Inclusive fitness and the sociobiology of the genome

Abstract

Inclusive fitness theory provides conditions for the evolutionary success of a gene. These conditions ensure that the gene is selfish in the sense of Dawkins (The selfish gene, Oxford University Press, Oxford, 1976): genes do not and cannot sacrifice their own fitness on behalf of the reproductive population. Therefore, while natural selection explains the appearance of design in the living world (Dawkins in The blind watchmaker: why the evidence of evolution reveals a universe without design, W. W. Norton, New York, 1996), inclusive fitness theory does not explain how. Indeed, Hamilton’s rule is equally compatible with the evolutionary success of prosocial altruistic genes and antisocial predatory genes, whereas only the former, which account for the appearance of design, predominate in successful organisms. Inclusive fitness theory, however, permits a formulation of the central problem of sociobiology in a particularly poignant form: how do interactions among loci induce utterly selfish genes to collaborate, or to predispose their carriers to collaborate, in promoting the fitness of their carriers? Inclusive fitness theory, because it abstracts from synergistic interactions among loci, does not answer this question. Fitness-enhancing collaboration among loci in the genome of a reproductive population requires suppressing alleles that decrease, and promoting alleles that increase the fitness of its carriers. Suppression and promotion are effected by regulatory networks of genes, each of which is itself utterly selfish. This implies that genes, and a fortiori individuals in a social species, do not maximize inclusive fitness but rather interact strategically in complex ways. It is the task of sociobiology to model these complex interactions.

Appendices

Appendix 1: Regression approach to the generalized Hamilton’s rule

Hamilton (1970) developed a more general notion of relatedness based on the Price equation (Price 1970), an approach developed by Queller (1992), Frank (1998), and many others. We can show that this approach can also be used to derive the Generalized Hamilton’s rule (suggested by Laurent Lehmann, personal communication).

Suppose in each period each individual is called upon once to help and once potentially to receive help. Only the individual with the helping allele actually helps. Let w _ij be the fitness of an individual where i = 1 if the individual has the helping allele, i = 0 otherwise, and j = 1 if the individual is helped, and j = 0 if not helped. Then we can write

$$w_{ij} = 1 - c i + b j - (1-i)\alpha - {\beta q},$$

(22)

where all parameters are as previously defined. The frequency p _ij of type ij is given by

$$\begin{aligned} p_{00} &= (1-q)\left(r + (1-r)(1-q)\right) \\ p_{01} &= p_{10}=(1-q)q(1-r) \\ p_{11} &= q\left(r + (1-r)q\right). \\ \end{aligned}$$

We can write the regression equations for w _ij as

$$\begin{aligned} w_{ij} &= \hat{w}_{ij} + \epsilon_{ij} \\ &= \gamma_0 + \gamma_1 i + \gamma_2 j + \epsilon_{ij}, \\ \end{aligned}$$

where $\hat{w}_{ij}$ is the additive fitness and $\epsilon_{ij}$ is an error term. For least squares estimation we minimize

$$\sum\limits_{i=0}^1\sum\limits_{j=0}^1 p_{ij}(w_{ij}-\hat{w}_{ij})^2 = \sum\limits_{i=0}^1\sum\limits_{j=0}^1 p_{ij}\epsilon_{ij}^2$$

with respect to γ ₀, γ ₁, and γ ₂. We find that

$$\begin{aligned} \gamma_0 &= 1-\alpha - q\beta \\ \gamma_1 &= \alpha - c \\ \gamma_2 &= b. \\ \end{aligned}$$

The additive portion of fitness f ₁ for the helping allele and f ₀ for the non-helping allele are then given by

$$\begin{aligned} f_0 &= \left(p_{01}\hat{w}_{01} + p_{00}\hat{w}_{00}\right)/(1-q) \\ f_1 &= \left(p_{11}\hat{w}_{11} + p_{10}\hat{w}_{10}\right)/q \\ \end{aligned}$$

The condition f ₁ > f ₀ then becomes, after some simplification,

$$b\hskip0.6pt r + \alpha > c,$$

which is the correct Generalized Hamilton’s rule expression. Note that br + α − c = γ ₂ r + γ ₁, which means that the reasoning leading to the Generalized Hamilton’s rule is the same as for the traditional Hamilton’s rule, only the parameters being altered.

Appendix 2: A model of free rider suppression

We can model the suppression of antisocial genes formally in terms of the Generalized Hamilton’s rule. Suppose a polluting allele x satisfies (18), so the GHR is simply Hamilton’s rule:

$$b_xr_x + \alpha_x > c_x.$$

(23)

Suppose further that an allele at locus y imposes a punishment p _y  =  − α _y  > 0 on each carrier of the x allele at a cost c _b to itself. The gain to copies of y from suppression of x is

$$b_y = \nu\alpha_x +q_x\beta_x,$$

where ν is the probability that the host of allele y does not contain a copy of the x allele. Allele y also receives the gain b _y so y’s net fitness sacrifice is c _y  − b _y . We assume the cost of punishing p _y is c _y  = κ p _y , for some κ > 0. The punishment p _y is dissipated, so it does not appear as a transfer of fitness to the y allele. Thus the GHR for the y allele becomes

$$(\nu\alpha_x + q_x\beta_x)(1+r_y) > \kappa p_y.$$

(24)

The effect of suppression on allele x is to increase the cost of x’s action from c _x to c _x  + q _y p _y . Thus the revised GHR for allele x becomes

$$b_x r_x + \alpha_x > c_x + q_yp_y.$$

(25)

If there is an interior equilibrium to this two-equation system, it is easy to check that the equilibrium values $q_x^*$ and $q_y^*$satisfy

$$q_x^* = \frac{p_yq_x\kappa}{(1+r_y)\left(\nu\alpha_x+q_x\beta_x\right)}$$

(26)

$$q_y^* = \frac{b_xr_x-c_x + \alpha_x^*}{p_y}.$$

(27)

Clearly if p _y is sufficiently large, the x allele will have a low but non-zero frequency in the population. Thus the GHR shows that an equilibrium level of an antisocial gene can be maintained, and the suppression is more complete the larger is the punishment p _y , the smaller the cost of punishing κ p _y , the higher the relatedness r _y at the suppressor gene, the stronger the pollution effect β _x , the stronger the thieving effect α _x , and the lower the probability ν that the suppressor locus contains a copy of the thieving allele. The GHR also shows that suppression will occur where there are no relatedness effect in the antisocial gene; i.e., when b _x  = 0 or r _x  = 0 but c _x  < 0, so the antisocial gene’s gain is purely in personal fitness.

Appendix 3: Hamilton’s seminal analysis

The original derivation of the inclusive fitness criterion in Hamilton (1964a) is more general than Hamilton’s rule as commonly expressed, and by modifying one of his assumptions, his derivation gives our Generalized Hamilton’s rule. Following Hamilton (but with simplified notation), suppose the population is diploid, and the alleles at the focal locus are numbered $k=1,\ldots,m.$ Let R _ij be the fitness increment (positive or negative) over baseline fitness unity conferred by an individual of genotype ij at the focal locus on members of the population who carry the i allele, and let T _ij be the total fitness increment conferred by an individual with genotype ij on the population. Let the frequency of genotype ij be p _ij , and let p _i be the frequency of allele i in the population. With random mating, we have p _ij  = p _i p _j . Hamilton (1964a) assumes this, but this plays no role in the analysis. The total fitness effect of one individual carrying allele i at the focal locus is then given by

$$T_i = \sum\limits_j p_{ij}T_{ij},$$

(28)

and the total fitness effect on allele i due to one carrier of allele i is given by

$$R_i = \sum\limits_j p_{ij}R_{ij}.$$

(29)

We then define

$$S_i = T_i - R_i$$

(30)

for $i = 1,\ldots,m,$ which Hamilton (1964a) calls the dilution effect for reasons discussed below. Note that the signs of R _i , S _i , and T _i are indeterminate, but if we assume sufficiently weak selection at this locus, which we do, and if $\overline{T}$ is the total increment in population fitness in one period, then

$$\overline{T} = \sum\limits_ip_iT_i > -1,$$

(31)

so population fitness $1+\overline{T}$ is strictly positive.

Following Hamilton, let us assume that the dilution effect S _i is uniformly distributed over the alleles at the focal locus, and let

$$\overline{S} = \sum_ip_iS_i$$

(32)

$$\overline{R} = \sum_ip_iR_i,$$

(33)

so $\overline{T} = \overline{R} + \overline{S}.$ Then the expression for an increase in the frequency of allele i is given by

$$\Updelta p_i = \frac{p_i + p_iR_i + p_i\overline{S}}{1 + \overline{T}}-p_i$$

(34)

$$= p_i\frac{1 + R_i + \overline{S}}{1+\overline{R} + \overline{S}}-p_i$$

(35)

$$= p_i\frac{R_i-\overline{R}}{1+\overline{T}}.$$

(36)

From this we get Hamilton’s rule, in the form that allele i will increase through natural selection exactly when

$$R_i > \overline{R}.$$

(37)

We can thus define the inclusive fitness of allele i as R _i . Because the rate growth of allele i is $g_i = \Updelta p_i/p_i,$ we observe that R _i  > R _j exactly when g _i  > g _j , so alleles at the focal locus are relatively successful in proportion to their inclusive fitness. If the R _i are frequency independent, we can then say that genes “maximize their inclusive fitness.”

As Hamilton stresses, the sign and magnitude of $\overline{S}$ do not affect (37), but only the rate at which the frequency of allele i changes in the population. It is for this reason that Hamilton calls $\overline{S}$ a ‘dilution’ effect.

Hamilton assumes without comment that R _i  > 0, but this need not be the case. For instance, suppose m = 2, R ₂₂ = 0, and R ₂₁ =  − α < 0; i.e., there are two alleles, and the heterozygote imposes a fitness loss α on the second allele. Then R ₂ =  − α p ₁₂, $\overline{R} = -\alpha p_2p_{12},$ and $R_i-\overline{R} = R_i+\alpha p_2p_{12},$ which can be positive even if R _i  < 0. This of course is the case of the antisocial allele.

In general, the assumption that $\overline{S}$ is uniformly distributed among alleles at the focal locus is overly restrictive. In biochemical terms it prevents using inclusive fitness theory to analyze segregation distortion and other allele actions that disfavor other alleles at the focal locus (Ratnieks 1988; Ratnieks and Reeves 1992; Burt and Trivers 2006), or to analyze social helping behaviors (b, r, c > 0) that involve imposing costs on non-relatives. So let us assume that allele i at the focal locus receives fraction $\gamma_i\overline{S},$ where γ _i  ≥ 0 and ∑ _i γ _i  = 1. Then (34) becomes

$$\Updelta p_i = \frac{p_i + p_iR_i + p_i\gamma_i\overline{S}}{1 + \overline{T}}-p_i$$

(38)

$$= p_i\frac{1 + R_i + \gamma_i\overline{S}}{1+\overline{R} + \overline{S}}-p_i$$

(39)

$$= p_i\frac{R_i-\overline{R}-(1-\gamma_i)\overline{S}}{1+\overline{T}}$$

(40)

It is clear that the revised condition for allele i to proliferate,

$$R_i - (1-\gamma_i)\overline{S} > \overline{R},$$

(41)

is the appropriate generalization of our GHR. Note that this inequality can be satisfied even if R _i  < 0, so the successful allele is uniformly harmful to carriers of the genome.

Note that this analysis does not depend on any particular notion of relatedness. However, Eq. (37) reduces to our expression for Hamilton’s rule (1) if we assume there are two alleles at the focal locus one of which is the wild type with zero fitness contribution and the other conferring fitness b on all individuals other than itself, with self-fitness increment −c. In this case R _i  = br − c, $\overline{R} = p_i(br-c),$ so (37) reduces to (1). In addition T = b − c and S = (1 − r)b, which are both positive when b > 0. This gives the standard contemporary interpretation of Hamilton’s rule. In particular, if the gene is indeed a helping gene (b > 0), some of the benefits to the recipient will be directed to non-carriers of the allele, so all members of the population gain from the helping behavior. Moreover, we get the GHR expression (18) with γ _i  = 0 and $\overline{S}=-\alpha.$

The dilution effect is important not because it affects the rate of change in the frequency of the focal allele, but because when $\overline{S} < 0,$ the success of the focal allele can come at the expense of a lower mean population fitness $1+\overline{T}$ even when $\overline{R} > 0.$ Indeed, the above analysis shows that the conditions for allele success and the conditions for contributing to the success of the reproductive population are distinct.

Appendix 4: Sex allocation and the phenotypic gambit

The following example, based on Charnov (1978), clarifies the meaning of conflict at the gene level in a sociobiological setting, deploying the phenotypic gambit to model both the behavior of the queen and the worker. The model shows how the resolution of this conflict deviates from the model of conflict based on divergence of genetic “interests” and inclusive fitness maximization as the conflict-resolving process. The genetic analysis in this case is sufficiently simple that a complete gene-level treatment is possible with the assumption that either the queen or the workers control the relative allocation of resources devoted to the production of male and female reproductives.

Consider a eusocial haplodiploid species where each colony has one queen, singly mated and mated for life. The workers, all female, raise the colony’s brood, which consists of male and female eggs due to queen oviposition, and male eggs due to worker oviposition. Workers cannot produce female zygotes because there are no males in the population except during the mating season.

We assume first that the queen controls the proportion of female and male reproductives, according to a preference that, however complexly regulated in the queen’s genome, can be represented by a single locus with alleles a and A, subject to Medelian segregation, the mutant allele A being dominant. We assume an aa queen prefers a female-male sex ratio of r, whereas an Aa or AA queen prefers a sex ratio $\hat{r}.$ We suppose that aa and a are fixed in the population and investigate the conditions for the A allele to invade (Charnov 1978). More precisely, we seek to specify a value of r that cannot be invaded by a distinct sex ratio $\hat{r}$ preferred by a mutant.

We denote the colonies by xyz, where $x,y,z\in \{a,A\},$ with xy being the queens type and z being her mate’s type. Because the mutant is rare, we can ignore colonies with more than one mutant type, and we can ignore females of type AA. This assumption is purely for ease of exposition, and cannot change the results, provided the mutant is sufficiently rare. The three remaining types are then aaa, Aaa, and aaA. Let n _xyz be the number of colonies of type xyz.

We assume the colony has one unit of resources to expend on raising reproductives, so an aa queen devotes a fraction r resources to gynes (female reproductives) and 1 − r to males (all of which are reproductive), while an Aa queen devotes $\hat{r}$ and $1-\hat{r}$ to gynes and males, respectively. We normalize the colony size, which we assume the same for all colonies, to unity. Let s _f and s _m be the expected number of gynes and males, respectively, that survive to reproductive age per unit of resource devoted to their care. We assume all surviving gynes found new colonies, and all surviving males compete equally successfully for mating opportunities. We also assume that in each period a colony has a probability p of persisting until the next period, and workers produce a fraction q of new males.

In the current period, an aaa colony produces r gynes of type aa and 1 − r males of type a, so the population produces n _aaa rs _f new gynes of type aa. An aaa colony passes n _aaa (1 − r)s _m males of type a and n _aaa (p + rs _f ) gynes of type aa to the next generation of reproductives. If $\epsilon$ is the fraction of A males at mating time, then this gives rise to $n_{aaa}(p+rs_f)(1-\epsilon)$ colonies of type aaa and $n_{aaa}rs_f\epsilon$ colonies of type aaA.

The aaA colony produces Aa gynes, with r resources devoted to gynes, and resources (1 − r) to males, all of which are of type a. Thus there are n _aaA rs _f new Aa gynes and n _aaA (1 − r)s _m new males in the population from aaA colonies. The queen produces a fraction (1 − q) of the males, all of which are a. The workers, which are Aa, produce a fraction q of the males, half of which are a and half are A. Thus aaA colonies contribute n _aaA (1 − r)qs _m /2 males of type A and n _aaA r s _f new Aa gynes to the next generation.

The Aaa colony produces half aa and half Aa gynes and a fraction $\hat{r}$ are devoted by the Aa queen to new gynes and $(1-\hat{r})$ to males. The gynes are half aa and half Aa, while the males are half a and half A. Thus Aaa colonies contribute $n_{Aaa}(p + \hat{r} s_f/2)$ gynes of type Aa and $n_{Aaa}(1-\hat{r})s_m/2$ males of type A to the next generation, a fraction q of which are from worker’s eggs.

If n′ is the next period frequency, we thus have

$$n_{aaa}^{\prime}=\left(p + rs_f-\epsilon\right)n_{aaa}$$

(42)

$$n_{aaA}^{\prime}=pn_{aaA} + rs_f\epsilon n_{aaa}$$

(43)

$$n_{Aaa}^{\prime} = n_{aaA}rs_f + \left(p+\frac{\hat{r} s_f}{2}\right)n_{Aaa},$$

(44)

where $\epsilon$ is the fraction of aa new gynes that mate with A males.

The number of new males of type a is n _aaa (1 − r)s _m . The new A males consist of those produced in Aaa and aaA colonies. In an Aaa colony, the queen produces a fraction (1 − q) of the males, half of which are A, so $n_{Aaa}(1-q)(1-\hat{r})s_m/2$ males are thus produced. The workers produce a fraction q of males, giving

$$n_{Aaa}(2(1-q) + q)(1-\hat{r})s_m/4) = n_{Aaa}(1-\hat{r})(2-q)s_m/4$$

new A males from Aaa colonies. In aaA colonies the queen devotes a fraction r to the production of males, but only the Aa workers, who produce a fraction r of the males, produce A males. This gives n _aaA (1 − r)qs _m /2 type A males.

The ratio of new A males to new a males is then

$$\epsilon = \frac{n_{Aaa}(1-\hat{r})(2-q)s_m/4}{n_{aaa}(1-r)} + \frac{n_{aaA}q(1-r)s_m}{2n_{aaa}(1-r)s_m}$$

(45)

$$= \frac{n_{Aaa}(1-\hat{r})(2-q)}{4n_{aaa}(1-r)} + \frac{n_{aaA}q}{2n_{aaa}}$$

(46)

We thus have

$$n_{aaa}^{\prime} = (p + rs_f - \epsilon)n_{aaa}$$

(47)

$$n_{aaA}^{\prime} = \left(p + \frac{rs_fq}{2}\right)n_{aaA} + \frac{rs_f(1-\hat{r})(2-q)}{4(1-r)}n_{Aaa}$$

(48)

$$n_{Aaa}^{\prime} = rs_fn_{aaA} + \left(p + \frac{\hat{r} s_f}{2}\right)n_{Aaa},$$

(49)

Equation (47) shows that when all reproductives are of type aa and a, the population grows at rate $p + rs_f - \epsilon- 1.$ The second and third equations are not linked with the first, and show the fate of an invasion of the population by a small number of mutant gynes of type Aa and males of type A. We can depict the dynamics of the invasion by a matrix equation

$$\left[\begin{array}{l} n_{aaA}^{\prime}\\ n_{Aaa}^{\prime}\\ \end{array}\right] = \left(\begin{array}{ll} p + \frac{rs_fq}{2} & \frac{rs_f(1-\hat{r})(2-q)}{4(1-r)}\\ rs_f & p + \frac{\hat{r} s_f}{2} \end{array}\right) \left[\begin{array}{l} n_{aaA}\\ n_{Aaa} \end{array}\right]$$

(50)

The matrix in (50) has positive entries, so it has a maximal real eigenvalue that represents the growth rate of the dynamical system (Elaydi 1999). If the incumbent population is incapable of being invaded, then the value of $\hat{r}$ at which this eigenvalue is maximized must be r.

After some rather tedious calculations, we find that the maximal eigenvalue occurs for $\hat{r}=r$ when r satisfies the equation

$$\frac{(2-q)(1-2r)s_f}{2(3-q)(1-r)} = 0,$$

(51)

which implies r = 1/2. This prediction of equal investment in males and females, first stated by Fisher (1915) for diploid species, is valid for haplodiploid as well, and does not depend on the fraction q of workers’s sons.

The inclusive fitness analysis of this model is much simpler (Trivers and Hare 1976). The queen is related 1/2 to her daughter queens, and each daughter queen produces one gyne, (1 − q) males, and q/2 males via her workers, for a total reproductive value of 1 + (1 − q) + q/2 = 1 + (1 + p)/2. The queen is related (1 − q) + q/2 = (2 − q)/2 to a male reproductive, and the male’s reproductive value per inseminated queen is the queen herself plus one half the number of males produced by her workers, which is q/2. The males reproductive value is thus (1 + q/2)x, where x is the ratio of gynes to males in the reproductive population. The population equilibrium occurs when the relatedness times reproductive value for males and gynes are equal:

$$\left(1-\frac{q}{2}\right)\left(1 + \frac{q}{2}\right)x = \frac{1}{2}\left(2 - \frac{q}{2}\right),$$

(52)

giving a sex ratio of

$$x = \frac{4-q}{4-q^2}.$$

(53)

This differs from the correct ratio of x = 1 by at most about 7 %.

Now let us assume that the workers rather than the queen control the allocation of resources to the reproductives. The aaA colony produces Aa workers, Aa gynes, and a males. The workers now devote $\hat{r}$ resources to gynes, and $(1-\hat{r})$ to males. Thus there are $n_{aaA}\hat{r} s_f$ new Aa gynes and $n_{aaA}(1-\hat{r})s_m$ new males in the population from aaA colonies. The aaA colonies thus contribute $n_{aaA}(1-\hat{r})qs_m/2$ males of type A and $n_{aaA}\hat{r} s_f$ new Aa gynes to the next generation. The Aaa colony produces half aa workers and half Aa workers, so the average workers devotes $\bar{r} = (r + \hat{r})/2$ resources to producing gynes and $(1-\bar{r})$ to producing males. The gynes are half aa and half Aa, while the males are half a and half A. Thus Aaa colonies contribute $n_{Aaa}(p + \bar{r} s_f/2)$ gynes of type Aa and $n_{Aaa}(1-\bar{r})s_m/2$ males of type A to the next generation, a fraction q of which are from worker’s eggs.

If $n^{\prime}$ is the next period frequency, we thus have

$$n_{aaa}^{\prime} = (p + rs_f)n_{aaa}$$

(54)

$$n_{aaA}^{\prime} = pn_{aaA} + rs_f\epsilon n_{aaa}$$

(55)

$$n_{Aaa}^{\prime} = \left(p + \frac{\bar{r} s_f}{2}\right)n_{Aaa} + n_{aaA}\hat{r} s_f,$$

(56)

where $\epsilon$ is the fraction of aa new gynes that mate with A males.

The new A males still consist of those produced in Aaa and aaA colonies. In an Aaa colony, the queen produces a fraction (1 − q) of the males, half of which are A, so $n_{Aaa}(1-q)(1-\bar{r})s_m/2$ are thus produced. The workers produce a fraction q of males, half of the workers are Aa, half of their male offspring are A, giving

$$n_{Aaa}(2(1-q) + q)(1-\bar{r})s_m/4) = (1-\bar{r})(2-q)s_m/4$$

new A males from Aaa colonies. In aaA colonies, the workers are Aa and half of them produce the fraction q of A males, or $n_{aaA}(1-\hat{r})qs_m/2$ A males.

The ratio of new A males to new a males is then

$$\epsilon = n_{Aaa}\frac{(1-\bar{r})(2-q)/4}{n_{aaa}(1-r)} + \frac{n_{aaA}q}{2n_{aaa}(1-r)}$$

(57)

$$= \frac{n_{Aaa}(1-\bar{r})(2-q)}{4n_{aaa}(1-r)} + \frac{n_{aaA}q(1-\hat{r})}{2n_{aaa}(1-r)}$$

(58)

We thus have

$$n_{aaa}^{\prime} = (p + rs_f)n_{aaa}$$

(59)

$$n_{aaA}^{\prime} = \left(p + \frac{rs_fq(1-\hat{r})}{2(1-r)}\right)n_{aaA} + \frac{rs_f(1-\bar{r})(2-q)}{4(1-r)}n_{Aaa}$$

(60)

$$n_{Aaa}^{\prime} = \left(p + \frac{\bar{r} s_f}{2}\right)n_{Aaa} + \hat{r} s_fn_{aaA},$$

(61)

Equation 59 shows that when all reproductives are of type aa and a, the population grows at rate p + rs _f  − 1. The second and third equations are not linked with the first, and show the fate of an invasion of the population by a small number of mutant gynes of type Aa and males of type A. We can depict the dynamics of the invasion by a matrix equation

$$\left[\begin{array}{l} n_{aaA}^{\prime}\\ n_{Aaa}^{\prime}\\ \end{array}\right] = \left(\begin{array}{ll} p + \frac{rs_fq(1-\hat{r})}{2(1-r)} & \frac{rs_f(1-\bar{r})(2-q)}{4(1-r)}\\ \hat{r} sf & p + \frac{\bar{r} s_f}{2} \end{array}\right) \left[\begin{array}{c} n_{aaA}\\ n_{Aaa} \end{array}\right]$$

(62)

The matrix in (62) has positive entries, so it has a maximal real eigenvalue that represents the growth rate of the dynamical system (Elaydi 1999). If the incumbent population is incapable of being invaded, then the value of $\hat{r}$ at which this eigenvalue is maximized must be r, in which case the eigenvalue must equal, p + rs _f  − 1, the growth rate of the incumbent population.

After some additional tedious calculations, we find that the maximal eigenvalue occurs for $\hat{r} = r$ when r satisfies the equation

$$\frac{6 - 8r + q(2r-3)}{4(3-q))(1-r))}sf = 0.$$

(63)

Solving for the value of r that cannot be invaded, we find

$$r = \frac{6-3q}{8-2q}.$$

(64)

This shows that when the queen produces all the males, the worker’s desired sex ratio is r = 3/4, while if the workers produce the males, the ratio is r = 1/2.

The inclusive fitness analysis of this model is similar (Trivers and Hare 1976). The workers are related 3/4 to their sisters, and related (1 − q)/4 + (3/8)q = (2 + q)/8 to their brothers. Equation (52) now becomes

$$\left(\frac{2+q}{8}\right)\left(1 + \frac{q}{2}\right)x = \frac{3}{4}\left(2 - \frac{q}{2}\right),$$

(65)

giving a sex ratio of

$$r = \frac{1}{1+x} = \frac{3(4-q)}{16+q(1-q)}.$$

(66)

This differ from the actual value (64) by at most 2.5 %.

Appendix 5: Hamilton’s rule in the diploid case

This section presents a diploid version of the Hamilton’s rule. This is often termed the regression approach in the literature, but there is in fact no statistical estimation involved in the derivation (Michod and Hamilton 1980). It will be of interest mainly to population biologists.

Consider a reproductive population X with individuals $\{X_i\in X|i=1,\ldots,n\}.$ Suppose the genome has a diploid autosomal locus with two alleles, s (selfish) which leads to a behavior that does not affect the fitness of other individuals, and a (altruistic), which leads its carrier X _i to incur an increased fitness cost c _i over that of the selfish allele, and to bestow fitness benefit b _i distributed over a subset Y _i of recipients. Suppose in addition that the altruistic allele has a social fitness effect β (pollution when β > 0 or a public good when β < 0) on both alleles. This cost may be intragenomic, borne by the carrier, or intergenomic, distributed over the population in some arbitrary manner.

Hamilton (1964a) assumes the social fitness effect is distributed uniformly over the genome. This is a significant limitation of his analysis because intragenomically, meiotic drive and other forms of segregation distortion, and socially, altruistic acts that are purchased in part by reducing the fitness of non-relatives, which we may call thieving effects, are of extreme importance, although the Inclusive Fitness Harmony Principle suggests that natural selection will limit their observed frequency. We can represent these thieving effects as transfers of fitness α from non-relatives to relatives.

Standard expositions of Hamilton’s rule take Y _i to be an individual. This, however, is a restrictive assumption because in many social species individuals interact in groups where it is difficult to apportion the benefit b _i among the various participants. Moreover, as we shall see, Hamilton’s rule does not depend on this assumption.

The genotypic value X ⁱ _g of X _i at the focal locus, the frequency of the focal allele at this locus, is 0, 1/2, and 1 for genotypes ss,  sa, and aa, respectively. The phenotypic value X ⁱ _p of X _i is 0, h, or 1 according as X _i is ss and never confers the benefit, is sa and confers the benefit with intensity h, or is aa and confers the benefit with intensity one. Here h can have any value, positive or negative, but if the allele effects are additive, then h = 1/2. Because there are 2n alleles at the focal locus in the population, the frequency of a is q _a  = ∑ _i X ⁱ _g /n. Let Y ⁱ _g be the mean genotype of members of Y _i .

The fitness cost to X _i in the current period is thus c _i X ⁱ _p , and the fitness gain to the recipients Y _i is b _i X ⁱ _p . The population in the next period is then

$$n(1-{\beta q}_a + (b-c)x_p)$$

(67)

where $x_p = \sum\nolimits_iX_p^i/n$ is the mean phenotype of the population, $b=\sum\nolimits_ib_iX_p^i/x_p$ is the mean benefit, and c = ∑ _i c _i X ⁱ _p /x _p is the mean cost. Note that because the thieving effect α is a within-population fitness transfer, it does not appear in (67). The number of donor alleles in the next period is

$$nq_a\left(1-{\beta q}_a + \alpha(1-q_a)\right) + \sum\limits_ib_iX_p^iY_g^i- \sum\limits_ic_iX_p^iX_g^i.$$

The increase in the frequency of the donor allele in the next period, writing the mean genotype of recipients as $q_a^y=\sum\nolimits_iY_g^i/n,$ is then given by

$$\begin{aligned} &\frac{nq_a\left(1-{\beta q}_a +\alpha(1-q_a)\right)+ \sum_ib_iX_p^iY_g^i- \sum_ic_iX_p^iX_g^i} {n\left(1 - {\beta q}_a + (b-c)x_p\right)} - q_a \\ &\quad =\frac{\left(\sum_ib_iX_p^iY_g^i - nbx_pq_a^y\right) + nq_a\alpha(1-q_a)} {n\left(1-{\beta q}_a + (b-c)x_p\right)} \\ &\quad - \frac{\left(\sum_ic_iX_p^iX_g^i-ncx_pq_a\right) + nbx_p(q_a-q_a^y)} {n\left(1-{\beta q}_a + (b-c)x_p\right)} \\ &\quad =\frac{\hbox{cov}(X_p^b,Y_g)-\,\hbox{cov}(X_p^c,X_g) + \alpha\hbox{var}(X_p) - bx_p(q_a-q_a^y)} {1 -{\beta q}_a+ (b-c)x_p}, \end{aligned}$$

(68)

where X ^b _p and X ^c _p are the variables b _i X ⁱ _p and c _i X ⁱ _p , respectively, and X _g is a binomial variable, so var(X _p ) = nq _a (1 − q _a ). Note that the expression (68) is positive, assuming weak selection, when

$$\frac{\hbox{cov}(X_p^b,Y_g)+\alpha\hbox{var}(X_p)-bx_p(q_a-q_a^y)} {\hbox{cov}(X_p^c,X_g)} > 1.$$

(69)

This inequality is the most general form of Hamilton’s rule, including both social fitness and thieving effects. If we assume donors distribute benefits that are, on average, independent from the allelic composition at the focal locus, i.e., q ^y _a  = q _a then (69) becomes

$$\,\hbox{cov}(X_p^b,Y_g) +\alpha\hbox{var}(X_p) >\,\hbox{cov}(X_p^c,X_g).$$

(70)

If we further assume that b _i  = b and c _i  = c for all individuals $i=1,\ldots,n,$ we get the expression:

$$\frac{b \,\hbox{cov}(X_p,Y_g)+\alpha \hbox{var}(X_p)}{\hbox{cov}(X_p,X_g)} > c.$$

(71)

Finally, if the effect of the altruistic allele is additive, so h = 1/2, then (71) becomes

$$b \frac{\hbox{cov}(X_p,Y_g)}{\hbox{var}(X_g)} > c-\alpha.$$

(72)

This is a standard expression for Hamilton’s rule (Michod and Hamilton 1980), except we have taken into account the thieving effect α (and the pollution/public good effect β, which does not appear in Hamilton’s rule). More generally, for arbitrary h, we have

$$br > cr^p-\alpha,$$

(73)

where

$$r = \frac{\hbox{cov}(X_p,Y_g)}{\hbox{var}(X_g)}$$

is the regression coefficient of Y _g on X _p , and r ^p is the regression coefficient of X _p on X _g :

$$r^p = \frac{\hbox{cov}(X_p,X_g)}{\hbox{var}(X_g)}.$$

It should be clear that, while we use mathematical terminology from statistical estimation theory, no statistical estimation is in fact involved.

To illustrate the increased generality of the form (70) of Hamilton’s rule, suppose the reproductive population is partitioned into social castes $\{Z^j\subset X|j=1,\ldots,m\},$ where caste j has frequency z _j in the population, and suppose members of the same caste j have the same costs c _j and benefits b _j . Let Y ^j be the weighted sum of $\{Y_i|X_i\in Z^j\},$ where each individual is weighted by the number of times the individual appears in the sum. Then we can write (70) as

$$\sum\limits_{j=1}^m c_j\left((b_j \,\hbox{cov}(Z^j_p,Y^j_g) -c_j \,\hbox{cov}(Z^j_p,Z^j_g)\right) + \alpha\,\hbox{var}(X_p) > 0.$$

(74)

Equation (74) shows that in general the social structure of the population allows a caste to be fundamentally altruistic in the sense that its net costs of helping exceed the net benefits that the caste contributes to the population. Because the inclusive fitness of caste j is

$$b_j \,\hbox{cov}(Z_p^j,Y_g^j) - c_j \,\hbox{cov}(Z_p^j,Z_g^j)< 0$$

(75)

it is then clear that caste j members would maximize their inclusive fitness by simply refusing to contribute to the social process. This shows that in a caste social structure, individuals do not maximize their inclusive fitness. Of course, if castes are genetically determined, then the partition $\{z_j|j=1,\ldots,m\}$ will be variable across periods and a fundamentally altruistic caste will become extinct in the long run. However, if castes are determined by developmental conditions (e.g., feeding in eusocial insects or socialization in humans), fundamentally altruistic castes can be maintained in the long run.

The sociobiological dynamics of Hamilton’s rule

The mapping X _i → Y _i , which we have taken as given, reflects the social structure of the reproductive population. This mapping does not presume any particular set of social relations of kinship, which is why we suggest that kin selection is in general an inappropriate description of inclusive fitness dynamics. Note that if the frequency of the a allele in the population does not affect the fitnesses of alleles at other loci in the genome, then the a allele will move to fixation in the population if Hamilton’s rule is satisfied, and will become extinct if the reverse inequality is satisfied. Ultimately, the focal locus will be heterozygous with zero probability.

With frequency dependence, when the focal allele becomes prevalent in the population, if b − c > 0, so the allele is beneficial to its carriers, there will be no selection at the level of the genome for genes that suppress the a allele at the focal locus, so the a allele will still move to fixation in the population. When the focal allele is prevalent and b − c < 0, there will be natural selection at other loci for genes that either alter the sociobiological mapping X _i → Y _i or otherwise suppress the a allele at the focal locus, so that Hamilton’s rule no longer holds for the antisocial allele. This is the essence of the Inclusive Fitness Harmony Principle. Of course there may be no likely mutation that suppresses an anti-social a allele, in which case the antisociality reflected in the behavior induced by the a allele will become ubiquitous in the population. natural selection does not guarantee optimality.

This phenomenon also represents a plausible counterexample to Fisher’s Fundamental Theorem (Ewens 1969; Price 1972; Frank and Slatkin 1992; Edwards 1994; Frank 1997): as an antisocial allele moves to fixation, the average fitness of population members declines. Some population biologists save Fisher’s theorem by calling this a transmission effect, and insisting that natural selection always produces fitness-enhancing gene frequency changes (Edwards 1994; Frank 1997; Gardner et al. 2011). This interpretation of natural selection should be avoided because it is arbitrary and difficult to understand for those who are not experts in population biology.

It follows that Hamilton’s rule is useful only in charting short-term genetic dynamics. Weak selection and additivity across loci are extremely powerful analytical tools, but in the long run changes in gene frequency at one locus are likely to induce compensatory and synergistic changes at other loci. Indeed, the very mapping X _i → Y _i on which Hamilton’s rule is based is itself coded in the core genome of the reproductive population, and hence in the long run is modified in the course of evolutionary selection and adaptation.

Altruism among relatives

A relative is a person “allied by blood$\ldots$ a kinsman” (Biology Online). The argument to this point has nothing to do with genealogy, and hence says nothing about altruism among family members. This is an attractive property of our exposition because in a highly social species, individuals interact frequently with non-relatives.

It remains to determine the exact relationship between the sociobiological conception (71) and the genealogical conception of relatedness. We follow Michod and Hamilton (1980), except that we assume the population is outbred at the focal locus. Suppose that each Y _i is an individual recipient, and all recipients have the same genealogical relationship to their donors (e.g., Y _i is a sibling of X _i ). Let p _xyzw be the joint distribution of genotypes xy for donor and zw for recipient where $x,y,z,w\in\{s,a\}.$ Let p ^x _ss , p ^x _as , and p ^x _aa be the marginal distribution of the genotypes ss,  sa, and aa for the donor (i.e., the fraction of these genotypes in the population), and similarly for p ^y _ss , p ^y _as , and p ^y _aa for the recipient.

We have

$$\begin{aligned} x_p &= hp_{as}^x + p_{aa}^x, \\ y_p &= hp_{as}^y + p_{aa}^y, \end{aligned}$$

because p ^x _as is the fraction of sa genotypes, their phenotypic value is h, and p _aa is the fraction of aa genotypes, which have phenotypic value one. Also,

$$p_{as}^x = 2q_nq_a$$

(76)

$$p_{aa}^x = q_a^2$$

(77)

To derive (76), note that either the paternal allele is s with probability q _n  = 1 − q _a and the second is a with probability q _a , or else the paternal allele is a with probability q _a and the second is s with probability q _n . The second equation is derived in a similar manner.

We thus have

$$x_p = 2hq_nq_a + q_a^2$$

(78)

$$y_p = 2hq_nq_a + q_a^2$$

(79)

Note that

$$\begin{aligned} x_g = & 1/2p_{as}^x + p_{aa}^x = q_a \\ y_g = & 1/2p_{as}^y + p_{aa}^y = q_a. \end{aligned}$$

To derive  cov(X _g , X _p ), note that

$$\begin{aligned} \sum_i X_p^iX_g^i/n =& hp_{as}^x/2 + p^x_{aa} \\ =& hq_nq_a + q_a^2 \end{aligned}$$

Given the values of p ^x _as and p ^x _aa from Eqs. (76) and (77), and after algebraic simplification, we find

$$\,\hbox{cov}(X_p,X_g) = q_nq_a\alpha/2,$$

(80)

where

$$\alpha = 2(h + q_a(1-2h).$$

(81)

Also,

$$\,\hbox{cov}(y_gx_p) = h p_{sasa}/2 + hp_{saaa} + p_{aasa}/2 + p_{aaaa} - y_gx_p.$$

Now let p ₁₁ be the probability X _i and Y _i share both alleles at the focal locus identically by descent, let p ₁₀ be the probability the share one allele at the focal locus identically by descent, and let p ₀₀ be the probability they share neither allele identically by descent. then we have

$$p_{asas} = 2q_nq_ap_{11} + q_nq_ap_{10} + 4q_n^2q_a^2p_{00}$$

(82)

$$p_{asaa} = q_aq_n^2p_{10} + 2q_nq_a^3p_{00}$$

(83)

$$p_{aaas} = q_nq_a^2p_{10} + 2q_nq_a^3p_{00}$$

(84)

$$p_{aaaa} = q_a^2p_{11} + q_a^3p_{10} + q_a^4p_{00}.$$

(85)

If we define f _XY as the probability that a random allele in X _i and a random allele in Y _i are identical by descent, then

$$f_{XY} = p_{11}/2 + p_{10}/4.$$

(86)

Then a little algebra shows that the r in Hamilton’s rule is given by

$$r = \frac{\hbox{cov}(X_pY_g)}{\hbox{cov}(X_pX_g)} = 2f_{XY}.$$

(87)

Note that r is then the expected number of copies of the focal allele in the recipient.

Consider, for instance, the case of siblings. The two share the same allele from the father with probability 1/2, and similarly for the mother. therefore p ₁₁ = 1/4, p ₁₀ = 1/2, and p ₀₀ = 1/4. Substituting these values in (82), we get

$$r = \frac{\hbox{cov}(Y_g,X_p)}{\hbox{cov}(X_g,X_p)} = \frac{1}{2}.$$

(88)

Thus the sociobiological definition of relatedness and the genealogical definition coincide.

The haploid form of sociobiological relatedness

We now show that the fraction in haploid expression of Hamilton’s rule (17) is precisely the sociobiological definition of relatedness. In this case, s is the selfish gene and a is the altruistic gene at the focal locus. The variance of X _g is now

$$\hbox{var}(X_g)=q_nq_a$$

and

$$\,\hbox{cov}(Y_gX_p) = p q_a - q_a^2$$

so

$$\frac{\hbox{cov}(Y_gX_p)}{\hbox{var}(X_g)} = \frac{p-q_a}{1-q_a},$$

which is equivalent to (17).