J. evol. Biol. 3: 501-512 (1991) 1010-061x/q1/03501-12$1.50+0.20/0 0 1991 BirkhPuser Verlag, Base1

Populational interactions among underdominant chromosome rearrangements help them to persist in small demes

Franc0 Spirito’, Carla Rossi and Marco Rizzoni’

’ Dipartimento di Genetica e Biologia Molecolare, Unioersittr di Roma I, “la Sapienza”, p. le A. More 5, 00185 Roma, Italy. ’ Dipartimento di Matematica, Unitlersitri di Roma II, “Towergata”, uia E. Carnecale 00173 Roma, Italy. ’ Dipartimento di Biologia, Uniwrsitci di Roma II, “Torcergata”, cia E. Carnevale 00173 Roma, Italy.

Key words: speciation; underdominant chromosome rearrangements; mathematical models.

Abstract

Populational interactions among unlinked chromosomal rearrangements with partial heterozygote sterility and multiplicative fitness were studied to verify whether they help such rearrangements to persist in small populations, in spite of a considerable migration rate with a large population. A deterministic island- continent model was studied by exact recurrence relations connecting gametic frequencies in successive generations.

The increase in the number of chromosomal rearrangements with the same sterility rate (s) causes an increase in the critical value of the migration rate (m,.) over which the chromosomal rearrangements are eliminated from the small popula- tion, mainly for medium s values (0.2 I s I 0.5) (“reciprocal stabilization” of the chromosomal rearrangements).

A comparison was made between the velocities of the fixation of a few rearrange- ments with high heterozygote sterility (low n/high S) and that of many rearrange- ments with low heterozygote sterility (high n/low s), to reach a given level of populational stability of the rearranged karyotype measured by m,. The former process was shown to be more rapid in small demes, the latter in large ones. However, the increase in the number of unlinked chromosomal rearrangements is more efficient in reducing the neutral gene flow than in increasing the populational stability of the rearranged karyotype.

501

502

Introduction

Spirito, Rossi and Riuoni

The hypothesis of “chromosomal speciation” (White, 1978) attributes a primary role to chromosomal rearrangements causing partial sterility of heterozygotes in triggering some speciation processes. According to this hypothesis such chromoso- mal rearrangements are fixed in small semi-isolated populations by genetic drift. Many rearrangements are often involved in “chromosomal speciation” (see Bush, 1981; White, 1978). Recently, mathematical models have been proposed which are based upon several underdominant chromosomal rearrangements; an increasing number of rearrangements with a decreasing value of partial heterozygote sterility requires a shorter time to reach the same degree of F, hybrid sterility as deme size increases (Walsh, 1982). Many closely linked rearrangements with very low het- erozygote sterility are a very efficient mechanism in reducing neutral gene flow (Bengtsson, 1985; Barton and Bengtsson, 1986).

However, a few rearrangements with a relevant heterozygote sterility (such as reciprocal translocations, inversions, centric fusions) often distinguish related taxa (see White, 1973; 1978). Therefore we focus our attention on rearrangements of this type and want to verify whether interactions among them in populations, after fixation, can contribute to their accumulation. It was assumed that fixation pro- cesses of single rearrangements took place in isolated populations and were independent of each other, without overlapping in time, in analogy to Lande (1979) and Walsh (1982).

Our aim was to verify whether several unlinked chromosome rearrangements with medium-high values of heterozygote sterility are more likely to persist in small populations than are single rearrangements, in the presence of a relevant migration rate with a large population having a “wild” karyotype (island-continent model). A deterministic model was used.

Unlinked rearrangements were studied in order to have a manageable model and because they are involved in many emblematic cases (e.g. the house mouse in Europe: Capanna et al, 1985; Gropp and Winking, 1981). An island-continent model simulates in a simple and realistic way a parapatric contact between a large population (continent) and a small one (island) in which new rearrangements can be fixed.

Methods

A deterministic model was studied with two populations, one of which (conti- nent) was infinitely larger than the other one (island). There is migration at a constant rate (m) each generation, measured as the fraction of diploid individuals of the island population which is exchanged with the continent population. Genera- tions are discrete and do not overlap. Chromosomal rearrangements are considered which cause partial heterozygote sterility, are unlinked and have multiplicative fitnesses. The value of partial heterozygote sterility (s) is the same for all the rearrangements involved. Selection acts after migration, given the features of the

Persistence of chromosome rearrangements 503

model. The island population is monomorphic for a number (n) of such chromoso- ma1 rearrangements and the continent one for the corresponding wild forms, when they come into contact.

As a consequence of the features of the model, all the chromosomal rearrange- ments always have the same frequency in the island population.

Our aim was to identify the equilibrium frequencies reached by the chromosomal rearrangements in the island population as a function of the values of m, s and n, and, in particular, to find the threshold critical value of m (m,.) over which the chromosomal rearrangements are eliminated. From this definition it ensues that m, is an index of the “populational stability” of the rearranged karyotype in an island-continent model.

Therefore the time course has been studied of the frequency of a chromosomal rearrangement (p) in the island population for different values of m, s and n by exact recurrence relations connecting gametic frequencies in successive generations. To obtain such relations we used the frequencies (p& of any gamete class carrying a given number (z) of chromosomal rearrangements; on this basis, assuming random mating, the “cumulative” frequencies were calculated for the zygote classes carrying a given number (j) of homozygous chromosomal rearrangements and a given number (i) of heterozygous chromosomal rearrangements. Taking into ac- count migration and fitness, the frequency of any gamete class with z chromosome rearrangements of the next generation (P;,,) could be calculated for any z as follows:

P’(7) =

where: n = total number of chromosomal rearrangements; I = number of chromosomes in a homozygous condition; j = number of chromosomal rearrangements in a homozygous condition; i = number of chromosomal rearrangements in a heterozygous condition; k = index of the term of the power (1 - s).

When z = 0 the term m must be added to the numerator to compute the newly immigrated gametes carrying the “wild” haplokaryotype.

The frequency of a chromosomal rearrangement (p) was calculated at any generation on the basis of the frequencies of the various gamete classes (p(,,) as follows:

(2)

504 Spirito, Rossi and Rizzoni

Numerical tests were performed using these relations for the following values of the parameters: m =0.1,0.2, 0.3, 0.4, 0.5, 0.6,0.7, 0.8, 0.9, 0.1;s =0.1,0.2,0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9; n = 1, 2, 3, 4, 5.

Results

Iden@cation of m,. cakes

The m, values have previously been determined analytically for single chromoso- mal rearrangements in a model consisting of two populations of equal size (Karlin and McGregor, 1972) and in an island continuous-time model (Lande, 1979). In our model (see Spirito, 1986) the following values of the equilibrium frequencies of a single chromosomal rearrangement (p) were found:

pi+: I- J

8m s(l -m)

(3) (locally stable)

p =:-i/G (3’) (unstable)

p = 0 (3”) (locally stable).

The value (3) is the one which is actually reached. From this expression the critical value m,. can be derived as:

m, - 81s (4).

For s = 1 the maximum value of m, is reached (m, max = 0.i). Numerical tests have been performed for 2-5 chromosomal rearrangements. The

p values actually reached are expressed in Table 1 (up to s = 0.5); their stability was analyzed only in the cases n = 2, 3; they were locally stable (see Appendix 1).

Further numerical tests have been performed to identify the values of m,, dividing by ten the interval between two successive m values previously tested. The maximum value of m to which a j value greater than zero corresponds has been accepted as m, (Fig. 1).

The increase in the number of chromosomal rearrangements (n) produces an increase in the values of p reached for the same m and s, and in the values of m, for the same s. An increase of jj has been shown for two underdominant loci in a simplified island model (Slatkin, 1982).

A noticeable increase in m,. values is observed for intermediate s values (0.2 I s I 0.5). The m, values are very close to m, max for high s values, also with a small number of rearrangements; further rearrangements do not have any relevant effect. The increase of m,. is very slight for small s values.

The jj values actually reached decrease as m increases for any pair of s and n values; these p values are always greater than 0.75 for any n value (when n = 1 it is easily demonstrated that the stable equilibrium value p # 0 is always greater than 0.75). A remarkable decrease in p values is found when m approaches m,..

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506 Spirito, Rossi and Rizzoni

Fig. 1. Critical migration rate (m,). over which underdominant chromosome rearrangements are eliminated from a small population (ordinate) as functions of the sterility rate of the heterozygotes for

single rearrangements (abscissa) (s) and of the number of unlinked rearrangements (different curves) (n). For n > I, m, values corresponding to s =O.l, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8 and 0.9 were empirically calculated (black dots) (see text), while the corresponding curves were calculated by interpolation. For n = 1, m, values corresponding to the same s values were analytically calculated (see text).

Alternatice ways to reach m,

One or more chromosomal rearrangements which significantly reduce het- erozygote fertility can be fixed with a non-negligible probability only in small and fully isolated populations (Bengtsson and Bodmer, 1976; Hedrick, 1981; Lande, 1979). In such conditions the rearranged karyotype can reach a certain degree of “populational stability” (m,). When such a population comes into contact with a much larger one with the “wild” karyotype, the rearranged karyotype can persist in the small population in spite of gene flow only if the migration rate is lower than

m, However, any given m, value can be reached in different ways: either with a few

chromosomal rearrangements each with a high heterozygote sterility rate (low n/high s), or with many rearrangements each with a low heterozygote sterility rate (high n/low s).

Our aim, in this section, is to compare the times required to reach given values of m, by taking either a low n/high s way or a high n/low s one in the full isolation phase in small populations. Therefore, the approximated formula of Walsh (1982) was used (see Appendix 2) to calculate the number of generations necessary to fix a sufficient number n of chromosomal rearrangements with the same s value so that a particular given value of m, is achieved. This formula is based upon the formula

Persistence of chromosome rearrangements 507

of Lande on the probability of fixation of underdominant alleles (1979). The approximated formula of Walsh, used here, is applicable to a phase of total isolation of the small population (m = 0) in which fixation processes take place; this phase preceeds contact with the large population described by our deterministic model.

The time between the appearance and the fixation of an underdominant chromo- somal rearrangement is negligible if compared to the time between the appearance of two successive rearrangements. In accordance with Lande (1979) and Walsh (1982) it is therefore assumed that, at most, one chromosomal rearrangement is present in a polymorphic condition in the small population.

The first step was to calculate the m, values for single underdominant chromoso- mal rearrangements (n = 1) with s = 0.1, 0.3 and 0.5, using equation (4). Subse- quently we calculated the s values which give place to these three m, values when 2, 3, 4 or 5 chromosomal rearrangements are involved. To calculate s values in the case of several chromosomal rearrangements, curves giving m,. values as functions of s were drawn for each n value by interpolating the results obtained in the numerical tests (Fig. 1).

To obtain a “smooth” interpolation of the observed data cubic splines were used, given by the formula: F(x) = ai + bix + cix2 + dix3. The coefficients were calculated for each 0.1 interval between two subsequent values of s previously tested. A continuous interpolating function with continuous first derivatives was then ob- tained. The normal smoothing procedure of the package Micro Speakeasy (Speakeasy Computing Corporation) was used.

The number of generations necessary to reach the three chosen m, values was calculated for the different pairs of n and s values and for several values of the effective deme size (N,). The number of generations was studied normalizing to the mutation rate. N, values were chosen starting from N, = 5 and multiplying it by two successively up to N, values such that Lande/Walsh approximation can still be used (see Appendix 2) and fixation probabilities are not yet negligible.

The main results are shown in Table 2. For the same m,. value a few chromoso- mal rearrangements with higher s are fixed in a shorter time than many rearrange- ments with lower s when N, is small (N, = 5); on the other hand, when N, is high many rearrangements require less time for fixation than few rearrangements. In the first case, a low n/high s way is more effective and in the second case, a high n/low s one. As the m,. value to be reached increases, the transition between the prevalence of these two ways takes place at lower N, values.

Discussion

These numerical results allow some conclusions within the constraints of the model and in the range of the values of the parameters actually tested.

The increase in the number of underdominant chromosomal rearrangements always produces an increase in the m, values; therefore, there is an increase in the range of values of the migration rate within which the rearranged karyotype can

508 Spirito. Rossi and Rizzoni

Table 2. Number of generations (normalized to the mutation rate) required to reach the same giben values of m, (critical value of the migration rate over which underdominant chromosomal rearrange- ments are eliminated from a small population) through the fixation of n (one to five) chromosomal

rearrangements with the same value of s (heterozygote sterility rate) in populations with different values of N, (effective deme size).

"‘ n 3 5 10 20 40 80

0.0123 I 0.1 1.573.100 2.026, IO’ 4.260.10° 2.364.10’ 9.312,122 2 0.0923 3.113~10° 3.859. IO’ 7.545.100 3.601.10’ 1.047.10~

3 0.092 4.663.10” 5.780.10” 1.127.10’ 5.346.10’ 1.536. IO’ 4 0.086 6.191.10” 7.444.100 1.366.10’ 5.770.10’ 1.039. IO? 5 0.0812 7.713~100 9.038.10” 1.586.10’ 6.101.10’ 1.146.10’

0.0361 I 0.3 2.883.10” 9.845.10’ 1.449.10’ 4.164.104 2 0.258 4.949.100 1.378.10’ 1.343.102 l.672.104

3 0.219 6.478, IO” 1.495.10’ 9.960.10’ 5.714~10’ 4 0.2 8.104.10’ 1.706.10’ 9.455.10’ 3.729. IO’ 5 0.179 9.503, IO” 1.805.10’ 8.158.10’ 2.123.10’

0.0588 I 0.5 6.438, IO0 5.806. IO’ 6.167.10’ 2 0.395 8.371~10’ 4.525, IO’ 1.698. IO3

3 0.322 9.413.10° 3.571.10’ 6.523.10* 4 0.284 1.087.10’ 3.435.10’ 4.318.102

5 0.253 1.213.10’ 3.301~10’ 3.067. IO2

persist in small populations when they are in contact with much larger populations with the wild karyotype (“reciprocal stabilization” of chromosomal rearrange- ments). An effect of this kind has been hypothesized by Barton (1983) on the basis of his analysis on multilocus clines.

This phenomenon is especially noticeable for s values corresponding to the chromosomal rearrangements involved in “chromosomal speciation” (White, 1978; see also Lande, 1979). Therefore, many chromosomal rearrangements of this type are required to allow the rearranged karyotype to persist under a strong migration pressure; such a phenomenon could contribute to the accumulation of such rearrangements, which is often observed in nature.

Barton (1983) analyzed multilocus clines for a very large number of underdomi- nant loci with a very weak selection on single loci and a very low recombination rate between adjacent loci, and he found two extreme patterns: when the selection/ recombination ratio is much higher than 1, the “effective” selection acting on single loci is comparable to the selection on heterozygotes for all the loci; when this ratio is much lower than 1, “effective” selection on single loci is slightly higher than the selection on heterozygotes for a single locus.

“Reciprocal stabilization” is very weak, in our model, for low s values, in particular for s = 0.1; this result is in agreement with Barton’s results corresponding to a low selection/recombination pattern, taking into account that the number of rearrangements analyzed in our model is small. For higher s values, “reciprocal

6

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510 Spirito, Rossi and Rizzoni

stabilization” is noticeable: in fact 2-5 chromosomal rearrangements lead to the same m, value as a single rearrangement with a heterozygote sterility rate which is significantly higher than that of one of the 2-5 rearrangements, even if it is lower than that of all the 2-5 rearrangements. This result could be explained by taking into account the fact that the selection/recombination ratio in our model for these s values is intermediate between those corresponding to Barton’s two extreme patterns.

Lande/Walsh approximation was used to compare the time required to reach the same m, values with a single rearrangement or with many rearrangements with the same s value. Within the limits of Lande/Walsh approximation (Appendix 2) the most rapid way to reach given values of m,. depends on the particular m,. value to be reached and on the effective deme size (N,). The way based upon a large number of rearrangements with a low s is more effective as N, and m, increase. However, it should be noted that many rearrangements with different s were neglected in this paper in the same way as they were by Walsh (1982). This procedure entails an underestimate of the role of the high n/low s way.

Different indexes were used to measure the degree of isolation due to underdom- inant mendelian factors: the sterility rate of heterozygotes for all the underdomi- nant factors (a) was suggested by Walsh (1982); the reduction of neutral gene flow between two populations, measured as the increase in the time or the reduction of the migration rate, is a measure of reproductive isolation (Barton and Bengtsson, 1986; Bengtsson, 1985; Spirit0 et al., 1983; 1987). In particular, as the migration rate tends to zero, the reduction of the migration rate (gff, Bengtsson, 1985) tends to (( 1 - s)/( 1 + s))“. For unlinked loci gff tends to ( 1 - a)’ as the number of underdominant loci (n) tends to infinity. In this paper the “populational stability” of the rearranged karyotype (m,,) is proposed.

The value of a which must be reached to obtain a given m, is always lower if one rearrangement is involved than if many are, in particular for medium-low a values; the values of a which must be reached to obtain a given gff are always higher if one rearrangement is involved rather than many; this effect is weak for low a values and becomes stronger for medium a values (Table 3) (see Bengtsson, 1985; Spirito, 1986). Therefore, single rearrangements with the same m, as n rearrangements are always fixed in a shorter time than those with the same a, which, in turn, are always fixed in a shorter time than those with the same gff. In conclusion, the increase in the number of chromosomal rearrangements with the same s is more efficient in reducing neutral gene flow than in increasing populational stability of the rear- ranged karyotype.

Acknowledgements

Many thanks are due to Mr. Giancarlo Baglioni for his excellent technical assistance in computer simulations. Many thanks are further due to Prof. Giuseppe Montalenti for his continuous help and suggestions. This work was supported by Progetto di Interesse Nazionale “Genetica Ecologica”

M.P.I.

Persistence of chromosome rearrangements

References

511

Barton, N. H. 1983. Multilocus clines. Evolution 37: 454-471. Barton, N. H., and B. 0. Bengtsson. 1986. The barrier to genetic exchange between hybridising

populations. Heredity 56: 3577376. Bengtsson, B. 0. 1985. The flow of genes through a genetic barrier. pp. 31-42. In P. J. Greenwood, P.

H. Harvey and M. Slatkin (eds.) Evolution essays in honour of John Maynard Smith. Cambridge University Press, Cambridge.

Bengtsson, B. O., and W. F. Bodmer. 1976. On the increase of chromosome mutation under random mating. Theor. Pop. Biol. 9: 260-281.

Bush, G. L. 1981. Stasipatric speciation and rapid evolution in animals. pp. 201-219. In W. R. Atchley and D. S. Woodsraft (eds.) Evolution and speciation: essays in honour of M. J. D. White. Cambridge University Press. Cambridge.

Capanna, E., M. Corti, and G. Nascetti. 1985. Role of contact areas in chromosomal speciation of the European long-tailed house mouse (Mus musculus domesticus). Boll. Zool. 52: 97~ 119.

Gropp, A., and H. Winking. 1981. Robertsonian translocation: cytology, meiosis. segregation patterns and biological consequences of heterozygosity. Symp. Zool. Sot. Lond. 47: 141-181.

Hedrick, P. W. 1981. The establishment of chromosomal variants. Evolution 35: 322-332.

Karlin, S., and J. McGregor. 1972. Application of the method of small parameters to multiniche population genetic models. Theor. Pop. Biol. 3: 1866209.

Kimura, M. 1962. On the probability of fixation of a mutant gene in a population. Genetics 47: 7133719.

Lande, R. 1979. Effective deme size during long-term evolution estimated from rates of chromosomal rearrangements. Evolution 33: 2344251.

Slatkin, M. 1982. Pleiotropy and parapatric speciation. Evolution 36: 263-270. Spirito. F. 1986. Reduction in gene flow among populations due to a reproductive isolating mechanism

based on a unifactorial or bifactorial heredity. Mem. Accad. Naz. Lincei, Cl. SC. Fis. Mat, e Nat.,

Ser. VIII. Vol. XVIII, Sez. III (5): 335-370. Spirito. F.. C. Rossi. and M. Rizzoni. 1983. Reduction of gene flow due to the partial sterility of

heterozygotes for a chromosome mutation. I. Studies on a “neutral” gene not linked to the chromosome mutation in a two population model. Evolution 37: 7855797.

Spirito, F., M. Rizzoni, E. Lolli, and C. Rossi. 1987. Reduction of neutral gene flow due to the partial

sterility of heterozygotes for a linked chromosome mutation. Theor. Pop. Biol. 31: 3233338. Walsh, J. B. 1982. Rate of accumulation of reproductive isolation by chromosome rearrangements.

Amer. Nat. 120: 510-532. White, M. J. D. 1973. Animal cytology and evolution. Cambridge University Press, Cambridge.

White, M. J. D. 1978. Modes of speciation. Freeman, San Francisco.

Received 15 September 1989;

accepted 2 August 1990. Corresponding Editor: V. Sbordoni

Appendix 1

In order to obtain the equilibrium frequencies of the chromosomal rearrange- ments, we worked, in this paper, with the simplified system of recurrence relations (I), with the constraint that, given the symmetry of the model, the frequencies of gametes carrying an equal number of chromosomal rearrangements are identical among them at each generation.

512 Spirito, Rossi and Rizzoni

However, for the analysis of the stability described in this appendix, we eliminated such a constraint and studied the systems of recurrence relations which take into account all the independent frequencies of the different kinds of gametes in order to analyze also the effects of non-symmetrical perturbations of such frequencies.

We restricted ourselves to the cases of 2 and 3 chromosomal rearrangements, because for higher numbers of rearrangements analysis would be very laborious. The system consists of 3 recurrence relations in the case of 2 chromosomal rearrangements, and in 7 relations in the case of 3.

We performed the linearization of the systems in a neighborhood of the equi- librium values found in the numerical tests which were described in the text. The matrices of the resulting systems were studied: the modules of the eigenvalues of these matrices were in all cases lower than 1; therefore all the observed equilibria are locally stable when 2-3 chromosomal rearrangements are studied.

Appendix 2

Lande (1979) used an approximated formula to calculate the probability of fixation of an allele newly arisen by mutation at an underdominant locus. This formula is based upon the general solution of the probability of fixation of a newly mutated allele given by Kimura ( 1962) through a procedure of diffusion approxi- mation. Walsh, following Lande, gave the formula to calculate the time necessary to fix a given number of newly mutated alleles for different underdominant loci with the same value of the coefficent of selection against heterozygotes (1982).

We used this formula, normalizing to the mutation rate:

(5)

(see text for symbols; rC,) = number of generations necessary to reach a; erf = the value of cumulated function of the standardized normal density evaluated at N;s).

This formula was obtained in a continuous time model; therefore fitnesses are measured in Malthusian parameters. For high values of coefficient of selection Wrightian and Malthusian fitnesses diverge.

Wrightian fitnesses were used in our model with non-overlapping generations. However, comparing results obtained with transition matrices with non-overlapping generations (Hedrick, 1981) and with Lande’s approximated formula, we could see that the divergence between the two estimates is not too large for the values of s and N, given in Table 2 (at most they differ by a factor of 2).