Quantifying loopy network architectures

Biology presents many examples of planar distribution and structural networks having dense sets of closed loops. An archetype of this form of network organization is the vasculature of dicotyledonous leaves, which showcases a hierarchically-nested architecture containing closed loops at many different levels. Although a number of methods have been proposed to measure aspects of the structure of such networks, a robust metric to quantify their hierarchical organization is still lacking. We present an algorithmic framework, the hierarchical loop decomposition, that allows mapping loopy networks to binary trees, preserving in the connectivity of the trees the architecture of the original graph. We apply this framework to investigate computer generated graphs, such as artificial models and optimal distribution networks, as well as natural graphs extracted from digitized images of dicotyledonous leaves and vasculature of rat cerebral neocortex. We calculate various metrics based on the Asymmetry, the cumulative size distribution and the Strahler bifurcation ratios of the corresponding trees and discuss the relationship of these quantities to the architectural organization of the original graphs. This algorithmic framework decouples the geometric information (exact location of edges and nodes) from the metric topology (connectivity and edge weight) and it ultimately allows us to perform a quantitative statistical comparison between predictions of theoretical models and naturally occurring loopy graphs.Comment: 17 pages, 8 figures. During preparation of this manuscript the authors became aware of the work of Mileyko at al., concurrently submitted for publicatio

A Computational Clonal Analysis of the Developing Mouse Limb Bud

A comprehensive spatio-temporal description of the tissue movements underlying organogenesis would be an extremely useful resource to developmental biology. Clonal analysis and fate mappings are popular experiments to study tissue movement during morphogenesis. Such experiments allow cell populations to be labeled at an early stage of development and to follow their spatial evolution over time. However, disentangling the cumulative effects of the multiple events responsible for the expansion of the labeled cell population is not always straightforward. To overcome this problem, we develop a novel computational method that combines accurate quantification of 2D limb bud morphologies and growth modeling to analyze mouse clonal data of early limb development. Firstly, we explore various tissue movements that match experimental limb bud shape changes. Secondly, by comparing computational clones with newly generated mouse clonal data we are able to choose and characterize the tissue movement map that better matches experimental data. Our computational analysis produces for the first time a two dimensional model of limb growth based on experimental data that can be used to better characterize limb tissue movement in space and time. The model shows that the distribution and shapes of clones can be described as a combination of anisotropic growth with isotropic cell mixing, without the need for lineage compartmentalization along the AP and PD axis. Lastly, we show that this comprehensive description can be used to reassess spatio-temporal gene regulations taking tissue movement into account and to investigate PD patterning hypothesis

Large-Scale Clonal Analysis Reveals Unexpected Complexity in Surface Ectoderm Morphogenesis

Background: Understanding the series of morphogenetic processes that underlie the making of embryo structures is a highly topical issue in developmental biology, essential for interpreting the massive molecular data currently available. In mouse embryo, long-term in vivo analysis of cell behaviours and movements is difficult because of the development in utero and the impossibility of long-term culture. Methodology/Principal Findings: We improved and combined two genetic methods of clonal analysis that together make practicable large-scale production of labelled clones. Using these methods we performed a clonal analysis of surface ectoderm (SE), a poorly understood structure, for a period that includes gastrulation and the establishment of the body plan. We show that SE formation starts with the definition at early gastrulation of a pool of founder cells that is already dorso-ventrally organized. This pool is then regionalized antero-posteriorly into three pools giving rise to head, trunk and tail. Each pool uses its own combination of cell rearrangements and mode of proliferation for elongation, despite a common clonal strategy that consists in disposing along the antero-posterior axis precursors of dorso-ventrally-oriented stripes of cells. Conclusions/Significance: We propose that these series of morphogenetic processes are organized temporally and spatially in a posterior zone of the embryo crucial for elongation. The variety of cell behaviours used by SE precursor cells indicates that these precursors are not equivalent, regardless of a common clonal origin and a common clonal strategy. Anothe

Simulation of Organ Patterning on the Floral Meristem Using a Polar Auxin Transport Model

An intriguing phenomenon in plant development is the timing and positioning of lateral organ initiation, which is a fundamental aspect of plant architecture. Although important progress has been made in elucidating the role of auxin transport in the vegetative shoot to explain the phyllotaxis of leaf formation in a spiral fashion, a model study of the role of auxin transport in whorled organ patterning in the expanding floral meristem is not available yet. We present an initial simulation approach to study the mechanisms that are expected to play an important role. Starting point is a confocal imaging study of Arabidopsis floral meristems at consecutive time points during flower development. These images reveal auxin accumulation patterns at the positions of the organs, which strongly suggests that the role of auxin in the floral meristem is similar to the role it plays in the shoot apical meristem. This is the basis for a simulation study of auxin transport through a growing floral meristem, which may answer the question whether auxin transport can in itself be responsible for the typical whorled floral pattern. We combined a cellular growth model for the meristem with a polar auxin transport model. The model predicts that sepals are initiated by auxin maxima arising early during meristem outgrowth. These form a pre-pattern relative to which a series of smaller auxin maxima are positioned, which partially overlap with the anlagen of petals, stamens, and carpels. We adjusted the model parameters corresponding to properties of floral mutants and found that the model predictions agree with the observed mutant patterns. The predicted timing of the primordia outgrowth and the timing and positioning of the sepal primordia show remarkable similarities with a developing flower in nature

Mechanical Stress Induces Remodeling of Vascular Networks in Growing Leaves

International audienceDifferentiation into well-defined patterns and tissue growth are recognized as key processes in organismal development. However, it is unclear whether patterns are passively, homogeneously dilated by growth or whether they remodel during tissue expansion. Leaf vascu-lar networks are well-fitted to investigate this issue, since leaves are approximately two-dimensional and grow manyfold in size. Here we study experimentally and computationally how vein patterns affect growth. We first model the growing vasculature as a network of viscoelastic rods and consider its response to external mechanical stress. We use the so-called texture tensor to quantify the local network geometry and reveal that growth is heterogeneous , resembling non-affine deformations in composite materials. We then apply mechanical forces to growing leaves after veins have differentiated, which respond by anisotropic growth and reorientation of the network in the direction of external stress. External mechanical stress appears to make growth more homogeneous, in contrast with the model with viscoelastic rods. However, we reconcile the model with experimental data by incorporating randomness in rod thickness and a threshold in the rod growth law, making the rods viscoelastoplastic. Altogether, we show that the higher stiffness of veins leads to their reorientation along external forces, along with a reduction in growth heterogeneity. This process may lead to the reinforcement of leaves against mechanical stress. More generally , our work contributes to a framework whereby growth and patterns are coordinated through the differences in mechanical properties between cell types

Multiscale modelling of auxin transport in the plant-root elongation zone

In the root elongation zone of a plant, the hormone auxin moves in a polar manner due to active transport facilitated by spatially distributed influx and efflux carriers present on the cell membranes. To understand how the cell-scale active transport and passive diffusion combine to produce the effective tissue-scale flux, we apply asymptotic methods to a cell-based model of auxin transport to derive systematically a continuum description from the spatially discrete one. Using biologically relevant parameter values, we show how the carriers drive the dominant tissue-scale auxin flux and we predict how the overall auxin dynamics are affected by perturbations to these carriers, for example, in knockout mutants. The analysis shows how the dominant behaviour depends on the cells' lengths, and enables us to assess the relative importance of the diffusive auxin flux through the cell wall. Other distinguished limits are also identified and their potential roles discussed. As well as providing insight into auxin transport, the study illustrates the use of multiscale (cell to tissue) methods in deriving simplified models that retain the essential biology and provide understanding of the underlying dynamics

Functional Specialization of the Plant miR396 Regulatory Network through Distinct MicroRNA–Target Interactions

MicroRNAs (miRNAs) are ∼21 nt small RNAs that regulate gene expression in animals and plants. They can be grouped into families comprising different genes encoding similar or identical mature miRNAs. Several miRNA families are deeply conserved in plant lineages and regulate key aspects of plant development, hormone signaling, and stress response. The ancient miRNA miR396 regulates conserved targets belonging to the GROWTH-REGULATING FACTOR (GRF) family of transcription factors, which are known to control cell proliferation in Arabidopsis leaves. In this work, we characterized the regulation of an additional target for miR396, the transcription factor bHLH74, that is necessary for Arabidopsis normal development. bHLH74 homologs with a miR396 target site could only be detected in the sister families Brassicaceae and Cleomaceae. Still, bHLH74 repression by miR396 is required for margin and vein pattern formation of Arabidopsis leaves. MiR396 contributes to the spatio-temporal regulation of GRF and bHLH74 expression during leaf development. Furthermore, a survey of miR396 sequences in different species showed variations in the 5′ portion of the miRNA, a region known to be important for miRNA activity. Analysis of different miR396 variants in Arabidopsis thaliana revealed that they have an enhanced activity toward GRF transcription factors. The interaction between the GRF target site and miR396 has a bulge between positions 7 and 8 of the miRNA. Our data indicate that such bulge modulates the strength of the miR396-mediated repression and that this modulation is essential to shape the precise spatio-temporal pattern of GRF2 expression. The results show that ancient miRNAs can regulate conserved targets with varied efficiency in different species, and we further propose that they could acquire new targets whose control might also be biologically relevant