NATURAL SELECTION OPPOSING ARTIFICIAL SELECTION: A TWO-LOCUS DETERMINISTIC SIMULATION

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NATURAL SELECTION OPPOSING ARTIFICIAL SELECTION:

A TWO-LOCUS DETERMINISTIC SIMULATION

FRANCIS MINVIELLE

Dkpartement de Zootechnie, Uniuersitk L a d , Qukbec, Canada GIK 7P4

Manuscript received November 3, 1980 Revised copy received February 4, 1981

ABSTRACT

A two-locus, two-allele metric trait was submitted to artificial truncation selection and to three types of opposing natural selection (two-locus extensions

of directional selection, overdominance and underdominance) by numerical simulation in a large random-mating population. Limits to selection were generally reached by generation 100. Intermediate selection plateaus were found, with minor genes, for all three modes of opposing natural selection, but they were least frequent with underdominance. Multiple outcomes were common. I n particular, fixation of the genotype favored by artificial selection was often associated with fixation of another genotype and/or with a central equilibrium; the end point actually reached depended on the genetic starting point of the simulation. I n general, when the alleles favored by truncation selection were combined (positive linkage disequilibrium) in the base population, or when the trait was determined by major genes, artificial selection would prevail. Limitations inherent to this type of work are discussed, and possible ave- nues for further work on the antagonism between artificial and natural selection are proposed.

U N T I L recently, the opposition between natural selection and artificial selection has been studied mainly with one-locus models of inheritance (e.g.,

JAMES

1962; VERGHESE 1974; CARMELLI and KARLIN 1975; MINVIELLE 1980;

NICHOLAS and

ROBERTSON

1980). These models were developed under various assumptions regarding some or all of the following: modes of natural and artificial selection, proportionate effect of the locus, environmental variability and population size. Since one-locus models cannot take into account the effects of linkage and epistasis, they have only limited usefulness in helping to interpret the results of selection experiments affected by, or designed to study, natural selection (e.g., VERGHESE and NORDSKOG 1968; KAUFMAN, ENFIELD and COM- STOCK 1977; MINVIELLE and GALL 1980).

In 1975,

KARLIN

and CARMELLI developed a two-locus model to examine the joint effects of natural and artificial selection. They found the limits of their selective process analytically and derived the conditions for central stable equilibria. However, their approach was criticized by

NICHOLAS

and

ROBERTSON

(1980) because their population was infinite, their quantitative trait was not influenced by the environment and they did not use the homeostatic model

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232 F. MINVIELLE

(LERNER 1954). But two-locus selection theory still relies on simplifying assumptions (KARLIN 1975) that, at the moment, forbid a general two-locus analyti- cal treatment of the conflict between natural and artificial selection.

A

different approach was taken for the present work: in an infinite population, the effects of varied opposing modes of selection, including the homeostatic model, on a trait determined by two loci and influenced by the environment were studied by deterministic computer simulations. Varied combinations of input parameters (natural viability matrix, pressure of artificial selection, proportionate effect of the locus and rate of recombination) and of starting points of the simulation

(gene frequencies and linkage disequilibrium) were used to reach robust con- clusions on the antagonism between natural and artificial selection.

METHODS

The model used i n the present study is a straightfsrward two-locus, two-allele extension of a one-locus model described previously (MINVIELLE 1980). At each locus independently, two alleles act additively with respect to a metric character. The genotypic values are + a ,

0, - a at locus A, and

+

b, 0,

-

b at locus B. Single-locus values are simply added to obtain the genotypic value of the corresponding two-locus genotype. Artificial selection favors the genotype A,A,B,B,, i.e., it has the highest genotypic value ( a

+

b ) . Opposing natural selection is imposed through symmetric viability matrices, both for simplicity and for discussion of the results within the framework of existing two-locus theory. Three modes of natural selection were used: ( 1 .) Two-locus overdominance: double heterozygotes A,A,B,B, have the best fitness (homeostatic model). (2.) Two-locus directional selection: homozygotes A,A,B,B, have the lowest fitness, and some other homozygotes have the highest fitness. (3.) Two-locus underdominance: double heterozygotes have the lowest fitness.

Theory and definitions: The extension of the one-locus definitions and calculations (MIN-

VIELLE 1980) to this two-locus work is direct, but recombination, r , and linkage disequilibrium,

D, between the two loci are new features in the two-locus study. Linkage disequilibrium a t generation t , D ( t ) , is defined as D ( t ) = ztt) z ( t ) - z(;t) zkt), where zl, x 2 , z3 and z p are the frequencies of the chromosomes A $ , , A,B,, A,B, and A$,, respectively. By simply extending FALCONER’S (1960) calculations of additive genetic variance to a two-locus setting, AVERY and HILL (1977) showed that; when there is no dominance; the additive variance becomes a:dd, = 2 a z p y ) [ l - ~ ( ~ ) ] f 2 b 2 p ( t ) [ l - p ( t ) ] + 4 a b D ( t ) , where D and the frequencies of genes A,

and B , , p.I and pe, have been measured at generation t, before selection.

The truncation point at the tth generation of artificial selection is defined as T ( t ) =

M ( t )

+

i’ o(t), where M is the population mean before selection, if is the standardized truncation point, and the total variance, U;, is the sum of the additive variance, utdd, and the environmental variance, uz Nine natural fitness values, kj, are specified in each natural viability matrix; coupling and repulsion heterozygotes have the same natural fitness. Let 2a/oE and 2b/uE be the relative effects of loci A and B.

I 4

E ’

Simulation: The life cycle was:

1

random truncation artificially natural

{

adult

--

progeny

----

selected ----+ adult. .

.

mating selection progeny selection

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233

successive generations, were smaller than 10-6 in absolute value. Obviously, unstable equilibria could not he characterized in this simulation study. Every ten generations and at the selection limit, the gene frequencies, additive variance, heritability, the phenotypic mean before selection, mean fitness, linkage disequilibrium and the phenotypic mean after selection were printed out. Major and minor genes were considered by giving values between 4 and 1/25 to the relative effects. Weak (i' = 0.1), moderate (i' = 1.0) or strong (i' = 2.0) artificial selection was applied. Recombination values between the two loci were taken between 0.5 (no linkage) and 0.1 (tight linkage). The fitness values in each natural viability matrix were chosen over the whole range of possible numerical values to create mild, as well as strong, opposing selection. Problems with only a single type of selection (natural selection o r artificial selection) acting on the population were analyzed also. These simulations were used as controls. Thirteen pairs of initial gene frequencies [ p ( O ) , p ' , o ) ] regularly spaced in the unit square were chosen. For each pair, the base

population was structured so that D ( 0 ) was maximum, min'mum or null in order to make up the 39 different simulation starting points per problem.

RESULTS

Overall, nine different kinds of outcomes have been observed. They can be grouped into central equilibria (all four chromosome types are present), side equilibria (only two chromosome types) and corner equilibria (one chromosome type). The first two categories correspond to nontrivial (i.e., U",,.

>

0) selection plateaus for the quantitative trait and the population fitness, and the last one implies fixation at both loci A and B. For a given problem, up to five different end points were reached, depending on the genetic starting point (gene frequencies and linkage disequilibrium). However, each selection limit was generally reached in less than 100 generations, and fixation of A,B, was usually observed when A and B were major genes.

Artificial selection versus natural overdominance: Fixation of A,B, was observed in only a few runs under weak opposing natural selection when the metric trait was determined by minor genes with small values for both p ( 1 ) and p ( 2 and with between-locus underdominance. By contrast, fixation of AIB, and central equilibria were frequent.

( 1 ) Between-locus underdominance: double homozygotes more fit than single homozygotes (ROUGHGARDEN 1979, p. 124). Unique central equilibria (Desu.

>

0) were found (in

5%

of the problems) with minor genes. Increased pressure of artificial selection displaced the equilibrium point towards fixation of A ] B ,

(60%). Two simultaneous central equilibria (10%) with Dequ. values of op- posite signs were also found with tight linkage only: when D(O) was maximum or when p(:) and p(;) had similar values, the equilibrium with positive Dequ, was reached; as the relative effects increased, this equilibrium (Dequ.

>

0) eventually became the A,B, corner still associated with the other (Dequ.

<

0) central equilibrium ( 5 % ) . Under weak natural selection, the selective process could also lead to the fixation of either A$, or another chromosome type (20%) depending on the genetic starting point of the simulation.

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234 F. MINVIELLE

and A,B,) were also found (10%) as the intermediate outcomes between the central equilibria and the fixation of A,B, (45

%).

None of the other three corner equilibria was observed.

( 3 ) Nonepistatic overdominance: multiplicative fitness: A single outcome was found for each problem. As the relative effects were increased, one could observe successively a central equilibrium

(Dequ.

= 0; 25%), a side equilibrium (A,B, and A,B, or A,B, and A B , ; 25%), and the corner equilibrium

(A,B,;

50%).

Artificial selection versus natural directional selection: Fixation of A,B, was observed either alone or associated with other corner equilibria; the corner equilibrium actually reached depended on the initial gene frequencies and on D(O) to a lesser extent. Central equilibria were observed as transitions between corner equilibria A,B, and

A,B,.

( 1 ) A , and B , recessive under natural selection: unique central equilibria (De,”.

<

0) were observed frequently (in 35% of the problems), independently of the genetic starting point and of the degree of linkage, for all of the natural viability matrices when A and B were minor genes. The central equilibria were little influenced by changes of

i’,

but they were eventually replaced by the

A,B,

end point (40%) for major genes. Transitions between these two main outcomes corresponded to side equilibria (20%), and the remaining three corner equilibria wererare (5%).

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A ,

and B, not recessive under natural selection: few (5%) symmetric central equilibria,

(Depu.

<

0), always associated with some corner equilibria were observed for incomplete dominance of A , and B,; similar initial frequencies

p ( i )

and p(;) yielded the central equilibrium. Fixation of A, B , either associated with the corner equilibria A,B, and/or A,B, (15%) or alone (35%) was the most frequent outcome. Under strong natural selection and with minor genes, fixation of A,B, (10%) was observed, sometimes in combination with the corner equilibria A,B, and/or A,B, (15%). Between these two extreme situations, the four corner equilibria could co-exist (15%) for minor genes and for complete dominance of A, and B,; the genetic starting point actually determined which corner end point would be reached. When the initial linkage disequilibrium was determi- nantal, an excess of the initial coupling gametes would yield the fixation of one coupling chromosome type. Also, side equilibria were reached

(5

%

) as transitions between a central and a corner end point.

Artificial selection versus underdominance: Few (in less than 5 % of the problems) central equilibria

(Dequ.

< 0) associated with the end point

A,B, and some other corner equilibrium(a) were found with the “most” minor genes and under fairly strong underdominance. Extreme initial gene frequencies generally led the population to the closest corner equilibrium. Intermediate gene frequencies yielded the central equilibrium if initial repulsion chromosomes were in excess, but fixation of A,B, (or A-B,) otherwise. Corner equilibria were very frequent.

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235 mined by the initial gene frequencies, but also (when they were intermediate) by the initial linkage disequilibrium.

DISCUSSION

Intermediate selection plateaus (i.e.7 central and side equilibria) have been found, independently of linkage but for minor genes mainly, with all three modes of natural selection (Table 1). However, the plateaus were more frequent under overdominance or directional selection than with underdominance. They may be classified as either structural or dynamic. The first class of plateaus follows directly from the mode of natural selection: a central stable equilibrium is built in the symmetric overdominance model of natural selection

(KARLIN

1975, 1979), and artificial selection merely displaces the equilibrium point. The second class is really created by the interaction between the two selective forces: for example, neither directional natural selection for A,B, nor mass selection for A,B, leads the population towards a stable intermediate equilibrium, but their combination may yield a nontrivial selection plateau. At this plateau, linkage disequilibrium was negative or null under natural directional selection and underdominance. With overdominance, it was generally close or equal to zero, but one high positive and one high negative value were always associated when two simultaneous central equilibria were observed.

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236 F. MINVIELLE

+ I l l

I I I I I I

I

I l + l + l

I

+ 1 + 1 + 1

+ I

I I + + + I

I I

I l l +

I I I I I I

I + + +

+ I

I

I +

.- v)

l + + l

l + l + l

I

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7 The present results cannot be deduced directly from homologous one-locus studies: opposing one-locus underdominance does not yield an intermediate selection plateau (MINVIELLE 1980), but two-locus underdominance does. In fact, the behavior of multigene systems cannot be readily predicted from single gene work, partly because the much-studied theoretical modes of action of single genes do not have clear-cut multilocus extensions. For example, NICHOLAS and ROBERT-

SON (1980) deduced from a one-locus (finite population size) study that a nontrivial plateau was unlikely to be reached for minor genes unless they were tightly linked. However, this work shows that a nontrivial intermediate plateau may indeed be reached for loosely linked or even independent minor genes. More complex polygenic traits exposed to a similar selective process could then reach a variety of selection limits (KARLIN and CARMELLI 1975).

It should be noted that in the present study, as in most of KARLIN’S two-locus work. natural viability matrices were given an arbitrary symmetric structure that may or may not represent an adequate approximation of reality. On the other hand, it would be a formidable task to explore numerically the effects of unconstrained viability matrices on the selective process. Moreover, general ref- erence points (the action of natural selection alone) would not be available for comparison. An alternative approach has been used recently for a model with one major gene by

KARLIN

and CARMELLI (1978). They considered natural fitness as a quantitative character and natural selection as a form of truncation (phenotypic) selection, the most efficient mode of directional selection (KIMURA and CROW 1978; CROW and KIMURA 1979). Homologous multilocus studies with situations of homeostatic, as well as directional, phenotypic natural selection might help to further analyze the conflict between selection for production characters and natural selection. Finally, a new approach by GINZBURG and

BRAUMAN

(1980). who have shown that selection tends to prevail over recombination as the number of genes that determine the character increases, might also be considered.

The help of FRANCINE FILLION in compiling the simulation runs is gratefully acknowledged.

LITERATURE CITED

AVERY, P . J. and W. G. HILL, 1977

CARMELLI, D. and S. KARLIN, 1975

CROW, J. F. and M. KIMURA, 1979

FALCONER, D. S., 1960

GINZBURG, L. R. and C. A. BRAUMANN, 1980 Multilocus population genetics: relative im- portance of selection and recombination. Theoret. Pop. Biol. 17 : 298-320.

JAMES, J. W., 1962 Conflict between directional and centripetal selection. Heredity 17: 487- 499.

KARLIN, S., 1975 General two-locus selection models: some objectives, results and interpre- tations. Theoret. Pop. Biol. 7: 364-398.

-

, 1979 Principles of polymorphism and epistasis for multilocus systems. Proc. Natl. Acad. Sci. U.S. 76: 541-515.

Variability in genetic parameters among small popula-

Some population genetics models combining artificial and

Efficiency of truncation selection. Proc. Natl. Acad. Sci. tions. Genet. Res. Camb. 29: 193-213.

natural selection pressures: I. One-locus theory. Theoret. Pop. Biol. 7 : 94-122.

U.S. 76: 396-399.

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238 F. MINVIELLE

KARLIN, S. and D. CARMELLI, 1975 Some population genetics models combining artificial and natural selection pressures: 11. Two-locus theory. Theoret. Pop. Biol. 7: 123-148. -,

1978 Evolutionary aspects and sensitivity studies of some major gene models. J. Theoret. Biol. 75: 197-222.

KAUFMAN, P. K., F. D. ENFIELD and R. E. COMSTOCK, 1977 Stabilizing selection for pupa weight in Tribolium castamum. Genetics 87: 327-341.

KIMURA, M. and J. F. CROW, 1978 Effect of overall phenotypic selection on genetic change at individual loci. Proc. Natl. Acad. Sci. US. 75: 6168-6171.

LERNER, I. M., 1954

MINVIELLE, F., 1980 A simulation study of truncation selection lor a quantitative trait opposed by natural selection. Genetics 9 4 : 989-1000.

MINVIELLE, F. and G. A. E. GALL, 1980 Artificial selection for 18-day pupa weight and opposing simulated natural selection in Tribolium castaneum. Theoret. Appl. Genet. 56: 49-55.

NICHOLAS, F. W. and A. ROBERTSON, 1980 The conflict between natural and artificial selection in finite populations. Theoret. Appl. Genet. 56: 57-64.

ROUGHGARDEN, J., 1979 Theory of Population Genetics and Evolutionary Ecology: An Intro- duction. Macmillan, New York.

VERGHESE, M. W., 1974 Interaction between natural selection for heterozygotes and directional selection. Genetics 76: 163-168.

VERGHESE, M. W. and A. W. NORDSKOG, 1968 Correlated response in reproductive fitness to Corresponding editor: B. S. WEIR

Genetic Homeostasis. Oliver and Boyd, Edinburg.