Evolution of functional traits(Joel Kingsolver, Biology)

• Traits as functions: functional, function-valued, infinite-dimensional

• A primer in evolutionary models:– Variation, inheritance, selection, evolution

• Approaches to analysing functional traits:– understanding genetic variation– Estimating selection– Predicting evolutionary responses & constraints

Examples of functional traits

• Functions of age: – growth trajectories; life history; aging

• Functions of environmental state: – Physiological ‘reaction norms’

• Descriptions of 2-D or 3-D shape, etc

Age-specific mortality rates (Drosophila) Pletcher et al, Genetics (1999)

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80

Weeks on Wheels

0

10

20

30

40

50

60

70

80

90

Wh

ee

l Re

volu

tio

ns

(k

ilom

ete

rs)

GS8 Nov. 20, 2000 2:49:35 PM

ControlSelected

0

0.02

0.04

0.06

0.08

10 15 20 25 30 35 40

Temperature ( C)

Me

an

Gro

wth

Ra

te (

g/g

/h) M. sexta

P. rapae

Evolution of quantitative traits: some basics

• Individual organism:– Phenotype: observable trait with value z– Genotype: genetic ‘type’ (usually inferred)

• Population: – Phenotypic variance, P = G + E– Genetic variance, G

• Evolution = change in mean trait value per generation, z

Evolution, in 3 easy steps

P

Trait value z

Fre

quen

cy

z

Evolution, in 3 easy steps (2)

Trait value z

Fre

quen

cy

z z’

Evolution, in 3 easy steps (3)

Trait value z

Fre

quen

cy

z z’

z

A simple evolutionary model

• Variation and inheritance– Variance: P = G + E

• Selection– Selection gradient: = P-1( z’ - z )

– Also: = d[ln(W)]/dz , where W = mean population fitness

• Evolutionary response z = G

Evolution on an adaptive landscape

Mean phenotype, z

Mea

n fi

tnes

s, W

z = G

z may be:

• a scalar

• a vector

• a function

z(t) = G(t,s) sds

Analysing genetics of functional traits

• Estimating G: an example

• Biological hypotheses about G: eigenfunction analysis

• G and the response to selection

Temperature & caterpillar growth rates: Thermal performance curves (TPCs)

z(t), where t = temperature

0

0.01

0.02

0.03

0.04

10 15 20 25 30 35

Temperature ( C)

Rel

ativ

e G

row

th R

ate

(g/g

/h)

GeneticVar-Cov for RGR

35 -0.214 0.735 1.613 3.947 23.094

29 -1.027 -1.026 3.725 14.393 3.947

23 0.043 -1.099 4.505 3.725 1.613

17 -0.229 3.156 -1.099 -1.026 0.735

11 1.255 -0.229 0.043 -1.027 -0.214

Temp 11 17 23 29 35

aa

11.17.

23.29.

35.11.

17.

23.

29.

35.

05

101520

11.17.

23.29.

35.

Genetic Covariance Function

11.17.

23.29.

35.11.

17.

23.

29.

35.

05

101520

11.17.

23.29.

35.

Full Fit Smooth Fit

Basis = orthogonal polynomials

Analysing genetics of functional traits

• Estimating G: an example

• Biological hypotheses about G: eigenfunction analysis

• G and the response to selection

aa

15 20 25 30 35

A

15 20 25 30 35

Temperature °C

C

15 20 25 30 35

B

Faster-slower

Hotter-colder

Generalist-specialist

aa

15 20 25 30 35

-0.2

-0.1

0.1

0.2 A

0

15 20 25 30 35

-0.2

-0.1

0.1

0.2 B

0

Temperature °C

15 20 25 30 35

-0.2

-0.1

0.1

0.2

0

C

Eig

enfu

ncti

on

Faster-slower

Hotter-colder

Generalist-specialist

aa

Leading and Second Eigenfunctions

15 20 25 30 35

-1

-0.5

0.5

1

15 20 25 30 35

-1

-0.5

0.5

1

Temperature °CTemperature °C

Full Fit Smooth Fit

00

Caterpillar growth rates

Temperature

Per

form

ance

Hotter-Colder II

-4

-2

0

2

4 -4

-2

0

2

4

-0.1-0.05

00.050.1

-4

-2

0

2

4

G-covariance function: Variation in TPC position

First eigenfunction (65%)

-3 -2 -1 0 1 2 3 4

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1

Second eigenfunction (25%)

-3 -2 -1 0 1 2 3 4

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1

Analysing genetics of functional traits

• Estimating G: an example

• Biological hypotheses about G: eigenfunction analysis

• G and the response to selection dssstGtz )(),()(

aa

15 20 25 30 35

-0.2

-0.1

0.1

0.2 A

0

15 20 25 30 35

-0.2

-0.1

0.1

0.2 B

0

Temperature °C

15 20 25 30 35

-0.2

-0.1

0.1

0.2

0

C

Eig

enfu

ncti

on

Faster-slower

Hotter-colder

Generalist-specialist

aa

Leading and Second Eigenfunctions

15 20 25 30 35

-1

-0.5

0.5

1

15 20 25 30 35

-1

-0.5

0.5

1

Temperature °CTemperature °C

Full Fit Smooth Fit

00

aaa

T015 20 25 30 35

-1

-0.5

0.5

1

Evolutionary Constraints: Identifying zero eigenfunctions

Selection and evolutionary response

• Estimating selection, (s): an example

• Predicting evolutionary responses

dssstGtz )(),()(

Selection on caterpillar growth rate TPCs

• Measure z(t) for a sample of individuals in the lab --> estimate P(s,t)

• Measure fitness of those individuals in the field

• Estimate (s) (cubic splines)

Selection on Growth Rate

-0.5

-0.3

-0.1

0.1

0.3

0.5

10 15 20 25 30 35

Temperature ( C)

Sele

cti

on

Gra

die

nt **

Evolutionary Response to Selection

aaa

15 20 25 30 35

-0.2

-0.1

0.1

0.2

0.3

0.4

T0

A

15 20 25 30 35

-15

-10

-5

5

0

z

T

B

15 20 25 30 35

5

10

15

20

25z

z’

T

C

0

Selection

Evolutionary response

Evolutionary change in one generation

Challenges

• Estimation methods for G

• Hypothesis testing of eigenfunctions

• Estimating zero eigenfunctions

• Estimation methods for • Predicting evolutionary responses