Two Geometrical Models of Branching of Botanical Trees

A botanical tree may be regarded as a system of axes which developby repeated bifurcation. A complicated tree can be describedby a few parameters of bifurcation which determine the geometryof the bifurcation process. A bifurcation model (H-model) haspreviously been proposed and shown to be useful for comparisonwith trees which have branch complexes which approach a horizontalplane (e.g. Terminalia). Another bifurcation model (P-model)is now proposed which is appropriate to branching systems inwhich successive branch planes are perpendicular to each other(e.g. Cameraria and Tabernaemontana of the Apocynaceae). Bymodification of the P-model to take into account a geotropiceffect, a more realistic branching model for one kind of treespecies is produced. The relationship among these geometricalmodels of branching is discussed and illustrated with computersimulations. computer simulations, tree crown geometry, branching patterns, bifurcation models, Terminalia, Cameraria latifolia, Tabernaemontana, sp     

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。     

四川大头茶的分枝率和顶芽动态

以四川大头茶植株的枝和顶芽为基本构件单元,对不同年龄级成熟植株的分枝率和枯芽数动态进行了研究。结果表明：四川大头茶成株任一种分枝率在种群内不同龄级植析间以及在同一植株内不同发育位置的主枝间无显著变化;而同一主枝内各类型分枝率并非按相同的几何及数增减,即枝内各类分枝率间具显著的差异。     

Bistability and oscillations in chemical reaction networks

Bifurcation theory is one of the most widely used approaches for analysis of dynamical behaviour of chemical and biochemical reaction networks. Some of the interesting qualitative behaviour that are analyzed are oscillations and bistability (a situation where a system has at least two coexisting stable equilibria). Both phenomena have been identified as central features of many biological and biochemical systems. This paper, using the theory of stoichiometric network analysis (SNA) and notions from algebraic geometry, presents sufficient conditions for a reaction network to display bifurcations associated with these phenomena. The advantage of these conditions is that they impose fewer algebraic conditions on model parameters than conditions associated with standard bifurcation theorems. To derive the new conditions, a coordinate transformation will be made that will guarantee the existence of branches of positive equilibria in the system. This is particularly useful in mathematical biology, where only positive variable values are considered to be meaningful. The first part of the paper will be an extended introduction to SNA and algebraic geometry-related methods which are used in the coordinate transformation and set up of the theorems. In the second part of the paper we will focus on the derivation of bifurcation conditions using SNA and algebraic geometry. Conditions will be derived for three bifurcations: the saddle-node bifurcation, a simple branching point, both linked to bistability, and a simple Hopf bifurcation. The latter is linked to oscillatory behaviour. The conditions derived are sufficient and they extend earlier results from stoichiometric network analysis as can be found in (Aguda and Clarke in J Chem Phys 87:3461–3470, 1987; Clarke and Jiang in J Chem Phys 99:4464–4476, 1993; Gatermann et al. in J Symb Comput 40:1361–1382, 2005). In these papers some necessary conditions for two of these bifurcations were given. A set of examples will illustrate that algebraic conditions arising from given sufficient bifurcation conditions are not more difficult to interpret nor harder to calculate than those arising from necessary bifurcation conditions. Hence an increasing amount of information is gained at no extra computational cost. The theory can also be used in a second step for a systematic bifurcation analysis of larger reaction networks. We have added a dedication of the paper to K. Gatermann.     

Gene trees and species trees are not the same

The relationship between species is usually represented as a bifurcating tree with the branching points representing speciation events. The ancestry of genes taken from these species can also be represented as a tree, with the branching points representing ancestral genes. The time back to the branching points, and even the branching order, can be different between the two trees. This possibility is widely recognized, but the discrepancies are often thought to be small. A different picture is emerging from new empirical evidence, particularly that based on multiple loci or on surveys with a wide geographical scope. The discrepancies must be taken into account when estimating the timing of speciation events, especially the more recent branches. On the positive side, the different timings at different loci provide information about the ancestral populations.     

Architectural evolution and its implications for domestication in grasses

BACKGROUND: The cereal crops domesticated from grasses provide a large percentage of the calories consumed by humans. Domestication and breeding in individual cereals has historically occurred in isolation, although this is rapidly changing with comparative genomics of the sequenced or soon-to-be sequenced genomes of rice, sorghum, maize and Brachypodium. Genetic information transferred through genomic comparisons is helping our understanding of genetically less tractable crops such as the hexaploid wheats and polyploid sugarcane, as well as the approx. 10 000 species of wild grasses. In turn, phylogenetic analysis helps put our knowledge of the morphology of cereal crops into an evolutionary context. GRASS ARCHITECTURE: Domestication often involves a change in the pattern and timing of branching, which affects both vegetative and inflorescence architecture, and ultimately yield. Cereal grasses exhibit two main forms of vegetative architecture: the pooid and erhartoid cereals such as wheat and rice have multiple basal tillers, while panicoid cereals such as maize, sorghum and the millets have few tillers or even only a single main stem. These differences are reflected in the differences between the wild species of pooid and some erhartoid grasses, which emphasize basal branching over axillary branching, and the panicoid grasses, where axillary branching is more frequently found. A combination of phylogenetic and genomic analysis is beginning to reveal the similarities and differences between different cereal crops, and relate these to the diversity of wild grasses to which they are related. Recent work on genes controlling branching emphasizes that developmental genetics needs to be viewed in both an evolutionary and ecological framework, if it is to be useful in understanding how morphology evolves. Increasingly, exploring the phylogenetic context of the crop grasses will suggest new ways to identify and create combinations of morphological traits that will best suit our future needs.     

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.     

民勤荒漠植物枝系构型的分类研究

荒漠植物的枝系构型因素包括各级分枝角度、各级分枝长度、枝径比、逐步分枝率和总体分枝率、分枝分维数和计盒维数等16个指标。采用组内欧式距离法进行聚类,把荒漠植物依构型指标分为4个类型：第一类型包括霸王（Zygophyllum xanthoxylum）、黄刺条（Caragana frutex）等14种荒漠植物;第二类型包括秦晋锦鸡儿（C．purdomii）、荒漠锦鸡儿（C．roborovskyi）等11种荒漠植物;第三类型包括网状沙拐枣（Calligonum cancellatum）、黄花草木樨（Melilotus suaveolens）等9种荒漠植物;第四类型包括扁果木蓼（Atraphaxis replicta）、洋白蜡（Frawinus amerirana）等14种荒漠植物。不同的枝系构型类型反映了不同荒漠植物对于空间资源与环境的长期适应对策。     

Basal EDRF activity helps to keep the geometrical configuration of arterial bifurcations close to the Murray optimum

We have used X-ray microangiography to investigate the hypothesis that the potent endogenous vasodilator endothelium-derived relaxing factor (EDRF) contributes to the maintenance of "optimality" in vascular branching by modulating the diameters of the parent (D0) and daughter (D1 and D2) arteries at bifurcations. Five anatomically different types of bifurcation were studied in buffer-perfused rabbit ear preparations both under resting conditions and after pharmacological constriction by 5-hydroxytryptamine (5HT). A range of flow rates (1-5 ml min-1) was employed as release of EDRF from endothelial cells is stimulated by shear stress. Experimental data obtained in the presence and absence of EDRF activity were compared with theoretical predictions in three ways. (1) Junction exponents (x) were determined at each bifurcation from the equation Dx1 + Dx2 = Dx0, and their frequency distributions constructed. Murray (1926a, Proc. natn. Acad. Sci., U.S.A. 12, 207-214; 1926b, J. gen. Physiol. 9, 835-841.) proposed that x will be exactly 3 if power losses and intravascular volume are minimized simultaneously. In unconstricted preparations, either in the presence or absence of EDRF activity, and in preparations constricted by 0.1 microM 5HT in the presence of EDRF activity, the modes and medians of the frequency distributions of x were found to be close to 3 at all flow rates. In contrast, in 0.1 microM 5HT-constricted preparations in the absence of EDRF activity, no single mode common to all flow rates was apparent and medians were significantly larger at all flow rates. (2) Theoretically "optimal" branching angles were derived from experimental diameter measurements using four mathematical models which minimize respectively the total surface area, total volume, total drag (shear stress) and total power losses at bifurcations (Murray, 1926b). These calculated branching angles were then compared with actual branching angles. EDRF activity was found to be necessary for accurate prediction of branching angles by the minimum volume and power loss models in 5HT-constricted but not in resting preparations. (3) For each model or "minimization principle", there is an optimal mathematical relationship between the junction exponent, x, and the angle between daughter arteries, psi 12, at a bifurcation (Roy & Woldenberg, 1982, Bull. math. Biol. 44, 349-360.) Experimentally determined values of x and psi 12 agreed closely with those predicted both by the minimum volume and the minimum power loss principles, except again in 5HT-constricted preparations in the absence of EDRF activity.(ABSTRACT TRUNCATED AT 250 WORDS)     

Mind and matter as asymptotically disjoint,inequivalent representations with broken time-reversal symmetry

Many philosophical and scientific discussions of topics of mind-matter research make implicit assumptions, in various guises, about the distinction between mind and matter. Currently predominant positions are based on either reduction or emergence, providing either monistic or dualistic scenarios. A more-involved framework of thinking, which can be traced back to Spinoza and Leibniz, combines the two scenarios, dualistic (with mind and matter separated) and monistic (with mind and matter unseparated), in one single picture. Based on such a picture, the transition from a domain with mind and matter unseparated to separate mental and material domains can be viewed as a result of a general kind of symmetry breaking, which can be described formally in terms of inequivalent representations. The possibility of whether this symmetry breaking might be connected to the emergence of temporal directions from temporally non-directed or even non-temporal levels of reality will be discussed. Correlations between mental and material aspects of reality could then be imagined as remnants of such primordial levels. Different conceivable types of inequivalent representations would lead to correlations with different characteristics.     

The receptor guanylyl cyclase Npr2 is essential for sensory axon bifurcation within the spinal cord

Sensory axonal projections into the spinal cord display a highly stereotyped pattern of T- or Y-shaped axon bifurcation at the dorsal root entry zone (DREZ). Here, we provide evidence that embryonic mice with an inactive receptor guanylyl cyclase Npr2 or deficient for cyclic guanosine monophosphate-dependent protein kinase I (cGKI) lack the bifurcation of sensory axons at the DREZ, i.e., the ingrowing axon either turns rostrally or caudally. This bifurcation error is maintained to mature stages. In contrast, interstitial branching of collaterals from primary stem axons remains unaffected, indicating that bifurcation and interstitial branching are processes regulated by a distinct molecular mechanism. At a functional level, the distorted axonal branching at the DREZ is accompanied by reduced synaptic input, as revealed by patch clamp recordings of neurons in the superficial layers of the spinal cord. Hence, our data demonstrate that Npr2 and cGKI are essential constituents of the signaling pathway underlying axonal bifurcation at the DREZ and neuronal connectivity in the dorsal spinal cord.     

Computer simulation of growth of anastomosing microvascular networks

Stochastic growth of polygonal microvascular networks was simulated on computer by dichotomous terminal branching and bridging (anastomosing with an existing segment). The model was applied to describe microvascular growth into a rectangular plane from the sides when vessels bifurcate in a probabilistic manner. The angle of bifurcation was drawn from a normal distribution, the mean of which was varied between 40 degrees and 80 degrees. The resulting networks contained an average of 88-104 nodes of which 30-38% were due to bridging. Number of nodes, number of branches, number of vascular polygons and a fractal dimension representing the density of nodes were calculated for each simulated network. Capillary density increased when mean angle of bifurcation was increased between 40 degrees and 80 degrees. Distributions of normalized vessel lengths and polygon shapes were compared with those of a mesenteric vascular network. The distributions were not found to be significantly different (p less than 0.05) for most values of the mean angle of bifurcation, matching best for the mean bifurcation angle of 50 degrees. Vascular polygons had an average shape between pentagonal and hexagonal for the mesenteric network as well as for all values of the mean bifurcation angle used in this study.     

Periodic flow at airway bifurcations. III. Energy dissipation

We measured the energy dissipation associated with large-amplitude periodic flow through airway bifurcation models. Each model consisted of a single asymmetric bifurcation with a different branching angle and area ratio, with each branch terminated into an identical elastic load. Sinusoidal volumetric oscillations were applied at the parent duct so that the upstream Reynolds number (Re) varied from 30 to 77,000 and the Womersley parameter (alpha) from 4 to 30. Pressures were measured continuously at the parent duct and at both terminals, and instantaneous branch flow rates were calculated. Time-averaged energy dissipation in the bifurcation was computed from an energy budget over a control volume integrated over a cycle and was expressed as a friction factor, F. We found that when tidal volume was small [ratio of tidal volume to resident (dead space) volume, VT/VD less than 1], F was independent of branching angle and fell with increasing alpha and VT/VD. When tidal volume was large (VT/VD greater than 1), F increased with increasing branching angle and varied less strongly with alpha and VT/VD. No simple benchmark flow represented the data well over the entire experimental range. This study demonstrates that only two nondimensional parameters, alpha and VT/VD, are necessary and are sufficient to describe time-averaged energy dissipation in a given bifurcation geometry during sinusoidal flow.     

沙质海岸黑松分枝格局特征及其抗风折能力分析

沙质海岸空间梯度上环境差异较大, 黑松(Pinus thunbergii)在长期的适应过程中树冠结构变化也很大。为揭示黑松树冠结构与环境间的适应机制, 在山东省胶南市灵山湾国家森林公园距海岸线0-50、200-250和400-450 m梯度内各设置1个样带, 记为带I、带II、带III, 采用枝构型的理论和方法, 对黑松的分枝格局进行了研究, 并采用模拟自然风法测定了黑松枝条的抗风折能力。结果表明: 1)在海岸梯度上黑松分枝格局差异较大, 随着距离海岸越来越远, 黑松各级枝的分枝长度、总体分枝率均逐渐增大, 而枝径比和逐步分枝率逐渐减小, 各级分枝角度表现为带I >带III >带II。2)带I树冠背风面与迎风面相比, 分枝长度、分枝角度、分枝数量、枝条干枯率分别是迎风面的1.62、1.38、2.65和0.59倍, 随着距离海岸越来越远, 这种不对称性逐渐减弱, 至带III树冠基本对称。3)海风是影响带I分枝角度偏转、枝条干枯和冠型不对称现象的主要原因。4)带III枝条的抗风折能力高于带I, 且两个样带模拟风速与枝条所承受的拉力之间的关系均符合逻辑斯蒂方程, 相关系数R²均达0.97以上。该研究揭示了不同海岸梯度上黑松分枝格局的形成机制及其抗风折能力, 可为沿海黑松防护林的合理经营提供科学依据。     

A genome evolution-based framework for measures of originality for clades

To evaluate the originality of a species for determining its conservation priority, most indices use the branching pattern and the branch length of a phylogenetic tree to represent the diversification pattern and the number of characters. One limitation of these indices is their lack of consideration of the dynamic process, such as character changes and distribution along lineages during evolution. In this study, we propose a robust framework incorporating the underlying dynamic processes under a framework of genome evolution to model character changes and distribution along different lineages in a given phylogenetic tree. Our framework provides a more transparent modeling, instead of the simple surrogates of branching pattern and branch length previously employed. Nonrandom extinction has been found to be clustered within old and species-poor clades, thus it is desirable to combine the evaluation of originality of clades, which will provide a more complete picture and a useful tool for setting global conservation priorities. Using a phylogenetic tree consisting of 70 species of New World terrestrial Carnivora, we demonstrate that the index derived from our framework can discern the difference in originality of clades. Moreover, we demonstrate that the originality of clades and species in a tree changes with different scenarios of dynamic processes, which were neglected by previous indices. We find that the originality of clades should be one of the criteria for setting global conservation priorities.     

On the divergence of species

Speciation may be considered as a kind of non-equilibrium phase transition which is caused by bifurcation at a critical point of instability. It can be shown that multiple branching of an evolutionary tree is highly improbable, since non-degenerate mathematical models describe only binary branchings.     

Blood flow downstream of a two-dimensional bifurcation

Many cardiovascular lesions such as aneurysms, intimal cushions, and atherosclerotic plaques tend to occur near bifurcations. This suggests that hemodynamic factors may be involved. Since measuring devices (such as anemometers) are still too large to allow local measurements of flow disturbances, we have attempted to predict the nature of these factors mathematically. Biological variables include pulsatile flow of a nonNewtonian fluid in distensible branching vessels with different angles and flow rates. Our initial analysis considers the flow in a two-dimensional bifurcation with a symmetrical flow divider perfused with steady flow at variable Reynolds numbers. At all flows, high shear forces develop on either side of the flow divider (i.e. at the apex of the bifurcation). With high flows, regions of sluggish or reverse flow develop near the outer walls of the bifurcation. The analysis confirms that the flow at the apex is quite different from that at the outer angles and that the latter varies more with flow rate than the former.     

山西濒危植物翅果油树植冠的构型分析

应用分形理论及方法,从分枝格局和冠幅扩展2个方面对山西翼城和乡宁的35株翅果油树(Elaeagnus mollis Diels)个体的植冠构型进行统计分析。结果表明,幼树和成树的总体分枝率和逐步分枝率有显著变化;一级枝的平均枝长和枝径有显著差异,但枝和叶的方位角及叶倾角差异不明显。不同发育阶段个体的冠幅变化较大,幼树冠幅的分形维数(2.0026)小于成树(2.2694)。翅果油树在不同生长发育阶段对生境变化有不同的构型策略。     

Drosophila Dscam is required for divergent segregation of sister branches and suppresses ectopic bifurcation of axons

Axon bifurcation results in the formation of sister branches, and divergent segregation of the sister branches is essential for efficient innervation of multiple targets. From a genetic mosaic screen, we find that a lethal mutation in the Drosophila Down syndrome cell adhesion molecule (Dscam) specifically perturbs segregation of axonal branches in the mushroom bodies. Single axon analysis further reveals that Dscam mutant axons generate additional branches, which randomly segregate among the available targets. Moreover, when only one target remains, branching is suppressed in wild-type axons while Dscam mutant axons still form multiple branches at the original bifurcation point. Taken together, we conclude that Dscam controls axon branching and guidance such that a neuron can innervate multiple targets with minimal branching.     

DiI-Labeling of DRG Neurons to Study Axonal Branching in a Whole Mount Preparation of Mouse Embryonic Spinal Cord

Here we present a technique to label the trajectories of small groups of DRG neurons into the embryonic spinal cord by diffusive staining using the lipophilic tracer 1,1''-dioctadecyl-3,3,3'',3''-tetramethylindocarbocyanine perchlorate (DiI)¹. The comparison of axonal pathways of wild-type with those of mouse lines in which genes are mutated allows testing for a functional role of candidate proteins in the control of axonal branching which is an essential mechanism in the wiring of the nervous system. Axonal branching enables an individual neuron to connect with multiple targets, thereby providing the physical basis for the parallel processing of information. Ramifications at intermediate target regions of axonal growth may be distinguished from terminal arborization. Furthermore, different modes of axonal branch formation may be classified depending on whether branching results from the activities of the growth cone (splitting or delayed branching) or from the budding of collaterals from the axon shaft in a process called interstitial branching² (Fig. 1).The central projections of neurons from the DRG offer a useful experimental system to study both types of axonal branching: when their afferent axons reach the dorsal root entry zone (DREZ) of the spinal cord between embryonic days 10 to 13 (E10 - E13) they display a stereotyped pattern of T- or Y-shaped bifurcation. The two resulting daughter axons then proceed in rostral or caudal directions, respectively, at the dorsolateral margin of the cord and only after a waiting period collaterals sprout from these stem axons to penetrate the gray matter (interstitial branching) and project to relay neurons in specific laminae of the spinal cord where they further arborize (terminal branching)³. DiI tracings have revealed growth cones at the dorsal root entry zone of the spinal cord that appeared to be in the process of splitting suggesting that bifurcation is caused by splitting of the growth cone itself⁴ (Fig. 2), however, other options have been discussed as well⁵.This video demonstrates first how to dissect the spinal cord of E12.5 mice leaving the DRG attached. Following fixation of the specimen tiny amounts of DiI are applied to DRG using glass needles pulled from capillary tubes. After an incubation step, the labeled spinal cord is mounted as an inverted open-book preparation to analyze individual axons using fluorescence microscopy.