Continuum theory of gene expression waves during vertebrate segmentation

Ripply suppresses Tbx6 to induce dynamic-to-static conversion in somite segmentation

The metameric pattern of somites is created based on oscillatory expression of clock genes in presomitic mesoderm. However, the mechanism for converting the dynamic oscillation to a static pattern of somites is still unclear. Here, we provide evidence that Ripply/Tbx6 machinery is a key regulator of this conversion. Ripply1/Ripply2-mediated removal of Tbx6 protein defines somite boundary and also leads to cessation of clock gene expression in zebrafish embryos. On the other hand, activation of ripply1/ripply2 mRNA and protein expression is periodically regulated by clock oscillation in conjunction with an Erk signaling gradient. Whereas Ripply protein decreases rapidly in embryos, Ripply-triggered Tbx6 suppression persists long enough to complete somite boundary formation. Mathematical modeling shows that a molecular network based on results of this study can reproduce dynamic-to-static conversion in somitogenesis. Furthermore, simulations with this model suggest that sustained suppression of Tbx6 caused by Ripply is crucial in this conversion.

Orchestration of tissue shape changes and gene expression patterns in development

In development, tissue shape changes and gene expression patterns give rise to morphogenesis. Understanding tissue shape changes requires the analysis of mechanical properties of the tissue such as tissue rigidity, cell influx from neighboring tissues, cell shape changes and cell proliferation. Local and global gene expression patterns can be influenced by neighbor exchange and tissue shape changes. Here we review recent studies on the mechanisms for tissue elongation and its influences on dynamic gene expression patterns by focusing on vertebrate somitogenesis. We first introduce mechanical and biochemical properties of the segmenting tissue that drive tissue elongation. Then, we discuss patterning in the presence of cell mixing, scaling of signaling gradients, and dynamic phase waves of rhythmic gene expression under tissue shape changes. We also highlight the importance of theoretical approaches to address the relation between tissue shape changes and patterning.

Positional information encoded in the dynamic differences between neighboring oscillators during vertebrate segmentation

A central problem in developmental biology is to understand how cells interpret their positional information to give rise to spatial patterns, such as the process of periodic segmentation of the vertebrate embryo into somites. For decades, somite formation has been interpreted according to the clock-and-wavefront model. In this conceptual framework, molecular oscillators set the frequency of somite formation while the positional information is encoded in signaling gradients. Recent experiments using ex vivo explants have challenged this interpretation, suggesting that positional information is encoded in the properties of the oscillators, independent of long-range modulations such as signaling gradients. Here, we propose that positional information is encoded in the difference in the levels of neighboring oscillators. The differences gradually increase because both the amplitude and the period of the oscillators increase with time. When this difference exceeds a certain threshold, the segmentation program starts. Using this framework, we quantitatively fit experimental data from in vivo and ex vivo mouse segmentation, and propose mechanisms of somite scaling. Our results suggest a novel mechanism of spatial pattern formation based on the local interactions between dynamic molecular oscillators.

Towards a physical understanding of developmental patterning

The temporal coordination of events at cellular and tissue scales is essential for the proper development of organisms, and involves cell-intrinsic processes that can be coupled by local cellular signalling and instructed by global signalling, thereby creating spatial patterns of cellular states that change over time. The timing and structure of these patterns determine how an organism develops. Traditional developmental genetic methods have revealed the complex molecular circuits regulating these processes but are limited in their ability to predict and understand the emergent spatio-temporal dynamics. Increasingly, approaches from physics are now being used to help capture the dynamics of the system by providing simplified, generic descriptions. Combined with advances in imaging and computational power, such approaches aim to provide insight into timing and patterning in developing systems.

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.

From local resynchronization to global pattern recovery in the zebrafish segmentation clock

Integrity of rhythmic spatial gene expression patterns in the vertebrate segmentation clock requires local synchronization between neighboring cells by Delta-Notch signaling and its inhibition causes defective segment boundaries. Whether deformation of the oscillating tissue complements local synchronization during patterning and segment formation is not understood. We combine theory and experiment to investigate this question in the zebrafish segmentation clock. We remove a Notch inhibitor, allowing resynchronization, and analyze embryonic segment recovery. We observe unexpected intermingling of normal and defective segments, and capture this with a new model combining coupled oscillators and tissue mechanics. Intermingled segments are explained in the theory by advection of persistent phase vortices of oscillators. Experimentally observed changes in recovery patterns are predicted in the theory by temporal changes in tissue length and cell advection pattern. Thus, segmental pattern recovery occurs at two length and time scales: rapid local synchronization between neighboring cells, and the slower transport of the resulting patterns across the tissue through morphogenesis.

Geometric models for robust encoding of dynamical information into embryonic patterns

During development, cells gradually assume specialized fates via changes of transcriptional dynamics, sometimes even within the same developmental stage. For anterior-posterior (AP) patterning in metazoans, it has been suggested that the gradual transition from a dynamic genetic regime to a static one is encoded by different transcriptional modules. In that case, the static regime has an essential role in pattern formation in addition to its maintenance function. In this work, we introduce a geometric approach to study such transition. We exhibit two types of genetic regime transitions arising through local or global bifurcations, respectively. We find that the global bifurcation type is more generic, more robust, and better preserves dynamical information. This could parsimoniously explain common features of metazoan segmentation, such as changes of periods leading to waves of gene expressions, ‘speed/frequency-gradient’ dynamics, and changes of wave patterns. Geometric approaches appear as possible alternatives to gene regulatory networks to understand development.

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.

Multiscale modeling of vertebrate limb development

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

Constraints on somite formation in developing embryos

Segment formation in vertebrate embryos is a stunning example of biological self-organization. Here, we present an idealized framework, in which we treat the presomitic mesoderm (PSM) as a one-dimensional line of oscillators. We use the framework to derive constraints that connect the size of somites, and the timing of their formation, to the growth of the PSM and the gradient of the somitogenesis clock period across the PSM. Our analysis recapitulates the observations made recently in ex vivo cultures of mouse PSM cells, and makes predictions for how perturbations, such as increased Wnt levels, would alter somite widths. Finally, our analysis makes testable predictions for the shape of the phase profile and somite widths at different stages of PSM growth. In particular, we show that the phase profile is robustly concave when the PSM length is steady and slightly convex in an important special case when it is decreasing exponentially. In both cases, the phase profile scales with the PSM length; in the latter case, it scales dynamically. This has important consequences for the velocity of the waves that traverse the PSM and trigger somite formation, as well as the effect of errors in phase measurement on somite widths.

The proneural wave in the Drosophila optic lobe is driven by an excitable reaction-diffusion mechanism

In living organisms, self-organised waves of signalling activity propagate spatiotemporal information within tissues. During the development of the largest component of the visual processing centre of the Drosophila brain, a travelling wave of proneural gene expression initiates neurogenesis in the larval optic lobe primordium and drives the sequential transition of neuroepithelial cells into neuroblasts. Here, we propose that this 'proneural wave' is driven by an excitable reaction-diffusion system involving epidermal growth factor receptor (EGFR) signalling interacting with the proneural gene l'sc. Within this framework, a propagating transition zone emerges from molecular feedback and diffusion. Ectopic activation of EGFR signalling in clones within the neuroepithelium demonstrates that a transition wave can be excited anywhere in the tissue by inducing signalling activity, consistent with a key prediction of the model. Our model illuminates the physical and molecular underpinnings of proneural wave progression and suggests a generic mechanism for regulating the sequential differentiation of tissues.

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.

Size-reduced embryos reveal a gradient scaling-based mechanism for zebrafish somite formation

Little is known about how the sizes of animal tissues are controlled. A prominent example is somite size, which varies widely both within an individual and across species. Despite intense study of the segmentation clock governing the timing of somite generation, how it relates to somite size is poorly understood. Here, we examine somite scaling and find that somite size at specification scales with the length of the presomitic mesoderm (PSM) despite considerable variation in PSM length across developmental stages and in surgically size-reduced embryos. Measurement of clock period, axis elongation speed and clock gene expression patterns demonstrate that existing models fail to explain scaling. We posit a 'clock and scaled gradient' model, in which somite boundaries are set by a dynamically scaling signaling gradient across the PSM. Our model not only explains existing data, but also makes a unique prediction that we confirm experimentally - the formation of periodic 'echoes' in somite size following perturbation of the size of one somite. Our findings demonstrate that gradient scaling plays a central role in both progression and size control of somitogenesis.

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.

Chemical event chain model of coupled genetic oscillators

We introduce a stochastic model of coupled genetic oscillators in which chains of chemical events involved in gene regulation and expression are represented as sequences of Poisson processes. We characterize steady states by their frequency, their quality factor, and their synchrony by the oscillator cross correlation. The steady state is determined by coupling and exhibits stochastic transitions between different modes. The interplay of stochasticity and nonlinearity leads to isolated regions in parameter space in which the coupled system works best as a biological pacemaker. Key features of the stochastic oscillations can be captured by an effective model for phase oscillators that are coupled by signals with distributed delays.

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.

Active hydrodynamics of synchronization and ordering in moving oscillators

The nature of emergent collective behaviors of moving interacting physical agents is a long-standing open issue in physical and biological systems alike. This calls for studies on the control of synchronization and the degree of order in a collection of diffusively moving noisy oscillators. We address this by constructing a generic hydrodynamic theory for active phase fluctuations in a collection of a large number of nearly-phase-coherent moving oscillators in two dimensions. Our theory describes the general situation where phase fluctuations and oscillator mobility mutually affect each other. We show that the interplay between the active effects and the mobility of the oscillators leads to a variety of phenomena, ranging from synchronization with long-range, nearly-long-range, and quasi-long-range orders to instabilities and desynchronization with short-range order of the oscillator phases. We highlight the complex dependences of synchronization on the active effects. These should be testable in wide-ranging systems, e.g., oscillating chemical reactions in the presence of different reaction inhibitors and facilitators, live oriented cytoskeletal extracts, and vertebrate segmentation clocks.

Determining the impact of cell mixing on signaling during development

Cell movement and intercellular signaling occur simultaneously to organize morphogenesis during embryonic development. Cell movement can cause relative positional changes between neighboring cells. When intercellular signals are local such cell mixing may affect signaling, changing the flow of information in developing tissues. Little is known about the effect of cell mixing on intercellular signaling in collective cellular behaviors and methods to quantify its impact are lacking. Here we discuss how to determine the impact of cell mixing on cell signaling drawing an example from vertebrate embryogenesis: the segmentation clock, a collective rhythm of interacting genetic oscillators. We argue that comparing cell mixing and signaling timescales is key to determining the influence of mixing. A signaling timescale can be estimated by combining theoretical models with cell signaling perturbation experiments. A mixing timescale can be obtained by analysis of cell trajectories from live imaging. After comparing cell movement analyses in different experimental settings, we highlight challenges in quantifying cell mixing from embryonic timelapse experiments, especially a reference frame problem due to embryonic motions and shape changes. We propose statistical observables characterizing cell mixing that do not depend on the choice of reference frames. Finally, we consider situations in which both cell mixing and signaling involve multiple timescales, precluding a direct comparison between single characteristic timescales. In such situations, physical models based on observables of cell mixing and signaling can simulate the flow of information in tissues and reveal the impact of observed cell mixing on signaling.

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.

Self-organization in development, regeneration and organoids

Self-organization of cells is a fundamental design principle in biology, yet the inherent non-linearity of self-organizing systems often poses significant challenges in deciphering the underlying mechanisms. Here, we discuss recent progress in this respect, focusing on examples from development, regeneration and organoid differentiation. Together, these three paradigms emphasize the active material properties of tissues that result from the functional coupling between individual cells as active units. Further, we discuss the challenge of obtaining reproducible outcomes on the basis of self-organizing systems, which development and regeneration, but not the current organoid culture protocols, achieve.

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.