Plant morphology is more than an attribute affecting plant organization, it is also dynamic. Developmentally, morphology reveals itself over the lifetime of a plant through varying rates of cell division, cell expansion, and anisotropic growth . Response to changes in environmental conditions further modulate the above mentioned parameters. Development is genetically programmed and driven by biochemical processes that are responsible for physical forces that change the observed patterning and growth of organs . In addition, external physical forces affect plant development, such as heterogeneous soil densities altering root growth or flows of air, water, or gravity modulating the bending of branches and leaves . Inherited modifications of development over generations results in the evolution of plant morphology . Development and evolution set the constraints for how the morphology of a plant arises, regardless of whether in a systematic, ecological, physiological, or genetic context . In 1790, Johann Wolfgang von Goethe pioneered a perspective that transformed the way mathematicians think about plant morphology: the idea that the essence of plant morphology is an underlying repetitive process of transformation .
The modern challenge that Goethe’s paradigm presents is to quantitatively describe transformations resulting from differences in the underlying genetic, developmental, dutch bucket hydroponic and environmental cues. From a mathematical perspective, the challenge is how to define shape descriptors to compare plant morphology with topological and geometrical techniques and how to integrate these shape descriptors into simulations of plant development.Several areas of mathematics can be used to extract quantitative measures of plant shape and morphology. One intuitive representation of the plant form relies on the use of skeletal descriptors that reduce the branching morphology of plants to a set of intersecting lines or curve segments, constituting a mathematical graph. These skeleton-based mathematical graphs can be derived from manual measurement or imaging data . Such skeletal descriptions can be used to derive quantitative measurements of lengths, diameters, and angles in tree crowns and roots, at a single time point or over time to capture growth dynamics . Having a skeletal description in place allows the definition of orders, in a biological and mathematical sense, to enable morphological analysis from a topological perspective .
Topological analyses can be used to compare shape characteristics independently of events that transform plant shape geometrically, providing a framework by which plant morphology can be modeled. The relationships between orders, such as degree of self similarity or self-nestedness are used to quantitatively summarize patterns of plant morphology. Persistent homology , an extension of Morse theory , transforms a given plant shape gradually to define self-similarity and morphological properties on the basis of topological event statistics. In the example in Figure 2B, topological events are represented by the geodesic distance at which branches are “born” and “die” along the length of the structure. In the 1980s, David Kendall defined an elegant statistical framework to compare shapes . His idea was to compare the outline of shapes in a transformation-invariant fashion. This concept infused rapidly as morphometrics into biology and is increasingly carried out using machine vision techniques . Kendall’s idea inspired the development of methods such as elliptical Fourier descriptors and new trends employing the Laplace Beltrami operator , both relying on the spectral decompositions of shapes . Beyond the organ level, such morphometric descriptors were used to analyze cellular expansion rates of rapidly deforming primordia into mature organ morphologies . From a geometric perspective, developmental processes construct surfaces in a three-dimensional space. Yet, the embedding of developing plant morphologies into a three dimensional space imposes constraints on plant forms. Awareness of such constraints has led to new interpretations of plant morphology that might provide avenues to explain symmetry and asymmetry in plant organs or the occurrence of plasticity as a morphological response to environmental changes .
Computer simulations use principles from graph theory, such as graph rewriting, to model plant morphology over developmental time by successively augmenting a graph with vertices and edges as plant development unfolds. These rules unravel the differences between observed plant morphologies across plant species and are capable of modeling fractal descriptions that reflect the repetitive and modular appearance of branching structures . Recent developments in functional-structural modeling abstract the genetic mechanisms driving the developmental program of tree crown morphology into a computational framework . Similarly, functional-structural modeling techniques are utilized in root biology to simulate the efficiency of nutrient and water uptake following developmental programs . Alan Turing, a pioneering figure in 20th-century science, had a longstanding interest in phyllotactic patterns. Turing’s approach to the problem was twofold: first, a detailed geometrical analysis of the patterns , and second, an application of his theory of morphogenesis through local activation and long range inhibition , which defined the first reaction diffusion system for morphological modeling. Combining physical experiments with computer simulations, Douady and Coudert subsequently modeled a diffusible chemical signal produced by a developing primordium that would inhibit the initiation of nearby primordia, successfully recapitulating known phyllotactic patterns in the shoot apical meristem , the number of floral organs , the regular spacing of root hairs , and the establishment of specific vascular patterns .A true synthesis of plant morphology, which comprehensively models observed biological phenomena and incorporates a mathematical perspective, remains elusive. In this section, we highlight current focuses in the study of plant morphology, including the technical limits of acquiring morphological data, phenotype prediction, responses of plants to the environment, models across biological scales, and the integration of complex phenomena, such as fluid dynamics, into plant morphological models. There are several technological limits to acquiring plant morphological data that must be overcome to move this field forward. One such limitation is the acquisition of quantitative plant images. Many acquisition systems do not provide morphological data with measurable units. Approaches that rely on the reflection of waves from the plant surface can provide quantitative measurements for morphological analyses. Time of flight scanners, dutch buckets system such as terrestrial laser scanning, overcome unitless measurement systems by recording the round-trip time of hundreds of thousands of laser beams sent at different angles from the scanner to the first plant surface within the line of sight . Leveraging the speed of light allows calculation of the distance between a point on the plant surface and the laser scanner. Laser scanning and the complementary, yet unitless, approach of stereovision both produce surface samples or point clouds as output. However, both approaches face algorithmic challenges encountered when plant parts occlude each other, since both rely on the reflection of waves from the plant surface . Radar provides another non-invasive technique to study individual tree and forest structures over wide areas. Radar pulses can either penetrate or reflect from foliage, depending on the selected wavelength . Although more compact and agile systems are being developed for precision forestry above- and below ground , their resolution is too low to acquire the detail in morphology needed to apply hierarchy or similarity oriented mathematical analysis strategies. Image acquisition that resolves occlusions by penetrating plant tissue is possible with X-ray and magnetic resonance imaging . While both technologies resolve occlusions and can even penetrate soil, their limitation is the requirement of a closed imaging volume.
Thus, although useful for a wide array of purposes, MRI and X-ray are potentially destructive if applied to mature plant organs such as roots in the field or tree crowns that are larger than the imaging volume . Interior plant anatomy can be imaged destructively using confocal microscopy and laser ablation or nano- or micro-CT tomography techniques, that are limited to small pot volumes, to investigate the first days of plant growth. One of the outstanding challenges in plant biology is to link the inheritance and activity of genes with observed phenotypes. This is particularly challenging for the study of plant morphology, as both the genetic landscape and morphospaces are complex: modeling each of these phenomena alone is difficult, let alone trying to model morphology as a result of genetic phenomena . Although classic examples exist in which plant morphology is radically altered by the effects of a few genes , many morphological traits have a polygenic basis . Quantitative trait locus analyses can identify the polygenic basis for morphological traits that span scales from the cellular to the whole organ level. At the cellular level, root cortex cell number , the cellular basis of carpel size , and epidermal cell area and number have been analyzed. The genetic basis of cellular morphology ultimately affects organ morphology, and quantitative genetic bases for fruit shape , root morphology , shoot apical meristem shape , leaf shape , and tree branching have been described. Natural variation in cell, tissue, or organ morphology ultimately impacts plant physiology, and vice versa. For example, formation of root cortical aerenchyma was linked to better plant growth under conditions of suboptimal availability of water and nutrients , possibly because aerenchyma reduces the metabolic costs of soil exploration. Maize genotypes with greater root cortical cell size or reduced root cortical cell file number reach greater depths to increase water capture under drought conditions, possibly because those cellular traits reduce metabolic costs of root growth and maintenance . The control of root angle that results in greater water capture in rice as water tables recede was linked to the control of auxin distribution . Similarly, in shoots, natural variation can be exploited to find genetic loci that control shoot morphology, e.g., leaf erectness . High-throughput phenotyping techniques are increasingly used to reveal the genetic basis of natural variation . In doing so, phenotyping techniques complement classic approaches of reverse genetics and often lead to novel insights, even in a well-studied species like Arabidopsis thaliana. Phenotyping techniques have revealed a genetic basis for dynamic processes such as root growth and traits that determine plant height . Similarly, high-resolution sampling of root gravitropism has led to an unprecedented understanding of the dynamics of the genetic basis of plasticity .Phenotypic plasticity is defined as the ability of one genotype to produce different phenotypes based on environmental differences and adds to the phenotypic complexity created by genetics and development. Trait variation in response to the environment has been analyzed classically using ‘reaction norms,’ where the phenotypic value of a certain trait is plotted for two different environments . If the trait is not plastic, the slope of the line connecting the points will be zero; if the reaction norm varies across the environment the trait is plastic and the slope of the reaction norm line will be a measure of the plasticity. As most of the responses of plants to their environment are nonlinear, more insight into phenotypic plasticity can be obtained by analyzing dose-response curves or dose-response surfaces . Seminal work by Clausen et al. demonstrated using several clonal species in a series of reciprocal transplants that, although heredity exerts the most measureable effects on plant morphology, environment is also a major source of phenotypic variability. Research continues to explore the range of phenotypic variation expressed by a given genotype in the context of different environments, which has important implications for many fields, including conservation, evolution, and agriculture . Many studies examine phenotypes across latitudinal or altitudinal gradients, or other environmental clines, to characterize the range of possible variation and its relationship to the process of local adaptation . Below-ground, plants encounter diverse sources of environmental variability, including water availability, soil chemistry, and physical properties like soil hardness and movement. These factors vary between individual plants and within an individual root system, where plants respond at spatio-temporal levels to very different granularity . Plasticity at a microenvironmental scale has been linked to developmental and molecular mechanisms . The scientific challenge here is to integrate these effects at a whole root system level and use different scales of information to understand the optimal acquisition in resource limited conditions . Since it is extremely difficult to examine complex interdependent processes occurring at multiple spatio-temporal scales, mathematical modeling can be used as a complementary tool with which to disentangle component processes and investigate how their coupling may lead to emergent patterns at a systems level .