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.     

Bicoid gradient formation mechanism and dynamics revealed by protein lifetime analysis

Embryogenesis relies on instructions provided by spatially organized signaling molecules known as morphogens. Understanding the principles behind morphogen distribution and how cells interpret locally this information remains a major challenge in developmental biology. Here, we introduce morphogen‐age measurements as a novel approach to test models of morphogen gradient formation. Using a tandem fluorescent timer as a protein age sensor, we find a gradient of increasing age of Bicoid along the anterior–posterior axis in the early Drosophila embryo. Quantitative analysis of the protein age distribution across the embryo reveals that the synthesis–diffusion–degradation model is the most likely model underlying Bicoid gradient formation, and rules out other hypotheses for gradient formation. Moreover, we show that the timer can detect transitions in the dynamics associated with syncytial cellularization. Our results provide new insight into Bicoid gradient formation and demonstrate how morphogen‐age information can complement knowledge about movement, abundance, and distribution, which should be widely applicable to other systems.     

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.     

The Decapentaplegic morphogen gradient: a precise definition

Two key processes are in the basis of morphogenesis: the spatial allocation of cell types in fields of na?ve cells and the regulation of growth. Both are controlled by morphogens, which activate target genes in the growing tissue in a concentration-dependent manner. Thus the morphogen model is an intrinsically quantitative concept. However, quantitative studies were performed only in recent years on two morphogens: Bicoid and Decapentaplegic. This review covers quantitative aspects of the formation and precision of the Decapentaplegic morphogen gradient. The morphogen gradient concept is transitioning from a soft definition to a precise idea of what the gradient could really do.     

Formation and maintenance of morphogen gradients: an essential role for the endomembrane system in Drosophila melanogaster wing development

As early as 1964 it was suggested that simple diffusion of morphogens away from their secretion source did not provide an adequate explanation for the formation and maintenance of morphogen gradients. Involvement of the endosome in morphogen distribution models provides an explanation for the slow, directional movement of morphogens, as well as their ability to form intracellular and extracellular gradients independent of morphogen production rates. Drosophila melanogaster morphogens Wg and Dpp form stable, steep, long-range gradients that specify the polarity of the wing disc. The process of endocytosis is imperative to the two central themes in gradient formation: active transport facilitating long-range signaling and degradation of morphogen to sustain gradient shape. This review investigates the endomembrane-mediated processes of re-secretion, degradation and argosome transport of Wg and Dpp in the hope that a better understanding of the endomembrane system will contribute to a more accurate and comprehensive model for morphogen gradient formation and maintenance.     

Formation and maintenance of morphogen gradients: An essential role for the endomembrane system in Drosophila melanogaster wing development

As early as 1964 it was suggested that simple diffusion of morphogens away from their secretion source did not provide an adequate explanation for the formation and maintenance of morphogen gradients. Involvement of the endosome in morphogen distribution models provides an explanation for the slow, directional movement of morphogens, as well as their abilty to form intracellular and extracellular gradients independent of morphogen production rates. Drosophila melanogaster morphogens Wg and Dpp form stable, steep, long-range gradients that specify the polarity of the wing disc. The process of endocytosis is imparative to the two central themes in gradient formation; active transport facilitating long-range signalling, and degradation of morphogen to sustain gradient shape. This review investigates the endomembrane mediated processes of re-secretion, degradation, and argosome transport of Wg and Dpp in the hope that a better understanding of the endomembrane system will contribute to a more accurate and comprehensive model for morphogen gradient formation and maintenance.     

Endocytosis of Fgf8 Is a Double-Stage Process and Regulates Spreading and Signaling

Tightly controlled concentration gradients of morphogens provide positional information and thus regulate tissue differentiation and morphogenesis in multicellular organisms. However, how such morphogenetic fields are formed and maintained remains debated. Here we show that fibroblast growth factor 8 (Fgf8) morphogen gradients in zebrafish embryos are established and maintained by two essential mechanisms. Firstly, Fgf8 is taken up into the cell by clathrin-mediated endocytosis. The speed of the uptake rate defines the range of the morphogenetic gradient of Fgf8. Secondly, our data demonstrate that after endocytosis the routing of Fgf8 from the early endosome to the late endosome shuts down signaling. Therefore, intracellular endocytic transport regulates the intensity and duration of Fgf8 signaling. We show that internalization of Fgf8 into the early endosome and subsequent transport towards the late endosome are two independent processes. Therefore, we hypothesize that Fgf8 receiving cells control both, the propagation width and the signal strength of the morphogen.     

Visualizing long-range movement of the morphogen Xnr2 in the Xenopus embryo

One way in which cells acquire positional information during embryonic development is by measuring the local concentration of a signaling factor, or morphogen, that is secreted by an organizing center . The ways in which morphogen gradients are established, particularly in vertebrates, remain obscure, although various suggestions have been made for the mechanisms by which signaling molecules traverse fields of cells. These include simple diffusion, "cytonemes", filopodia, "argosomes", and "transcytosis". In this study, we use a functional EGFP-tagged ligand to visualize long-range signaling in the Xenopus embryo in real time. Our results show that the TGF-beta family member Xnr2 is secreted efficiently from embryonic cells, and a new method of tissue recombination allows us to investigate the way in which the morphogen traverses multiple cell diameters. This reveals that Xnr2 exerts long-range effects by diffusing rapidly through the extracellular milieu of nonexpressing cells. No evidence has been obtained for long-range signaling through cytonemes, filopodia, argosomes, or transcytosis. In demonstrating that long-range signaling in the early Xenopus embryo occurs by diffusion rather than by these alternative routes, our results suggest that different morphogens in different developmental contexts use different means of transport.     

Robust Generation and Decoding of Morphogen Gradients

Morphogen gradients play a key role in multiple differentiation processes. Both the formation of the gradient and its interpretation by the receiving cells need to occur at high precision to ensure reproducible patterning. This need for quantitative precision is challenged by fluctuations in the environmental conditions and by variations in the genetic makeup of the developing embryos. We discuss mechanisms that buffer morphogen profiles against variations in gene dosage. Self-enhanced morphogen degradation and pre-steady-state decoding provide general means for buffering the morphogen profile against fluctuations in morphogen production rate. A more specific “shuttling” mechanism, which establishes a sharp and robust activation profile of a widely expressed morphogen, and enables the adjustment of morphogen profile with embryo size, is also described. Finally, we consider the transformation of the smooth gradient profile into sharp borders of gene expression in the signal-receiving cells. The integration theory and experiments are increasingly used, providing key insights into the system-level functioning of the developmental system.In order for a uniform field of cells to differentiate into a reproducible pattern of organs and tissues, cells need to receive information about their position within the field. During development, positional information is often conveyed by spatial gradients of morphogens (Wolpert 1989). In the presence of such gradients, cells are subject to different levels of morphogen, depending on their positions within the field, and activate, accordingly, one of several gene expression cassettes. The quantitative shape of the morphogen gradient is critical for patterning, with cell-fate boundaries established at specific concentration thresholds. Although these general features of morphogen-based patterning are universal, the range and form of the morphogen profile, and the pattern of induced target genes, vary significantly depending on the tissue setting and the signaling pathways used.The formation of a morphogen gradient is a dynamic process, influenced by the kinetics of morphogen production, diffusion, and degradation. These processes are tightly controlled through intricate networks of positive and negative feedback loops, which shape the gradient and enhance its reproducibility between individual embryos and developmental contexts. In the past three decades, many of the components comprising the morphogen signaling cascades have been identified and sorted into pathways, enabling one to start addressing seminal questions regarding their functionality: How is it that morphogen signaling is reproducible from one embryo to the next, despite fluctuations in the levels of signaling components, temperature differences, variations in size, or unequal distribution of components between daughter cells? Are there underlying mechanisms that assure a reproducible response? Are these mechanisms conserved across species, similar to the signaling pathways they control?In this review, we outline insights we gained by quantitatively analyzing the process of morphogen gradient formation. We focus on mechanisms that buffer morphogen profiles against fluctuations in gene dosage, and describe general means by which such buffering is enhanced. These mechanisms include self-enhanced morphogen degradation and pre-steady-state decoding. In addition, we describe a more specific “shuttling” mechanism that is used to generate a sharp and robust profile of a morphogen activity from a source that is broadly produced. We discuss the implication of the shuttling mechanism for the ability of embryos to adjust their pattern with size. Finally, we consider the transformation of the smooth gradient profile into sharp borders of gene expression in the signal-receiving cells.     

Activin as a morphogen in Xenopus mesoderm induction.

Activin, a member of the Transforming Growth Factor beta (TGF-beta) superfamily, can behave as a morphogen in cells of the early Xenopus embryo by inducing a range of mesodermal genes in a concentration-dependent manner. This review examines the behaviour of activin as it forms a morphogen gradient. It also discusses how a cell can perceive its position in a concentration gradient in order to activate appropriate mesodermal gene responses.     

Morphogens and cell survival during development

The notion of "morphogens" is an important one in developmental biology. By definition, a morphogen is a molecule that emanates from a specific set of cells that is present in a concentration gradient and that specifies the fate of each cell along this gradient. The strongest candidate morphogens are members of the transforming growth factor-beta (TGF-beta), Hedgehog (Hh), and Wnt families. While these morphogens have been extensively described as differentiation inducers, some reports also suggest their possible involvement in cell death and cell survival. It is frequently speculated that the cell death induction that is found associated with experimental removal of morphogens is the manifestation of abnormal differentiation signals. However, several recent reports have raised controversy about this death by default, suggesting that cell death regulation is an active process for shaping tissues and organs. In this review, we will present morphogens, with a specific emphasis on Sonic Hedgehog, a mammalian member of the Hh family, not as a positive regulators of cell differentiation but as key regulators of cell survival.     

Gradient formation of the TGF-beta homolog Dpp

Secreted morphogens such as the Drosophila TGF-beta homolog Decapentaplegic (Dpp) are thought to spread through target tissues and form long-range concentration gradients providing positional information. Using a GFP-Dpp fusion, we monitored a TGF-beta family member trafficking in situ throughout the target tissue and forming a long-range concentration gradient. Evidence is presented that long-range Dpp movement involves Dpp receptor and Dynamin functions. We also show that the rates of endocytic trafficking and degradation determine Dpp signaling range. We propose a model where the gradient is formed via intracellular trafficking initiated by receptor-mediated endocytosis of the ligand in receiving cells with the gradient slope controlled by endocytic sorting of Dpp toward recycling versus degradation.     

Is Chordin a morphogen?

Models employing gradients of diffusible morphogens are often invoked to explain regional specification in the early embryo. In these models, a concentration gradient of a secreted factor generates distinct cell fates, based on the differential response of cells to local concentrations of this morphogen. In recent years, there has been a great deal of focus on the factors involved in the dorsoventral patterning of the vertebrate mesoderm and ectoderm. Dorsoventral specification during amphibian gastrulation is regulated by the antagonistic actions of the Bone Morphogenetic Proteins (BMPs) and a number of dorsally expressed inhibitors, including the secreted factor, Chordin. In a recent paper, Blitz and colleagues explore the possibility that Chordin acts as a long-range factor in the amphibian gastrula embryo, to establish a functional gradient of BMP activity.     

BMP morphogen gradients in flies

Bone morphogenetic proteins (BMPs) act as morphogens to control patterning and growth in a variety of developing tissues in different species. How BMP morphogen gradients are established and interpreted in the target tissues has been extensively studied in Drosophila melanogaster. In Drosophila, Decapentaplegic (Dpp), a homologue of vertebrate BMP2/4, acts as a morphogen to control dorsal–ventral patterning of the early embryo and anterior–posterior patterning and growth of the wing imaginal disc. Despite intensive efforts over the last twenty years, how the Dpp morphogen gradient in the wing imaginal disc forms remains controversial, while gradient formation in the early embryo is well understood. In this review, we first focus on the current models of Dpp morphogen gradient formation in these two tissues, and then discuss new strategies using genome engineering and nanobodies to tackle open questions.     

Membrane-Associated Non-Receptors and Morphogen Gradients

A previously investigated basic model (System B) for the study of signaling morphogen gradient formation that allows for reversible binding of morphogens (aka ligands) with signaling receptors, degradation of bound morphogens and diffusion of unbound morphogens is extended to include the effects of membrane-bound non-signaling molecules (or non-receptors for short) such as proteoglycans that bind reversibly with the same morphogens and degrade them. Our main goal is to delineate the effects of the presence of non-receptors on the existence and properties of the steady-state concentration gradient of signaling ligand–receptor complexes. Stability of the steady-state morphogen gradients is established and the time to reach steady-state behavior after the onset of morphogen production will be analyzed. The theoretical findings offer explanations for observations reported in several previous experiments on Drosophila wing imaginal discs.     

Forming and Interpreting Gradients in the Early Xenopus Embryo

The amphibian embryo provides a powerful model system to study morphogen gradients because of the ease with which it is possible to manipulate the early embryo. In particular, it is possible to introduce exogenous sources of morphogen, to follow the progression of the signal, to monitor the cellular response to induction, and to up- or down-regulate molecules that are involved in all aspects of long-range signaling. In this article, I discuss the evidence that gradients exist in the early amphibian embryo, the way in which morphogens might traverse a field of cells, and the way in which different concentrations of morphogens might be interpreted to activate the expression of different genes.The idea that a morphogen gradient activates the expression of different genes at different concentrations was perhaps stated most clearly by Wolpert''s French flag model, in which a graded signal activates the expression of “blue,” “white,” and “red” genes at high, intermediate, and low concentrations (Wolpert 1969). Since that original work, great progress has been made in identifying morphogens and their target genes and it is now clear that the spatial pattern of gene expression in the developing embryo is frequently established by graded signals of this sort. But many questions remain, and in particular little is known about how gradients are established in the embryo with the necessary precision and how cells interpret different concentrations of morphogen to activate different genes. I discuss these issues with respect to mesoderm induction in the developing amphibian embryo.     

Morphogens, their identification and regulation

During the course of development, cells of many tissues differentiate according to the positional information that is set by the concentration gradients of morphogens. Morphogens are signaling molecules that emanate from a restricted region of a tissue and spread away from their source to form a concentration gradient. As the fate of each cell in the field depends on the concentration of the morphogen signal, the gradient prefigures the pattern of development. In this article, we describe how morphogens and their functions have been identified and analyzed, focusing on model systems that have been extensively studied.     

Regulating morphogen gradients in the Drosophila wing

During development, diffusible ligands, known as morphogens, are thought to move across fields of cells, regulating gene expression in a concentration dependent manner. The case for morphogens has been convincingly made for the Decapentapleigic (Dpp), Wingless (Wg) and Hedgehog (Hh) proteins in the Drosophila wing. In each case, the concentration of the morphogen's receptor plays an important role in shaping the morphogen gradient, through influencing ligand transport and/or stability. However, the relationships between each ligand/receptor pair are different. The role of heparan sulfated proteoglycans, endocytosis and novel exovesicles called argosomes in regulating morphogen distribution will also be discussed.     

Shaping Morphogen Gradients by Proteoglycans

During development, secreted morphogens such as Wnt, Hedgehog (Hh), and BMP emit from their producing cells in a morphogenetic field, and specify different cell fates in a direct concentration-dependent manner. Understanding how morphogens form their concentration gradients to pattern tissues has been a central issue in developmental biology. Various experimental studies from Drosophila have led to several models to explain the formation of morphogen gradients. Over the past decade, one of the main findings in this field is the characterization of heparan sulfate proteoglycan (HSPG) as an essential regulator for morphogen gradient formation. Genetic and cell biological studies have showed that HSPGs can regulate morphogen activities at various steps including control of morphogen movement, signaling, and intracellular trafficking. Here, we review these data, highlighting recent findings that reveal mechanistic roles of HSPGs in controlling morphogen gradient formation.Embryonic development involves many spatial and temporal patterns of cell and tissue organization. These patterning processes are controlled by gradients of morphogens, the “form-generating substances” (Tabata and Takei 2004; Lander 2007). Secreted morphogen molecules, including members of Wnt, Hedgehog (Hh), and transforming growth factor-β (TGF-β) families, are generated from organizing centers and form concentration gradients to specify distinct cell fates in a concentration-dependent manner. Understanding how morphogen gradients are established during development has been a central question in developmental biology. Over the past decade, studies in both Drosophila and vertebrates have yielded important insights in this field. One of the important findings is the characterization of heparan sulfate proteoglycan (HSPG) as an essential regulator for morphogen gradient formation. In this review, we first discuss various models for morphogen movement. Then, we focus on the functions of HSPGs in morphogen movement, signaling, and trafficking.     

Understanding how morphogens work

In this article, we describe the mechanisms by which morphogens in the Xenopus embryo exert their long-range effects. Our results are consistent with the idea that signalling molecules such as activin and the nodal-related proteins traverse responding tissue not by transcytosis or by cytonemes but by movement through the extracellular space. We suggest, however, that additional experiments, involving real-time imaging of morphogens, are required for a real understanding of what influences signalling range and the shape of a morphogen gradient.