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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
 
Curr Biol. Author manuscript; available in PMC 2008 April 17.
Published in final edited form as:
PMCID: PMC1914375
NIHMSID: NIHMS21859

Positive feedback loops as a flexible biological module

Summary

Background

Bistability in genetic networks allows cells to remember past events and make discrete decisions in response to graded signals. Bistable behavior can result from positive feedback, but feedback loops can have other roles in signal transduction as well.

Results

We introduced positive feedback into the budding yeast pheromone response to convert it into a bistable system. In the presence of feedback, transient induction with high pheromone levels caused persistent pathway activation, while at lower levels a fraction of cells became persistently active while the rest inactivated completely. We also generated mutations that quantitatively tuned the basal and induced expression levels of the feedback promoter and showed that they qualitatively changed the behavior of the system. Finally, we developed a simple stochastic model of our positive feedback system and showed the agreement between our simulations and experimental results.

Conclusions

The positive feedback loop can display several different behaviors, including bistability, and can switch between them as a result of simple mutations.

Introduction

In biology, transient stimuli often lead to long-lived responses, the most extreme example being the determination of cell fates in development. In other situations, a uniform stimulus induces individual cells to adopt one or the other of two distinct fates, for example when bacteriophage lambda infection induces some bacterial cells to lyse and others to integrate and stably harbor the viral genome. Both types of behavior reflect bistability, a phenomenon in which a network has two stable, self-perpetuating states and can be switched between them by appropriate inputs [1]. Bistability in a gene regulatory network could allow cells to stably maintain either of two distinct gene expression patterns, providing a mechanism by which they could remember past environmental conditions or intercellular signals [25]. For example, different cell fates in development could be stable states, with transient signals driving cells into different gene expression patterns that would then persist without further signaling.

Bistability in the cell cycle oscillator, a biochemical system, has been shown to promote irreversible transitions between distinct mitotic and interphase states [68]. Cell cycle bistability depends on positive feedback, in which the activity of a protein kinase (Cdk1) stimulates the activation of additional kinase molecules. Positive feedback is also necessary for bistable expression of metabolic genes, including those for Escherichia coli lactose utilization and budding yeast (Saccharomyces cerevisiae) galactose utilization [911]. The role of positive feedback in producing bistability is supported by theoretical models of regulatory networks, which suggest that multiple stable states can only arise when there is positive feedback [12]. In order to test these models, previous studies created artificial positive feedback by making a transcription factor bind to its own promoter [1316]. This simple self-activation architecture was fragile: bistability was seen only at intermediate concentrations of the relevant inducer, and single-cell measurements showed a heterogeneous response within a single population. Naturally-occurring positive feedback systems such as the budding yeast galactose response also required some continuing signal to maintain an active stable state. In contrast, neither natural nor synthetic versions of the classic prokaryotic genetic toggle switch, which employs two directly opposed repressors, required a maintenance signal for bistability [5, 17].

We began by asking how easily positive feedback could create bistability in an existing eukaryotic signaling pathway. To address this question, we chose the budding yeast mating pheromone response (see Fig. 1A), a well-studied signal transduction system that can be stimulated with an exogenous ligand (a factor) and involves a MAP kinase cascade [18]; this pathway is not bistable, but there are homologous metazoan MAP kinase pathways that are bistable [19, 20]. We hoped that the hyperbolic response of the pheromone signaling pathway would stabilize both the active and the inactive states, allowing either to persist indefinitely without exogenous signals. By manipulating the pheromone response, we confirmed theoretical predictions for the behavior of positive feedback systems. We compared three different genetic changes that add positive feedback to the pheromone response and found circumstances in which each produced bistability. To tune positive feedback until it yielded bistability, we created mutant promoters with quantitative changes in expression levels and showed that positive feedback using these altered promoters can qualitatively change the behavior of the system. Finally, we showed how stochastic cell-to-cell variability affects bistable behavior by comparing our experimental results to a simulation that incorporates gene expression noise.

Figure 1
Pheromone response pathway.(A) The wild-type pheromone signaling pathway. The inactive pathway is shown on the left, with the intact heterotrimeric G protein complex (Gpa1p, Ste4p, and Ste18p and the inactive MAPK cascade (Ste11p, Ste7p, and Fus3p, and ...

Results and Discussion

Positive feedback produces bistability or constitutive activity

We wanted to produce positive feedback with the properties of a typical eukaryotic signaling system. Expression of STE11ΔN, a dominant active allele of the MAP kinase kinase kinase, and overexpression of wild-type STE4, the β subunit of a trimeric G protein, both induce mating genes in the absence of pheromone [21, 22]. Transcription from the PFUS1 promoter is strongly induced by an active pheromoneresponse. We expressed a dominant active allele, such as PFUS1-STE11ΔN, from PFUS1, in addition to the endogenous copy of the gene. The endogenous copy is needed to maintain the expression of the signaling components and the activated or overexpressed allele produces the positive feedback that would be activated by pheromone-induced signaling. The wild-type pathway is induced by pheromone, but quickly inactivates when pheromone is removed. We reasoned that, in the absence of pheromone signal, there would be low expression from PFUS1 and the feedback construct would not affect the pathway Once the pheromone response was activated, however, it would induce the expression of the feedback protein, which would maintain the activity of the pathway without continued pheromone stimulation (see Fig. 1B). To test for bistability, we transiently treated cells with pheromone and monitored signaling activity with an unstable fluorescent protein expressed from an additional copy of the PFUS1 promoter. Cells without feedback showed only weak, residual reporter fluorescence 3.5 hours after pheromone removal, while cells expressing the activated MAP kinase kinase kinase from the FUS1 promoter (PFUS1-STE11ΔN) continued to have strong PFUS1 expression (see Fig. 2A-C). We also observed the persistently active PFUS1-STE11ΔN cells by videomicroscopy and found that they stopped dividing (see Fig. 2E)despite the fact that we had deleted FAR1, a gene required for pheromone-induced cell cycle arrest [23]. Another dominant active STE11 allele, STE11-4, causes arrest even in a far1Δ strain [24], so it is not surprising that STE11ΔN does so as well. We conclude that the pheromone response of cells with STE11ΔN-mediated positive feedback has two stable states, active and inactive, and that cells in the active state do not divide.

Figure 2
Effects of positive feedback on the pheromone response.(A) Diagram of the transient induction experiment used to test for bistability. (B) A strain with no positive feedback was treated with the timecourse in (A) and samples were measured by flow cytometry. ...

In contrast, cells overexpressing the beta subunit of the G protein from the FUS1 promoter (PFUS1-STE4) show strong reporter expression even in the absence of pheromone treatment (see Fig. 2D), but continue to divide despite this induced pheromone response (data not shown). A constitutively active pathway corresponds to a system with only a single, active stable state. The inactive state would be unstable if the pathway were so sensitive to STE4 overexpression that the basal transcription from PFUS1-STE4 was enough to partially activate the pathway, driving more STE4 expression and ultimately fully activating the pathway. To confirm that the inactive state of the pathway in PFUS1-STE4 cells was not stable, we replaced FUS3, one of the two partially redundant MAP kinase genes in the pheromone response pathway, with an allele (fus3-as) that is sensitive to Shokat’s Inhibitor (SI, 4-amino-1-tert-butyl-3-(1-naphthylmethyl)pyrazolo[3,4-d]pyrimidine) [25], which inhibits only specific, genetically modified protein kinases, and deleted the other MAP kinase, KSS1. This manipulation allowed us to reversibly inhibit the signaling pathway by adding SI and then ask whether PFUS1-STE4 cells would spontaneously reactivate the pheromone response when the inhibitor was removed (see Fig. 1B). Reporter expression resumed as soon as inhibition was relieved, confirming that the pathway is truly constitutive (see Fig. 2F-I) and that the basal expression of PFUS1-STE4 is enough to activate the feedback loop. We believe that the higher fluorescence of the PFUS1-STE4 cells during inhibition is a result of residual YFP and does not indicate actual activity of the pathway during inhibition. The activity of at least one of the MAP kinases FUS3 and KSS1 are absolutely required for pheromone signaling induced by either pheromone or STE4 overexpression [26], and inhibited fus3-as acts as a null allele in response to pheromone induction (data not shown).

Bistability requires low basal and high induced expression in the positive feedback loop

We asked if we could turn a constitutive response into a bistable one by making PFUS1 alleles that had lower basal expression but retained strong induced expression. We screened a library of mutagenized promoters by fluorescence activated cell sorting (FACS) and recovered numerous alleles with nearly undetectable basal expression and varying maximal induction levels (see Fig. 3A). We selected three for further analysis and found that each had several nucleotide substitutions (see Fig. S1). Mutational plasticity was previously observed in expression levels of a housekeeping gene’s promoter [27]. Our results suggest that the basal and induced expression level of PFUS1 can be tuned independently by mutation.

Figure 3
Positive feedback with low basal expression causes bistability.(A) PFUS1 alleles were fused to stable yEVenus and median fluorescence was measured by flow cytometry in uninduced samples and in those treated with saturating (6 μM) pheromone for ...

We found that driving STE11ΔN, STE4, or a third dominant active allele, STE5-CPRRAS2[28], from promoters with low basal expression caused bistability. The weakest of our alleles, PFUS1J1, had significantly lower induced expression than wild-type PFUS1, whose induced expression was strong enough to produce bistability in cells carrying a single copy of PFUS1-STE11ΔN. We found that positive feedback from one or two integrated copies of PFUS1J1-STE11ΔN was not strong enough to sustain activity after pheromone was removed, but a strain with a tandem array of three copies of the feedback construct was bistable (see Fig. 3B). This suggests that the increased STE11ΔN expression from multiple copies is necessary to maintain the active stable state when the promoter is weakened, demonstrating a qualitative change in the behavior of a positive feedback loop as a result of quantitative changes in transcriptional parameters. Decreasing induced expression levels roughly 2-fold by replacing wild-type PFUS1 with PFUS1J1 eliminates the persistent response to pheromone; the 1.5-fold change in dosage between two and three copies of PFUS1J1-STE11ΔN, which is analogous to carrying an extra copy of a chromosomein a diploid organism, restores bistability. We also noted that, while STE11ΔN expression can maintain the pheromone response in the absence of pheromone, STE11ΔN expression reduces the magnitude of the response during induction. Pheromone stimulation induces less reporter expression in either PFUS1-STE11ΔN or PFUS1J1-STE11ΔN than in wild-type cells, and the degree of PFUS1J1-STE11ΔN inhibitionis positively correlated with copy number. It seems likely that the constitutive Ste11ΔNp allele is less active than phosphorylated wild-type Ste11p, but competes for the same binding sites on Ste5p, explaining the dominant partial activation caused by STE11ΔN expression.

Positive feedback mediated by STE5-CPRRAS2 expression resulted in bistability similar to that seen from STE11ΔN (see Fig. 3C). However, when STE4 expression was used to provide positive feedback, raising the strength of induced expression caused a graded increase in the stability of the active state. We tested each of the three mutant promoters driving STE4, with one, two, or three integrated copies of the feedback construct. In all cases, the strains had low reporter expression in the absence of pheromone treatment, corresponding to an inactive stable state. When cells with positive feedback were induced with saturating concentrations of pheromone, all became fully active and remained active for longer than those without feedback. However, in some strains a significant fraction of the cells inactivated the response after pheromone was removed (see Fig. 4A-F). This resulted in a bimodal distribution of reporter expression, with some cells remaining as strongly fluorescent as they were during pheromone induction and other cells showing only residual levels of the reporter. We quantified the relative fraction of active and inactive cells in different strains and found that it correlated well with the strength of induced expression from the feedback construct (see Fig. 4G). The fraction of inactive cells changes over time, as some cells inactive soon after pheromone removal while others switch spontaneously over time. The fraction of active cells at a given timepoint is a measure of the stability of the active state, representing the likelihood of a cell becoming and remaining active. Self-perpetuating STE4 expression comprises the active state, explaining why cells with stronger feedback are both more likely to become active and less likely to subsequently inactivate the response. Induced expression levels are a strong predictor of the stability of the active state, but other factors, such as the expression at intermediate levels of signal or the dynamics of the response, may also play a role.

Figure 4
Bimodality from STE4-mediated positive feedback.(A)-(F) Wild-type cells as well as various PFUS1-STE4 integrants, whose copy number had been mea- sured by Southern blotting, were transiently induced with pheromone. The strength of positive feedback, corresponding ...

The different behaviors seen with STE4-mediated positive feedback as opposed to STE11ΔN- or STE5-CPRRAS2-mediated positive feedback (see Table 1) suggest that there are qualitative as well as quantitative differences between these feedback loops. Increasing STE11ΔN feedback produces an abrupt switch from no persistent response to stable long-term activation. In contrast, intermediate levels of STE4 feedback produce persistent but not permanent activation of the pheromone response. No- tably, overexpression of STE11ΔN and STE5-CPRRAS2 arrest the cell cycle in a far1Δ strain, while STE4overexpression does not. Progression through the cell cycle may play a role in the inactivation of the pheromone response in STE4 cells. This seems particularly likely because pheromone sensitivity varies over the course of the cell cycle and cells are most sensitive in G1, which is the arrest point of cells expressing STE11ΔN and STE5-CPRRAS2 [29, 30]. Cell cycle arrest may serve as an additional layer of positive feedback that stabilizes the active state by keeping cells sensitive to the feedback signal. When cells continue through the cell cycle, feedback that is strong enough to maintain the active state in G1 may be insu3cient in other parts of the cell cycle. Changes in sensitivity may explain why PFUS1-STE4 cells initially enter the active stable state but subsequently inactivate the pheromone response. We also observed that most of the PFUS1-STE4 cells that became inactive were the new daughter resulting from the asymmetric division of budding yeast (data not shown). Active, arrested PFUS1-STE11ΔN and PFUS1-STE5-CPRRAS2 cells do not produce daughters, further supporting the idea that the qualitative dif-ference between the feedback loops relates to their effects on the cell cycle. The converse possibility, that differences in active state stability between STE4 and STE11ΔN cause the different cell cycle arrest phenotypes, seemed less likely because even PFUS1J2-STE4×3 cells with a completely stable active state continue to divide.

Table 1
Phenotypes of positive feedback in the pheromone response. The phenotypes of strains with different positive feedback loops in the pheromone response. The positive feedback protein and the expression levels of the promoter driving the positive feedback ...

While cell cycle effects may explain the difference between STE4- and STE11ΔN-mediated positive feedback, they do not explain the heterogeneous response in a population of genetically identical cells with PFUS1-STE4. A recent study by Colman-Lerner et al. quantified variability in the pheromone response [31]. In addition to cell cycle effects, they found significant cell-to-cell variation in the strength of pheromone-induced transcription. Variability was dominated by differences in the overall activity of the transcriptional machinery between cells (extrinsic noise) rather than differences in the transcription of different copies of the gene in cells with identical overall transcriptional activity (intrinsic noise). Our data also supported extrinsic variation in the pheromone response, as cells with more integrated copies of the reporter had higher fluorescence but no less variation than those with fewer (see Fig. S2); intrinsic gene expression noise decreases as reporter copy number increases [32]. We incorporated cell- to-cell variability in a simulation of our positive feedback system. These simulations produced a bimodal fluorescence distribution similar to that seen for PFUS1-STE4 cells (see Fig. 4H-J), with weak positive feedback causing delayed inactivation after pheromone removal and stronger positive feedback causing an increasing fraction of persistently active cells. Thus, the known variability in the pheromone pathway can account for the bimodal population response we observe. However, our simulations did not capture the strong daughter bias amongst the inactivating cells despite incorporating asymmetric division (data not shown). This may point to a strong asymmetry in the inheritance of positive feedback protein or to some correlation between asymmetric division and variations in the pheromone response.

We next tuned the strength of positive feedback non-genetically by titrating SI, thereby partly blocking pheromone signaling. When we simulated the effects of inducing bistable cells and then releasing them into different concentrations of SI, we again saw a bimodal fluorescence distribution (see Fig. 5A). Stronger positive feedback, corresponding to a less inhibited pathway, resulted in a larger fraction of active cells. We observed the same e3ect experimentally–above 5 nM SI, there was a significant sub-population of entirely active cells (see Fig. 5B), and there is a gradual decrease in fluorescence intensity of the active state as well as a decrease in the fraction of active cells as the inhibitor concentration is raised. Active state fluorescence decreases because SI inhibits the pheromone response globally, including the induction of the fluorescent reporter. In contrast, genetic reduction of the strength of positive feedback, which does not affect other parts of the pathway, does not affect the fluorescence of the active state. In both cases, however, weakening positive feedback decreases the fraction of active cells.

Figure 5
Partial induction and inhibition of the bistable pheromone response.(A) Population distributions from stochastic simulations of pheromone induction and subsequent pathway inhibition. Simulations were performed with different levels of inhibition (shown ...

Bistability allows a mix of responses in a single population with genetically regulated ratios

The ability to produce multiple discrete cellular responses can be important for microbes, which use them to implement probabilistic survival strategies such as the generation of a small fraction of slow-growing, hardy, persister cells in a clonal population [33]. Distinct cell types in the pathogenic yeast Candida albicans are also discrete cellular responses under the control of a bistable positive feedback loop [34,35]. The bimodal reporter expression in PFUS1-STE4 strains shows how bistable gene regulatory networks can produce discrete responses that correspond to their distinct stable states. Furthermore, the spontaneous inactivation we observe corresponds to stochastic switching between responses, which would allow the descendents of a single cell to produce a repertoire of different behaviors. Both the ratio of different responses in a given environment and the rate of switching between them are key parameters in determining the fitness of a probabilistic survival strategy in a fluctuating environment [36]. In our system, these parameters are under genetic control, with point mutations in the feedback promoter changing the strength of feedback and thereby altering the stability of the active state.

Populations with a very stable active state may still show a mix of different responses when induced with a weak signal. We tested the effects of low concentrations of pheromone on cells with strong positive feedback. Cells carrying three copies of PFUS1J5-STE4 show almost complete activation in response to strong induction, but we found that they respond bimodally to lower levels of pheromone. The initial response of these cells to graded pheromone induction is graded reporter expression, with little difference between wild-type cells and those with positive feedback. At later times, cells with positive feedback showed different fractions of active and inactive cells depending on the strength of induction (see Fig. 5C). The levels of reporter expression in the active and inactive populations are independent of the level of pheromone treatment. After treatment with 1.0 nM pheromone, for instance, some cells become more fluorescent while others become less fluorescent, but none remain at the intermediate levels of reporter expression seen during induction. Our simulations also showed a graded early response followed by a bimodal fluorescence distribution at later timepoints. Bistable systems can gravitate towards discrete stable states in spite of significant stochastic variation. Strong positive feedback can create a bimodal response in weakly induced cells which is not associated with the unstable active state we saw in cells with weak feedback. Bistable systems have been proposed as a mechanism for responding to morphogen gradients, a context in which it is necessary to adopt discrete cell fates in response to graded induction (reviewed in [37, 38]). Our results demonstrate that positive feedback in the pheromone response can produce such discrete fates.

The bistable feedback loop can be activated without pheromone induction

In development, one pathway is often used to activate a pattern of gene expression while another is used to maintain it. In Drosophila, the pair-rule genes induce the initial expression of proteins like engrailed in the early stages of development, but a distinct positive feedback loop maintains their expression in later stages [3941]. We attempted to create an independent inducer that would confirm that the bistability we had engineered required continued transcription from the positive feedback promoter, rather than reflecting the inability to dilute or degrade an excess of a dominant active protein that had been made during the initial induction. We used the galactose-induced yeast promoter PGAL1 to drive expression of STE4. This allowed us to activate the mating response using only galactose, without any pheromone treatment (see Fig. 6A,B). Even cells without feedback showed a slower inactivation of the pheromone response after PGAL1 repression than after pheromone removal, reflecting the time needed to degrade the STE4 protein. However, only the strain which also had positive feedback showed long-term activation, demonstrating that transient STE4 overexpression cannot cause a persistent pheromone response–continuing positive feedback expression is needed for bistability. We did note that only half of the population remained in the active stable state, which we attribute to the fact that galactose induction requires growth conditions that weaken the pheromone response (data not shown).

Figure 6
Galactose induction of the active stable state.(A) Cells with gal1 Δ gal10Δ PACT1-GAL3 PGAL1-STE4, without positive feedback, were grown in YEP ra3nose, induced with 160 μM galactose, and transferred to YEP dextrose to inactivate ...

Our results show how adding positive feedback to an existing eukaryotic signaling pathway can convert it into a bistable system. Changing parameters such as basal and induced expression levels of the feedback promoter, which can be accomplished with a few point mutations, can switch between inducible, bistable, and constitutive responses. We also show how pathway noise, which is ubiquitous in biological systems, produces bimodal population response. The ratio of the two discrete responses was controlled by the strength of the positive feedback and of the inducing signal. Thus, positive feedback loops in a eukaryotic genetic network can act as an evolvable genetic circuit. Systems with positive feedback can produce a range of different behaviors, and mutations can tune the feedback loop to produce qualitative variations in phenotype.

Materials and Methods

Yeast Strains and Plasmid Construction

Standard microbial and genetic techniques were generally performed according to [42]. Strains used in this study are isogenic with the W303 ura3-1 ade2-1 his3-11,15 leu2-3,112 trp1-1 can1-100 background, all derived from parental strain yNTI1. Details of yeast strain and plasmid construction are given in supporting information. The full genotype of all yeast strains is given in Table S1, plasmids are described in Table S2, and oligos are listed in Table S3.

In order to analyze integrant copy number, yeast genomic DNA was extracted using freeze-thaw lysis followed by chloroform extraction [43]. Southern blotting was performed with depurination followed by neutral transfer onto Hybond-N+ (Amersham RPN203B) according to the manufacturer’s instructions. Probes were generated by PCR from plasmids using cloning primers and then labeled, hybridized, and visualized with the AlkPhos Direct system (Amersham RPN3690).

Pheromone Induction and Flow Cytometry

Cells were grown in YPD unless otherwise noted. Overnight cultures were diluted into fresh media and grown for 12 to 16 hours at 30 C, reaching a final density of 1 × 106 to 5 × 106 cells / ml. Cultures were diluted as necessary to maintain all samples at a cell density no greater than 1 × 107 cells / ml, and in each experiment all strains were diluted in parallel.

A stock of pheromone (Bio-Synthesis, Inc., Lewisville, TX) at 10 mg / ml in DMSO, corresponding to 6 mM, was used for all inductions. The pheromone stock was serially diluted into rich yeast media containing YEP when necessary. Pheromone was removed by washing cells at least twice in media, and cells were then resuspended in fresh media to continue growth.

A stock of Shokat’s Inhibitor at 10 mg / ml was generously provided by K. Shokat. When necessary, serial dilutions were made in YPD.

Cells were harvested by centrifugation and resuspended in 1× PBS+YNB (DIFCO Yeast Nitrogen Base without amino acids, #291920, 10× stock is 7.0 g in 100 ml water, filter sterilized). Cells were then sonicated 60 to 75 s in an ice-water bath in a 2” cup horn of a Branson Sonifier 250. Cells were analyzed on a DakoCytomation MoFlo, with 488 nm excitation at 80 to 100 mW. YFP fluorescence was measured in the FL1 channel (requires reflection from a 555 nm dichroic long pass mirror and from a 505 nm dichroic short pass mirror, and transmission through a 530 / 40 band-pass filter). The signal was recorded with the amplifier in log mode and the PMT voltage between 500 and 640. All samples in a given experiment were measured in parallel with the same instrument settings and fluorescence levels can be compared between different strains and timepoints. Fluorescence levels in each experiment were normalized based on the median fluorescence of an uninduced wild-type sample with the PFUS1-yEVenus-CLN2PEST reporter (yNTI83 or yNTI101) unless otherwise noted.

Details of strains and conditions used in each induction experiment are given in the supporting information.

Videomicroscopy

Videomicroscopy was performed according to (M. Piel and A. Murray, in preparation). Cells were grown in the same flow chamber and the two strains were distinguished by cell wall labeling as described. After establishing cells in DO -Met media at 25 C, the media flow was switched to Do -Met with saturating levels of pheromone for 2 hours of induction. The media was then switched back to Do -Met without pheromone and cells were followed for an additional 6 to 8 hours, at which point essentially all cells in the flow chamber had become dislodged.

Simulations

Lineages were modeled with exponential growth of the size of single cells followed by asymmetric division and the random selection of a mother or a daughter cell. Cell size during this deterministic growth was used to identify different cell cycle phases. Levels of fluorescent protein and of positive feedback protein were controlled by the rate of transcription from a pheromone-responsive promoter and first-order decay as well as dilution through cell division. Transcription from this promoter was blocked during the middle portion of the cell cycle or in the presence of SI. When transcription was not blocked, the effects of pheromone and of positive feedback protein were modeled with a Hill function with cooperativity ν = 2. The details of the simulation are given in the supporting information.

Supplementary Material

01

Acknowledgments

We thank M. Piel for assistance with videomicroscopy; M. Fischbach for assistance with preliminary experiments; all members of the Murray lab as well as N. Barkai, D. Fisher, R. Losick, and L. Lareau for helpful comments and suggestions; K. Shokat for the gift of Shokat’s Inhibitor; and M. Rose for the gift of plasmid pMR4280. Supported by NIH grants GM062566 and GM068763 (A.W.M.) and by a Howard Hughes Medical Institute pre-doctoral fellowship (N.T.I.).

Footnotes

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