Regulation of Cdc42 polarization by the Rsr1 GTPase and Rga1, a Cdc42 GTPase-activating protein, in budding yeast

Cell polarization generally occurs along a single axis in response to spatial cues. In the budding yeast Saccharomyces cerevisiae, the axis of polarized growth is determined by selection of a bud site, which is dependent on cell type – haploid a and α cells bud in the axial pattern, whereas diploid a/α cells bud in the bipolar pattern (Chant and Herskowitz, 1991; Freifelder, 1960; Hicks et al., 1977). Selection of a bud site depends on several genes, collectively called ‘BUD’ genes, which encode cell-type-specific cortical markers and components of the Rsr1 GTPase module – Rsr1/Bud1, its GTPase-activating protein (GAP) Bud2 and its GDP–GTP exchange factor (GEF) Bud5 (Bender and Pringle, 1989; Chant et al., 1991; Chant and Herskowitz, 1991; Park et al., 1993). These Bud proteins closely interact with the Cdc42 GTPase and its regulators to trigger bud emergence at a proper site (see Bi and Park, 2012 and references therein).

In the absence of spatial cues, yeast cells can still bud, albeit at a random site, through a process referred to as ‘symmetry breaking'. A large number of studies suggest that the mechanisms underlying symmetry breaking involve the actin cytoskeleton or the Cdc42 signaling network, which includes Bem1 and the Cdc42 GEF Cdc24 (Goryachev and Pokhilko, 2008; Irazoqui et al., 2003; Kozubowski et al., 2008; Wedlich-Soldner et al., 2003; Wedlich-Soldner et al., 2004). Guanine nucleotide dissociation inhibitor (GDI)- and endocytosis-mediated recycling of Cdc42 and negative feedback confer robust initiation of cell polarization (Howell et al., 2012; Jose et al., 2013; Marco et al., 2007; Ozbudak et al., 2005; Slaughter et al., 2009). These studies on symmetry breaking have revealed intricate crosstalk among polarity factors, although some aspects have been challenged (Freisinger et al., 2013; Kuo et al., 2014; Layton et al., 2011; Slaughter et al., 2013; Smith et al., 2013). Importantly, it remains unclear whether and how these mechanisms might be linked to spatial cues in wild-type cells.

During axial budding, a new bud forms at a site adjacent to the immediately preceding division site (Chant and Pringle, 1995) (Fig. 1A), which is marked with a transient cortical mark (i.e., ‘axial landmark’) that includes Bud3, Bud4, Axl1 and Axl2/Bud10. Septins, a conserved family of GTP-binding proteins, guides the assembly of the axial landmark, which localizes to the division site as uniform ring(s) (Chant et al., 1995; Fujita et al., 1994; Halme et al., 1996; Kang et al., 2012; Kang et al., 2013; Lord et al., 2002; Roemer et al., 1996; Sanders and Herskowitz, 1996). The axial landmark then interacts with Bud5 to direct axial budding in haploid cells (Kang et al., 2012; Kang et al., 2001; Marston et al., 2001). In addition to the axial landmark, Rga1 affects the selection of a proper bud site (Chen et al., 1996; Smith et al., 2002; Stevenson et al., 1995; Tong et al., 2007). Rga1 inhibits polarization of Cdc42 within the division site and thus prevents re-budding at the same bud site (Tong et al., 2007). Our recent study suggests that Cdc42 is activated in two steps in the G1 phase in haploid cells, first by Bud3 and then by Cdc24 (Kang et al., 2014). This stepwise activation of Cdc42 is likely to be important for linking the axial landmark to the establishment of polarity. Several questions still remain regarding the axial budding pattern. Importantly, how is a single new bud site established near the last division site even though the perimeter of the ring appears to be large enough to accommodate multiple sites (Fig. 1Ab)? How is the function of Rga1 coordinated with stepwise activation of Cdc42 in G1? How are other older bud sites that are adjacent to the last bud site excluded from being used again? Does Rga1 indeed localize to the division site, similar to the septin ring or the axial landmark (Caviston et al., 2003; Tong et al., 2007)? To address these questions, we examined Cdc42 polarization dynamics and the localization of Rga1 in haploid budding yeast.

Here, we show by using live-cell imaging that the Cdc42–GTP level fluctuates around the septin ring until the axis of Cdc42 polarization becomes stabilized in mid G1. We find that the development of the Cdc42–GTP cluster in the first step of G1 and its proper positioning depend on Rsr1 and Bud2. Importantly, we find that Rga1 exhibits a transient localization pattern during cytokinesis and G1, and that this distribution of Rga1 is necessary for dynamic Cdc42 polarization during axial budding. Mathematical modeling suggests that sequential positive-feedback and transient negative-feedback mechanisms are involved in directing Cdc42 polarization in the proper orientation.

Activation of Cdc42 by Bud3 is likely to be crucial for axial budding (Kang et al., 2014); however, it remains unclear how a single site of Cdc42 polarization is determined because Bud3 and other axial landmark proteins localize to the division site as a homogeneous ring (Fig. 1A). To gain insight into the mechanism underlying the axial budding pattern, we closely monitored the stabilization of the Cdc42 polarization site as a readout of bud-site selection. We used the p21-binding domain of Gic2 fused to tdTomato (PBD–RFP) as a marker for Cdc42–GTP, and we used green fluorescent protein (GFP)-tagged Cdc3 (Cdc3–GFP) to mark the bud neck and the division site. PBD specifically interacts with Cdc42–GTP and responds rapidly to changes in Cdc42–GTP in vivo (Ozbudak et al., 2005; Tong et al., 2007). Time-lapse imaging studies of haploid wild-type cells expressing PBD–RFP and Cdc3–GFP show that Cdc42–GTP starts to accumulate around the septin ring within a few minutes after the onset of cytokinesis (i.e. t=0 when the Cdc3–GFP ring split) at 22°C and 30°C (Kang et al., 2014). Interestingly, the site with the highest Cdc42–GTP level was established quickly, near the bud neck in mother cells, and the position remained stable at the same site where the new septin ring appeared (85%, n=27; Fig. 1B). By contrast, the Cdc42–GTP cluster appeared at multiple positions around the septin ring in daughter cells until the site became stabilized (100%, n=27). Although the axial budding pattern and the extent of the Cdc42–GTP cluster drifting were approximately the same at 22°C and 30°C (see Fig. 1B; data not shown), imaging at the lower temperature extended the time period in which the drifting of the Cdc42–GTP cluster occurred owing to slower growth, allowing easier visualization of the dynamics of Cdc42 polarization. This dynamic behavior of the Cdc42–GTP cluster was also observed more clearly by frequent image capturing. To better display the displacement of the Cdc42–GTP cluster, we generated kymographs of the bud-neck region in daughter cells (Fig. 1Bb). The maximum displacement of the Cdc42–GTP cluster in wild-type daughter cells in early G1 was estimated to be about twice the diameter of the Cdc3 ring (Fig. 1Bc; supplementary material Fig. S1A; see Materials and Methods).

Previous studies indicate that the G1 phase is partitioned into two steps, T1 and T2, by the exit of the transcriptional repressor Whi5 from the nucleus (Di Talia et al., 2007) and that the stepwise activation of Cdc42 correlates with this G1 partitioning (Kang et al., 2014). We thus asked whether the distinct Cdc42 polarization dynamics in daughter cells is temporally correlated with the G1 partitioning. Indeed, when PBD–RFP and Whi5–GFP were imaged in the same cells, the drifting of the Cdc42–GTP cluster in daughter cells was observed mainly during T1, sometimes including two Cdc42–GTP clusters of differing intensities (Fig. 1C, t=6 and t=20). Within a few minutes of Whi5 exiting the nucleus, a single site of Cdc42 polarization became stabilized (at t=22∼24; Fig. 1C), as shown by much smaller displacement of the Cdc42–GTP cluster during T2 compared to that during T1 (Fig. 1C; supplementary material Fig. S1A). This result could explain why there was little drifting of the Cdc42–GTP cluster in mother cells, which have a very short T1 phase. Taken together, these results suggest that the site of Cdc42 polarization keeps changing around the division site until it becomes stabilized, approximately at the beginning of T2.

Cdc42 can be directly activated by Bud3 in vitro in the absence of Rsr1 (Kang et al., 2014); however, the Rsr1 GTPase module is necessary for proper bud-site selection (Bi and Park, 2012). To better understand the role of Rsr1 in the dynamics of Cdc42 polarization, we performed time-lapse imaging of PBD–RFP and Cdc3–GFP in cells in which RSR1 or BUD2 had been deleted. Although weak, sporadic Cdc42–GTP elevations were observed in early G1, a strong Cdc42–GTP cluster developed in mid G1 in rsr1Δ or bud2Δ daughter cells (see below). Interestingly, once a strong Cdc42–GTP cluster developed, the axis of Cdc42 polarization did not change substantially in these cells (88%, n=33 for rsr1Δ; 95%, n=40 for bud2Δ); the displacement of the Cdc42–GTP cluster in rsr1Δ and bud2Δ cells was similar to that in wild-type cells during T2 (Fig. 2A,B; supplementary material Fig. S1A; Movie 1). We observed that some rsr1Δ cells occasionally abandoned a strong Cdc42–GTP cluster and then developed a new cluster at a distant site (12%, n=33), and this phenomenon was often observed in those rsr1Δ cells that were slower in undergoing bud emergence (see Discussion). Although these aberrant dynamics of the Cdc42–GTP cluster could be due to alteration of the cell cycle progression in some rsr1Δ cells, time-lapse imaging of rsr1Δ cells expressing Whi5–GFP indicated that the average duration of T1 and T2 in rsr1Δ daughter cells was similar to that of wild-type cells (Fig. 2C; supplementary material Table S1), despite cell-to-cell variation. Taken together, these results suggest that even in cells that bud in a random pattern, the axis of Cdc42 polarization is generally stabilized at approximately the beginning of T2.

Next, we determined Cdc42 polarization by quantifying the Cdc42–GTP clusters in rsr1Δ daughter cells over time using a threshold method, as previously described (Okada et al., 2013), and normalized the values to the intensity at t=0. In contrast to wild-type daughter cells, which exhibit two steps of Cdc42 polarization in G1 (Kang et al., 2014), the first step of Cdc42 polarization was not evident in the rsr1Δ daughter cells, despite some weak activation of Cdc42 in T1 (Fig. 2Da). When the highest level of the Cdc42–GTP cluster in each phase was plotted from individual cells (see Materials and Methods), the average peak levels of the Cdc42–GTP clusters were not significantly different between wild-type and rsr1Δ cells during T2, unlike during T1 (Fig. 2Db). Similar analyses of bud2Δ cells also indicated that Cdc42 was poorly polarized in early G1, whereas a strong Cdc42–GTP cluster developed in mid–late G1 in bud2Δ daughter cells (Fig. 2Db). Because the PBD–RFP protein levels were approximately the same in these strains (supplementary material Fig. S1B), the different fluorescence intensities of the Cdc42–GTP clusters among these cells during T1 is unlikely to be due to different levels of the PBD reporter. Instead, these results suggest that strong Cdc42 polarization in T1 cannot be established in the absence of Rsr1 or Bud2.

What might be involved in the displacement or fluctuation of the Cdc42–GTP cluster in T1? Because Cdc42 GAP(s) are likely to affect the dissociation rate of Cdc42 from the plasma membrane to the cytoplasm, we considered Rga1, which is the only Cdc42 GAP that specifically affects the axial budding pattern (Chen et al., 1996; Smith et al., 2002; Stevenson et al., 1995; Tong et al., 2007). Although it has been shown that Rga1 blocks Cdc42 polarization at the division site (Tong et al., 2007), it is unclear when Rga1 functions relative to the two steps of Cdc42 activation and whether Rga1 activity is involved in the fluctuating levels of Cdc42–GTP around the division site during T1. We thus examined how Cdc42 polarization is affected in the absence of Rga1 by using time-lapse imaging. We found that the Cdc42–GTP cluster developed at the central region of the Cdc3–GFP ring a few minutes after the onset of cytokinesis (t=0); this site of Cdc42 polarization appeared to be stable in daughter cells (100%, n=19) and most mother cells (95%, n=19) of an rga1Δ mutant (Fig. 3Aa). We estimated the displacement of the Cdc42–GTP cluster, as described above for wild-type cells (see Fig. 1Bc). The maximum displacement of the Cdc42–GTP cluster in the rga1Δ daughter cells was approximately 30% of that in wild-type cells in T1 (Fig. 3Ab; supplementary material Fig. S1A).

Imaging of PBD–RFP together with Whi5–GFP in rga1Δ cells revealed a considerable elevation of the Cdc42–GTP level during T1 and T2 (Fig. 3B; supplementary material Fig. S1C). When the Cdc42–GTP cluster around the bud-neck region was quantified, two distinct Cdc42–GTP waves in the G1 phase were less evident in rga1Δ cells (supplementary material Fig. S1Cb), although the level of the Cdc42–GTP cluster was slightly decreased in some cells in mid G1 (54%, n=24). When the highest levels of the Cdc42–GTP cluster in individual cells were plotted, the peak levels were approximately 3- to 5-fold higher in rga1Δ cells than those in wild-type cells in T1 and T2 (85%, n=20; supplementary material Fig. S2C).

To gain insight into the mechanism underlying the axial budding pattern, we employed mathematical modeling. We considered a generic model of particle density of the membrane-bound Cdc42 on a two-dimensional region of the plasma membrane with the axial landmark in the center as a ring (Fig. 4A). When we initially assumed a single positive-feedback loop continuously operating from early G1, we were unable to recapitulate the dynamic Cdc42 polarization that was observed in daughter cells (not shown; see below). We thus hypothesized that the process could be divided into two temporal phases, as suggested by the stepwise activation of Cdc42 in G1 (Kang et al., 2014) – in the first phase, the first positive-feedback loop and axial landmark might coexist; in the second phase, the stronger second positive-feedback event might replace the first one (see Materials or Methods). We assumed different lengths of the first phase – either 3 or 15 minutes corresponding to T1 in mother or daughter cells, respectively.

Stochastic noise was added to the levels of spatial cues and Cdc42 GAPs in our simulations. Because the dissociation rate of Cdc42 from the plasma membrane depends on its conversion to the GDP-bound state, the spatial distribution of Cdc42 GAPs is likely to be important for Cdc42 polarization. Indeed, when no Cdc42 GAP was assumed to be present, a Cdc42–GTP cluster developed within the previous division site in both mother and daughter cells (Fig. 4B), as observed in rga1Δ cells (Tong et al., 2007) (Fig. 3). Because previous studies indicate that Rga1 localizes to the mother-bud neck during cytokinesis, in a manner similar to that of the septin ring (Caviston et al., 2003; Tong et al., 2007), we considered the distribution of the Cdc42 GAP as a ring at the division site (Fig. 4C). Surprisingly, with this Cdc42 GAP distribution, the positive feedback and the competition (probably for the limited availability of feedback molecules involved in Cdc42 signaling) always induced a single Cdc42 GTP cluster within the previous division site (Fig. 4C) – a phenomenon rarely observed in wild-type cells. Thus, our modeling predicts that the fixed distribution of Cdc42 GAPs as a homogeneous ring at the division site leads to budding within the division site, contrary to a previous report (Tong et al., 2007) (see Discussion).

To test the prediction of our modeling, we wanted to obtain better spatiotemporal resolution of Rga1 localization during the M–G1 phase. We performed time-lapse imaging of GFP–Rga1 together with either Cdc3–mCherry or Myo1–mCherry as a marker for the position and timing of cytokinesis. Importantly, we observed two unique features of Rga1 localization that had not been previously appreciated. First, Rga1 localized to a site next to the bud neck in mother cells before the onset of cytokinesis (77%, n=17), and the new septin ring always appeared at a position on the other side of the bud neck (but still adjacent to the division site) (Fig. 5A; supplementary material Movie 2). Second, the intensity of GFP–Rga1 at the bud neck decreased during cytokinesis and then began to increase at the division site in G1 (100%; n=17). Similarly, GFP–Rga1 localized to a site next to the Myo1 ring (in addition to the bud neck), and line-scan analysis showed the asymmetric distribution of Rga1 between mother and daughter cells during cytokinesis and G1 (supplementary material Fig. S2A).

To visualize this Rga1 distribution more clearly, we took end-on views of large-budded or newly divided cells that had the rings of GFP–Rga1 and Myo1–mCherry (or Cdc3–mCherry) in the center of cells (as depicted in Fig. 1Ab). The GFP–Rga1 ring at the bud neck became fragmented as the Myo1 ring started to contract. The Rga1 ring became amorphous at the division site soon after completion of the Myo1 ring contraction and then formed a ring again (100%, n=5) (Fig. 5Ba). Consistent with these observations, the fragmented GFP–Rga1 distribution often appeared within the Cdc3–mCherry ring in newly born daughter cells (which have a bigger septin ring than mother cells) (100%, n=12; Fig. 5Bb). Taken together, our time-lapse imaging provides temporal resolution of the Rga1 distribution during cytokinesis and G1, although it was less clear in mother cells, probably because of their shorter G1 phase.

Is this newly identified pattern of Rga1 localization important for axial budding? We postulated that Rga1 localization to the bud scar(s) that are next to the last division site in mother cells prevents re-budding from these previous division sites. To test this idea, we stained cells expressing GFP–Rga1 with Calcofluor, which stains chitin-rich bud scars (i.e. all previous bud sites) and the bud neck. Indeed, GFP–Rga1 localized to the site(s) overlapping the bud scar(s) (Fig. 5Ca) or within the birth scar (i.e. the division site in daughter cells, which is devoid of chitin) (Fig. 5Cb) (100%, n=45), supporting our hypothesis.

Next, we tested whether limiting Rga1 localization only to the bud neck as a homogeneous ring could rescue bud-site selection defect of rga1Δ cells. We expressed the GAP domain of Rga1 as a fusion protein with Bud3 (Bud3–Rga1GAP), which localizes only to the bud neck as a homogenous double ring during M–G1 (Tong et al., 2007). This fusion is likely to have GAP activity towards Cdc42 because the same Rga1 GAP domain has been shown to accelerate GTP hydrolysis by Cdc42 in vitro (Tong et al., 2007). The peak levels of the Cdc42–GTP clusters in individual rga1Δ cells expressing Bud3–Rga1GAP were even slightly lower than those in wild type and much lower than those in rga1Δ cells (supplementary material Fig. S2C), indicating that Bud3–Rga1GAP has Cdc42 GAP activity in vivo. Nevertheless, expression of Bud3–Rga1GAP was unable to rescue the bud-site-selection defect of an rga1Δ mutant – when these cells were co-stained with Calcofluor and wheat germ agglutinin (WGA) conjugated to fluorescein isothiocyanate (WGA–FITC, which stains both bud scars and birth scars), almost all daughter cells budded within the birth scar (marked with an arrow in Fig. 6A). Even in mother cells that had more than one bud scar, these bud scars were observed within the birth scar (which often became enlarged, probably owing to repeated budding within the birth scar) (Fig. 6B). When we counted only those cells that had more than two bud scars (that were spaced separately), a significant number of rga1Δ cells exhibited the bipolar pattern (62%, n=50), as previously reported (Chen et al., 1996; Smith et al., 2002; Stevenson et al., 1995).

Because bud scars within a bud scar were sometimes less clear from static images (see Fig. 6Bii*), we performed time-lapse imaging to examine Cdc42 polarization in rga1Δ cells expressing Bud3–Rga1GAP–GFP after staining with Calcofluor. T1 and T2 in these cells lasted approximately the same amount of time as those in wild-type cells (supplementary material Table S1), and Cdc42 polarized within the Bud3–Rga1GAP ring in daughter cells (100%, n=22; supplementary material Fig. S2B) and either within the last division site (32%, n=22) or next to the bud neck (68%, n=22) in mother cells. The Cdc42–GTP cluster in the latter group overlapped with a bud scar (93%, n=15; supplementary material Fig. S2B), indicating that these cells budded at the old division site. These analyses indicate that Bud3–Rga1GAP is not functional in preventing re-budding at the previous bud sites, although the defect was less severe in mother cells. Taken together, these results suggest that the transient localization of Rga1 to the bud neck and to a site next to the bud neck is necessary for proper bud-site selection.

The temporal and spatial changes of Rga1 distribution are likely to contribute to the dissociation rate of Cdc42 from the plasma membrane. When the new localization patterns of Rga1 in mother and daughter cells were incorporated into our model settings, a strong Cdc42 cluster developed at a single site that was near to the division site (Fig. 7A). However, the Cdc42–GTP cluster exhibited little fluctuation or displacement, i.e. the simulation was unable to recapitulate the Cdc42–GTP dynamics in daughter cells.

We then considered negative feedback in our modeling because oscillation or fluctuation of biological systems often involves delayed negative-feedback regulation (Das et al., 2012; Howell et al., 2012; Ozbudak et al., 2005). We incorporated both Rga1 localization and delayed negative feedback, which were dynamic rather than time-invariant, into our model with two temporal phases – in the first phase, positive feedback and putative delayed negative feedback might coexist, whereas in the second phase, positive feedback might become stronger and the delayed negative feedback might diminish. Remarkably, our simulations with these settings indicated distinct Cdc42 polarization dynamics depending upon different durations of the first phase. When the first phase lasted for 15 min and localization of Cdc42 GAP resembled that of Rga1 in daughter cells, the position and level of the Cdc42–GTP cluster fluctuated at the division site during the first phase and then became stabilized at a single site (Fig. 7Ba). By contrast, when the first phase was assumed to last only approximately 3 minutes and GAP localization was similar to that of Rga1 in mother cells, robust Cdc42–GTP polarization was rapidly established at a single site adjacent to the previous division site with little drifting or fluctuation (Fig. 7Bb). Thus, these final modeling settings generated simulations that recapitulate the Cdc42–GTP dynamics observed in vivo.

To address the relevance of this two-phase system, we considered a single-phase system with persistent negative feedback and Rga1 distribution similar to those of either mother cells or daughter cells. In either case, a Cdc42 polarization axis could not be stabilized (supplementary material Fig. S3A), suggesting that the two phases with transient delayed negative feedback is necessary for stabilization of a single axis of Cdc42 polarization. We modeled another control case by switching conditions for two phases – i.e. the first phase involved a positive-feedback loop and the second phase involved a negative-feedback loop and a positive-feedback loop with lower strength. In this case, the Cdc42 cluster could not be stabilized but kept moving away from the previous division site (supplementary material Fig. S4A), similar to a single-phase system with persistent negative feedback (supplementary material Fig. S3A). In addition, if localized Rga1 activity was lacking (supplementary material Fig. S3B) or Rga1 was assumed to be a constant ring during G1 (Fig. 7C), then Cdc42 always polarized within the division site, even when negative feedback was included. These additional simulations suggest that the time-dependent Rga1 distribution and the transient negative feedback in the first phase are crucial for the establishment of an axial bud site. These in silico models are also consistent with our experimental observations with rga1Δ cells and rga1Δ cells that expressed the Bud3–Rga1GAP fusion protein (Figs 3 and 6), except for a minor discrepancy (see Discussion).

Despite a large number of studies on cell polarization, how the axis of cell polarity is determined in the proper orientation is not fully understood. In this study, we investigated when and how the axis of polarized growth is determined in haploid budding yeast by using quantitative microscopy and mathematical modeling. We show that Cdc42 is polarized in distinct dynamics in correlation with the two steps of the G1 phase – T1 and T2. Importantly, our data suggest that Cdc42 polarization in T1 depends on Rsr1, Bud2 and Rga1, in addition to the axial landmark in cells undergoing axial budding, and that the orientation of the polarity axis is determined at approximately the beginning of T2. Our computational modeling, enacted using live-cell imaging data, suggests that an axial bud site is established through a biphasic mechanism that involves sequential positive feedback and transient negative feedback.

Here, time-lapse analyses show that the Cdc42–GTP level fluctuates around the division site in wild-type haploid cells until the position of the Cdc42–GTP cluster becomes stabilized in T2. Surprisingly, even in rsr1Δ cells, the polarity axis remains relatively steady once the Cdc42–GTP cluster is formed at a random site in T2, although ‘relocation’ of the cluster is observed occasionally in rather slow growing cells. Thus, Cdc24 is likely to arrive at the already selected bud site (regardless of axial or random bud site) and then to trigger strong Cdc42 activation in T2. Although wild-type cells exhibit changes of the polarity axis around the bud-neck region during T1, we found that rsr1Δ or bud2Δ cells are unable to develop strong Cdc42–GTP clusters during this time window. However, it has been reported previously that cells lacking RSR1 exhibit dynamic changes of the polarity axis before selecting a single random site (Ozbudak et al., 2005). We are currently uncertain as to the reason for this discrepancy. Because no cell cycle marker (such as Whi5) was included in that previous report, it is unclear when (either early or mid–late G1) wandering of the Cdc42–GTP polar cap was observed in rsr1Δ cells. It is possible that the relocation of the polarity cluster (which we observed only in 12% of rsr1Δ cells) might be more common in certain strain backgrounds or might be sensitive to the level of the fusion protein(s) or imaging conditions. Our analyses used all chromosomal copies of fusion proteins expressed in haploid cells, whereas other studies of symmetry breaking have used expression of fusion proteins from plasmids or in diploid cells (Howell et al., 2012; Ozbudak et al., 2005) (see below). Based on our findings, we propose that a single bud site is determined at approximately the beginning of T2 and that the Rsr1 GTPase module is important in directing polarization of Cdc42 in T1.

Our time-lapse imaging provides better spatiotemporal resolution of Rga1 localization than previously studies. First, Rga1 exhibits a fragmented ring-like structure or amorphous distribution at the division site, rather than a septin-like ring. This first pattern might explain more easily how Rga1 could act as a ‘plug’ within the division site to clear active Cdc42 or to perturb Cdc42 activation around the division site. Second, Rga1 localizes to a site near the bud neck (in addition to the bud neck) in mother cells. Our analysis suggests that this second pattern is likely to be important for preventing re-budding from the older division site in mother cells. In contrast to the previous report (Tong et al., 2007), our experimental data and computational modeling indicate that if Rga1 GAP activity is limited only to the bud neck (and the division site) as a homogeneous ring, cells will bud within the previous bud site. We thus think that the previous study might have overlooked the defect in bud-site selection of cells expressing Bud3–Rga1GAP, because their analyses mainly relied on the number of bud scars without visualizing the position of birth scars. Because a birth scar is a frail structure that substantially fades after multiple cell divisions (Powell et al., 2003), it might also have been missed in some mother cells in the previous study. Importantly, because daughter cells exhibit only the transient pattern of Rga1 localization to the bud neck and to the division site, the phenotype (i.e. budding within the birth scar) of rga1Δ daughter cells expressing Bud3–Rga1GAP suggests that this transient distribution of Rga1 is crucial for axial budding. This Rga1 localization is also likely to contribute to fluctuation of the Cdc42 cluster during T1 because the displacement of the Cdc42–GTP cluster is diminished in rga1Δ cells. Interestingly, Aim44/Gps1, which also prevents re-budding at the previous bud site, localizes to the division site as a disk (rather than a ring) (Meitinger et al., 2013).

Our mathematical modeling suggests that sequential positive-feedback and negative-feedback regulation are involved in axial budding. Interestingly, modeling could recapitulate the distinct Cdc42 polarization dynamics when the duration of the first positive-feedback loop was altered in order to mimic the different lengths of T1 in mother and daughter cells. We thus extended our model of a biphasic mechanism of Cdc42 polarization in haploid cells (Kang et al., 2014) – the first phase might involve a positive-feedback loop, including the axial landmark and the Rsr1 GTPase module, and delayed negative feedback; the second phase might involve a stronger positive feedback response. The second feedback could be the same as that of the Bem1-mediated signaling network, which includes Cdc24, as proposed for symmetry breaking (Irazoqui et al., 2003; Wedlich-Soldner et al., 2004). A recent model has also suggested that, in addition to the Bem1-mediated feedback, transient or weak activity of a Cdc42 GEF is required for Cdc42 polarization in wild-type haploid cells (Wu et al., 2013).

Although our model suggests that transient negative feedback in T1 is involved in fluctuation of the Cdc42 polarity cluster in haploid cells, the underlying mechanism remains unclear. A recent study proposes a mechanism for negative feedback through phosphorylation of the Cdc42 GEF Cdc24 in the yeast polarity circuit (Kuo et al., 2014); however, this mechanism might not be relevant to our observation of Cdc42 dynamics in T1. Although that recent study and other studies of symmetry breaking often image polarity factors in diploid a/α cells, Cdc24 is sequestered in the nucleus through binding to Far1 in haploid cells (Shimada et al., 2000), and its relocation to the presumptive bud site is triggered by the cyclin-dependent kinase (CDK) Cdc28–Cln2 in T2 (Gulli et al., 2000). Because expression of FAR1 is repressed in a/α cells (Chang and Herskowitz, 1990), regulation of Cdc24 in diploid cells might be different from that in haploid cells. Moreover, we found that the axis of Cdc42 polarization becomes stable during T2, suggesting that the negative feedback involved in axial budding is dampened during T2. Because of this cell cycle timing, we postulate that Cdc24 is unlikely to be involved in the negative-feedback loop that we propose for axial budding of haploid cells.

Although our model of biphasic regulation of Cdc42 polarization is well supported by our experimental observations, we noticed a minor discrepancy. When the Rga1 GAP activity was limited to the division site as a homogeneous ring, our in silico modeling predicts the development of a Cdc42 cluster within the division site, regardless of the duration of the first phase. This prediction is consistent with in vivo observations in daughter cells but only partially with those in mother cells expressing Bud3–Rga1GAP. Simple modeling was used in this study; however, fine-tuning of polarity establishment could involve additional mechanisms, as previously reported (Klünder et al., 2013; Marco et al., 2007; Slaughter et al., 2009; Slaughter et al., 2013), and could thus require further investigation.

Standard methods of yeast genetics and DNA manipulation, as well as standard growth conditions, were used (Guthrie and Fink, 1991) unless indicated otherwise. All fusion proteins were expressed from the chromosomal loci, except Myo1–mCherry. See details in supplementary material Table S2.

Time-lapse imaging was performed essentially as previously described (Kang et al., 2014) using a spinning-disk confocal microscope (Ultra-VIEW VoX CSU-X1 system; PerkinElmer) equipped with a 100×, 1.4 NA Plan Apochromat objective lens (Nikon), 440-, 488-, 515- and 561-nm solid-state lasers (Modular Laser System 2.0; PerkinElmer), and a backthinned electron-multiplying charge-coupled device (CCD) camera (ImagEM C9100-13; Hamamatsu Photonics) on an inverted microscope (Ti-E; Nikon). Images were captured (nine stacks, 0.3 μm z steps) every minute at 22°C or every 2 minutes at 30°C on an agarose slab, except for those shown in Fig. 5B. To capture end-on views, the poly-lysine-coated glass bottomed dish (MatTek) was prepared using warm medium containing agarose (0.6%) that had been mixed with freshly grown cells. Three-color time-lapse imaging was performed after staining cells with Calcofluor White (see below) before mounting on a slab.

The Cdc42–GTP cluster was quantified by using a threshold method (Okada et al., 2013) and ImageJ software (National Institutes of Health) from summed images of five selected z-sections after background subtraction, as previously described (Kang et al., 2014). The intensity of PBD–RFP clusters at each time point was then normalized to that at t=0 (Fig. 2Da; supplementary material Fig. S1Cb), and the highest intensity of the PBD–RFP clusters of individual cells in T1 and T2 phases was plotted (Fig. 2Db; supplementary material Fig. S2C). Kymographs and heatmaps were generated from maximum intensity projection images of z-stacks using the multiple kymograph plugin and heatmap histogram plugin for ImageJ. To measure the displacement of the Cdc42–GTP cluster, the pixel position of the highest PBD–RFP intensity was measured from the kymograph and heat map of PBD–RFP. The measurement unit was converted from pixels to microns based on the size of box drawn on the image and the number of pixels in the kymograph. The maximum displacement of the Cdc42–GTP cluster was calculated by measuring the maximum distance between the pixels with the highest Cdc42–GTP intensity during each time window of T1 or T2; this is likely to be a slight underestimation because of the curvature of cell shape, particularly in the case of wild-type cells in T1. The GFP-fluorescence intensity at the bud neck and/or division site before and after the onset of cytokinesis of cells expressing GFP–Rga1 or Bud3–Rga1GAP–GFP was quantified using ImageJ, as previously described (Kang et al., 2012). This analysis suggests that the molar ratio of Bud3–Rga1GAP–GFP versus GFP–Rga1 is larger than 1 (supplementary material Fig. S2D). Representative images in figures were generated through maximum intensity projections of z-stacks using UltraView Vox software. The duration of T1 and T2 was determined by monitoring the localization of Whi5–GFP and was estimated to be similar for all strains examined. The duration of G1 in bud2Δ cells was estimated to be approximately the same as that of wild type based on the localization of Cdc3–GFP and bud emergence. Unexpectedly, bud2Δ in combination with the expression of Whi5–GFP resulted in a longer T1 phase, but the basis of this synthetic phenotype is currently unclear. Statistical differences between two sets of data were determined by a two-tailed Student's t-test using Excel (Microsoft).

Bud scars and birth scars were stained with Calcofluor White and WGA–FITC at the final concentration of 0.5 µg/ml and 100 µg/ml, respectively (Frydlová et al., 2009). Maximum intensity projections were generated from 15 stacks (0.3 µm z-steps) of images captured by the spinning-disk confocal microscope (see above) (Fig. 5C) or a microscope (E800; Nikon) fitted with a 100×1.3 NA oil Plan Fluor objective lens (Nikon), a CCD camera (ORCA-ER; Hamamatsu Photonics), and FITC–GFP and DAPI filters (Chroma Technology) and using SlideBook software (Intelligent Imaging Innovations) (note that Bud3–Rga1GAP–GFP was not easily detected under this imaging condition because of the relatively weak GFP signal; Fig. 6).

Preparation of yeast extracts and immunoblotting were performed as previously described (Kang et al., 2014). PBD–RFP was detected using antibodies against DsRED (Clontech Laboratories, Mountain View, CA) and Alexa Fluor 680 goat anti-rabbit IgG (Molecular Probes, Eugene, OR) secondary antibodies using the LI-COR Odyssey system (LI-COR Biosciences, Lincoln, NE). The relative amount of PBD–RFP was estimated using a non-specific cross reacting band on immunoblots and normalized against its level in the wild-type strain.

The particle density of the membrane-bound Cdc42 is denoted by a. The computational domain M is a two-dimensional region of cell membrane with the landmark cue at the center (Fig. 4Aa). Though curved in nature, this domain is taken as planar because only a small portion of the whole cell membrane is considered, and membrane-bound Cdc42 is assumed to appear only in this domain.

representing the average value of a over the membrane and bounded by one as long as the initial value is less than one.

The first term of the right-hand side of Eqn 1 represents the lateral surface diffusion of Cdc42 with the diffusion coefficient D_m. The function F is the rate coefficient for recruitment and activation of Cdc42 from cytoplasm to membrane, which depends on the level of the spatial landmark cue u and the particle density of the membrane-bound Cdc42 (Lo et al., 2013; Lo et al., 2014). The spatial function u represents the pre-localized signal, involving the axial landmark and the Rsr1 module. By Eqn 1, is the fraction of cytoplasm Cdc42. Cdc42 is assumed to be homogeneous in the cytoplasm, based on the fact that the diffusion in the cytoplasm is much faster than that on the membrane (Marco et al., 2007; Slaughter et al., 2009).

The k_off term in Eqn 1 is the disassociation rate of Cdc42 from the membrane to the cytoplasm. It depends on the location x on the plasma membrane and the value of a with a delay time of t₁, because the deactivation rate varies with the activation level of Rga1, which may be regulated by a (see supplementary material Table S4).

We assume that the dynamics of the feedback molecule is much faster than that of Cdc42, so the denominator, which involves the average value of a, represents the conservation of the total amount of feedback molecules (Lo et al., 2013; Lo et al., 2014).

In Eqn 2, we consider the feedback process in two temporal phases – first, a positive feedback involving the axial landmark and the Rsr1 module; and second, a stronger positive feedback involving the Cdc42-signaling network without the spatial cue. The first term of Eqn 2 represents the axial landmark and the Rsr1 module feedback with K₁ a normalizing factor. The cooperativity coefficient of the feedback (the exponent) is taken to be 2 because two actions are involved in the feedback loop – the recruitment of feedback molecules to the membrane and the binding between Cdc42 and feedback molecules. The second term of the right-hand side in Eqn 2 represents the feedback in the Cdc42-signaling network. Because the feedback is stronger than the Rsr1 module feedback, we apply a smaller normalizing factor K₂ (<K₁) here.

The spatial cue function u is assumed to be zero everywhere except in the region {0.5<|r|<0.65} (r is the distance from a point to the center of the domain), thus forming a ring with an inner radius of 0.5 μm and a thickness 0.15 μm (Fig. 4A). In the ring, the maximum value of u is assumed to be 8, which generated simulations most similar to the in vivo imaging data. The spatial cue is subject to 30% perturbation with uniform distribution. In the region {0.5<|r|<0.65}, the formula for spatial cue is u(x)=5.6+2.4δ(x), where δ(x) is a spatially uncorrelated random function from uniform distribution between 0 and 1, at each spatial point x; in other regions, u(x)=0. It is noteworthy that the conclusion from the outputs does not change, even when the maximum value is varied between 5 and 10.

The formula and parameters associated with the deactivation is listed in supplementary material Table S4.

The parameters are listed in supplementary material Table S3. Different settings on the deactivation term are used for each figure as listed in supplementary material Table S4.

We are grateful to E. Bi (University of Pennsylvania, Philadelphia, PA, USA) for providing strains and plasmids; P. J. Kang for discussion and comments on the manuscript; and C. E. Oakley and I. Kang for their help with proofreading.