developmental biology - Crans · 2014-04-01 · Over the past three decades, developmental biology has undergone a renaissance owing to the success of genetic and molecular approaches

Over the past three decades, developmental biology has undergone a renaissance owing to the success of genetic and molecular approaches in understanding many of the genes, signalling pathways and biochemical reactions that orchestrate the complex events of embryogenesis. Genetic screens have provided matching parts lists of phenotypes and molecules that define the genetic con-trol of early embryonic development. Classical concepts such as the embryological organizer now have molecular identities. The similarity of transcription factor and sig-nalling pathway biochemistry from worms to humans has revealed a stunning conservation in molecular and cellular mechanisms at this level, which has influenced our understanding of evolution and human medicine. However, much of this progress in developmental molec-ular genetics has been qualitative in nature, partly owing to technical limitations, but also as expected for a young discipline. As dynamic phenotypes are studied in more detail, and as the complexity of the details of embryogen-esis become clearer and their importance more widely appreciated, the explanatory power of simple informal models is reaching its limits. A quantitative approach has become essential.

The past decade has seen a phenomenal increase in the number of tools for the capture of quantitative data from living embryos. These include sophisticated microscopy methods, image analysis software and the use of vital fluorescent molecules, such as GFP and its derivatives, to probe gene function. This combination

of tools has allowed quantitative measurement of the dynamics of developmental processes at the molecular, cellular and tissue level. Examples include: the tem-poral and spatial changes in the expression levels and localization of specific molecules in cells or tissues; the position, shape and mechanical properties of cells; and the growth and morphology of tissues. Consequently, processes with detailed spatiotemporal dynamics, such as individual cell movement, genetic oscillations and molecular gradients in embryos, are becoming accessible to quantitative analysis.

Quantitative data analysis initially enables the pre-cise description of a given developmental phenomenon, regardless of whether we already possess a model for it. In some cases, an informal model exists that suggests the nature of the process being investigated (for exam-ple, the existence of a morphogen gradient), and this helps to determine how the data should be quantitated. The quantitative nature of such data may also reveal dynam-ics or structures that were not previously suspected, and so lead directly to the generation of a new hypothesis. The next stage is to evaluate the plausibility of a given mathematical model of the process, asking whether the model is consistent with the data. When more than one model is available, a key attribute of quantitative data is the power to distinguish between models. If it cannot, the iteration falls back to the design and execution of experiments that use perturbation or enhanced resolu-tion to make this discrimination. A quantitative model

*Max Planck Institute of Molecular Cell Biology and Genetics, D‑01307 Dresden, Germany.‡Department of Genetics, University of Cambridge, Cambridge, CB2 3EH, UK.§Department of Biochemistry, University of Geneva, 1211 Geneva, Switzerland.Correspondence to A.C.O. and C.‑P.H. e‑mails: oates@mpi‑cbg.de; heisenberg@mpi‑cbg.dedoi:10.1038/nrg2548Published online 7 July 2009

Morphogen gradientThe morphogen gradient model proposes that undifferentiated cells in a developing tissue acquire information about their position in the field by reading the concentration of a substance (a morphogen), which is distributed in a spatial gradient of concentration.

Quantitative approaches in developmental biologyAndrew C. Oates*, Nicole Gorfinkiel‡, Marcos González‑Gaitán§ and Carl‑Philipp Heisenberg*

Abstract | The tissues of a developing embryo are simultaneously patterned, moved and differentiated according to an exchange of information between their constituent cells. We argue that these complex self-organizing phenomena can only be fully understood with quantitative mathematical frameworks that allow specific hypotheses to be formulated and tested. The quantitative and dynamic imaging of growing embryos at the molecular, cellular and tissue level is the key experimental advance required to achieve this interaction between theory and experiment. Here we describe how mathematical modelling has become an invaluable method to integrate quantitative biological information across temporal and spatial scales, serving to connect the activity of regulatory molecules with the morphological development of organisms.

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Planar cell polarityThe polarization of many epithelial cells in the plane of the tissue.

Imaginal discEpithelial infoldings in the larvae of insects that are determined during the embryonic stage; they grow during the larval stage and finally develop into adult appendages during metamorphosis.

PhyllotaxisThe arrangement of leaves on the stem of a plant. Basic patterns are alternate, opposite, whorled and spiral.

Cell autonomousA genetic trait in multicellular organisms in which only genotypically mutant cells exhibit the mutant phenotype.

Bottom-up modellingA modelling approach wherein the microscopic dynamics of the individual constituents of a developing system is described as a function of the properties of each constituent and its relevant interactions with other constituents. Higher-level attributes of the system (emergent properties) are calculated from these interactions over time.

Markov chain modelA stochastic process such that, with the present state known, future states are independent of the past states. At each time step, the potential transition to the next state is drawn from a probability distribution.

Top-down modellingA modelling approach wherein an empirical relationship between observable parameters is defined by starting with the higher-level properties of the developing system that may have a collective or statistical character, for example, differentiated cell states, tissue deformation or oscillation period. A top-down model does not require detailed knowledge of lower-level processes, such as gene expression or function.

should also allow the estimation of unknown parameters from the fit to the data. This step is both an important internal consistency check and a specific prediction about the system.

This article examines several topics in develop-mental biology to which quantitative techniques and mathematical modelling have been applied with some success. These examples span a wide range of develop-mental processes in multicellular organisms and illus-trate diverse mathematical approaches to modelling. We first briefly discuss progress in three areas: planar cell polarity (PCP) and the geometric packing of epithelial cells in the Drosophila melanogaster wing imaginal disc, and phyllotaxis in Arabidopsis thaliana. We then focus in more detail on morphogen gradients, biomechanics and vertebrate segmentation, because these are strong exam-ples of the quantitative study of space, force and time in the embryo. Although morphogen gradients and biome-chanics are intrinsically biophysical topics, the emerging picture of somitogenesis borrows from a range of biolog-ical and biophysical models to explore new directions for research. We finish with key new technological advances and general principles for the expansion of quantitative approaches to developmental biology. Our thesis here is that dynamic, quantitative measurement is both timely, owing to recent advances in live imaging techniques, and essential, because it allows the iterative dialogue between theory and experiment that is required to understand the complexity of the developing embryo.

Recent progressPCP signalling. recent work on the establishment of PCP in D. melanogaster1 is a prominent example of the use of mathematical modelling to test the plausibility of a specific working hypothesis and to make predic-tions for subsequent experimentation. To explore how PCP proteins interact to polarize epithelia, the authors developed a reaction–diffusion model on the basis of their working hypothesis that PCP proteins function in a feedback loop to amplify an initial asymmetric cue. This feedback loop is thought to consist of the trans-membrane protein Frizzled (Fz) recruiting Dishevelled (DsH) to the plasma membrane in a cell autonomous manner, and recruiting Prickled (Pk) and strabismus (sTbm; also known as van Gogh, vAnG) to the mem-brane of an adjacent cell in a cell non-autonomous man-ner. The feedback loop is provided by Pk, which inhibits the ability of Fz to recruit DsH.

To model these protein interactions the authors used ten non-linear partial differential equations; the crucial parameters are the specific protein concentra-tions, reaction rates and diffusion constants, none of which has been experimentally determined. Instead, a computational approach that minimizes the difference between model performance and a set of experimental observations was used to find a set of parameters for which the model reproduces all the most characteristic PCP phenotypes, including domineering non-autonomy. These results suggest that the model is robust and thus is a plausible hypothesis for the yet unknown endogenous situation. Although it is not clear whether this model will

be useful beyond PCP in the wing imaginal disc of flies, the parameter search and evaluation techniques used are important research tools, with a wide applicability in bottom-up modelling approaches.

Emergence of geometric order in proliferating epithelia. mathematical modelling has also been used to explain the formation of a hexagonal cell shape in simple epithe-lia2. using a discrete Markov chain model the authors show that the distribution of polygonal cell shapes in epithelia can be explained as a simple consequence of cell prolifer-ation. based on several empirically derived rules for cell shape and cell division in epithelia — for example, that cells have a minimum of four sides and retain a com-mon junctional interface after division — a transition probability is derived for a cell to go from one polygonal state to another after one round of division. After only ten generations, an in silico epithelium of polygonal cells produces a high enrichment of hexagonal cells similar to those seen in naturally occurring epithelia.

The strength of this top-down modelling approach is that it does not depend on specific parameter settings and thus is generally applicable to epithelial tissues that share a simple set of shape and division rules. The disad-vantage, as with any top-down approach, is that because there is no molecular basis for any of the proposed rules, they are purely empirical and therefore cannot provide any mechanistic insights. Future work must seek the corresponding molecular mechanisms to gain a full understanding of the system.

Phyllotaxis in plants. Parameter quantitation from live imaging has been closely paired with mathematical modelling to explain phyllotaxis in plants3. It is known that peaks in the concentration of the plant hormone auxin determine the position of leaf and flower primor-dia in the shoot apical meristem, and that these auxin peaks are generated by the depletion of auxin from neighbouring cells. The authors used mathematical modelling to test the plausibility of their hypothesis that auxin depletion from surrounding cells is mediated by the auxin transporter PIn1, the subcellular localization of which is determined by the relative auxin concentra-tions between neighbouring cells. Consequently, PIn1 distribution becomes polarized towards higher auxin levels and thereby provides a positive feedback loop that allows cells with a relatively high auxin concentra-tion to increase their auxin content by influencing PIn1 distribution in adjacent cells.

On the basis of these assumptions, and from quan-titative measurements of PIn1 subcellular distribution in the shoot apical meristem of A. thaliana, the authors developed an auxin transport model that, when com-bined with a cell growth and elastic mechanics model, generated spiral phyllotactic-like patterns similar to those observed in plants. because this model is closely linked to the quantitation of key parameters, its general use is now testable by varying crucial parameter values and comparing the outcome of those variations between experiments and models. because of the unique role of the cell wall in plant growth, this particular multiscale

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Box 1 | The physical basis of exponential gradients in steady state

What does a steady state exponential gradient mean? If there is a localized source of production, the exponential gradient in the target tissue represents the steady state solution to a situation in which the morphogen moves non-directionally and is degraded while it moves17. These three features (localized source, non-directional movement and degradation) have been experimentally observed for the Decaptentaplegic (Dpp) morphogen in Drosophila melanogaster17, explaining why it is distributed as a steady state exponential gradient. This is illustrated in the figure using real data from a Dpp gradient; the green line is the exponential fit.

That the gradient is exponential implies that its distribution is captured by the equation λ , where C

(x) is the concentration at a distance x from

the source and from C0, which is the concentration at the

source boundary. λ is the decay length, defined as the position x at which the concentration equals C0/e. Thus, the shape

of an exponential gradient is described by two parameters, C0 and λ: C

0 is the gradient ‘amplitude’ and λ measures how

fast the concentration decays as one moves away from the source, that is, how steep the gradient is. However, a steady state exponential gradient does not mean that the key phenomena underlying its generation are always non-directional movement and degradation. For instance, biased movement can generate exponential gradients124, as can sequestration (instead of degradation)20,125: in the case of Bicoid, nuclear import immobilizes the morphogen and, together with diffusion, this could generate its exponential profile.

An interesting consequence of the description of the gradient as an exponential distribution that depends on non-directional movement and degradation is that λ in the steady state is related to the effective diffusion coefficient D and the effective degradation rate k by λ = √(D/k), and C

0 is related to D, k and the morphogen flux from the source (j

0)

by C0 = j

0/√Dk (REf. 17). These are simple equations that imply two properties: that the steepness of the gradient λ

depends exclusively on how the target cells ‘handle’ the morphogen, that is, on the morphogen speed that they can support (D) and how fast they can degrade it; and that the amplitude of the gradient C

0 depends on these two

parameters plus the production rate at the source, which influences the flux of molecules into the target tissue.This analysis provides a physical framework to explain gradient biogenesis: we need to know the speed of morphogen

movement, its degradation rate, and its rate of synthesis and secretion at the source. Unfortunately, these parameters cannot be inferred unambiguously from the steady state properties of the gradient (that is, C

0 and λ): a particular,

experimentally observed set of C0 and λ values (in wild type or mutant conditions) can be explained by infinite sets of

D, k and j0 values.

PrimordiaAn organ or tissue in its earliest recognizable stage of development. The leaf and flower primordia arise from the shoot apical meristem in a process that is regulated by the hormone auxin.

Apical meristemThe tissue that is found at the growth tip of plants. It consists of completely undifferentiated cells, and is equivalent to stem cells in animals.

Elastic mechanicsThe physical theory that deals with materials deforming under stress and returning to their original shape when the stress is removed, as typified by a spring.

Multiscale modellingThe integration of interactions between multiple levels of spatial or temporal organization, each with its own model substructure.

Genetic regulatory networkA common type of bottom-up model in which the emphasis is on rates of production of mRNA and proteins from genes in response to regulatory signals, leading to altered states of the network.

modelling architecture may not be applicable outside plants; however, the use of quantitative live imaging to determine model parameters is exemplary.

For reasons of space, we cannot cover all the recent activity in quantitative developmental biology, so we recommend several excellent reviews for distinct per-spectives on important additional topics, such as the segmentation gene regulatory network (Grn) and dors-oventral patterning in Drosophila species, and collective behaviour in single cell organisms4–7. We now turn to the first of our three detailed examples and discuss how mathematical modelling and quantitative experimenta-tion have allowed a deeper understanding of morphogen gradient formation.

Morphogen gradient formation and functionThere are many examples of morphogen gradients8,9 in development, including the transcription factor bicoid (bCD) in the early D. melanogaster embryo10 and secreted molecules, such as Decaptentaplegic (DPP), Activin, nodal, Wingless (WG), Hedgehog and Fibroblast growth factor 8, that are involved in patterning multiple tissues11,12.

The concept of the morphogen gradient is fundamen-tally quantitative: target genes respond to the morpho-gen above distinct concentration thresholds. However,

quantitative measurement and analysis of morphogen gradient formation and the response of target genes has only recently begun13–17. The use of mathematical mod-els has been instrumental in moving from the heuristic notion of a morphogen gradient to explicit descriptions about which properties of the system control the gra-dient. The key issues now being addressed by quan-titative methods include: the shape and dynamics of the concentration gradient; the molecular and cellular machineries that mediate the rate of morphogen produc-tion, spreading and degradation; the precision of target gene activation; and the relation between morphogen gradient precision and the accuracy of the resulting morphological pattern.

Shape and dynamics of the morphogen gradient. In several cases, for example, bCD and DPP, morpho-gens form exponential gradients14,15,17 and are in steady state17,18 (BOX 1). However, departures from this theme have also been reported19. Parameters that are defined by mathematical models as being important for under-standing gradient formation and its mode of action are the diffusion coefficient (D), degradation rate (k) and morphogen flux from the source (j0), all of which can be measured by pushing the system away from the steady state and allowing it to return. This has been achieved

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CytonemeA long, thin and polarized actin-based cytoplasmic extension with a diameter of approximately 0.2 μm that projects from a cell.

for three morphogens (bCD, DPP and WG) by using fluorescence recovery after photobleaching (FrAP)17,20 with functional GFP fusions. Eventually, these phe-nomenological parameters will need to be explained on a molecular and cellular basis.

several mechanisms of morphogen transport have been proposed, including passive spreading of molecules carried by cells as they divide, and long cellular proc-esses termed cytonemes that may transport morphogen molecules from a source to the distant target cells from which they extend. It has been proposed that secreted morphogens can be spread by diffusion through the extracellular milieu or be transported by reiterated rounds of endocytosis and resecretion of the morpho-gen in the target cells18,21–24. These extracellular diffusion and endocytosis–resecretion models have been studied theoretically and tested experimentally18,21,25. studies that have analysed the diffusion coefficient of DPP molecules after blocking endocytosis17,18 provide evidence in favour of the endocytosis–resecretion model. However, this is a topic of active debate in the field26. Endocytosis does not affect the speed of movement and degradation of WG17, suggesting that DPP and WG are transported by different mechanisms.

methods that allow specific, rather than global, inter-ference with the endocytosis and resecretion of particu-lar signalling molecules would help to resolve the true contribution of this mechanism to gradient formation. The diffusion coefficient of a morphogen is determined by intracellular and extracellular movement of these molecules: extracellular diffusion, endocytosis, endo-cytic trafficking and recycling are all involved. similarly, the degradation rate for morphogen molecules includes degradation in lysosomal compartments as well as the extracellular cleavage and destruction of the morpho-gen. Consequently, the difference between the extracel-lular diffusion and the endocytosis–resecretion models resides in the relative contribution of these different transport steps; further quantitative analysis will be necessary to resolve the issue.

Precision of the morphogen gradient. The significance of gradient precision was initially defined for the chemoat-traction of microorganisms27, but it is just as important for controlling target gene expression and morphologi-cal patterning in multicellular organisms. This problem has been best analyzed theoretically and experimentally in flies, for bCD in the early embryo and DPP in the developing wing14,28.

bCD specifies the position of target gap gene expres-sion in the embryo10. by measuring the variability of the profile from animal to animal the imprecision of the gradient was determined to be 10%, that is, a positional uncertainty of approximately one nucleus14,15. This pre-cise concentration is translated into a similarly precise readout by the activation of its target gene hunchback, suggesting a tight coupling between the precisions of morphogen gradient and target gene transcription. The bCD gradient is rapidly established after fertilization and then remains stable. It has been argued that simple cytoplasmic diffusion of bCD is not sufficient to achieve

such a precise concentration gradient, and that nuclear degradation of bCD might be implicated14,20. Differences in nuclear degradation of bCD have also been proposed to contribute to the effective scaling of the bCD gradient in species with different embryo lengths20,29. However, whether the rate of nuclear transport and degradation of bCD correlates with, and is responsible for, bCD gra-dient formation and scaling remains to be established. recent reports of active transport of bcd mrnA in the embryo30 are difficult to reconcile with existing work, but these observations, if validated, would have funda-mental implications for models of the bCD gradient that rely on diffusion.

mathematical modelling predicts that the imprecision of the DPP gradient in the wing imaginal disc has a local minimum close to the source, which increases at greater distances28 (fIG. 1a–d). The DPP concentration gradient and its signalling activity has a minimum positional uncertainty of approximately one cell (fIG. 1d), similar to that of bCD. The accuracy in positional information encoded by the DPP gradient is approximately three cells at the boundary of one of its target genes, spalt28, which is similar to the imprecision of the spalt expression range but is twice as high as the variability of the resulting mor-phological vein pattern of the wing. Thus, the precision of the DPP gradient alone is not sufficient to account for the precision of the resulting morphological structures, suggesting that later processes in development must refine the final morphological pattern15,28,31–35.

What could such precision-refining processes be? Temporal integration of morphogen concentration over time might account for positional information accu-racy19,36–38. Also, cross-repression of the downstream genes can also lead to sharpening and refinement of their expression domains32,34, suggesting that positional information also depends on regulatory interactions in the target tissue33. Thus, the emerging picture is that morphogen precision alone can not account for the pre-cision of the emerging morphological pattern, and that other mechanisms must aid this process33,35. In conclu-sion, the use of mathematical models has been essential to understand the precision of morphogen gradients, as well as their shape and dynamics.

Biomechanical properties of living organismsAlthough the field of developmental biology has recently been dominated by the search for differentially expressed genes that can turn on (or off) specific programmes of cell differentiation, the idea that mechanical forces also govern cell behaviour has seen a renewed interest39–41. morphogenesis is inherently a mechanical process in which tissues move and deform depending on the mechanical properties of their constituent cells and on the mechanical constraints imposed by their local microenvironment. Quantitation of mechanical cell and tissue properties is essential to study biomechan-ics in development. below we describe progress made in this field, with particular emphasis on the quantita-tive approaches used to elucidate mechanisms involved in mechanosensing and in mechanical control of tissue morphogenesis.

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Figure 1 | Precision of morphogen gradients. a | Imprecision of the morphogen gradient. The imprecision of the morphogen concentration increases with the distance from the source, although there is a local minimum at a close distance. b–d | Precision of the morphogen-dependent positional information. A group of naive cells expresses different target genes (yellow, black, blue, red and green) in response to the morphogen (b). Morphogen-dependent target gene expression mediates differentiation into different cell types. The morphogen allocates the cell types (for example, a hair cell) with maximal precision (approximately one cell) at a discrete distance from the source (seven cells to the left). Closer or further away from there, the positional information is coarse (c). After differentiation, however, the different cell types are positioned with very high precision (d).


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Focal adhesionsLarge, dynamic protein complexes at the cell cortex through which the actin cytoskeleton of a cell connects to the extracellular matrix and transmits force. It is typically coordinated by the binding of cellular integrin transmembrane proteins to the matrix, which also act as integrin-regulated signalling centres.

Adherens junctionsProtein complexes at cell–cell junctions in epithelial tissues that link the actin cytoskeleton across the tissue and transmit force. They are mediated by cadherin transmembrane protein binding to cadherins at an opposing adherens junction on a neighbouring epithelial cell, and can act as cadherin-regulated signalling centres.

Mechanosensing in single cells. When cells are cultured on substrates of varying geometries they acquire sizes and shapes that match the geometry of the underlying adhesive substrate. When micropatterned substrates are used to precisely control cell shape and spreading, cel-lular fate depends on cellular form: small and round cells enter apoptosis, whereas spread-out cells proliferate and differentiate42,43 (fIG. 2a,b). In a more specific example, cell shape influences the commitment and differentia-tion of human mesenchymal stem cells to adipocytes (if rounded in shape) or osteoblasts (when spread out)44.

Cells can respond not only to the shape but also to the stiffness of the substrate to which they adhere. For exam-ple, when mesenchymal stem cells are cultured onto soft, rigid or very rigid matrices to mimic the different elas-ticity of brain, muscle and bone, they differentiate into neurons, myoblasts and osteoblasts, respectively45. These findings suggest a crucial role for substrate geometry, area and stiffness in cell fate specification, but they do not reveal how these substrate properties influence cell fate.

little is known about how mechanical cues influ-ence the gene expression changes involved in cell fate specification, but force-induced changes in intracellular signalling and nuclear architecture that are mediated by focal adhesions and adherens junctions have been impli-cated. specifically, mechanical forces applied to the sur-face of endothelial cells through magnetic beads coated with the rGD cell-binding peptide from fibronectin can activate the cAmP signalling cascade and increase the phosphorylation levels of the CrEb transcription fac-tor46. similarly, force-controlled stem cell specification involves signalling through the small GTPase rhoA44,45. Together, these findings suggest that force application at adhesion sites can lead to alterations in intracellular signalling involved in gene expression and cell fate speci-fication. How forces are translated into biochemical sig-nalling is a field of active research and probably involves conformational changes of force-sensing molecules.

local forces can also generate structural changes in the nucleus: forces applied experimentally on the cell are transmitted to the nucleus by integrin adhesion mole-cules, which are coupled to the nuclear membrane by the actin and intermediate filament cytoskeleton47. However, whether these changes can modulate gene expression is an open question.

Mechanosensing in tissues and organisms. The studies reviewed above provide evidence that forces transmit-ted at adhesion sites influence cell fate determination, proliferation and differentiation; however, the role of this mechanism in the developing organism has only begun to be analysed. A first step in this direction has been to study the role of cell cluster shape on the spatial distribu-tion of cell proliferation in these clusters48. Cell cultures grown on micropatterned islands of defined size and shape show patterns of proliferation that are consistent with tractional stress being a regulator of proliferation49 (fIG. 2c,d). regions of high mechanical stress across the cell sheet were correlated with an increase in growth that depends on myosin contraction and cadherin-mediated adhesion between cells; therefore, mechanical tension

generated in the cytoskeleton and transmitted through cell junctions determines the spatial pattern of growth. local cell proliferation affects the shape of cell assemblies and therefore affects the distribution of stress, suggesting a dynamic interplay between shape, stress and prolifera-tion in multicellular assemblies. Whether mechani-cal tension has similar effects on cell proliferation in developing multicellular organisms is still unclear.

On an organismal level, mechanical stress might influence gene expression in the developing D. mela-nogaster embryo50,51. mechanical compression of the embryo directly induces the expression of the transcrip-tion factor twist, possibly through the transcriptional co-activator β-catenin50,51; this suggests that twist expres-sion in blastodermal cells in vivo can be modulated by

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Figure 2 | Mechanical cues in cell fate specification. Cells cultured on small microfabricated adhesive islands coated with extracellular matrix (ECM) undergo apoptosis (a), whereas cells cultured on large islands spread and proliferate (b).Colorimetric stacked image of mechanical stresses (c; as predicted by computational modelling) and cell proliferation (d) in aggregates cultured on round microfabricated ECM islands. Colorimetric scales are shown on the right depicting normalized tractional stress and proliferation rates. Scheme of the device used for mechanical deformation of Drosophila melanogaster embryos (e, left panel). The expression of Twist (blue), which in an embryo after a short 4 minute compression is restricted to the ventral side of the embryo, is expanded to the dorsal side after a 10 minute compression (e, right panel). The embryo is drawn with anterior to the left and dorsal at the top. Parts c and d are reproduced, with permission, from REf. 48 (2005) the National Academy of Sciences. Part e is modified, with permission, from REf. 51 (2003) Cell Press.

TensegrityDescribes structures that stabilize their shape by continuous tension, and includes pre-stressed and geodesic classes. In the pre-stressed class, a pre-existing tensile stress or isometric tension distributed among embedded compressive elements holds the joints in position. In the geodesic class, structural members are triangulated and oriented along minimal paths to geometrically constrain movement. for a cell, the internal pre-stressed cytoskeleton interconnects at the cell periphery with a highly elastic, geodesic cytoskeletal network directly beneath the plasma membrane.

Cell cortexA network of crosslinked actin filaments that is attached to the inner face of the plasma membrane and is able to contract through the action of myosin molecular motors.

local tissue compression as a result of morphogenetic movements of the blastoderm (fIG. 2e).

In an organism, forces must be integrated across different length scales. Tension integrity (tensegrity) architecture proposes that systems of interconnected extracellular matrix and cytoskeletal networks at dif-ferent length scales are in a state of isometric tension, allowing mechanotransduction processes to occur in a highly concerted and simultaneous manner52,53. How local structural changes in a tensegrity network lead to distinct biochemical responses and how these biochemi-cal responses feed back to the composition and function of the cytoskeletal and adhesion apparatus are yet to be understood.

Mechanical control of tissue morphogenesis. Cultured tissue fragments, similar to liquid droplets, tend to form spheres to minimize their surface area, and randomly mixed cell populations behave in a similar manner to phase separation of immiscible liquids: the cells become sorted into distinct cell populations54. measurements of tissue surface (interfacial) tensions55,56 are consistent with the mutual envelopment and sorting behaviour of the corresponding tissues; that is, the tissue with the lower surface tension envelops the tissue with higher surface

tension57. Although the molecular basis of surface ten-sion was originally attributed to the adhesive machinery of the cells54,58, recent data suggest a role for actomyosin-dependent cell contractility in generating the surface ten-sion of germ layer tissue during zebrafish gastrulation59.

How can differences in cell cortex tension between the different germ layer progenitor cells lead to differ-ences in tissue surface tension between the forming germ layers? Tissue surface tension is thought to result from the surface tension at the cell-to-medium inter-face and at the cell-to-cell interface. Cortex tension at the cell-to-medium interface promotes tissue surface ten-sion, whereas cortex tension at the cell-to-cell interface reduces it. Therefore, for differences in cortex tension of germ layer progenitors to correlate with differences in tis-sue surface tension of germ layer tissues, differential cor-tex tension has to be pronounced at the cell-to-medium interface and not at the cell-to-cell interface (fIG. 3). To evaluate the contribution of tissue surface tension to tissue stratification in vivo, it is essential to determine tissue surface tension in the endogenous environment, which for technical reasons is not yet possible.

Contact between different tissues is also important in modulating tissue surface tension in the axial meso-derm tissue of gastrulating Xenopus laevis embryos60.

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Figure 3 | tensile forces acting on germ layer organization. The spatial configuration of the of the germ layers within the germ ring margin (bottom left) is controlled by the germ layer tissue surface tension, which is determined by the difference between the interfacial tension at the cell-to-cell interface and the interfacial tension at the interface between cells and extracellular matrix (ECM) and/or the medium and/or the extraembryonic tissue. The micrograph shows a lateral view of the shield region of a wild type zebrafish embryo stained with an antibody raised against E-cadherin. The expanded section (bottom right) illustrates a magnification of the spatial configuration of the mesendoderm–ectoderm (blue–red) border. γ

EE, interfacial tension between two ectodermal cells;

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, interfacial tension between a mesodermal cell and the ECM and/or the medium and/or extraembryonic tissue. This figure is modified, with permission, from Nature Cell Biology REf. 59 (2008) Macmillan Publishers Ltd. All rights reserved.

AnisotropiesDifferences in the value of a physical property of a material when measured along different axes.

Finite-element modellingA numerical tool, widely used in engineering design and analysis, used to solve partial differential equations. It requires the subdivision of the system into discrete elements, which are analyzed separately in terms of the loads and displacements at the nodes.

Coating of this tissue with an epithelial cell layer confers an elongated shape to the otherwise spherical mesoder-mal tissue. This effect is interpreted as the ability of the epithelial layer to reduce the surface tension of the aggre-gates. How epithelial coating changes mesodermal tissue surface tension is unclear, although it is conceivable that cell cortex tension in mesoderm progenitors at the cell-to-medium interface is reduced following contact with the epithelium.

Mechanical control of epithelial morphogenesis. A simi-lar principle involving minimization of the overall sur-face energy of a tissue has been used to describe how the stable geometric topologies observed in different epithelia arise from the biophysical properties of indi-vidual cells. A combination of physical models derived from the cellular Potts model61 (BOX 2) and genetic and mechanical perturbation in D. melanogaster was used to simulate the final configuration of cone cells in the eye imaginal disc62,63 and the steady state configuration of epithelial cells in the wing disc64. In these studies, two-dimensional modelled epithelia have a surface energy that depends on the shape of the cells and their posi-tion in the tissue. by allowing the system to evolve to the nearest lower energy state after the introduction of random fluctuations in the position of cell boundaries63 or of random cell divisions64, conditions were identified for which modelled tissues reach stationary patterns that recapitulate the final configuration observed in vivo in wild type and perturbed situations.

An interesting outcome of this model is that local cell rearrangements, in which cells lose and gain neigh-bours (fIG. 4a), arise as a result of the force balance at cell junctions relaxing the tissue to a stable, low-energy state. However, some cell rearrangements occurring dur-ing convergent extension65–67, such as cell intercalations, are not passive responses but are active processes that rely on the planar subcellular localization of cytoskel-etal and adhesion proteins that gives directionality to the movement68–70 (fIG. 4b). Interestingly, theoretical models that include anisotropies in adhesive properties or cor-tical tension along different cell surfaces can account for the elongation of a tissue through cell intercala-tion, as observed during convergent extension71,72 as a consequence of energy minimization principles.

Another example in which the localized behaviour of cells affects the whole tissue is epithelial invagination, a process triggered by the actomyosin-dependent apical constriction of a small group of epithelial cells73–76. using finite element modelling of primary invagination in the sea urchin embryo, it has been proposed that specific changes in the contractile properties of a small group of cells trigger invagination as a consequence of the pas-sive elastic properties of the embryo77–79. similarly, by applying the principles of fluid motion to the study of tissue invagination in D. melanogaster80, it has been sug-gested that the local increase in actomyosin-dependent apical surface tension in ventral mesodermal cells is suf-ficient to reproduce the observed changes in cell shape and other cellular behaviours involved in mesoderm invagination (fIG. 4c).

Interestingly, recent work81 on the mechanism of apical contractions of mesodermal cells in ventral fur-row formation during D. melanogaster gastrulation has shown that these mesodermal cells undergo pulses of contraction interrupted by pauses, during which the cells stabilize their constricted state before reinitiating contraction. Contracting cells are being stretched by their neighbours, causing individual cells to contract asynchronously; nonetheless, a net tissue contraction is generated. This apical cell contraction is not driven by an actomyosin purse-string but by an actomyosin apical network, which is linked to the adherens junctions and pulls them inwards.

morphogenesis not only relies on local cell behav-iours but also on the specific mechanical properties of tissues, and on the forces and physical constraints

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a b c

Box 2 | The cellular Potts model

One of the most widely used mathematical models to simulate cell sorting behaviour and other features of biological tissues is the cellular potts model developed by Graner and Glazier126. In this model, tissues are represented on a two-dimensional lattice in which cells are composed of several contiguous lattice sites with the same identification number (in the example shown in the figure, the ‘tissue’ has 163 cells, each one occupying 25 lattice sites) and one out of several possible adhesive states (in this example, red or blue). The system tends to minimize the overall surface energy, reflecting its surface adhesiveness. This is driven by iteratively introducing random fluctuations in the position of cell boundaries in the lattice and recalculating the energy of the system after each modification. If the modification decreases the energy of the cell configuration, the new position is maintained (see the figure, part b). If the modification increases the energy of the system, the system returns to the previous configuration with a certain probability. These iterations are repeated until a steady state is reached, that is, a configuration that no longer changes (see the figure, part c). It is worth pointing out, however, that cellular potts models are not physical models of cells. The interfacial energy is not equivalent to how elastic structures store energy and how their deformations arise under stress. The cellular potts model has been used to reproduce the equilibrium configurations observed in a variety of biological phenomena, such as cell sorting, mesenchymal condensation, somitogenesis, convergent extension and interactions between different tissues.

Germ band extensionThe morphogenetic process that occurs shortly after gastrulation in long germ band insects, in which the body axis is elongated through extensive cell intercalation in the epidermal epithelium.

SomitesBilaterally symmetrical blocks of cells that are arranged in serial rows along the embryo. They give rise to the reiterated axial skeleton and associated musculature of the adult organism.

generated by the surrounding tissues and by other morphogenetic events65,82–87. During germ band exten-sion in D. melanogaster, for example, cell shape changes that contribute to the elongation and narrowing of the embryonic epidermis may be caused by mechanical stress from the invaginating mesoderm83. similarly, during dorsal closure in the embryo of D. melanogaster, the relative magnitude of the forces generated by the different tissues has been elucidated88,89. Analysis of the dynamic geometry of the closure as well as quantita-tion of concomitant cell shape changes suggest that the rate and pattern of individual cell behaviour depend on the mechanical constraints imposed by surrounding tis-sues84,90. These studies show that tissue morphogenesis is controlled by the close interaction between local cell behaviours and global morphogenetic events occurring in neighbouring tissues.

Somitogenesissomitogenesis is a dynamic and rhythmic morphologi-cal process that results in the tissue-level segmentation of the vertebrate embryonic body axis91. Somites (fIG. 5a) are formed as clusters of cells that segregate sequentially from the anterior end of the posteriorly extending presomitic mesoderm (Psm) (fIG. 5b) with a steady, species-specific rhythm and length that varies gradually along the body axis in most vertebrates. Only recently have quantitative measurements of these variables been made systemati-cally92,93: the dependence of somitogenesis period and somite length on axial position and temperature is the

quantitative foundation for the testing of mathematical models of the underlying mechanism.

The periodic nature of somitogenesis prompted Cooke and zeeman to propose the existence of an oscillatory mechanism in the Psm94. now supported by molecular data, this segmentation clock can be understood as a tissue-level, periodic pattern generator with three interacting elements: a population of auton-omously oscillating cells that are locally synchronized owing to intercellular communication and that are arrested by a wavefront of global signals at the anterior of the Psm. The movement of this system at the poste-rior end of the elongating embryo leaves a record of the oscillations, giving rise to the fixed spatial pattern of somites. Compared with morphogen gradients and bio-mechanics, quantitative study of the segmentation clock is still in its infancy, but a large body of theoretical work exists regarding the individual and collective behaviour of oscillatory systems, raising the possibility of a fruitful interaction between modelling and experiment.

What is the goal of this research? ultimately, one could conceive a multiscale model of the Psm in which the cell autonomous oscillations of the clock, their local phase coupling through cell–cell signalling and the tissue-level interactions leading to the arrest of the oscil-lators are detailed at an appropriate level and interact appropriately with other levels. Here we highlight both the first steps towards models for the cell autonomous oscillations, noting their incompleteness, and models for the local coupling that have already been success-fully tested with quantitative data. Testing of models of the global arrest wavefront95 and the integration of these different levels remain important challenges for the future.

Connecting molecular events with cellular oscillations. The first molecular evidence for the existence of the segmentation clock was a transcript expressed rhyth-mically in stripes across the chick Psm with a periodic-ity matching somitogenesis96 (fIG. 5d). This cyclic gene encodes a transcriptional repressor protein of the Hairy and Enhancer of split (Hes) family, suggesting that a cell autonomous transcriptional negative feedback loop could be the oscillatory mechanism97. by coupling the mouse cyclic Hes1 promoter to a destabilized luciferase reporter the first live images of the segmentation clock were seen in a mouse embryo, and quantitative dynamic data obtained from time-lapse movies of single dissoci-ated Psm cells revealed them to be cell autonomous but unstable oscillators98. Quantitative studies for the mech-anism of these oscillations have focused on bottom-up Grn models of the Hes feedback loop, but have yet to convincingly demonstrate anything more than their plausibility.

The most influential Grn model for the origin of Hes oscillations uses delay differential equations to describe the production and degradation of Hes mrnA and protein species, and the resulting autoregulation at the Hes promoter99–101 (fIG. 5c). It predicts that a single transcriptional repressor gene could form a negative feedback loop and generate stable oscillations if there

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a

b

c

Figure 4 | cell rearrangements and cell shape changes drive morphogenesis. a | During epithelial morphogenesis of Drosophila species, cells exchange neighbours through T1 and T2 transitions. In T1 transitions, the length of a contact domain between two adjacent cells (green cells) shrinks, while other boundaries increase until four cells meet at a point (green and blue cells). This can be then followed by the loss of contact between a pair of cells (green cells) while the other pair (blue cells) generate new contacts. In T2 transitions, a single cell (red cell) is lost from the tissue and new contacts are created between the cells surrounding the extruded cell (yellow cells). b | When neighbouring exchange occurs in a directed manner throughout an epithelium, tissue elongation results. For this to occur, cells need to polarize their adhesive (black arrows) and contractile (grey arrows) properties. c | Simulation of ventral furrow invagination in Drosophila species using finite-element modelling; dorsal is up. The increase in actomyosin cortical tension in ventral-most cells is sufficient to drive cellular and tissue deformations when three dimensions and the vitelline membrane are considered80. Part c is reproduced from REf. 80 (2008) Institute of Physics.

Mean fieldAn approximation from physics in which interactions between many components are replaced by interactions with a single component. Technically, all the components contribute to the generation of a mean field across the system, which in turn feeds back to each component to regulate its behaviour.

are significant delays in the loop. Choosing plausible delay and half-life values for the zebrafish Her genes (Hes gene orthologues) for analytic solutions and numerical simulations yielded a time period that was in reasonable agreement with the embryo99. Thus, these models demonstrated the plausibility of a Hes loop as the origin of cell autonomous oscillations in the Psm. The quantitative predictions of such models are limited because few of their parameter values have been directly measured in any species.

One notable exception is the study by Hirata et al.102, which was motivated by the modelling result that pro-tein stability should be crucial in setting the period and stability of the oscillations99. In mutant mice expressing a variant HEs7 with increased stability, cyclic gene oscil-lations in the Psm seemed to undergo amplitude damp-ing, as predicted by a simulation of the model using HEs7 stability values close to those from the experiment. The equally interesting prediction of the model that the period would lengthen by ~10% remains untested.

An important attempt to constrain the parameters of a Grn for the zebrafish segmentation clock by Giudicelli et al.103 was based on the idea that the stripes of cyclic gene expression (fIG. 5d) arise because oscillating cells slow down gradually, causing a frequency profile and cor-responding phase profile across the Psm96,104,105 (fIG. 5e). using a top-down approach, the phase profile could be approximately reconstructed from the wavelength of the cyclic gene expression stripes at various positions across the Psm. Combining this profile with quantitative infor-mation about gene expression allowed part of the delay in the transcription, and an upper bound to the lifetime of the mrnA, to be calculated. Although this elegant work produced a set of parameter value estimates that was close to those used for the initial model99, remain-ing uncertainty in the transcriptional delay is of similar magnitude to the experimental data. Furthermore, the consequences of the new larger experimentally measured values for transcript and protein life times have not been investigated in simulations. nevertheless, using tissue-level patterns to infer properties of the underlying cel-lular and molecular dynamics is an important approach that should feature in further studies.

Although these pioneering studies supported a delayed negative feedback model for the clock’s cell autonomous dynamics, Grn models are generally lim-ited when many of the parameters are tuned so that the model’s output most closely resembles the data. This raises the possibility that if the actual parameter val-ues (which are difficult and expensive to obtain) were known and the model re-simulated, the outcome would not agree quantitatively with experimental data. In con-clusion, no Grn model of the segmentation clock has been thoroughly quantitatively tested, and so the mecha-nism generating the cell autonomous oscillations is still an open question.

Local coordination of cellular oscillations. The unsta-ble nature of the cell autonomous oscillators98 high-lights the need to understand how the cells’ oscillations are coordinated in vivo to form a coherent tissue-level clock that ticks with the high precision of somitogen-esis93. The synchronization of Psm cells through inter-cellular communication94 is probably achieved in the presence of noise by coupling through Delta–notch sig-nalling106,107. That coupling can synchronize individual oscillators is an established principle, with well-studied examples including lasers, antenna arrays, chemical reactions, neuronal circuits, chirping crickets and flash-ing fireflies108–110. Grn models of Delta–notch coupling between two or several cells demonstrate the plausibility

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S = vT

S

Hes/her

AAAAA

AAAAATp

Tm

τm

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Determinedsegments Tail budSomites

Notochord

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τp A

PSM

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Gene expression waves Mean-field coupling

12 somites 19 somites

5 20 35Somite number

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0

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e (m

in)

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e (o

ne c

ycle

of c

lock

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e Spatially extended clock and wavefront model

Hes/her mRNA

27 somites Period = 24.5 ± 0.8 min(n = 9, T = 27.4 ± 0.1 °C)

a Embryonic somitogenesis

b

d Spatially coordinated genetic oscillations

c Minimal cell autonomous oscillator model

Hes/Her protein

Period (T)Segmentlength (S)

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Figure 5 | somitogenesis and the segmentation clock. a | The left-most four panels show a lateral view of somitogenesis in a zebrafish embryo, and highlight the sequential formation of segmental structures (arrows) from the posteriorly extending presomitic mesoderm (PSM; anterior is up). These four panels are taken from a time-lapse movie used to quantitate somitogenesis period93, and numbers indicate elapsed time in minutes. On the right hand side, the plot of somite number versus time elapsed illustrates the dynamics and precision of the process. The slope of the red line fitted to the data shows a constant period in the trunk, whereas the period increases gradually in the tail (T, temperature). b | A dorsal perspective of somitogenesis, showing a schematized anatomy of the posterior mesoderm (anterior is to the left). c | A schematic of the delayed feedback model for the cell autonomous genetic oscillations99 occurring throughout the PSM. This type of genetic feedback loop is proposed to control the periodicity of somitogenesis. Transcription, splicing and export of Hairy and Enhancer of Split (Hes) family genes (Her genes are the zebrafish orthologues) has a delay of T

m, and the stability

of the mRNA is given by the half-life τm

. The translation of protein, dimerization and DNA binding has a delay of T

p, and the protein stability is given by the half-life τ

p. Oscillation

occurs when (translation rate × transcription rate)/τmτ

p > 2p

0, and when this inequality is

satisfied the oscillation period is approximated by 2(Tm

+ Tp + τ

m + τ

p). d | Sequential

snapshots of the cyclic gene expression patterns in the zebrafish PSM, showing that oscillating cells are locally coordinated with their neighbours, but also display a tissue-scale spatial pattern over several hundred micrometers. The entire cycle of patterns repeats once for every newly forming somite98. The orientation is as in part b. e | A schematic of the spatial and temporal organization of the PSM during somitogenesis, as well as key properties and functions discussed in the main text. The orientation of the scheme is as in parts b and d. The PSM consists of cellular oscillators, the period (T) of which is set by an autonomous genetic oscillator, potentially similar to that illustrated in part c and indicated by the red sine curve. These noisy oscillators98 are synchronized with each other through Delta–Notch signalling106,111,112, indicated by grey arrows. This effect has been modelled as mean-field coupling between phase oscillators112, schematized in the posterior PSM and tail bud with dashed grey arrows. The entire tissue extends posteriorly with the velocity v, and as oscillating cells drift though the PSM they slow down along a frequency profile until arresting at the wavefront in the anterior PSM96,103,114. This gradual slowing gives rise to phase differences along the tissue; that is, the wave-like patterns of gene expression. The length of a segment (S) is given by the product of the velocity of the wavefront and the period of the clock (S = vT)94. Subsequent morphogenetic events not covered here produce the somites from the pattern laid down in the tissue by the moving segmentation clock. Part a is modified, with permission, from REf. 93 (2008) Wiley Interscience.

◀

Selective plane illumination microscopyAn approach that combines two-dimensional laser illumination with orthogonal camera-based detection, thereby obtaining high-resolution, optical sectioning throughout an entire embryo. Advantages include minimal phototoxicity and speeds capable of capturing dynamic phenomena.

Two photon laser scanning microscopyA fluorescence imaging technique that uses the simultaneous absorption of two low-energy photons to excite a fluorophore. The use of long wavelength excitation photons reduces scattering in biological material, allowing deeper tissue penetration. Additional advantages over conventional confocal microscopy are efficient light detection and reduced phototoxicity.

of such an interaction99,106,111; here we describe two recent quantitative tests of coupling models made with bottom-up and top-down approaches111,112.

In the bottom-up approach, simulations of a Grn model in which the synthesis and trafficking delays of Delta molecules and competitive interaction at cyclic gene promoters are explicitly considered were compared with transgenic reporter output levels in zebrafish embryos with compromised notch signalling111. This study shows good agreement between model and experiment, and nicely illustrates the strength of quantifying the signal of a reporter gene in live embryos. similar experiments could be attempted using the Hes1 transgenic reporter or a recent Lfng–YFP transgenic reporter113 in a Delta or notch mutant mouse embryo.

A top-down approach to understanding the synchro-nization of the Psm cells comes from the experimental and theoretical work of riedel-kruse et al.112. by abstract-ing the undoubtedly complex molecular details to three essential parameters — synchronization, coupling and noise — this approach adapts established physical theory to analyse the experimental results. The segmentation clock is described as a population of phase oscillators in mean field coupling, with an order parameter quantifying

the level of synchronization. Importantly, the change in synchrony over time is set by the balance of the coupling strength and the total noise in the system. using some simple assumptions about notch signalling, the bio-chemistry of this pathway was abstracted to the coupling strength among the cell autonomous oscillators. This is an example of a model architecture in which a phenom-enological term in a model (coupling strength) has been unpacked to provide molecular details for specific pro-teins (in notch receptor activation), thereby connecting theory with experiments.

by fitting the theory to quantitative measurements of synchrony decay time, the robustness of notch signalling to noise was determined as a free parameter, allowing both coupling strength and noise level to be estimated. As well as successfully describing the defect position for any given level of notch inhibition, the theory also pre-dicted the resynchronization of the clock following the restoration of coupling. However, only coupling strength was experimentally manipulated in this study; a simi-lar investigation of noise would deepen our confidence in this model. The work required to quantitatively test just one system parameter is a sobering example of the advantages of validating models with dozens of param-eters. In conclusion, both Grn and phenomenological phase oscillator models of Delta–notch coupling have received strong experimental support. by connecting the dynamics of synchronization in the segmentation clock to established theory, we can say that Psm cells synchronize much like lasers and fireflies.

Future directions. Although much has been learnt about the segmentation clock, we are still coming to grips with its mechanisms and dynamics. key future experiments will include generating live reporters for the oscillations in embryos of different species with sufficient temporal and spatial resolution to be able to follow the period, phase and amplitude of individual cells as they mix in the Psm, divide, couple with each other, slow down and finally arrest oscillation. Even in the absence of such reporters, time-lapse microscopy of embryos with perturbed segmentation will allow the precise measure-ment of the altered dynamics, enabling rival hypotheses about gene function to be distinguished114. To exploit the power of Grns in understanding mechanism, the kinetic parameters controlling their dynamics must be experimentally determined with sufficient precision. These data will first come from biochemical assays, but ideally must also be measured in transgenic embryos in which the dynamics of concentration, subcellular distribution and interactions of tagged proteins can be quantified. Although this discussion has ignored the differences between vertebrate species, it seems that the details of their molecular mechanisms do differ91. Whether the principles governing their operation will nevertheless be conserved remains to be seen.

outlookQuantitation has become increasingly common in developmental biology, but is it really necessary? Why quantify effects that are already obvious by simple visual

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inspection? We argue that no phenotype is ‘obvious’, and that, to describe a developmental phenotype, informa-tion is needed about the way its quality, its quantity and its variability change with time. Therefore, quan-titation is not an optional extra, but an integral part of developmental biology.

The ultimate aim of quantitative developmental biol-ogy should be to achieve a multiscale model of the bio-logical system of interest. An example that should inspire developmental biologists is the physiological model of the heart that has been developed by constant iteration of theory and experiment over the last four decades115. many groups working on physiological aspects of heart function at the molecular, cellular, organ and whole-body level have contributed to building an integrated series of models spanning from single ion transporters to blood flow, and the generation of the electrocardiogram. For example, stochastic models of ion transporter func-tion in cardiomyocytes explain the cell-level phenome-non of excitation–contraction coupling116, and models of coupled cardiomyocytes with anatomically accurate fibre orientation explain the way contraction affects how the wave of excitation spreads across the cardiac tissue117.

The development of powerful imaging techniques, such as selective plane illumination microscopy (sPIm) and two photon laser scanning microscopy, is particularly excit-ing. These tools have allowed the movement of a large number of cells to be recorded simultaneously during early development. Analysis of these movies using three-dimensional segmentation and tracking software has allowed the decomposition of complex cellular move-ments in D. melanogaster, zebrafish and chick gastrula-tion, and has provided information about the specific contribution of cell division, directed cell migration and cell intercalation to tissue morphogenesis85,118–123. As these data sets are integrated with genetic and bio-chemical experiments, the challenges and opportunities for mathematical modelling will only increase.

several lessons can be learned from the strengths and weaknesses of the studies discussed here. To express our outlook operationally, we have drawn up a list of rules that should be observed when conducting research in quantitative developmental biology (BOX 3). With quan-titation becoming commonplace, developmental biology may be slowly turning into an exact science, the aim of which, to cite the scottish physicist J. C. maxwell: “…is to reduce the problems of nature to the determination of quantities by operations with numbers.” We hope that this process will allow us to see deeper into the complex beauty of the developing embryo.

Box 3 | eight rules for research in quantitative developmental biology

Measure dynamicsStructure and function in embryos cannot be understood without inclusion of an accurate time axis. We need to ask questions about timing, duration and rates in the system with as much temporal resolution as appropriate. This is challenging because it requires observation and perturbation of growing embryos in conditions as near to physiological as possible. If this essential technical aspect does not appeal, we recommend the study of fossils, in which the rate of change is somewhat slower.

Understand variationWhen quantitating specific phenotypic traits in developmental biology, the high variability of measurements (that is, noise) seems to be a large obstacle. However, variation in genetic background and the fundamentally stochastic nature of molecular events are part of the fabric of biology — change in variation can itself be a phenotype. Standardization of assays and higher replicate numbers are required to detect the contribution from experimental error.

Develop technologyThe success of quantitation depends on the continuing development of new measurement and analysis tools. However, a study can be dramatically changed with much simpler innovations, such as writing a script to automatically measure the size of a gene expression domain in an image.

educate yourselfThe success of quantitation also depends on the ability and willingness of developmental biologists to engage with mathematical modelling, and to design their experiments with specific aspects of models in mind. Courses integrating biology with mathematics, physics, engineering or computer science are becoming more common at undergraduate levels, journals devoted to interdisciplinary approaches are growing, and conferences that include a mixture of themes are proliferating.

communicate with theoristsOne can ask whether models based on abstruse mathematics are useful. However, the real question is whether the model provides the best description of the biology. Although it is rare to find individuals that are equally competent in theory and experiment, in a truly interdisciplinary effort even complex models are practically useful as long as biologists and theorists can communicate. We recommend starting this communication before experiments begin.

start with detailed dataStart analyses at a point at which the data can be measured with the highest accuracy and precision.

chose the right level of abstractionThe choice of mathematical model should be tailored to the type of data that is obtainable. Although it is tempting to start with a molecular-scale mechanistic model featuring a favourite gene, it may be more realistic to start with a phenomenological model of the process at a cellular or tissue scale at which the number of parameters that can be measured experimentally closely matches the number in the model.

Do not over-quantitateFinally, quantitation does not automatically determine the validity of a conclusion. The significance of the statistical probabilities calculated from quantitative data depends on agreement in the scientific community. Furthermore, the degree of precision in the data needs only to be sufficient for testing the model. It is therefore important to see quantitation as one among many of the tools used to address the validity of a scientific conclusion.

1. Amonlirdviman, K. et al. Mathematical modeling of planar cell polarity to understand domineering nonautonomy. Science 307, 423–426 (2005).

2. Gibson, M. C., Patel, A. B., Nagpal, R. & Perrimon, N. The emergence of geometric order in proliferating metazoan epithelia. Nature 442, 1038–1041 (2006).

3. Jonsson, H., Heisler, M. G., Shapiro, B. E., Meyerowitz, E. M. & Mjolsness, E. An auxin-driven polarized transport model for phyllotaxis. Proc. Natl Acad. Sci. USA 103, 1633–1638 (2006).

4. Lewis, J. From signals to patterns: space, time, and mathematics in developmental biology. Science 322, 399–403 (2008).

5. Tomlin, C. J. & Axelrod, J. D. Biology by numbers: mathematical modelling in developmental biology. Nature Rev. Genet. 8, 331–340 (2007).

6. Reeves, G. T., Muratov, C. B., Schupbach, T. & Shvartsman, S. Y. Quantitative models of developmental pattern formation. Dev. Cell 11, 289–300 (2006).

7. Cooper, W. J. & Albertson, R. C. Quantification and variation in experimental studies of morphogenesis. Dev. Biol. 321, 295–302 (2008).

8. Turing, A. M. The chemical basis of morphogenesis. 1953. Philos. Trans. R. Soc. Lond. B 237, 37–72 (1952).

9. Wolpert, L. Positional information and the spatial pattern of cellular differentiation. J. Theor. Biol. 25, 1–47 (1969).

10. Driever, W. & Nusslein-Volhard, C. The bicoid protein determines position in the Drosophila embryo in a concentration-dependent manner. Cell 54, 95–104 (1988).

11. Gurdon, J. B., Harger, P., Mitchell, A. & Lemaire, P. Activin signalling and response to a morphogen gradient. Nature 371, 487–492 (1994).

12. Tabata, T. & Takei, Y. Morphogens, their identification and regulation. Development 131, 703–712 (2004).

R E V I E W S



13. Eldar, A. et al. Robustness of the BMP morphogen gradient in Drosophila embryonic patterning. Nature 419, 304–308 (2002).

14. Gregor, T., Tank, D. W., Wieschaus, E. F. & Bialek, W. Probing the limits to positional information. Cell 130, 153–164 (2007).

15. Houchmandzadeh, B., Wieschaus, E. & Leibler, S. Establishment of developmental precision and proportions in the early Drosophila embryo. Nature 415, 798–802 (2002).

16. Hufnagel, L., Teleman, A. A., Rouault, H., Cohen, S. M. & Shraiman, B. I. On the mechanism of wing size determination in fly development. Proc. Natl Acad. Sci. USA 104, 3835–3840 (2007).

17. Kicheva, A. et al. Kinetics of morphogen gradient formation. Science 315, 521–525 (2007).This paper uses FRAP to analyse the kinetic parameters of DPP and Wg morphogen gradient formation. They show that endocytosis is necessary for the diffusion and degradation of DPP, but not of Wg.

18. Entchev, E. V., Schwabedissen, A. & Gonzalez-Gaitan, M. Gradient formation of the TGF-β homolog Dpp. Cell 103, 981–991 (2000).

19. Bourillot, P. Y., Garrett, N. & Gurdon, J. B. A changing morphogen gradient is interpreted by continuous transduction flow. Development 129, 2167–2180 (2002).

20. Gregor, T., Wieschaus, E. F., McGregor, A. P., Bialek, W. & Tank, D. W. Stability and nuclear dynamics of the bicoid morphogen gradient. Cell 130, 141–152 (2007).

21. Lander, A. D., Nie, Q. & Wan, F. Y. Do morphogen gradients arise by diffusion? Dev. Cell 2, 785–796 (2002).

22. Ramirez-Weber, F. A. & Kornberg, T. B. Cytonemes: cellular processes that project to the principal signaling center in Drosophila imaginal discs. Cell 97, 599–607 (1999).

23. Strigini, M. & Cohen, S. M. Wingless gradient formation in the Drosophila wing. Curr. Biol. 10, 293–300 (2000).

24. Takei, Y., Ozawa, Y., Sato, M., Watanabe, A. & Tabata, T. Three Drosophila EXT genes shape morphogen gradients through synthesis of heparan sulfate proteoglycans. Development 131, 73–82 (2004).

25. Kruse, K., Pantazis, P., Bollenbach, T., Julicher, F. & Gonzalez-Gaitan, M. Dpp gradient formation by dynamin-dependent endocytosis: receptor trafficking and the diffusion model. Development 131, 4843–4856 (2004).

26. Belenkaya, T. Y. et al. Drosophila Dpp morphogen movement is independent of dynamin-mediated endocytosis but regulated by the glypican members of heparan sulfate proteoglycans. Cell 119, 231–244 (2004).

27. Berg, H. C. & Purcell, E. M. Physics of chemoreception. Biophys. J. 20, 193–219 (1977).

28. Bollenbach, T. et al. Precision of the Dpp gradient. Development 135, 1137–1146 (2008).

29. Gregor, T., Bialek, W., de Ruyter van Steveninck, R. R., Tank, D. W. & Wieschaus, E. F. Diffusion and scaling during early embryonic pattern formation. Proc. Natl Acad. Sci. USA 102, 18403–18407 (2005).

30. Spirov, A. et al. Formation of the bicoid morphogen gradient: an mRNA gradient dictates the protein gradient. Development 136, 605–614 (2009).

31. Houchmandzadeh, B., Wieschaus, E. & Leibler, S. Precise domain specification in the developing Drosophila embryo. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72, 061920 (2005).

32. Jaeger, J. et al. Dynamical analysis of regulatory interactions in the gap gene system of Drosophila melanogaster. Genetics 167, 1721–1737 (2004).

33. Jaeger, J. & Reinitz, J. On the dynamic nature of positional information. Bioessays 28, 1102–1111 (2006).

34. Jaeger, J., Sharp, D. H. & Reinitz, J. Known maternal gradients are not sufficient for the establishment of gap domains in Drosophila melanogaster. Mech. Dev. 124, 108–128 (2007).

35. Kerszberg, M. & Wolpert, L. Specifying positional information in the embryo: looking beyond morphogens. Cell 130, 205–209 (2007).

36. Bergmann, S. et al. Pre-steady-state decoding of the Bicoid morphogen gradient. PLoS Biol. 5, e46 (2007).

37. Bergmann, S., Tamari, Z., Schejter, E., Shilo, B. Z. & Barkai, N. Re-examining the stability of the Bicoid morphogen gradient. Cell 132, 15–17; author reply 17–8 (2008).

38. Dyson, S. & Gurdon, J. B. The interpretation of position in a morphogen gradient as revealed by occupancy of activin receptors. Cell 93, 557–568 (1998).

39. Ingber, D. E. Mechanical control of tissue morphogenesis during embryological development. Int. J. Dev. Biol. 50, 255–266 (2006).

40. Keller, R., Shook, D. & Skoglund, P. The forces that shape embryos: physical aspects of convergent extension by cell intercalation. Phys. Biol. 5, 15007 (2008).

41. Lecuit, T. & Lenne, P. F. Cell surface mechanics and the control of cell shape, tissue patterns and morphogenesis. Nature Rev. Mol. Cell Biol. 8, 633–644 (2007).

42. Chen, C. S., Mrksich, M., Huang, S., Whitesides, G. M. & Ingber, D. E. Geometric control of cell life and death. Science 276, 1425–1428 (1997).Provides direct experimental evidence that cell shape governs whether individual cells grow or die.

43. Singhvi, R. et al. Engineering cell shape and function. Science 264, 696–698 (1994).

44. McBeath, R., Pirone, D. M., Nelson, C. M., Bhadriraju, K. & Chen, C. S. Cell shape, cytoskeletal tension, and RhoA regulate stem cell lineage commitment. Dev. Cell 6, 483–495 (2004).

45. Engler, A. J., Sen, S., Sweeney, H. L. & Discher, D. E. Matrix elasticity directs stem cell lineage specification. Cell 126, 677–689 (2006).

46. Meyer, C. J. et al. Mechanical control of cyclic AMP signalling and gene transcription through integrins. Nature Cell Biol. 2, 666–668 (2000).

47. Maniotis, A. J., Chen, C. S. & Ingber, D. E. Demonstration of mechanical connections between integrins, cytoskeletal filaments, and nucleoplasm that stabilize nuclear structure. Proc. Natl Acad. Sci. USA 94, 849–854 (1997).

48. Nelson, C. M. et al. Emergent patterns of growth controlled by multicellular form and mechanics. Proc. Natl Acad. Sci. USA 102, 11594–11599 (2005).

49. Tan, J. L. et al. Cells lying on a bed of microneedles: an approach to isolate mechanical force. Proc. Natl Acad. Sci. USA 100, 1484–1489 (2003).

50. Desprat, N., Supatto, W., Pouille, P. A., Beaurepaire, E. & Farge, E. Tissue deformation modulates Twist expression to determine anterior midgut differentiation in Drosophila embryos. Dev. Cell 15, 470–477 (2008).

51. Farge, E. Mechanical induction of Twist in the Drosophila foregut/stomodeal primordium. Curr. Biol. 13, 1365–1377 (2003).

52. Ingber, D. E. Cellular mechanotransduction: putting all the pieces together again. FASEB J. 20, 811–827 (2006).

53. Sultan, C., Stamenovic, D. & Ingber, D. E. A computational tensegrity model predicts dynamic rheological behaviors in living cells. Ann. Biomed. Eng. 32, 520–530 (2004).

54. Steinberg, M. S. Reconstruction of tissues by dissociated cells. Some morphogenetic tissue movements and the sorting out of embryonic cells may have a common explanation. Science 141, 401–408 (1963).

55. Foty, R. A., Forgacs, G., Pfleger, C. M. & Steinberg, M. S. Liquid properties of embryonic tissues: measurement of interfacial tensions. Phys. Rev. Lett. 72, 2298–2301 (1994).

56. Foty, R. A. & Steinberg, M. S. The differential adhesion hypothesis: a direct evaluation. Dev. Biol. 278, 255–263 (2005).

57. Foty, R. A., Pfleger, C. M., Forgacs, G. & Steinberg, M. S. Surface tensions of embryonic tissues predict their mutual envelopment behavior. Development 122, 1611–1620 (1996).

58. Steinberg, M. S. Differential adhesion in morphogenesis: a modern view. Curr. Opin. Genet. Dev. 17, 281–286 (2007).

59. Krieg, M. et al. Tensile forces govern germ-layer organization in zebrafish. Nature Cell Biol. 10, 429–436 (2008).Provides experimental support for the hypothesis that cell sorting in vitro and in vivo is determined by differences in cell cortex tension and adhesion.

60. Ninomiya, H. & Winklbauer, R. Epithelial coating controls mesenchymal shape change through tissue-positioning effects and reduction of surface-minimizing tension. Nature Cell Biol. 10, 61–69 (2008).

61. Glazier, J. A. & Graner, F. Simulation of the differential adhesion driven rearrangement of biological cells. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 47, 2128–2154 (1993).

62. Hilgenfeldt, S., Erisken, S. & Carthew, R. W. Physical modeling of cell geometric order in an epithelial tissue. Proc. Natl Acad. Sci. USA 105, 907–911 (2008).

63. Kafer, J., Hayashi, T., Maree, A. F., Carthew, R. W. & Graner, F. Cell adhesion and cortex contractility determine cell patterning in the Drosophila retina. Proc. Natl Acad. Sci. USA 104, 18549–18554 (2007).

64. Farhadifar, R., Roper, J. C., Aigouy, B., Eaton, S. & Julicher, F. The influence of cell mechanics, cell–cell interactions, and proliferation on epithelial packing. Curr. Biol. 17, 2095–2104 (2007).

65. Glickman, N. S., Kimmel, C. B., Jones, M. A. & Adams, R. J. Shaping the zebrafish notochord. Development 130, 873–887 (2003).

66. Keller, R. et al. Mechanisms of convergence and extension by cell intercalation. Philos. Trans. R. Soc. Lond. B 355, 897–922 (2000).

67. Shih, J. & Keller, R. Cell motility driving mediolateral intercalation in explants of Xenopus laevis. Development 116, 901–914 (1992).

68. Bertet, C., Sulak, L. & Lecuit, T. Myosin-dependent junction remodelling controls planar cell intercalation and axis elongation. Nature 429, 667–671 (2004).Provides experimental evidence for a crucial function of junctional remodelling in directing cellular rearrangements underlying germ band extension during D. melanogaster gastrulation.

69. Blankenship, J. T., Backovic, S. T., Sanny, J. S., Weitz, O. & Zallen, J. A. Multicellular rosette formation links planar cell polarity to tissue morphogenesis. Dev. Cell 11, 459–470 (2006).

70. Classen, A. K., Anderson, K. I., Marois, E. & Eaton, S. Hexagonal packing of Drosophila wing epithelial cells by the planar cell polarity pathway. Dev. Cell 9, 805–817 (2005).

71. Rauzi, M., Verant, P., Lecuit, T. & Lenne, P. F. Nature and anisotropy of cortical forces orienting Drosophila tissue morphogenesis. Nature Cell Biol. 10, 1401–1410 (2008).

72. Zajac, M., Jones, G. L. & Glazier, J. A. Model of convergent extension in animal morphogenesis. Phys. Rev. Lett. 85, 2022–2025 (2000).

73. Costa, M. et al. A putative catenin–cadherin system mediates morphogenesis of the Caenorhabditis elegans embryo. J. Cell Biol. 141, 297–308 (1998).

74. Dawes-Hoang, R. E. et al. folded gastrulation, cell shape change and the control of myosin localization. Development 132, 4165–4178 (2005).

75. Haigo, S. L., Hildebrand, J. D., Harland, R. M. & Wallingford, J. B. Shroom induces apical constriction and is required for hingepoint formation during neural tube closure. Curr. Biol. 13, 2125–2137 (2003).

76. Kolsch, V., Seher, T., Fernandez-Ballester, G. J., Serrano, L. & Leptin, M. Control of Drosophila gastrulation by apical localization of adherens junctions and RhoGEF2. Science 315, 384–386 (2007).

77. Davidson, L. A., Koehl, M. A., Keller, R. & Oster, G. F. How do sea urchins invaginate? Using biomechanics to distinguish between mechanisms of primary invagination. Development 121, 2005–2018 (1995).

78. Davidson, L. A., Oster, G. F., Keller, R. E. & Koehl, M. A. Measurements of mechanical properties of the blastula wall reveal which hypothesized mechanisms of primary invagination are physically plausible in the sea urchin Strongylocentrotus purpuratus. Dev. Biol. 209, 221–238 (1999).

79. Odell, G. M., Oster, G., Alberch, P. & Burnside, B. The mechanical basis of morphogenesis. I. Epithelial folding and invagination. Dev. Biol. 85, 446–462 (1981).

80. Pouille, P. A. & Farge, E. Hydrodynamic simulation of multicellular embryo invagination. Phys. Biol. 5, 15005 (2008).

81. Martin, A. C., Kaschube, M. & Wieschaus, E. F. Pulsed contractions of an actin–myosin network drive apical constriction. Nature 457, 495–499 (2009).Provides experimental evidence that pulsed contractions of an apical actomyosin network drive apical cell constriction during ventral furrow formation in D. melanogaster.

82. Blanchard, G. B. et al. Tissue tectonics: morphogenetic strain rates, cell shape change and intercalation. Nature Methods 6, 458–464 (2009).

83. Butler, L. C. et al. Cell shape changes indicate a role for extrinsic tensile forces in Drosophila germband extension. Nature Cell Biol. (in the press).

84. Gorfinkiel, N., Blanchard, G. B., Adams, R. J. & Martinez-Arias, A. Mechanical control of global cell behaviour during dorsal closure in Drosophila. Development 136, 1889–1898 (2009).

R E V I E W S



85. McMahon, A., Supatto, W., Fraser, S. E. & Stathopoulos, A. Dynamic analyses of Drosophila gastrulation provide insights into collective cell migration. Science 322, 1546–1550 (2008).

86. Benko, R. & Brodland, G. W. Measurement of in vivo stress resultants in neurulation-stage amphibian embryos. Ann. Biomed. Eng. 35, 672–681 (2007).

87. Zhou, J., Kim, H. Y. & Davidson, L. A. Actomyosin stiffens the vertebrate embryo during crucial stages of elongation and neural tube closure. Development 136, 677–688 (2009).

88. Hutson, M. S. et al. Forces for morphogenesis investigated with laser microsurgery and quantitative modeling. Science 300, 145–149 (2003).Determines the relative contribution of forces generated in different tissues to dorsal closure in D. melanogaster.

89. Kiehart, D. P., Galbraith, C. G., Edwards, K. A., Rickoll, W. L. & Montague, R. A. Multiple forces contribute to cell sheet morphogenesis for dorsal closure in Drosophila. J. Cell Biol. 149, 471–490 (2000).

90. Peralta X. G. et al. Upregulation of forces and morphogenic asymmetries in dorsal closure during Drosophila development. Biophys. J. 92, 2583–2596 (2007).

91. Dequeant, M. L. & Pourquie, O. Segmental patterning of the vertebrate embryonic axis. Nature Rev. Genet. 9, 370–382 (2008).

92. Gomez, C. et al. Control of segment number in vertebrate embryos. Nature 454, 335–339 (2008).

93. Schroter, C. et al. Dynamics of zebrafish somitogenesis. Dev. Dyn. 237, 545–553 (2008).

94. Cooke, J. & Zeeman, E. C. A clock and wavefront model for control of the number of repeated structures during animal morphogenesis. J. Theor. Biol. 58, 455–476 (1976).

95. Goldbeter, A., Gonze, D. & Pourquie, O. Sharp developmental thresholds defined through bistability by antagonistic gradients of retinoic acid and FGF signaling. Dev. Dyn. 236, 1495–1508 (2007).

96. Palmeirim, I., Henrique, D., Ish-Horowicz, D. & Pourquie, O. Avian hairy gene expression identifies a molecular clock linked to vertebrate segmentation and somitogenesis. Cell 91, 639–648 (1997).

97. Hirata, H. et al. Oscillatory expression of the bHLH factor Hes1 regulated by a negative feedback loop. Science 298, 840–843 (2002).

98. Masamizu, Y. et al. Real-time imaging of the somite segmentation clock: revelation of unstable oscillators in the individual presomitic mesoderm cells. Proc. Natl Acad. Sci. USA 103, 1313–1318 (2006).Visualizes segmentation clock gene expression in live embryos and shows the noisy oscillations of isolated clock cells.

99. Lewis, J. Autoinhibition with transcriptional delay: a simple mechanism for the zebrafish somitogenesis oscillator. Curr. Biol. 13, 1398–1408 (2003).Describes the most influential gRN model for how oscillations of the segmentation clock may arise from delays in transcriptional negative feedback loops.

100. Monk, N. A. Oscillatory expression of Hes1, p53, and NF-κB driven by transcriptional time delays. Curr. Biol. 13, 1409–1413 (2003).

101. Jensen, M. H., Sneppen, K. & Tiana, G. Sustained oscillations and time delays in gene expression of protein Hes1. FEBS Lett. 541, 176–177 (2003).

102. Hirata, H. et al. Instability of Hes7 protein is crucial for the somite segmentation clock. Nature Genet. 36, 750–754 (2004).

103. Giudicelli, F., Ozbudak, E. M., Wright, G. J. & Lewis, J. Setting the tempo in development: an investigation of the zebrafish somite clock mechanism. PLoS Biol. 5, e150 (2007).Uses multiple techniques to estimate parameters, from the embryo, for a gRN model of the segmentation clock.

104. Jaeger, J. & Goodwin, B. C. A cellular oscillator model for periodic pattern formation. J. Theor. Biol. 213, 171–181 (2001).

105. Kaern, M., Menzinger, M. & Hunding, A. Segmentation and somitogenesis derived from phase dynamics in growing oscillatory media. J. Theor. Biol. 207, 473–493 (2000).

106. Horikawa, K., Ishimatsu, K., Yoshimoto, E., Kondo, S. & Takeda, H. Noise-resistant and synchronized oscillation of the segmentation clock. Nature 441, 719–723 (2006).

107. Jiang, Y. J. et al. Notch signalling and the synchronization of the somite segmentation clock. Nature 408, 475–479 (2000).

108. Winfree, A. T. The Geometry of Biological Time 2nd edn (Springer, New York, 2001).

109. Pikovsky, A., Rosenblum, M. & Kurths, J. Synchronization: a Universal Concept in Nonlinear Sciences (Cambridge Univ. Press, Cambridge, 2003).

110. Kuramoto, Y. Chemical Oscillations, Waves, and Turbulence (Courier Dover Publications, Mineola, 2003).

111. Ozbudak, E. M. & Lewis, J. Notch signalling synchronizes the zebrafish segmentation clock but is not needed to create somite boundaries. PLoS Genet. 4, e15 (2008).

112. Riedel-Kruse, I. H., Muller, C. & Oates, A. C. Synchrony dynamics during initiation, failure, and rescue of the segmentation clock. Science 317, 1911–1915 (2007).Applies a physical theory of coupled phase oscillators to mutant phenotypes to estimate Notch coupling strength and total noise in the segmentation clock.

113. Aulehla, A. et al. A β-catenin gradient links the clock and wavefront systems in mouse embryo segmentation. Nature Cell Biol. 10, 186–193 (2008).

114. Morelli, L. G. et al. Delayed coupling theory of vertebrate segmentation. HFSP J. 3, 55–66 (2009).

115. Noble, D. Modeling the heart — from genes to cells to the whole organ. Science 295, 1678–1682 (2002).

116. Clancy, C. E. & Rudy, Y. Linking a genetic defect to its cellular phenotype in a cardiac arrhythmia. Nature 400, 566–569 (1999).

117. Kohl, P. & Ravens, U. Cardiac mechano-electric feedback: past, present, and prospect. Prog. Biophys. Mol. Biol. 82, 3–9 (2003).

118. England, S. J., Blanchard, G. B., Mahadevan, L. & Adams, R. J. A dynamic fate map of the forebrain shows how vertebrate eyes form and explains two causes of cyclopia. Development 133, 4613–4617 (2006).

119. Keller, P. J., Schmidt, A. D., Wittbrodt, J. & Stelzer, E. H. Reconstruction of zebrafish early embryonic development by scanned light sheet microscopy. Science 322, 1065–1069 (2008).

120. Chuai, M. et al. Cell movement during chick primitive streak formation. Dev. Biol. 296, 137–149 (2006).

121. Newman, T. J. Grid-free models of multicellular systems, with an application to large-scale vortices accompanying primitive streak formation. Curr. Top. Dev. Biol. 81, 157–182 (2008).

122. Voiculescu, O., Bertocchini, F., Wolpert, L., Keller, R. E. & Stern, C. D. The amniote primitive streak is defined by epithelial cell intercalation before gastrulation. Nature 449, 1049–1052 (2007).

123. Bodenstein, L. & Stern, C. D. Formation of the chick primitive streak as studied in computer simulations. J. Theor. Biol. 233, 253–269 (2005).

124. Bollenbach, T., Kruse, K., Pantazis, P., Gonzalez-Gaitan, M. & Julicher, F. Morphogen transport in epithelia. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75, 011901 (2007).

125. Coppey, M., Berezhkovskii, A. M., Kim, Y., Boettiger, A. N. & Shvartsman, S. Y. Modeling the bicoid gradient: diffusion and reversible nuclear trapping of a stable protein. Dev. Biol. 312, 623–630 (2007).

126. Graner, F. & Glazier, J. A. Simulation of biological cell sorting using a two-dimensional extended Potts model. Phys. Rev. Lett. 69, 2013–2016 (1992).

AcknowledgementsWe thank A. Martinez-Arias for discussion and feedback dur-ing the preparation of this work, and members of our labora-tories for comments and helpful discussion. We are grateful to J. de Navacsués for discussion and help with BOX 2, J. Lewis, R. Kageyama, I. Riedel-Kruse, C. Eugster and D. Roellig for comments on an earlier version of the manu-script, and J.-L. Maitre for discussion. We also thank E. Farge and P.-A. Pouille for providing the images used in FIG. 4c.

FURTHeR inFoRMATionAndrew Oates’ homepage: http://www.mpi-cbg.de/research/research-groups/andrew-oates.htmlMarcos González-Gaitán’s homepage: http://www.unige.ch/sciences/biochimie/Research/MarcosGonzalez.html#2Carl-Philipp Heisenberg’s homepage: http://www.mpi-cbg.de/research/research-groups/carl-philipp-heisenberg.htmlNature Reviews Genetics Series on Modelling: http://www.nature.com/nrg/series/modelling/index.html

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