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Heterosis at the level of the chromosome and at the level of the gene
THEORETICAL
POPULATION
Heterosis
BIOLOGY
3, 491-506 (1972)
at the Level of the Chromosome Level of the Gene*
and at the
J. A. SVED School of Biological
Sciences, University
of Sydney, New South Wales, Australia
Received November
10, 1971
For a polymorphism to be maintained at a single locus in a small population through heterozygote advantage, symmetry or near symmetry of selection against the two homozygotes is important. However a consideration of chromosome segments rather than single loci suggests that this result does not necessarily extend to the situation where there are many linked loci, and suggests that in the extreme there may be some stability with dominance or partial dominance at individual loci rather than overdominance. It is shown algebraically that, if an intrinsically neutral locus is linked to a locus with dominance of selective values, the heterozygote at the neutral locus is expected to have an apparent selective value greater than or equal to the mean of the two homozygotes. Linkage of the neutral locus to a number of loci of this type is thus expected to lead to an apparent heterozygote advantage, which may lead to increased stability at the neutral locus. Computer simulation has been used to show that in a multiple-locus model with dominance of selective values there is increased stability at the loci themselves, and also at linked neutral loci. The stability is, however, considerably less than with an equivalent set of overdominant loci with the same overall heterozygote advantage. An excess of repulsion over coupling linkages (i.e., negative linkage disequilibrium) has been shown to be important in increasing stability with the dominance model, and it has been shown that the amount of negative disequilibrium in a population appears to increase slowly over time. Overall the results suggest that heterosis at the level of the chromosome does imply some stability regardless of whether this is attributable to dominance or overdominance at individual loci.
In the theory of the maintenance of single-locus polymorphisms in a finite population through heterozygote advantage (Robertson, 1962; Kimura, 1964; Ewens and Thomson, 1970), the symmetry of the disadvantage suffered by the two homozygotes constitutes a crucial factor. The greatest stability is achieved when the selective coefficients against the two homozygotes are equal. The stability diminishes with increasing asymmetry of the two selective coefficients * Computing Council.
costs were supported
by a grant from the Australian
491 Copyright All rights
0 1972 by Academic Press, Inc. of reproduction in any form reserved.
Research
Grants
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until in the limit, when one homozygote suffers no disadvantage compared to the heterozygote (dominance rather than overdominance), selection will tend to oppose rather than to favour the maintenance of variability. However, if we consider the problem of the maintenance of variability in a small population at the level of the chromosome rather than at the level of the gene, the situation is ostensibly quite different. The finite size of the population will lead to chromosome segments becoming homozygous, just as in the wellestablished cases of regular inbreeding systems (Fisher, 1949). If such chromosome segments are usually at a disadvantage, then it may be argued that this will lead to heterozygosity being favoured, just as in the case of heterozygote advantage at a single locus. This result at first sight seems contradictory to the single-locus results quoted above, as it is well known (Falconer, 1960, Chapter 14) that homozygous segments are at a disadvantage irrespective of whether there is dominance or overdominance at individual loci. This apparent contradiction is, however, removed if it is noted that the single-locus results of Robertson et al. can only be extended to cases of two or more loci if there is no correlation of gene frequencies between genes at linked loci (i.e., no linkage disequilibrium), a condition which is not fulfilled in a finite population. As shown by Sved (1971), homozygosity of chromosome segments must be accompanied by a correlation of gene frequencies. The crucial requirement for stability in the argument given above is that homozygous segments will “usually” contain genes which are disadvantageous in homozygous condition. The principal attempt of this paper is to quantify this argument for the case of linked loci at which there is dominance rather than overdominance of selective values. The mean length of homozygous segment surrounding a locus at which the genes are identical by descent is expected to be approximately (log N - 1)/2N (Sved, 1971), where N is the population size, so that the larger the population size the smaller such segments are expected to be. Similarly the more thinly spread the loci we are considering, the less the chance that any homozygous segment will contain disadvantageous genes. Ideally we would like to obtain expressions giving the stability of chromosome segments for a general distribution of selective values and population sizes. No such of two-locus expressions have been obtained, however, and a combination algebraic arguments and multiple-locus computer simulation have been used in an attempt to delineate the problem.
NEUTRAL Locus
LINKED TO SELECTED Locus
Some simple algebraic results may be obtained using a model modified from that of Sved (1968). We consider the simplest possible situation of an intrinsically
GENE
AND
CHROMOSOME
TABLE
493
HETEROSIS
I
Frequencies and Selective Values in Two-Locus Model Where the Frequencies of the Gametes AB, Ab, aB, and ab are x1 , x2 , x3 , and x4 , Respectively Frequency
Genotype
Selective value
ABlAB
1
ABlAb
1
ABlaB
1
ABlab
1
AblAb
l-s
AblaB
1
Ablab
l-s
aBlaB
1
aB/ab
1
ablab
l-s
selectively neutral locus (A) which is linked to a locus (B) at which recessive homozygotes are at a disadvantage s (Table I). Owing to their association with the B locus, the selective value of AA genotypes in the population, SAA , will be x12 * 1 + 2x,x, * 1 + x,2(1 - s) Xl2 + 2x,x, + x22 =1--s*
(x1 :zx2)2 *
Similarly x2x4
s,, = 1 - 5 . (Xl
+
X2)(%
+
x4)
and s,, = 1 -s
*
(xs T2x4)2 -
Thus
s,, - S& = s .
(x1 52x2)2 --s-x2-D
’
494
SVED
where
Similarly
s,, - s,, = (x1P-1.2;;.i3:x4)2. Thus SAa - SAA and SAa - S,, are of opposite sign, so that the heterozygote at the A locus will appear to lie between the two homozygotes in selective value. However the value will usually not be strictly intermediate, since
(SACA - SAA)+ (S.4,- Saa)= (x1 + ,$g
+ x4)2 >
which must be 3 0. The same type of result may be established in the more general case where the selective values of the three genotypes at the B locus are 1 + t : 1 : 1 - s. Then the overall heterozygote advantage at the A locus, as measured by 2S,, - S,, - S,, , is equal to
(s - t) . D2 (x1 + x2j2 6% + %I2 ’
t,
which is positive provided s > i.e., provided that there is some degree of dominance of the favourable over the unfavourable gene at the B locus. This conclusion is similar to that reached by Ohta and Kimura (1969) following an earlier numerical treatment by Maruyama and Kimura (1968). It is instructive to compare these results for a dominant linked locus with the case of a neutral gene linked to an overdominant locus (Sved, 1968; Ohta and Kimura 1969, 1970). The overall heterozygote advantage at the A locus, is dependent only on the overall herozygote advantage at 2s*, the B locus. Thus if, for example, the B gene is at a frequency of 0.5, then 20% disadvantage of bb compared to both BB and Bb would give the same value as 10% disadvantage of both BB and bb compared to Bb. Clearly, however, the same stability is not conferred on the A locus in both cases. In the first case there will be a selective pressure tending to reduce the frequency of whichever gene at the A locus is associated with the unfavourable gene at the B locus. In the second case, as argued by Sved (1968), selection will tend to oppose fluctuation in frequency of the A gene in either direction. The two cases only become comparable when we consider the model of a neutral locus linked to a number of selected loci. True associative overdominance (Frydenberg, 1963; Ohta and Kimura, 1970) will be produced at the neutral locus in the case of linked dominant loci if the genes at the A locus are positively
- SAA - s,, 3
GENE AND CHROMOSOME HETEROSIS
495
associated with some of the loci (i.e., positive D values), and negatively associated with others. On the other hand associative overdominance will be produced by a single linked overdominant locus, and the effect of many such linked loci will be cumulative.
POSITIW
AND NEGATIVE
DISEQUILIBRIUM
In considering the model of a neutral locus linked to a selected locus, the can only be applied arbitrarily. terms “positive” and “negative” disequilibrium However the expected value of D is equal to zero in this case, regardless of which are thought of as being the coupling genotypes and which the repulsion. Furthermore the selective consequences of positive and negative disequilibrium are essentially similar. These arguments no longer apply when selection is acting on both loci. An additional effect must now be taken into account, since linked deleterious genes in a finite population are more likely to be associated in repulsion than in coupling (Comstock and Robinson, 1952; Hill and Robertson, 1968; Robertson, 1970). This is because deleterious genes linked in repulsion produce a situation of heterozygote advantage (Jones, 1917) and thus tend to persist in the population, whereas linkage in coupling leads to rapid loss of the deleterious genes. Thus in a population which has existed for some time, the expected value of D will be negative, leading to greater stability than would otherwise be expected. It is important to note that the associative overdominance effect discussed above does not rely on this effect. It may, however, be enhanced by it. When more than two loci are considered the situation is more complicated, because there is an inherent limit to the overall amount of negative disequilibrium. For example, if there is complete negative disequilibrium between genes at the A and B loci, and similarly between genes at the B and C loci, then it follows that there must be complete positive disequilibrium between genes at the A and C loci. The situation is not symmetrical in this respect, in that it is possible to have complete positive disequilibrium between genes at a number of loci. As a measure of the overall amount of negative disequilibrium in the system it is convenient to use the value of D summed over all pairs of loci. The lowest value which C D may take is found by considering the case where there are only two chromosome types in the population, e.g., 1100010100..., and the complementary type 0011101011..., where “1” represents a dominant favorable gene, and “0” the recessive deleterious allele. If there are n loci overall and the two types of genes are distributed on the two chromosomes in the ratio i : n - i, then there are altogether (.$ + (“F~) p airs of loci in coupling, and i(n - i) pairs in repulsion. The maximum number of repulsion pairs occurs when i = n/2, assuming that 71is even (the situation is essentially similar for odd n),
496
SVED
and is equal to n2/4. The total number of coupling pairs in this case is n2/4 - n/2, so that the minimum value which CD can take is (l/4) . (-n/2) = -n/S. On the other hand, in the situation where the two gametic types are 11111111... and OOOOOOOO...,the (i) pairs of loci are all in coupling and the value of C D is (4W - 1). While this formulation may repulsion genotypes, this cannot genes are randomly distributed probability of having i dominant is
seem to indicate an excess of coupling over be true overall if the dominant and recessive over the two chromosomes. In this case the and n - i recessive genes on one chromosome
assuming equal frequencies of the two types of genes. Thus the total number of pairs of loci in coupling is equal to
i (9) i-0
which comes to (n/4)(n -
. ($”
[( ;,
+
(” ;
l), and the total number
q]y
of pairs in repulsion
is also
(n/4)@ - 1). Of course, in actual populations the situation of complete disequilibrium is not and the highest absolute values of D are possible if there is recombination, expected between the most closely linked loci. Under these circumstances the highest overall repulsion association is expected when there is a preponderance of the chromosome pairs 1010101010... and 0101010101... . This model is similar to that envisaged by Mather (1949) except that epistatic interaction was assumed to be the important causal factor in this discussion. In a balanced situation such as the above, if there is close linkage we might expect almost as much stability with the dominance as with the overdominance model. A computer comparison of the two models is included in the following section.
MULTIPLE-LOCUS
COMPUTER
SIMULATION
Several types of Monte Carlo simulation were carried out in order to demonstrate that fixation of variability is retarded in a system of closely linked selected loci. All of the runs to be reported were made with a monoecious population of size 50. The number of loci was chosen as 108, each locus having two alleles, and there was 0.1 o/0 recombination between adjacent loci, making a total map length of 10.7 cMs.
GENE
AND
CHROMOSOME
HETEROSIS
497
Choice of a suitable starting population is one of the most important features of multiple-locus simulation. If populations are started with unrealistic disequilibrium values it may take many generations to reach a stable value, during which time gene frequencies may fluctuate considerably. In the present investigation, populations were set up by a method described briefly by Sved (1971) to give values of r2 for all pairs of loci in approximate agreement with the formula
l/(1 + 4Nc). The method is a somewhat complicated one. Initially a single chromosome type and its complement were generated, and 50% of all chromosomes in the population were assigned one type and 50% the other. This produces 100% disequilibrium between all pairs of loci. It also sets all gene frequencies to 50%, which is convenient for making comparisons, although admittedly artificial. The alleles at each locus in the original chromosome type were chosen at random, thereby avoiding any systematic tendency for an excess of either coupling or repulsion types. However, as related later, the procedure led to considerable initial variability in C D values. To produce the required starting populations, the 100% disequilibrium values were now reduced to more realistic values by a recombination procedure. A pair of chromosomes was selected from the population, recombined at some random point, and then returned to the population. Repeated application of this procedure reduced the association between linked loci, and of course reduced it much more quickly for loosely linked than for tightly linked loci. However it was found that if chromosome pairs were chosen at random for the recombination procedure, the values of r2 for the more widely separated loci were reduced too rapidly in relation to those of closely linked loci. The procedure was therefore modified so that, as expected in actual populations, some chromosomes were recombined more often than others. Following the arguments of Sved (1971), a geometric distribution with parameter (1 - 1/2N) was used to describe the number of recombination points per chromosome. A series of long term runs was made in which all loci in a particular population were assigned equal selective differentials. For the overdominance model, selective values of approximately 0.99 : 1 : 0.99 were assigned at each locus. These values underestimate the actual selective intensity, since an additive model was used to avoid the complications of epistatic interactions (Franklin and Lewontin, 1970). The procedure used was to assign a relative selective value of unity to the highest expected level of heterozygosity in the population, and to assign a selective value of 1 - 0.01x to individuals heterozygous at x loci less than this level. For the dominance model, selective values of approximately 0.98 : 1 : 1 were assigned, so that the overall selective advantage of heterozygotes over homozygotes was the same in both cases. A third set of runs was made without selection. Results from the three models are given in Fig. 1, together with the expected
498
SWD
0
50
100 GCNERATION
150
200
FIG. 1. Percentage of loci fixed at various times in 108-10~~s model. The broken line gives the expected value without selection. Each of the other graphs is based on 6 replicate runs, and selective values are 0.99:1:0.99 for the overdominance model and 0.98:1:1 for the dominance model.
value for no selection. It may be seen that there is some stabilizing effect in the dominance runs, although considerably less than for the overdominance model. A second series of runs was made for comparison in which selection of twice the intensity was made to act at half the loci, the remainder being unselected. The 54 selected and unselected loci were assigned at random along the chromosome, except that the dominance and overdominance runs were made in pairs with the same pattern of selected and unselected loci. The proportion of fixed loci amongst the unselected loci was used as the indicator of instability in this case, thereby eliminating the effect of selection at the indicator loci themselves. As seen from Fig. 2, the results are comparable to those from the first series, although the differences are reduced, particularly between the dominance and unselected runs. While some stability is indicated for the dominance model, the above results give very little information about the extent of the stability. They also suffer from the disadvantage that after some period of time the overall levels of heterozygosity are much higher in the overdominance than in the dominance runs. At such times, therefore, the comparison between these two models is no longer strictly valid.
GENE
AND
CHROMOSOME
HETEROSIS
499
In order to obtain more precise information a more comprehensive series of short term runs was made. For these runs 12 loci spread evenly over the chromosome were kept selectively neutral, and the loss of heterozygosity at these loci was used as the indicator of instability. Also in these runs each starting population was run not once but a number of times to test the repeatability of results for a single population.
FIG. 2. Percentage of unselected loci fixed at various times in 108-10~~s model with 54 selected and 54 unselected loci. Selective values are 0.98:1:0.98 for the overdominance model and 0.96:1 :I for the dominance model.
Three selection models were again run; no selection, 1% symmetric, and 2% asymmetric. Each run lasted for only 5 generations, then gene frequencies were calculated at each locus, and the proportionate loss of heterozygosity estimated from the formula 1 - 4p(l - p). Ten populations were generated with linkage disequilibrium for each model, and twenty replicate runs were made for each population. The overall mean losses of variability for the three models were 4.71%, 2.63%, and 4.18% (Table II). The value for the unselected runs is in agreement with expectation from the formula (1 - 1/2N)5. The overdominance model has given a retardation factor (Robertson, 1962) of 1.8, while the retardation factor for the dominance model is only slightly greater than 1.1. Further insight into this question comes from the analysis of variance of results from the three models (Table II). Th e variability between populations
500
SVED
TABLE
II
Analysis of Variance of Proportionate Loss of Heterozygosity after 5 Generations in 10 Populations of Each Selective Regime, with 0.1 o/0 Recombination between Adjacent Loci
(df)
Unselected
Overdominance Dominance Mean square ( x 103)
Between populations
(9)
6.06 n.s.
1.47 as.
Between runs within populations
(190)
9.84
1.96
4.54
Between loci within populations
(110)
4.43 ns.
1.43 n.s.
6.32***
Runs x loci within populations
(2090)
3.56
1.29
2.74
26.93’**
Mean value ( x 102) 4.71
2.63
4.18
judged in relation to the variability between runs for a particular population is nonsignificant for the unselected and overdominance models, but highly significant for the dominance model. The composition of the starting population is therefore important in determining stability only in the case of the dominance model. As mentioned previously, dominant and recessive genes were assigned at random in setting up the most common chromosome type in the original populations. Regions of high positive disequilibrium will tend to occur by chance. This will not have any consequences for the overdominance model, but will lead to rapid loss of heterozygosity in these regions for the dominance model. Confirmation of this view comes from the fact that the between locus mean square is also nonsignificant for the unselected and overdominance models but significant for the dominance model. The comparison between the three models of the variability between runs within populations is also of interest. As seen from Table II, the dominance model is approximately intermediate between the other two models in this respect, which is one measure of the stability imparted by this type of selection. Short-term runs of the same type were also carried out for a model with 0.2% recombination between adjacent loci, and with all other parameters unchanged. The results are given in Table III, and show similar trends to the results in Table II. Further information of a different type comes from plotting the value of C D summed over all pairs of loci for a particular population against the overall loss of heterozygosity. This is done in Fig. 3 for 30 populations having dominance of
GENE
AND
CHROMOSOME
TABLE
501
HETEROSIS
III
Analysis of Variance of Short-Term Runs as in Table II, but with 0.2 y0 Recombination between Adjacent Loci
Cd.1
Unselected
10.09 ns.
Overdominance Dominance Mean square ( x 103)
Between populations
(9)
Between runs within populations
(190)
7.65
3.13
5.35
Between loci within populations
(110)
4.65 n.s.
2.83 n.s.
6.59***
Runs x loci within populations
(2090)
3.99
2.91
3.99
6.59 n.s.
28.66***
Mean value ( x 1Or)
4.85
3.85
4.12
FIG. 3. Proportionate loss of heterozygosity over 5 generations, plotted against CD summed over all pairs of selected loci. Gene frequencies were calculated at 12 selectively neutral loci spread evenly amongst 96 loci with selective values 0.98:1:1. All populations were started with linkage disequilibrium, 30 populations having the initial chromosome generated at random (closed circles), and 10 populations having a 1010101010... initial chromosome type.
502
SWD
selective value. From this it is seen that there is a high correlation between the two values (r = 0.92). It is apparent that a good deal of the overall variability between populations for the dominance model can be accounted for simply by the variation in overall C D values. Figure 3 also contains results from 10 populations in which the initial chromosome types were chosen as 1010101010... and 0101010101..., respectively. It is seen that the 2 D values for such populations are lower than for any of the cases where the initial chromosomes were assigned at random, although not much lower in some cases. The mean loss of heterozygosity from these 10 populations is 2.80%, which is not significantly different from the value found in the overdominance runs. The results from these runs can be regarded as the lower limit to be expected for the dominance model. In long-term runs with this type of dominance model the results have also been indistinguishable from those with the overdominance model. The high correlation between C D and the loss of heterozygosity opens up the possibility that if there is a bias towards negative values of D in natural populations, then the loss of heterozygosity for the dominance model will be much closer to the value for the overdominance model. For example, the dominance runs with negative values of CD gave an average loss of heterozygosity of 3.58%, which is considerably closer to the value for the overdominance model. As mentioned earlier, in a two-locus finite-population dominance model there is a tendency towards negative values of D. In view of the considerations of the previous section regarding the complications introduced by multiple-locus models, it is not certain to what extent this effect will apply for more than two loci. A number of runs were therefore set up to test for changes in C D values in populations run for a number of generations. Ten populations were generated in the manner described earlier in the section. Values of C D were calculated in these initial populations, and again after 20, 40, 60, 80, and 100 generations had elapsed. The results are given in Table IV, and show that on the average C D has fallen over the course of the simulation. Some of this fall might be attributable to a reduction in the absolute value of D, and this is allowed for in the last line of the table by calculating C D/C 1D j. It is seen that there was in fact a considerable bias towards positive values of D in the initial populations, which is a chance effect. The overall tendency appears to be a reduction in high positive C D values rather than a production of high negative values. In fact, in the two populations started with negative C D values there was no overall fall. On the whole it seems clear that there is a tendency towards negative disequilibrium but not a very pronounced one. The procedure described above was repeated for ten populations set up in linkage equilibrium, to test whether the buildup of negative disequilibrium might be inhibited in populations started with linkage disequilibrium. Again only a slow buildup of negative disequilibrium was observed.
GENE
AND
CHROMOSOME
TABLE
503
HETEROSIS
IV
Values of C D Summed over All Pairs of Loci at Various Times for 10 Populations Set Up with Linkage Disequilibrium (Selective values were 0.98:1:1 at each locus.)
Population 1.
0
20
Generation 40 60
80
3.80
7.60
2.56
4.33
-2.11
-1.11 -3.15
2.
3.70
-0.66
3.
9.90
-0.12
4.
4.50
-1.85
-1.71
-4.41
5.
- 5.70
-4.67
-1.80
-4.74
6.
1.60
-0.78
7.
1.60
8.
-5.55
6.96 -5.71
9.36
9.58
-2.08 4.30 -3.31 0.00
100 -1.63 -3.57 -0.77 -1.26 2.05
-4.05
-1.32
-4.23
-2.59
-2.53
-0.11
-3.60
-5.84
-2.65
-2.98
-2.89
-0.06
9.
4.15
4.41
0.24
1.66
10.
12.60
7.55
3.43
7.47
2.55
Ave.
3.06
0.81
0.17
- 0.26
0.01
-1.36
0.80
0.23
0.10
-0.12
0.00
-0.70
CD/ClDl(xlW
-0.23
-2.60 0.74
The results from the computing are therefore somewhat equivocal on this important point. It may be that the tendency towards negative D values is in fact sufficiently pronounced to lead to a significant excess of repulsion linkages in natural propulations. If this is so, then for a given overall level of heterozygosity the amount of stability given by the dominance model would be increased. Even if not, however, the results still suggest that there is some stability.
DISCUSSION
The results of this paper are perhaps of most relevance to the interpretation of inbreeding experiments. Several such experiments have been carried out to test the hypothesis of heterosis as a possible explanation for the maintenance of the large amounts of variability found in natural populations. As discussed by Sved and Ayala (1970), inbreeding experiments always yield information at the level of the chromosome rather than at the level of the gene. If the expectations from single-locus theory are accepted without reservation, this would mean that no stabilizing effect on polymorphic loci could be inferred even if high levels of heterosis are detected at the chromosome level, since this might be a mani-
504
WED
festation of dominance rather than overdominance at the gene level. However the results of this paper favour the suggestion that the interpretation in terms of stability is less dependent on symmetry of selection against the two homozygotes than would be suggested by single-locus theory. This strengthens the argument of Franklin and Lewontin (1970) that measurement and interpretation of selective values ought to be at the level of the chromosome rather than at the level of the gene. The dominance and overdominance hypotheses have previously been discussed in the context of hybrid vigour in corn. Crow (1948) and Fisher (1949) have argued that the advances made by hybridization could not be accounted for on the dominance hypothesis, assuming a balance between mutation and selection. The results of this paper, however, suggest the possibility that the consequences of dominance and overdominance may not be as different as previously supposed. The differences between the two hypotheses are in one sense still of practical importance. On the dominance hypothesis it would be possible, although not easy, to produce by directional selection the advances attained by hybridization. On the overdominance hypothesis this would not be possible. The results of some studies of directional selection in corn (see Sprague, 1967) suggest that the former hypothesis is still tenable. On the question of long-term stability it is far from clear that the model considered in this paper is of any significance. Even for the overdominance model there does not appear to be any long-term stability in a small population (Ohta and Kimura, 1970), and Sved (1968) h as argued that the effect of linked overdominant genes may only be of long-term significance in conjunction with overdominance at the locus itself. These considerations are much more critical for the dominance model, where no stable intermediate equilibrium is expected. Ohta and Kimura (1969) have argued that the equilibrium frequencies to be expected in natural populations are those given by the balance of mutation and selection. If this is the case, then the stabilizing effects discussed in this paper are so small as to be of no significance. Only if some transient equilibrium can be produced which is essentially unrelated to the mutation rate will the effects be of any significance. The computer models run with the dominance model, where unfavourable genes are started at high frequency, implicitly assume that such a situation is possible. In a sense, however, it is not crucial to the argument that such a situation can exist, since the principal argument being put forward is that if heterosis at the chromosome level is attributable to dominance rather than overdominance, nevertheless some stability is implied. Some attempt has been made to run long-term simulations of populations of size 50 with dominance of selective values and back and forward mutation. The equilibrium gene frequencies in these runs appeared to be only slightly greater than those predicted by the mutation-selection balance. It is clear that no stable
GENE AND CHROMOSOME HETEROSIS
505
effects can be produced by the dominance model in a simple population of this size. A model in which it seems possible that more substantial effects will be obtained is that of a subdivided population with restricted migration between isolates. This type of population structure provides a more favourable opportunity for random selective associations. The amount of linkage disequilibrium for all except the most closely linked loci will be determined by the size of the isolates rather than the size of the overall population. On the other hand, the migration between isolates ensures that there is a reasonable amount of homogeneity over the population. Preliminary computer simulation with this type of population structure and with overdominance of selective values appears to indicate that quite high levels of stability can be attained.
REFERENCES COMSTOCK, R. E., AND ROBINSON, H. F. 1952. Estimation of average dominance of genes, in “Heterosis” (J. Gowen, Ed.), pp. 494-516, Iowa State College Press, Ames. CROW, J. F. 1948. Alternative hypotheses of hybrid vigor, Genetics 33, 477-487. Ewwrs, W. J., AND THOMSON, G. 1970. Heterozygote selective advantage, Ann. Hum. Genet., London 33, 365-376. FALCONER, D. S. 1960. “Introduction to Quantitative Genetics,” Oliver and Boyd, Edinburgh. FISHER, R. A. 1949. “The Theory of Inbreeding,” Oliver and Boyd, Edinburgh. FRANKLIN, I. R., AND LEWONTIN, R. C. 1970. Is the gene the unit of selection ? Genetics 65, 707-734. FRYDENBERC, 0. 1963. Population studies of a lethal mutant in Drosophila melanogaster. I. Behaviour in populations with discrete generations, Hereditas 48, 89-l 16. HILL, W. G., AND ROBERTSON, A. 1968. Linkage disequilibrium in finite populations, Theor. Appl. Genet. 38, 226-231. JONES, D. F. 1917. Dominance of linked factors as a means of accounting for heterosis, Genetics 2, 466-479. KIMURA, M. 1964. Diffusion models in population genetics, J. Appl. Prob. 1, 177-232. MARUYAMA, T., AND KIMURA, M. 1968. Development of temporary overdominance associated with neutral alleles, Proc. Itint. Congr. Genet. 12th, 12.2.4. (Abstr.) MATHER, K. 1949. “Biometrical Genetics,” Dover, New York. OHTA, T., AND KI~XURA, M. 1969. Linkage disequilibrium at steady state determined by random genetic drift and recurrent mutation, Genetics 63, 229-238. OHTA, T., AND KIMURA, M. 1970. Development of associative overdominance through linkage disequilibrium in finite populations, Genet. Res. 16, 165-177. ROBERTSON, A. 1962. Selection for heterozygotes in small populations, Genetics 47, 1291-1300. ROBERTSON, A. 1970. A theory of limits in artificial selection with many linked loci, in “Mathematical Topics in Population Genetics” (K. Kojima, Ed.), pp. 246-288, Springer-Verlag, Berlin. SPRAGUE, G. F. 1967. Plant breeding, Ann. Rev. Genet. 1, 269-294.
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SVED, J. A. 1968. The stability of linked systems of loci with a small population size, Genetics 59, 543-563. SVED, J. A. 1971. Linkage disequilibrium and homozygosity of chromosome segments in finite populations, Theor. Pop. Biol. 2, 125-141. SVED, J. A., AND AYALA, F. J. 1970. A population cage test for heterosis in Drosophila pseudoobscura, Genetics 66, 97-I 13.
POPULATION
Heterosis
BIOLOGY
3, 491-506 (1972)
at the Level of the Chromosome Level of the Gene*
and at the
J. A. SVED School of Biological
Sciences, University
of Sydney, New South Wales, Australia
Received November
10, 1971
For a polymorphism to be maintained at a single locus in a small population through heterozygote advantage, symmetry or near symmetry of selection against the two homozygotes is important. However a consideration of chromosome segments rather than single loci suggests that this result does not necessarily extend to the situation where there are many linked loci, and suggests that in the extreme there may be some stability with dominance or partial dominance at individual loci rather than overdominance. It is shown algebraically that, if an intrinsically neutral locus is linked to a locus with dominance of selective values, the heterozygote at the neutral locus is expected to have an apparent selective value greater than or equal to the mean of the two homozygotes. Linkage of the neutral locus to a number of loci of this type is thus expected to lead to an apparent heterozygote advantage, which may lead to increased stability at the neutral locus. Computer simulation has been used to show that in a multiple-locus model with dominance of selective values there is increased stability at the loci themselves, and also at linked neutral loci. The stability is, however, considerably less than with an equivalent set of overdominant loci with the same overall heterozygote advantage. An excess of repulsion over coupling linkages (i.e., negative linkage disequilibrium) has been shown to be important in increasing stability with the dominance model, and it has been shown that the amount of negative disequilibrium in a population appears to increase slowly over time. Overall the results suggest that heterosis at the level of the chromosome does imply some stability regardless of whether this is attributable to dominance or overdominance at individual loci.
In the theory of the maintenance of single-locus polymorphisms in a finite population through heterozygote advantage (Robertson, 1962; Kimura, 1964; Ewens and Thomson, 1970), the symmetry of the disadvantage suffered by the two homozygotes constitutes a crucial factor. The greatest stability is achieved when the selective coefficients against the two homozygotes are equal. The stability diminishes with increasing asymmetry of the two selective coefficients * Computing Council.
costs were supported
by a grant from the Australian
491 Copyright All rights
0 1972 by Academic Press, Inc. of reproduction in any form reserved.
Research
Grants
492
WED
until in the limit, when one homozygote suffers no disadvantage compared to the heterozygote (dominance rather than overdominance), selection will tend to oppose rather than to favour the maintenance of variability. However, if we consider the problem of the maintenance of variability in a small population at the level of the chromosome rather than at the level of the gene, the situation is ostensibly quite different. The finite size of the population will lead to chromosome segments becoming homozygous, just as in the wellestablished cases of regular inbreeding systems (Fisher, 1949). If such chromosome segments are usually at a disadvantage, then it may be argued that this will lead to heterozygosity being favoured, just as in the case of heterozygote advantage at a single locus. This result at first sight seems contradictory to the single-locus results quoted above, as it is well known (Falconer, 1960, Chapter 14) that homozygous segments are at a disadvantage irrespective of whether there is dominance or overdominance at individual loci. This apparent contradiction is, however, removed if it is noted that the single-locus results of Robertson et al. can only be extended to cases of two or more loci if there is no correlation of gene frequencies between genes at linked loci (i.e., no linkage disequilibrium), a condition which is not fulfilled in a finite population. As shown by Sved (1971), homozygosity of chromosome segments must be accompanied by a correlation of gene frequencies. The crucial requirement for stability in the argument given above is that homozygous segments will “usually” contain genes which are disadvantageous in homozygous condition. The principal attempt of this paper is to quantify this argument for the case of linked loci at which there is dominance rather than overdominance of selective values. The mean length of homozygous segment surrounding a locus at which the genes are identical by descent is expected to be approximately (log N - 1)/2N (Sved, 1971), where N is the population size, so that the larger the population size the smaller such segments are expected to be. Similarly the more thinly spread the loci we are considering, the less the chance that any homozygous segment will contain disadvantageous genes. Ideally we would like to obtain expressions giving the stability of chromosome segments for a general distribution of selective values and population sizes. No such of two-locus expressions have been obtained, however, and a combination algebraic arguments and multiple-locus computer simulation have been used in an attempt to delineate the problem.
NEUTRAL Locus
LINKED TO SELECTED Locus
Some simple algebraic results may be obtained using a model modified from that of Sved (1968). We consider the simplest possible situation of an intrinsically
GENE
AND
CHROMOSOME
TABLE
493
HETEROSIS
I
Frequencies and Selective Values in Two-Locus Model Where the Frequencies of the Gametes AB, Ab, aB, and ab are x1 , x2 , x3 , and x4 , Respectively Frequency
Genotype
Selective value
ABlAB
1
ABlAb
1
ABlaB
1
ABlab
1
AblAb
l-s
AblaB
1
Ablab
l-s
aBlaB
1
aB/ab
1
ablab
l-s
selectively neutral locus (A) which is linked to a locus (B) at which recessive homozygotes are at a disadvantage s (Table I). Owing to their association with the B locus, the selective value of AA genotypes in the population, SAA , will be x12 * 1 + 2x,x, * 1 + x,2(1 - s) Xl2 + 2x,x, + x22 =1--s*
(x1 :zx2)2 *
Similarly x2x4
s,, = 1 - 5 . (Xl
+
X2)(%
+
x4)
and s,, = 1 -s
*
(xs T2x4)2 -
Thus
s,, - S& = s .
(x1 52x2)2 --s-x2-D
’
494
SVED
where
Similarly
s,, - s,, = (x1P-1.2;;.i3:x4)2. Thus SAa - SAA and SAa - S,, are of opposite sign, so that the heterozygote at the A locus will appear to lie between the two homozygotes in selective value. However the value will usually not be strictly intermediate, since
(SACA - SAA)+ (S.4,- Saa)= (x1 + ,$g
+ x4)2 >
which must be 3 0. The same type of result may be established in the more general case where the selective values of the three genotypes at the B locus are 1 + t : 1 : 1 - s. Then the overall heterozygote advantage at the A locus, as measured by 2S,, - S,, - S,, , is equal to
(s - t) . D2 (x1 + x2j2 6% + %I2 ’
t,
which is positive provided s > i.e., provided that there is some degree of dominance of the favourable over the unfavourable gene at the B locus. This conclusion is similar to that reached by Ohta and Kimura (1969) following an earlier numerical treatment by Maruyama and Kimura (1968). It is instructive to compare these results for a dominant linked locus with the case of a neutral gene linked to an overdominant locus (Sved, 1968; Ohta and Kimura 1969, 1970). The overall heterozygote advantage at the A locus, is dependent only on the overall herozygote advantage at 2s*, the B locus. Thus if, for example, the B gene is at a frequency of 0.5, then 20% disadvantage of bb compared to both BB and Bb would give the same value as 10% disadvantage of both BB and bb compared to Bb. Clearly, however, the same stability is not conferred on the A locus in both cases. In the first case there will be a selective pressure tending to reduce the frequency of whichever gene at the A locus is associated with the unfavourable gene at the B locus. In the second case, as argued by Sved (1968), selection will tend to oppose fluctuation in frequency of the A gene in either direction. The two cases only become comparable when we consider the model of a neutral locus linked to a number of selected loci. True associative overdominance (Frydenberg, 1963; Ohta and Kimura, 1970) will be produced at the neutral locus in the case of linked dominant loci if the genes at the A locus are positively
- SAA - s,, 3
GENE AND CHROMOSOME HETEROSIS
495
associated with some of the loci (i.e., positive D values), and negatively associated with others. On the other hand associative overdominance will be produced by a single linked overdominant locus, and the effect of many such linked loci will be cumulative.
POSITIW
AND NEGATIVE
DISEQUILIBRIUM
In considering the model of a neutral locus linked to a selected locus, the can only be applied arbitrarily. terms “positive” and “negative” disequilibrium However the expected value of D is equal to zero in this case, regardless of which are thought of as being the coupling genotypes and which the repulsion. Furthermore the selective consequences of positive and negative disequilibrium are essentially similar. These arguments no longer apply when selection is acting on both loci. An additional effect must now be taken into account, since linked deleterious genes in a finite population are more likely to be associated in repulsion than in coupling (Comstock and Robinson, 1952; Hill and Robertson, 1968; Robertson, 1970). This is because deleterious genes linked in repulsion produce a situation of heterozygote advantage (Jones, 1917) and thus tend to persist in the population, whereas linkage in coupling leads to rapid loss of the deleterious genes. Thus in a population which has existed for some time, the expected value of D will be negative, leading to greater stability than would otherwise be expected. It is important to note that the associative overdominance effect discussed above does not rely on this effect. It may, however, be enhanced by it. When more than two loci are considered the situation is more complicated, because there is an inherent limit to the overall amount of negative disequilibrium. For example, if there is complete negative disequilibrium between genes at the A and B loci, and similarly between genes at the B and C loci, then it follows that there must be complete positive disequilibrium between genes at the A and C loci. The situation is not symmetrical in this respect, in that it is possible to have complete positive disequilibrium between genes at a number of loci. As a measure of the overall amount of negative disequilibrium in the system it is convenient to use the value of D summed over all pairs of loci. The lowest value which C D may take is found by considering the case where there are only two chromosome types in the population, e.g., 1100010100..., and the complementary type 0011101011..., where “1” represents a dominant favorable gene, and “0” the recessive deleterious allele. If there are n loci overall and the two types of genes are distributed on the two chromosomes in the ratio i : n - i, then there are altogether (.$ + (“F~) p airs of loci in coupling, and i(n - i) pairs in repulsion. The maximum number of repulsion pairs occurs when i = n/2, assuming that 71is even (the situation is essentially similar for odd n),
496
SVED
and is equal to n2/4. The total number of coupling pairs in this case is n2/4 - n/2, so that the minimum value which CD can take is (l/4) . (-n/2) = -n/S. On the other hand, in the situation where the two gametic types are 11111111... and OOOOOOOO...,the (i) pairs of loci are all in coupling and the value of C D is (4W - 1). While this formulation may repulsion genotypes, this cannot genes are randomly distributed probability of having i dominant is
seem to indicate an excess of coupling over be true overall if the dominant and recessive over the two chromosomes. In this case the and n - i recessive genes on one chromosome
assuming equal frequencies of the two types of genes. Thus the total number of pairs of loci in coupling is equal to
i (9) i-0
which comes to (n/4)(n -
. ($”
[( ;,
+
(” ;
l), and the total number
q]y
of pairs in repulsion
is also
(n/4)@ - 1). Of course, in actual populations the situation of complete disequilibrium is not and the highest absolute values of D are possible if there is recombination, expected between the most closely linked loci. Under these circumstances the highest overall repulsion association is expected when there is a preponderance of the chromosome pairs 1010101010... and 0101010101... . This model is similar to that envisaged by Mather (1949) except that epistatic interaction was assumed to be the important causal factor in this discussion. In a balanced situation such as the above, if there is close linkage we might expect almost as much stability with the dominance as with the overdominance model. A computer comparison of the two models is included in the following section.
MULTIPLE-LOCUS
COMPUTER
SIMULATION
Several types of Monte Carlo simulation were carried out in order to demonstrate that fixation of variability is retarded in a system of closely linked selected loci. All of the runs to be reported were made with a monoecious population of size 50. The number of loci was chosen as 108, each locus having two alleles, and there was 0.1 o/0 recombination between adjacent loci, making a total map length of 10.7 cMs.
GENE
AND
CHROMOSOME
HETEROSIS
497
Choice of a suitable starting population is one of the most important features of multiple-locus simulation. If populations are started with unrealistic disequilibrium values it may take many generations to reach a stable value, during which time gene frequencies may fluctuate considerably. In the present investigation, populations were set up by a method described briefly by Sved (1971) to give values of r2 for all pairs of loci in approximate agreement with the formula
l/(1 + 4Nc). The method is a somewhat complicated one. Initially a single chromosome type and its complement were generated, and 50% of all chromosomes in the population were assigned one type and 50% the other. This produces 100% disequilibrium between all pairs of loci. It also sets all gene frequencies to 50%, which is convenient for making comparisons, although admittedly artificial. The alleles at each locus in the original chromosome type were chosen at random, thereby avoiding any systematic tendency for an excess of either coupling or repulsion types. However, as related later, the procedure led to considerable initial variability in C D values. To produce the required starting populations, the 100% disequilibrium values were now reduced to more realistic values by a recombination procedure. A pair of chromosomes was selected from the population, recombined at some random point, and then returned to the population. Repeated application of this procedure reduced the association between linked loci, and of course reduced it much more quickly for loosely linked than for tightly linked loci. However it was found that if chromosome pairs were chosen at random for the recombination procedure, the values of r2 for the more widely separated loci were reduced too rapidly in relation to those of closely linked loci. The procedure was therefore modified so that, as expected in actual populations, some chromosomes were recombined more often than others. Following the arguments of Sved (1971), a geometric distribution with parameter (1 - 1/2N) was used to describe the number of recombination points per chromosome. A series of long term runs was made in which all loci in a particular population were assigned equal selective differentials. For the overdominance model, selective values of approximately 0.99 : 1 : 0.99 were assigned at each locus. These values underestimate the actual selective intensity, since an additive model was used to avoid the complications of epistatic interactions (Franklin and Lewontin, 1970). The procedure used was to assign a relative selective value of unity to the highest expected level of heterozygosity in the population, and to assign a selective value of 1 - 0.01x to individuals heterozygous at x loci less than this level. For the dominance model, selective values of approximately 0.98 : 1 : 1 were assigned, so that the overall selective advantage of heterozygotes over homozygotes was the same in both cases. A third set of runs was made without selection. Results from the three models are given in Fig. 1, together with the expected
498
SWD
0
50
100 GCNERATION
150
200
FIG. 1. Percentage of loci fixed at various times in 108-10~~s model. The broken line gives the expected value without selection. Each of the other graphs is based on 6 replicate runs, and selective values are 0.99:1:0.99 for the overdominance model and 0.98:1:1 for the dominance model.
value for no selection. It may be seen that there is some stabilizing effect in the dominance runs, although considerably less than for the overdominance model. A second series of runs was made for comparison in which selection of twice the intensity was made to act at half the loci, the remainder being unselected. The 54 selected and unselected loci were assigned at random along the chromosome, except that the dominance and overdominance runs were made in pairs with the same pattern of selected and unselected loci. The proportion of fixed loci amongst the unselected loci was used as the indicator of instability in this case, thereby eliminating the effect of selection at the indicator loci themselves. As seen from Fig. 2, the results are comparable to those from the first series, although the differences are reduced, particularly between the dominance and unselected runs. While some stability is indicated for the dominance model, the above results give very little information about the extent of the stability. They also suffer from the disadvantage that after some period of time the overall levels of heterozygosity are much higher in the overdominance than in the dominance runs. At such times, therefore, the comparison between these two models is no longer strictly valid.
GENE
AND
CHROMOSOME
HETEROSIS
499
In order to obtain more precise information a more comprehensive series of short term runs was made. For these runs 12 loci spread evenly over the chromosome were kept selectively neutral, and the loss of heterozygosity at these loci was used as the indicator of instability. Also in these runs each starting population was run not once but a number of times to test the repeatability of results for a single population.
FIG. 2. Percentage of unselected loci fixed at various times in 108-10~~s model with 54 selected and 54 unselected loci. Selective values are 0.98:1:0.98 for the overdominance model and 0.96:1 :I for the dominance model.
Three selection models were again run; no selection, 1% symmetric, and 2% asymmetric. Each run lasted for only 5 generations, then gene frequencies were calculated at each locus, and the proportionate loss of heterozygosity estimated from the formula 1 - 4p(l - p). Ten populations were generated with linkage disequilibrium for each model, and twenty replicate runs were made for each population. The overall mean losses of variability for the three models were 4.71%, 2.63%, and 4.18% (Table II). The value for the unselected runs is in agreement with expectation from the formula (1 - 1/2N)5. The overdominance model has given a retardation factor (Robertson, 1962) of 1.8, while the retardation factor for the dominance model is only slightly greater than 1.1. Further insight into this question comes from the analysis of variance of results from the three models (Table II). Th e variability between populations
500
SVED
TABLE
II
Analysis of Variance of Proportionate Loss of Heterozygosity after 5 Generations in 10 Populations of Each Selective Regime, with 0.1 o/0 Recombination between Adjacent Loci
(df)
Unselected
Overdominance Dominance Mean square ( x 103)
Between populations
(9)
6.06 n.s.
1.47 as.
Between runs within populations
(190)
9.84
1.96
4.54
Between loci within populations
(110)
4.43 ns.
1.43 n.s.
6.32***
Runs x loci within populations
(2090)
3.56
1.29
2.74
26.93’**
Mean value ( x 102) 4.71
2.63
4.18
judged in relation to the variability between runs for a particular population is nonsignificant for the unselected and overdominance models, but highly significant for the dominance model. The composition of the starting population is therefore important in determining stability only in the case of the dominance model. As mentioned previously, dominant and recessive genes were assigned at random in setting up the most common chromosome type in the original populations. Regions of high positive disequilibrium will tend to occur by chance. This will not have any consequences for the overdominance model, but will lead to rapid loss of heterozygosity in these regions for the dominance model. Confirmation of this view comes from the fact that the between locus mean square is also nonsignificant for the unselected and overdominance models but significant for the dominance model. The comparison between the three models of the variability between runs within populations is also of interest. As seen from Table II, the dominance model is approximately intermediate between the other two models in this respect, which is one measure of the stability imparted by this type of selection. Short-term runs of the same type were also carried out for a model with 0.2% recombination between adjacent loci, and with all other parameters unchanged. The results are given in Table III, and show similar trends to the results in Table II. Further information of a different type comes from plotting the value of C D summed over all pairs of loci for a particular population against the overall loss of heterozygosity. This is done in Fig. 3 for 30 populations having dominance of
GENE
AND
CHROMOSOME
TABLE
501
HETEROSIS
III
Analysis of Variance of Short-Term Runs as in Table II, but with 0.2 y0 Recombination between Adjacent Loci
Cd.1
Unselected
10.09 ns.
Overdominance Dominance Mean square ( x 103)
Between populations
(9)
Between runs within populations
(190)
7.65
3.13
5.35
Between loci within populations
(110)
4.65 n.s.
2.83 n.s.
6.59***
Runs x loci within populations
(2090)
3.99
2.91
3.99
6.59 n.s.
28.66***
Mean value ( x 1Or)
4.85
3.85
4.12
FIG. 3. Proportionate loss of heterozygosity over 5 generations, plotted against CD summed over all pairs of selected loci. Gene frequencies were calculated at 12 selectively neutral loci spread evenly amongst 96 loci with selective values 0.98:1:1. All populations were started with linkage disequilibrium, 30 populations having the initial chromosome generated at random (closed circles), and 10 populations having a 1010101010... initial chromosome type.
502
SWD
selective value. From this it is seen that there is a high correlation between the two values (r = 0.92). It is apparent that a good deal of the overall variability between populations for the dominance model can be accounted for simply by the variation in overall C D values. Figure 3 also contains results from 10 populations in which the initial chromosome types were chosen as 1010101010... and 0101010101..., respectively. It is seen that the 2 D values for such populations are lower than for any of the cases where the initial chromosomes were assigned at random, although not much lower in some cases. The mean loss of heterozygosity from these 10 populations is 2.80%, which is not significantly different from the value found in the overdominance runs. The results from these runs can be regarded as the lower limit to be expected for the dominance model. In long-term runs with this type of dominance model the results have also been indistinguishable from those with the overdominance model. The high correlation between C D and the loss of heterozygosity opens up the possibility that if there is a bias towards negative values of D in natural populations, then the loss of heterozygosity for the dominance model will be much closer to the value for the overdominance model. For example, the dominance runs with negative values of CD gave an average loss of heterozygosity of 3.58%, which is considerably closer to the value for the overdominance model. As mentioned earlier, in a two-locus finite-population dominance model there is a tendency towards negative values of D. In view of the considerations of the previous section regarding the complications introduced by multiple-locus models, it is not certain to what extent this effect will apply for more than two loci. A number of runs were therefore set up to test for changes in C D values in populations run for a number of generations. Ten populations were generated in the manner described earlier in the section. Values of C D were calculated in these initial populations, and again after 20, 40, 60, 80, and 100 generations had elapsed. The results are given in Table IV, and show that on the average C D has fallen over the course of the simulation. Some of this fall might be attributable to a reduction in the absolute value of D, and this is allowed for in the last line of the table by calculating C D/C 1D j. It is seen that there was in fact a considerable bias towards positive values of D in the initial populations, which is a chance effect. The overall tendency appears to be a reduction in high positive C D values rather than a production of high negative values. In fact, in the two populations started with negative C D values there was no overall fall. On the whole it seems clear that there is a tendency towards negative disequilibrium but not a very pronounced one. The procedure described above was repeated for ten populations set up in linkage equilibrium, to test whether the buildup of negative disequilibrium might be inhibited in populations started with linkage disequilibrium. Again only a slow buildup of negative disequilibrium was observed.
GENE
AND
CHROMOSOME
TABLE
503
HETEROSIS
IV
Values of C D Summed over All Pairs of Loci at Various Times for 10 Populations Set Up with Linkage Disequilibrium (Selective values were 0.98:1:1 at each locus.)
Population 1.
0
20
Generation 40 60
80
3.80
7.60
2.56
4.33
-2.11
-1.11 -3.15
2.
3.70
-0.66
3.
9.90
-0.12
4.
4.50
-1.85
-1.71
-4.41
5.
- 5.70
-4.67
-1.80
-4.74
6.
1.60
-0.78
7.
1.60
8.
-5.55
6.96 -5.71
9.36
9.58
-2.08 4.30 -3.31 0.00
100 -1.63 -3.57 -0.77 -1.26 2.05
-4.05
-1.32
-4.23
-2.59
-2.53
-0.11
-3.60
-5.84
-2.65
-2.98
-2.89
-0.06
9.
4.15
4.41
0.24
1.66
10.
12.60
7.55
3.43
7.47
2.55
Ave.
3.06
0.81
0.17
- 0.26
0.01
-1.36
0.80
0.23
0.10
-0.12
0.00
-0.70
CD/ClDl(xlW
-0.23
-2.60 0.74
The results from the computing are therefore somewhat equivocal on this important point. It may be that the tendency towards negative D values is in fact sufficiently pronounced to lead to a significant excess of repulsion linkages in natural propulations. If this is so, then for a given overall level of heterozygosity the amount of stability given by the dominance model would be increased. Even if not, however, the results still suggest that there is some stability.
DISCUSSION
The results of this paper are perhaps of most relevance to the interpretation of inbreeding experiments. Several such experiments have been carried out to test the hypothesis of heterosis as a possible explanation for the maintenance of the large amounts of variability found in natural populations. As discussed by Sved and Ayala (1970), inbreeding experiments always yield information at the level of the chromosome rather than at the level of the gene. If the expectations from single-locus theory are accepted without reservation, this would mean that no stabilizing effect on polymorphic loci could be inferred even if high levels of heterosis are detected at the chromosome level, since this might be a mani-
504
WED
festation of dominance rather than overdominance at the gene level. However the results of this paper favour the suggestion that the interpretation in terms of stability is less dependent on symmetry of selection against the two homozygotes than would be suggested by single-locus theory. This strengthens the argument of Franklin and Lewontin (1970) that measurement and interpretation of selective values ought to be at the level of the chromosome rather than at the level of the gene. The dominance and overdominance hypotheses have previously been discussed in the context of hybrid vigour in corn. Crow (1948) and Fisher (1949) have argued that the advances made by hybridization could not be accounted for on the dominance hypothesis, assuming a balance between mutation and selection. The results of this paper, however, suggest the possibility that the consequences of dominance and overdominance may not be as different as previously supposed. The differences between the two hypotheses are in one sense still of practical importance. On the dominance hypothesis it would be possible, although not easy, to produce by directional selection the advances attained by hybridization. On the overdominance hypothesis this would not be possible. The results of some studies of directional selection in corn (see Sprague, 1967) suggest that the former hypothesis is still tenable. On the question of long-term stability it is far from clear that the model considered in this paper is of any significance. Even for the overdominance model there does not appear to be any long-term stability in a small population (Ohta and Kimura, 1970), and Sved (1968) h as argued that the effect of linked overdominant genes may only be of long-term significance in conjunction with overdominance at the locus itself. These considerations are much more critical for the dominance model, where no stable intermediate equilibrium is expected. Ohta and Kimura (1969) have argued that the equilibrium frequencies to be expected in natural populations are those given by the balance of mutation and selection. If this is the case, then the stabilizing effects discussed in this paper are so small as to be of no significance. Only if some transient equilibrium can be produced which is essentially unrelated to the mutation rate will the effects be of any significance. The computer models run with the dominance model, where unfavourable genes are started at high frequency, implicitly assume that such a situation is possible. In a sense, however, it is not crucial to the argument that such a situation can exist, since the principal argument being put forward is that if heterosis at the chromosome level is attributable to dominance rather than overdominance, nevertheless some stability is implied. Some attempt has been made to run long-term simulations of populations of size 50 with dominance of selective values and back and forward mutation. The equilibrium gene frequencies in these runs appeared to be only slightly greater than those predicted by the mutation-selection balance. It is clear that no stable
GENE AND CHROMOSOME HETEROSIS
505
effects can be produced by the dominance model in a simple population of this size. A model in which it seems possible that more substantial effects will be obtained is that of a subdivided population with restricted migration between isolates. This type of population structure provides a more favourable opportunity for random selective associations. The amount of linkage disequilibrium for all except the most closely linked loci will be determined by the size of the isolates rather than the size of the overall population. On the other hand, the migration between isolates ensures that there is a reasonable amount of homogeneity over the population. Preliminary computer simulation with this type of population structure and with overdominance of selective values appears to indicate that quite high levels of stability can be attained.
REFERENCES COMSTOCK, R. E., AND ROBINSON, H. F. 1952. Estimation of average dominance of genes, in “Heterosis” (J. Gowen, Ed.), pp. 494-516, Iowa State College Press, Ames. CROW, J. F. 1948. Alternative hypotheses of hybrid vigor, Genetics 33, 477-487. Ewwrs, W. J., AND THOMSON, G. 1970. Heterozygote selective advantage, Ann. Hum. Genet., London 33, 365-376. FALCONER, D. S. 1960. “Introduction to Quantitative Genetics,” Oliver and Boyd, Edinburgh. FISHER, R. A. 1949. “The Theory of Inbreeding,” Oliver and Boyd, Edinburgh. FRANKLIN, I. R., AND LEWONTIN, R. C. 1970. Is the gene the unit of selection ? Genetics 65, 707-734. FRYDENBERC, 0. 1963. Population studies of a lethal mutant in Drosophila melanogaster. I. Behaviour in populations with discrete generations, Hereditas 48, 89-l 16. HILL, W. G., AND ROBERTSON, A. 1968. Linkage disequilibrium in finite populations, Theor. Appl. Genet. 38, 226-231. JONES, D. F. 1917. Dominance of linked factors as a means of accounting for heterosis, Genetics 2, 466-479. KIMURA, M. 1964. Diffusion models in population genetics, J. Appl. Prob. 1, 177-232. MARUYAMA, T., AND KIMURA, M. 1968. Development of temporary overdominance associated with neutral alleles, Proc. Itint. Congr. Genet. 12th, 12.2.4. (Abstr.) MATHER, K. 1949. “Biometrical Genetics,” Dover, New York. OHTA, T., AND KI~XURA, M. 1969. Linkage disequilibrium at steady state determined by random genetic drift and recurrent mutation, Genetics 63, 229-238. OHTA, T., AND KIMURA, M. 1970. Development of associative overdominance through linkage disequilibrium in finite populations, Genet. Res. 16, 165-177. ROBERTSON, A. 1962. Selection for heterozygotes in small populations, Genetics 47, 1291-1300. ROBERTSON, A. 1970. A theory of limits in artificial selection with many linked loci, in “Mathematical Topics in Population Genetics” (K. Kojima, Ed.), pp. 246-288, Springer-Verlag, Berlin. SPRAGUE, G. F. 1967. Plant breeding, Ann. Rev. Genet. 1, 269-294.
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SVED, J. A. 1968. The stability of linked systems of loci with a small population size, Genetics 59, 543-563. SVED, J. A. 1971. Linkage disequilibrium and homozygosity of chromosome segments in finite populations, Theor. Pop. Biol. 2, 125-141. SVED, J. A., AND AYALA, F. J. 1970. A population cage test for heterosis in Drosophila pseudoobscura, Genetics 66, 97-I 13.