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A Glimpse of Developing Leaf Venation

This Demonstration gives a glimpse of the development process of leaf venation. Here, the venation is regarded as a transport network and all possible veins are built in advance. Our belief is simple: veins with higher usage are more important so they should grow; on the contrary, veins with lower usage are less important so they should shrink. Visually speaking, the thickness and transparency of individual veins will change during the simulation.

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Background
Being a quotidian tiny part of nature yet one of the most efficient organs ever developed, the plant leaf has been investigated for centuries, but not many of its secrets have been revealed. A leaf is sensitive to environmental changes and tends to maximize its internal efficiency in any bearable environment by developing an efficient transport network for water and nutrients. Setting aside species and the phenotypic plasticity of a leaf as a response to abiotic stress, the remaining factors of greatest concern that contribute to a leaf's shape normally involve water and light availability, and even microgravity [1]. Mathematically, leaf morphology has been quantified in various contexts, for example, taxonomy [2], differential growth and allometry [3, 4], leaf area estimation [4, 5], and the branching and development of veins [6] as summarized in [7], which re-examines the general applicability, significance, and usefulness of a previously proposed equation [8] relating the lengths of the secondary veins to the distances between the leaf tip and their insertions on the midrib [9], which tries to develop an efficient transport network of veins, offers another approach for the development of veins.
Incidentally, as a new approach for solving combinatorial optimization problems, it has been found that myxomycete plasmodia can find and flow to numerous spots of food with a maximum benefit/cost ratio [10, 11]. The authors adopted a fluid dynamics model to partly simulate the growth and shrinking of the myxomycete tubes observed in experiment [11]. This discovery is hoped to be of value for the topology control problem for self-organizing networks.
Algorithm
Essentially, leaf venation development is also a combinatorial optimization problem. But here, as a mere glimpse, I reduce the complexity so that it will not be combinatorial. My algorithm is mainly inspired by the idea of growth and shrinking introduced in [11], where growth refers to higher transparency and bigger line thickness, while shrinking refers to lower transparency and smaller line thickness. The algorithm's core is simple: all possible veins are built in advance, but their transparencies and thicknesses will change individually during the simulation, so that veins with higher usage will grow and veins with lower usage will shrink.
Step 1. Generate a leaf area and build a mesh to represent it.
The "superformula" introduced in [12] can generate a vast diversity of natural shapes, including leaf shapes:
,
where , , , , , and are parameters. In this Demonstration, a simpler formula is applied, however:
, , ,
where and are adjustable parameters.
Step 2. Name the node at the right-bottom corner of the leaf-mesh as the "root node" (RN), and name the segment between two directly linked nodes as the "path segment" (PS). Then, during each loop, find the shortest paths between the RN and all other nodes in the mesh; meanwhile, record the usage of each PS. Calculate the transparency and line thickness of each PS according to its usage, then use the results to render all the veins.
Designate as the set of all usages recorded, then define , . Corresponding to any , assign a normalized factor ,
If and , will take the form . Obviously, . Therefore, we can use to further define the transparency and thickness of the corresponding PS (vein).
Possible improvements
As mentioned above, this algorithm is not yet combinatorially optimal, because only those shortest paths connecting the RN and other nodes are considered; on the other hand in the transport system of real leaves, part of the nodes in the midrib do serve as secondary root nodes (SRN) and out of which new veins (secondary midribs) will grow. This leads to two new questions: (1) which nodes in the midrib should be considered as SRN? and (2) only some local nodes in the vicinity of a certain SRN should connect to it, but how to quantize the concept of "vicinity"? Further investigation is required to answer these questions.
Furthermore, transport performance alone will not suffice for building a real leaf's vein network, because the cost for constructing such a network and the network's resilience (fault tolerance) are also crucial to the adaptability of real leaves. As a negative example, this algorithm sometimes may build a leaf with two midribs very close to each other, which results in too much cost.
References
[1] G. W. Stutte, O. Monje, R. D. Hatfield, A–L. Paul, R. J. Ferl, and C. G. Simone, "Microgravity Effects on Leaf Morphology, Cell Structure, Carbon Metabolism and mRNA Expression of Dwarf Wheat," Planta, 224(5), 2006 pp. 1038–1049.
[2] R. J. Jensen, K. M. Ciofani, and L. C. Miramontes, "Lines, Outlines, and Landmarks: Morphometric Analyses of Leaves of Acer Rubrum, Acer Saccharinum (Aceraceae) and Their Hybrid," Taxon, 51, 2002 pp. 475–492.
[3] R. Melville, "Growth and Plant Systematics," Proceedings of the Linnean Society of London, 164(2), 1953 pp. 173–181.
[4] T. Verwijst and D–Z. Wen, "Leaf Allometry of Salix Viminalis during the First Growing Season," Tree Physiology, 16(7), 1996 pp. 655–660.
[5] A. Carbonneau, "Principes et méthodes de mesure de la surface foliaire. Essai de caractérisation des types de feuilles dans le genre Vitis," Annales de l'Amélioration des Plantes, 26, 1976 pp. 327–343.
[6] B. F. Slade, "Leaf Development in Relation to Venation, as Shown in Cercis Siliquastrum L., Prunus Serrulata Lindl. and Acer Pseudoplatanus L.," New Phytologist, 56, 1957 pp. 281–300.
[7] R. F. Burton, "The Mathematical Treatment of Leaf Venation: The Variation in Secondary Vein Length along the Midrib," Annals of Botany, 93(2), 2004 pp. 149–156.
[8] R. F. Burton, "The Mathematical Relationship between the Lengths of Lateral Nerves and Their Positions on a Leaf," New Phytologist, 59, 1960 pp. 48–52.
[9] Q. Xia, "The Formation of a Tree Leaf," ESAIM: COCV, 13, 2007 pp. 359–377.
[10] T. Nakagaki, H. Yamada, Á. Tóth, "Intelligence: Maze Solving by an Amoeboidorganism," Nature, 407, 2000 p. 470.
[11] A. Tero, et al., "Rules for Biologically Inspired Adaptive Network Design," Science, 327, 2010 pp. 439–442.
[12] J. Gielis, "A Generic Geometric Transformation That Unifies a Wide Range of Natural and Abstract Shapes," American Journal of Botany, 90, 2003 pp. 333–338.
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