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.

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

In many cases, the benefits of public goods go primarily to cells other than the producer. For example, in a

There is growing evidence from experiments (

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 (

(_{i} of public goods resulting from a single cooperator (center). In each case, the diffusion parameter is set as

To allow for a maximum variety of possible arrangements, we represent space as a weighted graph _{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}
_{ij} = 1, so that they represent relative frequencies of diffusion to each neighbor. Since we are modeling intercellular diffusion, we set _{ii} = 0 for each _{ij} = _{ji} for every pair

To characterize local structure, we introduce the

We consider two cells types: cooperators,

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 _{i} at each node

Above, _{i} = 0,1 indicates the current type, _{i} in _{i} represents utilization, −_{i} represents diffusion outward, and the remaining term represents diffusion inward.

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

Two key quantities in our analysis are the fractions, _{0} and _{1}, of public goods that are retained by its producer and the producer’s immediate neighbors, respectively (_{0} = _{i} and _{1} = Σ_{j}_{∈G}
_{ij}
_{j}.

Of the public good that a cooperator produces, a fraction _{0} is retained by the producer, a fraction _{1} is absorbed by each of the cooperator’s nearest neighbors, and the remainder diffuses to cells further away. (For graphs with unequal edge weights, _{1} is the edge-weighted average fraction received by each neighbor.) Cooperation is favored if _{0} + _{1}), that is, if the benefit _{0} received by producer, plus the average benefit _{1} received by each neighbor, exceeds the cost

Turning now to the dynamics of evolution, we suppose that the fecundity (reproductive rate) of cell _{i} = 1 + _{i} − _{i}. In words, each individual has baseline fitness 1, plus the benefit, _{i}, of public goods utilization, minus the cost, _{i} of public goods production. We suppose

Reproductions and deaths follow the Death–Birth update rule (

We quantify the evolutionary success of cooperation in terms of the fixation probabilities _{C} and _{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 _{C} > _{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 (

The assortment of cell types due to local reproduction can be studied using coalescing random walks (

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, _{0}, plus the (edge-weighted) average benefits received by each neighbor, _{1}, outweigh the cost _{1} is computed using the weights for the reproduction graph.)

Condition (_{0} (1 + 2λ) − 1] (‘Materials and methods’), showing how the success of cooperation depends on the relationship between the retention fraction _{0} and the diffusion parameter

Fraction of public goods retained by producer for different graph structures and diffusion rates

Graph structure | Fraction _{0} of public goods retained |
---|---|

Complete (well-mixed) | |

1D lattice | |

2D square lattice | |

These results are for large populations. Corrections for finite population size are given in

agm denotes the arithmetic-geometric mean.

This result applies to any mathematical lattice, including triangular and von Neumann lattices.

A Bethe lattice (a.k.a. infinite Cayley tree), is an infinite regular graph with no cycles. In the formula,

(_{0} in _{C} > _{D}), while a minus (−) indicates the opposite. In all cases the results were statistically significant (two-proportion pooled

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 _{ij} of public goods which, if produced by cell _{ij} as before. The diffusion fractions _{ij} are normalized so that ∑_{j}
_{ij} = 1 for each _{1} defined as _{1} = ∑_{j}
_{ij}
_{ij}.

Our results suggest three qualitative regimes for diffusible public goods scenarios. For

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 (

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

Our results help elucidate recent emiprical results on microbial cooperation in viscous environments. For example,

Here we have considered homotypic cooperation—cooperation within a single population.

Finally, our model can also represent the spread of behaviors via imitation on social networks (_{0} for the actor, and additionally generates further benefits that radiate outward according to some multiplier _{0}, second neighbors receive ^{2}_{0}, and so on. Education, for example, exhibits this kind of social multiplier in its effect on wages (_{0}/(1 −

In an alternate interpretation of our model, an action has benefits that radiate outward from the actor according to some multiplier

We obtain a recurrence relation for the stationary public goods distribution in a given state by setting

In particular, for a state in which only cell _{0} = 1 + _{1}. Combining this identity with (_{0} (1 + 2

We analyze the distribution of public goods and the assortment of cell types using the generating function for random walks (

Above,

We prove in

In particular, the fraction _{0} that a cooperator retains of its own public good can be written

Spatial assortment of types can be quantified using identity-by-descent IBD probabilities (

Considering the dynamics of Death–Birth updating, and applying established properties of generating functions, we derive (

To obtain the expressions in _{1},…,_{k} ∈ ^{n} with associated weights _{1},…,_{k}. The nodes of the lattice are all points of the form _{1},…,_{k} are integers. The edges from a node

Above,

The argument _{1},…,_{n}) of ^{n}. For example, for an

For a two-dimensional triangular lattice,

Similar expressions for other lattices, including the square lattice with von Neumann neighbors and lattices with unequal edge weights (e.g.,

We suppose that glucose uptake follows Michaelis–Menten kinetics, so that the uptake rate is given by _{max} is the maximal uptake rate, and _{max} ∼ 2 × 10^{7} molecules per second and

We calculate the lifetime

The diffusion length before uptake is calculated as ^{2}/sec in the colony environment. Combining with the above calculation of

Decay or escape of the public good can be incorporated into our model by adding a decay term to the right-hand side of

Above,

Defining the effective quantities _{i} by this same factor.

This paper was supported by the following grants:

We thank Andrea Velenich for obtaining images of

The authors declare that no competing interests exist.

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

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

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

eLife posts the editorial decision letter and author response on a selection of the published articles (subject to the approval of the authors). An edited version of the letter sent to the authors after peer review is shown, indicating the substantive concerns or comments; minor concerns are not usually shown. Reviewers have the opportunity to discuss the decision before the letter is sent (see

Thank you for sending your work entitled “Spatial dilemmas of diffusible public goods” for consideration at

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

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.”

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.

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.

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

We have added an exploration of the parallels of our work with that of

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