The joint effects of kin, multilevel selection and indirect genetic effects on response to genetic selection

Authors


Piter Bijma, Animal Breeding and Genomics Centre, Wageningen University, PO 338, 6700AH Wageningen, The Netherlands.
Tel.: +31 317 482731/335; fax: +31 317 483929; e-mail: piter.bijma@wur.nl

Abstract

Kin and levels-of-selection models are common approaches for modelling social evolution. Indirect genetic effect (IGE) models represent a different approach, specifying social effects on trait values rather than fitness. We investigate the joint effect of relatedness, multilevel selection and IGEs on response to selection. We present a measure for the degree of multilevel selection, which is the natural partner of relatedness in expressions for response. Response depends on both relatedness and the degree of multilevel selection, rather than only one or the other factor. Moreover, response is symmetric in relatedness and the degree of multilevel selection, indicating that both factors have exactly the same effect. Without IGEs, the key parameter is the product of relatedness and the degree of multilevel selection. With IGEs, however, multilevel selection without relatedness can explain evolution of social traits. Thus, next to relatedness and multilevel selection, IGEs are a key element in the genetical theory of social evolution.

Introduction

Social interactions among individuals are a key factor in evolution by natural selection (Darwin, 1859; Hamilton, 1964; Frank, 1998; Keller, 1999; Clutton-Brock, 2002; Rice, 2004). Such interactions lead to selection acting at multiple levels, which alters the outcome of natural selection within populations (Wilson, 1975; Wade, 1985; Frank, 1998; Keller, 1999; Rice, 2004). Two common approaches for modelling the evolutionary consequences of social interactions are kin selection models (wherein inclusive fitness is maximized) and levels-of-selection models.

Kin selection models centre on inclusive fitness and the direct fitness costs (c) and indirect fitness benefits (b) of social interactions among related individuals (Hamilton, 1964; Maynard Smith, 1964; Michod, 1982). The central parameter in kin selection models is the relatedness between interacting individuals (e.g. Michod, 1982; Queller, 1992b), which weights the indirect fitness benefits accruing to an individual from social interactions with its relatives. The genetic structure of the kin selection models is contained in the relatedness, which is averaged over all kin groups, each of which might have a different value for the relatedness as occurs, for example, when some queens have one mate, whereas others in the same population have several (Wade, 1984). A social behaviour evolves in these models when the fitness benefit weighed by relatedness exceeds fitness cost. This is the classic relationship known as Hamilton's rule, i.e. rb > c.

Levels-of-selection models centre on partitioning total selection pressure into components, one for each of the levels, most commonly, the individual and group level (Hamilton, 1975; Wilson, 1975; Wade, 1979, 1980, 1985; Keller, 1999; Traulsen & Nowak, 2006), although three levels of selection have been formally modelled (Wade, 1982). In these models, selection within groups opposes the evolution of a social behaviour because of its fitness costs, but selection among groups favours the behaviour because more social groups have higher mean fitness than less social groups. The behaviour evolves when selection among groups times the ‘group heritability’ exceeds the opposing selection within groups. Wade (1979, 1980, 1984), using a formal population genetic model, and Cheverud (1984) and Queller (1992a), both using a quantitative genetics approach, showed that this condition is Hamilton's rule. When groups exist transiently (e.g. during only one generation), the levels-of-selection approach is sometimes referred to as ‘new group selection’ (cf. discussion in West et al., 2007), ‘intra-demic selection’ (Wade, 1978), ‘trait group selection’ (Wilson, 1975, 1980), or ‘cohort selection’ (Mertz et al., 1984); not to be confused with Wynne-Edwardsian interpopulation selection (Wynne-Edwards, 1962).

Bourke & Franks (1995, p. 39) argued that ‘there is no fundamental clash between gene and colony-level selection’, and, later (p. 48), ‘…both the present ‘components of selection’ method of modelling the evolution of altruism, and the kin selection method, reach the same conclusion – namely that relatedness is necessary for a gene for altruism to spread’. More recently, however, Wilson & Holldobler (2005) have argued that kin and colony selection are different processes, whereas Lehmann et al. (2007) have argued that group selection and kin selection are two different modelling approaches applied to the same evolutionary process. Furthermore, West et al. (2007, 2008) have argued that there are many reasons to prefer kin selection to multilevel selection theory, given their mathematical equivalence.

Although a number of studies have shown that kin selection models and levels-of-selection models are equivalent descriptions of a single process (Hamilton, 1975; Grafen, 1984; Wade, 1985; Frank, 1986; Queller, 1992a; Dugatkin & Reeve, 1994; Gardner et al., 2007; Lehmann et al., 2007; West et al., 2007), the equivalence of both modelling approaches is still disputed, resulting in ongoing debate and confusion. For example, Traulsen & Nowak (2006) argued that kin selection models are inadequate for games in finite populations, but this claim was refuted by Lehmann et al. (2007).

A somewhat different approach to modelling social interactions has its roots in quantitative genetic models for maternally affected traits, which are widely used in agriculture (Dickerson, 1947; Willham, 1963; Crow & Aoki, 1982, 1984; Cheverud, 1984; Lynch & Walsh, 1998). In that approach, the trait value of an individual is modelled as the sum of a direct genetic effect due to the focal individual, and indirect genetic effects (IGE) due to other individuals interacting with the focal individual (Griffing, 1967, 1976, 1981a,b; Cheverud, 1984, 2003; Moore et al., 1997; Wolf et al., 1998; Agrawal et al., 2001; Rauter & Moore, 2002; Wolf, 2003; Muir, 2005; Bijma et al., 2007a,b). Henceforth, we will refer to these models as IGE models. The term ‘indirect genetic effects’ is used with a somewhat different meaning in these quantitative genetic models than the term, ‘indirect fitness effects’, used above in the kin selection models. An IGE is a heritable effect of one individual on the trait value of another individual. A well-known example of an IGE is the maternal genetic effect of a mother on preweaning growth rate of her offspring in a mammal. IGEs have also been referred to as ‘associative effects’ (Griffing, 1967). An appealing property of IGE models is that they describe social interactions within the conventional quantitative genetic framework, as described in Lush (1947), Cockerham (1954), Griffing (1967), Kempthorne (1969), Van Vleck (1970), Hill (1971), Hanrahan & Eissen (1973), Falconer & Mackay (1996) and Lynch & Walsh (1998). This makes IGE models empirically powerful, because they can be applied to field data using the flexible tools developed in livestock and plant breeding, such as animal models and restricted maximum-likelihood estimation (Patterson & Thompson, 1971; Kruuk, 2004; Muir, 2005; Bijma et al., 2007b).

The relationships between IGE models and kin and levels-of-selection models are unclear. Although kin and levels-of-selection models describe the same process, it is unclear whether they adequately address IGEs. Cheverud (2003), for example, used IGE models to argue that important results of kin selection theory, such as Hamilton's rule, are incomplete. An important distinction between kin and levels-of-selections models vs. IGE models is that there is only one trait in the former, a social behaviour which has fitness costs to the individual performer and fitness benefits to the recipients of the behaviour. The IGE models, by contrast, consider effects of social interactions on trait values, rather than directly on fitness. In IGE models, therefore, effects of social interactions on fitness work via effects on trait values. For example, genes in the maternal genome, i.e. maternal genetic effects, may affect body weight of offspring. As a result, mean body weight can increase in a population in response to positive selection even in the absence of direct additive genetic variation for offspring body weight, as long as there are additive maternal genetic effects. The resulting differences in body weight among individuals may subsequently affect mating success and thus fitness (in principle, IGE models could be applied directly to fitness, but that is not common).

Wolf et al. (1999) investigated the effects of interacting phenotypes on the selection differential. Hence, they considered the consequences of interacting phenotypes for selection, rather than for genetic response to selection. Their results show that social selection occurs when variation in fitness due to interacting phenotypes covaries with traits, resulting in a net force of selection acting on the interacting phenotypes.

Results of IGE models suggest that, when selection is applied to trait values, the response of the trait mean to selection depends on both the relatedness between interacting individuals and on the partitioning of selection pressure over multiple levels (Griffing, 1967, 1976; Bijma et al., 2007a). Hence, the IGE approach suggests that models of social selection should consider both relatedness and the levels of selection, rather than focusing on either relatedness or levels of selection. In the levels-of-selection models, the response to selection at the group level requires genetic variation among groups. Genetic variation among groups is equivalent to a positive genetic correlation between individuals within groups, which is the same as relatedness. Thus, relatedness also appears in some models of multilevel selection, because it affects the heritable variation among group mean trait values. For some kinds of social behaviours, therefore, the current view is that kin and level-of-selection models are equivalent and describe a single biological process (Gardner et al., 2007; West et al., 2007). However, IGE models address a broader range of social interaction effects than those considered by other approaches. We will show that kin and multilevel selection require both among-group selection and relatedness, whereas IGEs do not.

The objective of this work was to clarify the joint effects of relatedness, multilevel selection and IGEs on response to selection. In our opinion, the ongoing debate on equivalence of kin and levels-of-selection models is partly caused by the fact that levels-of-selection models tend to hide the relatedness component of response to selection, whereas kin selection models tend to hide the multilevel selection component of response to selection. We illustrate how the response to selection is naturally described by the combination of relatedness and the degree of multilevel selection, rather than by focusing on one or the other of the two factors. This result is obtained irrespective of the modelling approach used. Our results show that, in the absence of IGEs, the key parameter determining the impact of social interactions on response to selection is the product of relatedness and the degree of multilevel selection. In the presence of IGEs, however, the results can be very different in important ways that could contribute to rapid evolution, even in the absence of multilevel selection or relatedness.

This work consists of two sections. The first section considers relatedness and multilevel selection in the absence of IGEs. In this section, we use the approach of Queller (1992a) to demonstrate that, irrespective of the approach taken, the response to selection depends on the product of relatedness and the degree of multilevel selection. We use the formulations of Queller (1992a), because he is frequently cited as demonstrating the equivalence of levels-of-selection models and kin selection models (e.g. Gardner et al., 2007; Lehmann et al., 2007; West et al., 2007). Hence, the objective of this section was not to introduce new concepts, but to express a well-known existing result in a manner revealing the relatedness and levels-of-selection components of response. The second section builds on this foundation, but presents a more general approach including IGEs on trait values. This section clarifies the joint effects of IGEs, relatedness and levels of selection on response to selection.

Kin and multilevel selection without indirect genetic effects

Queller (1992a) studied the evolution of a social trait affecting fitness of the carrier of the trait and fitness of its partner(s). As in standard quantitative genetic models, the trait value of focal individual i was the sum of an additive genetic component, G, and a nonheritable component, E:

image(1)

In quantitative genetics, G is usually referred to as the ‘breeding value’ (Falconer & Mackay, 1996; Lynch & Walsh, 1998). Table 1 summarizes the notation used.

Table 1.   Notation key.
SymbolMeaning
i,j,nfocal individual, group member of focal individual, group size
Pitrait value of individual i
Giheritable component of trait value, ‘breeding value’ (absence of IGEs)
Einonheritable component of trait value (absence of IGEs)
Wipersonal fitness of individual i
inline imagemean trait value of the group, inline image
ΔPitrait value of i as deviation from group mean, inline image
βW D ,Pdirect selection gradient, regression coefficient of Wi on Pi
βW S ,Psocial selection gradient, regression coefficient of Wi on Pj
inline imagebetween-group selection gradient, regression coefficient of Wi on inline image
βWPwithin-group selection gradient, regression coefficient of Wi on ΔPi
inline imageresponse of mean trait value to genetic selection (absence of IGEs)
rrelatedness, correlation between genes in interacting individuals
gdegree of multilevel selection, g = βWS,P/βWD,P
AD,iheritable effect of i on its own trait value (direct genetic effect)
AS,iheritable effect of i on trait values of group members (indirect genetic effect)
WD,ieffect of genes in i on fitness of i
WS,ieffect of genes in i on individual fitness of group members of i
TBVitotal breeding value of i, TBVi = AD,i + (n−1)AS,i
inline imagevariance of TBVs among individuals = heritable variance in trait value
inline imageresponse of mean trait value to genetic selection (presence of IGEs)

Modelling approaches to phenotypic selection

Using a neighbour-modulated approach, personal fitness of focal individual i interacting with partner j is (combining eqns 10, 11 and 13 in Queller, 1992a; see also eqn 1 in Wolf et al., 1999)

image(2)

in which Pi is the trait value of focal individual i, Pj the trait value of its partner j, βWD,P the partial regression coefficient of Wi on Pi, βWS,P the partial regression coefficient of Wi on Pj and ei is a residual. Thus, the direct selection gradient βWD,P represents the direct effect of the trait value of an individual on its own fitness, and the social selection gradient βWS,P represents the social effect of the partner's trait value on the focal individual's fitness. In the special case that the trait considered is an altruistic behaviour, βWD,P is a measure of cost to the actor and βWS,P is a measure of benefit to the recipient. Although Queller (1992a) refers to eqn 2 as an inclusive fitness model, we prefer the term neighbour-modulated fitness model, because neighbour j modulates the personal fitness of focal individual i. Because focal i and partner j may be related genetically, eqn 2 is a neighbour-modulated model of a kin selection process. [Our notation differs from Queller's (1992a) notation for reasons of consistency with the second section of this work, which deals with IGEs. The relationship between both notations is: constant = αWx + αWy, βWD,P = βWx,Px, βWS,P = βWy,Py and ei = ɛWx + ɛWy.]

To estimate the social selection gradient, βWS,P, we would require multiple groups to determine the average effect of a social partner with trait value Pj on the fitness of the focal individual. So, although eqn 2 expresses individual fitness, it requires multiple groups to estimate one of its component parameters. [If there were only one group, then the term, βWS,PPj, would be indistinguishable from βWD,PPi. That is, as Rice (2004) pointed out, kin selection does not work without multiple groups of kin.] Moreover, note that the experimental design for estimating βWS,P does not require housing focal individuals with relatives. In fact, to reduce risk of confounding with other factors, is probably preferable to use groups composed of nonrelatives.

Queller (1992a) modelled the same process using a group selection model. In the group selection model, personal fitness of focal individual i in a group composed of individuals i and j was the sum of a component due to the mean trait value of the group, inline image, and a component due to the deviation of the trait value of i from the mean trait value of its group, inline image (Queller, 1992a, combining their eqns 20, 21 and 23),

image(3)

in which the between-group selection gradient inline image is the partial regression coefficient of focal individual fitness Wi on the group mean trait value, inline image, and the within-group selection gradient βWP is the partial regression coefficient of Wi on ΔPi. Thus, inline image represents the effect of the group on fitness of an individual, and βWPΔPi represents the within-group effect of the trait value of an individual on its fitness (rather than the direct effect across all groups as in eqn 2). Equation 3 is a levels-of-selection model, because it partitions individual fitness into a between-group component, inline image, that is common to all group members, and a within-group component βWPΔPi that is specific to each individual. [The relationship with Queller's (1992a) notation is: constant = αWg + αΔW and ei = ɛWg + ɛΔP.]

Although Queller (1992a) showed that both modelling approaches can be used to derive individual fitnesses and response to selection, he neither investigated the relationship between response to selection derived from both approaches nor proved that results of both approaches are equivalent. The formal relationship between the models follows from substituting inline image and inline image into eqn 3, which yields

image

Combining this result with eqn 2 shows that

image( (4a))
image( (4b))

which is equivalent to

image( (4c))
image( (4d))

Equations 4a–4d show that the multilevel model can be translated into the neighbour-modulated model, and vice versa, indicating that both models are indeed equivalent, as expected.

As βWD,P corresponds to cost, and βWS,P corresponds to benefit, eqns 4c and 4d represent the relationship between cost, benefit and multilevel selection, i.e. inline image and βWP = cb. (Note that real cost correspond to a negative value of c in those expressions.) Thus, the sum of cost and benefit corresponds to the strength of selection between groups, and the difference between cost and benefit corresponds to the strength of selection within groups. For groups of more than two individuals, this result generalizes to

image( (5a))
image( (5b))

as was found also by Wade (1980) using a one locus population genetic model, partitioning kin selection into within- and among-family components for families of n interacting full-sibs.

Genetic response to phenotypic selection

The equivalence of the fitness expressions in the kin selection models and levels-of-selection models does not imply that a single biological factor underlies social evolution. In the following, we show that the genetic response derived from either model is naturally expressed in terms of both relatedness and the degree of multilevel selection.

Using the group selection model, genetic response to phenotypic selection follows from substituting eqn 3 into inline image, giving (Price, 1970; eqn 25 of Queller, 1992a),

image

Substituting inline image into the first term gives

image

Next, substituting Pi = Gi + Ei, and using the standard quantitative genetic results that cov(G,E) = 0 and cov(Gi,Gj) = r var(G) (Falconer & Mackay, 1996; Lynch & Walsh, 1998), gives

image

so that

image

In this expression, r denotes genetic relatedness between i and j, which is the correlation between Gi and Gj. Thus, the first term of response to selection equals

image

Next, substituting inline image into the second term of response to selection gives

image

so that

image

Thus, the second term of response to selection equals

image

Total response is the sum of both components,

image(6)

This result shows that response to selection is fully determined by the combination of the multilevel selection gradients, inline image and βWP, and relatedness. The var(G) is merely a scaling factor, indicating that greater heritable variation yields greater response. The inline image in eqn 6 shows that relatedness increases the response to between-group selection, whereas the βWP(1−r) shows that relatedness decreases the response to within-group selection. Furthermore, we note that (1−r)var(G)/2 is the genetic variation within groups of pairs of individuals with genetic relationship r. Thus, as long as βWP is the same within all groups, selection within groups contributes βWP(1−r)var(G)/2 to the total genetic response to selection. Analogously, (1 + r)var(G)/2 is the genetic variation among groups of pairs of genetically related individuals, i.e. the variance of the mean trait values of groups, and inline image is the contribution of selection between groups to the total genetic response.

Alternatively, response to selection may be derived from the neighbour-modulated approach. Substituting eqn 2, inline image, so that

image

This result is equivalent to eqn 6, which follows from substituting eqns 4a and 4b,

image

and

image

yielding

image

which is eqn 6. The above, therefore, confirms that both modelling approaches are equivalent in terms of genetic response to selection. It also shows explicitly that, for both approaches to selection, there exist two distinct factors affecting genetic response to selection, namely the degree of multilevel selection, inline image vs. βWP, and relatedness within groups, r, which determines the relative apportionment of total genetic variation within and among groups.

In the following, we use eqn 6 to investigate the impact of multilevel selection and relatedness on response to selection. First, consider multilevel selection in the absence of relatedness. Substituting r = 0 into eqn 6 gives

image

Next, substituting inline image gives

image

This result may be recognized as the product of the phenotypic selection gradient and the genetic variance, which is a common way of expressing response to selection in the absence of social interactions (Lande, 1979). Substituting βWD,P = cov(Wi,Pi)/var(P) gives

image

where h2 denotes heritability (Falconer & Mackay, 1996). The covariance between an individual's fitness and its trait value equals the phenotypic selection differential in trait value, cov(Wi,Pi) = S (Robertson, 1966; Price, 1970). Thus, when r = 0, response to selection equals the product of heritability and the selection differential,

image

which is the well-known breeder's equation (e.g. Lynch & Walsh, 1998). Therefore, in the absence of kin, eqn 6 reduces to the breeder's equation, which expresses response in the absence of social interactions. In the absence of kin, therefore, multilevel selection does not alter response to selection.

Second, consider relatedness in the absence of multilevel selection. In the absence of multilevel selection, an individual's fitness is determined entirely by its own trait value, meaning that βWS,P = 0 (eqn 2). Substituting βWS,P = 0 into eqns 4c and 4d gives inline image and βWP = βWD,P. Response follows from substitution into eqn 6, giving

image

which as shown above, corresponds to the breeders equation

image

Therefore, in the absence of multilevel selection, response is independent of relatedness.

In conclusion, both modelling approaches show that social evolution differs from the breeders equation only when relatedness and multilevel selection occur simultaneously (the result is different when trait values are affected by IGEs, as we show in the next section). Thus, there are two distinct biological factors necessary for both the kin selection and the multilevel selection modelling approaches: (i) relatedness, being the correlation between genes in interacting individuals, r; and (ii) the multilevel selection process. In levels-of-selection models, relatedness is hidden in the between- and within-group heritabilities, whereas, in kin selection models, the multilevel selection process is hidden in the costs and benefits. Expressions such as eqn 6 integrate kin selection and levels-of-selection models, which reveals the need for both relatedness and multiple levels of selection. Such integration may be needed to bury the debate on equivalence of both modelling approaches.

The joint effect of relatedness, multilevel selection and indirect genetic effects

In the previous section, we used the trait model, Pi = Gi + Ei (eqn 1). This model ignores IGEs on trait value. Most kin and group selection models consider effects of social interactions only on fitness, not on trait values. This is probably because interest is often in the fitness effects of behavioural traits. The behaviour is usually assumed to affect fitness, but the behaviour itself is not directly affected by social interactions. (Reciprocal behaviours may represent an exception). By contrast, IGE models focus on the effect of social interactions on trait values of individuals, which may subsequently have direct and/or social effects on individual fitness (Willham, 1963; Griffing, 1967; Moore et al., 1997; Wolf et al., 1998; Rauter & Moore, 2002; Muir, 2005; Bijma et al., 2007a,b; Wolf et al., 1999).

Relevance of IGEs

The quantitative genetic model for maternally affected traits, which is widely used in agriculture, is a well-known example of an IGE model. In that model, the trait value of the offspring is the sum of a direct effect rooted in the offspring itself, and a maternal effect rooted in its mother. Both direct and maternal effects may have a heritable component and contribute to response to selection (Dickerson, 1947; Willham, 1963; Falconer, 1965; Cheverud, 1984, 2003; Kirkpatrick & Lande, 1989; Cheverud & Moore, 1994; Lynch & Walsh, 1998; Rauter & Moore, 2002; Bijma, 2006).

However, IGEs are not restricted to maternal effects. In many cases, social interactions will affect both trait values and fitness of individuals. The effects of social interactions on individual fitness will usually work via effects on individual traits that are important components of fitness, such as survival, fertility and fecundity, but probably also body weight or size. Blood sharing behaviour in vampire bats (Wilkinson, 1984), for example, affects not only fitness of individuals, but probably also their body weight. Pack-hunting behaviour in mammals (e.g. Creel & Creel, 1995) and fish will affect body weight and size of individuals. In plants, competition for sunlight and soil nutrients may affect fitness of individuals, but will also affect plant size and crop yield in agricultural varieties (Denison et al., 2003). A further example may be reciprocal behaviours, which depend on behaviours of conspecifics. Thus, for many traits, individual trait values will be affected by interactions among individuals.

Indirect genetic effects have substantial impact on both the magnitude and the direction of response to selection, particularly in the presence of multilevel selection (e.g. Griffing, 1967, 1976). Recent theoretical and empirical work shows that IGEs exist, and that they may increase heritable variance in trait value to levels threefold greater than the usual (direct) additive genetic variance (Bijma et al., 2007b; Bergsma et al., 2008; Ellen et al., 2008). Consequently, the presence of IGEs may allow populations to evolve much faster than suggested by classical theory. Therefore, in addition to the multilevel selection process and relatedness, models for prediction of response to selection should account for IGEs. Bijma et al. (2007a) present such a model, but leave unclear its relationship with common models of kin and multilevel selection. Moreover, the results presented in that work hide a striking symmetry in the effects of relatedness and multilevel selection, as we show below.

In the following, we briefly summarize the approach of Bijma et al. (2007a), and subsequently use that approach to clarify the joint consequences of IGEs, multilevel selection and relatedness on the evolution of trait values by natural selection.

Trait model

This section describes the IGE model for a quantitative trait. Consider a population split into many groups of n individuals each. Within groups, social interactions among individuals affect trait values. The observed trait value, Pi, of focal individual i is the sum of two unobserved components: a direct effect Di originating from the focal individual itself and the sum of social effects Sj of each of its n−1 group members j (Griffing, 1967; Wolf et al., 1998; Bijma et al., 2007a),

image(7)

This model applies to all individuals in the population. Each individual may affect both its own trait value and the trait values of its group members; there is no a priori categorization into actors and recipients. Thus, each individual carries two unobserved effects: a direct effect affecting its own trait value and a social effect affecting the trait values of each of its group members. Note that an individual's direct effect covers the full effect of the individual on its own trait value, irrespective of the cause of that effect. Direct and social effects may be partitioned into an additive genetic component (A), and a remaining nonheritable component, referred to as environment (E). Thus, trait values are given by (Griffing, 1967; Wolf et al., 1998; Bijma et al., 2007a)

image(8)

in which AD,i is additive genetic merit of individual i for direct effect, and AS,j is additive genetic merit of group member j for social effect. We will refer to AD as the direct genetic effect, and to AS as the IGE. An individual's IGE represents the mean heritable effect of that individual on the trait values of each of its group members. In the absence of IGEs, eqn 8 reduces to eqn 1.

Multilevel selection

We use eqn 2 to model the selection process. Generalizing eqn 2 to interactions among any number of individuals, n, yields (see also eqn 3 in Wolf et al., 1999)

image

in which βWD,P is the partial regression coefficient of fitness of an individual on its own trait value, βWS,P is the partial regression coefficient of fitness of an individual on the trait value of a group member, and the inline image is taken over the n−1 group members of i. Thus, βWD,PPi represents the direct effect of the trait value of i on fitness of i, and βWS,PPj represents the social effect of the trait value of j on fitness of i. Because any regression expression can be formulated either including or excluding an intercept, the constant can be dropped without altering the expression for Wi. Moreover, the expression may be reformulated into a term expressing the strength of direct selection, βWD,P, and a term expressing the relative strength of social selection vs. direct selection,

image(9)

giving

image(10)

Although eqn 10 is the result of a neighbour-modulated approach, it also has a group selection interpretation. In particular, the g is a measure of the degree of group selection relative to individual selection. A g = 0 corresponds to selection on individual trait value irrespective of the group, Wi = βWD,PPi + ei, and therefore represents the absence of multilevel selection. A g = 1 corresponds to selection on the sum of trait values of the entire group,

image

so that all individuals within a group have identical fitness (note that, in the second summation, the sum is taken over all n individuals in the group, rather than over the n−1 group members of i). In this case, selection must be entirely among groups, as there is no fitness variation among individuals within groups: the inline image has the same value for all n group members. Thus, g = 1 represents full between-group selection. A g = −1/(n−1) corresponds to selection of individuals on the deviation of their trait value from the mean trait value of their group:

image

where the sum is taken over all n group members. This expression is proportional to

image

so that selection occurs entirely within the group. In conclusion, therefore, g represents the degree of multilevel selection; g = 0 represents the absence of multilevel selection; g > 0 represents selection occurring at a level higher than the individual, i.e. between groups; g < 0 represents selection occurring at a level lower than the individual (i.e. within groups, also called ‘soft selection’; Wade, 1985). Note that g is undefined when there is social selection but no direct selection (βWD,P = 0). In that case, g = βWS,P/βWD,P yields a division by zero, indicating that the degree of between-group selection when compared with individual selection is infinitely large. For that case, one cannot use eqn 10, but

image

can still be used.

Consequences of IGEs and multilevel selection for individual fitness

Equation 10 specifies the action of multilevel selection on trait value. It describes how the trait value of an individual and the trait values of its group members combine to affect the fitness of the individual. With IGEs, however, not only fitness, but also the trait value of an individual depends on genes in its group members. For example, as shown in eqn 8, Pi in eqn 10 contains a component due to genes in group member j, AS,j. Similarly, Pj in eqn 10 has a component due to i, AS,i. In other words, the IGE of i affects the trait values of its group members (eqn 8), and subsequently the trait values of group members affect fitness of i due to multilevel selection (eqn 10). Graphically represented, inline image. Thus, the combination of IGEs and multilevel selection on trait value creates feedback of an individual's IGE on its own fitness, causing an individual's fitness to depend on its own IGE (see also Wolf et al., 1999).

Thus, the combination of IGEs with multilevel selection for trait value has consequences for the origin of the heritable components affecting individual fitness, which can be clarified by translating eqn 10 into a neighbour-modulated model for fitness. A neighbour-modulated fitness model is given by

image(11)

in which WD,i summarizes all components of Wi that originate from genes in focal individual i itself, WS,j summarizes all components of focal individual fitness Wi that originate from genes in group member j, irrespective of the paths via which those effects occur and ɛi summarizes all nonheritable components of Wi. Thus, eqn 11 traces fitness components back to the individuals from whose genes they originate.

To identify the origin of direct and social genetic effects underlying individual fitness, WD,i and WS,j, we combined eqns 8–11 (see the appendix for derivation). The result allows us to disentangle the fitness effects originating from IGEs from those originating from multilevel selection. Table 2 presents direct and social fitness effects for either the presence or absence of IGEs (inline image or inline image), and for either the presence or absence of multilevel selection (g = 0 or g ≠ 0). In the absence of IGEs, social effects on fitness occur only when there is multilevel selection (g ≠ 0). In the presence of IGEs, by contrast, there are always social effects on fitness, irrespective of whether or not there is multilevel selection on trait value (except when there is no selection at all, βWD,P = 0). This result shows that, with IGEs affecting trait values, the fitness of an individual depends on genes in other individuals (WS ≠ 0), even when the trait is not subject to multilevel selection (g = 0). In other words, even when an individual's fitness is determined entirely by its own trait value, it may still depend on genes in others because those genes may affect the individual's phenotypic trait value (inline image). This is important because it alters the response to selection, not only in magnitude, but potentially also in direction (see below). Thus, in addition to relatedness and multilevel selection, IGEs are a key element in social evolution. Note that results in Table 2 do not depend on relatedness, illustrating that relatedness in itself is not necessary for IGEs to affect selection.

Table 2.   Relationship between IGEs, multilevel selection, and direct and social fitness components*.
Level of selectionWD,iWS,i
  1. *Nonheritable terms are omitted from the table; see eqn 11 and below for interpretation of WD,i and WS,i.

  2. inline image corresponds to Pi = AD,i + ED,i.

  3. g = 0 corresponds to Wi = βWD,PPi + ei.

  4. §g > 0 corresponds to inline image.

  5. inline image corresponds to inline image.

 Absence of IGEs (inline image)†
g = 0‡βW D,PAD,i0
g ≠ 0§βW D,PAD,iβW D,PgAD,i
 Presence of IGEs (inline image
g = 0‡βW D,PAD,iβW D,PAS,i
g ≠ 0§βW D,P[AD,i + g(n−1)AS,i]βW D,P{AS,i + g [AD,i + (n−2)AS,i]}

Genetic response to multilevel selection with IGEs and relatedness

In this section, we consider the joint effect of multilevel selection, IGEs and relatedness on response to selection, working along the lines of Bijma et al. (2007a). Results will reveal a striking symmetry in the effects of multilevel selection and relatedness. Moreover, results will show that, with IGEs, multilevel selection in the absence of relatedness may reverse the direction of response to selection. Thus, when trait values are affected by IGEs, multilevel selection without kin selection may turn maladaption into adaptation.

The direct and indirect genetic effects represent the heritable components underlying phenotypic trait values (eqn 8). Taking the mean trait value over all individuals at two distinct points in time shows that response to genetic selection equals (see also Bijma et al., 2007a)

image(12)

Eqn 12 hints at a measure for the total heritable impact of an individual on the mean trait value of the population, which we refer to as the total breeding value (TBV). The total heritable impact of an individual on the population mean trait value is the sum of the individual's direct genetic effect and n−1 times the individual's IGE. Thus, for each individual, a TBV may be defined (Bijma et al., 2007a),

image(13)

The TBV should not be confused with the classical breeding value. The classical breeding value is the sum of average effects of genes in an individual on the trait value of that individual. It represents an individual's expected trait value given its genes (Fisher, 1930, 1941). The TBV, by contrast, represents the heritable impact of an individual on the mean trait value of the population, and therefore properly summarizes the ‘value of the individual for breeding’ in case trait values are affected by IGEs. It contains the component (n−1)AS,i, which is expressed not in the trait value of the individual itself, but in the trait values of its group members (Bijma et al., 2007a,b).

It follows from eqns 12 and 13 that response to selection equals the change in mean TBV of individuals,

image

Moreover, the total heritable variance in trait value is the variance of TBVs among individuals (Bijma et al., 2007a),

image(14)

Equation 14 may be interpreted as a generalization of the usual additive genetic variance in trait value, to account for IGEs. With IGEs, the inline image represents the heritable variance that is available for generating response to selection. Equation 14 shows that the presence of IGEs may substantially increase total heritable variance. This expectation has been confirmed by empirical work in domestic chicken and swine, which yielded values of total heritable variance two- to fourfold greater than conventional additive genetic variance (Bijma et al., 2007b; Bergsma et al., 2008; Ellen et al., 2008). Remarkably, when trait values are affected by IGEs, heritable variance in trait value may exceed the observed variance among trait values of individuals, indicating that ‘heritability’ can be greater than one (Bijma et al., 2007a; Bergsma et al., 2008). Thus, the presence of IGEs may allow populations to evolve much faster by means of natural selection.

Response to selection may be derived using Price's theorem (Price, 1970). Bijma et al. (2007a) presented an expression for response, but their formulation obscured a striking symmetry of the effects of relatedness and multilevel selection. The Appendix shows that eqns 5 and 7 of Bijma et al. (2007a) can be combined into

image(15)

Equation 15 is our most important result. It specifies the joint effect of IGEs, multilevel selection and relatedness on response to selection. Note that response to selection is entirely symmetric in the degree of multilevel selection, g, and relatedness, r; i.e. exchanging g and r in eqn 15 yields the same expression. The first term, inline image, represents the contribution of multilevel selection and relatedness to the response of mean phenotypic trait value to selection. It equals zero when g = r = 0. This term shows that multilevel selection and relatedness act directly on the total heritable variance in trait value. Because total heritable variance is non-negative, multilevel selection and relatedness always yield a response in the same direction as selection. Thus, multilevel selection and relatedness will lead to adaptation. The term (n−2)gr indicates that the effects of multilevel selection and relatedness amplify each other when the size of the groups is larger than two individuals. The (n−2) represents the number of individuals that two group members have in common (see eqns 17b of the Appendix for more detail).

The second term in eqn 15,

image

represents the complement of multilevel selection, (1−g), and relatedness, (1−r). It equals zero when selection is entirely between groups of fully related (e.g. clonal) individuals, in which case g = r = 1. The term

image

represents the response to individual selection among unrelated individuals (g = r = 0), and corresponds to the result of Griffing (1967). This term can be negative when direct and indirect genetic effects have a negative covariance (σADS < 0). A negative σADS indicates that individuals with positive effects on their own trait value have, on average, a negative effect on trait values of their group members, and vice versa. A negative value of σADS is likely, for example, when individuals compete for finite resources, and has been documented in the acorn ant, Leptothorax curvispinosa (Linksvayer, 2007), and in laying hens (Ellen et al., 2008). When the covariance between direct and indirect genetic effects is negative, individual selection among unrelated individuals will increase competition (inline image) and may yield a negative net response (Griffing, 1967). Therefore, when social effects affect trait values, natural selection can yield maladaptation (see also Griffing, 1967). This often occurs in plants with selection for increased individual yield – the individuals with the highest yield achieve this at the expense of the yield of their neighbours (discussed by Denison et al., 2003). It was this negative effect on selection response that motivated much of Griffing's work on ‘associative effects’ (IGEs). Plant breeders often prevent an increase in competition by selecting among plots of pure lines (i.e. clones). With selection among plots of pure lines, g = r = 1, so that response equals inline image, which has the same sign as the selection gradient. Thus, selection among plots of pure lines always yields a response in the same direction as selection.

Considering the first and second terms of eqn 15 together shows that relatedness and multilevel selection are mechanisms shifting the response to selection from a potentially negative value towards a positive value, i.e. from inline image towards inline image. Thus, relatedness and multilevel selection are mechanisms to prevent maladaptation of traits affected by competitive interactions (σADS < 0), as suggested by Wright (1969, p. 478–479).

To disentangle the effects of IGEs, relatedness and multilevel selection on the response to selection, Table 3 shows expressions for response when each factors is either present or absent. In the absence of IGEs, response to selection differs from the breeder's equation only when both g and r are nonzero, i.e. when gr ≠ 0. Thus, in the absence of IGEs, response to selection differs from classical theory only when both relatedness and multilevel selection are present.

Table 3.   Response to selection (inline image).
Level of selection*Unrelated group members (r = 0)Related group members (r ≠ 0)
  1. *A g = 0 corresponds to selection on individual trait value; g ≠ 0 corresponds to multilevel selection (eqn 10 and below).

  2. †This result corresponds to the breeders equation, inline image.

 Absence of IGEs (inline image)
g = 0inline imageinline image
g ≠ 0inline imageinline image
 Presence of IGEs (inline image)
g = 0inline imageinline image
g ≠ 0inline imageinline image

In the presence of IGEs (inline image), response to selection always differs from the breeders equation (Table 3). First consider the situation when both g and r are zero. In that case, there is no selection acting directly at the total heritable variance in trait value:

image

As a consequence, response can be opposite to the direction of selection, as shown by Griffing (1967). Griffing discussed this phenomenon in the context of ‘the risk of negative response to positive selection’. However, it can also be interpreted as ‘the opportunity of positive response to negative selection’. The latter is equivalent to the evolution of altruism, i.e. the evolution of a trait having fitness cost for the individual expressing the trait. For an altruistic trait, the direct selection gradient is a negative value by definition, βWD,P < 0. As response to selection equals inline image, positive response with βWD,P < 0 requires that inline image. This condition is equivalent to

image(16)

in which rDS is the (additive) genetic correlation between direct and indirect effects, rDS = σADS/(σADσAS). Condition 16 requires that direct and indirect genetic effects have a negative correlation, rDS < 0. If Condition 16 is satisfied and the trait is altruistic (βWD,P < 0), then the response in direct effect is negative, inline image, the response in indirect effect is positive, inline imageinline image, and the contribution of indirect effects to selection response exceeds the absolute contribution of direct effects to selection response, inline image. As a result, the response of trait value to selection is positive, inline image. In other words, the positively correlated response in indirect effects exceeds the negative response in direct effects, yielding a positive net response. The above demonstrates that IGEs create the opportunity for evolution of altruism, without the need for multilevel selection or relatedness.

Second, consider the situation when either g or r is nonzero. In contrast to the situation without IGEs, multilevel selection affects response also in the absence of relatedness (g ≠ 0 while r = 0), and relatedness affects response also in the absence of multilevel selection (r ≠ 0 while g = 0). Consider, for example, response in the absence of relatedness, which equals

image

In this expression, the term inline image can take negative values, whereas inline image is always a positive value. Multilevel selection, i.e. g > 0, has the effect to shift response towards the value inline image. As inline image is a positive value, the inline image always has the same sign as βWD,P, indicating that response is in the same direction as selection. Thus, when trait values are affected by IGEs, multilevel selection in the absence of relatedness can reverse the direction of response to selection. Because eqn 15 is symmetric in g and r, the same reasoning holds for relatedness in the absence of multilevel selection.

For applied evolutionary biologists, it may be helpful to express response to selection in terms of components that biologists tend to measure. Table 4, therefore, expresses response in terms of the underlying variance components, inline image, σADS and inline image, the selection gradients, βWD,P and βWS,P, relatedness and group size. Hence, results in Table 4 are equivalent to those in Table 3, but merely formulated differently.

Table 4.   Response to selection in the presence of IGEs (inline image), expressed as a function of underlying components.
Level of selection* Unrelated group members (r = 0) Related group members (r ≠ 0)
  1. *A g = 0 corresponds to selection on individual trait value; g ≠ 0 corresponds to multilevel selection (eqn 10 and below).

g = 0inline imageinline image
g ≠ 0inline imageinline image

Discussion

We have shown that both relatedness and the degree of multilevel selection are key factors in response to selection. In the absence of IGEs on trait values, response depends only on the product of relatedness and the degree of multilevel selection, emphasizing the need for both relatedness and multilevel selection, rather than either relatedness or multilevel selection. The result is fundamentally different, however, with IGEs on trait values. In that case, multilevel selection in the absence of relatedness can reverse the direction of response to selection, and vice versa (Table 3).

The central parameter in kin selection models is the relatedness among interacting individuals (Hamilton, 1964). Until now, no such parameter has been put forward for the multilevel selection process. Our results, in particular eqn 15, indicate that the degree of multilevel selection, g, is the central parameter describing the multilevel selection process. The g measures the degree of multilevel selection as the ratio of the selection gradient on trait values of group members over the selection gradient on individual trait value, g=βWS,P/βWD,P. There are striking parallels between g and relatedness. In particular, our expression for response to selection is symmetric with respect to relatedness and the degree of multilevel selection, indicating that both factors have exactly the same effect on response (eqn 15). Furthermore, g and r have the same natural bounds. A g = −1/(n−1) corresponds to selection of individuals based on the deviation of their trait value from the mean of the group, resulting in selection entirely within groups (full soft selection, see above). The expression −1/(n−1), however, also represents the lowest possible value of relatedness. A lower value would imply a negative variance of the genetic mean of the group, which is impossible. Moreover, g = 0 indicates the absence of multilevel selection, similar to r = 0 which indicates the absence of relatedness. Finally, g = 1 indicates full between-group selection, similar to r = 1 which indicates full relatedness. Hence, in genetic models describing social evolution, g is the natural partner of r.

Our main result, eqn 15, follows directly from eqn 5 of Bijma et al. (2007a). Equation 5 of Bijma et al. (2007a) generalizes by analogy to more than two levels of selection (eqn 8 of Bijma et al., 2007a). We therefore expect that our eqn 15 also generalizes to more than two levels of selection. A detailed treatment of more than two levels of selection, however, is outside the scope of the present article.

Recently, there has been debate about the usefulness of group selection (West et al., 2007; Wilson, 2007; West et al., 2008; Wilson & Wilson, 2008). In particular, West et al. (2008) state, ‘No group selection model has ever been constructed where the same result cannot be found with kin selection theory’. However, Griffing (1967) showed that, when traits values are affected by IGEs, selection among groups composed of unrelated individuals prevents the negative response to selection that can occur with selection among individuals. Griffing's result is a special case of our eqn 15 for r = 0 and g = 1. Note that there is no relatedness hidden in other model terms; r expresses the correlation between genes in interacting individuals, measured at the population level. Thus, r =0 truly means the absence of relatedness [our argument emphasizes that there is a distinction between kin selection (r > 0) and neighbour modulating of fitness in the absence of kin].

Lehmann et al. (2007) referred to group and kin selection as ‘two concepts but one process’. Our results, however, show that, in the absence of IGEs, there exists only a single concept, namely the combination of relatedness and multilevel selection as measured by the product gr (Table 3). In the presence of IGEs, however, there are both two concepts and two processes, as relatedness and multilevel selection can reverse the direction of response in isolation.

The focus on either relatedness or multilevel selection, instead of their combination, has led to a surplus of parameters in expressions for response to selection. Results of Queller (1992a), for example, consist of four parameters for each modelling approach, giving a total of eight parameters. Equation 15 show that, when relatedness and the degree of multilevel selection are combined into a single expression, response from both approaches is described by the same set of four parameters. Those parameters are: (i) heritable variance in trait value, inline image, which reduces to var(G) in the absence of IGEs; (ii) the overall strength of selection, βWD,P; (iii) relatedness among interacting individuals, r; and (iv) the degree of multilevel selection, g.

Moreover, the exclusive use of either relatedness or multilevel selection has lead to the proposition of new biological mechanism for the evolution of cooperation. For example, Griffin et al. (2004) distinguished between kin selection and the scale of competition. They studied the evolution of siderophore production in Pseudomonas aeruginosa in a 2 × 2 selection experiment, combining either high or low relatedness with either global or local competition. Siderophore production is an altruistic cooperative trait that is costly for the individual, but provides a benefit to the local group. Global competition was imposed by mixing cultures from all subpopulations before plating, allowing productivity of a subpopulation (i.e. group) to determine the contribution of a genotype to subsequent generations. Local competition was imposed by allowing every subpopulation to contribute an equal number of colonies to the next generation, so that the contribution of a genotype did not depend on productivity of the subpopulation. Thus, global competition corresponds to a certain degree of between-group selection, inline image, whereas local competition corresponds to pure within-group selection, inline image. The scale of competition, therefore, is not a novel mechanism, but merely represents the multilevel selection required in any kin selection process in the absence of IGEs.

Remarkably, in a commentary on the work of Griffin et al. (2004), Queller (2004) interpreted the experiment as a combination of group selection and the scale of competition. He argued that low relatedness corresponds to the presence of within-group selection, and high relatedness to the absence of within-group selection. However, relatedness in itself does not imply selection (e.g. Table 2). Instead, relatedness alters the outcome of multilevel selection processes. The natural way to interpret the experiment of Griffin et al. (2004), therefore, is as a combination of two levels of relatedness and two degrees of multilevel selection, yielding four different kin selection processes. The outcomes of those four processes are in agreement with predictions from eqns 6 and 15.

Appendix

Derivation of results in Table 2

Substituting eqn 8 into eqn 10, considering genetic terms only, yields

image

Grouping terms according to the individuals from which they originate yields

image

The first term represents the effect of genes in i on fitness of i; the second term represents the effect of genes in the group members of i on fitness of i. Equating this result to eqn 11 shows that

image( (17a))
image( (17b))

Equation 17a summarizes all heritable effects of genes in i on fitness of i. Equation 17b summarizes all heritable effects of genes in i on fitness of each of its group members. In eqn 17a, AD,i is the effect of i on its own fitness via the genes affecting its own trait value. The term g(n−1)AS,i represents the effect of i on its own fitness due to feedback of its IGE via the trait values of its n−1 group members. In eqn 17b, the AS,i represents the effect of i on fitness of its group members via the social effect of i on the trait values of those group members; the gAD,i represents the effect of the trait value of i on fitness of its group members, and the g(n−2)AS,i represents the effect of i on fitness of its group members occurring via the social effect of i on the trait values of the (n−2) group members that i and each of its group members have in common.

Derivation of eqn 15

The derivation of eqn 15 is the easiest starting with eqn 5 of Bijma et al. (2007a),

image

In this expression, ι/σC represents the selection gradient on individual trait value, ι/σC = βWD,P, and σP,TBV is the covariance between trait values and TBVs of individuals,

image

(eqn 7 of Bijma et al., 2007a). Using eqn 14,

image

the σP,TBV can be expressed as

image

Substituting this result into the expression for response yields

image

Collecting term in inline image and remaining terms yields

image

which is eqn 15.

Ancillary