Growth based morphogenesis of vertebrate limb bud

Mechanical Regulation of Limb Bud Formation

Early limb bud development has been of considerable interest for the study of embryological development and especially morphogenesis. The focus has long been on biochemical signalling and less on cell biomechanics and mechanobiology. However, their importance cannot be understated since tissue shape changes are ultimately controlled by active forces and bulk tissue rheological properties that in turn depend on cell-cell interactions as well as extracellular matrix composition. Moreover, the feedback between gene regulation and the biomechanical environment is still poorly understood. In recent years, novel experimental techniques and computational models have reinvigorated research on this biomechanical and mechanobiological side of embryological development. In this review, we consider three stages of early limb development, namely: outgrowth, elongation, and condensation. For each of these stages, we summarize basic biological regulation and examine the role of cellular and tissue mechanics in the morphogenetic process.

Bubble packing, eccentricity, and notochord development

This paper develops a theoretical basis for the observed relationship between cell arrangements in notochords and analog physical models, and the eccentricity of their cross sections. Three models are developed and analyzed, of the mechanics of cell packing in sheaths. The key ratios governing the packing patterns and eccentricity are cells per unit length λ, tension ratio Γ, and eccentricity e. For flexible and semi-flexible sheaths, the optimal packing pattern shifts from "bamboo", with a symmetric cross section, to "staircase", with an eccentric cross section, at a critical value λ = 1.13. In rigid tubes, this threshold is lowered as imposed eccentricity is increased. Patterns can be observed which are not optimal; pattern transitions may occur below or above the critical λ values. The eccentricity of staircase patterns in flexible and semi-flexible tubes is found to be dependent on the tension ratio Γ, increasing as sheath tension decreases relative to interior cell tension. A novel "serpentine" packing pattern appears for low Γ near the critical λ. The developmental utility of enforcing notochord eccentricity is discussed, as well as potential mechanisms for such control.

Quantification of cell behaviors and computational modeling show that cell directional behaviors drive zebrafish pectoral fin morphogenesis

MOTIVATION

Understanding the mechanisms by which the zebrafish pectoral fin develops is expected to produce insights on how vertebrate limbs grow from a 2D cell layer to a 3D structure. Two mechanisms have been proposed to drive limb morphogenesis in tetrapods: a growth-based morphogenesis with a higher proliferation rate at the distal tip of the limb bud than at the proximal side, and directed cell behaviors that include elongation, division and migration in a non-random manner. Based on quantitative experimental biological data at the level of individual cells in the whole developing organ, we test the conditions for the dynamics of pectoral fin early morphogenesis.

RESULTS

We found that during the development of the zebrafish pectoral fin, cells have a preferential elongation axis that gradually aligns along the proximodistal (PD) axis of the organ. Based on these quantitative observations, we build a center-based cell model enhanced with a polarity term and cell proliferation to simulate fin growth. Our simulations resulted in 3D fins similar in shape to the observed ones, suggesting that the existence of a preferential axis of cell polarization is essential to drive fin morphogenesis in zebrafish, as observed in the development of limbs in the mouse, but distal tip-based expansion is not.

AVAILABILITYAND IMPLEMENTATION

Upon publication, biological data will be available at http://bioemergences.eu/modelingFin, and source code at https://github.com/guijoe/MaSoFin.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Self-sustained planar intercalations due to mechanosignaling feedbacks lead to robust axis extension during morphogenesis

Tissue elongation is a necessary process in metazoans to implement their body plans that is not fully understood. Here we propose a mechanism based on the interplay between cellular mechanics and primordia patterning that results in self-sustained planar intercalations. Thus, we show that a location-dependent modulation of the mechanical properties of cells leads to robust axis extension. To illustrate the plausibility of this mechanism, we test it against different patterning models by means of computer simulations of tissues where we implemented mechano-signaling feedbacks. Our results suggest that robust elongation relies on a trade-off between cellular and tissue strains that is orchestrated through the cleavage orientation. In the particular context of axis extension in Turing-patterned tissues, we report that different directional cell activities cooperate synergetically to achieve elongation. Altogether, our findings help to understand how the axis extension phenomenon emerges from the dynamics of individual cells.

Mathematical modeling of chondrogenic pattern formation during limb development: Recent advances in continuous models

The phenomenon of chondrogenic pattern formation in the vertebrate limb is one of the best studied examples of organogenesis. Many different models, mathematical as well as conceptual, have been proposed for it in the last fifty years or so. In this review, we give a brief overview of the fundamental biological background, then describe in detail several models which aim to describe qualitatively and quantitatively the corresponding biological phenomena. We concentrate on several new models that have been proposed in recent years, taking into account recent experimental progress. The major mathematical tools in these approaches are ordinary and partial differential equations. Moreover, we discuss models with non-local flux terms used to account for cell-cell adhesion forces and a structured population model with diffusion. We also include a detailed list of gene products and potential morphogens which have been identified to play a role in the process of limb formation and its growth.

Multi-view light-sheet imaging and tracking with the MaMuT software reveals the cell lineage of a direct developing arthropod limb

During development, coordinated cell behaviors orchestrate tissue and organ morphogenesis. Detailed descriptions of cell lineages and behaviors provide a powerful framework to elucidate the mechanisms of morphogenesis. To study the cellular basis of limb development, we imaged transgenic fluorescently-labeled embryos from the crustacean Parhyale hawaiensis with multi-view light-sheet microscopy at high spatiotemporal resolution over several days of embryogenesis. The cell lineage of outgrowing thoracic limbs was reconstructed at single-cell resolution with new software called Massive Multi-view Tracker (MaMuT). In silico clonal analyses suggested that the early limb primordium becomes subdivided into anterior-posterior and dorsal-ventral compartments whose boundaries intersect at the distal tip of the growing limb. Limb-bud formation is associated with spatial modulation of cell proliferation, while limb elongation is also driven by preferential orientation of cell divisions along the proximal-distal growth axis. Cellular reconstructions were predictive of the expression patterns of limb development genes including the BMP morphogen Decapentaplegic.

During early life, animals develop from a single fertilized egg cell to hundreds, millions or even trillions of cells. These cells specialize to do different tasks; forming different tissues and organs like muscle, skin, lungs and liver. For more than a century, scientists have strived to understand the details of how animal cells become different and specialize, and have created many new techniques and technologies to help them achieve this goal.

Limbs – such as arms, legs and wings – form from small lumps of cells called limb buds. Scientists use the shrimp-like crustacean, Parhyale hawaiensis, to study development, including limb growth. This species is useful because it is easy to grow, manipulate and observe its developing young in the laboratory. Understanding how its limbs develop offers important new insights into how limbs develop in other animals too. Wolff, Tinevez, Pietzsch et al. have now combined advanced microscopy with custom computer software, called Massive Multi-view Tracker (MaMuT) to investigate this.

As limbs develop in Parhyale, the MaMuT software tracks how cells behave, and how they are organized. This analysis revealed that for cells to produce a limb bud, they need to split at an early stage into separate groups. These groups are organized along two body axes, one that goes from head to tail, and one that runs from back to belly. The limb grows perpendicular to these main body axes, along a new ‘proximal-distal’ axis that goes from nearest to furthest from the body. Wolff et al. found that the cells that contribute to the extremities of the limb divide faster than the ones that stay closer to the body. Finally, the results show that when cells in a limb divide, they mostly divide along the proximal-distal axis, producing one cell that is further from the body than the other. These cell activities may help limbs to get longer as they grow.

Notably, the groups of cells seen by Wolff et al. were expressing genes that had previously been identified in developing limbs. This helps to validate the new results and to identify which active genes control the behaviors of the analyzed cells.

These findings reveal new ways to study animal development. This approach could have many research uses and may help to link the mechanisms of cell biology to their effects. It could also contribute to new understanding of developmental and genetic conditions that affect human health.

Computer modeling in developmental biology: growing today, essential tomorrow

D'Arcy Thompson was a true pioneer, applying mathematical concepts and analyses to the question of morphogenesis over 100 years ago. The centenary of his famous book, On Growth and Form, is therefore a great occasion on which to review the types of computer modeling now being pursued to understand the development of organs and organisms. Here, I present some of the latest modeling projects in the field, covering a wide range of developmental biology concepts, from molecular patterning to tissue morphogenesis. Rather than classifying them according to scientific question, or scale of problem, I focus instead on the different ways that modeling contributes to the scientific process and discuss the likely future of modeling in developmental biology.

Collective cell movement promotes synchronization of coupled genetic oscillators

Collective cell movement is a crucial component of embryonic development. Intercellular interactions regulate collective cell movement by allowing cells to transfer information. A key question is how collective cell movement itself influences information flow produced in tissues by intercellular interactions. Here, we study the effect of collective cell movement on the synchronization of locally coupled genetic oscillators. This study is motivated by the segmentation clock in zebrafish somitogenesis, where short-range correlated movement of cells has been observed. We describe the segmentation clock tissue by a Voronoi diagram, cell movement by the force balance of self-propelled and repulsive forces between cells, the dynamics of the direction of self-propelled motion, and the synchronization of genetic oscillators by locally coupled phase oscillators. We find that movement with a correlation length of about 2 ∼ 3 cell diameters is optimal for the synchronization of coupled oscillators. Quantification of cell mixing reveals that this short-range correlation of cell movement allows cells to exchange neighbors most efficiently. Moreover, short-range correlated movement strongly destabilizes nonuniform spatial phase patterns, further promoting global synchronization. Our theoretical results suggest that collective cell movement may enhance the synchronization of the segmentation clock in zebrafish somitogenesis. More generally, collective cell movement may promote information flow in tissues by enhancing cell mixing and destabilizing spurious patterns.

Bayesian inference of whole-organ deformation dynamics from limited space-time point data

To understand the morphogenetic mechanisms of organ development and regeneration, it is essential to clarify the inter-hierarchical relationship between microscopic, molecular/cellular activities and organ-level tissue deformation dynamics. While the former have been studied for several decades, the latter - macroscopic geometrical information about physical tissue deformation - is often missing, especially for many vertebrates. This is mainly because live recording of detailed cell behaviors in whole tissues during vertebrate organogenesis is technically difficult. In this study, we have developed a novel method that combines snapshot lineage tracing with Bayesian statistical estimation to construct whole-organ deformation maps from landmark data on limited numbers of space-time points. Following the validation of the method using artificially generated data sets, we applied it to the analysis of tissue deformation dynamics in chick limb development. A quantitative tissue deformation map for St.23-St.24 has been constructed, and its precision has been proven by evaluating its predictive performance. Geometrical analyses of the map have revealed a spatially heterogeneous volume growth pattern that is consistent with the expression pattern of a major morphogen and anisotropic tissue deformation along an axis. Thus, our method enables deformation dynamics analysis in organogenesis using practical lineage marking techniques.

Digit patterning during limb development as a result of the BMP-receptor interaction

Turing models have been proposed to explain the emergence of digits during limb development. However, so far the molecular components that would give rise to Turing patterns are elusive. We have recently shown that a particular type of receptor-ligand interaction can give rise to Schnakenberg-type Turing patterns, which reproduce patterning during lung and kidney branching morphogenesis. Recent knockout experiments have identified Smad4 as a key protein in digit patterning. We show here that the BMP-receptor interaction meets the conditions for a Schnakenberg-type Turing pattern, and that the resulting model reproduces available wildtype and mutant data on the expression patterns of BMP, its receptor, and Fgfs in the apical ectodermal ridge (AER) when solved on a realistic 2D domain that we extracted from limb bud images of E11.5 mouse embryos. We propose that receptor-ligand-based mechanisms serve as a molecular basis for the emergence of Turing patterns in many developing tissues.

Mathematical modeling of vertebrate limb development

In this paper, we review the major mathematical and computational models of vertebrate limb development and their roles in accounting for different aspects of this process. The main aspects of limb development that have been modeled include outgrowth and shaping of the limb bud, establishment of molecular gradients within the bud, and formation of the skeleton. These processes occur interdependently during development, although (as described in this review), there are various interpretations of the biological relationships among them. A wide range of mathematical and computational methods have been used to study these processes, including ordinary and partial differential equation systems, cellular automata and discrete, stochastic models, finite difference methods, finite element methods, the immersed boundary method, and various combinations of the above. Multiscale mathematical modeling and associated computational simulation have become integrated into the study of limb morphogenesis and pattern formation to an extent with few parallels in the field of developmental biology. These methods have contributed to the design and analysis of experiments employing microsurgical and genetic manipulations, evaluation of hypotheses for limb bud outgrowth, interpretation of the effects of natural mutations, and the formulation of scenarios for the origination and evolution of the limb skeleton.

Wnt/planar cell polarity signaling: an important mechanism to coordinate growth and patterning in the limb

The limb is one of the premier models for studying how a simple embryonic anlage develops into complex three-dimensional form. One of the key issues in the limb field has been to determine how the limb becomes patterned along its proximal (shoulder/hip) to distal (digits) axis. For decades it has been known that the apical ectodermal ridge (AER) plays a crucial role in distal outgrowth and patterning of the vertebrate embryonic limb. Most studies have explored the relationship between the AER and the progressive assignment of cell fates to mesenchyme along the proximal to distal (PD) axis. Comparatively few, however, have examined the additional role of the AER to regulate distal outgrowth of the limb and how this growth may also influence pattern along the PD axis. Here, I will review key studies that explore the role of growth in limb development. In particular, I will focus on a recent flurry of papers that examine the role of the Wnt/planar cell polarity (PCP) pathway in regulating directed growth of the limb mesenchyme. Finally, I will discuss a potential mechanism that relates the AER to the Wnt/PCP pathway and how directed growth can play a role in shaping the limb along the PD axis.

Study on multicellular systems using a phase field model

A model of multicellular systems with several types of cells is developed from the phase field model. The model is presented as a set of partial differential equations of the field variables, each of which expresses the shape of one cell. The dynamics of each cell is based on the criteria for minimizing the surface area and retaining a certain volume. The effects of cell adhesion and excluded volume are also taken into account. The proposed model can be used to find the position of the membrane and/or the cortex of each cell without the need to adopt extra variables. This model is suitable for numerical simulations of a system having a large number of cells. The two-dimensional results of cell division, cell adhesion, rearrangement of a cell cluster, chemotaxis, and cell sorting as well as the three-dimensional results of cell clusters on the substrate are presented.

Multi-cellular logistics of collective cell migration

During development, the formation of biological networks (such as organs and neuronal networks) is controlled by multicellular transportation phenomena based on cell migration. In multi-cellular systems, cellular locomotion is restricted by physical interactions with other cells in a crowded space, similar to passengers pushing others out of their way on a packed train. The motion of individual cells is intrinsically stochastic and may be viewed as a type of random walk. However, this walk takes place in a noisy environment because the cell interacts with its randomly moving neighbors. Despite this randomness and complexity, development is highly orchestrated and precisely regulated, following genetic (and even epigenetic) blueprints. Although individual cell migration has long been studied, the manner in which stochasticity affects multi-cellular transportation within the precisely controlled process of development remains largely unknown. To explore the general principles underlying multicellular migration, we focus on the migration of neural crest cells, which migrate collectively and form streams. We introduce a mechanical model of multi-cellular migration. Simulations based on the model show that the migration mode depends on the relative strengths of the noise from migratory and non-migratory cells. Strong noise from migratory cells and weak noise from surrounding cells causes “collective migration,” whereas strong noise from non-migratory cells causes “dispersive migration.” Moreover, our theoretical analyses reveal that migratory cells attract each other over long distances, even without direct mechanical contacts. This effective interaction depends on the stochasticity of the migratory and non-migratory cells. On the basis of these findings, we propose that stochastic behavior at the single-cell level works effectively and precisely to achieve collective migration in multi-cellular systems.

Budding behaviors: Growth of the limb as a model of morphogenesis

Questions regarding morphogenesis have played second fiddle to those pertaining to pattern formation among the limb development set for some time. A recent series of publications has reinvigorated the search for mechanisms by which the limb bud arises, elongates and acquires its peculiar shape. While there are stage-specific variations, the theme that resonates across these studies is that mesoderm and cartilage cells in the limb bud exhibit polarity that drives directional movement and oriented division. Noncanonical Wnt signalling is important for these cell behaviors at all stages of limb development. While the emerging morphogenetic mechanisms underlying limb bud outgrowth are partly analogous to those of other developing structures, insights from the limb have the potential to reveal intriguing new mechanisms by which three dimensional mesoderm changes shape.

A computational clonal analysis of the developing mouse limb bud

A comprehensive spatio-temporal description of the tissue movements underlying organogenesis would be an extremely useful resource to developmental biology. Clonal analysis and fate mappings are popular experiments to study tissue movement during morphogenesis. Such experiments allow cell populations to be labeled at an early stage of development and to follow their spatial evolution over time. However, disentangling the cumulative effects of the multiple events responsible for the expansion of the labeled cell population is not always straightforward. To overcome this problem, we develop a novel computational method that combines accurate quantification of 2D limb bud morphologies and growth modeling to analyze mouse clonal data of early limb development. Firstly, we explore various tissue movements that match experimental limb bud shape changes. Secondly, by comparing computational clones with newly generated mouse clonal data we are able to choose and characterize the tissue movement map that better matches experimental data. Our computational analysis produces for the first time a two dimensional model of limb growth based on experimental data that can be used to better characterize limb tissue movement in space and time. The model shows that the distribution and shapes of clones can be described as a combination of anisotropic growth with isotropic cell mixing, without the need for lineage compartmentalization along the AP and PD axis. Lastly, we show that this comprehensive description can be used to reassess spatio-temporal gene regulations taking tissue movement into account and to investigate PD patterning hypothesis.

A comprehensive mathematical description of the growth of an organ can be given by the velocity vectors defining the displacement of each tissue point in a fixed coordinate system plus a description of the degree of mixing between the cells. As an alternative to live imaging, a way to estimate the collection of such velocity vectors, known as velocity vector field, is to use cell-labeling experiments. However, this approach can be applied only when the labeled populations have been grown for small periods of time and the tensors of the velocity vector field can be estimated directly from the shape of the labeled population. Unfortunately, most of the available cell-labeling experiments of developmental systems have been generated considering a long clone expansion time that is more suitable for lineaging studies than for estimating velocity vector fields. In this study we present a new computational method that allows us to estimate the velocity vector field of limb tissue movement by using clonal data with long harvesting time and a sequence of experimental limb morphologies. The method results in the first realistic 2D model of limb outgrowth and establishes a powerful framework for numerical simulations of limb development.

WNT5A/JNK and FGF/MAPK pathways regulate the cellular events shaping the vertebrate limb bud

BACKGROUND

The vertebrate limb is a classical model for understanding patterning of three-dimensional structures during embryonic development. Although decades of research have elucidated the tissue and molecular interactions within the limb bud required for patterning and morphogenesis of the limb, the cellular and molecular events that shape the limb bud itself have remained largely unknown.

RESULTS

We show that the mesenchymal cells of the early limb bud are not disorganized within the ectoderm as previously thought but are instead highly organized and polarized. Using time-lapse video microscopy, we demonstrate that cells move and divide according to this orientation. The combination of oriented cell divisions and movements drives the proximal-distal elongation of the limb bud necessary to set the stage for subsequent morphogenesis. These cellular events are regulated by the combined activities of the WNT and FGF pathways. We show that WNT5A/JNK is necessary for the proper orientation of cell movements and cell division. In contrast, the FGF/MAPK signaling pathway, emanating from the apical ectodermal ridge, does not regulate cell orientation in the limb bud but instead establishes a gradient of cell velocity enabling continuous rearrangement of the cells at the distal tip of the limb.

CONCLUSIONS

Together, these data shed light on the cellular basis of vertebrate limb bud morphogenesis and uncover new layers to the sequential signaling pathways acting during vertebrate limb development.

The Hemimelic extra toes mouse mutant: Historical perspective on unraveling mechanisms of dysmorphogenesis

Hemimelic extra toes (Hx) arose spontaneously as a dominant mutation in B10.D2/nSnJ mice in 1967. It specifically affects the appendicular skeleton, causing variable foreshortening of the tibia (radius) and preaxial polydactylism. Early anatomical studies revealed anterior overgrowth of the autopod, with decreased apoptosis and increased mitosis in the anterior apical ectodermal ridge and underlying mesenchyme; overextension of apoptosis in the central zeugopod accounted for hemimelia. The Hx mutant phenotype was coarsely mapped to mouse chromosome (Chr) 5 and closely linked to engrailed-2 (En2) and Sonic hedgehog (Shh). This region is syntenic to human Chr 7q36 that harbors several dominant mutations affecting the hand. High-resolution genome mapping identified the Hx mutation as a G --> A base pair transition within Intron 5 of the murine Lmbr1 locus. The critical effect is on a multifunctional conserved regulatory element that acts as a limb-specific, long-distance cis-acting enhancer of Shh expression. As such, the Hx mutant phenotype results from ectopic Shh signals at the anterior margin of the limb bud that directly or indirectly alter FGF4 signaling from the apical ectodermal ridge. Given significant advances in understanding of embryonic development in general and limb development in particular, this review article reveals how research that once attracted interest of teratologists has advanced across the decades to pinpoint a critical molecular lesion and reveal a potential mechanism of a specific malformation that is found commonly in experimental teratology.

The role of spatially controlled cell proliferation in limb bud morphogenesis

Although the vertebrate limb bud has been studied for decades as a model system for spatial pattern formation and cell specification, the cellular basis of its distally oriented elongation has been a relatively neglected topic by comparison. The conventional view is that a gradient of isotropic proliferation exists along the limb, with high proliferation rates at the distal tip and lower rates towards the body, and that this gradient is the driving force behind outgrowth. Here we test this hypothesis by combining quantitative empirical data sets with computer modelling to assess the potential role of spatially controlled proliferation rates in the process of directional limb bud outgrowth. In particular, we generate two new empirical data sets for the mouse hind limb--a numerical description of shape change and a quantitative 3D map of cell cycle times--and combine these with a new 3D finite element model of tissue growth. By developing a parameter optimization approach (which explores spatial patterns of tissue growth) our computer simulations reveal that the observed distribution of proliferation rates plays no significant role in controlling the distally extending limb shape, and suggests that directional cell activities are likely to be the driving force behind limb bud outgrowth. This theoretical prediction prompted us to search for evidence of directional cell orientations in the limb bud mesenchyme, and we thus discovered a striking highly branched and extended cell shape composed of dynamically extending and retracting filopodia, a distally oriented bias in Golgi position, and also a bias in the orientation of cell division. We therefore provide both theoretical and empirical evidence that limb bud elongation is achieved by directional cell activities, rather than a PD gradient of proliferation rates.

The role of spatially controlled cell proliferation in limb bud morphogenesis

Although the vertebrate limb bud has been studied for decades as a model system for spatial pattern formation and cell specification, the cellular basis of its distally oriented elongation has been a relatively neglected topic by comparison. The conventional view is that a gradient of isotropic proliferation exists along the limb, with high proliferation rates at the distal tip and lower rates towards the body, and that this gradient is the driving force behind outgrowth. Here we test this hypothesis by combining quantitative empirical data sets with computer modelling to assess the potential role of spatially controlled proliferation rates in the process of directional limb bud outgrowth. In particular, we generate two new empirical data sets for the mouse hind limb--a numerical description of shape change and a quantitative 3D map of cell cycle times--and combine these with a new 3D finite element model of tissue growth. By developing a parameter optimization approach (which explores spatial patterns of tissue growth) our computer simulations reveal that the observed distribution of proliferation rates plays no significant role in controlling the distally extending limb shape, and suggests that directional cell activities are likely to be the driving force behind limb bud outgrowth. This theoretical prediction prompted us to search for evidence of directional cell orientations in the limb bud mesenchyme, and we thus discovered a striking highly branched and extended cell shape composed of dynamically extending and retracting filopodia, a distally oriented bias in Golgi position, and also a bias in the orientation of cell division. We therefore provide both theoretical and empirical evidence that limb bud elongation is achieved by directional cell activities, rather than a PD gradient of proliferation rates.

Bare bones pattern formation: a core regulatory network in varying geometries reproduces major features of vertebrate limb development and evolution

BACKGROUND

Major unresolved questions regarding vertebrate limb development concern how the numbers of skeletal elements along the proximodistal (P-D) and anteroposterior (A-P) axes are determined and how the shape of a growing limb affects skeletal element formation. There is currently no generally accepted model for these patterning processes, but recent work on cartilage development (chondrogenesis) indicates that precartilage tissue self-organizes into nodular patterns by cell-molecular circuitry with local auto-activating and lateral inhibitory (LALI) properties. This process is played out in the developing limb in the context of a gradient of fibroblast growth factor (FGF) emanating from the apical ectodermal ridge (AER).

RESULTS

We have simulated the behavior of the core chondrogenic mechanism of the developing limb in the presence of an FGF gradient using a novel computational environment that permits simulation of LALI systems in domains of varying shape and size. The model predicts the normal proximodistal pattern of skeletogenesis as well as distal truncations resulting from AER removal. Modifications of the model's parameters corresponding to plausible effects of Hox proteins and formins, and of the reshaping of the model limb, bud yielded simulated phenotypes resembling mutational and experimental variants of the limb. Hypothetical developmental scenarios reproduce skeletal morphologies with features of fossil limbs.

CONCLUSIONS

The limb chondrogenic regulatory system operating in the presence of a gradient has an inherent, robust propensity to form limb-like skeletal structures. The bare bones framework can accommodate ancillary gene regulatory networks controlling limb bud shaping and establishment of Hox expression domains. This mechanism accounts for major features of the normal limb pattern and, under variant geometries and different parameter values, those of experimentally manipulated, genetically aberrant and evolutionary early forms, with no requirement for an independent system of positional information.

Mechanisms for split localization of Fgf10 expression in early lung development

In early lung development, epithelial tubes (lung buds) intrude into mesenchyme covered with pleural cells (lung border), and form tree-like networks, by means of repeated use of morphogenetic processes: "elongation," "terminal bifurcation," and "lateral budding." When a bud is elongating, a peak of Fgf10 expression is formed in the mesenchyme near the tip; whereas when terminal bifurcation and lateral budding occur, two separate peaks are formed instead. To explain the spatial pattern of Fgf10 expression, we developed a mathematical model for the regulation of Fgf10 expression with geometrical conditions including shapes of the lung buds and the lung border. Different localization patterns of Fgf10 expression can be explained by the geometrical conditions. Fgf10 expression has a single peak when a length between the tip of lung bud and the lung border is large. When the length is small, Fgf10 expression has two peaks, whose location depends on the curvature of lung border.