Green beards in the light of indirect genetic effects

Abstract The green‐beard effect is one proposed mechanism predicted to underpin the evolution of altruistic behavior. It relies on the recognition and the selective help of altruists to each other in order to promote and sustain altruistic behavior. However, this mechanism has often been dismissed as unlikely or uncommon, as it is assumed that both the signaling trait and altruistic trait need to be encoded by the same gene or through tightly linked genes. Here, we use models of indirect genetic effects (IGEs) to find the minimum correlation between the signaling and altruistic trait required for the evolution of the latter. We show that this correlation threshold depends on the strength of the interaction (influence of the green beard on the expression of the altruistic trait), as well as the costs and benefits of the altruistic behavior. We further show that this correlation does not necessarily have to be high and support our analytical results by simulations.


| INTRODUC TI ON
One of the explanations for the evolution of altruism is the socalled green-beard mechanism (Dawkins, 1976;Hamilton, 1964). In this concept, a single gene or several tightly linked genes encoding altruistic behavior need to meet three requirements: (a) cause its bearer to behave altruistically, (b) display an observable and distinctive trait (the "green beard"), and (c) recognize the signal and modify the behavior accordingly (Dawkins, 1976;Keller & Ross, 1998;Queller, 2008). An allele causing altruism has the capacity to recognize individuals with copies of itself, helping the individuals that carry them, and thus helping to propagate itself (Dawkins, 1976;Grafen, 1998). Therefore, the altruistic behavior of an individual is just a result of the selfish behavior of the green-beard gene (Dawkins, 1976).
A problem with the green-beard effect is that any mutant alleles that produce the advertising trait without providing the helping behavior-a cheater-will cause its bearer to have higher fitness and will, therefore, be selected for (Gardner & West, 2010). To avoid this, it is assumed that both the signaling and the altruistic trait must be encoded by the same gene or a set of very tightly linked genes. Therefore, so-called "green beards" are predicted to be rare.
Despite the initial skepticism about whether such situations occur in nature, green-beard-like genes have been documented in different species. Keller and Ross (1998) discovered a green-beard gene in the red fire ant, Solenopsis invicta, where a linked set of alleles causes workers to kill homozygous queens that lack the greenbeard allele, while not killing individuals that contain it. Workers appear to distinguish between carriers and noncarriers by a transferable odor cue (Grafen, 1998;Keller & Ross, 1998). The "green beard" is a chemical carried on the queens' cuticle. Keller and Ross (1998) showed that all components of a green-beard effect are present-a detectable phenotypic feature, the ability to recognize the feature, and differential responses toward individuals with and without the feature. Moreover, all these features are mediated by a group of closely linked genes.
Another example of green-beard scenario comes from yeast (Smukalla et al., 2008). Flocculation is a formation of flocs of flakes of yeast that helps to protect them from the damage from chemicals, for instance, alcohol. Flocculation is caused by a protein, a product of the gene FLO1. Cells that make this protein have to pay a cost: They grow more slowly than cells that do not express it. However, only cells expressing FLO1 can stay inside of the floc. Even though some of these cells on the outer side of the floc die, those inside the floc survive and pass the altruistic gene to the next generation. Cells without the FLO1 gene do not form flocs and die if exposed to harsh chemicals.
While several modelling approaches have been developed to study the evolution of altruism (mainly based on game theory), we find that a quantitative genetics framework based on indirect genetic effects is ideally suited to model the green-beard scenario.
One of the major advantages of IGE models is that they are based on parameters that can be readily determined (Bijma, 2010a;McGlothlin & Brodie, 2009). Here, we show that these models are ideally suited to the analysis of green-beard effects because the signaling-recognition mechanism in green beards is, by definition, captured by IGEs. The IGE models thus allow easy formalization of the green-beard scenario based on measurable parameters, such as the strength of selection and interaction strength (Bijma, 2010a;McGlothlin & Brodie, 2009) and allow to express benefit and cost for the interactants in these terms. McGlothlin et al. (2010) briefly investigated the green-beard scenario using IGE models and showed the conditions for altruism to evolve, in different scenarios: (a) when individuals assort randomly with respect to the signaling trait (a badge), and there is no genetic correlation between the traits, and when they assort nonrandomly, (b) with or (c) without the genetic correlation between the signaling and the altruistic trait. As the authors pointed out, the evolution of badge-based altruism is unlikely without any genetic covariance between the behavior and the signaling traits. Here, we build on their model and complement their results by investigating the fourth scenario: when individuals assort randomly (no relatedness), but there is a genetic correlation between the traits. We use an IGE model to specifically focus on the correlation between the signaling and altruistic trait required for the evolution of the latter. We argue that altruistic traits can evolve and persist even in random populations of unrelated individuals if the correlation between the signaling and altruistic trait is above a given threshold. We show that this correlation threshold depends on the strength of the interaction (influence of the green beard on the expression of the altruistic trait), as well as the strength of social and nonsocial selection and support our analytical results with agent-based simulations.

| I G E MODEL
To analyze the sufficient correlation between traits in the greenbeard scenario, we will use the multivariate model describing the phenotype of the focal individual (McGlothlin et al., 2010) where z i and a i are column vectors describing phenotypes and genotypes (additive genetic values) of the focal individual, respectively, e is a vector of residual values (environmental influences), and z′ denotes the mean phenotype of the focal individual's social partners in a group of size N. The square matrix describes IGEs.
Note that IGEs can be affected by interactions among individuals in a nonlinear way, described in (Trubenová & Hager, 2012), and capture epistasis.
We assume that the fitness of the focal individual is not only a function of its own phenotype, but is also affected by the phenotypes of others. The effect of an individual's own phenotype on its own fitness is described by a nonsocial selection gradient β N (Lande, 1979;Lande & Arnold, 1983;Moore et al., 1997), while the effect of interactant phenotypes on the focal individual's fitness is captured by a social selection gradient β S (Agrawal, 2001;Bijma & Wade, 2008;McGlothlin et al., 2010;Queller, 1992;Westneat, 2012;Wolf et al., 1999). Both selection gradients are column vectors with each (1) z i = a i + e + (N − 1) z � , F I G U R E 1 Scheme of the model. Individuals (green circles) interact (arrows) with each other within the group (black circles), but not between groups. Groups are of the same size and population (black square) size is constant element quantifying the fitness effect of a corresponding phenotypic trait. We further assume that groups are of the same size and the individuals do not interact between groups (Figure 1).
Following previous IGE models (Bijma & Wade, 2008;McGlothlin et al., 2010;Wolf et al., 1999), we define individual fitness as where the column vector β N is a nonsocial selection gradient describing the effect of an individual's traits on its own fitness, β S is a social selection gradient describing the effects of other traits on the fitness of the focal individual (Lande & Arnold, 1983;Frank, 1997;Wolf et al., 1999;McGlothlin et al., 2010, and α is a positive constant. The genotypic response to selection Δā is defined as the difference between the mean genotypic value of offspring and the mean genotypic value of the parental generation. To determine the genetic response to selection, we adapted the expression de- where N is the group size, r is relatedness between individuals, and G is an additive variance-covariance matrix (Lande, 1979).

| ANALY TIC AL RE SULTS
We apply the above-described IGE model to the green-beard scenario. We consider two traits of interest: a signaling trait and an altruistic trait (z s and z a , respectively), encoded by their corresponding genes (genotype a = [a s , a a ] T ). We further assume that the genotypic values are normally distributed around 0. The presence of a green beard (a signalling trait) positively enhances the expression of altruism in social partners, which can be captured in matrix form as where Ψ > 0. The altruistic behavior (phenotype) of each individual is given by the genotypic value of its altruistic trait (or, rather, "predisposition to altruistic behavior") mediated by the signaling of its social partners (Equation 1).
While the signaling trait has no direct influence on the fitness of any individual, the altruistic trait increases the direct fitness of others (social selection) and decreases the fitness of its bearer (nonsocial selection). Thus, we can write social selection and nonsocial selection gradients as β S = [0, β S ] T and β N = [0, β N ] T , respectively, where β S > 0 and β N < 0. The altruistic behavior is not directed at a specific individual-all social partners experience the same behavior from a particular individual.
As in randomly formed groups, the relatedness between interacting individuals is expected to be r = 0, the response to selection

(Equation 3) is simplified to
Filling in , β S , and β S for our specific scenario allows further simplification (Appendix A) and leads to where G 11 = var(a s ) is the genotypic variance of the signaling trait, is the genotypic variance of the altruistic trait, and G 12 = G 21 is the covariance between the two. For the altruistic trait to evolve, the response to selection of this trait must be positive, yielding the condition As the correlation coefficient is defined as = , we can express a threshold correlation coefficient between signaling and altruistic traits necessary for the latter to evolve Note that as β N has a negative sign, the threshold correlation coefficient is positive if Ψ is positive. If Ψ is negative, the correlation must be negative, and below the threshold. However, altruism can still evolve. Figure 2 shows the dependence of the correlation between traits necessary for the evolution of altruism, for different interaction strength Ψ and social selection strength β S . If this threshold is higher than 1, it means that the altruistic behavior cannot evolve in this particular circumstances (parameter space).

| S IMUL ATION RE SULTS
We support our analytical results discussed in the main manuscript with agent-based simulations carried out in MATLAB and Python.
We ran agent-based simulations of mN individuals randomly assorted into m groups consisting of N individuals each. Initial genotypic values of both the signaling and the altruistic genes are drawn from the bivariate normal distribution with mean of 0, standard deviation of 1/3, and specified correlation between the two. The altruistic behavior (phenotype) of each individual is given by the genotypic value of its altruistic gene (predisposition to altruistic behavior), as well as by the level of signaling of its social partners (Equation 1). The fitness of each individual is calculated using Equation 2. The population size remains constant, and the individuals contributing to the next generation are selected randomly with probability proportional to their fitness. We assume asexual populations with no recombination and no new mutations. The new generation of individuals is reshuffled and randomly assigned to groups.

| Response to selection
Equation 4 Figure 4 show that if the correlation is below the required threshold, the mean response to selection is negative, while if the correlation is above the threshold, the response is positive, as predicted by our theoretical results. This means that the mean genetic value of the altruistic gene (predisposition to altruistic behavior) has increased in the offspring generation. See Appendix B, Figure B1 for additional simulations of different parameter sets.

| Long-term evolution
To simulate long-term evolution, we simulated evolving populations for 500 generations, for various m (number of groups) and N (group size). As expected, after some time, fixation occurred, and genetic variance was lost from the population. Figure 5 shows the simulation F I G U R E 2 Threshold correlation between altruistic and signaling traits. The correlation necessary for the evolution of altruism in green beards depends on both interaction strength Ψ and the social selection gradient β. The stronger the interaction or social selection gradient, the lower the necessary correlation between the signaling and altruistic trait. However, when these are too low, the altruistic trait cannot evolve as the required threshold is higher than the maximum possible correlation. β N = −1, genetic variances of both traits are equal rameter set. Figure 6 shows the results of the simulations. We observed that while the initial correlation between the traits did not have any observable impact, the rate of invasions decreased with increasing number of groups. This could be expected, as the fixation probability of an allele decreases with the overall population size mN. However, this trend was not so clear with increasing group size, as the invasion rate slightly increased at the beginning. It should be noted, that our approach to study green-beard scenario is limited by the assumption of mediated, but nondiscriminate altruism toward an individual's social partners. Each individual's level of altruism is determined by its group, and all social partners experience the same benefit from a given individual. However, the same individual would behave differently in a different group. Therefore, it is rather a "group's green beard"

| D ISCUSS I ON
than an "individual's green beard" that plays a role, and thus, it could be argued that our model is not the green-beard model as originally suggested by Dawkins. One possibility to use IGEs while tending to this problem is to consider sequential pairwise interactions between individuals, rather than a simultaneous interaction (J. McGlothlin, personal communication).

| Response to selection
Equation 4

| Long-term evolution
A positive response to selection in one generation does not guarantee the long-term increase in the level of altruism, as the correlation between the traits will also develop with the evolution of the traits.
Our simulations of long-term evolution suggest that the response to selection is positive for a number of generations if the initial correlation is above the calculated threshold (Equation 7 Furthermore, altruistic behavior is mediated by the phenotypes of social partners but is nondiscriminatory-every social partner experiences the same benefit from a particular individual. However, "cheaters" can be simulated as individuals that have high genetic values for the signaling trait, but low genetic values of the altruistic trait. While the Janssen and Goldstone (2006) model is different from ours, some of the conclusions can be compared between the models.
In our model, the strong correlation between the traits also leads to quick fixation. However, we did not observe any effect of the initial correlation on the probability of cheater invasion. This is not surprising, as, after 500 generations, only one haplotype was left in the population in every trial. Furthermore, we observed that the increasing number of groups had a negative impact on the invasion rate. This is also to be expected, as the probability of fixation of a new mutation is inversely proportional to the size of the population. However, increasing the group size first lead to an increased invasion rate, then to a decreased rate. The peak of the invasion rate was observed when the number of individuals within one group was the same as the number of groups. This might be caused by the increased benefits that the cheater receives in bigger groups, partially compensating for the decreasing fixation probability.
In this study, we used IGE models to shed new light on the notorious problem of green-beard scenarios. We have shown that even if the green beard and the altruistic behavior are not encoded by the same gene, altruistic behavior can evolve if the correlation between the genotypic values of both traits is sufficiently high.
We complemented our analytical results with agent-based simulations and confirmed that even in the long term, altruistic behavior can evolve though the green-beard mechanism and persist in the population. Furthermore, our simulations show that such altruistic populations can be reasonably resistant to the invasion of cheaters.
However, we did not assume any recombination between the traits, which could increase the probability of cheaters invading. This, as well as other possible extensions including variable population sizes and mutation events, still need to be investigated.

ACK N OWLED G M ENTS
The

CO N FLI C T O F I NTE R E S T
None declared.
Data archival location: There are no data to be archived.

AUTH O R CO NTR I B UTI O N S
BT: analyzed the model and wrote the article; RH: supervised the analysis and wrote the article.

BEARD SCENARIO
We consider two traits of interest: a signaling and an altruistic trait (z s and z a , respectively), encoded by their corresponding genes (a s and a a , respectively). The presence of a green beard (a signaling trait) positively alters the expression of altruism in social partners, which can be captured in matrix form as = where Ψ> 0. The altruistic trait increases the direct fitness of others but decreases the fitness of its bearer, while the signaling trait has no direct influence on the fitness of any individual: β S = [0, β S ] and β N = [0, β N ], respectively, where β S > 0 and β N < 0.
In randomly formed groups, relatedness between interacting individuals is r = 0. Thus, using Equation 3, we can express the response to selection: Filling in , β S and β S for the green-beard scenario leads to as T gives a zero matrix. This further simplifies to that can also be expressed as (but only for these parameters) where G 11 = var (a s ) is the genotypic variance of the signaling trait, G 22 = var (a a ) is the genotypic variance of the altruistic trait and G 12 = G 21 is the covariance between the two.

APPENDIX B ADDITIONAL SIMULATIONS
We support our analytical results discussed in the main manuscript with agent-based simulations carried out in MATLAB or Python. To support our analytical derivation of the minimal correlation necessary for the altruistic trait to evolve, we calculated the mean response to selection for a range of correlation values as an average of 100 independent trials for each parameter set (each data point).
The response to selection in each trial was calculated as the difference between the mean genotypic value of the offspring and the parental generation. The simulations were carried out in Python. Figure B1 shows the results of simulations for various parameter sets. Figure B2 shows additional simulations of long-term evolution for various parameter sets. We simulated 20 trials for each parameter set. (A1)