  Bacterial quorum sensing is the communication that takes place between
bacteria as they secrete certain molecules into the intercellular medium that
later get absorbed by the secreting cells themselves and by others. Depending
on cell density, this uptake has the potential to alter gene expression and
thereby affect global properties of the community. We consider the case of
multiple bacterial species coexisting, referring to each one of them as a
genotype and adopting the usual denomination of the molecules they collectively
secrete as public goods. A crucial problem in this setting is characterizing
the coevolution of genotypes as some of them secrete public goods (and pay the
associated metabolic costs) while others do not but may nevertheless benefit
from the available public goods. We introduce a network model to describe
genotype interaction and evolution when genotype fitness depends on the
production and uptake of public goods. The model comprises a random graph to
summarize the possible evolutionary pathways the genotypes may take as they
interact genetically with one another, and a system of coupled differential
equations to characterize the behavior of genotype abundance in time. We study
some simple variations of the model analytically and more complex variations
computationally. Our results point to a simple trade-off affecting the
long-term survival of those genotypes that do produce public goods. This
trade-off involves, on the producer side, the impact of producing and that of
absorbing the public good. On the non-producer side, it involves the impact of
absorbing the public good as well, now compounded by the molecular
compatibility between the producer and the non-producer. Depending on how these
factors turn out, producers may or may not survive.
