Why do bacteria regulate public goods by quorum sensing?-How the shapes of cost and benefit functions determine the form of optimal regulation

Abstract

Many bacteria secrete compounds which act as public goods. Such compounds are often under quorum sensing (QS) regulation, yet it is not understood exactly when bacteria may gain from having a public good under QS regulation. Here, we show that the optimal public good production rate per cell as a function of population size (the optimal production curve, OPC) depends crucially on the cost and benefit functions of the public good and that the OPC will fall into one of two categories: Either it is continuous or it jumps from zero discontinuously at a critical population size. If, e.g., the public good has accelerating returns and linear cost, then the OPC is discontinuous and the best strategy thus to ramp up production sharply at a precise population size. By using the example of public goods with accelerating and diminishing returns (and linear cost) we are able to determine how the two different categories of OPSs can best be matched by production regulated through a QS signal feeding back on its own production. We find that the optimal QS parameters are different for the two categories and specifically that public goods which provide accelerating returns, call for stronger positive signal feedback.

Keywords: bacteria; cooperation; cost and benefit functions; public good production and regulation; quorum sensing.

Figure 1

Concave/convex benefit function. (A) The benefit is here a sigmoidal function: $b(x)={\beta }_1\frac{{(x∕K_E)}^h}{1+{(x∕K_E)}^h}+{\beta }_2\frac{(x∕K_E)}{1+(x∕K_E)}$, plotted for two different values of the exponent h = [1, 2], with x = E. The parameter h can be manipulated so that benefit initially decelerates (h = 1, concave, light green) or accelerates (h > 1, convex, dark green) with increasing concentration of public good. The remaining parameters are β₁ = 0.7, β₂ = 0.3, K_E = 25. (B) When the timescale of public good production and degradation are much faster than the timescale for growth of cells, public good concentration is proportional with production rate and population size, E ∝ σ_EN. Dark green, the convex benefit curve B_N(σ_E) = b(Nσ_E) is shown as a function of the common good production rate per cell σ_E for N = [1, 10, 25, 49, 100, 200]. Red, linear cost curve c(σ_E) = κσ_E, κ = 1.

Figure 2

Example of public goods with differently shaped benefit functions. Proteases which works extracellularly by degrading polymers that are too large to be transported over the cell membrane can either break the peptide bonds of the polymer starting from the end of the polymer (exoprotease) or target a range of specific types of peptide bond within the chain, effectively breaking bonds at random (endoprotease). In this scenario benefit is proportional to the probability of yielding “edible” pieces of polymer. (A1,A2) Benefit increases approximately linearly with exoprotease concentration resulting in an initially concave benefit function. (B1,B2) The probability of breaking the polymer at a site producing an piece small enough to transport over the cell membrane is low when enzyme concentration is low, but accelerates as endoprotease concentration increases, resulting in a initially convex benefit function.

Figure 3

Concave/convex benefit curves result in continuous/discontinuous optimal production curves respectively. (A,B) Green curves show optimal production rate of public good, ${\sigma }_E^{opt}$ as a function of population size, N for the concave/convex benefit functions shown in figure 1. The optimal value, ${\sigma }_E^{opt}(N)$, corresponds to the σ_E which maximizes Δg for N. The magnitude of Δg in the (N, σ_E)-space is shown by the colorbar. (A) In the case h = 1 the optimal production function is continuous and can be put in closed form: ${\sigma }_{E,h=1}^{opt}(N)=$ $max(\sqrt{(\beta K_E{\gamma }_E)∕(cN)}-(K_E{\gamma }_E)∕N,0)$, where β = β₁ + β₂. The critical population size above which public good production is nonzero, when h = 1 is: $N_{crit}=\frac{c}{\beta }{\gamma }_E{K}_E$. (B) When the benefit curve is convex, h = 2, the optimal production function is discontinuous. (C,D): Light/dark green curves show the effect on growth rate Δg for a population producing common good at exactly the optimal rate when the benefit curve is concave/convex respectively. Black dashed curves show the effect on growth rate Δg for a population producing common good at a constant rate ${\sigma }_E^{const}=1∕(N_{max}-N_{crit}){\Sigma }_{N_{crit}}^{N_{max}}{\sigma }_E^{opt}(N)$, equal to the average of the non-zero part of the optimal curve, (shown as black dashed lines in (A,B). (E) The cumulative fitness, $w\equiv ({\int }_1^{N_{max}}\Delta g(N)dN)∕w_{max}$, where $w_{max}\equiv {\int }_1^{N_{max}}\Delta g^{opt}(N)dN$, (def. in Equation fitnessDef) of different constant production strategies for a common good with concave (light green) and convex (dark green) benefit functions respectively. Note that in the case of the “concave common good” there is a range of different constant production rates which allows the population to perform better than a nonproducing population w > 0, however for the “convex common good” any constant production strategy will leed to worse fitness than that of a nonproducing population, w < 0 for all ${\sigma }_E^{const}>0$. (In this figure σ_E is given in units of κ∕γ_E, which was set to one).

Figure 4

Increasing the QS signal feedback exponent α causes hysteretic response to population changes. Steady state QS signal concentration, S^* as function of the population size N, plotted for four different values of the QS feedback exponent α = [1.0, 1.2, 1.5, 2.0]. When $\alpha >{\alpha }_c=\frac{2+{\sigma }_s^{max}+2\sqrt{1+{\sigma }_s^{max}}}{{\sigma }_s^{max}}\approx 1.2210$ the system will be bistable for a certain range of population sizes. (${\sigma }_S^{max}=100$, concentration of S is here given in units of ${\sigma }_S^{basal}∕{\gamma }_S$. ${\sigma }_S^{basal}$ is the basal production rate and γ_S is the degradation/depletion rate of the of signal, which is here set to 1.)

Figure 5

Quorum sensing parameters optimized for different ecological scenarios and different types of public goods. (A–C) Left side panels: Green curves shows optimal production rate, ${\sigma }_E^{opt}$, as a function of population size N. Black full line and dashed line show the production rate as a function of population size, N, when production is regulated by (A): a QS system with parameters such that cumulative fitness, w is optimized for a public good with the concave benefit function shown in Figure 1 (α = 1.1, K_S = 2200, ${\sigma }_E^{max}=0.33$), (B): a system with QS parameters for a public good with a convex benefit function maximizing cumulative fitness w in only “turn on” scenarios, (α = 2, K_S = 980, and ${\sigma }_E^{max}=0.43$), and (C): a QS system with parameters that maximize cumulative fitness w for a public good with a concave benefit function in both “turn on” and “turn off” scenarios, (α = 1.4, K_S = 2600, and ${\sigma }_E^{max}=0.6$). (A–C) Right side panels: Green lines show Δg^opt, the increase in growth rate achieved, as a function of N when producing public good using the optimal production curve ${\sigma }_E^{opt}$. Full black line shows Δg as a function of N for the QS regulated production curve marked “Turn on” in the plot to the right. Dashed black line shows Δg as a function of N for the QS regulated production curve marked “Turn off” in the plot to the left. The colored area underneath the Δg curves in the range N = [1, N_max], (N_max = 70) is proportional to the cumulative fitness, w (defined in Equations 7 and 8).