The clock and wavefront model revisited

A Cellular Potts Model of the interplay of synchronization and aggregation

We investigate the behavior of systems of cells with intracellular molecular oscillators ("clocks") where cell-cell adhesion is mediated by differences in clock phase between neighbors. This is motivated by phenomena in developmental biology and in aggregative multicellularity of unicellular organisms. In such systems, aggregation co-occurs with clock synchronization. To account for the effects of spatially extended cells, we use the Cellular Potts Model (CPM), a lattice agent-based model. We find four distinct possible phases: global synchronization, local synchronization, incoherence, and anti-synchronization (checkerboard patterns). We characterize these phases via order parameters. In the case of global synchrony, the speed of synchronization depends on the adhesive effects of the clocks. Synchronization happens fastest when cells in opposite phases adhere the strongest ("opposites attract"). When cells of the same clock phase adhere the strongest ("like attracts like"), synchronization is slower. Surprisingly, the slowest synchronization happens in the diffusive mixing case, where cell-cell adhesion is independent of clock phase. We briefly discuss potential applications of the model, such as pattern formation in the auditory sensory epithelium.

Unidirectional and phase-gated signaling synchronizes murine presomitic mesoderm cells

Oscillator systems achieve synchronization when oscillators are coupled. The presomitic mesoderm is a system of cellular oscillators, where coordinated genetic activity is necessary for proper periodic generation of somites. While Notch signaling is required for the synchronization of these cells, it is unclear what information the cells exchange and how they react to this information to align their oscillatory pace with that of their neighbors. Combining mathematical modeling and experimental data, we found that interaction between murine presomitic mesoderm cells is controlled by a phase-gated and unidirectional coupling mechanism and results in deceleration of their oscillation pace upon Notch signaling. This mechanism predicts that isolated populations of well-mixed cells synchronize, revealing a stereotypical synchronization in the mouse PSM and contradicting expectations from previously applied theoretical approaches. Collectively, our theoretical and experimental findings reveal the underlying coupling mechanisms of the presomitic mesoderm cells and provide a framework to quantitatively characterize their synchronization.

Synthetic spatial patterning in bacteria: advances based on novel diffusible signals

Engineering multicellular patterning may help in the understanding of some fundamental laws of pattern formation and thus may contribute to the field of developmental biology. Furthermore, advanced spatial control over gene expression may revolutionize fields such as medicine, through organoid or tissue engineering. To date, foundational advances in spatial synthetic biology have often been made in prokaryotes, using artificial gene circuits. In this review, engineered patterns are classified into four levels of increasing complexity, ranging from spatial systems with no diffusible signals to systems with complex multi-diffusor interactions. This classification highlights how the field was held back by a lack of diffusible components. Consequently, we provide a summary of both previously characterized and some new potential candidate small-molecule signals that can regulate gene expression in Escherichia coli. These diffusive signals will help synthetic biologists to successfully engineer increasingly intricate, robust and tuneable spatial structures.

A segmentation clock patterns cellular differentiation in a bacterial biofilm

Contrary to multicellular organisms that display segmentation during development, communities of unicellular organisms are believed to be devoid of such sophisticated patterning. Unexpectedly, we find that the gene expression underlying the nitrogen stress response of a developing Bacillus subtilis biofilm becomes organized into a ring-like pattern. Mathematical modeling and genetic probing of the underlying circuit indicate that this patterning is generated by a clock and wavefront mechanism, similar to that driving vertebrate somitogenesis. We experimentally validated this hypothesis by showing that predicted nutrient conditions can even lead to multiple concentric rings, resembling segments. We additionally confirmed that this patterning mechanism is driven by cell-autonomous oscillations. Importantly, we show that the clock and wavefront process also spatially patterns sporulation within the biofilm. Together, these findings reveal a biofilm segmentation clock that organizes cellular differentiation in space and time, thereby challenging the paradigm that such patterning mechanisms are exclusive to plant and animal development.

Morphogen-regulated contact-mediated signaling between cells can drive the transitions underlying body segmentation in vertebrates

We propose a unified mechanism that reproduces the sequence of dynamical transitions observed during somitogenesis, the process of body segmentation during embryonic development, that is invariant across all vertebrate species. This is achieved by combining inter-cellular interactions mediated via receptor-ligand coupling with global spatial heterogeneity introduced through a morphogen gradient known to occur along the anteroposterior axis. Our model reproduces synchronized oscillations in the gene expression in cells at the anterior of the presomitic mesoderm as it grows by adding new cells at its posterior, followed by travelling waves and subsequent arrest of activity, with the eventual appearance of somite-like patterns. This framework integrates a boundary-organized pattern formation mechanism, which uses positional information provided by a morphogen gradient, with the coupling-mediated self-organized emergence of collective dynamics, to explain the processes that lead to segmentation.

Time and space in segmentation

Arthropod segmentation and vertebrate somitogenesis are leading fields in the experimental and theoretical interrogation of developmental patterning. However, despite the sophistication of current research, basic conceptual issues remain unresolved. These include: (i) the mechanistic origins of spatial organization within the segment addition zone (SAZ); (ii) the mechanistic origins of segment polarization; (iii) the mechanistic origins of axial variation; and (iv) the evolutionary origins of simultaneous patterning. Here, I explore these problems using coarse-grained models of cross-regulating dynamical processes. In the morphogenetic framework of a row of cells undergoing axial elongation, I simulate interactions between an 'oscillator', a 'switch' and up to three 'timers', successfully reproducing essential patterning behaviours of segmenting systems. By comparing the output of these largely cell-autonomous models to variants that incorporate positional information, I find that scaling relationships, wave patterns and patterning dynamics all depend on whether the SAZ is regulated by temporal or spatial information. I also identify three mechanisms for polarizing oscillator output, all of which functionally implicate the oscillator frequency profile. Finally, I demonstrate significant dynamical and regulatory continuity between sequential and simultaneous modes of segmentation. I discuss these results in the context of the experimental literature.

Segmentation clock dynamics is strongly synchronized in the forming somite

During vertebrate somitogenesis an inherent segmentation clock coordinates the spatiotemporal signaling to generate segmented structures that pattern the body axis. Using our experimental and quantitative approach, we study the cell movements and the genetic oscillations of her1 expression level at single-cell resolution simultaneously and scale up to the entire pre-somitic mesoderm (PSM) tissue. From the experimentally determined phases of PSM cellular oscillators, we deduced an in vivo frequency profile gradient along the anterior-posterior PSM axis and inferred precise mathematical relations between spatial cell-level period and tissue-level somitogenesis period. We also confirmed a gradient in the relative velocities of cellular oscillators along the axis. The phase order parameter within an ensemble of oscillators revealed the degree of synchronization in the tailbud and the posterior PSM being only partial, whereas synchronization can be almost complete in the presumptive somite region but with temporal oscillations. Collectively, the degree of synchronization itself, possibly regulated by cell movement and the synchronized temporal phase of the transiently expressed clock protein Her1, can be an additional control mechanism for making precise somite boundaries.

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

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

Redox Is a Global Biodevice Information Processing Modality

Biology is well-known for its ability to communicate through (i) molecularly-specific signaling modalities and (ii) a globally-acting electrical modality associated with ion flow across biological membranes. Emerging research suggests that biology uses a third type of communication modality associated with a flow of electrons through reduction/oxidation (redox) reactions. This redox signaling modality appears to act globally and has features of both molecular and electrical modalities: since free electrons do not exist in aqueous solution, the electrons must flow through molecular intermediates that can be switched between two states - with electrons (reduced) or without electrons (oxidized). Importantly, this global redox modality is easily accessible through its electrical features using convenient electrochemical instrumentation. In this review, we explain this redox modality, describe our electrochemical measurements, and provide four examples demonstrating that redox enables communication between biology and electronics. The first two examples illustrate how redox probing can acquire biologically relevant information. The last two examples illustrate how redox inputs can transduce biologically-relevant transitions for patterning and the induction of a synbio transceiver for two-hop molecular communication. In summary, we believe redox provides a unique ability to bridge bio-device communication because simple electrochemical methods enable global access to biologically meaningful information. Further, we envision that redox may facilitate the application of information theory to the biological sciences.

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.

Information flow in the presence of cell mixing and signaling delays during embryonic development

Embryonic morphogenesis is organized by an interplay between intercellular signaling and cell movements. Both intercellular signaling and cell movement involve multiple timescales. A key timescale for signaling is the time delay caused by preparation of signaling molecules and integration of received signals into cells' internal state. Movement of cells relative to their neighbors may introduce exchange of positions between cells during signaling. When cells change their relative positions in a tissue, the impact of signaling delays on intercellular signaling increases because the delayed information that cells receive may significantly differ from the present state of the tissue. The time it takes to perform a neighbor exchange sets a timescale of cell mixing that may be important for the outcome of signaling. Here we review recent theoretical work on the interplay of timescales between cell mixing and signaling delays adopting the zebrafish segmentation clock as a model system. We discuss how this interplay can lead to spatial patterns of gene expression that could disrupt the normal formation of segment boundaries in the embryo. The effect of cell mixing and signaling delays highlights the importance of theoretical and experimental frameworks to understand collective cellular behaviors arising from the interplay of multiple timescales in embryonic developmental processes.

Recent advances in understanding vertebrate segmentation

Segmentation is the partitioning of the body axis into a series of repeating units or segments. This widespread body plan is found in annelids, arthropods, and chordates, showing it to be a successful developmental strategy for growing and generating diverse morphology and anatomy. Segmentation has been extensively studied over the years. Forty years ago, Cooke and Zeeman published the Clock and Wavefront model, creating a theoretical framework of how developing cells could acquire and keep temporal and spatial information in order to generate a segmented pattern. Twenty years later, in 1997, Palmeirim and co-workers found the first clock gene whose oscillatory expression pattern fitted within Cooke and Zeeman's model. Currently, in 2017, new experimental techniques, such as new ex vivo experimental models, real-time imaging of gene expression, live single cell tracking, and simplified transgenics approaches, are revealing some of the fine details of the molecular processes underlying the inner workings of the segmentation mechanisms, bringing new insights into this fundamental process. Here we review and discuss new emerging views that further our understanding of the vertebrate segmentation clock, with a particular emphasis on recent publications that challenge and/or complement the currently accepted Clock and Wavefront model.

Electrical Programming of Soft Matter: Using Temporally Varying Electrical Inputs To Spatially Control Self Assembly

The growing importance of hydrogels in translational medicine has stimulated the development of top-down fabrication methods, yet often these methods lack the capabilities to generate the complex matrix architectures observed in biology. Here we show that temporally varying electrical signals can cue a self-assembling polysaccharide to controllably form a hydrogel with complex internal patterns. Evidence from theory and experiment indicate that internal structure emerges through a subtle interplay between the electrical current that triggers self-assembly and the electrical potential (or electric field) that recruits and appears to orient the polysaccharide chains at the growing gel front. These studies demonstrate that short sequences (minutes) of low-power (∼1 V) electrical inputs can provide the program to guide self-assembly that yields hydrogels with stable, complex, and spatially varying structure and properties.

Stochastic models of gene transcription with upstream drives: exact solution and sample path characterization

Gene transcription is a highly stochastic and dynamic process. As a result, the mRNA copy number of a given gene is heterogeneous both between cells and across time. We present a framework to model gene transcription in populations of cells with time-varying (stochastic or deterministic) transcription and degradation rates. Such rates can be understood as upstream cellular drives representing the effect of different aspects of the cellular environment. We show that the full solution of the master equation contains two components: a model-specific, upstream effective drive, which encapsulates the effect of cellular drives (e.g. entrainment, periodicity or promoter randomness) and a downstream transcriptional Poissonian part, which is common to all models. Our analytical framework treats cell-to-cell and dynamic variability consistently, unifying several approaches in the literature. We apply the obtained solution to characterize different models of experimental relevance, and to explain the influence on gene transcription of synchrony, stationarity, ergodicity, as well as the effect of time scales and other dynamic characteristics of drives. We also show how the solution can be applied to the analysis of noise sources in single-cell data, and to reduce the computational cost of stochastic simulations.

A framework for quantification and physical modeling of cell mixing applied to oscillator synchronization in vertebrate somitogenesis

In development and disease, cells move as they exchange signals. One example is found in vertebrate development, during which the timing of segment formation is set by a 'segmentation clock', in which oscillating gene expression is synchronized across a population of cells by Delta-Notch signaling. Delta-Notch signaling requires local cell-cell contact, but in the zebrafish embryonic tailbud, oscillating cells move rapidly, exchanging neighbors. Previous theoretical studies proposed that this relative movement or cell mixing might alter signaling and thereby enhance synchronization. However, it remains unclear whether the mixing timescale in the tissue is in the right range for this effect, because a framework to reliably measure the mixing timescale and compare it with signaling timescale is lacking. Here, we develop such a framework using a quantitative description of cell mixing without the need for an external reference frame and constructing a physical model of cell movement based on the data. Numerical simulations show that mixing with experimentally observed statistics enhances synchronization of coupled phase oscillators, suggesting that mixing in the tailbud is fast enough to affect the coherence of rhythmic gene expression. Our approach will find general application in analyzing the relative movements of communicating cells during development and disease.

Modelling asymmetric somitogenesis: Deciphering the mechanisms behind species differences

Somitogenesis is one of the major hallmarks of bilateral symmetry in vertebrates. This symmetry is lost when retinoic acid (RA) signalling is inhibited, allowing the left-right determination pathway to influence somitogenesis. In all three studied vertebrate model species, zebrafish, chicken and mouse, the frequency of somite formation becomes asymmetric, with slower gene expression oscillations driving somitogenesis on the right side. Still, intriguingly, the resulting left-right asymmetric phenotypes differ significantly between these model species. While somitogenesis is generally considered as functionally equivalent among different vertebrates, substantial differences exist in the subset of oscillating genes between different vertebrate species. Variation also appears to exist in the way oscillations cease and somite boundaries become patterned. In addition, in absence of RA, the FGF8 gradient thought to constitute the determination wavefront becomes asymmetric in zebrafish and mouse, extending more anteriorly to the right, while remaining symmetric in chicken. Here we use a computational modelling approach to decipher the causes underlying species differences in asymmetric somitogenesis. Specifically, we investigate to what extent differences can be explained from observed differences in FGF asymmetry and whether differences in somite determination dynamics may also be involved. We demonstrate that a simple clock-and-wavefront model incorporating the observed left-right differences in somitogenesis frequency readily reproduces asymmetric somitogenesis in chicken. However, incorporating asymmetry in FGF signalling was insufficient to robustly reproduce mouse or zebrafish asymmetry phenotypes. In order to explain these phenoptypes we needed to extend the basic model, incorporating species-specific details of the somitogenesis determination mechanism. Our results thus demonstrate that a combination of differences in FGF dynamics and somite determination cause species differences in asymmetric somitogenesis. In addition,they highlight the power of using computational models as well as studying left-right asymmetry to obtain more insight in somitogenesis.

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.

The role of driver nodes in managing epileptic seizures: Application of Kuramoto model

Synchronization is an important global phenomenon which could be found in a wide range of complex systems such as brain or electronic devices. However, in some circumstances the synchronized states are not desirable for the system and should be suppressed. For example, excessively synchronized activities in the brain network could be the root of neuronal disorders like epileptic seizures. According to the controllability theory of the complex networks, a minimum set of driver nodes has the ability to control the entire system. In this study, we examine the role of driver nodes in suppressing the excessive synchronization in a generalized Kuramoto model, which consists of two types of oscillators: contrarian and regular ones. We used two different structural topologies: Barabási-Albert scale-free (BASF) network and Caenorhabditis elegans (C.elegans) neuronal network. Our results show that contrarian driver nodes have the sufficient ability to break the synchronized level of the systems. In this case, the system coherency level is not fully suppressed that is avoiding dysfunctions of normal brain functions which require the neuronal synchronized activities. Moreover, in this case, the oscillators grouped in two distinct synchronized clusters that could be an indication of chaotic behavior of the system known as resting-state activity of the brain.

Sequential pattern formation governed by signaling gradients

Rhythmic and sequential segmentation of the embryonic body plan is a vital developmental patterning process in all vertebrate species. However, a theoretical framework capturing the emergence of dynamic patterns of gene expression from the interplay of cell oscillations with tissue elongation and shortening and with signaling gradients, is still missing. Here we show that a set of coupled genetic oscillators in an elongating tissue that is regulated by diffusing and advected signaling molecules can account for segmentation as a self-organized patterning process. This system can form a finite number of segments and the dynamics of segmentation and the total number of segments formed depend strongly on kinetic parameters describing tissue elongation and signaling molecules. The model accounts for existing experimental perturbations to signaling gradients, and makes testable predictions about novel perturbations. The variety of different patterns formed in our model can account for the variability of segmentation between different animal species.

Positional information from oscillatory phase shifts : insights from in silico evolution

Complex cellular decisions are based on temporal dynamics of pathways, including genetic oscillators. In development, recent works on vertebrae formation have suggested that relative phase of genetic oscillators encode positional information, including differentiation front defining vertebrae positions. Precise mechanisms for this are still unknown. Here, we use computational evolution to find gene network topologies that can compute the phase difference between oscillators and convert it into a decoder morphogen concentration. Two types of networks are discovered, based on symmetry properties of the decoder gene. So called asymmetric networks are studied, and two submodules are identified converting phase information into an amplitude variable. Those networks naturally display a 'shock' for a well defined phase difference, that can be used to define a wavefront of differentiation. We show how implementation of these ideas reproduce experimental features of vertebrate segmentation.

Faster embryonic segmentation through elevated Delta-Notch signalling

An important step in understanding biological rhythms is the control of period. A multicellular, rhythmic patterning system termed the segmentation clock is thought to govern the sequential production of the vertebrate embryo's body segments, the somites. Several genetic loss-of-function conditions, including the Delta-Notch intercellular signalling mutants, result in slower segmentation. Here, we generate DeltaD transgenic zebrafish lines with a range of copy numbers and correspondingly increased signalling levels, and observe faster segmentation. The highest-expressing line shows an altered oscillating gene expression wave pattern and shortened segmentation period, producing embryos with more, shorter body segments. Our results reveal surprising differences in how Notch signalling strength is quantitatively interpreted in different organ systems, and suggest a role for intercellular communication in regulating the output period of the segmentation clock by altering its spatial pattern.

Rhythmic patterning governs the formation of somites in vertebrates, but how the period of such rhythms can be changed is unclear. Here, the authors generate a genetic model in zebrafish to increase DeltaD expression, which increases the range of Delta-Notch signalling, causing faster segmentation.

A Local, Self-Organizing Reaction-Diffusion Model Can Explain Somite Patterning in Embryos

During somitogenesis in embryos, a posteriorly moving differentiation front arrests the oscillations of "segmentation clock" genes, leaving behind a frozen, periodic pattern of expression stripes. Both mathematical theories and experimental observations have invoked a "clock and wavefront" model to explain this phenomenon, in which long-range molecular gradients control the movement of the front and therefore the placement of the stripes in the embryo. Here, we develop a fundamentally different model-a progressive oscillatory reaction-diffusion (PORD) system driven by short-range interactions. In this model, posterior movement of the front is a local, emergent phenomenon that, in contrast to the clock and wavefront model, is not controlled by global positional information. The PORD model explains important features of somitogenesis, such as size regulation, that previous reaction-diffusion models could not explain. Moreover, the PORD and clock and wavefront models make different predictions about the results of FGF-inhibition and tissue-cutting experiments, and we demonstrate that the results of these experiments favor the PORD model.

Continuum theory of gene expression waves during vertebrate segmentation

The segmentation of the vertebrate body plan during embryonic development is a rhythmic and sequential process governed by genetic oscillations. These genetic oscillations give rise to traveling waves of gene expression in the segmenting tissue. Here we present a minimal continuum theory of vertebrate segmentation that captures the key principles governing the dynamic patterns of gene expression including the effects of shortening of the oscillating tissue. We show that our theory can quantitatively account for the key features of segmentation observed in zebrafish, in particular the shape of the wave patterns, the period of segmentation and the segment length as a function of time.

Multiple roles of timing in somite formation

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

Nonlinear transient waves in coupled phase oscillators with inertia

Like the inertia of a physical body describes its tendency to resist changes of its state of motion, inertia of an oscillator describes its tendency to resist changes of its frequency. Here, we show that finite inertia of individual oscillators enables nonlinear phase waves in spatially extended coupled systems. Using a discrete model of coupled phase oscillators with inertia, we investigate these wave phenomena numerically, complemented by a continuum approximation that permits the analytical description of the key features of wave propagation in the long-wavelength limit. The ability to exhibit traveling waves is a generic feature of systems with finite inertia and is independent of the details of the coupling function.

Spatiotemporal oscillations of Notch1, Dll1 and NICD are coordinated across the mouse PSM

During somitogenesis, epithelial somites form from the pre-somitic mesoderm (PSM) in a periodic manner. This periodicity is regulated by a molecular oscillator, known as the ‘segmentation clock’, that is characterised by an oscillatory pattern of gene expression that sweeps the PSM in a caudal-rostral direction. Key components of the segmentation clock are intracellular components of the Notch, Wnt and FGF pathways, and it is widely accepted that intracellular negative-feedback loops regulate oscillatory gene expression. However, an open question in the field is how intracellular oscillations are coordinated, in the form of spatiotemporal waves of expression, across the PSM. In this study, we provide a potential mechanism for this process. We show at the mRNA level that the Notch1 receptor and Delta-like 1 (Dll1) ligand vary dynamically across the PSM of both chick and mouse. Remarkably, we also demonstrate similar dynamics at the protein level; hence, the pathway components that mediate intercellular coupling themselves exhibit oscillatory dynamics. Moreover, we quantify the dynamic expression patterns of Dll1 and Notch1, and show they are highly correlated with the expression patterns of two known clock components [Lfng mRNA and the activated form of the Notch receptor (cleaved Notch intracellular domain, NICD)]. Lastly, we show that Notch1 is a target of Notch signalling, whereas Dll1 is Wnt regulated. Regulation of Dll1 and Notch1 expression thus links the activity of Wnt and Notch, the two main signalling pathways driving the clock.

Genetic oscillations. A Doppler effect in embryonic pattern formation

During embryonic development, temporal and spatial cues are coordinated to generate a segmented body axis. In sequentially segmenting animals, the rhythm of segmentation is reported to be controlled by the time scale of genetic oscillations that periodically trigger new segment formation. However, we present real-time measurements of genetic oscillations in zebrafish embryos showing that their time scale is not sufficient to explain the temporal period of segmentation. A second time scale, the rate of tissue shortening, contributes to the period of segmentation through a Doppler effect. This contribution is modulated by a gradual change in the oscillation profile across the tissue. We conclude that the rhythm of segmentation is an emergent property controlled by the time scale of genetic oscillations, the change of oscillation profile, and tissue shortening.

Wnt-regulated dynamics of positional information in zebrafish somitogenesis

How signaling gradients supply positional information in a field of moving cells is an unsolved question in patterning and morphogenesis. Here, we ask how a Wnt signaling gradient regulates the dynamics of a wavefront of cellular change in a flow of cells during somitogenesis. Using time-controlled perturbations of Wnt signaling in the zebrafish embryo, we changed segment length without altering the rate of somite formation or embryonic elongation. This result implies specific Wnt regulation of the wavefront velocity. The observed Wnt signaling gradient dynamics and timing of downstream events support a model for wavefront regulation in which cell flow plays a dominant role in transporting positional information.

Mechanisms and constraints shaping the evolution of body plan segmentation

Segmentation of the major body axis into repeating units is arguably one of the major inventions in the evolution of animal body plan pattering. It is found in current day vertebrates, annelids and arthropods. Most segmented animals seem to use a clock-and-wavefront type mechanism in which oscillations emanating from a posterior growth zone become transformed into an anterior posterior sequence of segments. In contrast, few animals such as Drosophila use a complex gene regulatory hierarchy to simultaneously subdivide their entire body axis into segments. Here I discuss how in silico models simulating the evolution of developmental patterning can be used to investigate the forces and constraints that helped shape these two developmental modes. I perform an analysis of a series of previous simulation studies, exploiting the similarities and differences in their outcomes in relation to model characteristics to elucidate the circumstances and constraints likely to have been important for the evolution of sequential and simultaneous segmentation modes. The analysis suggests that constraints arising from the involved growth process and spatial patterning signal--posterior elongation producing a propagating wavefront versus a tissue wide morphogen gradient--and the evolutionary history--ancestral versus derived segmentation mode--strongly shaped both segmentation mechanisms. Furthermore, this implies that these patterning types are to be expected rather than random evolutionary outcomes and supports the likelihood of multiple parallel evolutionary origins.

Genetic analysis of vertebral regionalization and number in medaka (Oryzias latipes) inbred lines

Vertebral number is the most variable trait among vertebrates. In addition to the vertebral number, the ratio of abdominal to caudal vertebrae is a variable trait. The vertebral number and the ratio of abdominal to caudal vertebrae contribute to vertebrate diversity. It is very interesting to know how to determine the vertebral number and the ratio of abdominal to caudal vertebrae. In this study, we identify differences in the vertebral number and the ratio of abdominal vertebrae to vertebral number between two inbred lines of medaka, namely, Hd-rRII1 and Kaga. To identify the genetic factor of those differences, we performed quantitative trait locus (QTL) analysis for vertebral number and the ratio of abdominal vertebrae to vertebral number using 200 F₂ fish. Our results show a suggestive QTL of the ratio of abdominal vertebrae to vertebral number on chromosome 15, and five QTL of vertebral number on chromosomes 1, 10, 11, 17, and 23. The QTL on chromosome 15 contains hoxDb cluster genes. The QTL of vertebral number include some genes related to the segmentation clock and axial elongation. In addition, we show that the difference in vertebral number between two inbred lines is derived from differences in the anteroposterior length of somites. Our results emphasize that the developmental process should be considered in genetic analyses for vertebral number.

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.

Making waves: the rise and fall and rise of quantitative developmental biology

The tenth annual RIKEN Center for Developmental Biology symposium 'Quantitative Developmental Biology' held in March 2012 covered a range of topics from coat colour patterning to the mechanics of morphogenesis. The studies presented shared a common theme in which a combination of physical theory, quantitative analysis and experiment was used to understand a specific cellular process in development. This report highlights these innovative studies and the long-standing questions in developmental biology that they seek to answer.

Topology and dynamics of the zebrafish segmentation clock core circuit

By combining biochemical, embryological, and mathematical approaches, this work uncovers an important role for protein-protein interactions in determining the dynamics of the somite-forming segmentation clock in vertebrates.

During vertebrate embryogenesis, the rhythmic and sequential segmentation of the body axis is regulated by an oscillating genetic network termed the segmentation clock. We describe a new dynamic model for the core pace-making circuit of the zebrafish segmentation clock based on a systematic biochemical investigation of the network's topology and precise measurements of somitogenesis dynamics in novel genetic mutants. We show that the core pace-making circuit consists of two distinct negative feedback loops, one with Her1 homodimers and the other with Her7:Hes6 heterodimers, operating in parallel. To explain the observed single and double mutant phenotypes of her1, her7, and hes6 mutant embryos in our dynamic model, we postulate that the availability and effective stability of the dimers with DNA binding activity is controlled in a “dimer cloud” that contains all possible dimeric combinations between the three factors. This feature of our model predicts that Hes6 protein levels should oscillate despite constant hes6 mRNA production, which we confirm experimentally using novel Hes6 antibodies. The control of the circuit's dynamics by a population of dimers with and without DNA binding activity is a new principle for the segmentation clock and may be relevant to other biological clocks and transcriptional regulatory networks.

The segmented pattern of the vertebral column, one of the defining features of the vertebrate body, is established during embryogenesis. The embryo's segments, called somites, form sequentially and rhythmically from head to tail. The periodicity of somite formation is regulated by the segmentation clock, a genetic oscillator that ticks in the posterior-most embryonic tissue: for each tick of the clock, one new bilateral pair of segments is made. The period of the clock appears to determine the number and the length of segments, but what controls this periodicity? In this article, we have investigated the interactions of three transcription factors that form the core of the clock's regulatory circuit, and have measured how the period of segmentation changes when these factors are mutated alone or in combination. We find that these three factors contribute to a “dimer cloud” that contains all possible dimeric combinations; however, only two dimers in this cloud can bind DNA, which allows them to directly regulate the oscillatory gene expression that underpins the periodicity of segment formation. Nevertheless, a mathematical model of the clock's dynamics based on our experimental findings indicates that the non-DNA-binding dimers also influence the stability, and hence the function, of the two DNA-binding dimers controlling the segmentation clock's period. Such involvement of non-DNA-binding dimers is a novel regulatory principle for the segmentation clock, which might also be a general mechanism that operates in other biological clocks.

Optimal cellular mobility for synchronization arising from the gradual recovery of intercellular interactions

Cell movement and intercellular signaling occur simultaneously during the development of tissues, but little is known about how movement affects signaling. Previous theoretical studies have shown that faster moving cells favor synchronization across a population of locally coupled genetic oscillators. An important assumption in these studies is that cells can immediately interact with their new neighbors after arriving at a new location. However, intercellular interactions in cellular systems may need some time to become fully established. How movement affects synchronization in this situation has not been examined. Here, we develop a coupled phase oscillator model in which we consider cell movement and the gradual recovery of intercellular coupling experienced by a cell after movement, characterized by a moving rate and a coupling recovery rate, respectively. We find (1) an optimal moving rate for synchronization and (2) a critical moving rate above which achieving synchronization is not possible. These results indicate that the extent to which movement enhances synchrony is limited by a gradual recovery of coupling. These findings suggest that the ratio of time scales of movement and signaling recovery is critical for information transfer between moving cells.

Morphogen-based simulation model of ray growth and joint patterning during fin development and regeneration

The fact that some organisms are able to regenerate organs of the correct shape and size following amputation is particularly fascinating, but the mechanism by which this occurs remains poorly understood. The zebrafish (Danio rerio) caudal fin has emerged as a model system for the study of bone development and regeneration. The fin comprises 16 to 18 bony rays, each containing multiple joints along its proximodistal axis that give rise to segments. Experimental observations on fin ray growth, regeneration and joint formation have been described, but no unified theory has yet been put forward to explain how growth and joint patterns are controlled. We present a model for the control of fin ray growth during development and regeneration, integrated with a model for joint pattern formation, which is in agreement with published, as well as new, experimental data. We propose that fin ray growth and joint patterning are coordinated through the interaction of three morphogens. When the model is extended to incorporate multiple rays across the fin, it also accounts for how the caudal fin acquires its shape during development, and regains its correct size and shape following amputation.