Shape is one of the important characteristics for the structures observed in living organisms. Whereas biologists have proposed models where the shape is controlled on a molecular level [1], physicists, following Turing [2] and d'Arcy Thomson [3], have developed theories where patterns arise spontaneously [4]. Here, we propose a volume constraint that restricts the possible shapes of leaves. Focusing on palmate leaves, the central observation is that developing leaves first grow folded inside a bud, limited by the previous and subsequent leaves. We show that growing folded in this small volume controls globally the leaf development. This induces a direct relationship between the way it was folded and the final unfolded shape of the leaf. These dependencies can be approximated as simple geometrical relationships that we confirm on both folded embryonic and unfolded mature leaves. We find that independently of their position in the phylogenetic tree, these relationships work for folded species, but do not work for non-folded species. This steric constraint is a simple way to impose a global regulation for the leaf growth. Such steric regulation should be more general and considered as a new simple means of global regulation.
Deep Dive into Steric Constraints as a Global Regulation of Growing Leaf Shape.
Shape is one of the important characteristics for the structures observed in living organisms. Whereas biologists have proposed models where the shape is controlled on a molecular level [1], physicists, following Turing [2] and d’Arcy Thomson [3], have developed theories where patterns arise spontaneously [4]. Here, we propose a volume constraint that restricts the possible shapes of leaves. Focusing on palmate leaves, the central observation is that developing leaves first grow folded inside a bud, limited by the previous and subsequent leaves. We show that growing folded in this small volume controls globally the leaf development. This induces a direct relationship between the way it was folded and the final unfolded shape of the leaf. These dependencies can be approximated as simple geometrical relationships that we confirm on both folded embryonic and unfolded mature leaves. We find that independently of their position in the phylogenetic tree, these relationships work for folded s
Shape is one of the important characteristics for the structures observed in living organisms. Whereas biologists have proposed models where the shape is controlled on a molecular level [1], physicists, following Turing [2] and d'Arcy Thomson [3], have developed theories where patterns arise spontaneously [4]. Here, we propose a volume constraint that restricts the possible shapes of leaves. Focusing on palmate leaves, the central observation is that developing leaves first grow folded inside a bud, limited by the previous and subsequent leaves. We show that growing folded in this small volume controls globally the leaf development. This induces a direct relationship between the way it was folded and the final unfolded shape of the leaf. These dependencies can be approximated as simple geometrical relationships that we confirm on both folded embryonic and unfolded mature leaves. We find that independently of their position in the phylogenetic tree, these relationships work for folded species, but do not work for non-folded species. This steric constraint is a simple way to impose a global regulation for the leaf growth. Such steric regulation should be more general and considered as a new simple means of global regulation.
Leaves fascinate for their diversity. They can be simple, with lobes (palmate), with leaflets (compound), or dissected with holes. On one single plant, the shape of leaves vary, sometimes strongly (heterophilly). Despite this diversity, some common features are intuitively guessed. Until now, botanists have proposed two mechanisms to explain leaf forms. The first mechanism is based on localised enhancements and reductions of growth of the free margin of the embryonic leaf [5][6], which create the peaks and the valleys of the leaf border [7][8]. The second mechanism is the death of patches of cells (programmed cell death, PCD) that forms perforations in the leaf during the lamina development. When perforations are positioned near the leaf contour, the marginal tissue eventually breaks, as in Philodendron Monstrosa (Araceae, monocotyledon), resulting in a deeply dissected blade (pinnatisect) [9]. A particular case has been described for the dissected shape of palm leaves (Arecaceae, monocotyledons). The leaf first develops with many folds, where PCD eventually takes place, creating cuts [10].
These two mechanisms are general so that they can be tuned to reproduce the final shape of any leaf. They conceptually apply for a flat leaf during its expansion and do not take into account the actual geometry of the growing leaf inside the bud. Only the last case takes into account this geometry, through the folds. Folds have been recently highlighted for their mechanical importance in thin sheets [11][12] and are ascribed to play a role in the expansion of hornbeam leaves [13]. Focusing on palmate leaves, we found that most of them are first growing folded inside a bud.
Leaves are highly organised botanical elements. They originate from small groups of cells (primordia) protruding around the shoot apex (fig. 1a). From the beginning they present a fundamental asymmetry: the side turned toward the stem axis (adaxial) will become the smooth and shiny upper side of the leaf turned toward the light; the other side, turned toward outside (abaxial), present hairs and veins protruding and will become the lower side of the leaf.
Once the leaf has grown, it can be noticed that the final vein pattern is organised, hierarchical, and related to the leaf shape. In palmate leaves, each lobe corresponds to a major vein ending at its tip. Lobes and veins have the same hierarchy: beside the central lobe/vein stand symmetrically lateral lobes/veins that start from the same petiole origin. From these primary veins can originate secondary lobes/veins, and similarly for a rare third order (see figure 3).
In buds, leaves grow in a limited space defined by previous and following leaves. To keep on expanding its future lamina within this confined space, leaves either enroll (convolute, revolute or involute), or fold (plicate and palmate leaves) [14][15].
Our first remark is that in palmate leaves, folds are not irregular but strictly follow the leaf organisation (fig. 1f-g). Following the leaf asymmetry, the folds showing the abaxial epidermis outside (anticlinal folds) are very different than the ones showing the adaxial epidermis outside (synclinal folds). The anticlinal folds coincide with the main veins and follow their hierarchy (fig. 1f). This correlation between the veins and anticlinal folds is strictly inclusive: many veins do not correspond to any fold but an anticlinal fold always corresponds to a main vein. On the contrary, synclinal folds do not correspond to any major vein and are rather crossed only by the smallest ones (fig. 1g). Thus we call them “anti-veins”.
Our second and main observation is that the whole perimeter of folded leaves growing inside a bud is located at a particular place. For Palmat
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