Toward a quantification of self-similarity in plants

PLOS Computational Biology, 2015

The leaves of angiosperms contain highly complex venation networks consisting of recursively nested, hierarchically organized loops. We describe a new phenotypic trait of reticulate vascular networks based on the topology of the nested loops. This phenotypic trait encodes information orthogonal to widely used geometric phenotypic traits, and thus constitutes a new dimension in the leaf venation phenotypic space. We apply our metric to a database of 186 leaves and leaflets representing 137 species, predominantly from the Burseraceae family, revealing diverse topological network traits even within this single family. We show that topological information significantly improves identification of leaves from fragments by calculating a "leaf venation fingerprint" from topology and geometry. Further, we present a phenomenological model suggesting that the topological traits can be explained by noise effects unique to specimen during development of each leaf which leave their imprint on the final network. This work opens the path to new quantitative identification techniques for leaves which go beyond simple geometric traits such as vein density and is directly applicable to other planar or sub-planar networks such as blood vessels in the brain.

The New phytologist, 2018

Morphometrics, the assignment of quantities to biological shapes, is a powerful tool to address taxonomic, evolutionary, functional and developmental questions. We propose a novel method for shape quantification of complex modular architecture in thalloid plants, whose extremely reduced morphologies, combined with the lack of a formal framework for thallus description, have long rendered taxonomic and evolutionary studies extremely challenging. Using graph theory, thalli are described as hierarchical series of nodes and edges, allowing for accurate, homologous and repeatable measurements of widths, lengths and angles. The computer program MorphoSnake was developed to extract the skeleton and contours of a thallus and automatically acquire, at each level of organization, width, length, angle and sinuosity measurements. Through the quantification of leaf architecture in Hymenophyllum ferns (Polypodiopsida) and a fully worked example of integrative taxonomy in the taxonomically challen...

The geometries and topologies of leaves, flowers, roots, shoots, and their arrangements have fascinated plant biologists and mathematicians alike. As such, plant morphology is inherently mathematical in that it describes plant form and architecture with geometrical and topological techniques. Gaining an understanding of how to modify plant morphology, through molecular biology and breeding, aided by a mathematical perspective, is critical to improving agriculture, and the monitoring of ecosystems Frontiers in Plant Science | www.frontiersin.org 1 June 2017 | Volume 8 | Article 900 Bucksch et al. Plant Morphological Modeling is vital to modeling a future with fewer natural resources. In this white paper, we begin with an overview in quantifying the form of plants and mathematical models of patterning in plants. We then explore the fundamental challenges that remain unanswered concerning plant morphology, from the barriers preventing the prediction of phenotype from genotype to modeling the movement of leaves in air streams. We end with a discussion concerning the education of plant morphology synthesizing biological and mathematical approaches and ways to facilitate research advances through outreach, cross-disciplinary training, and open science. Unleashing the potential of geometric and topological approaches in the plant sciences promises to transform our understanding of both plants and mathematics.

Journal of Theoretical Biology, 2015

A method to compute the similarity between different plants is proposed, using features of a plant's topological structure and peripheral contour, as well as its geometry. The topological structures are described using tree graphs, and their similarity can be calculated based on the edit distance of these graphs. The peripheral contour of a plant is abstracted by its three-dimensional convex hull, which is projected in several directions. The similarity of the different projections is calculated by an algorithm to compute the similarity of two-dimensional shapes. The similarity of the geometrical detail is computed by considering the geometrical properties of different level branches. Finally the overall similarity between different plants is calculated by combining these different similarity measures. The validity of proposed method is evaluated by detailed experiments.

2010

In this paper we are interested in the problem of approximating trees by trees with a particular self-nested structure. Self-nested trees are such that all their subtrees of a given height are isomorphic. We show that these trees present remarkable compression properties, with high compression rates. In order to measure how far a tree is from being a self-nested tree, we then study how to quantify the degree of self-nestedness of any tree. For this, we define a measure of the self-nestedness of a tree by constructing a self-nested tree that minimizes the distance of the original tree to the set of self-nested trees that embed the initial tree. We show that this measure can be computed in polynomial time and depict the corresponding algorithm. The distance to this nearest embedding self-nested tree (NEST) is then used to define compression coefficients that reflect the compressibility of a tree.

The West, Brown, Enquist (WBE) model derives symmetrically self-similar branching to predict metabolic scaling from hydraulic conductance, K, (a metabolism proxy) and tree mass (or volume, V). The original prediction was K / V 0:75 . We ask whether trees differ from WBE symmetry and if it matters for plant function and scaling. We measure tree branching and model how architecture influences K, V, mechanical stability, light interception and metabolic scaling.

Plant morphology is inherently mathematical. The geometries of leaves and flowers and intricate topologies of the root have fascinated plant biologists and mathematicians alike. Beyond providing aesthetic inspiration, understanding plant morphology has become pressing in an era of climate change and a growing population. Gaining an understanding of how to modify plant architecture through molecular biology and breeding is critical to improving agriculture, and the monitoring of ecosystems and global vegetation is vital to modeling a future with fewer natural resources. In this white paper, we begin by summarizing the rich history and state of the art in quantifying the form of plants, mathematical models of patterning in plants, and how plant morphology manifests dynamically across disparate scales of biological organization. We then explore the fundamental challenges that remain unanswered concerning plant morphology, from the barriers preventing the prediction of phenotype from genotype to modeling the fluttering of leaves in a light breeze. We end with a discussion concerning the education of plant morphology synthesizing biological and mathematical approaches and ways to facilitate research advances through outreach, cross-disciplinary training, and open science. Never has the need to model plant morphology been more imperative. Unleashing the potential of geometric and topological approaches in the plant sciences promises to transform our understanding of both plants and mathematics.

Bulletin of Mathematical Biology, 2008

A major determinant of plant architecture is the arrangement of branches around the stem, known as phyllotaxis. However, the specific form of branching conditions is not known. Here we discuss this question and suggest a branching model which seems to be in agreement with biological observations.