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Conceived and designed the experiments: EJ. Performed the experiments: ML. Analyzed the data: ML SB EJ. Contributed reagents/materials/analysis tools: SB. Wrote the paper: ML SB EJ.

The authors have declared that no competing interests exist.

We explore the possible role of elastic mismatch between epidermis and mesophyll as a driving force for the development of leaf venation. The current prevalent ‘canalization’ hypothesis for the formation of veins claims that the transport of the hormone auxin out of the leaves triggers cell differentiation to form veins. Although there is evidence that auxin plays a fundamental role in vein formation, the simple canalization mechanism may not be enough to explain some features observed in the vascular system of leaves, in particular, the abundance of vein loops. We present a model based on the existence of mechanical instabilities that leads very naturally to hierarchical patterns with a large number of closed loops. When applied to the structure of high-order veins, the numerical results show the same qualitative features as actual venation patterns and, furthermore, have the same statistical properties. We argue that the agreement between actual and simulated patterns provides strong evidence for the role of mechanical effects on venation development.

Leaf venation patterns of most angiosperm plants are hierarchical structures that develop during leaf growth. A remarkable characteristic of these structures is the abundance of closed loops: the venation array divides the leaf surface into disconnected polygonal sectors. The initial vein generations are repetitive within the same species, while high-order vein generations are much more diverse but still show preserved statistical properties. The accepted view of vein formation is the auxin canalization hypothesis: a high flow of the hormone auxin triggers cell differentiation to form veins. Although the role of auxin in vein formation is well established, some issues are difficult to explain within this model, in particular, the abundance of loops of high-order veins. In this work, we explore the previously proposed idea that elastic stresses may play an important role in the development of venation patterns. This appealing hypothesis naturally explains the existence of hierarchical structures with abundant closed loops. To test whether it can sustain a quantitative comparison with actual venation patterns, we have developed and implemented a numerical model and statistically compare actual and simulated patterns. The overall similarity we found indicates that elastic stresses should be included in a complete description of leaf venation development.

For many years leaf venation motifs have marveled people, whether scientists or not. Venation patterns are different from one leaf to another, even in the same plant, but share some common features that are preserved throughout all angiosperm leaves

This leaf was subjected to a chemical treatment to remove all the soft tissues, leaving only the veins. The network-like structure as well as many open ends of the thinnest segments can be observed.

It has been argued that the vein architecture might ensure optimal water distribution

From a developmental perspective, the leaf venation is puzzling, too. Since the pioneering works of Sachs

These findings have led to models of venation formation based on a positive canalization feedback

An alternative model has been recently introduced by Feugier and Iwasaa

In general, we find that the modifications to the canalization hypothesis necessary to explain the existence of closed loops are not generic and rather unnatural, and the mechanism on which they are based require a lot of fine tuning.

Couder et al.

Evidence supporting this hypothesis is two-fold. On the one hand, micrographs taken in the early steps of leaf venation development show that in the first stages of differentiation, cells forming the procambium can be distinguished from the remaining cells by a mechanical distortion, consisting in a shrinkage of the cells perpendicular to the vein direction (see, for example, the images of Figure 2 of

Crack patterns on the surface of mud or other materials require the existence of two quasi-two dimensional layers of material, the substrate and the covering, the latter contracting with respect to the former upon desiccation. (A pioneering work by Skjeltorp and Meakin

The suggestion of Couder et al. on the importance of elastic factors in vein formation

In actual leaves, there is an obvious dependency between the morphology of veins and its rank in the venation structure. In other words, initial vein generations are strongly dependent on the form of the leaf and most probably, on genetic factors. It is this large-scale pattern that is repetitive within the same species and allows a broad leaf classification according to their venation patterns. It is also in these initial vein generations where the role of auxin is relatively well established. High order vein generations are much more isotropic, and much more universal in its statistical properties. It is to this stage that we intend to apply our model in its present form to compare statistical properties.

A comprehensive mathematical description of our model is given in the last section, but here we summarize the main hypotheses to ease the reading of this part. We assume that during growth, the inner cell layer (the mesophyll) is elastically attached to the epidermis. The epidermis is assumed to grow at a lower rate than mesophyll, and is otherwise supposed to be inert, i.e., it undergoes no deformations during growth. Due to the different growth rates of mesophyll and epidermis, compressive stresses develop in the mesophyll. Our main assumption is that the elastic properties of the mesophyll are such that this compressive stress can give rise to a shape change of the mesophyll cells. Such cells will acquire an elongated shape perpendicular to the main applied stress. These assumptions are basically equivalent to the description of collapsing surface layers presented in

To avoid an extremely uniform initial condition, we typically seed the simulation with a few large-scale veins that provide the initial veins of our numerical leaf. This first division is not significant in the statistical analysis we perform on the final patterns. We show results in which we prepare the system with tree-like thick initial veins, or divide the sample into two pieces.

When new veins are formed (upon increasing of η), they typically propagate rapidly through the system, reaching in most (but not all) cases an older vein, where they stop. This propagation, once triggered, occurs essentially at constant η, i.e., it is not driven by the growing itself.

A few snapshots during the numerical evolution are shown in

The values of the growing parameter, from top left to bottom right, are η = 1.2, 2.4, 3.6, and 4.8. The seed we use as the initial condition is shown in the first panel with a different color. The numerical lattice has 1024×1024 nodes.

The values of η and the system size are the same as in the previous figure. In both figures the hierarchical process can be clearly observed. Note also the open ends of some of the thinnest segments.

Before going to the quantitative characterization of the patterns obtained, two important features are worth noting. One is that in many cases several thin free-ended veins are observed. This also occurs in actual leaves and we propose an explanation in the next section. Another feature is that some minor veins are completely disconnected from other veins. They typically appear at the center of intact regions (where the stress is maximum), and seem unrealistic, since vein patterns in leaves are almost always connected. Although they might be due to an artifact in our simulations (in fact, the thickness of these disconnected veins is already comparable to our numerical discreteness), recall that our patterns are actually showing the places where the tension is high enough to generate collapsed cells that will eventually, but not necessarily, differentiate into veins. If the later differentiation process requires the canalization of a flux through the network of collapsed cells, differentiation of the disconnected segments into disconnected veins will not occur.

In order to test whether our simulation results are comparable with actual leaf patterns, we computed the vein width, length and angles from our simulation results, and compare them with data from actual leaves. The same numerical image processing technique was used for the two data sets; see a detailed explanation in

In

(A) Actual leaves. Each curve is the histogram of a given dycotiledon leaf:

Moving to the description of the results of

(A) Actual leaves. Histograms for the same three leaves showed in the previous figure. For all the leaves analyzed, a power decay with an exponent close to 3 is observed. Inset: Average over four leaves. A shoulder for thick veins can be observed in both figures. (B) Numerical leaves. Histograms for three different realizations. In the region of intermediate values of thickness, a power decay with an exponent close to 2 is obtained. Inset: Average for the same realizations as in the previous figure, showing a shoulder for the region of thick veins.

Each curve corresponds to one of the four stages of growth shown in

For intermediate values of thickness, the results of our model are compatible with a power decay of ^{−1}, justifying a more rapid decay for

Finally, we analyze the behavior of the angles between vein segments at the points where three vein segments meet. As pointed out in _{LS} is the angle between the thickest and the thinnest segments, α_{LI} is the angle between thick and intermediate segments, and α_{IS} is the angle between intermediate and thin segments. We calculated the averages of the three angles and plot them as a function of the ratio between the radius of the thinnest (R_{S}) and thickest (R_{L}) segments. The configuration of radii is well defined with the parameter R_{S}/R_{L} because the segment of intermediate radius has usually a value close to R_{L}. In _{S}/R_{L} close to one, all radii are almost equal and the three angles are near to 120 degrees. This describes a situation in which a vein has bifurcated into two. Since the three segments are then created almost simultaneously, the three radii are similar. On the other hand, R_{S}/R_{L} near to zero correspond to the case in which a thin vein reaches a thick one. In this case, the angle α_{LI} between thick and intermediate segments tends to be 180 degrees, meaning that the thick vein is almost unperturbed by the thin one. A continuous and rather linear variation is observed between these two extreme situations. Although the overall coincidence of measured angles in our simulations and in actual leaves is encouraging, a full understanding of the origin of a general relation between angles and radii is not achieved yet.

Angles between veins as a function of the ratio between the radius of the thinnest (R_{S}) and thickest (R_{L}) segments. The angle between thin and intermediate radius is labeled α_{IS}. The angle between thin and thick segments is α_{LS}, whereas the angle between thick and intermediate segments is α_{LI}. Isolated symbols are data obtained from actual leaves, and were taken from Figure 14 of

In our model, the free energy of a vein can be conceived as a interface energy between the two sectors into which the vein divides the leaf. In the case that all veins are of the same width, the minimization of this interface energy would give rise to a foam-like pattern with 120 degrees angles. However, irreversibility gives rise to the formation of veins of different thickness and free energy minimization produces angles whose values are correlated with the veins' age.

The ‘force model’ proposed in

In this paper we have set up a model to study leaf venation, which is based on the idea that venation patterns are strongly influenced by mechanical instabilities of the leaf, when the cellular layers of epidermis and mesophyll grow at different rates. We took a model that had been successfully applied to study phase separation process in alloys, added the interaction with a substrate, and made also the appropriate changes necessary to study the crucial effect of leaf growth. We claim that the properties of biological growth added to the characteristics of the model, explains the formation of a hierarchical structure with well defined statistical properties for different quantities. The results of the statistical analysis are in good agreement with results obtained in actual leaves. Our model explains the existence of abundant closed loops in venation patterns in a natural way. Moreover, some statistical features can be understood analyzing a very simple model of hierarchical division (see

A complete and realistic modeling also requires taking into account non-uniform and anisotropic growth, and probably genetic factors

However, it must be stressed that the existence of a instability is an assumption of our modeling, as we do not yet have a confirmation of its existence from a biological point of view. An in situ investigation of this collapse transition along the lines of the experiment made in

Our main assumption is that vein formation is triggered by the elastic collapse of cells of the mesophyll, growing at a larger rate than the (assumed rigid) epidermis to which they are attached. An appropriate approach would be to describe the mesophyll as an elastic layer with a highly non-linear behavior modeling an irreversible local collapse.

The natural way to theoretically describe the behavior of an elastic layer is by constructing a free energy in terms of the elastic displacement field,

(A) Mechanical analogy. Elastic stresses are accounted for by the springs indicated. Horizontal springs represent the cells of the mesophyll, and its deviation from its equilibrium length is a measure of the deformation energy of the cell. Vertical interlayer springs account for the interaction between mesophyll and epidermis. We suppose that the epidermis grows at a lower rate than the mesophyll, and thus the mismatch between layers will increase with time. A collapsed cell in this schema is represented by a horizontal spring suffering a stress higher than its elastic limit. Once this threshold is reached, the spring has a permanent deformation. (B) Representation of the mesophyll layer with a group of cells in the collapsed state. Note that the initial three-dimensional problem was reduced to two dimensions, as we only describe the intermediate plane where horizontal springs lie.

A free energy in terms of the elastic displacement field _{0} is a Ginzburg-Landau local free energy for Φ that has two different minima, representing the intact and collapsed states:^{2} is included to obtain smooth profiles of the fields by penalizing rapid spatial variations of Φ. It is introduced to make the behavior of the system almost isotropic and independent of the underlying numerical lattice. This term is also useful because allows the simulation of a continuous growth through the rescaling of the parameters, as will be explained later.

The parameter α is a measure of the coupling between the fields Φ and

The term _{el} is the usual elastic free energy density in the reference state in which Φ = 0, expressed in terms of the bulk and shear moduli, _{0}_{±} = ±(r_{0}/s_{0})^{1/2}. When these values are introduced in Equations 1 and 2 they define two different elastic states with different density and shear modulus, representing the intact and collapsed states of the cells in our model. The fact that the variable Φ is continuous, however, guarantees the possibility of a smooth transition between these states.

The only difference between these expressions and those in the works

A formal transformation in the model should be made before implementation in the computer. If in the free energy of Equation 1 we were able to integrate out the field Φ, we should end up with a non-linear elastic model written completely in terms of the displacement field _{1}, and an effective model in terms of Φ is obtained. The new model is non-linear and non-local in Φ, describing in an effective way the non-linear elastic behavior of the system. The free energy takes the form:_{ij} = ∂_{i}∂_{j}−(δ_{ij}/2) ∇^{2}, g_{E} = μ_{1} α^{2}/L_{0}^{2}, g_{L} = γ/L_{0}, L_{0} = _{0} in 2D, A_{ij} = 〈∇_{j} u_{i}〉, and

The main external condition that drives the evolution of the system is the fact that the leaf is growing. The natural way to model the growth (which mimics most closely the real situation) is to assume that, although the parameters of the model do not change upon growing, the linear dimension of the system ^{2} and γη^{2}. This means that changing

Note the scaling effect in the simulations: Decreasing

Our modeling is compatible with the hypothesis that when a new vein has been nucleated in an actual leaf, it will continue to grow at the same pace than the rest of the leaf. In particular its thickness should increase with time. In our modeling, due to our zooming out procedure this means that veins must preserve its width during the evolution and newer veins are progressively thinner than older ones. In order to achieve this, we have to avoid that the older (thicker) veins become thinner as the spatial scale in the system is changed. As we said, this implies a kind of irreversibility condition that guarantees that when a new vein was created, it is committed to grow at a fixed rate. The implementation of the irreversibility condition in the model is as follows. We include the condition that Φ (x,y) in the time step _{0}, namely, if at a certain stage of the simulation some point has a value Φ (x,y)>Φ_{0}, then this point is forced to remain with a value of Φ at least as large as Φ_{0}. Our numerical results indicate that the final patterns are reasonably independent on the value of the threshold we use to define each phase. Irreversibility is what stabilizes the existence of thick veins, as can be observed in _{0} = 2. Note in the bottom panel how the interface sharpness is greater (because of the increase in the effective

Values of Φ vs.

It is worth emphasizing the effect that the term that was used to generate irreversibility has on the simulations. In the absence of this term, the same parameters which lead to the snapshots of

Result of a simulation with the same parameters as in

We also include in our model a stochastic noise of small amplitude that helps to nucleate new veins. The evolution equation becomes dΦ/dt = −δ^{T}^{T}_{i}^{T}_{i}^{T}_{j}^{T}_{B} T_{ij}. The existence of random noisy effects on the growing of an actual leaf cannot be denied, and then our inclusion of a stochastic term in the evolution equation could be ultimately justified. However, we emphasize that we do not intend to model any precise physical process with this. We only want to include in a simple form the fact that there is some randomness in the nucleation events, which eventually make individual leaves of the same species to differ from one another. In order to be sure that the stochastic term does not introduce systematic spurious effects, we have explored the effect of the noise by applying it in three different ways: 1) a ‘static version’ in which the noisy term is included only in the initial condition, 2) a dynamic noise as described in the previous paragraph, and 3) an intermediate version, in which a fixed noisy landscape affect the leaf during its evolution. We found that the main characteristics of our patterns as well as its statistical properties are the same in the three cases. Then we present results only for the noisy dynamics, which in addition we consider to be the most realistic one, as fluctuations at the cellular level produced by discrete cellular division events can be considered as some sort of noise during the growing process.

A minimal model with scale invariance properties. We present here a toy model that has the minimal hierarchical properties we expect to obtain in the full simulation. It may be useful to better appreciate the results of the full modeling.

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Early discussions with M. Magnasco are greatly acknowledged.