Sasha Gimelfarb died on May 11, 2004

Sasha GimelfarbSasha Gimelfarbdied on May 11, 2004died on May 11, 2004

A Multilocus Analysis of A Multilocus Analysis of Frequency-Dependent Selection Frequency-Dependent Selection

on a on a QuantitativeQuantitative Trait Trait

Reinhard BürgerDepartment of Mathematics, University of Vienna

Frequency-dependent selection

has been invoked in the explanation of evolutionaryphenomena such as

• Evolution of behavioral traits

• Maintenance of high levels of genetic variation

• Ecological character divergence

• Reproductive isolation and sympatric speciation

Frequency-Dependent Selection Caused by

Intraspecific Competition

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)

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.

Ecological Assumptions

Fitness is determined by two components:

• by stabilizing selection on a quantitative trait,

and

• by competition among individuals of similar trait value,

,)(1)( 2 gsgS

.)(1),( 2hgahg

The strength of competition experienced byphenotype g (= genotypic value) for a givendistribution P of phenotypes is

where and VA denote the mean and (additive

genetic) variance of P.

],)[(1

)(),()(

A2 Vgga

hPhgg hP

g

If stabilizing selection acts independently ofcompetition, the fitness of an individual withphenotype g can be written as

),())(()( gSgNFgW P

where F(N) describes population growth accordingto N´=NF(N). (F may be as in the discrete logistic,the Ricker, the Beverton-Holt, the Hassell, or theMaynard Smith model.)

For weak selection ( , f = a/s constant),this fitness function is approximated by

,]))[((

)(1)()(

)(

A2

2

VggNs

gsNFgW

where is a compound measure for the strengthof frequency and density dependence relative tostabilizing selection, i.e., .

)(N

0)( 2 sO

fN NFNNF)(

)(')(

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 -i/2 and i/2. (This is general because the scaling constants can be absorbed by the position of the optimum and the strength of selection.)

Genetical and Ecological Dynamics

pi , pi´ : frequencies of gamete i in consecutive generations

Wjk : fitness of zygote consisting of gametes j, k

R(jki): probability that a jk-individual produces a

gamete of type i through recombination

W

kjkjjki ijkRppWWp

,

1)( '

WNN '

: mean fitness

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?

Numerical Results from a Statistical Approach (with A. Gimelfarb)

Strength of frequency dependence, f

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

Pol

ymor

phis

m

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

n = 2, = 0n = 3, = 0n = 4, = 0

Figure, poly, th=0

Strength of frequency dependence, f

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

Pol

ymor

phis

m

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

n = 2, = 0n = 3, = 0n = 4, = 0n = 2, = 0.25n = 3, = 0.25n = 4, = 0.25

Figure, poly,

th=0, 0.75

Figure: poly

Strength of frequency dependence, f

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Pol

ymor

phis

m

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0n = 2, r = 0.5n = 3, r = 0.5n = 4, r = 0.5n = 2, 0 < r < 0.01n = 3, 0 < r < 0.01n = 4, 0 < r < 0.01

Stabilizing Complicated DisruptiveSelection:

Quad. App.:

The Weak-Selection or

Linkage-Equilibrium Approximation

• 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 η 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.

iv

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.

Weak FDS

• If more than two loci contribute to the trait, then weak frequency dependence ( < 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.

Strong FDS

• Strong FDS ( > 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.

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).

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.

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.