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Embryonic development leads to the reproducible and ordered appearance of complexity from egg to adult. The successive differentiation of different cell types that elaborate this complexity results from the activity of gene networks and was likened by Waddington to a flow through a landscape in which valleys represent alternative fates. Geometric methods allow the formal representation of such landscapes and codify the types of behaviors that result from systems of differential equations. Results from Smale and coworkers imply that systems encompassing gene network models can be represented as potential gradients with a Riemann metric, justifying the Waddington metaphor. Here, we extend this representation to include parameter dependence and enumerate all three-way cellular decisions realizable by tuning at most two parameters, which can be generalized to include spatial coordinates in a tissue. All diagrams of cell states vs. model parameters are thereby enumerated. We unify a number of standard models for spatial pattern formation by expressing them in potential form (i.e., as topographic elevation). Turing systems appear nonpotential, yet in suitable variables the dynamics are low dimensional and potential. A time-independent embedding recovers the original variables. Lateral inhibition is described by a saddle point with many unstable directions. A model for the patterning of the Drosophila eye appears as relaxation in a bistable potential. Geometric reasoning provides intuitive dynamic models for development that are well adapted to fit time-lapse data.

The stereotyped arrangement of sensory bristles on the adult fly thorax arises from a self-organized process, in which inhibitory Notch signaling both delimits proneural stripes and singles out sensory organ precursor cells (SOPs). A dynamic balance between proneural factors and Enhancer of split-HLH (E(spl)-HLH) Notch targets underlies patterning, but how this is regulated is unclear. Here, were identify two classes of E(spl)-HLH factors, whose expression both precedes and delimits proneural activity, and is dependent on proneural activity and required for proper SOP spacing within the stripes, respectively. These two classes are partially redundant, since a member of the second class, that is normally cross-repressed by members of the first class, can functionally compensate for their absence. The regulation of specific E(spl)-HLH genes by proneural factors amplifies the response to Notch as SOPs are being selected, contributing to patterning dynamics in the notum, and likely operates in other developmental contexts., The patterning of sensory bristles on the dorsal thorax of flies is regulated by two transcription factor families but the dynamics of this regulation is unclear. Here, the authors visualize seven E(spl)-HLH proteins, showing their regulated expression promotes mutual inhibition by Notch during notum patterning.

Self-organization is pervasive in development, from symmetry breaking in the early embryo to tissue patterning and morphogenesis. For a few model systems, the underlying molecular and cellular processes are now sufficiently characterized that mathematical models can be confronted with experiments, to explore the dynamics of pattern formation. Here, we review selected systems, ranging from cyanobacteria to mammals, where different forms of cell-cell communication, acting alone or together with positional cues, drive the patterning of cell fates, highlighting the insights that even very simple models can provide as well as the challenges on the path to a predictive understanding of development.

Glutamate is a major excitatory neurotransmitter, and impaired glutamate clearance following synaptic release promotes spillover, inducing extra-synaptic signaling. The effects of glutamate spillover on animal behavior and its neural correlates are poorly understood. We developed a glutamate spillover model in Caenorhabditis elegans by inactivating the conserved glial glutamate transporter GLT-1. GLT-1 loss drives aberrant repetitive locomotory reversal behavior through uncontrolled oscillatory release of glutamate onto AVA, a major interneuron governing reversals. Repetitive glutamate release and reversal behavior require the glutamate receptor MGL-2/mGluR5, expressed in RIM and other interneurons presynaptic to AVA. mgl-2 loss blocks oscillations and repetitive behavior; while RIM activation is sufficient to induce repetitive reversals in glt-1 mutants. Repetitive AVA firing and reversals require EGL-30/Gαq, an mGluR5 effector. Our studies reveal that cyclic autocrine presynaptic activation drives repetitive reversals following glutamate spillover. That mammalian GLT1 and mGluR5 are implicated in pathological motor repetition suggests a common mechanism controlling repetitive behaviors., Katz and colleagues examine glutamate spillover effects on C. elegans behaviour. They show that impaired synaptic glutamate clearance in glial glutamate transporter mutants, causes presynaptic mgl-2/mGluR5 activation, generating postsynaptic neural activity oscillations driving repetitive behaviour.

Models of cell function that assign a variable to each gene frequently lead to systems of equations with many parameters whose behavior is obscure. Geometric models reduce dynamics to intuitive pictorial elements that provide compact representations for sparse in vivo data and transparent descriptions of developmental transitions. To illustrate, a geometric model fit to vulval development in Caenorhabditis elegans, implies a phase diagram where cell-fate choices are displayed in a plane defined by EGF and Notch signaling levels. This diagram defines allowable and forbidden cell-fate transitions as EGF or Notch levels change, and explains surprising observations previously attributed to context-dependent action of these signals. The diagram also reveals the existence of special points at which minor changes in signal levels lead to strong epistatic interactions between EGF and Notch. Our model correctly predicts experiments near these points and suggests specific timed perturbations in signals that can lead to additional unexpected outcomes., eLife digest At first, embryos are made up of identical cells. Then, as the embryo develops, these cells specialize into different types, such as heart and brain cells. Chemical signals sent and received by the cells are key to forming the right type of cell at the right time and place. The cellular machinery that produces and interprets these signals is exceedingly complex and difficult to understand. In the 1950s, Conrad Waddington presented an alternative way of thinking about how an unspecialized cell progresses to one of many different fates. He suggested visualizing the developing cell as a ball rolling along a hilly landscape. As the ball travels, obstacles in its way guide it along particular paths. Eventually the ball comes to rest in a valley, with each valley in the landscape representing a different cell fate. Although this “landscape model” is an appealing metaphor for how signaling events guide cell specialization, it was not clear whether it could be put to productive use. The egg-laying organ in the worm species Caenorhabditis elegans is called the vulva, and is often studied by researchers who want to learn more about how organs develop. The vulva develops from a small number of identical cells that adopt one of three possible cell fates. Two chemical signals, called epidermal growth factor (EGF) and Notch, control this specialization process. Corson and Siggia have now constructed a simple landscape model that can reproduce the normal arrangement of cell types in the vulva. When adjusted to describe the effect of genetic mutations that affect either EGF or Notch, the model could predict the outcome of mutations that affect both signals at once. The twists and turns of cell paths in the landscape could also account for several non-intuitive cell fate outcomes that had been assumed to result from subtle regulation of EGF and Notch signals. Landscape models should be easy to apply to other developing tissues and organs. By providing an intuitive picture of how signals shape cellular decisions, the models could help researchers to learn how to control cell and tissue development. This could lead to new treatments to repair or replace failing organs, making regenerative medicine a reality.

International audience; INTRODUCTION: Spatial patterning in developing multicellular organisms relies on positional cues and cell-cell interactions. Stereotyped sensory organ arrangements in Drosophila are commonly attributed to a prepattern that defines regions of neural competence. Notch-mediated interactions then isolate sensory organ precursor (SOP) cells from among the competent cells. In support of this view, prepattern factors direct the expression of proneural factors in discrete clusters and determine the location of large bristles on the dorsal thorax. However, no such prepattern is known to establish the proneural stripes that give rise to finer-bristle rows.RATIONALE: By analogy with reaction-diffusion systems, we wondered whether Notch-mediated cell-cell interactions might organize a pattern of proneural stripes. To explore a possible role for Notch in proneural patterning, we generated fluorescent reporters for the proneural factors Achaete and Scute, the ligand Delta, and the Notch early-response factor E(spl)m3-HLH, which antagonizes proneural activity. We observed expression of these reporters in live and fixed samples throughout early pupal development. In parallel, we developed a mathematical model for Notch-mediated patterning. In this abstract model, the dynamics of a cell is expressed in terms of just two variables, for the state of the cell and the level of signal it receives. The model incorporates a series of plausible assumptions that govern its patterning behavior: Cells, which adopt the SOP fate in the absence of signal and the alternative, epidermal fate under high enough signal, exhibit a bistable response under intermediate signal levels. Inhibitory signaling from a cell varies nonlinearly with cell state and reaches beyond immediate neighbors.RESULTS: Before the onset of proneural gene expression, a bimodal gradient of Delta expression drove Notch activity; as a result of cis-inhibition, Notch was activated only in regions of intermediate Delta levels. This defined an inhibitory template for a first series of three proneural stripes. The spatial pattern of Notch activity dynamically changed as these first proneural stripes emerged, forming a negative template for a second group of intercalating stripes. Eventually, each stripe resolved into a row of isolated SOPs through Notch signaling. Thus, Notch mediated both proneural stripe patterning and SOP selection via a self-organized process. Simulations of the model, with a time-dependent inhibitory gradient describing the proneural-independent signaling seen in vivo, recapitulated the sequential emergence and resolution of proneural stripes. In both model and experiments, mutual inhibition drove a gradual refinement of the proneural group, concomitant with the buildup of proneural activity. Cells on the sides of the stripes were excluded first, such that stripes narrowed before isolated SOPs emerged. In terms of the model, cell-intrinsic bistability allowed cells with higher proneural activity, at the center of the stripes, to evade levels of inhibition that were sufficient to exclude cells on the sides. Nonlinear signaling allowed a smooth transition from weak mutual inhibition within an extended proneural group to strong lateral inhibition from emerging SOPs, ensuring that attrition of the proneural group proceeds until only isolated cells remain. Finally, the model correctly predicted the outcome of perturbation experiments affecting the pattern of Notch activity, the level and activity of Delta, and the range of signaling.CONCLUSION:Our results show that self-organized Notch signaling can establish stripe and dot patterns on a tissue-wide scale. A transient spatial bias, mediated by an initial gradient of Delta, is transduced by cell-cell interactions to produce a finer pattern of proneural stripes and bristle rows. Input from extrinsic positional cues and self-organization, sometimes considered competing paradigms for fate patterning, combine during bristle development and operate through the same signal. Self-organized Notch dynamics may provide a flexible substrate to generate diverse patterns in response to varying inputs.

The emergence of spatial patterns in developing multicellular organisms relies on positional cues and cell-cell communication.

Developmental signaling networks are composed of dozens of components whose interactions are very difficult to quantify in an embryo. Geometric reasoning enumerates a discrete hierarchy of phenotypic models with a few composite variables whose parameters may be defined by in vivo data. Vulval development in the nematode Caenorhabditis elegans is a classic model for the integration of two signaling pathways; induction by EGF and lateral signaling through Notch. Existing data for the relative probabilities of the three possible terminal cell types in diverse genetic backgrounds as well as timed ablation of the inductive signal favor one geometric model and suffice to fit most of its parameters. The model is fully dynamic and encompasses both signaling and commitment. It then predicts the correlated cell fate probabilities for a cross between any two backgrounds/conditions. The two signaling pathways are combined additively, without interactions, and epistasis only arises from the nonlinear dynamical flow in the landscape defined by the geometric model. In this way, the model quantitatively fits genetic experiments purporting to show mutual pathway repression. The model quantifies the contributions of extrinsic vs. intrinsic sources of noise in the penetrance of mutant phenotypes in signaling hypomorphs and explains available experiments with no additional parameters. Data for anchor cell ablation fix the parameters needed to define Notch autocrine signaling.

Summary During development, cell-fate diversity can result from the unequal segregation of fate determinants at mitosis [1]. Polarization of the mother cell is essential for asymmetric cell division (ACD). It often involves the formation of a cortical domain containing the PAR complex proteins Par3, Par6, and atypical protein kinase C (aPKC) [1–5]. In the fly notum, sensory organ precursor cells (SOPs) divide asymmetrically within the plane of the epithelium and along the body axis to generate two distinct cells [6–12]. Fate asymmetry depends on the asymmetric localization of the PAR complex. In the absence of planar cell polarity (PCP), SOPs divide with a random planar orientation but still asymmetrically, showing that PCP is dispensable for PAR asymmetry at mitosis [6, 13–15]. To study when and how the PAR complex localizes asymmetrically, we have used a quantitative imaging approach to measure the planar polarization of the proteins Bazooka (Baz, fly Par3), Par6, and aPKC in living pupae. By using imaging of functional GFP-tagged proteins with image processing and computational modeling, we find that Baz, Par6, and aPKC become planar polarized prior to mitosis in a manner independent of the AuroraA kinase and that PCP is required for the planar polarization of Baz, Par6, and aPKC during interphase. This indicates that a "mitosis rescue" mechanism establishes asymmetry at mitosis in PCP mutants. This study therefore identifies PCP as the initial symmetry-breaking signal for the planar polarization of PAR proteins in asymmetrically dividing SOPs.

We study the dynamics of N point particles with a gravitational interaction. The divergence of the microcanonical partition function prevents this system from reaching equilibrium. Assuming a random diffusion in phase space we deduce a scaling law involving time, which is numerically checked for 3 interacting masses in a quadratic nonsymmetrical potential. This random walk on the potential energy scale is studied in some detail and the results agree with the numerics.

The packing of elastic bodies has emerged as a paradigm for the study of macroscopic disordered systems. However, progress is hampered by the lack of controlled experiments. Here we consider a model experiment for the isotropic two-dimensional confinement of a rod by a central force. We seek to measure how ordered is a folded configuration and we identify two key quantities. A geometrical characterization is given by the number of superposed layers in the configuration. Using temporal modulations of the confining force, we probe the mechanical properties of the configuration and we define and measure its effective compressibility. These two quantities may be used to build a statistical framework for packed elastic systems., 4 pages, 5 figures

We present a model for mechanically-induced pattern formation in growing biological tissues and discuss its application to the development of leaf venation networks. Drawing an analogy with phase transitions in solids, we use a phase field method to describe the transition between two states of the tissue, e.g. the differentiation of leaf veins, and consider a layered system where mechanical stresses are generated by differential growth. We present analytical and numerical results for one-dimensional systems, showing that a combination of growth and irreversibility gives rise to hierarchical patterns. Two-dimensional simulations suggest that such a mechanism could account for the hierarchical, reticulate structure of leaf venation networks, yet point to the need for a more detailed treatment of the coupling between growth and mechanical stresses., To appear in Philosophical Magazine. 18 pages, 8 figures

International audience; Development commonly involves an interplay between signaling, genetic expression and biophysical forces. However, the relative importance of these mechanisms during the different stages of development is unclear. Leaf venation networks provide a fitting context for the examination of these questions. In mature leaves, venation patterns are extremely diverse, yet their local structure satisfies a universal property: at junctions between veins, angles and diameters are related by a vectorial equation analogous to a force balance. Using a cell proliferation model, we reproduce in silico the salient features of venation patterns. Provided that vein cells are given different mechanical properties, tensile forces develop along the veins during growth, causing the network to deform progressively. Our results suggest that the local structure of venation networks results from a reorganization driven by mechanical forces, independently of how veins form. This conclusion is supported by recent observations of vein development in young leaves and by the good quantitative agreement between our simulations and data from mature leaves.

A central question in developmental biology is whether and how mechanical forces serve as cues for cellular behavior and thereby regulate morphogenesis. We found that morphogenesis at the Arabidopsis shoot apex depends on the microtubule cytoskeleton, which in turn is regulated by mechanical stress. A combination of experiments and modeling shows that a feedback loop encompassing tissue morphology, stress patterns, and microtubule-mediated cellular properties is sufficient to account for the coordinated patterns of microtubule arrays observed in epidermal cells, as well as for patterns of apical morphogenesis.

The structure of networks that provide optimal transport properties has been investigated in a variety of contexts. While many different formulations of this problem have been considered, it is recurrently found that optimal networks are trees. It is shown here that this result is contingent on the assumption of a stationary flow through the network. When time variations or fluctuations are allowed for, a different class of optimal structures is found, which share the hierarchical organization of trees yet contain loops. The transitions between different network topologies as the parameters of the problem vary are examined. These results may have strong implications for the structure and formation of natural networks, as is illustrated by the example of leaf venation networks., 4 pages, 4 figures

International audience

We address analytically and numerically the problem of crack path prediction in the model system of a crack propagating under thermal loading. We show that one can explain the instability from a straight to a wavy crack propagation by using only the principle of local symmetry and the Griffith criterion. We then argue that the calculations of the stress intensity factors can be combined with the standard crack propagation criteria to obtain the evolution equation for the crack tip within any loading configuration. The theoretical results of the thermal crack problem agree with the numerical simulations we performed using a phase field model. Moreover, it turns out that the phase-field model allows to clarify the nature of the transition between straight and oscillatory cracks which is shown to be supercritical., Comment: 19 pages, 8 figures

ResumeInitialement publie dans la revue Signs en 1987, l’article de Kristin Bumiller traduit dans le cadre de ce dossier combine une approche critique de l’ideologie du droit heritiere des Critical legal studies avec une demarche de sociologie empirique et comprehensive qui prefigure certains aspects du courant des Legal consciousness studies. En s’appuyant sur une enquete par entretiens approfondis, Bumiller restitue finement le sens que revet le droit pour des personnes qui estiment avoir subi des discriminations, et qui n’ont pas, pour la plupart, porte plainte devant les tribunaux. Elle livre une vision tres pessimiste du « modele de la protection juridique », incapable de proteger, selon elle, les victimes de discriminations, placees dans l’entrelacement des rapports sociaux de genre, de classe et de race. Elle montre que celles-ci anticipent les couts psychologiques et materiels d’une plainte ainsi que leurs faibles chances d’obtenir gain de cause, et preferent se penser au prisme valorisant de « survivantes » de la discrimination, plutot qu’a celui – socialement degradant – de victimes.

International audience; During embryonic development, digits gradually emerge in a periodic pattern. Although genetic evidence indicates that digit formation results from a self-organizing process, the underlying mechanisms are still unclear. Here, we find that convergent-extension tissue flows driven by active stresses underlie digit formation. These active stresses simultaneously shape cartilage condensations and lead to the emergence of a compressive stress region that promotes high activin/p-SMAD/SOX9 expression, thereby defining digit-organizing centers via a mechanical feedback. In Wnt5a mutants, such mechanical feedback is disrupted due to the loss of active stresses, organizing centers do not emerge, and digit formation is precluded. Thus, digit emergence does not result solely from molecular interactions, as was previously thought, but requires a mechanical feedback that ensures continuous coupling between phalanx specification and elongation. Our work, which links mechanical and molecular signals, provides a mechanistic context for the emergence of organizing centers that may underlie various developmental processes.

The way in which interactions between mechanics and biochemistry lead to the emergence of complex cell and tissue organization is an old question that has recently attracted renewed interest from biologists, physicists, mathematicians and computer scientists. Rapid advances in optical physics, microscopy and computational image analysis have greatly enhanced our ability to observe and quantify spatiotemporal patterns of signalling, force generation, deformation, and flow in living cells and tissues. Powerful new tools for genetic, biophysical and optogenetic manipulation are allowing us to perturb the underlying machinery that generates these patterns in increasingly sophisticated ways. Rapid advances in theory and computing have made it possible to construct predictive models that describe how cell and tissue organization and dynamics emerge from the local coupling of biochemistry and mechanics. Together, these advances have opened up a wealth of new opportunities to explore how mechanochemical patterning shapes organismal development. In this roadmap, we present a series of forward-looking case studies on mechanochemical patterning in development, written by scientists working at the interface between the physical and biological sciences, and covering a wide range of spatial and temporal scales, organisms, and modes of development. Together, these contributions highlight the many ways in which the dynamic coupling of mechanics and biochemistry shapes biological dynamics: from mechanoenzymes that sense force to tune their activity and motor output, to collectives of cells in tissues that flow and redistribute biochemical signals during development.

Epithelial tissues are ubiquitous in animal life, covering surfaces and separating body compartments in diverse organs and species. Epithelium is the first tissue to form, playing a key role in structuring the intricate steps of an organism's development. In such a complex system, the interplay between cell division, chemical signalling, and mechanical forces permits the emergence of different tissues with specific functions. To uncover the mechanisms at play in epithelial tissue mechanics, a useful tool is provided by vertex models, in which cells are idealized as juxtaposed polygons. Numerical simulations of cell division in vertex models yield geometrically irregular cells, similar to empirical observations in epithelia, even when cell mechanical properties are homogeneous. Nevertheless, existing theoretical analyses are mostly confined to the mechanics of regular hexagonal lattices. Here, we develop an analytical description of geometrically disordered vertex models. We first quantify, in numerical simulations, geometric and mechanical cell properties in the presence of diverse sources of disorder, including various division rules or relaxation in the presence of noise. We then develop a simple mean-field description able to account for these properties. This description is expanded to address the interaction of isolated and clustered cells with mechanical properties differing from the surrounding tissue relevant in a wide range of biological contexts. This allows us to bridge the gap between theory and experiments, quantitatively predicting how variation in forces may affect geometry and topology as well as fitting mechanistic model parameters to the observed geometry of cell membranes in a tissue. Finally, in a collaborative work with the Payre's lab (U. de Toulouse), we apply our analysis to account for the statistics of cell geometry and division dynamics in experiments performed on epidermis differentiation in the fly. We particularly focus on characterizing the mechanical properties of clustered clonal cells with distinct proliferation rates from their environment., Les tissus épithéliaux sont ubiquitaires dans les organismes vivants, couvrant des surfaces et séparant différents compartiments dans divers organes et espèces. L'épithélium est le premier tissu qui apparaît au cours du développement et il joue un rôle central pour en structurer les étapes. Dans ce processus complexe, les interactions entre les divisions cellulaires, la signalisation chimique et les forces mécaniques permettent l’émergence de différents tissus avec des fonctions diverses. Dans l’élucidation des mécanismes en jeu dans la mécanique des tissus épithéliaux, un outil utile est constitué par les modèles de vertex, dans lesquels les cellules sont représentées de façon simplifiée par des polygones. La simulation de la division cellulaire dans ces modèles, produit des tissus avec des cellules géométriquement irrégulières, semblables à celles observées dans les épithéliums naturels, même quand les propriétés mécaniques des différentes cellules sont identiques. Les analyses théoriques existantes sont cependant, pour la plupart, confinées aux réseaux réguliers hexagonaux. Dans ce travail, nous proposons une analyse théorique d’un modèle de vertex géométriquement désordonné. Nous commençons par quantifier à l’aide de simulations numériques, les propriétés géométriques et mécaniques des cellules, dans des tissus désordonnés produits par différentes règles de division cellulaires ou simplement par la relaxation en présence de bruit. Nous développons ensuite une analyse simple de type « champ moyen » pour décrire ces propriétés. Cette description est étendue au cas d’une cellule ou d’un clone de cellules interagissant avec le tissu environnant aux propriétés mécaniques différentes, une situation qui se retrouve dans différents contextes biologiques. Cela nous permet de jeter un pont entre la théorie et l’expérience et de prédire quantitativement comment cette variation mécanique affecte la géométrie et la topologie et d’ajuster un modèle mécanique aux observations empiriques de la géométrie des membranes cellulaires dans un tissu. Finalement, en collaboration avec le groupe de F. Payre (Université de Toulouse), nous appliquons cette analyse pour décrire la géométrie et la dynamique des divisions cellulaire au cours de la différenciation de l’épiderme de la mouche. Nous nous attachons particulièrement à caractériser les propriétés mécaniques d’un clone de cellules qui prolifère différemment du tissu environnant.