Branch mode selection during early lung development

Many organs of higher organisms, such as the vascular system, lung, kidney, pancreas, liver and glands, are heavily branched structures. The branching process during lung development has been studied in great detail and is remarkably stereotyped. The branched tree is generated by the sequential, non-random use of three geometrically simple modes of branching (domain branching, planar and orthogonal bifurcation). While many regulatory components and local interactions have been defined an integrated understanding of the regulatory network that controls the branching process is lacking. We have developed a deterministic, spatio-temporal differential-equation based model of the core signaling network that governs lung branching morphogenesis. The model focuses on the two key signaling factors that have been identified in experiments, fibroblast growth factor (FGF10) and sonic hedgehog (SHH) as well as the SHH receptor patched (Ptc). We show that the reported biochemical interactions give rise to a Schnakenberg-type Turing patterning mechanisms that allows us to reproduce experimental observations in wildtype and mutant mice. The kinetic parameters as well as the domain shape are based on experimental data where available. The developed model is robust to small absolute and large relative changes in the parameter values. At the same time there is a strong regulatory potential in that the switching between branching modes can be achieved by targeted changes in the parameter values. We note that the sequence of different branching events may also be the result of different growth speeds: fast growth triggers lateral branching while slow growth favours bifurcations in our model. We conclude that the FGF10-SHH-Ptc1 module is sufficient to generate pattern that correspond to the observed branching modes.     

A model for renal arterial branching based on graph theory

The kidney is one of the most complicated organs in terms of structure and physiology, in part because it is highly vascularized. The renal vascular development occurs through two mechanisms that sometimes overlap: vasculogenesis and angiogenesis. Here, we consider angiogenesis to model the renal arterial tree with the two processes of vascular angiogenesis: sprouting and splitting. We recognize the vessels are not tubes with ends that get glued but physiological factors are relevant into the vascular development. Our contribution integrates the graph theory and physiological information to derive a quantitative model for the vascular tree in the sense that the vertices and edges represent, respectively, a branching point and a vessel. From such a premise, development of the arterial vascular tree of the kidney is mathematically expressed, including physiological processes as the effect of the vascular endothelial growth factor (VEGF) on the vessel length. A definition of the graph is used to visualize the topology of vascular tree in kidney providing physiological information into the edges. Thus, renal arterial branching is modeled as a graph where edges are labeled and oriented.     

Branching tree model with fractal vascular resistance explains fractal perfusion heterogeneity

Perfusion heterogeneities in organs such as the heart obey a power law as a function of scale, a behavior termed "fractal." An explanation of why vascular systems produce such a specific perfusion pattern is still lacking. An intuitive branching tree model is presented that reveals how this behavior can be generated as a consequence of scale-independent branching asymmetry and fractal vessel resistance. Comparison of computer simulations to experimental data from the sheep heart shows that the values of the two free model parameters are realistic. Branching asymmetry within the model is defined by the relative tissue volume being fed by each branch. Vessel ordering for fractal analysis of morphology based on fed or drained tissue volumes is preferable to the commonly used Strahler system, which is shown to depend on branching asymmetry. Recently, noninvasive imaging techniques such as PET and MRI have been used to measure perfusion heterogeneity. The model allows a physiological interpretation of the measured fractal parameters, which could in turn be used to characterize vascular morphology and function.     

Modeling of Branching Patterns in Plants

A major determinant of plant architecture is the arrangement of branches around the stem, known as phyllotaxis. However, the specific form of branching conditions is not known. Here we discuss this question and suggest a branching model which seems to be in agreement with biological observations. Recently, a number of models connected with the genetic network or molecular biology regulation of the processes of pattern formation appeared. Most of these models consider the plant hormone, auxin, transport and distribution in the apical meristem as the main factors for pattern formation and phyllotaxis. However, all these models do not take into consideration the whole plant morphogenesis, concentrating on the events in the shoot or root apex. On the other hand, other approaches for modeling phyllotaxis, where the whole plant is considered, usually are mostly phenomenological, and due to it, do not describe the details of plant growth and branching mechanism. In this work, we develop a mathematical model and study pattern formation of the whole, though simplified, plant organism where the main physiological factors of plant growth and development are taken into consideration. We model a growing plant as a system of intervals, which we will consider as branches. We assume that the number and location of the branches are not given a priori, but appear and grow according to certain rules, elucidated by the application of mathematical modeling. Four variables are included in our model: concentrations of the plant hormones auxin and cytokinin, proliferation and growth factor, and nutrients—we observe a wide variety of plant forms and study more specifically the involvement of each variable in the branching process. Analysis of the numerical simulations shows that the process of pattern formation in plants depends on the interaction of all these variables. While concentrations of auxin and cytokinin determine the appearance of a new bud, its growth is determined by the concentrations of nutrients and proliferation factors. Possible mechanisms of apical domination in the frame of our model are discussed.     

Matrix GLA protein, an inhibitory morphogen in pulmonary vascular development

Deficiency of matrix GLA protein (MGP), an inhibitor of bone morphogenetic protein (BMP)-2/4, is known to cause arterial calcification and peripheral pulmonary artery stenosis. Yet the vascular role of MGP remains poorly understood. To further investigate MGP, we created a new MGP transgenic mouse model with high expression of the transgene in the lungs. The excess MGP led to a disruption of the pulmonary pattern of BMP-4, and resulted in significant morphological defects in the pulmonary artery tree. Specifically, the vascular branching pattern lacked characteristic side branching, whereas control lungs had extensive side branching accounting for as much as 40% of the vascular endothelium. The vascular changes could be explained by a dramatic reduction of phosphorylated SMAD1/5/8 in the alveolar epithelium, and in epithelial expression of the activin-like kinase receptor 1 and vascular endothelial growth factor, both critical in vascular formation. Abnormalities were also found in the terminal airways and in lung cell differentiation; high levels of surfactant protein-B were distributed in an abnormal pattern suggesting lost coordination between vasculature and airways. Ex vivo, lung cells from MGP transgenic mice showed higher proliferation, in particular surfactant protein B-expressing cells, and conditioned medium from these cells poorly supported in vitro angiogenesis compared with normal lung cells. The vascular branching defect can be mechanistically explained by a computational model based on activator/inhibitor reaction-diffusion dynamics, where BMP-4 and MGP are considered as an activating and inhibitory morphogen, respectively, suggesting that morphogen interactions are important for vascular branching.     

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.     

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.     

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     

High-resolution Time-lapse Imaging and Automated Analysis of Microtubule Dynamics in Living Human Umbilical Vein Endothelial Cells

The physiological process by which new vasculature forms from existing vasculature requires specific signaling events that trigger morphological changes within individual endothelial cells (ECs). These processes are critical for homeostatic maintenance such as wound healing, and are also crucial in promoting tumor growth and metastasis. EC morphology is defined by the organization of the cytoskeleton, a tightly regulated system of actin and microtubule (MT) dynamics that is known to control EC branching, polarity and directional migration, essential components of angiogenesis. To study MT dynamics, we used high-resolution fluorescence microscopy coupled with computational image analysis of fluorescently-labeled MT plus-ends to investigate MT growth dynamics and the regulation of EC branching morphology and directional migration. Time-lapse imaging of living Human Umbilical Vein Endothelial Cells (HUVECs) was performed following transfection with fluorescently-labeled MT End Binding protein 3 (EB3) and Mitotic Centromere Associated Kinesin (MCAK)-specific cDNA constructs to evaluate effects on MT dynamics. PlusTipTracker software was used to track EB3-labeled MT plus ends in order to measure MT growth speeds and MT growth lifetimes in time-lapse images. This methodology allows for the study of MT dynamics and the identification of how localized regulation of MT dynamics within sub-cellular regions contributes to the angiogenic processes of EC branching and migration.     

Distinct ECM mechanosensing pathways regulate microtubule dynamics to control endothelial cell branching morphogenesis

During angiogenesis, cytoskeletal dynamics that mediate endothelial cell branching morphogenesis during vascular guidance are thought to be regulated by physical attributes of the extracellular matrix (ECM) in a process termed mechanosensing. Here, we tested the involvement of microtubules in linking mechanosensing to endothelial cell branching morphogenesis. We used a recently developed microtubule plus end-tracking program to show that specific parameters of microtubule assembly dynamics, growth speed and growth persistence, are globally and regionally modified by, and contribute to, ECM mechanosensing. We demonstrated that engagement of compliant two-dimensional or three-dimensional ECMs induces local differences in microtubule growth speed that require myosin II contractility. Finally, we found that microtubule growth persistence is modulated by myosin II-mediated compliance mechanosensing when cells are cultured on two-dimensional ECMs, whereas three-dimensional ECM engagement makes microtubule growth persistence insensitive to changes in ECM compliance. Thus, compliance and dimensionality ECM mechanosensing pathways independently regulate specific and distinct microtubule dynamics parameters in endothelial cells to guide branching morphogenesis in physically complex ECMs.     

Agravitropic behaviour of roots of rapeseed (Brassica napus L.) transformed by Agrobacterium rhizogenes

Transgenic hairy roots of Brassica napus (cv. Omega) have been developed, using Agrobacterium rhizogenes strain AR 25, for use as a model system in the investigation of physiological and morphological differences between transgenic and normal roots. The basic parameters of growth and normal or altered gravitropical behaviour of hairy roots are for the first time presented in this paper together with an ultrastructural and morphological analysis of the root statocytes. The results obtained also represented the basis for the TRANSF0RM-experiment on the IML-2 mission performed onboard the Space Shuttle Columbia. Typical hairy root traits such as hormone-autonomous growth high growth rate, lateral branching, and changed/absence of gravitropism were detected. The transformed nature of the roots was confirmed by Southern blot analyses. The gravitropical behaviour of apices from hairy root cultures of this clone has been compared with root tips from normal seedlings. While the wild type roots curved progressively with increasing stimulation angles, the transformed roots showed no curvature when stimulated at 45 degrees, 90 degrees or 135 degrees on the ground. The morphology and ultrastructure of the root tip regions were examined by light microscopy and transmission electron microscopy. At the ultrastructural level no major differences could be detected between the roots studied. There was, however, a slight reduction in the starch content of most of the amyloplasts of the transgenic root tips, and the root cap was more V-shaped in the transgenic roots than in the wild type. Preliminary results from the Shuttle experiment TRANSFORM show a random distribution of amyloplasts in the root cells of both transformed and wild type root caps after 14 h on a 1xg centrifuge followed by 37 h in microgravity.     

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.     

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.     

The growth and form of tunnelling networks in ants

Many biological networks grow under strong spatial constraints, where the large-scale structure emerges from the extension, the branching and intersection of growing parts of the network. One example is provided by ant tunnelling networks, which represent the most common nest architecture in ants. Our goal was to understand how these network structures emerge from the tunnel growth dynamics. We used a standardized two-dimensional set-up shaped as a disk and studied the characteristics of tunnel growth in terms of initiation, propagation and termination of new digging sites and found that they can be described with simple probabilistic laws. We show that a model based on these simple laws and for which parameters were measured from the sand disks experiments can account for the emergence of several topological properties that were observed in experimental networks. In particular, the model accurately reproduced an allometric relation between the number of edges and the number of nodes, as well as an invariance of the node degree distribution. The model was then used to make predictions about the resulting networks' topology when the geometry of the sand substrate was shaped as a square. Experiments aimed at testing the model's predictions showed that the predictions were indeed validated. Both in the model and in the experiments, there was a similar trend for the node degree distribution tail to be steeper in the square sand patch than in the disk sand patch, while other characteristics such as the meshedness (i.e. how densely the network is internally connected) remained constant. Because network growth based on branching/fusion events is widespread in biological systems, this general model might provide useful insights for the study of other systems and, more generally, the evolution of spatial networks in biological systems.     

Consequences of root growth kinetics and vascular structure on the distribution of lateral roots

It has been proposed that the acropetal initiation of lateral roots is a built‐in process specified as part of the general process of cell division and differentiation in the parent root tip. Conversely, it is commonly reported that root branching is essentially a variable feature. In the present study, the interlateral distance along the parent root has been investigated using three banana varieties (Musa spp.) grown in two substrates. The pattern of lateral root initiation was obscured by variations of root growth patterns and vascular structure among roots, genotypes and substrates. A framework model is formulated showing the influence of growth pattern and vascular structure on branching density. The model raises a distinction between growth components which should not affect the branching density (i.e. rate of cell division) and which may affect it (i.e. size of mature cells and number of transverse divisions performed by cells executing their trajectory in the meristem). It appears also that lateral root density and root growth rate might be independently modulated by appropriate changes of root growth patterns, in banana and presumably many other taxa.     

Analysis of the plant architecture via tree-structured statistical models: the hidden Markov tree models

Plant architecture is the result of repetitions that occur through growth and branching processes. During plant ontogeny, changes in the morphological characteristics of plant entities are interpreted as the indirect translation of different physiological states of the meristems. Thus connected entities can exhibit either similar or very contrasted characteristics. We propose a statistical model to reveal and characterize homogeneous zones and transitions between zones within tree-structured data: the hidden Markov tree (HMT) model. This model leads to a clustering of the entities into classes sharing the same 'hidden state'. The application of the HMT model to two plant sets (apple trees and bush willows), measured at annual shoot scale, highlights ordered states defined by different morphological characteristics. The model provides a synthetic overview of state locations, pointing out homogeneous zones or ruptures. It also illustrates where within branching structures, and when during plant ontogeny, morphological changes occur. However, the labelling exhibits some patterns that cannot be described by the model parameters. Some of these limitations are addressed by two alternative HMT families.     

增温、施肥与种内竞争的交互作用对云杉根系属性的影响

增温、施肥与种内竞争的交互作用对云杉根系属性的影响 物种竞争、气温和土壤养分是青藏高原东部高寒地区影响树木生长的重要因素。虽然已开展了大量关于物种竞争、气温、施肥单因素对树木生长的影响研究,但关于这三者的交互作用对根系生长的影响还知之甚少。因此,本研究拟通过测量根系属性(细根长、根表面积、比根长、比表面积、根尖数、根系分支数等)、根生物量,以及根系养分吸收,研究施肥和增温对物种竞争的影响,并进一步探讨施肥、增温与物种竞争的交互作用对云杉(Picea asperata)生长的影响机制以及所采取的适应策略。研究结果表明,增温、施肥和竞争均提高了细根的氮、钾浓度,但并未影响细根生物量和根长、根表面积、根尖数和根分支数等根系特征。然而,无论是增温、施肥,或是它们的联合作用,与物种竞争进行交互时,均增加了根长、根表面积、根尖数、根系分支数和养分吸收。此外,施肥降低了根比表面积、比根长和单位面积的根尖数和根分支数,增温和竞争的交互作用使根比表面积、比根长下降,其他参数不受温度和竞争的影响。该结果表明,云杉在物种竞争、气候变暖、施肥及其交互作用下保持着保守的营养策略。该研究加强了对树木应对全球变化的生理和生态适应性的理解。     

Shape self-regulation in early lung morphogenesis

The arborescent architecture of mammalian conductive airways results from the repeated branching of lung endoderm into surrounding mesoderm. Subsequent lung's striking geometrical features have long raised the question of developmental mechanisms involved in morphogenesis. Many molecular actors have been identified, and several studies demonstrated the central role of Fgf10 and Shh in growth and branching. However, the actual branching mechanism and the way branching events are organized at the organ scale to achieve a self-avoiding tree remain to be understood through a model compatible with evidenced signaling. In this paper we show that the mere diffusion of FGF10 from distal mesenchyme involves differential epithelial proliferation that spontaneously leads to branching. Modeling FGF10 diffusion from sub-mesothelial mesenchyme where Fgf10 is known to be expressed and computing epithelial and mesenchymal growth in a coupled manner, we found that the resulting laplacian dynamics precisely accounts for the patterning of FGF10-induced genes, and that it spontaneously involves differential proliferation leading to a self-avoiding and space-filling tree, through mechanisms that we detail. The tree's fine morphological features depend on the epithelial growth response to FGF10, underlain by the lung's complex regulatory network. Notably, our results suggest that no branching information has to be encoded and that no master routine is required to organize branching events at the organ scale. Despite its simplicity, this model identifies key mechanisms of lung development, from branching to organ-scale organization, and could prove relevant to the development of other branched organs relying on similar pathways.     

A simulation study of the effects of architectural constraints and resource translocation on population structure and competition in clonal plants

(1) Spatially explicit simulation of clonal plant growth is used to determine how ramet-level traits affect ramet density, spatial pattern of ramets and competitive ability of a clonal plant. The simulation model used combines elements of (i) an individual-based model of plant interactions, (ii) an architectural model of clonal plant growth, and (iii) a model of resource translocation within a set of physiologically integrated plant individuals. (2) The effects of two groups of parameters were studied: growth and resource acquisition parameters (resource accumulation, density-dependence of resource accumulation, resource translocation between ramets) and architectural rules (branching angle and probability of branching, internode length). The model was parameterised by values approximating those of clonally growing grasses as closely as possible. The basic parameter values were chosen from a short-turf grassland. Sensitivity analysis was carried out on relevant parameters around three basic points in the parameter space. Both single-species and two-species systems were studied. (3) It is shown that increasing resource acquisition and growth parameters increase ramet density, genet number and competitive ability. Translocation parameters and architectural parameters modify the effects of resource acquisition and growth, but their effect in single-species stands was smaller. (4) The simulations of species with fixed ramet sizes showed that ramet density in single-species stands cannot be used for predicting competitive ability. Increase in resource acquisition and growth parameters was correlated with an increase in equilibrium ramet density and competitive ability. Increasing branching angle, branching probability or internode length lead to an increased competitive ability, but did not affect equilibrium ramet density. Change of architectural parameters could therefore affect competitive ability independently of their effect on the final ramet density. (5) Spatial pattern both in single-species and two-species stands was also highly parameter-dependent. Changes in architectural parameters and in translocation usually lead to pronounced change in the spatial pattern; change in growth and resource acquisition parameters generally had little effect on spatial pattern.