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)

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.

β_T = G^-1Δz_T

For these applications to be valid, the estimate of G must be representative of G over the time period in question. G must be stable.

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 or instability (Turelli, 1988).
• Stochastic, individual-based simulations provided a solution to the limitations of analytical theory.

Model details

• Direct Monte Carlo simulation with each gene and individual specified

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• Two traits affected by 50 pleiotropic loci

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• Additive inheritance with no dominance or epistasis

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• Allelic effects drawn from a bivariate normal distribution with means = 0, variances = 0.05, and mutational correlation r_μ = 0.0-0.9

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• Mutation rate = 0.0002 per haploid locus

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• Environmental effects drawn from a bivariate normal distribution with mean = 0, variances = 1

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

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

The Simulation Model (continued)

Population of

N adults

B * N Progeny

> N Survivors

Production of progeny

• Monogamy
• Mendelian assortment
• Mutation, Recombination

Gaussian selection

Random choice of N

individuals for the next

generation 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 on average single-generation changes in G, because we are interested in the effects of model parameters on relative stability of G

Mutation conventions

Mutational effect on trait 1

Mutational effect on trait 2

Mutational effect on trait 1

Mutational effect on trait 2

Selection conventions

Value of trait 1

Value of trait 1

Value of trait 2

Value of trait 2

Individual selection surfaces

Visualizing the G-matrix

Trait 1 genetic value

Trait 2 genetic value

G₁₁

G₂₂

G₁₂

Trait 1 genetic value

Trait 2 genetic value

eigenvector

eigenvalue

eigenvalue

φ

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

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

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

φ

0 Generations 2000

Stationary Optimum

(selectional correlation = 0, mutational correlation = 0)

Stronger correlational selection produces a more stable G-matrix

(selectional correlation = 0.75, mutational correlation = 0)

φ

0 Generations 2000

A high correlation between mutational effects produces stability

(selectional correlation = 0, mutational correlation = 0.5)

φ

0 Generations 2000

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

φ

0 Generations 2000

φ

0 Generations 2000

selectional correlation = 0.75, mutational correlation = 0.5

selectional correlation = 0.9, mutational correlation = 0.9

Misalignment causes instability

r_μ

r_μ

r_μ

r_μ

r_μ

Selectional correlation

Mean per-generation change in angle of the G-matrix

A larger population has a more stable G-matrix

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

Conclusions from a Stationary Optimum

• Correlational selection increases G-matrix stability, but not very efficiently unless selection is very strong.
• 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.

Average value of trait 1

Average value of trait 2

What happens when the optimum moves?

In the absence of mutational or selectional correlations, peak movement stabilizes the orientation of the G-matrix

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

Average value of trait 1

Average value of trait 1

Average value of trait 2

Average value of trait 2

Strong genetic correlations can produce a flying-kite effect

Direction of optimum movement 🡪

Reconstruction of net-β

More realistic models of movement of the optimum

(a) Episodic

(b) Stochastic

Trait 1 optimum

Trait 2 optimum

(every 100 generations)

Steadily moving optimum

Episodically moving optimum

G₁₁

G₂₂

β₁

β₂

Episodic, 250 generations

G₁₁

G₂₂

β₁

β₂

Steady, every generation

Generation

Average additive genetic variance (G₁₁ or G₂₂) or selection gradient (β₁ or β₂)

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

Steady movement, r_ω=0, r_μ=0

Stochastic, r_ω=0, r_μ=0, σ_θ=0.02

Static

optimum

Moving

optimum

Direction of peak movement

2

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

Episodic vs. stochastic

Episodic movement = smooth movement

Stochastic movement

Degree of correlational selection

Per generation change

in G angle

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

Episodic and stochastic peak movement increase the risk of population extinction

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.

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• Strong correlational selection and mutational correlations promote stability.

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• Movement of the optimum along genetic lines of least resistance promotes stability.

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• Alignment of mutation, selection and the G-matrix increase stability.

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

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• Both stochastic and episodic models of peak movement increase the risk of population extinction.

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.

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

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

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Using the Simulation Program

• A fully factorial approach is prohibitively time consuming, because the model has too many parameters

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• Think in terms of experimental design, with controls and treatments

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• Start with a core set of parameters and vary one or two parameters at a time

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• The default values provide a reasonable starting place (but possibly remove stochasticity, use a stationary optimum, and increase curvature of the selection surface by setting ω to 9 instead of 49)

If you make the number of loci too small, the program will probably crash because you won’t maintain any genetic variance

The sex ratio is equal and you’re changing the number of females

Each female has 2x this number of offspring

This mutation rate is 0.0002 – too small will result in no variance

How far does the optimum move for trait 1 and trait 2?

What generation (Move Peak Once) or how often (Move Peak Repeatedly) does the optimum move?

These parameters set the graphical window’s properties – experiment

Once – the bivariate optimum moves once at generation “Move peak at generation” by “Trait Optima Shift Units”. Repeatedly – the peak moves every “Move peak at generation” generations by the “Trait Optima Shift”.

If you want some generations of a moving optimum, with no data collected, set them here

Stochasticity in the position of the optimum – set this to zero for starters

Mutational variances for traits 1 and 2 (5 means 0.05)

Smaller values result in a steeper selection surface

How many simulation runs under these parameter values? Not the number of generations, which are set separately.

Mutational correlation – very important – can be between -100 and 100

Selectional correlation – set both values, generally to the same number

Save the contents of the text window

Run the simulation

Set parameters

Check this box to have the program save each run in a separate “.csv” file – uncheck and recheck the box to change filename

Additional Resources

• Tutorial on programming these sorts of models in C++:
• Free pdf at https://pipefishguysite.wordpress.com/jones-lab-publications/
• Hard copy from Amazon

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• Command-line version of the program and source code:
• https://github.com/JonesLabIdaho/GmatrixCommandLine

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