sequence, first the stylopod, then the zeugopod and finally the autopod (Tabin and Wolpert, 2007; Towers et al, 2012). The regulatory mechanism that specifies the three PD segments sequentially is not yet fully understood (Tabin and Wolpert, 2007; Roselló-Díez et al, 2014). Figure 1. A realistic data-driven model for PD patterning in the limb bud Each PD skeletal element expresses specific genes. Meis1 and Meis 2 are stylopod markers, Hoxa11 is the zeugopod marker, and Hoxa13 is the autopod marker. The process of mapping experimental gene expression data of Meis, Hoxa11 and Hoxa13 to the 2D limb bud model. The general regulatory model uses the RA and FGF morphogen gradients as inputs and should explain the expression patterns of the PD markers as outputs: Meis, Hoxa11 and Hoxa13 over time and space. An example of the simulated morphogen gradients of RA and FGF. The source of FGF signal (curved black lines) is combined from the FGF8 expression pattern, which is uniform along the AER, and the FG4 expression pattern, which is initially expressed in a small posterior region which is later expanded anteriorly. Solid interactions describe the "upstream" part of the circuit, which we take as given. The main hypothetical interactions to be explored in this study are shown as the dashed lines—the regulation of the Hox genes. Download figure Download PowerPoint During the development of the limb bud, a set of specific transcription factors marks the developing PD segments. Currently, the best known PD markers are homeobox genes: Meis1 and Meis2 (stylopod), Hoxa11 (zeugopod) and Hoxa13 (autopod) (Nelson et al, 1996; Tabin and Wolpert, 2007; Mercader et al, 2009) (Fig 1A). In the initial stages of limb development (around E9.25 stage), Meis1 and Meis2 are expressed across the entire limb bud. Successively, around E10.0, they are downregulated in the distal region where Hoxa11 is activated (Fig 1B). Finally, around E10.5, Hoxa13 starts to be expressed in a small distal posterior region and expands anteriorly and proximally with a simultaneous shrinking of Hoxa11, which then maintains a mutually exclusive pattern with Hoxa13 (Nelson et al, 1996; Tabin and Wolpert, 2007; Mercader et al, 2009). The question of how these three molecular zones are controlled by upstream signals has traditionally been split into two categories of model: (i) those in which cells make decisions by measuring the duration of exposure to a signal, or alternatively, (ii) those in which the decision is based on the strengths or levels of a signal. A classical model to explain the specification of the limb PD axis is the progress zone model (PZM). This model proposes that the progressive specification of the PD segments is based on a timing mechanism which is active in a distal region, called the progress zone (PZ), that is under the influence of the apical ectodermal ridge (AER) (Summerbell 1974). As the limb grows, cells exit the progress zone and assume different fates depending on the amount of time they received the "distalizing" fibroblast growth factor (FGF) signals coming from the AER (Niswander et al 1993; Fallon et al 1994). Under this hypothesis, it is the duration of exposure to FGF which determines how distal the cell fate will be—a longer duration specifies a more distal fate. It is important to note that this hypothesis does not specify the mechanism by which time is measured. It could be counting temporal oscillations (e.g. the cell cycle), or it could simply be the slow accumulation of a factor in the cells. Four different FGFs are expressed in the mouse AER, and although their precise role in PD patterning is still debated (Mariani et al, 2008; Yu and Ornitz, 2008), it is clear that they also play a major role in the physical outgrowth of the limb. FGF8 is expressed along all the AER and is the most essential for correct growth of the limb bud (Mariani et al, 2008). Fgf4,9,17 are functionally redundant and are expressed initially in a posterior region and then expand anteriorly (Mariani et al, 2008). A more recent model of PD patterning is the two-signal model (2SM) (Mercader et al, 2000). This model explains the specification of the stylopod by a mechanism based on two opposing signals: a distal FGF signal coming from the AER and a proximal signal coming from the body flank (Mercader et al, 2000; Cooper et al, 2011; Roselló-Díez et al, 2011). Experimental evidence suggests that retinoic acid (RA) plays the role of the proximal signal (Mercader et al, 2000; Cooper et al, 2011; Roselló-Díez et al, 2011) although a consensus on the importance of RA in PD patterning has not yet been reached (Cunningham and Duester, 2015). RA is synthesized in the lateral plate mesoderm (LPM) by the enzyme RALDH2 and diffuses into the mesenchyme (Yashiro et al, 2004). FGF signalling in the distal tissue promotes the expression of Cyp26b1, an enzyme that degrades RA to inactive forms (Probst et al, 2011)—thereby creating a gradient of RA along the PD axis (Yashiro et al, 2004). In the proximal tissue, RA upregulates the expression of the stylopod markers Meis1 and Meis2, while in the distal tissue, FGF signalling appears to downregulate them, probably through the promotion of Cyp26b1 (Mercader et al, 2000; Cooper et al, 2011; Roselló-Díez et al, 2011). In the 2SM, the stylopod–zeugopod boundary is thus explained as the tipping point between the proximal influence of RA and the distal influence of FGF, rather than as a timing mechanism. Recently, the idea of timing has again been revived (Roselló-Díez et al, 2014) in a study which showed that prematurely exposing distal tissue to the signalling environment typical for late distal tissue (i.e. low RA and high FGF) is unable to induce precocious expression of the distal marker Hoxa13. Moreover, it showed that reducing RA could only induce extra Hoxa13 expression after the developmental stage when Hoxa13 has been naturally activated. The fact that this could be overcome by promoting chromatin opening suggested that an epigenetic timing mechanism might also be involved (Roselló-Díez et al, 2014). It should be noted, however, that this proposal was not exactly the same as the PZM, as the authors were able to rule out a simple temporal integration of FGF signalling. In summary, the main alternative models for this process still focus on the distinction between measuring signal duration versus measuring signalling levels. Here, we use an approach combining parameter optimization and model selection to investigate these different hypotheses in an unbiased way. First, we map the gene expression of the PD markers over space and time into an accurate model of limb morphogenesis (Marcon et al, 2011). Then, we reverse-engineer the optimal gene network that controls the patterning of the PD markers by testing different networks that act downstream of the proposed FGF and RA gradients. Our method makes no assumption on which mechanistic concept underlies PD patterning (i.e. whether the system measures signalling levels, or measures time). We instead employ a step-by-step approach, starting from a basic network that describes the known regulation between the stylopod marker Meis and the RA/FGF signals (Mercader et al, 2000; Yashiro et al, 2004; Cooper et al, 2011; Roselló-Díez et al, 2011). We then test different ways in which Hoxa11 and Hoxa13 could be regulated by a combination of RA and FGF. We fit the different regulatory networks by inferring the optimal parameter values that can better reproduce the experimental wild-type expression patterns of Meis, Hoxa11 and Hoxa13. This systems biology approach allows us to identify which is the simplest network that can explain the known experimental data on the three PD markers. Finally, we use the model to investigate how the network relates to the different conceptual models that have been proposed to explain the specification of the PD segments. This allows to us determine to which extent a model based on measuring levels, versus measuring time, underlies PD patterning. Achieving this in a system with dynamically moving tissue does not alter the basic objective function (i.e. the method by which we calculate the difference between simulated and experimental data), but it allows us to find the model which explains the whole sequence of gene regulatory events despite the constantly changing shape of the growing tissue during this period. Results Optimizing the null hypotheses: RA- or FGF-dominant models To study different hypothetical PD patterning mechanisms, we simulated gene regulatory networks on a realistic growing model of limb development. This was done by using an accurate 2D numerical description of limb morphogenesis (Marcon et al, 2011) that has been previously generated by combining a morphometric analysis of limb shapes (Boehm et al, 2011) with clonal fate mapping data (Arques et al, 2007). The growth of the limb is represented by a 2D triangular mesh with anisotropic growth distortions, which is fully remeshed at 1-hour intervals over a 36-hour period to maintain high quality of the mesh throughout the simulation (Marcon et al, 2011). We used this framework to investigate how well different gene regulatory networks can pattern the PD markers from stage mE10:09 to stage mE11:12. To simulate the dynamics of the relative concentrations of the PD markers and morphogen gradients (FGF and RA), we employed reaction–diffusion partial differential equations (PDEs). These equations are solved on the growing triangular mesh using a finite-volume method. Our software uses precalculated remeshing and handles the redistribution of element contents into the new mesh at 1-h intervals, which is described in detail in Marcon et al (2011). The rate of change in the concentration of a given gene product G is given by three terms in equation 1: a production term PG, a diffusion term DG and a linear decay term λG. The production terms for all molecules vary in time and space. In the case of FGF and RA, these non-uniform production terms are mapped from experimental data, while for the other molecules, production rates are calculated from the regulatory dynamics of the network. (1) To obtain the experimental data needed for the reverse-engineering, we performed whole-mount in situ hybridizations (WMISH) of the main PD markers: Meis1, Hoxa11 and Hoxa13, from stage mE10:9 to stage mE11:12. These data are used to constrain the models and to implement the objective function for the optimization process. The Meis1 and Meis2 genes show similar expression patterns and patterning roles, so we chose Meis1 as representative of both. We mapped these data into the computational growth map by converting colour intensity values to relative and approximate molecular concentrations (see Materials and Methods, and Fig 1B). In situ hybridization is known to be a non-quantitative technique; however, previous studies in Drosophila (in a static 1D domain) have shown that successful reverse-engineering of gene regulatory networks does not require knowledge of the absolute concentrations of gene products (Jaeger et al, 2004; Crombach et al, 2012). Instead, it is the shapes of the gene expression domains which matter. To ensure that a non-quantitative read-out of the expression levels is not a problem for our study, we chose to perform an explicit non-linear sigmoid rescaling of the mapped values—thereby ensuring that the primary information in the data is about the shape of the expression domains rather than levels (see Materials and Methods for more details, and further discussion towards the end of the next results section). Our next goal was to optimize the parameter values of different gene regulatory networks to reproduce the wild-type gene expression patterns of Meis, Hoxa11 and Hoxa13 by interpreting the RA and FGF gradients (Fig 1C). As our model is a high-level one, we use a single variable to represent FGF signalling, even though it is composed of multiple FGF ligands. To take into account the different FGF sources, we calculated FGF production as a combination of two primary sources: (i) the FGF8 pattern is expressed earlier and more uniformly along all the AER and (ii) the patterns of FGF4, 9 and 17 are initiated slightly later with a clear posterior bias and then gradually expand anteriorly to produce more uniform distributions (Fig 1D) (Mariani et al, 2008). RA in the model is synthesized at a constant rate at the main body of the embryo and diffuses into the limb mesenchyme (Yashiro et al, 2004). The simulated FGF and RA gradients at different stages are shown in Fig 1D. As a starting point for the models, we used a simple regulatory network that includes a set of key molecular components and interactions that are well documented in literature. In particular, we considered the basic hypothesis that the distal FGF morphogen counteracts the action of a proximal RA gradient by induction of Cyp26b1, a RA-degrading enzyme (Mercader et al, 2000; Yashiro et al, 2004; Cooper et al, 2011, Roselló-Díez et al, 2011; Roselló-Díez et al, 2014). Moreover, we considered that the stylopod marker Meis1 (Mercader et al, 1999; Capdevila et al, 1999) is upregulated directly by RA in the proximal limb bud (Mercader et al, 2000). This basic regulatory network describing these interactions is shown as the solid interactions in Fig 1E. Using the simulated gradients, we explored the different ways in which FGF and RA could regulate the expression of the PD markers Meis, Hoxa11 and Hoxa13. How the FGF and RA signals control the distal markers Hoxa11 and Hoxa13 is still a matter of debate (Vargesson et al, 2001; Tabin and Wolpert, 2007; Roselló-Díez et al, 2014). It is theoretically possible that both of these opposing gradients directly feed into both genes—thus allowing different thresholds of signalling ratio to be encoded independently for each gene. However, since FGF is known to regulate RA (Mercader et al, 2000; Yashiro et al, 2004; Probst et al, 2011), it is also possible that each Hox gene is primarily regulated by one pathway or the other. We used our modelling approach to explore the hypothetical scenarios. In each model, the interactions were implemented with Hill functions that capture that main qualitative feature of gene regulation: a cooperative non-linear responses coupled with saturation (Goutelle et al, 2008). We define a model to describe the dynamics of FGF(F), RA(R), Cyp26b1(C) and Meis(M), equations (2–5). This is the "upstream" part of the network (mentioned above, Fig 1E), whose topology (regulatory circuit) is common to all models tested in this study. PF, PR, PC and PM are the corresponding production rates and λF , λR, λC and λM are the corresponding decay rates. DF and DR are the diffusion constants for the FGF and RA diffusible molecules, respectively. k's are specific constants of the Hill function relating to a threshold for activation or inhibition, μ's are Hill coefficients describing the steepness of the regulatory response, and c1 describes the linear strength by which Cyp26b1 degrades RA. (2) (3) (4) (5) The parameters of this system do not represent measurable biochemical constants but rather effective regulatory influences. One exception is the diffusion constants RA and FGF that are obtained from experimental estimates found in literature. For RA, we used a diffusion constant of D = 600 μm2/min, estimated from experiments in chick wing buds where a bead releasing RA was applied in the anterior wing bud region (Tickle et al, 1985). For FGF, we used a diffusion constant of D = 100 μm2/min, estimated using FRAP measurements in zebra fish (Müller et al, 2013). Starting from this upstream network, we explored how different models that interpret FGF and RA could regulate the expression of the PD markers Hoxa11 and Hoxa13. These different models correspond to different combinations of the dashed regulatory interactions in Fig 1E. Since there are two downstream genes, the minimal models all have two links, and there are consequently four possible minimal models, which we label A–D (and will be described in the subsequent sections one by one). To determine how well each model could theoretically explain the experimental data, we employed a parameter-fitting approach in which the parameters λF, λR, c1, μ and the various regulatory k's were optimized to give the best match between the simulated dynamic gene expression patterns and the experimental data. As far as we know, this is the first case of automated reverse-engineering (parameter optimization plus model selection) on a 2D growing/moving tissue. As experimental data, we used wild-type gene expression patterns of Meis, Hoxa11 and Hoxa13 (Fig 1B). For each model (e.g. Model A in Fig 2A), we employed a gradient descent optimization method (see Materials and Methods) in which the objective function consisted of an automatic scoring method which compared the predicted 2D patterns of each of the three genes with the digitized versions of the wild-type patterns (Fig 2B). The goal was not to search only for a good final result, but rather to find networks which recapitulate the entire developmental trajectory of dynamic expression patterns, so pattern scoring was performed for a series of time-points during development (not just the end-point of the simulations). The total score for a given simulation was defined as the sum of the scores for each gene product (Meis, Hoxa11, Hoxa13) and each experimental time-point. The score for a given gene product at a specific experimental time-point was defined as the sum of squared differences for each triangular element of the experimental and simulated concentrations (equation 6). Rather than the absolute values of concentrations at each position, it is the shapes of the expression domains that matter for the objective function, so we normalized both the experimental and predicted values by a non-linear rescaling process (see Materials and Methods) (6) Figure 2. Reverse-engineering (parameter optimization) of a regulatory network in a 2D growing domain An example of a regulatory network with a set of unknown parameters. The optimization algorithm iteratively finds a parameter set that minimizes the difference between the simulated and experimental gene expression patterns of Meis, Hoxa11 and Hoxa13 at different stages. Two examples of the simulation using random initial parameter values before optimization. The gradually improving predicted expression patterns over time and space, for different iterations of the algorithm. Meis is shown in red, Hoxa11 in green, and Hoxa13 in red. The experimentally mapped gene expression patterns are shown with the same colour code. Download figure Download PowerPoint For each model, the optimization process was run 27 times, starting from different initial parameter values which were partially random, but selected to be far away from each other in parameter space (see Materials and Methods). This was to ensure that a wide region of parameter space was being explored and to test whether good solutions were converging to the same global optimum, or whether conversely multiple local optima were being found. Two examples run with very different initial parameter values are shown in Fig 2C, indicating how far the optimizations start from a successful result. A summary of all initial parameter combinations is given in Supplementary Fig S1 and Supplementary Table S1. We first tested two simple models based on the idea that both Hoxa11 and Hoxa13 are regulated only by the RA gradient (Roselló-Díez et al, 2014) or only by the FGF gradient. In both models, we also include the previously documented inhibition between Hoxa11 by Hoxa13 (Nelson et al, 1996; Mercader et al, 2009). Since our molecular variables represent relative concentrations (rather than absolute concentrations), both the maximum production rates and the decay rates for Cyp26b1, Meis, Hoxa11 and Hoxa13 were fixed to the arbitrary value of 0.05/min (so that the maximum steady-state concentrations would be 1.0). The strength and speed of each gene regulatory interaction are represented by the steepness of the Hill functions. We used one Hill coefficient μ to govern the steepness of all regulatory interactions except for one exception. For the repression of Hoxa11 by Hoxa13, we allowed a distinct Hill coefficient , since this repression is experimentally seen to be extremely rapid [the expression patterns of the two genes become quickly mutually exclusive (Nelson et al, 1996; Mercader et al, 2009)]. Both μ and were free parameters to be optimized during the fitting process. In the first model (Model A, see Fig 2A), Hoxa11 and Hoxa13 are inhibited by the RA acid gradient and not directly regulated by FGF, as suggested in the most recent study (Roselló-Díez et al, 2014). Therefore, the expression of Hoxa11 should start when the RA concentration falls below a certain threshold value, and Hoxa13 should be initiated when the levels further decrease below a second threshold value. This model contains no direct activator of Hoxa11 or Hoxa13, only two different permissive conditions specified by the RA gradient. The regulatory inputs into the Hox genes are given by equations (7, 8) for Hoxa11() and Hoxa13(), where and are the corresponding production rates and and are the corresponding decay rates. k3, k4, k5, k6 and k7 are again specific constants for the corresponding Hill functions (see Fig 3A). (7) (8) Figure 3. Simulation results of Model A and Model B The regulatory network for Model A, and the simulated expression patterns of Meis shown in red, Hoxa11 in green, and Hoxa13 in red at successive time-points. The regulatory network Model B, and the simulated expression patterns with the same colour code. The same time-points as (A) and (B) showing the mapped experimental data. The last column for all rows shows the Hoxa11 pattern at the last time-point, using an intensity colour map to highlight the shape of the expression domain. Download figure Download PowerPoint In contrast, in the second model (Model B), both Hoxa11 and Hoxa13 are activated by Fgf but not regulated by RA. In equations (9, 10), we describe it is dynamics. (9) (10) We optimized the parameters of both models to fit the wild-type pattern of Meis, Hoxa11 and Hoxa13. An example of the iterative improvements in the gene expression patterns is shown for Model A in Fig 2D, which gradually converges to the most similar dynamic sequence of marker patterns. (The optimized values of all parameters for all models discussed in this paper are provided in Supplementary Table S2). After optimization, both Models A and B were able to recapitulate a number of features of the dynamic gene regulation of Hoxa11 and Hoxa13: the general positioning of the domains and the timing of their appearance (Fig 3). However, a more detailed examination of the patterns suggested that neither model was able to reproduce certain important qualitative features of the Hoxa11 pattern. In particular, the simulated Hoxa11 domains appeared as a curved stripe which is thicker in the middle than at the top or bottom (i.e. thicker at the centre than at the anterior or posterior ends of the domain). By contrast, the real expression pattern is thinner and weaker in the middle, giving a curved "dumb-bell" type of shape. We thus explored an alternative model which could rectify this problem. The value of 2D shape data: the crossover model A detailed analysis of the experimental Hoxa11 pattern revealed that its proximal boundary is more curved than the distal boundary (white dashed lines in Fig 4C). Moreover, the central part of the expression pattern is narrower and weaker than the anterior and posterior ends. A possible explanation for the failure of Models A and B to reproduce this expression pattern is that the Hox expression domains in these models are controlled only by one gradient in each case (the RA gradient for Model A and the FGF gradient for Model B). Examination of the shapes of these two gradients showed that they have different spatial profiles. The isoclines for RA (lines which connect all points with the same concentration) were less curved than those for FGF (Fig 4E and F). This is because the source of RA is essentially a straight line (the main body of the embryo), while FGFs are produced along a distal curved line that corresponds to the AER. Figure 4. Proximal and distal expression boundaries of Hoxa11 and the crossover model A, B. The predicted late expression patterns for Hoxa11 obtained using Model A or Model B. C–F. The real experimental pattern shows quite a different shape—a "dumb-bell" type of shape which is narrower in the middle. The proximal expression boundary is more curved than the distal boundary. Related