1. Functional Mapping of QTL and Recent Developments
Chang-Xing Ma
Department of Biostatistics
University at Buffalo
Rongling Wu
University of Florida

2. Outline Interval Mapping Functional Mapping Functional Mapping Demo
Recent Developments
Conclusion

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

4. Regression model for estimating the genotypic effect
Phenotype = Genotype + Error
yi = xij ei
xi is the indicator for QTL genotype
j is the mean for genotype j
ei ~ N(0, 2)

5. The genotypes for the trait are not observable and should be predicted from linked neutral molecular markers (M) M1
QTL M2
The genes that lead to the
phenotypic variation are called Quantitative Trait Loci (QTL) M3 .
Our task is to construct a statistical model that connects the QTL genotypes and marker genotypes through observed phenotypes Mm

6. Data Structure n = n22 + n21 + n20 + n12 + n00 + n02 + n01 + n00 1
Parents AA  aa
F Aa
F AA Aa aa
¼ ½ ¼
Data Structure
Subject
Marker (M)
Phenotype
Genotype frequency
M M … Mm
(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 3
Aa(1) Bb(1) y3
¼ ½ ¼ 4 y4
¼ ½ ¼ 5 y5 6
Aa(1) bb(0) y6 7
aa(0) Bb(1) y7 8
aa(0) bb(0) … y8
n = n22 + n21 + n20 + n12 + n00 + n02 + n01 + n00

7. Finite mixture model for estimating genotypic effects
yi ~ p(yi|,) = ¼ f2(yi) + ½ f1(yi) + ¼ f0(yi)
QTL genotype (j) QQ Qq qq
Code
where
fj(yi) is a normal distribution density
with mean j and variance 2
 = (2, 1, 0),  = (2)

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

9. We model the parameters contained within the mixture model using particular functions
QTL genotype frequency: j|i = gj(p)
Mean: j = hj(m)
Variance:  = l(v)
p contains the population genetic parameters
q = (m, v) contains the quantitative genetic parameters

10. j|i 2|22 1|12 r=a+b-2ab F2 QTL genotype frequency: M a Q b N QQ(2)
MM(2)
NN(2)
(1-r)2/4
1/4(1-a)2(1-b)2
1/2a(1-a)b(1-b)
1/4a2b2
Nn(1)
(1-r)r/2
1/2(1-a)2b(1-b)
1/2a(2b2-2b+1)(1-a)
1/2a2b(1-b)
nn(0)
r2/4
1/4(1-a)2b2
1/4a2(1-b)2
Mm(1)
1/2a(1-a)(1-b)2
1/2b(1-2a+2a2)(1-b)
1/2a(1-a)b2
½-(1-r)r
a(1-a)b(1-b)
1/2(2b2-2b+1)(1-2a+2a2)
mm(0)
2|22
1|12

11. Log-
Likelihood
Function

12. The EM algorithm E step M 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

13. Interval Mapping Program
- Type of Study
- Genetic Design

14. Interval Mapping Program
- Data and Options
Names of Markers (optional)
Cumulative Marker
Distance (cM)
Map Function
QTL Searching Step cM
Parameters Here for Simulation Study Only

15. Interval Mapping Program
- Data
Put Markers and Trait Data into box below OR

16. Interval Mapping Program
- Analyze Data
Trait:

17. Interval Mapping Program
- Profile

18. Interval Mapping Program
- Permutation Test
#Tests
Cut off Point at
Level Is
Based on
Tests.

19. Functional Mapping
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

20. Data Structure n = n22 + n21 + n20 + n12 + n00 + n02 + n01 + n00
Parents AA  aa
F Aa
F AA Aa aa
¼ ½ ¼
Subject
Marker (M)
Phenotype (y)
Genotype frequency
… m
… T
QQ(2) Qq(1) qq(0)
¼ ½ ¼ 1 …
y1(1) y1(2) … y1(T) 2
y2(1) y2(2) … y2(T) 3 …
y3(1) y3(2) … y3(T) 4
y4(1) y4(2) … y 4(T) 5
y5(1) y5(2) … y5(T) 6 …
y6(1) y6(2) … y6(T) 7 …
y7(1) y7(2) … y7(T) 8
y8(1) y8(2) … y8(T)
n = n22 + n21 + n20 + n12 + n00 + n02 + n01 + n00

21. The Finite Mixture Model
Observation vector, yi = [yi(1), …, yi(T)] ~ MVN(uj, )
Mean vector, uj = [uj(1), uj(2), …, uj(T)],
(Co)variance matrix,

22. 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

23. Stem diameter growth in poplar trees
Ma,
Casella
&
Wu:
Genetics
2002

24. Logistic Curve of Growth – A Universal Biological Law
Logistic Curve of Growth – A Universal Biological Law (West et al.: Nature 2001)
Logistic Curve of Growth – A Universal Biological Law
Instead of
estimating uj,
we estimate curve
parameters
q = (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 = 9
 10 =  3 = 9
 50 =  3 = 9

25. 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

26. Autoregressive model AR(1)
 =
q = (aj, bj, rj , ρ, σ2)

27. Box-Cox Transformation Differences in growth across ages
Untransformed
Log-transformed
Poplar
data

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

29. 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 ).
In the spring of 1988, a total of 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)

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

31. The dynamic pattern of QTL expression:

32. Functional Mapping - Data Genetic Design: Curve: Marker Place:
Time Point:
Parameters Here for Simulation Study
QTL Position:
Sample Size:
Curve Parameters:
Sigma^2:
Correlation rho:
Search Step:
cM Map Function:

33. Functional Mapping
- Data
Put Markers and Trait Data into box below OR

34. Functional Mapping
- Data Curves

35. Functional Mapping
- Profile
Initiate Values

36. Functional Mapping
- Profile

37. Functional Mapping
- Data Curves

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

39. transform-both-sides logistic model Developmental pattern of genetic effects
Wu,
Ma,
Lin,
Wang
&
Casella:
Biometrics
2004
Timing at which
the QTL is switched on

40. Functional mapping for epistasis in poplar
Wu,
Ma,
Lin
&
Casella
Genetics
2004
QTL 1
QTL 2

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

42. Genotype  environment interaction in rice
Zhao,
Zhu,
Gallo-Meagher
&
Wu:
Genetics
2004

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

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

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

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

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

48. 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

49. 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