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The relatedness parameter (r) in Hamilton's rule was introduced in 1922 by Sewall Wright as a coefficient of relationship that gives the probability that at a random locus, the alleles there will be identical by descent. Subsequent authors, including Hamilton, sometimes reformulate this with a regression, which, unlike probabilities, can be negative.
Here is my question:
- What are all the different definitions of relatedness in biology?
- how do we calculate them?
- what do they mean?
Is that really trying to imply that there are significantly different definitions? It seems to me the intent was just to put it in terms of an equation with more useful numbers. This terrifying paper from 1964 gives some of the follow-up works, in particular works from Cockerham in 1954 and a number of papers by Kempthorne, in particular this from 1955 and this from 1963. There is also Malecot's method of coancestry from 1948. (WorldCat.org link) The idea was mainly to look relatedness outside of the specific limitations present in Wright's work. Some of the math in those are more accessible than others…
Kin Selection - Hamilton's Rule
r = the genetic relatedness of the recipient to the actor, often defined as the probability that a gene picked randomly from each at the same locus is identical by descent. B = the additional reproductive benefit gained by the recipient of the altruistic act, C = the reproductive cost to the individual of performing the act.
This inequality is known as Hamilton's rule after W. D. Hamilton who published, in 1964, the first formal quantitative treatment of kin selection to deal with the evolution of apparently altruistic acts.
Originally, the definition for relatedness (r) in Hamilton's rule was explicitly given as Sewall Wright's coefficient of relationship: the probability that at a random locus, the alleles there will be identical by descent (Hamilton 1963, American Naturalist, p. 355). Subsequent authors, including Hamilton, sometimes reformulate this with a regression, which, unlike probabilities, can be negative. Regression analysis producing statistically significant negative relationships indicates that two individuals can be less genetically alike than two random ones on average (Hamilton 1970, Nature & Grafen 1985 Oxford Surveys in Evolutionary Biology). This has been invoked to explain the evolution of spiteful behaviours. Spiteful behavior defines an act (or acts) that results in harm, or loss of fitness, to both the actor and the recipient.
In the 1930s J.B.S. Haldane had full grasp of the basic quantities and considerations that play a role in kin selection. He famously said that, "I would lay down my life for two brothers or eight cousins". Kin altruism is the term for altruistic behaviour whose evolution is supposed to have been driven by kin selection.
Haldane's remark alluded to the fact that if an individual loses its life to save two siblings, four nephews, or eight cousins, it is a "fair deal" in evolutionary terms, as siblings are on average 50% identical by descent, nephews 25%, and cousins 12.5% (in a diploid population that is randomly mating and previously outbred). But Haldane also joked that he would truly die only to save more than a single identical twin of his or more than two full siblings.
In 2011, experimentalists found empirically that Hamilton's rule describes very accurately the conditions under which altruism emerged in simulated populations of foraging robots. The accuracy of this first quantitative corroboration of Hamilton's rule is all the more impressive given that Hamilton's model made several simplifications that did not apply to the foraging robots.
Read more about this topic: Kin Selection
Famous quotes containing the words hamilton and/or rule :
&ldquo In politics, as in religion, it is equally absurd to aim at making proselytes by fire and sword. Heresies in either can rarely be cured by persecution. &rdquo
&mdashAlexander Hamilton (1757)
&ldquo The first rule of venture capitalism should be Shoot the Inventor. &rdquo
&mdashRichard, Sir Storey (b. 1937)
Hamilton's rule and the causes of social evolution
Hamilton's rule is a central theorem of inclusive fitness (kin selection) theory and predicts that social behaviour evolves under specific combinations of relatedness, benefit and cost. This review provides evidence for Hamilton's rule by presenting novel syntheses of results from two kinds of study in diverse taxa, including cooperatively breeding birds and mammals and eusocial insects. These are, first, studies that empirically parametrize Hamilton's rule in natural populations and, second, comparative phylogenetic analyses of the genetic, life-history and ecological correlates of sociality. Studies parametrizing Hamilton's rule are not rare and demonstrate quantitatively that (i) altruism (net loss of direct fitness) occurs even when sociality is facultative, (ii) in most cases, altruism is under positive selection via indirect fitness benefits that exceed direct fitness costs and (iii) social behaviour commonly generates indirect benefits by enhancing the productivity or survivorship of kin. Comparative phylogenetic analyses show that cooperative breeding and eusociality are promoted by (i) high relatedness and monogamy and, potentially, by (ii) life-history factors facilitating family structure and high benefits of helping and (iii) ecological factors generating low costs of social behaviour. Overall, the focal studies strongly confirm the predictions of Hamilton's rule regarding conditions for social evolution and their causes.
Hamilton's inclusive fitness theory [1,2], now 50 years old, has had a revolutionary effect on our understanding of evolution following the Modern Synthesis of the mid-twentieth century. Many works, both specialist [3–6] and more general [7–11], have explained the basis and predictions of the theory, also known as kin selection theory. Conceptually, its fundamental contribution has been to identify genes as self-promoting strategists whose evolutionary interests are conditional on the relatedness class in which they reside [1,12–14]. Put more exactly, genes are selected to act as if they are maximizing their inclusive fitness [13–15]. This insight has substantially extended population genetics, the genetical theory of natural selection and the Modern Synthesis because it shows that natural selection on any gene depends on the gene's effects, or lack of effects, on the direct fitness (offspring number) of bearers of copies of itself. Conspecific individuals are not sealed off from one another in terms of fitness, and traditional ‘individual selection’ is, ultimately, gene selection [12,13]. All higher levels of organization, such as genomes, multicellular organisms and societies, arise through major transitions in evolution that are conditional on cooperating genes finding a coincidence of inclusive fitness interests in bringing them about [9,13,16–19].
A simple but powerful formalization of inclusive fitness theory is provided by Hamilton's rule [20,21]. This states that a gene for any social action will undergo selection when the sum of indirect fitness (rb) and direct fitness (c) exceeds zero, where r is the relatedness of the social actor and recipient and c and b are the changes brought about by the social action in the offspring numbers of, respectively, the actor and the recipient. From Hamilton's rule follow the well-known conditions for the four possible types of social action as defined by the signs of c and b, namely cooperation or mutual benefit (+, +), altruism (−, +), selfishness (+, −) and spite (−, −) [6,7,9]. Specifically, in its most celebrated application, Hamilton's rule states that altruism (net loss of direct fitness) is selected if rb–c > 0. By identifying this condition, inclusive fitness theory solved the problem of altruism [7,12]. Because of its grounding in fundamental theory, its incorporation of the four types of social action and its universal taxonomic scope, the theory provides the best current basis for a unified understanding of social evolution [5,15]. For example, it enables conflict within family groups and intragenomic conflict to be understood in the same terms [9,22]. When applied to the major transitions [9,16,17], it provides an explanation of the biological hierarchy itself.
The evidence for inclusive fitness theory is extensive, diverse and growing [8,10,11,23,24]. Nonetheless, explicit empirical tests of Hamilton's rule in natural populations are relatively few. Hamilton's rule predicts that each social action arises only under certain combinations of values of r, b and c [7,9]. Factors bringing about the required values of r, b and c within natural populations create conditions for social evolution. Variation in these values may then cause social evolution in the sense of making the difference (given appropriate genetic variation) between whether or not social behaviour undergoes selection. For example, Hamilton's rule finds that positive relatedness is a necessary condition for the evolution of altruism and that altruism evolves more readily when b is high and c is low. So, for a given relatedness structure, identifying factors affecting the relative values of b and c gives insight into the causes of altruism [21,25]. In this review, I consider results from two approaches to using Hamilton's rule and its predictions to investigate the causes of social evolution. First, I review studies that have empirically tested Hamilton's rule by estimating its parameters using genetic and demographic data from natural populations. Second, I review comparative phylogenetic analyses that have identified predicted genetic, life-history or ecological correlates of social evolution.
A consideration of studies that have tested Hamilton's rule with empirical data is worthwhile because, although several prominent studies have reported such tests, it is well known that, while measuring relatedness using molecular markers is fairly straightforward, estimating b and c in natural populations is far from easy . The impression has therefore arisen that empirical tests of Hamilton's rule are vanishingly scarce and that inclusive fitness theory's successful explanation of altruism relies simply on observing positive relatedness within social groups [26,27]. As will be shown, neither of these points is correct. However, to the best of my knowledge, no previous review has aimed to collate empirical tests of Hamilton's rule and systematically analyse the insights that they provide as regards the causes of social evolution. Crespi  highlighted the potential power of comparative phylogenetic analyses of the correlates of sociality to identify causes of social evolution operating over evolutionary time. But many relevant studies have appeared only recently as molecular phylogenies and advances in statistical methodology have become available and, again, a synthesis of the findings of such analyses has not been carried out. Overall, I seek to consider how tests of Hamilton's rule and comparative phylogenetic analyses of the correlates of sociality advance our knowledge of the causes of social evolution at ecological and evolutionary scales.
2. Empirical tests of Hamilton's rule in natural populations
A survey of the literature for studies estimating the parameters of Hamilton's rule using genetic and demographic data from natural populations reveals 12 studies that either have had this explicit aim or provide data permitting these parameters to be estimated (table 1 and figure 1 electronic supplementary material, table S1). The survey is not exhaustive and excludes some related studies. For example, in focusing on estimates of r, b and c, it excludes studies that test inclusive fitness theory by using empirical data to estimate inclusive fitness in other ways [42–46] or to test models of reproductive skew [47,48], which are derived from Hamilton's rule. In focusing on single species or populations, it excludes studies that test Hamilton's rule using correlations across species between social traits and relatedness, benefits or costs [49–52]. Finally, in focusing on common behaviours in natural populations, it excludes studies of rare behaviours  and recent applications of Hamilton's rule to social behaviour in humans  and robots . Excluding these studies is conservative, in that most of them support the predictions of inclusive fitness theory. The lack of many more studies estimating the parameters of Hamilton's rule in natural populations shows that, indeed, benefits and costs of social actions are difficult to measure in field settings (many of the focal studies involved painstaking fieldwork over multiple years). Nonetheless, such studies are evidently not as scarce as has sometimes been suggested and, though biased towards altruistic brood-rearing behaviour, cover a broad range of other behaviours, including egg dumping, cannibalism and cooperative lekking (table 1 and figure 1 electronic supplementary material, table S1).
Table 1. Studies parametrizing and testing Hamilton's rule with genetic and demographic data from natural populations. See electronic supplementary material, table S1 for an expanded version of the table (giving estimates of the relatedness, benefit and cost terms of Hamilton's rule in each study).
Figure 1. Hamilton's rule has been tested in a wide range contexts and organisms, including egg dumping, joining behaviour, cannibalism and cooperative lekking in, respectively (a–d): (a) Egg-plant lace bug, Gargaphia solani (image credit: copyright 2013 www.Croar.net) (b) Polistine wasp, Polistes dominulus (image credit: Andrew Bourke) (c) Tiger salamander larva, Ambystoma tigrinum (image credit: Kerry Matz) and (d) Wild turkey, Meleagris galloparvo (image credit: Tim Simos/National Wild Turkey Federation).
(a) Assumptions of empirical tests of Hamilton's rule
The studies included in the present survey (table 1 and electronic supplementary material, table S1) make a number of assumptions. First, in reaching their specific conclusions, they assume that the fitness accounting is complete, and that there are not alternative behavioural choices that occur at appreciable frequencies whose benefits and costs could not be estimated. An example of such an alternative is the behaviour within the eusocial Hymenoptera in which a subset of females enter diapause early instead of helping or nesting in the current year . Fitness returns from such behaviours may, indeed, be hard to measure (in this case, because they accrue in the following year), and to this extent the relevant analyses would be incomplete. But this would be true when attempting to apply empirical data from these systems in any sort of model. Second, more generally, applying Hamilton's rule uses the ‘phenotypic gambit’ , in which it is assumed that the exact genetic basis of the focal social behaviour (which is unknown in every case) is not such as to overturn the expectations based on Hamilton's rule. The phenotypic gambit is not an assumption of the field of social evolution alone but of behavioural ecology as a whole , and its justification comes from behavioural ecology's outstanding success as a research programme . Third, applying Hamilton's rule to data generally makes the assumptions that the social action has additive effects on fitness and that selection for the social action is weak [21,55]. When costs and benefits are estimated as offspring numbers averaged over the lifetimes of the actors and recipients, and when traits are close to equilibrium, these assumptions may be justified [3,21]. However, there are cases in which non-additivity affects the selective outcome [56,57]. Nonetheless, overall, the empirical application of Hamilton's rule is justified because it often appears robust to violations of these assumptions  and because it yields a valuable generality at the expense of an exactness that, in natural systems, is almost impossible to achieve [21,58]. Furthermore, applying Hamilton's rule to empirical data is particularly useful because, given it is explicitly based on fitness differences (in the b and c terms), Hamilton's rule compels investigators to analyse the central problem of why individuals exhibit one set of behaviours and not another [20,21].
(b) Conclusions from empirical tests of Hamilton's rule: forms of social behaviour and fulfilment of Hamilton's rule
Because of the interest in addressing the problem of altruism, empirical tests of Hamilton's rule have concentrated on cases where social behaviour aids recipients at what appears, a priori, to be a direct fitness cost to the actor. Of the 12 focal studies (table 1 and electronic supplementary material, table S1), 10 found that actors did indeed incur a direct fitness cost (negative c) and hence exhibited altruism even though social behaviour was facultative (electronic supplementary material, table S1). In two remaining cases (egg dumping in lace bugs and kin-discriminating cannibalism in larval salamanders nos. 1 and 10 in table 1 and electronic supplementary material, table S1), there was a direct fitness return of zero (c = 0). Of the 10 studies in which there was demonstrated altruism, five found that Hamilton's rule was quantitatively fulfilled, i.e. actor–recipient relatedness (r) and benefit to recipients (b) were both positive and high enough for the total indirect fitness benefit to outweigh the direct fitness cost (−c), such that rb −c > 0. These cases involved guarding in an allodapine bee, joining behaviour in polistine wasps, cooperative lekking in wild turkeys and helper behaviour in white-fronted bee-eaters (nos. 2, 8, 9, 11 and 12). In three cases, involving guarding in an allodapine bee and joining behaviour in polistine wasps (nos. 3, 5 and 7), Hamilton's rule was quantitatively fulfilled in some contexts (some years, some group sizes) and not others. In one case, again involving joining behaviour in a polistine wasp (no. 6), social behaviour was selectively neutral (rb − c = 0). In the single remaining case, involving worker behaviour in a halictid bee (no. 4), Hamilton's rule was not fulfilled (rb − c < 0). Overall, therefore, among 10 cases of demonstrated altruism, Hamilton's rule was quantitatively fulfilled in five cases and fulfilled in some contexts in a further three cases. Moreover, egg dumping in lace bugs and kin-discriminating cannibalism in larval salamanders (nos. 1 and 10) were each found to yield a positive indirect fitness benefit despite the lack of direct benefit (electronic supplementary material, table S1), again explaining the occurrence of these behaviours by Hamilton's rule.
(c) Conclusions from empirical tests of Hamilton's rule: relatedness
The focal studies show that there is considerable diversity in the mechanisms that generate positive actor–recipient relatedness in social systems. Standard mechanisms generating such relatedness are population viscosity (philopatry) and kin discrimination [7,9]. Population viscosity can arise through subsociality (group formation via parent–offspring association) or, provided aggregating individuals are relatives, through semisociality (group formation via aggregation of members of the same or mixed generations) . The focal studies include cases of subsociality (e.g. allodapine bee, halictid bee nos. 3 and 4 in table 1 and electronic supplementary material, table S1), cases of semisociality of relatives (e.g. polistine wasps nos. 5–9) and cases that probably involve a mixture of subsociality and semisociality (lace bug, allodapine bees, wild turkey, white-fronted bee-eater nos. 1, 2, 3, 11 and 12). The focal studies also include one clear case of kin discrimination at the individual level. The study of cannibalistic larval salamanders (no. 10) suggested that kin discrimination has evolved because it allows larvae to benefit from cannibalism while avoiding harm to coexisting relatives . This supports a general finding that kin discrimination evolves not as an automatic corollary of kin selection but specifically in social contexts in which it generates benefits [52,59].
(d) Conclusions from empirical tests of Hamilton's rule: benefit and cost
There are several conclusions to be drawn from the focal studies (table 1 and electronic supplementary material, table S1) as regards benefits and costs and their role in the causation of social evolution: