Rapid diffusion-state switching underlies stable cytoplasmic gradients in the Caenorhabditis elegans zygote

Robust spatiotemporal organization of mitotic events in mechanically perturbed C. elegans embryos

Early embryogenesis of the nematode Caenorhabditis elegans progresses in an autonomous fashion within a protective chitin eggshell. Cell-division timing and the subsequent mechanically guided positioning of cells is virtually invariant between individuals, especially before gastrulation. Here, we have challenged this stereotypical developmental program in early stages by mechanically perturbing the embryo without breaking its eggshell. Compressing embryos to about two-thirds of their unperturbed diameter only resulted in markedly slower cell divisions. In contrast, compressing embryos to half of their native diameter frequently resulted in a loss of cytokinesis, yielding a non-natural syncytium that still allowed for multiple divisions of nuclei. Although the orientation of mitotic axes was strongly altered in the syncytium, key features of division timing and spatial arrangement of nuclei remained surprisingly similar to those of unperturbed embryos in the first few division cycles. This suggests that few, very robust mechanisms provide a basic and resilient program for safeguarding the early embryogenesis of C. elegans.

Internal feedback circuits among MEX-5, MEX-6, and PLK-1 maintain faithful patterning in the Caenorhabditis elegans embryo

Proteins become asymmetrically distributed in the one-cell Caenorhabditis elegans embryo thanks to reaction-diffusion mechanisms that are often entangled in complex feedback loops. Cortical polarity drives the enrichment of the RNA-binding proteins MEX-5 and MEX-6 in the anterior cytoplasm through concentration gradients. MEX-5 and MEX-6 promote the patterning of other cytoplasmic factors, including that of the anteriorly enriched mitotic polo-like kinase PLK-1, but also contribute to proper cortical polarity. The gradient of MEX-5 forms through a differential-diffusion mechanism. How MEX-6 establishes a gradient and how MEX-5 and MEX-6 regulate cortical polarity is not known. Here, we reveal that the two MEX proteins develop concentration asymmetries via similar mechanisms, but despite their strong sequence homology, they differ in terms of how their concentration gradients are regulated. We find that PLK-1 promotes the enrichment of MEX-5 and MEX-6 at the anterior through different circuits: PLK-1 influences the MEX-5 gradient indirectly by regulating cortical polarity while it modulates the formation of the gradient of MEX-6 through its physical interaction with the protein. We thus propose a model in which PLK-1 mediates protein circuitries between MEX-5, MEX-6, and cortical proteins to faithfully establish and maintain polarity.

PLK-1 regulates MEX-1 polarization in the C. elegans zygote

The one-cell C. elegans embryo undergoes an asymmetric cell division during which germline factors such as the RNA-binding proteins POS-1 and MEX-1 segregate to the posterior cytoplasm, leading to their asymmetric inheritance to the posterior germline daughter cell. Previous studies found that the RNA-binding protein MEX-5 recruits polo-like kinase PLK-1 to the anterior cytoplasm where PLK-1 inhibits the retention of its substrate POS-1, leading to POS-1 segregation to the posterior. In this study, we tested whether PLK-1 similarly regulates MEX-1 polarization. We find that both the retention of MEX-1 in the anterior and the segregation of MEX-1 to the posterior depend on PLK kinase activity and on the interaction between MEX-5 and PLK-1. Human PLK1 directly phosphorylates recombinant MEX-1 on 9 predicted PLK-1 sites in vitro, four of which were identified in previous phosphoproteomic analysis of C. elegans embryos. The introduction of alanine substitutions at these four PLK-1 phosphorylation sites (MEX-1(4A)) significantly weakened the inhibition of MEX-1 retention in the anterior, thereby weakening MEX-1 segregation to the posterior. In contrast, mutation of a predicted CDK1 phosphorylation site had no effect on MEX-1 retention or on MEX-1 segregation. MEX-1(4A) mutants are viable and fertile but display significant sterility and fecundity defects at elevated temperatures. Taken together with our previous findings, these findings suggest PLK-1 phosphorylation drives both MEX-1 and POS-1 polarization during the asymmetric division of the zygote.

Truncated stochastically switching processes

There are a large variety of hybrid stochastic systems that couple a continuous process with some form of stochastic switching mechanism. In many cases the system switches between different discrete internal states according to a finite-state Markov chain, and the continuous dynamics depends on the current internal state. The resulting hybrid stochastic differential equation (hSDE) could describe the evolution of a neuron's membrane potential, the concentration of proteins synthesized by a gene network, or the position of an active particle. Another major class of switching system is a search process with stochastic resetting, where the position of a diffusing or active particle is reset to a fixed position at a random sequence of times. In this case the system switches between a search phase and a reset phase, where the latter may be instantaneous. In this paper, we investigate how the behavior of a stochastically switching system is modified when the maximum number of switching (or reset) events in a given time interval is fixed. This is motivated by the idea that each time the system switches there is an additive energy cost. We first show that in the case of an hSDE, restricting the number of switching events is equivalent to truncating a Volterra series expansion of the particle propagator. Such a truncation significantly modifies the moments of the resulting renormalized propagator. We then investigate how restricting the number of reset events affects the diffusive search for an absorbing target. In particular, truncating a Volterra series expansion of the survival probability, we calculate the splitting probabilities and conditional MFPTs for the particle to be absorbed by the target or exceed a given number of resets, respectively.

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.

Separable mechanisms drive local and global polarity establishment in the Caenorhabditiselegans intestinal epithelium

Apico-basolateral polarization is essential for epithelial cells to function as selective barriers and transporters, and to provide mechanical resilience to organs. Epithelial polarity is established locally, within individual cells to establish distinct apical, junctional and basolateral domains, and globally, within a tissue where cells coordinately orient their apico-basolateral axes. Using live imaging of endogenously tagged proteins and tissue-specific protein depletion in the Caenorhabditiselegans embryonic intestine, we found that local and global polarity establishment are temporally and genetically separable. Local polarity is initiated prior to global polarity and is robust to perturbation. PAR-3 is required for global polarization across the intestine but local polarity can arise in its absence, as small groups of cells eventually established polarized domains in PAR-3-depleted intestines in a HMR-1 (E-cadherin)-dependent manner. Despite the role of PAR-3 in localizing PKC-3 to the apical surface, we additionally found that PAR-3 and PKC-3/aPKC have distinct roles in the establishment and maintenance of local and global polarity. Taken together, our results indicate that different mechanisms are required for local and global polarity establishment in vivo.

How to assemble a scale-invariant gradient

Intracellular protein gradients serve a variety of functions, such as the establishment of cell polarity or to provide positional information for gene expression in developing embryos. Given that cell size in a population can vary considerably, for the protein gradients to work properly they often have to be scaled to the size of the cell. Here, we examine a model of protein gradient formation within a cell that relies on cytoplasmic diffusion and cortical transport of proteins toward a cell pole. We show that the shape of the protein gradient is determined solely by the cell geometry. Furthermore, we show that the length scale over which the protein concentration in the gradient varies is determined by the linear dimensions of the cell, independent of the diffusion constant or the transport speed. This gradient provides scale-invariant positional information within a cell, which can be used for assembly of intracellular structures whose size is scaled to the linear dimensions of the cell, such as the cytokinetic ring and actin cables in budding yeast cells.

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.

Anterior-posterior pattern formation in ciliates

As single cells, ciliates build, duplicate, and even regenerate complex cortical patterns by largely unknown mechanisms that precisely position organelles along two cell‐wide axes: anterior–posterior and circumferential (left–right). We review our current understanding of intracellular patterning along the anterior–posterior axis in ciliates, with emphasis on how the new pattern emerges during cell division. We focus on the recent progress at the molecular level that has been driven by the discovery of genes whose mutations cause organelle positioning defects in the model ciliate Tetrahymena thermophila. These investigations have revealed a network of highly conserved kinases that are confined to either anterior or posterior domains in the cell cortex. These pattern‐regulating kinases create zones of cortical inhibition that by exclusion determine the precise placement of organelles. We discuss observations and models derived from classical microsurgical experiments in large ciliates (including Stentor) and interpret them in light of recent molecular findings in Tetrahymena. In particular, we address the involvement of intracellular gradients as vehicles for positioning organelles along the anterior‐posterior axis.

Spatial localisation meets biomolecular networks

Spatial organisation through localisation/compartmentalisation of species is a ubiquitous but poorly understood feature of cellular biomolecular networks. Current technologies in systems and synthetic biology (spatial proteomics, imaging, synthetic compartmentalisation) necessitate a systematic approach to elucidating the interplay of networks and spatial organisation. We develop a systems framework towards this end and focus on the effect of spatial localisation of network components revealing its multiple facets: (i) As a key distinct regulator of network behaviour, and an enabler of new network capabilities (ii) As a potent new regulator of pattern formation and self-organisation (iii) As an often hidden factor impacting inference of temporal networks from data (iv) As an engineering tool for rewiring networks and network/circuit design. These insights, transparently arising from the most basic considerations of networks and spatial organisation, have broad relevance in natural and engineered biology and in related areas such as cell-free systems, systems chemistry and bionanotechnology.

Complex biomolecular networks are fundamental to the functioning of living systems, both at the cellular level and beyond. In this paper, the authors develop a systems framework to elucidate the interplay of networks and the spatial localisation of network components.

Long time behavior and stable patterns in high-dimensional polarity models of asymmetric cell division

Asymmetric cell division is one of the fundamental processes to create cell diversity in the early stage of embryonic development. During this process, the polarity formation in the cell membrane has been considered as a key process by which the entire polarity formation in the cytosol is controlled, and it has been extensively studied in both experiments and mathematical models. Nonetheless, a mathematically rigorous analysis of the polarity formation in the asymmetric cell division has been little explored, particularly for bulk-surface models. In this article, we deal with polarity models proposed for describing the PAR polarity formation in the asymmetric cell division of a C. elegans embryo. Using a simpler but mathematically consistent model, we exhibit the long time behavior of the polarity formation of a bulk-surface cell. Moreover, we mathematically prove the existence of stable polarity solutions of the model equation in an arbitrary high-dimensional domain and analyse how the boundary position of polarity domain is determined. Our results propose that the existence and dynamics of the polarity in the asymmetric cell division can be understood universally in terms of basic mathematical structures.

Describing the movement of molecules in reduced-dimension models

When addressing spatial biological questions using mathematical models, symmetries within the system are often exploited to simplify the problem by reducing its physical dimension. In a reduced-dimension model molecular movement is restricted to the reduced dimension, changing the nature of molecular movement. This change in molecular movement can lead to quantitatively and even qualitatively different results in the full and reduced systems. Within this manuscript we discuss the condition under which restricted molecular movement in reduced-dimension models accurately approximates molecular movement in the full system. For those systems which do not satisfy the condition, we present a general method for approximating unrestricted molecular movement in reduced-dimension models. We will derive a mathematically robust, finite difference method for solving the 2D diffusion equation within a 1D reduced-dimension model. The methods described here can be used to improve the accuracy of many reduced-dimension models while retaining benefits of system simplification.

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.

Construction of intracellular asymmetry and asymmetric division in Escherichia coli

The design principle of establishing an intracellular protein gradient for asymmetric cell division is a long-standing fundamental question. While the major molecular players and their interactions have been elucidated via genetic approaches, the diversity and redundancy of natural systems complicate the extraction of critical underlying features. Here, we take a synthetic cell biology approach to construct intracellular asymmetry and asymmetric division in Escherichia coli, in which division is normally symmetric. We demonstrate that the oligomeric PopZ from Caulobacter crescentus can serve as a robust polarized scaffold to functionalize RNA polymerase. Furthermore, by using another oligomeric pole-targeting DivIVA from Bacillus subtilis, the newly synthesized protein can be constrained to further establish intracellular asymmetry, leading to asymmetric division and differentiation. Our findings suggest that the coupled oligomerization and restriction in diffusion may be a strategy for generating a spatial gradient for asymmetric cell division.

PLK-1 Regulation of Asymmetric Cell Division in the Early C. elegans Embryo

PLK1 is a conserved mitotic kinase that is essential for the entry into and progression through mitosis. In addition to its canonical mitotic functions, recent studies have characterized a critical role for PLK-1 in regulating the polarization and asymmetric division of the one-cell C. elegans embryo. Prior to cell division, PLK-1 regulates both the polarization of the PAR proteins at the cell cortex and the segregation of cell fate determinants in the cytoplasm. Following cell division, PLK-1 is preferentially inherited to one daughter cell where it acts to regulate the timing of centrosome separation and cell division. PLK1 also regulates cell polarity in asymmetrically dividing Drosophila neuroblasts and during mammalian planar cell polarity, suggesting it may act broadly to connect cell polarity and cell cycle mechanisms.

Proofreading through spatial gradients

Key enzymatic processes use the nonequilibrium error correction mechanism called kinetic proofreading to enhance their specificity. The applicability of traditional proofreading schemes, however, is limited because they typically require dedicated structural features in the enzyme, such as a nucleotide hydrolysis site or multiple intermediate conformations. Here, we explore an alternative conceptual mechanism that achieves error correction by having substrate binding and subsequent product formation occur at distinct physical locations. The time taken by the enzyme–substrate complex to diffuse from one location to another is leveraged to discard wrong substrates. This mechanism does not have the typical structural requirements, making it easier to overlook in experiments. We discuss how the length scales of molecular gradients dictate proofreading performance, and quantify the limitations imposed by realistic diffusion and reaction rates. Our work broadens the applicability of kinetic proofreading and sets the stage for studying spatial gradients as a possible route to specificity.

Going with the flow: insights from Caenorhabditis elegans zygote polarization

Cell polarity is the asymmetric distribution of cellular components along a defined axis. Polarity relies on complex signalling networks between conserved patterning proteins, including the PAR (partitioning defective) proteins, which become segregated in response to upstream symmetry breaking cues. Although the mechanisms that drive the asymmetric localization of these proteins are dependent upon cell type and context, in many cases the regulation of actomyosin cytoskeleton dynamics is central to the transport, recruitment and/or stabilization of these polarity effectors into defined subcellular domains. The transport or advection of PAR proteins by an actomyosin flow was first observed in the Caenorhabditis elegans zygote more than a decade ago. Since then a multifaceted approach, using molecular methods, high-throughput screens, and biophysical and computational models, has revealed further aspects of this flow and how polarity regulators respond to and modulate it. Here, we review recent findings on the interplay between actomyosin flow and the PAR patterning networks in the polarization of the C. elegans zygote. We also discuss how these discoveries and developed methods are shaping our understanding of other flow-dependent polarizing systems. This article is part of a discussion meeting issue 'Contemporary morphogenesis'.

Subdiffusion-limited fractional reaction-subdiffusion equations with affine reactions: Solution, stochastic paths, and applications

In contrast to normal diffusion, there is no canonical model for reactions between chemical species which move by anomalous subdiffusion. Indeed, the type of mesoscopic equation describing reaction-subdiffusion systems depends on subtle assumptions about the microscopic behavior of individual molecules. Furthermore, the correspondence between mesoscopic and microscopic models is not well understood. In this paper, we study the subdiffusion-limited model, which is defined by mesoscopic equations with fractional derivatives applied to both the movement and the reaction terms. Assuming that the reaction terms are affine functions, we show that the solution to the fractional system is the expectation of a random time change of the solution to the corresponding integer order system. This result yields a simple and explicit algebraic relationship between the fractional and integer order solutions in Laplace space. We then find the microscopic Langevin description of individual molecules that corresponds to such mesoscopic equations and give a computer simulation method to generate their stochastic trajectories. This analysis identifies some precise microscopic conditions that dictate when this type of mesoscopic model is or is not appropriate. We apply our results to several scenarios in cell biology which, despite the ubiquity of subdiffusion in cellular environments, have been modeled almost exclusively by normal diffusion. Specifically, we consider subdiffusive models of morphogen gradient formation, fluctuating mobility, and fluorescence recovery after photobleaching (FRAP) experiments. We also apply our results to fractional ordinary differential equations.

PIG-1 MELK-dependent phosphorylation of nonmuscle myosin II promotes apoptosis through CES-1 Snail partitioning

The mechanism(s) through which mammalian kinase MELK promotes tumorigenesis is not understood. We find that the C. elegans orthologue of MELK, PIG-1, promotes apoptosis by partitioning an anti-apoptotic factor. The C. elegans NSM neuroblast divides to produce a larger cell that differentiates into a neuron and a smaller cell that dies. We find that in this context, PIG-1 MELK is required for partitioning of CES-1 Snail, a transcriptional repressor of the pro-apoptotic gene egl-1 BH3-only. pig-1 MELK is controlled by both a ces-1 Snail- and par-4 LKB1-dependent pathway, and may act through phosphorylation and cortical enrichment of nonmuscle myosin II prior to neuroblast division. We propose that pig-1 MELK-induced local contractility of the actomyosin network plays a conserved role in the acquisition of the apoptotic fate. Our work also uncovers an auto-regulatory loop through which ces-1 Snail controls its own activity through the formation of a gradient of CES-1 Snail protein.

Apoptosis is critical for the elimination of ‘unwanted’ cells. What distinguishes wanted from unwanted cells in developing animals is poorly understood. We report that in the C. elegans NSM neuroblast lineage, the level of CES-1, a Snail-family member and transcriptional repressor of the pro-apoptotic gene egl-1, contributes to this process. In addition, we demonstrate that C. elegans PIG-1, the orthologue of mammalian proto-oncoprotein MELK, plays a critical role in controlling CES-1^Snail levels. Specifically, during NSM neuroblast division, PIG-1^MELK controls partitioning of CES-1^Snail into one but not the other daughter cell thereby promoting the making of one wanted and one unwanted cell. Furthermore, we present evidence that PIG-1^MELK acts prior to NSM neuroblast division by locally activating the actomyosin network.

Transitions to slow or fast diffusions provide a general property for in-phase or anti-phase polarity in a cell

Cell polarity is an important cellular process that cells use for various cellular functions such as asymmetric division, cell migration, and directionality determination. In asymmetric cell division, a mother cell creates multiple polarities of various proteins simultaneously within her membrane and cytosol to generate two different daughter cells. The formation of multiple polarities in asymmetric cell division has been found to be controlled via the regulatory system by upstream polarity of the membrane to downstream polarity of the cytosol, which is involved in not only polarity establishment but also polarity positioning. However, the mechanism for polarity positioning remains unclear. In this study, we found a general mechanism and mathematical structure for the multiple streams of polarities to determine their relative position via conceptional models based on the biological example of the asymmetric cell division process of C. elegans embryo. Using conceptional modeling and model reductions, we show that the positional relation of polarities is determined by a contrasting role of regulation by upstream polarity proteins on the transition process of diffusion dynamics of downstream proteins. We analytically prove that our findings hold under the general mathematical conditions, suggesting that the mechanism of relative position between upstream and downstream dynamics could be understood without depending on a specific type of bio-chemical reaction, and it could be the universal mechanism in multiple streams of polarity dynamics of the cell.

Asymmetric cell division from a cell to cells: Shape, length, and location of polarity domain

Asymmetric cell division is one of the most elegant biological systems by which cells create daughter cells with different functions and increase cell diversity. In particular, PAR polarity in the cell membrane plays a critical role in regulating the whole process of asymmetric cell division. Numerous studies have been conducted to determine the underlying mechanism of PAR polarity formation using both experimental and theoretical approaches in the last 10 years. However, they have mostly focused on answering the fundamental question of how this exclusive polarity is established but the precise dynamics of polarity domain have been little notified. In this review, I focused on studies on the shape, length, and location of PAR polarity from a theoretical perspective that may be important for an integrated understanding of the entire process of asymmetric cell division.

Patterning and polarization of cells by intracellular flows

Beginning with Turing’s seminal work [1], decades of research have demonstrated the fundamental ability of biochemical networks to generate and sustain the formation of patterns. However, it is increasingly appreciated that biochemical networks both shape and are shaped by physical and mechanical processes [2, 3, 4]. One such process is fluid flow. In many respects, the cytoplasm, membrane and actin cortex all function as fluids, and as they flow, they drive bulk transport of molecules throughout the cell. By coupling biochemical activity to long range molecular transport, flows can shape the distributions of molecules in space. Here we review the various types of flows that exist in cells, with the aim of highlighting recent advances in our understanding of how flows are generated and how they contribute to intracellular patterning processes, such as the establishment of cell polarity.

Switching states: dynamic remodelling of polarity complexes as a toolkit for cell polarization

Polarity is defined by the segregation of cellular components along a defined axis. To polarize robustly, cells must be able to break symmetry and subsequently amplify these nascent asymmetries. Finally, asymmetric localization of signaling molecules must be translated into functional regulation of downstream effector pathways. Central to these behaviors are a diverse set of cell polarity networks. Within these networks, molecules exhibit varied behaviors, dynamically switching among different complexes and states, active versus inactive, bound versus unbound, immobile versus diffusive. This ability to switch dynamically between states is intimately connected to the ability of molecules to generate asymmetric patterns within cells. Focusing primarily on polarity pathways governed by the conserved PAR proteins, we discuss strategies enabled by these dynamic behaviors that are used by cells to polarize. We highlight not only how switching between states is linked to the ability of polarity proteins to localize asymmetrically, but also how cells take advantage of 'state switching' to regulate polarity in time and space.

Physical Chemistry of Cellular Liquid-Phase Separation

Compartmentalization of biochemical processes is essential for cell function. Although membrane-bound organelles are well studied in this context, recent work has shown that phase separation is a key contributor to cellular compartmentalization through the formation of liquid-like membraneless organelles (MLOs). In this Minireview, the key mechanistic concepts that underlie MLO dynamics and function are first briefly discussed, including the relevant noncovalent interaction chemistry and polymer physical chemistry. Next, a few examples of MLOs and relevant proteins are given, along with their functions, which highlight the relevance of the above concepts. The developing area of active matter and non-equilibrium systems, which can give rise to unexpected effects in fluctuating cellular conditions, are also discussed. Finally, our thoughts for emerging and future directions in the field are discussed, including in vitro and in vivo studies of MLO physical chemistry and function.

Membraneless organelles: P granules in Caenorhabditis elegans

Membraneless organelles are distinct compartments within a cell that are not enclosed by a traditional lipid membrane and instead form through a process called liquid-liquid phase separation. Examples of these non-membrane-bound organelles include nucleoli, stress granules, P bodies, pericentriolar material and germ granules. Many recent studies have used Caenorhabditis elegans germ granules, known as P granules, to expand our understanding of the formation of these unique cellular compartments. From this work, we know that proteins with intrinsically disordered regions (IDRs) play a critical role in the process of phase separation. IDR phase separation is further tuned through their interactions with RNA and through protein modifications such as phosphorylation and methylation. These findings from C elegans, combined with work done in other model organisms, continue to provide insight into the formation of membraneless organelles and the important role they play in compartmentalizing cellular processes.

Protein concentration gradients and switching diffusions

Morphogen gradients play a vital role in developmental biology by enabling embryonic cells to infer their spatial location and determine their developmental fate accordingly. The standard mechanism for generating a morphogen gradient involves a morphogen being produced from a localized source and subsequently degrading. While this mechanism is effective over the length and time scales of tissue development, it fails over typical subcellular length scales due to the rapid dissipation of spatial asymmetries. In a recent theoretical work, we found an alternative mechanism for generating concentration gradients of diffusing molecules, in which the molecules switch between spatially constant diffusivities at switching rates that depend on the spatial location of a molecule. Independently, an experimental and computational study later found that Caenorhabditis elegans zygotes rely on this mechanism for cell polarization. In this paper, we extend our analysis of switching diffusivities to determine its role in protein concentration gradient formation. In particular, we determine how switching diffusivities modifies the standard theory and show how space-dependent switching diffusivities can yield a gradient in the absence of a localized source. Our mathematical analysis yields explicit formulas for the intracellular concentration gradient which closely match the results of previous experiments and numerical simulations.

Single-molecule dynamics of the P granule scaffold MEG-3 in the Caenorhabditis elegans zygote

During the asymmetric division of the Caenorhabditis elegans zygote, germ (P) granules are disassembled in the anterior cytoplasm and stabilized/assembled in the posterior cytoplasm, leading to their inheritance by the germline daughter cell. P granule segregation depends on MEG (maternal-effect germline defective)-3 and MEG-4, which are enriched in P granules and in the posterior cytoplasm surrounding P granules. Here we use single-molecule imaging and tracking to characterize the reaction/diffusion mechanisms that result in MEG-3::Halo segregation. We find that the anteriorly enriched RNA-binding proteins MEX (muscle excess)-5 and MEX-6 suppress the retention of MEG-3 in the anterior cytoplasm, leading to MEG-3 enrichment in the posterior. We provide evidence that MEX-5/6 may work in conjunction with PLK-1 kinase to suppress MEG-3 retention in the anterior. Surprisingly, we find that the retention of MEG-3::Halo in the posterior cytoplasm surrounding P granules does not appear to contribute significantly to the maintenance of P granule asymmetry. Rather, our findings suggest that the formation of the MEG-3 concentration gradient and the segregation of P granules are two parallel manifestations of MEG-3′s response to upstream polarity cues.

PIE-1 Translation in the Germline Lineage Contributes to PIE-1 Asymmetry in the Early Caenorhabditis elegans Embryo

In the C. elegans embryo, the germline lineage is established through successive asymmetric cell divisions that each generate a somatic and a germline daughter cell. PIE-1 is an essential maternal factor that is enriched in embryonic germline cells and is required for germline specification. We estimated the absolute concentration of PIE-1::GFP in germline cells and find that PIE-1::GFP concentration increases by roughly 4.5 fold, from 92 nM to 424 nM, between the 1 and 4-cell stages. Previous studies have shown that the preferential inheritance of PIE-1 by germline daughter cells and the degradation of PIE-1 in somatic cells are important for PIE-1 enrichment in germline cells. In this study, we provide evidence that the preferential translation of maternal PIE-1::GFP transcripts in the germline also contributes to PIE-1::GFP enrichment. Through an RNAi screen, we identified Y14 and MAG-1 (Drosophila tsunagi and mago nashi) as regulators of embryonic PIE-1::GFP levels. We show that Y14 and MAG-1 do not regulate PIE-1 degradation, segregation or synthesis in the early embryo, but do regulate the concentration of maternally-deposited PIE-1::GFP. Taken together, or findings point to an important role for translational control in the regulation of PIE-1 levels in the germline lineage.