SCOPE 44 - Introduction of Genetically Modified Organisms into the Environment

6

Population Consequences of Genetic Design in Sexually Reproducing Organisms

FREDDY BUGGE CHRISTIANSEN
Department of Ecology and Genetics, University of Aarhus, Aarhus, Denmark
 
6.1 A NEW MUTANT
6.1.1 THE PROCESS OF RECOMBINATION
6.1.2 THE EFFECT OF FINITE POPULATION SIZE
6.1.3 THE INBREEDING DEPRESSION
6.1.4 GENETIC HITCH-HIKING
6.2 ASSOCIATIVE FITNESS EFFECTS
6.3 DISCUSSION
ACKNOWLEDGEMENTS
REFERENCES

The source of evolutionary change is heritable variation. This variation emerges as essentially random mutations in the genes recording the characteristics of organisms. In the Darwinian theory of evolution a basic assumption is that variation in a population is produced independently of natural selection, and so mutations are unrelated to the appearance, the phenotype, of the organism. This separation of the source of change, mutations, from the evolutionary force is intrinsic to Mendelian inheritance, in that indirect inheritance separates the hereditary material, the genes, from the phenotype of the organism. The neo-Darwinian theory of evolution, which is Darwin's theory including Mendelian inheritance, is to a large extent stretched over this framework, and it constitutes our main conception of the genetic basis of evolutionary change.

Exchange of genetic material among individuals within a species occurs by sexual processes, in higher organisms by sexual reproduction. Exchange of genetic material between organisms of different species rarely occurs in nature, but it may be possible through hybridization between closely related species with sexual reproduction. This kind of genetic exchange is unlikely for more distantly related species.

Genetic entities may, however, be infectiously transmitted from other organisms, a process known as horizontal transfer. This transfer represents a qualitative change in the rigid description of mutational change given above, in that genetic entities that are transmitted infectiously, with high probability, originate from species that survive and reproduce in the environment. The occurrence of horizontal transfer of genetic material may often be described formally as mutualistic symbiosis. For instance, the relationship between bacteria and plasmids conveying resistance to an environmental stress, such as the application of antibiotics, may be described as an interaction between a host and a parasite. The plasmids usually convey a cost to the host bacterium, so without the environmental stress bacterial cells which accidentally lose the plasmid are at an advantage, and the plasmid may be lost from the population. The incorporation and genomic transmission of pieces of foreign genetic material is in the classical genetic sense a mutation, and because the mutation occurs by horizontal transfer and transformation it depends on the transmissible DNA in the environment. Therefore, such a mutation may potentially be directed by the special demands posed by the environment.

Most of these observations are old and well established, but they have to a large extent been set aside as curiosities of procaryotic, or maybe unicellular, organisms. However, the discovery in higher organisms of transmitted DNA sequences that are clearly insertions of foreign origin takes the phenomenon of transformation out of the evolutionary curiosity shop. Nevertheless, it remains to be seen how important this phenomenon has been in the evolution of higher organisms and how frequently the nuclei of germ-line cells in natural populations are exposed to foreign DNA.

Mutation by transformation retains the property that the process of mutation is independent of natural selection; only the range of possible mutations may be influenced by the environment. The main property of this extension of the classical mutation concept is that quantum jumps in the range of capabilities of the organism are allowed; that is, new functions are not necessarily simple modifications of existing functions. Large changes in morphology or physiology are unlikely to be favoured by natural selection, so these are not the interesting jumps. Rather, the interesting transformational mutations are those that open new alleys in the further evolutionary development of the organism.

Genetic variation created by mutations is the source of evolutionary change in nature, and the same genetic variation has been the source for anthropogenic changes of domestic plants and animals by artificial selection. In this way the process of evolution and the process of plant and animal breeding is inherently the same as reflected in Darwin's original formulation of the theory of evolution by natural selection. A breeder chooses among the various evolutionary possibilities open to the particular organism when directed changes are sought, and the choices may produce processes that quantitatively differ from processes in nature. For instance, a plant breeder frequently employs polyploidizations of species or hybridizations of closely related species to change the available hereditary variation, and a contemporary breeder has techniques to facilitate formations of these changes. Many old domestic plants originate from these processes and many naturally occurring species show signs of the occurrence of these processes. Similarly, novel genetic variation may be found at a higher rate if the mutation rate is increased by chemical or radiative treatments, but again this is just a quantitative change of the naturally occurring process. The variation created by the breeder by these methods remains essentially random with respect to the goal of the breeder, as it is expressed in the artificial selection applied on the variation.

The possibility of directed transformation by the exposure of an organism to specific genetic material is a radical change compared to the situation in classical plant and animal breeding. The change is from quantitative management of the processes governing the creation of genetic variation to management of the quality of the genetic variation. A gene identified to have a certain function can be injected into individuals of a given species in a range of known or constructed allelic variants, and the performance of these different variants can be compared in order to form the basis for improvement of the species in the above sense. Therefore, the creation of variation in a breed can be made highly specific, so the production of variation and the artificial selection can both be oriented toward the goal of the breeder (Smith et al., 1987). Essentially the breeder can pick the variant that has the most desired property and use this for the production of an improved breed.

This breeding procedure is perceived to have two main advantages, both contributing to the excitement of the recombinant-DNA methodologies. First, the breadth of variation can be greatly extended over that of traditional breeding procedures. The available genetic variations transgress not only the variation present in the species but also the extended variation that might be mobilized from related species where hybridization is possible. The function of interest may, in principle, be conveyed by genes from any organism having this function. Second, the amount of breeding work needed to isolate the desirable variant in the breed is greatly reduced, because the variant classes are genetically discrete. The traditional breeding procedures utilize an elaborate crossing procedure to recover the desired properties of the original breed in an individual with the desired variant trait.

The first benefit should be gauged against the loss of variation in a species that accompanies any change brought about by breeding or by evolution by natural selection. The second benefit is too optimistic (Franklin, 1987), and it should be recognized as a recombinant-DNA breeding scenario which is basically the scenario of directed transformation in procaryotes. It focuses on the gene as the functional unit and assumes a direct connection between the presence of the gene and the expression of the desired quality. In diploid sexually reproducing species, of course, this has to be modified to take dominance relations into account, but this is a minor consideration because the scenario ideally refers to a true-breeding transformed variety. In self-fertilizing plants a couple of generations of selfing will produce such a variety, and the scenario is in general expected to work. However, the hidden assumption is that the transformation occurs in a functional transmissible genome, which in effect is assuming haploid individuals. In highly inbred species this may be a reasonable assumption, but in outbreeding plants and in animals the situation is not expected to be quite as simple.

6.1 A NEW MUTANT

A mutation is a rare event and for practical purposes each mutation is a unique event. Thus, a mutation of a gene in any organism occurs in a given cell with a given genome. Vegetative proliferation of this mutant cell is not only a proliferation of the mutant gene but also of the genome in which the mutation occurred. Therefore, in species that reproduce without sex and recombination, the proliferation of a given mutant results in loss of the preexisting variation in the population. For a mutation produced by recombinant-DNA methodology this may be overcome by transforming a large number of individuals, but in sexually reproducing organisms the uniqueness is a trap and the loss of variation inevitable. For simplicity let us restrict attention to diploid sexually reproducing organisms.

A mutation of a gene in a diploid organism occurs in a given cell in one of the two chromosomes carrying the gene. Only mutations in germ line cells are of interest from a genetic point of view, and we may assume accordingly that mutations only occur at the gamete stage. Then, after a mutation has occurred we expect to see it initially in heterozygote condition in one individual formed by the union of the gamete where the mutation occurred and another `normal' gamete. We now have two allelic forms of the gene, the old allele a and the mutated allele A, so the genotype of our unique individual is Aa.

No matter how we want to use this new variant, we need to build a homozygous AA stock. In self-compatible plants this may be done by selfing, where a quarter of the progeny is expected to be of type AA. Otherwise, we need more elaborate crossings. For instance, in animals we cross the unique individual Aa to any homozygote aa to get progeny half of which are of type Aa. After multiplication of the heterozygote genotype the desired homozygote type can be obtained from heterozygote by heterozygote matings. This procedure is elementary and seems uncomplicated. However, in every instance the AA individuals produced are highly inbred, so the stock of AA individuals will have lost a lot of the variation present in the original population. In addition, inbreed individuals of a predominantly outbreeding organism are in general weakened with lower fertility, and we have to take into account this phenomenon of inbreeding depression in the construction of breeds homozygote for the new mutant. Inbreeding depression also may serve to illustrate the magnitude of the loss of variation that occurs during the construction of the breed, in that the loss of performance due to inbreeding depression is linearly related to the increase of the inbreeding coefficient of the individual, i.e. to the increase in the probability that a random locus carries alleles that are identical by descent (see, for example, Falconer, 1981).

The amount of inbreeding produced by the proliferation of a new mutation may be reduced by repeated outcrossing of individuals carrying the mutant, i.e.. heterozygote Aa individuals, to population of aa individuals. This procedure would eventually produce a collection of Aa individuals that reflect the ideal situation that the A gene occurs in a population of genomes which represent the original population. However, the approach to this situation depends on the process of recombination, and this is a very slow process for genes linked to the new mutant A.

6.1.1 THE PROCESS OF RECOMBINATION

To evaluate the process of recombination suppose that an Aa individual is crossed to an aa individual and that one Aa individual is chosen among the offspring to repeat the procedure of outcrossing. Let the initial individual of this crossing scheme carry a stretch of chromosome of length R1 cM (or map units) to the left of the A gene which is transmitted from the original chromosome where the mutation occurred without being interrupted by an event of recombination. Similarly, let R2 cM to the right of A descend from the original mutant gamete:

(6.1)

Let r1 and r2 be the corresponding recombination frequencies, that is ri = Ri/100, i = 1,2. Further, let r'1 be the recombination frequency describing the length of the left conserved piece of chromosome after one generation. The expected value of r'1 is then

(6.2)

because with probability 1 - r1 no recombination occurs in the conserved piece (r'1 = r1) and with probability r, a recombination occurs leaving on average half the piece. Therefore,

(6.3)

and in the long run the length of the conserved piece of chromosome is expected to decrease by a factor of 1 - r1/2 every generation, which means a very slow change when the conserved piece becomes small. Similarly, for the right piece of chromosome and for the total length of the conserved piece of chromosome r = r1 + r2 we get

(6.4)

which if r1 and r2 are not too different is approximately given by E(r') (1 - r/4)r and if r1 or r2 is close to zero it is approximately given by E(r') (1 - r/2)r.

These deterministic calculations provide a crude picture of the stochastic process, but the order of magnitude of the effect turns out to be .correct. The maximal (or initial) length of the chromosome piece is r1, and we will assume that at most one chiasmata forms within this interval (complete interference). Let F(r) be the probability that the length of the left piece of chromosome in a given generation is less than the value r. Then the corresponding probability in the next generation is

F'(r) = F(r) + r[1- F(r)]

(6.5)

because a piece with length less than r stays shorter than r and a longer piece is changed into a piece less than r by a recombination within 100r crossover units from the A gene. The probability density function, f say, describing the length of the left piece of chromosome in a given generation is therefore given in the next generation by

(6.6)

If initially the length of the chromosome piece is r1, i.e. the initial probability density function, f, puts all of its mass at the point r1, then the probability density function, ft, describing the length of the left piece of chromosome after t generations is given by

ft(r)=(1-r)tf(r) + t(1-r)t-1

(6.7)

The mean length of the conserved piece of chromosome is then

(6.8)

and the variance is


(6.9)

The joint probability density of the left and right pieces, again assuming complete interference, can be found in a similar way as a function of the initial length of the chromosome piece to the left and right of the A gene.

Table 6.1 Conserved piece of chromosome after 10 generations for r1 = r2


Initial piece r

Distribution: mean ± s.d.

Equation (6.4) rate 1 - r/4


0.02 0.019 ±0.002 0.019
0.04 0.036 ±0.007 0.036
0.06 0.052 ±0.012 0.052
0.08 0.066 ±0.017 0.065
0.10 0.078 ±0.023 0.078
0.12 0.090 ±0.028 0.089
0.14 0.100 ±0.034 0.098
0.16 0.109 ±0.040 0.106
0.18 0.117 ±0.045 0.114
0.20 0.125 ±0.051 0.120

Tables 6.1 and 6.2 show the values of E(r(10)) in the simple case where r1 = r2 (Table 6.1) and in the case of initial asymmetry (Table 6.2). The mean values are compared to the deterministic prediction given by an iteration of equation (6.4) with the identification r' = E(r). Both tables show that the deterministic prediction gives a fair picture for judging the order of magnitude of the conserved chromosome piece. Therefore, the conclusion from the deterministic evaluation holds, in that the process of recombination is extremely slow in reducing and eventually removing the piece of chromosome associated with the new mutant. Ten generations of recombination only halves an initial piece of 20 cM around the new mutant. 

6.1.2 THE EFFECT OF FINITE POPULATION SIZE

The simple outcrossing scheme considered above replaces random chromosome segments flanking the conserved segment from the population of aa individuals. The A gene is placed in a random chromosome from the same population, so the procedure of recombination can restore the population variation around the A gene at the same time as the A gene is multiplied. However, the amount of variation restored by this procedure is very dependent on the number of independent outcrossing schemes used to produce the Aa individuals that founded the AA stock.

Table 6.2 Conserved piece of chromosome after 10 generations with initial asymmetry between the left and the right piece of chromosome: r1 ¹ r2 and r = 0.2


Initial
Mean Equation (6.4)
r1 r2

0.10 0.10 0.125 0.120
0.12 0.08 0.123 0.118
0.14 0.06 0.119 0.112
0.16 0.04 0.111 0.102
0.18 0.02 0.099 0.088
0.20 0.00 0.083 0.070

Within the conserved piece of the original chromosome containing the A gene every locus contains a copy of the allele carried in that chromosome. Any locus to the left of A which recombines with the A locus with probability r remain after t generations of outcrossing within the conserved piece with probability

(6.10)

when r < r1, where r1 as before is the initial length of the piece. Therefore, if we produce N independent Aa individuals by t generations of outcrossing, then we expect to get

Nt(r)=(1-r)t + [1-(1-r)t]N

(6.11)

independently derived copies of a gene that recombines with the A gene with probability r. Table 6.3 shows this effective number of genes produced from 100 lines of outcrossing in five and ten generations. For five generations of outcrossing the reduction is appreciable for loci within a stretch of about 10 cM around the A gene and after another five generations this region is about halved. Thus, even with very elaborate outcrossing schemes significant loss of variation is expected in the region surrounding the new mutant.

6.1.3 THE INBREEDING DEPRESSION

The loss of natural variation in the region surrounding the new mutant is a reflection of the inbreeding at the loci in that region due to the formation of the desired, but inbred, homozygote AA. This inbreeding is expected to produce inbreeding depression in predominantly outbreeding species, and this predicted effect will be compared to data from natural and experimental populations in the next section. The inbreeding effect is linked to the A gene, so the genotypes as and Aa will not be expected to show any effects, whereas the AA genotype is expected to have a decreased fitness due to the inbreeding depression. Thus, even after elaborate outcrossing the new mutant is expected to act like a recessive deleterious allele unless it has fitness effects of its own. The new mutant acts as a genetic marker for the piece of chromosome around it.

Table 6.3 Effective number of genes produced from 100 outcrossing lines 


Recombination Generation number
frequency r t = 5 t = 10

0 1 1
0.01 6 10
0.02 11 19
0.04 19 34
0.08 35 57
0.16 59 83
0.32 86 98

6.1.4 GENETIC HITCH-HIKING

The initial association of a new mutant with the genome in which it arises is a special case of the buildup of linkage disequilibrium in finite populations due to stochastic effects paralleling random genetic drift (Sved, 1971a; Weir and Cockerham, 1977). The persistence of an association after extensive outcrossing parallels the hitch-hiking effect of an advantageous gene on linked genes (Maynard Smith and Haig, 1974; Thomson, 1977), a powerful effect that can even carry recessive lethals to high frequencies (Wagener and Cavilli-Sforza, 1975). The considerations are also closely related to the theory of limits to artificial selection where effects of genetic linkage are taken into account (Hill and Robertson, 1966; Robertson, 1970).

6.2 ASSOCIATIVE FITNESS EFFECTS

Inbreeding depression of viability and fecundity due to homozygosity of chromosomes, or parts of chromosomes, is a well known phenomenon extensively studied in natural populations of Drosophila (Dobzhansky et al., 1963; Sved, 1971b; for a review see Lewontin, 1974). The effect of inbreeding in genetically marked chromosome pieces is well illustrated in many experimental Drosophila populations. In Drosophila melanogaster populations, founded from an inbred stock carrying the recessive ebony mutant allele e and from an inbred stock with the ebony wild-type allele +, Frydenberg (1964) observed initially (10-20 generations) a stable polymorphism indicating that the heterozygote e+ had a higher fitness than both homozygotes ee and ++. However, after some time (30-40 generations) the stability of the polymorphism apparently broke down, and eventually the ebony allele seemed to be on the way out of the populations (after 60-70 generations). Frydenberg interpreted these results as an effect of inbreeding and linkage in the sense discussed above. The e allele acted as a marker of a piece of chromosome from the inbred ebony stock, and the + allele marked a chromosome piece from the inbred wild-type stock, so fitness of the homozygote genotypes at the ebony locus were influenced by the inbreeding depression due to these linked pieces of chromosomes. The fitness of the heterozygote e+, on the other hand, would not be influenced by inbreeding depression, so this genotype is expected to be superior in fitness as long as the inbreeding depression is appreciable. Therefore, the initially stable polymorphism was due to associative overdominance at the ebony locus. As the recombination process reduced the length of the associated chromosome pieces the intrinsic fitness effect of the variation at the ebony locus played a larger role, and the e allele finally behaved as a deleterious allele.

Observations of associative overdominance of genetic markers derived from inbred stocks are very common. An especially interesting experiment is described by Barker (1977). He used two alleles w and wb1 at the white locus in Drosophila melanogaster in populations under conditions where the intrinsic fitness effects were minimized. The resulting associative effects gave an initial period of strong associative overdominance, but during the time following this period different populations showed different associative fitness effects. After 200 generations two populations remained polymorphic at different frequencies, and these polymorphisms lasted and changed slowly over the next 400 generations. These extended associative fitness effects are expected if the uniqueness of advantageous recombinant chromosomes are taken into account. A favoured recombinant chromosome may cause extensive hitch-hiking and buildup of novel associations between the genetic marker alleles and large pieces of chromosome.

In natural populations similar phenomena have been observed. Barker and East (1980) studied the effect of a release of genetically marked Drosophila buzzatii into an isolated natural population. The genetical markers were naturally occurring alleles at three enzyme loci obtained by 37 independent isolations from natural populations. The flies were produced in laboratory populations formed by intercrossing of these 37 lines. The subsequent release was successful in markedly changing the gene frequencies of the alleles in the natural population, but after the experimental release stopped the gene frequencies returned quite rapidly to the original values of the undisturbed population. In a later more extensive experiment at a different location (Barker et al., 1989) a locus perturbed in the same way as before was pushed almost to fixation for the released allele (Figure 6.1). The released allele was kept at a constant high frequency for a long time with intensive release, but again after the experimental release ceased the gene frequency returned quite rapidly to the original value of the undisturbed population (even though the released allele came from about double the number of isolations). The most immediate interpretation of these results is that the released allele is an associative deleterious recessive, as would be expected if it was associated with chromosome pieces showing inbreeding depression. This explains the long period with a high gene frequency as due to an immigration-selection balance equilibrium. Thus, either the number of independent isolations of the released allele is too low to average out the associative fitness effect or the proliferation of these flies in the laboratory builds associations between the markers and genes that are advantageous in the laboratory but disadvantageous in nature.

Figure 6.1 The change in allele frequencies at an enzyme locus in a natural population of Drosophila buzzatii due to experimental release of flies marked as homozygotes for a naturally occurring allele. The frequency of the allele used as a genetic marker is shown in two sections of the isolated population before (the first two samples), during (the next 13 samples) and after (the last nine samples) the period in which marked flies were released into the population. (Redrawn after Barker et al., 1989.) 

6.3 DISCUSSION

Recombinant-DNA methodology provides the breeder with increased genetic variation in the form of isolated genes. It does not provide an increase in the heritable phenotypic variation which is usually the source of improvement in outbreeding populations, especially of animals. The focus on single genetic elements in the breeding of an animal therefore goes counter to usual animal breeding practice. This may pose the problem that transfer of genes requires knowledge of single gene traits to make it worth the effort (Robertson, 1982), and our knowledge of genes involved in the heritable variation of desirable phenotypic traits is sparse. In addition, genes with a product that seems immediately related to some desirable character may not be the most important genes in the breeding for improvement of that character (Barker, 1985), so the loss of genetic variation in an animal breed following a genetic transformation is a very important cost to be considered in any breeding application of the methodologies of gene transfer. Loss of variation is of course a general concern in any animal breeding practice, and care is usually taken to avoid inbreeding with the concomitant inbreeding depression and general loss of genetic variation. However, the focus on one gene produces inbreeding around that gene even if carefully controlled outbreeding is performed. Further, the focus on one aspect of one trait carries with it the danger of underestimating the general inbreeding effects. Surely, a simplified picture of gene transfer as a quick and easy breeding scenario has to be modified.

The focus on single gene traits in a sense transforms animal breeding into plant breeding, where breeding procedures with a focus on genes are more widespread. In plants the breeding for a particular gene in an outbreeding plant also produces inbreeding effects, but extensive outcrossing and conservation of the original variation during breeding is more practical. Further, the more effective proliferation of plants allows a circumvention of inbreeding problems by producing commercial seeds by a hybridization procedure. However, it is not clear that these procedures in the long run are superior to procedures that are closer to the animal breeding techniques. The concept of uncomplicated and speedy gene transfer will probably be restricted to applications in selfing or in non-sexual plants.

The application of recombinant-DNA methodologies in plant and animal breeding provides exciting possibilities, but the complicated genetic structure of outbreeding populations entails limitations for the technique. However, the rDNA methodologies allow investigations with a focus on effects linked to single genes in relation to characters that show polygenic inheritance. Therefore, the techniques have a great potential as a tool to further the understanding of the genetic basis of these characters. In this way the methodology will be a potential tool for the long-term understanding of the processes in animal and plant breeding and therefore also for our understanding of the process of evolution.

The considerations here are also of relevance in the evaluation of potential applications of recombinant-DNA methodologies in transforming higher organisms in nature, e.g. to make natural enemies of pests more effective. This case is not very different from a usual animal breeding problem. However, one additional concern is appropriate here, namely the fate of a locally released organism with a special discrete genetic difference from the natural population. The experiments with Drosophila buzzatii described above show that the modified genes probably will be disadvantageous in nature due to associative fitness effects, so the modified type can be maintained in nature only by the pressure from recurrent immigration due to repeated release. This property is not intrinsic to the modified genes. Suitable recombinants, which are bound to occur in time, may free the modified genes from the breeding burden and permit their ultimate fate to be determined as a function of their intrinsic effects in the natural population.

ACKNOWLEDGEMENTS

The paper benefited from comments on a draft manuscript by Drs J.S.F. Barker and V. Loeschcke. The figure is drawn by Mr A. Jensen. 

REFERENCES

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Barker, J.S.F. (1985) Potential contributions of genetics to animal production. In: Proceedings of the 3rd AAAP Animal Science Congress, Korea, pp. 22-32.

 Barker, J.S.F. and East, P.D. (1980) Evidence for selection following perturbation of allozyme frequencies in a natural population of Drosophila. Nature 284, 166-8. 

Barker, J.S.F., East, P.D. and Christiansen, F.B. (1989) Estimation of migration from a perturbation experiment in natural populations of Drosophila buzzatii. Biol. J. Linnean Soc. 37, 311-34.

Dobzhansky, T., Spassky, B. and Tidwell, T. (1963) Genetics of natural populations. XXXII. Inbreeding and the mutational and balanced loads in natural populations of Drosophila pseudoobscura. Genetics 48, 361-73.

Falconer, D.S. (1981) Introduction to Quantitative Genetics, 2nd ed., Longman, London.

Franklin, I. (1987) Gene transfer for the Australian sheep industry. In: McGuirk, B.J. (Ed.) Merino Improvement Program in Australia, Australian Wool Corporation, Melbourne, pp. 495-509.

Frydenberg, O. (1964) Long-term instability of an ebony polymorphism in artificial populations of Drosophila melanogaster. Hereditas 51, 198-206.

Hill, W.G. and Robertson A. (1966) The effect of linkage on limits to artificial selection. Genet. Res. (Cambridge) 8, 269-94.

Lewontin, R.C. (1974) The Genetic Basis of Evolutionary Change, Columbia University Press, New York.

Maynard Smith, J. and Haig, J. (1974) The hitch-hiking effect of a favourable gene. Genet. Res. (Cambridge) 23, 23-35.

Robertson, A. (1970) A theory of limits in artificial selection with many linked loci. In: Kojima, K. (Ed.) Mathematical Topics in Population Genetics, Springer Verlag, New York, pp. 246-88.

Robertson, A. (1982) Genetic engineering in animal improvement. Proc. 2nd World Congr. on Genetics Applied to Livestock Production 6, 139-45.

Smith, C., Meuwissen, T.H.E. and Gibson, J.P. (1987) On the use of transgenes in livestock improvement. Animal Breeding Abstracts 55, 1-10.

Sved, J. (1971a) Linkage disequilibrium and homozygosity of chromosome segments in finite populations. Theoret. Population Biol. 2, 125-41.

Sved, J. (1971b) An estimate of heterosis in Drosophila melanogaster. Genet. Res. (Cambridge) 18, 97-105.

Thomson, G. (1977) The effect of a selected locus on linked neutral loci. Genetics 85, 753-88.

Wagener, D.K. and Cavalli-Sforza, L.L. (1975) Ethnic variation in genetic disease: possible roles of hitchhiking and epistasis. Am. J. Human Genet. 27, 348-64.

 Weir, B.S. and Cockerham, C.C. (1974) Behavior of pairs of loci in finite monoecious populations. Theoret. Population Biol. 6, 323-54.

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The electronic version of this publication has been prepared at
the M S Swaminathan Research Foundation, Chennai, India.