Transcript Simulation Studies of G-matrix Stability and Evolution Stevan J. Arnold Adam G. Jones (Oregon State Univ.) (Texas A&M Univ.) Reinhard Bürger (Univ.

Simulation Studies of G-matrix
Stability and Evolution
Stevan J. Arnold
Adam G. Jones
(Oregon State Univ.)
(Texas A&M Univ.)
Reinhard Bürger
(Univ. Vienna)
Overview
• Describe the rationale for the work.
• Outline the essential features of the
simulation model.
• Describe the main results from five
studies.
Rationale for simulation studies of
G-matrix stability and evolution
• Analytical results limited.
• In applying response to selection and drift
equations on evolutionary timescales,
useful to know the conditions under which
G is likely to be stable vs. unstable.
• Useful to understand the major features of
G evolution.
Overall idea of the simulations
• Set up conditions so that a G-matrix will evolve and equilibrate
under mutation-drift-selection balance.
• Characterize the shape, size and stability of the G-matrix at that
equilibrium.
• Use correlational selection to establish a selective line of least
resistance (45 deg line) with the expectation that mutation and G will
evolve towards alignment with that line.
• Use biologically realistic values for other parameters (mutation rates,
strength of stabilzing selection, effective population size).
• Determine the conditions under which the G-matrix is most and least
stable.
Model details
• Direct Monte Carlo simulation with each gene
and individual specified
• Two traits affected by 50 pleiotropic loci
• Additive inheritance with no dominance or
epistasis
• Allelic effects drawn from a bivariate normal
distribution with means = 0, variances = 0.05,
and mutational correlation rμ = 0.0-0.9
• Mutation rate = 0.0002 per haploid locus
• Environmental effects drawn from a bivariate
normal distribution with mean = 0, variances = 1
Mutation conventions
(b)
0.05 0 
M 
0.05
 0
r  0
Mutational effect on trait 1
Mutational effect on trait 2
Mutational effect on trait 2
(a)
 0.05 0.045
M 

0.045 0.05 
r  0.9
Mutational effect on trait 1
Arnold et al. 2008
More model details
• Discrete generations
• Life cycle: random sampling of breeding pairs
from survivors in preceding generation,
production of offspring (mutation &
recombination), viability selection (Gaussian).
• ‘Variances’ of Gaussian selection function = 0, 9,
49, or 99, with off-diagonal element adjusted so
that rω = 0.0-0.9
• Ne = 342, 683, 1366, or 2731
Selection conventions
0 
 .020
r  0
49 0 

 0 49

Value of trait 1
(b)
50 0 

 0 50
P
Average value of trait 1
(c)
Value of trait 2
 .020
 
 0
Average value of trait 2
Average value of trait 2
Value of trait 2
(a)
 .020  .023

 .023  .020
 
r  0.9
49 44

44 49

Individual
selection
surfaces
Value of trait 1
(d)
Adaptive
landscapes
50 44

44 50
P
Average value of trait 1
Arnold et al. 2008
Estimates of the strength of
stabilizing selection
observations
ofobservations
Numberof
Number
160
120
80
40
0
-160 -120 -80
-40
*
0
*
40
80
* 120
Strength of stabilizing
selection, 2
2
Data from Kingsolver et al. 2001
Simulation runs
• Initial burn-in period of 10,000 generations
• In each run, after burn-in, sample the next
2,000 – 10,000 generations with
calculation of output parameters every
generation
• 20 replicate runs
Measures of G-matrix stability
• Parameterization of the G-matrix: size (Σ =
sum of eigenvalues), eccentricity (ε =
ratio of eigenvalues), and orientation (φ =
angle of leading eigenvector).
• G-matrix stability: average per-generation
change relative to mean (ΔΣ, Δε) or on
original scale (Δφ in degrees).
Three measures of G-matrix stability
Change in size, ΔΣ
Change in eccentricity, Δε
Change in orientation, Δφ
Jones et al. 2003
Overview of simulation studies
• A single trait, stationary AL (Bürger & Lande
1994).
• Two traits, stationary AL (Jones et al. 2003).
• Two traits, moving adaptive peak (Jones et al.
2004).
• Two traits, evolving mutation matrix (Jones et al.
2007).
• Two traits, one way migration between
populations (Guillaume & Whitlock 2007).
• Two traits, fluctuation in orientation of AL (Revell
2007).
• Review of foregoing results (Arnold et al. 2008).
Evolution and stability of G when the
adaptive landscape is stationary: results
• Different aspects of stability react
differently to selection, mutation, and drift.
• The G-matrix evolves in expected ways to
the AL and the pattern of mutation.
Jones et al. 2003
The three stability measures
have different stability profiles
• Orientation: stability in increased by
mutational correlation, correlational
selection, alignment of mutation and
selection, and large Ne
• Eccentricity: stability in increased by
large Ne
• Size: stability in increased by large Ne
Jones et al. 2003
Mutational and selectional correlations stabilize
the orientation of the G-matrix
r
r
μ
Ne = 342
ω11=ω22=49
ω
0
0
0
0.75
0.50
0
0.50
0.75
0.90
0.90
Jones et al. 2003
The evolution of G reflects the patterns of mutation and selection
M P G
200
400
600
800
1000
Generation
1200 1400
1600
Arnold et al. 2008
The Flury hierarchy for G-matrix comparison
eigenvalues
eigenvectors
Equal
same
same
Proportional
proportional
same
CPC
different
same
Unrelated
different
different
Flury 1988, Phillips & Arnold 1999
Conservation of eigenvectors is a
common result in G-matrix comparisons
Experimental
treatments
Sexes
Conspecific
populations
Different
species
Equal
Proportional
Full CPC
Partial CPC
Unrelated
0
10
20
0
10
0
10
20
30
40
0
10
Number of comparisons
Arnold et al. 2008
Stability of G when the orientation of the
adaptive landscape fluctuates
• Fluctuation in orientation of the AL (rω )
has no effect on the stability of G-matrix
size or eccentricity.
• Fluctuation in orientation of the AL (rω )
affects the stability of G-matrix orientation
(larger fluctuations lead to more
instability).
Revell 2007
Evolution and stability of G when the peak of
the adaptive landscape moves at a constant
rate: simulation detail
• Direction of peak movement:
, , or
• Rate of peak movement: 0.008 phenotypic
standard deviations ( ≈ average rate in a
large sample of microevolutionary studies
compiled by Kinnison & Hendry 2001).
Jones et al. 2004
Evolution and stability of G when the peak of
the adaptive landscape moves at a constant
rate: results
• Evolution along a selective line of least
resistance (i.e., along the eigenvector
corresponding to the leading eigenvalue of the
AL) increased stability of the G-matrix
orientation.
• A continuously moving optimum can produce
persistent maladaptation for correlated traits: the
evolving mean never catches up with the moving
optimum.
• G elongates in the direction of peak movement
Jones et al. 2004
Average value of trait 2
Average value of trait 2
Peak movement along a selective line of least
resistance stabilizes the G-matrix
Average value of trait 1
Average value of trait 1
Arnold et al. 2008
The flying kite effect
rω = 0.0
rμ = 0.9
Jones et al. 2004
Evolution and stability of G with migration
between populations: simulation detail
• Life cycle: migraton, reproduction, viability
selection
• One way migration from a mainland pop
(constant N=104) to 5 island pops (each
with constant N=103)
• Island optima situated 5 environmental
standard deviations from the mainland
optimum at angles ranging from gmin to
gmax
• Migration rate varied from 0 to10-2
Guillaume & Whitlock 2007
Mainland→island migration model
islands 1-5
mainland
Guillaume & Whitlock 2007 model
Evolution and stability of G with migration
between populations: results
• Strong migration can affect all aspects of
the G-matrix (size, eccentricity and
orientation).
• Strong migration can stabilize the Gmatrix, especially if peak movement during
island–mainland differentiation is along a
selective line of least resistance.
Guillaume & Whitlock 2007
Effects of strong migration on the G-matrix
m = 0.01
Nm = 100
Guillaume & Whitlock 2007
G-matrix orientation stabilized by
strong migration: time series
rμ=rω=0
island
mainland
Guillaume & Whitlock 2007
Evolution and stability of G when the mutation
matrix evolves: simulation detail
• Each individual has a personal value for
the mutational correlation, rμ
• The value of rμ is determined by 10
additive loci, distinct from the 50 loci that
affect the two phenotypic traits
• rμ is transformed so that it varies between
-1 and +1
• No direct selection on rμ
Jones et al. 2007
Evolution and stability of G when the mutation
matrix evolves: results
• The M-matrix tends to evolve toward
alignment with the AL.
• An evolving M-matrix confers greater
stability on G than does a static mutational
process.
Jones et al. 2007
Individuals vary in the mutational
correlation parameter rμ
(b)
0.05 0 
M 
0.05
 0
r  0
Mutational effect on trait 1
Mutational effect on trait 2
Mutational effect on trait 2
(a)
 0.05 0.045
M 

0.045 0.05 
r  0.9
Mutational effect on trait 1
Mean Mutational Correlation
The M-matrix tends to evolve towards
alignment with the AL
0.6
0.5
0.4
0.3
0.2
0.1
0
15
20
25
30
35
40
45
50
Angle of Correlational Selection
Jones et al. 2007
Conclusions
• Simulation studies have successfully
defined the circumstances under which the
G-matrix is likely to be stable vs. unstable.
• They have also confirmed some
expectations about G-matrix evolution and
revealed new results.
• Simulation studies fill a void by providing a
conceptual guide for using the G-matrix in
various kinds of evolutionary applications.
Ongoing & future work
• Explore consequences of episodic vs. constant preak
movement.
• Assess the consequences of using other, nonGaussian
distributions for allelic effects
• Explore the consequence of dominance
• Explore the consequences of epistasis
Papers cited
•
•
•
•
•
•
•
•
Arnold et al. 2008. Evolution 62: 2451-2461.
Estes & Arnold 2007. Amer. Nat. 169: 227-244.
Hansen & Houle 2008. J. Evol. Biol. 21: 1201-1219.
Jones et al. 2003. Evolution 57: 1747-1760.
Jones et al. 2004. Evolution 58: 1639-1654.
Jones et al. 2007. Evolution 61: 727-745.
Guillaume & Whitlock. 2007. Evolution 61: 2398-2409.
Revell. 2007. Evolution 61: 1857-1872.
Acknowledgements
Russell Lande (University College)
Patrick Phillips (Univ. Oregon)
Suzanne Estes (Portland State Univ.)
Paul Hohenlohe (Oregon State Univ.)
Beverly Ajie (UC, Davis)