Functional Mapping of QTLand Recent DevelopmentsChang-Xing MaDepartment of BiostatisticsUniversity at Buffalocxma@buffalo.edu

Rongling WuUniversity of Florida

OutlineInterval MappingFunctional MappingFunctional Mapping DemoRecent DevelopmentsConclusion

Gene, Allele, Genotype, PhenotypeChromosomes from Father MotherGene A,with twoalleles Aand aGenotype PhenotypeAA 185 100AA 182 104Aa 175 103Aa 171 102aa 155 101aa 152 103 Height IQ

Regression model for estimatingthe genotypic effectPhenotype = Genotype + Error yi = xij + ei xi is the indicator for QTL genotypej is the mean for genotype jei ~ N(0, 2)

The genotypes for the trait are not observable and should be predicted from linked neutral molecular markers (M)M1M2M3MmQTL...Our task is to construct a statistical model that connects the QTL genotypes and marker genotypes through observed phenotypesThe genes that lead to the phenotypic variation are called Quantitative Trait Loci (QTL)

Data Structuren = n22 + n21 + n20 + n12 + n00 + n02 + n01 + n00Parents AA aa

F1 Aa

F2 AA Aa aa

SubjectMarker (M)PhenotypeGenotype frequency M1 M2 Mm (y)QQ(2) Qq(1) qq(0) 1 AA(2) BB(2) Y1 2 AA(2) BB(2) ...y2 3 Aa(1) Bb(1) ...y3 4 Aa(1) Bb(1) ...y4 5 Aa(1) Bb(1) ...y5 6 Aa(1) bb(0) ...y6 7 aa(0) Bb(1) ...y7 8 aa(0) bb(0) y8

Finite mixture model forestimating genotypic effects yi ~ p(yi|,) = f2(yi) + f1(yi) + f0(yi)QTL genotype (j) QQ Qq qq Code 2 1 0fj(yi) is a normal distribution densitywith 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).LikelihoodfunctionbasedonthemixturemodelL(, , |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 functionsp contains the population genetic parametersq = (m, v) contains the quantitative genetic parameters

F2 QTL genotype frequency:M a Q b Nr=a+b-2abj|i2|221|12

FreqQQ(2)Qq(1)qq(0)MM(2)NN(2)(1-r)2/41/4(1-a)2(1-b)21/2a(1-a)b(1-b)1/4a2b2Nn(1)(1-r)r/21/2(1-a)2b(1-b)1/2a(2b2-2b+1)(1-a)1/2a2b(1-b)nn(0)r2/41/4(1-a)2b21/2a(1-a)b(1-b)1/4a2(1-b)2Mm(1)NN(2)(1-r)r/21/2a(1-a)(1-b)21/2b(1-2a+2a2)(1-b)1/2a(1-a)b2Nn(1)-(1-r)ra(1-a)b(1-b)1/2(2b2-2b+1)(1-2a+2a2)a(1-a)b(1-b)nn(0)(1-r)r/21/2a(1-a)b21/2b(1-2a+2a2)(1-b)1/2a(1-a)(1-b)2mm(0)NN(2)r2/41/4a2(1-b)21/2a(1-a)b(1-b)1/4(1-a)2b2Nn(1)(1-r)r/21/2a2b(1-b)1/2a(2b2-2b+1)(1-a)1/2(1-a)2b(1-b)nn(0)(1-r)2/41/4a2b21/2a(1-a)b(1-b)1/4(1-a)2(1-b)2

Log-LikelihoodFunction

The EM algorithmM stepE stepIterations are made between the E and M steps until convergenceCalculate the posterior probability of QTL genotype j for individual i that carries a known marker genotype Solve the log-likelihood equations

- Type of StudyInterval Mapping Program- Genetic Design

- Data and OptionsNames of Markers (optional)Cumulative Marker Distance (cM)Interval Mapping ProgramMap FunctionParameters Here for Simulation Study OnlyQTL Searching Step cM

- DataInterval Mapping ProgramPut Markers and Trait Data into box below OR

- Analyze DataInterval Mapping ProgramTrait:

- ProfileInterval Mapping Program

- Permutation TestInterval Mapping Program#TestsCut off Point atLevel Is Based on Tests.

An innovative model for genetic dissection of complex traits by incorporating mathematical aspects of biological principles into a mapping frameworkFunctional MappingProvides a tool for cutting-edge research at the interplay between gene action and development

Data Structuren = n22 + n21 + n20 + n12 + n00 + n02 + n01 + n00Parents AA aa

F1 Aa

F2 AA Aa aa

SubjectMarker (M)Phenotype (y)Genotype frequency 1 2 m 1 2 TQQ(2) Qq(1) qq(0) 12 2 y1(1) y1(2) y1(T) 22 2 ...y2(1) y2(2) y2(T) 31 1 y3(1) y3(2) y3(T) 41 1 y4(1) y4(2) y 4(T) 51 1 y5(1) y5(2) y5(T) 61 0 y6(1) y6(2) y6(T) 70 1 y7(1) y7(2) y7(T) 80 0 ...y8(1) y8(2) y8(T)

The Finite Mixture ModelObservation vector, yi = [yi(1), , yi(T)] ~ MVN(uj, )Mean vector, uj = [uj(1), uj(2), , uj(T)], (Co)variance matrix,

Modeling the Mean VectorParametric 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

Stem diameter growth in poplar treesMa, Casella& Wu: Genetics2002

Logistic Curve of Growth A Universal Biological LawLogistic 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 curveparametersq = (aj, bj, rj)Number of parameters to be estimated in the mean vector

Time 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 MatrixStationary parametric approachAutoregressive (AR) model

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

Nonparametric approachLengendre function

Autoregressive model AR(1) = q = (aj, bj, rj , , 2)

Box-Cox TransformationDifferences in growth across agesUntransformedLog-transformedPoplardata

EM Algorithm (Ma et al 2002, Genetics)Estimate (aj, bj, rj; rho, sigma^2)

An example of a forest tree The study material used was derived from the triple hybridization of Populus (poplar). A Populus deltoides clone (designated I-69) was used as a female parent to mate with an interspecific P. deltoides x P. nigra clone (designated I-45) as a male parent (WU et al. 1992 ). In the spring of 1988, a total of 450 1-year-old rooted three-way hybrid seedlings were planted at a spacing of 4 x 5 m at a forest farm near Xuchou City, Jiangsu Province, China.The total stem heights and diameters measured at the end of each of 11 growing seasons are used in this example. A genetic linkage map has been constructed using 90 genotypes randomly selected from the 450 hybrids with random amplified polymorphic DNAs (RAPDs) (Yin 2002)

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

The dynamic pattern of QTL expression:

- DataFunctional MappingGenetic Design:Curve:Marker Place:Time Point:Parameters Here for Simulation StudyQTL Position:Curve Parameters:Sample Size:Sigma^2:Correlation rho:Search Step:cM Map Function:

- DataFunctional MappingPut Markers and Trait Data into box below OR

- Data CurvesFunctional Mapping

- ProfileFunctional MappingInitiate Values

- ProfileFunctional Mapping

- Data CurvesFunctional Mapping

Recent Developmentstransform-both-sides logistic model. Wu, Ma, et al Biometrics 2004Multiple genes Epistatic gene-gene interactions. Wu, Ma, et al Genetics 2004Multiple environments Genotype x environment Zhao,Zhu,Gallo-Meagher & Wu: Genetics 2004Multiple traits Trait correlations Zhao et al Biometrics 2005Genetype by Sex interactions - Zhao,Ma,Cheverud &Wu Physiological Genomics 2004

transform-both-sides logistic model Developmental pattern of genetic effectsWu,Ma,Lin,Wang&Casella:Biometrics2004Timing at which the QTL is switched on

Functional mapping for epistasis in poplarQTL 1QTL 2Wu,Ma,Lin&CasellaGenetics2004

Functional mapping for epistasis in poplar The growth curves of four different QTL genotypes for two QTL detected on the same linkage group D16

Genotype environment interaction in riceZhao,Zhu,Gallo-Meagher&Wu:Genetics2004

Plant height growth trajectories in rice affected by QTL in two contrasting environmentsRed: Subtropical HangzhouBlue: Tropical Hainan

QQqq

Functional mapping: Genotype sex interactionZhao,Ma,Cheverud&WuPhysiological Genomics2004

Red: Male miceBlue: Female mice

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

Functional mapping for trait correlationZhao,Hou,Littell&Wu: Biometricssubmitted

Growth trajectories for stem height and diameter affected by a pleiotropic QTLRed: DiameterBlue: Height

QQ Qq

Functional Mapping:toward high-dimensional biologyA new conceptual model for genetic mapping of complex traitsA systems approach for studying sophisticated biological problemsA framework for testing biological hypotheses at the interplay among genetics, development, physiology and biomedicine

Functional Mapping:Simplicity from complexityEstimating 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