Copyright 0 1983 by the Genetics Society of America
LINKAGE MODIFICATION WITH MIXED RANDOM MATING
AND SELFING: A NUMERICAL
STUDY
KENT E. HOLSINGER’ A N D MARCUS W. FELDMAN
Department of Biological Sciences, Stanford University, Stanford, Colifornio 94305
Manuscript received July 27, 1982 Revised copy accepted September 23, 1982
ABSTRACT
Although recombination cannot increase under conditions of random mating or complete selfing in regimes of constant selection, with mixed random mating and selfing, selection for increased recombination can occur. For some fitness regimes there may be selection for reduced recombination with both low and high degrees of selfing but selection for increased recombination with moderate degrees of selfing. With some fitness regimes there is a historical effect: depending on which equilibrium a population starts from, there may be selection for either increased or decreased recombination. In other cases the direction of selection may be determined by the present state of individuals within the population. If recombination is already fairly limited, there may be selection for further reduction. If recombination is already fairly frequent, there may be selection for increased recombination. For certain symmetric viability systems there may be an intermediate value of the recombination fraction between 0 and 0.5 toward which the population will evolve. Although it is not yet possible to classify precisely those fitness matrices that can exhibit selection for in- creased recombination, it does appear that selection for increased recombination can occur only if at least two of the double homozygotes are less fit than would be expected on the basis of a comparison of the fitnesses of the single and double heterozygotes on an additive scale.
N E 1
(1967, 1969) introduced a model for linkage modification in which alleles at a selectively neutral locus control the recombination fraction between two major loci that are under selection. His conclusion that a neutral modifier that reduces recombination between the two selected loci will be favored by natural selection in random mating populations was later confirmed by FELDMAN (1972), KARLIN and MCGRECOR (1972, 1974) and more generally by FELDMAN, CHRISTIANSEN and BROOKS (1980). The general result is that, under conditions of random mating, a new allele at a neutral locus that controls recombination between two selected loci will increase in frequency when rare from an equilibrium with respect to the selected loci if, and only if, it reduces the recombination fraction. (When the initial linkage disequilibrium is zero theleading eigenvalue of the gradient matrix is unity, and the linear analysis fails to determine the local stability of the original equilibrium.)
FELDMAN and BALKAU (1972) studied the same model under conditions of complete selfing. They again found that a new modifying allele would increase in frequency if, and only if, it reduced the recombination fraction between the
Current address: Department of Botany, University of California, Berkeley, CA 94720.
324 K. E. HOLSINGER AND M. W . FELDMAN
two selected loci. In the case in which the recombination fraction between the two selected loci was least in the modifier heterozygote, the recombination fraction between the modifier locus and the selected loci had to be less than a certain value for the modifying locus to remain polymorphic. With random mating, on the other hand, the recombination fraction between the modifier locus and the selected loci played no role in determining the stability or
instability of the initial equilibria. We obtained results analogous to these in our study of a similar model for the evolution of recombination in permanent translocation heterozygotes (HOLSINGER and FELDMAN 1982).
CHARLESWORTH, CHARLESWORTH and STROBECK (1977) made a numerical study
of NEI’S model under conditions of mixed random mating and selfing. They suggested that “increased levels of selfing on the whole favor a higher intensity of selection for reduced recombination.
. .”
In our study of the analogous model for the evolution of recombination in permanent translocation heterozygotes (HOLSINGER and FELDMAN 1982), we pointed out that this conclusion was partly a result of the set of recombination fractions chosen. Furthermore, in a later study CHARLESWORTH, CHARLESWORTH and STROBECK (1979) confirmed HOL- DEN’S (1979) surprising suggestion that for certain viability systems there can beselection for increased recombination with mixed random mating and selfing. Thus, both the intensity of selection on modifiers of recombination and its direction depend in a complicated way not only on the degree of selfing but also on the viability system and other parameters that will be discussed.
The results of the numerical study described here confirm and extend those of CHARLESWORTH, CHARLESWORTH and STROBECK (1979). First, for certain fitness
regimes there may be selection for reduced recombination under both low and high degrees of selfing but selection for increased recombination with moderate degrees of selfing. Second, for some fitness regimes there may be a historical effect: depending on which equilibrium a population starts from there may be selection for either increased or decreased recombination. Third, the direction in which the recombination fraction evolves may depend on the initial recom- bination fraction, with selection for reduced recombination when the initial recombination fraction is small and selection for increased recombination when the initial recombination fraction is larger. Finally, although it is not yet possible to classify precisely those fitness matrices that can exhibit selection for in- creased recombination, it does appear that selection for increased recombination can occur only if at least two of the double homozygotes are less fit than would be expected on the basis of a comparison of the fitnesses of the single and double heterozygotes on an additive scale.
THE MODEL
Consider three loci with two alleles each: A/a, B/b and M/m. The fitness of any particular genotype is determined only by the genotype at the A and B
LINKAGE MODIFICATION 325
fraction between A and B in mm genotypes. Unless otherwise specified we assume, for simplicity, that the ri are ordered either in order of increasing magnitude or of decreasing magnitude. R is the recombination fraction between
M and B in all genotypes. We assume that there is no interference so that the recombination fraction between M and A is ri(1 - R )
+
(1-
r J R , where ri is determined by the genotype at theM
locus. Assume that selfing occurs with frequency U and random union of gamets with frequency 1-
U.Imagine that the population is at a stable equilibrium, x*, where both A and
B
are polymorphic, but M is fixed on one allele, M say. A small amount of the alternative allele, m, is introduced, and we are interested in whether this new allele increases in frequency when rare. This corresponds to a local stability analysis of x * in the full 36-dimensional space. In terms of the stability toward the introduction of m, the general case where R>
0 requires an evaluation of the largest eigenvalue of a 26 x 26 matrix. We have, therefore, chosen to study this problem numerically. The procedures used are described in the APPENDIX.If rl
>
r2>
r3 and the new allele increases in frequency when rare, we say thatthere is selection for reduced recombination. Conversely, if
rl
<
rz<
r3 and the new allele increases in frequency when rare, we say that there is selection forincreased recombination.
RESULTS
In the following discussion it will be useful to consider a general fitness matrix of the sort illustrated in Table 1. The ~i measure the departure of the
double homozygotes from strict additivity. HOLDEN’S (1979) model (Table 2) is the special case where a1 = a2 =
81
=8 2
= 1 A b and KI = ~2 = ~3 = ~4 = 2b-
a-
1. The general two-locus symmetric viability model (Table 3) is the caseSince iterations made with numerous additive and multiplicative viability matrices of various sorts failed to reveal any instances of selection for increased recombination, we will present results for several other types of viability systems.
Completely symmetric viabilities: As suggested by HOLDEN (1979) and con- firmed by CHARLESWORTH, CHARLESWORTH and STROBECK (1979), selection for increased recombination can occur with completely symmetric fitnesses if k
>
0, where
k
= K I = K~ = ~3 = ~ 4 . There is no lower bound to the selfing rate forwhich selection for increased recombination can occur under these conditions. [Recall that when k
>
0 the coefficient of linkage disequilibrium, D, is zero at equilibrium with random mating and there is convergence to a neutral curve of equilibria with respect to the modifier locus (KARLIN and FELDMAN 1970; FELD-MAN 1972).] If the selfing frequency is too great, however, there will be selection
for reduced recombination.
A small change in the fitness matrix HOLDEN (1979) used to produce his Figure 4 (Table 4) preserves the feature of selection for increased recombination for a range of selfing frequencies, but introduces several interesting new features as well. For this fitness matrix K~ = 0.27, KZ = 0.35, ~3 = 0.26, and ~4 = 0.32. With this
326 K. E. HOLSINGER A N D M. W . FELDMAN
TABLE 1
A general two-locus viability matrix
TABLE 2
A completely symmetric two-locus viability matrix _ _ _ _ _ ~
AB OB A b ab
AB U b b 1
aB b a 1 b
A b b 1 U b
a b 1 b b U
TABLE 3
A general two-locus symmetric viability matrix
AB a B Ab ab
AB 1 - 6 1 - Y 1 - P 1
aB 1 - Y 1-(Y 1 1 - P
Ab 1 - P 1 1 - C U 1 - Y
a b 1 1 - P 1 - Y 1 - 6
TABLE 4
a. The two-locus viability matrix of HOLDEN (2979)
AB OB Ab ab
AB 0.5 0.9 0.9 1.0
aB 0.9 0.5 1.0 0.9
A b 0.9 1.0 0.5 0.9
a b 1.0 0.9 0.9 0.5
b. A two-locus viability matrix slightly altered from HOLDEN (1979)
AB UB Ab a b
~~~~
AB 0.52 0.91 0.88 1.0
A b 0.88 1.0 0.51 0.89
aB 0.91 0.48 1.0 0.92
a b 1.0 0.92 0.89 0.49
mating. Thus, for U = 0 (and, by continuity, when U is sufficiently small) we know that there is selection for reduced recombination, although it may be quite weak since the magnitude of
6
is not great. With r1 = 0.1, for example,6
=LINKAGE MODIFICATION 327
If the selfing frequency is as little as I%, however, there is selection for increased recombination, although it continues to be quite weak. Thus, a very small amount of selfing (one that would be very difficult'to detect experimen- tally) changes the direction of selection on modifiers. Of course, with the particular fitness matrix under consideration the strength of selection in one direction or the other is so small that it might not be an important factor in natural populations. Nevertheless, it is important to realize that a small amount of selfing (and presumably other forms of inbreeding) can significantly alter the predictions based on a consideration only of a random mating population. With this fitness matrix, if selfing occurs with a frequency of as little as 5% or more, the strength of selection in favor of increased recombinations is quite great.
Another surprising result is obtained from a consideration of even smaller selfing frequencies. When selfing occurs with a frequency of 0.1% and rl = 0.1,
the direction of selection depends on the recombination fraction between the modifier locus and the selected loci. If R is sufficiently small, there is selection for reduced recombination. If R is somewhat larger, there is selection for increased recombination. Thus, whether a new allele at the modifying locus will increase when rare may depend not only on how the recombination between A and B is altered by introduction of the new allele, but also on the recombination fraction between B and M. This is in some ways analogous to the situation with complete selfing when r1
>
r3>
rz. In that case whether a modifier polymorphism is maintained or fixation on the new allele occurs may depend on R. Even in that case, however the direction in which recombination between A and B will evolve does not depend on the magnitude of R.For U between 0.01 and 0.95 there is selection for increased recombination. As might be expected, a modifier polymorphism is possible if rz
>
rl, rz>
r3, and Ris sufficiently small [cf. HOLSINGER and FELDMAN (1982) where it is pointed out that the existence of a modifier polymorphism with mixed random mating and selfing may depend on RI. For U between 0.97 and 0.98 there is selection for reduced recombination. When U = 0.96 the situation is even more complex than that already described.
With U = 0.96 a new allele at the modifier locus that increases recombination between the selected loci apparently always increases when rare regardless of
R. It is possible, however, for a new allele that decreases recombination to increase when rare if R is sufficiently small. For example, if R = 0 and r1 = 0.05, both a new allele leading to rz = 0.06, r3 = 0.07 and a new allele leading to rz =
0.04, r3 = 0.03 increase when rare. How small R must be when rz
<
rl and 1 3<
rl in order to prevent elimination of the new allele depends on just how much recombination is reduced by the new allele. Thus, if rl = 0.1, rz = 0.05 and
R
=0.06, the new allele will be eliminated if r3 = 0.03, but not if r3 = 0.01.
General symmetric viabilities: For the general symmetric viability system there are two parameters that measure the departure of the fitnesses from strict additivity, c1 = I C ~ = K~ and e2 = KZ = I C ~ . On the basis of the results for completely
symmetric viabilities previously reported, we might expect that €1 or EZ or both
328 K. E. HOLSINGER A N D M. W. FELDMAN
When el and e2 were both positive, on the other hand, selection for increased recombination always occurred for some selfing frequency. One example of such a viability matrix will be found in Table 5. For this fitness matrix €1 = 0.05 and €2 = 0.1.
With random mating and this fitness matrix
6
# 0. There is, therefore, selection for reduced recombination when selfing is rare. When U = 0.2,however, there can be selection for increased recombination. For example, if rl
= 0.3, r2 = 0.35 and 1-3 = 0.4, the new allele is never eliminated, regardless of R.
On the other hand, if rl is somewhat smaller, the fate of the new allele may depend on R. For example, if rl = 0.1, r2 = 0.05 and r3 = 0.01, the new allele will be eliminated if R = 0.4 but will not be eliminated if R = 0.1.
This particular fitness matrix introduces a feature not found in other matrices studied. (Not even all symmetric fitness matrices show this property, and this particular one exhibits the property only for a certain range of selfing frequen- cies.) There seems to be an intermediate value toward which the recombination fraction will evolve for some selfing frequencies, provided that rl is not already too small. This value for U = 0.2 is apparently in the neighborhood of 0.42.
Other viabilities: Selection for increased recombination can occur with via- bilities other than the symmetric ones so far described. One example is the viability system discussed by FRANKLIN and FELDMAN (1977) (Table 6). For this fitness matrix K~ = 0.08, ~2 = 0.06694194, ~3 = 0.06694194 and ~4 = 0.02846597. With this viability system a
6
= 0 and a6
# 0 equilibrium are simultaneously stable under conditions of random mating when r1 is at least 0.005. With selfing, both of these equilibria continue to exist (at least until the selfing frequency is25%), although the one corresponding to
0
= 0 with random mating now hasfi
# 0. We will refer to the equilibrium corresponding to6
= 0 as2,
and the one with6
# 0 as9.
As would be expected from what is known about the evolution of recombi- nation under conditions of random mating, a new allele that alters the recom- bination fraction between A and B will increase when rare in the neighborhood of
9
only if it reduces recombination. For example, if cr = 0.01, r1 = 0.1, rz = 0.05 and r3 = 0.01, the new allele will increase when rare if the population is near9.
If the population is near f under these same conditions, however, the new allele will be eliminated. Thus, there is a historical effect. The direction in which recombination will evolve depends in this case on the nature of the initial equilibrium. KARLIN and MCGRECOR (1974) suggested the possibility that the fate of a neutral modifier could depend on the nature of the equilibrium at which it was introduced, but this is the first instance of which we are aware that this possibility has been demonstrated (with the exception of recombination modifiers with random mating whenfi
= 0 and6
# 0 equilibria are simulta- neously stable). A further complication is that if rl is sufficiently small when selfing is rare (less than about 0.004 for U = 0.01) there is selection for reducedLINKAGE MODIFICATION 329
TABLE 5
A symmetric fitness matrix that exhibits selection for increased recombination
AB OB Ab ab
AB 0.8 0.9 0.95 1.0
aB 0.9 0.75 1.0 0.95
Ab 0.95 1.0 0.75 0.9
ab ‘1.0 0.95 0.9 0.8
TABLE 6
The fitness matrix of FRANKLIN and FELDMAN (1977)
AB OB A b ab
AB 0.79440 0.82880 0.82880 0.79920
aB 0.82880 0.79020 0.79920 0.81410
Ab 0.82880 0.79920 0.79020 0.81410
ab 0.79920 0.81410 0.81410 0.80625
To get some idea of how common selection for increased recombination might be, a set of 75 randomly chosen viability matrices was studied. Of these
75 matrices, 14 allowed a stable two-locus polymorphism, and only one of these showed selection for increased recombination. It is shown in Table 7. For this matrix K~ = 0.30844204, ~2 = -0.32379242, K3 = 0.35378882 and K4 = -0.02064992.
With rl = 0.1,
6
# 0 for this viability system with random mating, and, as would be expected, there is selection for reduced recombination when the selfing frequency is not great (1% or less for example). When selfing is somewhat more frequent (10 to 20%), there is selection for increased recombination whenr1 is fairly large, but there is selection for reduced recombination when rl is small (-0.001 or less). Thus, the fate of a new allele at the modifier locus with moderate amounts of selfing depends not only on how it changes the recombi- nation fraction, but on what the initial recombination fraction was. Finally, when selfing is quite frequent (30 to 40%) there is selection for reduced recom- bination. For selfing rates greater than this the two-locus polymorphism is lost.
DISCUSSION
With both random mating and complete selfing the results on linkage modi- fication are quite straightforward. There is selection for reduced recombination. With complete random mating, the initial equilibrium is neutrally stable when
6
= 0, but there is never selection for increased recombination. The results for linkage modification with mixed random mating and selfing, on the other hand, are quite complex and difficult to interpret. Several new and unexpected features are introduced when linkage modification is studied under conditions of mixed random mating and selfing.One such feature is that selection for increased recombination can occur
330 K . E. HOLSINGER AND M. W. FELDMAN
TABLE 7
A randomly selected viability matrix that exhibits selection for increased recombination
AB O B Ab ob
AB 0.703656 0.884435 0.973709 0.882338
aB 0.884435 0.530650 0.882338 0.242858
Ab 0.973709 0.882338 0.681303 0.902093
a b 0.882338 0.242858 0.902093 0.280834
for certain viability systems the direction in which recombination will evolve may change as the selfing frequency is changed. For certain fitness matrices there is selection for reduced recombination when selfing is rare, selection for increased recombination with moderate degrees of selfing and selection for reduced recombination with still higher degrees of selfing.
A second new feature that is not found with either random mating or complete selfing (and not demonstrated for any other modifier problem of which we are aware) is that there can be a historical effect. Depending on the nature of the equilibrium that the population starts from, there may be Selection either for increased recombination or for decreased recombination. In other cases, the direction of selection may depend on the present state of the population. If recombination is fairly rare, there may be selection to reduce it further. If, on the other hand, recombination is fairly frequent there may be selection to further increase its frequency. Furthermore, whether a new allele at the modifier locus is eliminated or not may depend on the recombination fraction between the modifier locus and the selected loci, and the direction in which recombination between the selected loci will tend to evolve may also depend on the linkage relationships between the modifier locus and the selected loci. Finally, with certain viability systems there may he a value of the recom- bination fraction between 0 and 0.5 toward which the population evolves.
LEWONTIN (1971) and KARLIN and MCGREGOR (1974) pointed out that, if there is a two-locus polymorphism for tight linkage, the mean fitness decreases as recombination increases. Since FELDMAN (1972) had shown for a variety of special fitness regimes that there is always selection for reduced recombination, this (and other similar results) led KARLIN and MCGREGOR (1974) to propose their mean fitness principle: a new allele at a selectively neutral locus will increase in frequency when rare if it increases the equilibrium mean fitness of the population.
THOMSON and FELDMAN (1974) in a study of linkage modification and meiotic
LINKAGE MODIFICATION 331
segregation distortion poses problems, as pointed out by KARLIN and MCGREGOR (1974) and THOMSON and FELDMAN (1974), since there is no well-defined mean fitness in many of these cases.
In spite of these difficulties, the mean fitness principle can sometimes be useful. Indeed, it was the observation that for some parameter values in HOLDEN’S (1979) model the equilibrium mean fitness is an increasing function of recombination that led CHARLESWORTH, CHARLESWORTH and STROBECK (1979) and us to consider further the problem of linkage modification with mixed random mating and selfing. In some caes, e.g., for the fitness matrix in Table 5 where an intermediate optimum for recombination was found, the properties of the mean fitness seemed to coincide with the fate of new alleles at the modifier locus. In others, e.g., for the fitness matrix in Table 4b when U = 0.96 so that recombination could evolve in either direction, the properties of the mean fitness function did not seem at all informative.
Although it is not yet possible to determine analytically the fate of a new allele at the modifier locus for a given fitness matrix, a given set of rz’s, a given
R and a given selfing frequency, it is possible to give a general, although imprecise, characterization of those fitness regimes that can lead to selection for increased recombination. For the completely symmetric viability model the only cases allowing selection for increased recombination are when a condition HOLDEN (1979) calls negative disequilibrium potential exists (CHARLESWORTH, CHARLESWORTH and STROBECK 1979). It is quite easily shown that negative disequilibrium potential can never exist unless k
>
0, where k = K~ = KZ = KQ = K~ and that it always does exist when k>
0.From this we might expect that selection for increased recombination can occur only if at least some of the K, are positive. The results presented bear out this expectation. Selection for increased recombination was not found when fitnesses were chosen to be additive. In this case all of the K,’S = 0, by definition. Similarly, no evidence of selection for increased recombination was found with multiplicative fitnesses when there was overdominance at both loci. But for multiplicative fitnesses with overdominance at both loci it is quite easily demonstrated that all of the K ~ ’ S are negative.
On the other hand, in all cases in which selection for increased recombinatioin was found, at least two of the K~ were positive. In fact, the viability matrix in Table 7 was the only one found in which all four of the K~ were not positive. Furthermore, in all cases in which all four of the K~ were positive, we found selection for increased recombination. On the other hand, we did find cases in which two of the K~ were positive and there was no evidence of selection for reduced recombination. We are currently conducting a numerical study of the two-locus selection model with mixed random mating and selfing. The results of that study concerning the equilibrium structure and other properties of the pure selection model may shed more light on this subject.
Although the results presented here are strictly applicable only to systems with mixed random mating and selfing, we expect that similar phenomena may occur with other systems of inbreeding. Thus, the answer to TURNER’S (1967)
332 K. E. HOLSINGER AND M. W . FELDMAN
viability systems common in nature do not lead to selection for reduced recombination when small amounts of inbreeding occur. Of course, mechanical considerations, e.g., a breakdown of the meiotic mechanism, may be far more important and are difficult to incorporate in any population genetic model.
On the other hand cytological studies of several groups of flowering plants have shown that self-pollinating species often have a higher chiasma frequency than closely related outcrossing species (STEBBINS 1950,1958; GRANT 1958; LEWIS and JOHN 1963; ZARCHI et al. 1972), and it seems unlikely that the meiotic mechanism is under more severe constraints in self-pollinating species than in outcrossing species. For example, lines of Lolium perenne derived from several generations of complete selfing have reduced chiasma frequencies to such a point that univalents are present at meiosis (KARP and JONES 1982). In addition, that inbreeding may lead to selection for increased recombination is supported by SIMCHEN and CONNOLLY'S (1968) report that after inbreeding for 8-15 gener- ations, recombination within the A factor of Schizophyllum commune is signif- icantly higher in four of the inbred lines than in the original isolate.
This research has been supported in part by National Institutes of Health Grants GM10452-18 and GM28106, a National Institutes of Health training grant in integrative biology, and by a grant from the Andrew W. Mellon Foundation.
LITERATURE CITED
CHARLESWORTH, D., B. CHARLESWORTH, and C. STROBECK, 1977 Effects of selfing on selection for
CHARLESWORTH, D., B. CHARLESWORTH and C. STROBECK, 1979 Selection for recombination in
FELDMAN, M. W., 1972 Selection for linkage modification. I. Random mating populations. Theor.
FELDMAN, M. W. and B. BALKAU, 1972 Some results in the theory of three gene loci. pp. 357-383.
FELDMAN, M.
w.,
F. B. CHRISTIANSEN and L. D. BROOKS, 1980 Evolution of recombination in aFRANKLIN, I. R. and M. W. FELDMAN, 1977 Two loci with two alleles: linkage equilibrium and
GRANT, V., 1958 The regulation of recombination in plants. Cold Spring Harbor Symp. Quant.
New properties of the two-locus partial selfing model with selection.. Genetics
HOLSINGER, K. E. and M. W. FELDMAN, 1982 The evolution of recombination in permanent translocation heterozygotes. Theor. Pop. Biol., in press.
KARLIN, S. and D. CARMELLI, 1975 Numerical studies on two-loci selection models with general viabilities. Theor. Pop. Biol. 7 399-421.
KARLIN, S. and M. W. FELDMAN, 1970 Linkage and selection: two locus symmetric viability models. Theor. Pop. Biol. 1: 39-71.
KARLIN, S. and J. L. MCGREGOR, 1972 The evolutionary development of modifier genes. Proc. Natl. Acad. Sci. USA 69 3611-3614.
KARLIN, S. and J. L. MCGREGOR, 1974 Towards a general theory of modifie'r evolution. Theor. Pop. Biol. 5: 59-103.
recombination. Genetics 86 213-226.
partially self-fertilizing populations. Genetics 93: 237-244.
Pop. Biol. 3: 324-346.
In Population Dynamics, Edited by T. N. E. GREVILLE. Academic Press, New York.
constant environment. Proc. Natl. Acad. Sci. USA 77 4838-4841.
linkage disequilibrium can be simultaneously stable. Theor. Pop. Biol. 1 2 95-113.
Biol. 23: 337-363.
LINKAGE MODIFICATION 333
KARP, A. and R. N. JONES, 1982 Cytogenetics of Lolium perenne Part 1. Chiasma frequency
LEWIS, K. R. and B. JOHN, 1963 Chromosome Marker. pp. 281. London, J. & A. Churchill. LEWONTIN, R. C., 1971 The effect of genetic linkage on the mean fitness of a population. Proc.
NEI, M., 1967 Modification of linkage intensity by natural selection. Genetics 57: 625-642.
NEI, M., 1969 Linkage modification and sex difference in recombination. Genetics 63: 681-699.
SIMCHEN, G. and V. CONNOLLY, 1968 Changes in recombination frequency following inbreeding in Schizophyllum. Genetics 5 8 319-326.
STEBBINS, G. L., 1950 Variation and Evolution in Plants. pp. 180-181. New York, Columbia University Press.
STEBBINS, G. L., 1958 Longevity, habitat and release of genetic variability in the higher plants. Cold Spring Harbor Symp. Quant. Biol. 2 3 365-377.
THOMSON, G. J. and M. W. FELDMAN, 1974 Population genetics of modifiers of meiotic drive. 11.
Linkage modification in the segregation distortion system. Theor. Pop. Biol. 5: 155-162. TURNER, J. R. G., 1967 Why does the genotype not congeal? Evolution 21: 645-656. ZARCHI, Y., G. SIMCHEN, J. HILLEL and T. SCHAAP, 1972
variation in inbred lines. Theor. Appl. Genet. 6 2 177-183.
Natl. Acad. Sci. U.S.A 68: 984-986.
Chiasmata and the breeding system in wild
Corresponding editor: B. S. WEIR populations of diploid wheats. Chromosoma (Berl.) 3 8 77-84.
APPENDIX: NUMERICAL PROCEDURES
The iterations were begun with random initial frequencies, except that the modifying locus was fixed on the allele that corresponded to a recombination fraction of rl at the selected locus. Iterations were allowed to continue until the total absolute deviation between two successive runs was less than A random perturbation from this state on the order of 1% was then made to simulate the introduction of a new allele at the modifying locus. Iterations were then allowed to continue for a predetermined number of generations (usually 5000, but occasionally as many as 50,000 when the rate of change at the modifying locus was unusually slow) or until the total absolute deviation between two successive iterations was less than
Since we were unable to analyze the system, a number of indirect checks on the numerical accuracy of the program were run. Whenever selection resulted in fixation on one of the modifier classes (including the modifier heterozygote when R = 0), the two-locus genotype frequencies could be compared with those obtained from an independently written program for two loci undergoing selection with mixed random mating and selfing. In all cases checked the results were correct to 4
decimal places.
Under conditions of complete selfing with rl > r3 > r2 the value of R that determines whether a modifier polymorphism will result can be analytically determined (cf. FELDMAN and BALKAU 1972;
HOLSINGER and FELDMAN 1982). For rl = 0.3, r2 = 0.1, and r3 = 0.2 a modifier polymorphism exists at equilibrium if and only if R < 0.094248664. Numerical iterations indicated that for R < 0.094 a modifier polymorphism resulted. For R 3 0.095 the new allele became fixed.
Finally, the predictions of this program could be compared with those from the program used by HOLSINGER and FELDMAN (1982) simply by making the A locus lethal when homozygous. The genotype frequencies at a modifier polymorphism were correct to 4 decimal places for 8 combina- tions of U and R. Furthermore, for one fitness matrix and set of rz’s the existence of a modifier
