Generation model of positional values as cell operation during the development of multicellular organisms

Many conventional models have used the positional information hypothesis to explain each elementary process of morphogenesis during the development of multicellular organisms. Their models assume that the steady concentration patterns of morphogens formed in an extracellular environment have an important property of positional information, so-called “robustness”. However, recent experiments reported that a steady morphogen pattern, the concentration gradient of the Bicoid protein, during early Drosophila embryonic development is not robust for embryo-to-embryo variability. These reports encourage a reconsideration of a long-standing problem in systematic cell differentiation: what is the entity of positional information for cells? And, what is the origin of the robust boundary of gene expression? To address these problems at a cellular level, in this article we pay attention to the re-generative phenomena that show another important property of positional information, “size invariance”. In view of regenerative phenomena, we propose a new mathematical model to describe the generation mechanism of a spatial pattern of positional values. In this model, the positional values are defined as the values into which differentiable cells transform a spatial pattern providing positional information. The model is mathematically described as an associative algebra composed of various terms, each of which is the multiplication of some fundamental operators under the assumption that the operators are derived from the remarkable properties of cell differentiation on an amputation surface in regenerative phenomena. We apply this model to the concentration pattern of the Bicoid protein during the anterior-posterior axis formation in Drosophila, and consider the conditions needed to establish the robust boundary of the expression of the hunchback gene.     

Thresholds in development

The interpretation of gradients in positional information is considered in terms of thresholds in cell responses, giving rise to cell states which are discrete and persistent. Equilibrium models based on co-operative binding of control molecules do not show true thresholds of discontinuity, though with a very high degree of co-operativity they could mimic them; in any case they do not provide the cells with any memory of a transient signal. A simple kinetic model based upon positive feedback can account both for memory and for discontinuities in the pattern of cell states. The model is an example of a bistable control circuit, and transitions from one state to another may be brought about not only by morphogenetic signals, but also by disturbances in the parameters determining the kinetics of the system. This might explain some aspects of transdetermination in insects.An attempt is made to analyse the precision with which a spatial gradient of a diffusible morphogen could be interpreted by a kinetic threshold mechanism, in terms of the length of the field, the steepness of the concentration gradient, and the intrinsic random variability of cells. It is concluded that it would be possible to specify as many as 30 distinct cell states in a positional field 1 mm long with a concentration span of 10³. Mechanisms for reducing the positional error are considered.     

When are cellular oscillators sufficient for sequential segmentation?

Sequential segmentation during embryogenesis involves the generation of a repeated pattern along the embryo, which is concurrently undergoing axial elongation by cell division. Most mathematical models of sequential segmentation involve inherent cellular oscillators, acting as a segmentation clock. The cellular oscillation is assumed to be governed by the cell's physiological age or by its interaction with an external morphogen gradient. Here, we address the issue of when cellular oscillators alone are sufficient for predicting segmentation, and when a morphogen gradient is required. The key to resolving this issue lies in how cells determine positional information in the model - this is directly related to the distribution of cell divisions responsible for axial elongation. Mathematical models demonstrate that if axial elongation occurs through cell divisions restricted to the posterior end of the unsegmented region, a cell can obtain its positional information from its physiological age, and therefore cellular oscillators will suffice. Alternatively, if axial elongation occurs through cell divisions distributed throughout the unsegmented region, then positional information can be obtained through another mechanism, such as a morphogen gradient. Two alternative ways to establish a morphogen gradient in tissue with distributed cell divisions are presented - one with diffusion and the other without diffusion. Our model produces segment polarity and a distribution of segment size from the anterior-to-posterior ends, as observed in some systems. Furthermore, the model predicts segment deletions when there is an interruption in cell division, just as seen in heat shock experiments, as well as the growth and final shrinkage of the presomitic mesoderm during somitogenesis.     

Probing intrinsic properties of a robust morphogen gradient in Drosophila

A remarkable feature of development is its reproducibility, the ability to correct embryo-to-embryo variations and instruct precise patterning. In Drosophila, embryonic patterning along the anterior-posterior axis is controlled by the morphogen gradient Bicoid (Bcd). In this article, we describe quantitative studies of the native Bcd gradient and its target Hunchback (Hb). We show that the native Bcd gradient is highly reproducible and is itself scaled with embryo length. While a precise Bcd gradient is necessary for precise Hb expression, it still has positional errors greater than Hb expression. We describe analyses further probing mechanisms for Bcd gradient scaling and correction of its residual positional errors. Our results suggest a simple model of a robust Bcd gradient sufficient to achieve scaled and precise activation of its targets. The robustness of this gradient is conferred by its intrinsic properties of "self-correcting" the inevitable input variations to achieve a precise and reproducible output.     

Space-dependent cell determination under the control of a morphogen gradient

A model is proposed for space-dependent cell determination under the influence of a morphogen gradient. It provides an explanation of how groups of cells can be programmed in a particular direction and how a jump from one determination stage to the next can occur between them even though the controlling signal is of a smoothly graded morphogen concentration. Together with an earlier proposed mechanism for pattern formation, these models offer a complete system for the generation and interpretation of positional information. Each member of a set of structure-controlling genes is assumed to feed back onto its own activation such that a gene, once activated, remains in the activated state. A repressor, however, is produced by any activated gene of this set. This assures that only one gene of this set is active in one cell at any one time. A selective activation of a particular gene is possible if (i) the morphogen competes with the gene-produced, non-diffusible repressor, (ii) the feedback loops have some overlap and (iii) a hierarchy exists among the structure-controlling genes. The kinetics of this determination have all the properties demanded earlier from a study of the early insect development: It proceeds stepwise from determination for more anterior to more posterior structures until the gene that is activated corresponds to the local gradient level. A more anterior structure will be formed if the gradient is destroyed before the final determination level is reached. A more posterior structure will be formed after an additional increase of the morphogen concentration. After completion of the determination, the repressor concentration in each cell depends on which gene has become activated and it can be made roughly proportional to the morphogen concentration which the cell has seen. Therefore, a stable parameter (positional value) becomes available which can be used for further developmental decisions.     

Effects of species and habitat positional errors on the performance and interpretation of species distribution models

Aim A key assumption in species distribution modelling is that both species and environmental data layers contain no positional errors, yet this will rarely be true. This study assesses the effect of introduced positional errors on the performance and interpretation of species distribution models. Location Baixo Alentejo region of Portugal. Methods Data on steppe bird occurrence were collected using a random stratified sampling design on a 1‐km² pixel grid. Environmental data were sourced from satellite imagery and digital maps. Error was deliberately introduced into the species data as shifts in a random direction of 0–1, 2–3, 4–5 and 0–5 pixels. Whole habitat layers were shifted by 1 pixel to cause mis‐registration, and the cumulative effect of one to three shifted layers investigated. Distribution models were built for three species using three algorithms with three replicates. Test models were compared with controls without errors. Results Positional errors in the species data led to a drop in model performance (larger errors having larger effects – typically up to 10% drop in area under the curve on average), although not enough for models to be rejected. Model interpretation was more severely affected with inconsistencies in the contributing variables. Errors in the habitat layers had similar although lesser effects. Main conclusions Models with species positional errors are hard to detect, often statistically good, ecologically plausible and useful for prediction, but interpreting them is dangerous. Mis‐registered habitat layers produce smaller effects probably because shifting entire layers does not break down the correlation structure to the same extent as random shifts in individual species observations. Spatial autocorrelation in the habitat layers may protect against species positional errors to some extent but the relationship is complex and requires further work. The key recommendation must be that positional errors should be minimised through careful field design and data processing.     

The effect of positional error on fine scale species distribution models increases for specialist species

Species occurrences inherently include positional error. Such error can be problematic for species distribution models (SDMs), especially those based on fine-resolution environmental data. It has been suggested that there could be a link between the influence of positional error and the width of the species ecological niche. Although positional errors in species occurrence data may imply serious limitations, especially for modelling species with narrow ecological niche, it has never been thoroughly explored. We used a virtual species approach to assess the effects of the positional error on fine-scale SDMs for species with environmental niches of different widths. We simulated three virtual species with varying niche breadth, from specialist to generalist. The true distribution of these virtual species was then altered by introducing different levels of positional error (from 5 to 500 m). We built generalized linear models and MaxEnt models using the distribution of the three virtual species (unaltered and altered) and a combination of environmental data at 5 m resolution. The models’ performance and niche overlap were compared to assess the effect of positional error with varying niche breadth in the geographical and environmental space. The positional error negatively impacted performance and niche overlap metrics. The amplitude of the influence of positional error depended on the species niche, with models for specialist species being more affected than those for generalist species. The positional error had the same effect on both modelling techniques. Finally, increasing sample size did not mitigate the negative influence of positional error. We showed that fine-scale SDMs are considerably affected by positional error, even when such error is low. Therefore, where new surveys are undertaken, we recommend paying attention to data collection techniques to minimize the positional error in occurrence data and thus to avoid its negative effect on SDMs, especially when studying specialist species.     

Fast-tracking morphogen diffusion

The readout of morphogen concentrations has been proposed to be an essential mechanism allowing embryos to specify cell identities [Wolpert Trends Genet 12 (1996) 359], but theoretical and experimental results have led to conflicting ideas as to how useful concentration gradients can be established. In particular, it has been pointed out that some models of passive extracellular diffusion exhibit traveling waves of receptor saturation, inadequate for the establishment of positional information. Two alternative (but not mutually exclusive) models are proposed here, which are based on recent experimental results highlighting the roles of extracellular glycoproteins and morphogen oligomerization. In the first model, inspired from the interactions of Dally and Dally-like with Wingless and Decapentaplegic in the third-instar Drosophila wing disc, two morphogen populations are considered: one in a cell-membrane phase, and another one in an extracellular matrix phase, which does not interact with receptors; in the second model, inspired from biochemical studies of Sonic Hedgehog, morphogen oligomers are considered to diffuse freely without interacting with receptors. The existence of a dynamic sub-population of freely diffusing morphogen allows the system to establish a gradient of bound receptor that is suitable for the specification of positional information. Recent experimental results are discussed within the framework of these models, as well as further possible experiments. The role of Notum in the setup of the Wg gradient is also shown to be likely not to involve a gradient in Notum distribution, even though Notum is only expressed close to the source of Wg synthesis.     

Morphogen gradient formation and vesicular trafficking

Morphogens are secreted signaling molecules which form spatial concentration gradients while moving away from a restricted source of production. A simple model of gradient formation postulates that the morphogens dilute as they diffuse between cells. In this review we discuss recent data supporting the idea that movement of the morphogen could also occur via vesicular trafficking through the cells. We explore the implications of these results for the control of gradient formation and the determination of the gradient slope which ultimately encodes the coordinates of positional information.     

Positional information and patterning revisited

The concept of positional information proposes that cells acquire positional values as in a coordinate system, which they interpret by developing in particular ways to give rise to spatial patterns. Some of the best evidence for positional information comes from regeneration experiments, and the patterning of the leg and antenna in Drosophila and the vertebrate limb. Central problems are how positional information is set up, how it is recorded, and then how it is interpreted by the cells. A number of models have been proposed for the setting up of positional gradients, and most are based on diffusion of a morphogen and its interactions with extracellular molecules. It is argued that diffusion may not be reliable mechanism. There are also mechanisms based on timing. There is no good evidence for the quantitative aspects of any of the gradients and details how they are set up. The way in which a signalling gradient regulates differential gene expression in a concentration-dependent manner also raises several mechanistic issues.     

Use of retinoids to analyze the cellular basis of positional memory in regenerating amphibian limbs

Cells of the amphibian limb regeneration blastema inherit memories of their level of origin (positional memory) along the limb axes. These memories serve as boundaries of what is to be regenerated, thus preventing regeneration of any but the missing structures. Because of its importance in determining the boundaries of regenerate pattern, it is essential to understand the cellular and molecular basis of positional memory. One approach to this problem is to look for position-related differences in a cell or molecular property along a limb axis and then show, using an agent that modifies regenerate pattern, that the cell or molecular property and the pattern are coordinately modified. We have done this using retinoic acid (RA) as a pattern-modifying agent and an in vivo assay that detects position-related differences in a cell recognition-affinity property along the proximodistal (PD) axis of the regenerating axolotl limb. RA proximalizes positional memory in the PD axis, posteriorizes it in the anteroposterior axis, and ventralizes it in the dorsoventral axis. The level-specific PD cell recognition-affinity property is proximalized by RA, indicating that this property and positional memory are causally related. The effects of RA on positional memory may be mediated through a cellular RA-binding protein (CRABP), since the concentration of unbound (apo) CRABP molecules is highest during early stages of regeneration when the proximalizing effects of RA are greatest.     

Polyps, peptides and patterning

Peptides serve as important signalling molecules in development and differentiation in the simple metazoan Hydra. A systematic approach (The Hydra Peptide Project) has revealed that Hydra contains several hundreds of peptide signalling molecules, some of which are neuropeptides and others emanate from epithelial cells. These peptides control biological processes as diverse as muscle contraction, neuron differentiation, and the positional value gradient. Signal peptides cause changes in cell behaviour by controlling target genes such as matrix metalloproteases. The abundance of peptides in Hydra raises the question of whether, in early metazoan evolution, cell-cell communication was based mainly on these small molecules rather than on the growth-factor-like cytokines that control differentiation and development in higher animals.     

Theoretical aspects of stripe formation in relation to Drosophila segmentation

Many aspects of Drosophila segmentation can be discussed in one-dimensional terms as a linear pattern of repeated elements or cell states. But the initial metameric pattern seen in the expression of pair-rule genes is fully two-dimensional, i.e. a pattern of stripes. Several lines of evidence suggest a kinetic mechanism acting globally during the syncytial blastoderm stage may be responsible for generating this pattern. The requirement that the mechanism should produce stripes, not spots or some other periodic pattern, imposes preconditions on this act, namely (1) sharp anterior and posterior boundaries that delimit the pattern-forming region, and (2) an axial asymmetrizing influence in the form of an anteroposterior gradient. Models for Drosophila segmentation generally rely on the gradient to provide positional information in the form of concentration thresholds that cue downstream elements of a hierarchical control system. This imposes restrictions on how such models cope with experimental disturbances to the gradient. A shallower gradient, for example, means fewer pattern elements. This need not be the case if the gradient acts through a kinetic mechanism like reaction-diffusion that involves the whole system. It is then the overall direction of the gradient that is important rather than specific concentration values.(ABSTRACT TRUNCATED AT 250 WORDS)     

Making the body plan: precision in the genetic hierarchy of Drosophila embryo segmentation

We quantify fluctuations in protein expression for three of the segmentation genes in the fruit fly, Drosophila melanogaster. These proteins are representative members of the first three levels of a signalling hierarchy which determines the segmented body plan: maternal (Bicoid protein); gap (Hunchback protein); and pair-rule (Even-skipped protein). We quantify both inter-embryo and inter-nucleus (within a single embryo) variability in expression, especially with respect to positional specification by concentration gradient reading. Errors are quantified both early and late in cleavage cycle 14, during which the protein patterns develop, to study the dynamics of error transmission. We find that Bicoid displays very large positional errors, while expression of the downstream genes, Hunchback and Even-skipped, displays far more precise positioning. This is evidence that the pattern formation of the downstream proteins is at least partially independent of maternal signal, i. e. evidence against simple concentration gradient reading. We also find that fractional errors in concentration increase during cleavage cycle 14.     

Roles for the extracellular matrix in plant development and pollination: a special case of cell movement in plants.

Pattern formation in plants is now thought to be primarily dependent on positional information during development. We discuss the prevalent theories on how position is deciphered by cells in an organism and highlight the recent advances implicating molecules of the cell wall or extracellular matrix (ECM) in this process. We compare the functions of the ECM in plants and animals and describe the various cell and substrate adhesion molecules of the animal ECM which play a role in morphogenesis and cell movement. We propose that analogous molecules may occur in plants and provide evidence for the presence of a substrate adhesion molecule like vitronectin in plants and algae. We provide a model for how substrate adhesion molecules may be involved in a special case of cell movement in plants, pollination.     

Computer simulations of cell sorting due to differential adhesion

The actions of cell adhesion molecules, in particular, cadherins during embryonic development and morphogenesis more generally, regulate many aspects of cellular interactions, regulation and signaling. Often, a gradient of cadherin expression levels drives collective and relative cell motions generating macroscopic cell sorting. Computer simulations of cell sorting have focused on the interactions of cells with only a few discrete adhesion levels between cells, ignoring biologically observed continuous variations in expression levels and possible nonlinearities in molecular binding. In this paper, we present three models relating the surface density of cadherins to the net intercellular adhesion and interfacial tension for both discrete and continuous levels of cadherin expression. We then use then the Glazier-Graner-Hogeweg (GGH) model to investigate how variations in the distribution of the number of cadherins per cell and in the choice of binding model affect cell sorting. We find that an aggregate with a continuous variation in the level of a single type of cadherin molecule sorts more slowly than one with two levels. The rate of sorting increases strongly with the interfacial tension, which depends both on the maximum difference in number of cadherins per cell and on the binding model. Our approach helps connect signaling at the molecular level to tissue-level morphogenesis.     

Polarized signaling endosomes coordinate BDNF-induced chemotaxis of cerebellar precursors

During development, neural precursors migrate in response to positional cues such as growth factor gradients. However, the mechanisms that enable precursors to sense and respond to such gradients are poorly understood. Here we show that cerebellar granule cell precursors (GCPs) migrate along a gradient of brain-derived neurotrophic factor (BDNF), and we demonstrate that vesicle trafficking is critical for this chemotactic process. Activation of TrkB, the BDNF receptor, stimulates GCPs to secrete BDNF, thereby amplifying the ambient gradient. The BDNF gradient stimulates endocytosis of TrkB and associated signaling molecules, causing asymmetric accumulation of signaling endosomes at the subcellular location where BDNF concentration is maximal. Thus, regulated BDNF exocytosis and TrkB endocytosis enable precursors to polarize and migrate in a directed fashion along a shallow BDNF gradient.     

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.