Spatial dilemmas of diffusible public goods

RESEARCH ARTICLE elife.elifesciences.org Spatial dilemmas of diffusible public goods Benjamin Allen1,2*, Jeff Gore3, Martin A Nowak2,4,5 Department of Mathematics, Emmanuel College, Boston, United States; Program for Evolutionary Dynamics, Harvard University, Cambridge, United States; 3 Department of Physics, Massachusetts Institute of Technology, Cambridge, United States; 4Department of Mathematics, Harvard University, Cambridge, United States; 5 Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, United States 1 2 Abstract The emergence of cooperation is a central question in evolutionary biology. Microorganisms often cooperate by producing a chemical resource (a public good) that benefits other cells. The sharing of public goods depends on their diffusion through space. Previous theory suggests that spatial structure can promote evolution of cooperation, but the diffusion of public goods introduces new phenomena that must be modeled explicitly. We develop an approach where colony geometry and public good diffusion are described by graphs. We find that the success of cooperation depends on a simple relation between the benefits and costs of the public good, the amount retained by a producer, and the average amount retained by each of the producer’s neighbors. These quantities are derived as analytic functions of the graph topology and diffusion rate. In general, cooperation is favored for small diffusion rates, low colony dimensionality, and small rates of decay of the public good. DOI: 10.7554/eLife.01169.001 Introduction *For correspondence: Competing interests: The authors declare that no competing interests exist. Funding: See page 9 Received: 03 July 2013 Accepted: 03 November 2013 Published: 17 December 2013 Reviewing editor: Carl T Bergstrom, University of Washington, United States Copyright Allen et al. This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited. Public goods dilemmas are frequently observed in microbes. For example, the budding yeast Saccharomyces cerevisiae cooperates by producing the enzyme invertase, which hydrolyzes sucrose into monosaccharides, when yeast colonies are grown in glucose-limited media (Greig and Travisano, 2004; Gore et al., 2009). Other examples include the production of chemical agents that scavenge iron (Griffin et al., 2004; Buckling et al., 2007; Cordero et al., 2012; Julou et al., 2013), enable biofilm formation (Rainey and Rainey, 2003), eliminate competition (Le Gac and Doebeli, 2010), induce antibiotic resistance (Chuang et al., 2009; Lee et al., 2010), or facilitate infection of a host (Raymond et al., 2012). In many cases, the benefits of public goods go primarily to cells other than the producer. For example, in a S. cerevisiae population subject to continuous mixing, only ∼1% of monosaccharides are imported into the cell that hydrolyzes them, with the remainder diffusing away (Gore et al., 2009). Furthermore, production of public goods typically involves a metabolic cost, which may exceed the direct benefit to the producer. In this case, absent some mechanism to support cooperation (Nowak, 2006), public goods production is expected to disappear under competition from cheaters, resulting in the tragedy of the commons (Hardin, 1968). There is growing evidence from experiments (Griffin et al., 2004; Kümmerli et al., 2009; Julou et al., 2013; Momeni et al., 2013) and simulations (Allison, 2005; Misevic et al., 2012) that spatial or group clustering can support cooperation in microbial public goods dilemmas, although this effect depends on the nature of competition for space and resources (Griffin et al., 2004; Buckling et al., 2007). These findings agree with insights from mathematical models (Nowak and May, 1992; Durrett and Levin, 1994; Santos and Pacheco, 2005; Ohtsuki et al., 2006; Szabó and Fáth, 2007; Taylor et al., 2007; Perc and Szolnoki, 2008; Fletcher and Doebeli, 2009; Korolev and Nelson, 2011) suggesting Allen et al. eLife 2013;2:e01169. DOI: 10.7554/eLife.01169 1 of 11 Research article Ecology | Genomics and evolutionary biology eLife digest The natural world is often thought of as a cruel place, with most living things ruthlessly competing for space or resources as they struggle to survive. However, from two chimps picking the fleas off each other to thousands of worker ants toiling for the good of the colony, cooperation is fairly widespread in nature. Surprisingly, even single-celled microbes cooperate. Individual bacterial and yeast cells often produce molecules that are used by others. Whilst many cells share the benefits of these ‘public goods’, at least some cells have to endure the costs involved in producing them. As such, selfish individuals can benefit from molecules made by others, without making their own. However, if everyone cheated in this way, the public good would be lost completely: this is called the ‘public goods dilemma’. Allen et al. have developed a mathematical model of a public goods dilemma within a microbial colony, in which the public good travels from its producers to other cells by diffusion. The fate of cooperation in this ‘diffusible public goods dilemma’ depends on the spatial arrangement of cells, which in turn depends on their shape and the spacing between them. Other important factors include rates of diffusion and decay of the public good—both of which affect how widely the public good is shared. The model predicts that cooperation is favored when the diffusion rate is small, when the colonies are flatter, and when the public goods decay slowly. These conditions maximize the benefit of the public goods enjoyed by the cell producing them and its close neighbors, which are also likely to be producers. Public goods dilemmas are common in nature and society, so there is much interest in identifying general principles that promote cooperation. DOI: 10.7554/eLife.01169.002 that spatial structure can promote cooperation by facilitating clustering and benefit-sharing among cooperators. However, these mathematical results focus largely on pairwise interactions rather than diffusible public goods. On the other hand, previous theoretical works that specifically explore microbial cooperation (West and Buckling, 2003; Ross-Gillespie et al., 2007; Driscoll and Pepper, 2010) use a relatedness parameter in place of an explicit spatial model, obscuring the important roles of colony geometry and spatial diffusion in determining the success of cooperation. Results Here we present a simple spatial model of a diffusible public goods dilemma. Our model is inspired by the quasi-regular arrangements of cells in many microbial colonies (Figure 1A,B). The geometry of these arrangements depends on the shapes of cells and the dimensionality of the environment. For example, approximately spherical organisms such as (...truncated)