Nuclei determine the spatial origin of mitotic waves

From biological data to oscillator models using SINDy

Periodic changes in the concentration or activity of different molecules regulate vital cellular processes such as cell division and circadian rhythms. Developing mathematical models is essential to better understand the mechanisms underlying these oscillations. Recent data-driven methods like SINDy have fundamentally changed model identification, yet their application to experimental biological data remains limited. This study investigates SINDy's constraints by directly applying it to biological oscillatory data. We identify insufficient resolution, noise, dimensionality, and limited prior knowledge as primary limitations. Using various generic oscillator models of different complexity and/or dimensionality, we systematically analyze these factors. We then propose a comprehensive guide for inferring models from biological data, addressing these challenges step by step. Our approach is validated using glycolytic oscillation data from yeast.

Collective effects of cell cleavage dynamics

A conserved process of early embryonic development in metazoans is the reductive cell divisions following oocyte fertilization, termed cell cleavages. Cell cleavage cycles usually start synchronously, lengthen differentially between the embryonic cells becoming asynchronous, and cease before major morphogenetic events, such as germ layer formation and gastrulation. Despite exhibiting species-specific characteristics, the regulation of cell cleavage dynamics comes down to common controllers acting mostly at the single cell/nucleus level, such as nucleus-to-cytoplasmic ratio and zygotic genome activation. Remarkably, recent work has linked cell cleavage dynamics to the emergence of collective behavior during embryogenesis, including pattern formation and changes in embryo-scale mechanics, raising the question how single-cell controllers coordinate embryo-scale processes. In this review, we summarize studies across species where an association between cell cleavages and collective behavior was made, discuss the underlying mechanisms, and propose that cell-to-cell variability in cell cleavage dynamics can serve as a mechanism of long-range coordination in developing embryos.

Mitotic waves in frog egg extracts: Transition from phase waves to trigger waves

Cyclin-dependent kinase 1 (Cdk1) activity rises and falls throughout the cell cycle, a cell-autonomous process known as mitotic oscillations. These oscillators can synchronize when spatially coupled, providing a crucial foundation for rapid synchronous divisions in large early embryos like Drosophila (~ 0.5 mm) and Xenopus (~ 1.2 mm). While diffusion alone cannot achieve such long-range coordination, recent studies have proposed two types of mitotic waves, phase and trigger waves, to explain the phenomena. How the waves establish over time for efficient spatial coordination remains unclear. Using Xenopus laevis egg extracts and a Cdk1 FRET sensor, we observe a transition from phase waves to a trigger wave regime in an initially homogeneous cytosol. Adding nuclei accelerates such transition. Moreover, the system transitions almost immediately to this regime when externally driven by metaphase-arrested extracts from the boundary. Employing computational modeling, we pinpoint how wave nature, including speed-period relation, depends on transient dynamics and oscillator properties, suggesting that phase waves appear transiently due to the time required for trigger waves to entrain the system and that spatial heterogeneity promotes entrainment. Therefore, we show that both waves belong to a single biological process capable of coordinating the cell cycle over long distances.

Robust trigger wave speed in Xenopus cytoplasmic extracts

Self-regenerating trigger waves can spread rapidly through the crowded cytoplasm without diminishing in amplitude or speed, providing consistent, reliable, long-range communication. The macromolecular concentration of the cytoplasm varies in response to physiological and environmental fluctuations, raising the question of how or if trigger waves can robustly operate in the face of such fluctuations. Using Xenopus extracts, we found that mitotic and apoptotic trigger wave speeds are remarkably invariant. We derived a model that accounts for this robustness and for the eventual slowing at extremely high and low cytoplasmic concentrations. The model implies that the positive and negative effects of cytoplasmic concentration (increased reactant concentration vs. increased viscosity) are nearly precisely balanced. Accordingly, artificially maintaining a constant cytoplasmic viscosity during dilution abrogates this robustness. The robustness in trigger wave speeds may contribute to the reliability of the extremely rapid embryonic cell cycle.

Nuclear-cytoplasmic compartmentalization of cyclin B1-Cdk1 promotes robust timing of mitotic events

The cyclin-dependent kinase (Cdk1) oscillator is widely characterized in homogenized cytosolic extracts, leaving unclear the impact of nucleocytoplasmic compartmentalization. Here, by developing a Förster resonance energy transfer (FRET) biosensor, we track Cdk1 spatiotemporal dynamics in reconstituted cells with or without side by side and find compartmentalization significantly modulates clock properties previously found in bulk studies. Although nucleus-absent cells display highly tunable frequency, the nucleus-present cells maintain constant frequency against cyclin B1 variations. Despite high expression variability, cyclin degraded within the same duration, enabling a robust mitotic phase. Moreover, Cdk1 and cyclin B1 cycle rigorously out-of-phase, ensuring wide phase-plane orbits, essential for oscillation robustness. Although Cdk1 in homogeneous extracts is well known for delayed switch-like activation, we find active cyclin B1-Cdk1 accumulates in nuclei, without delay, until the nuclear envelope breakdown (NEB) when another abrupt activation triggers anaphase. Cdk1 biphasic activation and spatial compartmentalization may together coordinate the accurate ordering of different downstream events.

Manipulating the nature of embryonic mitotic waves

Early embryogenesis is characterized by rapid and synchronous cleavage divisions, which are often controlled by wave-like patterns of Cdk1 activity. Two mechanisms have been proposed for mitotic waves: sweep and trigger waves.1,2 The two mechanisms give rise to different wave speeds, dependencies on physical and molecular parameters, and spatial profiles of Cdk1 activity: upward sweeping gradients versus traveling wavefronts. Both mechanisms hinge on the transient bistability governing the cell cycle and are differentiated by the speed of the cell-cycle progression: sweep and trigger waves arise for rapid and slow drives, respectively. Here, using quantitative imaging of Cdk1 activity and theory, we illustrate that sweep waves are the dominant mechanism in Drosophila embryos and test two fundamental predictions on the transition from sweep to trigger waves. We demonstrate that sweep waves can be turned into trigger waves if the cell cycle is slowed down genetically or if significant delays in the cell-cycle progression are introduced across the embryo by altering nuclear density. Our genetic experiments demonstrate that Polo kinase is a major rate-limiting regulator of the blastoderm divisions, and genetic perturbations reducing its activity can induce the transition from sweep to trigger waves. Furthermore, we show that changes in temperature cause an essentially uniform slowdown of interphase and mitosis. That results in sweep waves being observed across a wide temperature range despite the cell-cycle durations being significantly different. Collectively, our combination of theory and experiments elucidates the nature of mitotic waves in Drosophila embryogenesis, their control mechanisms, and their mutual transitions.

Specificity of Nuclear Size Scaling in Frog Erythrocytes

In eukaryotes, the cell has the ability to modulate the size of the nucleus depending on the surrounding environment, to enable nuclear functions such as DNA replication and transcription. From previous analyses of nuclear size scaling in various cell types and species, it has been found that eukaryotic cells have a conserved scaling rule, in which the nuclear size correlates with both cell size and genomic content. However, there are few studies that have focused on a certain cell type and systematically analyzed the size scaling properties in individual species (intra-species) and among species (inter-species), and thus, the difference in the scaling rules among cell types and species is not well understood. In the present study, we analyzed the size scaling relationship among three parameters, nuclear size, cell size, and genomic content, in our measured datasets of terminally differentiated erythrocytes of five Anura frogs and collected datasets of different species classes from published papers. In the datasets of isolated erythrocytes from individual frogs, we found a very weak correlation between the measured nuclear and cell cross-sectional areas. Within the erythrocytes of individual species, the correlation of the nuclear area with the cell area showed a very low hypoallometric relationship, in which the relative nuclear size decreased when the cell size increased. These scaling trends in intra-species erythrocytes are not comparable to the known general correlation in other cell types. When comparing parameters across species, the nuclear areas correlated with both cell areas and genomic contents among the five frogs and the collected datasets in each species class. However, the contribution of genomic content to nuclear size determination was smaller than that of the cell area in all species classes. In particular, the estimated degree of the contribution of genomic content was greater in the amphibian class than in other classes. Together with our imaging analysis of structural components in nuclear membranes, we hypothesized that the observed specific features in nuclear size scaling are achieved by the weak interaction of the chromatin with the nuclear membrane seen in frog erythrocytes.

Waves in Embryonic Development

Embryonic development hinges on effective coordination of molecular events across space and time. Waves have recently emerged as constituting an ubiquitous mechanism that ensures rapid spreading of regulatory signals across embryos, as well as reliable control of their patterning, namely, for the emergence of body plan structures. In this article, we review a selection of recent quantitative work on signaling waves and present an overview of the theory of waves. Our aim is to provide a succinct yet comprehensive guiding reference for the theoretical frameworks by which signaling waves can arise in embryos. We start, then, from reaction-diffusion systems, both static and time dependent; move to excitable dynamics; and conclude with systems of coupled oscillators. We link these theoretical models to molecular mechanisms recently elucidated for the control of mitotic waves in early embryos, patterning of the vertebrate body axis, micropattern cultures, and bone regeneration. Our goal is to inspire experimental work that will advance theory in development and connect its predictions to quantitative biological observations. Expected final online publication date for the Annual Review of Biophysics, Volume 51 is May 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Yolk platelets impede nuclear expansion in Xenopus embryos

During metazoan early embryogenesis, the intracellular properties of proteins and organelles change dynamically through rapid cleavage. In particular, a change in the nucleus size is known to contribute to embryonic development-dependent cell cycle and gene expression regulation. Here, we compared the nuclear sizes of various blastomeres from developing Xenopus embryos and analyzed the mechanisms that control the nuclear expansion dynamics by manipulating the amount of intracellular components in a cell-free system. Nuclear expansion was slower in blastomeres from vegetal hemispheres during a longer interphase than in those from animal hemispheres. Furthermore, upon recapitulating interphase events by manipulating the concentration of yolk platelets, which are originally rich in the vegetal blastomeres, in cell-free cytoplasmic extracts, nuclear expansion and DNA replication became slower than that in normal yolk-free conditions. Under these conditions, the supplemented yolk platelets accumulated around the nucleus in a microtubule-dependent manner and impeded the organization of the endoplasmic reticulum network. Overall, we propose that yolk platelets around the nucleus reduce membrane supply from the endoplasmic reticulum to the nucleus, resulting in slower nuclear expansion and cell cycle progression in the yolk-rich vegetal blastomeres.

Attractive internuclear force drives the collective behavior of nuclear arrays in Drosophila embryos

The collective behavior of the nuclear array in Drosophila embryos during nuclear cycle (NC) 11 to NC14 is crucial in controlling cell size, establishing developmental patterns, and coordinating morphogenesis. After live imaging on Drosophila embryos with light sheet microscopy, we extract the nuclear trajectory, speed, and internuclear distance with an automatic nuclear tracing method. We find that the nuclear speed shows a period of standing waves along the anterior-posterior (AP) axis after each metaphase as the nuclei collectively migrate towards the embryo poles and partially move back. And the maximum nuclear speed dampens by 28-45% in the second half of the standing wave. Moreover, the nuclear density is 22-42% lower in the pole region than the middle of the embryo during the interphase of NC12-14. To find mechanical rules controlling the collective motion and packing patterns of the nuclear array, we use a deep neural network (DNN) to learn the underlying force field from data. We apply the learned spatiotemporal attractive force field in the simulations with a particle-based model. And the simulations recapitulate nearly all the observed characteristic collective behaviors of nuclear arrays in Drosophila embryos.

Analytical approximations for the speed of pacemaker-generated waves

In oscillatory media, waves can be generated by pacemaker regions which oscillate faster than their surroundings. In many chemical and biological systems, such waves can synchronize the whole medium and as such they are a means of transmitting information at a fixed speed over large distances. In this paper, we apply analytical tools to investigate the factors that determine the speed of these waves. More precisely, we apply singular perturbation and phase reduction methods to two types of negative-feedback oscillators, one built on underlying bistability and one including a time delay in the negative feedback. In both systems, we investigate the influence of time-scale separation on the resulting wave speed, as well as the effect of size and frequency of the pacemaker region. We compare our analytical estimates to numerical simulations which we described previously [J. Rombouts and L. Gelens, Phys. Rev. Research 2, 043038 (2020)2643-156410.1103/PhysRevResearch.2.043038].

Diffusive wave dynamics beyond the continuum limit

Scientists have observed and studied diffusive waves in contexts as disparate as population genetics and cell signaling. Often, these waves are propagated by discrete entities or agents, such as individual cells in the case of cell signaling. For a broad class of diffusive waves, we characterize the transition between the collective propagation of diffusive waves, in which the wave speed is well described by continuum theory, and the propagation of diffusive waves by individual agents. We show that this transition depends heavily on the dimensionality of the system in which the wave propagates and that disordered systems yield dynamics largely consistent with lattice systems. In some system dimensionalities, the intuition that closely packed sources more accurately mimic a continuum can be grossly violated.

Mitotic waves in an import-diffusion model with multiple nuclei in a shared cytoplasm

Nuclei import and export proteins, including cell cycle regulators. These import-export processes are modulated periodically by the cell cycle, for example due to the periodic assembly and breakdown of the nuclear envelope. As such, replicated DNA can be segregated between the two daughter cells and the proteins that were localized in the nucleus are free to diffuse throughout the cytoplasm. Here, we study a mathematical import-diffusion model to show how proteins, i.e. cell cycle regulators, could be redistributed in the cytoplasm by nuclei that periodically toggle between interphase and mitosis. We show that when the cell cycle period depends on the local concentration of regulators, the model exhibits mitotic waves. We discuss how the velocity and spatial origin of these mitotic waves depend on the different model parameters. This work is motivated by recent in vitro experiments reporting on mitotic waves in cycling cell-free extracts made with Xenopus laevis frog eggs, where multiple nuclei share the same cytoplasm. Such experiments have shown that nuclei act as pacemakers for the cell cycle and thus play an important role in collectively defining the spatial origin of mitotic waves.

Cell cycle control during early embryogenesis

Understanding the mechanisms of embryonic cell cycles is a central goal of developmental biology, as the regulation of the cell cycle must be closely coordinated with other events during early embryogenesis. Quantitative imaging approaches have recently begun to reveal how the cell cycle oscillator is controlled in space and time, and how it is integrated with mechanical signals to drive morphogenesis. Here, we discuss how the Drosophila embryo has served as an excellent model for addressing the molecular and physical mechanisms of embryonic cell cycles, with comparisons to other model systems to highlight conserved and species-specific mechanisms. We describe how the rapid cleavage divisions characteristic of most metazoan embryos require chemical waves and cytoplasmic flows to coordinate morphogenesis across the large expanse of the embryo. We also outline how, in the late cleavage divisions, the cell cycle is inter-regulated with the activation of gene expression to ensure a reliable maternal-to-zygotic transition. Finally, we discuss how precise transcriptional regulation of the timing of mitosis ensures that tissue morphogenesis and cell proliferation are tightly controlled during gastrulation.

Self-Organization and Information Processing: From Basic Enzymatic Activities to Complex Adaptive Cellular Behavior

One of the main aims of current biology is to understand the origin of the molecular organization that underlies the complex dynamic architecture of cellular life. Here, we present an overview of the main sources of biomolecular order and complexity spanning from the most elementary levels of molecular activity to the emergence of cellular systemic behaviors. First, we have addressed the dissipative self-organization, the principal source of molecular order in the cell. Intensive studies over the last four decades have demonstrated that self-organization is central to understand enzyme activity under cellular conditions, functional coordination between enzymatic reactions, the emergence of dissipative metabolic networks (DMN), and molecular rhythms. The second fundamental source of order is molecular information processing. Studies on effective connectivity based on transfer entropy (TE) have made possible the quantification in bits of biomolecular information flows in DMN. This information processing enables efficient self-regulatory control of metabolism. As a consequence of both main sources of order, systemic functional structures emerge in the cell; in fact, quantitative analyses with DMN have revealed that the basic units of life display a global enzymatic structure that seems to be an essential characteristic of the systemic functional metabolism. This global metabolic structure has been verified experimentally in both prokaryotic and eukaryotic cells. Here, we also discuss how the study of systemic DMN, using Artificial Intelligence and advanced tools of Statistic Mechanics, has shown the emergence of Hopfield-like dynamics characterized by exhibiting associative memory. We have recently confirmed this thesis by testing associative conditioning behavior in individual amoeba cells. In these Pavlovian-like experiments, several hundreds of cells could learn new systemic migratory behaviors and remember them over long periods relative to their cell cycle, forgetting them later. Such associative process seems to correspond to an epigenetic memory. The cellular capacity of learning new adaptive systemic behaviors represents a fundamental evolutionary mechanism for cell adaptation.

Dynamic bistable switches enhance robustness and accuracy of cell cycle transitions

Bistability is a common mechanism to ensure robust and irreversible cell cycle transitions. Whenever biological parameters or external conditions change such that a threshold is crossed, the system abruptly switches between different cell cycle states. Experimental studies have uncovered mechanisms that can make the shape of the bistable response curve change dynamically in time. Here, we show how such a dynamically changing bistable switch can provide a cell with better control over the timing of cell cycle transitions. Moreover, cell cycle oscillations built on bistable switches are more robust when the bistability is modulated in time. Our results are not specific to cell cycle models and may apply to other bistable systems in which the bistable response curve is time-dependent.

Many systems in nature show bistability, which means they can evolve to one of two stable steady states under exactly the same conditions. Which state they evolve to depends on where the system comes from. Such bistability underlies the switching behavior that is essential for cells to progress in the cell division cycle. A quick switch happens when the cell jumps from one steady state to another steady state. Typical of this switching behavior is its robustness and irreversibility. In this paper, we expand this viewpoint of the dynamics of the cell cycle by considering bistable switches which themselves are changing in time. This gives the cell an extra layer of control over transitions both in time and in space, and can make those transitions more robust. Such dynamically changing bistability can appear very naturally. We show this in a model of mitotic entry, in which we include a nuclear and cytoplasmic compartment. The activity of a crucial cell cycle protein follows a bistable switch in each compartment, but the shape of its response is changing in time as proteins are imported into and exported from the nucleus.

The nucleus serves as the pacemaker for the cell cycle

Mitosis is a dramatic process that affects all parts of the cell. It is driven by an oscillator whose various components are localized in the nucleus, centrosome, and cytoplasm. In principle, the cellular location with the fastest intrinsic rhythm should act as a pacemaker for the process. Here we traced the waves of tubulin polymerization and depolymerization that occur at mitotic entry and exit in Xenopus egg extracts back to their origins. We found that mitosis was commonly initiated at sperm-derived nuclei and their accompanying centrosomes. The cell cycle was ~20% faster at these initiation points than in the slowest regions of the extract. Nuclei produced from phage DNA, which did not possess centrosomes, also acted as trigger wave sources, but purified centrosomes in the absence of nuclei did not. We conclude that the nucleus accelerates mitotic entry and propose that it acts as a pacemaker for cell cycle.

Dynamics of diffusive cell signaling relays

In biological contexts as diverse as development, apoptosis, and synthetic microbial consortia, collections of cells or subcellular components have been shown to overcome the slow signaling speed of simple diffusion by utilizing diffusive relays, in which the presence of one type of diffusible signaling molecule triggers participation in the emission of the same type of molecule. This collective effect gives rise to fast-traveling diffusive waves. Here, in the context of cell signaling, we show that system dimensionality – the shape of the extracellular medium and the distribution of cells within it – can dramatically affect the wave dynamics, but that these dynamics are insensitive to details of cellular activation. As an example, we show that neutrophil swarming experiments exhibit dynamical signatures consistent with the proposed signaling motif. We further show that cell signaling relays generate much steeper concentration profiles than does simple diffusion, which may facilitate neutrophil chemotaxis.

Nuclei determine the spatial origin of mitotic waves

Traveling waves play an essential role in coordinating mitosis over large distances, but what determines the spatial origin of mitotic waves remains unclear. Here, we show that such waves initiate at pacemakers, regions that oscillate faster than their surroundings. In cell-free extracts of Xenopus laevis eggs, we find that nuclei define such pacemakers by concentrating cell cycle regulators. In computational models of diffusively coupled oscillators that account for nuclear import, nuclear positioning determines the pacemaker location. Furthermore, we find that the spatial dimensions of the oscillatory medium change the nuclear positioning and strongly influence whether a pacemaker is more likely to be at a boundary or an internal region. Finally, we confirm experimentally that increasing the system width increases the proportion of pacemakers at the boundary. Our work provides insight into how nuclei and spatial system dimensions can control local concentrations of regulators and influence the emergent behavior of mitotic waves.