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# Anisotropic growth shapes intestinal tissues during embryogenesis

Edited by David R. Nelson, Harvard University, Cambridge, MA, and approved April 29, 2013 (received for review October 5, 2012)

## Abstract

Embryogenesis offers a real laboratory for pattern formation, buckling, and postbuckling induced by growth of soft tissues. Each part of our body is structured in multiple adjacent layers: the skin, the brain, and the interior of organs. Each layer has a complex biological composition presenting different elasticity. Generated during fetal life, these layers will experience growth and remodeling in the early postfertilization stages. Here, we focus on a herringbone pattern occurring in fetal intestinal tissues. Common to many mammalians, this instability is a precursor of the villi, finger-like projections into the lumen. For avians (chicks’ and turkeys’ embryos), it has been shown that, a few days after fertilization, the mucosal epithelium of the duodenum is smooth, and then folds emerge, which present 2 d later a pronounced zigzag instability. Many debates and biological studies are devoted to this specific morphology, which regulates the cell renewal in the intestine. After reviewing experimental results about duodenum morphogenesis, we show that a model based on simplified hypothesis for the growth of the mesenchyme can explain buckling and postbuckling instabilities. Being completely analytical, it is based on biaxial compressive stresses due to differential growth between layers and it predicts quantitatively the morphological changes. The growth anisotropy increasing with time, the competition between folds and zigzags, is proved to occur as a secondary instability. The model is compared with available experimental data on chick’s duodenum and can be applied to other intestinal tissues, the zigzag being a common and spectacular microstructural pattern of intestine embryogenesis.

Growth instabilities in soft tissues are commonly observed in the human body, e.g., wrinkles on human skin (1⇓–3), mucosal buckling in tubular organs (4⇓–6), or convolutions in the brain (7, 8). Such instabilities are the final result of complex interactions between genetic, biochemical, and physical processes. To go from genotype to phenotype, that is, to understand the interplay between genetic pathways and the morphology of organs or simply renewal of organ tissues, is not an easy task because it requires the body remodeling induced by growth. Oversimplified growth processes in ideal geometries have been theoretically treated for skin, bones, and plants, for example. However, the best description for growing tissues remains a matter of debate, but clearly not a unique and universal description can be given for such wide domains, which cover among others morphogenesis (9, 10), embryogenesis (11, 12), and tumor growth (13). Independently of the biological events occurring at very small scales and at the origin of the embryo development, the complexity of living organs comes also from the stresses generated by the growth process itself. In a quasistatic description of the growth, the nontrivial final shape has to minimize the elastic energy taking into account the boundary constraints.

Here, we aim to describe the early stages of villi formation during chick duodenum growth (14⇓–16). Initially, the small cylinder interior covered by a bilayer made by the mesenchyme and the mucosal epithelium is smooth, and then folds appear, bulging into the lumen approximately during the midperiod between incubation and hatching (Figs. 1–3). These folds, called previlli, destabilize giving a zigzag pattern 2 d after. A few days later, the zigzag points become indented, and a more pronounced growth appears in the third dimension, giving protuberances called villi (17). During these different steps, the growth is continuous and highly regulated: during the folding and zigzag period, the increase in length is much bigger than that in the diameter, whereas the converse is observed during villi formation. The villi elongate rapidly before and after hatching. The same scenario occurs during turkey embryogenesis for which the precursor period of previlli has been less studied (18).

The appearance of the zigzag instability in intestine growth is common to many embryos such as humans (19, 20), avians (14⇓–16, 18), pisces (21, 22), and different parts of the small intestine (duodenum and jejenum). The large intestine does not present villi, which is explained by its different functions in physiology. For humans, however, the folding appears in the colon at 11 wk of gestation (19), giving also zigzags, but these structures disappear at 30 wk (20, 23). This pattern seems universal, although we have not found this instability for mice (24).

Focusing on the chick’s embryogenesis, in which zigzag instabilities are clearly established and more documented for our purpose, we report biological observations and measurements about the previllus formation found in the literature. We mention also some debates about the respective role of both layers: mesenchyme or epithelium in the villi formation process. Because part of the controversies comes from the lack of theoretical treatments, we propose a model of differential anisotropic growth in the mesenchyme to explain the observed patterns. Growth is taken anisotropic as the structure of the mesenchyme tissue. Based on simple biological assumptions, this model treats exactly the elastic stresses inside the tissue and is extended to a bilayer: mesenchyme–epithelium. Being completely analytical, it gives quantitative results about the pattern geometry, which may motivate new experiments. Moreover, it favors the scenario of differential growth in the mesenchyme for chicks, and it can be also applied to other growing layers where these folding and herringbone instabilities occur.

## Reported Experimental Observations

Although the embryonic development of chick’s duodenum (25) was described more than one century ago, there are very few quantitative descriptions of the development occurring in the cylindrical wall of the duodenum giving the villi evolution from the first days of incubation up to the hatching. Coulombre and Coulombre (14) reported the chronology of these changes during a period between 6 and 24 d of incubation, just before hatching. They measured the increase of length and radius of the duodenum cylinder, which presents three steps of rapid increase of length with relative stagnation of the radius followed by the converse. During the first period, with which this work is concerned (from the 6th [embryonic day (E6)] up to 13th [E13] day), the length of the duodenum varies more rapidly than the outer radius, both quite linearly in time. At the 8th day (E8), a first pair of previlli appears as axial folds, and their number increases up to 16 in a mathematical progression 2 by 2 up to the 13th (E13) day. From the microscopic and geometric view point, the duodenum can be seen as a cylindrical ring made of the mesenchyme tissue covered by two epithelia at the inner and outer surface (Fig. 1). The mesenchyme cells are circumferentially oriented and differentiate into circular smooth muscles around the 9th (E9) day. The axial observed folds at the mucosa seem to be related to this circular anisotropy of the mesenchyme structure. At E13, these folds exhibit a highly regular zigzag pattern, although this event does not seem to correspond to a change in the growth process itself. A change in the growth dynamics occurs later around E15 and each zigzag fold breaks at the points and separates into rows of small fingers, real precursors of the villi. Indeed, primary villi remain positioned on the zigzag lines. Using electron microscopy, Grey (15) confirmed these observations with perhaps a shift of 1 or 2 d in the chronology of the events. From his figures 6–8 in ref. 15, an estimation of 100 μm can be given for the distance between two rows of previlli, which reach an amplitude of 10 μm when the zigzag occurs. A few years later, Burgess (16), also with electron microscopy, reexamined the previlli formation in chick duodenum, exhibiting more clearly the relative size of the mesenchyme versus the mucosal epithelium, and favored the role of the epithelium growth, although he did not discard completely the mesenchyme–epithelium interaction. However, he did not give data specific to the mesenchyme. Although these authors presented different interpretations about the early stages of duodenum morphogenesis, they all believed in the role of active fibers or filaments for the folds and zigzags. Although Coulombre and Coulombre (14) believed in the active role of circumferential muscles in the mesenchyme, Burgess rejected this hypothesis: he destroyed these muscles and kept observing the folds. However, he detected intracellular microfilaments in the epithelium near the mucosa and explained the buckling by the active role of these filaments. This controversy does not seem to occur for the colon at least for humans (19) because here transmission electron microscopy experiments show clearly the subdominant role of the mucosal epithelium, which decreases in size during the fetal colon development and simply follows the mesenchyme transformation. We propose here a mechanical model in which the differential growth between elastic layers is responsible for the observed folding, and then of the zigzag of the previlli. For simplicity, we do not incorporate explicitly the specific elasticity of the fibers.

## Buckling and Postbuckling Mechanism

Several hypotheses have been advanced to explain villi formation. The different steps described for chick (14, 15) and turkey embryos (18) show that not a unique mechanism will explain the final shape of the villi, which when observed have a height of 100 μm at E9 and of 1,000 μm 12 d after hatching. The different scales involved, which vary by a factor of 10, show that we must distinguish between the role of mesenchyme from the role of the mucosal epidermis, which is also very active but probably not during the formation of previlli. In addition, for mammalian models such as mice, the structure of the mucosal epidermis, stratified or pseudostratified, has been debated recently (26), bringing a different hypothesis for the growth mechanism itself: addition of layers or intercalation. Focusing on the previlli formation, the folding and zigzag patterns are strongly reminiscent of buckling and secondary instabilities observed in material sciences (27, 28), studied theoretically with various elastic models such as Hookean elastic thin sheets (29, 30), poroelastic models (31), or finite elasticity for tissue (32⇓–34). To explain the dynamics of villi formation, we favor the biomechanical hypothesis of anisotropic growth of the mesenchyme, anisotropic because the structure of the mesenchyme is mostly oriented circumferentially. The growth is constrained by the adhesion to the serosal epithelium, itself probably attached to embryonic organs. The mucosal epidermis, attached to the mesenchyme, is free in the lumen, and we will consider it as a thin layer, which follows the structure of the mesenchyme (Fig. 1 and Fig. S1). The cause of the buckling is the growth process itself. Clearly, all of the components of the duodenum grow differently, leading to differential growth between the serosal mucosa and the mesenchyme. Differential constrained growth is at the origin of compressive stresses as shown in *SI Text*. However, these folds are destabilized into a zigzag, indicating the existence of transverse modes at this stage of the duodenum growth. This hypothesis is in agreement with Coulombre and Coulombre (14), who mentioned in the last sentence of their paper that duodenal villi are shaped by mechanical forces, our aim being to prove it as well as to quantify these forces and the resulting patterns.

## Formalism

Being the technical part of the paper, we present first the strategy for analyzing the fold existence, and then their loss of stability into a zigzag organization. After a mathematical description of volumetric growth, we justify our choice of the simpler planar geometry at the stage of zigzag appearance. Then we give the main lines of the elasticity of soft tissues, more details being given in *SI Text*. Having in mind that the zigzag is a perturbation of a primary instability, we perform a nonlinear treatment, which is presented at the end of this section.

Our formalism is based on morphological changes due to volumetric growth in a soft tissue. Anisotropic growth is represented by a tensor , a mathematical object that associates a characteristic positive number to three orthogonal principal directions. If growth is isotropic, the three eigenvalues are equal, anisotropic otherwise. As an example, we can choose the three directions of Fig. 1, each of them having a growth eigenvalue , . These directions are related to axis of symmetry of the material microscopic structure. To measure the eigenvalue , we need to define a small segment of length *L* parallel to one principal direction and follow its length at time *t* from the growth beginning. In other words, these eigenvalues are the relative length expansion in the direction *i*, at time *t*. When no growth occurs in one direction, . When two layers (*A*) and (*B*) are growing simultaneously but at different “speeds,” then one needs to consider the relative growth of the faster layer (*A*) to the slower layer , so . The relevant quantity is the differential growth eigenvalue: However, this is true only in a perfect space without any constraint and where an object can expand freely. In other words, these are preferential growth quantities imposed by the biology. Due to physical constraints such as integrity of the body or interplay between soft or rigid layers, a material cannot follow completely the rules of biology and these eigenvalues cannot be measured directly experimentally. However, they can be deduced a posteriori by the conclusions of the elastic model or by maintaining the biological growth process without the constraints, separating the layers in contact for example. Because constraints exist, such as simply the adhesion to a stiff substrate, stresses are induced inside the sample and our geometric deformation is the result of growth combined to elasticity. Another important ingredient in any calculation of elasticity is the geometry, which is more complicated for cylindrical growing layers than for horizontal layers. To simplify as much as possible the analysis, we focus on the period of interest for our purpose that is a network of 16 folds that first develop (their height increases), and then destabilize into a zigzag to finally show a herringbone pattern. To simplify the treatment, we first consider the curvature effect, making reference to previous works on cylindrical layers (33, 34).

Folding (formation of parallel folds along the axis of the cylinder) and segmentation (formation of circular folds in the orthoradial direction) of tubular tissues occurring during anisotropic growth process have been treated at the linear approximation in refs. 33 and 34. The elastic treatment is far from obvious, complicated by the geometry, the differential growth parameters being given by the growth tensor (*r* and *θ* denote the cylindrical coordinates). Based on ref. 33, the cascade of foldings between E8 and E12 is simply due to the decrease in the aspect ratio of the ring given by , being the thickness of the mesenchyme, the outer radius. doubles its size from 0.4 mm up to 0.8 mm. Unfortunately, we do not know the initial thickness of the mesenchyme and its evolution during the growth process. Based on our earlier treatment (figure 2 of ref. 34), the number of folds: 16, the vicinity of folding and segmentation events responsible on a postbuckling instability gives an estimation of and of order of the outer radius if we assume a small difference between and . From the theoretical viewpoint, the previlli rows can be considered as the first instability of the mesenchyme growth and the pronounced and regular zigzag as a secondary instability. To treat sequential instabilities like, first, the folding, and second, the zigzag, one needs a nonlinear treatment called postbuckling analysis that we restrict here to the first instability, treating the zigzag at linear order only. Due to technical complexity (see *SI Text* as an example), only very few works are concerned up until now with postbuckling phenomena in elastic pattern formations. This is why, for simplification, we will take into account the smallness of the aspect ratio and the relative large number of previlli to neglect the curvature of the duodenum assuming the outer serosal epithelium as a stiff planar substrate (Fig. 1).

In Cartesian coordinates, the deformation gradient that links the initial configuration to the current configuration is simply , where is the elastic strain tensor (see Fig. S2). Taking the initial thickness as the length unit, the mesenchyme rests on the plane and the mucosa concerns . We will choose the *Y* axis as the previlli axis, whereas the *X* axis corresponds to the circumference of the outer tube. To simplify, we choose each space independent, the local volume increase being . An essential property of living tissue is incompressibility, which means that loadings alone cannot modify the volume. In linear elasticity, this property corresponds to the limit for the Poisson coefficient. This strong property of the tissue, usually imposed in variational problems by the pressure, is hard to implement adding a new unknown function in strongly nonlinear Euler–Lagrange equations. As for fluids, we make the choice of a stream function, a technique that we develop in 2D for buckling (35). An extension of this technique in 3D is not obvious because one needs two different mathematical functions (called potentials) to solve the equations of elasticity with the boundary conditions, limiting a lot the advantage of this representation. However, in the linear and weakly nonlinear regime, we can restrict ourselves to a unique potential of gradient form (*SI Text*). Then any point of a sample is represented in the mixed coordinate system , a somewhat curious representation that allows to treat exactly and more easily the incompressibility constraint (*SI Text*). The potential links the coordinates of the initial configuration to the ones of the current configuration. It is decomposed into two contributions , representing the obvious elastic solution of a layer expanding only in the vertical direction, *ε* being a small-amplitude parameter that we aim to calculate. So the coordinates read as follows:giving, because of the incompressibility,The subscript means the derivative with respect to . Assuming the simplest elasticity model for the tissue, that is the neo-Hookean elasticity (36), the energy is given bywhere is scaled by μ the shear modulus of the growing layer, whereas is a capillary number related to a surface energy *σ* as . Let us stress that we work in the volume preserving ensemble of transformations, and then we do not need to add the pressure once we express by , where the index refers to the coordinates . The Euler–Lagrange equations in the bulk of the tissue is then as follows:where we use Einstein notations. We adopt increasing order for the coordinates, so and with . More explicitly written equations can be found in *SI Text*. The bottom layer is fixed to the basement membrane by anchoring proteins suppressing displacement and sliding during the growth, leading to the following:while the mucosal layer (*Z* = 1) is free from stresses (included capillarity), and we getwith the same convention for index and coordinates giving the sum of five terms. We first consider the simple growth of the layer and its first buckling mode. Due to the strong constraint at the bottom , the simplest solution is a global vertical displacement and . This leads to a diagonal biaxial compressive nominal stress (36) , stronger in the *X* than in the *Y* direction in favor of folds parallel to the *y* axis. This naive solution is expected to lose its stability and give rise to more complex patterns as the cylindrical buckling made of folds along the *Y* axis. This indicates that the elastic energy due to vertical growth along *Z* is no more a minimum and new patterns may minimize it. Let us first study the primary buckling choosing . Once two parameters and are introduced, the Euler–Lagrange equation derived to leading orders in simplifies and gives the unknown function :where the notation means the *n*th derivative of *h* with respect to *Z*. This equation has been solved in various contexts where the layer geometry is involved (35), and here we give the result:where *τ* is determined by the boundary conditions equations (Eq. **6**). Focusing on the physical results in the asymptotic limit of weak capillary energy and considering , we find that the control parameter involving only the growth eigenvalues in the horizontal plane drives the primary buckling when . Comparing both control parameters and for orthogonal directions proves that the folds aligning along the *y* axis correspond to the smallest growth eigenvalue . A nonlinear analysis beyond the quadratic order in the amplitude is found by an expansion of the elastic energy: , and by minimization we find the following:which corresponds to a supercritical bifurcation (37) (Eqs. **S27–S30**). Numerical values are given for simplicity, but all of the results are analytically derived. Notice that, when the parameter is small, it acts as a singular perturbation and its role consists in the selection of the wave number *k*. In *SI Text*, we show that this wave number, inverse of the wavelength of the folds, is given by the following:The profile of the pattern at the free border close to the threshold value is given by the following:where the length scale is the initial thickness. Combined measurements of the initial height of the mesenchyme and when folds appear allows us to estimate *J* while the measurement of fold amplitude gives . The wavelength itself is given by approximately four times the height of the layer when the folds appear divided by a logarithmic correction. We examine hereafter the growth conditions necessary for the emergence of zigzag patterns. They result from the superposition of folds of different orientations (29). At linear order patterns are independent and appear as a bifurcation of the base state. Zigzag patterns are due to this superposition, being represented mathematically by , satisfying the same differential equation (Eq. **7**) as previously, except and , substituting and , respectively, and recovering the pure stripe case if . To (respectively, ) is associated a wavelength (respectively, ) as indicated in Fig. 3 and . Taking into account the boundary conditions at the free border and defining the parameter , we find the threshold for the zigzag patterns as follows:As for Eq. **10** and shown in *SI Text*, surface tension selects the modulus of the wave vector *k* given by the same formula as Eq. **10**. For folds of arbitrary orientation, the control parameter is the same: it is not *J* but . For zigzag, the threshold is upper than for folds . Assuming a small mismatch between the two eigenvalues: , it becomes , showing that if the planar anisotropy is weak both thresholds are close to each other and can overlap. This overlap can also be obtained for finite anisotropy and long-wavelength instability along the *y* axis (small). Finally, far from threshold , both orientations *x* or *y* are equivalent and the wrinkling pattern (see Fig. S3) would exhibit a jog angle if the growth parameters increase smoothly without biological modifications. To describe the zigzag instability, we superpose and with into Eq. **3**. It reads as follows:with all of the coefficients being positive: , being a function of , *B* being a function of , and being a sharp peaked function around with maximum value 1. In *SI Text*, we show that, close to the zigzag threshold, is a tiny quantity and the zigzag patterns are preferred to stripes for finite value of by energy comparison.

## Discussion and Conclusion

Let us first sum up the main results of the model. It is based on a growing elastic layer, the mesenchyme bounded by two epithelia: the serosal one, considered as a stiff substrate, and the mucosal, which faces up the lumen. The three layers grow, but only the difference between growth induces elastic stresses responsible for buckling. As shown here, only the differential growth parallel to the plane of the substrate (quantified by ) drives the instability. We favor the growth of the mesenchyme because of the wavelength of the pattern, more compatible with its thickness. The thin mucosal epithelium follows the growth of the main layer, exhibiting the underlying instability shown in refs. 14⇓–16 and 18. It is considered as a surface effect, which can be evaluated by dimensionless analysis. Knowing the elastic shear modulus of each layer and (*e* for epithelial mucosa, *m* for the mesenchyme) and the thickness of each layer, and , we estimate . being weak, it acts as a regularizing factor of the Biot singularity (38) and fixes the wavelength at threshold, both for the folds (Eq. **S19**) and then for the zigzag (Eq. **12**). Once the thickness of the initial growing layer is known with the growth factors , our work allows us to estimate all of the observable quantities. Taking into account Eq. **11**, we derive the height of the growing layer when the folds appear, that is, . The threshold for folds is reached when . Because this quantity is growing with time, the observation of tiny folds gives the value of . Then the ratio between the fold amplitude and the layer thickness can be compared with the prediction of Eq. **11**. The wavelength between folds can be predicted quantitatively if we know the shear modulus of the layers, knowing that, for a monolayer epithelium, the ratio is of the order of 0.05. Taking this number for , we get for the logarithmic factor entering Eq. **10** an estimation of 6 or 7, so the wavelength is about assuming . The zigzag instability of the folds, which can be seen only in case of weak anisotropy, indicates that has reached the second threshold (Eq. **S36**). The wavelength of the zigzag decreases when increases (according to Eq. **S36**). Plate 2 of ref. 15 shows this evolution from E13 to E16. Ultimately, and only in case of weak anisotropy, the nonlinearities on this secondary instability will favor a nonlinear zigzag at with a resonant amplitude for as found also in material sciences for thin prestrained films stuck on a substrate (28).

For completeness, we have studied the possibility that the folding instability is driven by the growth inside the mucosal epithelium (*SI Text*). We then consider the case of a growing epithelium of thickness put on a thick substrate of arbitrary stiffness (see Fig. S4). We find that the predicted wavelength is a fraction of the epithelium thickness, the precise numerical prefactor being dependent on the ratio between the shear modulus of both layers. Our main conclusion is that the folding instability has a wavelength given by the height of the growing layer. Hereafter, we discuss these findings with the data available in the biomechanical literature.

Assuming that the growing layer is the mesenchyme (Coulombre and Coulombre hypothesis), the mucosal epithelium being only a soft monolayer, the axial direction of the ridges indicates that the differential growth in the orthoradial direction is larger than along the cylinder axis . Growth anisotropy is not astonishing because the mesenchyme presents a circumferential microstructure at E6 becoming smooth muscles at E8, whereas the mucosal epithelium increases but only in area by cell division. Moreover, the measurements in ref. 14 indicate an absolute and relative increase of the length much larger in comparison with the radius, showing that growth is more constrained in the radial and orthoradial direction compared with the cylinder axis direction. Estimation of the growth anisotropy can be done, noticing that it requires 1 or 2 d after 10 d of incubation of the chick’s embryo to present a zigzag pattern. Assuming a linear volumetric growth in time, that the ratio of both thresholds (folds and zigzags) can be estimated by the ratio of time appearance, we arrive at an anisotropy coefficient α of the order of 0.2, which is reasonable, although rather imprecise. We can argue also that the geometry of the cylinder favors axial buckling (that is, folding) in comparison with segmentation as shown by ref. 33. This is true for the first folds, but this orientation disappears as the number of folds increases.

A good way to compare directly to our theory will be to put the intestinal tissue directly on a solid substrate and to maintain growth in artificial conditions in vitro. This has been done in refs. 15 and 16, and the patterns persist. The wavelength of the folds with or without zigzag is about 100 μm, derived from the plates of refs. 14 and 15. In our model, this wavelength would correspond to a thickness of the mesenchyme of about *H*_{f} ∼ 150 μm. According to Grey (15), the crests rise to a height of about 10 μm above the floor of the gut. Taking Eq. **11**, the second contribution can be estimated to 0.56⋅100/2 ∼ 10 μm, again up to a numerical number, which is . The estimation for is consistent with the observed wrinkle amplitude, 10 μm. In summary, taking into account the incomplete information we have, the model is consistent with refs. 14 and 15. The viewpoint of Burgess (16) of a buckling driven mostly by the growth in the epithelium is not compatible with the previlli wavelength observations. We have precise information about the size of the cells in the mucosal epithelium from refs. 14 and 15. Their diameter is about 4 μm (a relatively small value for a cell diameter), but it is obvious that the crest of each previllus is covered by many rounded cells, indicating that the cell size does not give the wavelength scale. On the contrary, the microvilli that cover each cell can be the result of a compressive buckling instability occurring in the epithelium. Of course, our discussion is based on an epithelium made of one layer or few layers as it is commonly assumed for this system.

The mechanical model presented here explains the zigzag instability observed in growth processes. It has been inspired by the intestine embryogenesis. This instability seems to be common to different parts of the small and large intestines in many living species. Due to the lack of real quantitative measurements, which are probably hard to obtain except in the case of poultry, we simplify the model to its main ingredients. Focusing on the results, we avoid computational methods and use the theory of buckling and postbuckling to derive analytical results. Only a nonlinear treatment introducing growth anisotropy can explain the folding of the growing elastic layer followed by the zigzag instability of the folds. This furnishes quantitative results and we hope that they will be tested in future works concerning villi embryogenesis. From the mechanical viewpoint, our work can be compared with very recent experiments and analysis by Yin et al. (28). These authors show that a sequential wrinkling is necessary to observe ordered patterns going from straight to herringbone wrinkles. In their case, the patterns are derived via relaxation of a prestrained film put over a compliant substrate. Simultaneous release leads to disorganized labyrinth patterns. In our case, it is the growth anisotropy that ensures a sequential process favoring folds at first and then zigzag. It explains also why these patterns are so well structured for living samples (18). To understand villi embryogenesis is important. The complexity of the organization of these villi is part of the mechanisms that regulate and control the stem cells proliferation. Located in niche or crypt appearing quite at hatching for chicks, these cells are responsible of the renewal process of the intestine. The fact that stem cells are located in niches that appear once the previlli are well remodeled shows the strong interplay between mechanics, patterning, geometry, and biology (17). We have studied the first phase of previlli formation, which will give ultimately the finger-like villi. The following work is to understand the crypt formation in these tissues.

## Acknowledgments

We thank Diego V. Bohórquez for sending us a scanning electron micrograph to illustrate our work. We are very indebted to Pasquale Ciarletta for helpful discussions. F.J. was supported by China Scholarship Council (Student ID 2011621111).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: martine.benamar{at}lps.ens.fr.

Author contributions: M.B.A. and F.J. performed research, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1217391110/-/DCSupplemental.

Freely available online through the PNAS open access option.

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