Segmentation and somitogenesis derived from phase dynamics in growing oscillatory media

Influence of the Turing instability on the motion of domain boundaries

Turing's theory of pattern formation has provided crucial insights into the behavior of various biological, geographical, and chemical systems over the last few decades. Existing studies have focused on moving-boundary Turing systems for which the motion of the boundary is prescribed by an external agent. In this paper, we present an extension of this theory to a class of systems in which the front motion is governed by the physical processes that occur within the domain. Biological systems exhibiting apically dominant growth and corrosion of metals and alloys highlight some of the noteworthy examples of such systems. In this study, we characterize the nature of interaction between the moving front and the Turing-instability for both an activator-inhibitor and an activator-substrate model. Behavioral regimes of periodic, as well as nonperiodic (nonconstant), growth rates are obtained. Furthermore, the trends in the first show striking similarities with the cyclic-boundary-kinetics observed in experimental systems. In general, a stationary, periodic structure is also left behind the moving front. If the periodicity of the boundary kinetics agrees with the allowed range of the stable-periodic solutions, the pattern formed tends to persist. Otherwise, it evolves to a nearby energy-minimum either by peak-splitting, peak-decay, or by settling down to a spatially homogeneous state.

Spatial correlations in a finite-range Kuramoto model

We study spatial correlations between oscillator phases in the steady state of a Kuramoto model, in which phase oscillators that are randomly distributed in space interact with constant strength but within a limited range. Such a model could be relevant, for example, in the synchronization of gene expression oscillations in cells, where only oscillations of neighboring cells are coupled through cell-cell contacts. We analytically infer spatial phase-phase correlation functions from the known steady-state distribution of oscillators for the case of homogenous frequencies and show that these can contain information about the range and strength of interactions, provided the noise in the system can be estimated. We suggest a method for the latter, and also explore when correlations appear to be ergodic in this model, which would enable an experimental measurement of correlation functions to utilize temporal averages. Simulations show that our techniques also give qualitative results for the model with heterogenous frequencies. We illustrate our results by comparison with experimental data on genetic oscillations in the segmentation clock of zebrafish embryos.

Around the clock: gradient shape and noise impact the evolution of oscillatory segmentation dynamics

BACKGROUND

Segmentation, the subdivision of the major body axis into repeated elements, is considered one of the major evolutionary innovations in bilaterian animals. In all three segmented animal clades, the predominant segmentation mechanism is sequential segmentation, where segments are generated one by one in anterior-posterior order from a posterior undifferentiated zone. In vertebrates and arthropods, sequential segmentation is thought to arise from a clock-and-wavefront-type mechanism, where oscillations in the posterior growth zone are transformed into a segmental prepattern in the anterior by a receding wavefront. Previous evo-devo simulation studies have demonstrated that this segmentation type repeatedly arises, supporting the idea of parallel evolutionary origins in these animal clades. Sequential segmentation has been studied most extensively in vertebrates, where travelling waves have been observed that reflect the slowing down of oscillations prior to their cessation and where these oscillations involve a highly complex regulatory network. It is currently unclear under which conditions this oscillator complexity and slowing should be expected to evolve, how they are related and to what extent similar properties should be expected for sequential segmentation in other animal species.

RESULTS

To investigate these questions, we extend a previously developed computational model for the evolution of segmentation. We vary the slope of the posterior morphogen gradient and the strength of gene expression noise. We find that compared to a shallow gradient, a steep morphogen gradient allows for faster evolution and evolved oscillator networks are simpler. Furthermore, under steep gradients, damped oscillators often evolve, whereas shallow gradients appear to require persistent oscillators which are regularly accompanied by travelling waves, indicative of a frequency gradient. We show that gene expression noise increases the likelihood of evolving persistent oscillators under steep gradients and of evolving frequency gradients under shallow gradients. Surprisingly, we find that the evolutions of oscillator complexity and travelling waves are not correlated, suggesting that these properties may have evolved separately.

CONCLUSIONS

Based on our findings, we suggest that travelling waves may have evolved in response to shallow morphogen gradients and gene expression noise. These two factors may thus also be responsible for the observed differences between different species within both the arthropod and chordate phyla.

Travelling waves in somitogenesis: Collective cellular properties emerge from time-delayed juxtacrine oscillation coupling

The sculpturing of the vertebrate body plan into segments begins with the sequential formation of somites in the presomitic mesoderm (PSM). The rhythmicity of this process is controlled by travelling waves of gene expression. These kinetic waves emerge from coupled cellular oscillators and sweep across the PSM. In zebrafish, the oscillations are driven by autorepression of her genes and are synchronized via Notch signalling. Mathematical modelling has played an important role in explaining how collective properties emerge from the molecular interactions. Increasingly more quantitative experimental data permits the validation of those mathematical models, yet leads to increasingly more complex model formulations that hamper an intuitive understanding of the underlying mechanisms. Here, we review previous efforts, and design a mechanistic model of the her1 oscillator, which represents the experimentally viable her7;hes6 double mutant. This genetically simplified system is ideally suited to conceptually recapitulate oscillatory entrainment and travelling wave formation, and to highlight open questions. It shows that three key parameters, the autorepression delay, the juxtacrine coupling delay, and the coupling strength, are sufficient to understand the emergence of the collective period, the collective amplitude, and the synchronization of neighbouring Her1 oscillators. Moreover, two spatiotemporal time delay gradients, in the autorepression and in the juxtacrine signalling, are required to explain the collective oscillatory dynamics and synchrony of PSM cells. The highlighted developmental principles likely apply more generally to other developmental processes, including neurogenesis and angiogenesis.

Delta-Notch signalling in segmentation

Modular body organization is found widely across multicellular organisms, and some of them form repetitive modular structures via the process of segmentation. It's vastly interesting to understand how these regularly repeated structures are robustly generated from the underlying noise in biomolecular interactions. Recent studies from arthropods reveal similarities in segmentation mechanisms with vertebrates, and raise the possibility that the three phylogenetic clades, annelids, arthropods and chordates, might share homology in this process from a bilaterian ancestor. Here, we discuss vertebrate segmentation with particular emphasis on the role of the Notch intercellular signalling pathway. We introduce vertebrate segmentation and Notch signalling, pointing out historical milestones, then describe existing models for the Notch pathway in the synchronization of noisy neighbouring oscillators, and a new role in the modulation of gene expression wave patterns. We ask what functions Notch signalling may have in arthropod segmentation and explore the relationship between Notch-mediated lateral inhibition and synchronization. Finally, we propose open questions and technical challenges to guide future investigations into Notch signalling in segmentation.

Shear-flow-controlled mode selection in a nonlinear autocatalytic medium

The effect of shear flow on mode selection and the length scale of patterns formed in a nonlinear autocatalytic reaction-diffusion model is investigated. We predict analytically the existence of transverse and longitudinal modes. The type of the selected mode strongly depends on the difference in the flow rates of the participating species, quantified by the differential flow parameter. Spatial structures are obtained by varying the length scale of individual modes and superposing them via the differential flow parameter. Our predictions are in line with numerical results obtained from lattice Boltzmann simulations.

Modelling Delta-Notch perturbations during zebrafish somitogenesis

The discovery over the last 15 years of molecular clocks and gradients in the pre-somitic mesoderm of numerous vertebrate species has added significant weight to Cooke and Zeeman's 'clock and wavefront' model of somitogenesis, in which a travelling wavefront determines the spatial position of somite formation and the somitogenesis clock controls periodicity (Cooke and Zeeman, 1976). However, recent high-throughput measurements of spatiotemporal patterns of gene expression in different zebrafish mutant backgrounds allow further quantitative evaluation of the clock and wavefront hypothesis. In this study we describe how our recently proposed model, in which oscillator coupling drives the propagation of an emergent wavefront, can be used to provide mechanistic and testable explanations for the following observed phenomena in zebrafish embryos: (a) the variation in somite measurements across a number of zebrafish mutants; (b) the delayed formation of somites and the formation of 'salt and pepper' patterns of gene expression upon disruption of oscillator coupling; and (c) spatial correlations in the 'salt and pepper' patterns in Delta-Notch mutants. In light of our results, we propose a number of plausible experiments that could be used to further test the model.

Blocking and transmission of traveling flow-distributed-oscillation waves in an absolutely unstable flowing medium

For a flowing, self-oscillating medium, we study the competition between traveling flow-distributed-oscillation waves excited by periodic driving at the upstream boundary and bulk oscillations originating downstream from the boundary. As previously observed in the case of stationary driving, we find that there is a region in parameter space where boundary-driven traveling waves of sufficiently high amplitude can impose themselves on the entire medium despite the presence of an absolute instability, which otherwise tends to block information from upstream. For sufficiently low flow rates, however, the imposed waves are arrested at a nonlinear blocking transition. Unlike the stationary case, we find that the region of imposed waves extends well into regions where, according to the linear approximation, there should be no traveling waves at all. This suggests that the extinction of the traveling waves is analogous to a subcritical Hopf bifurcation.

Collective modes of coupled phase oscillators with delayed coupling

We study the effects of delayed coupling on timing and pattern formation in spatially extended systems of dynamic oscillators. Starting from a discrete lattice of coupled oscillators, we derive a generic continuum theory for collective modes of long wavelengths. We use this approach to study spatial phase profiles of cellular oscillators in the segmentation clock, a dynamic patterning system of vertebrate embryos. Collective wave patterns result from the interplay of coupling delays and moving boundary conditions. We show that the phase profiles of collective modes depend on coupling delays.

ONE-DIMENSIONAL MODEL OF SOMITIC CELLS POLARIZATION IN A BISTABILITY WINDOW OF EMBRYONIC MESODERM

The considerations are based on the understanding that somitic cells polarization in bistability window of embryonic (pre-somitic) mesoderm is a dynamical process. It occurs in the form of a polarization wavefront of somite cells spread in anterior–posterior direction of the embryonic mesoderm. It is assumed that a macroscopic cell polarization has a bistable behavior corresponding to the molecular mechanism of bistability window formation. Moreover this type of polarization is supposed to be transmittable to the other cells by contact interaction. At the end, a volume of polarized cells is taken, which is able to create mechanical tension in the volume of nonpolarized neighbor cells and to inhibit their polarization. On this basis we explore the leading aspect of somitogenesis robustness by considering a simple wavefront model of polarization and analyzing its propagation in terms of the standard methods of qualitative theory of differential equations. The obtained theoretical results are interpreted in the context of their possible experimental verification.

A spatio-temporal model of Notch signalling in the zebrafish segmentation clock: conditions for synchronised oscillatory dynamics

In the vertebrate embryo, tissue blocks called somites are laid down in head-to-tail succession, a process known as somitogenesis. Research into somitogenesis has been both experimental and mathematical. For zebrafish, there is experimental evidence for oscillatory gene expression in cells in the presomitic mesoderm (PSM) as well as evidence that Notch signalling synchronises the oscillations in neighbouring PSM cells. A biological mechanism has previously been proposed to explain these phenomena. Here we have converted this mechanism into a mathematical model of partial differential equations in which the nuclear and cytoplasmic diffusion of protein and mRNA molecules is explicitly considered. By performing simulations, we have found ranges of values for the model parameters (such as diffusion and degradation rates) that yield oscillatory dynamics within PSM cells and that enable Notch signalling to synchronise the oscillations in two touching cells. Our model contains a Hill coefficient that measures the co-operativity between two proteins (Her1, Her7) and three genes (her1, her7, deltaC) which they inhibit. This coefficient appears to be bounded below by the requirement for oscillations in individual cells and bounded above by the requirement for synchronisation. Consistent with experimental data and a previous spatially non-explicit mathematical model, we have found that signalling can increase the average level of Her1 protein. Biological pattern formation would be impossible without a certain robustness to variety in cell shape and size; our results possess such robustness. Our spatially-explicit modelling approach, together with new imaging technologies that can measure intracellular protein diffusion rates, is likely to yield significant new insight into somitogenesis and other biological processes.

Intercellular coupling regulates the period of the segmentation clock

BACKGROUND

Coupled biological oscillators can tick with the same period. How this collective period is established is a key question in understanding biological clocks. We explore this question in the segmentation clock, a population of coupled cellular oscillators in the vertebrate embryo that sets the rhythm of somitogenesis, the morphological segmentation of the body axis. The oscillating cells of the zebrafish segmentation clock are thought to possess noisy autonomous periods, which are synchronized by intercellular coupling through the Delta-Notch pathway. Here we ask whether Delta-Notch coupling additionally influences the collective period of the segmentation clock.

RESULTS

Using multiple-embryo time-lapse microscopy, we show that disruption of Delta-Notch intercellular coupling increases the period of zebrafish somitogenesis. Embryonic segment length and the spatial wavelength of oscillating gene expression also increase correspondingly, indicating an increase in the segmentation clock's period. Using a theory based on phase oscillators in which the collective period self-organizes because of time delays in coupling, we estimate the cell-autonomous period, the coupling strength, and the coupling delay from our data. Further supporting the role of coupling delays in the clock, we predict and experimentally confirm an instability resulting from decreased coupling delay time.

CONCLUSIONS

Synchronization of cells by Delta-Notch coupling regulates the collective period of the segmentation clock. Our identification of the first segmentation clock period mutants is a critical step toward a molecular understanding of temporal control in this system. We propose that collective control of period via delayed coupling may be a general feature of biological clocks.

Random cell movement promotes synchronization of the segmentation clock

In vertebrate somitogenesis, the expression of segmentation clock genes oscillates and the oscillation is synchronized over nearby cells. Both experimental and theoretical studies have shown that the synchronization among cells is realized by intercellular interaction via Delta-Notch signaling. However, the following questions emerge: (i) During somitogenesis, dynamic rearrangement of relative cell positions is observed in the posterior presomitic mesoderm. Can a synchronized state be stably sustained under random cell movement? (ii) Experimental studies have reported that the synchronization of cells can be recovered in about 10 or fewer oscillation cycles after the complete loss of synchrony. However, such a quick recovery of synchronization is not possible according to previous theoretical models. In this paper, we first show by numerical modeling that synchronized oscillation can be sustained under random cell movement. We also find that for initial perturbation, the synchronization of cells is recovered much faster and it is for a wider range of reaction parameters than the case without cell movement. When the posterior presomitic mesoderm is rectangular, faster synchronization is achieved if cells exchange their locations more with neighbors located along the longer side of the domain. Finally, we discuss that the enhancement of synchronization by random cell movement occurs in several different models for the oscillation of segmentation clock genes.

Harmonic resonant excitation of flow-distributed oscillation waves and Turing patterns driven at a growing boundary

We perform numerical studies of a reaction-diffusion system that is both Turing and Hopf unstable, and that grows by addition at a moving boundary (which is equivalent by a Galilean transformation to a reaction-diffusion-advection system with a fixed boundary and a uniform flow). We model the conditions of a recent set of experiments which used a temporally varying illumination in the boundary region to control the formation of patterns in the bulk of the photosensitive medium. The frequency of the illumination variations can select patterns from among the competing instabilities of the medium. In the usual case, the waves that are excited have frequencies (as measured at a constant distance from the upstream boundary) matching the driving frequency. In contrast to the usual case, we find that both Turing patterns and flow-distributed oscillation waves can be excited by forcing at subharmonic multiples of the wave frequencies. The final waves (with frequencies at integer multiples of the driving frequency) are formed by a process in which transient wave fronts break up and reconnect. We find ratios of response to driving frequency as high as 10.

Selection of flow-distributed oscillation and Turing patterns by boundary forcing in a linearly growing, oscillating medium

We studied the response of a linearly growing domain of the oscillatory chemical chlorine dioxide-iodide-malonic acid (CDIMA) medium to periodic forcing at its growth boundary. The medium is Hopf-, as well as Turing-unstable and the system is convectively unstable. The results confirm numerical predictions that two distinct modes of pattern can be excited by controlling the driving frequency at the boundary, a flow-distributed-oscillation (FDO) mode of traveling waves at low values of the forcing frequency f , and a mode of stationary Turing patterns at high values of f . The wavelengths and phase velocities of the experimental patterns were compared quantitatively with results from dynamical simulations and with predictions from linear dispersion relations. The results for the FDO waves agreed well with these predictions, and obeyed the kinematic relations expected for phase waves with frequencies selected by the boundary driving frequency. Turing patterns were also generated within the predicted range of forcing frequencies, but these developed into two-dimensional structures which are not fully accounted for by the one-dimensional numerical and analytical models. The Turing patterns excited by boundary forcing persist when the forcing is removed, demonstrating the bistability of the unforced, constant size medium. Dynamical simulations at perturbation frequencies other than those of the experiments showed that in certain ranges of forcing frequency, FDO waves become unstable, breaking up into harmonic waves of different frequency and wavelength and phase velocity.

Traveling wave formation in vertebrate segmentation

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

Delayed coupling theory of vertebrate segmentation

Rhythmic and sequential subdivision of the elongating vertebrate embryonic body axis into morphological somites is controlled by an oscillating multicellular genetic network termed the segmentation clock. This clock operates in the presomitic mesoderm (PSM), generating dynamic stripe patterns of oscillatory gene-expression across the field of PSM cells. How these spatial patterns, the clock's collective period, and the underlying cellular-level interactions are related is not understood. A theory encompassing temporal and spatial domains of local and collective aspects of the system is essential to tackle these questions. Our delayed coupling theory achieves this by representing the PSM as an array of phase oscillators, combining four key elements: a frequency profile of oscillators slowing across the PSM; coupling between neighboring oscillators; delay in coupling; and a moving boundary describing embryonic axis elongation. This theory predicts that the segmentation clock's collective period depends on delayed coupling. We derive an expression for pattern wavelength across the PSM and show how this can be used to fit dynamic wildtype gene-expression patterns, revealing the quantitative values of parameters controlling spatial and temporal organization of the oscillators in the system. Our theory can be used to analyze experimental perturbations, thereby identifying roles of genes involved in segmentation.

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

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

Practical Lessons from Theoretical Models about the Somitogenesis

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

Experimental steady pattern formation in reaction-diffusion-advection systems

We present here experimental evidence of a mechanism of a steady-chemical pattern formation called "flow-and-diffusion structures" (FDS). Experiments were performed using the photosensitive chlorine dioxide-iodine-malonic acid reaction, where the differential diffusion can be chemically controlled. Using the analogy between an advection boundary and a moving boundary, we obtain the formation of spatially periodic steady patterns, which matches all the previously theoretical predictions for FDS patterns. Numerical simulations are also reported in agreement with the experimental results.

Robustness and stability of flow-and-diffusion structures

Reaction-diffusion-advection systems have revealed an interesting variety of pattern formation mechanism during the last years. Inside this field, flow-and-diffusion structures (FDSs) appear as a generalization of the mechanism of spatial symmetry breaking for different diffusion coefficients and flow rates of activator and inhibitor. The recent experimental validation of FDSs situates these structures in the focus of the actual research. We will report here an experimental and numerical analysis of the theoretically predicted robustness of these flow-and-diffusion structures by using different boundary profiles of illumination used to obtain FDSs. The results here shown reveal important characteristics related with the coexistence and interaction between these structures.

Flow-distributed oscillation, flow-velocity modulation, and resonance

We examine the effects of a periodically varying flow velocity on the standing- and traveling-wave patterns formed by the flow-distributed oscillation mechanism. In the kinematic (or diffusionless) limit, the phase fronts undergo a simple, spatiotemporally periodic longitudinal displacement. On the other hand, when the diffusion is significant, periodic modulation of the velocity can disrupt the wave pattern, giving rise in the downstream region to traveling waves whose frequency is a rational multiple of the velocity perturbation frequency. We observe frequency locking at ratios of 1:1, 2:1, and 3:1, depending on the amplitude and frequency of the velocity modulation. This phenomenon can be viewed as a novel, rather subtle type of resonant forcing.

Pattern formation in a two-dimensional reaction-diffusion channel with Poiseuille flow

We study stationary patterns arising from a combination of flow and diffusion in a two-dimensional (2D) reaction-diffusion system in a channel with Poiseuille flow. Both transverse and longitudinal modes are investigated and compared with numerical computations.

Magnetic Resonance Imaging of Flow-Distributed Oscillations

The formation of stationary concentration patterns in a packed-bed reactor (PBR), using a manganese-catalyzed Belousov-Zhabotinsky (BZ) reaction in a mixed sulfuric-phosphoric acid medium, was studied using magnetic resonance imaging (MRI). The PBR was composed of a column filled with glass beads, which was fed by a continuous stirred tank reactor (CSTR). As the reactor is optically opaque, investigation of the three-dimensional (3D) structure of these reaction-diffusion-advection waves is not possible using conventional image capture techniques. MRI has been used to probe this system and the formation, 3D structure, and development of these waves has been studied. At reactor startup, traveling waves were observed. After this initial period the waves stabilized and became stationary. Once fixed, they were found to be remarkably stable. There was significant heterogeneity of the reaction fronts, which were not flat, as would be expected from a plug-flow reactor. Instead, the reaction wave fronts were observed to be conical in shape due to the local hydrodynamics of the bed and specifically the higher velocities and therefore lower residence times close to the wall of the reactor.

Pattern formation by boundary forcing in convectively unstable, oscillatory media with and without differential transport

Convectively unstable, open reactive flows of oscillatory media, whose phase is fixed or periodically modulated at the inflow boundary, are known to result in stationary and traveling waves, respectively. The latter are implicated in biological segmentation. The boundary-controlled pattern selection by this flow-distributed oscillator (FDO) mechanism has been generalized to include differential flow (DIFI) and differential diffusion (Turing) modes. Our present goal is to clarify the relationships among these mechanisms in the general case where there is differential flow as well as differential diffusion. To do so we analyze the dispersion relation for linear perturbations in the presence of periodic boundary forcing, and show how the solutions are affected by differential transport. We find that the DIFI and FDO modes are closely related and lie in the same frequency range, while the Turing mechanism gives rise to a distinct set of unstable modes in a separate frequency range. Finally, we substantiate the linear analysis by nonlinear simulations and touch upon the issue of competition of spatial modes.

Control of chemical pattern formation by a clock-and-wavefront type mechanism

The segmentation of many animals ranging from insects to mammals involves the sequential formation of stationary stripes of gene expression that are perpendicular to the growth axis of the developing embryo. This process has been accounted for by a variety of theoretical "clock-and-wavefront" type models that involve the arrest of an oscillation (the clock) at a moving boundary (the wavefront). Here, we demonstrate experimentally that progressive arrest of a homogeneous oscillation can control the symmetry as well as the wavelength of spatial structures in a chemical system. We show how a spontaneously formed, labyrinthine pattern can be converted into a pattern composed of ordered, parallel stripes and confirm a previously predicted proportionality between the wavelength and the period of the homogeneous oscillation. Our experiments provide the first experimental demonstration of a general mechanism for the control of pattern formation that has been hypothesized to operate in the context of biological morphogenesis.

A Notch feeling of somite segmentation and beyond

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

Segmentation and zooid formation in animals with a posterior growing region: the case for metabolic gradients and Turing waves

The discovery of periodic propagation of anteriorly moving pulses/stripes of gene expression in the presomitic mesoderm (PSM) of vertebrates has given new life to the clock and wavefront model, and other models of morphogenesis based on a molecular oscillator where the time periodicity is translated into spatial periodicity. Instead we suggest that segmentation, somitogenesis and metamerism in vertebrates and in invertebrates with a posterior growing region are based on a Turing-Child metabolic gradient that is progressively shifted posteriorly with the PSM as elongation, segmentation and somitogenesis proceed. This gradient corresponds to anteriorly propagating metabolic front in the PSM that drives the anteriorly propagating mRNA synthesis and which, together with mRNA degradation, explains stripe formation and spatial periodicity.The process of segmentation has been compared to zooid formation. We show that for annelids the metabolic profile behaves as a Turing field in the sense that an increase in the length of the system or a decrease of the Turing wavelength results in an additional peak in the posterior growing region as predicted by Turing theory. In particular, it is shown that the metabolic gradient that drives the segmentation is based on a Turing system.

General theory of nonlinear flow-distributed oscillations

We outline a general theory for the analysis of flow-distributed standing and traveling wave patterns in one-dimensional, open flows of oscillatory chemical media, emphasizing features that are generic to a variety of kinetic models. We draw particular attention to the cases far from a Hopf bifurcation and far from the so-called kinematic or zero-diffusion limit. We introduce a nonlinear formalism for both traveling and stationary waves and show that the wave forms and their amplitudes depend on a single reduced transport parameter that quantifies the departure from the kinematic limit. The nonlinear formalism can be applied to systems with more complex types of bifurcations (canards, period doublings, etc.).

Is the somitogenesis clock really cell-autonomous? A coupled-oscillator model of segmentation

A striking pattern of oscillatory gene expression, related to the segmentation process (somitogenesis), has been identified in chick, mouse, and zebrafish embryos. Somitogenesis displays great autonomy, and it is generally assumed in the literature that somitogenesis-related oscillations are cell-autonomous in chick and mouse. We point out in this article that there would be many biological reasons to expect some mechanism of coupling between cellular oscillators, and we present a model with such coupling, but which also has autonomous properties. Previous experiments can be re-interpreted in light of this model, showing that it is possible to reconcile both autonomous and non-autonomous aspects. We also show that experimental data, previously interpreted as supporting a purely negative-feedback model for the mechanism of the oscillations, is in fact more compatible with this new model, which relies essentially on positive feedback.

Coexistence of stationary and traveling waves in reaction-diffusion-advection systems

The flow- and diffusion-distributed structures (FDS) and the differential-flow instability (DIFI) are mechanisms that give rise to static and traveling waves in reactive flows with general, species-dependent transport terms. Here we consider a general framework which supports the simultaneous existence of FDS and DIFI patterns. We study the necessary conditions for each instability in general and compare them in order to derive their connection. The interaction between FDS and DIFI patterns gives rise to interesting wave behavior including stationary, upstream, and downstream traveling waves as well as an interesting regime where stationary and traveling waves coexist.

Hairy/E(spl)-related(Her) genes are central components of the segmentation oscillator and display redundancy with the Delta/Notch signaling pathway in the formation of anterior segmental boundaries in the zebrafish

We have examined the expression of a Hairy/E(spl)-related (Her) gene, her7, in the zebrafish and show that its expression in the PSM cycles similarly to her1 and deltaC. A decrease in her7 function generated by antisense oligonucleotides disrupts somite formation in the posterior trunk and tail, and disrupts the dynamic expression domains of her1 and deltaC, suggesting that her7 plays a role in coordinating the oscillations of neighboring cells in the presomitic mesoderm. This phenotype is reminiscent of zebrafish segmentation mutants with lesions in genes of the Delta/Notch signaling pathway, which also show a disruption of cyclic her7 expression. The interaction of HER genes with the Delta/Notch signaling system was investigated by introducing a loss of her7 function into mutant backgrounds. This leads to segmental defects more anterior than in either condition alone. Combining a decrease of her7 function with reduction of her1 function results in an enhanced phenotype that affects all the anterior segments, indicating that Her functions in the anterior segments are also partially redundant. In these animals, gene expression does not cycle at any time, suggesting that a complete loss of oscillator function had been achieved. Consistent with this, combining a reduction of her7 and her1 function with a Delta/Notch mutant genotype does not worsen the phenotype further. Thus, our results identify members of the Her family of transcription factors that together behave as a central component of the oscillator, and not as an output. This indicates, therefore, that the function of the segmentation oscillator is restricted to the positioning of segmental boundaries. Furthermore, our data suggest that redundancy between Her genes and genes of the Delta/Notch pathway is in part responsible for the robust formation of anterior somites in vertebrates.

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

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

A cellular oscillator model for periodic pattern formation

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