Moving optimum:

dynamics of linkage disequilibrium, genetic polymorphism,limiting behaviour of trajectories

  • In this paper, we describe some regularities of the behavior of linkage disequilibrium, D, under temporal environmental fluctuations in infinite panmictic populations with non-overlapping generations and diploid and haploid selecton regimes. Different types of environmental variation ware considered. Our analysis shows that under two-locus haploid selection with any period length, there exists an environmental state (e.s.) where D can change the sign (between periods), but not more than once. The direction of this change is fully determined by a special quantity, "integral fitness disequilibrium coefficient," 'delta', which is calculated from genotypic fitness in environmental states. If delta = 0, then two e.s. exist with only one change of the sign in each. Special types of environments (sub- or super-multiplicative) common in theoretical modelling can be mentioned where the sign of D between periods behaves rather simply for any considered e.s. It can be shown that under sufficiently small rates of recombination r < r' for any period length p in every e.s. the sign of D can change no more than once; the border r' is calculated as a function of fitness values which decreases with decreased differences between the fitnesses. The obtained results were applied to various types of fitness functions. Thus, cases of additive effects of fitness loci on the selected trait under various hypotheses about the selection regime in changing environment were considered in detail. In particular, we showed that the sign of D generated by selection is, in a sense, a nonsymmetrical function when one considers possible forms of fitness dependence on the selected trait: negative D are more probable. These results are also extended to the diploid case with an additively formed selected trait. In particular:

Theorem 7. With diploid selection for an additively formed trait in an environment with p states linkage disequilibrium D changes the sing no more then once, if the fitness function F is logariphmically convex. The change is possible only from plus to minus if F is log-convex up and from minus to plus if F is log-convex down

Kirzhner V., Korol A.B., Ronin Y. (1995). The dynamics of linkage disequilibrium under temporal environmental fluctuation. Two-locus selection. Theoretical Population Biology, 47, 257-276. (Article.popbiol)


  • To classify different types of cyclic selection, a measure of fitness disequilibrium was used, and a class of systems were considered where this measure has the same sign in all states (sign-concordant environments). The necessary conditions for existence of a fixed point (considering any moment within the period as a referring one) are obtained for sign-concordant systems. However, analytical study of such systems, in the case of selection for equal additive genes, and numerical testing of more general situations, allowed us to conclude that no polymorphism is possible. In the alternative class of sign-concordant systems, polymorphism is possible. However, we found that global stability is an exception rather than a rule for sign-nonconcordant systems. Massive numerical simulations of selection in a four-state environment were made for cycle lengths in the range 828 and with evenly distributed selection coefficients. The proportion of polymorphic regimes ranged up to about 1.5%, and was dependent on the recombination rate between the loci. It should be stressed, that polymorphism maintenance in the haploid systems, when it is possible, can not be considered as an effect derived from constant selection, or be a result of any hidden form of heterozygous advantage. In other words, polymorphism stability is causally connected with environmental fluctuations. Equally important is that this effect of fluctuations is only possible because of recombination: in single locus systems haploid cyclical selection is unable to produce protected polymorphism.

Kirzhner V., Korol A.B., Ronin Y.I. (1995). Cyclical environmental changes as factor maintaining genetic polymorphism. I. Two-locus haploid selection. Journal of Evolutionary Biology, 8, 93-120.(Article.evolbio)


  • Polymorphism maintenance due to stabilizing selection with moving optimum is the subject of this paper. It was shown that in case of two-locus additive control of the selected trait, global polymorphism is possible only when the geometric mean fitnesses of double homozygotes averaged over the period are lower than that of the single heterozygotes and of the double hetero-zygote (with a multiplier (1-r)p which depends on recombination rate r and period length p). But local stability of polymorphism cannot be excluded even if geometric mean fitnesses of all double homozygotes are higher than that of all heterozygotes. We proved, that for logarithmically convex fitness functions, cyclical changes of the optimum cannot help in polymorphism maintenance in case of additive control of the selected trait by two equal loci. However, within the same class of fitness functions, non-equal gene action and/or dominance effect for one or both loci may lead to local polymorphism stability with large enough polymorphism attracting domain. The higher the intensity of selection and closer the linkage between selected loci the larger is this domain. Note that even simple cyclical selection could result in two forms of polymorphic limiting behavior: (a) usually expected forced cycle with a period equal to that of environmental changes; (b) "supercycles" - non-dumping auto-oscillations with a period comprising of hundreds of forced oscillation periods.

Korol A., Kirzhner V., Ronin Y.I. (1996). Cyclical environmental changes as factor maintaining genetic polymorphism. II. Two-locus diploid selection. Evolution, 50, 1432-1441. (Article.evol96)


  • This paper deals with the problem of polymorphism maintenance in species coevolution mediated by selection for quantitative traits controlled by Mendelian genes. We showed here that the conditions for polymorphism maintenance in interacting species can be deduced from the behavior of the isolated partners in stable and changing environments. This allows also to address such difficult questions as evolution of sex and recombination, that can not be considered properly in non-Mendelian models. An abundance of polymorphic regimes was revealed in the proposed genetic model. The obtained results demonstrate a remarkable property of trait-dependent coevolution concerning the conditions for maintenance of genetic polymorphism: what seems to be more realistic, that is, non-equal gene effects and deviation from purely additive within-locus gene action, promotes polymorphism.

Kirzhner V., Korol A., Nevo E. (1999). Abudant multilocus polymorphisms caused by genomic interaction between species on trait-for-trait basis. Journal Theoretical Biology, 198,61-70. (Article.abudant)


  • We analysed a diploid population model with a mixed breeding system that includes panmixia and apomixis. Each individual produces a part (ss) of its progeny by random mating, the remainder (1-ss) being a result of precise copying (vegetative reproduction or apomixis) of the parental genotype. Both constant and periodically varying selection regimes were considered. In the main model, the selected trait was controlled by two diallelic additive or semidominant loci, A/a and B/b, whereas the parameter of breeding system (ss) was genotype-independent. A numerical iteration of the evolutionary equations were used to evaluate the proportion (V) of population trajectories converging to internal (polymorphic) fixed points. The results were the following. (a) A complex pattern of dependence of polymorphism stability on interaction among the breeding system, recombination rate, and the genetic architecture of the selected trait emerged. (b) The recombination provided some advantage to sex at intermediate period lengths and strong-to-moderate selection intensities. (c) The complex limiting behavior (CLB) was quite compatible with sexual reproduction, at least within the framework of pure genetic (not including variations in population density) models of multilocus varying selection.

Ryndin, A., Kirzhner, V. Nevo, E. and Korol, A. (2001) Polymorphism maintenance in populations with mixed random mating and apomixis subjected to stabilizing and cyclical selection. J. Theoretical Biol. 212, 2, 169-181. (Article.apomixis)


V.M. Kirzhner, A.V. Ryndin, E. Nevo, A.B. Korol (2003) Balancing Selection, Recombination Rate and Polymorphism: a Commentary. Preprint series: ESI preprints 1334. (http://www.esi.ac.at/preprints/ESI-Preprints.html)


Look also here item "Complex dynamics"