A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan...

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A Simulation-Based Approach to the Evolution of the G- matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ. Vienna)

Transcript of A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan...

Page 1: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

A Simulation-Based Approach to the Evolution of the G-matrix

Adam G. Jones (Texas A&M Univ.)

Stevan J. Arnold (Oregon State Univ.)

Reinhard Bürger (Univ. Vienna)

Page 2: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

β is a vector of directional selection gradients.

z is a vector of trait means.

G is the genetic variance-covariance matrix.

This equation can be extrapolated to reconstruct the history of selection:

It can also be used to predict the future trajectory of the phenotype.

Δz = G β

βT = G-1ΔzT

For this application of quantitative genetics theory to be valid, the estimate of G must be representative of G over the time period in question. G must be stable.

Page 3: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

Stability of G is an important question

- Empirical comparisons of G between populations within a species usually, but not always, produce similar G-matrices.

- Studies at higher taxonomic levels (between species or genera) more often reveal differences among G-matrices.

- Analytical theory cannot guarantee G-matrix stability (Turelli, 1988).

- Analytical theory also cannot guarantee G-matrix instability, and it gives little indication of how much G will change when it is unstable (and how important these changes may be for evolutionary inferences).

Page 4: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

Study Background and Objectives- It may be fair to say that analytical theory has reached

its limit on this topic.

- Stochastic computer models have been used successfully to study several interesting topics in single-trait quantitative genetics (e.g., maintenance of variation, population persistence in a changing environment).

- A decade ago, simulations had been applied sparingly to multivariate evolution and never to the issue of G-matrix stability.

- Our objective was to use stochastic computer models to investigate the stability of G over long periods of evolutionary time.

Page 5: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

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

• Gaussian individual selection surface, with a specified amount of correlational selection and ω = 9 or 49 (usually)

• Each simulation run equilibrated for 10,000 (non-overlapping) generations, followed by several thousand experimental generations

Page 6: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

Methods – The Simulation Model (continued)

Population ofN adults

B * N Progeny> N Survivors

Production of progeny- Monogamy- Mendelian assortment- Mutation, Recombination

Gaussian selection

Random choice of Nindividuals for the nextgeneration of adults

- Start with a population of genetically identical adults, and run for 10,000 generations to reach a mutation-selection-drift equilibrium.

- Impose the desired model of movement of the optimum.- Calculate G-matrix over the next several thousand generations (repeat 20

times).- We focus mainly on average single-generation changes in G, because

we are interested in the effects of model parameters on relative stability of G.

Page 7: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

Mutational effect on trait 1M

utat

iona

l effe

ct o

n tr

ait

2Mutational effect on trait 1

Mut

atio

nal e

ffect

on

trai

t 2

05.00

005.0M

0r 9.0r

05.0045.0

045.005.0M

Mutation conventions

Page 8: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

Value of trait 1 Value of trait 1

Val

ue o

f tr

ait

2

Val

ue o

f tr

ait

2

4944

4449

0r

490

049

9.0r

Individual selection surfaces

Selection conventions

Page 9: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

Visualizing the G-matrix

G = [ ]G11 G12

G12 G22

Trait 1 genetic value

Tra

it 2

gene

tic v

alue

G11

G22

G12

Trait 1 genetic value

Tra

it 2

gene

tic v

alue

eigenvector

eigen

value

eigenvalue

Page 10: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

φ

- We already know that genetic variances can change, and such changes will affect the rate (but not the trajectory) of evolution.

- The interesting question in multivariate evolution is whether the trajectory of evolution is constrained by G.

- Constraints on the trajectory are imposed by the angle of the leading eigenvector, so we focus on the angle φ.

Page 11: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

-90

-60

-30

0

30

60

90

0 Generations 2000

Stationary Optimum (selectional correlation = 0, mutational correlation = 0)

Page 12: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

Stronger correlational selection produces a more stable G-matrix(selectional correlation = 0.75, mutational correlation = 0)

-90

-60

-30

0

30

60

90

1

ω (trait 1) ω (trait 2) r (ω) r (μ) Δφ

49 49 0 0 9.1

49 49 0.25 0 9.2

49 49 0.50 0 8.9

49 49 0.75 0 7.8

49 49 0.85 0 5.4

49 49 0.90 0 4.3

φ

0 Generations 2000

Page 13: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

A high correlation between mutational effects produces stability(selectional correlation = 0, mutational correlation = 0.5)

-90

-60

-30

0

30

60

90

1

ω (trait 1) ω (trait 2) r (ω) r (μ) Δφ

49 49 0 0 9.9

49 49 0 0.25 7.9

49 49 0 0.50 3.6

49 49 0 0.75 1.5

49 49 0 0.85 1.1

49 49 0 0.90 0.9

φ

0 Generations 2000

Page 14: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

When the selection matrix and mutation matrix are aligned, G can be very stable

-90

-60

-30

0

30

60

90

1

-90

-60

-30

0

30

60

90

1

φ

0 Generations 2000

φ

0 Generations 2000

selectional correlation = 0.75, mutational correlation = 0.5

selectional correlation = 0.9, mutational correlation = 0.9

Page 15: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

Misalignment causes instability

0

2

4

6

8

10

12

-0.9 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 0.9

Selection correlation

Mea

n si

ngle

-gen

erat

ion

chan

ge in

G

-mat

rix

angl

e

ru = 0

ru = 0.25

ru = 0.50

ru = 0.75

ru = 0.90

Selectional correlation

Mea

n pe

r-ge

nera

tion

chan

ge in

ang

le

of t

he G

-mat

rix

Page 16: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

A larger population has a more stable G-matrix

Asymmetrical selection intensities or mutational variances produce stability without the need for correlations

ω (trait 1) ω (trait 2) r (ω) r (μ) N (e) Δφ

49 49 0 0 1366 8.8

49 49 0.5 0 1366 6.2

49 49 0 0.5 1366 2.7

49 49 0 0 2731 7.6

49 49 0.5 0 2731 2.3

49 49 0 0.5 2731 1.7

ω (trait 1) ω (trait 2) r (ω) r (μ) α (trait 1) α (trait 2) Δφ

49 49 0 0 0.05 0.05 9.9

49 49 0 0 0.05 0.03 7.2

49 49 0 0 0.05 0.02 3.8

49 49 0 0 0.05 0.01 1.9

99 99 0 0 0.05 0.05 9.6

99 4 0 0 0.05 0.05 3.8

Page 17: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

Conclusions from a Stationary Optimum- Correlational selection increases G-matrix stability, but not very

efficiently.

- Mutational correlations do an excellent job of maintaining stability, and can produce extreme G-matrix stability.

- G-matrices are more stable in large populations, or with asymmetries in trait variances (due to mutation or selection).

- Alignment of mutational and selection matrices increases stability.

- Given the importance of mutations, we need more data on mutational matrices.

- For some suites of characters, the G-matrix is probably very stable over long spans of evolutionary time, while for other it is probably extremely unstable.

Page 18: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

Average value of trait 1

Ave

rage

val

ue o

f tr

ait

2

What happens when the optimum moves?

Page 19: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

In the absence of mutational or selectional correlations, peak movement stabilizes the

orientation of the G-matrix

r rμ Δθ ΔG11 ΔG22

Δrg Δλ1

Δλ2 ΔΣ Δε Δφ

0 0 0.037 0.037 0.026 0.036 0.037 0.027 0.050 9.0

0 0 0.036 0.036 0.024 0.036 0.036 0.027 0.051 3.7

0 0 0.036 0.036 0.025 0.036 0.036 0.027 0.051 3.4

0 0 0.036 0.036 0.024 0.036 0.036 0.026 0.051 3.8

Page 20: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

Change in size, ΔΣ

Change in eccentricity, Δε

Change in orientation, Δφ

Three measures of G-matrix stability

Page 21: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

The three stability measures have different stability profiles

• Size: stability is increased by large Ne

• Eccentricity: stability is increased by large Ne

• Orientation: stability is increased by mutational correlation, correlational selection, alignment of mutation and selection, and large Ne

Page 22: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

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 genetic line of least resistance stabilizes the G-matrix

Page 23: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

Strong genetic correlations can produce a flying-kite effect

Direction of optimum movement

Page 24: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

Reconstruction of net-β

Page 25: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

More realistic models of movement of the optimum

0

3

6

9

0 3 6 9

0

3

6

9

0 3 6 9

(a) Episodic (b) Stochastic

Trait 1 optimum

Tra

it 2

op

timu

m

(every 100 generations)

Page 26: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

200 400 600 800 1000 1200 1400 1600

Generation GM

The evolution of G reflects the patterns of mutation and selection

Page 27: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

Steadily moving optimum

Page 28: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

Episodically moving optimum

Page 29: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

G11

G22 β1 β2

Episodic, 250 generations

G11

G22β1

β2

Steady, every generation

Generation

Ave

rag

e a

dd

itive

ge

ne

tic v

aria

nce

(G

11 o

r G

22)

or

sele

ctio

n

gra

die

nt

(β1 o

r β

2)

Cyclical changes in the genetic variance in response to episodic movement of the optimum

Page 30: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

Cyclical changes in the eccentricity and stability of the angle in response to episodic movement of the optimum

Page 31: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

Steady movement, rω=0, rμ=0 Stochastic, rω=0, rμ=0, σθ=0.02

Staticoptimum

Movingoptimum

Direction of peak movement

2

Effects of steady (or episodic) compared to stochastic peak movement

Page 32: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

Episodic vs. stochastic

0

2

4

6

8

10

-1.00 -0.50 0.00 0.50 1.00

r(µ) = 0 r(µ) = 0.25r(µ) = 0.50r(µ) = 0.75 r(µ) = 0.90

rμrμrμ

0

2

4

6

8

10

-1.00 -0.50 0.00 0.50 1.00

Direction of optimum movement

Episodic movement = smooth movement Stochastic movement

Degree of correlational selection

Per

gen

erat

ion

chan

gein

G a

ngle

Stochastic peak movement destabilizes G under stability-conferring parameter combinations and stabilizes G under destabilizing parameter combinations.

Page 33: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

Episodic and stochastic peak movement increase the risk of population extinction

Model of Peak Movement Parameters of selection and mutation

Mode of and interval between peak movement

(generations)

σ12 = σ2

2 Δθ r= 0.75

rμ = 0

r= 0.75

rμ = 0.5

r= 0.9

rμ = 0.9

r = -0.75

rμ = 0.5

Steady (1) 0 Steady (1) 0 Steady (1) 0

Episodic (100) 0 Episodic (100) 0 ex Episodic (100) 0 ex

Episodic (250) 0 ex ex Episodic (250) 0 ex ex ex ex Episodic (250) 0 ex ex ex

Stochastic (1) 0.01 ex Stochastic (1) 0.01 ex Stochastic (1) 0.01 ex

Stochastic (1) 0.02 ex ex ex Stochastic (1) 0.02 ex ex ex Stochastic (1) 0.02 ex ex ex

Page 34: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

Retrospective selection analysis underestimates β

Data from steady peak movement, but this result is general.Cause: selection causes skewed phenotypic distribution that retards response to selection.

Actual values Estimation from G Estimation from TG

rµ rω Δθ net Δθ1a net Δθ2 net-β1 net-β2 angle net-β1 net-β2 angle net-β1

0 0 28.3 28.3 62.4 60.2 44.0 48.5 47.1 44.2 47.8 0 0 40.0 0.0 85.8 -1.1 -0.7 67.8 -0.5 -0.4 69.3 0 0 28.3 -28.3 61.9 -61.8 -45.0 48.2 -48.9 -45.4 50.1

0.5 0 28.3 28.3 51.2 54.7 46.9 40.5 42.2 46.2 42.1 0.5 0 40.0 0.0 93.7 -20.9 -12.6 76.7 -22.6 -16.4 79.3 0.5 0 28.3 -28.3 80.5 -79.6 -44.7 68.0 -68.9 -45.4 68.7

0 0.75 28.3 28.3 57.9 55.8 43.9 44.3 46.2 46.2 49.9 0 0.75 40.0 0.0 119.8 -60.1 -26.6 90.4 -35.9 -21.7 99.1 0 0.75 28.3 -28.3 109.3 -108.5 -44.8 80.7 -80.9 -45.1 81.0

0.5 0.75 28.3 28.3 46.2 42.8 42.8 34.0 34.0 45.0 32.6 0.5 0.75 40.0 0.0 126.2 -78.1 -31.8 100.5 -59.7 -30.7 99.0 0.5 0.75 28.3 -28.3 123.6 -124.5 -45.2 100.3 -100.4 -45.0 99.2

0.5 -0.75 28.3 28.3 101.4 101.0 44.9 73.2 73.5 45.1 77.7 0.5 -0.75 40.0 0.0 131.6 35.4 15.1 101.5 9.0 5.1 106.9 0.5 -0.75 28.3 -28.3 83.9 -82.3 -44.4 71.9 -69.5 -44.0 68.1

Page 35: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

Conclusions(1) The dynamics of the G-matrix under an episodically or stochastically

moving optimum are similar in many ways to those under a smoothly moving optimum.

(2) Strong correlational selection and mutational correlations promote stability.

(3) Movement of the optimum along genetic lines of least resistance promotes stability.

(4) Alignment of mutation, selection and the G-matrix increase stability.

(5) Movement of the bivariate optimum stabilizes the G-matrix by increasing additive genetic variance in the direction the optimum moves.

(6) Both stochastic and episodic models of peak movement increase the risk of population extinction.

Page 36: A Simulation-Based Approach to the Evolution of the G-matrix Adam G. Jones (Texas A&M Univ.) Stevan J. Arnold (Oregon State Univ.) Reinhard Bürger (Univ.

Conclusions(7) Episodic movement of the optimum results in cycles in the additive genetic

variance, the eccentricity of the G-matrix, and the per-generation stability of the angle.

(8) Stochastic movement of the optimum tempers stabilizing and destabilizing effects of the direction of peak movement on the G-matrix.

(9) Stochastic movement of the optimum increases additive genetic variance in the population relative to a steadily or episodically moving optimum.

(10) Selection skews the phenotypic distribution in a way that increases lag compared to expectations assuming a Gaussian distribution of breeding values. This phenomenon also results in underestimates of net-β.

(11) Many other interesting questions remain to be addressed with simulation-based models.