Cell polarity refers to the asymmetric distribution of proteins and other molecules along a specified axis within a cell. Polarity establishment is the first step in many cellular processes. For example, directed growth or migration requires the formation of a cell front and back. In many cases, polarity occurs in the absence of spatial cues. That is, the cell undergoes symmetry breaking. Understanding the molecular mechanisms that allow cells to break symmetry and polarize requires computational models that span multiple spatial and temporal scales. Here, we apply a multiscale modeling approach to examine the polarity circuit of yeast. In addition to symmetry breaking, experiments revealed two key features of the yeast polarity circuit: bistability and rapid dismantling of the polarity site following a loss of signal. We used modeling based on ordinary differential equations (ODEs) to investigate mechanisms that generate these behaviors. Our analysis revealed that a model involving positive and negative feedback acting on different time scales captured both features. We then extend our ODE model into a coarse-grained reaction–diffusion equation (RDE) model to capture the spatial profiles of polarity factors. After establishing that the coarse-grained RDE model qualitatively captures key features of the polarity circuit, we expand it to more accurately capture the biochemical reactions involved in the system. We convert the expanded model to a particle-based model that resolves individual molecules and captures fluctuations that arise from the stochastic nature of biochemical reactions. Our models assume that negative regulation results from negative feedback. However, experimental observations do not rule out the possibility that negative regulation occurs through an incoherent feedforward loop. Therefore, we conclude by using our RDE model to suggest how negative feedback might be distinguished from incoherent feedforward regulation.

Cells use signaling pathways to receive and process information about their environment. These nonlinear systems rely on feedback and feedforward regulation to respond appropriately to changing environmental conditions. Mathematical models describing signaling pathways often lack predictive power because they are not trained on data that encompass the diverse time scales on which these regulatory mechanisms operate. We addressed this limitation by measuring transcriptional changes induced by the mating response in Saccharomyces cerevisiae exposed to different dynamic patterns of pheromone. We found that pheromone-induced transcription persisted after pheromone removal and showed long-term adaptation upon sustained pheromone exposure. We developed a model of the regulatory network that captured both characteristics of the mating response. We fit this model to experimental data with an evolutionary algorithm and used the parameterized model to predict scenarios for which it was not trained, including different temporal stimulus profiles and genetic perturbations to pathway components. Our model allowed us to establish the role of four architectural elements of the network in regulating gene expression. These network motifs are incoherent feedforward, positive feedback, negative feedback, and repressor binding. Experimental and computational perturbations to these network motifs established a specific role for each in coordinating the mating response to persistent and dynamic stimulation.

Cells polarize their growth or movement in many different physiological contexts. A key driver of polarity is the Rho GTPase Cdc42, which when activated becomes clustered or concentrated at polar sites. Multiple models for polarity establishment have been proposed. All of them rely on positive feedback to reinforce regions of high Cdc42 activity. Positive feedback can lead to bistability, a scenario in which cells can exist in either a polarized or unpolarized state under identical external conditions. Determining if the signaling circuit that drives Cdc42 polarity is bistable would provide important information about the mechanism that underlies polarity establishment and insights into the design features required for proper cellular function. We studied polarity establishment during the mating response of yeast. Using microfluidics to precisely control the temporal profile of mating pheromone and live-cell imaging to monitor the polarity process in single living cells, we found that the polarity circuit of yeast shows hysteresis, a characteristic feature of bistable systems. Our analysis also revealed that cells exposed to high pheromone concentrations rapidly lose polarity following a precipitous removal of pheromone. We used a reaction-diffusion model for polarity establishment to demonstrate that delayed negative regulation is sufficient to explain our experimental results. [Media: see text] [Media: see text] [Media: see text] [Media: see text].

Cells rely on mitogen-activated protein kinases (MAPKs) to survive environmental stress. In yeast, activation of the MAPK Hog1 is known to mediate the response to high osmotic conditions. Recent studies of Hog1 revealed that its temporal activity is subject to both negative and positive feedback regulation, yet the mechanisms of feedback remain unclear. By designing mathematical models of increasing complexity for the Hog1 MAPK cascade, we identified pathway circuitry sufficient to capture Hog1 dynamics observed in vivo. We used these models to optimize experimental designs for distinguishing potential feedback loops. Performing experiments based on these models revealed mutual inhibition between Hog1 and its phosphatases as the likely positive feedback mechanism underlying switch-like, dose-dependent MAPK activation. Importantly, our findings reveal a new signaling function for MAPK phosphatases. More broadly, they demonstrate the value using mathematical models to infer targets of feedback regulation in signaling pathways.### Competing Interest StatementThe authors have declared no competing interest.

The mating pathway in yeast Saccharomyces cerevisiae has long been used to reveal new mechanisms of signal transduction. The pathway comprises a pheromone receptor, a heterotrimeric G protein, and intracellular effectors of morphogenesis and transcription. Polarized cell growth, in the direction of a potential mating partner, is accomplished by the G-protein beta gamma subunits and the small G-protein Cdc42. Transcription induction, needed for cell-cell fusion, is mediated by G beta gamma and the mitogen-activated protein kinase (MAPK) scaffold protein Ste5. A potential third pathway is initiated by the G-protein alpha subunit Gpa1. Gpa1 signaling was shown previously to involve the F-box adaptor protein Dia2 and an endosomal effector protein, the phosphatidylinositol 3-kinase Vps34. Vps34 is also required for proper vacuolar sorting and autophagy. Here, using a panel of reporter assays, we demonstrate that mating pheromone stimulates vacuolar targeting of a cytoplasmic reporter protein and that this process depends on Vps34. Through a systematic analysis of F-box deletion mutants, we show that Dia2 is required to sustain pheromone-induced vacuolar targeting. We also found that other F-box proteins selectively regulate morphogenesis (Ydr306, renamed Pfu1) and transcription (Ucc1). These findings point to the existence of a new and distinct branch of the pheromone-signaling pathway, one that likely leads to vacuolar engulfment of cytoplasmic proteins and recycling of cellular contents in preparation for mating.

Mitogen-activated protein kinase (MAPK) pathways control many cellular processes, including differentiation and proliferation. These pathways commonly activate MAPK isoforms that have redundant or overlapping function. However, recent studies have revealed circumstances in which MAPK isoforms have specialized, nonoverlapping roles in differentiation. The mechanisms that underlie this specialization are not well understood. To address this question, we sought to establish regulatory mechanisms that are unique to the MAPK Fus3 in pheromone-induced mating and chemotropic fate transitions of the budding yeast Saccharomyces cerevisiae. Our investigations reveal a previously unappreciated role for inactive Fus3 as a potent negative regulator of pheromone-induced chemotropism. We show that this inhibitory role is dependent on inactive Fus3 binding to the α-subunit of the heterotrimeric G-protein. Further analysis revealed that the binding of catalytically active Fus3 to the G-protein is required for gradient tracking and serves to suppress cell-to-cell variability between mating and chemotropic fates in a population of pheromone-responding cells.

G protein-coupled receptor (GPCR) signaling is fundamental to physiological processes such as vision, the immune response, and wound healing. In the budding yeast Saccharomyces cerevisiae, GPCRs detect and respond to gradients of pheromone during mating. After pheromone stimulation, the GPCR Ste2 is removed from the cell membrane, and new receptors are delivered to the growing edge. The regulator of G protein signaling (RGS) protein Sst2 acts by accelerating GTP hydrolysis and facilitating pathway desensitization. Sst2 is also known to interact with the receptor Ste2. Here we show that Sst2 is required for proper receptor recovery at the growing edge of pheromone-stimulated cells. Mathematical modeling suggested pheromone-induced synthesis of Sst2 together with its interaction with the receptor function to reestablish a receptor pool at the site of polarized growth. To validate the model, we used targeted genetic perturbations to selectively disrupt key properties of Sst2 and its induction by pheromone. Together our results reveal that a regulator of G protein signaling can also regulate the G protein-coupled receptor. Whereas Sst2 negatively regulates G protein signaling, it acts in a positive manner to promote receptor retention at the growing edge.

Different environmental stimuli often use the same set of signaling proteins to achieve very different physiological outcomes. The mating and invasive growth pathways in yeast each employ a mitogen-activated protein (MAP) kinase cascade that includes Ste20, Ste11, and Ste7. Whereas proper mating requires Ste7 activation of the MAP kinase Fus3, invasive growth requires activation of the alternate MAP kinase Kss1. To determine how MAP kinase specificity is achieved, we used a series of mathematical models to quantitatively characterize pheromone-stimulated kinase activation. In accordance with the computational analysis, MAP kinase feedback phosphorylation of Ste7 results in diminished activation of Kss1, but not Fus3. These findings reveal how feedback phosphorylation of a common pathway component can limit the activity of a competing MAP kinase through feedback phosphorylation of a common activator, and thereby promote signal fidelity.

Ideal reporter genes for temporal transcription programmes have short half-lives that restrict their detection to the window in which their transcripts are present and translated. In an effort to meet this criterion for reporters of transcription in individual living cells, we adapted the ubiquitin fusion strategy for programmable N-end rule degradation to generate an N-degron version of green fluorescent protein (GFP) with a half-life of ~7 min. The GFP variant we used here (designated GFP*) has excellent fluorescence brightness and maturation properties, which make the destabilized reporter well suited for tracking the induction and attenuation kinetics of gene expression in living cells. These attributes are illustrated by its ability to track galactose- and pheromone-induced transcription in S. cerevisiae. We further show that the fluorescence measurements using the short-lived N-degron GFP* reporter gene accurately predict the transient mRNA profile of the prototypical pheromone-induced FUS1 gene.

All cells must detect and respond to changes in their environment, often through changes in gene expression. The yeast pheromone pathway has been extensively characterized, and is an ideal system for studying transcriptional regulation. Here we combine computational and experimental approaches to study transcriptional regulation mediated by Ste12, the key transcription factor in the pheromone response. Our mathematical model is able to explain multiple counterintuitive experimental results and led to several novel findings. First, we found that the transcriptional repressors Dig1 and Dig2 positively affect transcription by stabilizing Ste12. This stabilization through protein-protein interactions creates a large pool of Ste12 that is rapidly activated following pheromone stimulation. Second, we found that protein degradation follows saturating kinetics, explaining the long half-life of Ste12 in mutants expressing elevated amounts of Ste12. Finally, our model reveals a novel mechanism for robust perfect adaptation through protein-protein interactions that enhance complex stability. This mechanism allows the transcriptional response to act on a shorter time scale than upstream pathway activity.

The maintenance and detection of signaling gradients are critical for proper development and cell migration. In single-cell organisms, gradient detection allows cells to orient toward a distant mating partner or nutrient source. Budding yeast expand their growth toward mating pheromone gradients through a process known as chemotropic growth. MATα cells secrete α-factor pheromone that stimulates chemotropism and mating differentiation in MATa cells and vice versa. Paradoxically, MATa cells secrete Bar1, a protease that degrades α-factor and that attenuates the mating response, yet is also required for efficient mating. We observed that MATa cells avoid each other during chemotropic growth. To explore this behavior, we developed a computational platform to simulate chemotropic growth. Our simulations indicated that the release of Bar1 enabled individual MATa cells to act as α-factor sinks. The simulations suggested that the resultant local reshaping of pheromone concentration created gradients that were directed away from neighboring MATa cells (self-avoidance) and that were increasingly amplified toward partners of the opposite sex during elongation. The behavior of Bar1-deficient cells in gradient chambers and mating assays supported these predictions from the simulations. Thus, budding yeast dynamically remodel their environment to ensure productive responses to an external stimulus and avoid nonproductive cell-cell interactions.

Cell differentiation requires the ability to detect and respond appropriately to a variety of extracellular signals. Here we investigate a differentiation switch induced by changes in the concentration of a single stimulus. Yeast cells exposed to high doses of mating pheromone undergo cell division arrest. Cells at intermediate doses become elongated and divide in the direction of a pheromone gradient (chemotropic growth). Either of the pheromone-responsive MAP kinases, Fus3 and Kss1, promotes cell elongation, but only Fus3 promotes chemotropic growth. Whereas Kss1 is activated rapidly and with a graded dose-response profile, Fus3 is activated slowly and exhibits a steeper dose-response relationship (ultra-sensitivity). Fus3 activity requires the scaffold protein Ste5; when binding to Ste5 is abrogated, Fus3 behaves like Kss1, and the cells no longer respond to a gradient or mate efficiently with distant partners. We propose that scaffold proteins serve to modulate the temporal and dose-response behavior of the MAP kinase.

Fluorescent proteins are often used as reporters of transcriptional activity. Here we present a mathematical characterization of a novel fluorescent reporter that was recently engineered to have a short half-life (approximately 12 min). The advantage of this destabilized protein is that it can track the transient transcriptional response often exhibited by signaling pathways. Our mathematical model takes into account the maturation time and half-life of the fluorescent protein. We demonstrate that our characterization allows transient transcript profiles to be inferred from fluorescence data. We also investigate a stochastic version of the model. Our analysis reveals that fluorescence measurements can both underestimate and overestimate fluctuations in protein levels that arise from the stochastic nature of biochemical reactions.

Different environmental stimuli often use the same set of signaling proteins to achieve very different physiological outcomes. The invasive growth and mating pathways in yeast Saccharomyces cerevisiae each employ a mitogen-activated protein (MAP) kinase cascade that includes several common components (e.g. Ste11, Ste7), after which the pathways diverge leading to activation of different MAP kinases (Fus3 and Kss1, respectively). In order to determine how pathway specificity is achieved, we conducted a quantitative analysis of MAP kinase activity, and devised a series of six computational models that describe their behavior in vivo. As anticipated by one of the models, MAP kinase phosphorylation of shared pathway components results in diminished activation of one MAP kinase but not the other. These findings reveal how one MAP kinase can limit the activity of a competing MAP kinase and thereby ensure signal fidelity. Supported by National Institutes of Health GM073180 (H. G.) and GM073180 (T. E.) and the American Heart Association 0415413U (N. H.).

The 'programmable' features of the N-end rule degradation pathway and a ubiquitin fusion strategy were exploited to create a family of destabilized cyan fluorescent proteins (CFP) to be used as transcriptional reporters. The N-degron CFP reporters characterized in this report have half-lives of approximately 75, 50 and 5 min, but further modification of the N-degron signal sequences could readily generate additional variants within this range. These destabilized CFP reporters have been engineered into convenient plasmid constructs with features to enable their expression from upstream activating sequences of choice and to facilitate their targeted integration to the URA3-TIM9 intergenic region of chromosome V. The advantages and limitations of these reporters as temporal indicators of gene expression in living cells are illustrated by their application as reporters of galactose- and pheromone-induced transcription. The plasmid design we describe and the range of different stabilities that are theoretically feasible with this strategy make the N-degron CFP reporters easily adapted to a variety of applications.

The Ste12 transcription factor of Saccharomyces cerevisiae regulates transcription programs controlling two different developmental fates. One is differentiation into a mating-competent form that occurs in response to mating pheromone. The other is the transition to a filamentous-growth form that occurs in response to nutrient deprivation. These two distinct roles for Ste12 make it a focus for studies into regulatory mechanisms that impart biological specificity. The transient signal characteristic of mating differentiation led us to test the hypothesis that regulation of Ste12 turnover might contribute to attenuation of the mating-specific transcription program and restrict activation of the filamentation program. We show that prolonged pheromone induction leads to ubiquitin-mediated destabilization and decreased amounts of Ste12. This depletion in pheromone-stimulated cultures is dependent on the mating-pathway-dedicated mitogen-activated protein kinase Fus3 and its target Cdc28 inhibitor, Far1. Attenuation of pheromone-induced mating-specific gene transcription (FUS1) temporally correlates with Ste12 depletion. This attenuation is abrogated in the deletion backgrounds (fus3Delta or far1Delta) where Ste12 is found to persist. Additionally, pheromone induces haploid invasion and filamentous-like growth instead of mating differentiation when Ste12 levels remain high. These observations indicate that loss of Ste12 reinforces the adaptive response to pheromone and contributes to the curtailing of a filamentation response.

Mitogen-activated protein kinase kinase kinase-Ste11 (MAPKKK-Ste11), MAPKK-Ste7, and MAPK-Kss1 mediate pheromone-induced mating differentiation and nutrient-responsive invasive growth in Saccharomyces cerevisiae. The mating pathway also requires the scaffold-Ste5 and the additional MAPK-Fus3. One contribution to specificity in this system is thought to come from stimulus-dependent recruitment of the MAPK cascade to upstream activators that are unique to one or the other pathway. To test this premise, we asked if stimulus-independent signaling by constitutive Ste7 would lead to a loss of biological specificity. Instead, we found that constitutive Ste7 promotes invasion without supporting mating responses. This specificity occurs because constitutive Ste7 activates Kss1, but not Fus3, in vivo and promotes filamentation gene expression while suppressing mating gene expression. Differences in the ability of constitutive Ste7 variants to bind the MAPKs and Ste5 account for the selective activation of Kss1. These findings support the model that Fus3 activation in vivo requires binding to both Ste7 and the scaffold-Ste5 but that Kss1 activation is independent of Ste5. This scaffold-independent activation of Kss1 by constitutive Ste7 and the existence of mechanisms for pathway-specific promoter discrimination impose a unique developmental fate independently of any distinguishing external stimuli.