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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. Somite boundary determination by Ripply1/Ripply2-mediated removal of Tbx6 protein 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.

Bacterial cells maintain their characteristic cell size over many generations. Several rodshaped bacteria, such as Escherichia coli and the cyanobacteria Synechococcus elongatus, divide after adding a constant length to their length at birth. Through this division control known as the adder mechanism, perturbation in cell length due to physiological fluctuation decays over generations at a rate of 2−1 per cell division. However, previous experiments have shown that the circadian clock in cyanobacteria reduces cell division frequency at a specific time of day under constant light. This circadian gating should modulate the division control by the adder mechanism, but its significance remains unknown. Here we address how the circadian gating affects cell length, doubling time, and cell length stability in cyanobacteria by using mathematical models. We show that a cell subject to circadian gating grows for a long time, and gives birth to elongated daughter cells. These elongated daughter cells grow faster than the previous generation, as elongation speed is proportional to cell length and divide in a short time before the next gating. Hence, the distributions of doubling time and cell length become bimodal, as observed in experimental data. Interestingly, the average doubling time over the population of cells is independent of gating because the extension of doubling time by gating is compensated by its reduction in the subsequent generation. On the other hand, average cell length is increased by gating, suggesting that the circadian clock controls cell length. We then show that the decay rate of perturbation in cell length depends on the ratio of delay in division by the gating τG to the average doubling time τ0 as 2τG/τ0−1. We estimated τG ≈ 2.5, τ0 ≈ 13.6 hours, and τG/τ0 ≈ 0.18 from experimental data, indicating that a long doubling time in cyanobacteria maintains the decay rate similar to that of the adder mechanism. Thus, our analysis suggests that the acquisition of the circadian clock during evolution did not impose a constraint on cell size homeostasis in cyanobacteria.

Multiple feedback loops are often found in gene regulations for various cellular functions. In mammalian circadian clocks, oscillations of Period1 (Per1) and Period2 (Per2) expression are caused by interacting negative feedback loops (NFLs) whose protein products with similar molecular functions repress each other. However, Per1 expression peaks earlier than Per2 in the pacemaker tissue, raising the question of whether the peak time difference reflects their different dynamical functions. Here, we address this question by analyzing phase responses of the circadian clock caused by light-induced transcription of both Per1 and Per2 mRNAs. Through mathematical analyses of dual NFLs, we show that phase advance is mainly driven by light inputs to the repressor with an earlier expression peak as Per1, whereas phase delay is driven by the other repressor with a later peak as Per2. Due to the complementary contributions to phase responses, the ratio of light-induced transcription rates between Per1 and Per2 determines the magnitude and direction of phase shifts at each time of day. Specifically, stronger Per1 light induction than Per2 results in a phase response curve (PRC) with a larger phase advance zone than delay zone as observed in rats and hamsters, whereas stronger Per2 induction causes a larger delay zone as observed in mice. Furthermore, the ratio of light-induced transcription rates required for entrainment is determined by the relation between the circadian and light-dark periods. Namely, if the autonomous period of a circadian clock is longer than the light-dark period, a larger light-induced transcription rate of Per1 than Per2 is required for entrainment, and vice versa. In short, the time difference between Per1 and Per2 expression peaks can differentiate their dynamical functions. The resultant complementary contributions to phase responses can determine entrainability of the circadian clock to the light-dark cycle., Author summary Gene regulatory networks underlie various cellular functions and often include multiple feedback regulations. Multiple feedback loops confer robustness on cellular systems, but whether and how they can obtain different functions is unclear. To address this question, we analyze the phase responses of mammalian circadian rhythms to light signals. In mammals, dual negative feedback loops (NFLs) of Period1 (Per1) and Period2 (Per2) genes are responsible for rhythm generation. Light signals induce transcription of both Pers mRNAs, shifting the phase of clocks. We show that the time difference between expression peaks of two repressors, as in expression of Per genes, leads to functional differentiation: the NFL with an earlier expression peak of repressor, as Per1, contributes mainly to advancing the clock, whereas the other NFL with a later peak of repressor, as Per2, contributes to delaying the clock. These complementary contributions suggest that the ratio of light-induced transcription rates between two Per genes underlies the differences in phase responses between different mammalian species. Furthermore, the relation between the circadian and light-dark periods determines the ratio of light-induced transcription rates required for entrainment. Our study reveals a mechanism for functional differentiation of dual NFLs and its significance in circadian clock systems.

Rhythmic spatial gene expression patterns termed the segmentation clock regulate vertebrate body axis segmentation during embryogenesis. The integrity of these patterns requires local synchronization between neighboring cells by Delta-Notch signaling and its inhibition results in defective segment boundaries. The oscillating tissue deforms substantially throughout development, but whether such tissue-scale morphogenesis complements local synchronization during pattern generation and segment formation is not understood. Here, we investigate pattern recovery in the zebrafish segmentation clock by washing out a Notch inhibitor, allowing resynchronization at different developmental stages, and analyzing the recovery of normal segments. Although from previous work no defects are expected after recovery, we find that washing out at early stages causes a distinctive intermingling of normal and defective segments, suggesting unexpectedly large fluctuations of synchrony before complete recovery. To investigate this recovery behavior, we develop a new model of the segmentation clock combining key ingredients motivated by prior experimental observations: coupling between neighboring oscillators, a frequency profile, a gradient of cell mixing, tissue length change, and cell advection pattern. This model captures the experimental observation of intermingled normal and defective segments through the formation of persistent phase vortices of the genetic oscillators. Experimentally observed recovery patterns at different developmental stages are predicted by temporal changes of tissue-level properties, such as tissue length and cell advection pattern in the model. These results suggest that segmental pattern recovery occurs at two scales: local pattern formation and transport of these patterns through tissue morphogenesis, highlighting a generic mechanism of pattern dynamics within developing tissues.SIGNIFICANCEInteracting genetic oscillators can generate a coherent rhythm and a tissue-level pattern from an initially desynchronized state. Using experiment and theory we study resynchronization and pattern recovery of the zebrafish segmentation clock, which makes the embryonic body segments. Experimental perturbation of intercellular signaling with an inhibitor results in intermingled normal and defective segments. According to theory, this behavior may be caused by persistent local vortices scattered in the tissue during pattern recovery. Full pattern recovery follows dynamic global properties, such as tissue length and advection pattern, in contrast to other genetic oscillators in a static tissue such as circadian clocks. Our work highlights how dynamics of tissue level properties may couple to biochemical pattern formation in tissues and developing embryos.

Gear skiving has received considerable attention because of its high productivity, particularly in internal gear cutting, and skiving cutter design methods have been widely investigated. At the beginning of a cutter design process, a shaft angle, a center distance, and the number of teeth are examined as basic parameters for cutting gears. In current gear skiving, a shaft angle is generally set to be approximately 20°; however, the reason for selecting this value has not been clarified. In the present study, the validity of the value of shaft angle is discussed by calculating the cutting tool parameters, such as instantaneous rake angles, clearance angles, cut depths, and cutting speeds at continuously moving cutting points against various shaft angles. Therefore, the widely used value of shaft angle would result in moderate cutting tool parameters. In addition, the cutting force and cutter wear would be low when a cutter axis inclines opposite to the gear helix to be cut.

The Japan Ministry of Health, Labour and Welfare (MHLW) has published instructions for radiological protection against food after the Fukushima Daiichi nuclear power plant accident in 2011. Following the instructions, the export and consumption of food items identified as being contaminated were restricted for a certain period. We assessed the validity of the imposed restriction periods for two representative vegetables (spinach and cabbage) grown in Fukushima Prefecture from two perspectives: effectiveness for reducing dietary dose and economic efficiency. To assess effectiveness, we estimated the restriction period required to maintain consumers' dose below the guidance dose levels. To assess economic efficiency, we estimated the restriction period that maximizes the net benefit to taxpayers. All estimated restriction periods were shorter than the actual restriction periods imposed on spinach and cabbage from Fukushima in 2011, which indicates that the food restriction effectively maintained consumers' dietary dose below the guidance dose level, but in an economically inefficient manner. We also evaluated the response of the restriction period to the sample size for each weekly food safety test and the instructions for when to remove the restriction. Stringent MHLW instructions seemed to sufficiently reduce consumers' health risk even when the sample size for the weekly food safety test was small, but tended to increase the economic cost to taxpayers.

Positive and negative feedback loops are often present in regulatory networks for genetic oscillations. Relative time scales and integration of these feedback loops are key to robust oscillations in expression levels. Using examples from the circadian clock and synthetic genetic oscillators, we study positive and negative feedback loops interlocked at competitive binding sites. In the mammalian circadian clock, a key clock gene Bmal1 is regulated by the activator ROR and the repressor REV-ERB. Conversely, Bmal1 activates both of them, forming interlocked feedback loops. Previous experiments indicate that the activator and repressor compete for the same binding sites in the Bmal1 promoter. Transcription patterns predict that ROR peaks later than REV-ERB and, moreover, the peak phase difference between them is small. Using mathematical modeling we reveal an optimal ratio of dissociation constants of an activator and a repressor for the competitive binding sites to enhance the amplitude of Bmal1 oscillations. This optimal ratio arises only when the amplitude of the repressor is larger than that of the activator. Secondly, we reveal that the preference of binding sites for an activator and a repressor depends on their relative time scales. A previous study demonstrated that noncompetitive binding sites are preferable for synthetic genetic oscillators that comprise a fast activator and a slow repressor with a large time scale separation. Here we show that when their time scales are similar, competitive binding sites are more likely to generate oscillation than noncompetitive sites. In contrast, for a slow activator and a fast repressor with a small phase difference as in Bmal1 regulation, noncompetitive binding sites are advantageous for amplifying oscillations. Our results, therefore, predict that additional mechanisms are necessary to compensate the disadvantage of the Bmal1 promoter and further facilitate amplification under the regulation by ROR and REV-ERB.

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. Fil: Uriu, Koichiro. Kanazawa University; Japón Fil: Morelli, Luis Guillermo. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Parque Centenario. Instituto de Investigación en Biomedicina de Buenos Aires - Instituto Partner de la Sociedad Max Planck; Argentina. Institut Max Planck fur Molekulare Physiologie; Alemania

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. Fil: Petrungaro, Gabriela. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universitat zu Köln; Alemania. Universidad de Buenos Aires; Argentina Fil: Morelli, Luis Guillermo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Institut Max Planck fur Molekulare Physiologie; Alemania. Universidad de Buenos Aires; Argentina Fil: Uriu, Koichiro. Kanazawa University; Japón

Individual biological oscillators can synchronize to generate a collective rhythm. During vertebrate development, mobile cells exchange signals to synchronize a rhythmic pattern generator that makes the embryonic segments. Previous theoretical works have shown that cell mobility can enhance synchronization of coupled oscillators when signal exchange is instantaneous. However, in vertebrate segmentation, the exchange of signals is thought to comprise delays from signal sending and processing, which could alter the effect of mobility on synchronization. Here, we study synchronization dynamics of mobile phase oscillators in the presence of coupling delays. We find that mobility can speed up synchronization when coupling delays are present. We derive an analytical expression for the characteristic time of synchronization dynamics, which is in very good agreement with numerical simulations. This analytical expression suggests a subdivision of the mobility range into different dynamical regimes and reveals that, with delayed coupling, synchronization is enhanced at a lower mobility rate than with instantaneous coupling. We argue that these results may be relevant to the synchronization of mobile oscillators in vertebrate segmentation. Fil: Petrungaro, Gabriela. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Parque Centenario. Instituto de Investigación en Biomedicina de Buenos Aires - Instituto Partner de la Sociedad Max Planck; Argentina Fil: Uriu, Koichiro. Kanazawa University; Japón Fil: Morelli, Luis Guillermo. Institut Max Planck fur Molekulare Physiologie; Alemania. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Parque Centenario. Instituto de Investigación en Biomedicina de Buenos Aires - Instituto Partner de la Sociedad Max Planck; Argentina

We study the dynamics of mobile, locally coupled identical oscillators in the presence of coupling delays. We find different kinds of chimera states, in which coherent in-phase and anti-phase domains coexist with incoherent domains. These chimera states are dynamic and can persist for long times for intermediate mobility values. We discuss the mechanisms leading to the formation of these chimera states in different mobility regimes. This finding could be relevant for natural and technological systems composed of mobile communicating agents., 10 pages, 8 figures, draft version

The Japan Ministry of Health, Labour and Welfare (MHLW) has published instructions for radiological protection against food after the Fukushima Daiichi nuclear power plant accident in 2011. Following the instructions, the export and consumption of food items identified as being contaminated were restricted for a certain period. We assessed the validity of the imposed restriction periods for two representative vegetables (spinach and cabbage) grown in Fukushima Prefecture from two perspectives: effectiveness for reducing dietary dose and economic efficiency. To assess effectiveness, we estimated the restriction period required to maintain consumers' dose below the guidance dose levels. To assess economic efficiency, we estimated the restriction period that maximizes the net benefit to taxpayers. All estimated restriction periods were shorter than the actual restriction periods imposed on spinach and cabbage from Fukushima in 2011, which indicates that the food restriction effectively maintained consumers' dietary dose below the guidance dose level, but in an economically inefficient manner. We also evaluated the response of the restriction period to the sample size for each weekly food safety test and the instructions for when to remove the restriction. Stringent MHLW instructions seemed to sufficiently reduce consumers' health risk even when the sample size for the weekly food safety test was small, but tended to increase the economic cost to taxpayers.

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. Fil: Koichiro, Uriu. Riken; Japón Fil: Morelli, Luis Guillermo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; Argentina

Gear skiving is a technique proposed a long time ago for cutting internal gears at high productivity. Until recently, many problems have prevented its widespread use. With current technological breakthroughs, however, skiving is drawing attention again. The present paper describes cutting tool parameters, which could be vital for the optimum design of skiving cutters. Cutting tool parameters include depth of cut, rake angle, and clearance angle at each point on a cutting edge. They continuously change with progress in the cutting process. The parameters are defined on the basis of an oblique cutting model, which is a three-dimensional extension of an orthogonal cutting model. The example calculations in this study revealed the following features: Although rake angles almost always remain negative, clearance angles remain positive. At the points where clearance angles are large, depths of cut are large, but rake angles are small (i.e., largely negative). The decrease in shaft angle between the cutter and working blank axes increases depths of cut and clearance angles, while reducing rake angles (i.e., yields largely negative rake angles). Meanwhile, the increase in cutter tool face offset; i.e., the axial position of a tool face measured from a reference point on the conjugate pinion, narrows the area where depths of cut and clearance angles are small, but rake angles become largely negative. These parameters could be useful for evaluating tool cutting efficiencies in internal gear skiving.

In vertebrate somitogenesis, "segmentation clock" genes (her in zebrafish, hes in mouse, and hairy in chick) show oscillation, synchronized over nearby cells through intercellular interaction. In zebrafish, neighboring cells interact by Delta-Notch signaling to realize synchronization. Under Delta-Notch, however, a cell with a high expression of the segmentation clock gene tends to suppress its expression in adjacent cells, which might produce spatial heterogeneity instead of synchronized oscillation. Here we studied the conditions under which pre-somitic mesoderm cells show synchronized oscillation of gene expression mathematically. We adopted a model that explicitly considers the kinetics of the mRNA and proteins of the segmentation clock gene and cell-cell interaction via Delta-Notch signaling. From statistical study of a model with randomly generated parameters, we revealed how the likelihood that the system generates stable synchronized oscillation depends on the rate of each reaction in the gene-protein kinetics.

Erratum in :Correction: Object Segmentation and Ground Truth in 3D Embryonic Imaging.Bhavna R, Uriu K, Valentin G, Tinevez JY, Oates AC. PLoS One. 2016 Aug 16;11(8):e0161550. doi: 10.1371/journal.pone.0161550. eCollection 2016. PMID: 27529424 Free PMC article.; International audience; 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, morphogenetic processes are strictly coordinated in time. Cells in a developing tissue would need mechanisms for time-keeping. One such time-keeping mechanism is to use oscillations of gene expression. Oscillatory gene expression can be generated by transcriptional/translational feedback loops, usually referred to as a genetic oscillator. In this review article, we discuss genetic oscillators in the presence of developmental processes such as cell division, cell movement and cell differentiation. We first introduce the gene regulatory network for generating a rhythm of gene expression. We then discuss how developmental processes influence genetic oscillators. Examples include vertebrate somitogenesis and neural progenitor cell differentiation, as well as the circadian clock for comparison. To understand the behaviors of genetic oscillators in development, it is necessary to consider both gene expression dynamics and cellular behaviors simultaneously. Theoretical modeling combined with live imaging at single-cell resolution will be a powerful tool to analyze genetic oscillators in development. In this review article, we discuss genetic oscillators in the presence of developmental processes such as cell division, cell movement and cell differentiation. We first introduce the gene regulatory network for generating a rhythm of gene expression. We then discuss how developmental processes influence genetic oscillators. © 2016 Japanese Society of Developmental Biologists.

The objective of this study is to identify a procedure for determining sample size allocation for food radiation inspections of more than one food item to minimize the potential risk to consumers of internal radiation exposure. We consider a simplified case of food radiation monitoring and safety inspection in which a risk manager is required to monitor two food items, milk and spinach, in a contaminated area. Three protocols for food radiation monitoring with different sample size allocations were assessed by simulating random sampling and inspections of milk and spinach in a conceptual monitoring site. Distributions of (131)I and radiocesium concentrations were determined in reference to (131)I and radiocesium concentrations detected in Fukushima prefecture, Japan, for March and April 2011. The results of the simulations suggested that a protocol that allocates sample size to milk and spinach based on the estimation of (131)I and radiocesium concentrations using the apparent decay rate constants sequentially calculated from past monitoring data can most effectively minimize the potential risks of internal radiation exposure.

Cell movement and local intercellular signaling are crucial components of morphogenesis during animal development. Intercellular signaling regulates the collective movement of a cell population via direct cell–cell contact. Cell movement, conversely, can influence local intercellular signaling by rearranging neighboring cells. Here, we first discuss theoretical models that address how intercellular signaling regulates collective cell movement during development. Examples include neural crest cell migration, convergent extension, and cell movement during vertebrate axis elongation. Second, we review theoretical studies on how cell movement may affect intercellular signaling, using the segmentation clock in zebrafish as an example. We propose that interplay between cell movement and intercellular signaling must be considered when studying morphogenesis in embryonic development. Fil: Uriu, Koichiro. Riken; Japón Fil: Morelli, Luis Guillermo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; Argentina Fil: Oates, Andrew C.. University College London; Estados Unidos

We study the transient synchronization dynamics of locally coupled phase oscillators moving on a one-dimensional 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. Fil: Uriu, Koichiro. RIKEN Advanced Science Institute. Theoretical Biology Laboratory; Japan; Max Planck Institute of Molecular Cell Biology and Genetics; Alemania; Max Planck Institute for the Physics of Complex Systems; Alemania; Fil: Ares, Saúl. Max Planck Institute for the Physics of Complex Systems; Alemania; Consejo Superior de Investigaciones Científicas. Centro Nacional de Biotecnología. Logic of Genomic Systems Laboratory; España; Grupo Interdisciplinar de Sistemas Complejos; España; Fil: Oates, Andrew C.. Max Planck Institute of Molecular Cell Biology and Genetics; Alemania; Fil: Morelli, Luis Guillermo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires; Argentina; Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Física; Argentina; Max Planck Institute of Molecular Cell Biology and Genetics; Alemania

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., 18 single column pages + 1 table + 5 figures + Supporting Information

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.

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.

We study a three-variable Turing system with two kinds of cells and a diffusive chemical, considering the formation and maintenance of fish skin patterns with multiple pigment cells. The two types of cells are produced from undifferentiated cells. They inhibit the supply rate of the other cell type, forming local clusters of the same cell type. In addition, the cells of one type can be maintained only in the presence of a diffusive chemical produced by the other cell type, resulting in the coexistence of two cell types in heterogeneous spatial patterns. We assume linear interaction among cells and the chemical, and cell supply rates constrained to be positive or zero. We derive the condition for diffusion-driven instability. In one-dimensional model, we examine how the wavelength of the periodic pattern depends on parameters. In the two-dimensional model, we study the condition for spot, stripe or reversed spot pattern to emerge (pattern selection). We discuss heuristic criteria for the pattern selection.