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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1. 1. Allen B
   2. Tarnita C

   (2012) Measures of success in a class of evolutionary models with fixed population size and structure

   Journal of Mathematical Biology pp. 1–35.

   https://doi.org/10.1007/s00285-012-0622-x

   • Google Scholar
2. 1. Allen B
   2. Traulsen A
   3. Tarnita CE
   4. Nowak MA

   (2012) How mutation affects evolutionary games on graphs

   Journal of Theoretical Biology 299:97–105.

   https://doi.org/10.1016/j.jtbi.2011.03.034

   • Google Scholar
3. 1. Allison S

   (2005) Cheaters, diffusion and nutrients constrain decomposition by microbial enzymes in spatially structured environments

   Ecology Letters 8:626–635.

   https://doi.org/10.1111/j.1461-0248.2005.00756.x

   • Google Scholar
4. 1. Bala V
   2. Goyal S

   (1998) Learning from neighbours

   The Review of Economic Studies 65:595–621.

   https://doi.org/10.1111/1467-937X.00059

   • Google Scholar
5. 1. Bramoullé Y
   2. Kranton R

   (2007) Public goods in networks

   Journal of Economic Theory 135:478–494.

   https://doi.org/10.1016/j.jet.2006.06.006

   • Google Scholar
6. 1. Buckling A
   2. Harrison F
   3. Vos M
   4. Brockhurst MA
   5. Gardner A
   6. West SA
   7. Griffin A

   (2007) Siderophore-mediated cooperation and virulence in Pseudomonas aeruginosa

   FEMS Microbiology Ecology 62:135–141.

   https://doi.org/10.1111/j.1574-6941.2007.00388.x

   • Google Scholar
7. 1. Christakis NA
   2. Fowler JH

   (2007) The spread of obesity in a large social network over 32 years

   New England Journal of Medicine 357:370–379.

   https://doi.org/10.1056/NEJMsa066082

   • Google Scholar
8. 1. Chuang JS
   2. Rivoire O
   3. Leibler S

   (2009) Simpson’s paradox in a synthetic microbial system

   Science 323:272–275.

   https://doi.org/10.1126/science.1166739

   • Google Scholar
9. 1. Cordero OX
   2. Ventouras L-A.
   3. DeLong EF
   4. Polz MF

   (2012) Public good dynamics drive evolution of iron acquisition strategies in natural bacterioplankton populations

   Proceedings of the National Academy of Sciences USA 109:20059–20064.

   https://doi.org/10.1073/pnas.1213344109

   • Google Scholar
10. 1. Driscoll WW
    2. Pepper JW

    (2010) Theory for the evolution of diffusible external goods

    Evolution 64:2682–2687.

    https://doi.org/10.1111/j.1558-5646.2010.01002.x

    • Google Scholar
11. 1. Durrett R
    2. Levin S

    (1994) The importance of being discrete (and spatial)

    Theoretical Population Biology 46:363–394.

    https://doi.org/10.1006/tpbi.1994.1032

    • Google Scholar
12. 1. Fletcher JA
    2. Doebeli M

    (2009) A simple and general explanation for the evolution of altruism

    Proceedings of the Royal Society B: Biological Sciences 276:13–19.

    https://doi.org/10.1098/rspb.2008.0829

    • Google Scholar
13. 1. Glaeser EL
    2. Sacerdote BI
    3. Scheinkman JA

    (2003) The social multiplier

    Journal of the European Economic Association 1:345–353.

    https://doi.org/10.1162/154247603322390982

    • Google Scholar
14. 1. Gore J
    2. Youk H
    3. Van Oudenaarden A

    (2009) Snowdrift game dynamics and facultative cheating in yeast

    Nature 459:253–256.

    https://doi.org/10.1038/nature07921

    • Google Scholar
15. 1. Greig D
    2. Travisano M

    (2004) The prisoner’s dilemma and polymorphism in yeast suc genes

    Proceedings of the Royal Society of London. Series B: Biological Sciences 271:S25–S26.

    https://doi.org/10.1098/rsbl.2003.0083

    • Google Scholar
16. 1. Griffin A.
    2. West S.
    3. Buckling A

    (2004) Cooperation and competition in pathogenic bacteria

    Nature 430:1024–1027.

    https://doi.org/10.1038/nature02744

    • Google Scholar
17. 1. Hardin G

    (1968) The tragedy of the commons

    Science 162:1243–1248.

    https://doi.org/10.1126/science.162.3859.1243

    • Google Scholar
18. 1. Julou T
    2. Mora T
    3. Guillon L
    4. Croquette V
    5. Schalk IJ
    6. Bensimon D
    7. Desprat N

    (2013) Cell–cell contacts confine public goods diffusion inside Pseudomonas aeruginosa clonal microcolonies

    Proceedings of the National Academy of Sciences USA 110:12577–12582.

    https://doi.org/10.1073/pnas.1301428110

    • Google Scholar
19. 1. Kümmerli R
    2. Griffin AS
    3. West SA
    4. Buckling A
    5. Harrison F

    (2009) Viscous medium promotes cooperation in the pathogenic bacterium Pseudomonas aeruginosa

    Proceedings of the Royal Society B: Biological Sciences 276:3531–3538.

    https://doi.org/10.1098/rspb.2009.0861

    • Google Scholar
20. 1. Korolev KS
    2. Nelson DR

    (2011)

    Competition and cooperation in one-dimensional stepping-stone models

    Physical Review Letters 107:088103.

    • Google Scholar
21. Book

    1. Lawler G
    2. Limic V

    (2010)

    Random walk: a modern introduction

    New York: Cambridge University Press.

    • Google Scholar
22. 1. Le Gac M
    2. Doebeli M

    (2010) Environmental viscosity does not affect the evolution of cooperation during experimental evolution of colicigenic bacteria

    Evolution 64:522–533.

    https://doi.org/10.1111/j.1558-5646.2009.00814.x

    • Google Scholar
23. 1. Lee HH
    2. Molla MN
    3. Cantor CR
    4. Collins JJ

    (2010) Bacterial charity work leads to population-wide resistance

    Nature 467:82–85.

    https://doi.org/10.1038/nature09354

    • Google Scholar
24. 1. Lieberman E
    2. Hauert C
    3. Nowak M

    (2005) Evolutionary dynamics on graphs

    Nature 433:312–316.

    https://doi.org/10.1038/nature03204

    • Google Scholar
25. 1. Misevic D
    2. Frenoy A
    3. Parsons DP
    4. Taddei F

    (2012)

    Effects of public good properties on the evolution of cooperation

    Artificial Life 13:218–225.

    • Google Scholar
26. 1. Momeni B
    2. Waite A
    3. shou W

    (2013) Spatial self-organization favors heterotypic cooperation over cheating

    eLife 2:e00960.

    https://doi.org/10.7554/eLife.00960

    • Google Scholar
27. 1. Montroll E
    2. Weiss G

    (1965) Random walks on lattices. II

    Journal of Mathematical Physics 6:167.

    https://doi.org/10.1063/1.1704269

    • Google Scholar
28. 1. Nowak MA

    (2006) Five rules for the evolution of cooperation

    Science 314:1560–1563.

    https://doi.org/10.1126/science.1133755

    • Google Scholar
29. 1. Nowak MA
    2. May RM

    (1992) Evolutionary games and spatial chaos

    Nature 359:826–829.

    https://doi.org/10.1038/359826a0

    • Google Scholar
30. 1. Ohtsuki H
    2. Hauert C
    3. Lieberman E
    4. Nowak MA

    (2006) A simple rule for the evolution of cooperation on graphs and social networks

    Nature 441:502–505.

    https://doi.org/10.1038/nature04605

    • Google Scholar
31. 1. Perc M
    2. Szolnoki A

    (2008)

    Social diversity and promotion of cooperation in the spatial prisoner’s dilemma game

    Physical Review E 77:011904.

    • Google Scholar
32. 1. Rainey PB
    2. Rainey K

    (2003) Evolution of cooperation and conflict in experimental bacterial populations

    Nature 425:72–74.

    https://doi.org/10.1038/nature01906

    • Google Scholar
33. 1. Raymond B
    2. West SA
    3. Griffin AS
    4. Bonsall MB

    (2012) The dynamics of cooperative bacterial virulence in the field

    Science 337:85–88.

    https://doi.org/10.1126/science.1218196

    • Google Scholar
34. 1. Ross-Gillespie A
    2. Gardner A
    3. West SA
    4. Griffin AS

    (2007) Frequency dependence and cooperation: theory and a test with bacteria

    The American Naturalist 170:331–342.

    https://doi.org/10.1086/519860

    • Google Scholar
35. 1. Rousset F
    2. Billiard S

    (2000) A theoretical basis for measures of kin selection in subdivided populations: finite populations and localized dispersal

    Journal of Evolutionary Biology 13:814–825.

    https://doi.org/10.1046/j.1420-9101.2000.00219.x

    • Google Scholar
36. 1. Santos FC
    2. Pacheco JM

    (2005) Scale-free networks provide a unifying framework for the emergence of cooperation

    Physical Review Letters 95:98104.

    https://doi.org/10.1103/PhysRevLett.95.098104

    • Google Scholar
37. 1. Simpson EH

    (1949) Measurement of diversity

    Nature 163:688.

    https://doi.org/10.1038/163688a0

    • Google Scholar
38. 1. Szabó G
    2. Fáth G

    (2007) Evolutionary games on graphs

    Physics Reports 446:97–216.

    https://doi.org/10.1016/j.physrep.2007.04.004

    • Google Scholar
39. 1. Taylor PD
    2. Day T
    3. Wild G

    (2007) Evolution of cooperation in a finite homogeneous graph

    Nature 447:469–472.

    https://doi.org/10.1038/nature05784

    • Google Scholar
40. Book

    1. Wakeley J

    (2009)

    Coalescent theory: an introduction

    Greenwood Village, CO: Roberts & Company Publishers.

    • Google Scholar
41. 1. Waters CM
    2. Bassler BL

    (2005) Quorum sensing: cell-to-cell communication in bacteria

    Annual Review of Cell and Developmental Biology 21:319–346.

    https://doi.org/10.1146/annurev.cellbio.21.012704.131001

    • Google Scholar
42. 1. West SA
    2. Buckling A

    (2003) Cooperation, virulence and siderophore production in bacterial parasites

    Proceedings of the Royal Society of London. Series B: Biological Sciences 270:37–44.

    https://doi.org/10.1098/rspb.2002.2209

    • Google Scholar

Thank you for sending your work entitled “Spatial dilemmas of diffusible public goods” for consideration at eLife. Your article has been favorably evaluated by a Senior editor and 3 reviewers, one of whom, Carl Bergstrom, is a member of our Board of Reviewing Editors.

The editors and the reviewers discussed their comments before we reached this decision, and the Senior editor has assembled the following comments to help you prepare a revised submission.

Microbes frequently face public goods dilemmas and often these dilemmas involve the production of diffusible products secreted into the extracellular environment. It is an interesting and open question to determine when and how such behavior will be favored by natural selection. The manuscript provides a technically sound and elegant mathematical analysis of the problem based on an implicit graphical structure in the spatial organization of cells that make up a colony. The main mathematical result, which is inequality (2), is nice in its simplicity and how it incorporates the three different factors within a single representation.

The reviewers had two major concerns that need to be addressed before the manuscript can be accepted:

A) Whether the paper is of sufficient biological interest to merit publication in eLife. To this concern the following four comments/suggestions were provided by the reviewers:

1) While graphs provide a nice means of modeling some types of structure, one reviewer was less convinced that they are a natural way to model the structure of diffusing public goods. This approach that the authors have developed extensively over the years appears forced upon the biology of the problem rather than being an natural way to model natural interactions.

2) Do the results tell us much beyond what we already know in terms of the biological problem? For example, similar effects of the diffusion rate are already known from other models of public goods (some of which are cited), and the colony dimension results (which sounds really interesting at first) is also pretty obvious once it becomes clear what is meant by colony dimension. The main new insight about biology provided by the results is the role of the decay rate of the public good. To my knowledge at least, this idea has not previously been explored and it is clear that the tension between the various ways that decay rate enters the problem requires the sort of quantitative analysis presented here. Regardless, the result does seem like a rather modest advance in our understanding of the evolutionary interplay between public goods, diffusion, cooperation, etc.

3) The authors might also want to delve more deeply into the literature on public goods to better position their results within the existing literature. For example, there is good work by Brown, Taylor, Buckling, West, and others. Some of this is cited but not discussed in a very thorough way, and some is not even cited. A none-exhaustive list of other potentially useful papers include:

Buckling, A, Harrison, F, Vos, M, Brockhurst, MA, Gardner, A, West, SA & Griffin, AS. 2007 Siderophore-mediated cooperation and virulence in Pseudomonas aeruginosa. FEMS Microbiolol. Ecol. 62, 135-141. doi:10.1111/j.1574-6941.2007.00388.x

West, SA & Buckling, A. 2003 Cooperation, virulence and siderophore production in bacterial parasites. Proc. R. Soc. Lond. B 270, 37-44. doi:10.1098/rspb.2002.2209

Bramoulle, Y & Kranton, R. Public goods in networks. Journal of Economic Theory. 135 (1), 478-494

4) You may also be able to address this concern by referencing and coordinating the text of your paper with the parallel submission by Shou et al.

B) Avoid confusion about the use of the term “Bethe Lattice”. A reviewer provided the following commentary/suggestions:

“In order to avoid later confusion I suggest substituting the expression “Bethe lattice” or “locally Cayley tree structure” for “Cayley tree” through the whole text. For finite Cayley trees a relevant portion of the nodes are located on the periphery where each node has only one neighbor. This is the reason why the behavior of the Ising model on the Cayley tree is similar to those observed on the one-dimensional chain (no magnetic ordering at finite temperatures). On the contrary, the Ising model on Bethe lattice exhibits a mean-field type order-disorder phase transition (when increasing the temperature) that can be described exactly by several methods, e.g., by the cavity method or pair approximation [for details see the review by Dorogovtsev et al., Rev. Mod. Phys. 80 (2008) 1275-1335]. The concept of Bethe lattice neglects the effects of periphery and involves equivalence between the nodes, as it is assumed in the present work, too.”

The reviewers had two major concerns that need to be addressed before the manuscript can be accepted:

A) Whether the paper is of sufficient biological interest to merit publication in eLife. To this concern the following four comments/suggestions were provided by the reviewers:

1) While graphs provide a nice means of modeling some types of structure, one reviewer was less convinced that they are a natural way to model the structure of diffusing public goods. This approach that the authors have developed extensively over the years appears forced upon the biology of the problem rather than being an natural way to model natural interactions.

While they may have an abstract “flavor”, graphs are a very natural tool for representing a wide variety of spatial relationships. Compared, for example, to lattice models (an accepted tool of the field), graphs have more flexibility to represent the distinct patterns of cell arrangement that occur in microbial colonies. In this study we use weighted graphs to allow for different diffusion rates between different kinds of neighbors (e.g., lateral versus end-to-end). The symmetry assumptions correspond to the quasi-regular structures that are often found in colony interiors.

2) Do the results tell us much beyond what we already know in terms of the biological problem? For example, similar effects of the diffusion rate are already known from other models of public goods (some of which are cited), and the colony dimension results (which sounds really interesting at first) is also pretty obvious once it becomes clear what is meant by colony dimension. The main new insight about biology provided by the results is the role of the decay rate of the public good. To my knowledge at least, this idea has not previously been explored and it is clear that the tension between the various ways that decay rate enters the problem requires the sort of quantitative analysis presented here. Regardless, the result does seem like a rather modest advance in our understanding of the evolutionary interplay between public goods, diffusion, cooperation, etc.

In addition to our results on the effects of the decay rate, our model makes the unexpected prediction that the success of cooperation depends only on the amounts of public goods received by a cell and its immediate neighbors. Thus, even though public goods may be shared at arbitrarily large distances, the success of this behavior can be understood by examining neighbors at distance one.

3) The authors might also want to delve more deeply into the literature on public goods to better position their results within the existing literature. For example, there is good work by Brown, Taylor, Buckling, West, and others. Some of this is cited but not discussed in a very thorough way, and some is not even cited.

We thank the reviewers for the suggestions. We have incorporated the suggested references, along with others that have appeared recently. We now discuss these contributions in greater detail in the last paragraph of the Introduction. We have also incorporated a recent study of diffusible public goods by Julou et al. (2013) into our references and Discussion.

4) You may also be able to address this concern by referencing and coordinating the text of your paper with the parallel submission by Shou et al.

We have added an exploration of the parallels of our work with that of Momeni et al. (2013) at the second-to-last paragraph of the Discussion. Although the work of Momeni et al. (2013) concerns heterotypic cooperation—which is not directly represented in our model—we present a new argument that heterotypic cooperation in space also entails a kind of second-order homotypic cooperation, so that results from models like ours can also shed light on heterotypic cooperation, as investigated by Momeni et al. (2013).

B) Avoid confusion about the use of the term “Bethe Lattice”. A reviewer provided the following commentary/suggestions:

“In order to avoid later confusion I suggest substituting the expression “Bethe lattice” or “locally Cayley tree structure” for “Cayley tree” through the whole text. For finite Cayley trees a relevant portion of the nodes are located on the periphery where each node has only one neighbor. This is the reason why the behavior of the Ising model on the Cayley tree is similar to those observed on the one-dimensional chain (no magnetic ordering at finite temperatures). On the contrary, the Ising model on Bethe lattice exhibits a mean-field type order-disorder phase transition (when increasing the temperature) that can be described exactly by several methods, e.g., by the cavity method or pair approximation [for details see the review by Dorogovtsev et al., Rev. Mod. Phys. 80 (2008) 1275-1335]. The concept of Bethe lattice neglects the effects of periphery and involves equivalence between the nodes, as it is assumed in the present work, too.”

We apologize for this confusion. We now use the term Bethe lattice throughout.

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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