Short- and long-range effects of Sonic hedgehog in limb development

A mathematical framework for measuring and tuning tempo in developmental gene regulatory networks

Embryo development is a dynamic process governed by the regulation of timing and sequences of gene expression, which control the proper growth of the organism. Although many genetic programmes coordinating these sequences are common across species, the timescales of gene expression can vary significantly among different organisms. Currently, substantial experimental efforts are focused on identifying molecular mechanisms that control these temporal aspects. In contrast, the capacity of established mathematical models to incorporate tempo control while maintaining the same dynamical landscape remains less understood. Here, we address this gap by developing a mathematical framework that links the functionality of developmental programmes to the corresponding gene expression orbits (or landscapes). This unlocks the ability to find tempo differences as perturbations in the dynamical system that preserve its orbits. We demonstrate that this framework allows for the prediction of molecular mechanisms governing tempo, through both numerical and analytical methods. Our exploration includes two case studies: a generic network featuring coupled production and degradation, with a particular application to neural progenitor differentiation; and the repressilator. In the latter, we illustrate how altering the dimerisation rates of transcription factors can decouple the tempo from the shape of the resulting orbits. We conclude by highlighting how the identification of orthogonal molecular mechanisms for tempo control can inform the design of circuits with specific orbits and tempos.

Mechanistic regulation of planarian shape during growth and degrowth

Adult planarians can grow when fed and degrow (shrink) when starved while maintaining their whole-body shape. It is unknown how the morphogens patterning the planarian axes are coordinated during feeding and starvation or how they modulate the necessary differential tissue growth or degrowth. Here, we investigate the dynamics of planarian shape together with a theoretical study of the mechanisms regulating whole-body proportions and shape. We found that the planarian body proportions scale isometrically following similar linear rates during growth and degrowth, but that fed worms are significantly wider than starved worms. By combining a descriptive model of planarian shape and size with a mechanistic model of anterior-posterior and medio-lateral signaling calibrated with a novel parameter optimization methodology, we theoretically demonstrate that the feedback loop between these positional information signals and the shape they control can regulate the planarian whole-body shape during growth. Furthermore, the computational model produced the correct shape and size dynamics during degrowth as a result of a predicted increase in apoptosis rate and pole signal during starvation. These results offer mechanistic insights into the dynamic regulation of whole-body morphologies.

Non-canonical non-genomic morphogen signaling in anucleate platelets: a critical determinant of prothrombotic function in circulation

Circulating platelets derived from bone marrow megakaryocytes play a central role in thrombosis and hemostasis. Despite being anucleate, platelets express several proteins known to have nuclear niche. These include transcription factors and steroid receptors whose non-genomic functions are being elucidated in platelets. Quite remarkably, components of some of the best-studied morphogen pathways, namely Notch, Sonic Hedgehog (Shh), and Wnt have also been described in recent years in platelets, which regulate platelet function in the context of thrombosis as well as influence their survival. Shh and Notch pathways in stimulated platelets establish feed-forward loops of autocrine/juxtacrine/paracrine non-canonical signaling that helps perpetuate thrombosis. On the other hand, non-canonical Wnt signaling is part of a negative feedback loop for restricting platelet activation and possibly limiting thrombus growth. The present review will provide an overview of these signaling pathways in general. We will then briefly discuss the non-genomic roles of transcription factors and steroid receptors in platelet activation. This will be followed by an elaborate description of morphogen signaling in platelets with a focus on their bearing on platelet activation leading to hemostasis and thrombosis as well as their potential for therapeutic targeting in thrombotic disorders.

Cholesterol and Hedgehog Signaling: Mutual Regulation and Beyond

The Hedgehog (HH) signaling is one of the key agents that govern the precisely regulated developmental processes of multicellular organisms in vertebrates and invertebrates. The HH pathway in the receiving cell includes Patched1, a twelve-pass transmembrane receptor, and Smoothened, a seven-transmembrane G-protein coupled receptor (GPCR), and the downstream GLI family of three transcriptional factors (GLI1-GLI3). Mutations of HH gene and the main components in HH signaling are also associated with numerous types of diseases. Before secretion, the HH protein undergoes post-translational cholesterol modification to gain full activity, and cholesterol is believed to be essential for proper HH signaling transduction. In addition, results from recent studies show the reciprocal effect that HH signaling functions in cholesterol metabolism as well as in cholesterol homeostasis, which provides feedback to HH pathway. Here, we hope to provide new insights into HH signaling function by discussing the role of cholesterol in HH protein maturation, secretion and HH signaling transduction, and the potential role of HH in regulation of cholesterol as well.

Programming cell growth into different cluster shapes using diffusible signals

Advances in genetic engineering technologies have allowed the construction of artificial genetic circuits, which have been used to generate spatial patterns of differential gene expression. However, the question of how cells can be programmed, and how complex the rules need to be, to achieve a desired tissue morphology has received less attention. Here, we address these questions by developing a mathematical model to study how cells can collectively grow into clusters with different structural morphologies by secreting diffusible signals that can influence cellular growth rates. We formulate how growth regulators can be used to control the formation of cellular protrusions and how the range of achievable structures scales with the number of distinct signals. We show that a single growth inhibitor is insufficient for the formation of multiple protrusions but may be achieved with multiple growth inhibitors, and that other types of signals can regulate the shape of protrusion tips. These examples illustrate how our approach could potentially be used to guide the design of regulatory circuits for achieving a desired target structure.

Multiscale phenomena and patterns in biological systems: special issue in honour of Hans Othmer

This special issue on "Multiscale phenomena and patterns in biological systems" is an homage to the seminal contributions of Hans Othmer. He has remained at the forefront of multiscale modelling and pattern formation in biology for over half a century, developing models for molecular signalling networks, the mechanics of cellular movements, the interactions between multiple cells and their contributions to tissue patterning and dynamics. The contributions in this special issue follow Hans' legacy in using advanced mathematics to understand complex biological processes.

A mathematical model of the role of aggregation in sonic hedgehog signalling

Effective regulation of the sonic hedgehog (Shh) signalling pathway is essential for normal development in a wide variety of species. Correct Shh signalling requires the formation of Shh aggregates on the surface of producing cells. Shh aggregates subsequently diffuse away and are recognised in receiving cells located elsewhere in the developing embryo. Various mechanisms have been postulated regarding how these aggregates form and what their precise role is in the overall signalling process. To understand the role of these mechanisms in the overall signalling process, we formulate and analyse a mathematical model of Shh aggregation using nonlinear ordinary differential equations. We consider Shh aggregate formation to comprise of multimerisation, association with heparan sulfate proteoglycans (HSPG) and binding with lipoproteins. We show that the size distribution of the Shh aggregates formed on the producing cell surface resembles an exponential distribution, a result in agreement with experimental data. A detailed sensitivity analysis of our model reveals that this exponential distribution is robust to parameter changes, and subsequently, also to variations in the processes by which Shh is recruited by HSPGs and lipoproteins. The work demonstrates the time taken for different sized Shh aggregates to form and the important role this likely plays in Shh diffusion.

The sonic hedgehog (Shh) pathway is vital for normal development in a wide variety of species and its activity is strictly regulated to ensure correct spatiotemporal patterning of numerous developing tissues. Shh signalling requires the formation of Shh aggregates, formed on producing cells via a range of different mechanisms, that then diffuse to receiving cells. We formulate and analyse a mathematical model of the most well described mechanisms, namely monomer multimerisation, and recruitment of Shh by heparan sulfate proteoglycans and lipoproteins. Our results illustrate a distribution of the size and quantities of aggregates formed by these mechanisms. We found that as a consequence of competition between the mechanisms for Shh monomers the shape distribution of Shh aggregates resembles an exponential distribution. We also found the distribution to be robust to both parameter changes and variations to the processes by which mechanisms recruit Shh. We report that our approach and subsequent results demonstrate that these mechanisms act in synergy allowing Shh to aggregate in various quantities with diverse diffusive abilities. We postulate that this regulation contributes significantly to aid precision in signalling for Shh in areas of development.

Multiscale modeling of vertebrate limb development

We review the current state of mathematical modeling of cartilage pattern formation in vertebrate limbs. We place emphasis on several reaction-diffusion type models that have been proposed in the last few years. These models are grounded in more detailed knowledge of the relevant regulatory processes than previous ones but generally refer to different molecular aspects of these processes. Considering these models in light of comparative phylogenomics permits framing of hypotheses on the evolutionary order of appearance of the respective mechanisms and their roles in the fin-to-limb transition. This article is categorized under: Analytical and Computational Methods > Computational Methods Models of Systems Properties and Processes > Mechanistic Models Developmental Biology > Developmental Processes in Health and Disease Analytical and Computational Methods > Analytical Methods.

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.

Simulation of Morphogen and Tissue Dynamics

Morphogenesis, the process by which an adult organism emerges from a single cell, has fascinated humans for a long time. Modeling this process can provide novel insights into development and the principles that orchestrate the developmental processes. This chapter focuses on the mathematical description and numerical simulation of developmental processes. In particular, we discuss the mathematical representation of morphogen and tissue dynamics on static and growing domains, as well as the corresponding tissue mechanics. In addition, we give an overview of numerical methods that are routinely used to solve the resulting systems of partial differential equations. These include the finite element method and the Lattice Boltzmann method for the discretization as well as the arbitrary Lagrangian-Eulerian method and the Diffuse-Domain method to numerically treat deforming domains.

A Model of the Spatio-temporal Dynamics of Drosophila Eye Disc Development

Patterning and growth are linked during early development and have to be tightly controlled to result in a functional tissue or organ. During the development of the Drosophila eye, this linkage is particularly clear: the growth of the eye primordium mainly results from proliferating cells ahead of the morphogenetic furrow (MF), a moving signaling wave that sweeps across the tissue from the posterior to the anterior side, that induces proliferating cells anterior to it to differentiate and become cell cycle quiescent in its wake. Therefore, final eye disc size depends on the proliferation rate of undifferentiated cells and on the speed with which the MF sweeps across the eye disc. We developed a spatio-temporal model of the growing eye disc based on the regulatory interactions controlled by the signals Decapentaplegic (Dpp), Hedgehog (Hh) and the transcription factor Homothorax (Hth) and explored how the signaling patterns affect the movement of the MF and impact on eye disc growth. We used published and new quantitative data to parameterize the model. In particular, two crucial parameter values, the degradation rate of Hth and the diffusion coefficient of Hh, were measured. The model is able to reproduce the linear movement of the MF and the termination of growth of the primordium. We further show that the model can explain several mutant phenotypes, but fails to reproduce the previously observed scaling of the Dpp gradient in the anterior compartment.

DYNAMICS OF GLI REGULATION AND A STRATEGY TO CONTROL CANCEROUS SITUATION: HEDGEHOG SIGNALING PATHWAY REVISITED

The hedgehog signaling cascade generates highly diverse, fine-tuned responses in response to the external stimulus by the sonic hedgehog (SHH) protein. This is required for the flawless functioning of the cell, its development, survival and proliferation; maintained through production of Glioma protein (GLI) and transcriptional activation of its target genes. Any change in the behavior of GLI response by ectopic expression of SHH or mutations in the core pathway components may cause serious consequences in the cell fate through rapid, uncontrolled and elevated production of GLI. Here, we present a simple but extensive computational model that considers the detailed reaction mechanisms involved in the hedgehog signal transduction and provides a detailed insight into regulation of GLI. For the first time, by explicit involvement of suppressor of fused (SUFU) and Hedgehog interacting protein (HHIP) reaction kinetics in the model, we try to demonstrate the vital importance of HHIP and SUFU in maintaining the graded response of GLI in response to SHH. By performing parameter variations, we capture the conversion of a graded response of GLI to an ultrasensitive switch under SUFU-deficient conditions that might predispose abnormal embryonic development and the irreversible switching response of GLI that corresponds to signal-independent pathway activation observed in cancers.

Simulating tissue morphogenesis and signaling

During embryonic development tissue morphogenesis and signaling are tightly coupled. It is therefore important to simulate both tissue morphogenesis and signaling simultaneously in in silico models of developmental processes. The resolution of the processes depends on the questions of interest. As part of this chapter we introduce different descriptions of tissue morphogenesi s. In the simplest approximation tissue is a continuous domain and tissue expansion is described according to a predefined function of time (and possibly space). In a slightly more advanced version the expansion speed and direction of the tissue may depend on a signaling variable that evolves on the domain. Both versions will be referred to as "prescribed growth." Alternatively tissue can be regarded as incompressible fluid and can be described with Navier-Stokes equations. Local cell expansion, proliferation, and death are then incorporated by a source term. In other applications the cell boundaries may be important and cell-based models must be introduced. Finally, cells may move within the tissue, a process best described by agent-based models.

Mathematical modelling of digit specification by a sonic hedgehog gradient

BACKGROUND

The three chick wing digits represent a classical example of a pattern specified by a morphogen gradient. Here we have investigated whether a mathematical model of a Shh gradient can describe the specification of the identities of the three chick wing digits and if it can be applied to limbs with more digits.

RESULTS

We have produced a mathematical model for specification of chick wing digit identities by a Shh gradient that can be extended to the four digits of the chick leg with Shh-producing cells forming a digit. This model cannot be extended to specify the five digits of the mouse limb.

CONCLUSIONS

Our data suggest that the parameters of a classical-type morphogen gradient are sufficient to specify the identities of three different digits. However, to specify more digit identities, this core mechanism has to be coupled to alternative processes, one being that in the chick leg and mouse limb, Shh-producing cells give rise to digits; another that in the mouse limb, the cellular response to the Shh gradient adapts over time so that digit specification does not depend simply on Shh concentration.

Inter-dependent tissue growth and Turing patterning in a model for long bone development

The development of long bones requires a sophisticated spatial organization of cellular signalling, proliferation, and differentiation programs. How such spatial organization emerges on the growing long bone domain is still unresolved. Based on the reported biochemical interactions we developed a regulatory model for the core signalling factors IHH, PTCH1, and PTHrP and included two cell types, proliferating/resting chondrocytes and (pre-)hypertrophic chondrocytes. We show that the reported IHH-PTCH1 interaction gives rise to a Schnakenberg-type Turing kinetics, and that inclusion of PTHrP is important to achieve robust patterning when coupling patterning and tissue dynamics. The model reproduces relevant spatiotemporal gene expression patterns, as well as a number of relevant mutant phenotypes. In summary, we propose that a ligand-receptor based Turing mechanism may control the emergence of patterns during long bone development, with PTHrP as an important mediator to confer patterning robustness when the sensitive Turing system is coupled to the dynamics of a growing and differentiating tissue. We have previously shown that ligand-receptor based Turing mechanisms can also result from BMP-receptor, SHH-receptor, and GDNF-receptor interactions, and that these reproduce the wildtype and mutant patterns during digit formation in limbs and branching morphogenesis in lung and kidneys. Receptor-ligand interactions may thus constitute a general mechanism to generate Turing patterns in nature.

Structural and logical analysis of a comprehensive hedgehog signaling pathway to identify alternative drug targets for glioma, colon and pancreatic cancer

Hedgehog is an evolutionarily conserved developmental pathway, widely implicated in controlling various cellular responses such as cellular proliferation and stem cell renewal in human and other organisms, through external stimuli. Aberrant activation of this pathway in human adult stem cell line may cause different types of cancers. Hence, targeting this pathway in cancer therapy has become indispensable, but the non availability of detailed molecular interactions, complex regulations by extra- and intra-cellular proteins and cross talks with other pathways pose a serious challenge to get a coherent understanding of this signaling pathway for making therapeutic strategy. This motivated us to perform a computational study of the pathway and to identify probable drug targets. In this work, from available databases and literature, we reconstructed a complete hedgehog pathway which reports the largest number of molecules and interactions to date. Using recently developed computational techniques, we further performed structural and logical analysis of this pathway. In structural analysis, the connectivity and centrality parameters were calculated to identify the important proteins from the network. To capture the regulations of the molecules, we developed a master Boolean model of all the interactions between the proteins and created different cancer scenarios, such as Glioma, Colon and Pancreatic. We performed perturbation analysis on these cancer conditions to identify the important and minimal combinations of proteins that can be used as drug targets. From our study we observed the under expressions of various oncoproteins in Hedgehog pathway while perturbing at a time the combinations of the proteins GLI1, GLI2 and SMO in Glioma; SMO, HFU, ULK3 and RAS in Colon cancer; SMO, HFU, ULK3, RAS and ERK12 in Pancreatic cancer. This reconstructed Hedgehog signaling pathway and the computational analysis for identifying new combinatory drug targets will be useful for future in-vitro and in-vivo analysis to control different cancers.

Making sense-data-based simulations of vertebrate limb development

Limb bud development has been studied for decades and contributed a wealth of knowledge to our understanding of the molecular and cellular mechanisms that govern organogenesis in vertebrate embryos. However, the general regulatory paradigms that underlie the functional and structural organization of complex systems such as developing limb buds have remained largely elusive. A significant number of mathematical theories have been proposed to explain these developmental processes, but have rarely been validated by experimental analysis. In the age of systems biology, experimental and mathematical approaches have become interlinked and enable the experimental validation of computational models by molecular and genetic analysis. This in turn allows refinement of the mathematical simulations such that simulating limb bud development becomes increasingly more realistic. The resulting models not only detect inconsistencies in the interpretation of experimental data, but their predictive power facilitates identification of key regulatory interactions and definition of so-called core and accessory mechanisms. The ongoing integrative analysis of vertebrate limb organogenesis indicates that these network simulations may be suitable for in silico genetics, that is the computational modeling of complex loss-of-functions and gain-of-functions states. Such in silico genetic approaches will permit the simulation of complex mutant phenotypes tedious or impossible to generate using mouse molecular genetics.

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.

The importance of geometry in mathematical models of developing systems

Understanding the interaction between the spatial variation of extracellular signals and the interpretation of such signals in embryonic development is difficult without a mathematical model, but the inherent limitations of a model can have a profound impact on its utility. A central issue is the level of abstraction needed, and here we focus on the role of geometry in models and how the choice of the spatial dimension can influence the conclusions reached. A widely studied system in which the proper choice of geometry is critical is embryonic development of Drosophila melanogaster, and we discuss recent work in which 3D embryo-scale modeling is used to identify key modes of transport, analyze gap gene expression, and test BMP-mediated positive feedback mechanisms.

Computational and mathematical models of chondrogenesis in vertebrate limbs

The production of cartilage (chondrogenic patterning) in the limb is one of the best-studied examples of the emergence of form in developmental biology. At the core of the theoretical study is an effort to understand the mechanism that establishes the characteristic distribution of cartilage in the embryonic limb, which defines the future sites and shapes of bones that will be present in the mature limb. This review article gives an overview of the history and current state of a rich literature of mathematical and computational models that seek to contribute to this problem. We describe models for the mechanisms of limb growth and shaping via interaction with various chemical fields, as well as models addressing the intrinsic self-organization capabilities of the embryonic mesenchymal tissue, such as reaction-diffusion and mechanochemical models. We discuss the contributions of these models to the current understanding of chondrogenesis in vertebrate limbs, as well as their relation to the varied conceptual models that have been proposed by experimentalists.

The effect of the signalling scheme on the robustness of pattern formation in development

Pattern formation in development is a complex process which involves spatially distributed signals called morphogens that influence gene expression and thus the phenotypic identity of cells. Usually different cell types are spatially segregated, and the boundary between them may be determined by a threshold value of some state variable. The question arises as to how sensitive the location of such a boundary is to variations in properties, such as parameter values, that characterize the system. Here, we analyse both deterministic and stochastic reaction-diffusion models of pattern formation with a view towards understanding how the signalling scheme used for patterning affects the variability of boundary determination between cell types in a developing tissue.

The Sonic hedgehog gradient in the developing limb

A gradient of Sonic hedgehog (Shh) plays a major role in specifying the antero-posterior pattern of structures that develop in the distal part of the vertebrate limb, in particular, the antero-posterior pattern of the digits. Classical embryological experiments identified the polarizing region (or zone of polarizing activity, ZPA), a signaling region at the posterior margin of the early chick wing bud and, consistent with a model in which production of a diffusible morphogen specifies antero-posterior positional information, polarizing region signaling was shown to be dose dependent and long range. It is now well established that the vertebrate hedgehog gene, Sonic hedgehog (Shh), which encodes a secreted protein, is expressed in the polarizing region of the chick wing and that Shh signaling has the same characteristics as polarizing region signaling. Shh expression at the posterior of the early limb bud and the mechanism of Shh signal transduction are conserved among vertebrates including mammals. However, it is unlikely that a simple Shh gradient is responsible for digit pattern formation in mammalian limbs and there is still little understanding of how positional information specified by Shh signaling is encoded and translated into digit anatomy. Alterations in Shh signaling underlie some congenital limb abnormalities and also changes in timing and extent of Shh signaling appear to be related to the evolution of morphological diversity of vertebrate limbs.

Power spectra methods for a stochastic description of diffusion on deterministically growing domains

A central challenge in developmental biology is understanding the creation of robust spatiotemporal heterogeneity. Generally, the mathematical treatments of biological systems have used continuum, mean-field hypotheses for their constituent parts, which ignores any sources of intrinsic stochastic effects. In this paper we consider a stochastic space-jump process as a description of diffusion, i.e., particles are able to undergo a random walk on a discretized domain. By developing analytical Fourier methods we are able to probe this probabilistic framework, which gives us insight into the patterning potential of diffusive systems. Further, an alternative description of domain growth is introduced, with which we are able to rigorously link the mean-field and stochastic descriptions. Finally, through combining these ideas, it is shown that such stochastic descriptions of diffusion on a deterministically growing domain are able to support the nucleation of states that are far removed from the deterministic mean-field steady state.

Hedgehog signaling induced by breast cancer cells promotes osteoclastogenesis and osteolysis

Bone integrity is maintained by a dynamic equilibrium between the activities of bone-forming osteoblasts and bone-resorbing osteoclasts. Osteolytic lesions are a painful consequence of metastasis of breast cancer cells to bone in an overwhelming majority of breast cancer patients. Factors secreted by breast cancer cells propel a cascade of events that trigger osteoclastogenesis and elevated bone resorption. In the present study, we show that the Hedgehog (Hh) ligands secreted by breast cancer cells promote osteoclast differentiation and potentiate the activity of mature osteoclasts. Paracrine Hh signaling induced by breast cancer cells mediates a detrimental chain of events by the up-regulation of osteopontin (OPN), which in turn enhances osteoclastic activity by up-regulating cathepsin K and MMP9. Hh signaling is essential for osteoclasts because blocking the Hh pathway using the pharmacological Hh inhibitor, cyclopamine, results in an overall decrease in osteoclastogenesis and resorptive activity. Our studies suggest that inhibiting Hh signaling interferes with the ability of pre-osteoclasts to respond to the stimulatory effects of the breast cancer cells, indicating that Hh signaling is vital to osteoclast activity.

Analysis of stationary droplets in a generic Turing reaction-diffusion system

Solitonlike structures called "droplets" are found to exist within a paradigm reaction-diffusion model that can be used to describe patterning in a number of biological systems, for example, on the skin of various fish species. They have also been found in many other systems that can be modeled with a complex Ginzburg-Landau system. These droplets can be analyzed in the biological paradigm model because the system has two nonzero stable steady states that are symmetric; however, the asymmetric case is more challenging. We first review the properties of the paradigm system and then extend a recently developed perturbation technique [D. Gomila, J. Opt. B: Quantum Semiclassical Opt. 6, S265 (2004)] to investigate the weakly asymmetric case. We compare the results of our mathematical analysis with numerical simulations and show good agreement in the region where the assumptions hold.

Building a morphogen gradient without diffusion in a growing tissue

In many developmental systems, spatial pattern arises from morphogen gradients, which provide positional information for cells to determine their fate. Typically, diffusion is thought to be the mechanism responsible for building a morphogen gradient. An alternative mechanism is investigated here. Using mathematical modeling, we demonstrate how a non-diffusive morphogen concentration gradient can develop in axially growing tissue systems, where growth is due to cell proliferation only. Two distinct cases are considered: in the first, all cell proliferation occurs in a localized zone where active transcription of a morphogen-producing gene occurs, and in the second, cell proliferation is uniformly distributed throughout the tissue, occurring in both the active transcription zone and beyond. A cell containing morphogen mRNA produces the morphogen protein, hence any gradient in mRNA transcripts translates into a corresponding morphogen protein gradient. Proliferation-driven growth gives rise to both advection (the transport term) and dilution (a reaction term). These two key mechanisms determine the resultant mRNA transcript distribution. Using the full range of uniform initial conditions, we show that advection and dilution due to cell proliferation are, in general, sufficient for morphogen gradient formation for both types of axially growing systems. In particular, mRNA transcript degradation is not necessary for gradient formation; it is only necessary with localized proliferation for one special value of the initial concentration. Furthermore, the morphogen concentration decreases with distance away from the transcription zone, except in the case of localized proliferation with the initial concentration sufficiently large, when the concentration can either increase with distance from the transcription zone or sustain a local minimum. In both localized and uniformly distributed proliferation, in order for a concentration gradient to form across the whole domain, transcription must occur in a zone equal to the initial domain size; otherwise, it will only form across part of the tissue.

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.

Robustness of positional specification by the Hedgehog morphogen gradient

Spatial gradients of Hedgehog signalling play a central role in many patterning events during animal development, regulating cell fate determination and tissue growth in a variety of tissues and developmental stages. Experimental evidence suggests that many of the proteins responsible for regulating Hedgehog signalling and transport are themselves targets of Hedgehog signalling, leading to multiple levels of feedback within the system. We use mathematical modelling to analyse how these overlapping feedbacks combine to regulate patterning and potentially enhance robustness in the Drosophila wing imaginal disc. Our results predict that the regulation of Hedgehog transport and stability by glypicans, as well as multiple overlapping feedbacks in the Hedgehog response network, can combine to enhance the robustness of positional specification against variability in Hedgehog levels. We also discuss potential trade-offs between robustness and additional features of the Hedgehog gradient, such as signalling range and size regulation.

The Hedgehog signalling pathway as a therapeutic target in early breast cancer development

The Hedgehog (Hh) signalling pathway is a highly conserved developmental pathway, which plays critical roles in patterning of the embryo through epithelial to mesenchymal signalling and the maintenance of stem cells in the adult organism. There is increasing evidence that this pathway is dysregulated in many malignancies, including breast cancer. While there has been a significant decrease in mortality from breast cancer, a number of treatment challenges remain, particularly in those tumours which develop resistance to endocrine-based therapy, or which lack expression of hormone or c-erbB2/HER2 receptors. Therapeutic manipulation of the Hh pathway as a potential cancer therapy is attracting great interest, with preclinical studies and clinical trials underway in a range of malignancies. This review highlights important recent developments that affect the potential of the Hh pathway as a novel therapeutic target in early breast cancer.

Point mutation of Hoxd12 in mice

PURPOSE

Genes of the HoxD cluster play a major role in vertebrate limb development, and changes that modify the Hoxd12 locus affect other genes also, suggesting that HoxD function is coordinated by a control mechanism involving multiple genes during limb morphogenesis. In this study, mutant phenotypes were produced by treatment of mice with a chemical mutagen, N-ethyl-N-nitrosourea (ENU). We analyzed mutant mice exhibiting the specific microdactyly phenotype and examined the genes affected.

MATERIALS AND METHODS

We focused on phenotype characteristics including size, bone formation, and digit morphology of ENU-induced microdactyly mice. The expressions of several molecules were analyzed by genome-wide screening and quantitative real-time PCR to define the affected genes.

RESULTS

We report on limb phenotypes of an ENU-induced A-to-C mutation in the Hoxd12 gene, resulting in alanine-to-serine conversion. Microdactyly mice exhibited growth defects in the zeugopod and autopod, shortening of digits, a missing tip of digit I, limb growth affected, and dramatic increases in the expressions of Fgf4 and Lmx1b. However, the expression level of Shh was not changed in Hoxd12 point mutated mice.

CONCLUSION

These results suggest that point mutation rather than the entire deletion of Hoxd12, such as in knockout and transgenic mice, causes the abnormal limb phenotype in microdactyly mice. The precise nature of the spectrum of differences requires further investigation.

Identification of novel gene expression in healing fracture callus tissue by DNA microarray

Fracture healing requires controlled expression of thousands of genes. Only a small fraction of these genes have been isolated and fewer yet have been shown to play a direct role in fracture healing. The purpose of this study was threefold: (1) to develop a reproducible open femur model of fracture healing that produces consistent fracture calluses for subsequent RNA extraction, (2) to use this model to determine temporal expression patterns of known and unknown genes using DNA microarray expression profiling, and (3) to identify and validate novel gene expression in fracture healing. In the initial arm of the study, a total of 56 wild-type C57BL/6 mice were used. An open, stabilized diaphyseal femur fracture was created. Animals were killed at 1, 5, 7, 10, 14, 21, and 35 days after surgery and the femurs were harvested for analysis. At each time point, fractures were radiographed and sectioned for histologic analyses. Tissue from fracture callus at all stages following fracture yielded reproducibly large amounts of mRNA. Expression profiling revealed that genes cluster by function in a manner similar to the histologic stages of fracture healing. Based on the expression profiling of fracture tissue, temporal expression patterns of several genes known to be involved in fracture healing were verified. Novel expression of multiple genes in fracture callous tissue was also revealed including leptin and leptin receptor. In order to test whether leptin signaling is required for fracture repair, mice deficient in leptin or its receptor were fractured using the same model. Fracture calluses of mice deficient in both leptin or leptin receptor are larger than wild-type mice fractures, likely due to a delay in mineralization, revealing a previously unrecognized role of leptin signaling in fracture healing. This novel model of murine fracture repair is useful in examining both global changes in gene expression as well as individual signaling pathways, which can be used to identify specific molecular mechanisms of fracture healing.

Tooth morphogenesis in vivo, in vitro, and in silico

One of the aims of developmental biology is to understand how a single egg cell gives rise to the complex spatial distributions of cell types and extracellular components of the adult phenotype. This review discusses the main genetic and epigenetic interactions known to play a role in tooth development and how they can be integrated into coherent models. Along the same lines, several hypotheses about aspects of tooth development that are currently not well understood are evaluated. This is done from their morphological consequences from the model and how these fit known morphological variation and change during tooth development. Thus the aim of this review is two-fold. On one hand the model and its comparison with experimental evidence will be used to outline our current understanding about tooth morphogenesis. On the other hand these same comparisons will be used to introduce a computational model that makes accurate predictions on three-dimensional morphology and patterns of gene expression by implementing cell signaling, proliferation and mechanical interactions between cells. In comparison with many other models of development this model includes reaction-diffusion-like dynamics confined to a diffusion chamber (the developing tooth) that changes in shape in three-dimensions over time. These changes are due to mechanical interactions between cells triggered by the proliferation enhancing effect of the reactants (growth factors). In general, tooth morphogenesis can be understood from the indirect cross-regulation between extracellular signals, the local regulation of proliferation and differentiation rates by these signals and the effect of intermediate developing morphology on the diffusion, dilution, and spatial distribution of these signals. Overall, this review should be interesting to either readers interested in the mechanistic bases of tooth morphogenesis, without necessarily being interested in modeling per se, and readers interested in development modeling in general.

Biology by numbers: mathematical modelling in developmental biology

In recent years, mathematical modelling of developmental processes has earned new respect. Not only have mathematical models been used to validate hypotheses made from experimental data, but designing and testing these models has led to testable experimental predictions. There are now impressive cases in which mathematical models have provided fresh insight into biological systems, by suggesting, for example, how connections between local interactions among system components relate to their wider biological effects. By examining three developmental processes and corresponding mathematical models, this Review addresses the potential of mathematical modelling to help understand development.

Spatially patterned gene delivery for localized neuron survival and neurite extension

Natural tissues can have complex architectures, which arise in part from spatial patterns in gene expression. Regenerative strategies for damaged tissue must recreate these architectures to restore function. In this article, we demonstrate spatially controlled gene delivery from a substrate for directing cellular processes. Non-viral vectors were immobilized to substrates in linear patterns using microfluidic techniques, and cells cultured on the surface had localized gene expression within the cell population. Transfection was achieved in pattern widths as low as 100 mum, with efficiencies dependent on the microchannel treatment and vector concentration. The ability of patterned expression to localize cellular processes was investigated using a neuronal co-culture model. Patterned expression of the diffusible neurotrophic factor nerve growth factor initiated neuron survival and neurite out-growth primarily within the pattern, which decreased significantly in regions directly adjacent to the pattern. Primary neurite density was significantly greater on patterned substrates than on surfaces without patterns. This approach demonstrates the basic technology to create patterns of gene expression that can direct tissue formation and could be employed in regenerative strategies to recreate the complex cellular architectures observed in tissues.

Spatially patterned gene delivery for localized neuron survival and neurite extension

Natural tissues can have complex architectures, which arise in part from spatial patterns in gene expression. Regenerative strategies for damaged tissue must recreate these architectures to restore function. In this article, we demonstrate spatially controlled gene delivery from a substrate for directing cellular processes. Non-viral vectors were immobilized to substrates in linear patterns using microfluidic techniques, and cells cultured on the surface had localized gene expression within the cell population. Transfection was achieved in pattern widths as low as 100 mum, with efficiencies dependent on the microchannel treatment and vector concentration. The ability of patterned expression to localize cellular processes was investigated using a neuronal co-culture model. Patterned expression of the diffusible neurotrophic factor nerve growth factor initiated neuron survival and neurite out-growth primarily within the pattern, which decreased significantly in regions directly adjacent to the pattern. Primary neurite density was significantly greater on patterned substrates than on surfaces without patterns. This approach demonstrates the basic technology to create patterns of gene expression that can direct tissue formation and could be employed in regenerative strategies to recreate the complex cellular architectures observed in tissues.

Specification of optic nerve oligodendrocyte precursors by retinal ganglion cell axons

Cell fate commitment in the developing CNS frequently depends on localized cell-cell interactions. In the avian visual system the optic nerve oligodendrocytes are derived from founder cells located at the floor of the third ventricle. Here we show that the induction of these founder cells is directly dependent on signaling from the retinal ganglion cell (RGC) axons. The appearance of oligodendrocyte precursor cells (OPCs) correlates with the projection of RGC axons, and early eye removal dramatically reduces the number of OPCs. In vitro signaling from retinal neurites induces OPCs in responsive tissue. Retinal axon induction of OPCs is dependent on sonic hedgehog (Shh) and neuregulin signaling, and the inhibition of either signal reduces OPC induction in vivo and in vitro. The dependence of OPCs on retinal axonal cues appears to be a common phenomenon, because ocular retardation (or(J)) mice lacking optic nerve have dramatically reduced OPCs in the midline of the third ventricle.

Signal dynamics in Sonic hedgehog tissue patterning

During development, secreted signaling factors, called morphogens, instruct cells to adopt specific mature phenotypes. However, the mechanisms that morphogen systems employ to establish a precise concentration gradient for patterning tissue architecture are highly complex and are typically analyzed only at long times after secretion (i.e. steady state). We have developed a theoretical model that analyzes dynamically how the intricate transport and signal transduction mechanisms of a model morphogen, Sonic hedgehog (Shh),cooperate in modular fashion to regulate tissue patterning in the neural tube. Consistent with numerous recent studies, the model elucidates how the dynamics of gradient formation can be a key determinant of cell response. In addition,this work yields several novel insights into how different transport mechanisms or `modules' control pattern formation. The model predicts that slowing the transport of a morphogen, such as by lipid modification of the ligand Shh, by ligand binding to proteoglycans, or by the moderate upregulation of dedicated transport molecules like Dispatched, can actually increase the signaling range of the morphogen by concentrating it near the secretion source. Furthermore, several transcriptional targets of Shh, such as Patched and Hedgehog-interacting protein, significantly limit its signaling range by slowing transport and promoting ligand degradation. This modeling approach elucidates how individual modular elements that operate dynamically at various times during patterning can shape a tissue pattern.

Incredible journey: how do developmental signals travel through tissue?

How developmental signaling proteins traverse tissue during animal development, through or around tightly packed cells, remains an incompletely resolved mystery. Signaling protein movement is regulated to create gradients, control amounts, impose barriers, or provide direction. Signaling can be controlled by the rate of signal production, modification, active transport, trapping along the path, or by the properties of the receptor apparatus. Signals may move by diffusion outside cells, attached to migrating cells, attached to carrier molecules, through cells by transcytosis, along cell extensions, or in released membrane packets. Recent findings about the movement of Hedgehog, Wingless (Wnt), and TGF-beta signaling proteins have helped to clarify the molecular mechanisms used to ensure that developmental signals carry only good news.

Organogenesis of the exocrine gland

Morphogenesis of exocrine glands is a complex stepwise process of epithelial ingrowth, ductal elongation, ductal branching, and alveolar or acinar differentiation. Emerging from an increasing number of mouse gene knockout, dominant-negative, and antisense models is the identification of a remarkable collection of cell adhesion molecules, growth factors, and their receptors whose time-dependent contributions to glandular organogenesis are essential. Many have cryptically overlapping and interdependent but noncompensatory roles. Discoidin domain receptor 1 tyrosine kinase (DDR1) and the ErbB1 receptor of amphiregulin are, for example, required for ductal branching and elongation. Each is in turn dependent on the Wnt family of morphogenic factors for autophosphorylation or transactivation, respectively. Here we review the current cast of exocrine glandular morphogens, as a foundation for a global or systems biology appreciation of the interweaving signaling pathways that underlie mammalian glandular morphogenesis.

Interplay between activator-inhibitor coupling and cell-matrix adhesion in a cellular automaton model for chondrogenic patterning

We present a stochastic cellular automaton model for the behavior of limb bud precartilage mesenchymal cells undergoing chondrogenic patterning. This "agent-oriented" model represents cells by points on a lattice that obey rules motivated by experimental findings. The "cells" follow these rules as autonomous agents, interacting with other cells and with the microenvironments cell activities produce. The rules include random cell motion, production and lateral deposition of a substrate adhesion molecule (SAM, corresponding to fibronectin), production and release of a diffusible growth factor ("activator," corresponding to TGF-beta) that stimulates production of the SAM, and another diffusible factor ("inhibitor") that suppresses the activity of the activator. We implemented the cellular automaton on a two-dimensional (2D) square lattice to emulate the quasi-2D micromass culture extensively used to study patterning in avian limb bud precartilage cells. We identified parameters that produce nodular patterns that resemble, in size and distribution, cell condensations in leg-cell cultures, thus establishing a correspondence between in vitro and in silico results. We then studied the in vitro and in silico micromass cultures experimentally. We altered the standard in vitro micromass culture by diluting the initial cell density, transiently exposing it to exogenous activator, suppressing the inhibitor, and constitutively activating fibronectin production. We altered the standard in silico micromass culture in each case by changing the corresponding parameter. In vitro and in silico experiments agreed well. We also used the model to test hypotheses for differences in the in vitro patterns of cells derived from chick embryo forelimb and hindlimb. We discuss the applicability of this model to limb development in vivo and to other organ development.

A Dual Role for Hox Genes in Limb Anterior-Posterior Asymmetry

Anterior-to-posterior patterning, the process whereby our digits are differently shaped, is a key aspect of limb development. It depends on the localized expression in posterior limb bud of Sonic hedgehog (Shh) and the morphogenetic potential of its diffusing product. By using an inversion of and a large deficiency in the mouse HoxD cluster, we found that a perturbation in the early collinear expression of Hoxd11, Hoxd12, and Hoxd13 in limb buds led to a loss of asymmetry. Ectopic Hox gene expression triggered abnormal Shh transcription, which in turn induced symmetrical expression of Hox genes in digits, thereby generating double posterior limbs. We conclude that early posterior restriction of Hox gene products sets up an anterior-posterior prepattern, which determines the localized activation of Shh. This signal is subsequently translated into digit morphological asymmetry by promoting the late expression of Hoxd genes, two collinear processes relying on opposite genomic topographies, upstream and downstream Shh signaling.