Human Chromosomes Male Xy X y Female XX X XX Xy Daughter Son

Gene, Allele, Genotype, Phenotype Chromosomes from Father Mother Gene A, with two alleles A and a Genotype Phenotype AA AA Aa Aa aa aa Height IQ

Regression model for estimating the genotypic effect Phenotype = Genotype + Error y i = x i  j + e i x i is the indicator for QTL genotype  j is the mean for genotype j e i ~ 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 M1M1 M2M2 M3M3 MmMm 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 M 1 M 2 … M m Phenotype (y) QQ( 2 ) Qq( 1 ) qq( 0 ) ¼ ½ ¼ 1 AA(2) BB(2) …y1y1 ¼ ½ ¼ 2 AA(2) BB(2)...y2y2 ¼ ½ ¼ 3 Aa(1) Bb(1)...y3y3 ¼ ½ ¼ 4 Aa(1) Bb(1)...y4y4 ¼ ½ ¼ 5 Aa(1) Bb(1)...y5y5 ¼ ½ ¼ 6 Aa(1) bb(0)...y6y6 ¼ ½ ¼ 7 aa(0) Bb(1)...y7y7 ¼ ½ ¼ 8 aa(0) bb(0) …y8y8 ¼ ½ ¼ Data Structure n = n 22 + n 21 + n 20 + n 12 + n 00 + n 02 + n 01 + n 00 Parents AA  aa F 1 Aa F 2 AA Aa aa ¼ ½ ¼

Finite mixture model for estimating genotypic effects y i ~ p(y i | ,  ) = ¼ f 2 (y i ) + ½ f 1 (y i ) + ¼ f 0 (y i ) QTL genotype (j) QQ Qq qq Code f j (y i ) is a normal distribution density with mean  j and variance  2  = (  2,  1,  0 ),  = (  2 ) where

 j|i is the conditional (prior) probability of QTL genotype j (= 2, 1, 0) given marker genotypes for subject i (= 1, …, n). Likelihood function based on the mixture model L( , ,  |M, y)

QTL genotype frequency :  j|i = g j (  p ) Mean :  j = h j (  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

Log- Likelihood Function

The EM algorithm M step 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 …y 1 (1) y 1 (2) … y 1 (T) ¼ ½ ¼ y 2 (1) y 2 (2) … y 2 (T) ¼ ½ ¼ …y 3 (1) y 3 (2) … y 3 (T) ¼ ½ ¼ …y 4 (1) y 4 (2) … y 4 (T) ¼ ½ ¼ …y 5 (1) y 5 (2) … y 5 (T) ¼ ½ ¼ …y 6 (1) y 6 (2) … y 6 (T) ¼ ½ ¼ …y 7 (1) y 7 (2) … y 7 (T) ¼ ½ ¼ y 8 (1) y 8 (2) … y 8 (T) ¼ ½ ¼ Data Structure n = n 22 + n 21 + n 20 + n 12 + n 00 + n 02 + n 01 + n 00 Parents AA  aa F 1 Aa F 2 AA Aa aa ¼ ½ ¼

The Finite Mixture Model Observation vector, y i = [y i (1), …, y i (T)] ~ MVN(u j,  ) Mean vector, u j = [u j (1), u j (2), …, u j (T)], (Co)variance matrix,

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: Genetics 2002

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, u j = [u j (1), u j (2), …, u j (T)] = [,, …, ] Instead of estimating u j, we estimate curve parameters  m = (a j, b j, r j ) Number of parameters to be estimated in the mean vector Time points Traditional approach Our approach 5 3  5 = 15 3  3 =  10 = 30 3  3 =  50 =  3 = 9

Modeling the Variance Matrix Stationary parametric approach Autoregressive (AR) model Nonstationary parameteric approach Structured antedependence (SAD) model Ornstein-Uhlenbeck (OU) process Nonparametric approach Lengendre function

Differences in growth across ages UntransformedLog-transformed Poplar data

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

Developmental pattern of genetic effects Wu, Ma, Lin, Wang & Casella: Biometrics 2004 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 1 QTL 2 Wu, Ma, Lin & Casella Genetics 2004

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: Genetics 2004

Plant height growth trajectories in rice affected by QTL in two contrasting environments Red: Subtropical Hangzhou Blue: Tropical Hainan QQ qq

Functional mapping: Genotype  sex interaction Zhao, Ma, Cheverud & Wu Physiological Genomics 2004

Red: Male mice Blue: Female mice QQ Qq qq Body weight growth trajectories affected by QTL in male and female mice

Functional mapping for trait correlation Zhao, Hou, Littell & Wu: Biometrics submitted

Growth trajectories for stem height and diameter affected by a pleiotropic QTL Red: Diameter Blue: Height QQ Qq

Statistical Challenges Logistic curves provide a bad fit of the volume data! Stem volume = Coefficient  Height  Diameter 2

Modeling the mean vector using the Legendre function The general form of a Legendre polynomial of order r is given by the sum, where K = r/2 or (r-1)/2 is an integer. We have first few polynomials: P 0 (x) = 1 P 5 (x) = 1/8(63x 5 -70x 3 +15x) P 1 (x) = x P 6 (x) = 1/16(231x x x 2 -5) P 2 (x) = ½(3x 2 -1) P 3 (x) = ½(5x 3 -3x) P 4 (x) = 1/8(35x 4 -30x 2 +3)

New fit by the Legendre function Lin, Hou & Wu: JASA, under revision

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 2004 Lin & Wu: Pharmacogenomics Journal 2005