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.zebrafish | somitogenesis | Delta-Notch | mathematical models I n vertebrate development, somites bud off from the anterior end of the tissue not yet differentiated to somites, called the presomitic mesoderm (PSM), one by one moving posteriorly. The time interval between the formation of one somite and the next is almost constant during somitogenesis, and it is species-specific. In the PSM, there are segmentation clock genes with oscillating expression, and the timing of segmentation is considered to be controlled by the oscillatory expression of these genes because their period of oscillation is very close to the period of segmentation (1-5).The oscillatory expression of the segmentation clock genes is known to be caused by the negative feedback regulation by their own products (6-8). Neighboring cells are in contact with each other (9, 10). In the PSM, oscillatory expressions are synchronized among neighboring cells. This synchronized oscillation is necessary for normal segmentation, and disruption of the synchronization results in a defective somite boundary (11-13).Theoretical models of segmentation in vertebrates have been developed to explain the spatiotemporal periodicity of the segmentation process (14-18), the oscillatory expression of segmentation clock genes (19)(20)(21)(22)(23)(24), and the wave-like gene expression observed in the anterior PSM (23,(25)(26)(27)(28)(29). Previous theoretical studies have also addressed mechanisms of synchronization of the segmentation clock between cells (13,24,25,30). In zebrafish, the synchronization of the segment...

Computational approaches are breaking new ground in understanding how embryos form. Here, we discuss recent studies that couple precise measurements in the embryo with appropriately matched modeling and computational methods to investigate classic embryonic patterning strategies. We include signaling gradients, activator-inhibitor systems, and coupled oscillators, as well as emerging paradigms such as tissue deformation. Parallel progress in theory and experiment will play an increasingly central role in deciphering developmental patterning.

We study the transient synchronization dynamics of locally coupled phase oscillators moving on a onedimensional lattice. Analysis of spatial phase correlation shows that mobility speeds up relaxation of spatial modes and leads to faster synchronization. We show that when mobility becomes sufficiently high, it does not allow spatial modes to form and the population of oscillators behaves like a mean-field system. Estimating the relaxation timescale of the longest spatial mode and comparing it with systems with long-range coupling, we reveal how mobility effectively extends the interaction range.

Collective cell movement is a crucial component of embryonic development. Intercellular interactions regulate collective cell movement by allowing cells to transfer information. A key question is how collective cell movement itself influences information flow produced in tissues by intercellular interactions. Here, we study the effect of collective cell movement on the synchronization of locally coupled genetic oscillators. This study is motivated by the segmentation clock in zebrafish somitogenesis, where short-range correlated movement of cells has been observed. We describe the segmentation clock tissue by a Voronoi diagram, cell movement by the force balance of self-propelled and repulsive forces between cells, the dynamics of the direction of self-propelled motion, and the synchronization of genetic oscillators by locally coupled phase oscillators. We find that movement with a correlation length of about 2 ∼ 3 cell diameters is optimal for the synchronization of coupled oscillators. Quantification of cell mixing reveals that this short-range correlation of cell movement allows cells to exchange neighbors most efficiently. Moreover, short-range correlated movement strongly destabilizes nonuniform spatial phase patterns, further promoting global synchronization. Our theoretical results suggest that collective cell movement may enhance the synchronization of the segmentation clock in zebrafish somitogenesis. More generally, collective cell movement may promote information flow in tissues by enhancing cell mixing and destabilizing spurious patterns.

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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.

Many questions in developmental biology depend on measuring the position and movement of individual cells within developing embryos. Yet, tools that provide this data are often challenged by high cell density and their accuracy is difficult to measure. Here, we present a three-step procedure to address this problem. Step one is a novel segmentation algorithm based on image derivatives that, in combination with selective post-processing, reliably and automatically segments cell nuclei from images of densely packed tissue. Step two is a quantitative validation using synthetic images to ascertain the efficiency of the algorithm with respect to signal-to-noise ratio and object density. Finally, we propose an original method to generate reliable and experimentally faithful ground truth datasets: Sparse-dense dual-labeled embryo chimeras are used to unambiguously measure segmentation errors within experimental data. Together, the three steps outlined here establish a robust, iterative procedure to fine-tune image analysis algorithms and microscopy settings associated with embryonic 3D image data sets.

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.