Genetic Diversity in Partially Selfing Populations with the Stepping-Stone Structure

Heredity 77 (1996) 469—475 Received 20 October 1995

Genetic diversity in partially selfing populations with the stepping-stone structure

HIDENORI TACHIDA*t & HIROSHI YOSHIMARU tDepartment of Biology, Faculty of Science, Kyushu University 33, Fukuoka 812, Japan and tPopulation Genetics Laboratory, Forestry and Forest Products Research Institute, Tsukuba, Ibaraki 305, Japan

Amethod to compute identity coefficients of two genes in the stepping-stone model with partial selfing is developed. The identity coefficients in partially selfing populations are computed from those in populations without selfing as functions of s (selfing rate), m (migration rate), N (subpopulation size), n (number of subpopulations) and u (mutation rate). For small m, 1/N and u, it is shown that approximate formulae for the identity coefficients of two genes from different individuals are the same as those in random mating populations if we replace N in the latter with N(1 —s12).Thus,the effects of selfing on genetic variability are summarized as reducing variation within subpopulations and increasing differentiation among subpopulations by reducing the subpopulation size. The extent of biparental inbreeding as measured by the genotypic correlation between truly outcrossed mates was computed in the one-dimensional stepping-stone model. The correlation was shown to be independent of the selfing rate and starts to fall off as the migration rate increases when mN is larger than 0.1.

Keywords:biparentalinbreeding, fixation index, genetic variability, geographical differentiation, selfing, stepping-stone model.

Introduction Model analysis Manyplant species are partially self-fertilizing Genesare defined to be identical by descent if they (Brown, 1990; Murawski & Hamrick, 1991, 1992; have a common ancestor gene and there is no muta- Kitamura et a!., 1994). Thus, it is necessary to incor- tion in their descent from the common ancestor. We porate self-fertilization in the modelling of plant assume the selective neutrality of alleles (Kimura, genetic diversity. Pollak (1987) computed identity 1968) and let u be the mutation rate. Consider a coefficients of up to four genes in a partially selfing d-dimensional stepping-stone model in which each population without a geographicalstructure. subpopulation has the same size, N, and subpopula- Maruyama & Tachida (1992) analysed the island tions are arrayed in a d-dimensional torus (see model with partial selfing and derived formulae for Maruyama, 1977). Assume that generations are identity coefficients of pairs of genes in the equili- discrete. For simplicity, we consider only pollen brium state. Here, we develop a method to derive migration and the maternal gamete always comes equilibrium identity coefficients of two genes in from the same subpopulation, Let s and mp be the partially selfing populations with the stepping-stone selfing rate and pollen migration rate, respectively, structure (Kimura, 1953). It is shown that the in the population. Here, in order to simplify the subpopulation size is effectively reduced in partially expressions, s is defined not to include the prob- selfing populations resulting in an increase of differ- ability of selfing which occurs randomly in random entiation among subpopulations. monoecious mating. There are three ways in which the paternal gamete fertilizing the maternal gamete is chosen. (1) The paternal gamete comes from the *Correspondence same parent as that of the maternal gamete with a 1996 The Genetical Society of Great Britain. 469 470 H. TACHIDA & H. YOSHIMARU probability of s+(1—s—m)/N. (2) The paternal (Tachida, 1985; Slatkin, 1991; Slatkin & Voelm, gamete comes from a different parent in the same 1991). The time will be measured backwards in units subpopulation from that of a maternal gamete with of generations. Let T be the coalescence time for the a probability (1 —s —m)(1 —1/N). (3)The paternal two genes. Then, 0 is represented as gamete comes from a parent in one of the 2d neighbouring subpopulations with a probability of mp. oo = E[(1 _u)2T]. (2) Under these assumptions, the subpopulations are This coalescence time is divided into two parts equivalent to each other. Thus, the identity coeffi- (Slatkin, 1991); T1 when the ancestors of the two cient of a set of genes in the equilibrium state genes were in the same individual for the first time depends on the relative locations of the subpopula- (coalescence to an individual), and T2 for the two tions from which the genes are sampled. For genes in the same individual to coalesce. Therefore, example, the inbreeding coefficient, f, defined as the probability of two genes in an individual being iden- = E[(1 —u)2T'(1_u)2T2]. (3) tical by descent, is the same for all individuals. Because T1 and T2 are independent, this is further Define the following identity coefficients for pairs simplified to of genes where i.b.d. means identical by descent: Go =E[(1_u)2T]E[(1_u)2T21. (4) f= Prob[two genes in the same individual are i.b.d.]; To compute the first term of the right-hand side =Prob[twogenes in different individuals from the of (4), we first show that the distribution of T1 is the same subpopulations are i.b.d.J; and same as the coalescence time of two genes in a 0, = genes sampled from different sub- haploid population with the same population struc- Prob[two ture. Let m be the gamete migration rate in the populations i-steps apart are i.b.d.]. haploid population. The number of genes in each Here, i is a d-dimensional vector, the elements of subpopulation is N. If we take two genes from the which represent the numbers of steps the two subpo- same subpopulation, each gene is a migrant with a pulations are apart in respective dimensions. In the probability m. If the two genes come from the same following derivation, the identity coefficients of two subpopulation (nonmigrant), the probability of the genes from two different subpopulations one step two coming from the same individual is 1/N. Now we apart are important. We denote the average of the consider how the two genes come from the previous 2d identity coefficients for the neighbouring generation in the selfing population. Each one of the subpopulations by 01. two genes has either paternal or maternal origin Nowwe compare the equilibrium values of these with a probability one-half and is a migrant only identity coefficients. Because two uniting gametes in when it has paternal origin. Thus, each gene is a an individual in a subpopulation are from an indivi- migrant with a probability m1,12. If the two genes dual in the same subpopulation (selfing), two come from the same subpopulation, the probability different individuals in the same subpopulation, or of the two coming from the same individual is 1/N. one from the same subpopulation and the other If we replace the word individual with the word gene from the neighbouring subpopulation with a prob- in the above two descriptions, we can see that the ability s+(1—s—m)IN, (1—s—m)(1—1/N), or m, probability law for the coalescence to an individual respectively, and the probability of identity by in the selfing population is the same as that of the descent is (1 +f)12iftwo genes come from the same coalescence to a gene in a haploid population which individual, has the same geographical structure with m,I2 replaced by m, and the same subpopulation size, N. Thus, T1 has the same distribution as the coales- f= (lU)2{[S+(l5mP)](tf) cence time of two genes in the haploid population and the first term of (4) is expressed as +(1—s—m)1—— 00+m01 (1) E[(1 _u)2n1] g0(u), (5) where go(u) is the coancestry coefficient in the Next we consider O for two genes in different haploid population. Because the second term of the individuals in the same subpopulation. In the follow- right-hand side of (4) is (1 +f)/2, we obtain ing, we trace the descent of the two genes in the past and utilize the relationships between the coales- /1 +f cence of the genes and the identity coefficients (6)

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The same argument can be applied to the calcula- In the above expression, [i] represents the largest tion of 0, which is expressed as integer which does not exceed i. We can obtain the identity coefficients in the one-dimensional stepping- stone model with partial selfing by putting (10) and 0 =gj(u)), (7) (11) into (8), (6) and (7). Note that w is not a function of N. go(u) inthe two-dimensional step- where g1(u)s are the corresponding identity coeffi- ping-stone model is also written in the same form as cients for the haploid population. that in the one-dimensional stepping-stone model Putting (6) and (7) into (1) and solving for f, we with w independent of N. In fact, this is true for the obtain island model and a random mating population model. (1 —u)2A (s, m, N, u) In geographically structured populations such as = (8) —u)2A (s, mp, N, u) stepping stone models, inbreeding results not only 2—(1 from selfing but also from mating within the local with area. The latter inbreeding is called biparental inbreeding (Ennos & Clegg, 1982). This effect is conveniently measured by the biparental genotypic A(s, mp, N, u) =s+(1_s_m)[+(1_)o(u)] correlation, rb,betweentruly outcrossed mates (see Wailer & Knight, 1989) defined as +mpgi(u). (9) F0 (12) Otheridentity coefficients are obtained from f using Fsei (6)and (7). where F0, and Fseiarefixation indices of a truly Maruyama (1977) lists explicit expressions for outcrossed and a self-fertilized individual, respect- g1(u)s in the one- and two-dimensional stepping- ively. In terms of identity coefficients, these fixation stone models. For example, in the one-dimensional indices are represented as stepping stone model with n subpopulations, g, (u )s for our model are expressed as F0= , (13) (1—u)2w go(u) = and (10) N+(1—u)2w (1+f)12—O Fsei = (14) 1— —u)2(1 —g0(u)) [(n+1)/2J g,(u) =(1 N k0ak[i—(i—u)2k} whereO is the i.b.d. probability of two genes taken from two random individuals, respectively, in the 2irik population and expressed as xcos— (11) n - [(n+fl/2j 0=— with ni=O

[(n+1)/2] b —— ifi=Oorn/2 fi2otherwise. k=O ak[1—(l—u)k1 From (6) and (7), we obtain mp7 Ak= i——( go(u) —g(u) 2\\1_cos_)]fl/ rb= , (15) [ 1—g(u)

and where

(n (k=Oorn/2) 1 [(n+1)12] ak=n/i. (otherwise). bg1(u). n i=O

The Genetical Society of Great Britain, Heredity, 77, 469—475. 472 H. TACHIDA & H. YOSHIMARU

(1 —u)2w Set3 () 0.8 Substitutingthis into (18), with some rearrangement, we obtain rb Set 1 Set 2"\ (1 —u)2w 0.4 +0(m,u,1/N). (20) 2N(1 —s/2) + (1 —u)2w

0.2 This formula shows that 0 for a partially selfing population is computed by regarding N(1 —s/2) as the effective size of the subpopulation and using the 0 formula for the corresponding diploid population io- 0.001 0.01 0.1 m without selfing. This relationship was noted by p Golding & Strobeck (1980), and Pollak (1987) for Fig. I The biparental genotypic correlation, rb.One- populations without any geographical structures and dimensional stepping-stone model is assumed. The migra- by Maruyama & Tachida (1992) for the island tion rate m is changed. Other parameters are n =10, model. N =10,u =0.01in set 1, n =40,N =5,u =0.01in set 2 We can show that this relationship also holds for and n =100,N =5,u =0.004in set 3. other identity coefficients, 0,, if m0, u, 1/N are much smaller than one, considering the coalescence of the two genes in different subpopulations (Slatkin, 1991). Let Tb be the time when the ancestors of the Because gj(u)s are independent of the selfing rate, two genes were in the same subpopulation for the this equation shows that rb is independent of s. We first time and define T =T—Tb where T is the total numerically computed Tbforvarious n, N, u chang- coalescence time of the two genes as in the previous ing m0 and some of the results are shown in Fig. 1. argument. Then, we can write 0, as Note that although calculations are made for small N and large u, values would be similar when N and 0, =E[(1_u)2T]E[(1 _u)2T]. (21) u are changed keeping Nu constant. The genotypic Because the first ancestor genes which enter the correlation, rb, is a monotonic decreasing function of same subpopulation are in the same individual with mp and starts to decrease rapidly as m increases when mN is larger than 0.1. Waller & Knight a probability 1/N, the equation is expressed as (1989) estimated rb in Impatiens capensis and 1 [1+f obtained values ranging from 0.1—0.4. Such values 0, =E[(1 _u)2ThJ + (N— are expected with our parameter set 1 when nip is fairly large. = + If m, u, 1/N are much smaller than one and s, E[(1 — u)2T'] 00(1/N). simpler expressions for 0,s are possible. In this case, Note that because Tbdoesnot depend on the size, (8) is written as N, of the subpopulation, E[(1_.u)2T11 does not depend on it. This means that N enters into the A(s, mp, N, u) formula only through 0 and thus we can compute U +0(m0, u, 1/N) (16) 2—A(s, m, N, u) by regarding N(1 —s/2) as the effective size of the subpopulation. 17 How good is this approximation of regarding A (s, mp, N, u) =(1—s)go(u) +s, k N(1 —s/2) as the effective size of the subpopulation? where 0(m, u, 1/N) designates a function of the To check this, we numerically computed 0 for same order as those of mp, u and 1/N. Combining various values of N, m, n and some of the results the two relationships, 0 is represented as are shown in Fig. 2. The discrepancies between the exact and approximate values are very small even for go(u) the small N used in the computations. Even for 00= +0(m,u,1/N). '18" 2— [(1 —s)g0(u) +s] fairlylarge m, the difference is less than a few per cent. Thus, the approximation works very well for As noted above, go(u) is written as most parameter ranges encountered in nature.

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Set 3 we ignore mutation during the coalescence in an 0.9 individual, two random genes in one individual are identical by descent with a probability 0.8 [1 +sI(2—s)112= 1I(2—s). Ifwe regard this event of quickly becoming identical by descent as having 0.7 occurred in the parental generation, the probability 00 0.6 of the two parental genes being identical by descent is 0.5 1 0.4 1 1_ 2—sN 2N(1—s/2) (22) 0.3 This shows that N(1 —s/2) can be used as an effec- 0.2 tive number for partially selfing populations. Our 0.00 1 0.01 0.1 derivation in the previous section shows that this m p rough argument is actually applicable as an approx- Fig. 2. The exact and approximate values of the identity imation. Note that this argument does not rely on a coefficient, O, of two genes in different individuals in the specific geographical structure of the population. same subpopulation as functions of mp assuming the Thus, the method of using N(1 —s12) as the effective one-dimensional stepping-stone model. The straight lines size seems not to be restricted to symmetrical struc- are computed using the exact solution (6) with (8). tures such as the island and stepping-stone models Broken lines are computed using the approximate solu- and might be used, for example, for the cases where tion (20). Other parameters are n =10,N =10,u =0.01, parameters such as the selfing rate and subpopula- s =0.5in set 1, n =40,N =5,u =0.01,s =0.5in set 2 and tion size change among subpopulations (Karron et n = N =u = s = 100, 5, 0.004, 0.5in set 3. at., 1995). In our computation, only pollen migration was considered. This argument goes similarly if there is Discussion only seed migration. Let m be the seed migration Inthe present paper, we developed a method to rate. For the coefficient, f, for the two genes in the compute identity coefficients in stepping-stone same individual, the following equation holds in the models with partial selfing from those without equilibrium state: selfing. Because exact solutions are available for stepping-stone models (Maruyama, 1977), we could f= . (23) use those solutions to obtain the exact identity (1 _u)2[(s +_) _i+ (1 _s)(1_)o] coefficients in partially selfing populations. Although we assumed stepping-stone models in the derivation, Fortwo genes in different individuals, there are the critical assumption is that f is constant over the three cases with regard to whether they are migrants subpopulations. If this holds true, (7) holds and we or not. (1) Both are nonmigrant with a probability can compute the identity coefficients in the partially (1 —m)2. (2) One is migrant and the other is nonmi- selfing population in the same way. grant with a probability 2(1 —m)m. (3) Both are For small m, u, 1/N, we showed that N(1 —s/2) migrants with a probability m. Therefore, Os are can be used as the effective size of the subpopula- expressed as in (7) but in this case we use m5 as the tion. This relationship can be derived also by the migration rate to compute g,(u) in the corresponding following intuitive argument. Assume that we haploid population. If u, m, 1/N are much smaller sampled two genes from two different individuals in than one, we obtain the same approximation as that the same subpopulation. We consider the probability in the case of pollen migration in which (1 —s12)N of the two genes becoming identical by descent in can be used as the effective size of the sub- the parental generation. The two genes come from population. the same individual with a probability 1/N if we On the other hand, if both seed and pollen migra- ignore the higher order terms of mp and 1/N. Once tion occur, we can not just use g1(u) computed from the two genes are in an individual, they coalesce the corresponding haploid population to compute soon without going out of an individual (the coales- the exact solutions in partially selfing populations. cence in an individual) and become identical by For example, a gene may come from a subpopula- descent with a probability s/(2—s). Therefore, if tion two steps apart from the subpopulation in

The Genetical Society of Great Britain, Heredity, 77, 469—475. 474 H. TACHIDA & H. YOSHIMARU which it resides because the two types of migration corresponding identity coefficient, 0(u), by can occur for a gene in one generation. This never happens in the corresponding haploid population. 0(z) = 0(u(1 —z)). (24) However, if m, m, u, 1/N are much smaller than Thus, for example, the expected number of differ- one, the probability of such two-step migration is ences is very small and such events can be ignored. Then, we can use (1 —s/2)N as the effective number and E[kI = ®'(O). (25) compute the identity coefficients from the corre- This formula can be used to assess the magnitude of sponding haploid population with the migration rate nucleotide diversity (Nei & Li, 1979) in partially rn+m/2 as noted by Maruyama & Tachida (1992). selfing populations for which DNA data are accumu- From the above argurments, we can summarize lating (Clegg, 1990). Also, once E[kj is computed, the general effect of partial selfing on identities of we can compute the expected squared difference, two genes in different individuals as replacing the E[S],ofrepeat numbers between two genes in subpopulation size, N, in populations without selfing microsatellite loci assuming the stepwise mutation with N(1 —s/2). This reduction of the subpopulation model (Ohta & Kimura, 1973). Utilizing the argu- size reduces the variation within subpopulations and ment described in Garza et a!. (1995), we obtain promotes differentiation among subpopulations. E[S1 in terms of E[k] as Caballero & Hill (1992) derived a general formula to compute the variance effective number of nonran- E[SJ = E[k1cr, (26) dom mating populations. For a Poisson distribution where o is the variance of the change in allele size of family size and a specified system of partial given that a mutation occurs. Now microsatellite loci inbreeding, the effective size is represented as are found in a wide variety of plant species (e.g. Ne N/(1 + Fis) where F15 is Wright's F1 statistic. If Terauchi, 1994; Yang et al., 1994; Roder et a!., 1995) we substitute the equilibrium inbreeding coefficient, and they can be used to investigate microgeograph- s/(2—s), for infinite partially selfing populations ical genetic structures and mating systems of plant (see, for example, Hedrick & Cockerham, 1986) into populations. F15, we obtain Ne = N(1 —s/2). Thus, we can also reach the same conclusion using the argument of Caballero & Hill (1992). However, there are two Acknowledgements merits in our approach. First, the variance effective Wethank two anonymous reviewers for comments number is obtained by considering the change of the on the manuscript. This research was partially variance of the gene frequency whereas our supported by NIG Cooperative Research approach directly computes the equilibrium identity Programme, a grant from the Society for the Promo- coefficients under the pressure of mutation. There- tion of Genetics and a grant-in-aid from the Minis- fore, our approach is conceptually more direct. try of Education, Science and Culture of Japan. Secondly, our approach is based on the exact calcu- lation of Maruyama (1977) and as long as our concern is the identity coefficients, the result is References exact. If we ignore the second-order terms of ii, m, BROWN,A. F-I. D. 1990. Genetic characterization of plant and 1/N, the two approaches will give the same mating systems. In: Brown A. H. D., Clegg, M. T., second moments of gene frequencies with (1 —s/2)N Kahler, A. L. and Weir, B. S. (eds) Plant Population as an effective size of a subpopulation. However, if Genetics, Breeding, and Genetic Resources, pp. 145—162. some second-order terms cannot be neglected, there Sinauer Associates, Sunderland, MA. will be difference although it might not be so large CABALLERO, A. AND HILL, W. G. 1992. Effective size of (see Fig. 2). Thus, our approach, for example, can nonrandom mating populations. Genetics, 130, 909—916. provide the condition for the applicability of this CLEGG, M. T. 1990. Molecular diversity in plant popula- simple relationship, N = (1 —s/2)N,inselfing tions, In: Brown, A. H. D., Clegg, M. T., Kahler, A. L. populations. and Weir, B. S. (eds) Plant Population Genetics, Breed- Finally, our formula can be used to compute the ing, and Genetic Resources, pp. 98—115. Sinauer Asso- probability distribution of the number, k, of different ciates, Sunderland, MA. ENNOS, R. A. AND CLEGG, M. T. 1982. Effect of population sites between two sequences under the infinite site substructuring on estimates of outcrossing rate in plant model without recombination (Watterson, 1975). populations. Heredity, 48,283—292. Griffiths (1981) and Tachida (1985) showed that the GARZA, J. C., SLATKIN, M. AND FREIMER, N. B. 1995. Micro- generating function, 0(z), of k is related to the satellite allele frequencies in humans and chimpanzees,

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