A model of morphogenetic pattern formation

A model for the morphogenetic movement of surfaces composed of cellular monolayers is proposed. The cells are presumed joined at their lateral surfaces. An otherwise unspecified substance called a "morphogen" is introduced which is the agent of change in the individual cell (or cell-like region). The distribution of these cellular deformations define a surface (the middle surface, through the middle of the cell heights) via equations given for the Gauss and Mean curvatures of the surface defined at each point. The Gauss curvature as a function of the morphogen level determines the metric of the surface "g(u, v)" in conformal co-ordinates u, v. A unique equation for the morphogen distribution over the survace is presented which has the property of size invariance, that is, the model "regulates" without need of further arguments. The two resulting coupled equations for the metric and the morphogen, eqns (4) and (2), both non-linear equations, are to be solved self-consistently, once the individual cell deformation as a function of morphogen is given. The surface geometry determines the morphogen distribution, and the morphogen distribution in turn affects the surface geometry. Extension of the model to two or more morphogens is straightforward, and the key property of "regulation" or size invariance of the model is retained. Numerical integration of the two coupled equations is carried out in the case of axial symmetry, and the results presented by the case that individual cells deform by changing the ratio of their apical to basal areas, as well as their heights. Gastrulation in small regulating holoblastic eggs (e.g. starfish, sea urchin and amphioxus) is discussed in light of the present model.     

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.     

A mechanism for morphogen-controlled domain growth

Many developmental systems are organised via the action of graded distributions of morphogens. In the Drosophila wing disc, for example, recent experimental evidence has shown that graded expression of the morphogen Dpp controls cell proliferation and hence disc growth. Our goal is to explore a simple model for regulation of wing growth via the Dpp gradient: we use a system of reaction-diffusion equations to model the dynamics of Dpp and its receptor Tkv, with advection arising as a result of the flow generated by cell proliferation. We analyse the model both numerically and analytically, showing that uniform domain growth across the disc produces an exponentially growing wing disc.     

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.     

A Model for Selection of Eyespots on Butterfly Wings

Unsolved Problem

The development of eyespots on the wing surface of butterflies of the family Nympalidae is one of the most studied examples of biological pattern formation.However, little is known about the mechanism that determines the number and precise locations of eyespots on the wing. Eyespots develop around signaling centers, called foci, that are located equidistant from wing veins along the midline of a wing cell (an area bounded by veins). A fundamental question that remains unsolved is, why a certain wing cell develops an eyespot, while other wing cells do not.

Key Idea and Model

We illustrate that the key to understanding focus point selection may be in the venation system of the wing disc. Our main hypothesis is that changes in morphogen concentration along the proximal boundary veins of wing cells govern focus point selection. Based on previous studies, we focus on a spatially two-dimensional reaction-diffusion system model posed in the interior of each wing cell that describes the formation of focus points. Using finite element based numerical simulations, we demonstrate that variation in the proximal boundary condition is sufficient to robustly select whether an eyespot focus point forms in otherwise identical wing cells. We also illustrate that this behavior is robust to small perturbations in the parameters and geometry and moderate levels of noise. Hence, we suggest that an anterior-posterior pattern of morphogen concentration along the proximal vein may be the main determinant of the distribution of focus points on the wing surface. In order to complete our model, we propose a two stage reaction-diffusion system model, in which an one-dimensional surface reaction-diffusion system, posed on the proximal vein, generates the morphogen concentrations that act as non-homogeneous Dirichlet (i.e., fixed) boundary conditions for the two-dimensional reaction-diffusion model posed in the wing cells. The two-stage model appears capable of generating focus point distributions observed in nature.

Result

We therefore conclude that changes in the proximal boundary conditions are sufficient to explain the empirically observed distribution of eyespot focus points on the entire wing surface. The model predicts, subject to experimental verification, that the source strength of the activator at the proximal boundary should be lower in wing cells in which focus points form than in those that lack focus points. The model suggests that the number and locations of eyespot foci on the wing disc could be largely controlled by two kinds of gradients along two different directions, that is, the first one is the gradient in spatially varying parameters such as the reaction rate along the anterior-posterior direction on the proximal boundary of the wing cells, and the second one is the gradient in source values of the activator along the veins in the proximal-distal direction of the wing cell.     

The interaction of surface geometry with morphogens

Expressions are given for the Gauss and Mean curvatures of a surface of thickness h. The two curvatures, (K and H), which are given at each point of the middle surface, are adequate to describe the surface. The sheet thickness varies with position in the middle surface bisecting the apical and basal surfaces. The definitions of K and H are in terms of radii of curvature, but such radii are not appropriate variables for determining how morphogens in the surface may couple to the geometry. More suitable expressions are developed here. Two important geometrical constraints must be satisfied, namely the famous Gauss-Bonnet theorem, and an inequality stemming from the definition of the two curvatures. It is argued that these constraints are of great usefulness in determining the form of the coupling of morphogens to the geometry. In particular, when two key morphogens suffice to determine surface geometry, explicit expressions are suggested to determine both Gauss (K) and Mean (H) curvatures as functions of invariant morphogen densities.     

Modelling butterfly wing eyespot patterns

Eyespots are concentric motifs with contrasting colours on butterfly wings. Eyespots have intra- and interspecific visual signalling functions with adaptive and selective roles. We propose a reaction-diffusion model that accounts for eyespot development. The model considers two diffusive morphogens and three non-diffusive pigment precursors. The first morphogen is produced in the focus and determines the differentiation of the first eyespot ring. A second morphogen is then produced, modifying the chromatic properties of the wing background pigment precursor, inducing the differentiation of a second ring. The model simulates the general structural organization of eyespots, their phenotypic plasticity and seasonal variability, and predicts effects from microsurgical manipulations on pupal wings as reported in the literature.     

The role of a reaction-diffusion system in the formation of hair fibres

Epithelial cells destined to form the hair fibre begin to differentiate while still in the hair follicle bulb. The fibre cells continue to differentiate as they migrate out of the bulb and up the follicle towards the skin surface. The anatomy of the hair follicle and the different cell types observed within the follicle are briefly reviewed. A theoretical scheme for cell differentiation, capable of producing all the observed mature cell types, is presented. A major component of the scheme is a reaction-diffusion system of morphogens similar to that originally proposed by Turing (1952). The mathematical solution of the equations defining the reaction-diffusion system within the follicle bulb is discussed. The sequence of patterns in the spatial distribution of the morphogens expected in the hair follicle bulb is calculated and found to be in good agreement with the sequence of patterns of orthocortical and paracortical cells in the fibre cross section as the diameter of the fibre increases. The spatial patterns of the morphogens are also compared with the shape of the fibre cross section. It is concluded that a reaction-diffusion system may play a major role in the morphogenesis of hair fibres.     

A Flow Sensing Model for Mesenchymal Stromal Cells Using Morphogen Dynamics

The differentiation of mesenchymal stromal cells has been shown to be affected by many parameters such as morphogens, flow rate, medium viscosity, and shear stress when exposed to fluid flow. The mechanism by which these cells sense their environment is still under intense discussion. In particular, during flow chamber experiments, it is difficult to interpret the interplay of the above-mentioned parameters in the process of cell differentiation. In this work, we tested the hypothesis that the competition between diffusion and advection of paracrine morphogens could explain the dependency of the cell differentiation to the above-mentioned parameters. To evaluate this hypothesis, we developed a numerical model simulating a simplified version of the advection-diffusion-reaction of morphogens secreted by the cells within a flow chamber. The model predicted a sharp transition in the fraction of receptors bound to the morphogen. This transition was characterized by a new, dimensionless number depending on flow rate, flow viscosity, flow chamber dimensions, and morphogen decay rate. We concluded that the competition between diffusion and advection of paracrine morphogens can act as a probe for the cells to sense their pericellular environment.     

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.     

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 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.     

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.     

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.     

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.     

Novel brain wiring functions for classical morphogens: a role as graded positional cues in axon guidance

During embryonic development, morphogens act as graded positional cues to dictate cell fate specification and tissue patterning. Recent findings indicate that morphogen gradients also serve to guide axonal pathfinding during development of the nervous system. These findings challenge our previous notions about morphogens and axon guidance molecules, and suggest that these proteins, rather than having sharply divergent functions, act more globally to provide graded positional information that can be interpreted by responding cells either to specify cell fate or to direct axonal pathfinding. This review presents the roles identified for members of three prominent morphogen families--the Hedgehog, Wnt and TGFbeta/BMP families--in axon guidance, and discusses potential implications for the molecular mechanisms underlying their guidance functions.     

Patterned cell development in the secondary phloem of dicotyledonous trees: a review and a hypothesis

The secondary phloem of dicotyledonous trees and shrubs is constructed of sieve tube cells (S) and their companion cells, as well as parenchyma (P) and fibre (F) cells. Different species have characteristic sequences of these S, P and F cells within the radial files of their phloem. The sequences are recurrent, and are evidence of rhythmic cell determination and differentiation. A model was devised to account for the sequences found in various dicot tree species. It is based on the pattern of radial displacement of cells through a gradient of morphogen which supports secondary phloem development. According to this model, each tree species shows a particular pattern of post-mitotic cellular displacement along each radial file as a result of a corresponding sequence of periclinal division in the cambial initial and its descendents. The divisions and displacements ensure that at each timestep (equivalent to an interdivisional interval) each cell resides in a specific location within the morphogenic gradient. Cells then emerge from the post-mitotic zone of cell determination, having acquired different final positional values. These values lie above a series of thresholds that permit the respective determination and subsequent differentiation of one or other of the three cell types S, P and F. The recurrent nature of the sequences of the three cell types within each radial cell file, as well as their tangential banding, are a consequence of a shared rhythmic spatio-temporal pattern of periclinal cambial divisions. With a single set of morphogen parameters required for cell determination, and using three positions for cambial cell divisions, all the cellular sequences of secondary phloem illustrated in the literature can be accounted for.This is an invited article.     

Genetics of morphogen gradients

The organization of cells and tissues is controlled by the action of 'form-giving' signalling molecules, or morphogens, which pattern a developmental field in a concentration-dependent manner. As the fate of each cell in the field depends on the level of the morphogen signal, the concentration gradient of the morphogen prefigures the pattern of development. In recent years, molecular genetic studies in Drosophila melanogaster have allowed tremendous progress in understanding how morphogen gradients are formed and maintained, and the mechanism by which receiving cells respond to the gradient.     

Problems and paradigms: Morphogens and pattern formation

Morphogen gradient theories have enjoyed considerable popularity since the beginning of this century, but conclusive evidence for a role of morphogens in controlling multicellular development has been elusive. Recently, work on three secreted signalling proteins, Activin in Xenopus, and Wingless and Dpp in Drosophila, has stongly suggested that these proteins function as morphogens. In order to define a factor as a morphogen, it is necessary to show firstly, that it has a direct effect on target cells and secondly, that it affects the development of target cells in a concentration-dependent manner. With these criteria in mind, the evidence available for a variety of proposed morphogens is discussed. While the evidence is not conclusive in most of the cases considered, there is a strong case in favour of the three proteins mentioned above, which suggests that morphogens are potentially of general importance in controlling the development of multicellular organisms.     

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.