Branching characteristics of human coronary arteries

Branching angles and branch diameters were measured in a total of 850 arterial junctions in the coronary networks of two human hearts. Comparison is made with similar data obtained previously from the coronary networks of rats, and with what is considered to be optimum on theoretical grounds. It is concluded that the branching characteristics of the human coronary arteries are closer to the theoretical optimum than those of the coronary networks of rats. While the human data exhibit some departure from optimality and a good amount of scatter, these are well within levels observed elsewhere in the cardiovascular systems of man and animals, and considerably better than those found in the coronary networks of rats. The departure from optimality, in terms of physiological cost to the system, is within 5% for most data points.     

Cost analysis of arterial branching in the cardiovascular systems of man and animals

A new scheme is presented whereby data on arterial branching can be interpreted in terms of direct cost to the physiological system. The scheme makes it possible to assess, at a glance, the true degree of optimality of an arterial network. Departure from optimality is indicated in terms of cost, rather than in terms of the difference between theoretical and measured branching angles. The scheme is applied to several groups of biological data and new conclusions are reached with regard to their degrees of optimality.     

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.     

A two-phase growth strategy in cultured neuronal networks as reflected by the distribution of neurite branching angles

Neurite outgrowth and branching patterns are instrumental in dictating the wiring diagram of developing neuronal networks. We study the self-organization of single cultured neurons into complex networks focusing on factors governing the branching of a neurite into its daughter branches. Neurite branching angles of insect ganglion neurons in vitro were comparatively measured in two neuronal categories: neurons in dense cultures that bifurcated under the presence of extrinsic (cellular environment) cues versus neurons in practical isolation that developed their neurites following predominantly intrinsic cues. Our experimental results were complemented by theoretical modeling and computer simulations. A preferred regime of branching angles was found in isolated neurons. A model based on biophysical constraints predicted a preferred bifurcation angle that was consistent with this range shown by our real neurons. In order to examine the origin of the preferred regime of angles we constructed simulations of neurite outgrowth in a developing network and compared the simulated developing neurons with our experimental results. We tested cost functions for neuronal growth that would be optimized at a specific regime of angles. Our results suggest two phases in the process of neuronal development. In the first, reflected by our isolated neurons, neurons are tuned to make first contact with a target cell as soon as possible, to minimize the time of growth. After contact is made, that is, after neuronal interconnections are formed, a second branching strategy is adopted, favoring higher efficiency in neurite length and volume. The two-phase development theory is discussed in relation to previous results.     

A two‐phase growth strategy in cultured neuronal networks as reflected by the distribution of neurite branching angles

Neurite outgrowth and branching patterns are instrumental in dictating the wiring diagram of developing neuronal networks. We study the self‐organization of single cultured neurons into complex networks focusing on factors governing the branching of a neurite into its daughter branches. Neurite branching angles of insect ganglion neurons in vitro were comparatively measured in two neuronal categories: neurons in dense cultures that bifurcated under the presence of extrinsic (cellular environment) cues versus neurons in practical isolation that developed their neurites following predominantly intrinsic cues. Our experimental results were complemented by theoretical modeling and computer simulations. A preferred regime of branching angles was found in isolated neurons. A model based on biophysical constraints predicted a preferred bifurcation angle that was consistent with this range shown by our real neurons. In order to examine the origin of the preferred regime of angles we constructed simulations of neurite outgrowth in a developing network and compared the simulated developing neurons with our experimental results. We tested cost functions for neuronal growth that would be optimized at a specific regime of angles. Our results suggest two phases in the process of neuronal development. In the first, reflected by our isolated neurons, neurons are tuned to make first contact with a target cell as soon as possible, to minimize the time of growth. After contact is made, that is, after neuronal interconnections are formed, a second branching strategy is adopted, favoring higher efficiency in neurite length and volume. The two‐phase development theory is discussed in relation to previous results. © 2004 Wiley Periodicals, Inc. J Neurobiol, 2005     

Morphometric analysis of the distributing vessels of the kidney

Branching angles and branch diameters of the distributing vessels in the renal networks of rats were measured and the results are compared with data reported previously from the coronary network of the same species. Comparison is also made with what is known to be optimum on theoretical grounds to determine to what extent the branching characteristics of the renal network are governed by considerations of optimality, and to what extent they are affected by other considerations, relating particularly to the role that the network plays in the blood processing function of the kidney.     

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.     

Finding the optimal lengths for three branches at a junction

This paper presents an exact analytical solution to the problem of locating the junction point between three branches so that the sum of the total costs of the branches is minimized. When the cost per unit length of each branch is known the angles between each pair of branches can be deduced following reasoning first introduced to biology by Murray. Assuming the outer ends of each branch are fixed, the location of the junction and the length of each branch are then deduced using plane geometry and trigonometry. The model has applications in determining the optimal cost of a branch or branches at a junction. Comparing the optimal to the actual cost of a junction is a new way to compare cost models for goodness of fit to actual junction geometry. It is an unambiguous measure and is superior to comparing observed and optimal angles between each daughter and the parent branch. We present data for 199 junctions in the pulmonary arteries of two human lungs. For the branches at each junction we calculated the best fitting value of x from the relationship that flow ∞ (radius)^x. We found that the value of x determined whether a junction was best fitted by a surface, volume, drag or power minimization model. While economy of explanation casts doubt that four models operate simultaneously, we found that optimality may still operate, since the angle to the major daughter is less than the angle to the minor daughter. Perhaps optimality combined with a space filling branching pattern governs the branching geometry of the pulmonary artery.     

An Experimental Investigation of the Resistance of Model Root Systems to Uprooting

The architecture of a tree root system may influence its abilityto withstand uprooting by wind loading. To determine how theroot branching pattern may alter the anchorage efficiency ofa tree, artificial model root systems with different topologiesand branching angles were built. The root systems were embeddedat various depths in wet sand and the pull-out resistance measured.A model to predict the uprooting resistance from the data collectedwas designed, allowing predictions of anchorage strength withregards to architecture. The dominant factors influencing pull-outresistance were the depth and length of roots in the soil. Themost efficient type of branching pattern predicted by the programwas one with an increased number of roots deep in the soil.The optimum branching angle most likely to resist pull-out isa vertical angle of 90° between a lateral and the main axis.The predicted mechanically optimal radial angle between a lateralbranch and its daughter is between 0 and 20°. Values ofbranching angle are compared with those measured in real woodyroot systems of European larch and Sitka spruce. Root architecture; root anchorage; pull-out resistance; windthrow; Picea sitchensis ; Larix decidua     

Herbivory and the evolution of divaricate plants: Structural defences lost on an offshore island

Many island plants are characterized by unique morphology. For example, the high branching angles and small leaves of divaricate plants are a common feature of the New Zealand flora. The divaricate growth form may be an adaptation to deter browsing by extinct avian herbivores (moa); alternatively aspects of the insular climate may be responsible. However, our understanding of the selective pressures responsible for the high branching angles and small leaves of divaricate plants is incomplete. Here, I tested for differences in traits associated with the divaricate growth form between plants from Chatham Island and the New Zealand mainland. Moa never reached the Chatham Islands and its flora is derived from plants on mainland New Zealand. Therefore, I predicted Chatham Island plants to have lost morphological adaptations that may have deterred moa herbivory. Traits were quantified on 316 individuals in the field, allowing for 12 island‐mainland taxonomic comparisons. Chatham Island plants consistently produced smaller branching angles, larger leaves, shorter internodes and larger stems than related mainland plants. Results are therefore consistent with the hypothesis that selection for small leaves and high angled branching may be relaxed on the Chatham Islands due to an absence of moa. Smaller branching angles and larger leaves may offer a competitive advantage to Chatham Island plants.     

Scatter factor-dependent branching morphogenesis: structural and histological features

Branching morphogenesis is a multi-step process that controls the formation of polarised tubules starting from hollow cysts. Its execution entails a series of rate-limiting events which include reversible disruption of cell polarity, dismantling of intercellular contacts, acquisition of a motile phenotype, stimulation of cell proliferation, and final re-establishment of cell polarity for creation of the definitive structures. Branching morphogenesis takes place physiologically during development, accounting for the establishment of organs endowed with a ramified architecture such as glands, the respiratory tract and the vasculartree. In cancer, aberrant implementation of branching morphogenesis leads to deregulated proliferation, protection from apoptosis and enhanced migratory/invasive properties, which together exacerbate the aggressive features of neoplastic cells. Under both physiological and pathological conditions, branching morphogenesis is mainly accomplished by a family of growth factors known as scatter factors. In this review, we will summarise the current knowledge on the biological and functional roles of scatter factors during branching morphogenesis, with a special emphasis on the phenotypic (structural and histological) consequences of scatter factor activity in different tissues.     

On the cost functions for the control of the human arm movement

The aim of our investigation is to understand the mechanisms which control the movement of the human arm. The arm is here considered as a redundant system: the shoulder, elbow and wrist joints, which provide three degrees of freedom, combine to move the hand in a horizontal plane, i.e. a two dimensional space. Thus the system has one extra degree of freedom. Earlier investigations of the static situation led to the hypothesis that independent cost functions were attached to each of the three joints and that the configuration chosen for a given target position is that which provides the minimum total cost (Cruse 1986). The aim of the current investigation was to look for measurable values corresponding to the hypothetical cost functions. Experiments using pointers of different lengths attached to the hand showed that the strategy in choosing the joint angles are independent of the limb length. The muscle force necessary to reach a given angle is increased by a spring mounted across a joint. In this situation the angles of the loaded joint are changed for a given target point to give way to the force effect. This leads to the conclusion that the hypothetical cost functions are not independent of the physiological costs necessary to hold the joint at a given angle. The cost functions seem to depend on joint angle and on the force which is necessary to hold the joint in a given position. Cost functions are measured by psychophysical methods. The results showU-shaped curves which can be approximated by parabolas. The position of minimum cost (maximum comfort) for one joint showed no or weak dependency on the angles of the other joints. For each subject these psychophysical cost functions are compared with the hypothetical cost functions. The comparison showed reasonable agreement. This supports the assumption that the psychophysically measured comfort functions provide a measure for the hypothetical cost functions postulated to explain the targeting movements. Targeting experiments using a four joint arm which has two extra degrees of freedom showed a much larger scatter compared to the three joint arm. Nevertheless, the results still conform to the hypothesis that also in this case the minimum cost principle is applied to solve the redundancy problem. As the cost function for the whole arm shows a large minimum valley, quite a large range of arm positions is possible of about equal total costs. The scatter does not result from pure randomness but seems to be mainly produced by the fact that the angles at the end of the movement depend on the value of the joint angles at the beginning of the movement.     

Arterial bifurcations in the human retina

The branching angles and relative diameters of blood vessels in 51 arterial bifurcations in the retina of a normal human eye were measured. In eight other bifurcations, only the total branching angles were measured. The results are compared with theoretical predictions in an attempt to understand the physiological principles governing branching in the cardiovascular system.     

Orientational preferences shown by microspikes of growing nerve cells in vitro

The shapes of microspikes on single neurones cultured in vitro have been analysed with respect to angles of bending and branching. Characteristic frequency distributions were found in all of the four categories of angles. Certain `preferred' orientations of bending and branching were seen to center about 60°, 90°, and 120°. The possible cellular basis for such behavior is discussed, as well as the similarity of bending and branching in the axonal-microspike system of the single nerve cell to the analogous branching in the vertebrate blood vascular system.     

Relation of branching angles to optimality for four cost principles

The literature has suggested that branching angles depend on some principle of optimality. Most often cited are the minimization of lumen surface, volume, power and drag. The predicted angles depend on the principle applied, chi and alpha. Assuming flow o r chi, chi can be determined from r chi 0 = r chi 1 + r chi 2 when the radii of the parent (r0) major (r1) and minor (r2) daughters are known. The term alpha = r2/r1. Using different values for chi and alpha, we present graphs for the major and minor branching angles theta 1 and theta 2 and psi = theta 1 + theta 2 for each of the four optimization principles. Because psi is almost independent of alpha for values of chi and alpha found in 198 junctions taken from a human pulmonary artery, we are able to produce a plot of psi versus chi for each of the four principles on one graph. A junction can be provisionally classified as optimizing for a given principle if, knowing chi, the psi obs - psi pred is least for that principle. We find that this nomographic classification agrees almost perfectly with a previous classification based on a more exacting measure, the percent cost index I, where I = observed cost/minimum cost. We explain why this is to be expected in most but not all cases. First we generate a contoured percent cost surface of c = I - 100 around the optimally located junction, J, and superimpose a surface of equal angular deviations a = psi pred-psi obs. We find that c increases and a usually increases with distance from J as the actual junction moves along a straight line away from J. We then produce a plot of c versus a for two competing principles. A comparison of the principles demonstrates that, for most cases, a is smaller for the principle which has the smaller c value.     

Three-dimensional aspects of arterial branching

Arterial junctions give rise to different images when viewed from different directions. When a two-dimensional bifurcation is viewed in a direction other than normal to its branching plane, the branching angles will be distorted and the resulting picture will not be a true picture of that bifurcation. If the bifurcation is three-dimensional, some distortions will occur no matter which way the bifurcation is viewed. These distortions are analyzed for a wide range of situations and data is provided from which the corresponding errors can be estimated.     

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°。     

The cost of departure from optimal radii in microvascular networks

In the Murray optimality model of branching vasculatures, the radii of vessels are related to blood viscosity, vascular metabolic rate, and blood flow rate, in such a way as to minimize the total work (hydraulic and metabolic) of the system. The model predicts that flow is proportional to the cube of a vessel radius, and that at junctions the cube of the radius of the parent vessel equals the sum of the cubes of the daughter radii. In comparing real vasculatures to the Murray model, we have previously had no expressions for evaluating the apparent energy cost for departures from the optimal junction exponent of 3. Such expressions are derived here. They show that junction exponents, from about 1.5 to large positive values, are within 5% of the energy minimum. With the new equations, observed individual junctions or entire vascular trees can be compared, energy-wise, with the Murray optimum. Junctions in the transverse arteriolar trees of cat sartorius muscle were compared to the Murray optimality model, using these new expressions. The junction exponents for these small pre-capillary vessels had a broad range, with a median value greater than the Murray optimum of 3. The exponents were restricted, however, to values requiring, at individual junctions, little increase in energy. The majority of junctions had energy costs less than 1% above the Murray minimum. For entire trees involving many junctions the departures from optimality averaged less than 10%. Thus, while the branching geometry for these microvascular trees deviates significantly from the Murray optimum in the direction of larger daughter to parent ratios, the departures are small in energy terms.     

Protein three-dimensional structure generation with an empirical hydrophobic penalty function

Given current computational environments, it is worthwhile to establish amino acid residue-level functions which approximate protein folds quite well. Such functions must be the interim steps toward protein three-dimensional structure prediction, I have shown that an empirical hydrophobic penalty function of protein, derived from the number of residues in a sphere around each residue, could be utilized to distinguish the correctly folded structure from the incorrect ones. In order to assess the predictive power of the penalty function, I had generated conformations by randomly changing main chain dihedral angles, and applied the penalty function to them. If only a local region was allowed to change its conformation, native like structures could be generated within a reasonable computational time. In global simulations, however, a considerable number of nonnative conformations, which gave as small a penalty value as that of the native protein, were found. Although some of the conformations were compact and globular, they were quite different from the native structure in that they lacked most of the secondary structures. This result shows that the penalty function alone cannot define the native structure, and that substructure information may help the penalty function to reach the correctly folded structure.     

Properties and active center of the thermostable branching enzyme from Bacillus stearothermophilus.

Although the branching enzyme (EC 2.4.1.18) is a member of the alpha-amylase family, the characteristics are not understood. The thermostable branching enzyme gene from Bacillus stearothermophilus TRBE14 was cloned and expressed in Escherichia coli. The branching enzyme was purified to homogeneity, and various enzymatic properties were analyzed by our improved assay method. About 80% of activity was retained when the enzyme was heated at 60 degrees C for 30 min, and the optimum temperature for activity was around 50 degrees C. The enzyme was stable in the range of pH 7.5 to 9.5, and the optimum pH was 7.5. The nucleotide sequence of the gene was determined, and the active center of the enzyme was analyzed by means of site-directed mutagenesis. The catalytic residues were tentatively identified as two Asp residues and a Glu residue by comparison of the amino acid sequences of various branching enzymes from different sources and enzymes of the alpha-amylase family. When the Asp residues and Glu were replaced by Asn and Gln, respectively, the branching enzyme activities disappeared. The results suggested that these three residues are the catalytic residues and that the catalytic mechanism of the branching enzyme is basically identical to that of alpha-amylase. On the basis of these results, four conserved regions including catalytic residues and most of the substrate-binding residues of various branching enzymes are proposed.