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Functional Mapping A statistical model for mapping dynamic genes. Recall: Interval mapping for a univariate trait. Simple regression model for univariate trait. Phenotype = Genotype + Error y i = x i  j + e i. x i is the indicator for QTL genotype

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Functional mapping a statistical model for mapping dynamic genes

Functional MappingA statistical model for mapping dynamic genes

Recall: Interval mapping for a univariate trait

Simple regression model for univariate trait

Phenotype = Genotype + Error

yi = xij + ei

xiis the indicator for QTL genotype

jis the mean for genotype j

ei ~ N(0, 2)

! QTL genotype is unobservable (missing data)

A simulation example f2

Overall trait distribution




A simulation example (F2)

The overall trait distribution is composed of three distributions, each one coming from

one of the three QTL genotypes, QQ, Qq, and qq.

Solution consider a finite mixture model
Solution: consider a finite mixture model

With QQ=m+a, Qq=m+d, qq=m-a

We use finite mixture model for

estimating genotypic effects (F2)

yi ~ p(yi|,) = 2|if2(yi) + 1|if1(yi) + 0|if0(yi)

QTL genotype (j) QQQqqq

Code 210


  • fj(yi) is a normal distribution density

  • with mean jand variance 2

  • = (2, 1, 0)

    = QTL conditional probability given on flanking markers

Human development
Human Development

Robbins 1928, Human Genetics, Yale University Press

Tree growth
Tree growth

Looks mess, but there are simple rules underlying the complexity.

The dynamics of gene expression
The dynamics of gene expression

  • Gene expression displays in a dynamic fashion throughout lifetime.

  • There exist genetic factors that govern the development of an organism involving:

    • Those constantly expressed throughout the lifetime (called deterministic genes)

    • Those periodically expressed (e.g., regulation genes)

  • Also environment factors such as nutrition, light and temperature.

  • We are interested in identifying which gene(s) govern(s) the dynamics of a developmental trait using a procedure called Functional Mapping.

Stem diameter growth in poplar trees

Ma et al. (2002) Genetics

Mouse growth
Mouse growth

A: male; B: female









Developmental Pattern of Genetic Effects

Wu and Lin (2006) Nat. Rev. Genet.

Data structure1
Data Structure

Parents AA aa

F1Aa  Aa

F2AAAa aa

¼ ½ ¼

Mapping methods for dynamic traits
Mapping methods for dynamic traits

  • Traditional approach: treat traits measured at each time point as a univariate trait and do mapping with traditional QTL mapping approaches such as interval or composite interval mapping.

  • Limitations:

    • Single trait model ignores the dynamics of the gene expression change over time, and is too simple without considering the underlying biological developmental principle.

  • A better approach:Incorporate the biological principle into a mapping procedure to understand the dynamics of gene expression using a procedure calledFunctional Mapping(pioneeredby Wu and group).

A general framework pioneered by Dr. Wu and his colleagues, to map QTLs that affect the pattern and form of development in time course

- Ma et al., Genetics 2002

- Wu et al., Genetics 2004 (highlighted in Nature

Reviews Genetics)

- Wu and Lin, Nature Reviews Genetics 2006

While traditional genetic mapping is a combination between classic genetics and statistics, functional mapping combines genetics, statistics and biological principles.

Functional Mapping (FunMap)

Data structure for an F2 population

Phenotype Marker

_______________________________ ________________________________________

Sample y(1) y(2) … y(T) 1 2 … m


1 y11 y21 … yT1 1 1 … 0

2 y12 y22 … yT2 -1 1 … 1

3 y13 y23 … yT3 -1 0 … 1

4 y14 y24 … yT4 1 -1 … 0

5 y15 y25 … yT5 1 1 … -1

6 y16 y26 … yT6 1 0 … -1

7 y17 y27 … yT7 0 -1 … 0

8 y18 y28 … yT8 0 1 … 1

n y1n y2n … yTn 1 0 … -1

·There are nine groups of two-marker genotypes, 22, 21, 20, 12, 11, 10, 02, 01 and 00, with sample sizes n22, n21, …, n00;

·The conditional probabilities of QTL genotypes, QQ (2), Qq (1) and qq (0) given these marker genotypes 2i, 1i, 0i.

Univariate interval mapping

L(y) =

fj(yi) = j=2,1,0 for QQ, Qq, qq

The Lander-Botstein model estimates (2, 1, 0, 2, QTL position)

Multivariate interval mapping

L(y) =

 Vector y = (y1, y2, …, yT)

fj(yi) =


uj = (j1, j2, …, jT)

Residual variance-covariance matrix  =

The unknown parameters: (u2, u1, u0, , QTL position) [3T + T(T-1)/2 +T parameters]

Functional mapping the framework
Functional mapping: the framework

Observed phenotype:yi = [yi(1), …, yi(T)] ~ MVN(uj, )

Mean vector: uj = [μj(1), μj(2), …, μj(T)], j=2,1,0

(Co)variance matrix:

Functional Mapping

Functional mapping does not estimate (u2, u1, u0, ) directly, instead of the biologically meaningful parameters.

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

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

The finite mixture model
The Finite Mixture Model

Three statistical issues:

Modeling mixture proportions, i.e.,

genotype frequencies at a putative QTL

Modeling the mean vector

Modeling the (co)variance matrix

Modeling the developmental mean vector
Modeling the developmental 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)

    Spline techniques

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

Instead of

estimating mj,

we estimate curve


p= (aj, bj, rj)

Modeling the genotype-

dependent mean vector,

uj = [uj(1), uj(2),…, uj(T)]

= [ , , …, ]

Number of parameters to be estimated in the mean vector

Time points Traditional approach Our approach

5 3  5 = 15 3  3 = 9

10 3  10 = 30 3  3 = 9

50 3  50 = 150 3  3 = 9


Modeling the Covariance Matrix

  • Stationary parametric approach

    • Autoregressive (AR) model with log transformation

  • Nonstationary parameteric approach

    • Structured antedependence (SAD) model

    • Ornstein-Uhlenbeck (OU) process

Functional interval mappingL(y) = Vector y = (y1, y2, …, yk)f2(yi) = f1(yi) = f0(yi) = u2 = ( , ,…, )u1 = ( , , …, )u0 = ( , , …, )

The em algorithm
The EM algorithm

E step

Calculate the posterior probability of QTL genotype j

for individual i that carries a known marker genotype

M step

Solve the log-likelihood equations

Iterations are made between the E and M steps until convergence

Em continued
EM continued

The likelihood function:

Statistical derivations
Statistical Derivations

M-step: update the parameters (see Ma et al. 2002, Genetics for details)

Testing qtl effect global test
Testing QTL effect: Global test

  • Instead of testing the mean difference at every time points for different genotypes, we test the difference of the curve parameters.

  • The existence of QTL is tested by

  • H0 means the three mean curves overlap and there is no QTL effect.

  • Likelihood ratio test with permutation to assess significance.

    where the notation “~” and “^” indicate parameters estimated under the null and the alternative hypothesis, respectively.

Testing qtl effect regional test
Testing QTL effect: Regional test

  • Regional test: to test at which time period [t1,t2] the detect QTL triggers an effect, we can test the difference of the area under the curve (AUC) for different QTL genotype, i.e.,


  • Permutation tests can be applied to assess statistical significance.


  • Several real examples are used to show the utility of the functional mapping approach.

  • Application I is about a poplar growth data set.

  • Application II is about a mouse growth data set.

  • Application III is about a rice tiller number growth data set.

Application i a genetic study in poplars
Application I: A Genetic Studyin Poplars

Parents AA aa

F1Aa  AA





Stem diameter growth in poplar trees


Asymptotic growth


Initial growth


Relative growth rate

Ma, Casella & Wu,

Genetics 2002

Differences in growth across ages





Modeling the covariance structure
Modeling the covariance structure

Stationary parametric approach

First-order autoregressive model (AR(1))

Multivariate Box-Cox transformation to stabilize variance (Box and Cox, 1964

Transform-both-side (TBS) technique to reserve the interpretability of growth parameters (Carrol and Ruppert, 1984; Wu et al., 2004). For a log transformation (i.e., =0),

q= (,2)

Functional mapping incorporated by logistic curves and ar 1 model

Results by FunMap

Results by Interval mapping


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

FunMap has higher power to detect the QTL than the traditional interval mapping method does.

Ma, Casella & Wu,

Genetics 2002

Application II:

Mouse Genetic Study

Detecting Growth Genes

Data supplied by Dr. Cheverud at Washington University

Parents AA aa

F1Aa  Aa

F2AAAa aa

¼ ½ ¼

Body Mass Growth for Mouse

510 individuals measured

Over 10 weeks

Functional mapping genetic control of body mass growth in mice
Functional mappingGenetic control of body mass growth in mice

Zhao, Ma, Cheverud & Wu, Physiological Genomics


Application iii functional mapping of pcd qtl
Application III: functional mapping of PCD QTL

  • Rice tiller development is thought to be controlled by genetic factors as well as environments.

  • The development of tiller number growth undergoes a process called programmed cell death (PCD).

Parents AA aa






Joint model for the mean vector
Joint model for the mean vector

  • We developed a joint modeling approach with growth and death phases are modeled by different functions.

  • The growth phase is modeled by logistic growth curve to fit the universal growth law .

  • The dead phase is modeled by orthogonal Legendre function to increase the fitting flexibility.

Cui et al. (2006) Physiological Genomics

Advantages of functional mapping
Advantages of Functional Mapping

Incorporate biological principles of growth and development into genetic mapping, thus, increasing biological relevance of QTL detection

Provide a quantitative framework for hypothesis tests at the interplay between gene action and developmental pattern

- When does a QTL turn on?

- When does a QTL turn off?

- What is the duration of genetic expression of a QTL?

- How does a growth QTL pleiotropically affect developmental events?

The mean-covariance structures are modeled by parsimonious parameters, increasing the precision, robustness and stability of parameter estimation

Functional mapping toward high dimensional biology
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
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