Dynamics of Cdc42 network embodies a Turing-type mechanism of yeast cell polarity

Comparative live-cell imaging analyses of SPA-2, BUD-6 and BNI-1 in Neurospora crassa reveal novel features of the filamentous fungal polarisome

A key multiprotein complex involved in regulating the actin cytoskeleton and secretory machinery required for polarized growth in fungi, is the polarisome. Recognized core constituents in budding yeast are the proteins Spa2, Pea2, Aip3/Bud6, and the key effector Bni1. Multicellular fungi display a more complex polarized morphogenesis than yeasts, suggesting that the filamentous fungal polarisome might fulfill additional functions. In this study, we compared the subcellular organization and dynamics of the putative polarisome components BUD-6 and BNI-1 with those of the bona fide polarisome marker SPA-2 at various developmental stages of Neurospora crassa. All three proteins exhibited a yeast-like polarisome configuration during polarized germ tube growth, cell fusion, septal pore plugging and tip repolarization. However, the localization patterns of all three proteins showed spatiotemporally distinct characteristics during the establishment of new polar axes, septum formation and cytokinesis, and maintained hyphal tip growth. Most notably, in vegetative hyphal tips BUD-6 accumulated as a subapical cloud excluded from the Spitzenkörper (Spk), whereas BNI-1 and SPA-2 partially colocalized with the Spk and the tip apex. Novel roles during septal plugging and cytokinesis, connected to the reinitiation of tip growth upon physical injury and conidial maturation, were identified for BUD-6 and BNI-1, respectively. Phenotypic analyses of gene deletion mutants revealed additional functions for BUD-6 and BNI-1 in cell fusion regulation, and the maintenance of Spk integrity. Considered together, our findings reveal novel polarisome-independent functions of BUD-6 and BNI-1 in Neurospora, but also suggest that all three proteins cooperate at plugged septal pores, and their complex arrangement within the apical dome of mature hypha might represent a novel aspect of filamentous fungal polarisome architecture.

Excitable behavior can explain the "ping-pong" mode of communication between cells using the same chemoattractant

Here we elucidate a paradox: how a single chemoattractant-receptor system in two individuals is used for communication despite the seeming inevitability of self-excitation. In the filamentous fungus Neurospora crassa, genetically identical cells that produce the same chemoattractant fuse via the homing of individual cell protrusions toward each other. This is achieved via a recently described "ping-pong" pulsatile communication. Using a generic activator-inhibitor model of excitable behavior, we demonstrate that the pulse exchange can be fully understood in terms of two excitable systems locked into a stable oscillatory pattern of mutual excitation. The most puzzling properties of this communication are the sudden onset of oscillations with final amplitude, and the absence of seemingly inevitable self-excitation. We show that these properties result directly from both the excitability threshold and refractory period characteristic of excitable systems. Our model suggests possible molecular mechanisms for the ping-pong communication.

Membrane tension maintains cell polarity by confining signals to the leading edge during neutrophil migration

Little is known about how neutrophils and other cells establish a single zone of actin assembly during migration. A widespread assumption is that the leading edge prevents formation of additional fronts by generating long-range diffusible inhibitors or by sequestering essential polarity components. We use morphological perturbations, cell-severing experiments, and computational simulations to show that diffusion-based mechanisms are not sufficient for long-range inhibition by the pseudopod. Instead, plasma membrane tension could serve as a long-range inhibitor in neutrophils. We find that membrane tension doubles during leading-edge protrusion, and increasing tension is sufficient for long-range inhibition of actin assembly and Rac activation. Furthermore, reducing membrane tension causes uniform actin assembly. We suggest that tension, rather than diffusible molecules generated or sequestered at the leading edge, is the dominant source of long-range inhibition that constrains the spread of the existing front and prevents the formation of secondary fronts.

Morphogenesis and the cell cycle

Studies of the processes leading to the construction of a bud and its separation from the mother cell in Saccharomyces cerevisiae have provided foundational paradigms for the mechanisms of polarity establishment, cytoskeletal organization, and cytokinesis. Here we review our current understanding of how these morphogenetic events occur and how they are controlled by the cell-cycle-regulatory cyclin-CDK system. In addition, defects in morphogenesis provide signals that feed back on the cyclin-CDK system, and we review what is known regarding regulation of cell-cycle progression in response to such defects, primarily acting through the kinase Swe1p. The bidirectional communication between morphogenesis and the cell cycle is crucial for successful proliferation, and its study has illuminated many elegant and often unexpected regulatory mechanisms. Despite considerable progress, however, many of the most puzzling mysteries in this field remain to be resolved.

Spatial modeling of cell signaling networks

The shape of a cell, the sizes of subcellular compartments, and the spatial distribution of molecules within the cytoplasm can all control how molecules interact to produce a cellular behavior. This chapter describes how these spatial features can be included in mechanistic mathematical models of cell signaling. The Virtual Cell computational modeling and simulation software is used to illustrate the considerations required to build a spatial model. An explanation of how to appropriately choose between physical formulations that implicitly or explicitly account for cell geometry and between deterministic versus stochastic formulations for molecular dynamics is provided, along with a discussion of their respective strengths and weaknesses. As a first step toward constructing a spatial model, the geometry needs to be specified and associated with the molecules, reactions, and membrane flux processes of the network. Initial conditions, diffusion coefficients, velocities, and boundary conditions complete the specifications required to define the mathematics of the model. The numerical methods used to solve reaction-diffusion problems both deterministically and stochastically are then described and some guidance is provided in how to set up and run simulations. A study of cAMP signaling in neurons ends the chapter, providing an example of the insights that can be gained in interpreting experimental results through the application of spatial modeling.

Symmetry breaking and the establishment of cell polarity in budding yeast

Cell polarity is typically oriented by external cues such as cell-cell contacts, chemoattractants, or morphogen gradients. In the absence of such cues, however, many cells can spontaneously polarize in a random direction, suggesting the existence of an internal polarity-generating mechanism whose direction can be spatially biased by external cues. Spontaneous 'symmetry-breaking' polarization is likely to involve an autocatalytic process set off by small random fluctuations. Here we review recent work on the nature of the autocatalytic process in budding yeast and on the question of why polarized cells only develop a single 'front'.

A density-dependent switch drives stochastic clustering and polarization of signaling molecules

Positive feedback plays a key role in the ability of signaling molecules to form highly localized clusters in the membrane or cytosol of cells. Such clustering can occur in the absence of localizing mechanisms such as pre-existing spatial cues, diffusional barriers, or molecular cross-linking. What prevents positive feedback from amplifying inevitable biological noise when an un-clustered "off" state is desired? And, what limits the spread of clusters when an "on" state is desired? Here, we show that a minimal positive feedback circuit provides the general principle for both suppressing and amplifying noise: below a critical density of signaling molecules, clustering switches off; above this threshold, highly localized clusters are recurrently generated. Clustering occurs only in the stochastic regime, suggesting that finite sizes of molecular populations cannot be ignored in signal transduction networks. The emergence of a dominant cluster for finite numbers of molecules is partly a phenomenon of random sampling, analogous to the fixation or loss of neutral mutations in finite populations. We refer to our model as the "neutral drift polarity model." Regulating the density of signaling molecules provides a simple mechanism for a positive feedback circuit to robustly switch between clustered and un-clustered states. The intrinsic ability of positive feedback both to create and suppress clustering is a general mechanism that could operate within diverse biological networks to create dynamic spatial organization.

Mathematical analysis of steady-state solutions in compartment and continuum models of cell polarization

Cell polarization, in which substances previously uniformly distributed become asymmetric due to external or/and internal stimulation, is a fundamental process underlying cell mobility, cell division, and other polarized functions. The yeast cell S. cerevisiae has been a model system to study cell polarization. During mating, yeast cells sense shallow external spatial gradients and respond by creating steeper internal gradients of protein aligned with the external cue. The complex spatial dynamics during yeast mating polarization consists of positive feedback, degradation, global negative feedback control, and cooperative effects in protein synthesis. Understanding such complex regulations and interactions is critical to studying many important characteristics in cell polarization including signal amplification, tracking dynamic signals, and potential trade-off between achieving both objectives in a robust fashion. In this paper, we study some of these questions by analyzing several models with different spatial complexity: two compartments, three compartments, and continuum in space. The step-wise approach allows detailed characterization of properties of the steady state of the system, providing more insights for biological regulations during cell polarization. For cases without membrane diffusion, our study reveals that increasing the number of spatial compartments results in an increase in the number of steady-state solutions, in particular, the number of stable steady-state solutions, with the continuum models possessing infinitely many steady-state solutions. Through both analysis and simulations, we find that stronger positive feedback, reduced diffusion, and a shallower ligand gradient all result in more steady-state solutions, although most of these are not optimally aligned with the gradient. We explore in the different settings the relationship between the number of steady-state solutions and the extent and accuracy of the polarization. Taken together these results furnish a detailed description of the factors that influence the tradeoff between a single correctly aligned but poorly polarized stable steady-state solution versus multiple more highly polarized stable steady-state solutions that may be incorrectly aligned with the external gradient.

Neuronal (bi)polarity as a self-organized process enhanced by growing membrane

Early in vitro and recent in vivo studies demonstrated that neuronal polarization occurs by the sequential formation of two oppositely located neurites. This early bipolar phenotype is of crucial relevance in brain organization, determining neuronal migration and brain layering. It is currently considered that the place of formation of the first neurite is dictated by extrinsic cues, through the induction of localized changes in membrane and cytoskeleton dynamics leading to deformation of the cells' curvature followed by the growth of a cylindrical extension (neurite). It is unknown if the appearance of the second neurite at the opposite pole, thus the formation of a bipolar cell axis and capacity to undergo migration, is defined by the growth at the first place, therefore intrinsic, or requires external determinants. We addressed this question by using a mathematical model based on the induction of dynamic changes in one pole of a round cell. The model anticipates that a second area of growth can spontaneously form at the opposite pole. Hence, through mathematical modeling we prove that neuronal bipolar axis of growth can be due to an intrinsic mechanism.

Polarized growth in fungi: symmetry breaking and hyphal formation

Cell shape is a critical determinant for function. The baker's yeast Saccharomyces cerevisiae changes shape in response to its environment, growing by budding in rich nutrients, forming invasive pseudohyphal filaments in nutrient poor conditions and pear shaped shmoos for growth towards a partner during mating. The human opportunistic pathogen Candida albicans can switch from budding to hyphal growth, in response to numerous environmental stimuli to colonize and invade its host. Hyphal growth, typical of filamentous fungi, is not observed in S. cerevisiae. A number of internal cues regulate when and where yeast cells break symmetry leading to polarized growth and ultimately distinct cell shapes. This review discusses how cells break symmetry using the yeast S. cerevisiae paradigm and how polarized growth is initiated and maintained to result in dramatic morphological changes during C. albicans hyphal growth.

Limiting amounts of centrosome material set centrosome size in C. elegans embryos

BACKGROUND

The ways in which cells set the size of intracellular structures is an important but largely unsolved problem [1]. Early embryonic divisions pose special problems in this regard. Many checkpoints common in somatic cells are missing from these divisions, which are characterized by rapid reductions in cell size and short cell cycles [2]. Embryonic cells must therefore possess simple and robust mechanisms that allow the size of many of their intracellular structures to rapidly scale with cell size.

RESULTS

Here, we study the mechanism by which one structure, the centrosome, scales in size during the early embryonic divisions of C. elegans. We show that centrosome size is directly related to cell size and is independent of lineage. Two findings suggest that the total amount of maternally supplied centrosome proteins could limit centrosome size. First, the combined volume of all centrosomes formed at any one time in the developing embryo is constant. Second, the total volume of centrosomes in any one cell is independent of centrosome number. By increasing the amount of centrosome proteins in the cell, we provide evidence that one component that limits centrosome size is the conserved pericentriolar material protein SPD-2 [3], which we show binds to and targets polo-like kinase 1 [3, 4] to centrosomes.

CONCLUSIONS

We propose a limiting component hypothesis, in which the volume of the cell sets centrosome size by limiting the total amount of centrosome components. This idea could be a general mechanism for setting the size of intracellular organelles during development.

A common mechanism for protein cluster formation

Polarized states on the membranes are characterized by focal accumulation of proteins and lipids at local concentrations far exceeding their levels typically found outside of these dense clusters. Principles of thermodynamics argue that formation and maintenance of such structures require continuous expenditure of cellular energy to combat the effect of molecular diffusion that relentlessly dissipates the clusters in favor of the spatially homogeneous state. Small GTPases are known to play a crucial role in the formation of several such polarized states. Their ability to consume stored energy and convert it into a potentially useful work by cyclically hydrolyzing GTP and coupling to various effectors in a nucleotide-dependent way, makes them eligible candidates to fulfill the requirements for the molecules involved in the mechanisms responsible for the maintenance of polarized states. Consistently, continuous nucleotide cycling of small GTPases has been found required for the emergence of structures in several well characterized cases. Despite this general awareness, the detailed molecular mechanisms remain largely unknown. In a recent study, not directly involving small GTPases, we proposed a mechanism explaining the emergence and maintenance of the stable cell-polarity landmark that manifests itself as a protein cluster positioned on the plasma membrane at the growing ends of fission yeast cells. Unexpectedly, this study has suggested a number of striking parallels with the mechanisms based on the activity of small GTPases. These findings highlight common design principles of cellular pattern-forming mechanisms that have been mixed and matched in various combinations in the course of evolution to achieve the same desired outcome-tightly controlled in space and time formation of dense protein clusters.

Cla4 kinase triggers destruction of the Rac1-GEF Cdc24 during polarized growth in Ustilago maydis

In the dimorphic fungus Ustilago maydis, Rac1 and its activator Cdc24 are essential for hyphal tip growth. Rac1 is shown to stimulate Cla4 kinase, which in turn triggers destruction of Cdc24. Expression of stabilized Cdc24 interferes with cell polarization, indicating that negative feedback regulation of Cdc24 is critical for tip growth.

Dimorphic switching from budding to filamentous growth is a characteristic feature of many pathogenic fungi. In the fungal model organism Ustilago maydis polarized growth is induced by the multiallelic b mating type locus and requires the Rho family GTPase Rac1. Here we show that mating type–induced polarized growth involves negative feedback regulation of the Rac1-specific guanine nucleotide exchange factor (GEF) Cdc24. Although Cdc24 is essential for polarized growth, its concentration is drastically diminished during filament formation. Cdc24 is part of a protein complex that also contains the scaffold protein Bem1 and the PAK kinase Cla4. Activation of Rac1 results in Cla4-dependent degradation of the Rac1-GEF Cdc24, thus creating a regulatory negative feedback loop. We generated mutants of Cdc24 that are resistant to Cla4-dependent destruction. Expression of stable Cdc24 variants interfered with filament formation, indicating that negative feedback regulation of Cdc24 is critical for the establishment of polarized growth.

Modeling vesicle traffic reveals unexpected consequences for Cdc42p-mediated polarity establishment

BACKGROUND

Polarization in yeast has been proposed to involve a positive feedback loop whereby the polarity regulator Cdc42p orients actin cables, which deliver vesicles carrying Cdc42p to the polarization site. Previous mathematical models treating Cdc42p traffic as a membrane-free flux suggested that directed traffic would polarize Cdc42p, but it remained unclear whether Cdc42p would become polarized without the membrane-free simplifying assumption.

RESULTS

We present mathematical models that explicitly consider stochastic vesicle traffic via exocytosis and endocytosis, providing several new insights. Our findings suggest that endocytic cargo influences the timing of vesicle internalization in yeast. Moreover, our models provide quantitative support for the view that integral membrane cargo proteins would become polarized by directed vesicle traffic given the experimentally determined rates of vesicle traffic and diffusion. However, such traffic cannot effectively polarize the more rapidly diffusing Cdc42p in the model without making additional assumptions that seem implausible and lack experimental support.

CONCLUSIONS

Our findings suggest that actin-directed vesicle traffic would perturb, rather than reinforce, polarization in yeast.

Lateral dynamics of proteins with polybasic domain on anionic membranes: a dynamic Monte-Carlo study

Positively charged polybasic domains are essential for recruiting multiple signaling proteins, such as Ras GTPases and Src kinase, to the negatively charged cellular membranes. Much less, however, is known about the influence of electrostatic interactions on the lateral dynamics of these proteins. We developed a dynamic Monte-Carlo automaton that faithfully simulates lateral diffusion of the adsorbed positively charged oligopeptides as well as the dynamics of mono- (phosphatidylserine) and polyvalent (PIP(2)) anionic lipids within the bilayer. In agreement with earlier results, our simulations reveal lipid demixing that leads to the formation of a lipid shell associated with the peptide. The computed association times and average numbers of bound lipids demonstrate that tetravalent PIP(2) interacts with the peptide much more strongly than monovalent lipid. On the spatially homogeneous membrane, the lipid shell affects the behavior of the peptide only by weakly reducing its lateral mobility. However, spatially heterogeneous distributions of monovalent lipids are found to produce peptide drift, the velocity of which is determined by the total charge of the peptide-lipid complex. We hypothesize that this predicted phenomenon may affect the spatial distribution of proteins with polybasic domains in the context of cell-signaling events that alter the local density of monovalent anionic lipids.

PAR proteins diffuse freely across the anterior-posterior boundary in polarized C. elegans embryos

FRAP reveals that a stable PAR boundary requires balancing diffusive flux of PAR proteins between domains with spatial differences in PAR protein membrane affinities.

Polarization of cells by PAR proteins requires the segregation of antagonistic sets of proteins into two mutually exclusive membrane-associated domains. Understanding how nanometer scale interactions between individual PAR proteins allow spatial organization across cellular length scales requires determining the kinetic properties of PAR proteins and how they are modified in space. We find that PAR-2 and PAR-6, which localize to opposing PAR domains, undergo exchange between well mixed cytoplasmic populations and laterally diffusing membrane-associated states. Domain maintenance does not involve diffusion barriers, lateral sorting, or active transport. Rather, both PAR proteins are free to diffuse between domains, giving rise to a continuous boundary flux because of lateral diffusion of molecules down the concentration gradients that exist across the embryo. Our results suggest that the equalizing effects of lateral diffusion are countered by actin-independent differences in the effective membrane affinities of PAR proteins between the two domains, which likely depend on the ability of each PAR species to locally modulate the membrane affinity of opposing PAR species within its domain. We propose that the stably polarized embryo reflects a dynamic steady state in which molecules undergo continuous diffusion between regions of net association and dissociation.

Functional analysis of RhoGDI inhibitory activity on vacuole membrane fusion

RhoGDIs (Rho GDP-dissociation inhibitors) are the natural inhibitors of Rho GTPases. They interfere with Rho protein function by either blocking upstream activation or association with downstream signalling molecules. RhoGDIs can also extract membrane-bound Rho GTPases to form soluble cytosolic complexes. We have shown previously that purified yeast RhoGDI Rdi1p, can inhibit vacuole membrane fusion in vitro. In the present paper we functionally dissect Rdi1p to discover its mode of regulating membrane fusion. Overexpression of Rdi1p in vivo profoundly affected cell morphology including increased actin patches in mother cells indicative of polarity defects, delayed ALP (alkaline phosphatase) sorting and the presence of highly fragmented vacuoles indicative of membrane fusion defects. These defects were not caused by the loss of typical transport and fusion proteins, but rather were linked to the reduction of membrane localization and activation of Cdc42p and Rho1p. Subcellular fractionation showed that Rdi1p is predominantly a cytosolic monomer, free of bound Rho GTPases. Overexpression of endogenous Rdi1p, or the addition of exogenous Rdi1p, generated stable cytosolic complexes. Rdi1p structure-function analysis showed that membrane association via the C-terminal β-sheet domain was required for the functional inhibition of membrane fusion. Furthermore, Rdi1p inhibited membrane fusion through the binding of Rho GTPases independent from its extraction activity.

A comparison of mathematical models for polarization of single eukaryotic cells in response to guided cues

Polarization, a primary step in the response of an individual eukaryotic cell to a spatial stimulus, has attracted numerous theoretical treatments complementing experimental studies in a variety of cell types. While the phenomenon itself is universal, details differ across cell types, and across classes of models that have been proposed. Most models address how symmetry breaking leads to polarization, some in abstract settings, others based on specific biochemistry. Here, we compare polarization in response to a stimulus (e.g., a chemoattractant) in cells typically used in experiments (yeast, amoebae, leukocytes, keratocytes, fibroblasts, and neurons), and, in parallel, responses of several prototypical models to typical stimulation protocols. We find that the diversity of cell behaviors is reflected by a diversity of models, and that some, but not all models, can account for amplification of stimulus, maintenance of polarity, adaptation, sensitivity to new signals, and robustness.

Models at the single cell level

Many cellular behaviors cannot be completely captured or appropriately described at the cell population level. Noise induced by stochastic chemical reactions, spatially polarized signaling networks, and heterogeneous cell-cell communication are among the many phenomena that require fine-grained analysis. Accordingly, the mathematical models used to describe such systems must be capable of single cell or subcellular resolution. Here, we review techniques for modeling single cells, including models of stochastic chemical kinetics, spatially heterogeneous intracellular signaling, and spatial stochastic systems. We also briefly discuss applications of each type of model.

ASYMPTOTIC AND BIFURCATION ANALYSIS OF WAVE-PINNING IN A REACTION-DIFFUSION MODEL FOR CELL POLARIZATION

We describe and analyze a bistable reaction-diffusion (RD) model for two interconverting chemical species that exhibits a phenomenon of wave-pinning: a wave of activation of one of the species is initiated at one end of the domain, moves into the domain, decelerates, and eventually stops inside the domain, forming a stationary front. The second ("inactive") species is depleted in this process. This behavior arises in a model for chemical polarization of a cell by Rho GTPases in response to stimulation. The initially spatially homogeneous concentration profile (representative of a resting cell) develops into an asymmetric stationary front profile (typical of a polarized cell). Wave-pinning here is based on three properties: (1) mass conservation in a finite domain, (2) nonlinear reaction kinetics allowing for multiple stable steady states, and (3) a sufficiently large difference in diffusion of the two species. Using matched asymptotic analysis, we explain the mathematical basis of wave-pinning, and predict the speed and pinned position of the wave. An analysis of the bifurcation of the pinned front solution reveals how the wave-pinning regime depends on parameters such as rates of diffusion and total mass of the species. We describe two ways in which the pinned solution can be lost depending on the details of the reaction kinetics: a saddle-node or a pitchfork bifurcation.

From Physics to Pharmacology?

Over the last fifty years there has been an explosion of biological data, leading to the realization that to fully explain biological mechanisms it is necessary to interpret them as complex dynamical systems. The first stage of this interpretation is to determine which components (proteins, genes or metabolites) of the system interact. This is usually represented by a graph, or network. The behavior of this network can then be investigated using mathematical modeling. In vivo these biological networks show several remarkable (and seemingly paradoxical) properties including robustness, plasticity and sensitivity. Erroneous behavior of these networks is often associated with disease. Hence understanding the system-level properties can have important implications for the treatment of disease. Systems biology is an organized approach to quantitatively describe and elucidate the behavior of these complex networks. This review focuses on the progress and future challenges of a systems approach to biology.

A conceptual molecular network for chemotactic behaviors characterized by feedback of molecules cycling between the membrane and the cytosol

Cell chemotaxis has been characterized as the formation of a front-back axis that is triggered by a gradient of chemoattractant; however, chemotaxis is accompanied by more complicated behaviors. These include migration in a straight line with a stable axis [the stable single-axis (SSA) pattern] and repeated splitting of the leading edge of the cell into two regions, followed by the "choice" of one of these as the new leading edge [the split and choice (S&C) pattern]. Indeed, transition between these two behaviors can be observed in individual cells. However, the conceptual framework of the network of signaling molecules that generates these patterns remains to be clarified. We confirmed theoretically that a system that has positive and negative feedback loops involving the reciprocal cycling between the membrane and the cytosol of molecules that promote membrane protrusion or retraction generates SSA and S&C patterns of migratory behavior under similar conditions. We also predicted properties of the instabilities of such a system, which are essential for the generation of these behaviors, and we verified their existence in chemotaxing cells. Our research provides a simple model of network structure for chemotactic behaviors, including cell polarization.

A catalytic role for Mod5 in the formation of the Tea1 cell polarity landmark

Many systems regulating cell polarity involve stable landmarks defined by internal cues [1–5]. In the rod-shaped fission yeast Schizosaccharomyces pombe, microtubules regulate polarized vegetative growth via a landmark involving the protein Tea1 [6–9]. Tea1 is delivered to cell tips as packets of molecules associated with growing microtubule ends [10] and anchored at the plasma membrane via a mechanism involving interaction with the membrane protein Mod5 [11, 12]. Tea1 and Mod5 are highly concentrated in clusters at cell tips in a mutually dependent manner, but how the Tea1-Mod5 interaction contributes mechanistically to generating a stable landmark is not understood. Here, we use live-cell imaging, FRAP, and computational modeling to dissect dynamics of the Tea1-Mod5 interaction. Surprisingly, we find that Tea1 and Mod5 exhibit distinctly different turnover rates at cell tips. Our data and modeling suggest that rather than acting simply as a Tea1 receptor or as a molecular “glue” to retain Tea1, Mod5 functions catalytically to stimulate incorporation of Tea1 into a stable tip-associated cluster network. The model also suggests an emergent self-focusing property of the Tea1-Mod5 cluster network, which can increase the fidelity of polarized growth.

Signalling ballet in space and time

Although we have amassed extensive catalogues of signalling network components, our understanding of the spatiotemporal control of emergent network structures has lagged behind. Dynamic behaviour is starting to be explored throughout the genome, but analysis of spatial behaviours is still confined to individual proteins. The challenge is to reveal how cells integrate temporal and spatial information to determine specific biological functions. Key findings are the discovery of molecular signalling machines such as Ras nanoclusters, spatial activity gradients and flexible network circuitries that involve transcriptional feedback. They reveal design principles of spatiotemporal organization that are crucial for network function and cell fate decisions.

Developing stochastic models for spatial inference: bacterial chemotaxis

BACKGROUND

Biological systems are inherently inhomogeneous and spatial effects play a significant role in processes such as pattern formation. At the cellular level proteins are often localised either through static attachment or via a dynamic equilibrium. As well as spatial heterogeneity many cellular processes exhibit stochastic fluctuations and so to make inferences about the location of molecules there is a need for spatial stochastic models. A test case for spatial models has been bacterial chemotaxis which has been studied extensively as a model of signal transduction.

RESULTS

By creating specific models of a cellular system that incorporate the spatial distributions of molecules we have shown how the fit between simulated and experimental data can be used to make inferences about localisation, in the case of bacterial chemotaxis. This method allows the robust comparison of different spatial models through alternative model parameterisations.

CONCLUSIONS

By using detailed statistical analysis we can reliably infer the parameters for the spatial models, and also to evaluate alternative models. The statistical methods employed in this case are particularly powerful as they reduce the need for a large number of simulation replicates. The technique is also particularly useful when only limited molecular level data is available or where molecular data is not quantitative.

Cytoskeletal dynamics in fission yeast: a review of models for polarization and division

We review modeling studies concerning cytoskeletal activity of fission yeast. Recent models vary in length and time scales, describing a range of phenomena from cellular morphogenesis to polymer assembly. The components of cytoskeleton act in concert to mediate cell-scale events and interactions such as polarization. The mathematical models reduce these events and interactions to their essential ingredients, describing the cytoskeleton by its bulk properties. On a smaller scale, models describe cytoskeletal subcomponents and how bulk properties emerge.

A mechanism for the polarity formation of chemoreceptors at the growth cone membrane for gradient amplification during directional sensing

Accurate response to external directional signals is essential for many physiological functions such as chemotaxis or axonal guidance. It relies on the detection and amplification of gradients of chemical cues, which, in eukaryotic cells, involves the asymmetric relocalization of signaling molecules. How molecular events coordinate to induce a polarity at the cell level remains however poorly understood, particularly for nerve chemotaxis. Here, we propose a model, inspired by single-molecule experiments, for the membrane dynamics of GABA chemoreceptors in nerve growth cones (GCs) during directional sensing. In our model, transient interactions between the receptors and the microtubules, coupled to GABA-induced signaling, provide a positive-feedback loop that leads to redistribution of the receptors towards the gradient source. Using numerical simulations with parameters derived from experiments, we find that the kinetics of polarization and the steady-state polarized distribution of GABA receptors are in remarkable agreement with experimental observations. Furthermore, we make predictions on the properties of the GC seen as a sensing, amplification and filtering module. In particular, the growth cone acts as a low-pass filter with a time constant ∼10 minutes determined by the Brownian diffusion of chemoreceptors in the membrane. This filtering makes the gradient amplification resistent to rapid fluctuations of the external signals, a beneficial feature to enhance the accuracy of neuronal wiring. Since the model is based on minimal assumptions on the receptor/cytoskeleton interactions, its validity extends to polarity formation beyond the case of GABA gradient sensing. Altogether, it constitutes an original positive-feedback mechanism by which cells can dynamically adapt their internal organization to external signals.

Spatial aspects of intracellular information processing

The computational properties of intracellular biochemical networks, for which the cell is assumed to be a 'well-mixed' reactor, have already been widely characterized. What has so far not received systematic treatment is the important role of space in many intracellular computations. Spatial network computations can be divided into two broad categories: those required for essential spatial processes (e.g. polarization, chemotaxis, division, and development) and those for which space is simply used as an extra dimension to expand the computational power of the network. Several pertinent recent examples of each category are discussed that illustrate the often conceptually subtle role of space in the processing of intracellular information.

SIMULATING BIOCHEMICAL SIGNALING NETWORKS IN COMPLEX MOVING GEOMETRIES

Signaling networks regulate cellular responses to environmental stimuli through cascades of protein interactions. External signals can trigger cells to polarize and move in a specific direction. During migration, spatially localized activity of proteins is maintained. To investigate the effects of morphological changes on intracellular signaling, we developed a numerical scheme consisting of a cut cell finite volume spatial discretization coupled with level set methods to simulate the resulting advection-reaction-diffusion system. We then apply the method to several biochemical reaction networks in changing geometries. We found that a Turing instability can develop exclusively by cell deformations that maintain constant area. For a Turing system with a geometry-dependent single or double peak solution, simulations in a dynamically changing geometry suggest that a single peak solution is the only stable one, independent of the oscillation frequency. The method is also applied to a model of a signaling network in a migrating fibroblast.

Use of virtual cell in studies of cellular dynamics

The Virtual Cell (VCell) is a unique computational environment for modeling and simulation of cell biology. It has been specifically designed to be a tool for a wide range of scientists, from experimental cell biologists to theoretical biophysicists. The models created with VCell can range from the simple, to evaluate hypotheses or to interpret experimental data, to complex multilayered models used to probe the predicted behavior of spatially resolved, highly nonlinear systems. In this chapter, we discuss modeling capabilities of VCell and demonstrate representative examples of the models published by the VCell users.

Design of versatile biochemical switches that respond to amplitude, duration, and spatial cues

Cells often mount ultrasensitive (switch-like) responses to stimuli. The design principles underlying many switches are not known. We computationally studied the switching behavior of GTPases, and found that this first-order kinetic system can show ultrasensitivity. Analytical solutions indicate that ultrasensitive first-order reactions can yield switches that respond to signal amplitude or duration. The three-component GTPase system is analogous to the physical fermion gas. This analogy allows for an analytical understanding of the functional capabilities of first-order ultrasensitive systems. Experiments show amplitude- and time-dependent Rap GTPase switching in response to Cannabinoid-1 receptor signal. This first-order switch arises from relative reaction rates and the concentrations ratios of the activator and deactivator of Rap. First-order ultrasensitivity is applicable to many systems where threshold for transition between states is dependent on the duration, amplitude, or location of a distal signal. We conclude that the emergence of ultrasensitivity from coupled first-order reactions provides a versatile mechanism for the design of biochemical switches.

A theoretical model for ROP localisation by auxin in Arabidopsis root hair cells

Background

Local activation of Rho GTPases is important for many functions including cell polarity, morphology, movement, and growth. Although a number of molecules affecting Rho-of-Plants small GTPase (ROP) signalling are known, it remains unclear how ROP activity becomes spatially organised. Arabidopsis root hair cells produce patches of ROP at consistent and predictable subcellular locations, where root hair growth subsequently occurs.

Methodology/Principal Findings

We present a mathematical model to show how interaction of the plant hormone auxin with ROPs could spontaneously lead to localised patches of active ROP via a Turing or Turing-like mechanism. Our results suggest that correct positioning of the ROP patch depends on the cell length, low diffusion of active ROP, a gradient in auxin concentration, and ROP levels. Our theory provides a unique explanation linking the molecular biology to the root hair phenotypes of multiple mutants and transgenic lines, including OX-ROP, CA-rop, aux1, axr3, tip1, eto1, etr1, and the triple mutant aux1 ein2 gnom^eb.

Conclusions/Significance

We show how interactions between Rho GTPases (in this case ROPs) and regulatory molecules (in this case auxin) could produce characteristic subcellular patterning that subsequently affects cell shape. This has important implications for research on the morphogenesis of plants and other eukaryotes. Our results also illustrate how gradient-regulated Turing systems provide a particularly robust and flexible mechanism for pattern formation.

Singularity in polarization: rewiring yeast cells to make two buds

For budding yeast to ensure formation of only one bud, cells must polarize toward one, and only one, site. Polarity establishment involves the Rho family GTPase Cdc42, which concentrates at polarization sites via a positive feedback loop. To assess whether singularity is linked to the specific Cdc42 feedback loop, we disabled the yeast cell's endogenous amplification mechanism and synthetically rewired the cells to employ a different positive feedback loop. Rewired cells violated singularity, occasionally making two buds. Even cells that made only one bud sometimes initiated two clusters of Cdc42, but then one cluster became dominant. Mathematical modeling indicated that, given sufficient time, competition between clusters would promote singularity. In rewired cells, competition occurred slowly and sometimes failed to develop a single "winning" cluster before budding. Slowing competition in normal cells also allowed occasional formation of two buds, suggesting that singularity is enforced by rapid competition between Cdc42 clusters.

Symmetry breaking in the life cycle of the budding yeast

The budding yeast Saccharomyces cerevisiae has been an invaluable model system for the study of the establishment of cellular asymmetry and growth polarity in response to specific physiological cues. A large body of experimental observations has shown that yeast cells are able to break symmetry and establish polarity through two coupled and partially redundant intrinsic mechanisms, even in the absence of any pre-existing external asymmetry. One of these mechanisms is dependent upon interplay between the actin cytoskeleton and the Rho family GTPase Cdc42, whereas the other relies on a Cdc42 GTPase signaling network. Integral to these mechanisms appear to be positive feedback loops capable of amplifying small and stochastic asymmetries. Spatial cues, such as bud scars and pheromone gradients, orient cell polarity by modulating the regulation of the Cdc42 GTPase cycle, thereby biasing the site of asymmetry amplification.

Symmetry breaking: scaffold plays matchmaker for polarity signaling proteins

Many cell types can spontaneously polarize even in the absence of specific positional cues. In budding yeast, this symmetry-breaking polarization depends on a scaffold protein called Bem1p. A recent study defines Bem1p's molecular function during symmetry breaking.

Calling heads from tails: the role of mathematical modeling in understanding cell polarization

Theorists have long speculated on the mechanisms driving directed and spontaneous cell polarization. Recently, experimentalists have uncovered many of the mechanisms underlying polarization, enabling these models to be directly tested. In the process, they have demonstrated the explanatory and predictive value of these models and, at the same time, uncovered additional complexities not currently explained by them. In this review, we discuss some of main theories regarding cell polarization and highlight how the intersection of mathematical and experimental biology has yielded new insights into these mechanisms in the case of budding yeast and eukaryotic chemotaxis.

Symmetry-breaking polarization driven by a Cdc42p GEF-PAK complex

BACKGROUND

In 1952, Alan Turing suggested that spatial patterns could arise from homogeneous starting conditions by feedback amplification of stochastic fluctuations. One example of such self-organization, called symmetry breaking, involves spontaneous cell polarization in the absence of spatial cues. The conserved GTPase Cdc42p is essential for both guided and spontaneous polarization, and in budding yeast cells Cdc42p concentrates at a single site (the presumptive bud site) at the cortex. Cdc42p concentrates at a random cortical site during symmetry breaking in a manner that requires the scaffold protein Bem1p. The mechanism whereby Bem1p promotes this polarization was unknown.

RESULTS

Here we show that Bem1p promotes symmetry breaking by assembling a complex in which both a Cdc42p-directed guanine nucleotide exchange factor (GEF) and a Cdc42p effector p21-activated kinase (PAK) associate with Bem1p. Analysis of Bem1p mutants indicates that both GEF and PAK must bind to the same molecule of Bem1p, and a protein fusion linking the yeast GEF and PAK bypasses the need for Bem1p. Although mammalian cells lack a Bem1p ortholog, they contain more complex multidomain GEFs that in some cases can directly interact with PAKs, and we show that yeast containing an artificial GEF with similar architecture can break symmetry even without Bem1p.

CONCLUSIONS

Yeast symmetry-breaking polarization involves a GEF-PAK complex that binds GTP-Cdc42p via the PAK and promotes local Cdc42p GTP-loading via the GEF. By generating fresh GTP-Cdc42p near pre-existing GTP-Cdc42p, the complex amplifies clusters of GTP-Cdc42p at the cortex. Our findings provide mechanistic insight into an evolutionarily conserved pattern-forming positive-feedback pathway.