Fig 1.
Basics of yeast mating and model.
A) Mating as a developmental switch. Exposing haploid cells to pheromone makes them exit the cell division cycle, polarize towards, and fuse with a cell of the opposite mating type to form a diploid cell. Cells recover and resume budding if the pheromone signal disappears. The polarisome protein, Spa2 (shown in green and red), concentrates at the incipient bud site, at the bud tip as cells grow, at the bud neck during cytokinesis, and at the shmoo tip and fusion site during mating. We use the rate of accumulation of a fluorescent protein expressed from the FUS1 promoter, a gene induced by pheromone stimulation, as a readout of pheromone-induced signaling. B) The pheromone response pathway of a cells. α-factor diffuses through the cell wall and binds to the α-factor receptor (Ste2, a G protein-coupled receptor (GPCR)), which activates a trimeric G protein. The G protein recruits and activates two scaffolding proteins. One (Far1) recruits the actin polymerization machinery that leads to cell polarization and activates the kinase that activates the MAP kinase cascade. The other (Ste5) is responsible for the assembly of a MAP kinase cascade that activates the MAP kinase Fus3, leading to the induction of mating genes. Once phosphorylated by Fus3, Far1 arrests the cell cycle in G1. C) A two-dimensional model of cell polarization. The left panel shows a cell, the right a more detailed view introducing the various parameters in the model. Actin is polymerized into short filaments, that interact with each other and these are bundled together to form actin cables that cross the cell. The nucleation of filaments is proportional to both the local density of Cdc42 and to the concentration of pheromone. The endocytosis of markers at the membrane is described by very simple attachment/detachment dynamics (kon and koff) and their diffusion in the plane of the membrane is described by a diffusion constant, Dm, while their diffusion in the cytoplasm is described by a separate constant, Db. D) According to the fact that a polarisome occupies approximately 10% of the total membrane length, [12], we divide the membrane into 10 subregions (sectors). At each time point, κ(t,.), the parameter that describes the dynamics of the pheromone receptor, has a constant value within one segment. E) The temporal evolution of the effective pheromone receptor activity κ is ruled by an Ornstein-Uhlenbeck process. The parameter λ is the the noise damping. The parameter σ is the standard deviation of the instantaneous change of κ(t).
Table 1.
The sixteen parameters involved in the model with their definitions, their values, and the source from which the value is derived.
Fig 2.
The response of bar1Δ cells response to homogenous stimulation by α-factor.
A) Schematic of a device to produce a range of pheromone concentrations by using chaotic mixers in dilution chambers. The diagram shows a plan view (left) and cross sections at two magnifications (right). Structured micro channels allow fast mixing and thus permit serial dilutions in a small device. A lectin (concanavalin A) binds yeast cells to the coverslip that forms the roof of the chamber. They receive a constant flow of pheromone, allowing them to be exposed to concentrations that are stable over several hours and can be measured by quantifying the emission of a fluorescent dextran mixed with the pheromone (see Experimental Procedures in S1 Text and Fig A in S1 Text for more details). B) The behavior of cells in micro-channels at various pheromone concentrations. We assessed cell behavior at each pheromone concentration by following cells over time and overlaying differential interference contrast and Spa2-YFP images (see Fig B in S1 Text for images). The graph quantifies the bud/shmoo transition in spatially uniform fields of pheromone and summarizes data from about 4000 cells (on each curve (red and green), at least 160 cells (MP 384) were used to obtain the averages shown for each pheromone dose) in seven independent dilution chambers. The inset shows the standard deviation of the fraction of different events between experiments. The measured dissociation constant of α-factor from Ste2 is indicated (Kd) [41]. The half maximal point of the sigmoidal fit is 1.02 ± 0.03 nM and the Hill coefficient for the transition between budding and shmooing is 6.5 ± 0.6 (95% confidence interval). C) Quantification of polarization delays. Time was measured from the end of the first cytokinesis after the onset of pheromone treatment. Cells were considered polarized when a stable focused Spa2 cap was formed. Only those cells whose progenitors had completed cytokinesis during the first three hours after the start of pheromone treatment were considered. At least 52 cells (MP 384) were used to obtain the averages shown for each point. D and E) Cells arrested in G1 by G1 cyclin depletion were placed in exponential dilution chambers together with wild type cycling cells (using a fluorescent cell wall marker to distinguish the two strains). Experiments and measures are similar to those shown in B) and C). The transition to shmooing occurs at exactly the same concentration for G1 arrested and cycling cells (G1 arrested cells that do not shmoo grow isotropically, forming large spherical cells). Because Cln-depleted cannot leave G1 and cells arrested in G1 never polarize except when they form a shmoo, we do not show the delay before rebudding for wild type cells. The delay before shmooing is similar for cycling cells and for G1-arrested cells. In graph D) each point corresponds to an average value computed over 66 cells (MP 384 and MP 1333) and in graph E) over 52 cells (MP 384 and MP 1333).
Fig 3.
Model behavior and model parameters value selection.
A and B) Increasing the strength of pheromone stimulation (S) leads to increasing levels of spatial segregation and for a given S, depending on the spatial correlation length χ and the endocytosis rate koff the cell will polarize or not. Shown are kymographs from simulations (y axis, membrane position; x axis, time). Particle density is in absolute value. On the left we see an unpolarized cell and a polarized one on the right. C) Representation of mean time polarization isovalues for S = 1 nM as a function of the spatial correlation length χ and the endocytosis rate koff. If the numerical simulations did not lead to a polarized state before 10 hours, we considered the cell to be non-polarized. D) Representation of the cost function (labeled as fit quality) depending on χ and koff, allowing us to determine the optimal pair of values (χ, koff that fit the data (Fig 2C). E) For the optimal values of χopt = 2.5 × 105 μm2/s, s−1 and δopt = 0.2, we show the simulated timing of polarization for solutions to our model under varying pheromone concentration compared to the experimental data for polarization timing. The inset shows the standard deviation of the timing of polarization for solutions to our model under varying pheromone concentration compared to the experimental data for polarization timing. F) For the same optimal parameters
, we represent the numerical fraction of cells which polarize under varying pheromone concentration. Before fitting the model parameters with the data, we defined the numerical criterion for polarization. Following [42], we considered that polarization occurred when more than 50% of the total membrane protein pool was located in a window of 10% of the membrane. Hence, a numerical simulation corresponded to a polarization state if there existed a time for which more than 50% of the total membrane Cdc42 was located in less than 10% of the perimeter of the cell boundary (S1 Text).
Fig 4.
Response to pheromone gradients and model prediction.
A) Producing pheromone gradients in a laminar flow chamber. Pheromone mixed with a fluorescent dextran and a dilution buffer enters through two ports and diffusion between the two fluid streams creates a temporally stable, gradient (see Experimental Procedures in S1 Text, and Fig C in S1 Text for more details). The left hand view is from the top of the apparatus and right hand views are two different magnifications of a cross-section, showing the cells attached to the coverslip that forms the roof of the chamber. B) The transition between budding and shmooing, quantified as in Fig 2B. The lightly shaded, thick curves show the data from spatially uniform pheromone concentrations (Fig 2B) for comparison. For every cell, the difference in concentration between the two edges of the cell was > 5% (expressed relative to the mean concentration that the cell experienced). The inset shows the standard deviation. C) The accuracy of gradient detection as a function of pheromone concentration. Accuracy is defined as the mean cosine of the angle between the gradient and the line that connects the Spa2 polar cap to the center of the cell. D) The accuracy of gradient detection as a function of pheromone concentration for solutions to our model. For different damping coefficients λ, the gradient detection and the standard deviation were computed. We observe that the value of λ which fits the data the best is 101.25 which means 1/λopt ≈ 3 mn. E) We show the timing of polarization in representative numerical simulations of cells exposed to a uniform field and gradient field of α-factor (images are shown every 20 minutes). F) Comparing the timing of polarization in individual cells exposed to a uniform field (top) and a gradient (bottom) of α-factor. Images were taken every 20 minutes and pseudocolored to indicate the intensity of Spa2-YFP fluorescence. Note the small unstable Spa2 spots (arrowheads) that appeared in the homogenously stimulated cell long before a stable polar cap, which took four hours to develop. In contrast, in the gradient, a small Spa2 spot first appeared in the direction of the gradient and then gradually grew stronger allowing the cell polarize much faster.