The theory of tree bole and branch form

Stretched sapwood, ultra-widening permeability and ditching da Vinci: revising models of plant form and function

BACKGROUND

The mechanisms leading to dieback and death of trees under drought remain unclear. To gain an understanding of these mechanisms, addressing major empirical gaps regarding tree structure-function relations remains essential.

SCOPE

We give reasons to think that a central factor shaping plant form and function is selection simultaneously favouring constant leaf-specific conductance with height growth and isometric (1:1) scaling between leaf area and the volume of metabolically active sink tissues ('sapwood'). Sapwood volume-leaf area isometry implies that per-leaf area sapwood volumes become transversely narrower with height growth; we call this 'stretching'. Stretching means that selection must favour increases in permeability above and beyond that afforded by tip-to-base conduit widening ("ultra-widening permeability"), via fewer and wider vessels or tracheids with larger pits or larger margo openings. Leaf area-metabolically active sink tissue isometry would mean that it is unlikely that larger trees die during drought because of carbon starvation due to greater sink-source relationships as compared to shorter plants. Instead, an increase in permeability is most plausibly associated with greater risk of embolism, and this seems a more probable explanation of the preferential vulnerability of larger trees to climate change-induced drought. Other implications of selection favouring constant per-leaf area sapwood construction and maintenance costs are departure from the da Vinci rule expectation of similar sapwood areas across branching orders, and that extensive conduit furcation in the stem seems unlikely.

CONCLUSIONS

Because all these considerations impact the likelihood of vulnerability to hydraulic failure versus carbon starvation, both implicated as key suspects in forest mortality, we suggest that these predictions represent essential priorities for empirical testing.

Plant strategies for greatest height: tapering or hollowing

The tapered form and hollow cross-section of the stem and trunk of wild plants are rational mechanical approaches because they facilitate the plant simultaneously growing taller for photosynthesis and supporting its own weight. The purpose of this study is to clarify the advantages and disadvantages of tapering and hollowing from the perspective of the greatest probable height before self-buckling. We modelled woody plants using tapering or hollow cantilevers, formulated the greatest height before self-buckling, and derived a theoretical formula for the greatest probable height considering tapering and hollowing. This formula theoretically explains why almost all plants exhibit a tapered form: it allows for a greater height at an earlier growth stage than a hollow cross-section.

Mechanics-based classification rule for plants

The height of thick and solid plants, such as woody plants, is proportional to two-thirds of the power of their diameter at breast height. However, this rule cannot be applied to herbaceous plants that are thin and soft because the mechanisms supporting their bodies are fundamentally different. This study aims to clarify the rigidity control mechanism resulting from turgor pressure caused by internal water in herbaceous plants to formulate the corresponding scaling law. We modeled a herbaceous plant as a cantilever with the ground side as a fixed end, and the greatest height was formulated considering the axial tension force from the turgor pressure. The scaling law describing the relationship between the height and diameter in terms of the turgor pressure was theoretically derived. Moreover, we proposed a plant classification rule based on stress distribution.

Longevity of System Functions in Biology and Biomimetics: A Matter of Robustness and Resilience

Within the framework of a circular economy, we aim to efficiently use raw materials and reduce waste generation. In this context, the longevity of biomimetic material systems can significantly contribute by providing robustness and resilience of system functionality inspired by biological models. The aim of this review is to outline various principles that can lead to an increase in robustness (e.g., safety factor, gradients, reactions to environmental changes) and resilience (e.g., redundancy, self-repair) and to illustrate the principles with meaningful examples. The study focuses on plant material systems with a high potential for transfer to biomimetic applications and on existing biomimetic material systems. Our fundamental concept is based on the functionality of the entire system as a function of time. We use functionality as a dimensionless measure of robustness and resilience to quantify the system function, allowing comparison within biological material systems and biomimetic material systems, but also between them. Together with the enclosed glossary of key terms, the review provides a comprehensive toolbox for interdisciplinary teams. Thus, allowing teams to communicate unambiguously and to draw inspiration from plant models when developing biomimetic material systems with great longevity potential.

Rigidity control mechanism by turgor pressure in plants

The bodies of herbaceous plants are slender, thin, and soft. These plants support their bodies through the action of turgor pressure associated with their internal water stores. The purpose of this study was to apply the principles of structural mechanics to clarify the underlying mechanism of rigidity control that is responsible for turgor pressure in plants and the reason behind the self-supporting ability of herbaceous plants. We modeled a plant a horizontally oriented thin-walled cylindrical cantilever with closed ends enclosing a cavity filled with water that is acted on by its own weight and by internal tension generated through turgor pressure. We derived an equation describing the plant's consequent deflection, introducing a dimensionless parameter to express the decrease in deflection associated with the action of turgor pressure. We found that the mechanical and physical characteristics of herbaceous plants that would appear to be counter-productive from a superficial perspective increase the deflection decreasing effect of turgor pressure.

Mathematical modelling to determine the greatest height of trees

This study aimed to analyse the critical height of a column whose weight varies vertically in order to obtain a simple scaling law for a tree where the weight distribution considered. We modelled trees as cantilevers that were fixed to the ground and formulated a self-buckling problem for various weight distributions. A formula for calculating the critical height was derived in a simple form that did not include special functions. We obtained a theoretical clarification of the effect of the weight distribution of heavy columns on the buckling behaviour. A widely applicable scaling law for trees was obtained. We found that an actual tree manages to distribute the weight of its trunk and branches along its vertical extent in a manner that adequately secures its critical height. The method and findings of this study are applicable to a wide range of fields, such as the simplification of complicated buckling problems and the study of tree shape quantification.

Twist-to-Bend Ratios and Safety Factors of Petioles Having Various Geometries, Sizes and Shapes

From a mechanical viewpoint, petioles of foliage leaves are subject to contradictory mechanical requirements. High flexural rigidity guarantees support of the lamina and low torsional rigidity ensures streamlining of the leaves in wind. This mechanical trade-off between flexural and torsional rigidity is described by the twist-to-bend ratio. The safety factor describes the maximum load capacity. We selected four herbaceous species with different body plans (monocotyledonous, dicotyledonous) and spatial configurations of petiole and lamina (2-dimensional, 3-dimensional) and carried out morphological-anatomical studies, two-point bending tests and torsional tests on the petioles to analyze the influence of geometry, size and shape on their twist-to-bend ratio and safety factor. The monocotyledons studied had significantly higher twist-to-bend ratios (23.7 and 39.2) than the dicotyledons (11.5 and 13.3). High twist-to-bend ratios can be geometry-based, which is true for the U-profile of Hosta x tardiana with a ratio of axial second moment of area to torsion constant of over 1.0. High twist-to-bend ratios can also be material-based, as found for the petioles of Caladium bicolor with a ratio of bending elastic modulus and torsional modulus of 64. The safety factors range between 1.7 and 2.9, meaning that each petiole can support about double to triple the leaf's weight.

A predictive nondestructive model for the covariation of tree height, diameter, and stem volume scaling relationships

Metabolic scaling theory (MST) posits that the scaling exponents among plant height H, diameter D, and biomass M will covary across phyletically diverse species. However, the relationships between scaling exponents and normalization constants remain unclear. Therefore, we developed a predictive model for the covariation of H, D, and stem volume V scaling relationships and used data from Chinese fir (Cunninghamia lanceolata) in Jiangxi province, China to test it. As predicted by the model and supported by the data, normalization constants are positively correlated with their associated scaling exponents for D vs. V and H vs. V, whereas normalization constants are negatively correlated with the scaling exponents of H vs. D. The prediction model also yielded reliable estimations of V (mean absolute percentage error = 10.5 ± 0.32 SE across 12 model calibrated sites). These results (1) support a totally new covariation scaling model, (2) indicate that differences in stem volume scaling relationships at the intra-specific level are driven by anatomical or ecophysiological responses to site quality and/or management practices, and (3) provide an accurate non-destructive method for predicting Chinese fir stem volume.

Review: Wind impacts on plant growth, mechanics and damage

Land plants have adapted to survive under a range of wind climates and this involve changes in chemical composition, physical structure and morphology at all scales from the cell to the whole plant. Under strong winds plants can re-orientate themselves, reconfigure their canopies, or shed needles, leaves and branches in order to reduce the drag. If the wind is too strong the plants oscillate until the roots or stem fail. The mechanisms of root and stem failure are very similar in different plants although the exact details of the failure may be different. Cereals and other herbaceous crops can often recover after wind damage and even woody plants can partially recovery if there is sufficient access to water and nutrients. Wind damage can have major economic impacts on crops, forests and urban trees. This can be reduced by management that is sensitive to the local site and climatic conditions and accounts for the ability of plants to acclimate to their local wind climate. Wind is also a major disturbance in many plant ecosystems and can play a crucial role in plant regeneration and the change of successional stage.

Tree branching: Leonardo da Vinci's rule versus biomechanical models

This study examined Leonardo da Vinci's rule (i.e., the sum of the cross-sectional area of all tree branches above a branching point at any height is equal to the cross-sectional area of the trunk or the branch immediately below the branching point) using simulations based on two biomechanical models: the uniform stress and elastic similarity models. Model calculations of the daughter/mother ratio (i.e., the ratio of the total cross-sectional area of the daughter branches to the cross-sectional area of the mother branch at the branching point) showed that both biomechanical models agreed with da Vinci's rule when the branching angles of daughter branches and the weights of lateral daughter branches were small; however, the models deviated from da Vinci's rule as the weights and/or the branching angles of lateral daughter branches increased. The calculated values of the two models were largely similar but differed in some ways. Field measurements of Fagus crenata and Abies homolepis also fit this trend, wherein models deviated from da Vinci's rule with increasing relative weights of lateral daughter branches. However, this deviation was small for a branching pattern in nature, where empirical measurements were taken under realistic measurement conditions; thus, da Vinci's rule did not critically contradict the biomechanical models in the case of real branching patterns, though the model calculations described the contradiction between da Vinci's rule and the biomechanical models. The field data for Fagus crenata fit the uniform stress model best, indicating that stress uniformity is the key constraint of branch morphology in Fagus crenata rather than elastic similarity or da Vinci's rule. On the other hand, mechanical constraints are not necessarily significant in the morphology of Abies homolepis branches, depending on the number of daughter branches. Rather, these branches were often in agreement with da Vinci's rule.

How to become a tree without wood--biomechanical analysis of the stem of Carica papaya L

Carica papaya L. does not contain wood, according to the botanical definition of wood as lignified secondary xylem. Despite its parenchymatous secondary xylem, these plants are able to grow up to 10-m high. This is surprising, as wooden structural elements are the ubiquitous strategy for supporting height growth in plants. Proposed possible alternative principles to explain the compensation for lack of wood in C. papaya are turgor pressure of the parenchyma, lignified phloem fibres in the bark, or a combination of the two. Interestingly, lignified tissue comprises only 5-8% of the entire stem mass. Furthermore, the phloem fibres do not form a compact tube enclosing the xylem, but instead form a mesh tubular structure. To investigate the mechanism of papaya's unusually high mechanical strength, a set of mechanical measurements were undertaken on whole stems and tissue sections of secondary phloem and xylem. The structural Young's modulus of mature stems reached 2.5 GPa. Since this is low compared to woody plants, the flexural rigidity of papaya stem construction may mainly be based on a higher second moment of inertia. Additionally, stem turgor pressure was determined indirectly by immersing specimens in sucrose solutions of different osmolalities, followed by mechanical tests; turgor pressure was between 0.82 and 1.25 MPa, indicating that turgor is essential for flexural rigidity of the entire stem.

Integrative biomechanics for tree ecology: beyond wood density and strength

Functional ecology has long considered the support function as important, but its biomechanical complexity is only just being elucidated. We show here that it can be described on the basis of four biomechanical traits, two safety traits against winds and self-buckling, and two motricity traits involved in sustaining an upright position, tropic motion velocity (MV) and posture control (PC). All these traits are integrated at the tree scale, combining tree size and shape together with wood properties. The assumption of trait constancy has been used to derive allometric scaling laws, but it was more recently found that observing their variations among environments and functional groups, or during ontogeny, provides more insights into adaptive syndromes of tree shape and wood properties. However, oversimplified expressions have often been used, possibly concealing key adaptive drivers. An extreme case of oversimplification is the use of wood basic density as a proxy for safety. Actually, as wood density is involved in stiffness, loads, and construction costs, the impact of its variations on safety is non-trivial. Moreover, other wood features, especially the microfibril angle (MFA), are also involved. Furthermore, wood is not only stiff and strong, but it also acts as a motor for MV and PC. The relevant wood trait for this is maturation strain asymmetry. Maturation strains vary with cell-wall characteristics such as MFA, rather than with wood density. Finally, the need for further studies about the ecological relevance of branching patterns, motricity traits, and growth responses to mechanical loads is discussed.

Allometry and life history of tropical trees

The scaling of crown size and trunk diameter with tree height (allometry) was determined for 14 common species of the tropical wet lowland forest at La Selva, Costa Rica. The study showed that allometric differences between species are related to adult size, regeneration niche (gap vs. non-gap) and longevity, as follows: (1) adults of understorey species are larger crowned than similar statured (6–15 m) saplings of canopy trees; (2) species commonly found in gaps as saplings are somewhat larger crowned than shade-tolerant species over the 1–6 m height range; and (3) long-lived canopy species show greater increases in crown breadth with increasing height thandoshort-livedspecies.Trunkallometryisrelated to mechanical requirements for support, including the need to withstand greater wind forces in the upper canopy. The common canopy species, Pentaclethra macroloba, which comprises 40% of the basal area at La Selva, is particularly wide-crowned and thick-trunked at its maximum height. On the other hand, the comparatively narrower crowns and trunks of the other canopy species allow them to reach a given height with less biomass. These differences in allometry may influence tree density and forest structure at La Selva.

Toward extension of a single tree functional-structural model of Scots pine to stand level: effect of the canopy of randomly distributed, identical trees on development of tree structure

Functional-structural plant growth models (FSPMs) combine the description of the structure of plants and the resource acquisition and partitioning at a detailed architectural level. They offer a means to study tree and stand development on the basis of a structurally accurate description that combines resource capture at the same level of detail. We describe here how a 'shoot-based' individual tree model, LIGNUM of Scots pine (Pinus sylvestris L.) has been applied to a group of identical trees (forest). The model has been applied to isolated trees and saplings growing in forest gaps. First, we present the LIGNUM model and the changes necessary for simulation of a forest instead of individual trees. LIGNUM derives tree growth on the basis of a process-based model of tree carbon balance and the architectural development of the 3-D tree crown. The time step is 1 year. We realised the forest as consisting of individual Scots pine trees on a plot 17 × 17 m, but simplified the stand description by simulating the growth of only one tree in the middle of the plot and assumed that the other trees were identical to it at all times. The model produced results that are comparable with observations made in real Scots pine trees and tree stands in Finland. The simulations with variable values of the parameters controlling the foliage-sapwood relationship, amount of sapwood required below a point in a branch or a stem, and the senescence of sapwood showed how growth declines when the sapwood requirement in the branches and stem was high. In this case, the proportion of resources allocated to the needles became small and the needle mass was low.

Spiralling upward

Thin vertical leaves often manifest twist. Perhaps the most prominent example of this is in Typha sp., but such twist is also apparent in Narcissus, Pancratium and many other genera. Such a blade is often referred to as a "spiral leaf". We will indicate the mechanical advantage afforded to the leaf by this arrangement, i.e. that it permits the leaf to achieve a greater height without losing stability, that is bending over due to its own weight. We quantify this gain and show how by a simple experiment it can be shown that the advantage is indeed utilized in nature. Typha domingensis is offered as an example.

Growth and hydraulic (not mechanical) constraints govern the scaling of tree height and mass

The size-dependent variations of plant height L and mass M with respect to basal stem diameter D are important to the analysis of a broad range of ecological and evolutionary phenomena. Prior examination of some of the world's largest trees suggests that the scaling relationships L alpha D(2/3) and M alpha D(8/3) hold true, ostensibly as functional adaptations for mechanical stability. This concept remains engrained in the literature in the form of null hypotheses (or predictive models), despite numerous examples showing that the 2/3 and 8/3 rules are violated by small and intermediate-sized plants. Here, we present a growth-hydraulic model that provides more accurate and biologically realistic predictions of L and M. This model also sheds light on why L, D, and M scale differently across species and habitats as a result of differences in absolute size.

Computing factors of safety against wind-induced tree stem damage

The drag forces, bending moments and stresses acting on stems differing in size and location within the mechanical infrastructure of a large wild cherry (Prunus serotina Ehrh.) tree are estimated and used to calculate the factor of safety against wind-induced mechanical failure based on the mean breaking stress of intact stems and samples of wood drawn from this tree. The drag forces acting on stems are calculated based on stem projected areas and field measurements of wind speed taken within the canopy and along the length of the trunk. The bending moments and stresses resulting from these forces are shown to increase basipetally in a nearly log-log linear fashion toward the base of the tree. The factor of safety, however, varies in a sinusoidal manner such that the most distal stems have the highest factors of safety, whereas stems of intermediate location and portions of the trunk near ground level have equivalent and much lower factors of safety. This pattern of variation is interpreted to indicate that, as a course of normal growth and development, trees similar to the one examined in this study maintain a cadre of stems prone to wind-induced mechanical damage that can reduce the probability of catastrophic tree failure by reducing the drag forces acting on older portions of the tree. Comparisons among real and hypothetical stems with different taper experiencing different vertical wind speed profiles show that geometrically self-similar stems have larger factors of safety than stems tapering according to elastic or stress self-similarity, and that safety factors are less significantly influenced by the 'geometry' of the wind-profile.

The influence of gravity and wind on land plant evolution

Aspects of the engineering theory treating the elastic stability of vertical stems and cantilevered leaves supporting their own weight and additional wind-induced forces (drag) are reviewed in light of biomechanical studies of living and fossil terrestrial plant species. The maximum height to which arborescent species can grow before their stems elastically buckle under their own weight is estimated by means of the Euler-Greenhill formula which states that the critical buckling height scales as the 1/3 power of plant tissue-stiffness normalized with respect to tissue bulk density and as the 2/3 power of stem diameter. Data drawn from living plants indicate that progressively taller plant species employ stiffer and lighter-weight plant tissues as the principal stiffening agent in their vertical stems. The elastic stability of plants subjected to high lateral wind-loadings is governed by the drag torque (the product of the drag force and the height above ground at which this force is applied), which cannot exceed the gravitational bending moment (the product of the weight of aerial organs and the lever arm measured at the base of the plant). Data from living plants indicate that the largest arborescent plant species rely on massive trunks and broad, horizontally expansive root crowns to resist drag torques. The drag on the canopies of these plants is also reduced by highly flexible stems and leaves composed of tissues that twist and bend more easily than tissues used to stiffen older, more proximal stems. A brief review of the fossil record suggests that modifications in stem, leaf, and root morphology and anatomy capable of simultaneously coping with self-weight and wind-induced drag forces evolved by Devonian times, suggesting that natural selection acting on the elastic stability of sporophytes occurred early in the history of terrestrial plants.

Branching patterns, generating rules, and astogenetic trajectories in arborescent bryozoans

The terminal growth of bryozoans allows a reconstruction of their growth history and generating rules from the developed pattern. Three species of arborescent bryozoans share a bias in growth rate that favors reverser branches (those whose direction of growth is opposite that of their parent branch); this bias produces a common hummocky appearance to the top margin of the colony. Other differences in the growth rules, however, result in markedly different colony forms. For two species, the cue for splitting of branches seems to be the length of the branch; for the other, the cue seems to be the time since the last splitting. Simulations based on the distinction between reversing and nonreversing branches reproduce the hummocky pattern, but offer little insight into the underlying mechanism. Simulations based on occlusion, the line-of-sight blocking of the branches by each other, give similar patterns with fewer assumptions. These simulations also suggest that the pattern is due to a limited availability of nutrients to occluded growing tips, and that nutrients remain relatively localized within colonies. The actual determinants of colony form are likely to combine both nutrition and geometry.