The filling law: A general framework for leaf folding and its consequences on leaf shape diversity

Geometric entropy of plant leaves: A measure of morphological complexity

Shape is an objective characteristic of an object. A boundary separates a physical object from its surroundings. It defines the shape and regulates energy flux into and from an object. Visual perception of a definite shape (geometry) of physical objects is an abstraction. While the perceived geometry at an object's sharp interface (macro) creates a Euclidian illusion of actual shape, the notion of diffuse interfaces (micro) allows an understanding of the realistic form of objects. Here, we formulate a dimensionless geometric entropy of plant leaves (SL) by a 2-D description of a phase-field function. We applied this method to 112 tropical plant leaf images. SL was estimated from the leaf perimeter (P) and leaf area (A). It correlates positively with a fractal dimensional measure of leaf complexity, viz., segmental fractal complexity. Leaves with a higher P: A ratio have higher SL and possess complex morphology. The univariate cluster analysis of SL reveals the taxonomic relationship among the leaf shapes at the genus level. An increase in SL of plant leaves could be an evolutionary strategy. The results of morphological complexity presented in this paper will trigger discussion on the causal links between leaf adaptive stability/efficiency and complexity. We present SL as a derived plant trait to describe plant leaf complexity and adaptive stability. Integrating SL into other leaf physiological measures will help to understand the dynamics of energy flow between plants and their environment.

Large Growth Deformations of Thin Tissue Using Solid-Shells

Simulating large scale expansion of thin structures, such as in growing leaves, is challenging. Solid-shells have a number of potential advantages over conventional thin-shell methods, but have thus far only been investigated for small plastic deformation cases. In response, we present a new general-purpose FEM growth framework for handling a wide range of challenging growth scenarios using the solid-shell element. Solid-shells are a middle-ground between traditional volume and thin-shell elements where volumetric characteristics are retained while being treatable as a 2D manifold much like thin-shells. These elements are adaptable to accommodate the many techniques that are required for simulating large and intricate plastic deformations, including morphogen diffusion, plastic embedding, strain-aware adaptive remeshing, and collision handling. We demonstrate the capabilities of growing solid-shells in reproducing buckling, rippling, curling, and collision deformations, relevant towards animating growing leaves, flowers, and other thin structures. Solid-shells are compared side-by-side with thin-shells to examine their bending behavior and runtime performance. The experiments demonstrate that solid-shells are a viable alternative to thin-shells for simulating large and intricate growth deformations.

Within-individual leaf allometry and the evolution of leaf morphology: A multilevel analysis of leaf allometry in temperate Viburnum (Adoxaceae) species

A critical issue in evolutionary biology is understanding the relationship between macroevolutionary patterns of diversity and the origin of variation at the organismal level. Among-individual allometry, the relationship between the size and shape of a structure among organisms at a fixed developmental stage, is often similar to evolutionary allometry, the relationship between the size and shape of a structure among populations or species, and the genetic and developmental process that underlie allometric relationships at both levels are thought to influence evolutionary diversification. Metameric organisms present an additional level of allometry: the relationship between the size and shape of structures within individuals. We propose that within-individual allometry is also related to evolutionary diversification among metameric organisms. We explore this idea in temperate deciduous Viburnum (Adoxaceae) species that bear two types of leaves, that is, preformed and neoformed leaves, with contrasting patterns of development. Examination of within-individual, among-individual, among-population, and among-species allometry of leaf shape in both leaf types showed that the slopes of all allometric relationships were significantly different from isometry, and their sign was consistent across allometric hierarchies. Although the allometric slope of preformed leaves was constant across allometry levels, the allometric slope of neoformed leaves became increasingly steeper. We suggest that allometric variation underlying evolutionary diversification in metameric organisms may manifest among individuals and also among their repeated structures. Moreover, structures with contrasting patterns of development within metameric organisms can experience different degrees of developmental constraint, and this can in turn affect morphological diversification.

Evolution of the coniferous seed scale

BACKGROUND

The Florin model is the commonly accepted theory of coniferous seed scale evolution. It describes the derivation of extant seed scale morphology from the morphology of fossil conifers via the reduction of complex to simple axillary structures. In this framework the seed scale is composed of a reduced lateral shoot with fertile and sterile appendages which are interpreted as leaf homologues.

SCOPE

The Florin model has three crucial problems that we address here: (1) the original derivation series does not take the ontogeny of extant conifers into account, (2) it cannot explain the morphology of all extant conifers and (3) Taxaceae were originally excluded. Examination of seed cones of extant conifers shows that ovules occur in three different positions in the cone: in an axillary position, replacing a leaf or terminating the cone axis. By interpreting the fertile appendage or seed-bearing structure as a leaf, not all positions are possible. The exclusion of Taxaceae from conifers is in stark contrast to recent molecular phylogenetic studies, which include Taxaceae in conifers as sister to Cupressaceae. Therefore, the Florin model does not offer an adequate explanation for taxaceous morphology.

CONCLUSION

We conclude that the seed-bearing structure of conifers cannot be interpreted as homologous to a leaf. In the interpretation we present here, the seed-bearing structure is the modified funiculus of the ovule, multiples of which laterally fuse to form the seed scale. The seed scales of all extant conifers can be derived from a Cunninghamia-like morphology via fusion and reduction of individual funiculi.

Leaf morphogenesis: The multifaceted roles of mechanics

Plants produce a rich diversity of biological forms, and the diversity of leaves is especially notable. Mechanisms of leaf morphogenesis have been studied in the past two decades, with a growing focus on the interactive roles of mechanics in recent years. Growth of plant organs involves feedback by mechanical stress: growth induces stress, and stress affects growth and morphogenesis. Although much attention has been given to potential stress-sensing mechanisms and cellular responses, the mechanical principles guiding morphogenesis have not been well understood. Here we synthesize the overarching roles of mechanics and mechanical stress in multilevel and multiple stages of leaf morphogenesis, encompassing leaf primordium initiation, phyllotaxis and venation patterning, and the establishment of complex mature leaf shapes. Moreover, the roles of mechanics at multiscale levels, from subcellular cytoskeletal molecules to single cells to tissues at the organ scale, are articulated. By highlighting the role of mechanical buckling in the formation of three-dimensional leaf shapes, this review integrates the perspectives of mechanics and biology to provide broader insights into the mechanobiology of leaf morphogenesis.

Shaping Organs: Shared Structural Principles Across Kingdoms

Development encapsulates the morphogenesis of an organism from a single fertilized cell to a functional adult. A critical part of development is the specification of organ forms. Beyond the molecular control of morphogenesis, shape in essence entails structural constraints and thus mechanics. Revisiting recent results in biophysics and development, and comparing animal and plant model systems, we derive key overarching principles behind the formation of organs across kingdoms. In particular, we highlight how growing organs are active rather than passive systems and how such behavior plays a role in shaping the organ. We discuss the importance of considering different scales in understanding how organs form. Such an integrative view of organ development generates new questions while calling for more cross-fertilization between scientific fields and model system communities.

Examining the support-supply and bud-packing hypotheses for the increase in toothed leaf margins in northern deciduous floras

PREMISE

The proportion of woody dicots with toothed leaves increases toward colder regions, a relationship used to reconstruct past mean annual temperatures. Recent hypotheses explaining this relationship are that (1) leaves in colder regions are thinner, requiring thick veins for support and water supply, with the resulting craspedodromous venation leading to marginal teeth (support-supply hypothesis) or that (2) teeth are associated with the packing of leaf primordia in winter buds (bud-packing hypothesis).

METHODS

We addressed these hypotheses by examining leaf thickness, number of primordia in buds, growing season length (mean annual temperature, MAT), and other traits in 151 deciduous woody species using georeferenced occurrences and a Bayesian model controlling for phylogeny. We excluded evergreen species because longer leaf life spans correlate with higher leaf mass per area, precluding the detection of independent effects of leaf thickness on leaf-margin type.

RESULTS

The best model predicted toothed leaves with 94% accuracy, with growing season length the strongest predictor. Neither leaf thickness nor number of leaves preformed in buds significantly influenced margin type, rejecting the support-supply and bud-packing hypotheses.

CONCLUSIONS

A direct selective benefit of leaf teeth via a carbon gain early in the spring as proposed by Royer and Wilf (2006) would match the strong correlation between toothed species occurrence and short growing season found here using Bayesian hierarchical models. Efforts should be directed to physiological work quantifying seasonal photosynthate production in toothed and nontoothed leaves.

Fluttering of growing leaves as a way to reach flatness: experimental evidence on Persea americana

Simple leaves show unexpected growth motions: the midrib of the leaves swings periodically in association with buckling events of the leaf blade, giving the impression that the leaves are fluttering. The quantitative kinematic analysis of this motion provides information about the respective growth between the main vein and the lamina. Our three-dimensional reconstruction of an avocado tree leaf shows that the conductor of the motion is the midrib, presenting continuous oscillations and inducing buckling events on the blade. The variations in the folding angle of the leaf show that the lamina is not passive: it responds to the deformation induced by the connection to the midrib to reach a globally flat state. We model this movement as an asymmetric growth of the midrib, which directs an inhomogeneous growth of the lamina, and we suggest how the transition from the folded state to the flat state is mechanically organized.

Theoretical models for branch formation in plants

Various branch architectures are observed in living organisms including plants. Branch formation has traditionally been an area of interest in the field of developmental biology, and theoretical approaches are now commonly used to understand the complex mechanisms involved. In this review article, we provide an overview of theoretical approaches including mathematical models and computer simulations for studying plant branch formation. These approaches cover a wide range of topics. In particular, we focus on the importance of positional information in branch formation, which has been especially revealed by theoretical research in plants including computations of developmental processes.

On genes and form

The mechanisms by which organisms acquire their sizes and shapes through growth was a major focus of D'Arcy Thompson's book On Growth and Form. By applying mathematical and physical principles to a range of biological forms, Thompson achieved fresh insights, such as the notion that diverse biological shapes could be related through simple deformations of a coordinate system. However, Thompson considered genetics to lie outside the scope of his work, even though genetics was a growing discipline at the time the book was published. Here, we review how recent advances in cell, developmental, evolutionary and computational biology allow Thompson's ideas to be integrated with genes and the processes they influence to provide a deeper understanding of growth and morphogenesis. We consider how genes interact with subcellular-, cellular- and tissue-level processes in plants to yield patterns of growth that underlie the developmental and evolutionary shape transformations Thompson so eloquently described.

A common developmental program can produce diverse leaf shapes

Eudicot leaves have astoundingly diverse shapes. The central problem addressed in this paper is the developmental origin of this diversity. To investigate this problem, we propose a computational model of leaf development that generalizes the largely conserved molecular program for the reference plants Arabidopsis thaliana, Cardamine hirsuta and Solanum lycopersicum. The model characterizes leaf development as a product of three interwoven processes: the patterning of serrations, lobes and/or leaflets on the leaf margin; the patterning of the vascular system; and the growth of the leaf blade spanning the main veins. The veins play a significant morphogenetic role as a local determinant of growth directions. We show that small variations of this model can produce diverse leaf shapes, from simple to lobed to compound. It is thus plausible that diverse shapes of eudicot leaves result from small variations of a common developmental program.

Causes of ecological gradients in leaf margin entirety: Evaluating the roles of biomechanics, hydraulics, vein geometry, and bud packing

PREMISE OF THE STUDY

A recent commentary by Edwards et al. (Am. J. Bot. 103: 975-978) proposed that constraints imposed by the packing of young leaves in buds could explain the positive association between non-entire leaf margins and latitude but did not thoroughly consider alternative explanations.

METHODS

We review the logic and evidence underlying six major hypotheses for the functional significance of marginal teeth, involving putative effects on (1) leaf cooling, (2) optimal support and supply of the areas served by major veins, (3) enhanced leaf-margin photosynthesis, (4) hydathodal function, (5) defense against herbivores, and (6) bud packing.

KEY RESULTS

Theoretical and empirical problems undermine all hypotheses except the support-supply hypothesis, which implies that thinner leaves should have non-entire margins. Phylogenetically structured analyses across angiosperms, the El Yunque flora, and the genus Viburnum all demonstrate that non-entire margins are indeed more common in thinner leaves. Across angiosperms, the association of leaf thickness with non-entire leaf margins is stronger than that of latitude.

CONCLUSION

We outline a synthetic model showing how biomechanics, hydraulics, vein geometry, rates of leaf expansion, and length of development within resting buds, all tied to leaf thickness, drive patterns in the distribution of entire vs. non-entire leaf margins. Our model accounts for dominance of entire margins in the tropics, Mediterranean scrub, and tundra, non-entire margins in cold temperate deciduous forests and tropical vines and early-successional trees, and entire leaf margins in monocots. Spinose-toothed leaves should be favored in short-statured evergreen trees and shrubs, primarily in Mediterranean scrub and related semiarid habitats.

Evolutionary and Environmental Forces Sculpting Leaf Development

Leaf shape is spectacularly diverse. As a major component of plant architecture and an interface for light capture, gas exchange, and thermoregulation, the potential contributions of leaves to plant fitness are innumerable. Particularly because of their intimate association and interaction with the surrounding environment, both the plasticity of leaf shape during the lifetime of a plant and the evolution of leaf shape over geologic time are revealing with respect to leaf function. Leaf shapes arise within a developmental context that constrains both their evolution and environmental plasticity. Quantitative models capturing genetic diversity, developmental context, and environmental plasticity will be required to fully understand the evolution and development of leaf shape and its response to environmental pressures. In this review, we discuss recent literature demonstrating that distinct molecular pathways are modulated by specific environmental inputs, the output of which regulates leaf dissection. We propose a synthesis explaining both historical patterns in the paleorecord and conserved plastic responses in extant plants. Understanding the potential adaptive value of leaf shape, and how to molecularly manipulate it, will prove to be invaluable in designing crops optimized for future climates.

Folding to Curved Surfaces: A Generalized Design Method and Mechanics of Origami-based Cylindrical Structures

Origami structures enrich the field of mechanical metamaterials with the ability to convert morphologically and systematically between two-dimensional (2D) thin sheets and three-dimensional (3D) spatial structures. In this study, an in-plane design method is proposed to approximate curved surfaces of interest with generalized Miura-ori units. Using this method, two combination types of crease lines are unified in one reprogrammable procedure, generating multiple types of cylindrical structures. Structural completeness conditions of the finite-thickness counterparts to the two types are also proposed. As an example of the design method, the kinematics and elastic properties of an origami-based circular cylindrical shell are analysed. The concept of Poisson’s ratio is extended to the cylindrical structures, demonstrating their auxetic property. An analytical model of rigid plates linked by elastic hinges, consistent with numerical simulations, is employed to describe the mechanical response of the structures. Under particular load patterns, the circular shells display novel mechanical behaviour such as snap-through and limiting folding positions. By analysing the geometry and mechanics of the origami structures, we extend the design space of mechanical metamaterials and provide a basis for their practical applications in science and engineering.

A Mechanical Feedback Restricts Sepal Growth and Shape in Arabidopsis

How organs reach their final shape is a central yet unresolved question in developmental biology. Here we investigate whether mechanical cues contribute to this process. We analyze the epidermal cells of the Arabidopsis sepal, focusing on cortical microtubule arrays, which align along maximal tensile stresses and restrict growth in that direction through their indirect impact on the mechanical anisotropy of cell walls. We find a good match between growth and microtubule orientation throughout most of the development of the sepal. However, at the sepal tip, where organ maturation initiates and growth slows down in later stages, microtubules remain in a configuration consistent with fast anisotropic growth, i.e., transverse, and the anisotropy of their arrays even increases. To understand this apparent paradox, we built a continuous mechanical model of a growing sepal. The model demonstrates that differential growth in the sepal can generate transverse tensile stress at the tip. Consistently, microtubules respond to mechanical perturbations and align along maximal tension at the sepal tip. Including this mechanical feedback in our growth model of the sepal, we predict an impact on sepal shape that is validated experimentally using mutants with either increased or decreased microtubule response to stress. Altogether, this suggests that a mechanical feedback loop, via microtubules acting both as stress sensor and growth regulator, channels the growth and shape of the sepal tip. We propose that this proprioception mechanism is a key step leading to growth arrest in the whole sepal in response to its own growth.

The embryonic shoot: a lifeline through winter

The tiny vascular axis of the embryo emerges post-embryonically as an elaborate and critical infrastructure, pervading the entire plant system. Its expansive nature is especially impressive in trees, where growth and development continue for extended periods. While the shoot apical meristem (SAM) orchestrates primary morphogenesis, the vascular system is mapped out in its wake in the provascular cylinder, situated just below the emerging leaf primordia and surrounding the rib meristem. Formation of leaf primordia and provascular tissues is incompatible with the harsh conditions of winter. Deciduous trees of boreal and temperate climates therefore enter a survival mode at the end of the season. However, to be competitive, they need to maximize their growth period while avoiding cellular frost damage. Trees achieve this by monitoring photoperiod, and by timely implementation of a survival strategy that schedules downstream events, including growth cessation, terminal bud formation, dormancy assumption, acquisition of freezing tolerance, and shedding of leaves. Of central importance are buds, which contain an embryonic shoot that allows shoot development and elongation in spring. The genetic and molecular processes that drive the cycle in synchrony with the seasons are largely elusive. Here, we review what is known about the signals and signal conduits that are involved, the processes that are initiated, and the developmental transitions that ensue in a terminal bud. We propose that addressing dormancy as a property of the SAM and the bud as a unique shoot type will facilitate our understanding of winter dormancy.

Mechanical control of morphogenesis at the shoot apex

Morphogenesis does not just require the correct expression of patterning genes; these genes must induce the precise mechanical changes necessary to produce a new form. Mechanical characterization of plant growth is not new; however, in recent years, new technologies and interdisciplinary collaborations have made it feasible in young tissues such as the shoot apex. Analysis of tissues where active growth and developmental patterning are taking place has revealed biologically significant variability in mechanical properties and has even suggested that mechanical changes in the tissue can feed back to direct morphogenesis. Here, an overview is given of the current understanding of the mechanical dynamics and its influence on cellular and developmental processes in the shoot apex. We are only starting to uncover the mechanical basis of morphogenesis, and many exciting questions remain to be answered.

Widespread mechanosensing controls the structure behind the architecture in plants

Mechanical forces play an instructing role for many aspects of animal cell biology, such as division, polarity and fate. Although the associated mechanoperception pathways still remain largely elusive in plants, physical cues have long been thought to guide development in parallel to biochemical factors. With the development of new imaging techniques, micromechanics tools and modeling approaches, the role of mechanical signals in plant development is now re-examined and fully integrated with modern cell biology. Using recent examples from the literature, I propose to use a multiscale perspective, from the whole plant down to the cell wall, to fully appreciate the diversity of developmental processes that depend on mechanical signals. Incidentally, this also illustrates how conceptually rich this field is.

Evolution of leaf form correlates with tropical-temperate transitions in Viburnum (Adoxaceae)

Strong latitudinal patterns in leaf form are well documented in floristic comparisons and palaeobotanical studies. However, there is little agreement about their functional significance; in fact, it is still unknown to what degree these patterns were generated by repeated evolutionary adaptation. We analysed leaf form in the woody angiosperm clade Viburnum (Adoxaceae) and document evolutionarily correlated shifts in leafing habit, leaf margin morphology, leaf shape and climate. Multiple independent shifts between tropical and temperate forest habitats have repeatedly been accompanied by a change between evergreen, elliptical leaves with entire margins and deciduous, more rounded leaves with toothed or lobed margins. These consistent shifts in Viburnum support repeated evolutionary adaptation as a major determinant of the global correlation between leaf form and mean annual temperature. Our results provide a new theoretical grounding for the inference of past climates using fossil leaf assemblages.

Abaxial growth and steric constraints guide leaf folding and shape in Acer pseudoplatanus

PREMISE OF THE STUDY

How leaf shape is regulated is a long-standing question in botany. For diverse groups of dicotyledon species, lamina folding along the veins and geometry of the space available for the primordia can explain the palmate leaf morphology. Dubbed the kirigami theory, this hypothesis of fold-dependent leaf shape regulation has remained largely theoretical. Using Acer pseudoplatanus, we investigated the mechanisms behind the two key processes of kirigami leaf development.

METHODS

Cytological examination and quantitative analyses were used to examine the course of the vein-dependent lamina folding. Surgical ablation and tissue culturing were employed to test the effects of physical constraints on primordia growth. The final morphology of leaves growing without steric constraints were predicted mathematically.

KEY RESULTS

The cytological examination showed that the lamina's abaxial side along the veins grows substantially more than the adaxial side. The abaxial hypergrowth along the veins and the lamina extension correlated with the lamina folding. When a primordium was released from the physical constraints imposed by the other primordia, it rapidly grew into the newly available space, while maintaining the curvature inward. The morphology of such a leaf was predicted to lack symmetry in the lobe shapes.

CONCLUSIONS

The enhanced growth on the abaxial side of the lamina along the veins is likely to drive lamina folding. The surgical ablation provided clear support for the space-filling nature of leaf growth; thus, steric constraints play a role in determination of the shapes of folded leaves and probably also of the final leaf morphology.

Computational models of plant development and form

The use of computational techniques increasingly permeates developmental biology, from the acquisition, processing and analysis of experimental data to the construction of models of organisms. Specifically, models help to untangle the non-intuitive relations between local morphogenetic processes and global patterns and forms. We survey the modeling techniques and selected models that are designed to elucidate plant development in mechanistic terms, with an emphasis on: the history of mathematical and computational approaches to developmental plant biology; the key objectives and methodological aspects of model construction; the diverse mathematical and computational methods related to plant modeling; and the essence of two classes of models, which approach plant morphogenesis from the geometric and molecular perspectives. In the geometric domain, we review models of cell division patterns, phyllotaxis, the form and vascular patterns of leaves, and branching patterns. In the molecular-level domain, we focus on the currently most extensively developed theme: the role of auxin in plant morphogenesis. The review is addressed to both biologists and computational modelers.