tices of the following two main types. One culling criterion is to favor the phenotypes above a threshold value. In the other

Transcription

1 Proc. Nat. cad. Sci. US Vol. 7, No., pp , December 97 Some Population Genetic Models Combining rtificial and Natural Selection Pressures (culling/viability selection/stable equilibria/genotypic phenotypic associations) S. KRLIN* ND D. CRMELIt * Department of Mathematics, Stanford University, Stanford, California 9305; and Department of Pure Mathematics, Weizmann Institute of Science, Rehovot, Israel Contributed by Samuel Karlin, September 9, 97 BSTRCT The evolutionary behavior of a diploid population characterized by a trait determined at one or two major loci subject to the combined effects of artificial and natural selection pressures is investigated. number of different genotypic phenotypic associations are set forth, including additive allelic effects and additive loci effects with a' variety of culling programs. Threshold selection schemes as well as culling favoring intermediate phenotypic values are considered. For these formulations results are reported concerning the dynamic progress of the population, a delineation of the numbers and properties of the stable equilibria outcomes, and a discussion of their qualitative and quantitative dependence on the two kinds of selection forces. The level of culling can be used as a control through which natural selection parameters can be estimated. This paper reports results on the consequences of implementing some specific artificial selection programs on populations, taking account of certain natural genetic and ecological forces. We deal with a large diploid population characterized by a trait determined at several loci. To ease the exposition, the discussion will be in terms of two loci with two alleles at each locus, labeled,a and B,b respectively, involving the usual 0 genotypes. The genotypic phenotypic associations are described in two formulations. The first involves three phenotypes, Po, Pl, P, each determined by the cumulative heterozygosity, namely, PO B b ab ib b ab a P B B b ab b ab ab ab, P3 BLb Thus, all genotypes of phenotype class Po are pure homozygotes, those of class P are single heterozygotes, and those of class P are the double heterozygotes. The other phenotypic classification carries five distinguishable phenotypes determined by the additive allelic effects at the two loci where,b confer allelic value one and a,b value zero. Specifically, the 0 possible genotypes divide into five phenotypic classes as indicated. P0 P P P3 P i ab B b b B BT R ab lab ab { ab ab ab bl ab b B t each generation a fixed proportion c of the population is removed (i.e., culled by the breeder) in accordance with some order of preferences. We examined a variety of culling prac- [] II 77 tices of the following two main types. One culling criterion is to favor the phenotypes above a threshold value. In the other procedure intermediate phenotypic values are preferred. The objective was to evaluate the interaction of the coupled artificial and natural selection forces, the influence of dominance relations, recombination and linkage factors, mating structure and age effects (referring to the timing of culling). Note that the culling process, unlike viability or fertility selection, induces a form of frequency-dependent selection. We now highlight five specific models. Other multi-loci models, including the further interpretations and discussion of our findings, are elaborated in Karlin and Carmeli () and Carmeli and Karlin (3). Model I: Culling in favor of the pure homozygotes for a two-locus symmetric viability regime We assume the following set-up of selection forces: P P PO Phenotypes (defined in []) - (favoring Po) -a O< a,#< Fitness parameters of the above sort conform to the classical two-loci symmetric viability pattern [in this connection, see Lewontin and Kojima (), Bodmer and Felsenstein (5), Karlin and Feldman (6). ssuming culling performed on the new-born, the effective order of forces is as follows. random mating (or selfing - culling -- viability or assortative mating) selection. Thus, following mating of mature individuals for a prescribed culling rate c, individuals of phenotypes P are removed first. If c exceeds the frequency of such individuals in the population, then individuals of phenotype P are next culled, if available, etc. Viability selection acts on the population remaining after the implementation of culling. Model II: Threshold selection model with viability selection acting only through one locus The phenotypic value is determined as in [] (i.e., additive allelic contributions). The culling criterion is directional in favor of the phenotypes P3 and P, and the viability prescriptions are as follows, where s is the selection coefficient. I Some special culling models in a one-locus context were first introduced by Haldane ().

2 78 Genetics: Karlin and Carmeli Proc. Nat. cad. Sci. US 7 (97) a aa DIGRM. Genotypes: aa Viabilities: I Culling s.0 I: (i c -s3-cior (jj) C<l-L±L 3 S~~~s II:- II:I--< (8c) ssi>-adc and c- 0.8 ` 8s ' 3 f III: sa IV: I c s V: s<c, C>o BB C le V~~~~I IvI ijj z Z z Z" Z z t zzz-,ol--zzz III o 0,-- -,I-I- --I p- 0. O Partition of the selection parameter space corresponding to different evolutionary outcomes. Bb Manifestly, viability selection is acting here onl; y at the -a locus and for 0 < s < expresses heterozygote Eadvantage. In order to analyze these and related multi-loci models it becomes essential to ascertain the consequences of the combined culling and natural selection process for ahost of onelocus situations. We describe three relevant one -locus culling models of some independent interest. Some resulits and interpretations follow. Model III: Culling in favor of a dominant traiit Genotypes ( phenotypes) aa = (a,) case (a) case (b) + s (favoring ) Model IV: Culling in favor of the heterozygotte Genotypes = (phenotypes) bb In this case the three genotypes are distinguishable where disruptive viability selection is operating, usually in the guise of some micro-environmental adaptation germane for each homozygote. Model V: Culling for or against homozygotes Here, culling is assumed to favor intermediate or extreme values of the character selected for, with natural selection conferring disruptive or overdominance effects. SUMMRY OF RESULTS We begin with the one-locus models. Model III Case (a): Where the selection parameters satisfy c < s and if Po <, then fixation of the "aa" homozygote occurs. (po denotes the initial -frequency). For c < s and po > - (O < s < ) or for any initial conditions where c > s, the desired allele (0<s< O) s "" is fixed. Case (b): The population achieves a stable polymorphic Differential viability favors in case (a) the pheno equilibrium independent of the intensity of the culling rate. to the culling direction, while in case (b) nature selectionps For a range of small culling rates, it is possible that two simul- the trait to taneous polymorphic equilibria arise, and which is established is that of heterozygote advantage. In the latter cacse be subjected to culling selection may be constru depends on initial conditions. led as having pleiotropic expression. These selection schemes atre sometimes Model IV appropriate applied to certain populations characterized by The parameter range (c = culling rate, s = selection coefat one locus ficient) divides into five parts, each region conforming to a discrete or dichotomous characters controlled (for example, disease resistance, color or specia morph pat- distinctive qualitative evolutionary behavior. Diagrams and terns, and similarly). aa a a I-s O<S l.0 s delineate the final outcomes of the combined selection process. Model V Here culling favors the homozygote genotypes while natural -s (0 < < ) selection confers superior fitness value on the heterozygote (in favor of a) genotype. Results for the one-locus two-allele case are as follows: I

3 Proc. Nat. cad. Sci. US 7 (97) Models Combining rtificial and Natural Selection /+.$ = S CSE I: CSE II: Ivc- 7 (a) IF +v() < (b) IF _-s_ I + V-f (I -s) > WHERE P<Y<-VF ND Y (-Y P CSE III: + s- ( + s).-. &S(I-C) s. =I + S +V(I + O S) 8S o P I I < bi o P R 0.5 p p.0 p _ YI -P 0.5 p p Pi P.0 ~~~~~ CSE IV : (a) (b) ~~~~~ 0 p,' -p.5 P.0 L.I~. CSE V: 0 P* = FOR (a) ND (b) RE DEFINED IN CSE II. CONSULT THE DIGRM OF RESULTS FOR PRECISE DEFINITION OF CSES. DIGRM. Final results and domains of attraction. p, ply p are equilibrium points as explicitly displayed; Y is a value determined as indicated in (b). The region with an arrow pointing to an equilibrium indicates the domain of attraction to this point.. s a consequence of the combined effects of selection where c < there exist two interior unstable equilibrium points (pp) that delimit a symmetric interval about p* = (see Diagram 3). For all initial frequencies in the interval (p p) evolution to p* = occurs and otherwise fixation to the P~~~p "aa" or "" state transpires respectively, according as the initial frequency po of "", satisfies po <fp or po > p.. Where < c, i.e., with appropriately large culling rates, the population evolves to a pure homozygous state "" or "aa", respectively, according as the initial condition Po > or Po < holds. Diagram 3 describes the final outcomes. For the meaning of the arrows, consult the legend in Diagram. Model I. For a >,B but c < state (with equal gametic frequencies) is established independent of the recombination fraction.. Where P > a > # holds with tight linkage, then three a - a complete symmetric equilibrium stable polymorphic equilibria can arise for certain ranges of the culling rate c. 3. For P > a and c <, with both of small order, then un- symmetric stable equilibria appear.. Large culling rates yield a single segregating locus or fixation of one of the gametic types. Model II In this case natural selection acting only through one locus dominates and compels fixation of the second locus independent of the recombination rate. The only influence of linkage is revealed in the rate of attainment of the ultimate state such that tight linkage retards the rate of convergence. The final state reached coincides with the outcome in the corresponding one-locus model where culling favors the dominant phenotype and natural selection operates in the form of heterozygote advantage. DISCUSSION OF RESULTS There are many traits controlled principally at one, two, or few loci (sometimes also involving several modifier loci with small effects) on which artificial selection programs are implemented. These include one-locus traits related to disease resistance, especially in plant populations, for example, resistance to blue mold in tobacco (this is a 99% natural selfer), rust resistance in wheat (usually a dominant trait), resistance

4 730 Genetics: Karlin and Carmeli s 0 P P I/ P.0 I: c <,,.,.0,.< - I v MT. -, * I + II-_ II: c > P s p _ S 0 Pa./.0 i on DIGRM 3. II vs. susceptibility to barley yellow dwarf virus, and manyothers. Flower color is sometimes predominantly a one- or two-locus trait, usually entailing several alleles. Gossypol content in cotton is determined by or 3 principal loci. Coat color in many mammals involves often a few key loci [Searle (7)]. We have attempted quantitatively through a series of models to gain insights into the interaction between the usually antagonistic tendencies of artificial and natural selection, the influence of the imposed or inherent mating system, the relevance of timing in applications of the selection forces, the importance of multiple alleles, dominance relations, etc. We summarize some of the robust conclusions, implications and interpretations. I. Selection for a dominant or recessive trait (a) Sensitivity of outcomes to initial conditions: Where we assume that natural selection is expressed only through the phenotype and acts in the opposite direction to the culling incline, then fixation of the dominant or recessive type results, and which occurs depends critically on the initial composition of the population and the degree of culling compared to the magnitudes of the selection coefficients. In particular, deterioration (fixation of the inferior phenotype) may happen even if culling favors the dominant phenotype (see Model III) provided the initial population composition is mostly recessive. Such undesirable outcomes can only be overcome by devices such as hybridization (which effectively means altering the initial population makeup to include sufficiently many of the dominant type) or by substantially increasing the culling rate and possibly changing the mating system. High culling rates may not be feasible economically or desirable from other genetic considerations. The formal analysis in Carmeli and Karlin (3) partially provides means for appraising the costs incurred by such changes in the selection programs. (b) The manifestation of plateaus: In contrast to the above, if the natural selection regime is that of heterozygote advantage, then independent of the degree of culling, a "plateau" is attained, meaning that a stable polymorphism is established. Thus, natural selection always exercises an overwhelming effect over any degree of artificial selection for a dominant (or recessive) character and, even with extreme culling rates, fixation of the desirable allele cannot be totally achieved. (c) The possibility of two plateaus-(model III): In certain cases of natural selection favoring the heterozygote and culling favoring the dominant phenotype two stable plateaus are possible. In order to pass from the lower to the higher plateau (that with a larger phenotypic value) the frequency of the allele must be raised to a minimal threshold level and then the culling implemented at the appropriate rate. If a minimal phenotypic level is not available, then even some deterioration in the phenotypic level is conceivable and indeed occurs. (d) The age of the performance of culling and the nature of the mating system do not alter qualitatively the ultimate outcome as occurs in the cases of random mating. More specifically, (i) Culling at the adult stage lessens somewhat the impact of heterozygote advantage and apparently leads to a higher plateau. The result could be altered if differential fertility selection is important. (ii) ssortative mating, as anticipated, speeds the rate of fixation. II. Culling in favor of the heterozygote Simultaneous possibilities offixation and a plateau: n interesting and revealing example is the case where natural selection has both homozygotes more fit than the heterozygote. Three possible stable states, namely, fixation in either homozygote and a stable polymorphism, can now occur. Specifically, for certain orders of culling rates, fitness coefficients, and initial conditions, the phenotypic level could deteriorate, full fixation of the desired allele could be achieved, or a stable plateau could be reached. The artificial selection scheme does create some spurious excess of heterozygotes which should be recognized in interpreting the data. III. Evolution of the process in time Proc. Nat. cad. Sci. US 7 (97) Two surprising phenomena were observed for some specific ranges of selection and culling parameters, both in Models IV and V. (i) nonmonotonic changes in the gene frequency values in the process of convergence to a polymorphic equilibrium; and (ii) the existence of two noncontiguous domains of attraction to the pure "" state. The delicate influence of the nature of the initial composition of the population is underscored not only in the final states that are attained, but also reflected in the behavior pattern of the process in time. These findings suggest new diagnostic tools for discerning the relative influence of the forces involved, by studying the changes in time, as well as the stationary configuration of gene frequencies. gain, in the situation of opposing artificial and natural selection forces, we can obtain circumstances with deterioration, total improvement, and a plateau. When no dominance relations are present the selection and culling forces have equivalent strength and this contrasts with the situation of phenotypic dominance, where heterozygote advantage maintains a plateau independent of the degree of culling.

5 Proc. Nat. cad. Sci. US 7 (97) IV. The number of alleles n increasing number of competing alleles appears to slow the process in the attainment of its ultimate state. V. Culling on two-loci traits (a) In Model I of a two-loci trait, cases of three stable plateaus occur. Which "plateau" is reached is determined by initial conditions, the extent of linkage, and the special relations of the selection parameters [see Karlin and Carmeli (3) for full details]. (b) Equilibria with strong linkage disequilibrium and those with linkage equilibrium both exist even for tight linkage. (c) In the case that the double heterozygote is the desired kind, with no natural selection differences among genotypes, the effect of selfing can only be overcome by very strong artificial selection pressures (high culling order). The degree of culling to achieve its objective can be relaxed with weakening of linkage. The relevant comparison is r + (- r) < ( - c) indicating the extent of culling needed to prevent fixation. Where natural selection forces are also in force, the relationships are more involved. The general impact of linkage order seems to be complex. Tight linkage generally slows the pace of convergence to the ultimate state and there appear to be more possibilities for polymorphic plateaus with small positive recombination rates vis-d-vis free recombination. VI. Culling on a multi-loci trait with partial natural selection expression The main conclusion emanating from the models of Category II where natural selection operates through a set of loci labeled (L), while the culling policy (artificial selection) favors extremes of a phenotypic value determined by additive allelic effects from several loci including those of (L), is that the ultimate state of the population generally entails fixation of all loci other than those of (L). The above robust inference appears to apply for any threshold criteria with directional selection, provided the double heterozygotes correspond to intermediate values. On the other hand, if culling is directed to intermediate values with the double heterozygotes among the preferred phenotypes, culling can lead to a selection balance maintaining a polymorphic state. Models Combining rtificial and Natural Selection 73 Tight linkage enhances in these cases the departure from symmetric plateaus. Loose linkage leads to attainment of some of the boundary equilibria, involving segregation only of the group of loci (L) through which natural selection operates. Combined artificial and natural selection forces tend to produce more situations of fixation (or continued slow selection advance over longer durations) for phenotypic determinations based on additive allelic effects vis8--vis that of additive gene effects. Other multi-loci models including the full development of our results are presented in S. Karlin and D. Carmeli (). This research was supported in part by National Institutes of Health Grant USPHS Haldane, J. B. S. (96) "Some simple systems of artificial selection," J. Genet. 56, Karlin, S. & Carmeli, D. (975) "Some population genetic models combining artificial and natural selection pressures. II. Two locus theory," Theor. Pop. Biol., 6, to appear February Carmeli, D. & Karlin, S. (975) "Some population genetic models combining artificial and natural selection pressures. I. One locus theory," Theor. Pop. Biol., 6, to appear February Lewontin, R. C. & Kojima, K. (960) "The evolutionary dynamics of complex polymorphisms," Evolution, Bodmer, W. F. & Felsenstein, J. (967) "Linkage and selection: Theoretical analysis of the deterministic two locus random mating model," Genetics 57, Karlin, S. & Feldman, M. W. (970) "Linkage and selection: Two-locus symmetric viability model," Theor. Pop. Biol., Searle,. G. (968) Comparative Genetics of Coat Color in Mammals (cademic Press, New York). 8. Karlin, S. (968) "Equilibrium behavior of population genetic models with non-random mating," J. ppl. Prob. 5, 3-33; Lerner, I. M. (95) Genetic Homeostasis (Oliver and Boyd, Edinburgh). 0. Pollack, E. (966) "Some consequences of selection by culling when there is superiority of heterozygotes," Genetics 53, Spiess, E. B. (96) Papers on nimal Population Genetics (Little, Brown and Co., Inc., Boston, Mass.).. Thoday, J. M. (958) "Homeostasis in a selection experiment," Heredity, 0-6.