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.

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 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.

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 S. cerevisiae arrange themselves in a hexagonal lattice-like structure when the colony is constrained to a two-dimensional plane (Figure 1A). This differs from the arrangements of rod-shaped organisms such as the bacterium Escherichia coli (Figure 1B).

Figure 1

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To allow for a maximum variety of possible arrangements, we represent space as a weighted graph G (Figure 1C,D; Lieberman et al., 2005). Edges join cells to their neighbors, with edge weights e_ij proportional to the frequency of diffusion between neighboring cells. The graph structure thereby captures all features of cell arrangement that are relevant to the diffusion of public goods. The edge weights are normalized to satisfy Σ_j e_ij = 1, so that they represent relative frequencies of diffusion to each neighbor. Since we are modeling intercellular diffusion, we set e_ii = 0 for each i. We also suppose that G has bi-transitive symmetry (Taylor et al., 2007), which implies that space is homogeneous (i.e., that the colony looks the same from each cell). Our model therefore applies primarily to the interiors of colonies rather than their boundaries. Bi-transitive symmetry also requires that pairwise relationships are symmetric—in particular e_ij = e_ji for every pair i and j. This captures the reasonable assumption that public goods diffuse as frequently from cell i to cell j as they do from j to i.

To characterize local structure, we introduce the Simpson degree $\kappa ={\left({\displaystyle {\sum }_{j\in G}{e}_{ij}^2}\right)}^{-1}.$ This quantity can be understood as the Simpson diversity (Simpson, 1949) of neighbors per cell, and coincides with the usual notion of degree on regular unweighted graphs. By symmetry, κ does not depend on which vertex i is used in the above sum.

We consider two cells types: cooperators, C, that produce the public good, and defectors, D, that do not. These traits are passed to offspring upon reproduction. Production of the public good inflicts a cost c on its producer, and generates a total benefit b that is distributed among cells according to a diffusion process described below. Because our model is inspired by public goods that directly increase cell growth rate (such as hydrolyzed monosaccharides) it is less applicable to public goods with indirect benefits, such as quorum-sensing molecules (Waters and Bassler, 2005).

Cooperators produce one unit of public good per unit time. The public goods in the vicinity of a given cell either are utilized for the benefit of this cell or diffuse toward neighboring cells in proportion to edge weight. (The possibility of public goods decay is discussed below.) We quantify diffusion by the ratio λ of the diffusion rate to the utilization rate. The dynamics of the local public goods concentration ψ_i at each node i ∈ G are given by

Above, s_i = 0,1 indicates the current type, D or C respectively, of cell i. The term s_i in Equation 1 represents public goods production, −ψ_i represents utilization, −λψ_i represents diffusion outward, and the remaining term represents diffusion inward.

Equation 1 is equivalent to supposing that each particle of public good performs a random walk among cells (with step probabilities equal to edge weights), and has probability 1/(1+λ) of being utilized at each cell it encounters, including its producer. In this interpretation, λ equals the expected number of steps a particle travels before being utilized.

For most empirical systems, diffusion and utilization occur much faster than cell division. We therefore suppose that the local public goods concentrations ψ_i reach stationary equilibrium levels between reproductive events (‘Materials and methods’).

Two key quantities in our analysis are the fractions, ϕ₀ and ϕ₁, of public goods that are retained by its producer and the producer’s immediate neighbors, respectively (Figure 2). For a state in which only a single cell, i, is a cooperator, we have ϕ₀ = ψ_i and ϕ₁ = Σ_j∈G e_ij ψ_j.

Figure 2

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Turning now to the dynamics of evolution, we suppose that the fecundity (reproductive rate) of cell i is given by F_i = 1 + bψ_i − cs_i. In words, each individual has baseline fitness 1, plus the benefit, bψ_i, of public goods utilization, minus the cost, cs_i of public goods production. We suppose b > 0 and 0 < c < 1, so that benefits, costs, and overall fecundity are always positive. Some of our results apply to all such b and c values, while others apply only in the weak selection regime, b, c ≪ 1/κ.

Reproductions and deaths follow the Death–Birth update rule (Ohtsuki et al., 2006). At each time step, a cell is selected randomly to die, with uniform probability. A neighbor of the now-vacant position is randomly selected to reproduce, with probability proportional to fecundity times edge weight. The new offspring fills the vacancy. For the moment, we suppose that reproduction follows the same edge weights as diffusion (we will relax this assumption later). We also consider other update rules in Supplementary file 1.

We quantify the evolutionary success of cooperation in terms of the fixation probabilities ρ_C and ρ_D, defined as the probability that the cooperator or defector type, respectively, will fix, upon starting from a single mutant in a population initially of the opposite type. Cooperation is favored if ρ_C > ρ_D. This is equivalent to the condition that, for small mutation rates, cooperators have greater time-averaged frequency than would be expected from mutational equilibrium alone (Allen and Tarnita, 2012).

The assortment of cell types due to local reproduction can be studied using coalescing random walks (Wakeley, 2009; Allen et al., 2012), which represent the ancestral lineages of chosen individuals as the coalesce into the most recent common ancestor. By applying random walk theory to both diffusion and assortment, we are able to obtain exact conditions for the success of cooperation (‘Materials and methods’; Supplementary file 1).

We find that public goods cooperation is favored, for any graph and diffusion rate, if and only if

In words, cooperation is favored if, of the public goods a cooperator produces, the benefits received by the producer, bϕ₀, plus the (edge-weighted) average benefits received by each neighbor, bϕ₁, outweigh the cost c of production (Figure 2). This result is strikingly simple, given the complex patterns of public goods sharing that result from diffusion (Figure 1). Condition (2) holds for arbitrary selection strength on complete graphs and one-dimensional lattices, and for weak selection on other graphs. This condition also holds for a variety of other diffusion processes (Supplementary file 1)—including diffusion that follows a different graph structure from reproduction. (In this case, the neighbor average ϕ₁ is computed using the weights for the reproduction graph.)

Condition (2) can alternatively be expressed as b/c > λ/[ϕ₀ (1 + 2λ) − 1] (‘Materials and methods’), showing how the success of cooperation depends on the relationship between the retention fraction ϕ₀ and the diffusion parameter λ. We have derived this relationship exactly for simple graph structures (Table 1), and present a general method for obtaining this relationship in the ‘Materials and methods’. Figure 3A,B illustrates how the critical b/c ratios vary with the diffusion parameter λ and the graph topology.

Figure 3

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Above, we have assumed that diffusion and replacement are both described by the same graph structure. However, this may not be the case for all microbes. In E. coli colonies, for example, it is reasonable to conjecture that diffusion occurs more frequently among cells that have a long side in common, whereas replacement may occur more frequently among end-to-end neighbors (Figure 1A,C). Additionally, some systems may follow a public goods diffusion process other than that modeled by Equation 1. To account for these variations, we consider a more general model in which diffusion is described by the fractions ϕ_ij of public goods which, if produced by cell i, would be utilized by cell j. Probabilities of replacement are described by a graph with edge weights e_ij as before. The diffusion fractions ϕ_ij are normalized so that ∑_j ϕ_ij = 1 for each i, and they have the same symmetries as the replacement graph; within these restrictions, they may be specified arbitrarily. Remarkably, our main result, Equation 1, remains valid in this generalized setting, with the neighbor average ϕ₁ defined as ϕ₁ = ∑_j e_ij ϕ_ij.

Our results suggest three qualitative regimes for diffusible public goods scenarios. For λ ≪ 1, the benefits are almost all retained by producer, and production is favored whenever b/c > 1. Conversely, for λ ≪ 1, public goods are shared indiscriminately, and production is favored only if public goods are essential for survival, in which case b is effectively infinite. Between these extremes, public goods are shared locally, and the spatial arrangement of cells plays a critical role in the success of cooperation (Figure 3A). At the smaller end of this critical regime, the expansion $b/c>1+\lambda \left(\kappa -1\right)/\kappa +\mathcal{O}\left({\lambda }^2\right)$ of condition (2), derived in Supplementary file 1, shows how the difficulty of cooperation increases with the diffusion parameter λ and the Simpson degree κ. For the hydrolysis of monosaccharides in S. cerevisiae, we estimate λ ∼ 3 (‘Materials and methods’); thus we expect the success of invertase production to be strongly affected by colony geometry. Interestingly, this diffusion length is of the same order of magnitude as those reported in other recent experiments with diffusible public goods (Julou et al., 2013; Momeni et al., 2013).

Our model predicts that the advantage of cooperation decreases with colony dimensionality; for example, less cooperation would be expected in three-dimensional structures than in flat (2D) colonies (Figure 3A). It also predicts that cooperation becomes more successful with increased viscosity of the environment and/or rate of public goods utilization, both of which would decrease λ.

A more subtle question is how cooperation is affected if the public good may decay (or equivalently, escape the colony) instead of being utilized. Decay reduces the absolute amount of public goods to be shared, but also restricts this sharing to a smaller circle of neighbors; thus the net effect on cooperation is at first glance ambiguous. We show in the ‘Materials and methods’ that incorporating decay effectively decreases λ by a factor 1/(1 + d), reflecting the smaller neighborhood of sharing, and also effectively decreases b by the same factor, reflecting the diminished absolute amount of public goods. Here d represents the ratio of the decay rate to the utilization rate. Since the critical benefit-to-cost ratio always increases sublinearly with λ, the net effect is to make cooperation more difficult (see Figure 3C). Thus decay of the public good has a purely negative effect on cooperation.

Our results help elucidate recent emiprical results on microbial cooperation in viscous environments. For example, Kümmerli et al. (2009) found that increased viscosity promotes the evolution of siderophore production in Pseudomonas aeruginosa, while Le Gac and Doebeli (2010) found that viscosity had no effect on the evolution of colicin production in E. coli. In both cases, increased viscosity restricted cell movement, effectively leading to fewer neighbors per cell (lower graph degree). The crucial difference lies in the effect on public goods diffusion. In the study of Kümmerli et al. (2009), the diffusion rate decreased significantly as viscosity increased, while for Le Gac and Doebeli (2010), the diffusion rate remained large even with high viscosity. Thus the divergent outcomes can be understood as a consequence of differences in the diffusion rate, captured in our model by λ.

Here we have considered homotypic cooperation—cooperation within a single population. Momeni et al. (2013), published previously in eLife, investigate heterotypic cooperation between distinct populations of S. cerevisiae, in the form of exchange of essential metabolites. Type R produces adenine and requires lysine, type G produces lysine and requires adenine, and type C (a cheater) requires adenine but does not produce adenine. While such heterotypic cooperation is not incorporated in our model, the results are qualitatively similar, in that spatial structure promoted the cooperative strategies G and R over the cheater C. This similarity can be understood by noting that heterotypic cooperation also entails a form of second-order homotypic cooperation. For example, G-cells aid nearby R-cells, which in turn aid nearby G-cells, so the benefit produced by a G-cell indirectly aids other G-cells nearby. Thus the conclusion that spatial structure aids cooperative strategies can apply to heterotypic cooperation as well.

Finally, our model can also represent the spread of behaviors via imitation on social networks (Bala and Goyal, 1998; Bramoullé and Kranton, 2007; Christakis and Fowler, 2007). Suppose an action generates a benefit b₀ for the actor, and additionally generates further benefits that radiate outward according to some multiplier m, so that first neighbors receive a combined benefit mb₀, second neighbors receive m²b₀, and so on. Education, for example, exhibits this kind of social multiplier in its effect on wages (Glaeser et al., 2003). This effect can be captured using the parameter change b = b₀/(1 − m), λ = m/(1 − m). For non-well-mixed social networks, the action becomes more likely to spread as the multiplier increases, and can spread even if there is a net cost to the actor (Figure 4).

Figure 4

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Author details

1. Benjamin Allen

   1. Department of Mathematics, Emmanuel College, Boston, United States
   2. Program for Evolutionary Dynamics, Harvard University, Cambridge, United States

   Contribution

   BA, Conception and design, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article

   For correspondence

   benjcallen@gmail.com

   Competing interests

   The authors declare that no competing interests exist.

2. Jeff Gore

   Department of Physics, Massachusetts Institute of Technology, Cambridge, United States

   Contribution

   JG, Conception and design, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article

   Competing interests

   The authors declare that no competing interests exist.

3. Martin A Nowak

   1. Program for Evolutionary Dynamics, Harvard University, Cambridge, United States
   2. Department of Mathematics, Harvard University, Cambridge, United States
   3. Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, United States

   Contribution

   MAN, Conception and design, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article

   Competing interests

   The authors declare that no competing interests exist.

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