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.     

植物分枝模型

从生物体总是最有效地利用物质的思想出发,对植物分枝形状建立了一个数学模型。该模型认为,当主干与侧枝的截面积之间存在类似平行四边形法则的关系时,分枝的体积取极小值。该模型揭示植物分枝形态不仅符合力学平衡的原则,在进化上也有显著生物学意义。根据实测数据提出了偏移度的概念,认为分枝形态建成与个体内部枝条相互作用有关,植物分枝取向受空间效应影响时仍满足体积最小的原则。由于此模型具有随机性,遵循此模型的分枝可呈现千变万化的形式。模型将分枝角度与分枝截面积有机结合起来,可作为计算机模拟植物的方法。     

Can morphogenesis be understood in terms of physical rules?

Because the morphogenesis of biological systems is not fully understood, researches from various points of view are necessary. The present author has recently made computer simulations with his colleagues to construct branching systems of human organs, such as the lung airway and the liver blood vessels. In the simulations certain rules are assumed to govern bifurcating processes of the systems. These rules are expressed in terms of physical and geometrical concepts, such as minimum energy consumption and uniform filling of branches in the space of organs. Results of computer simulation are quite similar to real structures. However, actual mechanisms of morphogenesis, i.e. effects of genes or proteins, are not considered in these studies. In this article, the present work is discussed in relation to the concept of biological pattern formation by Meinhardt and a recent study by Miura and Shiota on lung growth.     

A model of apical bifurcation applicable to trees and other organisms

An elementary model to describe the branching process in tree-like structures is proposed. It is assumed that when the apex reaches a certain size, it divides into two apices. The relative sizes of the two apices depends upon a branching parameter λ, which is the critical parameter of the model. With this constraint, and with some supplementary assumptions about growth, a computer is used to generate branched structures, and these are analyzed using Strahler's method of ordering. Diverse tree-like structures can be obtained by simply varying the parameter λ, and these resemble natural structures when compared on the basis of Strahler's technique.     

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.     

A computer program for characterizing root system branching patterns

A computer program is presented which measures the length, branching patterns and distribution of link length within a root system. The program skeletonizes digitized images of root systems, loads these images into a binary tree data structure and uses this data structure to characterize the root systems. Measurements of the root length and topological parameters of root systems of Senecio vulgaris made by hand and by computer program were linearly related, with r² values greater than 0.99 in all cases.     

Modeling of branching structures of plants

Previous studies of branching structures generally focused on arteries. Four cost models minimizing total surface area, total volume, total drag and total power losses at a junction point have been proposed to study branching structures. In this paper, we highlight the branching structures of plants and examine which model fits data of branching structures of plants the best. Though the effect of light (e.g. phototropism) and other possible factors are not included in these cost models, a simple cost model with physiological significance, needs to be verified before further research on modeling of branching structures is conducted. Therefore, data are analysed in this paper to determine the best cost model. Branching structures of plants are studied by measuring branching angles and diameters of 234 junctions from four species of plants. The sample includes small junctions, large junctions, two- and three-dimensional junctions, junctions with three branches joining at a point and those with four branches joining at a point. First, junction exponents (x) were determined. Second, log-log plots indicate that model of volume minimization fits data better than other models. Third, one-sided t -tests were used to compare the fitness of four models. It is found that model of volume minimization fits data better than other cost models.     

BRANCH GEOMETRY AND EFFECTIVE LEAF AREA: A STUDY OF TERMINAUA-BRANCHING PATTERN. 1. THEORETICAL TREES

Single lateral branches and branch tiers of Terminalia catappa L. are simulated and drawn by computer. Leaf clusters on the branches are approximated by discs, and the effective leaf areas are determined by use of Dirichlet domains. Theoretical optimal branching angles which produce the maximum effective leaf area are obtained from simulations. Symmetrical and asymmetrical branching angles are contrasted; the latter characterize real trees. Varying leaf disc radius and ratio of branch-unit lengths affects optimal branching angles, as does the symmetry of a tier of five branches. Leaf area indices for individual branches and branch tiers are given for all simulations. The number of branches in a tier has a major effect on leaf area index and effective leaf area. The theoretical optimal branching angles of many simulations are very close to the values observed in real trees of T. catappa. We conclude that the observed branching angles and number of branches in a tier of this species optimize light interception within constraints of a fixed pattern of branching, one that is widespread among tropical trees.     

Branching and self-organization in marine modular colonial organisms: a model

Despite the universality of branching patterns in marine modular colonial organisms, there is neither a clear explanation about the growth of their branching forms nor an understanding of how these organisms conserve their shape during development. This study develops a model of branching and colony growth using parameters and variables related to actual modular structures (e.g., branches) in Caribbean gorgonian corals (Cnidaria). Gorgonians exhibiting treelike networks branch subapically, creating hierarchical mother-daughter relationships among branches. We modeled both the intrinsic subapical branching along with an ecological-physiological limit to growth or maximum number of mother branches (k). Shape is preserved by maintaining a constant ratio (c) between the total number of branches and the mother branches. The size frequency distribution of mother branches follows a scaling power law suggesting self-organized criticality. Differences in branching among species with the same k values are determined by r (branching rate) and c. Species with rr/2 or c>r>0). Ecological/physiological constraints limit growth without altering colony form or the interaction between r and c. The model described the branching dynamics giving the form to colonies and how colony growth declines over time without altering the branching pattern. This model provides a theoretical basis to study branching as a simple function of the number of branches independently of ordering- and bifurcation-based schemes.     

The Morphological Identity of Insect Dendrites

Dendrite morphology, a neuron's anatomical fingerprint, is a neuroscientist's asset in unveiling organizational principles in the brain. However, the genetic program encoding the morphological identity of a single dendrite remains a mystery. In order to obtain a formal understanding of dendritic branching, we studied distributions of morphological parameters in a group of four individually identifiable neurons of the fly visual system. We found that parameters relating to the branching topology were similar throughout all cells. Only parameters relating to the area covered by the dendrite were cell type specific. With these areas, artificial dendrites were grown based on optimization principles minimizing the amount of wiring and maximizing synaptic democracy. Although the same branching rule was used for all cells, this yielded dendritic structures virtually indistinguishable from their real counterparts. From these principles we derived a fully-automated model-based neuron reconstruction procedure validating the artificial branching rule. In conclusion, we suggest that the genetic program implementing neuronal branching could be constant in all cells whereas the one responsible for the dendrite spanning field should be cell specific.     

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.     

Dendrimers in gene transfection

Dendrimers are a new class of nanocomposite materials. They are branching polymers whose structure is formed by monomeric subunit branches diverging to all sides from a central nucleus. The type of nucleus, attached monomers, and functional groups can be chosen during synthesis, which produces dendrimers of definite size, shape, density, polarity, branch mobility, and solubility. This review deals with problems of dendrimer molecular structures and capability of in vitro, in vivo, ex vivo, and in situ transfection of genetic material. Advantages and shortcomings of different types of dendrimers in this respect are discussed.     

Growth of branched actin networks against obstacles

A method for simulating the growth of branched actin networks against obstacles has been developed. The method is based on simple stochastic events, including addition or removal of monomers at filament ends, capping of filament ends, nucleation of branches from existing filaments, and detachment of branches; the network structure for several different models of the branching process has also been studied. The models differ with regard to their inclusion of effects such as preferred branch orientations, filament uncapping at the obstacle, and preferential branching at filament ends. The actin ultrastructure near the membrane in lamellipodia is reasonably well produced if preferential branching in the direction of the obstacle or barbed-end uncapping effects are included. Uncapping effects cause the structures to have a few very long filaments that are similar to those seen in pathogen-induced "actin tails." The dependence of the growth velocity, branch spacing, and network density on the rate parameters for the various processes is quite different among the branching models. An analytic theory of the growth velocity and branch spacing of the network is described. Experiments are suggested that could distinguish among some of the branching models.     

Angles of branching and diameters of branches in the human bronchial tree

The principle of minimal work requires that the conducting airways of the human lung should have a maximum radius for minimal resistance to gas flow. At the same time there is a requirement that the airways should have a minimal volume for economy of space. These two opposing requirements have been investigated mathematically, and a method for calculating the angle of branching which produces minimal volume has been derived. The relationship of the radii of the parent and daughter branches to produce minimal resistance has been similarly defined. By measurement of a bronchial cast from a human lung the extent to which the predicted optimum structure is realized in practice has been shown. The change in structure associated with change of function at the transition from conducting airway to diffusion zone has been demonstrated.     

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     

樟子松人工林分枝结构的分析

基于对6块樟子松(Pinus sylvestris var. mongolica)人工林固定标准地中的30株样木枝解析调查数据,通过分析不同林分、不同大小林木1级枝和2级枝的分枝概率、分枝格局和分枝角度,揭示了樟子松人工林树冠的分枝结构特点。研究结果表明：樟子松人工林1级枝和2级枝的平均分枝数量分别为3.84个和2.80个,两者分枝概率均呈正态分布;1级和2级枝条在光照条件好的几个区间（方位角46°～225°）分布较多,1级枝条的水平分布遵从均匀分布,而2级枝条则不遵从均匀分布;树冠上层枝条的分枝角度略小于树冠中、下层,上层平均分枝角度为45.6°,而中下层平均分枝角度都为49.4°。不同大小林木的1级枝分枝结构规律表明：Ⅰ级木和Ⅴ级木的每轮平均分枝数非常接近,分别为3.89和3.94个,比Ⅲ级木每轮分枝数大0.5个左右;1级枝水平分布在各区间内(45°间隔)相差在0.24%～2.81%之间,方差分析结果表明枝条水平分布与林木大小无关;不同大小林木的分枝角度有所差别,Ⅰ级木、Ⅲ级木和Ⅴ级木的平均分枝角度分别为48.5°、42.2°和50.7°。     

Prediction of geometrically feasible three-dimensional structures of pseudoknotted RNA through free energy estimation

Accurate free energy estimation is essential for RNA structure prediction. The widely used Turner''s energy model works well for nested structures. For pseudoknotted RNAs, however, there is no effective rule for estimation of loop entropy and free energy. In this work we present a new free energy estimation method, termed the pseudoknot predictor in three-dimensional space (pk3D), which goes beyond Turner''s model. Our approach treats nested and pseudoknotted structures alike in one unifying physical framework, regardless of how complex the RNA structures are. We first test the ability of pk3D in selecting native structures from a large number of decoys for a set of 43 pseudoknotted RNA molecules, with lengths ranging from 23 to 113. We find that pk3D performs slightly better than the Dirks and Pierce extension of Turner''s rule. We then test pk3D for blind secondary structure prediction, and find that pk3D gives the best sensitivity and comparable positive predictive value (related to specificity) in predicting pseudoknotted RNA secondary structures, when compared with other methods. A unique strength of pk3D is that it also generates spatial arrangement of structural elements of the RNA molecule. Comparison of three-dimensional structures predicted by pk3D with the native structure measured by nuclear magnetic resonance or X-ray experiments shows that the predicted spatial arrangement of stems and loops is often similar to that found in the native structure. These close-to-native structures can be used as starting points for further refinement to derive accurate three-dimensional structures of RNA molecules, including those with pseudoknots.     

LOPAL and SCAMP: techniques for the comparison and display of protein structures

This paper describes two computer programs designed to assist in the comparison of protein structures. LOPAL (LOoP ALignment) applies a dynamic programming algorithm to the comparison of regions of protein three-dimensional (3D) structure and gives a similarity score and suggested sequence alignment with that score. SCAMP (Structure Comparison and Alignment of Multiple Proteins) is an interactive graphics program for the Evans and Sutherland PS300 graphics terminal that allows the simultaneous display, manipulation and pairwise least-squares fitting of up to nine independent structures. Together, LOPAL and SCAMP provide an integrated system for characterizing structural similarities in proteins with the aim of improving the accuracy of predicted protein structures. An application of these programs to loop regions in the immunoglobulin constant domains is illustrated.     

兴安落叶松分枝格局的分形特征

对于树木分枝格局分形特征的定量描述,可以加深对树木生长过程的理解。本文采用分形几何学对兴安落叶松(Larix gmelini)的分枝格局进行研究,结果表明1)兴安落叶松分枝格局是一种分形结构,存在自相似性。2)兴安落叶松分枝格局的分形维数介于1.4~1.7之间,揭示了它的结构复杂性程度和占据生态空间、利用生态空间的能力。分形维数在树木光合作用及生长发育研究中是一个有用的参数。     

Use of the smith degradation in the study of the branching pattern in the complex-type carbohydrate units of glycoproteins

The site of substitution of the peripheral branches on the core mannose-residues in the complex-type carbohydrate units of glycoproteins has been studied by Smith degradation. By using glycopeptides of known structure, it is shown that the site of attachment of the (1→4)-linked, third branch to the mannose core in tri- and tetra-antennary structures may be conveniently demonstrated by this technique. A critical factor was found to be the concentration of acid used for cleavage of the periodate-oxidized product. This finding could explain discrepancies concerning the previously published structures of fetuin glycopeptides. The technique was applied for study of the branching pattern in human transferrin, porcine thyroglobulin, and rat plasma and brain glycoproteins. The results support the concept that, in glycoproteins from most sources, the third (1→4)-linked branch is attached to the (1→3)-linked mannose residue of the core portion in the complex-type structures. An indication for a novel type of branching pattern in human transferrin was, however, also obtained.