J. theor Biol. (1978) 73,679-686

Coefficients of Relationship and Coefficients of Relatedness in Kin Selection: A Covariance Form for the RHO Formula M. J. ORLOVE School of Biological Sciences, University of Sussex, Falmer, Brighton, BNl 9QG, Sussex, England AND CONSTANCE L. WOOD Biometrics Unit, Warren Hall, Cornell University, Ithaca, New York 14853, U.S.A. (Received 18 January 1977, and in revisedform 24 February 1978) Hamilton used the coefficient of relationship to predict the direction of selectionacting on “genesfor altruism” in a theoretical model. The sense (direction) of the inequality determined the direction of selection. But, paradoxically, Hamilton pointed out that traditional formulaefor the coefficient of relationship cannot be evaluated during directional selection. Hamilton escapedthe paradox by assumingvery weak selection. Later papers,including three by Hamilton, used other coefficientsin placeof the coefficientof relationshipin the inequality. Thusinbreedingand directional selectioncould be handledin exact models.This paper tries to clarify the differencesbetweenthe various coefficientsand to point out an error in one of the author’s previousworks: Namely to statethat there is analytical proof that the regressioncoefficient is a special case of a coefficient called p whereasthe earlier paper claimed only numerical verification waspossible. Sincethe symbol “p” is also usedto denote the correlation coefficient, it is suggestedhere that the p of kin selectionbe given a new name. Hamilton (1964a,b) stated that altruism will be selected for if K > l/r and selected against if K < l/r, where: K Gain in beneficiary’s Darwinian fitness (1) = Loss in altruist’s Darwinian fitness and r is the coefficient of relationship (fraction of genesidentical by descent) 0022-5193/78/0821-0679 $02.00/O

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between them. If we tag a small (almost zero) random fraction of altruism genes in a simulated population, then: r = PROB

potential altruist has ( tagged gene

I

potential beneficiary has ). a tagged gene too

(2)

The term “coefficient of relationship” and “fraction of genes identical by descent” are treated as equivalent (Hamilton, 1964; Henderson, 1976; Crozier, 1970). The former implies evaluation as a correlation coefficient, and the latter implies evaluation as a conditional probability as in equation (2). If we evaluate our coefficient anew each generation using up-to-date parameters from the population, then the correlation coefficient no longer equals the conditional probability in many cases. Since such up-to-date evaluation is necessary in considerations of kin selection (Orlove, 1978), it will be used throughout this discussion. Under up-to-date evaluation, r as defined in equation (2) has the properties claimed for Y as defined as a correlation coefficient by Wright (1922). We are certain of this claim for relationships between siblings, and certain the claims of equation (7) are general for all relationships, as claimed by Hamilton (1975) for his coefficient. There would appear to be exceptions to Hamilton’s K-r rule so other coefficients have been used in place of r in Hamilton’s inequality, e.g. B, the regression coefficient of relatedness (Hamilton 1971, 1972, 1975; Orlove, 197&b). Hamilton chose the term relatedness to distinguish B from r, the coefficient of relationship. However, recently Dawkins (1976, p. 98) ignored this distinction. When B was used in place of r in Hamilton’s inequalities, fewer exceptions occurred. When B was evaluated from up-to-date data from the population rather than from pedigree positions alone, still fewer exceptions occurred. To deal with these remaining exceptions another coefficient, called p, was used in place of r in Hamilton’s inequalities (Orlove 1975u,b). p was defined such that no exceptions could occur. The symbol “p” was chosen because it is the Greek counterpart of the letter r. The symbol “p” is also used to denote the correlation coefficient. This is regrettable since whenever mating is random, the correlation coefficient is equivalent to B. It is hoped that the p of kin selection be given a new name. Here no new name occurs. “p” will be used for the new coefficient, and the correlation coefficient will be referred to by name as “the correlation coefficient”. During polymorphisms maintained at a constant frequency, the correlation coefficient, B, and p are equivalent. But if the frequency is intermediate, they deviate from r (contrary to popular belief). In spite of these popular beliefs, r and the correlation coefficient are not equivalent at intermediate frequencies, even though the population is in

COEFFICIENTS

OF

RELATEDNESS

681

Hardy-Weinberg equilibrium at the time of zygote formation, except for the special case of a neutral equilibrium existing simultaneously with additive gene effects within a locus. This is because of an effective departure from Hardy-Weinberg equilibrium at the time of gamete production (in the previous generation) due to heterozygote advantage or homozygote advantage. For example, if two sibs have a homozygous parent, there is a 50% probability they will receive genes from that parent which are alike in state (but not identical by descent). This will increase the correlation coefficient but will not increase r. For non-directional selection the stability, or instability, is determined by the slope of the curve, p as a function of gene frequency, at the point where the curve intersects the horizontal line whose equation is p = l/K, see Fig. 1. Concave-up curves show heterozygote advantage. Concavedown curves show homozygote advantage (by “advantage” we mean advantage to reproductive success). Unlike classical models, the stability of polymorphism in the kin selection model is not determined by heterozygote advantage or homozygote advantage (Orlove, 1975u). These results are easily accepted if we realize that the fitnesses of the diploid zygotes are frequency dependent. We can consider each potential altruist and its potential beneficiary as a temporary “colony” composed of two individuals. Representing each colony as a point on a graph (see Hamilton in Orlove, 1975a): Let N = the total number of points in the population. Let X, = the X-co-ordinate of the ith point, i.e. the fraction of its genes that are altruism genes in the ith potential altruist. Let Yi = the Y-co-ordinate of the ith point, i.e. the fraction of its genes which are altruism genes in the ith potential beneficiary. B is the slope of the best-fitting straight line (the regression line) through the cluster of Npoints, as determined by the least-squares method. This gives:

which becomes (4) (It will become apparent why explicit representation of second degree terms and variance is being avoided.) p = B except when there is : (1) directional selection, (2) a lack of additive gene effects within a locus, and (3) sex-limited selection, acting on the altruism

682

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of setection-

Y p 15.75

! vde diploid

I 0

t

t

I

03

I Frequency

I

0,5 of the altruism

I

I

07

I

I

I

I.0

gene

FIG. 1. Stability of polymorphisms in altruism among siblings. If the l/K line is in view on this graph, then the population will be confined to the immediate oroximitv of one of the curves. It will move along the curve in the direction indicated by the arrow on the same side of the l/K line as the population. If the l/K line does not intersect the curve, then fixation results. The drawing is schematic: the selhshness-dominant curve has a minimum which is slightly further from the codominance (i.e. additive gene effects) line than the maximum of the altruismdominant curve; all the maxima and minima occur at frequencies of the altruism gene slightly less than 075; the maxima and minima are never more than a distance of 0.03 from the codominance line. The correlation coefficient, B and p are, for all practical purposes, equivalent when the l/K line intersects one of these curves. Y remains on the line labeled “COD~MINANGE" during all degrees of dominance, except in the case where the altruism gene was rare when the initial mutation event, simulated by the “tagging” [see equation (2)], occurred. However, p is on the codominance line only during codommance. When Ql/Qz approaches ~0 the other curves come closer to the additive-gene-effects line, and coincide with it when QJQa = ~0. This accounts for the result by Scudo & Ghiselin (1975) that K at equilibrium = l/O*75 during dominance. This finding of equilibrium K = l/O.75 during dominance and its apparent conflict with Move (1975) made Crozier (1977) question briefly the use of mathematical models. Levitt (1975) page 1535 contrasts his kin selection model with a reciprocal altruism model by stating that the kin selection model can have a stable polymorphism only if both flxation points are unstable. He seems to give the impression this is a general difference between kin selection and reciprocal altruism, the latter being characterized by a “threshold phenomenon”. An examination of this figure will show that kin selection can produce a “threshold phenomenon” with a stable and unstable polymorphism,, and one stable and one unstable fixation point. The threshold phenomenon being that once an altruism allele is made common enough by some external force (e.g. drift, hitch-hiking), i.e. once it passes a threshold frequency, it can increase in frequency on its own steam.

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gene. All three factors must exist simultaneously for p not to be equivalent to B (Orlove, 1975a,b). The second of these papers stated that the equivalence of p and B during additive gene effects within a locus can be demonstrated numerically, but not analytically. This is in error and below follows a proof that p = B during additive gene effects within a locus. Let F(Z) = the phenotype of a potential altruist with the fraction, 2, of its genes as altruism alleles, i.e. the fraction of its resources used selfishly. Let G(Z) = the “genotype” of an animal with the fraction, Z, of its genes as altruism alleles, i.e. the fraction of its gametes carrying the selfishness gene. Then for the genotypes: AA

AS

112 l/2 0

ss 01 1 1

for Mendelian segregation for additive gene effects for altruism dominant

Let X and Y take on the significance they did in the discussion on regression above. Then the regression of the genotype of the potential beneficiary on the genotype of the potential altruist is:

B = cov CGCO WN WV CG03, GWI ’

(5)

The regression of the phenotype of the potential beneficiary on the genotype of the potential altruist is:

B = cov EGtX), F(Y)1 -_

cov CG(-XI, GCO1 ’

(6)

and we shall see:

cov I3 (XL F O’II ’ = cov [G(X), F(X)]

’

(7)

When there are additive gene effects (and Mendelian segregation) F = G and therefore p = B. Animal breeders evaluating the worthiness of bulls and roosters in programs to improve milk and egg production, consider directional selection on sex-limited characters exhibiting dominance. Should the breeders be criticized for using B instead of p in their calculations ? Since they only use the numerators of the regression coefficients in their calculations involving the “A matrix” (Henderson 1976), p and B give identical results so long as the

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formulae are evaluated from up-to-date parameters at the beginning of each generation and the regression is phenotype on genotype or genotype on phenotype. (These last two regressions are equivalent if for every X0, Y” colony there is a Y”, X0 colony. This is a safe assumption to make (Orlove, 1975a). This is so because under these circumstances p and B have identical numerators. Thanks to-Helen Orlove, Milton Orlove, John Maynard Smith, Walter Federer, and Sheila Laurence.

REFERENCES CROZIER, R. H. (1977).Ann. Rev. Entomol. 22,263-268. DAWKINS, R. (1976).T/zeSelfish Gene. NewYork: Oxford UniversityPress. HAMILMN, W. D. (1964u). J. theor. Biol. 7,l. HAMETON, W. D. (1964b).J. theor.Biol. 7, 17. HAMILTON, W. D. (1971).In Man and Beast: Comparative Social Behavior (J. P. Eisen-

berg& W. S.Dillon, eds),ch. 11,pp. 57-91.Washington, D. C.: Smithsonian Press.

HAMILTON, W. D. (1972).A. Rev. ecol. Syst. 3, 193-232. HAMILTON, W. D. (1975). In Biosociul Anthropology (R. Fox, ed.), pp. 133-155. London:

Malaby Press. HENDERSON, C. R. (1976). Biometrics. 32, 69. L~vmr, P. R. (1975). Proc. rut. Acad. Sci. 72,453l. ORLOVE, M. J. (1975a). J. theor. Biol. 49, 289. ORLOVE, M. J. (19756), J. theor. Biol. 55,547. ORLOVE, M. J. (1978). In The Biology and Systematics of Colonial Organisms. B. Rosen&

G. Larwood,cds.London,NewYork: AcademicPress. SCUDO, F. M. & GHISELIN, M.T. (1975). J. Gene& 62, 1.

APPENDIX

To derive the formula for p, appearing in equation (7), which looks like the regression formula. As in Orlove (1975a,b): Let W = the number of offspring produced by a completely selfish potential altruist. Let 0 (zero) = the number of offspring produced by a completely altruistic potential altruist. Let QI = the number of offspring produced by an unaided potential beneficiary. Let Q,+Q, = the number of offspring produced by a completely-aided potential beneficiary. Hamilton’s K = Q,/ W. The W is for worker and the Q is for queen. Let q’ = the frequency of the selfishness allele in the gametes entering next generation, i.e. produced in this generation.

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W iil J’tX3 G(X3+ Q1 i$l GtX3 + Q2iil Cl- F( J’JI G(X3

4’=

(Al) wit1 JIXJ + QI i$l 1 + Q2,tl Cl- fTY,)I

’

There are two ways to make the altruism allele and the selfishness allele equally fit : (1) A cessation of selection, e.g. W = 0, Q, = 1, Q2 = 0, and (2) non-directional selection, i.e. W = p, Q, = anything, Q, = 1. If values of 0 c W < 1 bother you, just imagine W, Q,, and Q,, each multiplied by the same arbitrarily-large constant. The constant cancels out in the formulae. Given that (W, Ql, Q2) = (0, 1, 0), then 2 ax,) (A2 q’= i=lN We want to define p such that when Q,/W > l/p the altruism allele will increase in frequency and when Q,/ W < l/p the altruism allele will decrease. A decrease in frequency is delined as the evaluated q’ coming out greater than the right-hand side of equation (A2). In equations (A2) and (A3) both alleles are equally fit. Letting the right-hand side of equation (A2) equal to the righthand side of equation (A3) yields equation (A4), which is the definition of p. This is the most crucial step in the whole argument but it requires too much space to justify it fully here. The justification is given in full in terms reasonable to the layman in Orlove (1978). Given that, (W, Q,, Q2) = (p, Q1, 1)

P iil J’(XMX3 + Ql ,tl G(XJ + i$l Cl- JIr,>l @X3 q’ =

C43)

P i$l F(XJ + QI ifl 1 + i$l Cl-I;(Y,)I 2 GtX3

i=l ~

N

P,il J’(XJ ‘3x3 + QI itl G(X3 f igl P-J’(YJl c(XJ

(A4)

=

Pi=l$ FtX3 + QlN+ ,gl Cl- F(Y31

**-Pi=l 5 JYX3 i=li G(Xd/N+ QI fl G(XJ+ fl Cl-J’(YJl jcI G(XdIN = Pfl J’(X3 GtX3 + QI i$l G(X3+ itl Cl- J’tYJI G(Xd* The Q, terms cancel.

(A51

686

M.

-‘* P ii1

I;(xJ

GtX3

J.

ORLOVE

- P itl

JIXJ

AND i$l

C.

L.

WOOD

GtXd/N

Gw

= ii1 Cl-F(YdI ,gI WXJIN- i$ Cl-JTYdl GGJ *‘aP ,fl J’(XJ G(XJ- i$l Rx3 i$l G(X,)/N] [ = it1 Cl -FCYJI ,t G(XiIIN-

@7)

ii ‘Ll- WJI

G(XJ

... p = i$l Cl-~(J’Jl ,iI G(XJIN- i$I Cl-F(Ydl G(X3 CW

,gl f”(XJ G(XJ - itl F(X3 ii G(XJ/N

.

Here are two identities we need. jl

C1-F(Yi)I

= N - ii

Ftyi)

i$l Cl- J’(Ydl c(XJ = iiI G(X3- ,iI W’d G(XJ k:. p =

5 HY,)] i& G(XJ/N- [flGtX,) T$ F(YJl G(Xi)]

i=l

(All) i$l F(XJ WJ

*-. p

-

- i$l F(XJ it1 G(Xi)IN

i G(XJ- $ F(Yi)ii G(XJIN- ii G(XJ+ ,g F(YJG(XJ

i=l

--ifl li(xJ

GtX3 - iil PC;3 ii

GtXJ/‘N

IfI F(YJ ‘3x3 - i$ F(YJ i$ G(XdIN **- p = i’N1 iJIl f’(xJ G(xJ - ,fl Ftx3 i$ G(XJIN

cov L-‘3-9, F( Y)l ‘**’ = cov [G(X), F(X)] which, in the notation

(A12)

W3)

C-414)

of Orlove (1975b), becomes P

= cov (I 4 J, Jsd K) cov (I 3 J, Isd J) *

W 5)