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Functional Mapping A statistical model for mapping dynamic genes

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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 = xij + ei

xiis the indicator for QTL genotype

jis the mean for genotype j

ei ~ N(0, 2)

! QTL genotype is unobservable (missing data)

Overall trait distribution

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

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

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

where

- fj(yi) is a normal distribution density
- with mean jand variance 2
- = (2, 1, 0)
= QTL conditional probability given on flanking markers

Robbins 1928, Human Genetics, Yale University Press

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

- 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

A: male; B: female

Developmental Pattern of Genetic Effects

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

Parents AA aa

F1Aa Aa

F2AAAa aa

¼ ½ ¼

- 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

PhenotypeMarker

_______________________________________________________________________

Sampley(1)y(2)…y(T)12…m

_____________________________________________________________________________________

1y11y21…yT111…0

2y12y22…yT2-11…1

3y13y23…yT3-10…1

4y14y24…yT41-1…0

5y15y25…yT511…-1

6y16y26…yT610…-1

7y17y27…yT70-1…0

8y18y28…yT801…1

ny1ny2n…yTn10…-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) =

Vectors

uj = (j1, j2, …, jT)

Residual variance-covariance matrix =

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

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

Three statistical issues:

Modeling mixture proportions, i.e.,

genotype frequencies at a putative QTL

Modeling the mean vector

Modeling the (co)variance matrix

- 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

Example: Stem diameter growth in poplar trees

Ma, et al.

Genetics

2002

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

Instead of

estimating mj,

we estimate curve

parameters

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 = ( , , …, )

Estimation

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

The likelihood function:

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

- 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.

- 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.,
where

- 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.

Parents AA aa

F1Aa AA

BC AAAa

½½

Genetic

design

Stem diameter growth in poplar trees

a:

Asymptotic growth

b:

Initial growth

r:

Relative growth rate

Ma, Casella & Wu,

Genetics 2002

Differences in growth across ages

Untransformed

Log-transformed

Poplar

data

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)

Results by FunMap

Results by Interval mapping

QTL

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

Zhao, Ma, Cheverud & Wu, Physiological Genomics

2004

- 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

F1Aa

DH AAaa

½½

Genetic

design

- 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

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

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

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