The Role of Elastic Stresses on Leaf Venation Morphogenesis

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.


A minimal model with scale invariance properties
In view of the rather complicated technical characteristics of the full model we present in this paper, it may be appropriate to present here a toy model that has the minimal hierarchical properties we expect to obtain in the full simulation. As a price for its simplicity, this toy model produces patterns that are unrealistic in its global appearance. However, clarifying the hierarchical properties of this toy model may be important to better appreciate the results of the full modeling.
The toy model is defined as follows. Let us consider a rectangular surface.
In the beginning, it is assumed that this surface has a very small area and represents the germ of the leaf that will grow. When its (linear) size reaches the critical value l c , a new vein of unitary width appears, dividing the original surface in two. The system continues to grow isotropically, and every time a sector free of veins reaches a (horizontal or vertical) length l c , a new vein is nucleated, dividing this sector in two. We consider that the new vein does not appear necessarily in the middle of the sector that is divided, but in an arbitrary position, with some probability distribution (most probably in the middle, and less toward the borders). This eliminates the existence of four veins junctions, which are rarely observed in real leaves. A few steps of this process are illustrated in Fig. 1. In this figure, all stages have been plotted as of the same size, i.e., we use the same 'zooming out' procedure as in the full This very simple model admits and equally simple calculation of the statistics of segments lengths and widths. In fact, first of all, it is easy to see that the typical length L(w) of a segments of width w is independent of w, as segments get divided by thinner ones, independently of its width. This is true of course if the model is iterated infinitely. Otherwise we should get a cutoff at low w, with L(w) going to zero for w going to zero.
Another interesting result is the scaling law of the number of segments N (w) of a given width w. The total length of segments of width w is roughly 1/w, as they appeared to divide a pattern with typical size ∼ l c in pieces of smaller size. Since at the end the mean length of segments is independent of w, the number of segments of width w is ∼ w −1 . However, in this estimation the implicit assumption is made that sectors are progressively divided in halves.
If we want to go to a continuous description, this has to be taken into account, The present simple model and its expected statistical behavior is a good benchmark to validate the numerical algorithms for segment location and counting we use in the full simulation. To do this we have run different configurations of the model and made the counting of segments length and width using the full machinery that has been explained in detail in Ref. (1). The results can be seen in Fig. 2. We confirm that in this toy model the mean length of segments is independent of its width, and the number of segments with a given width is 3 N (w) ∼ w −2 . These results are consequence of the hierarchical way in which the patterns is constructed, and form the basis on which the results of the full simulations can be analyzed.