Mechanochemical Symmetry Breaking in Hydra Aggregates

Tissue morphogenesis comprises the self-organized creation of various patterns and shapes. Although detailed underlying mechanisms are still elusive in many cases, an increasing amount of experimental data suggests that chemical morphogen and mechanical processes are strongly coupled. Here, we develop and test a minimal model of the axis-defining step (i.e., symmetry breaking) in aggregates of the Hydra polyp. Based on previous findings, we combine osmotically driven shape oscillations with tissue mechanics and morphogen dynamics. We show that the model incorporating a simple feedback loop between morphogen patterning and tissue stretch reproduces a wide range of experimental data. Finally, we compare different hypothetical morphogen patterning mechanisms (Turing, tissue-curvature, and self-organized criticality). Our results suggest the experimental investigation of bigger (i.e., multiple head) aggregates as a key step for a deeper understanding of mechanochemical symmetry breaking in Hydra.     

Activator-inhibitor dynamics of vertebrate limb pattern formation

The development of the vertebrate limb depends on an interplay of cellular differentiation, pattern formation, and tissue morphogenesis on multiple spatial and temporal scales. While numerous gene products have been described that participate in, and influence, the generation of the limb skeletal pattern, an understanding of the most salient feature of the developing limb--its quasiperiodic arrangement of bones, requires additional organizational principles. We review several such principles, drawing on concepts of physics and chemical dynamics along with molecular genetics and cell biology. First, a "core mechanism" for precartilage mesenchymal condensation is described, based on positive autoregulation of the morphogen transforming growth factor (TGF)-beta, induction of the extracellular matrix (ECM) protein fibronectin, and focal accumulation of cells via haptotaxis. This core mechanism is shown to be part of a local autoactivation-lateral inhibition (LALI) system that ensures that the condensations will be regularly spaced. Next, a "bare-bones" model for limb development is described in which the LALI-core mechanism is placed in a growing geometric framework with predifferentiated "apical," differentiating "active," and irreversibly differentiated "frozen" zones defined by distance from an apical source of a fibroblast growth factor (FGF)-type morphogen. This model is shown to account for classic features of the developing limb, including the proximodistal (PD) emergence over time of increasing numbers of bones. We review earlier and recent work suggesting that the inhibitory component of the LALI system for condensation may not be a diffusible morphogen, and propose an alternative mechanism for lateral inhibition, based on synchronization of oscillations of a Hes mediator of the Notch signaling pathway. Finally, we discuss how viewing development as an interplay between molecular-genetic and dynamic physical processes can provide new insight into the origin of congenital anomalies.     

A model of gradient interpretation based on morphogen binding

We propose a model in which pattern formation is controlled by several concentration gradients of “morphogens” and by allosteric proteins which bind them. In this model, each protein can bind up to two molecules of each morphogen and has an “active state” when one molecule of each morphogen is bound. The concentration of the active state of such a “morphogen binding protein” varies with position in a way that depends on the values given the binding constants. In a contour map of the active state concentration, the contours can have a variety of simple shapes.Simply-shaped regions of cell differentiation can be defined directly by concentration contours of a morphogen binding protein using a threshold-sensing mechanism. More complex shapes may be generated using several proteins and a “winner-take-all” rule according to which each protein specifies some particular sort of cell differentiation and the differentiation of cells in any position is governed by the protein with the highest active state concentration.We present an application of our model to the vertebrate limb skeleton; we use the “winner-take-all” mechanism and thirteen morphogen binding proteins, eleven of which specify cartilage formation. In this model we use one morphogen binding protein to specify the shaft of a typical long bone and one for each epiphysis. Our model is reasonably successful in imitating the in vivo positions and orientations of developing bones and in generating simple, plausible-looking articular surfaces.In addition to the morphogen-binding model we propose a mechanism which could transform morphogen-binding patterns into high-amplitude patterns capable of controlling the activity of structural genes. This “amplifying mechanism” can account for two previously unexplained features of limb skeletal development: the early formation of the diffusely-bounded “scleroblastema” in the limb bud and the center-to-edge gradations in cartilage formation rate which are later seen within individual chondrification foci.A simple modification of the morphogen-binding model provides an explanation for the general anatomical phenomenon of metamerism: The model can account for the formation of inexactly repeating patterns (such as the pattern of the vertebral column) and suggests a mechanism by which such patterns could (1) evolve from exactly repeating patterns, and (2) acquire, in further evolution, a high degree of specialization of the individual repeating units.The most promising approach for testing the morphogen-binding model would appear to involve experiments in which cytoplasm is transferred between cells at various stages of pattern development. Support for the model could also come from the discovery of certain kinds of hereditary limb defects.     

Improving the understanding of cytoneme-mediated morphogen gradients by in silico modeling

Morphogen gradients are crucial for the development of organisms. The biochemical properties of many morphogens prevent their extracellular free diffusion, indicating the need of an active mechanism for transport. The involvement of filopodial structures (cytonemes) has been proposed for morphogen signaling. Here, we describe an in silico model based on the main general features of cytoneme-meditated gradient formation and its implementation into Cytomorph, an open software tool. We have tested the spatial and temporal adaptability of our model quantifying Hedgehog (Hh) gradient formation in two Drosophila tissues. Cytomorph is able to reproduce the gradient and explain the different scaling between the two epithelia. After experimental validation, we studied the predicted impact of a range of features such as length, size, density, dynamics and contact behavior of cytonemes on Hh morphogen distribution. Our results illustrate Cytomorph as an adaptive tool to test different morphogen gradients and to generate hypotheses that are difficult to study experimentally.     

Implications of diffusion and time-varying morphogen gradients for the dynamic positioning and precision of bistable gene expression boundaries

The earliest models for how morphogen gradients guide embryonic patterning failed to account for experimental observations of temporal refinement in gene expression domains. Following theoretical and experimental work in this area, dynamic positional information has emerged as a conceptual framework to discuss how cells process spatiotemporal inputs into downstream patterns. Here, we show that diffusion determines the mathematical means by which bistable gene expression boundaries shift over time, and therefore how cells interpret positional information conferred from morphogen concentration. First, we introduce a metric for assessing reproducibility in boundary placement or precision in systems where gene products do not diffuse, but where morphogen concentrations are permitted to change in time. We show that the dynamics of the gradient affect the sensitivity of the final pattern to variation in initial conditions, with slower gradients reducing the sensitivity. Second, we allow gene products to diffuse and consider gene expression boundaries as propagating wavefronts with velocity modulated by local morphogen concentration. We harness this perspective to approximate a PDE model as an ODE that captures the position of the boundary in time, and demonstrate the approach with a preexisting model for Hunchback patterning in fruit fly embryos. We then propose a design that employs antiparallel morphogen gradients to achieve accurate boundary placement that is robust to scaling. Throughout our work we draw attention to tradeoffs among initial conditions, boundary positioning, and the relative timescales of network and gradient evolution. We conclude by suggesting that mathematical theory should serve to clarify not just our quantitative, but also our intuitive understanding of patterning processes.     

Coding design of positional information for robust morphogenesis

Robust positioning of cells in a tissue against unavoidable noises is important for achieving normal and reproducible morphogenesis. The position in a tissue is represented by morphogen concentrations, and cells read them to recognize their spatial coordinates. From the engineering viewpoint, these positioning processes can be regarded as an information coding. Organisms are conjectured to adopt good coding designs with high reliability for a given number of available morphogen species and their chemical properties. To answer, quantitatively, the questions of how good coding is adopted, and subsequently when, where, and to what extent each morphogen contributes to positioning, we need a way to evaluate the goodness of coding. In this article, by introducing basic concepts of computer science, we mathematically formulate coding processes in morphogen-dependent positioning, and define some key concepts such as encoding, decoding, and positional information and its precision. We demonstrate the best designs for pairs of encoding and decoding rules, and show how those designs can be biologically implemented by using some examples. We also propose a possible procedure of data analysis to validate the coding optimality formulated here.     

Mathematical Model of the Formation of Morphogen Gradients Through Membrane-Associated Non-receptors

The importance of morphogens is a central concept in developmental biology. Multiple-fate patterning and the robustness of the morphogen gradient are essential for embryo development. The ways by which morphogens diffuse from a local source to form long distance gradients can differ from one morphogen to the other, and for the same morphogen in different organs. This paper will study the mechanism by which morphogens diffuse through the aid of membrane-associated non-receptors and will investigate how the membrane-associated non-receptors help the morphogen to form long distance gradients and to achieve good robustness. Such a mechanism has been reported for some morphogens that are rapidly turned over. We will establish a set of reaction-diffusion equations to model the dynamical process of morphogen gradient formation. Under the assumption of rapid morphogen degradation, we discuss the existence, uniqueness, local stability, approximation solution, and the robustness of the steady-state gradient. The results in this paper show that when the morphogen is rapidly turned over, diffusion of the morphogen through membrane-associated non-receptors is a possible strategy to form a long distance multiple-fate gradient that is locally stable and is robust against the changes in morphogen synthesis rate.     

Insights into the Mechanisms of Membrane Curvature and Vesicle Scission by the Small GTPase Sar1 in the Early Secretory Pathway

The small GTPase protein Sar1 is known to be involved in both the initiation of COPII-coated vesicle formation and scission of the nascent vesicle from the endoplasmic reticulum. The molecular details for the mechanism of membrane remodeling by Sar1 remain unresolved. Here, we show that Sar1 transforms synthetic liposomes into structures of different morphologies including tubules and detached vesicles. We demonstrate that Sar1 alone is competent for vesicle scission in a manner that depends on the concentration of Sar1 molecules occupying the membrane. Sar1 molecules align on low-curvature membranes to form an extended lattice. The continuity of this lattice breaks down as the curvature locally increases. The smallest repeating unit constituting the ordered lattice is a Sar1 dimer. The three-dimensional structure of the Sar1 lattice was reconstructed by substituting spherical liposomes with galactoceramide lipid tubules of homogeneous diameter. These data suggest that Sar1 dimerization is responsible for the formation of constrictive membrane curvature. We propose a model whereby Sar1 dimers assemble into ordered arrays to promote membrane constriction and COPII-directed vesicle scission.     

Drosophila glypicans Dally and Dally-like shape the extracellular Wingless morphogen gradient in the wing disc

Drosophila Wingless (Wg) is the founding member of the Wnt family of secreted proteins. During the wing development, Wg acts as a morphogen whose concentration gradient provides positional cues for wing patterning. The molecular mechanism(s) of Wg gradient formation is not fully understood. Here, we systematically analyzed the roles of glypicans Dally and Dally-like protein (Dlp), the Wg receptors Frizzled (Fz) and Fz2, and the Wg co-receptor Arrow (Arr) in Wg gradient formation in the wing disc. We demonstrate that both Dally and Dlp are essential and have different roles in Wg gradient formation. The specificities of Dally and Dlp in Wg gradient formation are at least partially achieved by their distinct expression patterns. To our surprise, although Fz2 was suggested to play an essential role in Wg gradient formation by ectopic expression studies, removal of Fz2 activity does not alter the extracellular Wg gradient. Interestingly, removal of both Fz and Fz2, or Arr causes enhanced extracellular Wg levels, which is mainly resulted from upregulated Dlp levels. We further show that Notum, a negative regulator of Wg signaling, downregulates Wg signaling mainly by modifying Dally. Last, we demonstrate that Wg movement is impeded by cells mutant for both dally and dlp. Together, these new findings suggest that the Wg morphogen gradient in the wing disc is mainly controlled by combined actions of Dally and Dlp. We propose that Wg establishes its concentration gradient by a restricted diffusion mechanism involving Dally and Dlp in the wing disc.     

Formation of a functional morphogen gradient by a passive process in tissue from the early Xenopus embryo

In early development much of the cellular diversity and pattern formation of the embryo is believed to be set up by morphogens. However, for many morphogens, including members of the TGF-beta superfamily, the mechanism(s) by which they reach distant cells is unknown. We have used immunofluorescence to detect, at single cell resolution, a morphogen gradient formed across vertebrate tissue. The TGF-beta ligand is distributed in a gradient visible up to 7 cell diameters (about 150-200 microm) from its source, and is detectable only in the extracellular space. This morphogen gradient is functional, since we demonstrate activation of a high response gene (Xeomes) and a low-response gene (Xbra) at different distances from the TGF-beta source. Expression of the high affinity type II TGF-beta receptor is necessary for detection of the gradient, but the shape of the gradient formed only depends in part on the spatial variation in the amount of receptor. Finally, we demonstrate that the molecular processes that participate in forming this functional morphogen gradient are temperature independent, since the gradient forms to a similar extent whether the cells are maintained at 4 degrees C or 23 degrees C. In contrast, TGF-beta1 internalisation by cells of the Xenopus embryo is a temperature-dependent process. Our results thus suggest that neither vesicular transcytosis nor other active processes contribute to a significant extent to the formation of the morphogen gradient we observe. We conclude that, in the model system used here, a functional morphogen gradient can be formed within a few hours by a mechanism of passive diffusion.     

Mechanogenetic Coupling of Hydra Symmetry Breaking and Driven Turing Instability Model

The freshwater polyp Hydra can regenerate from tissue fragments or random cell aggregates. We show that the axis-defining step (“symmetry breaking”) of regeneration requires mechanical inflation-collapse oscillations of the initial cell ball. We present experimental evidence that axis definition is retarded if these oscillations are slowed down mechanically. When biochemical signaling related to axis formation is perturbed, the oscillation phase is extended and axis formation is retarded as well. We suggest that mechanical oscillations play a triggering role in axis definition. We extend earlier reaction-diffusion models for Hydra regrowth by coupling morphogen transport to mechanical stress caused by the oscillations. The modified reaction-diffusion model reproduces well two important experimental observations: 1), the existence of an optimum size for regeneration, and 2), the dependence of the symmetry breaking time on the properties of the mechanical oscillations.     

The Role of Endocytosis during Morphogenetic Signaling

Morphogens are signaling molecules that are secreted by a localized source and spread in a target tissue where they are involved in the regulation of growth and patterning. Both the activity of morphogenetic signaling and the kinetics of ligand spreading in a tissue depend on endocytosis and intracellular trafficking. Here, we review quantitative approaches to study how large-scale morphogen profiles and signals emerge in a tissue from cellular trafficking processes and endocytic pathways. Starting from the kinetics of endosomal networks, we discuss the role of cellular trafficking and receptor dynamics in the formation of morphogen gradients. These morphogen gradients scale during growth, which implies that overall tissue size influences cellular trafficking kinetics. Finally, we discuss how such morphogen profiles can be used to control tissue growth. We emphasize the role of theory in efforts to bridge between scales.A fundamental challenge in biology is to understand how morphologies and complex patterns form in multicellular systems by the collective organization of many cells. Cells divide and undergo apoptosis, and they communicate via signaling pathways that use molecules as information carriers. In tissues, large-scale patterns of gene expression emerge from the coordinated signaling activity and response of many cells. The establishment of such patterns is often guided by long-range concentration profiles of morphogens. Cell divisions and cell rearrangements must be coordinated over large distances to achieve specific tissue sizes and shapes. To unravel how molecular processes and interactions can eventually be responsible for the formation of structures and patterns in tissues during development, it is important to study processes at different scales and understand how different levels of organization are connected. Such an approach becomes strongest if it involves a combination of quantitative experimental studies with theory.In the present article, we discuss several such approaches on different scales with a particular emphasis on theory. Starting from the kinetic and dynamic properties of endosomal networks inside a cell, we discuss transport processes in a tissue that can be related to kinetic trafficking parameters. Such transport processes are then responsible for the formation of graded morphogen concentration profiles. To permit scalable patterns in tissues of different sizes, it has been suggested that morphogen gradients scale during growth. This can be achieved on the tissue level by feedback systems that are sensitive to tissue size and regulate, for example, morphogen degradation. Finally, morphogen gradients that scale with tissue size can provide a system to robustly organize cell division in a large tissue and generate homogeneous growth. Theory can play an important role to bridge scales and understand how molecular and cellular processes can control pattern formation and tissue growth on larger scales.Morphogens are signaling molecules that are secreted in specific regions of developing tissues and can induce signaling activity far from their source. They typically form graded concentration profiles and therefore endow cells with positional information (cells can obtain information about their position in a tissue). Thus, they can guide cells to differentiate into complex morphological patterns. Morphogens also control cell growth and cell division. Because they control both patterning and growth, they may play a key role to coordinate these two processes. Such coordination is important because the size of morphological patterns must adjust during growth, whereas growth influences such patterns. A well-studied morphogen is Decapentaplegic (Dpp), which controls morphogenesis in the imaginal wing disc of developing Drosophila. Consequently, mutations in Dpp or defects in the trafficking pathways that control its graded concentration profiles and signaling affect the formation and structure of the adult wing.The study of morphogens was traditionally approached from a genetic perspective: Which gene products behave like morphogens? Which mutants affect patterning and growth? The realization that morphogens typically operate by a gradient of concentration raised the question of how morphogen gradients are generated. It became clear that the cellular trafficking of morphogens is a key issue for the generation of morphogen profiles. Morphogens are secreted ligands that bind receptors in the plasma membrane. The secretion of the ligands and the concentrations of receptor, ligand, and receptor/ligand complex at the plasma membrane are governed by their trafficking in the cell by vesicular transport. In particular, it was shown that trafficking through the endocytic pathway has an important impact on the formation of morphogen gradients (reviewed in Gonzalez-Gaitan 2003; see Bökel and Brand 2014). This is, to a large extent, how the cells respond to morphogens and contribute to set their local concentrations. To understand functions of morphogens in a tissue, we need to study how the gradient is formed. This, in turn, requires insights into morphogen trafficking through the endocytic pathway. The problem of morphogen behavior, therefore, becomes a problem spanning several levels of complexity: the organ level, the tissue level, the cell level, the organelle level, and the molecular level. Theoretical approaches motivated by physics combined with quantitative experimental approaches provide an ideal framework to understand how these different levels of complexity are intertwined.Two recent discoveries highlighted such integration. (1) The observation that profiles of the morphogen Dpp scale during growth, which implies that the rate of Dpp degradation mediated by the endocytic pathway of each of the cells in the tissue depends on the size of the overall tissue. This suggests that two levels of complexity are linked because cellular trafficking receives cues about the global tissue size. (2) As a result of the changes of the degradation rate that leads to gradient scaling, cells receive an increasing level of signaling. This, in turn, can be used by the cells to decide when to divide. This regulation again involves two levels of complexity because regulation at the endocytic pathway determines the growth properties of the tissue and, ultimately, its final size.In the following, we discuss quantitative approaches to study cellular signaling processes on different scales. Here, the aim is to understand how patterns on large scales can emerge during development from molecular processes and signaling pathways that involve endocytosis and cellular trafficking. We begin by describing trafficking of ligands in the endocytic pathway. We then consider the situation of a morphogen ligand and its impact in gradient formation. Subsequently, we discuss how gradient scaling might be realized. Finally, we discuss how such scaling processes play an important role in the regulation of morphogenetic growth.     

On the role of glypicans in the process of morphogen gradient formation

Glypicans are cell surface molecules that influence signaling and gradient formation of secreted morphogens and growth factors. Several distinct functions have been ascribed to glypicans including acting as co-receptors for signaling proteins. Recent data show that glypicans are also necessary for morphogen propagation in the tissue. In the present study, a model describing the interaction of a morphogen with glypicans is formulated, analyzed and compared with measurements of the effect of glypican Dally-like (Dlp) overexpression on Wingless (Wg) morphogen signaling in Drosophila melanogaster wing imaginal discs. The model explains the opposing effect that Dlp overexpression has on Wg signaling in the distal and proximal regions of the disc and makes a number of quantitative predictions for further experiments. In particular, our model suggests that Dlp acts by allowing Wg to diffuse on cell surface while protecting it from loss and degradation, and that Dlp rather than acting as Wg co-receptor competes with receptors for morphogen binding.     

Dpp gradient formation by dynamin-dependent endocytosis: receptor trafficking and the diffusion model

Developing cells acquire positional information by reading the graded distribution of morphogens. In Drosophila, the Dpp morphogen forms a long-range concentration gradient by spreading from a restricted source in the developing wing. It has been assumed that Dpp spreads by extracellular diffusion. Under this assumption, the main role of endocytosis in gradient formation is to downregulate receptors at the cell surface. These surface receptors bind to the ligand and thereby interfere with its long-range movement. Recent experiments indicate that Dpp spreading is mediated by Dynamin-dependent endocytosis in the target tissue, suggesting that extracellular diffusion alone cannot account for Dpp dispersal. Here, we perform a theoretical study of a model for morphogen spreading based on extracellular diffusion, which takes into account receptor binding and trafficking. We compare profiles of ligand and surface receptors obtained in this model with experimental data. To this end, we monitored directly the pool of surface receptors and extracellular Dpp with specific antibodies. We conclude that current models considering pure extracellular diffusion cannot explain the observed role of endocytosis during Dpp long-range movement.     

Reaction–Diffusion Systems and External Morphogen Gradients: The Two-Dimensional Case,with an Application to Skeletal Pattern Formation

We investigate a reaction–diffusion system consisting of an activator and an inhibitor in a two-dimensional domain. There is a morphogen gradient in the domain. The production of the activator depends on the concentration of the morphogen. Mathematically, this leads to reaction–diffusion equations with explicitly space-dependent terms. It is well known that in the absence of an external morphogen, the system can produce either spots or stripes via the Turing bifurcation. We derive first-order expansions for the possible patterns in the presence of an external morphogen and show how both stripes and spots are affected. This work generalizes previous one-dimensional results to two dimensions. Specifically, we consider the quasi-one-dimensional case of a thin rectangular domain and the case of a square domain. We apply the results to a model of skeletal pattern formation in vertebrate limbs. In the framework of reaction–diffusion models, our results suggest a simple explanation for some recent experimental findings in the mouse limb which are much harder to explain in positional-information-type models.