I. theor. Biol. (1975) 49, 289-310

A Model of Kin Selection not Invoking Coefficients of Relationship M. J. ORLOVE

School of Biological Sciences, University of Sussex, Falmer, Brighton BNI 90G, Sussex, England (Received 8 January 1974, and in revisedform 5 July 1974) The pioneeringwork of Hamilton assumedthe coefficient of relationship easyto compute and usedit to predict the direction of selectionacting on “genesfor altruism” in a theoretical model. A similar model is presented herewhich doesnot require the coefficient of relationship asexplicit data, in order to predict the direction of selection. The model presentedimplicitly defines the coefficient of relationship and is reducible to two equations. Inverting one of these produces an explicit formula for the coefficient of relationship. This formula has a domain of input beyond traditional formulae for the coefficient of relationship. Theseinclude: (1) Populations undergoing selection during half dominance and populationsnot undergoingselectionwhile altruism is either dominant or recessive. (2) Populationsundergoingselectionon an altruism genewhile that gene is either dominant or recessive. The formula agreeswith Hamilton’s regressioncoefficient of relatedness when dealing with the category (1) populations, but not when dealing with the category (2) populations. The biological significance of the model is mainly: (a) Traditional coefficients of relationship apply in the absenceof selection only when: (i) the genefor altruism, or its alternate allele, is infinitesimally rare; (ii) half dominanceprevails, i.e. the heterozygote uses half of its resourcesselfishly and the other half altruistically; (iii) the beneficiary’s fitness is only infinitesimally augmented by the altruism, e.g. when the beneficiary would have had an infinite number of children anyway. (b) At other timesin the absenceof selectionthe coefficient of relationship alters so as to: (i) deviate most from traditional coefficientswhen the frequency of the altruism gene is approximately one-half; (ii) favour selectionfor the dominant allele; (iii) account for the fact that increasing the number of offspring which the beneficiary has anyway, always helps the recessiveallele. Rapid selectionagainsta dominant altruism genecausesthe coefficient of relationship, as defined here, to approach unity as the frequency of the altruism geneapproacheszero. This result and all departuresof the coefficient defined here frum Hamilton’s regressioncoefficient, during 289

290

M.

J.

ORLOVE

selection with gene dominance, can be accounted for as a hysteresis effect due to the fact there is no selection in the males. This model may be generalized to one which sheds light on the selective advantages of sexual reproduction and it is proposed here that self-dilution (which includes altruism and sexual reproduction) may be as important an asset to the animal as self-replication in evolution. The significance of this on existing definitions of altruism is discussed. 1. Intmduction

The term “kin selection” was coined by Maynard Smith (1964). It denotes selection on genes which cause copies of themselves to come into existence, not by fissioning but by causing already existing replicas of themselves to fission, or by enhancing the probability of survival of existing replicas. Thus a gene whose expression causes an animal to help his relatives could be selected for because replicas of the gene in the relatives would be helped as well. Although Haldane (1955) was aware of the concept, Wells, Wells & Huxley (1931) gave a remarkably intuitive account of it. They explained how genes expressed only in the phenotype of sterile worker bees could evolve and be “kept up to the mark by natural selection”. Their explanation likened queens and males to the germ plasm, and workers to the soma of an animal. Their argument is not unique as a “super-organism” model for the insect society, but it showed an early insight into the way workers (soma) expressing a gene could affect the fitness of that gene due to the influence of the worker’s (soma’s) behaviour on the welfare of the queen (germ plasm) who possesses replicas of the gene. Parallels to kin selection in other disciplines such as book-keeping and oriental philosophy will be left to the end of this paper. Hamilton (1964) gave a complicated treatment of kin selection although he didn’t use the term. He examined the fact that the worker and queen (or their counterparts in any animal society) are often not genetically identical. His argument is beyond the scope of this paper. His conclusions are important and are the basis for this paper. 2. Hamilton’s

Results

An altruist is defined as an individual which increases the fitness of other members of its species. An altruist is on a limited resource budget. There is an “opportunity cost” of a unit of the altruist’s fitness in terms of similar units of the fitness of a potential beneficiary, i.e. a gain in fitness to a beneficiary is (often) accompanied by a loss in fitness on the part of the benefactor.

KIN

The relationship

SELECTION

is represented by the scalar K where K = Gain in beneficiary’s fitness Loss in altruist’s fitness ’

291

(1)

K will often take values other than unity due to individual variation in the relative efficiencies of animals for doing different things. Thus, one animal may be so fecund as to lay more eggs than she can rear, while her sister may be less fecund and be limited by how many eggs she can lay. If the latter rears nieces instead of daughters she could enjoy a value of K greater than unity. Later we will make our model simpler to analyse by assuming differences in fitness to be solely due to differences in fecundity. But this is only for the sake of simplicity and so assuming equal survivorship for altruist and beneficiary we rephrase : Consequent increase in number of offspring produced by beneficiary~. K = -.-- ---~-.--~..~---,--~ ~-__---...(2) Number of offsprmg given up by altruist ’ Ceteris parabus animals are expected to have evolved such that each individual behaves as if weighing K against r in the face of an opportunity to be altruistic, where r is the fraction of genes identical by descent between the potential altruist and the potential beneficiary. When K > l/r we expect the animal to behave altruistically, and then K < l/v we expect him to behave selfishly. Various procedures exist for calculating r (Hamilton, 1964, 1971, 1972; Falconer, 1961; Malecot, 1969; Kempthorne, 1957; Crozier, 1970; Wright, 1959). In most cases the results are the same. The value of r between two animals depends only on their relative positions in a pedigree. Falconer’s approximation depends on how many generations back one is willing to consider the pedigree. Hamilton (1971), Trivers (1971) and Orlove (unpublished) contend that real altruists, including humans as well as social insects, are facultative, i.e. behaving altruistically, or not, depending on the circumstances of their environments. But genetical selection may determine the thresholds of such facultative behaviour. Yet for didactic purposes we can produce a model in which altruism is obligate and controlled only by the genotype of the potential altruist. Furthermore we can assume that all potential altruists share the same value of K, and the same relative positions, on their respective pedigrees, toward their respective potential beneficiaries. Using our model in a simulation will show that the condition: 1 K= (3) r as predicted from the pedigree positions alone’

292

M.

1.

ORLOVE

only sometimes prevents selection, that is, in the absence of selection, or at gene frequencies near fixation (i.e. altruism common or altruism rare) or at intermediate gene frequencies with additive gene effects. Hamilton (1964, p. 4) stated that r was, as of yet, unevaluated for cases during selection. So rather than using the evidence from simulations to refute Hamilton’s K-r rule, we will assume the K-r rule to be a law, i.e. a tautology, and define r as the value of l/K which keeps the gene frequency constant. In the case of sex-linked genes we will read “effective gene frequency” for “gene frequency” in the last sentence. The effective gene frequency is the true measure of gene frequency for a sex-linked gene and equals 2 x [gene frequency in females] + gene frequency in males (4) 3 (Wright, 1959). The traditional procedures for computing r were intended for studying inbreeding. Definitions of r which preserve tautologies about

(IO) ~0000125

,

I 0250

I 0375

(2)

,

0500

0625

I 0750

I 0875

(0) 1000

x FIG. 1. Let each point on the graph represent a pair of potential interactants. The X co-ordinate = the fraction of genes that are altruism alleles in the potential altruist. The Y co-ordinate = the fraction of genes that are altruism alleles in the potential beneficiary. The population is a cluster of such points. The slope of the regression line is a coefficient of relatedness (Hamilton, pers. comm.) represented by the scalar B. The matrix Ba,b described by Hamilton (1971, 1972) is an estimate of this scalar, based on a pedigree. Two points can have the same co-ordinates. The numbers in brackets by each point show how many times the point occurs in the population. The ratio here represents Hardy-Weinberg equilibrium with gene frequency = f.

KIN

SELECTION

293

inbreeding may clash with definitions of r which preserve tautologies about kin selection. However unlikely such a clash may be, I think it wise to call my coefficient p rather than r. p = l/K (in the absence of selection). (5) For similar reasons Hamilton (1971, 1972) calls his coefficient B and a “coefficient of relatedness”. It is interesting that r is a correlation coefficient and that B is the corresponding regression coefficient on the same data (Hamilton, pers. comm.); thus B is sometimes called the “regression coefficient of relatedness”. Hamilton (1964, p. 18) suggested that the true r is a regression coefficient but at that time was not aware of the import of his statement (see Fig. 1). 3. A Model of Kin Selection

Various exact models already exist for simulating selection for altruism, (Levins, 1970; Eshel, 1972; Boorman & Levitt, 1973), but none of these well define K and although they show altruism can be favourably selected, they do not lend themselves as tests of the K-r rule, and thus leave p ill defined, if not undefined. Our didactic model will consist of a population of idealized wasps. In wasps and other insects in the order Hymenoptera, all genes are sexlinked, i.e. the males are haploid, arise from unfertilized eggs, and produce their sperm by mitosis. Thus all the sperm from one male are identical. Sexlinked heredity results in higher values of mean r between females than would exist in autosomal heredity. Thus some investigators (Maynard Smith, 1966; Lin & Michener, 1972; West, 1967; West-Eberhard, 1974; Wilson, 1971) have emphasized or criticized the use of r more than the use of Kin explaining the polyphyletic origin of sociality in the Hymenoptera. It should be noted that the K-r rule should be important in considerations of all intra-specific interactions, and our choice of a sex-linked locus model in this paper is for the sole purpose of keeping the algebra simple, and should not imply that kin selection is only important in male-haploid species. The corresponding equations for the male-diploid version are much more long-winded. The model wasps most closely approximate paper wasps of the genus Polistes discussed in relation to kin selection by Hamilton (1964); West (1967) ; and West-Eberhard (1969, 1974). Polistes wasps found colonies either in groups or sibling females or singly. When a group founds a colony, a “pecking order” arises in which the top female serves as queen and the others (often called auxiliaries) serve as workers. In allusion to a social insect colony we can call a potential altruist a worker and her potential beneficiary a queen.

294

M.

J.

ORLOVE

In honeybees and many other social insects whether an individual becomes a worker or a queen is environmentally determined. It is not at all far fetched to assume that in most animals, the opportunity to be altruistic and the possession of any genetic proclivity to be altruistic, when such an opportunity arises, are independent events. This assumption becomes necessary in a model used to test the K-r rule by comparing l/K with pedigree predicted r. This is because we wish to know the mean r for the population, say, between sisters. So all genotypes are given an equal opportunity to be altruistic, but each altruist must choose her beneficiary, only because she is her sister, i.e. from at random among her sisters. In our model this condition will be realized by assuming that either member of a pair of potential interactants is equally likely to be “queen” but as soon as one is chosen as “queen” the other must be a “worker”. With this constraint we can say 4’ = F(W, Q1, Qd (6) where: 4’ = The frequency of the gene for altruism in the ova entering the next generation (i.e. generated this generation), W = the number of offspring produced by a selfish worker. An altruistic worker has zero offspring, Q1 = the number of offspring produced by an unaided queen, QZ = the extra offspring produced by an aided queen. (A queen with a selfish worker has Q, offspring. A queen with an altruistic worker has Q, + Q, offspring. Notice K = QJW.), and the genetic make-up of the population this generation is incorporated in F. We can now define p without stipulating whether “effective gene frequency” or “gene frequency” is the appropriate phrase in our definition. This is done by saying: WJX,O)=W,QI, 1) (7) and solving for p. The left side of equation (7) simulates killing all workers at birth. Since the altruism gene is expressed in the phenotypes of workers only, there is no selection on the altruism gene. The right side of equation (7) simulates a cessation in selection due to the fact K = l/p. 4. A Rigorous Statement of the Model The model is limited to interactions between sibling Hymenoptera (p is as if all genes are sex-linked) like West’s Polistes waspi. Females are polyphenic, i.e. each female is of an altruism-expressing phenotype (worker) or of a phenotype which doesn’t express the altruism gene (queen). The model

KIN

295

SELECTION

considers selection on a single locus with two alleles. The phenotype with regard “worker” and “queen” is either environmentally induced at random or controlled by a locus not linked to the one under consideration. As far as the locus under consideration is concerned, any genotype is equally frequent in the worker as it is in the queen, and a (female) zygote has the same probability of becoming a queen as it has of becoming a worker (50 %), regardless of genotype. The gene under consideration is expressed in the worker phenotype only and controls altruism of the worker to a sibling queen. Mating is random in this model. Like the model of Hamilton (1964) this one involves an infinite population with non-overlapping generations and is subject to the resulting limitations. 5. Deriving the Model The allele for altruism is denoted by 0. The allele for selfishness is denoted by 1. Each colony has a worker and a queen so we can find the genotype of the different kinds of colonies by combining the genotype of a worker and a queen (each colony has one worker and one queen). (See Table 1). TABLE

Genotypes of worker Worker 00 E 01 01 01 11 11 11

1 and queen QUtS2.U

00 01 11 00 01 11 00 01 11

Since the locus is sex-linked, two of these colony genotypes cannot exist, namely 00 11 and 1100. Thus we can represent the colony genotypes as three columns of numbers since one gene is always identical between the worker and the queen. This gene will go in the middle column and represent the genotype of the male (hymenopteran males are haploid) who is the father of the worker and queen. The left column represents the contribution the worker gets from the mother and the right column represents the contribution the queen gets from the mother (see Fig. 2). Figure 2 shows the eight kinds T,B.

20

296

M. J. ORLOVE Queen

Worker

own tnmme

to worker

ovull to

I

sperm J

becomequeen K

0 0 0

0 0 I

0 I I

I 0 0

0 I 0 I 0 I

I

I

0

I ‘Jlbrker

I

genotype I

’

QJeen perotype

I

FIG. 2. The eight kinds of colonies which can exist. Left column = I; middle column = J; right column = K.

of colonies which can exist. The fact that there are eight kinds of colonies shows that there are three bits (binary digits) of information specifying the genotype of a colony. There are one locus, three genomes and two alleles or 1 x 23 kinds of colonies. This is what Li (1955, p. 18) means by three independent genes existing in a system of this sort. There is a correspondence between the four column and three column system (see Fig. 3). Worker 00

Queen -

00 NUll

00

*

0

0

0

0

I

01

0

I

o*

01

0

I

I

I

0

0

I

I

I

I

I

I

I

I

I

I

4 columns FIG.

Queen

0

0

Nul I

IMorker

3 columns

3. The correspondence between the four column and three column system,

KIN

SELECTION

297

Let N be the vector which describes the population. If the worker of a particular colony is selfish then her queen goes unaided. The two live separately on different nests but we assume for now that we know that they would have lived on the same nest had the worker been altruistic. Thus a colony with a selfish worker is called a dissociated colony but its genotype is counted. Thus N1, J, k is the number of (associated or dissociated) colonies of genotype I, J, K, i.e. the worker is ZJ and the queen is JK. No, o,. is the number of colonies where both the worker and queen are homozygous for altruism. Ni, i, 1 is the number of associated colonies that would exist, where both the worker and queen are homozygous for selfishness, if the effect of the selfishness gene were nullified. We can treat N as a row vector whose single subscript is written in base 2. Thus N could be written as Nnr; M = 0, 1, 2, 3, 4, 5, 6, 7 but instead it is written N,, J, K; I= 0, 1; J= 0, 1; K= 0, 1. M= 4xI+2xJ+K. N, is written N,, ,,, i, N6 is written N,,,,o, etc. We want to multiply N by the transformation matrix T to get N’ (which is N in the next generation). T can be similarly treated as an 8 x 8 matrix whose base,, subscripts go from O-7, but we chose to write them in binary form, assigning to each binary digit the status of variables in their own right. Thus an entry in T is called T,,,, R; II, 8, y and represents the number of colonies of genotype a, /3, y formed by the female offspring produced by one colony of genotype I, J, K. The dot product formula can be represented as:

Let C$= the frequency of sperm carrying the altruism gene. Let SD = the fraction of her resources used selfishly by a heterozygous worker. SD = 0 for altruism dominant. SD = 1 for selfishness dominant. SD = f for additive gene effects. We can assume single insemination of females, so two sisters have the same father. Even in the multiply-inseminated queen honeybee Apis me&f&a L. the wads of sperm from different males do not mix much in the spermathaeca, thus two females the same age are likely to have the same father (Taber, 1955). Assuming random pairing of sisters to form colonies is the same as assuming random pairing of the ova before fertilization. Making all the assumptions thus presented plus assuming Mendelian population genetics we can fill in T by hand. Figure 4 shows T filled in. “1, J” and “a, j?” are

298

M. 0 0 0

T 4 J, K

J.

0 1 0

:, 0

0 0

0 0

0 1

tQl+Qa)x4 tQl + Qa> x d/4

tQ1+Q:x4/4

0

1

0

0

1

1

(WxSD+Ql +Qax(l--SD>> x 414 WxSDxq5/4

(WxSD+Q, + Qa x (1 -SD)) x $14 Wx SD x d/4

(WxSD/4+Qx + Qa x (1 -SD))

WxSD/4x4

c$xSD+Q,

(WxSD+Ql ,~4xtl

100

1

0

1

+Qax(l-SD>>

x 414 1 1

1 1

0 1

Ql x 414

--SD))

Ql x 414 0

0 FIG.

ORLOVE

4. A

tQl+QdxU-41 (Ql+Qd x (1 - $))I4 tWxSD+Ql -t Qa x (1 -SD)) x (1 -d)/4 W x SD x (l-9914

1 1 0

tQl+ Qi x (1 -h/4 (WxSD+Ql + Qa x (1 --SD)) x (1 -o/4 WxSDx(l-&/4

(WxSD/4+Q1 ; t:;; -SD))

WxSD/4x(l--$

tWxSD+Ql + Qa x (1 -SD)) x (1 -d/4 Q1x(l--&I4 0

(WxSDSQl + Qa x (1 --SD)) x (1 -d)/4 Q1xtl-d/4 0

transformation matrix:

TI.I.R;~.~,~

is the

number

worker genotypes and “J, K” and “B, X” are queen genotypes. Queenness strikes at random. Either member of a sibling pair could have been the queen thus : N I,J.K -= NK,J,P (8) We need not assume this to make the model work. It is a consequence of T as shown in Fig. 4. Simulating a population using T will show this. If we start with a population in which N I,J, x # NK,J,I this situation will disappear in one generation; N;, J, x will equal N;P, ,, r. However, we can exploit this fact in order to re-work T into a form more compatable with algebraic manipulation. In its new form, however, Twill give identical results in simulation as before, but the initial population must have Nr, J, x = Nx, J, *. T is reworked by transferring all terms in Q, and Q, from their respective “I, J, K” positions to the corresponding “K, J, i” positions in the same column. T, in its new form, is shown in Fig. 5. This is the form we will expect T to be in when we refer to it from now on. T now represents the mean number of offspring produced by a female of genotype I, J when the other member of her colony is of genotype J, K. If we limit ourselves to the use of constants as subscripts, we might as well use scalars and dispense with subscripts altogether. Furthermore the length of the ensuing equations would render them too unwieldly to present here. Therefore we want a formula for TI, J, x; =,#, ,, as a function of Z, J, K, a, /?, and y. This formula will introduce a notation in which our subscripts can be

KIN

0

SELECTION

1 0 1

0 1

tQl+Q~~4/4

Mh+&w4

(WxSDS-Q,

;g4x (1--SD))

(WxSD+Q, -kk4xtf--sD))

WxSDx4/4

(WxSD/4+Qx

WxSD/4x4

txFXtl-SD)) WxSD/4x4

(WxSD+Q, + 91 x (1 --SD)) x 414 QI x 414 0

(WxSD+Q, ; g4x (1 -SD))

yw&4/4;4 1

299

0 1 1

1 1 1

tQl+Q: x (1 -dl/4 (WxSD+Ql

CQl+QP) x(1--0/4

(WxSD+Ql +QaxU--SD),

+ Qz x (1--SD)) x (l--98/4

WxSD/4x(l---/)

x(1--1)/4 (WxSD/4+Q, + Qz x (1 --SD)) x(1-4) WxSD/4x(l-4)

WxsD+Ql +Qax(l--SD)) x (l-4)/4 Q1x(l--O/4 0

(WxSD+Q, +QaxU-SD)) x(1-4)/4 Uf’+Q1/4)~(1-4) tW+QdxU-4,

WxSDx(l-&/4

T

1, J, K 0 0 0 001 0

1 0

0 1 1 1 0 0 1 0

; ; y

of a, B, y colonies produced per Z, J, K colony per generation.

variables, and in which the minimum amount of changes are made in the symbolism to generalize this model to other cases. The method for deriving the formula for TI, J, K; a, ), y from T as shown in Fig. 5 is vaguely analogous to the method shown by Dockter & Steinhauer (1973) for deriving a Boolean expression from its corresponding truth table. A simplified version of this method is presented in the following short example. Definition: Boolean expression. An expression equal to 0 or 1. It may be an arithmetic expression, a simple variable, or a statement or equation surrounded by brackets. If the statement or equation is true then the expression equals one, if the statement or equation is false then the expression equals zero, (Berry, 1969; Iverson, 1962). Examples

Example 1

Value 1 0

; = 4-1) (7 = 2+2)

0

(I am a bee)

0

(7-6)

1

1

1

300 T 1, J,

K

0 0 0 0 0 1

M. J. ORLOVE 0 0 0

; Y

1 0 0

(Q~+Qz)xd (81+Qs

0 0

x(1-SD))xdi

0 1 0 0 1 1 100

(WxSD+Qx

(WxSD+Ql

+ Qa x (1 ---SW

+ Qz x (1 --SW

x 414 (WxSD+Qd x d/4

x 414 WxSD+Qd x 414 (WxSD+Ql

(WXSD+QI

(wkSD+Q, +Qzx(l-=W

1 1 0

x 414

FIG. 5. The reworked

(WxSDSQ,

..

+Qzx(l--so> x(1-&/4

tWxSD+Qd x u-99/4 (WXSDSQI

+ Qz, x d/4

+ Qd x (1 +I/4

+ Qz) x Cl- 4)/4

(WxSDSQ, + Qa x (1 --SD))

(WxSD+Ql +Q~.x(l--sD))

0

1 1 1

0 0

(WxSD+Q, +Qzx(l-SD))

x 414

0

1 1 0

(QI + Qz) x (1-O (Ql+Qz

x(1 -SD)> x u-4 (WxSDSQ, +Qax(l-SD)) x (l-&/4 (Wx SD+Qd x (1 - W4

kw; ,“i7$tQ1

1 0 1

0 1 0

0

x (l-&/4

0 transformation

0 0

x (l-N4

0 0

matrix; T I,I,K:z, #,I = the mean number of colonies of

Simple example: find an expression equal to I/?- Xl for - co I X I co ; /I = 0, 1, using the operations +, -, and x . p-x/ Forall

Y=O,l;(Y=

=xx(p=o)+(1-x)x(#I=1). l)=

p-x1

(8.1)

Yand(Y=O)=(l-Y)

=Xx(1-B)+(l-X)x/?.

(8.2)

This expression can be simplified and indeed the method is justified only when the expressions generated can be simplified algebraically. Such is the case with the expression for TI, J, R; LI,8, ?. The equation we set up is : T I,J,K;~,B,Y

=

T o,o,o;o,o,ox(z=O)x(J=O)x(K=O)x(cr=0)x(j?=o)x(y=0) +To,o,o:o,o,I

+G,o,o;o,I.o

x (I = 0) x (J = 0) x (K = 0) x (oz= 0) x (j3 = 0) x (y = 1) x (I = 0) x (J = 0) x (K = 0) x (cl = 0) x (j? = 1) x (y = 0)

+ . .. ... ... ... ... ... .. .... ... ... ... ... ... ... .. .... ... .. .... ... .. ... ... ... ... ... ... .... .. .. . ... .. .... .. ... ... ... .. .... ... .. .... ... .. .... .. ... .... .. .... ... ... ... .. ... .... .. ... .... .... . ... .. .... .. ... .... .. ... ... ... ... ... ... .... .. ... .... .. ... ... ... .... .. ... .... .. .... .. ... ....

... ... .. .... ... .. .... .. ... ... ... ... .... .. ... ... ... ... ... .. .... ... ... ... ... ... ... .... ... .... x (I = 1) x (J = 1) x (K = 1) x (c1= 1) x (/I = 1) x (y = 1). +TI,I,I;I,I,I

(9)

KIN

301

SELECTION

0 0 1

1 0 1

0 1 1

1 1 1

0 0

0 0

0 0

0 0

(WxSD+Ql + Qa x (1 --SD)) x 414 (WxSD+Ql>

(WxSDfQl + Qz x (1 ---SD))

x /I4

X(1--6)/4

x(1+)/4

(WxSD+Qd

(WxSD+Qd

(WxSD+Qd

x 414

x d/4

x (1 -h/4

x(1--dY4

(WxSD+Ql + QzJ x 414 (WxSD+Q, i--4” (1 -SD))

(WxSDfQl + Qz> x (WxSD+Q,

d/4

;g4”

(1 ---SD))

(WxSD+Q, +Qz)x(l-O/4 (WxSD+Q, + Qz x (1 ---SD))

(WxSD+Ql + Qd x (l-014 (WxSDfQl +02x(1--SD)) x (l-&/4 (W+Qx+Qz x(1-SD)) x(1-4) (W+Qdx(l-4)

0

(W+Ql+Qa x(1-.SD))x4

0

W+QJx4

(WxSD+Ql + 122 x (1 --SD))

x (1 --d/4 0 0

;

T

Y I, J, K

0 0 0 0 0 1 0

(WxSD+Ql + Qa x (1 --SD))

1 0

0 1 1 1 0 0 1 0

1 1 0 1 1 1

genotype a, 8, y formed by the offspring of a female of genotype I, J when her pair-mate

is J, K.

If we simplify this we arrive at: T I,J.K;Q,B,Y

=

(Wx(SDx(Z-J)2+Zx.Z)+Q,+Q2x(l-(SDx(J-K)2+.Z~K))) xIjck$lx

i

i

A=0

B=O

[(r-(a-(zxA+.Zx(1-A)))2) x(1-(y-(zxB+Jx(1-B)))2)]/4.

(10) N represents numbers rather than frequencies, but this can be remedied by dividing the vector by its sum. 4’ is calculated by dividing the number of altruism-carrying ova (ova represented as zeros) by the total number of ova (males come from unfertilized eggs in the Hymenoptera). This is done by the formula:

i

i

I=0

i

K=O

J=O

.____ 2x

j.

j.

i

(Wx(SDx(Z-J)2+Z~.Z)+Q1+Q2~(1-(SD

A=0 x(~-K)2+~xK)))xN,,,Kx(1-(zxA+~x(1-A)))

K~o(~x(sDx(Z-J)2+ZxJ)+Ql+e2 X(1-(SDX(J-K)2+JXK)))XNr,J,K

1

I

(11)

1

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Now the model is sufficiently well defined to allow simulations Some form of shorthand notation is necessary

2. 3. 4. 5. and

“I SD J” means “SD x (Z-J)’ + I x J” “J SD K” means “SD x (J - K)2 + J x K” “IV’ means “Nr,J,K” “ ” means subtraction from one, Z SD J = 1 -I

6. “I”

in a computer.

SD J

means “(ZxA+Jx(l-A))“.

Thus

C[J=Nml Substituting for p gives

=,io j.

j:, “~~[(l-(sox(J-K)‘fJxK))

xN,,,,.x(l-(ZxA+Jx(l-A)))]. equation (11) for equation (6) into equation (7) and solving

If we copy out the algebra used to derive equation (12) from equation (7) and replace “p” with “l/K” and replace all the equal signs with greater-than signs, then copy this and replace the greater-than signs with less-than signs, we will have composed the proof that the inequality properties of the K-r rule apply to p, i.e. altruism is selected for when K > l/p and is selected against when K < l/p. It can be noticed that all the Qi’s have cancelled so there are no Q,‘s in equation (12). This does not mean that Q, does not affect the value of p. What it does mean is that the value of Q, this generation is not needed in order to compute p this generation. Numerical simulations show that the value of Q, this generation affects the distribution of the colony genotype frequencies next generation which in turn affects the value of p for that generation. 6. Resdts of Numerical

Simulations

If (IV, Q,, Q,) = (0, X, 0) for X > 0, an equilibrium will be approached. This equilibrium will be reached in two generations if the initial gene frequencies are the same in both sexes. If the two sexes differ in initial gene

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frequencies, then the equilibrium will be approached gradually as the two frequencies go through the damped oscillations around the effective gene frequency characteristic of a sex-linked locus in the absence of selection, Wright (1959). This equilibrium is the one we would expect among triploid units if we allowed a population in Hardy-Weinberg equilibrium to produce diploid entities only by selfing (by selfing we mean random pairing of gametes from the same individual) and then allowing these diploid entities to become triploid entities by fertilizing them with sperm from the population at large. It is only at this equilibrium that p and B will equal the r predicted from a pedigree, where B is Hamilton’s regression coefficient of relatedness. This equilibrium will in general be slightly deviated from, with Q, very high, or gene frequencies near fixation (of either allele) in the absence of selection. With additive gene effects (half dominance, i.e. SD = f), this equilibrium will also exist at intermediate gene frequencies with (IV, Q,, QJ = (p, X, 1) where p will be O-75 in our special case. With steady state equilibria reached due to making (W, Q,, Q,) = (p, X, 1) each generation for SD # $, p > 0.75 for SD < 3, and p < 0.75 for SD > 3. The maximum deviation

F+o. 6. Steady state equilibria. The many layers in this surface represent different values of SD. Only the top layer is shown in full. The equilibria are stable when the slope of the surface in the B, 4 plane is negative, and unstable when said slope is positive.

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occurs at frequencies of the altruism gene slightly less than 3, the exact position of the maximum deviation changing slightly with different values of SD. The deviations of p from O-75 decrease as Q,/Q, increases or as ISD-tl decreases. The positions of these equilibria are shown in Fig. 6. We should notice that these equilibria: (1) are such that p = B, i.e. Hamilton’s regression coefficient of relatedness; (2) are such that no matter what value of SD we supply to equation (12) the evaluated p comes out the same, and it is the true value of SD which gives rise to the colony-genotype frequency distribution which affects p; (3) they are approached at a rate of about one decimal place per generation. During selection, p alters only slightly from the equilibria shown in Fig. 6 if the selection is slow. During rapid selection for altruism the function of p against effective gene frequency becomes very close to linear, see Fig. 7. During rapid selection against altruism, with SD = 0, p goes very close to unity. This can be explained by a hysteresis effect, i.e. if altruism is being rapidly selected against in the 000 079

-

0’18

-

0.77

-

0.72

-

FIG. 7. Coefficient of relationship as defined in this paper, p, plotted against effective gene frequency, 4.. The graph is the plane QI = 0, SD = 0 from the space represented in Fig. 6. All that remains of the four-dimensional surface is the arch-shaped curve. The almost linear cluster of dots is a population undergoing rapid selection for altruism (w, QI, Qd = O-&O, 1).

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females, but not in the males, then the males will serve as a reservoir for the altruism gene. The ova will be much freer of the altruism gene than the sperm fertilizing them. So most of the altruism genes in females will occur in heterozygotes, who in the male haploid case will be members of colonies in which the other member is a heterozygote too. Normally p # B during selection. Updating K each generation such that K = l/p will cause the effective gene frequency to remain constant at all times during and after the generation in which the scheme is started. When p # B, updating K with values l/B each generation causes the effective gene frequency to go through damped oscillations, see Table 2. This cannot be explained in a simple way as a result of the hysteresis effect, since p = B during selection when SD = 0.5 and the hysteresis effect is still there. Substituting 0.5 for SD into equation (12) renders it a formula for B. P x B during selection when gene frequencies are near fixation or when selection is slow. p = B during selection: (1) If SD = 0.5. (2) If the hysteresis effect is removed from the model by changing it such that only virgin females survive to the onset of the next generation. These TABLE

Attempting

2

to stop selection by updating K each generation. The updating starts in generation 10. 1 using K = l/B. 2 using K = l/p Effective 1

O-I40329218107 0.5113387619303 057527279638 0.626304275755 0.6680380798497 0.7025137716597 0.7019338271057 0~702190652611 0.7020526253243 0.7021245698797 0.7020866281627 0.7021065206443 0*702096059018 0~702101552002 0.7020986654047 0~7021001816533

gene frequency 2 0440329218107 05113387619303 0.57527279638 0.626304275755 * 06680380798497 0.7025137716597 0.7025137716597 0.7025137716597 0.70251377166 0.7025137716593 0.7025137716593 0.702513771659 0.7025137716593 0.7025137716593 0.702513771659 0.7025137716587

Generation number 5 6 7 8 9 10 11 12

:i 15 16 17 18 19 20

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produce sons by arrhenotokous parthenogenesis, the females then mate at random with the generation of their sons to produce the next generation’s females. This is implemented by substituting 4’ for (p into equation (10) and leaving the other equations unchanged. This suggests that for animals in which there is no sexual dimorphism with regard to the incidence of altruism, e.g. termites, p = B. This may also be the case in Pharaoh’s ant Monomorium pharaoh, whose colonies are sometimes founded by workers carrying a young larva which becomes a virgin queen who mates with her parthenogenetically-produced sons (Wilson, 1971) but for most Hymenoptera the model presented here is more realistic. 7. Definitions of Altruism In the sixties Hamilton (1964), West (1967), and others, defined altruism as decrementing one’s personal fitness to increment the fitness of others. This was often taken to mean losing descendants in order to gain additional collateral relatives. Such an interpretation would preclude as altruism such cases as: (1) maternal care, (2) producing fewer daughters in order to have more grandchildren, (3) raising daughters instead of sisters in a male-haploid species. A more fitting definition for altruism would be: To prevent or discontinue the existence of individuals in whom one’s genotype is less diluted, in order to bring about or continue the existence of individuals in whom one’s genotype is more diluted. Dilution is used in the sense of Hamilton (1964) and is not with water but with random genes from the population at large. It seems plausible, if not obvious, that selection for altruism depends on K and r, or B or p, regardless of the positions on a pedigree. Helping a sister (r = 0.5 for male-diploidy) at one’s expense is as adaptive as helping a daughter (r = 0.5) of the same reproductive value. In this context raising sisters in a male-haploid species, r = 0.75 instead of daughters, r = 0.5, is selfish, not altruistic. More significant is the fact that giving up a twin to raise children is altruism, so is the similar case of abandoning asexual reproduction for sexual reproduction. This suggests three possibilities : (1) All altruism may be selected for even more readily than K and p predict due to the advantages of recombination. Just as it is fitter to produce ova which becomes zygotes, r = 0.5, than it is to produce eggs by the thelytoky even when K = 1, so too, may it be fitter to raise nieces instead of daughters with lower values of K than we might expect. (2) Values of SD < 0.5 may make sexual reproduction more adaptive than otherwise expected.

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(3) Obligate thelytokes may benefit the most from the variation among them due to mutation by being altruistic to their collateral relatives. Given the existence of some heterozygosity due to mutation in an asexual or thelytokous species, one can regress one genotype on another. Between very close relatives B would equal one, between members of pairs picked randomly from the population, B = 0. Somewhere in between B must equal 3. Since sexual reproduction is selected for with K = I and r = 3, altruism might be selected for with K = 1 and r = 3 as well. It is easy to imagine that some values for population size, viscosity of the population, and mutation rate must exist such that altruism to a nearest neighbour with K = 1 could evolve in an asexual or thelytokous species, or between members of adjacent inbred colonies with a viscous dispersion of colonies in time and space. Trivers (1971) considers as altruism only those cases involving interactions between non-related individuals. He feels, that since helping relatives already benefits the helper by incrementing his inclusive fitness, there is no altruism. Trivers’ “ altruism” to non-relatives is selected for due to the eventual reciprocation of the beneficiary. In making this point Trivers fails to see the fact that there exist two systems of language for describing selection on altruism. These are mentioned briefly in the literature already (Hamilton, 1964, p. 3-5, 1970; Stern, 1970). Both “languages” say the same thing, but say it in different ways. One involves correlated events the other cause and effect. An individual who is reared by someone other than its parents, cannot be counted as an increment to the fitness of both the parents and the altruist. If it is counted as an increment of fitness to the altruist, then to be consistent, all other altruistically-reared offspring are counted as increments to the fitnesses of those who reared them. If it is counted as an increment to the fitness of its parent then all altruistically-reared offspring must be so counted. If an altruistically-reared individual is counted as belonging to the fitness of the altruist then the size of the increment equals p, where p is the coefficient of relationship between the parent of the altruistically-reared individual and the altruist who did the rearing. If the increment in fitness belongs to the parent then every offspring counts as an increment of value unity. In this way of looking at it, the altruist gains nothing as a consequence of his altruism but correlated with it is a probability p that someone will rear one of his offspring, or an offspring of someone of his genotype. In the other way of looking at it, he gains an immediate increment p, and if someone does rear extra offspring for him, these don’t increment his fitness. Thus an individual’s expected fitness is a constant function of his genotype regardless of which language system we use.

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Assortative mating has a similar effect as inbreeding, so by analogy recognition of altruists by their altruism should have a similar effect as that of recognition of altruists by their closeness to oneself (if oneself is already an altruist) on a pedigree. Limiting one’s altruism to fellow altruists is the mechanism which causes selection for altruism. Maynard Smith (pers. comm.) calls this assortative altruism. Emlen (1970) looked at assortative altruism, and considered the increments in fitness belonged to the altruist. Trivers (1971) looked at assortative altruism and considered the increments in fitness belonged to the parent. Altruism to km and to potentially altruistic non-kin can be looked at in both ways. This doesn’t alter, in any way, the validity (or invalidity) of Trivers’ other conclusions. Lin & Michener (1972, p. 142) felt that kin selection was not important in the evolution of sociality in communal sweat bees. This is because the altruists, themselves, are reciprocated and reproduce eventually. The fact that many worker social insects have the apparatus to reproduce, and sometimes use it, is taken to suggest an assortative altruism route to sociality. When a polyphenism exists between altruists and beneficiaries it is easier to count increments in fitness as belonging to the altruists. When there is no such polyphenism, or if the polyphenism is ill-defined, then we may wish to count the increments in fitness as belonging to the parents. This doesn’t change the significance of instantaneous values of K and p, nor does it say whether altruists are recognized by their altruistic behaviour, or by their proximity on a pedigree. Confusing assortative altruism with a lack of a reproductive division of labour is like confusing asexual reproduction with isogamy. Now we have two ways to compute an altruist’s fitness. These give the same result for expected fitness, but an individual’s estimated fitness may be expected to be different when computed by these two methods. Both estimates are based on physical reality. One counts how many children one has produced, the other counts genes identical by descent to oneself produced. Many people have observed that a person receives good or bad retribution according to the goodness or badness of his own deeds. This retribution may be a purposeful reciprocation on the part of others, or may be a consequence of the same laws of psychology, biology, physics, chemistry, etc. which caused the original act. In this case there will be a correlation between the nature of acts a man commits to others and those others commit to him. This is called the Law of Karma and is important in Hinduism and Buddhism. Chinese Buddhists call it “Yin kuo hsiin huan”, which means “the circulation of cause and effect”. The Buddhists like to think in terms of cause and effect, although they understand well the principles of correlation which make the Law of Karma work. Still they find it desirable to say a

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man’s good deed causes a good thing to happen to him. If the retribution is at the moment unfulfilled, the good thing is the good “Karma” which the man has gained. Some western thinkers find it just as hard to accept replication of genes identical by descent in a collateral relative as a fair compensation for a failure of reproduction, as they find good “Karma” as fair compensation for a failure of a beneficiary to reciprocate. A third system analogous to the other two is quite acceptable in western society, that is double entry book-keeping. A man who has credit is quite happy. Credit is analogous to good “Karma” or an inclusive fitness higher than your classical fitness. A debit is analogous to bad “Karma” or an inclusiveness fitness lower than your classical fitness. It is interesting that giving to charity increases your credits and becoming wealthier by accident increases your debits. These credits and debits are to “nature”, and are supposed to cancel to zero on the average, and are treated as if they were retribution for one another. The author would like to thank the following persons: Dr L. C. Cole, Dr Melvin Kreithen, Dr Henry D. Block, Dr Howard Holland, Dr Frederick E. Warburton, Dr Simon A. Levin, Mrs Seamon, Mrs White, Miss C. M. Collins, Mrs Van Order, Dr W. T. Federer, Mr J. Coiket, Dr Bruce Wallace, Dr William L. Brown, Jr, Dr Peter Brussard, Dr John Emlen, Dr Vernon Wright, Dr George Murdie, Miss C. Martinez Clark, Dr Eong Bok Tay, Prof. J. Maynard Smith, Prof. T. R. E. Southwood, Dr W. D. Hamilton, Dr M. J. West Eberhard, Mr C. Gliddon, MS H. Pope and Mrs J. Atherton. REFERENCES BERRY, P. (1969). APL\360 Primer, form number GH20-0689. Program numbers 5734, 5736 x II.IBM. BOORMAN, S. A. & Lm, P. R. (1973). Proc. natn. Acud. Sci. U.S.A. 70, 187. CROZIER, R. H. (1970). Am. Nat. 104, 216. DOICTI?R,F. &S TElNHAmR, C. J. (1973). Digital electronics, pp. 100-101. London, Basingstoke: Macmillan. EMLEN, J. M. (1970). Ecobgy 51,588. ETHEL, I. (1972). Z?teor. Pop. Biol. 3,258. FALCONER, P. S. (1961). Introduction to Quantitative Genetics. London: Oliver and Boyd. HALDANE, J. B. S. (1955). New Biol. l&44. HAMILTON, W. D. (1964). J. theor. Biol. 7, 1, 17. HAMILTON, W. D. (1970). Nature, Land. 228, 1218. HMLTON, W. D. (1971). Chapter II, pp. 57-91. In Man and Beast: Comparative Social Behuviour (J. P. Eisenberg & W. S. Dillon, eds), chap. 2, pp. 57-91. Washington, D.C.: Smithsanian Press. HAMXLTON, W. D. (1972). A. Rev. Ecol. Syst. 3,193. IvERsoN, K. E. (1962). A Programing Language. New York: John Wiley and Sons, Inc. KEMPTHORNE, 0. (1957). An Introduction to Genetic Statistics. New York: John Wiley and Sons, Inc. LEVINS, R. (1970). In Some Mathematical Questions in Biology, pp. 77-107. Providence, Rhode Island: American Mathematical Society.

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LI, C. C. (1955). Population Genetics, pp. 18-19. Chicago: University of Chicago Press. LIN, N. & MICHENER, C. D. (1972). Q. Rev. &of. 47, 131. MALECOT, G. (1969). The Mathematics of Heredity. San Francisco: W. H. Freeman. MAYNARD SMITH, J. (1964). Nature, Lo&. 201, 1145. MAYNARD SMITH, J. (1966). Tie Theory of Evolution. London: Penguin Books Limited. ORLO~E, M. J. (unpublished). A contribution to the biology of the great carpenter bee Xyfocopu virginicu (L.) (Anthophoridae: Apoidea: Hymenoptera); geographic variation in courtship: mimicry: rearing methods. STERN, J. T. JR. (1970). Evol. Biology 4,39. TABER, S. (1955). J. econ. Ent. 48, (5), 22. Tmv~as, R. L. (1971). Q. Rev. Biol. 46, 35. WELLS, H. G., WELLS, G. P. & HUXLEY, J. (1931). The Science of Life. London: Casseil. WEST, M. J. (1967). Science, N. Y. 157, 1584. WEST-EBERHARD, M. J. (1969). Misc. PubI. Mus. Zool. Univ. Mich. 140, 1. WEST-EBERHARD, M. J. (1974). Q. Rev. Bill. (in press). WIL.KIN, E. 0. (1971). Tie Insect Societies. Harvard: The Belknap Press of Harvard University Press. WRIGHT, S. (1959). Genetics of Populations. In Encyclopuediu Britannica. London: Encyclopaedia Britannica Ltd.