A cell cycle model for somitogenesis: mathematical formulation and numerical simulation

Spatiotemporal control of pattern formation during somitogenesis

Spatiotemporal patterns widely occur in biological, chemical, and physical systems. Particularly, embryonic development displays a diverse gamut of repetitive patterns established in many tissues and organs. Branching treelike structures in lungs, kidneys, livers, pancreases, and mammary glands as well as digits and bones in appendages, teeth, and palates are just a few examples. A fascinating instance of repetitive patterning is the sequential segmentation of the primary body axis, which is conserved in all vertebrates and many arthropods and annelids. In these species, the body axis elongates at the posterior end of the embryo containing an unsegmented tissue. Meanwhile, segments sequentially bud off from the anterior end of the unsegmented tissue, laying down an exquisite repetitive pattern and creating a segmented body plan. In vertebrates, the paraxial mesoderm is sequentially divided into somites. In this review, we will discuss the most prominent models, the most puzzling experimental data, and outstanding questions in vertebrate somite segmentation.

The Clockwork Embryo: Mechanisms Regulating Developmental Rate

Organismal development requires the reproducible unfolding of an ordered sequence of discrete steps (cell fate determination, migration, tissue folding, etc.) in both time and space. Here, we review the mechanisms that grant temporal specificity to developmental steps, including molecular clocks and timers. Individual timing mechanisms must be coordinated with each other to maintain the overall developmental sequence. However, phenotypic novelties can also arise through the modification of temporal patterns over the course of evolution. Two main types of variation in temporal patterning characterize interspecies differences in developmental time: allochrony, where the overall developmental sequence is either accelerated or slowed down while maintaining the relative duration of individual steps, and heterochrony, where the duration of specific developmental steps is altered relative to the rest. New advances in in vitro modeling of mammalian development using stem cells have recently enabled the revival of mechanistic studies of allochrony and heterochrony. In both cases, differences in the rate of basic cellular functions such as splicing, translation, protein degradation, and metabolism seem to underlie differences in developmental time. In the coming years, these studies should identify the genetic differences that drive divergence in developmental time between species.

Sequential changes in cellular properties accompanying amniote somite formation

Somites are transient structures derived from the pre-somitic mesoderm (PSM), involving mesenchyme-to-epithelial transition (MET) where the cells change their shape and polarize. Using Scanning electron microscopy (SEM), immunocytochemistry and confocal microscopy, we study the progression of these events along the tail-to-head axis of the embryo, which mirrors the progression of somitogenesis (younger cells located more caudally). SEM revealed that PSM epithelialization is a gradual process, which begins much earlier than previously thought, starting with the dorsalmost cells, then the medial ones, and then, simultaneously, the ventral and lateral cells, before a somite fully separates from the PSM. The core (internal) cells of the PSM and somites never epithelialize, which suggests that the core cells could be 'trapped' within the somitocoele after cells at the surfaces of the PSM undergo MET. Three-dimensional imaging of the distribution of the cell polarity markers PKCζ, PAR3, ZO1, the Golgi marker GM130 and the apical marker N-cadherin reveal that the pattern of polarization is distinctive for each marker and for each surface of the PSM, but the order of these events is not the same as the progression of cell elongation. These observations challenge some assumptions underlying existing models of somite formation.

Cellular aspects of somite formation in vertebrates

Vertebrate segmentation, the process that generates a regular arrangement of somites and thereby establishes the pattern of the adult body and of the musculoskeletal and peripheral nervous systems, was noticed many centuries ago. In the last few decades, there has been renewed interest in the process and especially in the molecular mechanisms that might account for its regularity and other spatial-temporal properties. Several models have been proposed but surprisingly, most of these do not provide clear links between the molecular mechanisms and the cell behaviours that generate the segmental pattern. Here we present a short survey of our current knowledge about the cellular aspects of vertebrate segmentation and the similarities and differences between different vertebrate groups in how they achieve their metameric pattern. Taking these variations into account should help to assess each of the models more appropriately.

A mechanical model of early somite segmentation

Somitogenesis is often described using the clock-and-wavefront (CW) model, which does not explain how molecular signaling rearranges the pre-somitic mesoderm (PSM) cells into somites. Our scanning electron microscopy analysis of chicken embryos reveals a caudally-progressing epithelialization front in the dorsal PSM that precedes somite formation. Signs of apical constriction and tissue segmentation appear in this layer 3-4 somite lengths caudal to the last-formed somite. We propose a mechanical instability model in which a steady increase of apical contractility leads to periodic failure of adhesion junctions within the dorsal PSM and positions the future inter-somite boundaries. This model produces spatially periodic segments whose size depends on the speed of the activation front of contraction (F), and the buildup rate of contractility (Λ). The Λ/F ratio determines whether this mechanism produces spatially and temporally regular or irregular segments, and whether segment size increases with the front speed.

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Dorsal pre-somitic mesoderm of chicken embryos epithelializes before somite formation

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Dorsal epithelium shows signs of apical constriction and early segmentation

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A mechanical instability model can reproduce sequential segmentation

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A single ratio describes spatial and temporal patterns of segmentation

A Tribute to Lewis Wolpert and His Ideas on the 50th Anniversary of the Publication of His Paper 'Positional Information and the Spatial Pattern of Differentiation'. Evidence for a Timing Mechanism for Setting Up the Vertebrate Anterior-Posterior (A-P) Axis

This article is a tribute to Lewis Wolpert and his ideas on the occasion of the recent 50th anniversary of the publication of his article 'Positional Information and the Spatial Pattern of Differentiation'. This tribute relates to another one of his ideas: his early 'Progress Zone' timing model for limb development. Recent evidence is reviewed showing a mechanism sharing features with this model patterning the main body axis in early vertebrate development. This tribute celebrates the golden era of Developmental Biology.

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.

Some Questions and Answers About the Role of Hox Temporal Collinearity in Vertebrate Axial Patterning

The vertebrate anterior-posterior (A-P = craniocaudal) axis is evidently made by a timing mechanism. Evidence has accumulated that tentatively identifies the A-P timer as being or involving Hox temporal collinearity (TC). Here, I focus on the two current competing models based on this premise. Common features and points of dissent are examined and a common model is distilled from what remains. This is an attempt to make sense of the literature.

Cell cycle regulation of oscillations yields coupling of growth and form in a computational model of the presomitic mesoderm

A striking example of coupling between growth and form occurs during the segmentation of the vertebrate embryo. During segmentation, pairs of segments, one on either side of the anterior-posterior axis, bud off from the presomitic mesoderm (PSM) at regular intervals in time. In the clock and wavefront model, a multicellular oscillator regulates the time at which the next pair of segments form whilst a wavefront regulates their spatial location. In most mathematical models of segmentation, it is assumed that cells in the PSM are oscillators that have a constant natural frequency. Based on recent experimental findings, here we propose a model in which the natural oscillation frequency of each PSM cell is a function of its position in the cell cycle. Given adequate oscillator coupling and that cells in the PSM are randomly distributed in the cell cycle, we find that the emergent oscillator frequency is a weighted average of the constituent oscillator frequencies with the weightings dependent on the fraction of cells in a given cell cycle state. Here, we show that such a model can allow for coupling between pattern formation and growth rate in PSM tissue.

Multiple roles of timing in somite formation

During development, vertebrate embryos produce serially repeated elements, the somites, on each side of the midline. These generate the vertebral column, skeletal musculature and dermis. They form sequentially, one pair at a time, from mesenchymal tissue near the tail. Somite development is a complex process. The embryo must control the number, size, and timing of somite formation, their subdivision into functional regions along three axes, regional identity such that somites develop in a region-specific way, and interactions with neighbouring tissues that coordinate them with nearby structures. Here we discuss many timing-related mechanisms that contribute to set up the spatial pattern.

Chick stem cells: current progress and future prospects

Chick embryonic stem cells (cESCs) can be derived from cells obtained from stage X embryos (blastoderm stage); these have the ability to contribute to all somatic lineages in chimaeras, but not to the germ line. However, lines of stem cells that are able to contribute to the germ line can be established from chick primordial germ cells (cPGCs) and embryonic germ cells (cEGCs). This review provides information on avian stem cells, emphasizing different sources of cells and current methods for derivation and culture of pluripotent cells from chick embryos. We also review technologies for isolation and derivation of chicken germ cells and the production of transgenic birds.

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Chick embryonic stem cells (cESCs) can be derived from a variety of sources.

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cESCs can contribute to all somatic cell types but not to the germ line.

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germ cells can be isolated from early embryos, embryonic blood and gonads.

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germ cells can establish self-renewing lines and contribute to the germline.

Waves and patterning in developmental biology: vertebrate segmentation and feather bud formation as case studies

In this article we will discuss the integration of developmental patterning mechanisms with waves of competency that control the ability of a homogeneous field of cells to react to pattern forming cues and generate spatially heterogeneous patterns. We base our discussion around two well known patterning events that take place in the early embryo: somitogenesis and feather bud formation. We outline mathematical models to describe each patterning mechanism, present the results of numerical simulations and discuss the validity of each model in relation to our example patterning processes.

Computational systems biology of the cell cycle

One of the early success stories of computational systems biology was the work done on cell-cycle regulation. The earliest mathematical descriptions of cell-cycle control evolved into very complex, detailed computational models that describe the regulation of cell division in many different cell types. On the way these models predicted several dynamical properties and unknown components of the system that were later experimentally verified/identified. Still, research on this field is far from over. We need to understand how the core cell-cycle machinery is controlled by internal and external signals, also in yeast cells and in the more complex regulatory networks of higher eukaryotes. Furthermore, there are many computational challenges what we face as new types of data appear thanks to continuing advances in experimental techniques. We have to deal with cell-to-cell variations, revealed by single cell measurements, as well as the tremendous amount of data flowing from high throughput machines. We need new computational concepts and tools to handle these data and develop more detailed, more precise models of cell-cycle regulation in various organisms. Here we review past and present of computational modeling of cell-cycle regulation, and discuss possible future directions of the field.

Traveling wave formation in vertebrate segmentation

In vertebrate somitogenesis, "segmentation clock" genes (such as her in zebrafish, hairy in chick, and hes in mouse) show oscillation, synchronized over nearby cells through cell-cell interaction. The locations of high gene expression appear with regular intervals and move like a wave from posterior to anterior with the speed slowing down toward the anterior end. We analyze traveling wave pattern of her gene expression when there is an anterior-posterior gradient of one of the reaction rates in the gene-protein kinetics. We adopt a model which includes the kinetics of mRNA and proteins of her gene in each cell and cell-cell interaction by Delta-Notch system explicitly. We show that the observed spatio-temporal pattern can be explained if mRNA degradation, protein translation, protein transportation to nucleus occurs faster, or mRNA transcription, Delta protein synthesis occurs slower in posterior than in anterior regions. All of these gradients are those that produce longer periodicity of oscillation of clock gene expression in the anterior than in the posterior. Based on this result, we derive a mathematical formula for how the peak of gene expression moves along the pre-somitic mesoderm.

Adding adhesion to a chemical signaling model for somite formation

Somites are condensations of mesodermal cells that form along the two sides of the neural tube during early vertebrate development. They are one of the first instances of a periodic pattern, and give rise to repeated structures such as the vertebrae. A number of theories for the mechanisms underpinning somite formation have been proposed. For example, in the "clock and wavefront" model (Cooke and Zeeman in J. Theor. Biol. 58:455-476, 1976), a cellular oscillator coupled to a determination wave progressing along the anterior-posterior axis serves to group cells into a presumptive somite. More recently, a chemical signaling model has been developed and analyzed by Maini and coworkers (Collier et al. in J. Theor. Biol. 207:305-316, 2000; Schnell et al. in C. R. Biol. 325:179-189, 2002; McInerney et al. in Math. Med. Biol. 21:85-113, 2004), with equations for two chemical regulators with entrained dynamics. One of the chemicals is identified as a somitic factor, which is assumed to translate into a pattern of cellular aggregations via its effect on cell-cell adhesion. Here, the authors propose an extension to this model that includes an explicit equation for an adhesive cell population. They represent cell adhesion via an integral over the sensing region of the cell, based on a model developed previously for adhesion driven cell sorting (Armstrong et al. in J. Theor. Biol. 243:98-113, 2006). The expanded model is able to reproduce the observed pattern of cellular aggregates, but only under certain parameter restrictions. This provides a fuller understanding of the conditions required for the chemical model to be applicable. Moreover, a further extension of the model to include separate subpopulations of cells is able to reproduce the observed differentiation of the somite into separate anterior and posterior halves.

Somite cell cycle analysis using somite-staging to measure intrinsic developmental time

Somite stages were employed as units of intrinsic developmental time to measure cell doubling rate and other cell cycle parameters of chick forelimb level somites. Somite cell nuclei doubled over an interval corresponding to approximately 7+ somite stages (7+ ss; approximately 11 hr) and approximately 24 new primary myotome cells are born per somite stage ( approximately 16/hr). FACS analysis of DNA content in dissociated paraxial mesoderm cells indicated that slightly more than half are in G1/G0 phase of the cell cycle and that the average combined length of the S phase and G2 phase intervals is approximately 3 ss ( approximately 4.5 hr). A wavefront of increased mitotic nuclei per segment coincident with somite budding potentially reflects a surge in the number of cells entering S phase 3 ss earlier as each PSM segment becomes unresponsive to FGF signaling as it passes through the determination front.

Cell cycle progression is required for zebrafish somite morphogenesis but not segmentation clock function

Cell division, differentiation and morphogenesis are coordinated during embryonic development, and frequently are in disarray in pathologies such as cancer. Here, we present a zebrafish mutant that ceases mitosis at the beginning of gastrulation, but that undergoes axis elongation and develops blood, muscle and a beating heart. We identify the mutation as being in early mitotic inhibitor 1 (emi1), a negative regulator of the Anaphase Promoting Complex, and use the mutant to examine the role of the cell cycle in somitogenesis. The mutant phenotype indicates that axis elongation during the segmentation period is driven substantially by cell migration. We find that the segmentation clock, which regulates somitogenesis, functions normally in the absence of cell cycle progression, and observe that mitosis is a modest source of noise for the clock. Somite morphogenesis involves the epithelialization of the somite border cells around a core of mesenchyme. As in wild-type embryos, somite boundary cells are polarized along a Fibronectin matrix in emi1(-/-). The mutants also display evidence of segment polarity. However, in the absence of a normal cell cycle, somites appear to hyper-epithelialize, as the internal mesenchymal cells exit the core of the somite after initial boundary formation. Thus, cell cycle progression is not required during the segmentation period for segmentation clock function but is necessary for the normal segmental arrangement of epithelial borders and internal mesenchymal cells.

Coordinated action of N-CAM, N-cadherin, EphA4, and ephrinB2 translates genetic prepatterns into structure during somitogenesis in chick

During gastrulation in vertebrates, mesenchymal cells at the anterior end of the presomitic mesoderm (PSM) periodically compact, transiently epithelialize and detach from the posterior PSM to form somites. In the prevailing clock-and-wavefront model of somitogenesis, periodic gene expression, particularly of Notch and Wnt, interacts with an FGF8-based thresholding mechanism to determine cell fates. However, this model does not explain how cell determination and subsequent differentiation translates into somite morphology. In this paper, we use computer simulations of chick somitogenesis to show that experimentally-observed temporal and spatial patterns of adhesive N-CAM and N-cadherin and repulsive EphA4-ephrinB2 pairs suffice to reproduce the complex dynamic morphological changes of somitogenesis in wild-type and N-cadherin (-/-) chick, including intersomitic separation, boundary-shape evolution and sorting of misdifferentiated cells across compartment boundaries. Since different models of determination yield the same, experimentally-observed, distribution of adhesion and repulsion molecules, the patterning is independent of the details of this mechanism.

Mathematical models for somite formation

Somitogenesis is the process of division of the anterior-posterior vertebrate embryonic axis into similar morphological units known as somites. These segments generate the prepattern which guides formation of the vertebrae, ribs and other associated features of the body trunk. In this work, we review and discuss a series of mathematical models which account for different stages of somite formation. We begin by presenting current experimental information and mechanisms explaining somite formation, highlighting features which will be included in the models. For each model we outline the mathematical basis, show results of numerical simulations, discuss their successes and shortcomings and avenues for future exploration. We conclude with a brief discussion of the state of modeling in the field and current challenges which need to be overcome in order to further our understanding in this area.

From segment to somite: segmentation to epithelialization analyzed within quantitative frameworks

One of the most visually striking patterns in the early developing embryo is somite segmentation. Somites form as repeated, periodic structures in pairs along nearly the entire caudal vertebrate axis. The morphological process involves short- and long-range signals that drive cell rearrangements and cell shaping to create discrete, epithelialized segments. Key to developing novel strategies to prevent somite birth defects that involve axial bone and skeletal muscle development is understanding how the molecular choreography is coordinated across multiple spatial scales and in a repeating temporal manner. Mathematical models have emerged as useful tools to integrate spatiotemporal data and simulate model mechanisms to provide unique insights into somite pattern formation. In this short review, we present two quantitative frameworks that address the morphogenesis from segment to somite and discuss recent data of segmentation and epithelialization.

The genetics and embryology of zebrafish metamerism

Somites are the most obvious metameric structures in the vertebrate embryo. They are mesodermal segments that form in bilateral pairs flanking the notochord and are created sequentially in an anterior to posterior sequence concomitant with the posterior growth of the trunk and tail. Zebrafish somitogenesis is regulated by a clock that causes cells in the presomitic mesoderm (PSM) to undergo cyclical activation and repression of several notch pathway genes. Coordinated oscillation among neighboring cells manifests as stripes of gene expression that pass through the cells of the PSM in a posterior to anterior direction. As axial growth continually adds new cells to the posterior tail bud, cells of the PSM become relatively less posterior. This gradual assumption of a more anterior position occurs over developmental time and constitutes part of a maturation process that governs morphological segmentation in conjunction with the clock. Segment morphogenesis involves a mesenchymal to epithelial transition as prospective border cells at the anterior end of the mesenchymal PSM adopt a polarized, columnar morphology and surround a mesenchymal core of cells. The segmental pattern influences the development of the somite derivatives such as the myotome, and the myotome reciprocates to affect the formation of segment boundaries. While somites appear to be serially homologous, there may be variation in the segmentation mechanism along the body axis. Moreover, whereas the genetic architecture of the zebrafish, mouse, and chick segmentation clocks shares many common elements, there is evidence that the gene networks have undergone independent modification during evolution.

Can tissue surface tension drive somite formation?

The prevailing model of somitogenesis supposes that the presomitic mesoderm is segmented into somites by a clock and wavefront mechanism. During segmentation, mesenchymal cells undergo compaction, followed by a detachment of the presumptive somite from the rest of the presomitic mesoderm and the subsequent morphological changes leading to rounded somites. We investigate the possibility that minimization of tissue surface tension drives the somite sculpting processes. Given the time in which somite formation occurs and the high bulk viscosities of tissues, we find that only small changes in shape and form of tissue typically occur through cell movement driven by tissue surface tension. This is particularly true for somitogenesis in the zebrafish. Hence it is unlikely that such processes are the sole and major driving force behind somite formation. We propose a simple chemotactic mechanism that together with heightened adhesion can account for the morphological changes in the time allotted for somite formation.

Travelling gradients in interacting morphogen systems

Morphogen gradients are well known to play several important roles in development; however the mechanisms underlying the formation and maintenance of these gradients are often not well understood. In this work, we investigate whether the presence of a secondary morphogen can increase the robustness of the primary morphogen gradient to perturbation, thereby providing a more stable mechanism for development. We base our model around the interactions of Fibroblast Growth Factor 8 and retinoic acid, which have been shown to act as morphogens in many developmental systems. In particular, we investigate the formation of opposing gradients of these morphogens along the antero-posterior axis of vertebrate embryos, thereby controlling temporal and spatial aspects of axis segmentation and neuronal differentiation.

Practical Lessons from Theoretical Models about the Somitogenesis

Vertebrae and other mammalian repetitive structures are formed from embryonic organs called somites. Somites arise sequentially from the unsegmented presomitic mesoderm (PSM). In mice, a new bilateral pair of somites arise every two hours from the rostral PSM. On the other hand, cells are added to the caudal side of the PSM due to cell proliferation of the tail bud. Somite formation correlates with cycles of cell-autonomous expression in the PSM of genes like Hes7. Because the somitogenesis is a highly dynamic and coordinated process, this event has been subjected to extensive theoretical modeling. Here, we describe the current understanding about the somitogenesis in mouse embryos with an emphasis on insights gained from computer simulations. It is worth noting that the combination of experiments and computer simulations has uncovered dynamical properties of the somitogenesis clock such as the transcription/translation delays, the half-life and the synchronization mechanism across the PSM. Theoretical models have also been useful to provide predictions and rigorous hypothesis about poorly understood processes such as the mechanisms by which the temporal PSM oscillations are arrested and converted into an spatial pattern. We aim at reviewing this theoretical literature in such a way that experimentalists might appreciate the resulting conclusions.

Coupling cellular oscillators: A mechanism that maintains synchrony against developmental noise in the segmentation clock

A unique feature of vertebrate segmentation is its strict periodicity, which is governed by the segmentation clock consisting of numerous cellular oscillators. These cellular oscillators, driven by a negative-feedback loop of Hairy transcription factor, are linked through Notch-dependent intercellular coupling and display the synchronous expression of clock genes. Combining our transplantation experiments in zebrafish with mathematical simulations, we review how the cellular oscillators maintain synchrony and form a robust system that is resistant to the effects of developmental noise such as stochastic gene expression and active cell proliferation. The accumulated evidence indicates that the segmentation clock behaves as a "coupled oscillators," a mechanism that also underlies the synchronous flashing seen in fireflies.

Transient block of receptor may be a mechanism controlling unidirectional propagation of signaling

In tissue development, juxtacrine signaling often propagates across cells, carrying and delivering temporal and spatial information for cells to make correct patterning. Observed complex and accurate tissue patterning indicates that signaling propagation via ligand-receptor interactions is precisely controlled. It is important and interesting to reveal the possible control mechanisms. The directionality of signaling in cells, which is a common issue for all intercellular signaling pathways, is a critical aspect. To understand the propagation of Notch signaling in presomitic mesoderm cells in the mouse, a novel method is used to build a multicellular model to simulate Notch signaling. Simulation reveals that the transient block of Notch by Notch induced Lfng and the delayed removal of the block by another Notch induced protein Hes7 may explain the observed unidirectional propagation of Notch signaling in these cells. Both mutation in and overexpression of lfng cause the same signaling profile in the tissue, due to the inappropriate timing of Notch signaling block by Lfng. The reverse Notch/Delta signaling quickly develops into reciprocating signaling among cells, causing irregular expression of cyclic genes. Irregular Notch signaling in cells would change their response to the positional information provided by the Fgf8 gradient, resulting in disordered and irregular somite segmentation. As Notch signaling is highly conserved, we hypothesize that the mechanism of controlling unidirectional propagation of signaling in cells by transient receptor block may exist in other tissues and in other vertebrates. Our simulation results also suggest that segmentation clock and unidirectional propagation may be inherently coupled in Notch signaling.

Unraveling the nature of the segmentation clock: Intrinsic disorder of clock proteins and their interaction map

Vertebrate segmentation has been proved to be under a strict temporal control governed by a biological clock, known as the segmentation clock. The present experimental evidence suggests that the segmentation clock initiates and maintains its periodic cycle by the periodic activation or inhibition of the Notch signaling pathway as well as the periodic autoregulation of the cyclic genes themselves. In this paper, we investigate the structural and evolutionary properties of the Notch pathway proteins involved in the mice segmentation clock and computationally identify the interaction map within the Notch signaling pathway. The results of our analysis strongly indicate that most of the pathway proteins are intrinsically disordered and that the mechanism of their interaction likely involves helical molecular recognition elements, short loosely structured segments within disordered regions which are directly involved in protein-protein interactions. Predicted interactions are in agreement with gene knock-out studies available in the literature.

A clock and wavefront mechanism for somite formation

Somitogenesis, the sequential formation of a periodic pattern along the antero-posterior axis of vertebrate embryos, is one of the most obvious examples of the segmental patterning processes that take place during embryogenesis and also one of the major unresolved events in developmental biology. In this article, we develop a mathematical formulation of a new version of the Clock and Wavefront model proposed by Pourquié and co-workers (Dubrulle, J., McGrew, M.J., Pourquié, O., 2001. FGF signalling controls somite boundary position and regulates segmentation clock control of spatiotemporal Hox gene activation. Cell 106, 219-232). Dynamic expression of FGF8 in the presomitic mesoderm constitutes the wavefront of determination which sweeps along the body axis interacting as it moves with the segmentation clock to gate cells into somites. We also show that the model can mimic the anomalies formed when progression of the wavefront is disturbed and make some experimental predictions that can be used to test the hypotheses underlying the model.

A mathematical investigation of a Clock and Wavefront model for somitogenesis

Somites are transient blocks of cells that form sequentially along the antero-posterior axis of vertebrate embryos. They give rise to the vertebrae, ribs and other associated features of the trunk. In this work we develop and analyse a mathematical formulation of a version of the Clock and Wavefront model for somite formation, where the clock controls when the boundaries of the somites form and the wavefront determines where they form. Our analysis indicates that this interaction between a segmentation clock and a wavefront can explain the periodic pattern of somites observed in normal embryos. We can also show that a simplification of the model provides a mechanism for predicting the anomalies resulting from perturbation of the wavefront.

Computational pharmacokinetics during developmental windows of susceptibility

Computational modeling has an increasing role in analyses of biological effects, including how the body handles chemicals (i.e., pharmacokinetics or toxicokinetics) and how the body responds to chemicals (i.e., pharmacodynamics or toxicodynamics). Pharmacokinetic models increasingly describe not just adult humans and animals, but also changes with age and life stage (e.g., pregnancy and fetal exposures, lactational exposures, and childhood growth). Physiologically based pharmacokinetic models provide an important route to estimate the potential changes in internal dose that may occur throughout the life cycle. These models require inputs describing changes in physiology, metabolism, and exposure with age and life stage. A particular challenge exists when the "equivalent" developmental period in the rodents and humans differs (e.g., early postnatal in rats and in utero in humans) such that the "equivalent" window of susceptibility to toxic effects of the chemical may involve substantially different exposures (e.g., lactational versus placental transfer). Pharmacodynamic modeling could similarly address changes with age, but few such models currently exist. The growth of systems biology is anticipated to change this over the coming decade.

A Notch feeling of somite segmentation and beyond

Vertebrate segmentation is manifested during embryonic development as serially repeated units termed somites that give rise to vertebrae, ribs, skeletal muscle and dermis. Many theoretical models including the "clock and wavefront" model have been proposed. There is compelling genetic evidence showing that Notch-Delta signaling is indispensable for somitogenesis. Notch receptor and its target genes, Hairy/E(spl) homologues, are known to be crucial for the ticking of the segmentation clock. Through the work done in mouse, chick, Xenopus and zebrafish, an oscillator operated by cyclical transcriptional activation and delayed negative feedback regulation is emerging as the fundamental mechanism underlying the segmentation clock. Ubiquitin-dependent protein degradation and probably other posttranslational regulations are also required. Fgf8 and Wnt3a gradients are important in positioning somite boundaries and, probably, in coordinating tail growth and segmentation. The circadian clock is another biochemical oscillator, which, similar to the segmentation clock, is operated with a negative transcription-regulated feedback mechanism. While the circadian clock uses a more complicated network of pathways to achieve homeostasis, it appears that the segmentation clock exploits the Notch pathway to achieve both signal generation and synchronization. We also discuss mathematical modeling and future directions in the end.

Cell dynamics during somite boundary formation revealed by time-lapse analysis

We follow somite segmentation in living chick embryos and find that the shaping process is not a simple periodic slicing of tissue blocks but a much more carefully choreographed separation in which the somite pulls apart from the segmental plate. Cells move across the presumptive somite boundary and violate gene expression boundaries thought to correlate with the site of the somite boundary. Similarly, cells do not appear to be preassigned to a given somite as they leave the node. The results offer a detailed picture of somite shaping and provide a spatiotemporal framework for linking gene expression with cell movements.

Intercellular adhesion and cancer invasion: a discrete simulation using the extended Potts model

We develop a discrete model of malignant invasion using a thermodynamic argument. An extension of the Potts model is used to simulate a population of malignant cells experiencing interactions due to both homotypic and heterotypic adhesion while also secreting proteolytic enzymes and experiencing a haptotactic gradient. In this way we investigate the influence of changes in cell-cell adhesion on the invasion process. We demonstrate that the morphology of the invading front is influenced by changes in the adhesiveness parameters, and detail how the invasiveness of the tumour is related to adhesion. We show that cell-cell adhesion has less of an influence on invasion compared with cell-medium adhesion, and that increases in both proteolytic enzyme secretion rate and the coefficient of haptotaxis act in synergy to promote invasion. We extend the simulation by including proliferation, and, following experimental evidence, develop an algorithm for cell division in which the mitotic rate is explicitly related to changes in the relative magnitudes of homotypic and heterotypic adhesiveness. We show that although an increased proliferation rate usually results in an increased depth of invasion into the extracellular matrix, it does not invariably do so, and may, indeed, cause invasiveness to be reduced.

Models for pattern formation in somitogenesis: a marriage of cellular and molecular biology

Somitogenesis, the process by which a bilaterally symmetric pattern of cell aggregations is laid down in a cranio-caudal sequence in early vertebrate development, provides an excellent model study for the coupling of interactions at the molecular and cellular level. Here, we review some of the key experimental results and theoretical models related to this process. We extend a recent chemical pre-pattern model based on the cell cycle Journal of Theoretical Biology 207 (2000) 305-316, by including cell movement and show that the resultant model exhibits the correct spatio-temporal dynamics of cell aggregation. We also postulate a model to account for the recently observed spatio-temporal dynamics at the molecular level.

A cellular oscillator model for periodic pattern formation

In this paper, we present a model for pattern formation in developing organisms that is based on cellular oscillators (CO). An oscillatory process within cells serves as a developmental clock whose period is tightly regulated by cell autonomous or non-autonomous mechanisms. A spatial pattern is generated as a result of an initial temporal ordering of the cell oscillators freezing into spatial order as the clocks slow down and stop at different times or phases in their cycles. We apply a CO model to vertebrate somitogenesis and show that we can reproduce the dynamics of periodic gene expression patterns observed in the pre-somitic mesoderm. We also show how varying somite lengths can be generated with the CO model. We then discuss the model in view of experimental evidence and its relevance to other instances of biological pattern formation, showing its versatility as a pattern generator.

The making of the somite: molecular events in vertebrate segmentation

The reiterated structures of the vertebrate axial skeleton, spinal nervous system and body muscle are based on the metameric structure of somites, which are formed in a dynamic morphogenetic process. Somite segmentation requires the activity of a biochemical oscillator known as the somite-segmentation clock. Although the molecular identity of the clock remains unknown, genetic and experimental evidence has accumulated that indicates how the periodicity of somite formation is generated, how the positions of segment borders are determined, and how the rostrocaudal polarity within somite primordia is generated.

The murine winged helix transcription factors, Foxc1 and Foxc2, are both required for cardiovascular development and somitogenesis

The murine Foxc1/Mf1 and Foxc2/Mfh1 genes encode closely related forkhead/winged helix transcription factors with overlapping expression in the forming somites and head mesoderm and endothelial and mesenchymal cells of the developing heart and blood vessels. Embryos lacking either Foxc1 or Foxc2, and most compound heterozygotes, die pre- or perinatally with similar abnormal phenotypes, including defects in the axial skeleton and cardiovascular system. However, somites and major blood vessels do form. This suggested that the genes have similar, dose-dependent functions, and compensate for each other in the early development of the heart, blood vessels, and somites. In support of this hypothesis, we show here that compound Foxc1; Foxc2 homozygotes die earlier and with much more severe defects than single homozygotes alone. Significantly, they have profound abnormalities in the first and second branchial arches, and the early remodeling of blood vessels. Moreover, they show a complete absence of segmented paraxial mesoderm, including anterior somites. Analysis of compound homozygotes shows that Foxc1 and Foxc2 are both required for transcription in the anterior presomitic mesoderm of paraxis, Mesp1, Mesp2, Hes5, and Notch1, and for the formation of sharp boundaries of Dll1, Lfng, and ephrinB2 expression. We propose that the two genes interact with the Notch signaling pathway and are required for the prepatterning of anterior and posterior domains in the presumptive somites through a putative Notch/Delta/Mesp regulatory loop.

Segmentation and somitogenesis derived from phase dynamics in growing oscillatory media

The formation of spatially repetitive structures along the growth axis of a developing embryo is a common theme in developmental biology. Here we apply the novel flow-distributed oscillator (FDO) mechanism of wave pattern formation to the problem of axial segmentation in general and to somitogenesis in particular. We argue that the conditions for formation of FDO waves are satisfied during somitogenesis in the chick and mouse and that the waves of gene expression observed in these species arise from phase dynamics in a growing oscillatory medium. We substantiate this claim by showing that the FDO mechanism allows the waves to be mimicked by an inorganic experiment and that it predicts a wavelength that coincides with that observed experimentally. To see whether the FDO mechanism is compatible with other aspects of somitogenesis, we construct an FDO-based model of somitogenesis and successfully test it against a number of experimental observations, including the effect of heat shock. Our analysis provides a rigorous physical basis for the hypothesis that the phase dynamics of a segmental clock controls important stages of segmentation during somitogenesis in the chick and mouse as well as in other organisms that undergo segmentation during their axial growth.