Gene mobility and the concept of relatedness

Abstract

Cooperation is rife in the microbial world, yet our best current theories of the evolution of cooperation were developed with multicellular animals in mind. Hamilton’s theory of inclusive fitness is an important case in point: applying the theory in a microbial setting is far from straightforward, as social evolution in microbes has a number of distinctive features that the theory was never intended to capture. In this article, I focus on the conceptual challenges posed by the project of extending Hamilton’s theory to accommodate the effects of gene mobility. I begin by outlining the basics of the theory of inclusive fitness, emphasizing the role that the concept of relatedness is intended to play. I then provide a brief history of this concept, showing how, over the past fifty years, it has departed from the intuitive notion of genealogical kinship to encompass a range of generalized measures of genetic similarity. I proceed to argue that gene mobility forces a further revision of the concept. The reason in short is that, when the genes implicated in producing social behaviour are mobile, we cannot talk of an organism’s genotype simpliciter; we can talk only of an organism’s genotype at a particular stage in its life cycle. We must therefore ask: with respect to which stage(s) in the life cycle should relatedness be evaluated? For instance: is it genetic similarity at the time of social interaction that matters to the evolution of social behaviour, or is it genetic similarity at the time of reproduction? I argue that, strictly speaking, it is neither of these: what really matters to the evolution of social behaviour is diachronic genetic similarity between the producers of fitness benefits at the time they produce them and the recipients of those benefits at the end of their life-cycle. I close by discussing the implications of this result. The main payoff is that it makes room for a possible new mechanism for the evolution of altruism in microbes that does not require correlated interaction among bearers of the genes for altruism. The importance of this mechanism in nature remains an open empirical question.

Appendices

Appendix 1: Diachronic relatedness in a Prisoner’s dilemma

In the main text I argue that, when organisms engage in plasmid conjugation following pairwise social interaction, the best measure of $r$ is a diachronic variant of the usual regression definition. This appendix supports that argument by providing a formal analysis of a simple model of pairwise cooperation. This is not intended to provide a realistic model of microbial cooperation; it is a simple model designed to make the conceptual issues surrounding the definition of $r$ as clear as possible.

Basic setup

We consider an infinite population of asexually reproducing organisms. The population is sorted at random into pairs, and each pair plays a one-shot two-player Prisoner’s Dilemma with the following payoff matrix:

	COOPERATE	DEFECT
COOPERATE	$B-C$	$-C$
DEFECT	$B$	$0$

We assume that the population is under weak viability selection only, so that the payoffs can be interpreted as marginal costs and benefits to the viability of the players.

A fraction $f_X$ of organisms bear a plasmid, $X$, that reliably causes them to cooperate. Let $g_{Xi}(t)$ represent the genic value of the ith organism with respect to $X$ at time $t$, such that $g_{Xi}(t)=1$ for an organism that carries $X$ at $t$, and $g_{Xi}(t)=0$ for an organism that does not. Meanwhile, a fraction $f_Y$ of organisms bear a different plasmid, $Y$, that has no fitness-relevant phenotypic effects on its host. Let $g_{Yi}(t)$ represent the genic value of the ith organism with respect to $Y$ at time $t$, such that $g_{Yi}(t)=1$ for an organism that carries $Y$ at $t$, and $g_{Yi}(t)=0$ for an organism that does not. Both $X$ and $Y$ may in principle be carried by the same organism. We assume, however, that the two plasmids are transmitted independently of each other, so that there is no correlation, positive or negative, between possessing $X$ and possessing $Y$.

After playing the game and conferring payoffs, the players conjugate with probability $\pi$. During conjugation, a bearer of plasmid $X$ will transmit the plasmid to its social partner (if it is initially a non-bearer) with transmission probability $\lambda _X$. The $\lambda _X$ parameter may be interpreted as quantifying the infectivity of $X$. Correspondingly, a bearer of plasmid $Y$ will transmit the plasmid to its social partner (if it is initially a non-bearer) with transmission probability $\lambda _Y$. The $\lambda _Y$ parameter may be interpreted as quantifying the infectivity of $Y$.

Next, there is a global competition phase, in which all individuals compete for representation in the next generation, and individuals are killed off with a probability equal to their net viability (i.e. individuals who received $B$ without paying $C$ are least likely to be killed off; individuals who paid $C$ without receiving $B$ are most likely). Finally, at the end of the life-cycle, the surviving cells divide to produce the next generation. We assume that all plasmids are transmitted from a parent cell to its daughter cells with perfect fidelity. In reality, plasmids are sometimes ‘lost’ during cell division (i.e. they are absent from one of the daughter cells). We neglect this phenomenon here, because it would complicate the analysis without affecting the main conceptual conclusions.

Because individuals are paired up at random in the social phase, there is no positive assortment at $t_A$. We thus intuitively expect that defectors will outperform cooperators, and that plasmids (such as $X$) that encode cooperation will decrease in frequency at the expense of those (such as $Y$) that do not. But, as we will see, the possibility of plasmid transfer after social interaction complicates the picture.

Modified price equation

We can employ the Price equation (Price 1970, 1972) to study the conditions under which $X$ will spread more rapidly than $Y$, or vice versa. The standard Price equation describes the change in the additive genetic value of some trait between an ancestor population and a descendant population. These populations may represent consecutive generations in a discrete generations model, or appropriately chosen earlier and later census points in an overlapping generations model. Price showed that this change could be decomposed into a covariance term and an expectation term:

$$\begin{aligned} \Delta \overline{g_i} = \frac{1}{\overline{w}}\left[ {\mathrm{Cov}}({w_i},g_i) +{\mathrm{E}}(w_i\Delta g_i)\right] \end{aligned}$$

(7)

where $w_i$ is the fitness of the ith individual, conceptualized as its total contribution to the descendant population, and $g_i$ is its additive genetic value for the character of interest.

The standard version of the Price equation is not ideal in the present context, for two main reasons. Firstly, the genetic value in the Cov term is not indexed to any particular time in the life-cycle, so its value will be indeterminate in cases in which an organism’s genotype is altered during its life-cycle by plasmid transfer. For our purposes, we need to specify the time in the life-cycle at which $g_i$ is to be evaluated. Secondly, both terms in the standard Price equation are affected by fitness differences; the equation thus splits the effects of differential fitness between its two terms. For our purposes, it will be convenient to capture all the effects of differential fitness in a single term. We overcome both these drawbacks by employing a modified version of the Price equation first derived by Frank (1998):

$$\begin{aligned} \Delta \overline{g_i} = \frac{1}{\overline{w}}\left[ {\mathrm{Cov}}(w_i,g'_i)\right] +{\mathrm{E}}(\Delta g_i) \end{aligned}$$

(8)

where ${g'}_i$ is the transmitted genetic value of the ith individual with respect to the character of interest, i.e. the average genetic value of its descendants in the next generation.

In our model, in which we assume that plasmids are transmitted with perfect fidelity from parent cells to daughter cells, $g'_i$ can be identified with ${g}_i(t_T)$, the terminal genetic value of the ith individual at the end of its life-cycle (that is, at either the time of cell death or the time of cell division, depending on the organism’s fate). $\Delta {g}_i$ can then be interpreted as the average change in an organism’s genetic value between the beginning and end of its own life-cycle, while $\Delta \overline{g_i}$ can be interpreted as the change in the population mean of $g$ during a single iteration of the life-cycle. Hence:

$$\begin{aligned} \Delta \overline{g_i} = \frac{1}{\overline{w}}\left[ {\mathrm{Cov}}(w_i,{g}_i(t_T))\right] +{\mathrm{E}}(\Delta g_i) \end{aligned}$$

(9)

This is a particularly useful version of the Price equation in a microbial context, and has been employed to this end in recent work by Mc Ginty et al. (2013).

Fitness function

In the Price formalism, the fitness of the ith individual $(w_i)$ is defined as its total contribution to the descendant population. In a discrete generations model, this is simply its total number of offspring. In our simple model, we can write this as a function of net viability, plus a residual component representing the deviation from expectation:

$$\begin{aligned} w_i=2\left( V-C_ig_{Xi}(t_A)+B\hat{g}_{Xi}(t_A)\right) +\epsilon \end{aligned}$$

(10)

As explained in the basic setup, the net viability of an organism depends on a baseline component plus the net payoff it receives in the social phase. This payoff in turn depends on the genetic values of itself and its social partner at the time of action; hence the genetic values in the fitness function should be evaluated at $t_A$. Note that every organism will deviate from expectation one way or the other, since it determinately either will or will not divide, and consequently its realized value for $w_i$ will be either 2 or 0. This is unproblematic, provided the deviation from expectation does not co-vary with genetic value.

The intergenerational change in the frequency of $X$

We can derive an expression for the intergenerational change in the frequency of plasmid $X$ by substituting Eq. (10) into Eq. (9), yielding:

$$\begin{aligned} \Delta \overline{g}_{Xi} = \frac{2}{\overline{w}}\left[ -C{\mathrm{Cov}}(g_{Xi}(t_A),{g}_{Xi}(t_T)) +B{\mathrm{Cov}}(\hat{g}_{Xi}(t_A),{g}_{Xi}(t_T))\right] +{\mathrm{E}}(\Delta g_{Xi}) \end{aligned}$$

(11)

Note that both $V$ and $\epsilon$ have now dropped out of the analysis, because neither co-varies with genetic value. We can rearrange 11 as follows:

$$\begin{aligned} \Delta \overline{g}_{Xi} = \frac{2}{\overline{w}}\left[ \left( rB -C\right) {\mathrm{Cov}}(g_{Xi}(t_A),{g}_{Xi}(t_T))\right] +{\mathrm{E}}(\Delta g_{Xi}) \end{aligned}$$

(12)

where $r$ is identified with the following covariance ratio:

$$\begin{aligned} r=\frac{{\mathrm{Cov}}\left( g_{Xi}(t_T),\hat{g}_{Xi}(t_A)\right) }{{\mathrm{Cov}}\left( g_{Xi}(t_T), g_{Xi}(t_A)\right) } \end{aligned}$$

(13)

We can therefore see that the expression for the change in frequency of $X$ contains a Hamilton’s rule-like component, albeit one that requires an unorthodox definition or $r$ (more on this below). However, we can see that it also contains a separate, second component, ${\mathrm{E}}(\Delta g_{Xi})$, which reflects the fact that $X$ may also spread partly in virtue of its infectivity, irrespective of its effects on its hosts fitness (cf. Smith 2001; Giraud and Shykoff 2011).

The intergenerational change in the frequency of $Y$

The equation describing the change in frequency of plasmid $Y$ is much simpler. $Y$ has no fitness-relevant phenotypic effect on its host organism, and it is uncorrelated with the fitness-relevant plasmid $X$. Consequently, the presence or absence of $Y$ does not co-vary with $w_i$. Hence ${\mathrm{Cov}}(w_i,{g}_{Yi}(t_T))=0$, and only the second term in our modified Price equation remains:

$$\begin{aligned} \Delta \overline{g}_{Yi} = {\mathrm{E}}(\Delta g_{Yi}) \end{aligned}$$

(14)

This reflects the fact that $Y$ is selectively neutral. If $Y$ spreads, it can only be as a result of its infectivity.

When will $X$ outperform $Y$?

By combining (12) with (14), we can obtain a condition under which the intergenerational change in the frequency of $X$ is greater than the intergenerational change in the frequency of $Y$:

$$\begin{aligned} \Delta \overline{g}_{Xi}>\Delta \overline{g}_{Yi} \iff \frac{2}{\overline{w}}\left[ \left( rB -C\right) {\mathrm{Cov}}(g_{Xi}(t_A),{g}_{Xi}(t_T))\right] >{\mathrm{E}}(\Delta g_{Yi})-{\mathrm{E}}(\Delta g_{Xi}) \end{aligned}$$

(15)

If we assume that $X$ and $Y$ are initially equal in their frequency $(f_X=f_Y)$ and in their infectivity $(\lambda _X=\lambda _Y)$, it follows that ${\mathrm{E}}(\Delta g_{Yi}) = {\mathrm{E}}(\Delta g_{Xi})$. This leaves the following simplified condition:

$$\begin{aligned} \Delta \overline{g}_{Xi}>\Delta \overline{g}_{Yi} \iff \frac{2}{\overline{w}}\left[ \left( rB -C\right) {\mathrm{Cov}}(g_{Xi}(t_A),{g}_{Xi}(t_T))\right] >0 \end{aligned}$$

(16)

On the further but relatively mild assumption that ${\mathrm{Cov}}(g_{Xi}(t_A),{g}_{Xi}(t_T))>0$ (i.e. an organism’s genotype at the time of action co-varies positively with its genotype at the end of its life-cycle), we can derive a condition under which $X$ will outperform $Y$ that is closely analogous to Hamilton’s rule in its traditional form:

$$\begin{aligned} \Delta \overline{g}_{Xi}>\Delta \overline{g}_{Yi} \iff rB-C>0 \end{aligned}$$

(17)

This expression can be interpreted as a statement of the conditions under which a plasmid that encodes a social phenotype will outperform a plasmid that is identical in its infectivity but neutral with regard to its host’s fitness. Importantly, the significance of this result is not limited to cases in which two very similar plasmids are actually competing. More generally, the expression can be interpreted as a statement of the conditions under which encoding a social phenotype is evolutionarily advantageous for a plasmid in a two-player Prisoner’s Dilemma. If $rB - C < 0$, then the actual spread of the plasmid will be slower than the rate that would have been observed if the plasmid had been phenotypically inert, so the plasmid gains no advantage by encoding the social phenotype. By contrast, if $rB - C > 0$, then the plasmid will spread more rapidly than an otherwise identical but phenotypically inert plasmid would have done. In this situation, encoding the social phenotype is advantageous for the plasmid, despite the deleterious effect it has on the host.

The derivation of this condition could not proceed without the unorthodox definition of relatedness given in (13), on which $r$ is defined a measure of the diachronic genetic similarity between actors at the time of action and recipients at the end of their life-cycle. Adopting this diachronic measure is therefore essential if we want to arrive at a Hamilton’s rule-like condition that describes when encoding a social phenotype is evolutionarily advantageous for a plasmid in this model. Essentially, this is because in this model the advantageousness (or otherwise) of encoding the social phenotype is largely dependent on the probability ($\pi \lambda _X$) that the plasmid responsible will be able to ‘jump ship’ to the recipient after causing the altruistic act. Our diachronic measure of $r$, in contrast to more traditional measures, takes this ‘ship jumping’ phenomenon into account.

The dependence of $r$ on $\pi \lambda _X$

To calculate the precise dependence of $r$ on $\pi \lambda _X$, we begin by re-writing our covariance ratio as a ratio of regression coefficients:

$$\begin{aligned} r=\frac{{\mathrm{Cov}}\left( {g_{Xi}}(t_T),{\hat{g}_i}(t_A)\right) }{{\mathrm{Cov}}\left( g_{Xi}(t_T), g_{Xi}(t_A)\right) }=\frac{\beta _{{\hat{g}_{Xi}}(t_A),{g_{Xi}}(t_T)}}{\beta _{g_{Xi}(t_A),{g_{Xi}}(t_T)}} \end{aligned}$$

(18)

We then compute the regression coefficients from the conditional expected values:

$$\begin{aligned} \beta _{{\hat{g}_{Xi}}(t_A),{g_{Xi}}(t_T)}={\mathrm{E}}(\hat{g}_{Xi}(t_A)|{g_{Xi}}(t_T)=1)-{\mathrm{E}}(\hat{g}_{Xi}(t_A)|{g_{Xi}}(t_T)=0)=\pi \lambda _X\end{aligned}$$

(19)

$$\beta _{{g_{{Xi}} (t_{A} ),g_{{Xi}} (t_{T} )}} = E(g_{{Xi}} (t_{A} )|g_{{Xi}} (t_{T} ) = 1) - E(g_{{Xi}} (t_{A} )|g_{{Xi}} (t_{T} ) = 0) = 1 - (1 - f_{X} )\pi \lambda _{X}$$

(20)

This yields:

$$\begin{aligned} r=\frac{\pi \lambda _X}{1-(1-f_X)\pi \lambda _X} \end{aligned}$$

(21)

From which we see that in this model, all else being equal, the diachronic assortment between altruists $(r)$ increases with increasing probability of conjugation between social partners after social interaction $(\pi )$ and with increasing probability of plasmid transmission when they conjugate $(\lambda _X)$.

Appendix 2: Diachronic relatedness in a public goods game

We turn now to a linear public goods game, a somewhat more realistic model of microbial cooperation. The analysis of this game largely parallels that of “Appendix 1”. The aim is to derive a diachronic version of the whole-group relatedness (Eq. (3) in the main text) that provides the best measure of $r$ in models in which plasmid transfer follows public goods production.

Basic setup

Consider, then, an infinite population of asexually reproducing organisms sorted randomly into groups of size $N$. As before, a fraction $f_X$ of organisms bear a plasmid, $X$, that reliably causes them to produce a public good, while a fraction $f_Y$ bear a different plasmid, $Y$, that has no fitness-relevant phenotypic effects on its host. Genetic values are defined as before.

At $t_A$, the organisms play a linear public goods game. Bearers of $X$ produce a marginal viability benefit, $B$, that is shared evenly among the members of their social group. Each group member (including the producer) thus receives a total benefit $(G_{Xi}(t_A))B$, where $G_{Xi}(t_A)$ is the local (within-group) frequency of public good producers at $t_A$. Each producer incurs a viability cost $C$ as a consequence of producing the public good.

As in the previous model, players may conjugate after the game. Each organism may conjugate at most once, and does so with probability $\pi$. These conjugating pairs are drawn at random from the group, and it is assumed that $N$ is sufficiently large the frequencies of various possible pairs match the background frequencies in the global population. During conjugation, a bearer of plasmid $X$ will transmit the plasmid to its social partner (if it is initially a non-bearer) with transmission probability $\lambda _X$. Correspondingly, a bearer of plasmid $Y$ will transmit the plasmid to its social partner (if it is initially a non-bearer) with transmission probability $\lambda _Y$.

There follows a global competition phase, in which all individuals compete for representation in the next generation of groups, and individuals are killed off with a probability equal to their net viability. Finally, at the end of the life-cycle, the surviving cells divide to produce the next generation. We again assume that all plasmids are transmitted from a parent cell to its daughter cells with perfect fidelity.

Because groups are formed randomly, there is no positive assortment between the group members at $t_A$. We thus intuitively expect that public goods production will be undermined by free-riding, and that plasmids (such as $X$) that produce public goods will decrease in frequency at the expense of those (such as $Y$) that free-ride. Again, however, we find that allowing for the possibility of plasmid transfer after social interaction complicates the picture.

Equations for gene frequency change

As before, we take an appropriately rearranged Price equation as our starting point:

$$\begin{aligned} \Delta \overline{g_i} = \frac{1}{\overline{w}}\left[ {\mathrm{Cov}}(w_i,{g}_i(t_T))\right] +{\mathrm{E}}(\Delta g_i) \end{aligned}$$

(10)

The fitness function for this game is as follows:

$$\begin{aligned} w_i=2\left( V-C_ig_{Xi}(t_A)+BG_{Xi}(t_A)\right) +\epsilon \end{aligned}$$

(22)

Substituting (22) into (9), we obtain:

$$\begin{aligned} \Delta \overline{g}_{Xi} = \frac{2}{\overline{w}}\left[ -C{\mathrm{Cov}}(g_{Xi}(t_A),{g}_i(t_T)) +B{\mathrm{Cov}}(G_{Xi}(t_A),{g}_{Xi}(t_T))\right] +{\mathrm{E}}(\Delta g_{Xi}) \end{aligned}$$

(23)

which can be re-written as:

$$\begin{aligned} \Delta \overline{g}_{Xi} = \frac{2}{\overline{w}}\left[ \left( r_GB -C\right) {\mathrm{Cov}}(g_{Xi}(t_A),{}_{Xi}(t_T))\right] +{\mathrm{E}}(\Delta g_{Xi}) \end{aligned}$$

(24)

where $r_G$ is now identified with the following covariance ratio:

$$\begin{aligned} r_G=\frac{{\mathrm{Cov}}\left( g_{Xi}(t_T),G_{Xi}(t_A)\right) }{{\mathrm{Cov}}\left( g_{Xi}(t_T), g_{Xi}(t_A)\right) } \end{aligned}$$

(25)

Plainly, this covariance ratio is very similar to that given in Eq. (13); the only difference is that a whole-group genetic value has replaced the social partner’s genetic value in the numerator. The relationship between these covariance ratios is thus closely analogous to the relationship between the ‘whole-group relatedness’ (3) and the standard ‘others-only’ regression measure of $r$ (2). We can think of the above ratio as a diachronic variant of the whole-group relatedness, designed to accommodate the effects of gene mobility occuring after social interaction.

Plasmid $Y$, as before, is phenotypically inert, so its change in frequency is again given simply by:

$$\begin{aligned} \Delta \overline{g}_{Yi} = {\mathrm{E}}(\Delta g_{Yi}) \end{aligned}$$

(26)

When will $X$ outperform $Y$?

From here on, the analysis is virtually identical to that of “Appendix 1”. Combining (24) with (26), we see that $X$ outperforms $Y$ in this model under the following conditions:

$$\begin{aligned} \Delta \overline{g}_{Xi}>\Delta \overline{g}_{Yi} \iff \frac{2}{\overline{w}}\left[ \left( r_GB -C\right) {\mathrm{Cov}}(g_{Xi}(t_A),{g}_{Xi}(t_T))\right] >{\mathrm{E}}(\Delta g_{Yi})-{\mathrm{E}}(\Delta g_{Xi}) \end{aligned}$$

(27)

In the special case in which plasmids $X$ and $Y$ are equal in their frequency and in their degree of infectivity, and on the further assumption that ${\mathrm{Cov}}\left( G_{Xi}(t_T),G_{Xi}(t_A)\right) >0$ (i.e. a social group’s average genetic value at $t_A$ co-varies positively with its average genetic value at the end of the life-cycle), we can again derive a Hamilton’s rule-like condition under which $X$ will be selectively favoured over $Y$:

$$\begin{aligned} \Delta \overline{g}_{Xi}>\Delta \overline{g}_{Yi} \iff r_GB-C>0 \end{aligned}$$

(28)

This expression can be interpreted as a statement of the conditions under which encoding public goods production is evolutionarily advantageous for a plasmid in a public goods game. If $r_GB - C < 0$, then the actual spread of the plasmid will be slower than the rate that would have been observed if the plasmid had been phenotypically inert, so the plasmid gains no advantage by encoding the social phenotype. By contrast, if $r_GB - C > 0$, then the plasmid will spread more rapidly than an otherwise identical but phenotypically inert plasmid would have done. In this situation, producing the public good is advantageous for the plasmid, despite the deleterious effect it has on the host.

As before, the derivation of this result relies on our adopting an unorthodox, diachronic definition of $r$. This time, $r$ quantifies the fraction of the overall diachronic covariance between genotypes at $t_A$ and genotypes at $t_T$ that can be accounted for by covariance between the earlier and later group genetic values. It is a natural extension of the ‘whole-group relatedness’ to the diachronic case, and this extension is needed in order to capture the diachronic genetic correlations between group members generated after the social phase by plasmid transfer.

The dependence of $r_G$ on $\pi \lambda _X$

When (as in our model) all groups are of equal size $N$, there is a close relationship between the whole-group relatedness and the standard (‘others-only’) regression definition (Pepper 2000):

$$\begin{aligned} r_G=\frac{1+(N-1)r}{N} \end{aligned}$$

(29)

The same relationship holds between our modified diachronic measures for the whole-group and standard (‘others-only’) relatedness (i.e. (25) and (13) respectively).

In our model, each organism conjugates once (at most) with a randomly selected member of its randomly formed group, and $N$ is sufficiently large that the frequencies of the various types of pair in each group match the background frequencies in the population. Consequently, we can treat these pairs as if they had been drawn at random from the global population. The probability that they conjugate successfully is $\pi$, and the probability that an $X$-bearer will transfer $X$ in this event is $\lambda _X$. Given these assumptions, the value of $r$ will be the same as it was in the previous game. We can therefore combine (21) with (29) to obtain an expression for $r_G$:

$$\begin{aligned} r_G=\frac{1+(N-1)({\pi \lambda _X}/{(1-\pi \lambda _X(1-f_X)))}}{N} \end{aligned}$$

(30)

This shows that, all else being equal, increasing $\pi$ or $\lambda _X$ increases the diachronic whole-group relatedness at the plasmid locus.