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.

Intermittent ERK oscillations downstream of FGF in mouse embryonic stem cells.

Signal transduction networks generate characteristic dynamic activities to process extracellular signals and guide cell fate decisions such as to divide or differentiate. The differentiation of pluripotent cells is controlled by FGF/ERK signaling. However, only a few studies have addressed the dynamic activity of the FGF/ERK signaling network in pluripotent cells at high time resolution. Here, we use live cell sensors in wild-type and Fgf4-mutant mouse embryonic stem cells to measure dynamic ERK activity in single cells, for defined ligand concentrations and differentiation states. These sensors reveal pulses of ERK activity. Pulsing patterns are heterogeneous between individual cells. Consecutive pulse sequences occur more frequently than expected from simple stochastic models. Sequences become more prevalent with higher ligand concentration, but are rarer in more differentiated cells. Our results suggest that FGF/ERK signaling operates in the vicinity of a transition point between oscillatory and non-oscillatory dynamics in embryonic stem cells. The resulting heterogeneous dynamic signaling activities add a new dimension to cellular heterogeneity that may be linked to divergent fate decisions in stem cell cultures.

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.

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.

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.

Interplay between intercellular signaling and cell movement in development.

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.

Patterning embryos with oscillations: structure, function and dynamics of the vertebrate segmentation clock.

The segmentation clock is an oscillating genetic network thought to govern the rhythmic and sequential subdivision of the elongating body axis of the vertebrate embryo into somites: the precursors of the segmented vertebral column. Understanding how the rhythmic signal arises, how it achieves precision and how it patterns the embryo remain challenging issues. Recent work has provided evidence of how the period of the segmentation clock is regulated and how this affects the anatomy of the embryo. The ongoing development of real-time clock reporters and mathematical models promise novel insight into the dynamic behavior of the clock.

Topology and dynamics of the zebrafish segmentation clock core circuit.

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.

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.

Collective cell movement promotes synchronization of coupled genetic oscillators.

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.

Synchronization dynamics in the presence of coupling delays and phase shifts.

In systems of coupled oscillators, the effects of complex signaling can be captured by time delays and phase shifts. Here, we show how time delays and phase shifts lead to different oscillator dynamics and how synchronization rates can be regulated by substituting time delays by phase shifts at a constant collective frequency. For spatially extended systems with time delays, we show that the fastest synchronization can occur for intermediate wavelengths, giving rise to novel synchronization scenarios.

Adult-born granule cells improve stimulus encoding and discrimination in the dentate gyrus.

Heterogeneity plays an important role in diversifying neural responses to support brain function. Adult neurogenesis provides the dentate gyrus with a heterogeneous population of granule cells (GCs) that were born and developed their properties at different times. Immature GCs have distinct intrinsic and synaptic properties than mature GCs and are needed for correct encoding and discrimination in spatial tasks. How immature GCs enhance the encoding of information to support these functions is not well understood. Here, we record the responses to fluctuating current injections of GCs of different ages in mouse hippocampal slices to study how they encode stimuli. Immature GCs produce unreliable responses compared to mature GCs, exhibiting imprecise spike timings across repeated stimulation. We use a statistical model to describe the stimulus-response transformation performed by GCs of different ages. We fit this model to the data and obtain parameters that capture GCs' encoding properties. Parameter values from this fit reflect the maturational differences of the population and indicate that immature GCs perform a differential encoding of stimuli. To study how this age heterogeneity influences encoding by a population, we perform stimulus decoding using populations that contain GCs of different ages. We find that, despite their individual unreliability, immature GCs enhance the fidelity of the signal encoded by the population and improve the discrimination of similar time-dependent stimuli. Thus, the observed heterogeneity confers the population with enhanced encoding capabilities.

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.

Collective modes of coupled phase oscillators with delayed coupling.

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

Effective description of bistability and irreversibility in apoptosis.

Apoptosis is a mechanism of programmed cell death in which cells engage in a controlled demolition and prepare to be digested without damaging their environment. In normal conditions, apoptosis is repressed until it is irreversibly induced by an appropriate signal. In adult organisms, apoptosis is a natural way to dispose of damaged cells and its disruption or excess is associated with cancer and autoimmune diseases. Apoptosis is regulated by a complex signaling network controlled by caspases, specialized enzymes that digest essential cellular components and promote the degradation of genomic DNA. In this work, we propose an effective description of the signaling network focused on caspase-3 as a readout of cell fate. We integrate intermediate network interactions into a nonlinear feedback function acting on caspase-3 and introduce the effect of pro-apoptotic stimuli and regulatory elements as a saturating activation function. We show that activation dynamics in the theory is similar to previously reported experimental results. We compute bifurcation diagrams and obtain cell fate maps describing how stimulus intensity and feedback strength affect cell survival and death fates. These fates overlap within a bistable region that depends on total caspase concentration, regulatory elements, and feedback nonlinearity. We study a strongly nonlinear regime to obtain analytical expressions for bifurcation curves and fate map boundaries. For a broad range of parameters, strong stimuli can induce an irreversible switch to the death fate. We use the theory to explore dynamical stimulation conditions and determine how cell fate depends on stimulation temporal patterns. This analysis predicts a critical relation between transient stimuli intensity and duration to trigger irreversible apoptosis. We derive an analytical expression for this critical relation, valid for short stimuli. Our description provides distinct predictions and offers a framework to study how this signaling network processes different stimuli to make a cell fate decision.

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.

Synchronization dynamics of mobile oscillators in the presence of coupling delays.

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.

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.

Segmentation of the zebrafish axial skeleton relies on notochord sheath cells and not on the segmentation clock.

Segmentation of the axial skeleton in amniotes depends on the segmentation clock, which patterns the paraxial mesoderm and the sclerotome. While the segmentation clock clearly operates in teleosts, the role of the sclerotome in establishing the axial skeleton is unclear. We severely disrupt zebrafish paraxial segmentation, yet observe a largely normal segmentation process of the chordacentra. We demonstrate that axial entpd5+ notochord sheath cells are responsible for chordacentrum mineralization, and serve as a marker for axial segmentation. While autonomous within the notochord sheath, entpd5 expression and centrum formation show some plasticity and can respond to myotome pattern. These observations reveal for the first time the dynamics of notochord segmentation in a teleost, and are consistent with an autonomous patterning mechanism that is influenced, but not determined by adjacent paraxial mesoderm. This behavior is not consistent with a clock-type mechanism in the notochord.