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.

Describing the movement of molecules in reduced-dimension models.

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

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.

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.

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.

UNC-6 (netrin) stabilizes oscillatory clustering of the UNC-40 (DCC) receptor to orient polarity.

The receptor deleted in colorectal cancer (DCC) directs dynamic polarizing activities in animals toward its extracellular ligand netrin. How DCC polarizes toward netrin is poorly understood. By performing live-cell imaging of the DCC orthologue UNC-40 during anchor cell invasion in Caenorhabditis elegans, we have found that UNC-40 clusters, recruits F-actin effectors, and generates F-actin in the absence of UNC-6 (netrin). Time-lapse analyses revealed that UNC-40 clusters assemble, disassemble, and reform at periodic intervals in different regions of the cell membrane. This oscillatory behavior indicates that UNC-40 clusters through a mechanism involving interlinked positive (formation) and negative (disassembly) feedback. We show that endogenous UNC-6 and ectopically provided UNC-6 orient and stabilize UNC-40 clustering. Furthermore, the UNC-40-binding protein MADD-2 (a TRIM family protein) promotes ligand-independent clustering and robust UNC-40 polarization toward UNC-6. Together, our data suggest that UNC-6 (netrin) directs polarized responses by stabilizing UNC-40 clustering. We propose that ligand-independent UNC-40 clustering provides a robust and adaptable mechanism to polarize toward netrin.

Interaction between bud-site selection and polarity-establishment machineries in budding yeast.

Saccharomyces cerevisiae yeast cells polarize in order to form a single bud in each cell cycle. Distinct patterns of bud-site selection are observed in haploid and diploid cells. Genetic approaches have identified the molecular machinery responsible for positioning the bud site: during bud formation, specific locations are marked with immobile landmark proteins. In the next cell cycle, landmarks act through the Ras-family GTPase Rsr1 to promote local activation of the conserved Rho-family GTPase, Cdc42. Additional Cdc42 accumulates by positive feedback, creating a concentrated patch of GTP-Cdc42, which polarizes the cytoskeleton to promote bud emergence. Using time-lapse imaging and mathematical modelling, we examined the process of bud-site establishment. Imaging reveals unexpected effects of the bud-site-selection system on the dynamics of polarity establishment, raising new questions about how that system may operate. We found that polarity factors sometimes accumulate at more than one site among the landmark-specified locations, and we suggest that competition between clusters of polarity factors determines the final location of the Cdc42 cluster. Modelling indicated that temporally constant landmark-localized Rsr1 would weaken or block competition, yielding more than one polarity site. Instead, we suggest that polarity factors recruit Rsr1, effectively sequestering it from other locations and thereby terminating landmark activity.

Daughter cell identity emerges from the interplay of Cdc42, septins, and exocytosis.

Asymmetric cell division plays a crucial role in cell differentiation, unequal replicative senescence, and stem cell maintenance. In budding yeast, the identities of mother and daughter cells begin to diverge at bud emergence when distinct plasma-membrane domains are formed and separated by a septin ring. However, the mechanisms underlying this transformation remain unknown. Here, we show that septins recruited to the site of polarization by Cdc42-GTP inhibit Cdc42 activity in a negative feedback loop, and this inhibition depends on Cdc42 GTPase-activating proteins. Combining live-cell imaging and computational modeling, we demonstrate that the septin ring is sculpted by polarized exocytosis, which creates a hole in the accumulating septin density and relieves the inhibition of Cdc42. The nascent ring generates a sharp boundary that confines the Cdc42 activity and exocytosis strictly to its enclosure and thus clearly delineates the daughter cell identity. Our findings define a fundamental mechanism underlying eukaryotic cell fate differentiation.

Tracking shallow chemical gradients by actin-driven wandering of the polarization site.

BACKGROUND: Many cells are remarkably proficient at tracking very shallow chemical gradients, despite considerable noise from stochastic receptor-ligand interactions. Motile cells appear to undergo a biased random walk: spatial noise in receptor activity may determine the instantaneous direction, but because noise is spatially unbiased, it is filtered out by time averaging, resulting in net movement upgradient. How nonmotile cells might filter out noise is unknown. RESULTS: Using yeast chemotropic mating as a model, we demonstrate that a polarized patch of polarity regulators "wanders" along the cortex during gradient tracking. Computational and experimental findings suggest that actin-directed membrane traffic contributes to wandering by diluting local polarity factors. The pheromone gradient appears to bias wandering via interactions between receptor-activated Gßγ and polarity regulators. Artificially blocking patch wandering impairs gradient tracking. CONCLUSIONS: We suggest that the polarity patch undergoes an intracellular biased random walk that enables noise filtering by time averaging, allowing nonmotile cells to track shallow gradients.

Mechanistic mathematical model of polarity in yeast.

The establishment of cell polarity involves positive-feedback mechanisms that concentrate polarity regulators, including the conserved GTPase Cdc42p, at the "front" of the polarized cell. Previous studies in yeast suggested the presence of two parallel positive-feedback loops, one operating as a diffusion-based system, and the other involving actin-directed trafficking of Cdc42p on vesicles. F-actin (and hence directed vesicle traffic) speeds fluorescence recovery of Cdc42p after photobleaching, suggesting that vesicle traffic of Cdc42p contributes to polarization. We present a mathematical modeling framework that combines previously developed mechanistic reaction-diffusion and vesicle-trafficking models. Surprisingly, the combined model recapitulated the observed effect of vesicle traffic on Cdc42p dynamics even when the vesicles did not carry significant amounts of Cdc42p. Vesicle traffic reduced the concentration of Cdc42p at the front, so that fluorescence recovery mediated by Cdc42p flux from the cytoplasm took less time to replenish the bleached pool. Simulations in which Cdc42p was concentrated into vesicles or depleted from vesicles yielded almost identical predictions, because Cdc42p flux from the cytoplasm was dominant. These findings indicate that vesicle-mediated delivery of Cdc42p is not required to explain the observed Cdc42p dynamics, and raise the question of whether such Cdc42p traffic actually contributes to polarity establishment.

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.