Simulation Studies of G-matrix Stability and Evolution

Stevan J. Arnold (Oregon State Univ.)

Adam G. Jones (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

Mutational effect on trait 1

Mut

atio

nal e

ffec

t on

tra

it 2

Mutational effect on trait 1

Mut

atio

nal e

ffec

t on

tra

it 2

(a) (b)

05.00

005.0M

0r 9.0r

05.0045.0

045.005.0M

Mutation conventions

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

Average value of trait 1 Average value of trait 1

Value of trait 1 Value of trait 1

Val

ue

of

trai

t 2

Val

ue

of

trai

t 2

4944

4449

Ave

rag

e va

lue

of

trai

t 2

Ave

rag

e va

lue

of

trai

t 2

500

050P

5044

4450P

020.0

0020.

020.023.

023.020.(a)

(b)

(c)

(d)

0r

490

049

9.0rIndividual selection surfaces

Adaptive landscapes

Selection conventions

Arnold et al. 2008

Estimates of the strength of stabilizing selection

0

40

80

120

160

-160 -120 -80 -40 0 40 80 120

2

Num

ber

of o

bser

vatio

ns

Strength of stabilizing selection, 2

Num

ber

of o

bser

vatio

ns

Strength of stabilizing selection,

Num

ber

of o

bser

vatio

ns

0

40

80

120

160

-160 -120 -80 -40 0 40 80 120

Num

ber

of o

bser

vatio

ns

* **

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).

Change in size, ΔΣ

Change in eccentricity, Δε

Change in orientation, Δφ

Three measures of G-matrix stability

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

rμ rω

0 0

0 0.75

0.50 0

0.50 0.75

0.90 0.90

Mutational and selectional correlations stabilize the orientation of the G-matrix

Jones et al. 2003

Ne = 342ω11=ω22=49

200 400 600 800 1000 1200 1400 1600

Generation GPM

The evolution of G reflects the patterns of mutation and selection

Arnold et al. 2008

Equal

Proportional

CPC

Unrelated

Flury 1988, Phillips & Arnold 1999

eigenvalues eigenvectors

same same

proportional same

different same

different different

The Flury hierarchy for G-matrix comparison

0 10 20 0 10 20 30 40 0 100 10

Experimental treatments

Conspecificpopulations

Different species

Sexes

Equal

Proportional

Full CPC

Partial CPC

Unrelated

Number of comparisons

Arnold et al. 2008

Conservation of eigenvectors is a common result in G-matrix comparisons

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 1 Average value of trait 1

Ave

rage

val

ue o

f tr

ait

2

Ave

rage

val

ue o

f tr

ait

2

Peak movement along a selective line of least resistance stabilizes the G-matrix

Arnold et al. 2008

The flying kite effect

Jones et al. 2004

rμ = 0.9

rω = 0.0

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

g maxg

min

mainland

islands 1-5

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 G-matrix, 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

Guillaume & Whitlock 2007

m = 0.01

Nm = 100

G-matrix orientation stabilized by strong migration: time series

Guillaume & Whitlock 2007

rμ=rω=0

mainland

island

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

Mutational effect on trait 1

Mut

atio

nal e

ffec

t on

tra

it 2

Mutational effect on trait 1

Mut

atio

nal e

ffec

t on

tra

it 2

(a) (b)

05.00

005.0M

0r 9.0r

05.0045.0

045.005.0M

Individuals vary in the mutational correlation parameter rμ

Jones et al. 2007

The M-matrix tends to evolve towards alignment with the AL

0

0.1

0.2

0.3

0.4

0.5

0.6

15 20 25 30 35 40 45 50

Angle of Correlational Selection

Mea

n M

uta

tio

nal

Co

rrel

atio

n

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)