Mechanical stability of trees under dynamic loads

Tree stability in windstorms and tree failure are important issues in urban areas where there can be risks of damage to people and property and in forests where wind damage causes economic loss. Current methods of managing trees, including pruning and assessment of mechanical strength, are mainly based on visual assessment or the experience of people such as trained arborists. Only limited data are available to assess tree strength and stability in winds, and most recent methods have used a static approach to estimate loads. Recent research on the measurement of dynamic wind loads and the effect on tree stability is giving a better understanding of how different trees cope with winds. Dynamic loads have been measured on trees with different canopy shapes and branch structures including a palm (Washingtonia robusta), a slender Italian cypress (Cupressus sempervirens) and trees with many branches and broad canopies including hoop pine (Araucaria cunninghamii) and two species of eucalypt (Eucalyptus grandis, E. teretecornus). Results indicate that sway is not a harmonic, but is very complex due to the dynamic interaction of branches. A new dynamic model of a tree is described, incorporating the dynamic structural properties of the trunk and branches. The branch mass contributes a dynamic damping, termed mass damping, which acts to reduce dangerous harmonic sway motion of the trunk and so minimizes loads and increases the mechanical stability of the tree. The results from 12 months of monitoring sway motion and wind loading forces are presented and discussed.     

Quantifying the degree of self-nestedness of trees: application to the structural analysis of plants

In this paper, we are interested in the problem of approximating trees by trees with a particular self-nested structure. Self-nested trees are such that all their subtrees of a given height are isomorphic. We show that these trees present remarkable compression properties, with high compression rates. In order to measure how far a tree is from being a self-nested tree, we then study how to quantify the degree of self-nestedness of any tree. For this, we define a measure of the self-nestedness of a tree by constructing a self-nested tree that minimizes the distance of the original tree to the set of self-nested trees that embed the initial tree. We show that this measure can be computed in polynomial time and depict the corresponding algorithm. The distance to this nearest embedding self-nested tree (NEST) is then used to define compression coefficients that reflect the compressibility of a tree. To illustrate this approach, we then apply these notions to the analysis of plant branching structures. Based on a database of simulated theoretical plants in which different levels of noise have been introduced, we evaluate the method and show that the NESTs of such branching structures restore partly or completely the original, noiseless, branching structures. The whole approach is then applied to the analysis of a real plant (a rice panicle) whose topological structure was completely measured. We show that the NEST of this plant may be interpreted in biological terms and may be used to reveal important aspects of the plant growth.     

Arterial branching within the confines of fractal L-system formalism

Parametric Lindenmayer systems (L-systems) are formulated to generate branching tree structures that can incorporate the physiological laws of arterial branching. By construction, the generated trees are de facto fractal structures, and with appropriate choice of parameters, they can be made to exhibit some of the branching patterns of arterial trees, particularly those with a preponderant value of the asymmetry ratio. The question of whether arterial trees in general have these fractal characteristics is examined by comparison of pattern with vasculature from the cardiovascular system. The results suggest that parametric L-systems can be used to produce fractal tree structures but not with the variability in branching parameters observed in arterial trees. These parameters include the asymmetry ratio, the area ratio, branch diameters, and branching angles. The key issue is that the source of variability in these parameters is not known and, hence, it cannot be accurately reproduced in a model. L-systems with a random choice of parameters can be made to mimic some of the observed variability, but the legitimacy of that choice is not clear.     

Further characterization of a brain high molecular weight actin-binding protein (BABP): interaction with brain actin and ultrastructural studies

Crude actomyosin fraction from porcine brain contained a large amount of high molecular weight actin-binding protein (BABP). The molar ratio of BABP to actin (BABP/actin) in the fraction was estimated to be 0.22. From this fraction, BABP and actin were solubilized with a molar ratio of 0.25, suggesting the existence of an interaction between BABP and brain actin. BABP was finally purified to 90% purity. The purified BABP was negatively stained and observed by electron microscopy; it appeared to be a slender, flexible, two-stranded molecule whose contour length was about 200 nm. The structure was very similar to those of fodrin and other high molecular weight actin-binding proteins such as filamin, spectrin, and ABP. Lattice cage-like structures composed of BABP molecules were occasionally observed at high BABP concentrations. The addition of BABP to actin filaments resulted in the appearance of many branching, filamentous bundles. The electron microscopic observations suggested that a single BABP molecule could crosslink actin filaments, that is, one BABP molecule has two actin binding sites.     

Branch geometry in Cornus kousa (Cornaceae): computer simulations

Computer simulations similar to actual trees were constructed using simple branching rules. Branch orientation with respect to the direction of gravity was a fundamental consideration. In Cornus kousa BUERG. ex HANCE, several types of branches develop from winter buds, varying from orthotropic shoots to plagiotropic ones. Based on actual observations and measurements of branching structures with a wide range of orientations, we made a flexible geometrical model consisting of five forking branches that varied in outgrowth depending on the direction of the shoot with respect to gravity. Repetition of the branching by computer generated a realistic tree pattern, which was close to the shape of a young C. kousa tree. Reproductive shoots seem to be under a branching rule that was a modification of vegetative branching, although the reproductive branch size was considerably smaller than the vegetative one, and reproductive branching was bifurcated instead of five-forked. We conclude that all branchings in orthotropic and plagiotropic shoots in the vegetative phase and shoots in the reproductive phase are formed under the same branching rule, but each has different parameter values.     

Structure and Function in Homodimeric Enzymes: Simulations of Cooperative and Independent Functional Motions

Large-scale conformational change is a common feature in the catalytic cycles of enzymes. Many enzymes function as homodimers with active sites that contain elements from both chains. Symmetric and anti-symmetric cooperative motions in homodimers can potentially lead to correlated active site opening and/or closure, likely to be important for ligand binding and release. Here, we examine such motions in two different domain-swapped homodimeric enzymes: the DcpS scavenger decapping enzyme and citrate synthase. We use and compare two types of all-atom simulations: conventional molecular dynamics simulations to identify physically meaningful conformational ensembles, and rapid geometric simulations of flexible motion, biased along normal mode directions, to identify relevant motions encoded in the protein structure. The results indicate that the opening/closure motions are intrinsic features of both unliganded enzymes. In DcpS, conformational change is dominated by an anti-symmetric cooperative motion, causing one active site to close as the other opens; however a symmetric motion is also significant. In CS, we identify that both symmetric (suggested by crystallography) and asymmetric motions are features of the protein structure, and as a result the behaviour in solution is largely non-cooperative. The agreement between two modelling approaches using very different levels of theory indicates that the behaviours are indeed intrinsic to the protein structures. Geometric simulations correctly identify and explore large amplitudes of motion, while molecular dynamics simulations indicate the ranges of motion that are energetically feasible. Together, the simulation approaches are able to reveal unexpected functionally relevant motions, and highlight differences between enzymes.     

Branching structure in arborescent animals: Models of relative growth

Many invertebrate animals belonging to diverse phyla grow as regularly branching structures with the general appearance of miniature trees. If it is assumed that regularity of branching implies regularity in growth, models can be mathematically derived to depict growth of such a structure as a set of changing morphologic properties. Modes of growth, branching properties, and growth models can be expected to differ markedly from one major taxonomic group to another. Nevertheless, these properties can furnish a useful basis for comparing adaptive morphologies and underlying mechanical designs not only among arborescent animals, but with arborescent plants as well.Branching structures of some cheilostome bryozoans with rigidly erect, arborescent growth habits are inferred to result from continuous growth at steadily increasing numbers of growing tips through a process of repeated bifurcation and lengthening. In a model of continuous growth, the pattern by which the number of growing tips increases can be shown to be a generalized mathematical series, of which the Fibonacci series and a geometric series are two special cases. The quantities which determine the series can be calculated from measurable properties of the branching structure: lengths of paired branch portions ending in growing tips (relative growth ratio), lengths of paired branch portions between bifurcations (mean link length and link-length ratio), and numbers of branch portions belonging to different orders (branching ratio). Data for eight species of cheilostome bryozoans indicate, with high levels of confidence, that measurable branching properties and the models of relative growth inferred from them are species-specific. This specificity and a tendency to adhere to characteristic values of branching properties during growth are apparently direct expressions of internal control in these bryozoans.     

Morphometric analysis of root systems: application of the technique and influence of soil fertility on root system development in two herbaceous species

Abstract. A method for describing root systems based on geomorphological techniques developed for river systems is described. Root systems, in common with other natural branching structures (rivers, bronchioles, trees), appear to obey Morton's Law of Branching: there is a constant ratio, the bifurcation or branching ratio, R_b, between the number of branches of a given order, N_u , and that of the next order. N_u+1 , In experiments where Poa annua , and Rumex cripus , were grown at two levels of fertility, the first-order roots (the youngest members in this system) were generally unresponsive to fertility, and differences in the root systems were largely the result of changes in the second-order roots, those formed at the junction of two first-order roots. These differences were reflected in the branching ratio, R_b Although it is possible to explain these results by a stochastic model of branch development, the R_b values for roots are higher than for other natural branching structures, and higher than the random model predicts. It is possible that a model based on optimum exploration of space may be more appropriate and provide a key to the factors governing root branching patterns.     

On fractal properties of arterial trees

The question of fractal properties of arterial trees is considered in light of data from the extensive tree structure of the right coronary artery of a human heart. Because of the highly non-uniform structure of this tree, the study focuses on the purely geometrical rather than statistical aspects of fractal properties. The large number of arterial bifurcations comprising the tree were found to have a mixed degree of asymmetry at all levels of the tree, including the depth of the tree where it has been generally supposed that they would be symmetrical. Cross-sectional area ratios of daughter to parent vessels were also found to be highly mixed at all levels, having values both above and below 1.0, rather than consistently above as has been generally supposed in the past. Calculated values of the power law index which describes the theoretical relation between the diameters of the three vessel segments at an arterial bifurcation were found to range far beyond the two values associated with the cube and square laws, and not clearly favoring one or the other. On the whole the tree structure was found to have what we have termed "pseudo-fractal" properties, in the sense that vessels of different calibers displayed the same branching pattern but with a range of values of the branching parameters. The results suggest that a higher degree of fractal character, one in which the branching parameters are constant throughout the tree structure, is unlikely to be attained in non-uniform vascular structures.     

Temperature Fluctuation in Wintering Trees

To investigate the mechanism of frost damage in trees, the temperature fluctualions in the stems and leaves of some wintering trees were recorded with copperconstantan thermocouples. In Sapporo, even the trunk of large elm trees with diameter of 86 cm are frozen during the winter. In a Kalopanax trunk with a diameter of 13.5 cm, the bark temperature on the south side which is exposed to direct sunshine reaches nearly 20°C about midday in midwinter; while, on the north side, the temperature remains nearly the same as the environmental temperature (0 to -5°C). The rise in the bark temperature in trees is considerably affected by factors such as the intensity of sunshine, the environmental temperature, the diameter of the trunk, the side of the trunk which the bark is on, the height above the ground, and the colour of the bark surface, etc. This rise is far less in small twigs, slender stems, and small leaves than in large ones. The south side of the bark 10 to 15 cm above the snow surface or above the ground in a slender stem is exposed to a remarkable fluctuation in temperature, especially when the ground is covered with snow. Even in northern trees, the cortical cells on the south side of trunks and twigs are less resistant to freezing than those on the north.     

A physics-based link model for tree vibrations

• Premise of the study: A new mathematical model for the vibration of trees is presented for developing a more thorough understanding of the underlying structure of the response. It may be used, for example, to assess the stability of a tree or to interpret experimental data. • Methods: A model is developed for the motion of the trunk and its N number of branches. The spatial distribution and initial orientation of the branches are left for the user to prescribe. A Newtonian analysis yields (N + 1) nonlinear, coupled differential equations that, when solved, describe the response of the trunk and each branch. After the model is linearized near equilibrium, the natural frequencies and vibration mode shapes are found. Closed-form expressions for the response (i.e., the actual time histories) are then obtained using modal analysis. Numerical solutions are also found; these are used to validate the analytical solutions and to serve as a means for considering large amplitude vibrations. • Key results: A new physics-based model is described. For small motion, the tree response may be constructed from the individual mode shapes and frequencies. Also demonstrated are the limitations of the linear theory as well as numerical solutions that can be obtained when trunk/branch amplitudes are large. • Conclusions: The model presented here incorporates critical physics into a model that describes tree vibrations. It also sheds light on the underlying structure of the vibration response in terms of the modal nature of the solution. Limitations to the linear solutions are demonstrated and discussed.     

Flutter in collapsible tubes: a theoretical model of wheezes

A mathematical analysis of flow through a flexible channel is examined as a model of flow-induced flutter oscillations that pertain to the production of wheezing breath sounds. The model provides predictions for the critical fluid speed that will initiate flutter waves of the wall, as well as their frequency and wavelength. The mathematical results are separated into linear theory (small oscillations) and nonlinear theory (larger oscillations). Linear theory determines the onset of the flutter, whereas nonlinear theory determines the relationships between the fluid speed and both the wave amplitudes and frequencies. The linear theory predictions correlate well with data taken at the onset of flutter and flow limitation during experiments of airflow in thick-walled collapsible tubes. The nonlinear theory predictions correlate well with data taken as these flows are forced to higher velocities while keeping the flow rate constant. Particular ranges of the parameters are selected to investigate and discuss the applications to airway flows. According to this theory, the mechanism of generation of wheezes is based in the interactions of fluid forces and friction and wall elastic-restoring forces and damping. In particular, a phase delay between the fluid pressure and wall motion is necessary. The wave speed theory of flow limitation is discussed with respect to the specific data and the flutter model.     

Analysis of a data set of paired uncomplexed protein structures: new metrics for side-chain flexibility and model evaluation

We compiled and analyzed a data set of paired protein structures containing proteins for which multiple high-quality uncomplexed atomic structures were available in the Protein Data Bank. Side-chain flexibility was quantified, yielding a set of residue- and environment-specific confidence levels describing the range of motion around chi1 and chi2 angles. As expected, buried residues were inflexible, adopting similar conformations in different crystal structure analyses. Ile, Thr, Asn, Asp, and the large aromatics also showed limited flexibility when exposed on the protein surface, whereas exposed Ser, Lys, Arg, Met, Gln, and Glu residues were very flexible. This information is different from and complementary to the information available from rotamer surveys. The confidence levels are useful for assessing the significance of observed side-chain motion and estimating the extent of side-chain motion in protein structure prediction. We compare the performance of a simple 40 degrees threshold with these quantitative confidence levels in a critical evaluation of side-chain prediction with the program SCWRL.     

Theoretical relationship between the optimal models of the vascular tree

Two models of optimal branching structure of the vascular tree are compared. Murray’s minimum work model derived from minimum energy loss due to flow and volume in the duct system is proved to be included as a mathematical group in the authors’ model defined by the minimum volume under determinant pressure, flow and position at the terminals. The problem about heterotypical trees which are identical at the terminal conditions but different in the topological order of branch combinations are discussed, applying the results of analyses on the equivalent duct of uniform terminal pressure trees. It is proved that the minimum work tree has the least energy loss compared with its heterotypical minimum volume trees and is a better model of branching structure of the vascular tree.     

The effect of airway wall motion on surfactant delivery

Soluble surfactant and airway surface liquid transport are examined using a mathematical model of Marangoni flows which accounts for airway branching and for cyclic airway stretching. Both radial and longitudinal wall strains are considered. The model allows for variation of the amplitude and frequency of the motion, as may occur under a variety of ventilatory situations occurring during surfactant replacement therapy. The soluble surfactant dynamics of the thin fluid film are modeled by linear sorption. The delivery of surfactants into the lung is handled by setting the proximal boundary condition to a higher concentration compared to the distal boundary condition. Starting with a steady-state, nonuniform, surfactant distribution, we find that transport of surfactant into the lung is enhanced for increasing strain amplitudes. However, for fixed amplitude, increasing frequency has a smaller effect. At small strain amplitudes, increasing frequency enhances transport, but at large strain amplitudes, increasing cycling frequency has the opposite effect.     

Stem and root anatomical correlations with life form diversity, ecology, and systematics in Moringa (Moringaceae)

Four life forms (habits) are identified in the 13 species of Moringa (bottle trees, sarcorhizal trees, slender trees, and tuberous shrubs) which are examined for wood anatomical correlations with habit, ecology, and systematic. Wood anatomy is similar within habit classes except for the sarcorhizal trees. The four bottle tree species and M. arborea (one of the sarcorhizal trees) are characterized by bands of confluent paratracheal parenchyma alternating with bands of libriform fibres, some of which may be parenchyma-like. The other sarcorhizal tree, M. ruspoliana , is characterized by alternating bands of parenchyma-like and long, slender libriform fibres. Root secondary xylem of all these species is characterized by bands of parenchyma and fibres. Slender trees do not show bands of fibres of different shapes and have fibrous roots with less parenchyma than the other species. Tuberous shrubs have stems mostly composed of long, slender fibres and large underground tubers mostly composed of parenchyma. Quantitative trends between ecologically different localities include wider vessel elements and higher conductive area in moister localities. Wood anatomy provides characters that are of potential phylogenetic utility at a variety of levels of relationship. Based on wood anatomy and geography, the most likely sister taxon to Moringa is Cylicomorpha (Caricaceae).     

LIGNUM: A Tree Model Based on Simple Structural Units

The model LIGNUM treats a tree as a collection of a large numberof simple units which correspond to the organs of the tree.The model describes the three dimensional structure of the treecrown and defines the growth in terms of the metabolism takingplace in these units. The activities of physiological processescan be explicitly related to the tree structures in which theyare taking place. The time step is 1 year. The crown of the model tree consists of tree segments, branchingpoints and buds. Each pair of tree segments is separated bya branching point. The buds produce new tree segments, branchingpoints and buds. The tree segments contain wood, bark and foliage.A model tree consisting of simple elements translates convenientlyto a list structure: the computer program implementing LIGNUMtreats the tree as a collection of lists. The annual growth of the tree is driven by available photosyntheticproducts after respiration losses are accounted for. The photosyntheticrate of foliage depends on the amount of light. The amount ofphotosynthates allocated to the growth of new tree segmentsis controlled by the light conditions and the amount of foliageon the mother tree segment. In principle, the biomass relationshipsof the tree parts follow the pipe model hypothesis. The orientationof new tree segments results from the application of constantbranching angles. LIGNUM has been parametrized for young Scotspine (Pinus sylvestrisL.) trees. However, the model is generic;with a change of parameter values and minor modifications itcan be applied to other species as well. Growth model; object-oriented modelling; tree architecture; branching structure; Pinus sylvestrisL.; developmental morphology and physiology; photosynthesis; respiration     

Reprint of: Tree-tree interactions and crown complementarity: the role of functional diversity and branch traits for canopy packing

Previous studies have shown that tree species richness increases forest productivity by allowing for greater spatial complementarity of tree crowns (crown complementarity), which in turn results in more densely packed canopies. However, the mechanisms driving crown complementarity in tree species mixtures remain unclear. Here, we take advantage of a high-resolution, three-dimensional terrestrial laser scanning approach in the context of a large-scale biodiversity-ecosystem functioning experiment in subtropical China (BEF-China) to quantify the extent to which functional dissimilarity and divergences in branch traits between neighbouring trees affect crown complementarity at the scale of tree species pairs (i.e., two adjacent trees). Overall, we found no support that functional dissimilarity (divergence in morphological flexibility, specific leaf area and wood density) promotes crown complementarity. However, we show that the effects of functional dissimilarity (the plasticity of the outer crown structure) on crown complementarity vary in their magnitude and importance depending on branch trait divergences. Firstly, crown complementarity tended to be highest for tree species pairs that strongly differed in their functional traits, but were similar in branch density. In contrast, heterospecific pairs with a low functional trait divergence benefitted the most from a large difference in branch density compared with pairs characterised by a large functional dissimilarity. Secondly, the positive effects of increasing divergence in branching intensity (the plasticity of the inner crown structure) on crown complementarity became most important at low levels of functional dissimilarity, i.e. when species pairs were similar in their branch packing and vice versa. This suggests that species mixing allows trees to occupy canopy space more efficiently mainly due to phenotypic changes associated with crown morphology and branch plasticity. Our findings highlight the importance of considering outer and inner crown structures (e.g. branching architecture) to deepen our understanding of tree-tree interactions in mixed-species communities.     

Tree-tree interactions and crown complementarity: The role of functional diversity and branch traits for canopy packing

Previous studies have shown that tree species richness increases forest productivity by allowing for greater spatial complementarity of tree crowns (crown complementarity), which in turn results in more densely packed canopies. However, the mechanisms driving crown complementarity in tree species mixtures remain unclear. Here, we take advantage of a high-resolution, three-dimensional terrestrial laser scanning approach in the context of a large-scale biodiversity-ecosystem functioning experiment in subtropical China (BEF-China) to quantify the extent to which functional dissimilarity and divergences in branch traits between neighbouring trees affect crown complementarity at the scale of tree species pairs (i.e., two adjacent trees). Overall, we found no support that functional dissimilarity (divergence in morphological flexibility, specific leaf area and wood density) promotes crown complementarity. However, we show that the effects of functional dissimilarity (the plasticity of the outer crown structure) on crown complementarity vary in their magnitude and importance depending on branch trait divergences. Firstly, crown complementarity tended to be highest for tree species pairs that strongly differed in their functional traits, but were similar in branch density. In contrast, heterospecific pairs with a low functional trait divergence benefitted the most from a large difference in branch density compared with pairs characterised by a large functional dissimilarity. Secondly, the positive effects of increasing divergence in branching intensity (the plasticity of the inner crown structure) on crown complementarity became most important at low levels of functional dissimilarity, i.e. when species pairs were similar in their branch packing and vice versa. This suggests that species mixing allows trees to occupy canopy space more efficiently mainly due to phenotypic changes associated with crown morphology and branch plasticity. Our findings highlight the importance of considering outer and inner crown structures (e.g. branching architecture) to deepen our understanding of tree-tree interactions in mixed-species communities.     

Emergence of matched airway and vascular trees from fractal rules

The bronchial, arterial, and venous trees of the lung are complex interwoven structures. Their geometries are created during fetal development through common processes of branching morphogenesis. Insights from fractal geometry suggest that these extensively arborizing trees may be created through simple recursive rules. Mathematical models of Turing have demonstrated how only a few proteins could interact to direct this branching morphogenesis. Development of the airway and vascular trees could, therefore, be considered an example of emergent behavior as complex structures are created from the interaction of only a few processes. However, unlike inanimate emergent structures, the geometries of the airway and vascular trees are highly stereotyped. This review will integrate the concepts of emergence, fractals, and evolution to demonstrate how the complex branching geometries of the airway and vascular trees are ideally suited for gas exchange in the lung. The review will also speculate on how the heterogeneity of blood flow and ventilation created by the vascular and airway trees is overcome through their coordinated construction during fetal development.