Growth velocities of branched actin networks

The growth of an actin network against an obstacle that stimulates branching locally is studied using several variants of a kinetic rate model based on the orientation-dependent number density of filaments. The model emphasizes the effects of branching and capping on the density of free filament ends. The variants differ in their treatment of side versus end branching and dimensionality, and assume that new branches are generated by existing branches (autocatalytic behavior) or independently of existing branches (nucleation behavior). In autocatalytic models, the network growth velocity is rigorously independent of the opposing force exerted by the obstacle, and the network density is proportional to the force. The dependence of the growth velocity on the branching and capping rates is evaluated by a numerical solution of the rate equations. In side-branching models, the growth velocity drops gradually to zero with decreasing branching rate, while in end-branching models the drop is abrupt. As the capping rate goes to zero, it is found that the behavior of the velocity is sensitive to the thickness of the branching region. Experiments are proposed for using these results to shed light on the nature of the branching process.     

Non-axillary branching in the palms Eugeissona and Oncosperma (Arecaceae)

FISHER, J. B., GOH, C. J. & RAO, A. N., 1989. Non-axillary branching in the palms Eugeissona and Oncosperma (Arecaceae). The south-east Asian palms, Eugeissona (Calamoideae) and Oncosperma (Arecoideae) are multiple-stemmed. The morphology and development of branching in two species of each genus were examined in Singapore, Borneo, and the Malay Peninsula. Cultivated seedling and adult plants of 0. tigillarium were also observed in Florida. A new shoot arises most often from a longitudinal abaxial groove at the base of an enclosing leaf sheath. In some instances, especially in E. tristis , the enclosing leaf has two equal, adjacent grooves such that any distinction between an original mother shoot and a lateral daughter shoot is impossible. No axillary buds occur in Eugeissona which is hapaxanthic. In Oncosperma , which is pleonanthic, axillary buds are absent from young pre-flowering stems, but an inflorescence bud occurs in the axil of each leaf in older aerial stems. Early ontogenetic stages of vegetative branching, as seen in sectioned apices, indicate that a new vegetative shoot is present on the abaxial base of the first (youngest) leaf primordium. There is no ontogenetic evidence for the displacement of an originally axillary meristem as previously described for the palm Salacca (Calamoideae). Shoot development in Eugeissona is interpreted as a putative dichotomy of the apical meristem in which the meristem centres commonly develop unequally. In Oncosperma the smaller sucker bud meristem may be described as an abaxial leaf base bud, but ontogenetic variations indicate this form of branching is close to dichotomous branching. These new examples of non-axillary branching are compared to similar cases previously reported for palms and other monocotyledons.     

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.     

Hoxa11 and Hoxd11 regulate branching morphogenesis of the ureteric bud in the developing kidney

Hoxa11 and Hoxd11 are functionally redundant during kidney development. Mice with homozygous null mutation of either gene have normal kidneys, but double mutants have rudimentary, or in extreme cases, absent kidneys. We have examined the mechanism for renal growth failure in this mouse model and find defects in ureteric bud branching morphogenesis. The ureteric buds are either unbranched or have an atypical pattern characterized by lack of terminal branches in the midventral renal cortex. The mutant embryos show that Hoxa11 and Hoxd11 control development of a dorsoventral renal axis. By immunohistochemical analysis, Hoxa11 expression is restricted to the early metanephric mesenchyme, which induces ureteric bud formation and branching. It is not found in the ureteric bud. This suggests that the branching defect had been caused by failure of mesenchyme to epithelium signaling. In situ hybridizations with Wnt7b, a marker of the metanephric kidney, show that the branching defect was not simply the result of homeotic transformation of metanephros to mesonephros. Absent Bf2 and Gdnf expression in the midventral mesenchyme, findings that could by themselves account for branching defects, shows that Hoxa11 and Hoxd11 are necessary for normal gene expression in the ventral mesenchyme. Attenuation of normal gene expression along with the absence of a detectable proliferative or apoptotic change in the mutants show that one function of Hoxa11 and Hoxd11 in the developing renal mesenchyme is to regulate differentiation necessary for mesenchymal-epithelial reciprocal inductive interactions.     

A computational model of dendrite elongation and branching based on MAP2 phosphorylation

We introduce a new computational model of dendritic development in neurons. In contrast to previous models, our model explicitly includes cellular mechanisms involved in dendritic development. It is based on recent experimental data which indicates that the phosphorylation state of microtubule-associated protein 2 (MAP2) may play a key role in controlling dendritic elongation and branching (Audesirk et al., 1997). Dephosphorylated MAP2 favours elongation by promoting microtubule polymerization and bundling, whilst branching is more likely to occur when MAP2 is phosphorylated and microtubules are spaced apart. In the model, the rate of elongation and branching is directly determined by the ratio of phosphorylated to dephosphorylated MAP2. This is regulated by calmodulin-dependent protein kinase II (CaMKII) and calcineurin, which are both dependent on the intracellular calcium concentration. Results from computer simulations of the model suggest that the wide variety of branching patterns observed among different cell types may be generated by the same underlying mechanisms and that elongation and branching are not necessarily independent processes. The model predicts how the branching pattern will change following manipulations with calcium, CaMKII and MAP2 phosphorylation.     

Dynamic modeling of branching morphogenesis of ureteric bud in early kidney development

In the early kidney development, a simple epithelial tube called ureteric bud is derived from the intermediate mesoderm and undergoes a complex process of growth and terminal bifid branching. The branching of the ureteric bud is achieved by different cellular behaviors including cell proliferation and chemotaxis. In this paper, we examine how the branching morphology depends on different physical or chemical factors by constructing a cell-based model to describe the simple tube branching in the early kidney development. We conclude that a proper balance between growth speed of epithelial sheet due to cell proliferation and cell mobility due to chemotaxis is necessary to realize the development of normal Y-shaped pattern. When cell proliferation is fast compared to chemotaxis, kinked pattern is formed, and when cell proliferation is slow, bloated pattern is formed. These are consistent with experimental observations in different morphological anomalies of mutants. We show that the different branching patterns are accurately predicted by growth-chemotaxis ratio.     

Quantifying the branching frequency of virtual filamentous microbes using fractal analysis

The productivity of an industrial fermentation process involving a filamentous microbe is heavily dependent on the morphological form adopted by the organism. The development of systems capable of rapidly and accurately characterizing morphology within a given process represents a significant challenge, as the complex phenotypes that are manifested are not easily quantified. Conventional parameters employed in these analyses are often of limited value, as they reveal little about the branching behavior of the organism; an important consideration given the demonstrated link between branching frequency and metabolite production. In this study, the influence of branching behavior on the spatial distribution of mycelia grown in silico is examined through fractal analysis. It is demonstrated that fractal dimension, quantified based on the frequency distribution of parameterized boundary curves, and lacunarity act as robust estimators of branching behavior. The analysis can, in theory, be applied to any morphological form, providing universally applicable process parameters for more complete data acquisition. Biotechnol. Bioeng. 2013; 110: 437–447. © 2012 Wiley Periodicals, Inc.     

Opposing gradients of ephrin-As and EphA7 in the superior colliculus are essential for topographic mapping in the mammalian visual system

During development of the retinocollicular projection in mouse, retinal axons initially overshoot their future termination zones (TZs) in the superior colliculus (SC). The formation of TZs is initiated by interstitial branching at topographically appropriate positions. Ephrin-As are expressed in a decreasing posterior-to-anterior gradient in the SC, and they suppress branching posterior to future TZs. Here we investigate the role of an EphA7 gradient in the SC, which has the reverse orientation to the ephrin-A gradient. We find that in EphA7 mutant mice the retinocollicular map is disrupted, with nasal and temporal axons forming additional or extended TZs, respectively. In vitro, retinal axons are repelled from growing on EphA7-containing stripes. Our data support the idea that EphA7 is involved in suppressing branching anterior to future TZs. These findings suggest that opposing ephrin-A and EphA gradients are required for the proper development of the retinocollicular projection.     

Formation of the branching pattern of blood vessels in the wall of the avian yolk sac studied by a computer simulation

The present study was performed to provide data to support the notion previously believed but not proved experimentally or theoretically, that blood vessels are formed by the selection of capillaries in the network. In an attempt to understand the mechanism of formation of blood vessel branching structures, the transformation of a capillary network to a branching system in the wall of quail yolk sac was successively recorded by a series of photographs, and a computer simulation was carried out for the process of in vivo vascularization based on the photographs. The simulation demonstrated that a positive feedback system participated in the formation of a branching structure. That is, vessels which had been much used were enlarged, whereas less used vessels were reduced in their size and finally extinguished. The enlarged vessels became major components of the branching system. As the body of an embryo grew, it was observed that polygonal capillary networks enlarged, which led each polygon of the network to divide into a few finer polygons. Then, some of the capillary vessels were again selected and formed a branching system. This process repeated during the body growth, indicating that the vascular system developed adaptively to the body growth. A region where the growth was fast, received much blood flow and produced finer networks of capillaries. Thus, it was experimentally demonstrated for the first time that capillaries in the network are successively selected by a positive feedback mechanism and form blood vessels.     

Adaptive dynamics in diploid, sexual populations and the evolution of reproductive isolation

Evolutionary branching is the process whereby an initially monomorphic population evolves to a point where it undergoes disruptive selection and splits up into two phenotypically diverging lineages. We studied evolutionary branching in three models that are ecologically identical but that have different genetic systems. The first model is clonal, the second is sexual diploid with additive genetics on a single locus and the third is like the second but with an additional locus for mate choice. Evolutionary branching occurred under exactly the same ecological circumstances in all three models. After branching the evolutionary dynamics may be qualitatively different. In particular, in the diploid, sexual models there can be multiple evolutionary outcomes whereas in the corresponding clonal model there is only one. We showed that evolutionary branching favours the evolution of (partial) assortative mating and that this in turn effectively restores the results from the clonal model by rendering the alternative outcomes unreachable except for the one that also occurs in the clonal model. The evolution of assortative mating during evolutionary branching can be interpreted as the initial phase of sympatric speciation with phenotypic divergence and partial reproductive isolation.     

Evolutionary Branching in a Finite Population: Deterministic Branching vs. Stochastic Branching

Adaptive dynamics formalism demonstrates that, in a constant environment, a continuous trait may first converge to a singular point followed by spontaneous transition from a unimodal trait distribution into a bimodal one, which is called “evolutionary branching.” Most previous analyses of evolutionary branching have been conducted in an infinitely large population. Here, we study the effect of stochasticity caused by the finiteness of the population size on evolutionary branching. By analyzing the dynamics of trait variance, we obtain the condition for evolutionary branching as the one under which trait variance explodes. Genetic drift reduces the trait variance and causes stochastic fluctuation. In a very small population, evolutionary branching does not occur. In larger populations, evolutionary branching may occur, but it occurs in two different manners: in deterministic branching, branching occurs quickly when the population reaches the singular point, while in stochastic branching, the population stays at singularity for a period before branching out. The conditions for these cases and the mean branching-out times are calculated in terms of population size, mutational effects, and selection intensity and are confirmed by direct computer simulations of the individual-based model.     

Lung regions differently modulate bronchial branching development and extracellular matrix plays a role in regulating the development of chick embryo whole lung

Normal branching development is dependent on the correlation between cells and extracellular matrix. In this interaction glycosaminoglycans, cytokines and growth factors play a fundamental role. In order to verify the distribution and influence of extracellular matrix and related enzymes on chick embryo lung development, 6 day-old whole lungs were maintained in vitro with testicular hyaluronidase, beta-N-acetyl-D-glucosaminidase and chondrotinase ABC or in linkage with apical, medial and caudal lung regions of 6-day development before and after enzyme treatment. In a separate lung region beta-N-acetyl-D-glucosaminidase and hyaluronidase were determined. Our data show that the whole lung cultures increase bronchial branching development when the medial region is admixed separately, while the separate apical or caudal regions or apical combined with caudal region do not affect bronchial branching development. The enzyme treatment of medial region prevents the branching development in associated whole lung. The bronchial branching development of whole lung cultured in medium containing the enzymes related to glycosaminoglycans turnover is significantly altered. In conclusion, these data show that the different influence of separate apical, medial, caudal lung regions on bronchial branching development is related to the extracellular matrix composition.     

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.     

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     

Branch architecture of the fly larval abdominal serotonergic neurons

In the metazoan central nervous system (CNS), serotonergic neurons send projections throughout the synaptic neuropil. Little is known about the rules that govern these widespread neuromodulatory branching patterns. In this study, we utilize the Drosophila as a model to examine serotonergic branching. Using single cell GFP labeling we show that within each segment of the Drosophila ventral nerve cord (VNC), each of two serotonergic neurons tiles distinct innervation patterns in the contralateral neuropil. In addition, branches extend only a short distance from the target segment. Through ablation-mediated isolation of serotonergic cells, we demonstrate that the distinct areas of innervation are not maintained through competition between neighboring like-serotonergic neurites. Furthermore, the basic branching pattern of serotonergic neurons within the neuropil remains unchanged despite alterations of initial axonal trajectories.     

Tests of a cellular model for constant branch distribution in the filamentous fungus Neurospora crassa

The growth of mycelial fungi is characterized by the highly polarized extension of hyphal tips and the formation of subapical branches, which themselves extend as new tips. In Neurospora crassa, tip growth and branching are crucial elements for this saprophyte in the colonization and utilization of organic substrates. Much research has focused on the mechanism of tip extension, but a cellular model that fully explains the known phenomenology of branching by N. crassa has not been proposed. We described and tested a model in which the formation of a lateral branch in N. crassa was determined by the accumulation of tip-growth vesicles caused by the excess of the rate of supply over the rate of deposition at the apex. If both rates are proportional to metabolic rate, then the model explains the known lack of dependence of branch interval on growth rate. We tested the model by manipulating the tip extension rate, first by shifting temperature in both the wild type and hyperbranching (colonial) mutants and also by observing the behavior of both tipless colonies and colonyless tips. We found that temperature shifts in either direction result in temporary changes in branching. We found that colonyless tips also pass through a temporary transition phase of branching. The tipless colonies produced a cluster of new tips near the point of damage. We also found that branching in colonial mutants is dependent on growth rate. The results of these tests are consistent with a model of branching in which branch initiation is controlled by the dynamics of tip growth while being independent of the actual rate of this growth.     

Self-organization of the vascular system in plant leaves: inter-dependent dynamics of auxin flux and carrier proteins

The vegetative hormone Auxin is involved in vascular tissues formation throughout the plant. Trans-membrane carrier proteins transporting auxin from cell to cell and distributed asymmetrically around each cell give to auxin a polarized movement in tissues, creating streams of auxin that presume future vascular bundles. According to the canalization hypothesis, auxin transport ability of cells is thought to increase with auxin flux, resulting in the self-enhancement of this flux along auxin paths. In this study we evaluate a series of models based on canalization hypothesis using carrier proteins, under different assumptions concerning auxin flux formation and carrier protein dynamics. Simulations are run on a hexagonal lattice with uniform auxin production. A single cell located in the margin of the lattice indicates the petiole, and acts as an auxin sink. The main results are: (1) We obtain branching auxin distribution patterns. (2) The type of self-enhancement described by the functional form of the carrier proteins regulation responding to the auxin flux intensity in different parts of a cell, has a strong effect on the possibility of generating the branching patterns. For response functions with acceleration in the increase of carrier protein numbers compared to the auxin flux, branching patterns are likely to be generated. For linear or decelerating response functions, no branching patterns are formed. (3) When branching patterns are formed, auxin distribution greatly differs between the case in which the number of carrier proteins in different parts of a cell are regulated independently, and the case in which different parts of a cell compete for a limited number of carrier proteins. In the former case, the auxin level is lower in veins than in the surrounding tissue, while in the latter, the auxin is present in greater abundance in veins. These results suggest that canalization is a good candidate for describing plant vein pattern formation.     

Tests of a Cellular Model for Constant Branch Distribution in the Filamentous Fungus Neurospora crassa

The growth of mycelial fungi is characterized by the highly polarized extension of hyphal tips and the formation of subapical branches, which themselves extend as new tips. In Neurospora crassa, tip growth and branching are crucial elements for this saprophyte in the colonization and utilization of organic substrates. Much research has focused on the mechanism of tip extension, but a cellular model that fully explains the known phenomenology of branching by N. crassa has not been proposed. We described and tested a model in which the formation of a lateral branch in N. crassa was determined by the accumulation of tip-growth vesicles caused by the excess of the rate of supply over the rate of deposition at the apex. If both rates are proportional to metabolic rate, then the model explains the known lack of dependence of branch interval on growth rate. We tested the model by manipulating the tip extension rate, first by shifting temperature in both the wild type and hyperbranching (colonial) mutants and also by observing the behavior of both tipless colonies and colonyless tips. We found that temperature shifts in either direction result in temporary changes in branching. We found that colonyless tips also pass through a temporary transition phase of branching. The tipless colonies produced a cluster of new tips near the point of damage. We also found that branching in colonial mutants is dependent on growth rate. The results of these tests are consistent with a model of branching in which branch initiation is controlled by the dynamics of tip growth while being independent of the actual rate of this growth.     

Observations of the Effect of Water on the Hyphal Apices of Fusarium oxysporum

When colonies of Fusarium oxysporum, growing on plates of mineral-sucroseagar, are flooded with the mineral-sucrose solution, withoutadded agar, or with solutions of any of the constituents ofthe mineral-sucrose mixture at a concentration of 0.076 M theleading hyphal apices at the agar surface continue to grow onunchecked. If, however, the colonies are flooded with solutionsof decreasing and increasing molarity from 0.076 M an increasingproportion of the leading hyphal apices at the agar surfacestop growing, and branch subterminally. In distilled water about50 per cent. of the apices branch and this branching is precededby swelling, whereas in 0.5 M sucrose more than 90 per cent.of the apices branch and the branching is not accompanied byswelling. In the distilled water those hyphae which do not branchswell a little and grow on from the apex within 40 seconds. When hyphal apices are flooded with distilled water for from10 to 40 seconds and then transferred to mineral-sucrose solutionmore than 90 per cent, of the hyphal apices branch, whereasflooding with distilled water for 60 seconds or longer givesthe same percentage of branched apices as does flooding withdistilled water alone. It is shown that swelling and branching of the hyphal apex arenot causally related but that branching always occurs followingarrestment of the hyphal apex for more than 60 seconds. It issuggested that the phenomena reported can be explained in termsof an irreversible change in the apical cap of the arrestedhypha such that continued extension can no longer take placein this region and fresh outlets for growth must then be foundsubterminally. Such a mechanism, however triggered, could accountfor a wide variety of morphogenetic forms in the fungi.