Human Chromosomes

Male Xy

X yFemaleXX X XX Xy Daughter Son

Gene, Allele, Genotype, Phenotype

Chromosomes from Father Mother

Gene A,with twoalleles Aand a

Genotype Phenotype

AA 185 100AA 182 104

Aa 175 103Aa 171 102

aa 155 101aa 152 103

Height IQ

Regression model for estimatingthe genotypic effect

Phenotype = Genotype + Error yi = xij + ei

xi is the indicator for QTL genotypej is the mean for genotype j

ei ~ N(0, 2)

The genotypes for the trait are not observable and should be predicted from linked neutral molecular markers (M)

Uniqueness for our genetic problem

M1

M2

M3

Mm

QTL

.

.

. Our task is to construct a statistical model that connects the QTL genotypes and marker genotypes through observed phenotypes

The genes that lead to the phenotypic variation are called Quantitative Trait Loci (QTL)

Subject

Marker (M) Genotype frequency

M1 M2 … Mm

Phenoty

pe (y)

QQ(2) Qq(1) qq(0)

¼ ½ ¼

1 AA(2) BB(2) … y1

¼ ½ ¼

2 AA(2) BB(2) ... y2

¼ ½ ¼ 3 Aa(1) Bb(1) ... y

3¼ ½ ¼

4 Aa(1) Bb(1) ... y4

¼ ½ ¼5 Aa(1) Bb(1) ... y

5¼ ½ ¼

6 Aa(1) bb(0) ... y6

¼ ½ ¼7 aa(0) Bb(1) ... y

7¼ ½ ¼

8 aa(0) bb(0) … y8

¼ ½ ¼

Data Structure

n = n22 + n21 + n20 + n12 + n00 + n02 + n01 + n00

Parents AA aa

F1 Aa

F2 AA Aa aa ¼ ½ ¼

Finite mixture model forestimating genotypic effects

yi ~ p(yi|,) = ¼ f2(yi) + ½ f1(yi) + ¼ f0(yi)QTL genotype (j) QQ Qq qq Code 2 1 0

fj(yi) is a normal distribution densitywith mean j and variance 2

= (2, 1, 0), = (2)

where

n

iiiiiii

n

iiii

n

iiii

n

iiii

n

iiii

yfyfyf

yfyfyf

yfyfyf

yfyfyf

yfyfyf

1|||

100|00|00|

121|21|21|

122|22|22|

141

21

41

)()()(

)()()(

...

)()()(

)()()(

)()()(

00

21

22

0011222

001122

001122

001122

012

j|i is the conditional (prior) probability of QTL genotype j (= 2, 1, 0) given marker genotypes for subject i (= 1, …, n).

Likelihoodfunctionbasedonthemixturemodel

L(, , |M, y)

QTL genotype frequency: j|i = gj(p)Mean: j = hj(m)Variance: = l(v)

We model the parameters contained within the mixture model using particular functions

p contains the population genetic parameters

q = (m, v) contains the quantitative genetic parameters

n

i jijijqp yfyL

1

2

0| )(log),|,(log MΘΘ

n

iij

qp

ij

ij

ij

n

iij

qp

ij

ijj ijij

ijij

n

ij ijij

ijij

j ijij

ij

qp

yf

yfyf

yf

yf

yf

yf

yf

yL

qp

ij

1

|

|

|

1

|

|

2

0 |

|

12

0 |

|

2

0 |

)(log1

)(log1

)(

)(

)(

)(

)(

)(

)|,(log

|

ΘΘ

ΘΘ

ΘΘ

ΘΘ

Log-LikelihoodFunction

The EM algorithm

2

0 |

||

)(

)(

j ijij

ijijij

yf

yf

M step 0)|,(

yL qp ΘΘ

E step

Iterations are made between the E and M steps until convergence

Calculate the posterior probability of QTL genotype j for individual i that carries a known marker genotype

Solve the log-likelihood equations

Three statistical issues

Modeling mixture proportions, i.e., genotype frequencies at a putative QTL

Modeling the mean vector

Modeling the (co)variance matrix

An innovative model for genetic dissection of complex traits by incorporating mathematical aspects of biological principles into a mapping framework

Functional Mapping

Provides a tool for cutting-edge research at the interplay between gene action and development

Developmental Pattern of Genetic Effects

Subject

Marker (M) Phenotype (y) Genotype frequency

1 2 … m

1 2 … T

QQ(2) Qq(1) qq(0)

¼ ½ ¼

1 2 2 … y1(1) y

1(2) … y

1(T) ¼ ½ ¼

2 2 2 ... y2(1) y

2(2) … y

2(T) ¼ ½ ¼

3 1 1 … y3(1) y

3(2) … y

3(T) ¼ ½ ¼

4 1 1 … y4(1) y

4(2) … y

4(T)

¼ ½ ¼

5 1 1 … y5(1) y

5(2) … y

5(T) ¼ ½ ¼

6 1 0 … y6(1) y

6(2) … y

6(T) ¼ ½ ¼

7 0 1 … y7(1) y

7(2) … y

7(T) ¼ ½ ¼

8 0 0 ... y8(1) y

8(2) … y

8(T) ¼ ½ ¼

Data Structure

n = n22 + n21 + n20 + n12 + n00 + n02 + n01 + n00

Parents AA aa

F1 Aa

F2 AA Aa aa ¼ ½ ¼

The Finite Mixture Model

n

iiiiiii

qp

fff

L

1||| )()()(

),|,,(

yyy

yMΘΘ

0011222

Observation vector, yi = [yi(1), …, yi(T)] ~ MVN(uj, )

Mean vector, uj = [uj(1), uj(2), …, uj(T)],

(Co)variance matrix,

)()2,()1,(

),2()2()1,2(

),1()2,1()1(

2

2

2

TTT

T

T

Σ

Modeling the Mean Vector

• Parametric approach Growth trajectories – Logistic curve HIV dynamics – Bi-exponential function Biological clock – Van Der Pol equation Drug response – Emax model

• Nonparametric approach Lengedre function (orthogonal polynomial) B-spline (Xueli Liu & R. Wu: Genetics, to be submitted)

Stem diameter growth in poplar trees

Ma, Casella& Wu: Genetics2002

Logistic Curve of Growth – A Universal Biological Law

Logistic Curve of Growth – A Universal Biological Law (West et al.: Nature 2001)

Modeling the genotype-dependent mean vector,uj = [uj(1), uj(2), …, uj(T)]

= [ , , …, ]

Instead of estimating uj, we estimate curveparametersm = (aj, bj, rj)jr

j

j

eb

a1 jr

j

j

eb

a2

1 jTr

j

j

eb

a1

Number of parameters to be estimated in the mean vectorTime points Traditional approach Our approach 5 3 5 = 15 3 3 = 910 3 10 = 30 3 3 = 950 3 50 = 150 3 3 = 9

Modeling the Variance Matrix

Stationary parametric approachAutoregressive (AR) model

Nonstationary parameteric approachStructured antedependence (SAD) modelOrnstein-Uhlenbeck (OU) process

Nonparametric approachLengendre function

Differences in growth across agesUntransformed Log-transformed

Poplardata

Functional mapping incorporated by logistic curves and AR(1) model

QTL

Developmental pattern of genetic effects

Wu,Ma,Lin,Wang&Casella:Biometrics2004

Timing at which the QTL is switched on

The implications of functional mapping for high-dimensional biology

• Multiple genes – Epistatic gene-gene interactions

• Multiple environments – Genotype environment interactions

• Multiple traits – Trait correlations• Multiple developmental stages• Complex networks among genes,

products and phenotypes

High-dimensional biology deals with

Functional mapping for epistasis in poplar

QTL 1QTL 2

Wu,Ma,Lin&CasellaGenetics2004

The growth curves of four different QTL genotypes for two QTL detected on the same linkage group D16

Genotype environment interaction in rice

Zhao,Zhu,Gallo-Meagher&Wu:Genetics2004

Plant height growth trajectories in rice affected by QTL in two contrasting environments

Red: Subtropical HangzhouBlue: Tropical Hainan

QQqq

Functional mapping: Genotype sex interaction

Zhao,Ma,Cheverud&WuPhysiological Genomics2004

Red: Male mice

Blue: Female mice

QQQqqq

Body weight growth trajectories affected by QTL in male and female mice

Functional mapping for trait correlation

Zhao,Hou,Littell&Wu: Biometricssubmitted

Growth trajectories for stem height and diameter affected by a pleiotropic QTL

Red: DiameterBlue: Height

QQ Qq

Statistical Challenges

Logistic curves provide a bad fit of the volume data!

Stem volume=CoefficientHeightDiameter2

Modeling the mean vector using the Legendre function

The general form of a Legendre polynomial of order r is given by the sum,

K

k

kr

r

kr x

krkrkkr

xP0

2

)!2()!(!2)!22(

)1()(

where K = r/2 or (r-1)/2 is an integer. We have first few polynomials:

P0(x) = 1 P5(x) = 1/8(63x5-70x3+15x)P1(x) = x P6(x) = 1/16(231x6-315x4+105x2-5)P2(x) = ½(3x2-1)P3(x) = ½(5x3-3x)P4(x) = 1/8(35x4-30x2+3)

New fit by the Legendre function

Lin,Hou&Wu:JASA,underrevision

Functional Mapping:toward high-dimensional biology

• A new conceptual model for genetic mapping of complex traits

• A systems approach for studying sophisticated biological problems

• A framework for testing biological hypotheses at the interplay among genetics, development, physiology and biomedicine

Functional Mapping:Simplicity from complexity

• Estimating fewer biologically meaningful parameters that model the mean vector,

• Modeling the structure of the variance matrix by developing powerful statistical methods, leading to few parameters to be estimated,

• The reduction of dimension increases the power and precision of parameter estimation

Prospects

Teosinte and Maize

Teosinte branched 1(tb1) is found to affect the differentiation in branch architecture from teosinte to maize (John Doebley 2001)

Biomedical breakthroughs in cancer, next?

Single Nucleotide Polymorphisms (SNPs)

no cancer

cancer

Liu, Johnson, Casella & Wu:Genetics 2004Lin & Wu: PharmacogenomicsJournal 2005