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Microbial cooperation typically consists in the sharing of secreted metabolites (referred to as public goods) within the community. Although public goods generally promote population growth, they are also vulnerable to exploitation by cheating mutants, which no longer contribute, but still benefit from the public goods produced by others. Although previous studies have identified a number of key factors that prevent the spreading of cheaters, little is known about how these factors interact and jointly shape the evolution of microbial cooperation. Here, we address this issue by investigating the interaction effects of cell diffusion, cell density, public good diffusion and durability (factors known to individually influence costs and benefits of public goods production) on selection for cooperation. To be able to quantify these effects across a wide parameter space, we developed an individual-based simulation platform, consisting of digital cooperator and cheater bacteria inhabiting a finite two-dimensional continuous toroidal surface. Our simulations, which closely mimic microbial microcolony growth, revealed that: (i) either reduced cell diffusion (which keeps cooperators together) or reduced public good diffusion (which keeps the public goods closer to the producer) is not only essential but also sufficient for cooperation to be promoted; (ii) the sign of selection for or against cooperation can change as a function of cell density and in interaction with diffusion parameters; and (iii) increased public goods durability has opposing effects on the evolution of cooperation depending on the level of cell and public good diffusion. Our work highlights that interactions between key parameters of public goods cooperation give rise to complex fitness landscapes, a finding that calls for multifactorial approaches when studying microbial cooperation in natural systems.

This article gives a brief overview of some recent progress in the characterization and parametrization of density matrices of finite dimensional systems. We discuss in some detail the Bloch-vector and Jarlskog parametrizations and mention briefly the coset parametrization. As applications of the Bloch parametrization we discuss the trace invariants for the case of time dependent Hamiltonians and in some detail the dynamics of three-level systems. Furthermore, the Bloch vector of two-qubit systems as well as the use of the polarization operator basis is indicated. As the main application of the Jarlskog parametrization we construct density matrices for composite systems. In addition, some recent related articles are mentioned without further discussion.

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