Sensory input attenuation allows predictive sexual response in yeast

Animals are known to adjust their sexual behaviour depending on mate competition. Here we report similar regulation for mating behaviour in a sexual unicellular eukaryote, the budding yeast Saccharomyces cerevisiae. We demonstrate that pheromone-based communication between the two mating types, coupled to input attenuation by recipient cells, enables yeast to robustly monitor relative mate abundance (sex ratio) within a mixed population and to adjust their commitment to sexual reproduction in proportion to their estimated chances of successful mating. The mechanism of sex-ratio sensing relies on the diffusible peptidase Bar1, which is known to degrade the pheromone signal produced by mating partners. We further show that such a response to sexual competition within a population can optimize the fitness trade-off between the costs and benefits of mating response induction. Our study thus provides an adaptive explanation for the known molecular mechanism of pheromone degradation in yeast.


Supplementary Figure 4.
Transcriptional regulation and putative cell-wall associated fraction of Bar1 do not affect pathway response. a. Bar1 expression is less sensitive to αfactor than P FUS1 -GFP expression. A strain carrying both the P FUS1 -GFP reporter (blue) and an mCherry-tagged functional version of Bar1 (red) was exposed to different α-factor concentrations. Fluorescence was measured using microscopy (see Supplementary Fig. 1). For the P FUS1 -GFP reporter, fluorescence of unstimulated cells was subtracted as in all other experiments with purified α-factor. For the Bar1-mCherry fusion, the autofluorescence of cells without mCherry was subtracted instead, to highlight the background value of Bar1 expression in unstimulated cells. For both reporters, the values were normalized to the respective maximal response. Bar1-mCherry fusion protein is induced at much higher pheromone concentrations than the P FUS1 -GFP reporter, suggesting that production of Bar1 remains at a basal level in the studied range of the mating response. b. Bar1 induction by αfactor is not important in determining the sex-ratio response. The BAR1 promoter was replaced with a strong constitutive yeast promoter (P TEF ) and the response to density and sex ratio in the mixed population of MATα and MATa cells was measured using microscopy and compared with the wild-type response at different values of ρ T (OD 600 ) of 0.18 (red), 0.54 (green) and 4.9 (blue). c. Cell-wall associated Bar1 has no effect on the observed regulation.
Plots show the P FUS1 -GFP response kinetics in wild-type (blue) and bar1∆ (red) MATa cells mixed in equal proportion and exposed to different α-factor concentrations (indicated above each plot, in nM). The global extracellular pool of Bar1 is shared by both cell populations, however the cell-wall associated activity is exclusive to the wild-type. The cell-wall associated fraction of Bar1 appears to play no significant role in the response attenuation, because the mixed populations of the wild-type and bar1∆ MATa cells that share a common pool of diffusive Bar1 show identical responses to purified α-factor (compare to differences observed in Supplementary Fig. 1 (1-f) of the remaining haploid population (1-g). Shown example illustrates a mating situation with a partner cell fraction θ α =0.5 with a total cell density ρ T =2. In both cases the mating efficiency is g=0.5, half of the original haploid population will mate. However, in the case of the bar1∆ population, (the density-sensing) strategy, the fitness reduction of the remaining haploid cells is heavier, resulting in a lower average fitness of the bar1∆ population. b. The fractional sensing strategy of wt cells compared to a hypothetical case of density sensors with adjustable sensitivity (see Supplementary Methods). The calculated fitness landscape (at λ=2.1) of the fractional (wild type) sensor with two density sensor strategies (with different values of c, c=1, c=e 6.85 ) is shown in panel on the left. For the chosen population distribution parameters of γ=8 (e -8 ≤ρ T ≤e 8 , log-uniformly distributed) and σ θ =0.2, the density sensor (red fitness landscape) with c=e 6.85 has the optimal value of c (c max ), shown by a red dot in the panel on the right. This is due to its strong amplification of the response at lower densities leading to higher population fitness. However, this is at the expense of heavy fitness reduction at low θ α values when ρ T is high, resulting in a mean fitness <W Δ > that is still lower than !"# ! !! ! ! at different levels of λ (value of diploidy) and γ (defining the interval of ρ T values in which these are log-uniformly distributed). Values smaller than 1 are in blue and not coloured gradually. On the right we have the mean of plotted as a function of λ. b. The ratio !"# ! !! ∆ ! plotted at different levels of λ, comparing the wt fractional sensor strategy with the density sensor (bar1∆, in the case of a truncated Gaussian distribution for θ α and a log-uniform distribution of ρ T . The value r is the average of !"# ! !! ∆ ! plotted at different levels of λ, comparing the wild type fractional sensor strategy with the density sensor (bar1∆, in the case of a truncated Gaussian distribution for θ α and a log-normal distribution of ρ T . The rightmost panel shows <  rtTA-S2, tetO7 driven mCherry a P FUS1 -and P YLR194C -fluorescent protein reporters were genomically integrated by means of integrative plasmids based on the pRS30x series 2 , P tetO7 -fluorescent protein reporters were genomically integrated by single-copy integrating plasmids based on the pNH series 3  The concentration dynamics of α-factor and the enzyme Bar1 in a homogeneous mixed population of MATα and MATa cells can be described by the following ordinary differential where α(t) and b(t) are the concentrations of α-factor and Bar1, respectively; ρ α and ρ a are the number of MATα and MATa cells per unit of volume; ν 1 and ν 2 are per cell production rates of α-factor and Bar1, respectively; and κ is the rate constant of Bar1-dependent α-factor degradation. Here we note that the α-factor degradation follows first-order kinetics, which is justified because the K M of Bar1 (30 µM) 4 is much higher than the sensitive range of the pheromone response ( Supplementary Fig. 1). We further neglect spontaneous degradation of Bar1 and of α-factor, as both are negligible on the time-scale of the experiment. This simplifies the two equations above to The system of Equations I and II has the exact solution where c 1 = 1.253 υ 1 1 κ υ 2 , c 2 = 0.707 κ υ 2 ρ a and erfi is the imaginary error function. This solution has non-monotonic time dependence, falling to zero after reaching a maximum α max = c ρ α ρ a

(IV)
Where c is a combination of kinetic constants only, c = 1.253 υ 1 1 κ υ 2 0.6105 (irrespective of the value of c 2 ).

Model 1 (constant production rate of pheromones)
We fit our model treating both α-factor and GFP level as dynamical variables where the dynamics of α-factor is described by Equation III and the dynamics of GFP can be described by the ordinary differential equation Here δ GFP is the parameter for first-order GFP degradation (and dilution), experimentally estimated as δ GFP ≈ 0.02 min -1 (AB, unpublished). This parameter is allowed to vary in a narrow range [0.01; 0.03] around the experimentally determined value. The EC 50 value (2 nM) is derived from the experimentally measured dose-dependence of reporter induction upon stimulation with synthetic α-factor ( Supplementary Fig. 1). The Hill coefficient H was introduced here to allow for sigmoidality of the response ( Supplementary Fig. 1), and is allowed to vary between 1 and 3. The maximal GFP production rate V max and the basal rate V 0 are allowed to vary within 10 -4 ≤ V max ≤ 10 -3 (AU min -1 ). The ratio (V 0 +V max )/δ GFP corresponds to the maximal fluorescence value measured in flow cytometry (0.044 AU). The parameters ν 1 and ν 2 κ (Equations I-III) are allowed to vary within the constraints 10 -9 ≤ ν 1 ≤ 10 -5 (pmol min -1 ) and 10 -16 ≤ ν 2 κ ≤ 10 -10 (L min -2 ), respectively.
The objective function to be minimized to fit these parameters is the weighted sum of squared residuals between the data points (GFP data ) and the model outputs (GFP mod ): The parameter values from the fitting to the GFP values from flow cytometry experiments at t = 135 min (Fig. 2)

Model 2 (mutual pheromone induction)
The production of pheromones is known to be mutually inducible 5,6 , meaning that stimulation of MATa cells with α-factor induces production of a-factor and vice versa. We assumed that this induction shows the following behavior and that pheromone induction has similar dose-dependence as P FUS1 -GFP. We thus fix EC 50 to 2 nM as above, but let its Hill coefficient, fold-change parameter Φ and the basal production rate ν vary. For simplicity, we assume that a-and α-factor induction follow identical dependence, except for the absolute level of ν (ν 1 : α-factor basal production rate, ν 3 : a-factor basal production rate). The dynamics of the α-and a-factor induction can then be described as Since we focus on the early time points of the response, we further neglect spontaneous Model 2 produced only marginally better fits, the simpler Model 1 was used in the main text to minimize the number of free parameters.

Mating probability model
To schematically describe the probability of mating, we assume a simple scenario where collision of cells lead to the irreversible formation of a mating pair, which can be described by mass action kinetics as: where ρ m is the concentration of mating pairs. We can use the conservation relation ρ α (t = 0) + ρ a (t = 0) = ρ a (t) + ρ α (t) + 2ρ m (t) to obtain the analytical solution for the fraction of mated The stationary solution of this equation is Equation X allows to calculate the fraction of mated MATa cells at different time points as a function of population parameters.

Comparison of fractional and absolute sensing strategies
To perform a schematic cost-benefit comparison between the wild-type strategy of fractional (sex ratio) sensing and the bar1∆ strategy of absolute sensing (or density sensing) of mating partners, we consider the fitness effect of these two regulation strategies on an initial population of haploid MATa cells encountering different amounts of partner MATα cells ( Supplementary Fig. 8a). A fraction of the MATa population will mate (benefit), whereas the fitness of MATa cells that are stimulated and induce the mating response but do not mate is reduced, e.g. due to a transient cell-cycle arrest (cost, f). The cellular response f is a schematic representation of our experimental data: for the sex-ratio sensor (wild type), the response becomes invariant to total density over a reference value (defined as ρ T =1), and simply equals the partner cell fraction θ α (See Equation XII below). In contrast, the density sensor (bar1Δ) simply follows the absolute abundance of partner cells (θ α ρ T ), going into saturation for ρ T >1 (See Equation XIII below). We assume that the efficiency of mating (g, the fraction of the initial MATa population that forms diploid cells) is proportional to the level of response induction in MATa cells, limited by the abundance of MATα cells as we observed experimentally (Fig. 3c). The resulting description for the wild-type and bar1∆ MATa cells is Wild type (fractional sensor)

bar1Δ (density sensor)
where f(θ α ,ρ T ) is the relative response [0,1] (that is equivalent to the cost); and g(θ α ,ρ T ) is the relative mating efficiency [0,1]. Note that here we do not consider regulation of MATα cells.
In general form the fitness equation for the population is The contribution of diploid cells to the population fitness is the fraction of mated cells (g), scaled by a parameter λ, representing the relative advantage of diploidy. The contribution of the remaining haploid cells to the population fitness is again their fraction in the total population (1-g) times their fitness, which is proportionally reduced with the level of induction (1-f).
As in our model above (Equations XII and XIII) f WT (θ α ,ρ T ) = g WT (θ α ,ρ T ) = g ∆ (θ α ,ρ T ), these functions can be simply replaced by g(θ α ,ρ T ). The fitness of the population (W) for the wild type or bar1Δ strategy (at a particular total cell density and partner cell fraction) is,

From Equation XIII
we can see that f ∆ ≥ g, therefore W WT ≥ W ∆ is always true in the current model. Whatever distribution θ α and ρ T have, this will also be true for the mean fitness values over these distributions, i.e. <W WT > ≥ <W ∆ >. With the maximal mating efficiency (g in Equations XII, XIII) limited as above (based on our experimental data, Fig. 3c), the higher induction of bar1Δ cells at higher population densities cannot yield higher benefits, but will result in higher cost. Therefore the population fitness of density sensing bar1Δ cells will always be lower.
To make a more general comparison, we consider that cells using the density sensing (bar1Δ) strategy could have evolved a different strategy, adjusting their response sensitivity to achieve a higher fitness. Then the response, mating efficiency and fitness of the density sensor are Here the response of the density sensor cells is scaled by a parameter c that can be optimally adjusted to maximize fitness of the density sensors, as illustrated in Supplementary Fig. 8b.
We compare this strategy to a fractional sensor (Equation XII).
The population fitness W was calculated above at a particular value of the population parameters θ α and ρ T . In reality, these population parameters would assume different probability distributions, and we therefore need to calculate a mean fitness value <W> over these distributions. We first explore the two limiting cases of no variation in these two parameters and a uniform distribution for both. We then consider the intermediate case of normal distributions of varying width.

No variation or uniform distributions for population parameters
In these two limiting cases analytical solutions for the mean fitness can be obtained. In the first limiting case, if there is no variation in θ α and ρ T (θ α =0.5 and ρ T =1), then In the second and third case (of <W ∆ >) it is easy to see that <W Δ > is smaller than <W WT >. In the first case of c<1, for <W ∆ > > 1, c+2λ>4 has to be true, but for <W ∆ > > <W WT > the condition is c+2λ<3, which cannot be both true. Consequently, the density sensing strategy is either identical (c=1) to the wild type, or performs worse.
In the second limiting case we assume that θ α is uniformly distributed within the interval [0,1], whereas ρ T is also uniformly but logarithmically distributed in the interval [e -γ e γ ].
The mean of a function f(x) over an interval [e -γ e γ ], with logarithmically spaced (with uniform probability) x values is: Integrating over the distributions, the equations for mean fitness have analytical solutions, which are, respectively: where r=max(-γ, min(γ, -ln(c))). (XIX).
For any value of γ (defining variability of total density values) and λ we take the density sensor strain with the highest mean fitness (an optimal value of c) and compare it to the mean fitness of the wild type by taking the ratio This analysis shows that the fractionalsensing strategy outperforms the density-sensing strategy, as long as γ (total density variation) exceeds a minimal value and the advantage of diploidy (λ) is moderate (Supplementary Fig.   9a).

Normal distribution for θ α and log-uniform distribution for ρ T
For the intermediate case we assume that the mean of θ α is 0.5 and the distribution is a truncated Gaussian, as values are only possible in the range [0,1].
The total densities we use are log-uniformly distributed as in the previous example (Equation !"# ! !! ! ! > is >1) over a wide range of σ θ and γ, with the difference generally growing with σ θ and γ (Fig. 4d,e and Supplementary Fig. 9b).

Normal distribution for θ α , log-normal distribution for ρ T
Alternatively, for total densities we can also use a lognormal distribution with the median at ρ T =1: We can then calculate the mean fitness of the population in a certain environment: For any two distributions of the population parameters (defined by σ θ and σ ρ ) we again take the density sensor strain with the highest mean fitness (an optimal value of c) and compare it to the mean fitness of the fractional sensor. As in the case of the uniform distribution above, we observed that at intermediate values of λ the wild type strategy performs better (i.e., the ratio ! !" !"# ! !! ! ! is >1) over a wide range of σ θ and σ ρ . (Supplementary Fig. 9c).

Comparison of fractional sensing with a constant-investment strategy
We can also compare the fractional sensing strategy to one where the level of induction is constant and not regulated. In this case the level of induction f is a constant, ! = , 0 ≤ ≤ 1, and mating efficiency g is ! = , ≤ ! ! , > ! , which yields the fitness function: Calculating the mean fitness again the same way as before at a certain distribution of θ α and ρ T , we have (using a log-normal and a truncated Gaussian distribution):

Comparison of density-independent fractional sensing (wt) strategy with a constantinvestment strategy
We assume here constant investment irrespective of the total cell density. Therefore we first make the comparison with a fractional sensing strategy that is also completely densityindependent and has the fitness equation: First we compare the two strategies in the limiting cases of no variation or a uniform distribution of θ α . For a fixed θ α =0.5, we obtain The fitness function W const is evidently smaller than W WT in the case of c>0.5 and identical to W WT if c=0.5.
In the case of c<0.5, for W const >1 we need λ>1.5. The roots of W WT -W const =0 are c=0.5 and c=0.5 (3-2λ), and between these values of c, W WT -W const >0. Therefore if there is no variation in θ α , the constant investment strategy is identical to the wild type regulation if c=0.5, or is worse if c has any other value.
If θ α is uniformly distributed, the equations for mean fitness are For <W const > > <W WT > to be true, < !!!"!!!! ! !!! ! !(!!!) ! . But <W const > also needs to be larger than 1 to be a viable strategy of investment of resources into mating, and the condition for this is > !!!!!!! ! !!! . But for 0<c<1, these two conditions cannot be true at the same time, as . Therefore the constant investment strategy always performs poorer than regulated fractional investment under a uniform distribution of the partner cell fraction.

Comparison of density-dependent fractional sensing (wt) strategy with a-constant investment strategy
Alternatively, we can compare the fitness of the constant investment strategy to the densitydependent fractional (wild type) strategy by again taking the ratio ! !" !"# ! !! !"#$% ! as a function of σ θ and σ ρ , and at different λ values. Again, at each value of σ θ , σ ρ and λ the best-performing 'constant investor' (highest <W const >) is compared to the fitness of the fractional sensor. A constant investment strategy performs poorer when the partner cell fraction has higher variation ( Supplementary Fig. 9d). As above at intermediate λ values the fractional sensor strategy outperforms the constant investment strategy (Supplementary Fig. 9d).