Cooperation in the microbial world is abundant, mostly through excreted products benefiting not only the producer but other individuals, too. Microbial communities often rely on the production of metabolites or matrix substances that serve as common goods for the group as a whole. However, sacrificing valuable resources by the individual for the good of the group is a risky investment: cheaters may take advantage of honest cooperators by contributing less (or nil) to the common effort while they still enjoy the benefit. This means that selfish cheaters (“laggards”) have a growth advantage compared to cooperative producers, which, in the long run, leads to the tragedy of the commons [1] and the ultimate collapse of cooperation [2, 3].

In the face of the obvious fitness advantages of cheating in a cooperative group, it is puzzling how cooperation evolves and is maintained even in species with evolved mechanisms to avoid fraud. Aimed reward and punishment, straightforward antidotes to cheating, work only if group members can distinguish each other personally and remember the record of past actions of every group member back into a non-zero length of time. This is rarely the case even in vertebrate species and much less in unicellular organisms with no brain or memory at all. Prokaryotes are therefore the least expected to harvest the benefits of cooperative group actions, lacking sophisticated mechanisms of partner recognition and record-keeping.

Yet, there is an astonishing diversity and abundance of examples of genuine cooperation within and even between different prokaryotic strains producing public goods [4]. These include the excretion of exoproducts like luciferin [5], exoenzymes [6], bacteriocins [7, 8], siderophores [9, 10], virulence factors [11], and biofilm matrix substances [12], to mention just the most obvious forms of microbial cooperation. The actual functions of such different cooperative features may be connected in diverse combinations within the same strain, opening a wide range of complex microbial social strategies yet to be explored [10].

Most known forms of prokaryotic cooperation are threshold-limited: a certain number of nearby cooperators must all act simultaneously for the collective benefit of cooperation to exceed its individual costs [13,14,15]. Thus, cooperating individuals have to coordinate their actions within a narrow spatiotemporal range, i.e., to synchronously express and excrete public goods in close proximity to one another.

Any mechanism ensuring that cooperators interact with other cooperators at a probability higher than their proportion within the population increases the chance of persistent cooperation. On the other hand, more frequent cooperator-cheater interactions increase the probability of cheater takeover. Since prokaryotic gene expression patterns are clonally passed down the generations, regardless of whether they are genetically or epigenetically determined, see, e.g., [16]) with little room for phenotypic plasticity, the mechanism maintaining cooperation must always be a variant of kin selection. However, the actual form this mechanism takes may vary substantially.

The most trivial of such mechanisms is "environmental viscosity" [17]. This means physical constraints limiting the mobility of individuals, thereby keeping the offspring adjacent to their parent and ensuring the overwhelming dominance of intraclonal interactions and effective kin selection thereof. It has been repeatedly shown that sufficiently high environmental viscosity can indeed maintain cooperation and prevent the invasion of cheaters in standard public good games [18, 19]. Cooperators benefit from population viscosity in threshold public goods games as well [20], but they can stably coexist with cheaters even if the interaction size is limited and the population is perfectly mixed in this model context [21].

However, viscosity is rarely high enough in natural microbial habitats to effectively prevent cheater invasion from outside and/or constrain cheating mutants within. Therefore, in less viscous environments, it may be of substantial selective advantage for the individuals to be able to size up the local density of potential cooperators and make actual cooperation dependent on that. This may prevent wasting valuable resources on futile attempts to cooperate locally when lacking sufficient cooperator density.

Quorum sensing (QS [22]) is a simple genetic switching mechanism of communication-aided cooperation that may have evolved to provide this kind of phenotypic flexibility for unicells. It consists of a constitutively expressed signal, a membrane receptor, and an expression-excretion mechanism for cooperation (Fig. 1). The QS switch triggers the transcription of certain genes upon sensing a sufficient number (a “quorum”) of cooperators in the neighborhood. The quorum is sensed by the capture of a sufficient number of signal molecules by a specific, dedicated membrane receptor, which then transmits the signal through an intracellular signal channel (often using cAMP) to the chromosome and activates the cooperation genes. Gram-positive bacteria use small autoinducing peptides as QS signals [23], whereas Gram-negative bacteria [24, 25] and archaea [26, 27] usually excrete N-acyl homoserine lactones for the same purpose. The fundamental signaling mechanism is the same in both cases. Quorum sensing has been discovered in almost any bacterial strain in which it was looked for, some strains utilizing multiple different QS systems [28], sometimes in synergy with each other. For example, the two principal QS systems (las and rhl) regulating the expression of virulence factors in Pseudomonas aeruginosa have been shown to form a reciprocal signaling network that synergistically enhances and tunes the strain’s reactivity to its physical and “social” environment.

Fig. 1

A simplified representation of quorum sensing regulated cooperation for the production of common goods collectively utilized by bacteria having access to them. The three basic genetic components of the QS-regulated cooperative system are (a) the signaling component (pink), which comprises the genes for signal molecule production and excretion of signal molecules (stars); (b) the signal detection and transduction system (green), including genes for the signal receptor and the second messenger system; (c) the cooperation genes (blue) which are transcribed and expressed (and products externalized) if the extracellular concentration of signal molecules is sufficiently high. Red asterisks signify component events of signal detection: signal capture and signal transduction

There are different interpretations of the function of quorum sensing (see [29]). The first interpretation is that it aims at sizing up the local density of potential cooperators [30,31,32], whereas the second one is that it serves to assess the diffusibility of exoproducts in the environment [33]. These two functions are difficult to disentangle as both high cell density and low diffusion can lead to a high local concentration of the QS signal. There is some indication that some bacterial strains might be able to distinguish these two types of information using combinatorial QS signals, i.e., by using two different signal molecules with covarying decay and autoinduction rates [29].

Either way, as QS is a communication system, it is prone to defection or deceit by another type of cheater: individuals with a silent set of cooperation genes but capable of sending out false signals of intent for cooperation (lying) may still enjoy the full benefit of the public good produced by nearby cooperators responding to the false signal. A textbook example of such defectors (“liars”) is the lasR mutant of Pseudomonas aeruginosa that does not respond to QS signals of the wild type and, consequently, does not cooperate in producing an important public good, a protease exoenzyme virulence factor [3]. The cheating mutant has been shown to enjoy a substantial reproductive advantage over cooperative strains [34] for the obvious reason of not carrying the metabolic burden of cooperation (exoenzyme production).

A more nuanced way of cheating is increasing signal production while evolving a higher threshold value for cooperation. Brown and Johnstone in a seminal model of QS were able to show that increasing conflict of interest (decreasing relatedness) favors such “coercive” variants [35]. These can manipulate older strains with lower thresholds into increased production of the public good by mimicking a higher cooperator density with the increased signal production. In silico study of QS found that such coercive variants are more likely to emerge in genetically mixed populations with decreased relatedness [36].

If, however, the signal cannot be switched off, cheating is expected to be less deleterious for cooperation. For example, the signal may be the public good itself, as in the case of lactic acid bacteria producing the bacteriocin called nisin. Nisin production is QS-regulated [37], but the QS signal is nisin itself, so bacteria producing nisin are also signaling. Cooperators always produce nisin at a low expression level, advertising their willingness to cooperate, and they express and excrete nisin at an elevated level when a quorum is reached. However, the cooperation signal can be faked by non-cooperators: the bacteriocin gene can be expressed constitutively at a low level to produce the signal, but cheaters never express it at sufficiently high levels to considerably contribute to the common good. Therefore, individuals capable only of low-level nisin expression are actually liars.

Another type of cheater is a cooperator that expresses more nisin in its signaling state than the normal signal level but less than the cooperation level. By issuing extra signals above the basic expression level of honest cooperators, these cheaters gain a fitness benefit by increasing the local signal concentration which, therefore, may reach the quorum threshold and convince more cooperators to join in the common effort, possibly in vain. This is equivalent to promising more cooperation than actually provided—a milder version of the liar strategy.

To study the dynamical and evolutionary properties of QS-regulated cooperation, it is sufficient to assume that the individuals clonally inherit three fundamental QS-related properties: (1) signaling (S) or not (s) the intention/ability to cooperate, (2) responding (R) or not (r) to above-quorum signal levels, and (3) cooperating (C) or not (c) when a quorum is reached (see Fig. 1). Each of these properties may be determined by a number of different genes, but the only trait we consider relevant is the functionality of the corresponding gene set as a unit. Therefore, we assume three loci with two functional alleles on each, which allows for \(^=8\) different genotypes. For QS to hold, cooperation (i.e., the expression of the C allele) is assumed to be conditional on the presence of a critical number of signaler individuals (the quorum, i.e., those harboring S and/or C) within the interaction neighborhood of individuals possessing both C and R. In other words, cooperators capable of detecting the signal will cooperate only above the critical local quorum and mute their cooperation gene otherwise. Non-responder cooperators cooperate unconditionally, and they may issue the signal either at the normal expression level or at an elevated one.

The question we aim to answer is whether cue-driven (e.g., nisin type) cooperation and/or communication can be maintained in a population of quorum sensing microbes in the face of all possible mutations allowing cheater strategies, assuming different costs of cooperation, signaling, and signal detection/response. Here, we will scrutinize a family of models built on the above assumptions, allowing all three possible types of cheaters to appear. The models are built on the individual-based approach of Czárán and Hoekstra [38], extending it both in scope and methodology of representation. For a deeper insight into the coexistence dynamics of the various strategies, we derive analytical (1) mean-field (MF) and (2) configuration-field (CF) [39] approximations besides the corresponding individual-based (3) non-spatial and (4) spatial stochastic simulations of the threshold public goods game (TPPG).

Model basics

We used four different model types: MF approximation, CF approximation, non-spatial and spatial (on-lattice) agent-based models. They share the same basic assumptions (explained below) but gradually relax crucial simplifications while also losing analytical tractability. For the specifics of the different approaches, see Methods and models; for their parameters, see Table 1.

Table 1 Model variables and parameters used throughout this studyMean-field model (MF)

In Additional file 1: Appendix 1 [40], we construct the mean-field version of the model, with the assumptions that population size (\(P\)), interacting group size (\(N\)) and the cooperation threshold \(\kappa\) are all very large, while \(N/P\ll 1\) and \(\kappa /N\to _\). In this limit, all the dynamical effects are averaged across the entire habitat so that the fitnesses of the strategies depend only on the average frequencies of the strains present.

Configuration-field model (CF)

Next, we assume that the population is still very large (infinite), but individuals form random interacting groups of finite size \(N\). While in the previous section we considered \(N\) to be so large that each interacting group consists of strategies in exact proportion to their global frequencies in the population, now we assume that \(N\) is smaller, and thus different interaction groups with different configurations of strategies are formed in an inherently stochastic manner, due to sampling errors. The two players participating in an elementary game step are randomly chosen members of their own interaction groups (both of size N) that are drawn at random from the population. The overall fitness for each of the eight strategies is calculated as the weighted average of its local fitness in all possible configurations of the interaction group around a focal individual of the given strategy (for more details, see Additional file 1: Appendix 1).

Agent-based models

To address the effect of finite interacting groups, along with the spatial constraints arising from limited agent mobility and local (neighborhood-) interactions, we have developed an agent-based simulation implementation of the configuration-field model (agent-based nonspatial model) and a spatially explicit lattice version of it (agent-based spatial model).

Figure 2 depicts the relationship of these models. The configuration field model is a technical tool that allows the relaxation of constraining assumptions one-by-one from the mean-field model to the lattice-based individual model. The CF (compared to MF) represents the relaxation of the assumption of infinitely large interaction groups and hence can enable compositional variance. The agent-based lattice model introduces spatial correlations (that are missing both from MF and CF).

Fig. 2

The relation between different model types: MF, CF, and agent-based (well mixed and lattice). The CF (compared to MF) represents the relaxation of the assumption of infinitely large groups, hence the introduction of compositional variance. The agent-based lattice model introduces spatial correlations

Strategy set

Individual behaviors (strategies) are determined by three “functional genes” (heritable traits possibly encoded by a number of genes each) which control cooperation (C), extra signaling (S), and signal detection and response (R) (see Fig. 3). Each of these genes can be in one of two states (i.e., they have two “alleles”): they may be active (denoted by bold capitals: C, S, R) or inactive (bold minuscules: c, s, r). Note that an active cooperation gene (C) provides two things in the model: (i) the public good and (ii) a baseline (cost-free) signal level (hence “cue-based” cooperation). Accordingly, there are eight possible strategies (“phenotypes”). “Lazy” (La: csr) never issues or detects the quorum signal and does not cooperate. “Trusty” (Tr: Csr) cooperates unconditionally, as it does not communicate and neither gives nor listens to signals. “Bouncer” (Bo: CSr) is also an unconditional cooperator issuing extra signals but not listening to them. We assume that the extra signal expression doubles the signal level so that a Bo individual counts as two signalers in its interaction group. “Smart” (Sm: CsR) is a cooperator that detects the quorum signal and cooperates if the signal level in its immediate vicinity exceeds the quorum threshold. “Nerd” (Ne: CSR) is a quorum-sensitive cooperator that always produces an extra signal dose. “Liar” (Li: cSr) issues the quorum signal but never cooperates. “Curious liar” (Cl: cSR) acts like Liar but also detects the signal. Finally, “Voyeur” (Vo: csR) only detects the signal and never cooperates.

Fig. 3

Strategy set of the QS model. Strategies are listed in boxes; their genotypes are denoted in bold typeface and their metabolic costs in parentheses. Capital letters in genotypes indicate expressed “genes”; minuscules indicate inactive alleles. \(_\) is the baseline metabolic cost paid by everyone, \(c\) is the cost of cooperation, \(\theta =1\) if the quorum threshold is reached (otherwise \(\theta =0\)), \(s\) is the signal production cost, and \(r\) is the signal-detection cost. Underlined strategies are context dependent, capable of switching to cooperation when a signal quorum is reached. The individual strategies are characterized as follows: Lazy does nothing; Voyeur detects signal but does not cooperate; Liar signals but does not cooperate; Curious liar produces and detects signal but does not cooperate; Trusty does not communicate but always cooperates and thus also issues the QS signal; Smart detects the signal and cooperates if the signal level exceeds quorum; Bouncy issues extra signal but does not listen to it; Nerd produces the extra signal, detects the signal and cooperates if the signal level exceeds quorum

Metabolic costs

Each player invests a fixed metabolic effort \(_\) into its own maintenance. This baseline metabolic burden is the same for all strategies. Cooperation (C), the emission of extra signal molecules (S), and the production, maintenance, and operation of the signal response system (R) are all metabolically costly; the corresponding \(c\), \(s\), and \(r\) costs are added to the baseline metabolic burden of the players expressing them, to yield the total metabolic cost of the corresponding genotype. Cooperators always express the public good at a low level, which constitutes an inevitable cue of cooperation. The extra signal molecules and the signal response system are always expressed in all the players carrying the active C and/or R genes for these functions. The high-level expression of the cooperation gene C means cooperation, which may be conditional on the concentration of nearby signal molecules (i.e., the number of signaling neighbors), provided that they express the signal response system R and thus they are capable of detecting the signal. Cooperating players also issue the cooperation cue at no additional cost, i.e., the (honest) signal cost of cooperators is included in c. The condition for the high-level expression of the cooperation gene in conditional cooperators (the Smart and the Nerd strategies) is that the number of signal doses within the QS neighborhood exceeds the QS threshold \(Q\). Obviously, non-expressed cooperation genes carry no metabolic cost. Figure 3 summarizes the total metabolic costs of the strategies with the local signal levels (i.e., the number of signal doses within the QS neighborhood) below and above the quorum signal threshold.

Cooperation benefit and fitness

The metabolic cost of a player with at least \(\kappa\) active cooperators in its own interaction group is reduced by a factor \(0<b<1\), which is the cooperation benefit. The benefit reduces the metabolic cost of the individual in a multiplicative manner. Notice that the cooperation threshold \(\kappa\) is not equivalent with the quorum signal threshold \(Q\)—these two thresholds are different, even if their values are the same numerically. \(Q\) is the minimum number of quorum signal doses within the interaction group of a conditional cooperator that is sufficient to switch its C gene on, whereas \(\kappa\) is the minimum number of active cooperators within a group necessary for members of the group to enjoy the cooperation benefit. We assume \(\kappa =Q\) throughout this study, implying that this relation is the evolutionary optimum for cooperators: any deviation by switching on cooperation too late or too early is selected against. The fitness \(_\) of a player \(i\) is linearly decreasing with its actual metabolic cost \(_\), that is:

$$_=_-_= _-\left(1-_ b\right)\left(_+_}}_+_+_\right),$$

(1)

where \(_=c\) if the cooperation gene can be expressed in player \(i\), otherwise \(_=0\); similarly, \(_\) and \(_\) are the corresponding costs of issuing an extra signal and responding to a quorum of signals by player \(i\). \(_=1\) if the number of active cooperators in the interacting group that \(i\) belongs to is at least \(\kappa\), otherwise \(_=0\), and \(_=1\) if the number of signal doses (i.e., the number of cooperators plus the number of extra signalers) is at least \(Q\) within the group of \(i\), or if \(i\) is an unconditional cooperator; \(_=0\) otherwise.

The metabolic cost of various strategies

Trusty always cooperates and signals, and thus it always pays the cost of cooperation (\(c\)), but it enjoys the cooperation benefit (\(b\)) only if the number of cooperators in its interacting group exceeds the critical threshold \(\kappa\). Bouncer behaves as Trusty, except that it produces twice as many signal molecules as Trusty at some extra cost \(s\). Smart cooperates (and pays the cost \(c\)) only if the number of signal doses is at least \(Q\) in its interacting group but operating the response system listening to the signal of the others implies a small cost \(r\). Nerd is a conditional cooperator like Smart, producing an extra dose of signal at the extra cost \(s\). The strategy Liar never cooperates and does not listen to the quorum signal, but it does produce it, trying to induce conditional cooperators in its interacting group, again at the cost \(s\). This model assumes that the quorum signal is the product of cooperation; thus, the emission of a single dose of it is free of charge in all potential and actual cooperators, but they can produce an additional dose at a cost. Non-cooperators pay even for the first signal dose if they produce it, so that the Curious liar strategy both producing and detecting the signal pay the corresponding costs \(s\) and \(r\); the Voyeur strategy pays only \(r\) for detecting the signal. Figure 3 summarizes the metabolic costs and benefits for all possible genotypes in interacting groups with the number of actual cooperators below and above the cooperation threshold \(\kappa\).

Reproduction, mutation, evolution

Reproduction takes place during pairwise interactions between individuals, following the rules of probabilistic imitation dynamics: player \(i\) has a chance of occupying the site of its opponent \(j\) with its own offspring. This has probability \(_\), proportional to the relative fitness (cost advantage) of \(i\), as \(\Delta _=\frac_-_}_}\), where \(\Delta _=c+s+r+b _\) is the largest possible cost difference (between an individual expressing all functional genes but not receiving benefit, and another one expressing none of the genes but receiving the full benefit). Then:

where \(\sigma\) is the strength of selection. Obviously, \(_=1-_.\)

We assume that, during reproduction, any of the three functional loci (C, S, and R) may mutate from its functional to its inactive form in the offspring, and back-mutations are also allowed, with each of the six possible mutation events occurring at its own specific rate (possibly zero). The dynamical equilibria and the trajectories of the resulting selection processes are the primary targets of this study, discussed in the four different modeling approaches.

Temporal resolution of the models

The MF approximation is continuous in time, implemented as a set of ordinary differential equations (ODE-s), one equation for each of the eight strategies, with their interactions depending on the actual overall densities of the strategies (cf. Supplement p.1). The CF approximation is also a set of ODE-s comprising a single differential equation for each strategy, but the interaction terms are dependent on the weighted average frequencies of all possible interaction neighborhood configurations (cf. Supplement p.6). The agent-based models assume discrete interaction events in continuous time: each agent participates in a single interaction event per unit time but in a random order. Since we are interested only in the stationary states of the models (in terms of strategy frequencies), the possible differences in their temporal resolution are indifferent.