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Is there a cost of meiosis in life history?
J. theor. Biol. (1985) 116, 613-623
Is There a Cost of Meiosis in Life History? ROBERT A B U G O V
Department of Ecology and Evolutionary Biology, Northwestern University, Evanston, Illinois 60201, U.S.A.t and Laboratory of Genetics, 509 Genetics Building, University of Wisconsin, Madison, Wisconsin 53706, U.S.A. (Received 14 February 1985, and in revised form 1 May 1985) Individuals who survive pass their entire genome to the next time period, while those who reproduce sexually pass only half of their genome. This observation has led to a "cost of meiosis" argument for life history, in which increased sexual reproduction can evolve only if the incremental gain in number of offspring more than twice exceeds the concomitant loss to survival. The age-structured genetic model developed here indicates that present measures for the cost of meiosis in life history obtain only if the number of progeny each living individual produces per unit time is independent of his or her age. For systems which meet this biologically important assumption, an approach to life history is delineated which bypasses biological restrictions from traditional models of age-structured selection.
In his classic treatise on the evolution of sex, Williams (1975) noted that alleles for sexual reproduction bear a 50% risk o f loss during the formation of meiotically produced gametes. But alleles which successfully express asexual reproduction do not encounter this risk: sexual reproduction should therefore be able to endure only if offspring produced sexually are at least twice as fit as those produced asexually. Waller & Green (1981) have expressed surprise that William's "cost of meiosis" argument, summarized above, has not had a great impact on life history theory. In particular, they intuitively argued that a gene which allocates more to sexual reproduction should be favored only if the gain in sexually produced offspring is more than twice the concomitant loss to survival and asexual reproduction, since each sexually p r o d u c e d offspring contains only one-half o f the parental genotype. This appears to contrast with current life history theory (reviewed in Charlesworth, 1980), which seems to give equal weight to female survival and fertility. Nevertheless, Charnov, Bull & Mitchell-Olds (1981) have corroborated Waller & Green's t Present address. 613
0022-5193/85/200613+ 11 $03.00/0
© 1985 Academic Press Inc. (London) Ltd
614
R. ABUGOV
argument with a preliminary genetic analysis in which the recursion equations were limited to fixation equilibria in a two stage life history when only one of the life history stages (the youngest) is present. The present analysis extends that of Charnov et al. (1981) to provide a global analysis o f gene frequency change with multiple age classes. In the process, an alternative to traditional models for selection with age structure will be delineated.
Analysis The analysis will first use Price's (1970, 1972) algebra of covariance to determine general conditions under which results from single age class models apply when selection acts in populations which are age structured. Next, global analysis for a single age class model is performed, and the earlier results are applied to produce an age structured model. SELECTION
IN SINGLE
VS M U L T I P L E
AGE
CLASS MODELS
Consider a "progenitor" population composed of individuals of various ages at time t who, via survival and reproduction, give rise to a " p r o g e n y " population at time t + 1. Progeny, here has a special meaning: it includes not only an individual's offspring, but also includes himself, if he survives. This definition is useful in a genetic sense because, when the generations are not discrete, survival provides a means of transmitting genes to the next time period. Also define the frequency of allele Ai at a single locus as the frequency of that gene in the various age classes, averaged over individuals from all age classes. Thus, if we let xt be the frequency of allele As at time t, if we let Co be the relative frequency of age class a at time t, and if we let xi,, be the relative frequency of allele A~ within age class a at time t, then
x, = Z Cox,.a.
( 1)
a
Let x~ be the frequency of allele Ai in the population at time t + 1. It will be the number of A, alleles produced by the progenitor population of equation ( 1 ), divided by the total number of alleles the progenitor population produces. Thus if we let x[o be the frequency of allele A; among the progeny from age class a, and if we let Go be the number o f genes each gene from age class " a " passes on during this time interval, we obtain
Z CoGox',,o x', Y c . Q a
a
(2)
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After defining AX~.a=X~.a-x~.~, the covariance algebra of Price (1970 equation (4), 1972 equation (All)) can be used (see Appendix 1 here) to show that
Z CaG. ZXx~.,, Ax, -- ~
t COV (G, x,)
Z CoGa
(3)
Z CoGo ' a
where the covariance is the product of the deviations averaged over all age classes. After defining the average change in gene frequency as weighted by progeny number to be
CaG~ Axi.~ AX
i __ a
Z CaGa a
and after defining d as Za CoGo, we can rewrite equation (3) as ~x~ = (,~x~.) 4 COV (~ (G, x~)
(4)
We thus see that selection with age structure may be divided into two components: selection within age classes, and covariance between gene frequency and progeny number. As an intuitive example, let us say that older maples produce more progeny than younger maples and that, because of previous selection on survival, older maples also have a higher than average frequency of allele A~. Then even if selection is completely relaxed for a time unit (with Axe,, = 0 for all age classes), gene frequency will increase because progenitors of higher gene frequency (i.e. older maples) tend to produce more offspring. We shall now perform a more detailed analysis to develop an overall picture of age-structured selection as defined in equation (4). SELECTION WITHIN SINGLE AGE CLASSES In this section we shall analyze the first component of equation (4), i.e. the average from the various age classes of the gene frequency difference between progenitors and progeny. As suggested by equation (3), we shall first find the AX~,a for any given age class and then average that result over all age classes. The treatment here will include non-random mating with respect to age, maternal and paternal effects of fertility, sexual selection, and differential survival according to sex, genotype, and age. Throughout the analysis, we shall assume that selection is weak in a population which
616
R.
ABUGOV
is without assortative mating and which is comprised of relatively few age classes, assumptions which render terms on the order of the selection coefficient squared negligible as compared to other terms which appear in the recursion equations for gene frequency change. We shall first derive the equations for dioecious populations and follow with a treatment of selection with monoecy.
Selection within a single age class: Survival Let N(a) be the total number of individuals in age class a, within which proportion F~ are female and proportion M~ are male. Further, let W~,~ and W~.~ be the survival rates of AiA~ among females and males within that age class, and let x~j.abe the ordered frequency of genotype A~Aj within that age class. Then the total number of adult "progeny" of ordered genotype AiAj who survive into the next age class will be
No( a + 1) = N ( a )( Maxo.,~W~.a + Faxij.aW~,a).
(5)
Define W~a = ~j Xij, a Wij, a/Xi, a, and define N~ = 2 Y.j N~. Then summing equation (5) over all j shows that the number of A~ alleles which survive into the next age class is t d~ N,(a + 1) = 2N(a)x,.,~(MaWi.,~ + Fo Wi.,,).
(6)
Defining the mean of any variable K over both sexes a s / ¢ = MK¢'+ FK ~, we obtain N~(a + 1) = 2N(a)xo~VV~.,, (7) The total number of alleles surviving to the next age class can be gained by summing equation (7) over all i, to yield
N'r(a + 1) = 2 N ( a ) IV~
(8)
where l~z~= ~ x~.al~'~.a.
Selection within a single age class: Fertility Consider a female within any particular age class a. She may mate with males of various ages and genotypes, and we can define the "mating types" among the females of a particular age class according to the genotype of the female and the age and genotype of her mate. Any particular mating type can thus be denoted by AkAt × AjA,,~, where the first genotype is that of the female, and where the " a " denotes the age of the male. Further, define f~a.j,,~ as the frequency of mating type AkA~ x A j A ~ among the females from our fixed age class. Then
f~,.j,, = ~ fkl,jma a
(9)
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is the frequency of mating type A k A t ×A~Am among the females of our fixed age class. Now, let bkij,.a be the total number of offspring per mating type A k A I x AjA.,~, equal to the number of gametes each individual in that mating passes on to his or her offspring. Also, let Q~.k~ be the frequency of males of age " a " among the mates of the A k A t females from our fixed age class. Then the mean number of gametes a female involved in that mating type passes on will be b 9kl,i,. = ~ Qa, ktbkld,.~ .
(10)
a
The total number of gametes that females from a given age class produce via AgA~ × AjA~., matings is then 9
(11)
N ( a ) F f kld,. b kl,i,. .
Now, define Xi.kOm as the probability that any given gamete passed to an offspring from mating type A k A ~ × A~A~ contains allele A~. Then the total number of Ai alleles passed down to the next generation from consideration of females of age a and their mates is 2N(a)F
~ f ~gom b k l j m X i ,
kljm .
kl, j m
A "2" precedes the N ( a ) here because each diploid offspring contains two alleles. The total number of Ai alleles passed down to offspring which are attributed to the female parent will be half of this, equal to b 9i,a -- - N ( a ) F
E f 9kombkomX~,kli,. , kl, jrn
(12)
with the other half being attributable to the male. We may abbreviate the mating type designations kl, j m with r's, shortening equation (12) to b ~i.~ - N ( a ) F S " ,., jr.~ f~ ~. l, ~ . .x. . .
(13)
r
The notation of equation (13) will be used henceforth because it is consistent with previous related studies (Abugov, 1983, 1985). A similar treatment, but for males, yields bq| , a = N ( a ) M
'~ E f '~ ,.~b,.,x,.~.
(14)
r
Note that the "b," takes into account that males of our particular age class may father more than one set of brood, since it counts the number of offspring fathered.
618
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A B U G O V
Summing equations (13) and (14) shows that the total number of alleles passed to newborns from age class a is
b,~) = N ( a ) Y, [La + O( ¢ ) ]xi,~( Fb ~. + Mb ~o).
Ai
(15)
r
The f~.a term here is that expected from the Hardy Weinberg distribution. The O(sr) term, then, is on the order of the selective coefficient, and it results from selection induced sex specific differences in genotypic frequencies. Averaging the br.a's in equation (15) over both sexes yields
bi(a) = N ( a ) E [La + O(~)]xi.f~,~.
(16)
r
The total number of alleles passed to the newborns from both males and females here is bi(a) summed over all i which, from equation (15), is
b(a) = N(a) Y. [f~,,, + O(~')]6r,,,.
(17)
r
Fertility and survival selection combined within a single age class Summing equations (7) and (13) shows that the number of which originate from age class a is
N~(a)=2N(a)x,,aff'~,~+ N ( a ) ~,[f~,~+O(~)]xh,br, a.
Ai alleles (18)
r
A "2" comes into the survival but not the fertility term because each parent donates only one allele to each of its offspring, while each survivor carries both alleles. The total number of alleles passed down to the next time period from this age class is the sum of equations (8) and (17), which is
2N(a) ffVa+{ N(a) E [La + O(r~)]6.,a}. The relative frequency of given by
(19)
Ai among the progeny of age class a is therefore
2Xi,alYCha+2 [f.,a + O(~) ]Xi,rG~,. t Xi,
a
__ --
r
2 fro +y~ [f~.o + o(¢)]6r, o r
(20)
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The change in gene frequency, x~,.- x~.~, is then 2x,.a( Wi,~ - lYga)+ ~Aabr, a(X~.r- Xi,a) r
ax,,o =
2 ~i,o + £
~ 0 ( ~ ~)
(21)
where the average birth rate for each age class is defined as 6~ =E.f~..br,~. Assuming weak selection with only two alleles at a locus, we may, from Wright (1937), rewrite the term 1 - xi,. dlg'~ Wi.a - IVa as - 2 dxi and from Abugov (1983), we can rewrite
Y.Labr..(x,.-x,.)r
1 - xi.. dFg. as
4
dxi
Substituting these terms into equation (21) yields x,,~ (1 - x,.~) [4 d if'. + d/~ ] Ax""-4(2lg'.+/~.) L ~ ~X/.J+O(~2)'
(22)
or
x,,~(1 - x~o) d4 if'. +/~ dx, + O(~2),
Ax,.a - 4(2 if'. + 6~)
(23)
where, provided selection is weak, the O(~"z) term is negligible compared to the first term which is of order O(~). The birth rate /~ in equation (23) is defined as the number of gametes an average individual passes to the next generation which, provided the sexes are equal in abundance, equals the average number of offspring per family which, in turn, equals the average number of offspring per female. But the average number of gametes passed per individual must be twice the number of offspring per individual; there is one offspring per two gametes. Thus a diploid individual who passes on one gamete will contribute half an individual to the next generation. We can therefore define B as the average number of offspring per individual, equal to (½)b, and after substituting into equation (23) the resultant identity/~= 2/~, we obtain x,a(1-x,~) d ( 2 W ~ + / ~ ) ~_O(~2).
(24)
dx, The treatment for monoecy will exactly parallel that for dioecy, except that the number of males will equal the number of females which, in turn, will equal the number of matings. Thus F and M in equations (11) through
620
R. ABUGOV
(17) will be unity in monoecy, and b per individual will be the sum b~+b '~ of his contributions via male and female function. Also, there will be no differentiation between male and female survival rates in monoecy, so equation (6) can be written as Ni.a = Ni(a)W o . These two observations allow us to directly show that equation (23) will hold for monoecy as well as dioecy. Again, the number of young born per individual will be only half the number of gametes passed per monoecious individual; thus equation (24) also holds with monoecy. E Q U A T I O N S FOR G E N E F R E Q U E N C Y C H A N G E
We shall now use the equations for within age class gene frequency change to derive the expectations of gene frequency change seen in the first terms of equations (3) and (4). First, note that 2Wa+bo in (23) is the number of alleles passed, GQ, per individual in age class a. Substituting equation (23) into (3) and rearranging thus yields C xlo(1-x,a) d(2Wo+Bo) + COV (G, xi) I- O(~'2).
(25)
a
Applying to the first term on the right-hand side of equation (25) the fact that the expectation of a product equals the product of the expectations plus their covariance (Feller, 1968), and suppressing the i's in xi of equation (25) yields Ax= x(1--x) d(2I~/+/~) ÷COV(G,x) ~_O(~.2)' 4( if' +/~) dx Z CaGo
(26)
a
which, after noting that the number of gene copies Go passed by an average individual from age class " a " equals 2Wo + ba or 2( Wo + Bo), yields x ( 1 - x ) d(2ff'+B) +COV(W+B,x) Ax = 2(I~/+/~) dx 2( I~+/~) +0(62)"
(27)
For the limiting case in which age specific differences in progeny number per individual are small, on the order of the selection coefficients, equation (27) becomes x(1 - x ) d(2 ~z+/~) Ax = 2( if'+/~) dx + O(~'2). (28) Equation (27) defines a Wrightian adaptive topography for life history which is directly analogous to the cost of meiosis, with the maximization
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principle weighting increases in survival twice as greatly as increases in sexual reproduction. Discussion
The adaptive topography equation derived here implies that evolution will tend to maximize 2 ~'+/~, suggesting that increased sexual reproduction will evolve only if the incremental gain in number of offspring more than twice exceeds the concomitant loss to survival. Important to the accuracy of this topography is the assumption that the number of progeny each living individual produces per unit time is independent of his or her age. With this important caveat, the cost of meiosis analogy does appear to play a role in the evolution of life history, with survival, which accurately copies the entire genome. Corroborating this result, Scott Gleeson (pers. comm.) has carried out a preliminary analysis which suggests that a cost of meiosis is implicit in the maximization of population growth upon which traditional models for the evolution of life history (see Schaffer, 1981) are based. It may be that double weighting for survival was not noted by earlier workers because maximization principles for life history have traditionally been couched in terms of population growth and Fisher's (1958) reproductive value rather than directly in terms of survival and reproduction. In the process of deriving the equations, a model which provides an alternative to traditional models for selection with age structure has been delineated. This new model relies on assumptions which are very different from those of the traditional age structured models (Haldane, 1926; Norton, 1928; Anderson & King, 1970; Charlesworth, 1970, 1974; Pollak & Kempthorne, 1971; summarized in Charlesworth, 1980, p 126) which form the basis of present theories of life history (e.g. Charnov & Schaffer, 1973 ; Schaffer, 1974, 1981; Taylor et al., 1974; Michod, 1979) and, because the assumptions of a theory can limit its strict applicability to biological systems, it seems worthwhile to briefly compare the assumptions of the two theories. Those of what we may call the "Norton-Charlesworth'" theory are (Charlesworth, 1980, p. 126): (i) fecundity of any given mating determined solely by age and genotype of the female, (ii) mating random with respect to age and genotype, (iii) primary sex ratio constant and independent of age and genotype, (iv) age specific survival rates equal in males and females, (v) age specific fecundity rates equal in males and females, (vi) selection weak, and (vii) age distribution stable. Assumptions of the theory developed here are: (i) assortative mating with respect to genotype barred, (ii) progeny production by any given genotype at most only slightly dependent on age, (iii) selection weak, and (iv) age distribution stable. With many empirical
622
R. ABUGOV
systems m a t c h i n g n e i t h e r class o f a s s u m p t i o n s , the e m p i r i c i s t ' s best strategy will be to use the t h e o r y which best m a t c h e s the b i o l o g y of his p a r t i c u l a r system. I am pleased to thank James F. Crow, who provided lab space and discussions while this research progressed. I am also grateful to Charles Cotterman, Carter Denniston, Bill Engels, Scott Gleeson, Craig Pease, Don Waller, and an anonymous reviewer for useful comments during various phases of this study. This research was supported by Grant DEB 8213975 from the National Science Foundation.
REFERENCES ABUGOV, R. (1983). Am. Nat. 21, 880. ABUGOV, R. (1985). J. theor. Biol. (in press). ANDERSON, W. W. & KING, C. E. (1970). Proc. hath. Acad. Sci. U.S.A. 66, 780. CHARLESWORTH, B. (1970). Theor. Pop. Biol. 1, 352. CHARLESWORTH, B. (1974). Theor. Pop. Biol. 6, 108. CHARLESWORTH, B. (1980). Evolution in age-structured populations. Cambridge: Cambridge University Press. CHARNOV, E. L. & SCHAr:FER, W. M. (1973). Am. Nat. 107, 791. CHARNOV, E. L., BULL, J. J. & MITCHELLOLDS, S. T. (1981). Am. Nat. 117, 814. FELLER, W. (1968). An introduction to probability theory and its applications, 2nd edn. New York: Wiley. FISHER, R. A. (1958). The genetical theory of natural selection, 2nd edn. New York: Dover Press. HALDANE, J. B. S. (1926). Proc. Cambr. Philosoph. Soc. 23, 607. KEYFITZ, N. (1977). Introduction to the mathematics o f population. Reading: Addison Wesley. MICHOD, R. E. (1979). Am. Nat. 113, 531. NORTON, H. T. J. (1928). Proc. London Mathern. Soc., Set. 2, 28, 1. POLLAK, E. & KEMPTHORNE, O. (1971). Theor. Pop. BioL 2, 351. PRICE, G. R. (1970). Nature 227, 520. PRICE, G. R. (1972). Ann. Hum. Genet. 35, 485. SCHAFFER, W. M. (1974). Ecology 55, 291. SCHAFFER, W. M. (1981). Ecology 62, 1683. TAYLOR, H. M., GOURLEY, R. S., LAWRENCE,C. E. & KAPLAN, R. S. (1974). Theor. Pop. Biol. 5, 104. WALLER, D. M. & GREEN, D. (1981). Am. Nat. 117, 810. WILLIAMS,G. C. (1975). Sex and evolution. Princeton, New Jersey: Princeton University Press.
APPENDIX 1 Ax~ is d e f i n e d as x ~ - x~. S u b s t i t u t i n g i n t o this i d e n t i t y from e q u a t i o n s (1) a n d (2) yields
Axi - ~
ECoPo tl
~. C . x , : . o
(A1)
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By the definition of covariance, COV ( X Y ) Feller, 1968). Thus
= E(XY)
- E(X)E(Y),
(see
or
Z CoGox,.o COV ( G, x~) Yo Cox,.o- E CoQ E CoG°
(A3)
Substituting this into equation (A1), assuming the population is at stable age distribution (Keyfitz, 1977) so that Co is constant in time for all a, and rearranging yields
Zo CoGo[x',,o-x,,o] COV(G,x,) Axi --
Z CoOo a
Z CoOo
(A4)
o
After defining Axi,~ as x~,o-x~,o equation (A4) directly yields equation (3).
Is There a Cost of Meiosis in Life History? ROBERT A B U G O V
Department of Ecology and Evolutionary Biology, Northwestern University, Evanston, Illinois 60201, U.S.A.t and Laboratory of Genetics, 509 Genetics Building, University of Wisconsin, Madison, Wisconsin 53706, U.S.A. (Received 14 February 1985, and in revised form 1 May 1985) Individuals who survive pass their entire genome to the next time period, while those who reproduce sexually pass only half of their genome. This observation has led to a "cost of meiosis" argument for life history, in which increased sexual reproduction can evolve only if the incremental gain in number of offspring more than twice exceeds the concomitant loss to survival. The age-structured genetic model developed here indicates that present measures for the cost of meiosis in life history obtain only if the number of progeny each living individual produces per unit time is independent of his or her age. For systems which meet this biologically important assumption, an approach to life history is delineated which bypasses biological restrictions from traditional models of age-structured selection.
In his classic treatise on the evolution of sex, Williams (1975) noted that alleles for sexual reproduction bear a 50% risk o f loss during the formation of meiotically produced gametes. But alleles which successfully express asexual reproduction do not encounter this risk: sexual reproduction should therefore be able to endure only if offspring produced sexually are at least twice as fit as those produced asexually. Waller & Green (1981) have expressed surprise that William's "cost of meiosis" argument, summarized above, has not had a great impact on life history theory. In particular, they intuitively argued that a gene which allocates more to sexual reproduction should be favored only if the gain in sexually produced offspring is more than twice the concomitant loss to survival and asexual reproduction, since each sexually p r o d u c e d offspring contains only one-half o f the parental genotype. This appears to contrast with current life history theory (reviewed in Charlesworth, 1980), which seems to give equal weight to female survival and fertility. Nevertheless, Charnov, Bull & Mitchell-Olds (1981) have corroborated Waller & Green's t Present address. 613
0022-5193/85/200613+ 11 $03.00/0
© 1985 Academic Press Inc. (London) Ltd
614
R. ABUGOV
argument with a preliminary genetic analysis in which the recursion equations were limited to fixation equilibria in a two stage life history when only one of the life history stages (the youngest) is present. The present analysis extends that of Charnov et al. (1981) to provide a global analysis o f gene frequency change with multiple age classes. In the process, an alternative to traditional models for selection with age structure will be delineated.
Analysis The analysis will first use Price's (1970, 1972) algebra of covariance to determine general conditions under which results from single age class models apply when selection acts in populations which are age structured. Next, global analysis for a single age class model is performed, and the earlier results are applied to produce an age structured model. SELECTION
IN SINGLE
VS M U L T I P L E
AGE
CLASS MODELS
Consider a "progenitor" population composed of individuals of various ages at time t who, via survival and reproduction, give rise to a " p r o g e n y " population at time t + 1. Progeny, here has a special meaning: it includes not only an individual's offspring, but also includes himself, if he survives. This definition is useful in a genetic sense because, when the generations are not discrete, survival provides a means of transmitting genes to the next time period. Also define the frequency of allele Ai at a single locus as the frequency of that gene in the various age classes, averaged over individuals from all age classes. Thus, if we let xt be the frequency of allele As at time t, if we let Co be the relative frequency of age class a at time t, and if we let xi,, be the relative frequency of allele A~ within age class a at time t, then
x, = Z Cox,.a.
( 1)
a
Let x~ be the frequency of allele Ai in the population at time t + 1. It will be the number of A, alleles produced by the progenitor population of equation ( 1 ), divided by the total number of alleles the progenitor population produces. Thus if we let x[o be the frequency of allele A; among the progeny from age class a, and if we let Go be the number o f genes each gene from age class " a " passes on during this time interval, we obtain
Z CoGox',,o x', Y c . Q a
a
(2)
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After defining AX~.a=X~.a-x~.~, the covariance algebra of Price (1970 equation (4), 1972 equation (All)) can be used (see Appendix 1 here) to show that
Z CaG. ZXx~.,, Ax, -- ~
t COV (G, x,)
Z CoGa
(3)
Z CoGo ' a
where the covariance is the product of the deviations averaged over all age classes. After defining the average change in gene frequency as weighted by progeny number to be
CaG~ Axi.~ AX
i __ a
Z CaGa a
and after defining d as Za CoGo, we can rewrite equation (3) as ~x~ = (,~x~.) 4 COV (~ (G, x~)
(4)
We thus see that selection with age structure may be divided into two components: selection within age classes, and covariance between gene frequency and progeny number. As an intuitive example, let us say that older maples produce more progeny than younger maples and that, because of previous selection on survival, older maples also have a higher than average frequency of allele A~. Then even if selection is completely relaxed for a time unit (with Axe,, = 0 for all age classes), gene frequency will increase because progenitors of higher gene frequency (i.e. older maples) tend to produce more offspring. We shall now perform a more detailed analysis to develop an overall picture of age-structured selection as defined in equation (4). SELECTION WITHIN SINGLE AGE CLASSES In this section we shall analyze the first component of equation (4), i.e. the average from the various age classes of the gene frequency difference between progenitors and progeny. As suggested by equation (3), we shall first find the AX~,a for any given age class and then average that result over all age classes. The treatment here will include non-random mating with respect to age, maternal and paternal effects of fertility, sexual selection, and differential survival according to sex, genotype, and age. Throughout the analysis, we shall assume that selection is weak in a population which
616
R.
ABUGOV
is without assortative mating and which is comprised of relatively few age classes, assumptions which render terms on the order of the selection coefficient squared negligible as compared to other terms which appear in the recursion equations for gene frequency change. We shall first derive the equations for dioecious populations and follow with a treatment of selection with monoecy.
Selection within a single age class: Survival Let N(a) be the total number of individuals in age class a, within which proportion F~ are female and proportion M~ are male. Further, let W~,~ and W~.~ be the survival rates of AiA~ among females and males within that age class, and let x~j.abe the ordered frequency of genotype A~Aj within that age class. Then the total number of adult "progeny" of ordered genotype AiAj who survive into the next age class will be
No( a + 1) = N ( a )( Maxo.,~W~.a + Faxij.aW~,a).
(5)
Define W~a = ~j Xij, a Wij, a/Xi, a, and define N~ = 2 Y.j N~. Then summing equation (5) over all j shows that the number of A~ alleles which survive into the next age class is t d~ N,(a + 1) = 2N(a)x,.,~(MaWi.,~ + Fo Wi.,,).
(6)
Defining the mean of any variable K over both sexes a s / ¢ = MK¢'+ FK ~, we obtain N~(a + 1) = 2N(a)xo~VV~.,, (7) The total number of alleles surviving to the next age class can be gained by summing equation (7) over all i, to yield
N'r(a + 1) = 2 N ( a ) IV~
(8)
where l~z~= ~ x~.al~'~.a.
Selection within a single age class: Fertility Consider a female within any particular age class a. She may mate with males of various ages and genotypes, and we can define the "mating types" among the females of a particular age class according to the genotype of the female and the age and genotype of her mate. Any particular mating type can thus be denoted by AkAt × AjA,,~, where the first genotype is that of the female, and where the " a " denotes the age of the male. Further, define f~a.j,,~ as the frequency of mating type AkA~ x A j A ~ among the females from our fixed age class. Then
f~,.j,, = ~ fkl,jma a
(9)
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MEIOSIS
617
is the frequency of mating type A k A t ×A~Am among the females of our fixed age class. Now, let bkij,.a be the total number of offspring per mating type A k A I x AjA.,~, equal to the number of gametes each individual in that mating passes on to his or her offspring. Also, let Q~.k~ be the frequency of males of age " a " among the mates of the A k A t females from our fixed age class. Then the mean number of gametes a female involved in that mating type passes on will be b 9kl,i,. = ~ Qa, ktbkld,.~ .
(10)
a
The total number of gametes that females from a given age class produce via AgA~ × AjA~., matings is then 9
(11)
N ( a ) F f kld,. b kl,i,. .
Now, define Xi.kOm as the probability that any given gamete passed to an offspring from mating type A k A ~ × A~A~ contains allele A~. Then the total number of Ai alleles passed down to the next generation from consideration of females of age a and their mates is 2N(a)F
~ f ~gom b k l j m X i ,
kljm .
kl, j m
A "2" precedes the N ( a ) here because each diploid offspring contains two alleles. The total number of Ai alleles passed down to offspring which are attributed to the female parent will be half of this, equal to b 9i,a -- - N ( a ) F
E f 9kombkomX~,kli,. , kl, jrn
(12)
with the other half being attributable to the male. We may abbreviate the mating type designations kl, j m with r's, shortening equation (12) to b ~i.~ - N ( a ) F S " ,., jr.~ f~ ~. l, ~ . .x. . .
(13)
r
The notation of equation (13) will be used henceforth because it is consistent with previous related studies (Abugov, 1983, 1985). A similar treatment, but for males, yields bq| , a = N ( a ) M
'~ E f '~ ,.~b,.,x,.~.
(14)
r
Note that the "b," takes into account that males of our particular age class may father more than one set of brood, since it counts the number of offspring fathered.
618
R.
A B U G O V
Summing equations (13) and (14) shows that the total number of alleles passed to newborns from age class a is
b,~) = N ( a ) Y, [La + O( ¢ ) ]xi,~( Fb ~. + Mb ~o).
Ai
(15)
r
The f~.a term here is that expected from the Hardy Weinberg distribution. The O(sr) term, then, is on the order of the selective coefficient, and it results from selection induced sex specific differences in genotypic frequencies. Averaging the br.a's in equation (15) over both sexes yields
bi(a) = N ( a ) E [La + O(~)]xi.f~,~.
(16)
r
The total number of alleles passed to the newborns from both males and females here is bi(a) summed over all i which, from equation (15), is
b(a) = N(a) Y. [f~,,, + O(~')]6r,,,.
(17)
r
Fertility and survival selection combined within a single age class Summing equations (7) and (13) shows that the number of which originate from age class a is
N~(a)=2N(a)x,,aff'~,~+ N ( a ) ~,[f~,~+O(~)]xh,br, a.
Ai alleles (18)
r
A "2" comes into the survival but not the fertility term because each parent donates only one allele to each of its offspring, while each survivor carries both alleles. The total number of alleles passed down to the next time period from this age class is the sum of equations (8) and (17), which is
2N(a) ffVa+{ N(a) E [La + O(r~)]6.,a}. The relative frequency of given by
(19)
Ai among the progeny of age class a is therefore
2Xi,alYCha+2 [f.,a + O(~) ]Xi,rG~,. t Xi,
a
__ --
r
2 fro +y~ [f~.o + o(¢)]6r, o r
(20)
THE
COST
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619
MEIOSIS
The change in gene frequency, x~,.- x~.~, is then 2x,.a( Wi,~ - lYga)+ ~Aabr, a(X~.r- Xi,a) r
ax,,o =
2 ~i,o + £
~ 0 ( ~ ~)
(21)
where the average birth rate for each age class is defined as 6~ =E.f~..br,~. Assuming weak selection with only two alleles at a locus, we may, from Wright (1937), rewrite the term 1 - xi,. dlg'~ Wi.a - IVa as - 2 dxi and from Abugov (1983), we can rewrite
Y.Labr..(x,.-x,.)r
1 - xi.. dFg. as
4
dxi
Substituting these terms into equation (21) yields x,,~ (1 - x,.~) [4 d if'. + d/~ ] Ax""-4(2lg'.+/~.) L ~ ~X/.J+O(~2)'
(22)
or
x,,~(1 - x~o) d4 if'. +/~ dx, + O(~2),
Ax,.a - 4(2 if'. + 6~)
(23)
where, provided selection is weak, the O(~"z) term is negligible compared to the first term which is of order O(~). The birth rate /~ in equation (23) is defined as the number of gametes an average individual passes to the next generation which, provided the sexes are equal in abundance, equals the average number of offspring per family which, in turn, equals the average number of offspring per female. But the average number of gametes passed per individual must be twice the number of offspring per individual; there is one offspring per two gametes. Thus a diploid individual who passes on one gamete will contribute half an individual to the next generation. We can therefore define B as the average number of offspring per individual, equal to (½)b, and after substituting into equation (23) the resultant identity/~= 2/~, we obtain x,a(1-x,~) d ( 2 W ~ + / ~ ) ~_O(~2).
(24)
dx, The treatment for monoecy will exactly parallel that for dioecy, except that the number of males will equal the number of females which, in turn, will equal the number of matings. Thus F and M in equations (11) through
620
R. ABUGOV
(17) will be unity in monoecy, and b per individual will be the sum b~+b '~ of his contributions via male and female function. Also, there will be no differentiation between male and female survival rates in monoecy, so equation (6) can be written as Ni.a = Ni(a)W o . These two observations allow us to directly show that equation (23) will hold for monoecy as well as dioecy. Again, the number of young born per individual will be only half the number of gametes passed per monoecious individual; thus equation (24) also holds with monoecy. E Q U A T I O N S FOR G E N E F R E Q U E N C Y C H A N G E
We shall now use the equations for within age class gene frequency change to derive the expectations of gene frequency change seen in the first terms of equations (3) and (4). First, note that 2Wa+bo in (23) is the number of alleles passed, GQ, per individual in age class a. Substituting equation (23) into (3) and rearranging thus yields C xlo(1-x,a) d(2Wo+Bo) + COV (G, xi) I- O(~'2).
(25)
a
Applying to the first term on the right-hand side of equation (25) the fact that the expectation of a product equals the product of the expectations plus their covariance (Feller, 1968), and suppressing the i's in xi of equation (25) yields Ax= x(1--x) d(2I~/+/~) ÷COV(G,x) ~_O(~.2)' 4( if' +/~) dx Z CaGo
(26)
a
which, after noting that the number of gene copies Go passed by an average individual from age class " a " equals 2Wo + ba or 2( Wo + Bo), yields x ( 1 - x ) d(2ff'+B) +COV(W+B,x) Ax = 2(I~/+/~) dx 2( I~+/~) +0(62)"
(27)
For the limiting case in which age specific differences in progeny number per individual are small, on the order of the selection coefficients, equation (27) becomes x(1 - x ) d(2 ~z+/~) Ax = 2( if'+/~) dx + O(~'2). (28) Equation (27) defines a Wrightian adaptive topography for life history which is directly analogous to the cost of meiosis, with the maximization
THE
COST
OF
MEIOSIS
621
principle weighting increases in survival twice as greatly as increases in sexual reproduction. Discussion
The adaptive topography equation derived here implies that evolution will tend to maximize 2 ~'+/~, suggesting that increased sexual reproduction will evolve only if the incremental gain in number of offspring more than twice exceeds the concomitant loss to survival. Important to the accuracy of this topography is the assumption that the number of progeny each living individual produces per unit time is independent of his or her age. With this important caveat, the cost of meiosis analogy does appear to play a role in the evolution of life history, with survival, which accurately copies the entire genome. Corroborating this result, Scott Gleeson (pers. comm.) has carried out a preliminary analysis which suggests that a cost of meiosis is implicit in the maximization of population growth upon which traditional models for the evolution of life history (see Schaffer, 1981) are based. It may be that double weighting for survival was not noted by earlier workers because maximization principles for life history have traditionally been couched in terms of population growth and Fisher's (1958) reproductive value rather than directly in terms of survival and reproduction. In the process of deriving the equations, a model which provides an alternative to traditional models for selection with age structure has been delineated. This new model relies on assumptions which are very different from those of the traditional age structured models (Haldane, 1926; Norton, 1928; Anderson & King, 1970; Charlesworth, 1970, 1974; Pollak & Kempthorne, 1971; summarized in Charlesworth, 1980, p 126) which form the basis of present theories of life history (e.g. Charnov & Schaffer, 1973 ; Schaffer, 1974, 1981; Taylor et al., 1974; Michod, 1979) and, because the assumptions of a theory can limit its strict applicability to biological systems, it seems worthwhile to briefly compare the assumptions of the two theories. Those of what we may call the "Norton-Charlesworth'" theory are (Charlesworth, 1980, p. 126): (i) fecundity of any given mating determined solely by age and genotype of the female, (ii) mating random with respect to age and genotype, (iii) primary sex ratio constant and independent of age and genotype, (iv) age specific survival rates equal in males and females, (v) age specific fecundity rates equal in males and females, (vi) selection weak, and (vii) age distribution stable. Assumptions of the theory developed here are: (i) assortative mating with respect to genotype barred, (ii) progeny production by any given genotype at most only slightly dependent on age, (iii) selection weak, and (iv) age distribution stable. With many empirical
622
R. ABUGOV
systems m a t c h i n g n e i t h e r class o f a s s u m p t i o n s , the e m p i r i c i s t ' s best strategy will be to use the t h e o r y which best m a t c h e s the b i o l o g y of his p a r t i c u l a r system. I am pleased to thank James F. Crow, who provided lab space and discussions while this research progressed. I am also grateful to Charles Cotterman, Carter Denniston, Bill Engels, Scott Gleeson, Craig Pease, Don Waller, and an anonymous reviewer for useful comments during various phases of this study. This research was supported by Grant DEB 8213975 from the National Science Foundation.
REFERENCES ABUGOV, R. (1983). Am. Nat. 21, 880. ABUGOV, R. (1985). J. theor. Biol. (in press). ANDERSON, W. W. & KING, C. E. (1970). Proc. hath. Acad. Sci. U.S.A. 66, 780. CHARLESWORTH, B. (1970). Theor. Pop. Biol. 1, 352. CHARLESWORTH, B. (1974). Theor. Pop. Biol. 6, 108. CHARLESWORTH, B. (1980). Evolution in age-structured populations. Cambridge: Cambridge University Press. CHARNOV, E. L. & SCHAr:FER, W. M. (1973). Am. Nat. 107, 791. CHARNOV, E. L., BULL, J. J. & MITCHELLOLDS, S. T. (1981). Am. Nat. 117, 814. FELLER, W. (1968). An introduction to probability theory and its applications, 2nd edn. New York: Wiley. FISHER, R. A. (1958). The genetical theory of natural selection, 2nd edn. New York: Dover Press. HALDANE, J. B. S. (1926). Proc. Cambr. Philosoph. Soc. 23, 607. KEYFITZ, N. (1977). Introduction to the mathematics o f population. Reading: Addison Wesley. MICHOD, R. E. (1979). Am. Nat. 113, 531. NORTON, H. T. J. (1928). Proc. London Mathern. Soc., Set. 2, 28, 1. POLLAK, E. & KEMPTHORNE, O. (1971). Theor. Pop. BioL 2, 351. PRICE, G. R. (1970). Nature 227, 520. PRICE, G. R. (1972). Ann. Hum. Genet. 35, 485. SCHAFFER, W. M. (1974). Ecology 55, 291. SCHAFFER, W. M. (1981). Ecology 62, 1683. TAYLOR, H. M., GOURLEY, R. S., LAWRENCE,C. E. & KAPLAN, R. S. (1974). Theor. Pop. Biol. 5, 104. WALLER, D. M. & GREEN, D. (1981). Am. Nat. 117, 810. WILLIAMS,G. C. (1975). Sex and evolution. Princeton, New Jersey: Princeton University Press.
APPENDIX 1 Ax~ is d e f i n e d as x ~ - x~. S u b s t i t u t i n g i n t o this i d e n t i t y from e q u a t i o n s (1) a n d (2) yields
Axi - ~
ECoPo tl
~. C . x , : . o
(A1)
THE
COST
OF
623
MEIOSIS
By the definition of covariance, COV ( X Y ) Feller, 1968). Thus
= E(XY)
- E(X)E(Y),
(see
or
Z CoGox,.o COV ( G, x~) Yo Cox,.o- E CoQ E CoG°
(A3)
Substituting this into equation (A1), assuming the population is at stable age distribution (Keyfitz, 1977) so that Co is constant in time for all a, and rearranging yields
Zo CoGo[x',,o-x,,o] COV(G,x,) Axi --
Z CoOo a
Z CoOo
(A4)
o
After defining Axi,~ as x~,o-x~,o equation (A4) directly yields equation (3).