PAR-3 oligomerization may provide an actin-independent mechanism to maintain distinct par protein domains in the early Caenorhabditis elegans embryo

Order from chaos: cellular asymmetries explained with modelling

Molecules inside cells are subject to physical forces and undergo biochemical interactions, continuously changing their physical properties and dynamics. Despite this, cells achieve highly ordered molecular patterns that are crucial to regulate various cellular functions and to specify cell fate. In the Caenorhabditis elegans one-cell embryo, protein asymmetries are established in the narrow time window of a cell division. What are the mechanisms that allow molecules to establish asymmetries, defying the randomness imposed by Brownian motion? Mathematical and computational models have paved the way to the understanding of protein dynamics up to the 'single-molecule level' when resolution represents an issue for precise experimental measurements. Here we review the models that interpret cortical and cytoplasmic asymmetries in the one-cell C. elegans embryo.

Oligomerization of peripheral membrane proteins provides tunable control of cell surface polarity

Asymmetric distributions of peripheral membrane proteins define cell polarity across all kingdoms of life. Non-linear positive feedback on membrane binding is essential to amplify and stabilize these asymmetries, but how specific molecular sources of non-linearity shape polarization dynamics remains poorly understood. Here we show that the ability to oligomerize, which is common to many peripheral membrane proteins, can play a profound role in shaping polarization dynamics in simple feedback circuits. We show that size-dependent binding avidity and mobility of membrane-bound oligomers endow polarity circuits with several key properties. Size-dependent membrane binding avidity confers a form of positive feedback on the accumulation of oligomer subunits. Although insufficient by itself, this sharply reduces the amount of additional feedback required for spontaneous emergence and stable maintenance of polarized states. Size-dependent oligomer mobility makes symmetry breaking and stable polarity more robust with respect to variation in subunit diffusivities and cell sizes, and slows the approach to a final stable spatial distribution, allowing cells to "remember" polarity boundaries imposed by transient external cues. Together, these findings reveal how oligomerization of peripheral membrane proteins can provide powerful and highly tunable sources of non-linear feedback in biochemical circuits that govern cell surface polarity. Given its prevalence and widespread involvement in cell polarity, we speculate that self-oligomerization may have provided an accessible path to evolving simple polarity circuits.

Delineating mechanisms and design principles of Caenorhabditis elegans embryogenesis using in toto high-resolution imaging data and computational modeling

The nematode (roundworm) Caenorhabditis elegans is one of the most popular animal models for the study of developmental biology, as its invariant development and transparent body enable in toto cellular-resolution fluorescence microscopy imaging of developmental processes at 1-min intervals. This has led to the development of various computational tools for the systematic and automated analysis of imaging data to delineate the molecular and cellular processes throughout the embryogenesis of C. elegans, such as those associated with cell lineage, cell migration, cell morphology, and gene activity. In this review, we first introduce C. elegans embryogenesis and the development of techniques for tracking cell lineage and reconstructing cell morphology during this process. We then contrast the developmental modes of C. elegans and the customized technologies used for studying them with the ones of other animal models, highlighting its advantage for studying embryogenesis with exceptional spatial and temporal resolution. This is followed by an examination of the physical models that have been devised—based on accurate determinations of developmental processes afforded by analyses of imaging data—to interpret the early embryonic development of C. elegans from subcellular to intercellular levels of multiple cells, which focus on two key processes: cell polarization and morphogenesis. We subsequently discuss how quantitative data-based theoretical modeling has improved our understanding of the mechanisms of C. elegans embryogenesis. We conclude by summarizing the challenges associated with the acquisition of C. elegans embryogenesis data, the construction of algorithms to analyze them, and the theoretical interpretation.

Modeling protein dynamics in Caenorhabditis elegans embryos reveals that the PLK-1 gradient relies on weakly coupled reaction-diffusion mechanisms

Intracellular gradients have essential roles in cell and developmental biology, but their formation is not fully understood. We have developed a computational approach facilitating interpretation of protein dynamics and gradient formation. We have combined this computational approach with experiments to understand how Polo-Like Kinase 1 (PLK-1) forms a cytoplasmic gradient in Caenorhabditis elegans embryos. Although the PLK-1 gradient depends on the Muscle EXcess-5/6 (MEX-5/6) proteins, we reveal differences in PLK-1 and MEX-5 gradient formation that can be explained by a model with two components, PLK-1 bound to MEX-5 and unbound PLK-1. Our combined approach suggests that a weak coupling between PLK-1 and MEX-5 reaction–diffusion mechanisms dictates the dynamic exchange of PLK-1 with the cytoplasm, explaining PLK-1 high diffusivity and smooth gradient.

Protein gradients have fundamental roles in cell and developmental biology. In the one-cell Caenorhabditis elegans embryo, the mitotic Polo-Like Kinase 1 (PLK-1) forms an anterior-rich cytoplasmic gradient, which is crucial for asymmetric cell division and embryonic development. The PLK-1 gradient depends on the RNA-binding Muscle-EXcess-5 protein (MEX-5), whose slow-diffusing complexes accumulate in the anterior via a reaction–diffusion mechanism. Here, we combine experiments and a computational approach to investigate the dynamics of PLK-1 gradient formation. We find that the gradient of PLK-1 initiates later, is less steep, and forms with slower dynamics than does the MEX-5 gradient. The data show that PLK-1 diffuses faster than MEX-5 in both anterior and posterior cytoplasmic regions. Our simulations suggest that binding to slow-diffusing MEX-5 is required for PLK-1 gradient formation, but that a significant fraction of unbound PLK-1 is necessary to justify the different gradient dynamics. We provide a computational tool able to predict gradient establishment prior to cell division and show that a two-component, bound and unbound, model of PLK-1 dynamics recapitulates the experimental observations.

Biophysical Models of PAR Cluster Transport by Cortical Flow in C. elegans Early Embryogenesis

The clustering of membrane-bound proteins facilitates their transport by cortical actin flow in early Caenorhabditis elegans embryo cell polarity. PAR-3 clustering is critical for this process, yet the biophysical processes that couple protein clusters to cortical flow remain unknown. We develop a discrete, stochastic agent-based model of protein clustering and test four hypothetical models for how clusters may interact with the flow. Results show that the canonical way to assess transport characteristics from single-particle tracking data used thus far in this area, the Péclet number, is insufficient to distinguish these hypotheses and that all models can account for transport characteristics quantified by this measure. However, using this model, we demonstrate that these different cluster-cortex interactions may be distinguished using a different metric, namely the scalar projection of cluster displacement on to the flow displacement vector. Our results thus provide a testable way to use existing single-particle tracking data to test how endogenous protein clusters may interact with the cortical flow to localize during polarity establishment. To facilitate this investigation, we also develop both improved simulation and semi-analytic methodologies to quantify motion summary statistics (e.g., Péclet number and scalar projection) for these stochastic models as a function of biophysical parameters.

Cytoskeletal symmetry breaking in animal cells

Symmetry breaking is a crucial step in structure formation and function of all cells, necessary for cell movement, cell division, and polarity establishment. Although the mechanisms of symmetry breaking are diverse, they often share common characteristics. Here we review examples of nematic, polar, and chiral cytoskeletal symmetry breaking in animal cells, and analogous processes in simplified reconstituted systems. We discuss the origins of symmetry breaking, which can arise spontaneously, or involve amplification of a pre-existing external or internal bias to the whole cell level. The underlying mechanisms often involve both chemical and mechanical processes that cooperate to break symmetry in a robust manner, and typically depend on the shape, size, or properties of the cell's boundary.

A balance between antagonizing PAR proteins specifies the pattern of asymmetric and symmetric divisions in C. elegans embryogenesis

Coordination between cell differentiation and proliferation during development requires the balance between asymmetric and symmetric modes of cell division. However, the cellular intrinsic cue underlying the choice between these two division modes remains elusive. Here, we show evidence in Caenorhabditis elegans that the invariable lineage of the division modes is specified by the balance between antagonizing complexes of partitioning-defective (PAR) proteins. By uncoupling unequal inheritance of PAR proteins from that of fate determinants during cell division, we demonstrate that changes in the balance between PAR-2 and PAR-6 can be sufficient to re-program the division modes from symmetric to asymmetric and vice versa in two daughter cells. The division mode adopted occurs independently of asymmetry in cytoplasmic fate determinants, cell-size asymmetry, and cell-cycle asynchrony between sister cells. We propose that the balance between PAR proteins represents an intrinsic self-organizing cue for the specification of the two division modes during development.

The Role of Cytoplasmic MEX-5/6 Polarity in Asymmetric Cell Division

In the process of asymmetric cell division, the mother cell induces polarity in both the membrane and the cytosol by distributing substrates and components asymmetrically. Such polarity formation results from the harmonization of the upstream and downstream polarities between the cell membrane and the cytosol. MEX-5/6 is a well-investigated downstream cytoplasmic protein, which is deeply involved in the membrane polarity of the upstream transmembrane protein PAR in the Caenorhabditis elegans embryo. In contrast to the extensive exploration of membrane PAR polarity, cytoplasmic polarity is poorly understood, and the precise contribution of cytoplasmic polarity to the membrane PAR polarity remains largely unknown. In this study, we explored the interplay between the cytoplasmic MEX-5/6 polarity and the membrane PAR polarity by developing a mathematical model that integrates the dynamics of PAR and MEX-5/6 and reflects the cell geometry. Our investigations show that the downstream cytoplasmic protein MEX-5/6 plays an indispensable role in causing a robust upstream PAR polarity, and the integrated understanding of their interplay, including the effect of the cell geometry, is essential for the study of polarity formation in asymmetric cell division.

The first steps in the life of a worm: Themes and variations in asymmetric division in C. elegans and other nematodes

Starting with Boveri in the 1870s, microscopic investigation of early embryogenesis in a broad swath of nematode species revealed the central role of asymmetric cell division in embryonic axis specification, blastomere positioning, and cell fate specification. Notably, across the class Chromadorea, a conserved theme emerges-asymmetry is first established in the zygote and specifies its asymmetric division, giving rise to an anterior somatic daughter cell and a posterior germline daughter cell. Beginning in the 1980s, the emergence of Caenorhabditis elegans as a model organism saw the advent of genetic tools that enabled rapid progress in our understanding of the molecular mechanisms underlying asymmetric division, in many cases defining key paradigms that turn out to regulate asymmetric division in a wide range of systems. Yet, the consequence of this focus on C. elegans came at the expense of exploring the extant diversity of developmental variation exhibited across nematode species. Given the resurgent interest in evolutionary studies facilitated in part by new tools, here we revisit the diversity in this asymmetric first division, juxtaposing molecular insight into mechanisms of symmetry-breaking, spindle positioning and fate specification, with a consideration of plasticity and variability within and between species. In the process, we hope to highlight questions of evolutionary forces and molecular variation that may have shaped the extant diversity of developmental mechanisms observed across Nematoda.

Mechanochemical Control of Symmetry Breaking in the Caenorhabditis elegans Zygote

Cell polarity is the asymmetric organization of cellular components along defined axes. A key requirement for polarization is the ability of the cell to break symmetry and achieve a spatially biased organization. Despite different triggering cues in various systems, symmetry breaking (SB) usually relies on mechanochemical modulation of the actin cytoskeleton, which allows for advected movement and reorganization of cellular components. Here, the mechanisms underlying SB in Caenorhabditis elegans zygote, one of the most popular models to study cell polarity, are reviewed. A zygote initiates SB through the centrosome, which modulates mechanics of the cell cortex to establish advective flow of cortical proteins including the actin cytoskeleton and partitioning defective (PAR) proteins. The chemical signaling underlying centrosomal control of the Aurora A kinase–mediated cascade to convert the organization of the contractile actomyosin network from an apolar to polar state is also discussed.

Quantifying the roles of space and stochasticity in computer simulations for cell biology and cellular biochemistry

Most of the fascinating phenomena studied in cell biology emerge from interactions among highly organized multimolecular structures embedded into complex and frequently dynamic cellular morphologies. For the exploration of such systems, computer simulation has proved to be an invaluable tool, and many researchers in this field have developed sophisticated computational models for application to specific cell biological questions. However, it is often difficult to reconcile conflicting computational results that use different approaches to describe the same phenomenon. To address this issue systematically, we have defined a series of computational test cases ranging from very simple to moderately complex, varying key features of dimensionality, reaction type, reaction speed, crowding, and cell size. We then quantified how explicit spatial and/or stochastic implementations alter outcomes, even when all methods use the same reaction network, rates, and concentrations. For simple cases, we generally find minor differences in solutions of the same problem. However, we observe increasing discordance as the effects of localization, dimensionality reduction, and irreversible enzymatic reactions are combined. We discuss the strengths and limitations of commonly used computational approaches for exploring cell biological questions and provide a framework for decision making by researchers developing new models. As computational power and speed continue to increase at a remarkable rate, the dream of a fully comprehensive computational model of a living cell may be drawing closer to reality, but our analysis demonstrates that it will be crucial to evaluate the accuracy of such models critically and systematically.

A hybrid stochastic-deterministic mechanochemical model of cell polarization

Polarization is a crucial component in cell differentiation, development, and motility, but its details are not yet well understood. At the onset of cell locomotion, cells break symmetry to form well-defined cell fronts and rears. This polarity establishment varies across cell types: in Dictyostelium discoideum cells, it is mediated by biochemical signaling pathways and can function in the absence of a cytoskeleton, while in keratocytes, it is tightly connected to cytoskeletal dynamics and mechanics. Theoretical models that have been developed to understand the onset of polarization have explored either signaling or mechanical pathways, yet few have explored mechanochemical mechanisms. However, many motile cells rely on both signaling modules and actin cytoskeleton to break symmetry and achieve a stable polarized state. We propose a general mechanochemical polarization model based on coupling between a stochastic model for the segregation of signaling molecules and a simplified mechanical model for actin cytoskeleton network competition. We find that local linear coupling between minimally nonlinear signaling and cytoskeletal systems, separately not supporting stable polarization, yields a robustly polarized cell state. The model captures the essence of spontaneous polarization of neutrophils, which has been proposed to emerge due to the competition between frontness and backness pathways.

Geometric cues stabilise long-axis polarisation of PAR protein patterns in C. elegans

In the Caenorhabditis elegans zygote, PAR protein patterns, driven by mutual anatagonism, determine the anterior-posterior axis and facilitate the redistribution of proteins for the first cell division. Yet, the factors that determine the selection of the polarity axis remain unclear. We present a reaction-diffusion model in realistic cell geometry, based on biomolecular reactions and accounting for the coupling between membrane and cytosolic dynamics. We find that the kinetics of the phosphorylation-dephosphorylation cycle of PARs and the diffusive protein fluxes from the cytosol towards the membrane are crucial for the robust selection of the anterior-posterior axis for polarisation. The local ratio of membrane surface to cytosolic volume is the main geometric cue that initiates pattern formation, while the choice of the long-axis for polarisation is largely determined by the length of the aPAR-pPAR interface, and mediated by processes that minimise the diffusive fluxes of PAR proteins between cytosol and membrane.

Intracellular morphogens: Specifying patterns at the subcellular scale

The notion that graded distributions of signals underlie the spatial organization of biological systems has long been a central pillar in the fields of cell and developmental biology. During morphogenesis, morphogens spread across tissues to guide development of the embryo. Similarly, a variety of dynamic gradients and pattern-forming networks have been discovered that shape subcellular organization. Here we discuss the principles of intracellular pattern formation by these intracellular morphogens and relate them to conceptually similar processes operating at the tissue scale. We will specifically review mechanisms for generating cellular asymmetry and consider how intracellular patterning networks are controlled and adapt to cellular geometry. Finally, we assess the general concept of intracellular gradients as a mechanism for positional control in light of current data, highlighting how the simple readout of fixed concentration thresholds fails to fully capture the complexity of spatial patterning processes occurring inside cells.

A cell size threshold limits cell polarity and asymmetric division potential

Reaction-diffusion networks underlie pattern formation in a range of biological contexts, from morphogenesis of organisms to the polarisation of individual cells. One requirement for such molecular networks is that output patterns be scaled to system size. At the same time, kinetic properties of constituent molecules constrain the ability of networks to adapt to size changes. Here we explore these constraints and the consequences thereof within the conserved PAR cell polarity network. Using the stem cell-like germ lineage of the C. elegans embryo as a model, we find that the behaviour of PAR proteins fails to scale with cell size. Theoretical analysis demonstrates that this lack of scaling results in a size threshold below which polarity is destabilized, yielding an unpolarized system. In empirically-constrained models, this threshold occurs near the size at which germ lineage cells normally switch between asymmetric and symmetric modes of division. Consistent with cell size limiting polarity and division asymmetry, genetic or physical reduction in germ lineage cell size is sufficient to trigger loss of polarity in normally polarizing cells at predicted size thresholds. Physical limits of polarity networks may be one mechanism by which cells read out geometrical features to inform cell fate decisions.

The importance of mechanical constraints for proper polarization and psuedo-cleavage furrow generation in the early Caenorhabditis elegans embryo

Intracellular polarization, where a cell specifies a spatial axis by segregation of specific factors, is a fundamental biological process. In the early embryo of the nematode worm Caenorhabditis elegans (C. elegans), polarization is often accompanied by deformations of the cortex, a highly contractile structure consisting of actin filaments cross-linked by the motor protein myosin (actomyosin). It has been suggested that the eggshell surrounding the early embryo plays a role in polarization although its function is not understood. Here we develop a mathematical model which couples a reaction-diffusion model of actomyosin dynamics with a phase field model of the cell cortex to implicitly track cell shape changes in the early C. elegans embryo. We investigate the potential rigidity effect of the geometric constraint imposed by the presence and size of the eggshell on polarization dynamics. Our model suggests that the geometric constraint of the eggshell is essential for proper polarization and the size of the eggshell also affects the dynamics of polarization. Therefore, we conclude that geometric constraint on a cell might affect the dynamics of a biochemical process.

Polarization, whereby molecules and proteins are asymmetrically distributed throughout the cell, is a vital process for many cellular functions. In the early C. elegans embryo the asymmetric distribution of cell cytoskeleton during the initiation of polarization leads to asymmetric contractions which are higher in the anterior and lower in the posterior of a cell. The C. elegans embryo is surrounded by a rigid body, the eggshell, which functions in numerous cell processes. We investigate the structural support of eggshell during the establishment phase by tracking the moving cell surface. We incorporate protein dynamics involved in polarization into the membrane evolution. We conclude that eggshell might have a role in cell polarization by preventing the distortion of cell surface.

Axon Regeneration: Antagonistic Signaling Pairs in Neuronal Polarization

Genome-wide screens, proteomics, and candidate-based approaches have identified numerous genes associated with neuronal regeneration following central nervous system (CNS) injury. Despite significant progress, functional recovery remains a challenge, even in model systems. Neuronal function depends on segregation of axonal versus dendritic domains. A key to functional recovery may lie in recapitulating the developmental signals that instruct axon specification and growth in adult neurons post-injury. Theoretically, binary activator-inhibitor elements operating as a Turing-like system within neurons can specify axonal versus dendritic domains and promote axon growth. We review here various molecules implicated in axon specification that function as signaling pairs driving neuronal polarization and axon growth.

The PAR proteins: from molecular circuits to dynamic self-stabilizing cell polarity

PAR proteins constitute a highly conserved network of scaffolding proteins, adaptors and enzymes that form and stabilize cortical asymmetries in response to diverse inputs. They function throughout development and across the metazoa to regulate cell polarity. In recent years, traditional approaches to identifying and characterizing molecular players and interactions in the PAR network have begun to merge with biophysical, theoretical and computational efforts to understand the network as a pattern-forming biochemical circuit. Here, we summarize recent progress in the field, focusing on recent studies that have characterized the core molecular circuitry, circuit design and spatiotemporal dynamics. We also consider some of the ways in which the PAR network has evolved to polarize cells in different contexts and in response to different cues and functional constraints.

Laminins in Epithelial Cell Polarization: Old Questions in Search of New Answers

Laminin, a basement membrane protein discovered in 1979, was shortly thereafter implicated in the polarization of epithelial cells in both mammals and a variety of lower organisms. To transduce a spatial cue to the intrinsic polarization machinery, laminin must polymerize into a dense network that forms the foundation of the basement membrane. Evidence suggests that activation of the small GTPase Rac1 by β1-integrins mobilizes laminin-binding integrins and dystroglycan to consolidate formation of the laminin network and initiate rearrangements of both the actin and microtubule cytoskeleton to help establish the apicobasal axis. A key coordinator of spatial signals from laminin is the serine-threonine kinase Par-1, which is known to affect dystroglycan availability, microtubule and actin organization, and lumen formation. The signaling protein integrin-linked kinase (ILK) may also play a role. Despite significant advances, knowledge of the mechanism by which assembled laminin produces a spatial signal remains fragmentary, and much more research into the complex functions of laminin in polarization and other cellular processes is needed.

Many roads to symmetry breaking: molecular mechanisms and theoretical models of yeast cell polarity

Mathematical modeling has been instrumental in identifying common principles of cell polarity across diverse systems. These principles include positive feedback loops that are required to destabilize a spatially uniform state of the cell. The conserved small G-protein Cdc42 is a master regulator of eukaryotic cellular polarization. Here we discuss recent developments in studies of Cdc42 polarization in budding and fission yeasts and demonstrate that models describing symmetry-breaking polarization can be classified into six minimal classes based on the structure of positive feedback loops that activate and localize Cdc42. Owing to their generic system-independent nature, these model classes are also likely to be relevant for the G-protein–based symmetry-breaking systems of higher eukaryotes. We review experimental evidence pro et contra different theoretically plausible models and conclude that several parallel and non–mutually exclusive mechanisms are likely involved in cellular polarization of yeasts. This potential redundancy needs to be taken into consideration when interpreting the results of recent cell-rewiring studies.

A Single-Cell Biochemistry Approach Reveals PAR Complex Dynamics during Cell Polarization

Regulated protein-protein interactions are critical for cell signaling, differentiation, and development. For the study of dynamic regulation of protein interactions in vivo, there is a need for techniques that can yield time-resolved information and probe multiple protein binding partners simultaneously, using small amounts of starting material. Here we describe a single-cell protein interaction assay. Single-cell lysates are generated at defined time points and analyzed using single-molecule pull-down, yielding information about dynamic protein complex regulation in vivo. We established the utility of this approach by studying PAR polarity proteins, which mediate polarization of many animal cell types. We uncovered striking regulation of PAR complex composition and stoichiometry during Caenorhabditis elegans zygote polarization, which takes place in less than 20 min. PAR complex dynamics are linked to the cell cycle by Polo-like kinase 1 and govern the movement of PAR proteins to establish polarity. Our results demonstrate an approach to study dynamic biochemical events in vivo.

Protein clustering for cell polarity: Par-3 as a paradigm

The scaffold protein Par-3 ( Drosophila Bazooka) is a central organizer of cell polarity across animals. This review focuses on how the clustering of Par-3 contributes to cell polarity. It begins with the Par-3 homo-oligomerization mechanism and its regulation by Par-1 phosphorylation. The role of polarized cytoskeletal networks in distributing Par-3 clusters to one end of the cell is then discussed, as is the subsequent maintenance of polarized Par-3 clusters through hindered mobility and inhibition from the opposite pole. Finally, specific roles of Par-3 clusters are reviewed, including the bundling of microtubules, the cortical docking of centrosomes, the growth and positioning of cadherin–catenin clusters, and the inhibition of the Par-6–aPKC kinase cassette. Examples are drawn from Drosophila, Caenorhabditis elegans, mammalian cell culture, and biochemical studies.

Mechanisms of Cell Polarization

Cell polarization is a key step in the migration, development, and organization of eukaryotic cells, both at the single cell and multicellular level. Research on the mechanisms that give rise to polarization of a given cell, and organization of polarity within a tissue has led to new understanding across cellular and developmental biology. In this review, we describe some of the history of theoretical and experimental aspects of the field, as well as some interesting questions and challenges for the future.

Agent-based modeling: case study in cleavage furrow models

The number of studies in cell biology in which quantitative models accompany experiments has been growing steadily. Roughly, mathematical and computational techniques of these models can be classified as "differential equation based" (DE) or "agent based" (AB). Recently AB models have started to outnumber DE models, but understanding of AB philosophy and methodology is much less widespread than familiarity with DE techniques. Here we use the history of modeling a fundamental biological problem-positioning of the cleavage furrow in dividing cells-to explain how and why DE and AB models are used. We discuss differences, advantages, and shortcomings of these two approaches.

Dynamic Opposition of Clustered Proteins Stabilizes Cortical Polarity in the C. elegans Zygote

Dynamic maintenance of cell polarity is essential for development and physiology. Here we combine experiments and modeling to elucidate mechanisms that maintain cortical polarity in the C. elegans zygote. We show that polarity is dynamically stabilized by two coupled cross-inhibitory feedback loops: one involves the oligomeric scaffold PAR-3 and the kinase PAR-1, and the other involves CDC-42 and its putative GAP CHIN-1. PAR-3 and CDC-42 are both required locally to recruit PAR-6/PKC-3, which inhibits PAR-1 (shown previously) and inhibits local growth/accumulation of CHIN-1 clusters. Conversely, PAR-1 inhibits local accumulation of PAR-3 oligomers, while CHIN-1 inhibits CDC-42 (shown previously), such that either PAR-1 or CHIN-1 can prevent recruitment of PAR-6/PKC-3, but loss of both causes complete loss of polarity. Ultrasensitive dependence of CHIN-1 cluster growth on PAR-6/PKC-3 endows this core circuit with bistable dynamics, while transport of CHIN-1 clusters by cortical flow can stabilize the AP boundary against diffusive spread of PAR-6/PKC-3.

Quantitative analysis and modeling probe polarity establishment in C. elegans embryos

Cell polarity underlies many aspects of metazoan development and homeostasis, and relies notably on a set of PAR proteins located at the cell cortex. How these proteins interact in space and time remains incompletely understood. We performed a quantitative assessment of polarity establishment in one-cell stage Caenorhabditis elegans embryos by combining time-lapse microscopy and image analysis. We used our extensive data set to challenge and further specify an extant mathematical model. Using likelihood-based calibration, we uncovered that cooperativity is required for both anterior and posterior PAR complexes. Moreover, we analyzed the dependence of polarity establishment on changes in size or temperature. The observed robustness of PAR domain dimensions in embryos of different sizes is in agreement with a model incorporating fixed protein concentrations and variations in embryo surface/volume ratio. In addition, we quantified the dynamics of polarity establishment over most of the viable temperatures range of C. elegans. Modeling of these data suggests that diffusion of PAR proteins is the process most affected by temperature changes, although cortical flows appear unaffected. Overall, our quantitative analytical framework provides insights into the dynamics of polarity establishment in a developing system.

Self-organization and advective transport in the cell polarity formation for asymmetric cell division

Anterior-Posterior (AP) polarity formation of cell membrane proteins plays a crucial role in determining cell asymmetry, which depends not only on the several genetic process but also biochemical and biophysical interactions. The mechanism of AP formation of Caenorhabditis elegans embryo is characterized into the three processes: (i) membrane association and dissociation of posterior and anterior proteins, (ii) diffusion into the membrane and cytosol, and (iii) active cortical and cytoplasmic flows induced by the contraction of the acto-myosin cortex. We explored the mechanism of symmetry breaking and AP polarity formation using self-recruitment model of posterior proteins. We found that the AP polarity pattern is established over wide range in the total mass of polarity proteins and the diffusion ratio in the cytosol to the membrane. We also showed that the advective transport in both membrane and cytosol during the establishment phase affects optimal time interval of establishment and positioning of the posterior domain, and plays a role to increase the robustness in the AP polarity formation by reducing the number of posterior domains for the sensitivity of initial conditions. We also demonstrated that a proper ratio of the total mass to cell size robustly regulate the length scale of the posterior domain.

Protein abundance may regulate sensitivity to external cues in polarized cells

Cell polarization is a ubiquitous process which results in cellular constituents being organized into discrete intracellular spatial domains. It occurs in a variety of cell types, including epithelial cells, immune system cells and neurons. A key player in this process is the Par protein family whose asymmetric localization to anterior and posterior parts of the cell is crucial for proper division and cell fate specification. In this paper, we explore a stochastic analogue of the temporal model of Par protein interactions first developed in Dawes & Munro (Dawes and Munro 2011 Biophys. J. 101, 1412-1422. (doi:10.1016/j.bpj.2011.07.030)). We focus on how protein abundance influences the behaviour of both the deterministic and stochastic versions of the model. In Dawes & Munro (2011), it was found that bistable behaviour in the temporal model of Par protein led to the existence of complementary domains in the corresponding spatio-temporal model. Here, we find that the corresponding temporal stochastic model permits switching behaviour (the model solution 'jumps' between steady states) for lower protein abundances, whereas for higher protein abundances the stochastic and deterministic models are in good agreement (the model solution evolves to one of two steady states). This led us to the testable hypothesis that cells with lower abundances of Par protein may be more sensitive to external cues, whereas cells with higher abundances of Par protein may be less sensitive to external cues. In order to gain more control over the precise abundance of Par protein, we proposed and explored a second model (again, examining both deterministic and stochastic versions) in which the total number of Par molecules is conserved. We found that this model required an additional dimerization reaction in the cytoplasm in order for bistable and switching behaviour to be found. Once this additional reaction was included, we found that both the first and second models gave qualitatively similar results but in different regions of the parameter space, suggesting a further regulatory mechanism that cells could potentially use to modulate their response to external signals.

Cytoplasmic localization and asymmetric division in the early embryo of Caenorhabditis elegans

During the initial cleavages of the Caenorhabditis elegans embryo, a series of rapid and invariant asymmetric cell divisions pattern the fate, size, and position of four somatic blastomeres and a single germline blastomere. These asymmetric divisions are orchestrated by a collection of maternally deposited factors that are initially symmetrically distributed in the newly fertilized embryo. Maturation of the sperm-derived centrosome in the posterior cytoplasm breaks this symmetry by triggering a dramatic and highly stereotyped partitioning of these maternal factors. A network of conserved cell polarity regulators, the PAR proteins, form distinct anterior and posterior domains at the cell cortex. From these domains, the PAR proteins direct the segregation of somatic and germline factors into opposing regions of the cytoplasm such that, upon cell division, they are preferentially inherited by the somatic blastomere or the germline blastomere, respectively. The segregation of these factors is controlled, at least in part, by a series of reaction-diffusion mechanisms that are asymmetrically deployed along the anterior/posterior axis. The characterization of these mechanisms has important implications for our understanding of how cells are polarized and how spatial organization is generated in the cytoplasm. For further resources related to this article, please visit the WIREs website.

PAR polarity: from complexity to design principles

The par-titioning-defective or PAR proteins comprise the core of an essential cell polarity network that underlies polarization in a wide variety of cell types and developmental contexts. The output of this network in nearly every case is the establishment of opposing and complementary membrane domains that define a cell׳s polarity axis. Yet, behind this simple pattern is a complex system of interactions, regulation and dynamic behaviors. How these various parts combine to generate polarized patterns of protein localization in cells is only beginning to become clear. This review, part of the Special Issue on Cell Polarity, aims to highlight several emerging themes and design principles that underlie the process of cell polarization by components of the PAR network.

Actomyosin regulation and symmetry breaking in a model of polarization in the early Caenorhabditis elegans embryo : symmetry breaking in cell polarization

Polarization, whereby a cell defines a spatial axis by segregating specific determinants to distinct regions, is an essential and highly conserved biological process. The process of polarization is initiated by a cue that breaks an initially symmetric distribution of determinants, allowing for a spatially asymmetric redistribution. The nature of this cue is currently not well understood. Utilizing the conservation of polarization process and its determinants, we theoretically investigate the nature of the cue and the regulation of contractility that enables the establishment of polarity in early embryos of the nematode worm Caenorhabditis elegans (C. elegans). Our biologically based model, which consists of coupled partial differential equations, suggests that a biochemical but not mechanical cue is sufficient for symmetry breaking, and inhibition of contractile elements by specific determinants is needed for sustained spatial redistribution.

The PAR network: redundancy and robustness in a symmetry-breaking system

To become polarized, cells must first 'break symmetry'. Symmetry breaking is the process by which an unpolarized, symmetric cell develops a singularity, often at the cell periphery, that is used to develop a polarity axis. The Caenorhabditis elegans zygote breaks symmetry under the influence of the sperm-donated centrosome, which causes the PAR polarity regulators to sort into distinct anterior and posterior cortical domains. Modelling analyses have shown that cortical flows induced by the centrosome combined with antagonism between anterior and posterior PARs (mutual exclusion) are sufficient, in principle, to break symmetry, provided that anterior and posterior PAR activities are precisely balanced. Experimental evidence indicates, however, that the system is surprisingly robust to changes in cortical flows, mutual exclusion and PAR balance. We suggest that this robustness derives from redundant symmetry-breaking inputs that engage two positive feedback loops mediated by the anterior and posterior PAR proteins. In particular, the PAR-2 feedback loop stabilizes the polarized state by creating a domain where posterior PARs are immune to exclusion by anterior PARs. The two feedback loops in the PAR network share characteristics with the two feedback loops in the Cdc42 polarization network of Saccharomyces cerevisiae.

Cortical geometry may influence placement of interface between Par protein domains in early Caenorhabditis elegans embryos

During polarization, proteins and other polarity determinants segregate to the opposite ends of the cell (the poles) creating biochemically and dynamically distinct regions. Embryos of the nematode worm Caenorhabditis elegans (C. elegans) polarize shortly after fertilization, creating distinct regions of Par protein family members. These regions are maintained through to first cleavage when the embryo divides along the plane specified by the interface between regions, creating daughter cells with different protein content. In wild type single cell embryos the interface between these Par protein regions is reliably positioned at approximately 60% egg length, however, it is not known what mechanisms are responsible for specifying the position of the interface. In this investigation, we use two mathematical models to investigate the movement and positioning of the interface: a biologically based reaction-diffusion model of Par protein dynamics, and the analytically tractable perturbed Allen-Cahn equation. When we numerically simulate the models on a static 2D domain with constant thickness, both models exhibit a persistently moving interface that specifies the boundary between distinct regions. When we modify the simulation domain geometry, movement halts and the interface is stably positioned where the domain thickness increases. Using asymptotic analysis with the perturbed Allen-Cahn equation, we show that interface movement depends explicitly on domain geometry. Using a combination of analytic and numeric techniques, we demonstrate that domain geometry, a historically overlooked aspect of cellular simulations, may play a significant role in spatial protein patterning during polarization.

Structural insights into the intrinsic self-assembly of Par-3 N-terminal domain

Par-3, the central organizer of the Par-3/Par-6/atypical protein kinase C complex, is a multimodular scaffold protein that is essential for cell polarity establishment and maintenance. The N-terminal domain (NTD) of Par-3 is capable of self-association to form filament-like structures, although the underlying mechanism is poorly understood. Here, we determined the crystal structure of Par-3 NTD and solved the filament structure by cryoelectron microscopy. We found that an intrinsic "front-to-back" interaction mode is important for Par-3 NTD self-association and that both the lateral and longitudinal packing within the filament are mediated by electrostatic interactions. Disruptions of the lateral or longitudinal packing significantly impaired Par-3 NTD self-association and thereby impacted the Par-3-mediated epithelial polarization. We finally demonstrated that a Par-3 NTD-like domain from histidine ammonia-lyase also harbors a similar self-association capacity. This work unequivocally provides the structural basis for Par-3 NTD self-association and characterizes one type of protein domain that can self-assemble via electrostatic interactions.

PAR-2, LGL-1 and the CDC-42 GAP CHIN-1 act in distinct pathways to maintain polarity in the C. elegans embryo

In the one-cell C. elegans embryo, polarity is maintained by mutual antagonism between the anterior cortical proteins PAR-3, PKC-3, PAR-6 and CDC-42, and the posterior cortical proteins PAR-2 and LGL-1 on the posterior cortex. The mechanisms by which these proteins interact to maintain polarity are incompletely understood. In this study, we investigate the interplay among PAR-2, LGL-1, myosin, the anterior PAR proteins and CDC-42. We find that PAR-2 and LGL-1 affect cortical myosin accumulation by different mechanisms. LGL-1 does not directly antagonize the accumulation of cortical myosin and instead plays a role in regulating PAR-6 levels. By contrast, PAR-2 likely has separate roles in regulating cortical myosin accumulation and preventing the expansion of the anterior cortical domain. We also provide evidence that asymmetry of active CDC-42 can be maintained independently of LGL-1 and PAR-2 by a redundant pathway that includes the CDC-42 GAP CHIN-1. Finally, we show that, in addition to its primary role in regulating the size of the anterior cortical domain via its binding to PAR-6, CDC-42 has a secondary role in regulating cortical myosin that is not dependent on PAR-6.

Cell polarity: mechanochemical patterning

Nearly every cell type exhibits some form of polarity, yet the molecular mechanisms vary widely. Here we examine what we term 'chemical systems' where cell polarization arises through biochemical interactions in signaling pathways, 'mechanical systems' where cells polarize due to forces, stresses and transport, and 'mechanochemical systems' where polarization results from interplay between mechanics and chemical signaling. To reveal potentially unifying principles, we discuss mathematical conceptualizations of several prototypical examples. We suggest that the concept of local activation and global inhibition - originally developed to explain spatial patterning in reaction-diffusion systems - provides a framework for understanding many cases of cell polarity. Importantly, we find that the core ingredients in this framework - symmetry breaking, self-amplifying feedback, and long-range inhibition - involve processes that can be chemical, mechanical, or even mechanochemical in nature.

Cell polarity: quantitative modeling as a tool in cell biology

Among a number of innovative approaches that have modernized cell biology, modeling has a prominent yet unusual place. One popular view is that we progress linearly, from conceptual to ever more detailed models. We review recent discoveries of cell polarity mechanisms, in which modeling played an important role, to demonstrate that the experiment-theory feedback loop requires diverse models characterized by varying levels of biological detail and mathematical complexity. We argue that a quantitative model is a tool that has to fit an experimental study, and the model's value should be judged not by how complex and detailed it is, but by what could be learned from it.

Anterior PAR proteins function during cytokinesis and maintain DYN-1 at the cleavage furrow in Caenorhabditis elegans

PAR proteins are key regulators of cellular polarity and have links to the endocytic machinery and the actin cytoskeleton. Our data suggest a unique role for PAR proteins in cytokinesis. We have found that at the onset of cytokinesis, anterior PAR-6 and posterior PAR-2 proteins are redistributed to the furrow membrane in a temporal and spatial manner. PAR-6 and PAR-2 localize to the furrow membrane during ingression but PAR-2-GFP is distinct in that it is excluded from the extreme tip of the furrow. Once the midbody has formed, PAR-2-GFP becomes restricted to the midbody region (the midbody plus the membrane flanking it). Depletion of both anterior PAR proteins, PAR-3 and PAR-6, led to an increase in multinucleate embryos, suggesting that the anterior PAR proteins are necessary during cytokinesis and that PAR-3 and PAR-6 function in cytokinesis may be partially redundant. Lastly, anterior PAR proteins play a role in the maintenance of DYN-1 in the cleavage furrow. Our data indicate that the PAR proteins are involved in the events that occur during cytokinesis and may play a role in promoting the membrane trafficking and remodeling events that occur during this time.

Organelle growth control through limiting pools of cytoplasmic components

The critical importance of controlling the size and number of intracellular organelles has led to a variety of mechanisms for regulating the formation and growth of cellular structures. In this review, we explore a class of mechanisms for organelle growth control that rely primarily on the cytoplasm as a 'limiting pool' of available material. These mechanisms are based on the idea that, as organelles grow, they incorporate subunits from the cytoplasm. If this subunit pool is limited, organelle growth will lead to depletion of subunits from the cytoplasm. Free subunit concentration therefore provides a measure of the number of incorporated subunits and thus the current size of the organelle. Because organelle growth rates are typically a function of subunit concentration, cytoplasmic depletion links organelle size, free subunit concentration, and growth rates, ensuring that as the organelle grows, its rate of growth slows. Thus, a limiting cytoplasmic pool provides a powerful mechanism for size-dependent regulation of growth without recourse to active mechanisms to measure size or modulate growth rates. Variations of this general idea allow not only for size control, but also cell-size-dependent scaling of cellular structures, coordination of growth between similar structures within a cell, and the enforcement of singularity in structure formation, when only a single copy of a structure is desired. Here, we review several examples of such mechanisms in cellular processes as diverse as centriole duplication, centrosome and nuclear size control, cell polarity, and growth of flagella.