Model description

We consider a population of asexual individuals. We assume that each individual hosts one of two microbe types: microbes of type α manipulate their host to act altruistically, while microbes of type β have no effect on behaviour. Individuals interact in pairs, with a prisoner’s dilemma payoff24 (Fig. 1a): a host acting altruistically pays a fitness cost 1>c>0, and the recipient gains a benefit b>c. During host interaction, microbes can be transmitted between the interacting hosts with probabilities T α and T β . T α represents the probability of microbes of type α being transmitted to the other host, replacing the resident microbes, and likewise for T β (Fig. 1b). This direct link between interaction and the possibility for horizontal transmission is at the core of our model and differs from all related works25,26. At the end of each generation, individuals reproduce according to their fitness, microbes are vertically transmitted from one generation to the next, and the offspring generation replaces the parent generation.

Figure 1: Interactions among pairs yield fitness change and a chance for horizontal transmission. (a) Payoff matrix. An individual carrying microbes of type α acts altruistically: in each interaction it pays a fitness cost c, and its partner receives a fitness benefit b. Microbe β does not affect behaviour. (b) When two individuals interact, their fitness changes according to the payoff matrix. In addition, when the interacting individuals host different microbes, horizontal transmission may occur. With probability T α , microbe α is transmitted to the other host and establishes, replacing β. With probability T β , microbe β is transmitted to the other host and establishes, replacing α. Transmission and establishment of one microbe is independent of the other microbe, and when both occur, they occur simultaneously. Full size image

We first investigate the special case where hosts behave altruistically only when carrying microbe α, there is no intrinsic cost to carrying a microbe, and offspring always inherit their parent’s microbe (all three assumptions are relaxed below, yielding the same qualitative results). We compare the evolution of microbe-induced altruism with the classical case of altruism encoded genetically in the host, with perfect vertical transmission, no horizontal transmission, neglecting mutations, and using the same parameters b and c.

Fully mixed populations

Consider an infinite, fully mixed population, that is, each individual has the same probability of interaction with any other individual in the population. Proportion p of the individuals host microbe α, and proportion q=1−p host microbe β. In each generation the population is randomly divided into pairs in which a single interaction occurs, with potential for microbe horizontal transmission (Fig. 1). After interaction, individuals reproduce according to their fitness, which is determined by the interactions they had. Microbe-induced altruism spreads when p, the proportion of hosts carrying microbe α, increases from one generation to the next. This happens when (see Methods):

Under equal horizontal transmission (T α =T β =T), condition (1) reduces to a simpler form . A more general condition for the increase in p under relaxed model assumptions are detailed in the Methods, and further investigated in Supplementary Notes 1 and 2.

A few insights arise from condition (1). First, this condition does not depend on the proportion of altruists. This means that if (1) is satisfied, hosts carrying α will increase in proportion in the next generation, regardless of their current proportion in the population. That is, altruism will take over the population, even from rarity. Second, when T α =0, condition (1) is never satisfied. That is, microbe-induced altruism cannot evolve in the absence of horizontal transmission of microbe α. Analogously, altruism encoded in the host genes, which also does not transmit horizontally, cannot evolve in such fully mixed populations (see Methods and previous works2). Third, condition (1) shows resemblance to Hamilton’s rule which considers the relatedness between donor and recipient, r, defined as the probability that two alleles drawn at random from the two individuals are identical by descent27. According to Hamilton’s rule4, altruistic behaviour towards kin is favoured if r·b>c, that is, if the product of the benefit to the recipient, b, and the relatedness between donor and recipient, r, is greater than the cost to the donor, c. In the case of microbe-induced altruism, the spread from rarity of an altruism-inducing microbe can be described using the relatedness of the microbes of the two interacting hosts. While the identity of a host genotype is stable within a generation, the identity of its microbes may change: a rare altruism-inducing microbe can meet a relative (with probability zero in our fully mixed model) or infect the individual it meets and turn its microbes into relatives (with probability T α ). Thus, with probability T α , manipulation by α microbes causes their host to help another host that now (after the interaction) carries relatives of the manipulating microbe α. Furthermore, the altruism-inducing microbe may be replaced because of infection of its host by a different microbe with probability T β , and in that case the cost paid by the host has no effect on the original microbe α. Finally, the factor (T β −T α ) represents the direct horizontal transmission disadvantage of T α during interaction.

Solving condition (1) shows that the critical value of b/c needed for the evolution of microbe-induced altruism decreases with increasing horizontal transmission probability (Fig. 2, solid lines). In other words, horizontal transmission of microbes helps the establishment of altruism in the host population. This is true even when the horizontal transmission probability of α is lower than that of β, corresponding to a within-host disadvantage for α (Fig. 2, solid red lines). When the horizontal transmission probability of α is higher, corresponding to the case that the altruistic behaviour increases the rate of transmission (for example, feeding), altruism evolves more easily (Fig. 2, solid blue lines).

Figure 2: Horizontal transmission facilitates the fixation of altruism-inducing microbes. Using condition (1) we calculate the minimal b/c value that allows the fixation of microbe α for different values of the cost c (subfigures a, b, c, with c=0.01, c=0.05 and c=0.2, respectively), T α , T β and vertical transmission VT. For all c and VT values, the critical b/c value decreases with increasing horizontal transmission, even when T α <T β and vertical transmission is imperfect (VT<1). When the horizontal transmission probabilities are equal T α =T β =T (green solid lines), the condition for the spread of altruism becomes , for any VT>0 (see Methods for details). Thus, the line depends only on T and is identical in all three subplots. However, the altruism-inducing bacteria spreads more slowly when VT<1 (Supplementary Note 1, Rate of α’s spread as a function of vertical transmission). As c increases (from a to c), the fitness effect of interaction on vertical transmission increases, diminishing the relative effect of imbalance between the horizontal transmission rates. The effect of imperfect vertical transmission (VT<1), is opposite, diminishing the effect of fitness differences on vertical transmission, thus giving more weight to imbalance between the horizontal transmission rates (compare red and blue solid lines to dashed lines). Presented are b, c parameters within the range of the prisoner’s dilemma (namely, b>c). All curves have an asymptote at T α =0, namely altruism cannot evolve without horizontal transmission. Similarly, altruism cannot evolve in such a fully mixed population when it is encoded in the host’s genome (see Methods). Full size image

Condition (1) was derived under the simplifying assumption of perfect vertical transmission. Relaxing this assumption, we generalized the model to assume imperfect vertical transmission of microbes, where with probability VT an offspring inherits its parent’s microbe, and with probability 1−VT it inherits a random microbe from the parent population. We find that horizontal transmission facilitates the evolution of microbe-induced altruism even when vertical transmission is far from perfect (Fig. 2, dashed lines).

So far, we considered microbe-induced altruism in the absence of altruism induced by host genes. Extending our model, we consider a population that is polymorphic with respect to both altruism-inducing host genes and altruism-inducing microbes, and find that all our results hold: Altruism encoded genetically in the host does not evolve, irrespective of the presence of microbe-induced altruism, while the evolution of microbe-induced altruism is independent of the presence of altruism encoded in the host’s genes (Supplementary Note 3). Our model is also robust to the addition of a baseline level of host altruism (Supplementary Figure 2).

Spatially structured populations

One key explanation for the evolution of altruism relies on the existence of spatial structure28,29,30,31. In classical studies, individuals interact only with neighbours, which are more likely to be related to them, and therefore altruists are more likely to interact with altruists. In addition, the probability of repeating interactions with the same individual increases significantly compared with a fully mixed population. Both characteristics generate a higher potential for benefit to altruists, and allow altruism encoded in an individual’s genome to evolve under certain parameters. We thus used simulations to investigate whether microbe-induced altruism further widens the parameter range allowing the evolution of altruism in a spatially structured population, compared with classical altruism encoded in the host’s genome. By studying spatial models, we extend our analysis to populations that are subject to drift, local interactions, local transmissions, and limited dispersal.

The spatial simulation consists of a 2D 100 × 100 lattice grid, where each site is inhabited by an individual host. Individuals carry either microbe α, which drives them to behave altruistically, or microbe β, which does not. During a generation every individual initiates K interactions, each with a neighbour randomly chosen from its immediate neighbours (eight unless at the lattice edge; see Methods). To eliminate possible effects of the order of the interactions, each generation is divided into K iterations over all individuals, where the order of the individuals initiating the interaction is randomized. The fitness of an individual is the sum of the payoffs it received from all its interactions according to the payoff matrix (Fig. 1), normalized by the number of actual interactions the individual had. In addition to fitness change, an interaction may also result in microbe horizontal transmission, with probabilities T α , T β as in the analytical model. Once all interactions are completed, reproduction takes place. Each site in the next generation grid is inhabited by a copy of the fittest individual in the neighbourhood consisting of this site and its immediate neighbours. The offspring inherits the microbe of its parent with probability VT. With probability 1−VT it obtains the microbe of a randomly chosen individual in that neighbourhood (see Methods).

Our results show that similarly to the case of a fully mixed population, horizontal microbe transmission significantly extends the conditions allowing the evolution and maintenance of altruism. When individuals initiate one interaction per generation (K=1), microbe-induced altruism spreads in the population for a wide range of b/c values, including a range of stable polymorphism (Fig. 3a). In contrast, altruism encoded in the host’s genome does not persist even for high values of b/c (‘Gen’ column in Fig. 3a). The parameter range allowing the evolution of altruism in the spatial model shows good agreement with the analytical results for a fully mixed population (see dashed line in Fig. 3a). Assuming that the vertical transmission of microbes is imperfect (VT<1) somewhat narrows the parameter range allowing the evolution of microbe-induced altruism (Fig. 3b), since it reduces the advantage of altruism-inducing microbes, which is based on enhancing the vertical transmission of the microbes in the recipient host. To compare with previous works that have shown that an allele for altruism can persist in a spatial model30, we set the number of interactions per individual, K, to 8, and reset VT to 1. Indeed, for this case, altruism encoded in the host genes can persist for sufficiently high b/c values (’Gen’ column in Fig. 3c), but the parameter range allowing persistence is wider for microbe-induced altruism, and widens with horizontal transmission probability (T=T α =T β ) (Fig. 3c). As in the case of a single interaction, imperfect vertical transmission has a mild effect on the parameter range allowing the evolution of microbe-induced altruism (Fig. 3d). Note that, as expected, when vertical transmission is perfect, microbe-induced altruism with zero horizontal transmission (T=0) is identical to the case of altruism encoded in the host genes (Fig. 3a,c), whereas for imperfect vertical transmission (VT<1), this is not necessarily the case (Fig. 3d).

Figure 3: Microbe-induced altruism vs. altruism encoded in the host’s genes in a spatial Prisoners’ Dilemma scenario. For microbe-induced altruism (matrix part of each sub-plot): hosts carrying either microbes of type α or β are placed on a 100 × 100 lattice grid. Hosts carrying microbe α initially inhabit 5% of the sites, chosen in random positions in the lattice. The final proportion of hosts that carry microbe α is plotted (colour-coded) as a function of horizontal transmission probability T α =T β =T and b/c values, for (a) K=1, VT=1, (b) K=1, VT=0.75, (c) K=8, VT=1 and (d) K=8, VT=0.75, where K is the number of interactions each host initiates per generation and VT is the vertical transmission of microbes. For altruism encoded in the host’s genes (the first column in each plot, named ‘Gen’): hosts carrying an allele for altruistic behaviour initially inhabit 5% of the sites, chosen at random. The final proportion of altruists is plotted (similarly colour-coded) as a function of b/c for the same K as described above. This is the classical case of altruism encoded genetically in the host, where vertical transmission is perfect, and no horizontal transmission occurs. Each cell in the plots represents the mean of at least 100 runs (see Methods for stopping criteria). For comparison with the analytic result of microbe-induced altruism, we plot in (a) the b/c threshold derived from the analytical model, for the case of K=1 (the dashed line, plotted only in the range where the y scale is linear), as plotted in the green lines of Fig. 2a–c. As for the non-linear part of the y-axis, we get from the analytical model that for T=0.1, 0.01, 0.001 the critical b/c values are 9, 99, and 999 respectively. For the case we get very similar results (Supplementary Figure 4). We use c=0.05 throughout the simulation runs. Full size image

Finally, we tested if microbe-induced altruism can evolve from extreme rarity: we started with only a central 2 × 2 patch of individuals carrying microbe α while the rest of the population hosts microbe β. Figure 4 plots the proportion of runs in which p reached 0.05 (complementing the analysis presented in Fig. 3a, where the starting proportion is 5%) for various parameters, and shows that microbe-induced altruism can increase in frequency even from extreme rarity, while altruism induced by the host genes cannot (see also Supplementary Videos 1 and 2) .