The distribution of genetic variance across phenotypic space and the response to selection - Mark...

The distribu,on of gene,c variance across phenotypic space and the response to selec,on

Mark Blows

School of Biological Sciences University of Queensland

Nothing would happen without these guys at UQ

•  Katrina McGuigan and Emma Hine –  Long-‐term collaborators on mul,variate responses to selec,on and the gene,c analysis of high dimensional traits

•  Steve Chenoweth, Julie Collet and ScoI Allen –  Collabora,on on the gene,c analysis of gene expression

Framework for today’s talk: Understanding the distribu,on of gene,c variance

How does gene,c covariance (pleiotropy) change the availability of

gene,c variance? (a bit of theory and data)

How does pleiotropy influence the response to selec,on in

small sets of traits? (a selec,on experiment)

How widespread is pleiotropy among small sets of traits?

(some random matrix theory)

What is the phenome-‐wide extent of pleiotropy?

(gene,c analysis of 1000s of gene expression traits)

The geometry of gene,c varia,on and mul,variate evolu,on

A covariance matrix of gene,c rela,onships among

traits

1v 1,2cov … 1,ncov2v 2,ncov

nv

!

"

######

$

%

&&&&&&

G

∆z = Gβ

€

β1β 2β n

#

$

% % % %

&

'

( ( ( (

A vector of the strength of selec,on ac,ng on mul,ple

traits

β

How does it all fit together? Spectral decomposi,on of Lande’s equa,on

ββββ TTT1 λλλ nnn222maxmax ggggggΔz +++== …G

Projec,on of the direc,on of selec,on along the first eigenvector of G, weighted by its eigenvalue

Consequence 1 If gene,c variance is unevenly distributed in G (eigenvalues vary greatly in size), the response to selec,on will o\en be biased

away from β

Consequence 2 The response of individual traits may be in a direc,on opposite to the selec,on applied on them, par,cularly when β is in a direc,on with low gene,c variance

What does the distribu,on of gene,c variance look like for func,onally related traits?

€

1 −0.52 −0.52−0.52 1 −0.45−0.52 −0.45 1

#

$

% % %

&

'

( ( (

Toy example: A nearly singular

G matrix

Time (minutes)8.0 8.5 9.0 9.5 10.0 10.5

30

34

38

42

46

Sig

nal s

treng

th (p

A)

5,9-C24

5,9-C25

9-C259-C26

2-Me-C26

5,9-C27

5,9-C29

2-Me-C28

2-Me-C30

Time (minutes)8.0 8.5 9.0 9.5 10.0 10.5

30

34

38

42

46

Sig

nal s

treng

th (p

A)

5,9-C24

5,9-C25

9-C259-C26

2-Me-C26

5,9-C27

5,9-C29

2-Me-C28

2-Me-C30

99% of the gene,c variance in the 10 Drosophila wing traits explained by 5 gene,cally independent traits

(McGuigan and Blows 2007, Evolu,on)

98% of the gene,c variance in 9 cu,cular hydrocarbons is contained

in 5 significant dimensions (Van Homrigh et al 2007, Current Biology)

High dimensional selec,on experiment: will the en,re phenotypic space respond?

Select along all 8 gene,c eigenvectors of the 8 dimensional

phenotypic space Time (minutes)

8.0 8.5 9.0 9.5 10.0 10.5

30

34

38

42

46

Sig

nal s

treng

th (p

A)

5,9-C24

5,9-C25

9-C259-C26

2-Me-C26

5,9-C27

5,9-C29

2-Me-C28

2-Me-C30

Time (minutes)8.0 8.5 9.0 9.5 10.0 10.5

30

34

38

42

46

Sig

nal s

treng

th (p

A)

5,9-C24

5,9-C25

9-C259-C26

2-Me-C26

5,9-C27

5,9-C29

2-Me-C28

2-Me-C30

Index T1 0.659 T2 0.209 T3 0.052 T4 0.316 T5 0.325 T6 0.528 T7 -‐0.116 T8 0.146

Design of selec,on experiment •  3 replicate popula,ons for each

of the 8 selec,on indices •  Select for 6 genera,ons •  50% trunca,on selec,on •  2 control lines

Selec,on along all eight gene,c eigenvectors

Distribu,on of standing gene,c variance in the base popula,on

✓

✓

✓

✓

✓

?

?

? Hine et al 2014, American Naturalist

Black indicates a trait evolved in the direc,on opposite to its selec,on gradient

95% Bayesian uncertainty intervals

How to compare base G and the response: The realised G matrix

Δz11Δz12Δz13Δz21Δz22Δz23

"

#

$$$$$$$$$

%

&

'''''''''

=

w11 w12 w13 0 0 0

0 w11 0 w12 w13 0

0 0 w11 0 w12 w13

w11 w12 w13 0 0 0

0 w11 0 w12 w13 0

0 0 w11 0 w12 w13

"

#

$$$$$$$$$

%

&

'''''''''

v1cov12cov13v2

cov23v3

"

#

$$$$$$$$$

%

&

'''''''''

Response to selec,on of traits

Example for 3 traits and 2 selec,on indices (popula,ons)

Selec,on weights

Realised gene,c (co)variances

Note the overes,ma,on of the gene,c variance in the base popula,on

Mul,variate gene,c variances successfully predicted responses

Hine et al 2014, American Naturalist

Can we make any generaliza,ons about the spectral distribu,on of gene,c variance?

40 5-‐d G matrices From Pitchers et al (2014)

Yes, gene,c “nearly-‐null” spaces may be common But, how do we know if the exponen,al decline in eigenvalues is more than expected in the absence of gene,c covariance? In other words, what is the null model for mul,variate quan,ta,ve gene,cs?

Recent colla,on of all es,mated G matrices by Pitchers et al 2014, Phil. Trans. R. Soc. B

6

5

4

3

2

1

0

-1

-2

-3

-4

Ge

ne

tic V

aria

nce

1 2 3 4 5Eigenvector

Life History - 2

Morphology - 32

Sexually Selected - 7

61 2 3 4 5

Eigenvector

6

4

2

0

-2

-4

Ge

ne

tic V

aria

nce

Life History - 9

Morphology - 26


10

12

-6

761 2 3 4 5Eigenvector

6

5

4

3

2

1

0

-1

-2

-3

Ge

ne

tic V

aria

nce

7Life History - 4

Morphology - 11


(b)

(a)

(c)

Blows and McGuigan 2015, Molecular Ecology

Random Matrix Theory: The quarter-‐circle law for P matrices

BA

1 2 3 4 5 6 7 8 9 10

Trait

1.0

0.6

0.8

1.2

1.4

1.6

VP

0.5

1.0

1.5

0.00.6 0.8 1.0 1.2 1.4 1.6

Eigenvalues

De

nsity

10 traits drawn from N(0,1), with 250 individuals measured, and the iden,ty matrix as the covariance matrix

The bulk and edge behaviour of eigenvalues of symmetrical matrices follow some surprisingly universal laws

Marchenko-‐Pastur distribu,on

Individual trait

1st eigenvalue


SNP relatedness matrices will conform to this distribu,on

The need for a null model in mul,variate quan,ta,ve gene,cs

€

1v 1,2cov … 1,ncov2v 2,ncov

nv

"

#

$ $ $ $ $ $

%

&

' ' ' ' ' '

1.  10 traits drawn at random from N(0,1), with the iden,ty matrix as the covariance matrix

2.  Experimental design; 50 lines, 5 individuals per line.

3.  Use mul,variate linear model to es,mate among-‐line G

for 200 replica,ons of a 10-‐trait G matrix:

Eigenvector1 2 3 4 5 6 7 8 9 10

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Eig

enva

lue

AB

-5 -4 -3 -2 -1 0 54321 6

600

0

500

300

200

100

400

Fre

quency

TW Statistic

Clearly, there is a need for determining how real data deviate from the random expecta,on for the spectral distribu,on

If, Then,

The leading eigenvalue of G and the Tracy-‐Widom Law

200 leading eigenvalues from the simulated G (10 of 200 significantly deviate from random)

Caveat Analy,cal expressions for the centering and scaling parameters of the TW distribu,on for variance-‐component matrices are unknown at present. Approxima,on method of Saccen, et al 2011 used.

10000 samples from the Tracy-‐Widom distribu,on for the leading eigenvalue

5% cut-‐off value of 0.979

-5 -4 -3 -2 -1 0 54321

600

0

500

300

200

100

400

Fre

quency

TW Statistic


Using the TW distribu,on for real data: 40 5-‐

dimensional G matrices

-5 -4 -3 -2 -1 0 54321 6 7 8

600

0

500

300

200

100

400

Fre

qu

en

cy

70

-4 -3 -2 -1 0 43210

60

50

40

30

20

10

Fre

qu

en

cy

1451cases

BA

40 leading eigenvalues from es,mated G in Pitchers et al (2014) data set

(26 significantly deviate from random expecta,on)

Grey bars: 10000 samples from the TW distribu,on Open bars: 10000 simulated values of the TW sta,s,c for the leading eigenvalues of 5-‐d correla,on matrices with random off-‐diagonal elements

Analy,cal steps: 1.  Simulate the structure of the real

matrices (here, 5-‐d correla,on matrices)

2.  Scale leading eigenvalues to the TW distribu,on to establish scaling parameters

3.  Use scaling parameters to adjust observed leading eigenvalues


What is the extent of gene,c covariance across the phenome?

Grey bars: TW distribu,on Open bars: Simulated 5 traits sets Dark bars: leading eigenvalue from 5-‐d G

Gene,c analysis of gene expression; 30 inbred lines derived from a natural popula,on of D. serrata (data from McGuigan et al 2014, Gene,cs)

Vast majority (95%) of 5-‐trait G had a leading eigenvalue larger than expected by chance

Approach: randomly allocate 8750 expression traits with significant gene,c variance at 5% FDR to one of 1756 5-‐trait sets

-4 -3 -2 -1 0 43210

60

40

20

Fre

quency

TW Statistic-5

80

100

120

Fre

quency

0.0 1.00.80.60.40.2Broad Sense Heritability

0

100

200

300

400

500

BA

What is the extent of muta,onal covariance? Muta,onal pleiotropy in gene expression

8,000

600

500

400

300

200

100

00.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Count

ofT

raits

Among-Line Variance

~

A

B

600

500

400

300

200

100

0

Co

un

to

fT

raits

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Among-Line Variance

Average = 0.344

Average = 0.033

•  Most traits (71%) unaffected by muta,on in 27 genera,ons

•  3385 traits (29%) with non-‐zero VM •  Mean muta,onal heritability of 0.001 (0.1%

of the phenotypic variance generated by muta,on each genera,on)

•  Randomly allocate the 3385 traits to 5-‐trait sets

•  21% of trait sets displayed significant muta,onal gene,c covariance (at 5% FDR)

•  Suggests a muta,on affects 70 traits on average (assuming modules of equal size)

McGuigan et al 2014, Gene,cs

Standing VG

Muta,onal VM

Univariate parameters

Evidence for muta,onal pleiotropy

Gene,c analysis of 41 muta,on accumula,ons lines of D. serrata

Why do we see widespread gene,c covariance among small sets of random traits?

G =

B1,1 B1,2 B1,m

B2,2

Bm,m

!

"

#####

$

%

&&&&&

K =

B1,1 0 0

B2,2

0Bm,m

!

"

#####

$

%

&&&&&

Gnk =

B1,11/2 0 0

B2,21/2 0 0

0Bm,m

1/2 0 0

!

"

######

$

%

&&&&&&

B1,11/2 B2,2

1/2 Bm,m1/2

0 0 0 0 0 0 0

!

"

#####

$

%

&&&&&

Take the 8750 expression traits, arranged in the 5x5 matrices Bi,j. There are too many elements in G to es,mate (3.83 x 107)

Es,mate only the 5x5 principal submatrices along the diagonal of K. Only need to es,mate 223125 (0.6%) elements

Complete an approxima,on of the 8750-‐d G using a geometric mean approach

What does the eigenstructure of BIG G reveal?

Blows et al 2015, American Naturalist

Pleiotropy exists among a large number of traits

500

400

300

200

100

0

-0.36 -0.24 -0.12 0.00 0.240.12Trait Loading on gmax

Fre

qu

en

cy

BA

0 5 10 15 20 25 30 35-15 -10 -50

10

20

2

4

6

8

12

14

16

18

22

Fre

qu

en

cy

Sum of +/- Loadings

Null distribu,on of loadings on the leading eigenvector

Large number of traits influenced in the same direc,on by gmax

Trait contribu,ons to the leading eigenvector (gmax)

Consistent with the ubiquitous presence of correlated responses to selec,on


Gene enrichment analysis of GO terms 500

400

300

200

100

0

-0.36 -0.24 -0.12 0.00 0.240.12Trait Loading on gmax

Fre

qu

en

cy

BA

0 5 10 15 20 25 30 35-15 -10 -50

10

20

2

4

6

8

12

14

16

18

22

Fre

qu

en

cy

Sum of +/- Loadings

100 genes with lowest contribu,ons to gmax

2 GO terms enriched:

transferase ac,vity (GO:0016740) cell surface (GO:0009986)

100 genes with highest contribu,ons to gmax

31 GO terms enriched:

Captured many processes related to regula,on of gene expression, including transcrip,on factors


PLEIOTROPY

1. Reduces the availability of gene,c variance in some dimensions, resul,ng in gene,c nearly-‐null subspaces

3. Gene,c nearly-‐null subspaces likely to be common, but more work in RMT needed

2. Response to selec,on governed by the spectral distribu,on of gene,c variance

4. Muta,onal pleiotropy is widespread among expression traits for selec,on to act upon

5. Pleiotropy can be among a very large number of traits of disparate func,on

Date post:	05-Aug-2015
Category:	Science
Upload:	australian-bioinformatics-network
View:	142 times
Download:	2 times