Frequency-Dependent Selection on a Polygenic Trait

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Sasha Gimelfarb died on May 11, 2004
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A Multilocus Analysis of Frequency-Dependent
Selection on a Quantitative Trait

• Reinhard Bürger
• Department of Mathematics, University of Vienna

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Frequency-dependent selection

• has been invoked in the explanation of
  evolutionary
• phenomena such as
• Evolution of behavioral traits
• Maintenance of high levels of genetic variation
• Ecological character divergence
• Reproductive isolation and sympatric speciation

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Frequency-Dependent Selection Caused by
Intraspecific Competition
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Intraspecific competition mediated by a
quantitative trait under stabilizing selection

• Bulmer (1974, 1980)
• Slatkin (1979, 1984)
• Christiansen Loeschcke (1980), Loeschcke
  Christiansen (1984)
• Clarke et al (1988), Mani et al (1990)
• Doebeli (1996), Dieckmann Doebeli (1999)
• Matessi, Gimelfarb, Gavrilets (2001)
• Bolnick Doebeli (2003)
• Bürger (2002a,b), Bürger Gimelfarb (2004)

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The General Model

• A randomly mating, diploid population with
  discrete generations and equivalent sexes is
  considered.
• Its size is large enough that random genetic
  drift can be ignored.
• Viability selection acts on a quantitative trait.
• Environmental effects are ignored (in particular,
  GxE interaction). Therefore, the genotypic value
  can be identified with the phenotype.

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Ecological Assumptions

• Fitness is determined by two components
• by stabilizing selection on a quantitative trait,

• and
• by competition among individuals of similar
  trait value,

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• The strength of competition experienced by
• phenotype g ( genotypic value) for a given
• distribution P of phenotypes is

where and VA denote the mean and
(additive genetic) variance of P.
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• If stabilizing selection acts independently of
• competition, the fitness of an individual with
• phenotype g can be written as

where F(N) describes population growth
according to NNF(N). (F may be as in the
discrete logistic, the Ricker, the Beverton-Holt,
the Hassell, or the Maynard Smith model.)
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• For weak selection ( , f a/s
  constant),
• this fitness function is approximated by

where is a compound measure for the
strength of frequency and density dependence
relative to stabilizing selection, i.e.,
.
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Genetic Assumptions

• The trait is determined by additive contributions
  from n diploid loci, i.e., there is neither
  dominance nor epistasis.
• At each locus there are two alleles. The allelic
  effects at locus i are -gi/2 and gi/2. (This is
  general because the scaling constants can be
  absorbed by the position of the optimum and the
  strength of selection.)

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Genetical and Ecological Dynamics

• pi , pi frequencies of gamete i in
  consecutive generations
• Wjk fitness of zygote consisting of
  gametes j, k
• R(jk-gti) probability that a jk-individual
  produces a
• gamete of type i through
  recombination

mean fitness
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Issues and problems to be addressed

• What aspects of genetics and what aspects of
  ecology are relevant, and under what conditions?
• When does FDS have important consequences for the
  genetic structure of a population?
• How does FDS affect the genetic structure?
• How much genetic variation is maintained by this
  kind of FDS?

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Numerical Results from a Statistical Approach
(with A. Gimelfarb)
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Figure, poly, th0
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Figure, poly, th0, 0.75
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Figure poly
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The Weak-Selection orLinkage-Equilibrium
Approximation
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• The structure is the same as in Turelli and
  Barton 2004 (but ). The proofs of the
  results below use their results.
• The population is assumed to be in demographic
  equilibrium, i.e., N and ? are treated as
  constants.
• All models of intraspecific competition and
  stabilizing selection I know have the above
  differential equation as their weak-selection
  approximation.
• Arbitrary population regulation, i.e., with a
  unique stable carrying capacity, is admitted.

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General Conclusions

• The amount and properties of variation maintained
  depend in a nearly threshold-like way on ?, the
  strength of frequency and density dependence
  relative to stabilizing selection.
• This critical value is independent of the number
  of loci and, apparently, also of the linkage map.

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Weak FDS

• If more than two loci contribute to the trait,
  then weak frequency dependence (? lt 1) can
  maintain significantly more genetic variance than
  pure stabilizing selection, but still not much.
  The more loci, the larger this effect.
• FDS of such strength does not induce a
  qualitative change in the equilibrium structure
  relative to pure stabilizing selection.
• Such FDS does not lead to disruptive selection,
  rather, stabilizing selection prevails.

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Strong FDS

• Strong FDS (? gt 1) causes a qualitative change in
  the genetic structure of a population by inducing
  highly polymorphic equilibria in positive linkage
  disequilibrium.
• In parallel, such FDS induces strong disruptive
  selection, the fitness differences between
  phenotypes maintained in the population being
  much larger than under pure stabilizing
  selection.

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Disruptive Selection

• Therefore, disruptive selection should be easy to
  detect empirically whenever FDS is strong enough
  to affect the equilibrium structure
  qualitatively.
• Its strength (the curvature of the fitness
  function at equilibrium) is s(? 1).

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When Genetics Matters

• The degree of polymorphism maintained by strong
  FDS depends on the number of loci and the
  distribution of their effects.
• Models based on popular symmetry assumptions,
  such as equal locus effects or symmetric
  selection, are often not representative (they
  maintain more polymorphism).
• Linkage becomes important only if tight. It
  produces clustering of the phenotypic
  distribution. Otherwise, the LE-approximation
  does a very good job.

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Outlook

• Include assortative mating to study the
  conditions leading to divergence within a
  population (work in progress ? K. Schneider).
• Determine the conditions under which sympatric
  speciation is induced by intraspecific
  competition.