THEORETICAL

Conditions

POPULATION

BIOLOGY

10,

1-9 (1976)

for Genetic Polymorphism in Social Hymentopera under Selection at the Colony Level* R. H. CROZIER+ Department

of Zoology,

University

of Georgia,

Athens,

Georgia

AND P. c. CONSUL* Department of Mathematics, University of Calgary, Received

Statistics Calgary, September

and Computing Science, Alberta, Canada 4, 1974

Selection at the colony level in social Hymenoptera with colonies containing single, once-mated queens is examined under a simple two-allele model. The condition for balanced polymorphism is x

> 2VZ/(V

+ l),

where V is the fitness of colonies with all workers homozygous and X that of colonies with both heterozygous and homozygous workers, relative to the fitness of colonies with all workers heterozygous. For certain fitness combinations satisfying the above relationship and characterized by values of V and X much lower than one, iteration reveals the development of stable limit cycles of allele frequencies rather than convergence to an equilibrium point. Addition of a third allele, or overlap between generations, eliminates these cycles. Queen-level overdominance is sufficient but not necessary for balanced polymorphism when V < 1, is both sufficient and necessary when V = 1, and is necessary but not sufficient when V > 1. Colony-level selection is a potentially powerful force maintaining genetic variation in populations of social insects, but does not imply correspondence between queen and worker genotype frequencies.

* This work was supported by USNSF grant GB 38778 awarded to the senior author. + Present address: School of Zoology, University of New South Wales, Kensington, N.S.W. 2033, Australia. * Collaborated in this work during sabbatical leave spent in part at the Computer Center, University of Georgia.

1 Copyright 0 1976by Academic Press,Inc. All rights of reproduction in cby form reserved.

CROZIER AND CONSUL INTRODUCTION

In Wilson’s (1968) analogy, the colony of an ant or other social insect species can be thought of asa fortified factory dedicatedto the production of queensand maleswhich will disperseand found other such factories. In many speciesthe bulk of the colony is made up of workers which make no or very little direct contribution to the next generation (Wilson, 1971; Crozier, 1975). However, these nonreproductive individuals do have a crucial impact in that it is through their efforts that the colony-factory is able to survive and rear reproductives. The genetic composition of this work force will therefore be a potentially crucial factor in determining the successor failure of the colony, and we can distinguish another level of selectionin social insects-that of the colony. If colony-level selectionis a potentially significant component of evolutionary processesin social insects,what will be the dynamics of selection at this level? A specific question, which we treat below, is that of the conditions for balanced polymorphism due to selectionat the colony level. Such selectionis equivalent to selectiondiffering with specificmating combinations.The casefor male-diploids, such as termites, falls under the general casefor mating-type selection, which appears to be intractable (Bodmer, 1965). However, the male-haploid case pertaining to the Hymenoptera is tractable under a simple model.

THE MODEL

We consider a speciesin which each colony is the progeny of a single, oncemated queen. All fertile eggsare laid by the queen, the workers having no descendants. Males are, of course,haploid and impaternate. There are probably many specieswith identical or similar characteristics, although other life-cycles also abound (Wilson, 1971; Crozier, 1975). A single,two-allele locusis subject to colony-level selectionunder the following schemeof relative fitnesses: Males A, 4 -_____ 44 Queens A,A, 4%

v, 1 Xl X2 1 vz

Colonies resulting from A,A, x A, matings have the samefitness as those from Adz x A, matings becausethey both produce worker forces consisting entirely of A,A, individuals. Generationsare discrete.

COLONY-LEVEL

3

SELECTION

The analysis below is that of the special case where VI = Va = V and X, = X, = X. This case brings out certain biologically significant features of the model and colony-level selection; the general case appears to have no simple solution.

THE

ANALYSIS

Let Q and 1 - Q, respectively, be the frequencies of A, and A, males. Let R, 2S, and T denote the frequencies of A,A, , A,A,, and A,A, females, respectively. Finally, let W be the average fitness of the population, here defined as C fiwi , where fi is the frequency of colony type i and wi is its fitness as defined above. At equilibrium, the stationary genotypic frequencies are given by: R

=

QW' w+ sx)

=

(RV

+

i.e., P =

-Q>

T = (TV + SX)(l w Q

R

,

W+SX) w

i.e., T )

,

(1)

= TV+SV w

(2)

’ 1-Q SX)Q + (R + S-W - Q>= R + (R + sx)(l - Q) W

i.e., m

Q

= -

w

R

RfSX w

-

1-Q

’ (3)

’

and

,,=(R+SX)(l-Q)+(T+SX)Q

=Q-R+(~+SX)J?

W

w

.

Since R + 2s + T = 1, the above relation becomes 1- Q By subtracting

T = (T

+Gx'Q ,

.

(4)

(1) from (3) and dividing by R, (1 -QQ)R

By subtracting

T+SX i.e., L...I$L _ _ ; = w

-

1-v Q(l !- Q) = w

’

(5)

(2) from (4) and dividing by T, --1-Q

QT

1-v Q(lyQ)

=w’

(6)

4

CROZIER

Evidently,

AND

CONSUL

by (5) and (6)

Q”

Now, dividing have

1-

=

Q

(1 which

2kQ25’+ XL1 - HQ2 + (1 - Q)“H 2k( 1 -

By cross multiplying

Q)"V +

2Q)WQ(l

I-

-

Q) V

+

we get

(8’

two independent 2Q = 0,

- k(Q2 + (1 - Q)“}] ’

X[l

and simplifying,

gives the following

= ’ (saY)* c7)

:Q)2

the values of R, T, and S from (7), we

(I) by (2) and substituting

8

&

+ (1 -

Q>“)X>-

X] =

0

cases for equilibrium:

i.e. Q = 1 - Q = =$. .. .

(8)

and

k = X/M

(9)

where M = 2Q(l Case 1.

- Q) V + [l - 2Q(1 - Q)]X.

(10)

When Q = 4, the Eqs. (7) provide R=Tz+-S

which obtain

(11)

make Eqs. (1) and (3) identical with Eqs. (2) and (4), respectively. a relationship between V and X, we divide (1) by (3) and obtain

To

RV+SX R+SX I.e.,

which

gives the quadratic

equation

X * (S/R)2 - (X - V)(S/R) - 1 = 0. Since a negative value of (S/R) is inadmissible, feasible solution, namely:

the above equation

S/R = ((X - V) + ((X - V)* + 4X)la)/2X.

(12) has only one

(13)

COLONY-LEVEL

5

SELECTION

This solution holds for all positive, nonzero, values of X and V. Therefore, from (13), (8), and (ll), populations with allele frequencies of 0.5 in both sexes will maintain these frequencies, in the absence of disturbing factors, for any feasible set of colony-fitness values. However, as will be made clear below, this point is one of stable equilibrium only for particular values of X and I’. Case II. When R = X/M where M is given by Eq. (lo), the values of R, T, and S are given by (7) and (9) as R = Q=X/M

(14)

T = (1 - Q)2X/M

(15)

S = Q(l - Q)V/M.

(16)

Further, by (2) and (3), T TV’SX’

Q--R R+SX= Substituting

from (14), (15), and (16) and simplifying

QX+(l

M-QX -Q)VX

1-Q

=

= (1 -Q)V+QV

By (10) and by cross multiplication,

-.1-Q V

the above relation gives

(1 - Q)[2QV2 + VX - 2QXV - QX - (1 - Q) VX] = 0, i.e., (1 -Q)Q[2V2..

Q = 0,

Q = 1,

VX-X] or

=0 X = 2V2/(V + 1).

(17)

It can be easily seen from (14), (15) and (16) that the first two are the trivial degenerate solutions. Equation (17) yields a curve with two sections. Only that portion of the curve where both X and V are positive is in the “real world” and need be considered further. There are no obtainable unique values for R, S, T, and Q from (17), there thus being an infinite array of stationary genotype frequencies for sets of fitnesses satisfying this solution. However, iteration based on equations (l), (2), and (3) reveals that (17) defines the boundary of a region of fitness values yielding stable polymorphisms (Fig. l), so that the condition for balanced polymorphism under this model can be seen to be x > 2vqv

+ 1).

(18)

Iteration reveals an unexpected feature in the fitness space, namely, a region

1.0

23

.6

X .4

.2

(

FIG. 1. Relationship of colony fitness morphism resulting in either an equilibrium frequency. Further discussion in text.

values with occurrence gene frequency or stable

of balanced polyoscillations of gene

.65 0 FEMALE 60

A MALE

GENE GENE

FREQUENCY FREOUENCY

t

1

0 OVERALL

GENE

FREQUENCY

GENERATION FIG. 2. 0.4 in both

The course of selection sexes. Further explanation

when V = X = 0.1 and initial in text.

gene frequencies

are

COLONY-LEVEL

SELECTION

7

yielding stable limit cycles rather than a single equilibrium point. The result of an iteration using some fitness values within this region is shown in Fig. 2. Stable limit cycles of gene and genotype frequency due to colony-level selection in social Hyrnenoptera seem likely to be rare for two reasons. FirstIy, the values of V and X required are extreme (Fig. 2). Secondly, slight modifications of the model eliminate such cycling. Thus, iterations using fitnesses from the stablelimit zone but involving a carry-over of 10% of colonies from one generation to the next, or involving three alleles rather than two, did not show stable limit cycles. Three-allele iteration indicates that, whereas two-allele systems under this mode1 have either stable equilibria with allele frequencies of 0.5 in both sexes or stable limit cycles around this point, n-allele systems have stable equilibria with each allele having a frequency of I/n. Considering the possible extreme values of the homozygote queen fitnesses, overdominance at the queen level can be defined as either: (a) X greater than the mean of the homozygote fitness components, X > $( V + 1); or (b) V

i.e.,

X greater than the highest homozygote fitness component, i.e., when landwherrY> 1,X> V.

When I/ == I, (1/2)(V + 1) = 2P/(V

+ 1) = 1

so that queen-level overdominance is both a necessary and a sufficient condition for balanced polymorphism. When V < 1, let V = 1 - k, where 0 < k < 1. Therefore, by definition: 1 - &k > (1 ~- @)a > (1 - k)2. Dividing

by 1 - +k, 1 > 1 - (l/2) k > 2(1 - k)2/(2 - k).

Thus, under both overdominance definitions, when V < 1, queen-level overdominance is a sufficient but not necessary condition for polymorphism. When V > 1, let V = 1 + c, where c > 0. Then, by definition: (1 + d2 > (1 + c)(l + $4 > (1 + w. Dividing

by 1 + +c, 2(1 + W(2

+ 4 > 1 + c > (l/2)(2 + 4.

Therefore, when V > 1, queen-level overdominance is a necessary but not a sufficient condition for polymorphism under either definition of overdominance.

8

CROZIER

AND

CONSUL

DISCUSSION

Under the colony-level selection model considered, the fitness of a homozygote queen depends on the genotype of the male with whom she mates. The queen homozygote fitnesses are frequency-dependent, approaching (V + I)/2 as the allele frequency approaches 0.5. The behaviour of n-allele systems in equalizing allele frequencies at l/n under conditions equivalent to (18) shows that colony-level selection resembles the case of sex-determining loci (Crozier, 1971) in its characteristics and is a potentially-powerful force maintaining genetic variability, both directly and through associative effects. Although the colonies of most social insects are perennial entities, some, such as those of cool-temperate social wasps, last but one season (Wilson, 1971). Stable-limit allele-frequency cycles are thus more likely to occur in these social insects than in others, but the extreme fitness values required for stable oscillations (Fig. l)make it unlikely thatsuch cycling will be commonly found eveninthiscase. Queen-level heterosis undercolony-level selectioncould arisethrough a number of different mechanisms. Where worker task specialization has a genetic basis, the colonies founded by heterozygous queens would have a broader range of worker competencies than monomorphic colonies and hence higher fitness. Another possibility would arise when different worker genotypes have different susceptibilities to different environments. For example, one genotype might be highly susceptible to dessication, another to cold. Where colonies face both challenges during their lifespan, those comprised of two worker genotypes are more likely to weather every crisis with an adequate worker force than those comprised of just one. Under neither of these possibilities is there any guarantee that queen-level heterosis would be reflected in detectable heterozygote excess among workers. Thus, in colonies of heterozygous queens, their heterozygote worker offspring might be more responsive to tasks such as defense and foraging than homozygous workers, leading to a deficiency of heterozygotes in such colonies. The lack of any necessity for correspondence between worker and queen heterozygote excess under colony-level selection underscores the conclusion (Crozier, 1970) that the practice of using worker genotype frequencies as a basis for speculation about the mode of selection operating (Johnson et al., 1969; Tomaszewski et al., 1973) is erroneous. In fact, one striking case of noncorrespondence between worker and queen genotype frequencies has been reported (Crozier, 1973). Although analytical solutions do not seem readily obtainable, colony-level selection would seem likely to be important in more complex social systems than examined here. This possibility raises a number of interesting suggestions. Where polyandry occurs, there should be an optimum mix of male genotypes for each queen genotype. In the case of multiple-queened colonies, certain combinations would yield colonieswith higher fitnessesthan others.

COLONY-LEVEL

SELECTION

9

Colony-level selection can be tested for readily, given an appropriate species and appropriate circumstances. Such a situation would be one where a population of colonies can be monitored as it ages, and any differential mortality between colony types detected.

REFERENCES BODMER, W. F. 1965. Differential fertility in population genetics models, Genetics 51, 41 l-424. CROZIER, R. H. 1970. On the potential for genetic variability in haplodiploidy, Genetica 51, 551-556. CROZIER, R. H. 1971. Heterozygosity and sex-determination in haplo-diploidy, Amer. Natur. 105, 339-412. CROZIER, R. H. 1973. Apparent differential selection at an isozyme locus between queens and workers of the ant Aphaenogaster rudis, Genetics 73, 313-3 18. CROZIER, R. H. 1975. “Animal cytogenetics 3 Insecta (7), Hymenoptera,” Gebrtider Borntraeger Verlagsbuchandlung, Stuttgart. JOHNSON, F. M., SCHAFFER, H. E., GILLASPY, J. E., AND ROCKWOOD, E. S. 1969. Isozyme genotype-environment relationships in natural populations of the harvester ant, Pogonomyrmex barbatw, from Texas, Biochem. Genet. 3, 429-450. TOMASZEWSKI, E., SCHAFFFZR, H. E., AND JOHNSON, F. M. 1973. Isozyme genotypeenvironment associations in natural populations of the harvester ant, Pogonomyrmex badius, Genetics 75, 405-421. WILSON, E. 0. 1968. The ergonomics of caste in the social insects, Amer. Natur. 102, 41-66. WILSON, E. 0. 1971. “The Insect Societies,” Harvard Univ. Press, Cambridge, Massachusetts.