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

0.05 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 r = 0.0-0.9 Ne = 342, 683, 1366, or 2731 Selection conventions

r 0 49 0 0 49 Value of trait 1 (b) 50 0 P 0 50 Average value of trait 1 (c)

Value of trait 2 0 .020 .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 P 44 50 Average value of trait 1 Arnold et al. 2008 Estimates of the strength of stabilizing selection Numberof ofobservations observations 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. 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 ( = sum of eigenvalues), eccentricity ( =

ratio of eigenvalues), and orientation ( = angle of leading eigenvector). G-matrix stability: average per-generation change relative to mean (, ) or on original scale ( in degrees). Three measures of G-matrix stability Change in size, Change in eccentricity, Change in orientation, Jones et al. 2003 Overview of simulation studies A single trait, stationary AL (Brger & 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

Ne = 342 11=22=49 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 (r ) has no effect on the stability of G-matrix size or eccentricity. Fluctuation in orientation of the AL (r ) 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

r = 0.0 r = 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 gm g m ax Mainlandisland migration model in 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 islandmainland 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 r=r=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, r The value of r is determined by 10 additive loci, distinct from the 50 loci that affect the two phenotypic traits

r is transformed so that it varies between -1 and +1 No direct selection on r 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 r

(b) 0 0.05 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)