Mutual inactivation of Notch receptors and ligands facilitates developmental patterning

Developmental patterning requires juxtacrine signaling in order to tightly coordinate the fates of neighboring cells. Recent work has shown that Notch and Delta, the canonical metazoan juxtacrine signaling receptor and ligand, mutually inactivate each other in the same cell. This cis-interaction generates mutually exclusive sending and receiving states in individual cells. It generally remains unclear, however, how this mutual inactivation and the resulting switching behavior can impact developmental patterning circuits. Here we address this question using mathematical modeling in the context of two canonical pattern formation processes: boundary formation and lateral inhibition. For boundary formation, in a model motivated by Drosophila wing vein patterning, we find that mutual inactivation allows sharp boundary formation across a broader range of parameters than models lacking mutual inactivation. This model with mutual inactivation also exhibits robustness to correlated gene expression perturbations. For lateral inhibition, we find that mutual inactivation speeds up patterning dynamics, relieves the need for cooperative regulatory interactions, and expands the range of parameter values that permit pattern formation, compared to canonical models. Furthermore, mutual inactivation enables a simple lateral inhibition circuit architecture which requires only a single downstream regulatory step. Both model systems show how mutual inactivation can facilitate robust fine-grained patterning processes that would be difficult to implement without it, by encoding a difference-promoting feedback within the signaling system itself. Together, these results provide a framework for analysis of more complex Notch-dependent developmental systems.     

Intra-membrane ligand diffusion and cell shape modulate juxtacrine patterning

A key problem in developmental biology is how pattern and planar polarity are transmitted in epithelial structures. Examples include Drosophila neuronal differentiation, ommatidia formation in the compound eye, and wing hair polarization. A key component for the generation of such patterns is direct cell-cell signalling by transmembrane ligands, called juxtacrine signalling. Previous models for this mode of communication have considered homogeneous distributions in the cell membrane, and the role of polarity has been largely ignored. In this paper we determine the role of inhomogeneous protein and receptor distributions in juxtacrine signalling. We explicitly include individual membrane segments, diffusive transport of proteins and receptors between these segments, and production terms with a combination of local and global responses to ligand binding. Our analysis shows that intra-membrane ligand transport is vital for the generation of long wavelength patterns. Moreover, with no ligand transport, there is no pattern formation for lateral induction, a process in which receptor activation up-regulates ligand production. Biased production of ligand also modulates patterning bifurcations and predicted wavelengths. In addition, biased ligand and receptor trafficking can lead to regular polarity across a lattice, in which each cell has the same orientation-directly analogous to patterns of hairs in the Drosophila wing. We confirm the trends in pattern wavelengths previously observed for patterns with cellular homogeneity-lateral inhibition tends to give short-range patterns, while lateral induction can give patterns with much longer wavelengths. Moreover, the original model can be recovered if intra-membrane bound receptor diffusion is included and rapid equilibriation between the sides is considered. Finally, we consider the role of irregular cell shapes and waves in such networks, including wave propagation past clones of non-signalling cells.     

Mathematical modelling of juxtacrine patterning

Spatial pattern formation is one of the key issues in developmental biology. Some patterns arising in early development have a very small spatial scale and a natural explanation is that they arise by direct cell—cell signalling in epithelia. This necessitates the use of a spatially discrete model, in contrast to the continuum-based approach of the widely studied Turing and mechanochemical models. In this work, we consider the pattern-forming potential of a model for juxtacrine communication, in which signalling molecules anchored in the cell membrane bind to and activate receptors on the surface of immediately neighbouring cells. The key assumption is that ligand and receptor production are both up-regulated by binding. By linear analysis, we show that conditions for pattern formation are dependent on the feedback functions of the model. We investigate the form of the pattern: specifically, we look at how the range of unstable wavenumbers varies with the parameter regime and find an estimate for the wavenumber associated with the fastest growing mode. A previous juxtacrine model for Delta-Notch signalling studied by Collier et al. (1996, J. Theor. Biol. 183, 429–446) only gives rise to patterning with a length scale of one or two cells, consistent with the fine-grained patterns seen in a number of developmental processes. However, there is evidence of longer range patterns in early development of the fruit fly Drosophila. The analysis we carry out predicts that patterns longer than one or two cell lengths are possible with our positive feedback mechanism, and numerical simulations confirm this. Our work shows that juxtacrine signalling provides a novel and robust mechanism for the generation of spatial patterns.     

Continuum limits of pattern formation in hexagonal-cell monolayers

Intercellular signalling is key in determining cell fate. In closely packed tissues such as epithelia, juxtacrine signalling is thought to be a mechanism for the generation of fine-grained spatial patterns in cell differentiation commonly observed in early development. Theoretical studies of such signalling processes have shown that negative feedback between receptor activation and ligand production is a robust mechanism for fine-grained pattern generation and that cell shape is an important factor in the resulting pattern type. It has previously been assumed that such patterns can be analysed only with discrete models since significant variation occurs over a lengthscale concomitant with an individual cell; however, considering a generic juxtacrine signalling model in square cells, in O’Dea and King (Math Biosci 231(2):172–185 2011), a systematic method for the derivation of a continuum model capturing such phenomena due to variations in a model parameter associated with signalling feedback strength was presented. Here, we extend this work to derive continuum models of the more complex fine-grained patterning in hexagonal cells, constructing individual models for the generation of patterns from the homogeneous state and for the transition between patterning modes. In addition, by considering patterning behaviour under the influence of simultaneous variation of feedback parameters, we construct a more general continuum representation, capturing the emergence of the patterning bifurcation structure. Comparison with the steady-state and dynamic behaviour of the underlying discrete system is made; in particular, we consider pattern-generating travelling waves and the competition between various stable patterning modes, through which we highlight an important deficiency in the ability of continuum representations to accommodate certain dynamics associated with discrete systems.     

EGFR-dependent network interactions that pattern Drosophila eggshell appendages

Similar to other organisms, Drosophila uses its Epidermal Growth Factor Receptor (EGFR) multiple times throughout development. One crucial EGFR-dependent event is patterning of the follicular epithelium during oogenesis. In addition to providing inductive cues necessary for body axes specification, patterning of the follicle cells initiates the formation of two respiratory eggshell appendages. Each appendage is derived from a primordium comprising a patch of cells expressing broad (br) and an adjacent stripe of cells expressing rhomboid (rho). Several mechanisms of eggshell patterning have been proposed in the past, but none of them can explain the highly coordinated expression of br and rho. To address some of the outstanding issues in this system, we synthesized the existing information into a revised mathematical model of follicle cell patterning. Based on the computational model analysis, we propose that dorsal appendage primordia are established by sequential action of feed-forward loops and juxtacrine signals activated by the gradient of EGFR signaling. The model describes pattern formation in a large number of mutants and points to several unanswered questions related to the dynamic interaction of the EGFR and Notch pathways.     

A model of pattern formation based on signaling pathways

A model of pattern formation in the early embryo is presented. It is motivated by the necessary interaction of kinases and phosphatases, and in particular by the family of Wnt kinase and RPTP phosphatase signaling pathways. It is seen as complementary to the more short-range patterning of the Delta/Notch (or juxtacrine) signaling pathway.     

Oscillations and patterns in spatially discrete models for developmental intercellular signalling

We extend previous models for nearest neighbour ligand-receptor binding to include both lateral induction and inhibition of ligand and receptor production, and different geometries (strings of cells and hexagonal arrays, in addition to square arrays). We demonstrate the possibility of lateral inhibition giving patterns with a characteristic length scale of many cell diameters, when receptor production is included. In contrast, lateral induction combined with inhibition of receptor synthesis cannot give rise to a patterning instability under any circumstances. Interesting new dynamics include the analytical prediction and consequent numerical observation of spatiotemporal oscillations, this depends crucially on the production terms and on the relationship between the decay rates of ligand and free receptor. Our approach allows for a detailed comparison with the model for Delta-Notch interactions of Collier et al. [4], and we find that a formal reduction may be made only when the ligand receptor binding kinetics are very slow. Without such very slow receptor kinetics, spatial pattern formation via lateral inhibition in hexagonal cellular arrays requires significant activation of receptor production, a feature that is not apparent from previous analyses.Send offprint requests to:Markus R. Owen     

A model of primitive streak initiation in the chick embryo

Initiation of the primitive streak in avian embryos provides a well-studied example of a pattern-forming event that displays a striking capacity for regulation. The mechanisms underlying the regulative properties are, however, poorly understood and are not easily accounted for by traditional models of pattern formation, such as reaction-diffusion models. In this paper, we propose a new activator-inhibitor model for streak initiation. We show that the model is consistent with experimental observations, both in its pattern-forming properties and in its ability to form these patterns on the correct time-scales for biologically realistic parameter values. A key component of the model is a travelling wave of inhibition. We present a mathematical analysis of the speed of such waves in both diffusive and juxtacrine relay systems. We use the streak initiation model to make testable predictions. By varying parameters of the model, two very different types of patterning can be obtained, suggesting that our model may be applicable to other processes in addition to streak initiation.     

Lateral induction by juxtacrine signaling is a new mechanism for pattern formation

Many signaling molecules in epithelia are now known to function in a membrane-bound form, binding to receptors on immediately neighbouring cells. This "juxtacrine" mode of communication has been well studied in the case of lateral inhibition, where ligand binding at the cell surface downregulates ligand and receptor expression, and is known to generate spatial patterns with a wavelength of exactly two cells. However, recent evidence shows that a number of juxtacrine signals can lead to the opposite phenomenon of lateral induction. Here, we use mathematical modeling to show that such positive feedback, in combination with juxtacrine communication, provides a novel mechanism for the generation of spatial patterns, with wavelengths that vary with parameters and can be many cell lengths.     

Type specification and spatial pattern formation of floral organs: A dynamic development model

A mathematical model simulating spatial pattern formation (positioning) of floral organs is proposed. Computer experiment with this model demonstrated the following sequence of spatial pattern formation in a typical cruciferous flower: medial sepals, carpels, lateral sepals, long stamens, petals, and short stamens. The positioning was acropetal for the perianth organs and basipetal for the stamens and carpels. Organ type specification and positioning proceed non-simultaneously in different floral parts and organ type specification goes ahead of organ spatial pattern formation. Computer simulation of flower development in several mutants demonstrated that the AG and AP2 genes determine both organ type specification and formation of the zones for future organ development. The function of the AG gene is to determine the basipetal patterning zones for the development of the reproductive organs, while the AP2 gene maintains proliferative activity of the meristem establishing the acropetal patterning zone for the development of the perianth organs.     

Sdf1a patterns zebrafish melanophores and links the somite and melanophore pattern defects in choker mutants

Pigment pattern formation in zebrafish presents a tractable model system for studying the morphogenesis of neural crest derivatives. Embryos mutant for choker manifest a unique pigment pattern phenotype that combines a loss of lateral stripe melanophores with an ectopic melanophore ;collar' at the head-trunk border. We find that defects in neural crest migration are largely restricted to the lateral migration pathway, affecting both xanthophores (lost) and melanophores (gained) in choker mutants. Double mutant and timelapse analyses demonstrate that these defects are likely to be driven independently, the collar being formed by invasion of melanophores from the dorsal and ventral stripes. Using tissue transplantation, we show that melanophore patterning depends upon the underlying somitic cells, the myotomal derivatives of which--both slow--and fast-twitch muscle fibres--are themselves significantly disorganised in the region of the ectopic collar. In addition, we uncover an aberrant pattern of expression of the gene encoding the chemokine Sdf1a in choker mutant homozygotes that correlates with each aspect of the melanophore pattern defect. Using morpholino knock-down and ectopic expression experiments, we provide evidence to suggest that Sdf1a drives melanophore invasion in the choker mutant collar and normally plays an essential role in patterning the lateral stripe. We thus identify Sdf1 as a key molecule in pigment pattern formation, adding to the growing inventory of its roles in embryonic development.     

Interactions between the Foot and the Head Patterning Systems in Hydra vulgaris

The Cnidarian, hydra, is an appealing model system for studying the basic processes underlying pattern formation. Classical studies have elucidated much basic information regarding the role of development gradients, and theoretical models have been quite successful at describing experimental results. However, most experiments and computer simulations have dealt with isolated patterning events such as the dynamics of head regeneration. More global events such as interactions among the head, bud, and foot patterning systems have not been extensively addressed. The characterization of monoclonal antibodies with position-specific labeling patterns and the recent cloning and characterization of genes expressed in position-specific manners now provide the tools for investigating global interactions between patterning systems. In particular, changes in the axial positional value gradient may be monitored in response to experimental perturbation. Rather than studying isolated patterning events, this approach allows us to study patterning over the entire animal. The studies reported here focus on interactions between the foot and the head patterning systems in Hydra vulgaris following induction of a foot in close proximity to a head, axial grafting of a foot closer to the head, or doubling the amount of basal tissue by lateral grafting of an additional peduncle-foot onto host animals. Resulting positional value changes as monitored by antigen (TS19) and gene (ks1 and CnNK-2) expression were assessed in the foot, head, and intervening tissue. The results of the experiments indicate that positional values changed rapidly, in a matter of hours, and that there were reciprocal interactions between the foot and the head patterning systems. Theoretical interpretations of the results in the form of computer simulations based on the reaction-diffusion model are presented and predict many, but not all, of the experimental observations. Since the lateral grafting experiment cannot, at present, be simulated, it is discussed in light of what has been learned from the axial grafting experiments and their simulations.     

A two-step patterning process increases the robustness of periodic patterning in the fly eye

Complex periodic patterns can self-organize through dynamic interactions between diffusible activators and inhibitors. In the biological context, self-organized patterning is challenged by spatial heterogeneities (‘noise’) inherent to biological systems. How spatial variability impacts the periodic patterning mechanism and how it can be buffered to ensure precise patterning is not well understood. We examine the effect of spatial heterogeneity on the periodic patterning of the fruit fly eye, an organ composed of ～800 miniature eye units (ommatidia) whose periodic arrangement along a hexagonal lattice self-organizes during early stages of fly development. The patterning follows a two-step process, with an initial formation of evenly spaced clusters of ～10 cells followed by a subsequent refinement of each cluster into a single selected cell. Using a probabilistic approach, we calculate the rate of patterning errors resulting from spatial heterogeneities in cell size, position and biosynthetic capacity. Notably, error rates were largely independent of the desired cluster size but followed the distributions of signaling speeds. Pre-formation of large clusters therefore greatly increases the reproducibility of the overall periodic arrangement, suggesting that the two-stage patterning process functions to guard the pattern against errors caused by spatial heterogeneities. Our results emphasize the constraints imposed on self-organized patterning mechanisms by the need to buffer stochastic effects. Author summary Complex periodic patterns are common in nature and are observed in physical, chemical and biological systems. Understanding how these patterns are generated in a precise manner is a key challenge. Biological patterns are especially intriguing, as they are generated in a noisy environment; cell position and cell size, for example, are subject to stochastic variations, as are the strengths of the chemical signals mediating cell-to-cell communication. The need to generate a precise and robust pattern in this ‘noisy’ environment restricts the space of patterning mechanisms that can function in the biological setting. Mathematical modeling is useful in comparing the sensitivity of different mechanisms to such variations, thereby highlighting key aspects of their design.We use mathematical modeling to study the periodic patterning of the fruit fly eye. In this system, a highly ordered lattice of differentiated cells is generated in a two-dimensional cell epithelium. The pattern is first observed by the appearance of evenly spaced clusters of ～10 cells that express specific genes. Each cluster is subsequently refined into a single cell, which initiates the formation and differentiation of a miniature eye unit, the ommatidium. We formulate a mathematical model based on the known molecular properties of the patterning mechanism, and use a probabilistic approach to calculate the errors in cluster formation and refinement resulting from stochastic cell-to-cell variations (‘noise’) in different quantitative parameters. This enables us to define the parameters most influencing noise sensitivity. Notably, we find that this error is roughly independent of the desired cluster size, suggesting that large clusters are beneficial for ensuring the overall reproducibility of the periodic cluster arrangement. For the stage of cluster refinement, we find that rapid communication between cells is critical for reducing error. Our work provides new insights into the constraints imposed on mechanisms generating periodic patterning in a realistic, noisy environment, and in particular, discusses the different considerations in achieving optimal design of the patterning network.     

Radial patterning of Arabidopsis shoots by class III HD-ZIP and KANADI genes

BACKGROUND: Shoots of all land plants have a radial pattern that can be considered to have an adaxial (central)-abaxial (peripheral) polarity. In Arabidopsis, gain-of-function alleles of PHAVOLUTA and PHABULOSA, members of the class III HD-ZIP gene family, result in adaxialization of lateral organs. Conversely, loss-of-function alleles of the KANADI genes cause an adaxialization of lateral organs. Thus, the class III HD-ZIP and KANADI genes comprise a genetic system that patterns abaxial-adaxial polarity in lateral organs produced from the apical meristem. RESULTS: We show that gain-of-function alleles of REVOLUTA, another member of the class III HD-ZIP gene family, are characterized by adaxialized lateral organs and alterations in the radial patterning of vascular bundles in the stem. The gain-of-function phenotype can be obtained by changing only the REVOLUTA mRNA sequence and without changing the protein sequence; this finding indicates that this phenotype is likely mediated through an interference with microRNA binding. Loss of KANADI activity results in similar alterations in vascular patterning as compared to REVOLUTA gain-of-function alleles. Simultaneous loss-of-function of PHABULOSA, PHAVOLUTA, and REVOLUTA abaxializes cotyledons, abolishes the formation of the primary apical meristem, and in severe cases, eliminates bilateral symmetry; these phenotypes implicate these three genes in radial patterning of both embryonic and postembryonic growth. CONCLUSIONS: Based on complementary vascular and leaf phenotypes of class III HD-ZIP and KANADI mutants, we propose that a common genetic program dependent upon miRNAs governs adaxial-abaxial patterning of leaves and radial patterning of stems in the angiosperm shoot. This finding implies that a common patterning mechanism is shared between apical and vascular meristems.     

Induction and axial patterning of the neural plate: Planar and vertical signals

In this review I summarize recent findings on the contributions of different cell groups to the formation of the basic plan of the nervous system of vertebrate embryos. Midline cells of the mesoderm—the organizer, notochord, and prechordal plate—and midline cells of the neural ectoderm—the notoplate and floor plate—appear to have a fundamental role in the induction and patterning of the neural plate. Vertical signals acting across tissue layers and planar signals acting through the neural epithelium have distinct roles and cooperate in induction and pattern formation. Whereas the prechordal plate and notochord have distinct vertical signaling properties, the initial anteroposterior (A-P) pattern of the neural plate may be induced by planar signals originating from the organizer region. Planar signals from the notoplate may also contribute to the mediolateral (M-L) patterning of the neural plate. These and other findings suggest a general view of neural induction and axial patterning. © 1993 John Wiley & Sons, Inc.     

Patterning of the avian intermediate mesoderm by lateral plate and axial tissues

Amniote kidney tissue is derived from the intermediate mesoderm (IM), a strip of mesoderm that lies between the somites and the lateral plate. While much has been learned concerning the later events which regulate the differentiation of IM into tubules and other types of kidney tissue, much less is known concerning the earlier events which regulate formation of the IM itself. In the current study, the chick pronephros was used as a model system to identify tissues that play a role in patterning the IM and the critical time periods during which such patterning events take place. Explant studies revealed that the prospective pronephric IM is already specified to express kidney genes by stage 6, shortly after its gastrulation through the primitive streak, and earlier than previously reported. Transplant and explant experiments revealed that the lateral plate contains an activity that can repress IM formation in tissues that are already specified to express IM genes. In contrast, Hensen's node can promote formation of IM in the lateral plate. Paraxial tissues (presomitic mesoderm plus neural plate and notochord) were found to influence the morphogenesis of the nephric duct, but did not induce IM tissue to an appreciable extent. Combining lateral plate and paraxial tissue in vivo or in vitro led to induction of IM genes in the paraxial mesoderm but not in the lateral plate mesoderm. Based on these results and those of others, we propose a two-step model for the patterning of the IM. While tissue is still in the primitive streak, the prospective IM is relatively uncommitted. By stage 6, shortly after cells leave the primitive streak, a field of cells is generate which is specified to give rise to IM (Step 1). Subsequently, competing signals from the lateral plate and axial tissues modulate the number of cells that commit to an IM fate (Step 2).     

Agent-based simulation of notch-mediated tip cell selection in angiogenic sprout initialisation

Angiogenic sprouting requires functional specialisation of endothelial cells into leading tip cells and following stalk cells. Experimental data illustrate that induction of the tip cell phenotype is dependent on the protein VEGF-A; however, the process of tip cell selection is not fully understood. Here we introduce a hierarchical agent-based model simulating a suggested feedback loop that links VEGF-A tip cell induction with delta-like 4 (Dll4)/notch-mediated lateral inhibition. The model identifies VEGF-A concentration, VEGF-A gradients and filopodia extension as critical parameters in determining the robustness of tip/stalk patterning.The behaviour of the model provides new mechanistic insights into the vascular patterning defects observed in pathologically high VEGF-A, such as diabetic retinopathy and tumour angiogenesis. We investigate the role of cell morphology in tip/stalk patterning, highlighting filopodia as lateral inhibition amplifiers. The model has been used to make a number of predictions, which are now being tested experimentally, including: (1) levels of Dll4/VEGFR-2, or related downstream proteins, oscillate in synchrony along a vessel in high VEGF environments; (2) a VEGF gradient increases tip cell selection rate.     

High‐resolution computational imaging of leaf hair patterning using polarized light microscopy

The leaf hairs (trichomes) on the aerial surface of many plant species play important roles in phytochemical production and herbivore protection, and have significant applications in the chemical and agricultural industries. Trichome formation in the model plant Arabidopsis thaliana also presents a tractable experimental system to study cell differentiation and pattern formation in plants and animals. Studies of this developmental process suggest that trichome positioning may be the result of a self‐forming pattern, emerging from a lateral inhibition mechanism determined by a network of regulatory factors. Critical to the continued success of these studies is the ability to quantitatively characterize trichome pattern phenotypes in response to mutations in the genes that regulate this process. Advanced protocols for the observation of changes in trichome patterns can be expensive and/or time consuming, and lack user‐friendly analysis tools. In order to address some of these challenges, we describe here a strategy based on polarized light microscopy for the quick and accurate measurement of trichome positions, and provide an online tool designed for the quantitative analyses of trichome number, density and patterning.