Statistical Challenges for Studying the Evolution of Function Valued Traits

Patrick A. Carter

School of Biological Sciences

Washington State University

Interesting Problems in Evolutionary Biology

• How do organisms evolve?– What processes are involved (selection, genetic drift);

how do they interact?

• What is the role of genetic variation in evolution?– How does genetic variation shape evolution?– How does evolution alter genetic variation?

• What influences the tempo of evolution?– Constraints

• My lab investigates these questions at the both the genetic and physiological levels.

Basic Quantitative Genetics• Most traits show continuous variation

and are influenced by many genes.• If family relationships are known, the

phenotypic variance can be partitioned:– VP = VG+VE

• VG also can be partitioned:– VG = VA + VD + VI

• VA is additive genetic variance (variance in breeding values)– One trait: simple variance– Multiple traits: G matrix– FV Trait: G function

Trait

Fre

qu

ency

S

Interesting Questions about the Evolution of FV Traits

• Have we estimated the phenotypic function in the most meaningful way biologically?

• Has the phenotypic function evolved in response to selection, and in the way we predicted?

• Has the underlying genetic variance-covariance function evolved in response to selection?

Statistical Challenges

• Register the curves in a biologically meaningfully way.

• Compare mean trajectories from different populations with different evolutionary histories.

• Compare G functions from different populations with different evolutionary histories.

Registration

• How do we align curves that contain variation in multiple points of biological interest?

• Growth curves of larval insects:– Hatch– High Growth– Peak = hormonal shift– Wandering phase = loss of body mass– Pupation = end of larval phase

Mean Phenotype by Half-Sib Family

0

50

100

150

200

250

300

350

1 6 11 16 21

Days

Mass (

µg

)M

ass

(µg)

Mass Vs. Age By Half-Sib Family (Warped Ages)

0

0.5

1

1.5

2

2.5

3

0 5 10 15 20 25

Age (Days, Warped)

log(

Mas

s, g

)

Population Comparisons

• Evolutionary biologists frequently want to compare characteristics of populations with different evolutionary histories, especially:– Mean population phenotypes (has selection

changed the trait?)– Genetic variances and covariances (has

selection eroded the variances and covariances as alleles become fixed?)

Statistical Challenges

• Formally compare mean phenotypic curves– Most experimental designs are nested, with

replicate lines nested within experimental selection group. Replicates lines can provide information about genetic drift.

• Formally compare G functions (and their eigenfunctions) from different populations.

Control LinesFemales Females

Selected Lines

Acknowledgements

• Students: Ted Morgan, Steph Kane, Greg Ragland, Drew Reinbold, Kristy Bellinger, Kristen Irwin, Anna Heink

• Collaborators: Fun Value Group• Funding: NSF (DEB 0083638, DEB

0105079, EF 0328594), National Institute of Mathematical and Biological Synthesis

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Age (days)

0

100

200

300

400

Bod

y M

ass

(mic

rogr

ams)

Overall Mean PhenotypeOverall Mean Phenotype

Mean Phenotype by Half-Sib Family

0

50

100

150

200

250

300

350

1 6 11 16 21

Days

Mass (

µg

)M

ass

(µg)

Additive Genetic Covariance FunctionAdditive Genetic Covariance Function

0 2 4 6 8 10 12 14 16 18 20 22

Age (days)

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Eig

enfu

nctio

ns

EF1EF2EF3

EigenfunctionsEigenfunctions

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Age (days)

0

50

100

150

200

250

300

350

400

Bod

y M

ass

( m

icro

gra

ms)

Predicted Response to SelectionCurrent Mean Function

Response to SelectionResponse to Selection

Statistical Challenges

• Formally compare growth curves in selected vs. non-selected populations.

• Compare selection responses: along axis of major variation vs. axis in “nearly null space”.

• Formally compare G functions from different populations.

0 2 4 6 8 10 12 14 16

Generation

2

4

6

8

10

12

14

16

18

Whe

el R

evol

utio

ns (

kilo

met

ers)

Mean of Days 5 + 6 in Females

GS2 Sept. 19, 2001 1:14:49 PM

ControlSelected

5 15 25 35 45 55 65 75 85

Age (weeks)

18

20

22

24

26

28

30

32

34

36

38

40

Bo

dy

Ma

ss

(g

)

Active Control

Active Selection

Sedentary Control

Sedentary Selection

Females

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Age (weeks)

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

Wh

ee

l-R

evo

lutio

ns

(km

)Mean Wheel-Running Activity

Females

Control LinesSelected Lines