Stable Heterogeneity for the Production of Diffusible Factors in Cell Populations

Citation: Archetti M ( Stable Heterogeneity for the Production of Diffusible Factors in Cell Populations Marco Archetti 0 James A.R. Marshall, University of Sheffield, United Kingdom 0 School of Biological Sciences, University of East Anglia , Norwich , United Kingdom The production of diffusible molecules that promote survival and growth is common in bacterial and eukaryotic cell populations, and can be considered a form of cooperation between cells. While evolutionary game theory shows that producers and non-producers can coexist in well-mixed populations, there is no consensus on the possibility of a stable polymorphism in spatially structured populations where the effect of the diffusible molecule extends beyond one-step neighbours. I study the dynamics of biological public goods using an evolutionary game on a lattice, taking into account two assumptions that have not been considered simultaneously in existing models: that the benefit of the diffusible molecule is a non-linear function of its concentration, and that the molecule diffuses according to a decreasing gradient. Stable coexistence of producers and non-producers is observed when the benefit of the molecule is a sigmoid function of its concentration, while strictly diminishing returns lead to coexistence only for very specific parameters and linear benefits never lead to coexistence. The shape of the diffusion gradient is largely irrelevant and can be approximated by a step function. Since the effect of a biological molecule is generally a sigmoid function of its concentration (as described by the Hill equation), linear benefits or strictly diminishing returns are not an appropriate approximations for the study of biological public goods. A stable polymorphism of producers and non-producers is in line with the predictions of evolutionary game theory and likely to be common in cell populations. - Data Availability: The authors confirm that all data underlying the findings are fully available without restriction. All relevant data are within the paper and its Supporting Information files. Funding: This work was supported by the Natural Environment Research Council grant NE/H015701/1 (www.nerc.ac.uk). The funder had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests: The author has declared that no competing interests exist. Cooperation for the production of diffusible molecules is commonly observed in cell populations, from bacteria to eukaryotes [1]: bacteria, for example, produce molecules that contribute to population growth (like pyocyanin [2] and pyoverdine [3]), that enable the buildup of biofilms [4] or that confer resistance to antibiotics [5]; yeast cells produce invertase that catalyzes the hydrolysis of sucrose [6], and cancer cells produce growth factors that contribute to tumour expansion [7]. Because the effect of diffusible molecules is not limited to the producer cells, a mutant cell not producing the molecule can still benefit from the presence of its neighbour producers. The freerider advantage enjoyed by non-producer cells may lead to an increase in their frequency in the population and drive producers to extinction, with a consequent reduction in average fitness for the population - similar to what is often referred to as the tragedy of the commons [8]. It is understood, however, that because this free-rider advantage is frequency-dependent, if the benefit conferred by the public good is non-linear, the dynamics is generally more complex and in well-mixed populations it can lead to a stable coexistence of producers and non-producers [9]. Whether this is also the case in spatially structured populations, however, is unclear. In the study of public goods games in spatially structured populations it is usually assumed [10] that an individuals action affects only the fitness of individuals one node away and that an individuals fitness is the sum of all the payoffs accumulated in all the groups she belongs to (all the groups formed by the one-step neighbours of her one-step neighbours). This is reasonable for interactions in human social networks, but not for cellular networks, in which molecules typically diffuse beyond a cells one-step neighbours, and in which the benefit for a cell is a function of the number of producer cells within the diffusion range of the molecule. In order to study diffusible public goods, therefore, one must decouple the interaction neighbourhood (the group playing the game, defined by the diffusion range of the molecule) and the update neighbourhood (the one-step neighbours). While such models have been used to study a simple twoperson game with a linear benefit function (the prisoners dilemma) on a regular lattice [11,12] only recently it has been used to study the dynamics of multi-player public goods games (which are appropriate for the study of biological molecules) and there seems to be no consensus on the conclusions of these studies. Borenstein et al. [13] showed that in a 2-D model with diffusion and linear benefits producers and non-producers can never coexist. Scheuring [14] showed, instead, stable coexistence in a 1-D model with concave benefits (diminishing returns) and even Figure 1. Realistic Hill coefficients lead to coexistence of producers and non-producers. For different benefit functions B(x) and gradients of diffusion G(i), the fraction of producers over time is show for c = 0.05 and c = 0.15. The lattices show the population after 1000 generations per cell. A: Linear benefit (s = 1, h = 0.5) with a diffusion gradient (z = 3, d = 0, D = 7). B: Sigmoid benefit (s = 20, h = 0.5) with no diffusion gradient (z = 1000, d = 3, D = 6). C: Sigmoid benefit (s = 20, h = 0.5) with a diffusion gradient (z = 3, d = 0, D = 7). doi:10.1371/journal.pone.0108526.g001 (although rarely) with linear benefits, depending on the initial conditions of the system. Archetti [15] showed that coexistence is the typical outcome of the dynamics in a 2-D model with diffusion, but did not take into account the fact that the efficacy of the diffusible molecule may decline with the distance form the source. Allen et al. [16] studied a model with diffusion and linear benefits, but did not investigate the possibility of coexistence of the two types, since in their finite stochastic population one of the strategies eventually goes to fixation (it is known, however, that in the presence of a stochastically stable polymorphism, coexistence in large populations is possible since fixation time increases exponentially with population size [17]). A number of different assumptions in these studies [1316] can account for the different conclusions about the possibility of a stable polymorphism. I will analyse two assumptions that seem the most prominent differences in the 2-D models described above: the shape of the diffusion gradient of the molecule (the efficacy of the molecule as a function of t (...truncated)