Neutral gene flow in the presence of a selected gene with random or assortative mating

THEORETICAL

Neutral

POPULATION

BIOLOGY

35, 295-306 (1989)

Gene Flow in the Presence of a Selected with Random or Assortative Mating

Gene

FRANCO SPIRITO Department of Genetics and Molecular Biology, Faculty of hf. F. N. Sciences, “La Sapienza” University, 00185 Rome, Italy

Received March 15. 1988

Neutral gene flow in an island model in the presence of a gene subject to selection has been analysed. The selection regime on the gene is such that the maintenance of a stable interpopulation differentiation at this locus is allowed for very low values of the migration rate. Mating at this locus may be random or assortative. Many models of zygotic or prezygotic isolating mechanisms with unifactorial inheritance are represented by this genetic model. Two general solutions for migration rates tending toward zero have been obtained for the case of migration preceding selection and vice versa. It is found that the zygotic isolating mechanisms and one type of prezygotic isolating mechanisms have the same qualitative and quantitative effects for the same values of selection coefficients and of recombination frequencies between the selected gene and the neutral gene. A second type of prezygotic mechanism behaves in a different way: the reduction of gene flow caused by such mechanisms is also a function of the “mating coefficients” of the various genotypes. 0 1989 Academic Press, Inc.

INTRODUCTION

The study of neutral gene exchange between two groups of individuals which have differing forms of partial reproductive isolating mechanisms is an important issue which must be thoroughly investigated in order to gain further insight into speciation processes.In recent years, this problem has been tackled in several theoretical works. Most of the results have been obtained for cases of reproductive isolation caused by a single Mendelian factor (pair of alleles of a gene or pair of alternative chromosomal forms) (Bengtsson, 1974, 1985; Barton, 1979; Petry, 1983; Spirit0 et al., 1983; Spirito, 1986, 1987; Spirit0 et al., 1987) but some studies have focused on cases of isolation with multifactorial inheritance (Barton, 1983; Bengtsson, 1985; Barton and Bengtsson, 1986; Spirito, 1986). Barring processesof sympatric speciation in the strict sense,the most interesting situations to analyse are those in which the neutral gene flow occurs in the presence of a stable interpopulation differentiation for one or more gene 295 0040-5809189$3.00 Copyright 0 1989 by Academic Press, Inc. All rights of reproduction in any lorm reserved.

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loci subject to selection. In the case of a single selected locus, three main sorts of selection regimes are able to maintain a situation of stable interpopulation differentiation at the locus in question: (i) the homozygote for one of the two alleles is favoured in the first population and the homozygote for the other allele in the second population; (ii) the heterozygote is disadvantaged; (iii) the selection coefficients are frequencydependent in such a way that the genotype which is very frequent is advantaged. In this paper, I have analysed the reduction of neutral gene flow caused by a single biallelic gene locus subject to selection in an island model, for very small migration rates. The neutral gene can have any recombination frequency with the selected gene. The approach used is similar to that of Bengtsson (1974, 1985) and Spirit0 (1986) and consists in the search for a limiting solution for migration rates tending toward zero in an island model. The methodology used is similar to that of Spirit0 (1986). Analytic solutions have been obtained that are valid both in the case of random mating (the selected gene causes zygotic isolation) and in the case of assortative mating (the selected gene causes prezygotic isolation).

THE MODEL

Two autosomal gene loci are considered. The gene ,4,/A* is subject to selection; the mating of the three genotypes at this locus may be random or assortative. The gene locus B,/B, is neutral and has a recombination frequency R with respect to A ,/A,. A deterministic two-population model is considered: population 1 (island population) is initially composed only of individuals A i A i B, B, ; population 2 (continent population) is infinitely larger and is initially composed only of individuals A,A, B, B,. The frequencies of A i and B, alleles in population 1 have been denoted as p and P, respectively. A fraction m of population 1 is exchanged with population 2 at each generation. Migration occurs in the diploid phase and may take place in a period of individual life which is before or after the period in which selection acts. Generations are discrete and not overlapping. The selection regime on gene Al/A, is arbitrary provided that it is such that the allele A2 is counterselected in population 1 when allele A, is very rare. The litnesses in population 2 are not important given the mathematical structure of the model: in fact population 2 is infinitely larger than population 1 and therefore it remains monomorphic for A,, its composition not being modified by the inflow of genes from population 1. However, from a biological point of view, the situations that are interesting are those in which the A, allele is counterselected in population 2 when Al is very rare in it. Only in these situations, in fact, can there be a stable differentiation

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in two-population models for the gene A,/A, when the ratio between the sizes of the two populations is not infinite. An analysis of the stability of the equilibria of gene frequency of a selectedlocus in models constituted by two populations of the same size was made by Karlin and McGregor (1972). In the model described above the values of the migration reduction equivalent (MRE) have been studied. MRE is a measure of the effect of reduction of neutral gene flow between the populations. By definition MRE = m*/m, where m* is the migration rate for which there is, in the absence of selection on A ,/A,, the same quantity of gene flow for the gene B,/B* as that observed for the migration rate m in the presenceof selection on AI/A, (Spirit0 et al., 1983). It must be observed that the index MRE studied here is equivalent to the gff index of Bengtsson (1985).

THE GENERAL SOLUTIONS

The aim is to calculate the limiting value of MRE for m -+ 0 in the model described above. The approach used consists in following the fate of the m alleles B, entering population 1 at a given generation. Account is taken of the fact that the regimes of selection of interest to us are such that for m + 0 the equilibrium values of A,(p) tend toward one. Moreover, for m + 0, it can be assumed that the m alleles B, entering population 1 have an insignificant probability of leaving it in the course of the first generations. The calculation of the effective migration rate (m* = m . MRE) for very small m values is, in our model, therefore equivalent to the calculation of the final value of the frequency of B, in a model that assumesa single burst of immigration (see Spirito, 1986). Following this approach general solutions have been obtained both for the case in which migration precedes selection and for the case in which selection precedes migration. The only assumptions made in deriving solutions are that the limits for p + 1 of the relative fitnesses and of the “mating coefficients” of genotypes A, A,, A 1A,, and A2A, can be calculated. The limiting values of the relative fitnesses of A, A,, A, AZ, and A2 A, are denoted, respectively, as w,, w,,, and W, (w, = 1, wb < 1, W,< 1; wi, and/or w, are less than 1). The “mating coefficients” of genotype A, A, are defined as the probabilities that an individual Al AI mates with an individual A, A,, A, A,, or A,A,. The limiting values for p --) 1 are denoted as h,,, hab, and h,,. Similarly, the mating coefficients of genotype A, AI are the probabilities that A, A, mates with A, A i, A, A,, or A, A2 and their limiting values for p + 1 have been denoted as hba, h,, and hbc, respectively. The probabilities for p + 1 that A2A, mates with A,A,, A,A,, or A,A2 are h,,, hcb, h,,. The procedure for obtaining the solutions is the following. A B, allele

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that has entered population 1 may be present in one of four different conditions. Condition 1 (u, ) is the presence of a B, allele in phase A i B, in a homozygote A i A, (A, B,/A ,-); condition 2 (uZ) is the presence of a B2 allele in phase A,B, in a heterozygote A, A, (A,BJA,-); condition 3 (Q) is its presence in phase A, B, in a heterozygote A, A, (A, B,/A2-); condition 4 (04) is its presence in phase A2 B, in a homozygote A,A, (A,B,/A,-). Moreover, I have denoted as u5 the state which represents the B, alleles eliminated in the population by selection on Al/A, in the presence of gametic disequilibrium between A,/A, and B,/B,. I have indicated as T, the probability that a B, allele present at a given generation in ui will be in uj at the next generation. (It must be noted that, given the deterministic structure of the model, these probabilities correspond to the proportions of B, alleles effectively transferred from ui to uj each generation). The matrix showing, for p -+ 1, the values of constants T, is 1 w,W,,+

T=

0 hd

wd-R)(h,,+fM 0

[

~d-R)h

0 wAh+M

+ fM

wtW,,+Ih,d wcv,, + t&J

0

0 ~‘b(l-R)(thbb+hbc)l-wb

wd-Wh,+hJ 0

0

0

wJ#hb+hJ l-w, wc(t~cb+~cc) l-w,

0

0

1

It must be observed that T,, is equal to 1 as a consequence of the fact that h,, = 1, hab= 0, and h,, = 0 when there is random mating or positive assortative mating. Let us first consider the case in which migration precedes selection. The system is described by a live-state Markov chain (i.e., the four states u1, u2, u3, 04, and the state u5 which represents the eliminated B, alleles). Initially, all B, alleles entering the population are in u,; finally, they will be found in u, or u5, which are the two absorbing states. It is of interest to calculate how many B2 alleles which are initially in uq will reach ui, since this value corresponds to lim, _ 0 MRE. By following the standard procedure which consists in solving the appropriate system of equations, we obtain lim

MRE

=

m-0

(Tdl

-

T.tJ)-cp

l-$

’

(1)

where cp= Tz, + (T,, Tx,)/(l - T,,)

ti = T,, + Ta . Tdl + Tn. T,, . T&l

- TM) + TX . T,,l(l - T,,) - Tdl

- TM)-

Let us now consider the case in which selection precedes migration. The only difference from the previous case is that, at the first generation, all the

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m alleles B, are transmitted to the offspring instead of a fraction w,. All that must be done to obtain the solution in this case is to divide the previous solution by w,. We obtain lim MRE=wu

- T‘l‘J).(P

m-0

wc.(l-@)

(2) .

The analysis of (1) and (2) shows that the value of MRE is very low when at least one of these conditions is satisfied: (i) the value of wb (and/or w, when (1) is applied) is very low; (ii) the value of R is very low; (iii) the value of h,, is close to 1 (with w, < 1). It is important to observe that for h,, = 1, h,, = 0, h,, =O, h,, = 1, h,, = 0, h,, = 0, a condition which occurs when mating is random or also in several models of assortative mating, relations (1) and (2) can be written, respectively, in the following simplified forms: TN lim MRE=-----= m-0

I-

w;w,,.R I-wi,-(l-R)

. 7-21 T22

(1’)

and lim MRE= m-0

T42.

Tz,

wb

.R

~;(l-T~~)=l-w~~(l-R)~

(2’)

For situations of selection with constant coefficients and random mating, solutions (1’) and (2’) are equivalent to the various solutions already obtained in several special casesin island models and in models which consist in two populations of equal size (Bengtsson, 1974, 1985; Petry, 1983; Spirit0 et al., 1987).

APPLICATION OF THE GENERAL SOLUTIONS TO SPECIAL CASES OF ASSORTATIVE MATING

This section treats the application of the general solutions for m + 0 in special cases of genes subject to selection with assortative mating (prezygotic isolation). Three different models have been studied of genes which cause assortative mating and which are subject to frequency-dependent selection (the very frequent genotype is advantaged) as a consequence of the mating pattern. These three casesare described by Karlin and Scudo (1969) together with other cases either subject to different selection regimes or selectively neutral. They can be considered cases of sexual isolating mechanisms, according to Dobzhansky’s definition (Dobzhansky, 1970). The gene A ,/A,

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determines the preferential choice by the females of males of the same genotype. For a detailed description of the model seethe paper cited above. Only a short description of the caseswill be given. Case 1. This model refers to animals with a sex-ratio of 1: 1 and which form permanent bonds. Parameters CI,j, and y describe the “tendency” to assort of genotypes A I A r, A r A z, and A 2A 2, respectively. A fraction u of the females of genotype A,A, tend to mate with males of the same genotype, while the complementary fraction mate at random. /I and y have a similar meaning for genotypes A r A, and A, A,. Furthermore, it is assumed that females practicing random mating pair earlier than assorting females and that c(= y > p. From this it ensues that only a fraction of the assorting A, A, and A,A, females can pair with proper male partners; the remaining females do not contribute to the next generation. Case 2. The model is similar to that of the previous case. The only difference is that c(= y < j3. Case 3. This case represents the “mass-action” model. The chance of a given type of mating is assumed to be proportional to the product of the densities of the two kinds of individuals which can perform the mating. Table I shows the relative frequencies of the different mating types (Ml, M2, M3, M4, M5, M6) as a function of the frequencies of genotypes A, A r, A, A,, and A2 A2 (X, Y, and 2) and of ~1,/I, y, in the three cases. For Cases 1, 2, and 3 it is possible, on the basis of the values of Ml, M2, M3, M4, M5, M6 in each case, to calculate the limits for p + 1 of the relative litnesses and of the mating coefficients of the three genotypes. Table II gives the general formulas valid for all three cases and the particular values obtained by applying these formulas to the three cases. It may be observed that in Case 3 the mating coefficients for p + 1 have the same values as when mating is random. The application of the solutions of the limiting values of MRE given in the preceding section allows the expected MRE values to be calculated for small values of m in the three cases.In all three cases selection on Al/A2 is a consequence of the mating pattern and therefore acts at the time of reproduction. So the solution that must be used is (1 )-or its simplified form ( 1’) in Case 3--which is valid for migration preceding selection. A check of the goodness of this solution as an approximation of the MRE values for small m was made by performing several numerical tests using a computer, for R = 0.5. Tests were run in Cases 1, 2, and 3 for several sets of values of u, /?, and y. For each numerical test, the MRE was estimated from the number of generations taken for the frequency of P to fall to 4, using 4 = (1 -m*)” and MRE = m*/m. Table III shows the results of numerical tests. The value of MRE observed in each numerical test is given,

I

Frequency in Case 1

Ml=[X’(l-a)+X(a-Y(n-B))]/N M2 = [2XY(l -a/2 -,9/2)1/N M3 = [2X2( 1 - a)]/N M4= [YZ(l -B)+/WJ/N MS = [2YZ( 1 -a/2 - /I/2)1/N M6 = [Z’(l - a) + Z(a - Y(a - j?))J/N

Mating type

A,A, xA,A, AIA, xA,A2 A,A,xAd2 A,A,xA,A, At-%x Ad, AAxA,A,

Ml = [X2( 1 - a) + aXJ/N’ M2= [ZXY(l -a/2-#I/2)1/N M3 = [2X2( 1 - a)]/N’ M4= [Y2(l -a)+aY]/N’ M5=[2YZ(l-a/2-/I/2)]/N’ h46 = [Z2( 1 -a) + aZ]/N’

Frequency in Case 2

Frequency in Case 3

Models of

MI =X2/N” M2 = 2XY( 1 -a/2 - /?/2)/N” M3 = 2XZ( I- a/2 - y/2 )fN” A44 = Y2/Nn M5=2YZ(l-/3/2-y/2)/N” M6 = Z’/N”

Relative Frequencies (Ml, M2, M3, M4, MS, M4) of the Various Mating Types for the Different a Gene (AI/A,) with Assortative Mating (Cases 1, 2, and 3)

TABLE

i-4 zm w

F

2

z!i u 2

g

3

2

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together with the equilibrium value of the frequency of A,(p). It can be seen that, other things being equal, for smaller m the equilibrium frequency of the A, allele (p) is closer to 1 and the MRE value is closer to the limiting value calculated with (1). The agreement between the MRE values observed for the smallest m and the MRE values for m + 0 calculated with (1) is very good in all the analysed cases.Finally, it can be observed that for small m values-as would be expected on the basis of solution (1)-results of assortative mating in Case 3 are very similar to those obtained for a zygotic isolating mechanism with the same coefftcients of selection (see Spirito, 1986) whereas in Case 1 and in Case 2 the reduction of gene flow is stronger than that observed for the same coefftcients of selection when mating is random. TABLE II Limiting Values of Fitnesses and Mating Coeficients for p -t 1 in Cases 1, 2, and 3 General formulas

Values in Case 1

Values in Case 2

Values in Case 3

1

1

1

1

wb

(M2+2.M4+M5).X d’-“I(2.Ml+M2+M3).Y

WC

(M3+M5+2.M6).X b’:(2.Ml+M2+M3).Z

1

1

l-?-g?

h aa

2.Ml :?12,Ml+M2+M3

1

1

1

h ab

M2 ;?12.Ml+M2+M3

0

0

0

h ac

M3 lim .-.12,Ml+M2+M3

0

0

0

h ba

M2 Iim p-l M2+2.M4+M5

2-a-/? 2-a++3

2-a-p 2+a-B

1

2.M4 ;t? M2+2.M4+M5

28 2-a+p

2a 2+a-/3

0

0

0

0

l-a

l-a

1

0

0

0

a

a

0

hbb

h,

M5 lim p-+,M2+2.M4+M5

h ca

M3 f:M3+M5+2.M6

h cb

,,-I M3+M5+2.M6

h cc

2.M6 lim p-1, M3+M5+2.M6

lim

M5

l-y- a-B

I-c(+p

1-B-a

2

2

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TABLE III Values of MRE and j Obtained for the Different Models of a Selected Gene with Assortative Mating m = 0.004

m = 0.001

m = 0.00025

0.971 (-) 0.970 (-) 0.969 t-1 0.712 (0.974) 0.692 (0.963) 0.684 (0.929)

0.920 (0.977) 0.918 (0.976) 0.916 (0.975) 0.677 (0.994) 0.649 (0.992) 0.618 (0.985)

0.908 (0.995) 0.906 (0.995) 0.904 (0.994) 0.669 (0.999) 0.639 (0.998)

0.971 (-) 0.970 l--j 0.969 t-1 0.708 (0.977) 0.683 (0.973) 0.654 (0.966)

0.920 (0.977) 0.918 (0.976) 0.916 (0.975) 0.676 (0.995) 0.647 (0.994) 0.612 (0.992)

0.908 (0.995) 0.908 (0.995) 0.905 (0.995) 0.669 (0.999) 0.639 (0.999) 0.603 (0.998)

0.905

0.846 (0.900) 0.856 (0.892) 0.867 (0.876) 0.410 (0.987) 0.414 (0.987) 0.417 (0.987)

0.823 (0.981) 0.825 (0.981) 0.826 (0.980) 0.403 (0.997) 0.403 (0.997)

0.816 (0.995) 0.817 (0.995) 0.817 (0.995) 0.401 (0.999) 0.401 (0.999) 0.401 (0.999)

0.814

m-t0

Case 1 a=y=o,1;p=o a=y=O.15;~=0.05 a=y=O.2;~=0.1 a = y = 0.4; p = 0 a=y=0.6;/l=0.2 a = y = 0.8; /I = 0.4 Case 2 a = y = 0; p = 0.1 a=y=0.05;~=0.15 a=y=0.1;p=0.2 a = y = 0; p = 0.4 a = y = 0.2; /I = 0.6 a=y=0.4;/3=0.8

Case 3 a=O;B=O.l;y=0.2 a = 0.05; /I = 0.05; y = 0.15

a=O.l;/l=O;y=O.l a = 0; /I = 0.4; y = 0.8 a = 0.2; /I = 0.2; y = 0.6

a = 0.4; j3= 0; y = 0.4

0.905 0.902 0.900 0.667 0.636 0.600

(Ez)

0.902 0.900 0.667 0.636 0.600

0.814 0.814 0.400 0.400 0.400

Note. The values of MRE and the values of p (in brackets) observed in numerical tests for the various values of m and the various sets of values of a, 8, y, are shown. The MRE values for m + 0 have been calculated using solution (1). It must be noted that in some tests the A, allele is eliminated, for m = 0.004. In Cases 1 and 2, the first three sets of values of a, p, y are such that W,= 1, wb = 0.95, w, = 1; the other three sets of values are such that w, = 1, wb = 0.8, W,= 1. In Case 3, the first three sets are such that W,= 1, ~3~= 0.95, w, = 0.9; the other three sets are such that: W, = 1, wb = 0.8, wc= 0.6.

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A general solution has been obtained which allows the reduction of neutral gene flow for small migration rates in an island model in the presence of many different models of unifactorial isolating mechanisms to be quantified. An interesting feature of this solution is its applicability to a very broad range of different biological situations. The neutral gene flow in the presence of a gene subject to selection of different kinds with random or assortative mating is described by this solution. The only condition which is necessary for the solution to be valid is that the equilibrium frequency of one of the two alleles of the selected gene should tend toward one in the island population (while its frequency is zero in the continent population) when the migration rate tends toward zero. The following biological conclusions can be drawn from the application of the general solution. The effect of reduction of gene flow when mating is random (zygotic isolation) is a function of the recombination frequency between the selected gene and the neutral gene and of the litnesses of the genotypes for m + 0. The effect is drastic only for very high values of the selection coefficients and/or for very small values of the recombination frequency. This biological conclusion has already been reached by several authors (Barton, 1979; Bengtsson’l974, 1985; Spirit0 et al., 1987; Spirito, 1986; Petry, 1983). It must be observed that all solutions previously described in the literature for various special casesof selection at one locus with constant coefficients and random mating in two-population models (Bengtsson, 1974, 1985; Petry, 1983; Spirito, 1986; Spirit0 et al., 1987) are included in the general solution for m -+ 0 described in this paper. As to the reduction of gene flow in the presence of selected genes with assortative mating (prezygotic isolation), it is necessary to distinguish among different models of assortative mating. A first type of model (see Case 3) gives results similar to those observed when mating is random. The reduction of gene flow is a function only of the recombination frequency and of the selection coefficients and is quantitatively equal to that observed in the casesof zygotic isolation for the same values of these parameters. A second type of model (see Cases 1 and 2) gives different results. In these casesthe gene flow is also a function of the values of mating coefficients for m -+ 0. In models of this type, a drastic reduction of gene flow is possible also for high recombination frequencies and small selection coefficients when the mating coefficient h,, is close to 1 (and w, < 1). An important issue which must be addressed in order to fully understand the biological significance of these results is the degree of realism achieved by models of assortative mating of the second type. A model of assortative mating belongs to the second type when it is such that the few homozygotes for a very rare allele and the few heterozygotes are able to perform assortative

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mating by finding a mate of the same genotype from among a great number of individuals which are homozygous for the frequent allele. Such a condition is probably .very rarely achieved and models of assortative mating of the second type are presumably valid only in a minority of cases (see also the comments of Scudo and Karlin (1969) on this subject). However, the issue is open to debate and deserves to be throughly discussed. There are few precedents in the literature with which to compare the results here obtained in cases of assortative mating. In the few papers which describe the behaviour of a neutral gene in the presence of a gene causing assortative mating, genes with assortative mating which are not subject to selection are considered (Gregorius, 1980a, 1980b; Spirito, 1987). The only analysis of the reduction of gene flow caused by a selected gene with assortative mating was made only recently (Spirito, 1987; see Appendix). This earlier study refers to a gene causing temporal or ecological isolation. Although the biological model is different, the mathematical formalization is equivalent to that of Case 3 described in this paper for /I = 0 and c1= y = z (z is the parameter used in the preceding study).

ACKNOWLEDGMENTS Many thanks are due to Professors G. Montalenti and M. Rizzoni for their critical reading of the manuscript. I also thank Dr. G. Spirit0 for his helpful mathematical suggestions.

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