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

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.