The cells of fungi and other eukaryotic organisms are generally asymmetric in shape and structure. This is possible because newly-formed cells can establish front and back ends that influence how each cell grows and develops. A protein called Cdc42 – which can cycle between an active and inactive form – establishes this cell “polarity”. The front of the cell is specified by the accumulation of active Cdc42 proteins in a small area of the membrane that surrounds the cell.

However, it is not clear why the accumulation of active Cdc42 only happens in one place. Budding yeast is widely used as a model to study cell polarity. This yeast reproduces through the budding of new daughter cells out of one end of the mother cell. To achieve this, the mother establishes a single front end through the accumulation of active Cdc42. This marks the site where the daughter cell will emerge.

Previous research has shown that during the early stages of cell polarization, Cdc42 clusters start to form at several places on the inner surface of the membrane. Only a single cluster emerges as the winner, while the others lose and shrink away. How does this happen? One possibility is that Cdc42 clusters all compete with each other to attract polarity proteins from the cell interior, so that when a protein falls off one cluster it can be picked up by another.

Another possibility is that the winning cluster acquires a factor that stabilizes it, while the other clusters do not. Here, Wu et al. use microscopy and mathematical modelling to distinguish between these two possible scenarios in budding yeast. The experiments show that the speed of competition can be manipulated by altering how quickly key polarity proteins move between the membrane and the interior of the cell. Cells with slowed competition can maintain multiple fronts for long enough to make several buds at the same time.

Wu et al.’s findings suggest that the formation of a single front in budding yeast is due to competition for polarity factors. The next challenge will be to understand how some cells in animals and other fungi manage to suppress this competition in order to produce more than one front.

Differentiated cells exhibit a stunning variety of morphologies that enable specialized cell-specific functions. Morphological diversity emerges, in part, from specialization of cortical domains, which are often demarcated by the local accumulation of active GTPases. Among the best-understood cortical specification events is the establishment of cell polarity, wherein local accumulation of a cortical Rho-family GTPase (Cdc42, Rac, or Rop depending on the organism) creates a region destined to become the 'front' (Etienne-Manneville, 2004; Park and Bi, 2007; Yang and Lavagi, 2012). For some cells, restricting polarity to a single front is absolutely imperative: for example, a migrating leukocyte with two fronts would split itself apart (Houk et al., 2012). However, other cells routinely specify more than one front: for example, neurons can grow several neurites simultaneously, each with a front-like tip (Dotti et al., 1988). Similar phenomena occur in plants and fungi, raising the question of how different cell types generate the correct number of fronts (Wu and Lew, 2013). Here we focus on the mechanism whereby budding yeast cells guarantee that they only establish a single polarity site, growing one and only one bud.

Polarity establishment is thought to occur through a cooperative process involving positive feedback, which allows localized fluctuations in concentration to set off growth of a cluster of polarity factors to establish a front (Bi and Park, 2012; Johnson et al., 2011). But if stochastic effects can trigger production of a front, what restricts cells to form only one front? A potential mechanism involves competition between different fronts for a common pool of polarity factors (Goryachev and Pokhilko, 2008; Howell et al., 2009). The strongest experimental support for this competition hypothesis comes from studies of 're-wired' yeast cells that were engineered to use a synthetic polarity factor created from a fusion between two endogenous proteins (Howell et al., 2009). In that system, many cells were observed to initially form two fronts (cortical sites enriched for the synthetic protein). In the majority of cells that developed two fronts, one front then grew stronger while the other concurrently grew weaker and disappeared. When a cell initially developed only one front, that front never shrank or disappeared, suggesting that in the two-front cells, growth of the 'winning' front was responsible for the disappearance of the 'losing' front, as predicted by the competition hypothesis. In a few cells, the two initial polarity sites did eventually grow into buds, indicating that competition is not fully effective in re-wired cells.

Whether competition is responsible for the uniqueness of the front in yeast with a natural (as opposed to synthetic) polarity system is not known. Although we detected initial development of two or more polarity clusters prior to establishment of a single front (Howell et al., 2012; Wu et al., 2013), others did not (Klunder et al., 2013). Moreover, even when a transient multi-cluster intermediate was observed, the process whereby such early intermediates were resolved to a single front remained unclear. Unlike in the strains with a synthetically rewired polarity pathway (Howell et al., 2009), in the natural system early polarity clusters were observed to disappear spontaneously even when there was no other cluster present (Howell et al., 2012). Thus, the disappearance of a cluster could not be unambiguously attributed to the presence of a competing cluster in the same cell.

Why would some polarity clusters spontaneously disappear? This behavior was traced to a negative feedback loop in the yeast polarity circuit (Howell et al., 2012; Kuo et al., 2014). As the combination of positive and negative feedback can yield a pulse generator (Brandman and Meyer, 2008), it could be that stochastic fluctuations routinely trigger growth of a cluster by positive feedback followed by cluster dissolution due to negative feedback. But if that is the case, then why don’t ALL polarity clusters disappear? Why does one and only one cluster remain stable following the initial dynamic behavior? One possibility is that during their brief existence, initial (unstable) polarity clusters have a chance to capture a critical stabilizing factor. Then, once a lucky cluster had captured the stabilizer, all other clusters would be doomed to disappear. Like the competition hypothesis, the stabilizer hypothesis can explain resolution of a multi-cluster intermediate to a final single-front state. Indeed, some models in the field posit that actin cables play roles analogous to the stabilizer, reinforcing polarity clusters and protecting them from dissolution (Freisinger et al., 2013; Wedlich-Soldner et al., 2003).

Yeast actin is organized into two distinct types of structures. Actin cables are bundles of parallel actin filaments nucleated by formins: their primary role is to enable myosin-driven delivery of cargo towards the bud (Pruyne et al., 2004). Actin patches are assemblies of branched actin filaments nucleated by the Arp2/3 complex: their primary role is to promote internalization of endocytic vesicles (Kaksonen et al., 2006). Both actin cable-mediated traffic of secretory vesicles and actin patch-mediated endocytosis have been proposed to stabilize and reinforce polarity clusters (Freisinger et al., 2013; Jose et al., 2013; Marco et al., 2007; Slaughter et al., 2009; Wedlich-Soldner et al., 2004). When yeast cells were treated with Latrunculin to depolymerize actin, polarity clusters were observed to serially assemble and disassemble, sometimes relocating from one site to another, to a much greater degree than seen in untreated cells (Howell et al., 2012; Okada et al., 2013; Wedlich-Soldner et al., 2004; Wu et al., 2013). This observation is consistent with a potential 'stabilizer' role for actin: in cells with two polarity clusters, the first one to capture some actin structure may be stabilized and persist while the other disappears due to negative feedback.

We now report experiments that distinguish between the competition and stabilizer hypotheses. Our findings suggest that uniqueness of the yeast front is due to competition for polarity factors, and not to a downstream stabilizer. We show that the speed of competition can be manipulated by altering the rates at which key polarity factors exchange between membrane and cytoplasm, and that cells with slowed competition can maintain multiple fronts for long enough to make two, three, or even four buds simultaneously. Our findings provide insight into the mechanism of competition, uncovering how yeast cells can guarantee the uniqueness of the front.

In wild-type yeast cells, polarization is biased towards specific sites by a system of inherited bud-site-selection landmarks (Bi and Park, 2012). Localized landmarks influence the site of polarization through the Ras-family GTPase Rsr1, and polarity clusters tend to form near the poles (Wu et al., 2013). Because polarity factors also accumulate at the cytokinesis site (which overlaps one pole), some polarity clusters are difficult to quantify separate from the cytokinesis signal. In the absence of Rsr1, polarity clusters can form over much of the cell surface (Bender and Pringle, 1989; Howell et al., 2012), allowing easier imaging of the resolution from >1 cluster to a single cluster. For that reason, our experiments were carried out in rsr1 mutant strains.

Because GFP-tagged Cdc42 is not fully functional (Freisinger et al., 2013; Howell et al., 2012; Watson et al., 2014), we adapted a strategy recently shown to produce a functional internal mCherry-tagged Cdc42 in S. pombe (Bendezu et al., 2015). Although more functional than GFP-Cdc42 at single copy, this probe was still not fully functional in S. cerevisiae (Figure 1A,B). Thus, when possible we used fluorescently tagged Bem1 as a functional marker for polarity clusters. Bem1 is a scaffold protein that participates in positive feedback (Kozubowski et al., 2008) and accumulates at the same sites as Cdc42 with very similar timing (Howell et al., 2012); when a losing cluster disassembles, Cdc42 and Bem1 disappear in concert (Figure 1C) (Video 1).

Figure 1

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Testing candidate stabilizers

According to the stabilizer hypothesis, the difference between a polarity cluster that persists and a cluster that disappears is that the persistent 'winning' cluster acquires a stabilizer, while the disappearing 'losing' cluster does not. Thus, simultaneous imaging of a polarity marker and the stabilizer should reveal the recruitment of the stabilizer to one but not both clusters (Figure 2A).

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We initially focused on actin cables and actin patches as candidate stabilizers. Actin cables are difficult to visualize directly in live cells (Huckaba et al., 2004), so we used two surrogate markers to report cable nucleation and subsequent vesicle delivery by cables. Spa2 is a regulator of the formin Bni1, which nucleates actin cables (Evangelista et al., 2002; Sagot et al., 2002; Sheu et al., 1998); Spa2 recruitment to the polarity site occurs via both actin-dependent and actin-independent routes (Ayscough et al., 1997). Sec4 is a secretory vesicle-associated Rab-family GTPase, which polarizes as vesicles are delivered on actin cables to the polarity site (Mulholland et al., 1997; Schott et al., 2002; Walch-Solimena et al., 1997). Spa2-mCherry and GFP-Sec4 both became detectable at the polarity site within about 1 min after Bem1 became detectable (Figure 2B). We found that when cells formed two polarity clusters, Spa2 and Sec4 generally accumulated at both sites (Figure 2C) (Video 2). That is, both the 'winner' (W) and the 'loser' (L) recruited vesicles (and presumably actin cables), indicating that actin cable recruitment does not guarantee persistence of the polarity cluster. Hence, actin cables are unlikely to act as the hypothesized stabilizer.

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Actin patches were visualized using the patch marker Abp1 (Drubin et al., 1988; Kaksonen et al., 2003). Actin patches were initially distributed randomly around the cortex (with some concentration at the old cytokinesis site), and then clustered at the polarity site several minutes after Bem1 became detectable (Figure 2D). In most cells that formed two polarity clusters, actin patches remained randomly distributed until one of the clusters had disappeared (Figure 2E) (Video 3). As neither the winner nor the loser accumulated actin patches during the relevant timeframe, actin patches are also unlikely to act as the stabilizer.

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In addition to actin structures, polarity sites acquire a ring of septin filaments, which then grow to form a very stable hourglass structure at the mother-bud neck (McMurray and Thorner, 2009; Oh and Bi, 2011). Thus, we considered the possibility that the septin ring might act as a stabilizer. We visualized septin structures using the functional septin probe Cdc3-mCherry (Caviston et al., 2003). Septins assembled into a ring around the polarity site several minutes after Bem1 became detectable (Figure 2F). In cells that formed two polarity clusters, septins were not readily detectable at either cluster until after one cluster disappeared in most cells (Figure 2G). However, we occasionally detected septins at both clusters before one cluster disappeared (Video 4). Thus, septins also seem unlikely to act as the stabilizer. Indeed, it has been suggested that septins contribute to negative feedback and cluster destabilization by recruiting Cdc42-directed GAPs (Okada et al., 2013).

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These findings do not exclude the possibility that some other stabilizer is recruited only to the winning cluster. However, the experiments discussed below allow us to address this possibility more definitively.

Testing the competition model: reducing polarity protein mobility

If polarity clusters compete with each other for a common pool of polarity factors, then competition would involve transfer of components from the losing cluster to the winning cluster via the cell interior (Figure 3A). In this scenario, the relevant factors must exchange dynamically between the cluster and the cell interior on a timescale that is rapid relative to the time it takes to resolve the multi-cluster intermediate. Indeed, fluorescence recovery after photobleaching (FRAP) experiments indicate that polarity factors exchange in and out of clusters on a 2–4 s timeframe (Freisinger et al., 2013; Slaughter et al., 2009; Wedlich-Soldner et al., 2004), whereas multi-cluster resolution occurs on a 1–2 min timeframe (Howell et al., 2012). If the exchange of relevant polarity factors in and out of the clusters were to be slowed, then resolution of multicluster intermediates should also occur more slowly. To test this prediction, we generated strains in which Cdc42, or its guanine nucleotide exchange factor (GEF) Cdc24, or the scaffold protein Bem1, exchanged more slowly between membrane and cytoplasm.

Figure 3

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Our strategy was based on the expectation that membrane-cytoplasm exchange of a prenyl-anchored protein would be slow compared to that due to the very transient interaction of cytoplasmic proteins with membrane factors. Cdc42 itself is attached to the membrane by a C-terminal polybasic-prenyl motif, but GDP-Cdc42 exchange is rapid due to dedicated factors called Rho guanine nucleotide dissociation inhibitors (Rho-GDIs) (Johnson et al., 2009; Michaelson et al., 2001). We confirmed previous reports (Freisinger et al., 2013; Slaughter et al., 2009) that in the absence of the sole yeast Rho-GDI, Rdi1, exchange of Cdc42 in and out of the polarity cluster was much slower (Figure 3B,C), while levels of total Cdc42 were similar in wild-type and rdi1Δ cells (Figure 3D). Biochemical experiments indicated that Cdc42 was able to exchange between different lipid vesicles in vitro even in the absence of a GDI (Johnson et al., 2009), and there was still a substantial pool of Cdc42 in the cytoplasm of rdi1Δ mutants lacking a GDI, as detected either by fractionation (Tiedje et al., 2008) or fluorescence correlation spectroscopy (Das et al., 2012). Thus, we anticipated that the slowed Cdc42 dynamics were due to slower exchange of Cdc42 between membrane and cytoplasm, and we fused the Cdc42 polybasic-prenyl motif to the C-termini of Cdc24 and Bem1 (hereafter Cdc24-CAAX and Bem1-CAAX: Figure 4A) in order to slow the exchange of these proteins. However, others have argued that in the absence of the GDI, Cdc42 is 'locked on' to cellular membranes, and that the observed exchange of Cdc42 in and out of the polarity site is due to actin-mediated vesicle trafficking (Freisinger et al., 2013; Slaughter et al., 2009). Thus, we first investigated whether Bem1-CAAX would polarize using membrane-cytoplasm exchange or vesicle trafficking.

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In previous work, we fused Bem1 to the exocytic v-SNARE Snc2, a transmembrane protein that becomes polarized by a combination of directed exocytosis, slow diffusion, and efficient endocytosis (Howell et al., 2009; Valdez-Taubas and Pelham, 2003). This fusion protein was able to replace endogenous Bem1, but created a situation in which formin-nucleated actin cables and actin patch-mediated endocytosis became essential for polarization, because the Bem1-Snc2 protein could only traffic on vesicles and not through the cytoplasm (Chen et al., 2012; Howell et al., 2009). Unlike Bem1-Snc2, however, we found that Bem1-CAAX did not require the formin Bni1 (Figure 4B,C) or F-actin (Figure 4D) in order to polarize. The finding that Bem1-CAAX polarizes in these situations implies that its mobility is not dependent on actin-mediated vesicle traffic.

In a parallel approach to the same question, we used the 'anchor away' (Haruki et al., 2008) system to ask whether Bem1-CAAX was 'locked on' to membranes. This system is based on the ability of the drug rapamycin to induce a stable interaction between FKBP (Fpr1 in yeast) and the FKBP-binding domain (FRB) of Tor1. We fused two tandem copies of FKBP to the ribosomal protein Rpl13A, and two tandem copies of FRB to Bem1-CAAX. Upon addition of rapamycin, this should induce binding of Bem1-CAAX to ribosomes. If Bem1-CAAX is able to exchange between membrane and cytoplasm, then rapamycin should trap it in the cytoplasm, resulting in a loss of Bem1-CAAX from the polarity site. On the other hand, if Bem1-CAAX were locked onto membranes, then rapamycin should not affect Bem1-CAAX localization (though some ribosomes might become attached to the membrane). We found that rapamycin led to a rapid loss of detectable Bem1-CAAX from the polarity site in all cells (Figure 4E), providing independent evidence that Bem1-CAAX exchanges between membrane and cytoplasm. In aggregate, these experiments indicate that the polybasic-prenyl motif slows but does not eliminate membrane-cytoplasm exchange, and that it is valid to use rdi1Δ mutants as a way to slow exchange of Cdc42, and Bem1-CAAX and Cdc24-CAAX as a way to slow exchange of Bem1 and Cdc24, between membrane and cytoplasm.

Strains in which Cdc24-CAAX replaced endogenous Cdc24 exhibited very poor viability (Figure 5A). Given recent findings that Cdc24 GEF activity can be inhibited by multisite phosphorylation occurring at the membrane (Kuo et al., 2014), we wondered whether the Cdc24-CAAX might be nonfunctional due to enhanced inhibitory phosphorylation. Indeed, a mostly nonphosphorylatable Cdc24^38A-CAAX was viable (Figure 5A), although the cells were slower-growing and temperature-sensitive (Figure 5B). In contrast, cells in which Bem1-CAAX replaced Bem1 were fully viable and grew well at all temperatures (Figure 5B), so in most subsequent experiments we used Bem1-CAAX.

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Cdc24^38A-CAAX and Bem1-CAAX were expressed at comparable levels to Cdc24 and Bem1, respectively (Figure 5C). Bem1-CAAX displayed stronger plasma membrane association than Bem1 (Figure 5D), and Bem1-CAAX exchange in and out of the polarity site was slower than that of Bem1, as assessed by FLIP or FRAP (Figure 5E,F). Bem1-CAAX clusters grew more slowly than Bem1 clusters, and failed to show the characteristic overshoot before reaching their final intensity (Figure 5G). This finding suggests that membrane-cytoplasm exchange of Bem1 can (when slowed) become rate-limiting for the growth of polarity clusters. These strains display slowed exchange of key polarity factors between the polarity clusters and the cell interior, allowing us to ask how slowing exchange affects competition between polarity clusters.

Slowing the exchange of polarity proteins prolongs competition

To test whether slow exchange of polarity factors would delay competition, we conducted time-lapse imaging of the strains discussed above. When two or more polarity clusters formed in any of the slow-exchange strains, the clusters tended to persist for prolonged periods compared to wild-type cells (Figure 6A–D). Prolonged coexistence could be documented with any of several polarity probes, including GFP-Cdc42, Bem1-GFP, Spa2-mCherry, and PBD-tdTomato (a probe for GTP-Cdc42) (Tong et al., 2007) (Figure 6A–C). Similar phenotypes were observed for a strain in which Cdc42 was mutated so as to reduce interaction with Rdi1 (Lin et al., 2003) (Figure 6D). Quantification revealed a heterogeneous range of coexistence times, with average intervals changing from ~1.5 min in control strains to ~7 min in slow-exchange strains (Figure 6E). The coexistence interval could be subdivided into two phases: an initial 'growth'' phase in which two or more clusters all grew in intensity, and a 'competition' phase in which 'losing' clusters shrank and disappeared. Both the growth and competition intervals were longer in slow-exchange strains than in wild-type controls (Figure 6F). Thus, slowing the exchange of polarity factors extended the time necessary to resolve multi-cluster intermediates, consistent with a model in which clusters compete for shared components.

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Prolonged competition allows formation of more septin rings and buds

The prolonged competition observed in slow-exchange strains allowed us to ask whether late-arriving factors such as septins are recruited to one or more of the competing clusters. In several cases both winning and losing clusters acquired septin rings (Figure 7A) (Video 5). However, the presence of septins did not prevent cluster disassembly, and the septin ring also disappeared when a cluster lost the competition (Figure 7A). Because we never (n>200) observed disassembly of a septin-containing cluster in cells that did not have another cluster present, it would appear that septin disassembly does not occur spontaneously, and therefore that the disappearance of 'losing' clusters is due to the presence of another cluster, consistent with the competition hypothesis.

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In all of the slow-exchange strains, we also encountered cells that formed two buds at the same time (Figure 7B,C) (Video 6). Simultaneous formation of two buds has been documented previously for rdi1Δ mutants, although those investigators had a somewhat different interpretation as to the cause of multibudding (Freisinger et al., 2013) (see Discussion). Buds could be similar (Figure 7B, cell 1; 7C, cell 1) or dissimilar (Figure 7B, cells 2,3; 7C, cell 2) in size, but both buds always emerged at about the same time. This observation indicates that the size difference does not arise because one bud gets a head start; rather, in those cases with different-sized buds competition had proceeded to form unequal clusters at the time of bud emergence, giving one bud a growth advantage. In a few cases, the smaller bud ceased growing (Figure 7B, cells 2,3), suggesting that competition continued even after bud emergence, leaving an abandoned bud. We never (n>200) saw a bud stop growing in cells that had only a single bud, suggesting that abandonment of the bud is due to competition with another bud. These findings indicate that the presence of actin and septin structures is unable to stabilize a cluster against competition, arguing strongly against the stabilizer hypothesis.

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Additive effects of combining slow-exchange genotypes

We combined the slow-exchange genotypes discussed above to investigate the effects of simultaneously slowing the exchange of combinations of Cdc42, Cdc24, and Bem1. We were able to combine rdi1Δ mutants with either Cdc24^38A-CAAX or Bem1-CAAX, but combination of Cdc24^38A-CAAX with Bem1-CAAX proved lethal (Figure 8A). rdi1Δ BEM1-CAAX strains displayed multibudded cells at increased frequency (Figure 8B), as did rdi1Δ CDC24^38A-CAAX strains (though the latter were too sick for accurate quantification). The frequency of multibudded cells in viable strains rose to almost 40% (Figure 8B), and some cells grew three or four buds simultaneously (Figure 8C-E) (Video 7). As discussed above, in a few instances the smallest bud ceased growing, suggesting that competition can continue after bud emergence.

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As DNA replication only generates two copies of the genome, cells making more than one bud are unable to pass on a full genetic complement to each daughter. Imaging slow-exchange strains carrying a fluorescent histone revealed that multibudded cells generated anucleate (Figure 8F, cell 1) or aneuploid (Figure 8F, cells 2 and 3) progeny in which a mother and bud appeared to fight over the daughter nuclei (Video 8). This observation is rather surprising, and the mechanism by which chromosomes attached to a single spindle pole end up on different sides of the neck remains to be elucidated.

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Video 9

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Mechanism of competition in a computational model

A variety of simple computational models based on biochemical aspects of Rho-family GTPase behavior have illustrated how such GTPases might polarize spontaneously (Mori et al., 2008; Otsuji et al., 2007; Semplice et al., 2012). Like earlier Turing-type models (Gierer and Meinhardt, 1972; Turing, 1952), some of these can generate and maintain more than one peak of polarity factors in sufficiently large domains. However, a bottom-up model describing the activities and interactions of the yeast Cdc42, Cdc24, Bem1, and GDI proteins displays competition between polarity clusters for all parameters examined thus far (Goryachev and Pokhilko, 2008; Howell et al., 2012; Howell et al., 2009; Savage et al., 2012). In this model, whose elements have considerable experimental support (Kozubowski et al., 2008), clustering of Cdc42 occurs through a positive feedback loop involving a cytoplasmic complex that contains Bem1 and the GEF Cdc24. Cortical GTP-Cdc42 recruits Bem1-Cdc24 complexes from the cytoplasm, which then load neighboring Cdc42 with GTP, leading to further Bem1-Cdc24 recruitment and Cdc42 activation (Figure 9A). Additional Cdc42 is delivered to the growing cluster from the cytoplasm by the GDI, as well as by other pathways (Johnson et al., 2011). Because of positive feedback, stochastic activation of a small amount of Cdc42 somewhere on the membrane leads to further accumulation of active Cdc42 until depletion of the cytoplasmic pools of polarity proteins stops the process. With suitable parameter choices, the system develops a stable polarized peak of GTP-Cdc42: diffusion, inactivation, and release of Cdc42 into the cytoplasm is counteracted by recruitment of more Cdc42 to the peak from the cytoplasmic GDI-bound pool (Figure 9B). As discussed above, FRAP experiments confirm that apparently stable polarized peaks are indeed maintained by very dynamic recycling of the Cdc42, Bem1, and Cdc24.

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The model can be manipulated into generating two peaks if they are initiated with identical stimuli at diametrically opposite poles of the cell. However, this situation is unstable, as the addition of infinitesimally small noise leads to a stable single-peak steady state (Figure 9C) (Video 9). At either the two-peak (unstable) or one-peak (stable) steady state, Bem1-Cdc24 complexes and Cdc42 recycle between the peak(s) and the cytoplasm. The net transfer of polarity factors from the 'losing' to the 'winning' peak occurs without significant changes in the cytoplasmic concentrations of Cdc42 and Bem1-GEF for most of the competition time course (Figure 9D).

To understand why the two-peak state is unstable, we investigated what happens at the center of each peak when one peak acquires more Cdc42 and Bem1-GEF than the other. We first consider the Bem1-GEF complex. The larger peak has a higher concentration of GTP-Cdc42, which can bind Bem1-GEF from the cytoplasm: this gives it an advantage over the smaller peak in recruiting Bem1-GEF (Figure 9E). To evaluate loss of Bem1-GEF from the peak, we started with an arbitrary amount of Cdc42-Bem1-GEF, and ran simulations to monitor the loss of Bem1-GEF from the membrane over time, for different values of GTP-Cdc42 (Figure 9F). With higher levels of GTP-Cdc42 (i.e. for larger peaks), it takes longer for Bem1-GEF complexes to detach from the membrane, because when a complex detaches from one molecule of GTP-Cdc42 it is more likely to bind to another GTP-Cdc42 rather than release into the cytoplasm. From these data we extracted the half-life for membrane-bound Bem1-GEF (dwell time), which increased linearly with GTP-Cdc42 (Figure 9G). See the Materials and methods for a quasi-steady state approximation demonstrating this effect of GTP-Cdc42 on the Bem1-GEF dwell time.

Now consider the recruitment/removal of Cdc42. To compute the dwell time of Cdc42 as a function of the membrane-bound Bem1-GEF concentration, we used a similar approach as described above for computing the Bem1-GEF loss according to the governing equations (Figure 9H, inset). Delivery of Cdc42 from the cytoplasm by the GDI is unaffected by protein concentrations at the membrane, so a similar amount of Cdc42 will be delivered to the center of each peak from the cytoplasm. However, because the larger peak has more Bem1-GEF, GDP-Cdc42 in a larger peak is converted more rapidly to GTP-Cdc42. Because the GDI only extracts GDP-Cdc42, more GEF activity translates to a reduced loss of Cdc42 to the cytoplasm, and hence a longer dwell time (Figure 9H). See the Materials and methods for a quasi-steady state approximation demonstrating how Cdc42 dwell time is related to GEF activity. In summary, the larger peak has an edge in recruiting Bem1-GEF complexes and in retaining both Cdc42 and Bem1-GEF. Thus, the net flux of both species from the cytoplasm to the center of the peak is greater for larger peaks. This mismatch in recruitment and retention for peaks of different sizes provides a mechanism that promotes competition.

Another mechanism that contributes to competition in the model is based on lateral diffusion of polarity factors in the plane of the membrane. As a peak grows or shrinks, its 'waistline' also grows or shrinks in parallel (Figure 9C). We define the 'waistline' as the circle at which Cdc42 concentration is half-maximal (i.e. circle diameter is the peak width at half-height) (Figure 9I, inset). (The following qualitative argument is not sensitive to the exact definition of the waistline). Monitoring the dissipative flux of Cdc42 due to diffusion across the waistline, we see that a larger peak does not lose as great a proportion of its Cdc42 content as does a smaller peak (Figure 9I). Thus, diffusion provides a more powerful dissipative effect for the smaller peak, favoring the larger peak in a competition scenario (Howell et al., 2009).

If the diffusional flux of Cdc42 out of the peak is plotted on the same graph as the net recruitment rate of Cdc42 from the cytoplasm into the peak (defined as the area within the waistline), then the intersections of the curves represent steady states, where there is no net change in Cdc42 concentration and the peak size remains constant (Figure 9J). From this graph, which is derived from the full simulation of competition in Video 9, it is easy to understand why the two-peak solution is unstable. The steady state with two peaks of equal size corresponds to the middle intersection point on the flux plot (Figure 9J). If the peaks become slightly unequal, then the diffusional loss is greater than Cdc42 recruitment for the smaller peak (left of intersection point), causing this peak to shrink. However, for the larger peak (right of intersection point), the Cdc42 recruitment flux is greater than the diffusional flux, and this peak grows until the system reaches the one-peak steady state.

In summary, a computational model based on the behavior of the core polarity factors displays competitive behavior because a larger peak has advantages both in terms of reducing diffusional losses and improved recruitment and retention of factors from the cytoplasm. Thus, in a cell with unequal polarity clusters, the largest will grow at the expense of the others.

Substrate depletion and negative feedback

As polarity factors are recruited to one or more peaks, the cytoplasmic levels of the polarity factors decline, and it is this substrate depletion from the cytoplasm that ultimately stops clusters from growing further. From Figure 9D, it is apparent that the cytoplasmic levels of polarity factors at the one-peak steady state are slightly lower than those at the two-peak steady state. Thus, once a single peak has been consolidated, the levels of cytoplasmic factors are too low to support a second peak.

Because substrate depletion is what limits growth in the model, each peak at the two-peak steady state has a lower polarity protein content compared to the single peak that emerges from competition. However, in cells this is rarely the case: instead, the winning peak goes on to shed some polarity factors, and in some cases displays oscillations in polarity protein content or even disappears altogether, leading to polarization elsewhere (Howell et al., 2012). This behavior has been traced to a negative feedback loop that operates via inhibitory phosphorylation of the GEF Cdc24 (Kuo et al., 2014), reducing the level of active GEF available for positive feedback.

Mutants in which the GEF is nonphosphorylatable (CDC24^38A) largely short-circuit the major negative feedback mechanism, although a slower negative feedback may also occur via Cdc42-directed GAPs (Kuo et al., 2014; Okada et al., 2013). In CDC24^38A mutants, polarity clusters showed competition on similar timescales as that observed in wild-type cells (Kuo et al., 2014). Moreover, in CDC24^38Ardi1Δ mutants we observed slow competition and formation of two-budded cells (Figure 10A). As predicted by the substrate depletion scenario, cells that made a single bud developed polarity clusters with a higher polarity protein content than those in cells that made two buds (Figure 10B). Thus, competition in cells lacking negative feedback proceeds in a manner consistent with the critical features of the model: insatiable positive feedback combined with substrate depletion. The observation that competition proceeds similarly after eliminating a major negative feedback pathway suggests that negative feedback does not greatly affect the competition process in yeast cells.

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In computational models that incorporate negative feedback as well as positive feedback, simulations indicate that although competition can proceed in much the same way as discussed above, it is possible to specify parameter values in such a way that the two-peak steady state becomes stable (Howell et al., 2012). The basis for this switch in behavior is currently unclear, but likely reflects situations in which the negative feedback loop is sufficiently strong to neutralize the advantage of the larger peak in recruiting polarity factors.

Emergence of polarity clusters from stochastic fluctuations

The model simulations in Figure 9 were initiated at a two-peak steady state. If instead simulations are initiated at the homogeneous steady state by addition of random noise, then several small clusters begin to grow and eventually compete, leaving a single winner (Goryachev and Pokhilko, 2008). However, a recent modeling study challenged the idea that cluster competition is relevant to yeast polarity establishment, concluding instead that only a single peak of Cdc42, Bem1, and Cdc24 would emerge from initial random noise (Klunder et al., 2013). In those simulations, starting noisy distributions of polarity factors merged to form a single very broad but very shallow peak covering an entire hemisphere, which then grew into a single focused peak. The authors used linear stability analysis to demonstrate that only the first mode had a positive growth rate, implying that only a single cluster would grow from the homogeneous steady state.

We sought to understand why the different models predicted different behaviors. Although the models are broadly similar and deal with molecular interactions among the same polarity factors, they differ both in the details of how the protein interactions are modeled (Figure 11A) and in parameter values (Figure 11B). Here we show that the discrepancy stems mainly from how those parameters affect competition versus merging of polarity clusters.

Figure 11

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A significant difference between the two models concerns the protein concentrations (Figure 11B). In one study (Klunder et al., 2013), these were based on estimated molecule numbers per haploid cell as measured by quantitative Western blotting (Ghaemmaghami et al., 2003). However, those numbers were applied to a model sphere with volume 258 fL, whereas haploid yeast cells have an average volume of 44 fL (Klis et al., 2014). We found that if the molecule numbers were adjusted to account for this volume discrepancy, then higher modes also had a positive growth rate in the linear stability analysis (Figure 11C). Thus, with more protein the same model would often yield more than one initial cluster, which would then show competition.

Another difference between the models concerns the estimate of the diffusion constant for membrane-bound species: 0.0025 μm²/s in one study (Goryachev and Pokhilko, 2008) and 0.03 μm²/s in the other (Klunder et al., 2013). We repeated the linear stability analysis using different values for the diffusion constant, and found that with slower diffusion, higher modes now had a positive growth rate (Figure 11D). Combining slow diffusion with higher protein concentrations had a synergistic effect (Figure 11E).

These findings demonstrate that the number of clusters likely to emerge from initial noise depends on parameter values. In particular, when polarity concentrations are very low and diffusion is fast in this model, the small initial clusters will tend to merge together before growing to form a single detectable peak. Slower diffusion, or the more powerful positive feedback that occurs in the model when polarity factors are more abundant, can lead to growth of separate clusters before they have a chance to merge. Resolution of those clusters then occurs by competition in both models.

A prediction of these computational findings is that if polarity factor concentrations were lowered, then multi-cluster intermediates should be less prevalent. To test this prediction, we imaged diploid strains in which one copy of CDC42 or CDC24 was deleted. Western blotting showed that hemizygotes contained half as much Cdc42 or Cdc24 as homozygotes (Figure 11F). We monitored polarity establishment in these strains using a Bem1-GFP probe, whose abundance was similar in all strains (Figure 11F). Whereas we detected more than one initial cluster in about 50% of wild-type cells, multi-cluster intermediates were detected in only 25% of CDC24 hemizygotes and 5% of CDC42 hemizygotes (n>100 cells for each strain) (Figure 11G). No multi-cluster instances were observed in cells doubly hemizygous for both CDC42 and CDC24 (n=73) (Figure 11G). In separate experiments, we detected 30% fewer instances of multicluster intermediates in BEM1-GFP/bem1Δ hemizygotes than in BEM1-GFP/BEM1-GFP homozygotes. Thus, multiple clusters are less frequent in cells that express lower levels of polarity factors.

Most polarized cells generate only one front. Our findings indicate that in yeast, this rule is enforced by a greedy competition between potential polarity sites to accumulate polarity factors.

We detected multiple polarity clusters as an intermediate stage in polarity establishment in approximately 50% of cells under our imaging conditions (see also Howell et al., 2012). As clusters can occur anywhere on the cell surface and resolution to a single cluster is typically rapid, the frequency with which we detect such intermediates will clearly depend on the spatiotemporal resolution at which imaging is conducted. This may explain why another recent study detected many fewer multi-cluster intermediates when imaging only the medial planes of large cells (Klunder et al., 2013). In addition, there may be strain background differences in multi-cluster frequency, as we found that two-fold reductions in polarity factor abundance reduced the incidence of multi-cluster intermediates considerably.

Why would polarity protein abundance be correlated with the incidence of multi-cluster intermediates? When polarity factor concentration is low, small initial clusters grow more slowly, perhaps allowing more time for diffusion-based merging of nearby clusters to form a broad and shallow single peak (Klunder et al., 2013). However, it is unclear whether merging is sufficient to explain the reduced incidence of multi-cluster intermediates. In our slow-exchange mutant strains, clusters often co-existed in close proximity for prolonged periods. This suggests that merging is inefficient, presumably because diffusion is very slow in the yeast plasma membrane (Valdez-Taubas and Pelham, 2003). An alternative hypothesis is that whereas the models display hair-trigger Turing instability, in the cells it may take more than just a tiny asymmetry to set off growth of a cluster. Indeed, inclusion of negative feedback can produce this effect in the model (Howell et al., 2012). If stochastic events need to cross some threshold of local polarity factor concentration in order to trigger growth of a cluster, then the frequency of such stochastic events may be quite sensitive to polarity protein concentration.

Given that cells frequently develop more than one initial cluster of polarity factors, there must be a mechanism to eliminate excess clusters so that only one persists to form the front. We suggest that this mechanism involves competition between polarity clusters for components including Cdc42, Bem1, and Cdc24. Each of these factors exchanges constantly and rapidly (2–5 s half-time) between the polarity cluster and the cell interior (Freisinger et al., 2013; Slaughter et al., 2009; Wedlich-Soldner et al., 2004). Polarity factors released from one cluster may be captured by another, and if larger clusters have an advantage in recruiting and retaining such factors, then they would grow at the expense of smaller clusters. Consistent with that hypothesis, we found that slowing the exchange of Cdc42, Bem1, or Cdc24 in and out of the clusters resulted in correspondingly slower resolution of multi-cluster intermediates, leading to the occasional formation of more than one bud. Combinatorial slowing of polarity factor exchange had additive effects, yielding strains that frequently made more than one bud.

The finding that rdi1Δ mutants occasionally make two buds was also reported recently by another group (Freisinger et al., 2013). Those authors suggested that in the absence of RDI1, polarity establishment occurs through a pathway involving actin cables, and that once actin cables attach at a particular site the cell is committed to making a bud there. Our data argue against this hypothesis: we found that several polarity clusters could recruit actin cables (as judged by delivery of vesicles) in both wild-type and rdi1Δ mutant cells, but this did not prevent elimination of most clusters. Slow resolution of multi-cluster intermediates also continued in rdi1Δ and other slow-exchange strains after septins had been recruited, and even in some instances after buds had begun to grow. Thus, it would seem that neither actin cables nor indeed any other 'stabilizer' acts to lock in a polarity site in cells that have other polarity sites. Notably, slow-exchange mutant cells that developed only one cluster never eliminated that cluster. These findings imply that it is the presence of a competing cluster that promotes the dissolution of other clusters in the same cell.

In principle, clusters could actively inhibit other clusters in the same cell, rather than simply competing for shared components. Indeed, this type of mechanism has been proposed to explain why neutrophils maintain only one front (Houk et al., 2012). As in yeast, more than one front can transiently co-exist in neutrophils. Each front promotes actin polymerization and membrane protrusion, leading to increased membrane tension, which in turn appears to inhibit GTPase activation. Thus, tension promoted by a dominant GTPase cluster actively extinguishes smaller clusters. This seems unlikely to account for the elimination of excess Cdc42 clusters in yeast. First, membrane tension in yeast (and other walled cells) is determined by turgor pressure rather than actin polymerization. Second, if a yet to be identified inhibition mechanism was functioning in yeast, it is not obvious why slowing the exchange of polarity factors would counteract it. Thus, the simplest hypothesis is that elimination of excess clusters reflects the depletion of polarity factors from losing clusters as they are acquired by a competing cluster.

As the slow-exchange mutant cells could make up to four buds simultaneously without overexpression or any increase in ploidy, all cell components required for budding must be present in considerable excess of what is required to make a functioning polarity site. If there are sufficient polarity factors to make several functional fronts, why is it that in wild-type cells, competition continues until there is only a single winner? Analyses of a computational model incorporating some of the known interactions among yeast polarity factors suggests that a larger cluster would have significant advantages over a smaller cluster in both recruiting and retaining polarity factors. The insatiably acquisitive nature of this competitive process would lead to an inexorable rich-get-richer spiral in which the winning cluster starves all others of polarity factors. This behavior has clear parallels (though with differences in mechanism) to coarsening phenomena in physics (Semplice et al., 2012).

Is competition for polarity factors also relevant to other situations in which cells generate a single front? In plant roots, each trichoblast cell polarizes to grow a single root hair (Cole and Fowler, 2006). Root hair outgrowth is regulated by the GTPase Rop2, a member of a plant-specific 'Rop' family closely related to Cdc42 and Rac GTPases (Jones et al., 2002). Strikingly, mutations in a plant GDI gene lead to the frequent production of multiple growing root hair sites in a single cell (Carol et al., 2005). We speculate that competition between polarity sites for Rop2 may ensure that only one root hair grows per cell. By analogy to our findings for yeast, slowing the exchange of Rop2 may impair competition in that system, allowing more than one site to initiate tip growth.

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Author details

1. Chi-Fang Wu

   Department of Pharmacology and Cancer Biology, Duke University Medical Center, Durham, United States

   Contribution

   CFW, Conception and design, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article

   Competing interests

   The authors declare that no competing interests exist.

2. Jian-Geng Chiou

   Department of Pharmacology and Cancer Biology, Duke University Medical Center, Durham, United States

   Contribution

   JGC, Conception and design, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article

   Competing interests

   The authors declare that no competing interests exist.

3. Maria Minakova

   Department of Pharmacology, University of North Carolina at Chapel Hill, Chapel Hill, United States

   Contribution

   MM, Conception and design, Analysis and interpretation of data, Drafting or revising the article

   Competing interests

   The authors declare that no competing interests exist.

4. Benjamin Woods

   Department of Pharmacology and Cancer Biology, Duke University Medical Center, Durham, United States

   Contribution

   BW, Conception and design, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article

   Competing interests

   The authors declare that no competing interests exist.

5. Denis Tsygankov

   Department of Pharmacology, University of North Carolina at Chapel Hill, Chapel Hill, United States

   Contribution

   DT, Analysis and interpretation of data, Drafting or revising the article, Contributed unpublished essential data or reagents

   Competing interests

   The authors declare that no competing interests exist.

6. Trevin R Zyla

   Department of Pharmacology and Cancer Biology, Duke University Medical Center, Durham, United States

   Contribution

   TRZ, Generated yeast strains, Conception and design, Contributed unpublished essential data or reagents

   Competing interests

   The authors declare that no competing interests exist.

7. Natasha S Savage

   Institute of Integrative Biology, University of Liverpool, Liverpool, United Kingdom

   Contribution

   NSS, Conception and design, Analysis and interpretation of data, Drafting or revising the article

   Competing interests

   The authors declare that no competing interests exist.

8. Timothy C Elston

   Department of Pharmacology, University of North Carolina at Chapel Hill, Chapel Hill, United States

   Contribution

   TCE, Conception and design, Analysis and interpretation of data, Drafting or revising the article

   Competing interests

   The authors declare that no competing interests exist.

9. Daniel J Lew

   Department of Pharmacology and Cancer Biology, Duke University Medical Center, Durham, United States

   Contribution

   DJL, Conception and design, Analysis and interpretation of data, Drafting or revising the article

   For correspondence

   daniel.lew@duke.edu

   Competing interests

   The authors declare that no competing interests exist.

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