Mechanistic mathematical model of polarity in yeast

Cdc42 mobility and membrane flows regulate fission yeast cell shape and survival

Polarized exocytosis induced by local Cdc42 GTPase activity results in membrane flows that deplete low-mobility membrane-associated proteins. A reaction-diffusion particle model comprising Cdc42 positive feedback activation, hydrolysis by GTPase-activating proteins (GAPs), and flow-induced displacement by exo/endocytosis shows that flow-induced depletion of low mobility GAPs promotes polarization. We modified Cdc42 mobility in Schizosaccharomyces pombe by replacing its prenylation site with 1, 2 or 3 repeats of the Rit C-terminal membrane-binding domain (ritC), yielding alleles with progressively lower mobility and increased flow-coupling. While Cdc42-1ritC cells are viable and polarized, Cdc42-2ritC polarize poorly and Cdc42-3ritC are inviable, in agreement with model's predictions. Deletion of Cdc42 GAPs restores viability to Cdc42-3ritC cells, verifying the model's prediction that GAP deletion increases Cdc42 activity at the expense of polarization. Our work demonstrates how membrane flows are an integral part of Cdc42-driven pattern formation and require Cdc42-GTP to turn over faster than the surface on which it forms.

Redundancy and the role of protein copy numbers in the cell polarization machinery of budding yeast

How can a self-organized cellular function evolve, adapt to perturbations, and acquire new sub-functions? To make progress in answering these basic questions of evolutionary cell biology, we analyze, as a concrete example, the cell polarity machinery of Saccharomyces cerevisiae. This cellular module exhibits an intriguing resilience: it remains operational under genetic perturbations and recovers quickly and reproducibly from the deletion of one of its key components. Using a combination of modeling, conceptual theory, and experiments, we propose that multiple, redundant self-organization mechanisms coexist within the protein network underlying cell polarization and are responsible for the module's resilience and adaptability. Based on our mechanistic understanding of polarity establishment, we hypothesize that scaffold proteins, by introducing new connections in the existing network, can increase the redundancy of mechanisms and thus increase the evolvability of other network components. Moreover, our work gives a perspective on how a complex, redundant cellular module might have evolved from a more rudimental ancestral form.

Particle-based simulations reveal two positive feedback loops allow relocation and stabilization of the polarity site during yeast mating

Many cells adjust the direction of polarized growth or migration in response to external directional cues. The yeast Saccharomyces cerevisiae orient their cell fronts (also called polarity sites) up pheromone gradients in the course of mating. However, the initial polarity site is often not oriented towards the eventual mating partner, and cells relocate the polarity site in an indecisive manner before developing a stable orientation. During this reorientation phase, the polarity site displays erratic assembly-disassembly behavior and moves around the cell cortex. The mechanisms underlying this dynamic behavior remain poorly understood. Particle-based simulations of the core polarity circuit revealed that molecular-level fluctuations are unlikely to overcome the strong positive feedback required for polarization and generate relocating polarity sites. Surprisingly, inclusion of a second pathway that promotes polarity site orientation generated relocating polarity sites with properties similar to those observed experimentally. This pathway forms a second positive feedback loop involving the recruitment of receptors to the cell membrane and couples polarity establishment to gradient sensing. This second positive feedback loop also allows cells to stabilize their polarity site once the site is aligned with the pheromone gradient.

Cdc42 mobility and membrane flows regulate fission yeast cell shape and survival

Local Cdc42 GTPase activation promotes polarized exocytosis, resulting in membrane flows that deplete low-mobility membrane-associated proteins from the growth region. To investigate the self-organizing properties of the Cdc42 secretion-polarization system under membrane flow, we developed a reaction-diffusion particle model. The model includes positive feedback activation of Cdc42, hydrolysis by GTPase-activating proteins (GAPs), and flow-induced displacement by exo/endocytosis. Simulations show how polarization relies on flow-induced depletion of low mobility GAPs. To probe the role of Cdc42 mobility in the fission yeast Schizosaccharomyces pombe, we changed its membrane binding properties by replacing its prenylation site with 1, 2 or 3 repeats of the Rit1 C terminal membrane binding domain (ritC), yielding alleles with progressively lower unbinding and diffusion rates. Concordant modelling predictions and experimental observations show that lower Cdc42 mobility results in lower Cdc42 activation level and wider patches. Indeed, while Cdc42-1ritC cells are viable and polarized, Cdc42-2ritC polarize poorly and Cdc42-3ritC is inviable. The model further predicts that GAP depletion increases Cdc42 activity at the expense of loss of polarization. Experiments confirm this prediction, as deletion of Cdc42 GAPs restores viability to Cdc42-3ritC cells. Our combined experimental and modelling studies demonstrate how membrane flows are an integral part of Cdc42-driven pattern formation.

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.

Targeted secretion: Myosin V delivers vesicles through formin condensates

Secretory vesicles are often delivered to very specific targets, like pre-synaptic terminals or cell tips, to focus exocytosis. New work suggests that a biomolecular condensate focuses actin filaments that deliver incoming vesicles through the condensate to the plasma membrane.

Chemotropism and Cell-Cell Fusion in Fungi

Fungi exhibit an enormous variety of morphologies, including yeast colonies, hyphal mycelia, and elaborate fruiting bodies. This diversity arises through a combination of polar growth, cell division, and cell fusion. Because fungal cells are nonmotile and surrounded by a protective cell wall that is essential for cell integrity, potential fusion partners must grow toward each other until they touch and then degrade the intervening cell walls without impacting cell integrity. Here, we review recent progress on understanding how fungi overcome these challenges. Extracellular chemoattractants, including small peptide pheromones, mediate communication between potential fusion partners, promoting the local activation of core cell polarity regulators to orient polar growth and cell wall degradation. However, in crowded environments, pheromone gradients can be complex and potentially confusing, raising the question of how cells can effectively find their partners. Recent findings suggest that the cell polarity circuit exhibits searching behavior that can respond to pheromone cues through a remarkably flexible and effective strategy called exploratory polarization.

A positive feedback loop involving the Spa2 SHD domain contributes to focal polarization

Focal polarization is necessary for finely arranged cell-cell interactions. The yeast mating projection, with its punctate polarisome, is a good model system for this process. We explored the critical role of the polarisome scaffold protein Spa2 during yeast mating with a hypothesis motivated by mathematical modeling and tested by in vivo experiments. Our simulations predicted that two positive feedback loops generate focal polarization, including a novel feedback pathway involving the N-terminal domain of Spa2. We characterized the latter using loss-of-function and gain-of-function mutants. The N-terminal region contains a Spa2 Homology Domain (SHD) which is conserved from yeast to humans, and when mutated largely reproduced the spa2Δ phenotype. Our work together with published data show that the SHD domain recruits Msb3/4 that stimulates Sec4-mediated transport of Bud6 to the polarisome. There, Bud6 activates Bni1-catalyzed actin cable formation, recruiting more Spa2 and completing the positive feedback loop. We demonstrate that disrupting this loop at any point results in morphological defects. Gain-of-function perturbations partially restored focal polarization in a spa2 loss-of-function mutant without restoring localization of upstream components, thus supporting the pathway order. Thus, we have collected data consistent with a novel positive feedback loop that contributes to focal polarization during pheromone-induced polarization in yeast.

Chemotactic movement of a polarity site enables yeast cells to find their mates

How small eukaryotic cells can interpret dynamic, noisy, and spatially complex chemical gradients to orient growth or movement is poorly understood. We address this question using Saccharomyces cerevisiae, where cells orient polarity up pheromone gradients during mating. Initial orientation is often incorrect, but polarity sites then move around the cortex in a search for partners. We find that this movement is biased by local pheromone gradients across the polarity site: that is, movement of the polarity site is chemotactic. A bottom-up computational model recapitulates this biased movement. The model reveals how even though pheromone-bound receptors do not mimic the shape of external pheromone gradients, nonlinear and stochastic effects combine to generate effective gradient tracking. This mechanism for gradient tracking may be applicable to any cell that searches for a target in a complex chemical landscape.

Cell polarisation in a bulk-surface model can be driven by both classic and non-classic Turing instability

The GTPase Cdc42 is the master regulator of eukaryotic cell polarisation. During this process, the active form of Cdc42 is accumulated at a particular site on the cell membrane called the pole. It is believed that the accumulation of the active Cdc42 resulting in a pole is driven by a combination of activation–inactivation reactions and diffusion. It has been proposed using mathematical modelling that this is the result of diffusion-driven instability, originally proposed by Alan Turing. In this study, we developed, analysed and validated a 3D bulk-surface model of the dynamics of Cdc42. We show that the model can undergo both classic and non-classic Turing instability by deriving necessary conditions for which this occurs and conclude that the non-classic case can be viewed as a limit case of the classic case of diffusion-driven instability. Using three-dimensional Spatio-temporal simulation we predicted pole size and time to polarisation, suggesting that cell polarisation is mainly driven by the reaction strength parameter and that the size of the pole is determined by the relative diffusion.

Coordinating cell polarization and morphogenesis through mechanical feedback

Many cellular processes require cell polarization to be maintained as the cell changes shape, grows or moves. Without feedback mechanisms relaying information about cell shape to the polarity molecular machinery, the coordination between cell polarization and morphogenesis, movement or growth would not be possible. Here we theoretically and computationally study the role of a genetically-encoded mechanical feedback (in the Cell Wall Integrity pathway) as a potential coordination mechanism between cell morphogenesis and polarity during budding yeast mating projection growth. We developed a coarse-grained continuum description of the coupled dynamics of cell polarization and morphogenesis as well as 3D stochastic simulations of the molecular polarization machinery in the evolving cell shape. Both theoretical approaches show that in the absence of mechanical feedback (or in the presence of weak feedback), cell polarity cannot be maintained at the projection tip during growth, with the polarization cap wandering off the projection tip, arresting morphogenesis. In contrast, for mechanical feedback strengths above a threshold, cells can robustly maintain cell polarization at the tip and simultaneously sustain mating projection growth. These results indicate that the mechanical feedback encoded in the Cell Wall Integrity pathway can provide important positional information to the molecular machinery in the cell, thereby enabling the coordination of cell polarization and morphogenesis.

Fission Yeast Polarization: Modeling Cdc42 Oscillations, Symmetry Breaking, and Zones of Activation and Inhibition

Cells polarize for growth, motion, or mating through regulation of membrane-bound small GTPases between active GTP-bound and inactive GDP-bound forms. Activators (GEFs, GTP exchange factors) and inhibitors (GAPs, GTPase activating proteins) provide positive and negative feedbacks. We show that a reaction-diffusion model on a curved surface accounts for key features of polarization of model organism fission yeast. The model implements Cdc42 membrane diffusion using measured values for diffusion coefficients and dissociation rates and assumes a limiting GEF pool (proteins Gef1 and Scd1), as in prior models for budding yeast. The model includes two types of GAPs, one representing tip-localized GAPs, such as Rga3; and one representing side-localized GAPs, such as Rga4 and Rga6, that we assume switch between fast and slow diffusing states. After adjustment of unknown rate constants, the model reproduces active Cdc42 zones at cell tips and the pattern of GEF and GAP localization at cell tips and sides. The model reproduces observed tip-to-tip oscillations with periods of the order of several minutes, as well as asymmetric to symmetric oscillations transitions (corresponding to NETO "new end take off"), assuming the limiting GEF amount increases with cell size.

Size-Regulated Symmetry Breaking in Reaction-Diffusion Models of Developmental Transitions

The development of multicellular organisms proceeds through a series of morphogenetic and cell-state transitions, transforming homogeneous zygotes into complex adults by a process of self-organisation. Many of these transitions are achieved by spontaneous symmetry breaking mechanisms, allowing cells and tissues to acquire pattern and polarity by virtue of local interactions without an upstream supply of information. The combined work of theory and experiment has elucidated how these systems break symmetry during developmental transitions. Given that such transitions are multiple and their temporal ordering is crucial, an equally important question is how these developmental transitions are coordinated in time. Using a minimal mass-conserved substrate-depletion model for symmetry breaking as our case study, we elucidate mechanisms by which cells and tissues can couple reaction-diffusion-driven symmetry breaking to the timing of developmental transitions, arguing that the dependence of patterning mode on system size may be a generic principle by which developing organisms measure time. By analysing different regimes of our model, simulated on growing domains, we elaborate three distinct behaviours, allowing for clock-, timer- or switch-like dynamics. Relating these behaviours to experimentally documented case studies of developmental timing, we provide a minimal conceptual framework to interrogate how developing organisms coordinate developmental transitions.

Regulation of Cdc42 for polarized growth in budding yeast

The Rho GTPase Cdc42 is a central regulator of cell polarity in diverse cell types. The activity of Cdc42 is dynamically controlled in time and space to enable distinct polarization events, which generally occur along a single axis in response to spatial cues. Our understanding of the mechanisms underlying Cdc42 polarization has benefited largely from studies of the budding yeast Saccharomyces cerevisiae, a genetically tractable model organism. In budding yeast, Cdc42 activation occurs in two temporal steps in the G1 phase of the cell cycle to establish a proper growth site. Here, we review findings in budding yeast that reveal an intricate crosstalk among polarity proteins for biphasic Cdc42 regulation. The first step of Cdc42 activation may determine the axis of cell polarity, while the second step ensures robust Cdc42 polarization for growth. Biphasic Cdc42 polarization is likely to ensure the proper timing of events including the assembly and recognition of spatial landmarks and stepwise assembly of a new ring of septins, cytoskeletal GTP-binding proteins, at the incipient bud site. Biphasic activation of GTPases has also been observed in mammalian cells, suggesting that biphasic activation could be a general mechanism for signal-responsive cell polarization. Cdc42 activity is necessary for polarity establishment during normal cell division and development, but its activity has also been implicated in the promotion of aging. We also discuss negative polarity signaling and emerging concepts of Cdc42 signaling in cellular aging.

Mechanistic insights into actin-driven polarity site movement in yeast

Directed cell growth or migration are critical for the development and function of many eukaryotic cells. These cells develop a dynamic "front" (also called "polarity site") that can change direction. Polarity establishment involves autocatalytic accumulation of polarity regulators, including the conserved Rho-family GTPase Cdc42, but the mechanisms underlying polarity reorientation remain poorly understood. The tractable model yeast, Saccharomyces cerevisiae, relocates its polarity site when searching for mating partners. Relocation requires polymerized actin, and is thought to involve actin-mediated vesicle traffic to the polarity site. In this study, we provide a quantitative characterization of spontaneous polarity site movement as a search process and use a mechanistic computational model that combines polarity protein biochemical interactions with vesicle trafficking to probe how various processes might affect polarity site movement. Our findings identify two previously documented features of yeast vesicle traffic as being particularly relevant to such movement: tight spatial focusing of exocytosis enhances the directional persistence of movement, and association of Cdc42-directed GTPase-Activating Proteins with secretory vesicles increases the distance moved. Furthermore, we suggest that variation in the rate of exocytosis beyond simple Poisson dynamics may be needed to fully account for the characteristics of polarity site movement in vivo.

How Diffusion Impacts Cortical Protein Distribution in Yeasts

Proteins associated with the yeast plasma membrane often accumulate asymmetrically within the plane of the membrane. Asymmetric accumulation is thought to underlie diverse processes, including polarized growth, stress sensing, and aging. Here, we review our evolving understanding of how cells achieve asymmetric distributions of membrane proteins despite the anticipated dissipative effects of diffusion, and highlight recent findings suggesting that differential diffusion is exploited to create, rather than dissipate, asymmetry. We also highlight open questions about diffusion in yeast plasma membranes that remain unsolved.

Ratiometric GPCR signaling enables directional sensing in yeast

Accurate detection of extracellular chemical gradients is essential for many cellular behaviors. Gradient sensing is challenging for small cells, which can experience little difference in ligand concentrations on the up-gradient and down-gradient sides of the cell. Nevertheless, the tiny cells of the yeast Saccharomyces cerevisiae reliably decode gradients of extracellular pheromones to find their mates. By imaging the behavior of polarity factors and pheromone receptors, we quantified the accuracy of initial polarization during mating encounters. We found that cells bias the orientation of initial polarity up-gradient, even though they have unevenly distributed receptors. Uneven receptor density means that the gradient of ligand-bound receptors does not accurately reflect the external pheromone gradient. Nevertheless, yeast cells appear to avoid being misled by responding to the fraction of occupied receptors rather than simply the concentration of ligand-bound receptors. Such ratiometric sensing also serves to amplify the gradient of active G protein. However, this process is quite error-prone, and initial errors are corrected during a subsequent indecisive phase in which polarity clusters exhibit erratic mobile behavior.

Cells use surface receptors to decode spatial information from chemical gradients, but accurate decoding is hampered by small cell size and the presence of molecular noise. This study shows that yeast cells decode pheromone gradients by measuring the local ratio of bound to unbound receptors. This mechanism corrects for uneven receptor density at the surface and amplifies the gradient transmitted to downstream components.

Self-organization principles of intracellular pattern formation

Dynamic patterning of specific proteins is essential for the spatio-temporal regulation of many important intracellular processes in prokaryotes, eukaryotes and multicellular organisms. The emergence of patterns generated by interactions of diffusing proteins is a paradigmatic example for self-organization. In this article, we review quantitative models for intracellular Min protein patterns in Escherichia coli, Cdc42 polarization in Saccharomyces cerevisiae and the bipolar PAR protein patterns found in Caenorhabditis elegans. By analysing the molecular processes driving these systems we derive a theoretical perspective on general principles underlying self-organized pattern formation. We argue that intracellular pattern formation is not captured by concepts such as ‘activators’, ‘inhibitors’ or ‘substrate depletion’. Instead, intracellular pattern formation is based on the redistribution of proteins by cytosolic diffusion, and the cycling of proteins between distinct conformational states. Therefore, mass-conserving reaction–diffusion equations provide the most appropriate framework to study intracellular pattern formation. We conclude that directed transport, e.g. cytosolic diffusion along an actively maintained cytosolic gradient, is the key process underlying pattern formation. Thus the basic principle of self-organization is the establishment and maintenance of directed transport by intracellular protein dynamics.

This article is part of the theme issue ‘Self-organization in cell biology’.

Minimal in vitro systems shed light on cell polarity

Cell polarity - the morphological and functional differentiation of cellular compartments in a directional manner - is required for processes such as orientation of cell division, directed cellular growth and motility. How the interplay of components within the complexity of a cell leads to cell polarity is still heavily debated. In this Review, we focus on one specific aspect of cell polarity: the non-uniform accumulation of proteins on the cell membrane. In cells, this is achieved through reaction-diffusion and/or cytoskeleton-based mechanisms. In reaction-diffusion systems, components are transformed into each other by chemical reactions and are moving through space by diffusion. In cytoskeleton-based processes, cellular components (i.e. proteins) are actively transported by microtubules (MTs) and actin filaments to specific locations in the cell. We examine how minimal systems - in vitro reconstitutions of a particular cellular function with a minimal number of components - are designed, how they contribute to our understanding of cell polarity (i.e. protein accumulation), and how they complement in vivo investigations. We start by discussing the Min protein system from Escherichia coli, which represents a reaction-diffusion system with a well-established minimal system. This is followed by a discussion of MT-based directed transport for cell polarity markers as an example of a cytoskeleton-based mechanism. To conclude, we discuss, as an example, the interplay of reaction-diffusion and cytoskeleton-based mechanisms during polarity establishment in budding yeast.

Exploration and stabilization of Ras1 mating zone: A mechanism with positive and negative feedbacks

In mating fission yeast cells, sensing and response to extracellular pheromone concentrations occurs through an exploratory Cdc42 patch that stochastically samples the cell cortex before stabilizing towards a mating partner. Active Ras1 (Ras1-GTP), an upstream regulator of Cdc42, and Gap1, the GTPase-activating protein for Ras1, localize at the patch. We developed a reaction-diffusion model of Ras1 patch appearance and disappearance with a positive feedback by a Guanine nucleotide Exchange Factor (GEF) and Gap1 inhibition. The model is based on new estimates of Ras1-GDP, Ras1-GTP and Gap1 diffusion coefficients and rates of cytoplasmic exchange studied by FRAP. The model reproduces exploratory patch behavior and lack of Ras1 patch in cells lacking Gap1. Transition to a stable patch can occur by change of Gap1 rates constants or local increase of the positive feedback rate constants. The model predicts that the patch size and number of patches depend on the strength of positive and negative feedbacks. Measurements of Ras1 patch size and number in cells overexpressing the Ras1 GEF or Gap1 are consistent with the model.

Unicellular fission yeasts mate by fusing with partners of the opposite mating type. Each pair member grows towards its selected partner that signals its presence through secreted pheromone. The process of partner selection occurs through an exploratory patch (containing activated signaling protein Cdc42 and upstream regulator Ras1) that assembles and disassembles on the cell cortex, stabilizing in regions of higher opposite pheromone concentration. We present a computational model of the molecular mechanisms driving the dynamical pattern of patch exploration and stabilization. The model is based on reaction and diffusion along the curved cell membrane, with diffusion coefficients measured experimentally. In the model, a positive Ras1 activation feedback loop generates a patch containing most of the activating protein (Ras1 GEF). The fast diffusing inhibitor Gap1 that is recruited locally from the cytoplasm spreads on the cell membrane, limiting patch size and causing its decay. Spontaneous reinitiation of Ras1 activation elsewhere on the cortex provides a mechanism for exploration. Transition of the system’s behavior to that of a single stable patch is possible upon simulated pheromone sensing. The computational model provides predictions for the number of patches and patch size dependence on parameters that we tested experimentally.

The effect of cell geometry on polarization in budding yeast

The localization (or polarization) of proteins on the membrane during the mating of budding yeast (Saccharomyces cerevisiae) is an important model system for understanding simple pattern formation within cells. While there are many existing mathematical models of polarization, for both budding and mating, there are still many aspects of this process that are not well understood. In this paper we set out to elucidate the effect that the geometry of the cell can have on the dynamics of certain models of polarization. Specifically, we look at several spatial stochastic models of Cdc42 polarization that have been adapted from published models, on a variety of tip-shaped geometries, to replicate the shape change that occurs during the growth of the mating projection. We show here that there is a complex interplay between the dynamics of polarization and the shape of the cell. Our results show that while models of polarization can generate a stable polarization cap, its localization at the tip of mating projections is unstable, with the polarization cap drifting away from the tip of the projection in a geometry dependent manner. We also compare predictions from our computational results to experiments that observe cells with projections of varying lengths, and track the stability of the polarization cap. Lastly, we examine one model of actin polarization and show that it is unlikely, at least for the models studied here, that actin dynamics and vesicle traffic are able to overcome this effect of geometry.

Exocytosis-coordinated mechanisms for tip growth underlie pollen tube growth guidance

Many tip-growing cells are capable of responding to guidance cues, during which cells precisely steer their growth toward the source of guidance signals. Though several players in signal perception have been identified, little is known about the downstream signaling that controls growth direction during guidance. Here, using combined modeling and experimental studies, we demonstrate that the growth guidance of Arabidopsis pollen tubes is regulated by the signaling network that controls tip growth. Tip-localized exocytosis plays a key role in this network by integrating guidance signals with the ROP1 Rho GTPase signaling and coordinating intracellular signaling with cell wall mechanics. This model reproduces the high robustness and responsiveness of pollen tube guidance and explains the connection between guidance efficiency and the parameters of the tip growth system. Hence, our findings establish an exocytosis-coordinated mechanism underlying the cellular pathfinding guided by signal gradients and the mechanistic linkage between tip growth and guidance.

Cell Polarity in Yeast

A conserved molecular machinery centered on the Cdc42 GTPase regulates cell polarity in diverse organisms. Here we review findings from budding and fission yeasts that reveal both a conserved core polarity circuit and several adaptations that each organism exploits to fulfill the needs of its lifestyle. The core circuit involves positive feedback by local activation of Cdc42 to generate a cluster of concentrated GTP-Cdc42 at the membrane. Species-specific pathways regulate the timing of polarization during the cell cycle, as well as the location and number of polarity sites.

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.

Pulses of Ca2+ coordinate actin assembly and exocytosis for stepwise cell extension

Many eukaryotic cells grow by extending their cell periphery in pulses. The molecular mechanisms underlying this process are not yet fully understood. Here we present a comprehensive model of stepwise cell extension by using the unique tip growth system of filamentous fungi. Live-cell imaging analysis, including superresolution microscopy, revealed that the fungus Aspergillus nidulans extends the hyphal tip in an oscillatory manner. The amount of F-actin and secretory vesicles (SV) accumulating at the hyphal tip oscillated with a positive temporal correlation, whereas vesicle amounts were negatively correlated to the growth rate. The intracellular Ca²⁺ level also pulsed with a positive temporal correlation to the amount of F-actin and SV at the hyphal tip. Two Ca²⁺ channels, MidA and CchA, were needed for proper tip growth and the oscillations of actin polymerization, exocytosis, and the growth rate. The data indicate a model in which transient Ca²⁺ pluses cause depolymerization of F-actin at the cortex and promote SV fusion with the plasma membrane, thereby extending the cell tip. Over time, Ca²⁺ diffuses away and F-actin and SV accumulate again at the hyphal tip. Our data provide evidence that temporally controlled actin polymerization and exocytosis are coordinated by pulsed Ca²⁺ influx, resulting in stepwise cell extension.

Polarity establishment by Cdc42: Key roles for positive feedback and differential mobility

Cell polarity is fundamental to the function of most cells. The evolutionarily conserved molecular machinery that controls cell polarity is centered on a family of GTPases related to Cdc42. Cdc42 becomes activated and concentrated at polarity sites, but studies in yeast model systems led to controversy on the mechanisms of polarization. Here we review recent studies that have clarified how Cdc42 becomes polarized in yeast. On one hand, findings that appeared to support a key role for the actin cytoskeleton and vesicle traffic in polarity establishment now appear to reflect the action of stress response pathways induced by cytoskeletal perturbations. On the other hand, new findings strongly support hypotheses on the polarization mechanism whose origins date back to the mathematician Alan Turing. The key features of the polarity establishment mechanism in yeasts include a positive feedback pathway in which active Cdc42 recruits a Cdc42 activator to polarity sites, and differential mobility of polarity "activators" and "substrates."

Scaffold-mediated gating of Cdc42 signalling flux

Scaffold proteins modulate signalling pathway activity spatially and temporally. In budding yeast, the scaffold Bem1 contributes to polarity axis establishment by regulating the GTPase Cdc42. Although different models have been proposed for Bem1 function, there is little direct evidence for an underlying mechanism. Here, we find that Bem1 directly augments the guanine exchange factor (GEF) activity of Cdc24. Bem1 also increases GEF phosphorylation by the p21-activated kinase (PAK), Cla4. Phosphorylation abrogates the scaffold-dependent stimulation of GEF activity, rendering Cdc24 insensitive to additional Bem1. Thus, Bem1 stimulates GEF activity in a reversible fashion, contributing to signalling flux through Cdc42. The contribution of Bem1 to GTPase dynamics was borne-out by in vivo imaging: active Cdc42 was enriched at the cell pole in hypophosphorylated cdc24 mutants, while hyperphosphorylated cdc24 mutants that were resistant to scaffold stimulation displayed a deficit in active Cdc42 at the pole. These findings illustrate the self-regulatory properties that scaffold proteins confer on signalling pathways.

DOI:http://dx.doi.org/10.7554/eLife.25257.001

Parallel Actin-Independent Recycling Pathways Polarize Cdc42 in Budding Yeast

The highly conserved Rho-family GTPase Cdc42 is an essential regulator of polarity in many different cell types. During polarity establishment, Cdc42 becomes concentrated at a cortical site, where it interacts with downstream effectors to orient the cytoskeleton along the front-back axis. To concentrate Cdc42, loss of Cdc42 by diffusion must be balanced by recycling to the front. In Saccharomyces cerevisiae, the guanine nucleotide dissociation inhibitor (GDI) Rdi1 recycles Cdc42 through the cytoplasm. Loss of Rdi1 slowed but did not eliminate Cdc42 accumulation at the front, suggesting the existence of other recycling pathways. One proposed pathway involves actin-directed trafficking of vesicles carrying Cdc42 to the front. However, we found no role for F-actin in Cdc42 concentration, even in rdi1Δ cells. Instead, Cdc42 was still able to exchange between the membrane and cytoplasm in rdi1Δ cells, albeit at a reduced rate. Membrane-cytoplasm exchange of GDP-Cdc42 was faster than that of GTP-Cdc42, and computational modeling indicated that such exchange would suffice to promote polarization. We also uncovered a novel role for the Cdc42-directed GTPase-activating protein (GAP) Bem2 in Cdc42 polarization. Bem2 was known to act in series with Rdi1 to promote recycling of Cdc42, but we found that rdi1Δ bem2Δ mutants were synthetically lethal, suggesting that they also act in parallel. We suggest that GAP activity cooperates with the GDI to counteract the dissipative effect of a previously unappreciated pathway whereby GTP-Cdc42 escapes from the polarity site through the cytoplasm.

Modelling of Yeast Mating Reveals Robustness Strategies for Cell-Cell Interactions

Mating of budding yeast cells is a model system for studying cell-cell interactions. Haploid yeast cells secrete mating pheromones that are sensed by the partner which responds by growing a mating projection toward the source. The two projections meet and fuse to form the diploid. Successful mating relies on precise coordination of dynamic extracellular signals, signaling pathways, and cell shape changes in a noisy background. It remains elusive how cells mate accurately and efficiently in a natural multi-cell environment. Here we present the first stochastic model of multiple mating cells whose morphologies are driven by pheromone gradients and intracellular signals. Our novel computational framework encompassed a moving boundary method for modeling both a-cells and α-cells and their cell shape changes, the extracellular diffusion of mating pheromones dynamically coupled with cell polarization, and both external and internal noise. Quantification of mating efficiency was developed and tested for different model parameters. Computer simulations revealed important robustness strategies for mating in the presence of noise. These strategies included the polarized secretion of pheromone, the presence of the α-factor protease Bar1, and the regulation of sensing sensitivity; all were consistent with data in the literature. In addition, we investigated mating discrimination, the ability of an a-cell to distinguish between α-cells either making or not making α-factor, and mating competition, in which multiple a-cells compete to mate with one α-cell. Our simulations were consistent with previous experimental results. Moreover, we performed a combination of simulations and experiments to estimate the diffusion rate of the pheromone a-factor. In summary, we constructed a framework for simulating yeast mating with multiple cells in a noisy environment, and used this framework to reproduce mating behaviors and to identify strategies for robust cell-cell interactions.

One of the riddles of Nature is how cells interact with one another to create complex cellular networks such as the neural networks in the brain. Forming precise connections between irregularly shaped cells is a challenge for biology. We developed computational methods for simulating these complex cell-cell interactions. We applied these methods to investigate yeast mating in which two yeast cells grow projections that meet and fuse guided by pheromone attractants. The simulations described molecules both inside and outside of the cell, and represented the continually changing shapes of the cells. We found that positioning the secretion and sensing of pheromones at the same location on the cell surface was important. Other key factors for robust mating included secreting a protein that removed excess pheromone from outside of the cell so that the signal would not be too strong. An important advance was being able to simulate as many as five cells in complex mating arrangements. Taken together we used our novel computational methods to describe in greater detail the yeast mating process, and more generally, interactions among cells changing their shapes in response to their neighbors.

Coordinated process of polarized growth in filamentous fungi

Filamentous fungi are extremely polarized organisms, exhibiting continuous growth at their hyphal tips. The hyphal form is related to their pathogenicity in animals and plants, and their high secretion ability for biotechnology. Polarized growth requires a sequential supply of proteins and lipids to the hyphal tip. This transport is managed by vesicle trafficking via the actin and microtubule cytoskeleton. Therefore, the arrangement of the cytoskeleton is a crucial step to establish and maintain the cell polarity. This review summarizes recent findings unraveling the mechanism of polarized growth with special emphasis on the role of actin and microtubule cytoskeleton and polarity marker proteins. Rapid insertions of membranes via highly active exocytosis at hyphal tips could quickly dilute the accumulated polarity marker proteins. Recent findings by a super-resolution microscopy indicate that filamentous fungal cells maintain their polarity at the tips by repeating transient assembly and disassembly of polarity sites.

A Predictive Model for Yeast Cell Polarization in Pheromone Gradients

Budding yeast cells exist in two mating types, a and α, which use peptide pheromones to communicate with each other during mating. Mating depends on the ability of cells to polarize up pheromone gradients, but cells also respond to spatially uniform fields of pheromone by polarizing along a single axis. We used quantitative measurements of the response of a cells to α-factor to produce a predictive model of yeast polarization towards a pheromone gradient. We found that cells make a sharp transition between budding cycles and mating induced polarization and that they detect pheromone gradients accurately only over a narrow range of pheromone concentrations corresponding to this transition. We fit all the parameters of the mathematical model by using quantitative data on spontaneous polarization in uniform pheromone concentration. Once these parameters have been computed, and without any further fit, our model quantitatively predicts the yeast cell response to pheromone gradient providing an important step toward understanding how cells communicate with each other.

Role of Polarized G Protein Signaling in Tracking Pheromone Gradients

Yeast cells track gradients of pheromones to locate mating partners. Intuition suggests that uniform distribution of pheromone receptors over the cell surface would yield optimal gradient sensing. However, yeast cells display polarized receptors. The benefit of such polarization was unknown. During gradient tracking, cell growth is directed by a patch of polarity regulators that wanders around the cortex. Patch movement is sensitive to pheromone dose, with wandering reduced on the up-gradient side of the cell, resulting in net growth in that direction. Mathematical modeling suggests that active receptors and associated G proteins lag behind the polarity patch and act as an effective drag on patch movement. In vivo, the polarity patch is trailed by a G protein-rich domain, and this polarized distribution of G proteins is required to constrain patch wandering. Our findings explain why G protein polarization is beneficial and illuminate a novel mechanism for gradient tracking.

Polarity establishment requires localized activation of Cdc42

Positive feedback by localized activation of Cdc42 drives polarity establishment in the budding yeast Saccharomyces cerevisiae.

Establishment of cell polarity in animal and fungal cells involves localization of the conserved Rho-family guanosine triphosphatase, Cdc42, to the cortical region destined to become the “front” of the cell. The high local concentration of active Cdc42 promotes cytoskeletal polarization through various effectors. Cdc42 accumulation at the front is thought to involve positive feedback, and studies in the budding yeast Saccharomyces cerevisiae have suggested distinct positive feedback mechanisms. One class of mechanisms involves localized activation of Cdc42 at the front, whereas another class involves localized delivery of Cdc42 to the front. Here we show that Cdc42 activation must be localized for successful polarity establishment, supporting local activation rather than local delivery as the dominant mechanism in this system.

Cdc42 and Cellular Polarity: Emerging Roles at the Golgi

Cdc42 belongs to the Rho family of small GTPases and plays key roles in cellular events of polarity. This role of Cdc42 has typically been attributed to its function at the plasma membrane. However, Cdc42 also exists at the Golgi complex. Here we summarize major insights that have been gathered in studying the Golgi pool of Cdc42 and propose that Golgi-localized Cdc42 enables the cell to diversify the function of Cdc42, which in some cases represents new roles and in other cases acts to complement the established roles of Cdc42 at the plasma membrane. Studies on how Cdc42 acts at the Golgi also suggest key questions to address in the future.

Role of competition between polarity sites in establishing a unique front

Polarity establishment in many cells is thought to occur via positive feedback that reinforces even tiny asymmetries in polarity protein distribution. Cdc42 and related GTPases are activated and accumulate in a patch of the cortex that defines the front of the cell. Positive feedback enables spontaneous polarization triggered by stochastic fluctuations, but as such fluctuations can occur at multiple locations, how do cells ensure that they make only one front? In polarizing cells of the model yeast Saccharomyces cerevisiae, positive feedback can trigger growth of several Cdc42 clusters at the same time, but this multi-cluster stage rapidly evolves to a single-cluster state, which then promotes bud emergence. By manipulating polarity protein dynamics, we show that resolution of multi-cluster intermediates occurs through a greedy competition between clusters to recruit and retain polarity proteins from a shared intracellular pool.

Superresolution microscopy reveals a dynamic picture of cell polarity maintenance during directional growth

Polar (directional) cell growth, a key cellular mechanism shared among a wide range of species, relies on targeted insertion of new material at specific locations of the plasma membrane. How these cell polarity sites are stably maintained during massive membrane insertion has remained elusive. Conventional live-cell optical microscopy fails to visualize polarity site formation in the crowded cell membrane environment because of its limited resolution. We have used advanced live-cell imaging techniques to directly observe the localization, assembly, and disassembly processes of cell polarity sites with high spatiotemporal resolution in a rapidly growing filamentous fungus, Aspergillus nidulans. We show that the membrane-associated polarity site marker TeaR is transported on microtubules along with secretory vesicles and forms a protein cluster at that point of the apical membrane where the plus end of the microtubule touches. There, a small patch of membrane is added through exocytosis, and the TeaR cluster gets quickly dispersed over the membrane. There is an incessant disassembly and reassembly of polarity sites at the growth zone, and each new polarity site locus is slightly offset from preceding ones. On the basis of our imaging results and computational modeling, we propose a transient polarity model that explains how cell polarity is stably maintained during highly active directional growth.

Spontaneous cell polarization: Feedback control of Cdc42 GTPase breaks cellular symmetry

Spontaneous polarization without spatial cues, or symmetry breaking, is a fundamental problem of spatial organization in biological systems. This question has been extensively studied using yeast models, which revealed the central role of the small GTPase switch Cdc42. Active Cdc42-GTP forms a coherent patch at the cell cortex, thought to result from amplification of a small initial stochastic inhomogeneity through positive feedback mechanisms, which induces cell polarization. Here, I review and discuss the mechanisms of Cdc42 activity self-amplification and dynamic turnover. A robust Cdc42 patch is formed through the combined effects of Cdc42 activity promoting its own activation and active Cdc42-GTP displaying reduced membrane detachment and lateral diffusion compared to inactive Cdc42-GDP. I argue the role of the actin cytoskeleton in symmetry breaking is not primarily to transport Cdc42 to the active site. Finally, negative feedback and competition mechanisms serve to control the number of polarization sites.

Local perturbation analysis: a computational tool for biophysical reaction-diffusion models

Diffusion and interaction of molecular regulators in cells is often modeled using reaction-diffusion partial differential equations. Analysis of such models and exploration of their parameter space is challenging, particularly for systems of high dimensionality. Here, we present a relatively simple and straightforward analysis, the local perturbation analysis, that reveals how parameter variations affect model behavior. This computational tool, which greatly aids exploration of the behavior of a model, exploits a structural feature common to many cellular regulatory systems: regulators are typically either bound to a membrane or freely diffusing in the interior of the cell. Using well-documented, readily available bifurcation software, the local perturbation analysis tracks the approximate early evolution of an arbitrarily large perturbation of a homogeneous steady state. In doing so, it provides a bifurcation diagram that concisely describes various regimes of the model's behavior, reducing the need for exhaustive simulations to explore parameter space. We explain the method and provide detailed step-by-step guides to its use and application.

Quantitative analysis of membrane trafficking in regulation of Cdc42 polarity

Vesicle delivery of Cdc42 has been proposed as an important mechanism for generating and maintaining Cdc42 polarity at the plasma membrane. This mechanism requires the density of Cdc42 on secretory vesicles to be equal to or higher than the plasma membrane polarity cap. Using a novel method to estimate Cdc42 levels on post-Golgi secretory vesicles in intact yeast cells, we: (1) determined that endocytosis plays an important role in Cdc42's association with secretory vesicles (2) found that a GFP-tag placed on the N-terminus of Cdc42 negatively impacts this vesicle association and (3) quantified the surface densities of Cdc42 on post-Golgi vesicles which revealed that the vesicle density of Cdc42 is three times more dilute than that at the polarity cap. This work suggests that the immediate consequence of secretory vesicle fusion with the plasma membrane polarity cap is to dilute the local Cdc42 surface density. This provides strong support for the model in which vesicle trafficking acts to negatively regulate Cdc42 polarity on the cell surface while also providing a means to recycle Cdc42 between the cell surface and internal membrane locations.

Bud3 activates Cdc42 to establish a proper growth site in budding yeast

Cell polarization occurs along a single axis that is generally determined by a spatial cue, yet the underlying mechanism is poorly understood. Using biochemical assays and live-cell imaging, we show that cell polarization to a proper growth site requires activation of Cdc42 by Bud3 in haploid budding yeast. Bud3 catalyzes the release of guanosine diphosphate (GDP) from Cdc42 and elevates intracellular Cdc42-guanosine triphosphate (GTP) levels in cells with inactive Cdc24, which has as of yet been the sole GDP-GTP exchange factor for Cdc42. Cdc42 is activated in two temporal steps in the G1 phase: the first depends on Bud3, whereas subsequent activation depends on Cdc24. Mutational analyses suggest that biphasic activation of Cdc42 in G1 is necessary for assembly of a proper bud site. Biphasic activation of Cdc42 or Rac GTPases may be a general mechanism for spatial cue-directed cell polarization in eukaryotes.

Inhibitory GEF phosphorylation provides negative feedback in the yeast polarity circuit

Cell polarity is critical for the form and function of many cell types. During polarity establishment, cells define a cortical "front" that behaves differently from the rest of the cortex. The front accumulates high levels of the active form of a polarity-determining Rho-family GTPase (Cdc42, Rac, or Rop) that then orients cytoskeletal elements through various effectors to generate the polarized morphology appropriate to the particular cell type [1, 2]. GTPase accumulation is thought to involve positive feedback, such that active GTPase promotes further delivery and/or activation of more GTPase in its vicinity [3]. Recent studies suggest that once a front forms, the concentration of polarity factors at the front can increase and decrease periodically, first clustering the factors at the cortex and then dispersing them back to the cytoplasm [4-7]. Such oscillatory behavior implies the presence of negative feedback in the polarity circuit [8], but the mechanism of negative feedback was not known. Here we show that, in the budding yeast Saccharomyces cerevisiae, the catalytic activity of the Cdc42-directed GEF is inhibited by Cdc42-stimulated effector kinases, thus providing negative feedback. We further show that replacing the GEF with a phosphosite mutant GEF abolishes oscillations and leads to the accumulation of excess GTP-Cdc42 and other polarity factors at the front. These findings reveal a mechanism for negative feedback and suggest that the function of negative feedback via GEF inhibition is to buffer the level of Cdc42 at the polarity site.

Establishment of a robust single axis of cell polarity by coupling multiple positive feedback loops

Establishment of cell polarity—or symmetry breaking—relies on local accumulation of polarity regulators. Although simple positive feedback is sufficient to drive symmetry breaking, it is highly sensitive to stochastic fluctuations typical for living cells. Here, by integrating mathematical modelling with quantitative experimental validations, we show that in the yeast Saccharomyces cerevisiae a combination of actin- and guanine nucleotide dissociation inhibitor-dependent recycling of the central polarity regulator Cdc42 is needed to establish robust cell polarity at a single site during yeast budding. The guanine nucleotide dissociation inhibitor pathway consistently generates a single-polarization site, but requires Cdc42 to cycle rapidly between its active and inactive form, and is therefore sensitive to perturbations of the GTPase cycle. Conversely, actin-mediated recycling of Cdc42 induces robust symmetry breaking but cannot restrict polarization to a single site. Our results demonstrate how cells optimize symmetry breaking through coupling between multiple feedback loops.

A positive feedback loop which results in localized accumulation of the small GTPase Cdc42 generates cell polarity in budding yeast; however, such loops are inherently susceptible to noise. Here the authors demonstrate how two pathways that mediate Cdc42 recycling work together to ensure the robustness of symmetry breaking.

GDI-mediated cell polarization in yeast provides precise spatial and temporal control of Cdc42 signaling

Cell polarization is a prerequisite for essential processes such as cell migration, proliferation or differentiation. The yeast Saccharomyces cerevisiae under control of the GTPase Cdc42 is able to polarize without the help of cytoskeletal structures and spatial cues through a pathway depending on its guanine nucleotide dissociation inhibitor (GDI) Rdi1. To develop a fundamental understanding of yeast polarization we establish a detailed mechanistic model of GDI-mediated polarization. We show that GDI-mediated polarization provides precise spatial and temporal control of Cdc42 signaling and give experimental evidence for our findings. Cell cycle induced changes of Cdc42 regulation enhance positive feedback loops of active Cdc42 production, and thereby allow simultaneous switch-like regulation of focused polarity and Cdc42 activation. This regulation drives the direct formation of a unique polarity cluster with characteristic narrowing dynamics, as opposed to the previously proposed competition between transient clusters. As the key components of the studied system are conserved among eukaryotes, we expect our findings also to apply to cell polarization in other organisms.

Cell polarization is a fundamental cellular process that defines a single orientation axis within prokaryotic or eukaryotic cells and is a prerequisite for developmental processes such as cell migration, proliferation or differentiation. In the yeast Saccharomyces cerevisiae cell polarization determines the position of a new growth or bud site. Although many studies have focused on identifying polarity regulators and their interactions, the fundamental mechanisms and features of cell polarity still remain controversial. Here, we develop a detailed mathematical model of diffusion-driven cell polarization, which we verify experimentally. We show that this polarization mechanism provides precise spatial and temporal control of signals, which determine the place of a new growth site. Changes induced by the cell cycle allow simultaneous switch-like regulation of polarization and activation of the GTPase Cdc42, the central polarity regulator which initiates formation of a new bud. This regulation drives direct formation of a unique Cdc42 cluster with characteristic narrowing dynamics and robustly narrow spatial focus. Hence, our analysis reveals fundamental design principles that allow cell polarization to reliably initiate developmental processes at a specific time and place. As the key components of the studied system are conserved among eukaryotes, we expect our findings also to apply to cell polarization in other organisms.

Beyond symmetry-breaking: competition and negative feedback in GTPase regulation

Cortical domains are often specified by the local accumulation of active GTPases. Such domains can arise through spontaneous symmetry-breaking, suggesting that GTPase accumulation occurs via positive feedback. Here, we focus on recent advances in fungal and plant cell models - where new work suggests that polarity-controlling GTPases develop only one 'front' because GTPase clusters engage in a winner-takes-all competition. However, in some circumstances two or more GTPase domains can coexist, and the basis for the switch from competition to coexistence remains an open question. Polarity GTPases can undergo oscillatory clustering and dispersal, suggesting that these systems contain negative feedback. Negative feedback may prevent polarity clusters from spreading too far, regulate the balance between competition and coexistence, and provide directional flexibility for cells tracking gradients.