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Conceived and designed the experiments: LM CGA MST JS. Performed the experiments: LM CGA. Analyzed the data: LM JS. Contributed reagents/materials/analysis tools: LM. Wrote the paper: LM JS.

The authors have declared that no competing interests exist.

A comprehensive spatio-temporal description of the tissue movements underlying organogenesis would be an extremely useful resource to developmental biology. Clonal analysis and fate mappings are popular experiments to study tissue movement during morphogenesis. Such experiments allow cell populations to be labeled at an early stage of development and to follow their spatial evolution over time. However, disentangling the cumulative effects of the multiple events responsible for the expansion of the labeled cell population is not always straightforward. To overcome this problem, we develop a novel computational method that combines accurate quantification of 2D limb bud morphologies and growth modeling to analyze mouse clonal data of early limb development. Firstly, we explore various tissue movements that match experimental limb bud shape changes. Secondly, by comparing computational clones with newly generated mouse clonal data we are able to choose and characterize the tissue movement map that better matches experimental data. Our computational analysis produces for the first time a two dimensional model of limb growth based on experimental data that can be used to better characterize limb tissue movement in space and time. The model shows that the distribution and shapes of clones can be described as a combination of anisotropic growth with isotropic cell mixing, without the need for lineage compartmentalization along the AP and PD axis. Lastly, we show that this comprehensive description can be used to reassess spatio-temporal gene regulations taking tissue movement into account and to investigate PD patterning hypothesis.

A comprehensive mathematical description of the growth of an organ can be given by the velocity vectors defining the displacement of each tissue point in a fixed coordinate system plus a description of the degree of mixing between the cells. As an alternative to live imaging, a way to estimate the collection of such velocity vectors, known as velocity vector field, is to use cell-labeling experiments. However, this approach can be applied only when the labeled populations have been grown for small periods of time and the tensors of the velocity vector field can be estimated directly from the shape of the labeled population. Unfortunately, most of the available cell-labeling experiments of developmental systems have been generated considering a long clone expansion time that is more suitable for lineaging studies than for estimating velocity vector fields. In this study we present a new computational method that allows us to estimate the velocity vector field of limb tissue movement by using clonal data with long harvesting time and a sequence of experimental limb morphologies. The method results in the first realistic 2D model of limb outgrowth and establishes a powerful framework for numerical simulations of limb development.

The cellular processes by which a field of cells develops into a spatially-organized tissue have traditionally been split into two distinct questions: pattern formation and morphogenesis. The first focuses on the regulatory mechanisms underlying spatial and temporal cell fate specification. The second focuses on the cellular behaviors that physically drive growth and shaping of multicellular structures. While these two processes can indeed be considered to be conceptually separated, in practice they usually occur simultaneously and are believed to be tightly coordinated. The vertebrate limb is an excellent model system to study how these two processes work in combination

Several studies in chick

Discrepancies between different interpretations of tissue movement data are due to a few specific limitations of previous studies which we aim to address through the modeling framework presented here. Firstly, quantitative details matter. Although alternative hypotheses may be qualitatively different from each other (such as the Progress Zone Model (PZM)

As an important step beyond the general overview given by previous fate maps, a formal numerical description of limb tissue movements over time would be of immense help to analyze the complex morphogenesis of the limb. Such a description could be defined by the velocity vectors defining the displacement of each tissue point in a fixed coordinate system. Ideally, the collection of these vectors, known as velocity vector field, would be directly measured by tracking tissue points during growth by time-lapse imaging. However, despite recent advances in live imaging of the mammalian limb bud

Due to this lack of quantitative data on tissue movements, most of the existing 2D mathematical models of limb growth have been used as theoretical tools to explore possible cellular hypothesis explaining limb outgrowth

The paper is organized as follows. In the first two sections of the results we introduce the new computational approach that was developed to explore different limb tissue movement maps matching experimental change in limb morphology. In the subsequent two sections we present a mouse clonal analysis of early limb development (from 9 to 12 days pf.) and we show how the tissue movement map that better matched the distribution and shapes of clones was selected. In the following section, the tissue movement map is modified to match the experimental degree of cell mixing observed in the experimental clones. In the last section of the results we characterize the tissue movement map and relate it to traditional PD patterning hypothesis. Finally in the

We present here a new computational method that estimates the velocity vector field of limb tissue movement by using 2 experimental constraints: a sequence of experimental limb morphologies, and long-term clonal data. The main idea underlying the method is to generate a set of hypothetical velocity vector fields that are consistent with the first constraint (the experimental morphological changes), and then to select the one responsible for real limb outgrowth by comparing simulated fate maps with the second constraint (the experimental clonal data). In this respect our approach is analogous to a “reverse-engineering' method, as the data cannot lead to a direct calculation of tissue movements but can only constrain the possible forward simulations. The resulting velocity vector field was then used to derive the tensors of the velocity gradient which describe the local behaviors of the tissue movement. As an application of the model, a reverse version of the tissue movement map was generated to provide a relative estimate of the degree of mixing between the progenitors of three PD segments. Finally, by mapping a time course of Hoxd13 gene expression into the model, we were able reveal the contribution of tissue movement to the expansion of the Hoxd13 domain.

The shape of the limb bud at any point in time can be defined by a spline curve, but clear morphological features along this line are absent – the limb displays a smooth rounded shape. This lack of landmarks means that even a very precise knowledge of the shapes over time is insufficient to define the underlying tissue movements. This is true both for the internal tissue and also the limb boundary itself, as a particular point of tissue (or landmark) could slide along the boundary spline without altering the shape.

(A) A simple semi circle (first column) grows into a defined shape (second column). The two shapes are aligned at their left boundary (third column). (B) A velocity vector field pointing in the distal direction with a vector magnitude distribution that leads to a uniform expansion. (C–E) A variety of velocity vector fields which can all create the same boundary shape change. The first magnitude distribution (C) defines uniform expansion, the second (D) defines a greater distal expansion and the third (E) a greater proximal expansion.

For this study we had to develop software that would allow the exploration of a set of possible velocity vector fields that can each reproduce the same observed boundary changes. In particular, different tissue movement maps are equivalent to considering the 2D limb shape as a rubber sheet or mesh, with different distributions of elastic deformation (e.g.. the various mesh deformations shown in

(A) A sequence of limb photos at different developmental stages. (B)The chronological sequence of limb morphologies derived from

For the first step, in effect we must define a series of landmarks which explicitly map points in each boundary shape to their positions in the next shape (equivalent to controlling the blue and red triangles in

For the second step we had to devise a way of defining the internal tissue movements (the velocity vector field), and in particular a method for exploring some variations on these maps. The internal movements must be consistent with a given boundary mapping (defined above) and we therefore chose to calculate internal velocity vectors using an edge spring analogy algorithm which can smoothly propagate a given set of displacements from the boundary into the internal points of each mesh. Spring analogy algorithms are a popular approach to deform mesh elements by modeling edges as lineal tension springs

Our spring analogy method was implemented as follows: given a mesh

the displacement of all the boundary points on the left-most boundary are set to zero (representing the deep internal tissue of the body) and the displacements of the remaining mesh boundary points is set to the velocity vectors calculated with the radial basis function interpolation described above in equation (1):

The following iterative formula is used to equilibrate the forces

Eventually the new vertex positions are calculated as:

A variety of alternative movement maps which all fit to the given boundary displacements, can now easily be generated by altering the spatial distribution of the stiffness coefficient

In conclusion, we can define a velocity vector field for each mesh in the chronological sequence (the red arrows in

In this section we wish to generate virtual clone experiments, which can later be compared to real clonal data. Although the velocity vector fields defined above are smooth across time, to create virtual clones the resulting hour-by-hour mesh deformations must be linked together to allow tracking the fates of individual tissue regions over time.

Each of the 72 velocity vector fields defines how to deform a limb mesh in order to match the following mesh in the sequence, and the complete set of fields describes a hypothetical computational tissue movement map that matches the entire sequence of experimental morphologies. To track a region of tissue over the full 72 hours, we must determine how the triangular elements of each mesh will map to the different set of triangles of the 1-hour older mesh. In particular, we must calculate a triangle-interpolation map from mesh to mesh. A graphical representation of this process is provided in

Repeating this operation for each pair of contiguous meshes, we generate a correspondence map that defines the fate of each triangle of the first mesh in the sequence (stage E9) with respect to a set of triangles on the last mesh in the sequence (stage E12). This map is a computational implementation of an experimental fate map. The interpolation is conservative and is based upon the velocity vector fields that define the whole virtual tissue movement. A virtual fate map can be performed by marking a triangle with a “virtual clonal dye” – in practice by assigning the triangle a probability of one and then following the evolution of the probability distribution over time, see

This approach has also a more general application, since it defines the basis for any kind of numerical simulation on a growing triangular mesh representing limb growth. Indeed, by interpolating numerical values on a newly generated mesh at every hour of development we are effectively implementing a global re-meshing scheme that avoids the large element deformations that a grid would undergo over the whole 72 hours of limb development. It is well known that frequent re-meshing introduces a source of spatial diffusion in numerical solutions

In this section we present a mouse clonal analysis of early hind limb development, which will subsequently be used to compare with hypothetical virtual clones. These experimental results are compared with previously published fate maps in chick and implications on PD patterning are discussed.

We used the tamoxifen inducible Cre-line presented in

(A) Four clones showing the quantification of the AP and PD clone lengths. (B) In order to compensate for the variation in developmental stage between different embryos each limb was staged and the tamoxifen injection time was adjusted accordingly. Large triangles represent the AP and PD clone expansion over space and time. PD and AP lengths were mapped at E12 (red line) considering prospective (dotted line) or retrospective lengths. (C) Each rectangle represents the AP and PD length of one clone. Clones were clustered into two groups: isotropically expanding clones, with comparable AP and PD length (blue rectangles), and an-isotropically expanding clones having the PD length greater than AP length (red and green rectangles). (D) A graph showing the degree of clone anisotropy in the limb, PD length over the AP length. Blue means low anisotropy and red high anisotropy. (E) Top: In-situ of Sox9, a known early skeletal marker showing the position of the three PD segments (S = stylopod, Z = zeugopod, A = autopod) Bottom: 16 clones showing the degree of overlap between clones spanning across different PD segments.

Now that we have (i) a method for generating virtual clones on hypothetical growth maps, and (ii) a suitable set of experimental clone data, the next task is to use the latter to infer a biologically-realistic growth map for the mouse limb bud. This task can be split into two parts: firstly defining a suitable set of hypothetical velocity maps, and secondly developing a method to systematically compare each map against the experimental clone data.

The space of all possible movement maps is highly multidimensional and potentially very large, (as exemplified in

Using the two levels of manual control described in a previous section of results (the boundary control splines, and the spring stiffness distribution) we were able to create a collection of 9 maps which represent the main plausible asymmetries in limb bud development (

(A) Initial conditions used for the comparison between the tissue movement maps. Clones are positioned on a grid along the AP and PD axis and are colored according to the PD position, from proximal to distal: blue, green,red, green and blue. (B) Virtual fate maps resulting from 9 different maps obtained combining different stiffness coefficient distributions and spline curves (described in more detail in the main text). The left column shows the control spline curves. The stiffness of the distal springs is increasing from left to right. The tissue movement map outlined in red (Map6) is the one that best matched the mouse clonal data.

We explored a number of scaling functions,

The second step was to evaluate which of the 9 hypothetical tissue movement maps best fitted the experimental data. We mapped the 13 clone pictures having better contrast and best capturing the main features of the clonal data set onto the last triangular mesh in the sequence (stage E12). This was done by manually aligning the limb morphologies of the thresholded clone pictures on the last mesh boundary. Results are shown in

Given a set of triangles representing a virtual clone

(A) The left column shows pictures of experimental clones. The remaining columns show numerical comparisons between the experimental clones (white shapes) and the best matching virtual clone (colored contour lines). The comparison is made for three different maps (Map1, Map6, Map7). The contour lines define three regions of probability of the virtual clones: the area enclosed by the red line contain the

As mentioned in the previous sections, the hourly global re-meshing process introduces an inherent source of diffusion to the numerical simulation, which we consider equivalent to the redistribution in the density of labeled cells. This is in agreement with our clonal data that clearly shows a decrease of labeled cell density during early phases of clone expansion resulting from the mixing between labeled and non-labeled mesenchymal cells, see

(A) A picture showing the degree of cell mixing observed at early times after clone induction. (B) The first column shows 3 experimental clones used for this analysis. The second column shows the estimated probability distributions of experimental clones obtained by a mean filter. The second and the third row of this column show a quantification of the overlap between pairs of experimental clone distributions – C6-C7 and C6-C2. The number in white is the score representing the amount of overlap. In the three right-hand columns the overlap between the correspondent virtual clones from Map6 is calculated by considering different amount of additional diffusion: no additional diffusion (first column), a diffusion constant of 0.03 (second column) and a diffusion constant of 0.08 (third column). It can be seen that addition of some diffusion improves the score (compare with the “Estimated overlap” column), while too much extra diffusion makes the scores worse again.

First, we estimated the spatial probability distribution of three different clones. This was done by applying a mean filter to the experimental clones as they were mapped into the mesh at stage E12. The mean filter averaged the value of each triangle with its direct neighbors and normalized the overall spatial distribution to 1. By iteratively applying the filter we smooth the distribution of the labeled triangles until the data did not present any spatial discontinuities. The estimated probability distributions of three experimental clones is presented in the second column of

Secondly, we defined a score to quantify the degree of overlap between clone probabilities distributions. Given two clones

We therefore introduced an additional diffusion term that would model a higher degree of cell mixing. The quantification of the clone overlap was repeated multiple times with different degrees of extra-diffusion. In this way we were able to provide a rough estimate of the diffusion constant that better fitted the experimental clone overlap (see the two columns on the right in

In conclusion, the refined version of Map6 with the extra-diffusion not only matched the shape and distribution of the clonal data but was also matched the relative positing and spatial extension of the clones. A qualitative comparison between the experimental and the virtual clones obtained with this map is shown in

(A) A direct comparison between experimental clones (white background) and simulated clones (black background). Experimental clones have been thresholded and the clone shape has been highlighted. (B) A collection of clones matching the distribution and shape of the experimental clonal data. Blue clones expand isotropically on the PD and the AP axis, and green and red clones expand more on the PD axis. On the left, the initial conditions of the fates are shown.

In this section we present some applications of the growth model that highlight the power of mathematical modeling in characterizing limb outgrowth in space and time.

A first interesting application of the model was to derive and visualize the local tissue behaviors that contributed to the global tissue movement responsible for limb outgrowth. In the model, local tissue behaviors can be represented by the growth tensors associated with each mesh triangle. Tensors were derived from the spatial gradient of the velocity vector field and provided three useful pieces of information: tissue growth rate, anisotropy and rotation

(A) The velocity vector field of the map that best recapitulates limb tissue movement. Velocities have been normalized for clarity. (B) A heat map visualizing the expansion rate of each triangle. Red corresponds to high expansion rate (low cell cycle time of 10h) and blue to low expansion rate (high cell cycle 42h). Average cell cycle times of distal (1/3 of the PD axis from the tip) and proximal parts (remaining 2/3 of the PD axis) are shown for each time point. (C) Ellipses visualizing the anisotropy and rotation of different parts of the tissue. During an initial phase of development the anisotropy is relatively uniform (until stage mE10.18) and oriented towards the distal tip of the limb. After this stage the anisotropy is non-uniformly distributed, and is higher in the region under the influence of the AER. In the central sub-ridge region the direction of the anisotropy is parallel to the AER, while in more anterior or posterior regions it is perpendicular to the AER.

The second application of the model focused on the PD patterning of the limb. In particular we used our model to address a matter of debate for the last two decades: that is to identify at which stage the three PD segments of the limb can be specified. The problem has been addressed numerous times in the chick by creating fate maps to study two different aspects: the degree of mixing between cells of the prospective segments, and to follow the lineage of cells expressing markers of the three PD segments, like Hoxa13 and Hoxd13 for the autopod. An early study using both approaches

Considering some of the controversies mentioned above we decided to use our model in two ways: firstly to give an estimation of the degree of mixing along the PD axis, and secondly to estimate the extent to which limb tissue movement could be responsible for the expansion of the Hoxd13 domain, one of the known distal marker of the autopod.

For the first question we used the model to compute a reverse version of the tissue movement map. This would allow us to start by marking the three PD segments at the oldest timepoint (E12), and then work backwards to determine which regions of the young limb bud could contribute to the 3 segments. Importantly, this is not equivalent to running a clonal experiment backwards in time. As in a traditional heat diffusion problem, individual virtual clones cannot be reverse simulated to discover where they came from. On the contrary, if a clone was reverse simulated from its final spatial distribution back to the young limb bud the corresponding region on the young shape will be proportionally larger than the final clone. This is clearly the opposite of the normal forward simulation, which starts with a region much smaller than the final clone – a single triangle. This distinction is explained in more detail in

The positions of the PD segments at E12 were determined by the expression of the Sox9 skeletal marker, see

(A) A reverse tissue movement map was calculated in order to identify the progenitor regions for the three PD segments. In the graphs, the stylopod is highlighted in red, the zeugopod in green and the autopod in blue. (B) On the top, the initial position of the three PD segments is specified as shown by an in situ hybridization at stage E12 of the Sox9 skeletal marker, on the bottom. (C) Graphs showing the retrospective probability to belong to the three segments along the proximal distal axis. The regions having a high probability to belong to more than one segment are highlighted with diagonal black lines.

For the second analysis of PD patterning we investigated the possible contribution of tissue movement to the known expansion of the Hoxd13 distal marker. First we mapped into the model a gene expression time course of Hoxd13 that was obtained from in-situ hybridization at 7 different developmental stages, see

(A) Top row: a sequence of 7 in-situ hybridizations of Hoxd13 with corresponding stage given by the morphometric staging system

In conclusion, our model predicts that the degree of mixing observed in mouse is too high to support the Early Specification Model as a realistic description of PD region specification. Moreover we have also shown that the expansion of the Hoxd13 domain, one of the genes proposed as a distal marker of the limb, cannot be explained considering tissue movement only but has to involve active up-regulation in at least two distinct phases of the development.

In this study we present a novel computational method which combines a sequence of experimental 2D limb morphologies and clonal data to estimate a comprehensive description of the tissue movement map responsible for limb morphogenesis. We present a mouse clonal analysis of early hind limb development and show how this allows us to estimate a 2D descriptive model of limb outgrowth that fits the experimental data. In practice, our approach is a reverse-engineering method. It is important to note that the spring analogy algorithm is used as a convenient tool for creating a variety of different hypothetical growth maps, but is not employed to represent the mechanical properties of the tissue.

A major advantage of our model over previous fate maps is the resulting comprehensive prediction of tissue movements over time and space. Previously, the behavior of a point of tissue had to be inferred by manual comparison to its closest experimental clone. By contrast, in our new map the movement of every piece of tissue is described numerically across the whole period of development. A related advantage is the temporal accuracy – the state of any hypothetical clone can be predicted at any intermediate time point – not only at the beginning or end of a virtual clonal experiment. The spatio-temporal comprehensiveness of the model gives it the power to make more concrete predictions about PD patterning. To the best of our knowledge, this is the first comprehensive 2D model of limb outgrowth derived from experimental data.

Many aspects of our clonal analysis agree with previous results in mouse

Another interesting observation regards the general construction of our model. By representing the local density of labeled cells as a probability distribution which can diffuse through a smoothly deforming mesh, we shows that biologically-realistic tissue movements can be captured through the combination of anisotropic velocity vector field, with isotropic diffusion. This could suggest that the cellular properties which govern mixing, such as cell-cell adhesion, may not themselves display any cell polarity. In other words, it is theoretically plausible that cells are subject to two types of activity: directional movements (such as oriented cell divisions or convergent extension) which are responsible for the tissue-level shape changes, and non-directional cell mixing. However, in reality, alternative scenarios may also be equally compatible with our model. For example, it is likely that oriented movements naturally lead to the intercalation and therefore to the mixing of cells, such that directional movement and cell mixing cannot be conceptually uncoupled.

Finally, we used the model to clarify the relation between mouse limb tissue movement and the existing PD patterning hypothesis. Firstly we showed, by using a reverse version of the model, that there is a considerable degree of mixing between the progenitors of the three PD segments (

To conclude, the software that we developed will allow us to easily integrate, inside a realistic 2D model of limb growth, numerical simulations of gene regulatory networks and morphogen gradients taking a big step forward in the study of limb development by using a systems biology approach.

All animals were handled in strict accordance with good animal practice as defined by the relevant national and/or local animal welfare bodies, and all animal work was approved by the appropriate committee.

The clonal data was produced using the tamoxifen inducible Cre-line presented in

72 limb morphologies were extracted from an extended version of the standard morphological trajectory presented in

The Hoxd13 gene expression time course in

Software to generate the virtual tissue movement maps was written in Java and used the free visualization library vtk

Standard morphological trajectory. The 72 experimental limb bud morphologies describing mouse hind-limb development from stage E9 to stage E12.

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Analysis of the clonal data (PD and AP length measurement). The clonal data that was generated using the tamoxifen inducible CRE transgenic mouse line. Clones are divided in two groups: isotropically expanding clones and an-isotropically expanding clones. The PD and AP clone lengths relative to the maximum PD and AP length of the limb are measured. Colored triangles represent the AP and PD clone expansion.

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Proliferation patterns of the maps shown in

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Experimental clone registration. The collection of 13 experimental clones that were mapped into the last triangular mesh of the sequence (stage E12).

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Clone scores of the tissue movement maps in

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Description of the tissue movement maps in

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Simulation with a refined triangular mesh. The 13 virtual clone positions in Map6 that best matched the experimental data were used to simulate virtual clones on a version of Map6 with a refined mesh.

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Forward and backward maps. A description of the different information that can be extrapolated from a virtual tissue movement map: fate maps vs progenitor regions. The PD segment progenitor prediction shown in

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Mesh deformation video. Part of the sequence of triangular meshes that was derived from the experimental limb bud morphologies. Each mesh is deformed to match the next mesh in the sequence from which the simulation continues.

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Virtual fate map video. A video showing a virtual fate map. The spatial probability distribution of the fate is colored with a heat map, red represents a high probability and blue a low probability. A discrete number of triangle-interpolation steps can be appreciated in the video.

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Video of the simulation in

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We wish to thank Laura Quintana for the photos of the Hoxd13 in-situs patterns, Jelena Raspopovic for the Sox9 in-situ, Bernd Boehm for help with the limb bud shapes and Jim Swoger for the useful discussion.