The Textural Aspects of Vessel Formation during Embryo Development and Their Relation to Gastrulation Movements

Abstract

We have investigated the microscopic physical inhomogeneity (“texture”) of the avian embryo in vivo by shadowgraph. This noninvasive technique allows one to correlate the shape of blood vessels to the physical, micro-structural, pattern that exists in the embryo prior to vessel appearance. Before any vessel forms, vascular paths are present and are prepatterned, by fields of cellular orientations and lumen anisotropies. We find the origin of this prepattern in the movements of the embryo during gastrulation, and the related deformation and force field, which establish both the animal and vascular pattern.

Introduction

During early development of a vertebrate embryo, the formation of the vasculature proceeds most generally through two steps (Fig. 1). First a plexus (a lattice) of capillaries forms and is then remodelled into a branched network. It has long been argued that mechanical factors play a role in vessel development and, in particular, in the maturation process of capillaries¹ (Risau, 1997). Forces induce tissue deformations and flows that depend on the elasto-visco-plastic properties of the living material.

Figure 1.

A pictorial view of the mechanism of formation of vascular trees. Remodelling of vessels implies elimination of some strands, fusion of others, diameter enlargement, but also, physical lateral displacements of the tubes (like the snaking of a garden hose).

On fundamental grounds, there exist essentially two kinds of stresses : shear stress (stretch forces per unit area), and compressive stress, akin to pressure (pressure = force per unit surface). When in the presence of a fluid or an incompressible solid,² compressive stress can be identified as a pressure.

Mechanical forces play an important role in embryo development, as well as in vascular development.^2,3 The aim of this article is to show how deformation fields during embryonic development serve as template for the vascular development. In other words, the purpose is to show how vascular adaptation to mechanical forces is related to embryonic adaptation to mechanical forces.

It is well proven that shear stress and transmural pressure induce adaptation of smaller vessels,^4–8 such that vessels develop or regress in accordance to the mechanical forces which they bear, and not only in response to morphogen gradients such as VEGF.^9,10 It is possible to simulate the formation of blood vessel architecture with some accuracy in a uniform medium,¹¹ when including both the effect of shear stress due to blood flow inside the vessels, and the visco-plastic deformations induced by the pressure in the tissue surrounding the blood vessels. In such models, vessel diameters adapt, but, at the same time, blood vessels are deformed, and are even displaced like hoses by the tissue expansion (and by the associated stresses). The progressive deformation and displacement of vessels tends to bring them along lines of steepest descent of stress.¹¹

However, in addition to the effect of flow forces, and to the effect of the homogeneous push-pull forces in the tissue, one can observe that the microscopic pattern of blood vessels may vary considerably from one region to a another region nearby. For example, vessels may exhibit a stratified aspect, or a honeycomb structure, as if vessels were decorating some underlying physical structure and were strictly following some underlying template. Since the remodelling of vessels acts on a capillary bed, we hypothesize that the very structure of the capillary bed will have an impact on the final shape of the vessels. In particular, vessels of the skin or of muscle have a stratified aspect, which seems to adapt itself to the stratified aspect of the tissue in which they are situated.¹²

Therefore, the textural properties of the tissue (its microscopic physical organization), which are themselves produced by mechanics, eventually impose their pattern on blood vessels, via a remodelling process which gives a huge role to mechanical forces. Our hypothesis is that growth of the embryo induces a stress pattern. Individual cells align according to this stress pattern, which in turn aligns the structural biopolymers of the cells and matrix with the stress field. Blood islands appear and “decorate” that pattern. Remodelling of the vessels and further growth of the embryo align the vessels more strictly along specific stress gradient lines. A simple way of understanding this rationale is to consider tree rings: the growth of plants induces pressure fields that are arranged as concentric rings, even in the absence of a cellular organization. However, the cell organization espouses the stress map and forms itself into conspicuous circles of cells. The tree venation eventually follows the cellular organization and follows stress gradients (e.g., rays, as in fruits). However, in animals, not all stresses are compressive, and the tissue-flow field in the embryo is not radial as in tree trunks,² therefore, it is not as easy to recognize the stress map in animals as it is in plants.

The concept may apply to the direction chosen by the dorsal aortas, to the appearance of a ladder pattern of blood vessels in somite intervals, to the fanning pattern of vessels in the brain, etc. Tumors have a more erratic vascular pattern than normal tissue, and this fact may well be linked to their chaotic texture.¹²

If the formation by remodelling of mature vessels through a lattice of small capillaries is a self-organized biomechanical process, the shape of the underlying lattice would be expected to play an important role. This is known in statistical physics as the effect of a “relevant parameter”.¹³ In the simplest of all processes, the “dendritic growth” anisotropies of the lattice lead to organized branching patterns (like snowflakes), while a purely random lattice leads to fractal self-organized irregular trees.¹³ Consider a capillary lattice: if the initial lattice of capillaries has a definite textural property (orientational order, lumen gradients or anisotropies, etc.), these will imply specific routes to the future vessels, or specific shapes of vascular trees after the self-organization process is finished.

One may wonder how mechanical forces may explain so many different situations as honeycomb vessels, elongated strands, fractal-like bushy vessels etc. It should first be said that it is even more difficult to explain so many different situations by chemical concentrations (because these quantities are scalar). In addition, the hypothesis presented here does not require the endothelial cells to be different in any respect. If endothelial cells adapt to the mechanical context, then, so many different situations might well be explained by the mechanical field, if one explains properly why the mechanical fields are different in all these situations. For example : is there a mechanical model able to explain at the same time the crab-like aspect of vitelline arteries of an avian yolk-sac, and the radiating aspect of vessels in the Zona Pellucida, around the head?

We report in this article an in vivo shadowgraph study of an embryonic tissue, in relation to the establishment of the vascular pattern. We describe in the supplementary material the principle of the shadowgraph. This image analysis reveals that the microscopic physical textures of the different regions of the embryo are quite different, and that they relate directly to the different shapes of the (future) vessels which appear in the different regions. Therefore, development of vessels is, at least in this respect, organized by the initial tissue texture.

We then show that a recent model of gastrulation movements predicts qualitatively, but correctly, the force and flow fields in the embryo, and that this force field positions the cells in the young embryo with an initial texture which relates directly to the prepattern of blood vessels. In summary : there is a cellular flow on the top layer of the embryo (epiblast) which generates the radiating material properties of the vessels in the central part of the embryo, as the epiblast converges towards the center of the blastodisc. After ingression, a similar flow migrates away from the center of the blastodisc, thus generating the crab-like pattern of vessels in the yolk-sac as it expands. The physics is the same, the flow is similar, but whether the flow is converging or diverging suffices to induce a completely different tissue texture, and hence, different vascular shapes.

The blood vessels, which appear almost random at the stage when blood islands form, actually deform progressively until their final pattern matches the force field in the tissue, which also is the field of cellular flow. In simple terms : starting from an initial irregular lattice, the incremental deformations induced by development progressively add up to align the lattice and therefore vessels along steepest stress gradients (which is the stretch force).

Essential Background: Early Embryology of the Chick

During early development, chicken embryos form a blastodisc, which is a flat disk of organized cells, 2 layers thick.¹⁴ The top layer is called epiblast, the bottom layer hypoblast. There exist several data bases dedicated to chicken embryo development, for example Geisha (http://geisha.biosci.arizona.edu/data/anatomy.php) In the first day after the egg is laid, the embryo undergoes gastrulation. A fissure opens in the center of the blastodisc through which cells ingress, to form the mesoderm. At the stage that interests us (between day 1 and 4 of incubation), the embryo body occupies the center of a region called Zona-Pellucida (ZP). The ZP has a very conspicuous figure 8 shape (Fig. 2).

Figure 2.

(A) A typical 1.5 days chicken embryo. The antero-posterior axis lies vertical, head to the top, tail to the bottom. The AP axis of the embryo is visible in the center. The Figure 8 is the Zona Pellucida, and the large roundish pancake which contains blood islands is the Zona Opaca. The vessels form mostly in the Zona Opaca. The dark spots are the first blood islands in the ZO. The entire field of view is 7.5 millimeters. (B) Shadowgraph image of the embryo of (A), showing an overal radiating texture of relief in the yolk-sac, diverging from the embryo, which constitutes a prepattern. Please remember that this is an in vivo direct image with no preparation, therefore this image has nothing to do with cell staining. It shows the physical corrugations associated to the textural differences between blood islands and mesoderm cells. (C) A typical vascular pattern in a 3.5 days chick embryo. The Zona Pellucida is the yellowish area around the embryo (it appears yellow because it is quite transparent, and the yolk underneath is what gives it its color). The Zona Opaca is the whiter area in which one sees the large crab-like vessels. The field of view is 3.7cms.

The Zona Pellucida is surrounded by the Zona Opaca (ZO) of the yolk-sac. The yolk-sac produces the blood islands which form the initial capillary lattice, the first of these blood islands appear for example in Figure 2A, towards the edge of the large circular frontier of the yolk-sac. As the heart starts to beat, the yolk-sac is perfused and blood vessels form across the capillary plexus by progressive transformation (“remodelling”) of small capillary strands under the action of the flow (Le Noble et al., 2003) (Fig. 1). After four days a clear vascular pattern is formed (Fig. 2), seemingly very complex, which we wish to understand. The shadowgraph image of the embryo in (Fig. 2A) will be discussed below.

Materials and Methods

The Shadowgraph Technique.

The Shadowgraph technique is a well know method in physics.¹⁷ It makes use of a parallel beam of light (Supplementary Fig. 1). Optical studies use generally shadowgraphs to explore gradients of refractive index. For example, shining a parallel light through a warmed gas reveals the gradients of temperature and the flows. We do not use the shadowgraph to image light transmitted through the embryo, but to image light that has been reflected. In this case, a parallel beam of light is shone on the surface of the embryo, and the reflection is observed in the direction of specular reflection. Since the embryo is not at all flat, there exists only one direction of observation, which may be difficult to find. Once this direction of specular reflection is found, details of the surface will appear as brighter or darker, depending on the surface curvature. A small concave feature, if approximated by a paraboloid, will show a bright spot in the centre of a dark halo (Supplementary Fig. 2). By this means the details of the surface are visible. When the subject is not observed under a strict specular reflection, interesting shadowing effects may be obtained, which reveal the surface corrugations. With this technique, we have studied the surface of chicken embryo yolk-sacs, and their vasculatures.

There exists a number of problems with the shadowgraph technique. First, the tissue must be flat, in average. Since the yolk is not so flat, the imaging is not possible as soon as the embryo lies on a very curved part of it. The imaging is much better when the embryo lies on the top of the yolk. Therefore, a large number of eggs, perhaps 50%, cannot be studied because the embryo is not on top.

The shadowgraph effect, being a focalisation-defocalisation effect, can only be observed in the focal plane of the corrugations, therefore, not all magnifications are available for the image, and corrugations of different sizes will appear in different planes.

Another problem lies in dessication of the embryo. We found that the images were much better if the embryo was left a few minutes to dry off because it lets the vitelline membrane stick to the surface, so that the underneath relief appears in all its detail. This relief is linked to the softness of the tissue to which the vitelline membrane sticks, in a way which we did not try to quantify. However, the fact that the mesoderm appears convex, while the blood islands appear concave shows that the area of the blood islands is more compliant, which is not surprising. However, if the embryo dessicates too much it may die, or start to fold by buckling. We also found an interesting oscillatory behavior of the tissue, which will be presented elsewhere.

Eggs preparation.

Eggs were purchased from Couvoir de l'Osier in Louvigné-du-désert (France). These were incubated at 37°C for up to five days of incubation. Eggs were opened and transferred to Petri dishes at day 1 or 2, and then studied either by classical optical microscopy, or by shadowgraph. The best method to get good shadowgraph images is to incubate the eggs horizontally for one day, then to open a small hole on top and a small hole on the side with a needle. Up to 10ml of of albumin is then sucked from the hole on the side with a syringe, and the shell is cut off over approx. One-third of the egg diameter. This technique allows one to find almost 50% of eggs with the yolk sac laying on top, and flat.

Numerical simulation of yolk-sac texture.

In order to predict the shape of the yolk-sac, we model the flow of the epiblast on top, and of the mesoderm underneath. This can be done by the following procedure. We know that the initial blastodisc is a circle (Supplementary Fig. 3A).

We know qualitatively the flow (Supplementary Fig. 3B).² It is therefore possible to follow the fate of the blastodisc in the flow. In order to do so explicitely, we displace points of the epiblast with a simple iterative algorithm:

• x_n+1=x_n+dt.U_x(x_n, y_n)

• y_n+1=y_n+ dt.U_y(x_n, y_n)

where dt is a small time step, and U_x, U_y the coordinates of the speed in the deformation field of the blastodisc. If (x₁, y₁) is any point in the initial blastodisc, (x_n, y_n) will be the new position of this point at time ndt.

This gives a typical blastodisc deformation as shown in (Supplementary Fig. 4).

But this cannot be continued forever: we know that the epiblast invaginates by diving underneath and making a U-turn (in the perpendicular direction). However, the motion continues underneath. We therefore split the simulated epiblast in two along the invagination line, then unfold the invagination line and watch the continuation of the flow as it occurs underneath. We do this on the two halves of the embryo, and reconstruct the yolk-sac underneath by pasting the fates of the two halves. This gives the result of (Fig. 11) (see paper). To do this the flow of (Fig. 3B) is therefore used on the epiblast up to the vertical line located in the middle of the embryo, where the epiblast dives underneath by making a U turn in the plane perpendicular to the plane of the sheet. We then calculate the motion of the mesoderm, in the following way : when particles of the epiblast arrive at the edge along the antero-posterior axis (the dark vertical line “primitive streak” in Fig. 11), they ingress, by making a U turn in the vertical plane. This is modelled by considering the fluid speed as making a mirror symmetry along the vertical axis. The flow is simply continued but only the sources of flow x₀ and y₀ on their sides are considered in the formula giving the flow of cells, since when the epiblast is split in two, the left and right parts do not interact. Why the flow of cells has the shape of vortices is explained in.² The result of this calculation is that the deterministic mosaic present in the epiblast is actually deterministically deformed during the invagination, and adopts a deterministic fate. The fate of the epiblast gives the texture of the yolk-sac, which will influence the shape of the blood vessels.

Figure 11.

Starting from one half of the embryo (left), we calculate the fate of the lines underneath, after they have reached the primitive streak and made a U turn (right). This is done by calculating step by step the speed, at the point where they are, and displacing them by a small increment (this was done by Kaleidagraph). We see deterministic lines which reform underneath (right). They form a very nonintuitive, although deterministic, image of the lines on top, after the epiblast has moved towards the streak, made a U turn, and then has grown again away from the streak. Please note that the ingressing mesoderm makes spontaneously a Figure 8 in its middle. (More about this calculation in Materials and Methods section).

Figure 3.

(A) An early blastodisc, and its primitive streak. The grainy structure around the blastodisc shows the fatty vesicles of the yolk-sac (B and C) Reconstruction of the surface profile by analysis of the signal along the transverse cut of the lips of the streak.

Results

A considerable number of details of the surface of the growing embryo and its yolk-sac can be observed by the shadowgraph technique.

This technique differs from direct visual inspection by optical microscopy, and with classical stainings, in three major ways. Firstly, the features observed are true verticall reliefs in the tissue, above the average plane of the yolk-sac. Therefore, any swelling, bump or cluster will appear in the image. This relief relates to the compression field in the yolk-sac, since any force distribution in the surface will induce a deformation in the vertical direction. This vertical inhomogeneity is a small variation over an overall flat average of the yolk-sac.

Secondly, the shadowgraph technique is performed in vivo. Thirdly, at early stages, many blood vessels or capillaries of the yolk-sac are not perfused with actual blood, but with plasma (the erythrocytes being scarce). These capillaries or vessels appear extremely faintly by direct optical imaging. The shadowgraph makes no distinction between vessels perfused with plasma or with blood: what matters is the geometry of the surface around the vessels, not the nature of the liquid inside them.

When imaged at an early stage of development (on the first day), the embryo clearly shows the primitive streak, i.e., the crack through which the mesoderm has ingressed (Fig. 3).¹⁵ At somewhat later stages, one can easily recognize the neural crest folds, the streak through which the mesoderm cells have ingressed, and the limit of the progressing mesoderm underneath. Hensen's node and the cephalic bulge are also easily recognized (Fig. 4).

Figure 4.

Shadowgraph of an embryo during ingression of the yolk-sac (the ingressing mesoderm), showing the formation of the neural folds, and Hensen's node. The head is towards the top, and the tail towards the bottom.

If we look at a region close to the edge of the yolk-sac of a 30 hrs embryo, by optical microscopy, we clearly see the blood islands (Fig. 5A). If we study exactly the same region by shadowgraph, we immediately discover a considerable relief associated with the cavities and bumps of the mesoderm (Fig. 5B). The relief which is imaged is that of the mesoderm around the vessels. We note that many of the capillary cavities visible in shadowgraph appear in the conventional image to be devoid of erythrocytes. The shadowgraph also reveals the fatty vesicles of the yolk located outside the yolk-sac; these appear as a finely grained medium in the very top of Figure 5B, also visible in Figure 3A.

Figure 5.

(A) Optical microscopy of the yolk-sac edge in a 30 hrs embryo, close to the sinus vein. (B) Same region in shadowgraph. The mesoderm in between the blood islands appears in relief.

A typical large field view of a 1.5 days old embryo (Fig. 2B) shows a distribution of bumps and lines in the yolk-sac, away from the embryo, prior to vessel formation. These vertical inhomogeneities reflect pressure inhomogeneities in the growing yolk-sac. It is important to note that the blood islands of the yolk-sac (Fig. 2A) are mostly visible distally, towards the edge, while it may be expected that there are capillary vessels or cavities already present more proximally. Indeed, the shadowgraph shows that the texture of the proximal mesoderm is similar to that of the distal mesoderm.

Figure 2B reveals the structure of the different parts of the embryo: the embryo itself, the ZP and ZO have different thicknesses and corrugations. The embryonic tissues and the ZP form a set of concentric figure 8 patterns. Far away, the yolk-sac is a disc with a circular boundary. The boundary separating the zona opaca and the zona pellucida forms a large figure eight. Closer to the embryo, the tissue reliefs form also figure eights, which are more slender and elongated along the Antero-Posterior axis, as appears clearly around the tail bud.

These images show that we are able to detect the texture of the tissue prior to vessel formation. The appearance of the blood islands create an apparent randomness in the yolk-sac. In the litterature, the capillary plexus is always referred as random. It is then natural to ascribe the complexity of the initial condition for the vascular remodelling to this randomness. But actually, the physical texture of the yolk-sac and of the capillaries which form inside, is not random, as we show next.

Order in the yolk-sac.

When the blood vessels of an embryo yolk-sac are inspected carefully, one finds different regions that have marked differences in vessel shape. In the main part of the yolk-sac, i.e., in the entire region outside the figure 8 formed by the Zona Pellucida, the vascular pattern has a tendency to form vessels radiating away from the embryo, with a very conspicuous rainbow-like or crab-like pattern of the main vitelline arteries (Fig. 2C). Locally, at higher magnification than Figure 2C, one observes a random lattice inside the ZO, with small anastomoses (loops) (Fig. 6A and B, top), while next to the ZO, the ZP exhibits different vessels.

Figure 6.

(A) Image of the transition between the Zona Pellucida and the Zona Opaca at day 3. (B) Detail of (A). This image shows that vessels may have a completely different shape in areas which are very close to each other.

In the Zona Pellucida, vessels have a very striking shape. Although they form at the same time as the other vessels, they are straight (Fig. 6B, bottom), and very regularly organized, radiating away from the embryo, in an approximately parallel arrangement. This is all the more evident as the vessels become chaotic with many anastomoses as soon as they reach the ZO. Figure 6A and B are taken very close to each other, inside the yolk-sac. The two regions show a large difference in vessel shape, at the same developmental stage. May it be that the Zona Pellucida and the Zona Opaca of the yolk-sac have different blood vessels because of a preexisting textural difference?

When we inspect the shadowgraph of the yolk-sac (Fig. 2B and Fig. 7), we find that the small random anastomoses of the Zona Opaca are present in the texture before vascular formation. In addition, away from the embryo, prior to vessel formation, in the region where the dorsal aortas enter the yolk-sac (presumptive navel), we observe a radiating texture (Fig. 7 arrow), with small bowing successions of anastomoses which already suggest the future shape of the large crab-like vessels. Hence, although it is extremely “noisy”, or in other words, randomized by the presence of blood islands, there exists something like a prepattern for the vessels in the yolk-sac, in the shape of squarish anastomoses which follow curved paths. If we magnify the shadowgraph image in that area, we find a very clear topology of pools of blood radiating outwards. This is also visible to the right of the embryo (Fig. 2B). In the Zona Opaca, the pools of blood prepare a path for the future vitelline arteries.

Figure 7.

Magnification of the shadowgraph image Figure 2B, in the region of the presumptive vitelline arteries. The line to the bottom right is the neural crest. The image shows a capillary network with a clear centrifugal orientation. (See also right side of body in Fig. 2B).

Turning to the Zona Pellucida (Fig. 8), we find a much more organized texture of lines radiating away, like rays, around the head of the embryo, although these images are obtained before any vessel exists. Therefore, the texture of the tissue, revealed by the shadowgraph technique, contains a geometrical prepattern, before formation of the blood vessels, and this prepattern is not the same everywhere. This prepattern belongs to the tissue; it is somewhat disturbed and randomized by the process of island formation, but there remains large scale tendencies, which are amplified during the further growth and remodelling of the embryo. In particular, the initial randomness is progressively smoothed out, to form blood vessels which are much more straight and linear than the initial anastomoses. Remodelling of the vessels is therefore not only a matter of diameter adaptation.

Figure 8.

Image of a 2 days embryo showing the texture of the ZP. Straight lines with very little textural noise appear radiating from the embryo body, while the rest of the yolk-sac (Zona Opaca) has a much less marked radiating texture, because of the geometrical noise in the plexus.

As another example, Figure 9 shows the yolk-sac in the caudal area at the stage where it carries a plexus, and the next day, when the vessels have remodelled.

Figure 9.

(A) The caudal area of a chick embryo, at the stage of the plexus, and the next day (B). Same vessels, the next day. (A) Shows the top left quarter of (B). (C) Analysis of the capillaries in the caudal region (this image is a part of image (A), rotated so that the main direction of the capillaries lies horizontal) reveals an anisotropy of the capillary lattice which favors paths radiating outwards. The lumens are in the ratio 1.8 denser in one direction than in the other, a fact easily felt by the naked eye (1.8 = the ratio of the peaks in one direction over the peaks in the other direction).

Visual inspection shows that the capillaries are already oriented in the directions that the vessels will take. This can be quantified by averaging the vessel area in a square box along a projective direction parallel or perpendicular to what is, apparently, the favored direction (Fig. 9C). We find that there exists an average modulation in both directions with well defined peaks. However the peaks of the direction radiating outwards are larger and less noisy than peaks at 90°. The lumen density is almost twice as big in one direction than in the other, and the resistance of the plexus is anisotropic : the hydrodynamic resistance will be about ten times greater in one direction than in the other (hydrodynamic resistance varies like the fourth power of the radius). This will favor those directions, and a radial progression of the vessel remodelling will be observed, with an additional stochastic noise. This means that, from a physical and mathematical point of view, the lumen of the capillaries is not a scalar quantity. At the same location, the lumen diameter depends strongly on the direction from which it is considered. So is the force field, as we shall explain in conclusion.

Discussion

From all these observations, one may therefore conclude that the textural properties of the tissue play a role in the establishment of almost all vessels, by creating a preferential path for vessels. But why is there such a texture in the yolk-sac and in the embryo in the first place? Why, for example, should the tissue exhibit alignments?

If we analyze the vessel pattern more carefully, we find the following important feature. In addition to the lines radiating away from the embryo at the tail, the head, and in the region where the dorsal aortas flow into the yolk-sac (presumptive navel), we find a very clear rotational tendency of the vessels in the region of the presumptive hindlimbs (lateral plate) (Fig. 10), and also in the region of the presumptive forelimbs (data not shown). These curved paths follow the figure 8 deformation of the blastodisc when it forms the tail bud. If looked globally, the prepattern of vessels in the embryo and yolk sac has radial centripetal orientations around the head, circular orientations around the tail bud and lateral plates of the presumptive limbs, longitudinal orientations along the back-bone, and centrifugal orientations in the yolk-sac which radiate away from the region of aortas (origin of vitelline arteries).

Figure 10.

Region of the hindlimbs showing a rotational tendency of the plexus, prior to vessel formation. VA, vitelline arteries. DA, dorsal arteries. (Right, a schematic representation of the situation, the dashed line shows the capillary lattice orientation). The trajectories intersect the antero-posterior axis at right angles, at regular intervals, where vessels belonging to somite intervals will form. These features are also visible in Le Noble 2005, Figure 2E. Similar effects exist in the forelimb region (data not shown).

This situation has been actually described recently in a hydrodynamic model of gastrulation by one of us.^2,16 The origin of these textures lies in the gastrulation motions. Indeed, during the gastrulation motion, a pair of contra-rotative dipoles² (Supplementray Fig. 3), deforms the blastodisc around a hyperbolic point located in the center of the vortices. As a consequence of this flow, the embryo is stretched in the antero-posterior direction, contracted in the perpendicular direction and forms convergent sets of lines on each side of the flanks, these lines wind around two vortices at the caudal half, and two at the anterior half, with mirror symmetry around the hyperbolic point, which is the location of the presumptive navel.

This describes the shape of the embryo texture on the epiblast. But along the “primitive streak” the epiblast invaginates to create the yolk-sac mesoderm. In this mesoderm, blood vessels of the yolk-sac will form, and they will be influenced by the orientational fields in it. The fundamental difference between the Zona Opaca and the Zona Pellucida is that the Zona Pellucida is formed from epiblast still in its phase of convergence towards the Antero-Posterior axis, this is why it has the shape of a constricted eight,² while the Zona Opaca is formed from the mesoderm which has already ingressed, and forms progressively a circle as it migrates away. From the model presented in,² one can predict the shape of the texture in the yolk-sac, and the fact that the yolk-sac will indeed progressively form a circle.

To model this situation, we need to follow the fate of cells ingressing into the primitive streak. This can be done mathematiclly almost exactly and it gives the pattern given in Figure 11. A technical explanation of the simulation is given in the Materials and methods section III. The flow of mesoderm which invaginates (Fig. 11B right) takes a round shape as it migrates away from the embryo; the shape of the yolk-sac exhibits a peculiar figure 8. This figure 8 is centered around the vitelline arteries, as actually observed. The mosaic inside the yolk-sac is radial, close to the embryo, but more circular away, a fact which corresponds to the structure of the main vessels of the true yolk-sac. If we follow the fate of the initial lines of the epiblast inside the yolk-sac, we find that the initial mosaic is modified. The cells of the mosaic progressively align with the flow field in the yolk-sac.

This mathematical model, formally deterministic, gives the shape of the observed mosaic, and hence of the future vessels, in the yolk-sac: the radial and orthoradial pattern of the morula/blastula takes the shape of rainbows or curled rays forming a “crab-like” pattern inside the yolk-sac.

Therefore, the model predicts correctly the texture in the ectoderm (ZP) and in the ZO, and on the flanks of the embryo, these are just the fate of the blastodisc mosaic, after passing through a deterministic deformation. The shape of the vessels in the ZP (straight lines converging towards the head) simply reflects the shape of the tissue on the epiblast (the not invaginated part of the blastodisc). Vessels can be viewed as “decorating” the deformed mosaic, under the influence of compression and stretch forces.

If one continues this logical line of thought, the vessels passing through somite intervals explore a preexisting mosaic, somewhat randomized by the process of blood island formation, but which is progressively straightened by the compression and stretch forces, and so do the dorsal aortas. The ladder distribution of vessels on the back of the embryo is just a reflection of the stress field. Actually, the “ladder” distribution of vessels of the somites is already visible in the mesoderm prior to appearance of the somites (Fig. 10A, one notices the regular intervals of blood islands between vessels along the presumptive tail). In this view, the ladder vessels around the somites is not a ladder that forms bar by bar, but a complete, although somewhat random ladder which becomes more regular with time under compression and stretch forces.

In more general terms, all main vessels might simply follow the deformed paths of a preexisting mosaic, present at the earliest times of development, and reinforced by mechanical forces. This mosaic is just the mosaic of the morula but deformed by the gastrulation stress fields. This may be how genetics predetermines simply the initial frame of the vascular pattern.¹⁰

An interesting consequence of the model is that the tissue located in the region of the presumptive limbs is in fact rotated, as appears obviously by the shape of the vessels. This is a consequence of the vortex motions of the blastodisc. One may think that this is the origin of the Apical Ectodermal Ridge (AER) at the tips of the limbs.¹⁵ The AER is described in biology as the source of tissue growth for the limbs. It is described as a crescent along which the tissue of the limb grows outwards to form the limb paddle. Eventually, the AER is just the edge of the tip of the limb paddle. If we look at the geometry of the problem, we see that, if a piece of blastodisc is rotated, as is the tissue in the center of the vortices of (Supplemental Figs. 3 and 4) and as seen in Figure 11A, the growth direction on the flanks and on the turned piece of tissue become antagonistic and a bud will protrude by buckling along the edge where the two tissues with opposite orientations collide.

The next, and final, question one can ask is why do vessels follow the texture? From the images above a simple mechanism may be inferred. The progressive centrifugal growth of the entire yolk-sac requires a physical contact between the moving mesoderm cells. As a consequence, if a lattice of capillaries exists, whose minute anastomoses surround clusters of mesoderm, the strands oriented perpendicular to the solid flow of the cells are squeezed such that the centrifugal solid migration is at all possible. By Newton's laws, the squeeze exerted on a microscopic piece of tissue is the stress gradient. Both growth speed and squeeze are larger in the direction of the highest stress gradient. This is because the squeeze is the elastic component of the deformation (it is a deformation), while the growth is the visco-plastic component of the deformation (it is a deformation rate). In the direction perpendicular to the growth direction, the growth rate is smaller, so the stress gradient is lower, and therefore the liquid is expelled from the strands perpendicular to the solid migration towards the strands parallel to the solid migration. Endothelial cells themselves are stretched along these lines.

A magnified view of the mesoderm clusters shows indeed that there is a tighter contact in the direction (centrifugal) corresponding to the yolk-sac expansion (Fig. 7, right), than perpendicularly, which is also what the direct inspection of the capillary plexus shows. Eventually, the vessels dress a path which is a compromise between the existing texture and the path of smallest orthogonal stress gradient (the vessels form in the least squeezed direction). The texture itself progressively alines with the lines of highest stress gradient. This situation may be found in other organs, possibly all of them.

Note

Supplementary information can be found at www.landesbioscience.com/journals/organogenesis/supplement/fleuryORG3-1-supp.pdf.