Leonardo's rule, self-similarity, and wind-induced stresses in trees

Mathematical definition and rules of the splitting/merging patterns in bundles of human peripheral nerve segment

Accurately measuring the spatial extension distance of nerve bundles in completing a split/merge is impossible because no clear mathematical definition exists for the starting and ending positions in nerve-bundle splitting/merging. We manually count the number of nerve-bundle splits/merges in long nerve segments, which is labor-intensive, inefficient, and prone to counting errors. Currently, the mathematics are unclear for the nerve-bundle diameter before and after splitting/merging. This paper explores these problems and proposes nerve-bundle splitting/merging rules. Based on the method of defining the beginning and ending positions of nerve-bundle splitting/merging, we explored the mathematical law of equivalent diameter of nerve bundles before and after splitting/merging. The experimental results revealed that the moving average of circularity of nerve bundle accurately defines the beginning and ending positions of nerve-bundle splitting/merging. The diameter of the nerve bundles before and after split/merge approximately conforms to the principles of the Da Vinci formula. The proposed automatic counting algorithm based on centroid offset matching obtains the number of split/merged nerve bundles in the sequence scan images with 100 % accuracy. The mathematical definition of the starting and ending positions of nerve-bundle splitting/merging proposed in this paper is accurate and strict and is the foundation of subsequent research. The proposed automatic counting algorithm based on centroid offset matching (ACA-COM) can accurately and efficiently count the number of times the nerve bundles split and merge in sequential images. The mathematical law satisfied by the diameter of the nerve bundles before and after splitting/merging reflects that the nerve bundles tend to have better capability to resist breaking.

Modeling Xylem Functionality Aspects

Depending on the questions to be answered, water flow in the xylem can be modelled following different approaches with varying spatial and temporal resolution. When focussing on the influence of hydraulic architecture upon flow dynamics, distribution of water potentials in a tree crown or questions of vulnerability of the hydraulic system, functional-structural plant models, which link representations of morphological structure with simulated processes and with a virtual environment, can be a promising tool. Such a model will then include a network of idealized xylem segments, each representing the conducting part of a stem or branch segment, and a numerical machinery suitable for solving a system of differential equations on it reflecting the hydrodynamic laws, which are the basis of the broadly accepted cohesion-tension theory of water flow in plants. We will discuss functional-structural plant models, the simplifications that are useful for hydraulic simulations within this framework, the deduction of the used differential equations from basic physical conservation laws, and their numerical solution, as well as additional necessary models of radiation, photosynthesis, and stomatal conductance. In some supplementary notes, we are shortly addressing some related questions, for example, about root systems or about the relation between macro-scale hydraulic parameters and fine-grained (anatomical) xylem structure.

Engineering Themes in Plant Forms and Functions

Living structures constantly interact with the biotic and abiotic environment by sensing and responding via specialized functional parts. In other words, biological bodies embody highly functional machines and actuators. What are the signatures of engineering mechanisms in biology? In this review, we connect the dots in the literature to seek engineering principles in plant structures. We identify three thematic motifs-bilayer actuator, slender-bodied functional surface, and self-similarity-and provide an overview of their structure-function relationships. Unlike human-engineered machines and actuators, biological counterparts may appear suboptimal in design, loosely complying with physical theories or engineering principles. We postulate what factors may influence the evolution of functional morphology and anatomy to dissect and comprehend better the why behind the biological forms.

Fractal Branched Microwires of Organic Semiconductor with Controlled Branching and Low-Threshold Amplified Spontaneous Emission

Fractals are quite normal in nature. However, fractal self-assembly of organic semiconductors remains challenging. Herein, we develop a facile solution assembly route to access organic microwires (MWs) comprising an oligo(p-phenylenevinylene) derivative (OPV-A) with and without branching. Instead of kinetically controlled β-OPV-A microrods (MRs), thermodynamically favored α-OPV-A gives fractal branching MW patterns. As-prepared 9,10-dicyanoanthracene (DCA) alloyed assemblies function as seeds to allow for the heteroepitaxial growth of branching α-OPV-A MWs via either coassembly or two-step seeded growth. Consequently, fractal MWs with single- and multisite growth were both achieved, accompanied by tailorable branching densities and hierarchies. Thermodynamic control and a well-matched epitaxial relationship should be crucial to the formation of fractal MW patterns. Importantly, the aligned α-OPV-A MW array functions as a multichannel optical gain medium and exhibits low-threshold amplified spontaneous emission (ASE). The present work deepens the research into fractal self-assembly of functional organic semiconductors.

On the integration of LiDAR and field data for riparian biomass estimation

The role of vegetation in supporting life on Earth is widely known. Nevertheless, the relevance of riparian corridors has been overlooked for a long time, leading to a dramatic reduction of vegetated buffers alongside them. Vegetation monitoring systems, including those for biomass estimation, are required to manage riparian corridors properly. Field surveys may support monitoring, but their usefulness is reduced by numerous drawbacks, therefore needing coupling with other data sources. The present work shows how Light Detection And Ranging (LiDAR) datasets can integrate targeted field measurements to estimate above-ground biomass at temperate or boreal latitudes and generate accurate biomass maps over large areas. By referring to the case study of the Orco river (northwest Italy), we defined a technique to reconstruct the geometry of an individual shrub from LiDAR point clouds. We tested the technique by comparing field measurements with Terrestrial and Airborne Laser Scanner data (TLS and ALS, respectively), assessing the former's superiority but the broader range of applicability of the latter. After these preliminary tests, we coupled the presented technique with a literature algorithm for individual tree detection, providing a more generalized procedure for the overall mapping and budgeting of riparian biomass based on ALS data. We applied the procedure to a fluvial bar of the Orco river, achieving a quantitative assessment of the shrub and tree biomass budget for 2019 and 2021 and visualizing the changes that occurred in that period. These results allowed us to shed light on the prevailing natural and anthropogenic processes in the investigated area and provide insights into the strengths and weaknesses of the proposed procedure.

Branching pattern of flexible trees for environmental load mitigation

Wind-induced stress is the primary mechanical cause of tree failures. Among different factors, the branching mechanism plays a central role in the stress distribution and stability of trees in windstorms. A recent study showed that Leonardo da Vinci's original observation, stating that the total cross section of branches conserved across branching nodes is the optimal configuration for resisting wind-induced damage in rigid trees, is correct. However, the breaking risk and the optimal branching pattern of trees are also a function of their reconfiguration capabilities and the processes they employ to mitigate high wind-induced stress hotspots. In this study, using a numerical model of rigid and flexible branched trees, we explore the role of flexibility and branching patterns of trees in their reconfiguration and stress mitigation capabilities. We identify the robust optimal branching mechanism for an extensive range of tree flexibility. Our results show that the probability of a tree breaking at each branching level from the stem to terminal foliage strongly depends on the cross section changes in the branching nodes, the overall tree geometry, and the level of tree flexibility. Three response categories have been identified: the stress concentration in the main trunk, the uniform stress level through the tree's height, and substantial stress localization in the terminal branches. The reconfigurability of the tree determines the dominant response mode. The results suggest a very similar optimal branching law for both flexible and rigid trees wherein uniform stress distribution occurs throughout the tree's height. An exception is the very flexible branched plants in which the optimal branching pattern deviates from this prediction and is strongly affected by the reconfigurability of the tree.

Branched droplet clusters and the Kramers theorem

Scaling laws inherent for polymer molecules are checked for the linear and branched chains constituting two-dimensional (2D) levitating microdroplet clusters condensed above the locally heated layer of water. We demonstrate that the dimensionless averaged end-to-end distance of the droplet chain r[over ¯] normalized by the averaged distance between centers of the adjacent droplets l[over ¯] scales as r[over ¯]/l[over ¯]∼n^{0.76}, where n is the number of links in the chain, which is close to the power exponent ¾, predicted for 2D polymer chains with excluded volume in the dilution limit. The values of the dimensionless Kuhn length b[over ̃]≅2.12±0.015 and of the averaged absolute value of the bond angle of the droplet chains |θ|[over ¯]=22.0±0.5^{0} are determined. Using these values we demonstrate that the predictions of the Kramers theorem for the gyration radius of branched polymers are valid also for the branched droplets' chains. We discuss physical interactions that explain both the high value of the power exponent and the applicability of the Kramers theorem including the effects of the excluded volume, surrounding droplet monomers, and the prohibition of extreme values of the bond angle.

Experimental evidence for logarithmic fractal structure of botanical trees

The area-preserving rule for botanical trees by Leonardo da Vinci is discussed in terms of a very specific fractal structure, a logarithmic fractal. We use a method of the numerical Fourier analysis to distinguish the logarithmic fractal properties of the two-dimensional objects and apply it to study the branching system of real trees through its projection on the two-dimensional space, i.e., using their photographs. For different species of trees (birch and oak) we observe the Q^{-2} decay of the spectral intensity characterizing the branching structure that is associated with the logarithmic fractal structure in two-dimensional space. The experiments dealing with the side view of the tree should complement the area preserving Leonardo's rule with one applying to the product of diameter d and length l of the k branches: d_{i}l_{i}=kd_{i+1}l_{i+1}. If both rules are valid, then the branch's length of the next generation is sqrt[k] times shorter than previous one: l_{i}=sqrt[k]l_{i+1}. Moreover, the volume (mass) of all branches of the next generation is a factor of d_{i}/d_{i+1} smaller than previous one. We conclude that a tree as a three-dimensional object is not a logarithmic fractal, although its projection onto a two-dimensional plane is. Consequently, the life of a tree flows according to the laws of conservation of area in two-dimensional space, as if the tree were a two-dimensional object.

Gigantic floating leaves occupy a large surface area at an economical material cost

The giant Amazonian waterlily (genus Victoria) produces the largest floating leaves in the plant kingdom. The leaves' notable vasculature has inspired artists, engineers, and architects for centuries. Despite the aesthetic appeal and scale of this botanical enigma, little is known about the mechanics of these extraordinary leaves. For example, how do these leaves achieve gigantic proportions? We show that the geometric form of the leaf is structurally more efficient than those of other smaller species of waterlily. In particular, the spatially varying thickness and regular branching of the primary veins ensures the structural integrity necessary for extensive coverage of the water surface, enabling optimal light capture despite a relatively low leaf biomass. Leaf gigantism in waterlilies may have been driven by selection pressures favoring a large surface area at an economical material cost, for outcompeting other plants in fast-drying ephemeral pools.

Role of the crown in tree resistance against high winds

Rather than using wooden sticks to simulate the breakage of trees in high winds as in most research, we employ fresh samples from camphor and Formosa gum with branches and leaves to certify the crucial role of the tree crown. By using a blowdown wind tunnel with a maximum wind speed of 50 m/s, we purposely reduce the number of leaves and show that the drag force will drop by as much as two thirds when half pruned. Based on real observations, we model the leaf by an open and full cone in the presence of light and strong winds, and calculate how their corresponding cross-sectional area A and drag force F vary with wind speed v. Different slopes before and after the formation of a full cone are predicted and confirmed when these two quantities are plotted in full-log scale. Compared to the empirical value, our simple model gave α=2/5 and 2/3 for A∝v^{-α} and β=4/5 and 2/3 for F∝v^{β} at low and high winds. Discrepancies can be accounted for by including further details, such as the reorientation of open cones and the movement of branches.

Functional morphology of plants - a key to biomimetic applications

Learning from living organisms has emerged from a mainly curiosity-driven examination, where helpful functions of biological structures have been copied, into systematic biomimetic approaches that transfer a targeted function and its underlying principles from the biological model to a technical product. Plant biomimetics is based on functional morphology, which combines the knowledge gained from the morphology, anatomy and mechanics of plants and makes a statement about their form-structure-function relationship. Since the functional morphology of plants has become key to biomimetic applications, we present its central role in deciphering the functional principles that can be applied to engineering solutions. We consider that the future of biomimetics will include bioinspired developments that will contribute to better sustainability than that achieved by conventional products.

Measuring the Contribution of Leaves to the Structural Complexity of Urban Tree Crowns with Terrestrial Laser Scanning

Trees have a fractal-like branching architecture that determines their structural complexity. We used terrestrial laser scanning technology to study the role of foliage in the structural complexity of urban trees. Forty-five trees of three deciduous species, Gleditsia triacanthos, Quercus macrocarpa, Metasequoia glyptostroboides, were sampled on the Michigan State University campus. We studied their structural complexity by calculating the box-dimension (Db) metric from point clouds generated for the trees using terrestrial laser scanning, during the leaf-on and -off conditions. Furthermore, we artificially defoliated the leaf-on point clouds by applying an algorithm that separates the foliage from the woody material of the trees, and then recalculated the Db metric. The Db of the leaf-on tree point clouds was significantly greater than the Db of the leaf-off point clouds across all species. Additionally, the leaf removal algorithm introduced bias to the estimation of the leaf-removed Db of the G. triacanthos and M. glyptostroboides trees. The index capturing the contribution of leaves to the structural complexity of the study trees (the ratio of the Db of the leaf-on point clouds divided by the Db of the leaf-off point clouds minus one), was negatively correlated with branch surface area and different metrics of the length of paths through the branch network of the trees, indicating that the contribution of leaves decreases as branch network complexity increases. Underestimation of the Db of the G. triacanthos trees, after the artificial leaf removal, was related to maximum branch order. These results enhance our understanding of tree structural complexity by disentangling the contribution of leaves from that of the woody structures. The study also highlighted important methodological considerations for studying tree structure, with and without leaves, from laser-derived point clouds.

How neurons exploit fractal geometry to optimize their network connectivity

We investigate the degree to which neurons are fractal, the origin of this fractality, and its impact on functionality. By analyzing three-dimensional images of rat neurons, we show the way their dendrites fork and weave through space is unexpectedly important for generating fractal-like behavior well-described by an ‘effective’ fractal dimension D. This discovery motivated us to create distorted neuron models by modifying the dendritic patterns, so generating neurons across wide ranges of D extending beyond their natural values. By charting the D-dependent variations in inter-neuron connectivity along with the associated costs, we propose that their D values reflect a network cooperation that optimizes these constraints. We discuss the implications for healthy and pathological neurons, and for connecting neurons to medical implants. Our automated approach also facilitates insights relating form and function, applicable to individual neurons and their networks, providing a crucial tool for addressing massive data collection projects (e.g. connectomes).

Switch of fractal properties of DNA in chicken erythrocytes nuclei by mechanical stress

The small-angle neutron scattering (SANS) on the chicken erythrocyte nuclei demonstrates the bifractal nature of the chromatin structural organization. Use of the contrast variation (D_{2}O-H_{2}O) in SANS measurements reveals the differences in the DNA and protein arrangements inside the chromatin substance. It is the DNA that serves as a framework that constitutes the bifractal behavior showing the mass fractal properties with D=2.22 at a smaller scale and the logarithmic fractal behavior with D≈3 at a larger scale. The protein spatial organization shows the mass fractal properties with D≈2.34 throughout the whole nucleus. The borderline between two fractal levels can be significantly shifted toward smaller scales by centrifugation of the nuclei disposed on the dry substrate, since nuclei suffer from mechanical stress transforming them to a disklike shape. The height of this disk measured by atomic force microscopy (AFM) coincides closely with the fractal borderline, thus characterizing two types of the chromatin with the soft (at larger scale) and rigid (at smaller scale) properties. The combined SANS and AFM measurements demonstrate the stress induced switch of the DNA fractal properties from the rigid, but loosely packed, mass fractal to the soft, but densely packed, logarithmic fractal.

An Integrated Methodology to Study Riparian Vegetation Dynamics: From Field Data to Impact Modeling

Riparian environments are highly dynamic ecosystems that support biodiversity and numerous services and that are conditioned by anthropogenic activities and climate change. In this work, we propose an integrated methodology that combines different research approaches-field studies and numerical and analytical modeling-in order to calibrate an ecohydrological stochastic model for riparian vegetation. The model yields vegetation biomass statistics and requires hydrological, topographical, and biological data as input. The biological parameters, namely, the carrying capacity and the flood-related decay rate, are the target of the calibration as they are related to intrinsic features of vegetation and site-specific environmental conditions. The calibration is here performed for two bars located within the riparian zone of the Cinca River (Spain). According to our results, the flood-related decay rate has a spatial dependence that reflects the zonation of different plant species over the study site. The carrying capacity depends on the depth of the phreatic surface, and it is adequately described by a right-skewed curve. The calibrated model well reproduces the actual biogeography of the Cinca riparian zone. The overall percentage absolute difference between the real and the computed biomass amounts to 9.3% and 3.3% for the two bars. The model is further used to predict the future evolution of riparian vegetation in a climate-change scenario. The results show that the change of hydrological regime forecast by future climate projections may induce dramatic reduction of vegetation biomass and strongly modify the Cinca riparian biogeography.

From Dendrochronology to Allometry

The contribution of tree-ring analysis to other fields of scientific inquiry with overlapping interests, such as forestry and plant population biology, is often hampered by the different parameters and methods that are used for measuring growth. Here I present relatively simple graphical, numerical, and mathematical considerations aimed at bridging these fields, highlighting the value of crossdating. Lack of temporal control prevents accurate identification of factors that drive wood formation, thus crossdating becomes crucial for any type of tree growth study at inter-annual and longer time scales. In particular, exactly dated tree rings, and their measurements, are crucial contributors to the testing and betterment of allometric relationships.

Mechanics of a plant in fluid flow

Plants live in constantly moving fluid, whether air or water. In response to the loads associated with fluid motion, plants bend and twist, often with great amplitude. These large deformations are not found in traditional engineering application and thus necessitate new specialized scientific developments. Studying fluid-structure interaction (FSI) in botany, forestry, and agricultural science is crucial to the optimization of biomass production for food, energy, and construction materials. FSIs are also central in the study of the ecological adaptation of plants to their environment. This review paper surveys the mechanics of FSI on individual plants. I present a short refresher on fluid mechanics then dive into the statics and dynamics of plant-fluid interactions. For every phenomenon considered, I examine the appropriate dimensionless numbers to characterize the problem, discuss the implications of these phenomena on biological processes, and propose future research avenues. I cover the concept of reconfiguration while considering poroelasticity, torsion, chirality, buoyancy, and skin friction. I also assess the dynamical phenomena of wave action, flutter, and vortex-induced vibrations.

Feeling Stressed or Strained? A Biophysical Model for Cell Wall Mechanosensing in Plants

Mechanical signals have recently emerged as a major cue in plant morphogenesis, notably influencing cytoskeleton organization, gene expression, protein polarity, or cell division. Although many putative mechanosensing proteins have been identified, it is unclear what mechanical cue they might sense and how this would occur. Here we briefly explain the notions of mechanical stress and strain. We present the challenges to understand their sensing by plants, focusing on the cell wall and the plasma membrane, and we review putative mechanosensing structures. We propose minimal biophysical models of mechanosensing, revealing the modes of mechanosensing according to mechanosensor lifetime, threshold force for mechanosensor dissociation, and type of association between the mechanosensor and the cell wall, as the sensor may be associated to a major load-bearing structure such as cellulose or to a minor load-bearing structure such as pectins or the plasma membrane. Permanent strain, permanent expansion, and relatively slow variations thereof are sensed in all cases; variations of stress are sensed in all cases; permanent stress is sensed only in the following specific cases: sensors associated to minor load-bearing structures slowly relaxing in a growing wall, long-lived sensors with high dissociation force and associated to major-load-bearing structures, and sensors with low dissociation force associated to major-load-baring structures behaving elastically. We also find that all sensors respond to variations in the composition or the mechanical properties of the cell wall. The level of sensing is modulated by the properties of all of mechanosensor, cell wall components, and plasma membrane. Although our models are minimal and not fully realistic, our results yield a framework to start investigating the possible functions of putative mechanosensors.

The pipe model theory half a century on: a review

Background

More than a half century ago, Shinozaki et al. (Shinozaki K, Yoda K, Hozumi K, Kira T. 1964a. A quantitative analysis of plant form - the pipe model theory. I. Basic analyses. Japanese Journal of Ecology B: 97-105) proposed an elegant conceptual framework, the pipe model theory (PMT), to interpret the observed linear relationship between the amount of stem tissue and corresponding supported leaves. The PMT brought a satisfactory answer to two vividly debated problems that were unresolved at the moment of its publication: (1) What determines tree form and which rules drive biomass allocation to the foliar versus stem compartments in plants? (2) How can foliar area or mass in an individual plant, in a stand or at even larger scales be estimated? Since its initial formulation, the PMT has been reinterpreted and used in applications, and has undoubtedly become an important milestone in the mathematical interpretation of plant form and functioning.

Scope

This article aims to review the PMT by going back to its initial formulation, stating its explicit and implicit properties and discussing them in the light of current biological knowledge and experimental evidence in order to identify the validity and range of applicability of the theory. We also discuss the use of the theory in tree biomechanics and hydraulics as well as in functional-structural plant modelling.

Conclusions

Scrutinizing the PMT in the light of modern biological knowledge revealed that most of its properties are not valid as a general rule. The hydraulic framework derived from the PMT has attracted much more attention than its mechanical counterpart and implies that only the conductive portion of a stem cross-section should be proportional to the supported foliage amount rather than the whole of it. The facts that this conductive portion is experimentally difficult to measure and varies with environmental conditions and tree ontogeny might cause the commonly reported non-linear relationships between foliage and stem metrics. Nevertheless, the PMT can still be considered as a portfolio of properties providing a unified framework to integrate and analyse functional-structural relationships.

Wind loads and competition for light sculpt trees into self-similar structures

Trees are self-similar structures: their branch lengths and diameters vary allometrically within the tree architecture, with longer and thicker branches near the ground. These tree allometries are often attributed to optimisation of hydraulic sap transport and safety against elastic buckling. Here, we show that these allometries also emerge from a model that includes competition for light, wind biomechanics and no hydraulics. We have developed MECHATREE, a numerical model of trees growing and evolving on a virtual island. With this model, we identify the fittest growth strategy when trees compete for light and allocate their photosynthates to grow seeds, create new branches or reinforce existing ones in response to wind-induced loads. Strikingly, we find that selected trees species are self-similar and follow allometric scalings similar to those observed on dicots and conifers. This result suggests that resistance to wind and competition for light play an essential role in determining tree allometries.

Tree branches follow allometric scalings between length, thickness and dry mass. Here, Eloy and colleagues develop a functional-structural model that shows how such allometries in tree architecture can emerge through evolution as a result of competition for light, wind biomechanics, and wind sensing.

The hydraulic architecture of Ginkgo leaves

PREMISE OF THE STUDY

The hydraulics of xylem has been widely studied in numerous species and organ types. However, comparatively little is known about how phloem and xylem are hydraulically coupled or about many of the basic structural properties of phloem (such as conducting cell numbers and conductive areas), which nevertheless have direct bearing on understanding phloem loading and unloading.

METHODS

Using a combination of light, epifluorescence, confocal, and transmission electron microscopy, we quantified the hydraulic architecture of Ginkgo biloba leaf laminae and examined the scaling relationships between phloem and xylem in five fully mature leaves.

KEY RESULTS

The conductive areas and lengths of sieve cells and tracheids increase basipetally toward the petiole in a manner that is consistent with Münch's pressure flow hypothesis for phloem transport. This trend holds true for individual veins, the sum of conductive areas across all veins at any distance from the petiole, and for individual sieve cells and tracheids. Further, the conductive areas of phloem and xylem are isometrically correlated across the entire vasculature of the leaf lamina. The data for conducting cell areas do not conform with the predictions of the hydraulic models of da Vinci and Murray.

CONCLUSIONS

The scaling of Ginkgo lamina hydraulics complies with that observed in leaves of other gymnosperms and most angiosperms and is inconsistent with theoretical models that assume that the volume of transported incompressible fluids is conserved.

High-Resolution Laser Scanning Reveals Plant Architectures that Reflect Universal Network Design Principles

Transport networks serve critical functions in biological and engineered systems, and yet their design requires trade-offs between competing objectives. Due to their sessile lifestyle, plants need to optimize their architecture to efficiently acquire and distribute resources while also minimizing costs in building infrastructure. To understand how plants resolve this design trade-off, we used high-precision three-dimensional laser scanning to map the architectures of tomato, tobacco, or sorghum plants grown in several environmental conditions and through multiple developmental time points, scanning in total 505 architectures from 37 plants. Using a graph-theoretic algorithm that we developed to evaluate design strategies, we find that plant architectures lie along the Pareto front between two simple length-based objectives-minimizing total branch length and minimizing nutrient transport distance-thereby conferring a selective fitness advantage for plant transport processes. The location along the Pareto front can distinguish among species and conditions, suggesting that during evolution, natural selection may employ common network design principles despite different optimization trade-offs.

Additive scaling law for structural organization of chromatin in chicken erythrocyte nuclei

Small-angle neutron scattering (SANS) on nuclei of chicken erythrocytes demonstrates the cubic dependence of the scattering intensity Q^{-3} in the range of momentum transfer Q∈10^{-3}-10^{-2}nm^{-1}. Independent spin-echo SANS measurements give the spin-echo function, which is well described by the exponential law in a range of sizes (3×10^{2})-(3×10^{4}) nm. Both experimental dependences reflect the nature of the structural organization of chromatin in the nucleus of a living cell, which corresponds to the correlation function γ(r)=ln(ξ/r) for r<ξ, where ξ=(3.69±0.07)×10^{3} nm, the size of the nucleus. It has the specific scaling property of the logarithmic fractal γ(r/a)=γ(r)+ln(a), i.e., the scaling down by a gives an additive constant to the correlation function, which distinguishes it from the mass fractal, which is characterized by multiplicative constant.

Convergence in leaf size versus twig leaf area scaling: do plants optimize leaf area partitioning?

BACKGROUND AND AIMS

Corner's rule states that thicker twigs bear larger leaves. The exact nature of this relationship and why it should occur has been the subject of numerous studies. It is obvious that thicker twigs should support greater total leaf area ([Formula: see text]) for hydraulical and mechanical reasons. But it is not obvious why mean leaf size ([Formula: see text]) should scale positively with [Formula: see text] We asked what this scaling relationship is within species and how variable it is across species. We then developed a model to explain why these relationships exist.

METHODS

To minimize potential sources of variability, we compared twig properties from six co-occurring and functionally similar species: Acer grandidentatum, Amelanchier alnifolia, Betula occidentalis, Cornus sericea, Populus fremontii and Symphoricarpos oreophilus We modelled the economics of leaf display, weighing the benefit from light absorption against the cost of leaf tissue, to predict the optimal [Formula: see text] combinations under different canopy openings.

KEY RESULTS

We observed a common [Formula: see text] by [Formula: see text] exponent of 0.6, meaning that [Formula: see text]and leaf number on twigs increased in a specific coordination. Common scaling exponents were not supported for relationships between any other measured twig properties. The model consistently predicted positive [Formula: see text] by [Formula: see text] scaling when twigs optimally filled canopy openings. The observed 0·6 exponent was predicted when self-shading decreased with larger canopy opening.

CONCLUSIONS

Our results suggest Corner's rule may be better understood when recast as positive [Formula: see text] by [Formula: see text] scaling. Our model provides a tentative explanation of observed [Formula: see text] by [Formula: see text] scaling and suggests different scaling may exist in different environments.

How do plants read their own shapes?

Contents 333 I. 333 II. 334 III. 334 IV. 336 336 References 337 SUMMARY: Although the sensing of shape and deformation was historically involved in the control of animal locomotion, it is now increasingly being incorporated in developmental biology. Proprioception, the perception of the self, is particularly key to the question of the reproducibility of shapes: the many regulators of growth may lead to a large array of geometries, but shape sensing restricts these diverse outputs to a limited number of forms. Mechanistically, and in addition to geometrical feedback onto the diffusion and transport of molecular factors, we highlight the role of shape-derived mechanical stress and strain in this process. Through examples at the cell, tissue and organism scales, it appears that such mechanical feedback adds robustness to morphogenesis. Interestingly, synergies exist between shape sensing and response to external cues, such as wind and gravity. Understanding the molecular basis of proprioception is now within reach and opens up many avenues for an integrative view of development.

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.

Critical wind speed at which trees break

Data from storms suggest that the critical wind speed at which trees break is constant (≃42m/s), regardless of tree characteristics. We question the physical origin of this observation both experimentally and theoretically. By combining Hooke's law, Griffith's criterion, and tree allometry, we show that the critical wind speed indeed hardly depends on the height, diameter, and elastic properties of trees.

Optical Coherence Tomography Angiography Signs of Vascular Abnormalization With Antiangiogenic Therapy for Choroidal Neovascularization

PURPOSE

To investigate the vascular appearance of choroidal neovascularization (CNV) treated with recurrent intravitreous anti-vascular endothelial growth factor (VEGF) injections, which have been proposed to cause transient vascular normalization along with decreased vascularity and leakage.

DESIGN

Retrospective case series with perspective on the topic.

METHODS

Patients with treated CNV secondary to age-related macular degeneration from a community-based retinal referral practice were evaluated with optical coherence tomography angiography employing split-spectrum amplitude decorrelation. The choroidal neovascular morphology of the 17 eyes of 14 consecutive patients was described.

RESULTS

The mean age of the patients, 8 men and 6 women, was 78.4 (standard deviation ± 9.3) years. The mean greatest linear dimension of the lesion was 3600 μm. The mean number of anti-VEGF injections was 47 (±21). The vascular diameter of the vessels in the CNV appeared large even in small lesions, with feeder vessels approaching the size of the major arcade vessels of the retina. The vessels had few branch points and many vascular anastomotic connections among larger vessels. There was a paucity of capillaries visualized within the lesions.

CONCLUSIONS

The findings of this study do not support the hypothesis of vascular normalization in eyes receiving recurrent periodic antiangiogenic treatment. The observed "abnormalization" of the vessels may be explained by periodic pruning of angiogenic vascular sprouts by VEGF withdrawal in the face of unimpeded arteriogenesis. As the eye is a readily accessible VEGF laboratory, features expressed therein may also apply to neovascularization elsewhere in the body, such as in tumors.

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.

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.

Avoiding steric congestion in dendrimer growth through proportionate branching: a twist on da Vinci's rule of tree branching

Making defect-free macromolecules is a challenging issue in chemical synthesis. This challenge is especially pronounced in dendrimer synthesis where exponential growth quickly leads to steric congestion. To overcome this difficulty, proportionate branching in dendrimer growth is proposed. In proportionate branching, both the number and the length of branches increase exponentially but in opposite directions to mimic tree growth. The effectiveness of this strategy is demonstrated through the synthesis of a fluorocarbon dendron containing 243 chemically identical fluorine atoms with a MW of 9082 Da. Monodispersity is confirmed by nuclear magnetic resonance spectroscopy, mass spectrometry, and small-angle X-ray scattering. Growing different parts proportionately, as nature does, could be a general strategy to achieve defect-free synthesis of macromolecules.