EVOLUTION ALONG SELECTIVE LINES OF LEAST RESISTANCE*

Stevan J. ArnoldOregon State University

*ppt available on Arnold’s website

Overview• A visualization of the selection surface tell us more than directional selection gradients can.• Selection surfaces and inheritance matrices have

major axes (leading eigenvectors).• Peak movement along these axes could account for adaptive radiation.• We can test for different varieties of peak movement using MIPoD, a software package.• A test case using MIPoD:

the evolution of vertebral numbers in garter snakes.• Conclusions

Directional selection gradients and what they tell us

2211 zzw

1. Suppose you have data on the fitness (w) of each individual in a sample and measurements of values for two traits (z1 and z2) .

2. You can fit a planar selection surface to the data, which has two regression slopes, β1 and β2:

3. The two slopes are called directional selection gradients. They measure the force of directional selection and can be used to predict the change in the trait means from one generation to the next: e.g.,

2121111 GGz

Lande & Arnold 1983

Stabilizing selection gradients and what they tell us

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1. You can fit a curved (quadratic) selection surface to the data using a slightly more complicated model:

2. γ11 and γ22 measure the force of stabilizing (disruptive) selection and

are called stabilizing selection gradients.

3. γ12 measures the force of correlational selection and is called a

correlational selection gradient.

4. The two kinds of selection gradients can be used to predict how muchthe inheritance matrix, G, is changed by selection (within a generation):

GGG Ts )(

Lande & Arnold 1983

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

020.0

0020.

020.023.

023.020.(a)

(b)

(c)

(d)

0r9.0r

500

050P

5044

4450P

Individual selection surfaces

Adaptive landscapes

Selection surfaces and adaptive landscapes have a major axis, ωmax

Arnold et al. 2008

ωmax

490

049

The G-matrix also has a major axis, gmax

Arnold et al. 2008

gmax

Average value of trait 1

Ave

rage

val

ue o

f tr

ait

2

Peak movement along ωmax could account for correlated evolution: how can we test for it?

Arnold et al. 2008

MIPoD: Microevolutionary Inference from

Patterns of Divergence

P. A. Hohenlohe & S. J. Arnold

American Naturalist March 2008Software available online

MIPoD: what you can get

Input:• phylogeny• trait values• selection surface (≥1)• G-matrix (≥1)

• Ne

Output:• Test for adaptive,

correlated evolution• Tests for diversifying and

stabilizing selection• Tests for evolution along

genetic lines of least resistance

• Tests for evolution along selective lines of least resistance

neutral process model

A test case using MIPoDThe evolution of vertebral numbers in

garter snakes: a little background

bodytail

Phylogeny of garter snake species based on four mitochondrial genes; vertebral counts on museum specimens

de Queiroz et al. 2002

body tail vertebral counts

190K generations

4.5 Mya ≈ 900,000 generations ago

Observe correlated evolution of body and tail vertebral numbers in garter snakes

50

60

70

80

90

100

110

120 130 140 150 160 170 180

body vertebrae

tail

ve

rte

bra

e

Hohenlohe & Arnold 2008

Correlated evolution: described with a 95%confidence ellipse with a major axis, dmax

50

60

70

80

90

100

110

120 130 140 150 160 170 180

body vertebrae

tail

ve

rte

bra

e

dmax

Hohenlohe & Arnold 2008

An adaptive landscape vision of the radiation: a population close to its adaptive

peak

50

60

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90

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110

120 130 140 150 160 170 180

body vertebrae

tail

ve

rte

bra

e

ωmax

An adaptive landscape vision of the radiation:peak movement principally along a selective line of

least resistance

50

60

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90

100

110

120 130 140 150 160 170 180

body vertebrae

tail

ve

rte

bra

e

ωmax

Arnold et al. 2001

Vertebral numbers may be an adaptation to vegetation density

Jayne 1988, Kelley et al. 1994

MIPoD

Input:• phylogeny of garter snake species • mean numbers of body and tail vertebrae• selection surfaces (2)• G-matrices (3)

• Ne estimates

Output: uses a neutral model to assess the importance and kind of selection

Hohenlohe & Arnold 2008

Arnold 1988

+1σ +2σ-1σ-2σ 0

+1σ

+2σ

-1σ

-2σ

0

body vertebrae

tail

vert

ebra

e

selective line of least resistance

ωmax

Field growth rate as a function of vertebral numbers

+1σ +2σ-1σ 0-2σ

+1σ

+2σ

-1σ

-2σ

0

body vertebrae

tail

vert

ebra

e

ωmax

selective line of least resistance

Arnold & Bennett 1988

Crawling speed as a function of vertebral numbers

165 175

65

75

85

95

155

body vertebrae

tail

vert

ebra

e

Humboldt T. elegans

Lassen T. elegans

Similar G-matrices in three poplations, two species

Dohm & Garland 1993, Phillips & Arnold 1999

T. sirtalis

165 175

65

75

85

95

155

body vertebrae

tail

vert

ebra

e

Humboldt T. elegans

Lassen T. elegans

Schluter’s conjecture: population differentiation occurs along a genetic line of least resistance, gmax

Similar G-matrices in three poplations, two species

Dohm & Garland 1993, Phillips & Arnold 1999

gmax

T. sirtalis

Estimates of Ne for two species from microsatellite data: average Ne ≈ 500

Manier & Arnold 2005

T. elegans T. sirtalis

Neutral model for a single trait: specifies the distribution of the trait means as replicate lineages

diverge

• Trait means normally distributed with variance proportional to elapsed time, t, and genetic variance, G, and

• inversely proportional to Ne

Lande 1976mean body vertebrae

Pro

babi

lity

t=200

t=1,000

t=5,000

t=20,000 generations

h2 = 0.4Ne = 1000

Neutral model for two traits: as replicate populationsdiverge, the cloud of trait means is bivariate normal• Size: proportional to elapsed time and the size of the average

G-matrix , inversely proportional to Ne

• Shape: same as the average G-matrix

• Orientation: same as the average G-matrix, dmax = gmax

Lande 1979

t=200

t=1,000

t=5,000 generations

body vertebrae

tail

vert

ebra

e

Neutral model: equation format

• One trait, replicate lineages

• Multiple traits, replicate lineages

• Multiple traits, lineages on a phylogeny

• D(t) = G(t/Ne )

• D(t) = G(t/Ne )

• A(t) = G(T/Ne )

Neutral model: specifies a trait distribution at time t

• Trait means are normally distributed with mean μ and variance-covariance A

• Using that probability, we can write a likelihood expression

• Using that expression, we can test hypotheses with likelihood ratio tests

A

AP

mxn

T

)2(

)]())(2/1(exp[)(

1

Neutral model: specifies a trait distribution at time t

• Trait means are normally distributed with mean μ and variance-covariance GT/Ne

• Using that probability, we can write a likelihood expression

• Using that expression, we can test hypotheses with likelihood ratio tests

emxn

eT

NGT

NGTP

/()2(

)]()/())(2/1(exp[)(

1

Hypothesis testing in the MIPoD maximum likelihood framework

bold = parameters estimated by maximum likelihood (95% confidence interval)

Size: we observe too little divergence

Implication: some force (e.g., stabilizing selection) has constrained divergence

P < 0.0001

body vertebrae

tail

vert

ebra

e

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

Hohenlohe & Arnold 2008

Shape:divergence is more elliptical than we expect

Implication: the restraining force acts more strongly along PCII than along PCI

P = 0.0122

body vertebrae

tail

vert

ebra

e

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

Hohenlohe & Arnold 2008

Orientation: the main axis of divergence is tilted down more than we expect

Implication: the main axis of divergence is not a genetic line of least resistance

P = 0.0001

body vertebrae

tail

vert

ebra

e

d max

g max

Hohenlohe & Arnold 2008

Divergence occurs along a selective line of least resistance

Implication: adaptive peaks predominantly move along a selective line of least resistance

Yes, P = 0.2638

No, P = 0.0003

tail

vert

ebra

e

body vertebrae

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

max (growth)

max

(spe

ed)

d max

Hohenlohe & Arnold 2008

Arnold 1988

+1σ +2σ-1σ-2σ 0

+1σ

+2σ

-1σ

-2σ

0

body vertebrae

tail

vert

ebra

e

Field growth rate as a function of vertebral numbers

ωmax

coincideswith dmax

An adaptive landscape vision of the radiation:peaks move along a selective line of least

resistance in the garter snake case

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120 130 140 150 160 170 180

body vertebrae

tail

ve

rte

bra

e

Hohenlohe & Arnold 2008

General conclusions

• Using estimates of the selection surface, the G-matrix, Ne , and a phylogeny enables us to visualize the adaptive landscape and to assess the role that it plays in adaptive radiation.

• Need empirical tests for homogeneity of selection surfaces.

• Need a ML hypothesis testing framework that explicitly incorporates a model of peak movement.

Acknowledgements

Lynne Houck (Oregon State Univ.)Russell Lande (Imperial College)Albert Bennett (UC, Irvine)Charles Peterson (Idaho State Univ.)Patrick Phillips (Univ. Oregon)Katherine Kelly (Ohio Univ.)Jean Gladstone (Univ. Chicago)John Avise (UC, Irvine)Michael Alfaro (UCLA)Michael Pfrender (Univ. Notre Dame)Mollie Manier (Syracuse Univ.)Anne Bronikowski (Iowa State Univ.)Brittany Barker (Univ. New Mexico)

Adam Jones (Texas A&M Univ.)Reinhard Bürger (Univ. Vienna)Suzanne Estes (Portland State Univ.)Paul Hohenlohe (Oregon State Univ.)Beverly Ajie (UC, Davis)Josef Uyeda (Oregon State Univ.)

References* • Lande & Arnold 1983 Evolution 37: 1210-1226.• Arnold et al. 2008 Evolution 62: 2451-2461.• Hohenlohe & Arnold 2008 Am Nat 171: 366-385.• de Queiroz et al. 2002 Mol Phylo Evol 22:315-329.• Estes & Arnold 2007 Am Nat 169: 227-244.• Arnold et al. 2001 Genetica 112-113:9-32.• Jayne 1988• Kelley et al. 1994 Func Ecol 11:189-198.• Arnold 1988 in Proc. 2nd Internat Conf Quant Genetics• Arnold & Bennett 1988 Biol J Linn Soc 34:175-190.• Phillips & Arnold 1999 Evolution 43:1209-1222.• Dohm & Garland 1993 Copeia 1993: 987-1002.• Manier et al. 2007 J Evol Biol 20:1705-1719. • Lande 1976 Evolution 30:314-334.• Lande 1979 Evolution 33: 402-416.•Arnold & Phillips 1999 Evolution 43: 1223-__________________________________________________* Many are available as pdfs on Arnold’s website