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Viability-fecundity selection at a single multiallelic locus in random mating populations: Protectedness of an allele
GREGORIUS
Lehrstuhl fiir Forstgenetik
AND
MARTIN
ZIEHE
und Forstpfunrenriichtung
der Universltiit
GBttingen,
Biisgenweg 2, 3400 Giittingen- Weende, Federul Republic of Germuq Received 15 September
1981; revised 1 March 1982
ABSTRACT To simplify
comprehension
on the genetic structure, such as average fitness
of the effect of combined
viability
and fecundity
selection
emphasis is put on the definition of those population parameters, of genotypes or alleles, which provide insight into the principles
which govern the system’s dynamics. With the help of these parameters, it is possible to point out the resemblance of the model presented here to the known models of pure viability selection. Protectedness of alleles is considered in two steps: (1) treatment of the possibility that alleles become extinct in finite time, and (2) provided sudden extinction cannot occur, derivation of conditions for the initial increase of a new allele. The range of relevance
of these results
factorizations of fecundity tive and the additive.
is demonstrated
by applying
them to the most frequently
into the male and female contributions,
namely
used
the multiplica-
INTRODUCTION The concept of fitness when applied to Mendelian populations relies on the specification of the expected number of offspring produced by the members of the population and therefore basically refers to heterosexual pairs of individuals. This fact was stressed by several authors, who tried to point out the difficulties in biological interpretation, estimation and, as a consequence, the evolutionary significance of individual fitnesses (see e.g. Prout [ 141, Kempthome and Pollak [6], Mandel [8], Roux [ 151). Here, fitness is termed “individual” if it appears sensible to assume that the expected number of offspring is a function of the genotypes of individuals. In this case, the effect of selection, if it occurs, is frequently attributed to selection with respect to viability only and thus disregards differences in fecundity between pairs (we find it advantageous to apply the term fecundity only to pairs of individuals and not to single individuals, as is sometimes done).
MA THEMA
TICA L BIOSCIENCES
6 1:29-50
QElsevier Science Publishing Co., Inc., 1982 52 Vanderbilt Ave., New York, NY 10017
29
(I 982)
OO25-5564/82/07029+
22$02.75
30
HANS-ROLF
GREGORIUS
AND
MARTIN
ZIEHE
If differences in fecundity are to be taken into account, the number of offspring an individual can have depends on its capacity to survive to the reproductive age, as well as on its reproductive capacity and the extent to which this can be realized by potential mating partners, which themselves are subjected to these two selective forces. Since, however, a mating results in the same number of offspring for both parents, fitness should not be treated as a property of a single individual, but rather of a pair of individuals. To make this difference apparent, we shall speak of “pair fitness” as opposed to individual fitness. The argument that consideration of individual fitnesses does not suffice to characterize the change in genotypic and allelic structures in populations is presented in the works of Bodmer [l] and Mandel [8]. Both these authors, dealing with models in which pair fitness is assumed to be the product of differential individual male and female fitnesses, showed that the existence and stability of equilibrium frequencies cannot simply be described in terms of heterozygote advantage, as is possible in the classical model of pure viability selection without sex differences. Instead, it proved necessary to consider the relationship between pairs of individual fitnesses. In a more explicit manner, the necessity of considering pair fitnesses was demonstrated by Hadeler and Liberman [5], who treated the situation of symmetric fecundities. The assumption that viability-fecundity selection does not follow the laws that are expressed in the various versions of Fisher’s fundamental theorem of natural selection was confirmed by Pollak [ 12, 131. He showed that average population fitness (fecundity) need not increase along population trajectories, nor must these trajectories converge to equilibrium states. Moreover, he believes it a wasted effort to search for measures of fitness that are capable of reflecting basic characteristics of the dynamics of genetic structures, which is easily understood if one considers the possibility of the existence of a multiplicity of equilibrium states having all kinds of stability properties. This makes it in fact very difficult to find answers even to the most basic questions, such as the existence of conditions for the persistence of genetic polymorphisms. One of the present authors [3] suggested that in such a situation it might be more appropriate to view the problem of persistence of alleles via consideration of repulsivity properties of those population states that represent extinction of alleles. In the current technical literature, this kind of analysis is usually performed by finding conditions for the initial increase in frequency of newly introduced alleles (see e.g. [2,10,7,8]). The main aim of the present paper will be to try to apply the above idea to viability-fecundity selection at a single, multiallelic locus within random mating populations reproducing in nonoverlapping generations. In order to
ALLELE
PROTECTION
WITH
FECUNDITY
SELECTION
31
be able to point out the resemblance of this model to the known characteristics of “classical” single locus selection theory, particular care will be taken in developing and representing parameters which are related to the classical ones in that they refer to individual fitnesses and provide insight into the mode of action of viability-fecundity selection. NOTATION
AND BASIC ASSUMPTIONS
= probability of survival to the reproductive age of the ordered female and male genotype A,A,, respectively. Assumption: sz = sz, s;y = sj?. = fecundity (expected number of offspring at the zygotic stage) of a mating between a female genotype A,A, and a male genotype
w,/;kl
p,(k) P
pk
PO
W(P) ci$[(P)
A/J,. Assumption: brlik, = bJlik, = b ,,:, k, i.e., the contribution of a genotype does not depend on its ordering. =sz br,;k, skT is the “pair fitness” of a female genotype A,A, and a male genotype A,A,. From the above assumptions it follows that = wz],/k. w,,; kl = Wjl;kl =relative frequency of the ordered genotype A,A, at the zygotic stage of the population. This frequency is assumed to be the same among males and females (no sex ratio distortion). =zJP,,,p;x : =z,PJ,,p, : =+(pp + p,““) are the allelic frequencies of A, within the female contribution to the zygote population, the male contribution to the same, and the total zygote population, respectively. = i( P,k + Pkr)/pk, where pk # 0, is the frequency of the allele A, among all genotypes carrying the allele A,. = the set of all nonnegative n X n matrices with sum of components equal to 1. Here n is the number of alleles considered, and a matrix PE P represents a genotypic structure at the zygotic stage of a population, such that the element P,, of P gives the frequency of the ordered genotype A, A,. P will be conceived of as a simplex in the Banach space of all real square matrices of order n. = the set of all genotypic structures PE $ for which pk = 0, i.e. which do not contain the allele A,. = the set of all genotypic structures PE P for which p, > 0 for i=l,..., n, i.e. which contain all of the n alleles. =B ,,,, k, ,P,, w,,;~, Pkl, i.e. the average population (pair) fitness for a given genotypic structure P. =C,,jPi,wk ,:,,, EL?(P) : = C,,,Pl,~,,ik, are the average fitnesses of
HANS-ROLF
32
GREGORIUS
AND
MARTIN
ZIEHE
the female and male genotype A,A, for a given genotypic structure P for the male and female populations, respectively. Wk,(P) : = f[ i$( P)+ i$T( P)] is the average fitness of the genotype A,A, in the total population of males and females. ii$( P) : =Zjg,( P)p,( k), ic”( P) : = X,ii$( P)p,( k) are the average fitnesses of the allele A, among females and males, respectively, for a given genotypic structure P with pk > 0. EIJP) := f[ i$( P)+ i$f( P)] is the average fitness of the allele A, in the total population of males and females. Frequencies
in the next generation
will be denoted by primes, i.e. P,‘,, pk, etc.
THE MODEL Viability-fecundity selection will be considered for random mating Mendelian populations of (in effect) infinite size, which reproduce in nonoverlapping generations. Genotypic frequencies are observed at the zygotic stage in each generation. Since we do not require the viabilities to be independent of sex and the fecundities to be interchangeable with respect to the sexes of the pair of genotypes involved in a mating, genotypic frequencies must refer to ordered genotypes (i.e., in general the frequency of A,A, is not equal to that of A,A, , where it is always assumed that the first gene is derived from the female and the second from the male parent of the individual having this genotype). On the other hand, viabilities and contributions to fecundities are assumed to be independent of whether an allele in a genotype originates from the female or male parent. Under these conditions and the assumption that selection acts on a single, multiallelic locus, Roux [15] derived recurrence equations for genotypic frequencies in successive generations: PL,= &
Ii2 :(P,i+P,,).w,,;,,.:(P,,+p,,). r,j
(1)
Naturally, this relationship is well defined only for genotypic structures PEP with W(P) > 0. Consequently, Equation (1) specifies a mapping G,, which is defined on the set of genotypic structures P,:
= {PEPliiqP)>O}
and assumes values in P, i.e.
such that, given the genotypic
structure
PE P,
for one generation,
P’ =
ALLELE
PROTECTION
WITH
FECUNDITY
33
SELECTION
G,(P) is the genotypic structure in the next generation. The index W is used to indicate the selection regime of wik;,, ‘S to which the transition refers. In order to point out the resemblance of the information contained in Equation (1) to the results of “classical” selection theory, which is largely concerned with the influence of selection on the change in allelic frequencies, the notations defined in the previous section will be employed. For this purpose let PE P, and pk > 0 (i.e. P @ pk). Then $( P),i$%( P), and iCk(P) are well defined. Applying Equation (I), it is easily seen that summation of PLI and P/k over I yields
The last of the three equations has a structure similar to that known from the selection model with sex independent viabilities and no fecundity differences, in that it states that the frequency of an allele increases or decreases from one generation to the next according as its average fitness is greater or less than the average population fitness. The difference between the two models is to be found in the fact that for sex independent viability selection, Hardy-Weinberg proportions are attained after the first generation and consequently the average allelic as well as the average population fitness can be assumed to be functions of the allelic frequencies (allelic structure), while for general viability-fecundity selection these parameters must be considered as functions of the genotypic structure. This is also the reason why, in the present model, identical average allelic fitness for all alleles is only a necessary but not a sufficient condition for a genotypic structure to be at a nontrivial equilibrium state with respect to its allelic structure. The asymmetry of pair fitnesses with respect to sex implies asymmetry of genotypic structures (i.e. asymmetry of the matrices P) in the course of generations. However, using most of the common techniques for identifying genotypes, discrimination between the male and female contributions to a genotype is not possible. Therefore, it is of some relevance to know whether Equation (1) can be altered in a way that results in a recurrence relationship that does not depend on the ordering of genotypes and, at the same time, refers to the population trajectories defined by Equation (1). For this purpose consider the symmetrized versions of W and P given by
In matrix
Ij,j:=t(P,,+P,,).
Wik;,l ..=f(wLk;,,+wJ,;J
and
notation,
version
the symmetrized
j
of P can be written
p
34
HANS-ROLF
: = i( P + P’),
and
thus
GREGORIUS
where P’ denotes the transposition
G,(P)
= G,(p),
GT)
=+[G,(P)+
AND MARTIN
ZIEHE
of P. It follows that
G,(P)T]
= G,(P)
=
G&P).
Hence, Equation (la) defines the same population trajectory as Equation (1) provided one is only interested in observing frequencies of unordered genotypes (i.e., the frequency of the unordered heterozygote is A,A, = 2p,k, and clearly for homozygotes p,, = P,,). Since the major aim of this paper is to investigate the problem of protectedness of alleles, and since pk = Z,pk,, most of the considerations in the following sections will refer to the symmetrized version of the selection model. To prevent possible misunderstandings, it will be mentioned that, in general, an equilibrium structure P of G, (i.e. G,(P) = P), which must be symmetric (P = PT), need not be an equilibrium structure of G,, but in the following generation P’ = G,( P’), as can be deduced easily from Equation (1). PERSISTENCE ATIONS
OF ALLELES
OVER A FINITE
NUMBER
OF GENER-
The fact that some of the pair fitnesses wk,;,, may be equal to zero, for instance due to mating incompatibilities, could lead to W(P) = 0 for certain genotypic structures P, either immediately for the first or for some later generation. In terms of the dynamical system defined by the iterates GrWof G W (t : = number of generations), this situation means that for certain initial conditions a trajectory is defined for only a finite time interval, while the biological interpretation is that the population dies out after a finite number of generations. Therefore, it is of fundamental significance for all advanced considerations to characterize genotypic structures which, as initial conditions, guarantee that the population will not become extinct after a finite number of generations. Hence, such initial structures must at least belong to the set P,. The following proposition gives sufficient conditions for population persistence: PROPOSITION
1
A population starting with PE P, remains there indefinitely if there exist indices i, j, k, 1 and u, v E {i, k}, r, SE { j, I} such that j,k~,k;,l&> 0, W,,:,, > 0, and G,:r;ks> 0. Proof. From Equation (1) it follows that P,‘,, P,;, PL,, PL, are all positive. applyHence, W(P’) > I: r,sE C,,II&‘,.$ir;ks&s > 0, i.e. P’E P,. Furthermore,
ALLELE
PROTECTION
ing Equation
WITH
FECUNDITY
SELECTION
35
(la),
and
which implies ~,~w,~;,,$ > 0, i.e. W(P”) > 0 and thus P”E P,. Moreover, with this result we have arrived at the situation with which we started, and W consequently G’,(P) E P, for all generations t. From the above proof technique it can be easily deduced that if Remark. the conditions of Proposition 1 are satisfied, the allelic frequencies of A,, A,, A,, and A, remain positive over the course of the generations. The statement of Proposition 1 refers only to certain genotypic structures with positive average population fitness, and it is therefore not yet clear which properties a selection regime W must have to assure that population persistence is guaranteed for each structure in P,. This question, which amounts to the investigation of the invariance of P, (i.e., PE P, implies G,+,(P) E pw), deserves a comprehensive answer, since our interest is focused on the development of genotypic structures in existing populations, i.e. those having positive average fitness. PROPOSITION
2
P, is invariant (with respect to G,) if and only if for each positive symmetrized pair fitness 15,~;~~ at least one of the following is fulfilled:
(i) there exist indices r, u E (i, k} and s, v E {I, j} such that KJ~:,;.” > 0, (ii> (iii)
W,k;,k w,,;,,
’
O,
>
0.
Pk,@kr.,,P,I> 0, and let the above condition be Proof. Let PEP,, fulfilled. Then, analogously to the proof of Proposition 1, p,‘, , p,‘,, FL,, FL, are all positive and therefore
which is positive if (i) is fulfilled. Otherwise, if for instance Wik;,k> 0, then F,; > 0 and W(P’) 2 F,:w,~;,~ > 0. Thus, P’E P,, i.e., P, is invariant. For
HANS-ROLF
36
GREGORIUS
AND MARTIN ZIEHE
the other direction let LP,,, be invariant and Grk;,, > 0. Choose P such that p,, ,i,, are positive and P,k + E’;,= 1; hence PE P,. Since by assumption P’E P,, we have W(P’) > 0, and consequently the above condition must be fulfilled, since the only positive genotypic frequencies are p,‘,, p,‘,, FL-,, FL,, and possibly P,:, P&, q;, P;l, p,I,, 4;. The proof by contradiction is straightforward. n Remark. The condition in Proposition 2 says that if a cross has positive pair fitness, then either among its progeny at least one cross has positive pair fitness or there exists a cross with positive pair fitness between two individuals both having the same genotype which is identical to one of the parental genotypes. The first is always true if the parents share one allele, since the resulting progeny contains both genotypes of the parents. If the parents are homozygous with respect to different alleles, the first condition requires that crosses between the resulting heterozygotes have positive pair fitness. With these two propositions, the basis is provided from which the real problem of this section, namely the persistence of alleles for a finite number of generations, can be attacked in a meaningful way. From Equation (2) it follows that, given a genotypic structure PE IFD,with pk > 0, the allele A, is present in the next generation if and only if iGJ P) > 0. The set of genotypic structures having this property for all alleles will be denoted by pk, i.e.,
P&:=
{PEP’I
Gk(P)>Ofork=l,...,n)
Hence, lPL is the analogue to P, if instead of population persistence the persistence of polymorphisms at least to the next generation is to be considered. Clearly, lPb is empty if and only if for at least one k, Gikr:j, = 0 for all i, j, 1. Moreover, Pk c P,n P” and, provided P& is nonempty, $ ‘, lFJ and Pk have the topological property of being open sets in P, the cl:ure of which equals all of P. According to the remark following Proposition 1, it is possible to specify conditions under which a population starting with a certain genotypic structure exists at any finite time and maintains a certain number and kind of its initially present alleles. Now the question is which properties of the selection regime W must be realized so that all of the initially present alleles are maintained. This again requires a search for invariance conditions, now with reference to the set PL. In order to avoid an extensive graph theoretical treatment of the subject, we shall confine ourselves to the proof of a sufficient condition for the invariance of Ph, which shows some resemblance to that of Proposition 2.
ALLELE
PROTECTION
PROPOSITION
WITH
FECUNDITY
SELECTION
37
3
Let oD& be nonempty. Then IF’; is invariant (with respect to G,) iffor each k for which 1.5~~; jl > 0, either there exist indices r, s E {j, f} and u E {i, k} with wkr.su > 0, or w~~.~,> 0.
Proof. The proof follows exactly the same pattern as with Proposition 2, the only difference being that the single steps must be applied to iGkin place n of w. The condition in Proposition 3 says that, if a cross has positive Remark. pair fitness, then, among its progeny, crosses with positive pair fitness can be formed such that each of the parental alleles is involved in at least one of these crosses. Otherwise, if not all of the parental alleles are involved in a progeny crosses, it is required that the parental genotype containing nonrepresented allele have positive pair fitness when crossed with a mate carrying the same genotype. As was already stated in the remarks following Proposition 2, the condition is automatically fulfilled for parents which share one allele. Moreover, the conditions of Proposition 3 imply those of Proposition 2 and therefore the invariance of P,. The true reason justifying consideration of invariance for P& originates from the fact that the loss of one allele in a single step generally implies that for the succeeding generations the dynamics of the genotypic structure obey completely different laws due to the reduced selection regime which is active. In particular, this can lead to the situation in which an allele can be established in the initial but not in the reduced system and vice versa. Therefore, in order to investigate the problem of the protectedness of an allele (with respect to a selection regime) it must be known whether or not IP; itself, or at least a subset of Pk, is invariant. The derivations of the following section will consequently be built upon this idea. PROTECTEDNESS
OF AN ALLELE
In its probably most general form, the concept of the protectedness of an -allele with respect to a given selection regime refers to the idea that population trajectories starting with low frequency for the allele under consideration move in the direction of increasing this frequency. For the this means that starting with PEP& allele A,, using our terminology, sufficiently close to P,, the genotypic structures in the succeeding generations move away from P,. In some cases, however, it is not necessary to assume that this property holds for all PE P& in the vicinity of P,, since it might for instance be known that after the first generation Hardy-Weinberg proportions are attained (as is true for viability selection with no sex differences). Consequently, it would suffice to consider only Hardy-Weinberg
HANS-ROLF
38
GREGORIUS
AND
MARTIN
ZIEHE
structures in the vicinity of P,. Therefore, it might occasionally be convenient to restrict the considerations to population trajectories staying within an invariant subset of Pg. In accordance with common usage, we shall treat the problem of protectedness as one of the initial increase in frequency of a new allele. For this purpose the term “protected” will be used in the sense described in the following definition: DEFINITION
Let B c IFDLbe a set of genotypic structures which is invariant under G,, and let lk denote the closure of B. Then the allele A, is called protected within B if either the intersection of b with P, is empty or there exists a neighborhood UJ of P, such that PE B 17111implies pi > pk. It might be of interest to note that the last condition in the above definition specifies the allelic frequency pk as a function of the Ljapunov type, which in turn implies that the set B n P, of genotypic structures is “strictly repulsive” for the trajectories in 8, as was recently shown by one of the present authors [3]. This definition has the further advantage that it applies more generally to the idea of asymptotic persistence of an allele if B has the additional property that, irrespective of the genotypic structure from Pk with which the population starts, it enters B after a finite number of generations. If it is not possible to extract a priori from a particular type of selection regime some information which justifies consideration of a restricted range of genotypic structures within which the essential part of the development of a population takes place, one is forced to take into account all structures in P& when investigating the protectedness of an allele. Since this situation can be thought to be the basis from which further specialization should proceed, we shall try to characterize not only what is sufficient but also what is necessary for protectedness. THEOREM
I
Let P& be nonempty and invariant under G,. Then, in order for the allele A, to be protected within p&, it is sufficient that Wkr(P) > W(P) and it is necessatythatWk,(P)~iC(P)foraNPEF’kandi~l,...,n. Proof. Let Wkr(.)- W( -) be positive on P, for all i. Then the continuity of w~,( .)- W( .) on P and the compactness of P, imply the existence of a neighborhood U of P, on which Wki( .) - W( .) is positive for all i. Hence, for PE POflP,,
ALLELE
PROTECTION
WITH
and thus from Equation
FECUNDITY
SELECTION
39
(2) ,_
Pkw,(P) ‘Pk.
Pk-
W(P)
For the other direction recall that P& is dense in P and P, is convex and compact. Therefore, U-lcan be assumed to be convex, and for each i there exists PE P On UJ such that p,( k) = 1 [i.e. Pj( k) = 0 forj # i]. Consequently, forP*EP,,thestraightlineP(a):=(l-a)P*+aPbelongstoIFD’nUforO < crG 1, and it is easily verified that Wk(P(a)) = Wki( P( a)). Since lp’: is dense in P, it follows that O
1
Let B be any subset of Pk which is invariant under G,. Then the allele A, is protected within B if 3, ,.,, + )?lk,.rs> 2@,,:,, for all j and all i, I, r, s # k. Proof.
For PE B n P, one obtains
wk,(p)-w(p)=
2
f(~k~;,,+~k,;r.~-2~,,:rr)P,,P,>o
i,l,r,s#k
for all j. Hence, application assertion.
of the remark following Theorem
1 proves the n
The condition of Corollary 1 states a kind of triangular relationship between pair fitnesses: for a genotype containing the allele A,, the arithmetic mean of the symmetrized pair fitnesses belonging to crosses with any two genotypes both not containing this allele exceeds the symmetrized pair fitness of these two genotypes. In particular, this implies that a cross between a possessor of allele A, and an individual not having the allele is superior to a cross of this individual with itself, so to say. In Theorem 1 and the subsequent remark, the relationship Gkr( P) > W(P) was employed for PE Pk. Here W(P) refers to the average fitness of a population not containing the allele A,, and Wki( P) measures the average fitness of the genotype A,A, if it is newly introduced into the population.
40
HANS-ROLF
GREGORIUS
AND
MARTIN
ZIEHE
Hence iCkr(P) > W(P) for all i (thus including k) gives expression to the superiority of the newly introduced allele A, over the resident ones. Now, from the “classical” theory of viability selection it is known that the establishment of an allele is already guaranteed if only its heterozygote appearance is superior to the other genotypes. Consequently, it is reasonable to look for classes of selection regimes for which conditions of protectedness can be restricted to heterozygote superiority only, i.e., iGkr(P) > W(P) for all i with i # k. The following theorem accounts for such conditions. THEOREM
2
Let B c P g be invariant under G,, pk( k) tends to zero
and suppose that as
PE B approaches P, .
Then A, is protected within II3 if Fkr( P) > W(P) for all PE b nP, with i # k.
(*) and all i
Proof. From the assumption, it follows that there exists a neighborhood U’ of l&n P, on which iGk,(.) - W( .) is positive for all i # k and it holds that p,(k)#l. Hence, for PEBnU’,
W,(P)- W(P)= 2 [W,i(P>-W(P)] P,(k)
+L(P)Without
loss of generality,
W’)] p,(k).
assume UJ’ to be compact, and furthermore
a:=m;inpiC”~,[F~,(P)-W(P)]
and
let
b: =;z;,[i&(P)-W(P)];
rfk
then a is positive. If b 2 0, then Wk(P) - W(P) > 0 and hence pi > pk , which would end the proof without having used the condition (*). For b < 0 one has a/( a - b) > 0, and by the condition (*) there exists a neighborhood U of 8 n P, such that U c U’, and for PE UJn B it holds that pk( k) < a/( a - b). Hence, for such P,
~a-[l-Pk(k)]+[Wkk(P)-W(P)]Pk(k) aa.[l-p,(k)]+b.p,(k)=a-p,(k)(a-b) >a-a=O.
ALLELE
PROTECTION
WITH
FECUNDITY
SELECTION
41
The following example will serve to demonstrate the wide applicability of the condition (*): Consider a selection regime in which all symmetrized pair let B: = G,(P~); then Ph is fitnesses Gtki;,[ are positive. Furthermore, invariant, B C IPL, and B is also invariant. Making use of a : = max,,, Gkj; rk, b: =min, ,,Gkj ;,,, a,: =min,3j~)k,;,k, and b,: =maxi ,,,, Gkj;,[, oneobtains for PE Pb’
and similarly p;( k) 2 pk. a, / b,. This shows that pk tends to zero if and only if pL( k) does, and in an analogous manner it is verified that pk tends to zero if and only if p; does. Consequently, B = G,(P~) satisfies the condition (*) whenever the symmetrized selection regime @ is positive. Since by definition of 5, after the first generation all population trajectories enter B and remain there indefinitely, persistence of an allele is warranted if the condition of protectedness in Theorem 2 is fulfilled for this allele within B. Further examples of sets B satisfying the condition (*) will be found in the following sections about multiplicative and additive pair fitnesses. To close the general considerations of viability-fecundity selection, the analogue of Corollary 1 under the assumptions of Theorem 2 will be stated without giving the straightforward proof. COROLLARY
2
If Gklirs + SkJii, > 2 I?,,:~~for all i, j, 1, r, s # k, then the allele A, is protected within any invariant subset B of P& which satisfies the condition (*).
The triangular relationship between pair fitnesses refers now only to heterozygotes, i.e., for a genotype heterozygous for the allele A,, the arithmetic mean of the symmetrized pair fitnesses belonging to crosses with any two genotypes both not containing this allele exceeds the symmetrized pair fitness of these two genotypes. For the particular case of two alleles, the above corollary assumes a very simple form which also allows for a comparison with already existing results. PROPOSITION
4
Consider the case of two alleles and a positive symmetrized selection regime. Then the allele A, (i = 1 or i = 2) is protected if R ,,;,, > w,,;,, , where j # i ( j = 1 or j = 2). Consequently, the biallelic polymorphism is protected if 6,2;22 > w22;22 and 3,,;, 1> w, ,;, , Zf Si ,;,, < w,,; jj (j # i), then the extinction state for A, (i.e. p, = 0) is locally attractive, and therefore Ai is not protected.
42
HANS-ROLF
GREGORIUS
AND MARTIN
ZIEHE
The last statement in this proposition follows easily from the preceding derivations. This result generalizes the findings obtained by Hadeler and Liberman [5] for a symmetric selection regime in which NJ,,;,, = w22:2z= : a and G,z:,, = 3,,.,, = : p. As these authors have proved, cr
/3 correspond to protectedness and local attractiveness, respectively, of the fixation states of both alleles. This coincides with the present result. Another interesting interpretation of Proposition 4 emerges if our model is viewed in the framework of mating system theory. For this purpose, we assume that each individual has the same probability distribution with respect to the number of encounters of potential mates and that the encounters occur at random. Furthermore, given that a pair of genotypes which are potential mates have actually encountered each other, we assign to this pair a certain probability of mating. Excluding viability and fecundity selection, the w’s in Equation (1) may now be interpreted as mating probabilities. Consequently, the biallelic polymorphism is protected if both homozygotes prefer, on the average, to mate with the heterozygote rather than with their own respective types; the remaining mating preferences are then irrelevant. In the opposite case, i.e. if each homozygote prefers its own type to the heterozygote, the polymorphism is not protected, irrespective of the remaining mating preferences. This is the same conclusion Moore [9] arrived at for a more specific model of the same type. MULTIPLICATIVE
PAIR FITNESSES
The most common way to discriminate between the contributions of each of the two parents to their offspring production is to assume that a multiplicative factorization of fecundity is possible. In terms of pair fitnesses this implies the representation
of the female where w,: and w,-F are conceived of as the contribution genotype A,A, and the male genotype A,A, to their pair fitness and are therefore individual fitnesses. Such a factorization is realized for instance in all models based on the biological situation of differential gamete production in the sexes (mostly referring to plants) and random union of the produced gametes as opposed to random mating of diploid individuals. It was pointed out by several authors (see e.g. Bodmer [l] or Kempthorne and Pollak [6]) that multiplicative models are analytically equivalent to models of differential viability selection in the sexes with no fecundity differences. A comprehensive analytical treatment of this model for the case of two alleles was given by Mandel [8]. However, there are practically no
ALLELE
PROTECTION
WITH
FECUNDITY
43
SELECTION
results available for the multiple allele case. Therefore, in this section we shall try to demonstrate how the multiplicative model fits into the general statements on viability-fecundity selection and apply the findings of the previous section. Making use of the quantities G’(P): = Z,,,w,TP,, and Wom(P): = 2, jw,F P,,, one obtains K$( P) = w$3y
P),
i$(
P) = w;yiq
w(P)=wq(P)iF~x(P).
qyP)=w~(P)~w;~pj(k),
G,,(P)=W”“(P)~W;p,(k),
j
J
and from Equation
P),
(2)
Consequently, Equation (1) now reads PL, = pz’.~;~. The last equation tells us that, after the first generation, the genotypic frequencies are always equal to the product of two allelic frequencies which, in general, originate from two different allelic structures. The set of these genotypic structures will be denoted P x, i.e. PX:={PEPIP,,=pT.pJafori,j=l
,..., n},
and it has the property that G&P,) c Px . For brevity, we leave it to the reader to apply the results of Propositions 1, 2 and 3 to the situation of multiplicative pair fitnesses. The above statements on the multiplicative model suggest restricting the investigation of protectedness of alleles to invariant subsets of P $l P x. In this case the results of Theorem 2 apply, since for PE P”nPX one obtains
p,(k)
= 2 “” p; + py
<
and consequently the condition (*) is met on the set P’II Px of genotypic structures. For the two allele case and under the assumptions of Corollary 2 (with B c Phn P x ) it follows that the allele A,, say, is protected within B if w,~.~~> w22;22,which for the multiplicative model, is equivalent to +(wzwiF ‘3x. 9 + WI2 w22) ’ w*$*?. This inequality, provided wz~wi?: > 0, is in turn equivalent to w;“,/ wz + WY?/ wzy > 2. Mandel [8], using a mathematically less
44
HANS-ROLF
GREGORIUS
AND MARTIN
ZIEHE
elementary method of proof, arrived at the conclusion that this condition is even necessary and sufficient for a newly arising allele to initially increase in frequency. More generally speaking, the inequality Gki( P) > W(P) applied in Theorem 2 can, for the multiplicative model, be given the representation 9 Wkr
9lL Wkr
WF(P)
w3”(P)
-+-
‘2,
which is assumed to hold for all P from B n Pkn P x (I3 an invariant subset from tP;n P x ) and all i # k in order to guarantee that A, is protected within 8. Without further characterization of B it is very difficult, if not impossible, to specify relationships between the individual fitnesses implying the validity of the above inequality and which go much beyond the comparatively weak statement already contained in Corollary 2, namely that
for all i, j, I, r, s # k. Since, however, a satisfactorily comprehensive treatment of the multiple allele multiplicative model with respect to the problem of protectedness would be beyond the scope of the present paper, the last result will close this section. ADDITIVE
PAIR FINESSES
The second commonly used method of reducing pair fitnesses to individual fitness structures results from the assumption that the pair fitness is the sum of the female and male contributions, i.e., w,~;~, allows the decomposition
The first analytical treatment of the additive model in the two allele case was that of Penrose [ 111. Roux [ 151 gave the formal extension to the multiallelic case and additionally demonstrated with respect to the symmetrized version that the additive model is analytically equivalent to a special case of the multiplicative one. Still, as indicated in this section, the additive model, as opposed to the multiplicative, allows a more detailed treatment. The biological interpretation of the additive pair fitness model may be found in constant survival probabilities for all genotypes and additive fecundity for pairs. In addition to the motivations for the treatment of additive fecundities as they are known from the current literature (for example [ 1 l]), there is a further reason for analysing this model. Mating
45
ALLELE PROTECTION WITH FECUNDITY SELECTION
systems with performance conceived of sex. Using the xi,jpt,wsT9
w
random contact of the individuals and probabilities for the of mating which are determined by one sex can be analytically as models of additive fecundity with no contribution from one additive decomposition, -JR-(P):
it is again possible to define G’(P):
=
= IX,,,Pijw,y, so that w(P)=w~7(P)+wyP).
Then the average fitness components
acquire the representation i$yP)=w~~(P)+w;~,
i$,(P)=wuX(P)+w~, w,,(P) i$( P) = w”x( P) + &I,(
=G(P)+
Wk,:kA iiFqP)=W”(P)+&,(k)w~,
k)w$,
I
w,(p) =4 w(p)+ ~Pl(+kl;k,
[
I
provided
P E P ‘.
1
These equations show the same structure as in the multiplicative that multiplication is replaced by addition. The recurrence equations (1) and (1 a) reduce to
case, except
Remark. Since in the additive case W(P) = Zk, ,Pk,wkItk,, the recurrence equations for pk and jk, and therefore the population trajectories in the symmetrized version only depend on the pair fitnesses wkl; k, of genotypically identical female and male individuals. Employing (2), the equations for the genotypic frequencies in the next generation may be written as
p;, = P?P, + FL,
=
P;P,
+
P;““Pk P;Pk
-
PkP,?
-PkPI.
In order to illustrate more clearly the impact of additivity in pair fitness on the genotypic structure, consider the following representation referring to the deviation of the genotypic structure after selection from the Hardy-Weinberg
46
proportions
HANS-ROLF
in the symmetrical
GREGORIUS
AND MARTIN
ZIEHE
version:
Therefore, after the first generation, the homozygote frequencies for all alleles are less than or equal to the corresponding Hardy-Weinberg frequencies : P-$Gp;:
forallk.
If we adapt the DeFinetti diagram to the two allele case for a graphical demonstration of the symmetrized genotypic frequencies after one generation, G&P) is a set of genotypic frequencies at or above the Hardy-Weinberg parabola (compare Figure 1).
FIG. I. DeFinetti diagram with genotypic trajectories starting from the margin of the triangle. The lower left and lower right vertices of the triangles respectively represent the situations in which only A, or only A, are present in the population, and the parabolas within the triangles represent populations having Hardy-Weinberg proportions. Three cases are considered.
ALLELE
PROTECTION
WITH
FECUNDITY
(b) W*=
(bl
SELECTION
(A:;::;,)
47
48
HANS-ROLF
GREGORIUS
AND MARTIN ZIEHE
In view of our interest in invariant and attractive structures with respect to G, and G,, we define pf
.. - { PE PI there exists a vector CJof allelic frequencies
sets of genotypic such that P = p”qT
+qp%T + 4PT- qqT)
and p+ := {PE Pith ere exists a vector q of allelic frequencies
such that P = pqT
+4PT-qqT} (={PEP+IP=PT}). P+ and 6” have the properties G,(P,) cP+ and G&P,) cP’ (q represents the vector of allelic frequencies in the preceeding generation). In particular, Pkk < pi remains valid for all PE p +. The importance of the described way of proceeding in terms of average fitnesses of alleles may be demonstrated by the following Proposition. PROPOSITION
5
pkkl= pkp, for all k, I and Ek( P) = W(p) f or all k is necessary and sufficient for the existence of an equilibrium point FE P” with respect to G,.
Proof.
Suppose Wk(F) = W(p) for all k. Then
and if pk, = pkp, for all k, I, then $, = Fk, follows for all k, I. Otherwise suppose FL, = pk,, for all k, I. Then as stated in the first section, p; = pk for all k implies the second condition Wk(i) = W(P) for all k and therefore & = pkpI. With our assumption P’k, - Pkl, - we get the first condition ek,, = n PkPP Remark. Using the necessary and sufficient condition for equilibria in the symmetrized additive model, we are able to get an analytical formulation which allows computation of the set of nontrivial equilibria. Both equalities result in the system of equations hWk,:kl
=
w(F).
The computation of the set of nontrivial equilibrium points follows the method demonstrated by Gregorius [4] in the case of constant viability selection. A simple consequence is the existence of only one single nontrivial equilibrium point if and only if the matrix W’, which consists of pair fitnesses wk:: = wk,;k, of genotypically identical pairs, is regular and W*- ‘e
ALLELE PROTECTION WITH FECUNDITY SELECTION
49
(where e is the vector of unity entries) has only positive components. If a single nontrivial equilibrium point exists, it may be computed by ( eTW* - ‘e)) ’ W*- ‘e, where e’ is the corresponding transposed vector to e [161. The authors suspect that in the symmetrized version, after a few generations every genotypic population trajectory enters a small neighborhood of the set of Hardy-Weinberg frequencies and remains there indefinitely. The size of this neighborhood depends on W*. This suspicion was supported by several computer simulations for the two allele case. See Figure 1. Reverting to the application of the derived general results about protectedness of alleles, we notice that in P”
p;(k)
_
2 -
2P;yi-p:<2P,,
and therefore P On P + fulfils the condition (*) in Theorem 2 and Corollary 2. It is easy to reduce the sufficient condition given in Corollary 2, for the protectedness of A, in P&nP+ in the case of additive pair fitnesses, to the form
IlllnWk,:kJ > max r s w~$.~~. , j#k
r#!f#s
For a system with more than two alleles, the sufficient condition for at most one allele. The stronger condition in Theorem 2 leads to
mmwk,;
i /fk
k,
>
can be valid
W(P). sup PEP,n(P%P+)
The authors did not succeed in finding an even stronger applicable version and think that this once again stresses the necessity to search for further restricted invariant and attractive sets B which allow more powerful estimations of suppEB W(P). The previous analytical considerations supported by Figure 1 suggest that these sets IEB can possibly be chosen in the vicinity of the set of Hardy-Weinberg proportions. As follows from Proposition 4, which states that all equilibrium points show Hardy-Weinberg proportions, this would certainly be true, even for arbitrary numbers of alleles, if it could be proved that the limit sets of population trajectories only consist of equilibrium points.
50
HANS-ROLF
GREGORIUS
AND MARTIN
ZIEHE
This investigation was financially supported by a Heisenberg Fellowship and by Grant Gr 435/2 from the Deutsche Forschungsgemeinschaft, Bonn- Bad Godesberg. REFERENCES I
W. F. Bodmer,
Differential
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settings, Gottingen, Research Notes in Forest Genetics, No. I (1979). K. P. Hadeler and U. Liberman, Selection models with fertility differences,
6
Biol. 2: 19-32 (1975). 0. Kempthome and E. Pollak, Concepts
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7
64:125-145 (1970). S. P. H. Mandel, A note on the initial
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8
(1963). S. P. H. Mandel,
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sexes, Heredity 26149-63 (1971). W. S. Moore, A single locus mass-action model of assortative mating, with comments on the process of speciation, Heredity 42(2): 173- 186 (1979). P. A. Parsons, The initial progress of new genes with viability differences between sexes
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L. S. Penrose,
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Genetics 90:383-389 (1978). E. Pollak, Some models of genetic selection, Biometrics 35: 119- 137 (I 979). T. Prout, The estimation of fitnesses from population data, Generics 63:949-967
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and with sex linkage, The
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(I 969). 15
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C. Z. Roux, Fecundity differences between mating pairs for a single autosomal locus, sex differences in viability and nonoverlapping generations, Theoret. Popukafion Biol. 12:1-9 (1977). G. M. Tallis, Equilibria
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