Geometry of gene regulatory dynamics.

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.

Geometry, epistasis, and developmental patterning.

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.

Turning a plant tissue into a living cell froth through isotropic growth.

The forms resulting from growth processes are highly sensitive to the nature of the driving impetus, and to the local properties of the medium, in particular, its isotropy or anisotropy. In turn, these local properties can be organized by growth. Here, we consider a growing plant tissue, the shoot apical meristem of Arabidopsis thaliana. In plants, the resistance of the cell wall to the growing internal turgor pressure is the main factor shaping the cells and the tissues. It is well established that the physical properties of the walls depend on the oriented deposition of the cellulose microfibrils in the extracellular matrix or cell wall; this order is correlated to the highly oriented cortical array of microtubules attached to the inner side of the plasma membrane. We used oryzalin to depolymerize microtubules and analyzed its influence on the growing meristem. This had no short-term effect, but it had a profound impact on the cell anisotropy and the resulting tissue growth. The geometry of the cells became similar to that of bubbles in a soap froth. At a multicellular scale, this switch to a local isotropy induced growth into spherical structures. A theoretical model is presented in which a cellular structure grows through the plastic yielding of its walls under turgor pressure. The simulations reproduce the geometrical properties of a normal tissue if cell division is included. If not, a "cell froth" very similar to that observed experimentally is obtained. Our results suggest strong physical constraints on the mechanisms of growth regulation.

HaloTag-based reporters for sparse labeling and cell tracking.

Multiscale analysis of morphogenesis requires to follow and measure in real-time the in vivo behaviour of large numbers of individual cells over long period of time. Despite recent progress, the large-scale automated tracking of cells in developing embryos and tissues remains a challenge. Here we describe a genetic tool for the random and sparse labelling of individual cells in developing Drosophila tissues. This tool is based on the conditional expression of a nuclear HaloTag protein that can be fluorescently labelled upon the irreversible binding of a cell permeable synthetic ligand. While the slow maturation of genetically encoded fluorescent renders the tracking of individual cells difficult in rapidly dividing tissues, nuclear HaloTag proteins allowed for rapid labelling of individual cells in cultured imaginal discs. To study cell shape changes, we also produced an HaloTag version of the actin-bound protein LifeAct. Since sparse labelling facilitates cell tracking, nuclear HaloTag reporters will be useful for the single-cell analysis of fate dynamics in Drosophila tissues cultured ex vivo.

Fluctuations and redundancy in optimal transport networks.

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.

A tensile ring drives tissue flows to shape the gastrulating amniote embryo.

Tissue morphogenesis is driven by local cellular deformations that are powered by contractile actomyosin networks. How localized forces are transmitted across tissues to shape them at a mesoscopic scale is still unclear. Analyzing gastrulation in entire avian embryos, we show that it is driven by the graded contraction of a large-scale supracellular actomyosin ring at the margin between the embryonic and extraembryonic territories. The propagation of these forces is enabled by a fluid-like response of the epithelial embryonic disk, which depends on cell division. A simple model of fluid motion entrained by a tensile ring quantitatively captures the vortex-like "polonaise" movements that accompany the formation of the primitive streak. The geometry of the early embryo thus arises from the transmission of active forces generated along its boundary.

In silico leaf venation networks: growth and reorganization driven by mechanical forces.

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.

Self-Organization in Pattern Formation.

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.

Regulation of Notch output dynamics via specific E(spl)-HLH factors during bristle patterning in Drosophila.

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.

Glutamate spillover in C. elegans triggers repetitive behavior through presynaptic activation of MGL-2/mGluR5.

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.

Glia Modulate a Neuronal Circuit for Locomotion Suppression during Sleep in C. elegans.

Glia have been suggested to regulate sleep-like states in vertebrates and invertebrates alike. In the nematode Caenorhabditis elegans, sleep is associated with molting between larval stages. To understand if glia modulate neural circuits driving sleep in C. elegans larvae, we ablated the astrocyte-like CEPsh glia. We found that glia-ablated animals exhibit episodes of immobility preceding sleep, prolonged sleep, molting-independent short-duration locomotory pausing, and delayed development. CEPsh glia ensheath synapses between the sleep-associated ALA neuron and its postsynaptic partner AVE, a major locomotion interneuron. While AVE calcium transients normally correlate with head retraction, glia ablation results in prolonged calcium transients that are uncoupled from movement. Strikingly, all these glia ablation defects are suppressed by the ablation of ALA. Our results suggest that glia attenuate sleep-promoting inhibitory connections between ALA and AVE, uncovering specific roles for glia in sleep behavior. We propose that similar mechanisms may underlie glial roles in sleep in other animals.

Gene-free methodology for cell fate dynamics during development.

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.

Self-organized Notch dynamics generate stereotyped sensory organ patterns in Drosophila.

The emergence of spatial patterns in developing multicellular organisms relies on positional cues and cell-cell communication. Drosophila sensory organs have informed a paradigm in which these operate in two distinct steps: Prepattern factors drive localized proneural activity, then Notch-mediated lateral inhibition singles out neural precursors. Here we show that self-organization through Notch signaling also establishes the proneural stripes that resolve into rows of sensory bristles on the fly thorax. Patterning, initiated by a gradient of Delta ligand expression, progresses through inhibitory signaling between and within stripes. Thus, Notch signaling can support self-organized tissue patterning as a prepattern is transduced by cell-cell interactions into a refined arrangement of cellular fates.

Planar Cell Polarity Breaks the Symmetry of PAR Protein Distribution prior to Mitosis in Drosophila Sensory Organ Precursor Cells.

During development, cell-fate diversity can result from the unequal segregation of fate determinants at mitosis. 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). 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. 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. 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.

Developmental patterning by mechanical signals in Arabidopsis.

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.