Disruptive selection on a continuous multilocus trait

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Disruptive selectionon a continuous multi-locus
trait

• Carlo Matessi
• Istituto di Genetica Molecolare - Consiglio
  Nazionale delle Ricerche - Pavia - Italy
• Marco Archetti
• Dept. Biologie, Ecologie et Evolution -
  Université de Fribourg - Switzerland
• Alexander Gimelfarb
• San Francisco - U.S.A.

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Sasha died in San Francisco last May. This
presentation is dedicated to him.
Sasha Gimelfarb
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Food specializations
(Smith T.B., 1987, 1993)
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Predatorial tactics and food types
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Sympatric speciation ?
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Questions about discrete adaptive polymorphisms

• Can they be generated and maintained stably
  throughout evolution by frequency dependent
  disruptive selection?
• What prevents appearance of intermediate
  variation between extreme forms , given multiple
  alleles and loci?
• Under what conditions such polymorphisms lead to
  sympatric speciation?

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The Point of View of Long-Term Evolution

• Adaptive evolution of a population is viewed as
  a succession of transient equilibrium states
  produced by the short term demographic dynamics.
• Transit along this succession is driven by
  mutation, causing the appearance of new types in
  a resident population. Natural selection
  determines whether such mutations are eliminated
  or become established (invasion).
• A long term equilibrium is therefore a state,
  either monomorphic or polymorphic, that cannot be
  invaded by any mutation.

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Frequency dependent disruptive selection

• A continuous trait (strategy) with values in
  -1,1
• Random pairwise contests where y against x gets
  payoff
• v(y,x) 1ay2(ab)xybx2 , 0 lt a lt b
  v(x,x) ? 1

• Evolutionary singularity at x 0 ("PEAST"
  see Christiansen, 1991 or "Branching Point")

• Fitness of y in a population of mean m and
  variance s2 is
• w(y,m,s2) Ev(y,x)y 1ay2(ab)myb(m2s2)
• Mean fitness is therefore
• W 1 (ab)s2
• and is maximized in a population containing only
  the two extreme types -1 and 1 at equal
  frequencies, in which s2 1.
• This fitness representation approximates any
  specific ecological situation causing disruptive
  selection, as long as trait values in the
  population are close to x

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Summary

• Primary trait controlled by two (non additive)
  loci
• simulation of long term dynamics
• outline of possible polymorphic long term
  equilibria
• coevolution of modifiers
• assortment
• recombination
• variability of expression
• likelihood that such modifiers actually evolve

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The genetic model

• Loci A and B with alleles A1, , Am and B1, ,
  Bn (m, n 20), and recombination rate r

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Simulation of Long Term Evolution
1. Start from a monomorphic resident population
of phenotype x, taken at random in -1,1
3. This population, with phenotypic table X ,Y,
is iterated according to the exact, deterministic
two-locus recurrence equations (short term
dynamics), under frequency dependent disruptive
selection and random mating, until equilibrium
is reached.
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Two types of mutations
Pattern mutations. Each element of Y is chosen at
random in -1,1, subject to the limited effect
constraint.
Scale mutations. (i) Choose at random a resident
allele of the mutating locus, e.g., A1. (ii)
Mutant trait values, yj kl , y kl, result
from a linear transformation of the trait values
induced by A1, x1j kl meanyj kl kl
meanx1j kl kl d , d -0.5,0.5 ?
j yj kl - meanyj kl kl c x1j kl - mean
x1j kl kl , c 0,1 or 1,1.5 with
equal probability ? j meany kl kl
meany1 kl kl d y kl - meany kl kl
c y1 kl - meany1 kl kl
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Parameter Values
24 parameter sets considered i.e., all
combinations of a/b 0.1 , 0.9 b 0.1 , 0.5 ,
1 r 0.01 , 0.1 , 0.25 , 0.5
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Results - 50 Pattern and 50 Scale Mutations
r 0.01
r 0.5
a/b 0.1 b 0.1
phenotypic variance
a/b 0.9 b 1
number of mutation events
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Phenotypic Variances at the End
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Phenotypic Distribution at the End
mean of 120 runs 5 replicates for each of the
24 parameter sets
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Maximal Phenotypic Tables

• In 88 cases out of 120, the final population is
  either exactly or approximately at a state
  consisting of 2 alleles per locus, with a
  phenotypic table that contains only the trait
  values -1, 0 , 1, a class of tables that we may
  call maximal.

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Coevolution of modifiers
1. Modifiers of mating behavior, increasing or
decreasing strength of assortative mating with
respect to the primary trait (possibly leading to
sympatric speciation).
2. Modifiers of linkage, increasing or decreasing
the rate of recombination, r, between the A and
B loci (possibly resulting in the formation of a
supergene).
3. Modifiers of expression of the primary trait,
increasing or decreasing the non genetic
variability of the trait (parameter n).
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The simulation procedure
1. With a given probability (e.g., 10), a
mutation affects a locus of the modifier
phenotype instead of the primary trait. 2. With
equal probabilities this mutation increases or
decreases, of a small fixed amount, d, the trait
value, m, of the modifier in the current
population. 3. The dominant eigenvalue, l, of
the linearized recurrence equations for the short
term dynamics of the (rare) mutants is evaluated.
If l gt 1 the mutant invades and the trait value
of the modifier is changed to m' m d.
Otherwise the modifier remains unchanged.
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Model of Assortative Mating(a particular case of
Gavrilets Boake, 1998)

• Given that a female of (primary) trait value x
  has encountered a male of trait value y, the
  probability that she accepts to mate with this
  male is
• p(x,y) exp-S(x-y)2 , S strength of
  assortment , 0 S lt 8.
• When a male is rejected, the female searches for
  an other male, till one is found that is
  suitable. Hence, each female is certain to mate
  and there is no cost of assortment to females.

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Results for Assortment
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Results for Recombination
a/b 0.1 , b 1 , r 0.5
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Variable expression of the primary trait

• Suppose that the trait value, x, is the sum of a
  genotypic value, X, and a random error due to
  various disturbances
• Assume that x has Beta distribution over -1,1,
  with mean X and variance (1-X2)/(1n)

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Results for Variability of Expression
a/b 0.1 b 0.1 r 0.5
Nexp-? 0 lt N 1
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Comparison of invasion eigenvalues
Small mutational step at least 100 steps
required to cover entire range assortment 0
S 1 recombination 0.5 r 0 variability 0
lt N 1
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Comparison of invasion eigenvalues
Large mutational step at least 50 steps required
to cover entire range assortment 0 S
1 recombination 0.5 r 0 variability 0 lt N
1
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Effect of strength of selection
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Conclusions I

• Disruptive selection for a continuous trait
  creates adaptive polymorphisms that are
  intrinsically stable in the long term. In the
  meanwhile it raises the variance, and hence the
  mean fitness of the population.
• Only if the primary trait is controlled by a
  single locus maximal variance is reached and the
  polimorphism consists of two discrete and extreme
  forms ("branching").
• If several loci are involved, polymorphism
  takes instead the form of a continuous
  distribution, the variance of which increases
  with increasing strength of selection and
  decreasing recombination, but stays far below its
  maximum level.
• In these cases there remains selective pressure
  to increase phenotypic variance. Such pressure
  could in principle produce discrete morphs by
  acting on appropriate modifiers that might be
  available to coevolve with the primary trait.

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Conclusions II

• Modifiers that could have such effect in all
  circumstances are (1) increase of variability of
  expression of the primary loci, and (2) reduction
  of recombination. In case (1) the genetic
  contribution to polimorphism is likely to be
  moderate or even negligible. In case (2) the
  support of the polymorphism is fully genetic and
  would appear as a super-gene.
• Also modifiers of assortment could have the same
  effect, leading eventually to sympatric
  speciation, but only if strength of selection is
  high. Otherwise, assortment might not even invade
  random mating, or if invading it might
  precipitate destruction of polymorphism.
• However, none of these modifier effects is
  likely to actually evolve. Any modifier is most
  likely to carry with itself some intrinsic cost,
  inducing a negative density independent selective
  component that, even if quite small, can easily
  overcome the tiny frequency dependent fitness
  advantage carrried by the modifier.
• In this respect, of the three types of modifiers
  considered, modifiers of assortment are the most
  unlikely to evolve, while modifiers of
  variability are the least unlikely.