Joint Analysis of Multiple Gene Expression Traits to Map Expression Quantitative Trait Loci. Jessica Mendes Maia

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1 ABSTRACT MAIA, JESSICA M. Joint Analysis of Multiple Gene Expression Traits to Map Expression Quantitative Trait Loci. (Under the direction of Professor Zhao-Bang Zeng). The goal of this dissertation is to address the issue of how to meaningfully find quantitative trait loci (QTL) for correlated traits. It has been shown in the literature that a joint QTL analysis of multiple traits can have more power and be more precise than single trait QTL analysis when traits are correlated. Phenotypic correlation arises from environmental correlation, genetic correlation, or both. We wish to characterize the extent of the genetic correlation among traits. First, we use a canonical transformation, in the form of principal component analysis, to combine many correlated traits into one, and apply single trait QTL analysis to it. We analyzed two different data sets: one from Saccharomyces cerevisiae, and another from eucalyptus. The traits analyzed in both data sets were gene expression levels generated in microarray experiments. Subsequently, we implemented a novel multiple trait mapping method based on Multiple Interval Mapping to functionally related clusters previously studied in Saccharomyces cerevisiae. Treating RNA abundance as a phenotypic trait, we quantified the extent of the phenotypic variance due to genetic variance, and found additional QTL, previously undetected, which were functionally related to the clusters being studied. The last part of our research contains a study of QTL for individual amino acid biosynthetic pathways of Saccharomyces cerevisiae. In the first part of this chapter, we look at the QTL topology for all individual amino acid biosynthetic pathways, finding a major transcriptional regulatory region for traits in these pathways. In the second part, we look at the QTL topology of some individual amino acid biosynthetic pathways in detail, paying close attention to the pleiotropic QTL in each of them.

2 Joint Analysis of Multiple Gene Expression Traits to Map Expression Quantitative Trait Loci by Jessica Mendes Maia A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Bioinformatics Raleigh, North Carolina 2007 Approved By: Dr. Trudy Mackay Dr. Jung-Ying Tzeng Dr. Zhao-Bang Zeng Chair of Advisory Committee Dr. Dahlia Nielsen

3 ii Dedication To Karthik

4 iii Biography Jessica Maia was born in Long Beach, California. She grew up in Brazil, in the small historic town of Mariana, in the state of Minas Gerais. She has two younger brothers, Eric and Alexandre. In 2005, she married Karthik Sundaramoorthy.

5 iv Acknowledgements I would like to thank my advisor, Dr. Zhao-Bang Zeng, and the members of my committee: Dahlia Nielsen, Trudy MacKay, and Jung-Ying Tzeng for fruitful discussions and academic direction which lead me to complete this dissertation.

6 v Contents List of Figures List of Tables ix xi 1 Introduction Foreword QTL Analysis Review Genetic Maps QTL Studies Yeast Expression QTL Studies Eucalyptus Expression QTL Studies Review of Quantitative Trait Locus Analysis Methods Single Trait QTL Mapping Methods Single Marker Analysis Interval Mapping Composite Interval Mapping Multiple Interval Mapping Threshold for QTL detection Thesis chapters References

7 vi 2 Using Principal Component Analysis for Expression Quantitative Trait Locus Mapping Abstract Introduction Materials and Methods Expression Data Sets Linkage map Principal component Analysis Clustering Analysis Gene annotation QTL Analysis Eucalyptus Results Clustering and QTL Analyses Eucalyptus Cluster Annotation Eucalyptus Principal Component Analysis Results Yeast Results Discussion Acknowledgements References Appendix Multiple Trait Multiple Interval Mapping Abstract Introduction MT-MIM Model Likelihood Parameter Estimation

8 vii 3.5 MT-MIM Strategy Model Selection Hypothesis testing Data Analysis Cluster E Cluster G Cluster H Discussion Materials and Methods Linkage map Data Missing Markers QTL Analysis Acknowledgements References Appendix Genetic Correlation and Heritability Cluster E - Genetic Correlations Cluster G - Genetic Correlations Cluster H - Genetic Correlations Multiple Trait Multiple Interval Mapping for Pathway Analysis in Yeast Abstract Introduction QTL Analysis Distribution of QTL per Pathway Shared QTL per Pathway

9 viii Leucine Pathway Isoleucine Pathway Arginine Pathway Lysine Pathway Methionine Pathway The Role of GCN Phenotypic and Genetic Correlation Discussion Materials and Methods Data Pathways QTL Analysis References Appendix Pathway Gene Fuction Genes in Chromosome III Hotspot

10 ix List of Figures Figure 1.1 Eucalyptus cross design Figure 2.1 QTL locations for all eucalyptus traits Figure 2.2 QTL locations for all eucalyptus genes Figure 2.3 QTL locations for principal components of all eucalyptus genes.. 56 Figure 2.4 QTL location for the first 12 principal components of all genes Figure 2.5 QTL locations for genes in cluster G Figure 2.6 QTL locations for genes in cluster G Figure 2.7 QTL locations for genes in cluster G Figure 2.8 QTL locations for genes in cluster G Figure 2.9 QTL locations for principal components of cluster G Figure 2.10 QTL locations for genes in cluster G Figure 2.11 QTL locations for genes in cluster G Figure 2.12 QTL location for principal components of cluster G Figure 2.13 QTL locations for genes in cluster G Figure 2.14 QTL location for principal components of cluster G Figure 2.15 QTL locations for genes in cluster G Figure 2.16 QTL location for principal components of cluster G Figure 2.17 QTL locations for genes in cluster G Figure 2.18 QTL location for principal component 6 of cluster G Figure 2.19 QTL locations for genes in cluster G

11 x Figure 2.20 QTL location for principal component 7 of cluster G Figure 2.21 QTL locations for genes in cluster G Figure 2.22 QTL location for principal components of cluster G Figure 2.23 QTL locations for genes in cluster G Figure 4.1 Distribution of QTL per chromosome Figure 4.2 Length and position of pathway QTL in chromosome III Figure 4.3 Phenotypic correlations Figure 4.4 Genetic correlations for pathway traits Figure 4.5 Genetic to phenotypic variance ratio for traits in each pathway

12 xi List of Tables Table 1.1 Backcross QTL and marker genotype frequencies and effects Table 1.2 QTL genotype probabilities given three marker genotype classes Table 1.3 QTL genotype probabilities given two marker genotype classes Table 2.1 Distribution of QTL for each eucalyptus cluster Table 2.2 Principal components per cluster with at least one QTL Table 2.3 Principal component QTL of traits with QTL located in a particular linkage group Table 2.4 Cluster B QTL location Table 2.5 Cluster E QTL location Table 2.6 Cluster F QTL location Table 2.7 Cluster H QTL location Table 3.1 Example: QTL positions Table 3.2 Example (continued): close linkage model Table 3.3 Example (continued): pleiotropic model Table 3.4 MIM effects - cluster E Table 3.5 MT-MIM cluster E initial model Table 3.6 Cluster E MT-MIM and MIM additive effects Table 3.7 Cluster E heritabilities Table 3.8 MIM effects - cluster G

13 xii Table 3.9 MT-MIM cluster G initial model Table 3.10 Cluster G MT-MIM and MIM additive effects Table 3.11 Cluster G heritabilities Table 3.12 MIM H cluster Table 3.13 Cluster H initial MT-MIM model Table 3.14 Cluster H - MT-MIM and MIM additive effects Table 3.15 Cluster H heritabilities Table 3.16 Cluster E genetic correlations Table 3.17 Cluster G genetic correlations - Part I Table 3.18 Cluster G genetic correlations - Part II Table 3.19 Cluster H genetic correlations - Part I Table 3.20 Cluster H genetic correlations - Part II Table 4.1 Distribution of QTL per pathway Table 4.2 Percentage and number of QTL per chromosome Table 4.3 Number of shared QTL per pathway

14 1 Chapter 1 Introduction 1.1 Foreword Recently, quantitative trait locus analysis has been applied to microarray experiments, treating RNA transcript abundance as a phenotypic trait (Brem et al., 2002; Yvert et al. 2003; Schadt et al. 2003; Kirst et al. 2005; Li et al. 2006). A quantitative trait locus (QTL) for a gene expression trait is a regulatory region which has a polymorphism in the segregating population. Expression QTL studies, which examine thousands of gene expression levels, differ drastically from previous QTL studies which analyzed fewer number of traits. The increase in the number of phenotypes presents new challenges to the realm of quantitative trait locus mapping methods such as automating QTL software to perform thousands of genotype vs. phenotype associations, and establishing QTL detecting thresholds which take into account the multiplicity of hypothesis testing. In addition, groups of gene expression levels tend to be highly correlated and this information should be taken

15 2 into account when finding QTL for correlated traits. This dissertation addresses the issue of finding quantitative trait loci for correlated gene expression traits. It has been shown in the literature that a joint QTL analysis of multiple traits can have more power and be more precise than single trait QTL analysis when traits are correlated (Jiang and Zeng 1995; Knott and Haley 2000; Sorensen et al. 2003). We approach the challenge of finding common expression QTL (eqtl) among traits in two ways. First we find QTL for each expression trait individually, and scan the genome for shared QTL among traits. Some of these shared QTL represent transcriptional regulatory regions common to many traits. We then group gene expression levels which share a QTL according to function or using cluster analysis. Given groups of related genes, most of which share a QTL, we use a novel multiple trait QTL mapping method to estimate genetic correlation among traits due the QTL they share. Our second approach involves reducing the number of expression traits by dimension reduction. We reduce the number of gene expression levels using principal components analysis. We then apply single trait QTL mapping to these principal component traits, hoping that the QTL we find for these principal components have similar location to QTL which were found for each of the expression traits individually. 1.2 QTL Analysis Review Using fine-scale molecular maps to find regions in the genome associated with a trait of interest has been done successfully for many years (Lander and Botstein 1989; Haley and Knott 1992; Jansen 1993; Zeng 1994). In this section, we review a few QTL studies and some quantitative trait locus mapping methods, starting with the simplest method, which is single marker analysis. First, we describe how to obtain a genetic map.

16 Genetic Maps The goal of quantitative trait locus studies is to find regions of the genome associated with a trait of interest. For these types of studies, measurements of the trait of interest (phenotypic measurements) are needed as well as a genetic map. Genetic maps show the order of markers on a chromosome, and the distance between markers as a fraction of the recombination frequencies between them. One of the most common mapping functions were introduced by Haldane (1919) and Kosambi (1944). Haldane s mapping function assumes that crossovers occur randomly and independently from one another. Haldane s mapping function is given by: m = ln(1 2c), 2 where c is the observed recombination frequency, and m is the map distance in Morgans. Kosambi s mapping function allows for small interference and is given by: m = 1 ( 1 + 2c ) 4 ln, 1 2c where m and c are the same as in the Haldane s mapping function QTL Studies QTL studies have been done on a number of traits in different species such as bristle number in Drosophila (Payne 1918; Thoday 1961), leanness in pigs (Smith and Bampton 1977), seed and pigment weight in beans (Sax 1923), heterosis in maize (Stuber et al. 1992), prolificacy in sheep (Pipe and Bindon 1982), among many others. The common thread about these traditional QTL studies is that the number of traits being studied is small. More recently, QTL mapping methods have been used to find transcriptional regulatory regions, where thousands of gene expression levels are treated as phenotypic traits. QTL

17 4 mapping has been applied to microarray experiments in order to better understand the nature of regulatory regions of gene expression levels (Brem et al. 2002; Schadt et al. 2003; Li et al. 2006). Using QTL mapping with mrna abundance is treated as a phenotypic trait, one can identify gene expression regulatory regions for each trait separately. This allows for the study of patterns of cis vs. trans regulation in the entire data set (Yvert et al. 2003; Kirst et al. 2005). Some expression QTL studies have a goal to better understand transcriptional regulatory regions. For example, Brem and colleagues (2002) and Yvert and colleagues (2003) studied transcriptional regulatory patterns in yeast, revealing whether cis or trans regulation is more prevalent. This issue is at the heart of whether transcriptional regulation occurs at the site of the gene whose RNA abundance is being studied (cis-regulation), or at some other regulatory region (trans-regulation). Gibson and Weir (2005) discuss some quantitative aspects of eqtl studies and summarize the extent of cis vs. trans regulation in various experiments. Other eqtl papers tie the eqtl regions found by analyzing gene expression levels, to a phenotypic trait such as lignin biosynthesis in eucalyptus (Kirst et al. 2004), or fat pad mass and obesity in mice (Schadt et al. 2003). For example, Kirst and colleagues (2004) applied correlation analysis to each of the gene expression levels and diameter growth. Some expression traits highly correlated with diameter growth also shared QTL with the diameter growth trait. Li et al. (2006) studied eqtl by environment interactions in C. elegans. RNA abundance was measured in 80 samples at two different temperatures, at which there would be differences in body size, lifespan and other characteristics. Li and collaborators (2006) found that a significantly greater percentage of trans-acting genes showed eqtl by environment interaction compared to cis-acting genes. For the remainder of this chapter, we will describe in detail expression QTL studies

18 5 for eucalyptus and yeast, which are two data sets which will be studied further in this dissertation Yeast Expression QTL Studies The budding yeast data set which we used in our analysis was published in several installments (Brem et al. 2002; Yvert et al. 2003; Brem and Kruglyak 2005). Each installment expanded the number of segregants in the population. Subsequently, there have been several studies which analyzed this same data set. Next we will describe the first two papers which made the expression traits publicly available (Brem et al. 2002; Yvert et al. 2003) and some others which we find relevant. Brem and colleagues (2002) studied transcriptional regulation in Saccharomyces cerevisiae. Gene expression levels and marker genotypes were observed on two parental strains and their progeny. The parental strains are from a laboratory strain (BY) and a wild strain (RM). The progeny consists of 40 haploid samples. Gene expression levels for 6,215 genes or expression traits were measured in the parental strains with 6 replications. Twenty five percent of genes were differentially expressed between the parental strains at a p-value < These results were found using the Wilcoxon-Mann-Whitney test, permuting the data set to obtain a significance threshold. Gene expression levels were also observed for 40 haploid progeny samples obtained from the cross between the parental strains. Heritability for these expression traits was computed for the progeny as a function of the ratio of the parental expression variance to the segregant expression variance with the formula: (segregant variance - parent variance)/segregant variance. The expression traits were shown to have median heritability of 84%. To find genome locations associated with the expression traits, Brem and colleagues

19 6 (2002) performed single marker analysis on 6,215 traits with 3,312 markers. A total of 570 expression traits showed associated with one or more markers with a p-value < Twenty percent of differentially expressed genes showed linkage to at least one marker with a p-value < There are 262 gene expression traits which are linked to at least one marker in the genome (p-value < ) which are not differentially expressed between parental strains (p-value < 0.005). Brem and colleagues (2002) stipulate that these linkages could be false positives; or that in a given parent, many alleles with opposing effects regulate gene expression, which would lower the expression difference between parents; or that difference in expression levels between parents exists but there is lack of power to detect QTL. There are 1220 expression traits differentially expressed between parents (p-value < 0.005) which are not linked to any marker. Brem and colleagues (2002) argue that is because transcription is regulated by multiple loci, each with a small effect. For only about 20% of the genes differentially expressed is the single marker effect big enough to be detected. Thirty-six percent of 570 traits with an eqtl are cis-regulated. Brem and colleagues define trait to be cis-regulated if the marker linked to it is within 10kb of that trait. In the second yeast expression QTL paper published by the same group as Brem et al. (2002), Yvert and colleagues (2003) expanded the initial data set, by increasing the number of segregants in the cross to 86 samples. Yvert and colleagues (2003) found expression QTL, using single marker analysis, for the expression of all genes in the yeast genome. They found that 75% of all QTL were trans-acting QTL, and that most of these QTL were not enriched for transcriptional factors. In addition, Yvert and colleagues (2003) used hierarchical clustering to define gene expression clusters. Yvert and colleagues (2003) focused on clusters in which gene expression levels had a pair wise correlation greater than No clusters of more than 2

20 7 genes are expected to have pair wise correlation greater than by chance. In chapter 3, we re-analyze a subset of clusters shown in this paper, finding additional QTL and genes under the QTL peaks which have similar functions to genes in a cluster. Yvert and colleagues (2003) positionally cloned two trans-acting regulators; each regulator contains a polymorphism and affects the transcription of functionally related genes in that cluster. Brem and Kruglyak (2005) conclude that most QTL they detect for the yeast data set they use have weak effects. The data set is the same as in Brem et al. (2002) and Yvert et al. (2003), except that the number of segregants was increased to 112 samples and the number of markers is 2,957. Single marker analysis was performed to detect QTL, and permutations were used to declare QTL at 5% false discovery rate (Storey and Tibshirani 2003). Brem and Kruglyak (2005) claim to find epistasis for about 16% of highly heritable transcripts (h 2 > 0.687). The heritability of each transcript, h 2, was computed as h 2 = (σs 2 σ2 p )/σ2 p, where σp 2, σ2 s are the pooled variance among parental measurements, and the phenotypic variance among segregants. Storey and collaborators (2005) used data from the same yeast experiment as Brem and Kruglyak (2005) to come up with a new scheme to estimate epistasis between QTL. Storey et al. (2005) performed QTL analysis of 6,216 yeast expression traits, using 3,312 markers and 112 haploid segregants. Storey et al. (2005) claim their sequential search is more powerful to find main QTL effects and epistatic effects as compared to an exhaustive 2-dimentional scan. The exhaustive 2-dimentional scan works the following way: for every expression trait, every pair of markers is fitted to a model which includes two QTL main effects and an epistatic effect. The model is: expression= baseline level+locus1 effect + locus2 effect + epistatic effect + noise. (1.1) In contrast with the 2-dimentional scan, Storey et al. (2005) used a sequential genome scan to incorporate one QTL at a time in the model. The three models being considered

21 8 are: M0: expression=baseline level + noise M1: expression=baseline level + locus1 + noise M2: expression=baseline level + locus1 + locus2 + epistasis + noise. In step one, one selects the QTL which shows the most improvement in the goodness of fit of model M1 compared with model M0. Then Storey et al. (2005) select a second QTL which is the one that shows the greatest improvement in the goodness of fit of M2 compared with model M1. A total of 170 QTL pairs were found to be significant under model M2 at a false discovery rate of 10%. Zou (2006) using the same data set, was able to find more gene expression traits which had 2 QTL only but less epistatic interactions than Storey et al. (2005) by performing a sequential genome scan slightly different than what is presented in Storey et al. (2005). Zou s method used Multiple Interval Mapping (Kao et al. 1999) on one expression trait at a time. Zou s strategy is as follows: (1) search for the first QTL and add it into the model if its effect is significant at type I error rate of 10%, obtained through permutations; (2) if a QTL is found in step 1, add one QTL at a time into the model conditional on the existing QTL in the model (given that the QTL effect is greater than the threshold); (3) search for epistatic interactions between QTL found in steps 1 and 2; (4) delete QTL from the model that are not significant at a type I error rate of 5%. The main difference between the models of Zou (2006) and Storey et al. (2005) is that Zou searches for epistatic interactions only after all the QTL main effects have been added into the model. In addition, QTL are only added into the model if their effects are significant. Surprisingly, the number of expression traits controlled by only 2 QTL at a false discovery rate of 10% found by Zou is 729 compared with 170 of Storey et al. (2005). Besides the yeast data set, in this dissertation, we apply single and multiple quantitative

22 9 trait locus analysis to genes expression levels of a eucalyptus inbred cross. The next section describes some results for this data set Eucalyptus Expression QTL Studies Kirst and colleagues (2005) studied transcriptional regulation of genes in an Eucalyptus pseudobackcross: E. grandis F1 hybrid (E. grandis E. globulus). E. globulus has high wood density and relatively slow growth; E. grandis has lower wood density but faster growth. Crosses between these two strains have shown ample genetic and phenotypic variation (Kirst et al. 2005). The pseudobackcross is shown in Figure 1.1:

23 10 FIGURE 1.-- Mating design of the E. grandis pseudobackcross mapping population Kirst, M. et al. Genetics 2005;169: Copyright 2007 by the Genetics Society of America Figure 1.1: Eucalyptus cross design. Kirst and colleagues (2005) set out to discover the transcriptional regulation differences between individuals in the same species and individuals in two related species. They used the two marker maps to find expression QTL for 91 progeny samples. One map is that of the F1 hybrid (tree BBT01058) and the other of the E. grandis backcross parent (tree ) (Figure 1.1). In the E.grandis marker map, there were a total of 96 AFLP fragments, in 12 major linkage groups; in the F1 hybrid map, there were a total of 122 fragments which mapped to 11 linkage groups.

24 11 Of the 2,608 genes considered, 1373 (53%) were differentially expressed. Using Composite interval mapping (Zeng 1994), Kirst and colleagues (2005) identified eqtl for 811 genes using the F1 hybrid map, and 451 eqtl using the E.grandis map. These eqtl were significant using a type I error rate of 10% obtained through permutations. Combining the eqtl data from both maps, 1067 traits had a total of 1655 eqtl. A total of 821 gene expression traits had only one significant eqtl. Kirst and colleagues (2005) estimated epistatic interaction via Multiple Interval Mapping (Kao et al. 1999). Epistatic interactions were significant for 310 genes in the F1 hybrid map, and 285 genes in the E. grandis map. A total of 195 genes had eqtl in both marker maps, and 13 of these genes were located in homologous regions. This suggests that most eqtl were trans-acting. Kirst et al. (2005) argue that if cis-regulation were more prevalent, then we would see more homologous eqtl. This result is the similar to yeast eqtl study of Yvert et al. (2003) which found that trans-regulation is more prevalent than cis-regulation in yeast. Transcriptional hotspots were found using both eucalyptus maps. In another paper, Kirst et al. (2004) tied gene expression regulatory regions to regions which regulate growth variation, previously detected by QTL mapping. The experimental cross is the same one described previously in Kirst et al. (2005). QTL analysis for diameter growth and for each of the 2,608 genes was done for 91 samples using Composite Interval Mapping (Zeng 1993, 1994). Two significant QTL (experiment α = 0.01) for diameter growth were identified. Subsequently, Kirst and colleagues (2004) applied correlation analysis to each of the 2,608 gene expression levels and diameter growth. A total of 37 gene expression levels were correlated with growth (individual test significant threshold of ), most of which were negatively correlated with diameter growth. It turns out these 37 genes were mostly involved in lignin biosynthesis and the phenyl-

25 12 propanoid pathways. High lignin content in a tree can be detrimental to growth (Kirst et al. 2004). Then Kirst and colleagues (2004) confirmed that diameter growth and lignin content were negatively correlated by sampling 8 individuals from the backcross progeny. Kirst and colleagues (2004) also found common QTL for diameter growth and expression levels of genes in lignin biosynthesis. 1.3 Review of Quantitative Trait Locus Analysis Methods 1.4 Single Trait QTL Mapping Methods Single Marker Analysis To find associations between a trait and a marker in a population, with marker classes M/M, M/m, and m/m at a given loci, one can perform a parametric test such as the t-test or a non-parametric test such as the Wilcoxon-Mann-Whitney test. These tests can find significant difference between trait means of different marker groups. Let µ MM, µ Mm, and µ mm be the observed trait means for individuals with marker genotypes M/M, M/m, and m/m at a given locus respectively. And let n MM, n Mm, and n mm be the sample size of the marker classes, and s 2 MM, s2 Mm, and s2 mm be the sample variance for each class. Next we will give the t statistic for a backcross and an F 2 population (Zeng 2000). For an F 2 population, there are three marker classes MM, Mm, and mm. The test for the

26 13 additive marker effect is: t = µ MM µ mm, where (1.2) s 2 1 ( n MM + 1 n mm ) (1.3) s = (n MM 1)s 2 MM + (n mm 1)s 2 mm. (1.4) n MM + n mm 2 The test for dominance effect in the F 2 population is: t 2 = µ Mm µ MM /2 µ mm /2, where (1.5) s 2 ( 1 n Mm + 1 4n MM + 1 4n mm ) (1.6) s 2 = (n MM 1)s 2 MM + (n Mm 1)s 2 Mm + (n mm 1)s 2 mm. (1.7) n MM + n Mm + n mm 3 In a backcross population, there are only two marker classes, denoted here by MM and Mm. The test statistic for a backcross population is: t = µ MM µ Mm, (1.8) s 2 ( 1 n Mm + 1 n MM ) where s 2 = (n MM 1)s 2 MM + (n Mm 1)s 2 Mm. (1.9) n MM + n Mm 2 Single marker analysis for QTL mapping has been used for many years. Two problems with single marker analysis are: it does not estimate QTL position and the difference in trait means of marker classes is confounded with the recombination frequency between the QTL and its flanking markers. For example, consider a backcross population, and let r be the recombination frequency between the QTL genotype (QQ or Qq), and the marker genotype (MM or Mm). This next table will show putative QTL and marker genotypes, and QTL frequencies and effects:

27 14 Table 1.1: Backcross QTL and marker genotype frequencies and effects (Zeng 2000) QQ Qq MM Frequency 1 r r Effect µ + a µ + d Mm Frequency r 1 r Effect µ + a µ + b Below we will see how the difference in the trait means between marker classes in a backcross population is confounded by the recombination frequency r: µ MM µ Mm = [(1 r)(µ + a) + r(µ + d)] r[(r(µ + a) + (1 r)(µ + d)] (1.10) = (1 2r)(a d). (1.11) Next we will find an improvement to single marker analysis, by a QTL mapping method named interval mapping Interval Mapping The precision of quantitative trait locus mapping methodology has increased significantly since Lander and Botstein (1989) proposed Interval Mapping (IM), which can estimate a QTL effect based on a marker map. Nonetheless, interval mapping was the foundation for future QTL mapping methods, and the paper which introduced it (Lander and Botstein 1989) was very influential. First, let s establish the possible probabilities of a QTL genotype based on its flanking marker genotypes. Let r Mi Q be the recombination fraction between marker M i and QTL Q and let r Mi M i+1 be the recombination fraction between markers M i and M i+1. The next table will describe QTL genotype probabilities given their flanking marker genotypes for a

28 15 population with three marker classes such as an F 2 population. Table 1.2: QTL genotype probabilities given marker genotypes (Jiang and Zeng 1995) Marker Genotype QQ Qq qq M i M i M i+1 M i M i M i M i+1 m i+1 1 p p 0 M i M i m i+1 m i+1 (1 p) 2 2p(1 p) p 2 M i m i M i+1 M i+1 p 1 p 0 M i m i M i+1 m i+1 δp(1 p) 1 2δp(1 p) δp(1 p) M i m i m i+1 m i p p m i m i M i+1 M i+1 p 2 2p(1 p) (1 p) 2 m i m i M i+1 m i+1 0 p 1 p m i m i m i+1 m i where p = r Mi Q/r Mi M i+1, and δ = rm 2 i M i+1 /[(1 r Mi M i+1 ) 2 + rm 2 i M i+1 ]. Double recombination is ignored. For a backcross population, the putative QTL genotype can take on two values, QQ and Qq. In the next table we will show the probability of a QTL genotype given the genotype of its flanking markers, assuming that double recombination is ignored.

29 16 Table 1.3: QTL genotype probabilities given marker genotypes (Zeng 2000) Marker Genotype QQ Qq (1 r Mi Q)(1 r QMi+1 ) r Mi Qr QMi+1 M i M i M i+1 M i+1 1 r Mi 1 M i+1 1 r Mi 0 M i+1 M i M i m i+1 M i+1 (1 r Mi Q)r QMi+1 r Mi M i+1 m i M i M i+1 M i+1 r Mi Q(1 r QMi+1 ) m i M i m i+1 M i+1 r Mi M i+1 1 p p r Mi Qr QMi+1 1 r Mi M i+1 0 r Mi Q(1 r QMi+1 ) r Mi M i+1 (1 r Mi Q)r QMi+1 r Mi M i+1 p 1 p (1 r Mi Q)(1 r QMi+1 ) 1 r Mi M i+1 1 where p = r Mi Q/r Mi M i+1. For a backcross population, let y j be the phenotypic trait measurement for individual i; b be the effect of a single allele substitution at the QTL; x be an indicator random variable of the QTL genotype; b 0 be the mean of the model, and e j be a random variable which follows a normal distribution N(0,σ 2 ). Then interval mapping s linear model is as follows: y j = b 0 + bx j + e j. (1.12) The likelihood equation for interval mapping s the linear model (equation 1.12) is: n L(b 0, b, σ 2 ) = [G j (0)L j (0) + G j (1)L j ] where j = 1,, n. (1.13) j=1 The likelihood function is given by L j (x) = z((y j (b 0 + bx j)), σ 2 ), where z(x, σ 2 ) = (2πσ 2 ) 1/2 exp( x 2 /2σ 2 ). The function G j (x) represents the probability of the QTL genotype x given the flanking marker genotypes (Table 1.3). In the case of a backcross, the QTL genotypes QQ and Qq correspond to x values of 1 and 0 respectively. Lander and Botstein (1989) use the maximum likelihood analysis to obtain estimates of the model parameters. One disadvantage of interval mapping is that the additive effects estimated by interval mapping can be biased if there are more than one QTL in a linkage group (Jansen 1993;

30 17 Zeng 1994; Kao et al. 1999). In addition, interval mapping is not an interval test, that is, if there is a QTL, regions linked to it might appear to be significant even when there is no QTL present in those locations (Zeng 1994) Composite Interval Mapping An improvement of the precision and estimates of QTL effects of Interval Mapping came about when Jansen (1993) and Zeng (1994) independently used regression methods which could take into account the presence of other QTL effects into a linear model. Composite interval mapping (CIM), as labeled by Zeng (1994), used covariate markers to estimate QTL position and additive effects. The linear model for composite interval mapping is given by y j = µ + b x j + b k x jk + e j, (1.14) k i,i+1 where: y j is the trait value of the j th individual, µ is the mean of model, b is the effect of the QTL expressed as a difference in effects between the homozygous and heterozygote QTL genotype classes QQ and Qq, x j is an indicator random variable taking value 0 or 1 with probability depending on the genotypes of the flanking marker genotypes and the position of putative QTL (Table 1.3), b k is the partial regression coefficient of the phenotype y on the k th marker conditional on all other markers, x jk is an indicator random variable for the j th individual and k th marker genotype which takes values 0 or 1 depending on whether the maker type is homozygote or heterozygote, and e j is a random variable which we assume follows a normal N(0,σ 2 ) distribution.

31 18 The known parameters are x jk, b 0, y j. Using maximum likelihood analysis, the parameters b, b k, σ 2 are estimated. Composite interval mapping, unlike interval mapping, is an interval test. If linked markers to the QTL are included in the summation in equation 1.14, CIM is able to control for the effects of other linked QTL in the model (Zeng 1993, 1994). In addition, if epistasis is ignored, the partial regression coefficient b k, depends only on the QTL located in the marker interval being tested for the presence of a QTL (Zeng 1993, 1994). The likelihood for a backcross population using composite interval mapping is: L(b, B, σ 2 ) = n [p 1j φ( y j X j B b ) + p 0j φ( y j X j B )], (1.15) σ σ j=1 where X j B = µ + k b kx jk, p 1j is the probability that the marker x j is homozygous, and p 0j is the probability that the marker x j is heterozygous. Maximum likelihood analysis is used to find estimates for model parameters are computed using the expectation maximization (EM) algorithm. Defining a threshold to add or delete a QTL for this QTL mapping method is non-trivial because the test statistic under the null hypothesis of no QTL is not known. A threshold for QTL detection depends on the number of markers included into the linear regression model, the size of the QTL interval being tested in terms of the genetic distance, and on the sample size. In his 1994 paper, Zeng suggested that for a large sample size, and when not too many markers are fitted into the model, that the value of χ 2 α/m,2 can be used as an approximation for the 100α% threshold value when there are M intervals in the genome in some marker scenarios. Another way of finding an appropriate threshold to declare a QTL is to use permutations. Churchill and Doerge (1994) suggested simulating the null hypothesis of no QTL by permuting the trait values among individuals in the segregating population, while keeping

32 19 the maker genotypes for each individual fixed Multiple Interval Mapping Kao et al. (1999) proposed a linear model which can fit multiple QTL into a model, estimating both additive and epistatic effects of QTL which affect a given trait. They named their model Multiple Interval Mapping (MIM). MIM can be more precise and powerful than CIM. The statistical backcross model for MIM (Kao et al. 1999) for trait y, individual i, and m QTL (Q 1,, Q m ), can be written in the form: m m y i = µ + a r x ir + δ rk (w rk x ir x ik ) + ε i, (1.16) where µ is the mean of the model, a r is the additive effect of QTL r, x ir r=1 r k represents the putative QTL genotype for individual i, QTL r, δ rk is the indicator variable for epistasis between QTL r and QTL k, w rk is the epistatic effect between QTL r and QTL k, and ε i is the error term for individual i, which we assume is distributed as N(0, σ 2 ). The likelihood equation for MIM with a model n samples, m QTL (Q 1,, Q m ), located in positions (p 1,, p m ) for θ = (p 1,, p m, a 1,, a m,, w jk,, σ 2 ) is: L(θ X, Y ) = n 2 m i=1 j=1 p ij φ((y i µ ij )/σ), (1.17) where p ij is a variable containing information about the probability of QTL genotypes. There are different ways of searching for QTL with MIM. One way is an iterative procedure which adds one QTL at a time in the model based on the significance of its marginal

33 20 effect, then tests for epistasis between QTL in the model. After that one would then re-test QTL in the model for significance, and then optimize QTL positions. These steps would be performed until no QTL can be added into the model according to a particular QTL detection threshold or stopping criteria such as the information criteria (IC) (Stuart and Ord 1991; Miller 1990). MIM has some advantages over other QTL mapping methods because unlike interval mapping and composite interval mapping, with MIM epistatic interactions can be modeled explicitly; MIM can also give better QTL position estimates because it searches simultaneously for QTL in multiple marker intervals. 1.5 Threshold for QTL detection In classical QTL analysis, the number of traits analyzed is small compared to the thousands and sometimes tens of thousands of traits analyzed in expression QTL studies. Even though the statistical methodology of traditional QTL studies is used in the realm of eqtl studies, figuring out significance levels for various QTL detecting methods is non-trivial. One way of finding an appropriate threshold to declare a QTL is to use permutations. Churchill and Doerge (1994) suggested simulating the null hypothesis of no QTL by permuting the trait values among individuals in the segregating population, while keeping the maker genotypes for each individual fixed. One would then record the test values with the permuted samples and compare them to test values obtained by applying a QTL mapping method to the data of interest. Zou et al. (2004) proposed a re-sampling method using the result that the score statistic is asymptotically equivalent to the likelihood ratio statistic. This method has a much lower computation burden than the permutation scheme proposed by Churchill and Doerge (1994) to compute QTL detection thresholds.

34 21 In order to address this issue of testing many null hypotheses, Storey and Tibshirani (2003) came up with an estimate of the positive false discovery rate (FDR). With this FDR estimate, one can compute the expected proportion of genes which falsely were declared to have one or more QTL, given the total number of expression traits with at least one QTL. The FDR estimate of Storey and Tibshirani (2003) is applicable to eqtl studies where thousands of QTL are claimed to be significant. Zou (2006) adapted Storey s false discovery rate methodology to multiple interval mapping (MIM). Model selection is an active research area in QTL methodology development. Finding an appropriate threshold level for which to add or delete QTL into a model is still very challenging. 1.6 Thesis chapters In this first chapter, we described a few expression QTL studies, focusing on yeast and eucalyptus expression QTL experiments. Then we reviewed a few QTL mapping methods such as single marker analysis, interval mapping, composite interval mapping and multiple interval mapping. In the second chapter, we test the usefulness of principal component analysis in the realm of QTL mapping. We transform the original traits using principal component analysis, and examine whether the QTL found for these principal component traits match the location of QTL for individual traits. We analyzed two different data sets: one from Saccharomyces cerevisiae, and another from eucalyptus. The traits analyzed in both data sets were gene expression levels generated in microarray experiments. In the third chapter, we implement a novel multiple trait mapping method based on multiple interval mapping. We propose a way to find an initial model and to test for pleiotropy vs. close linkage. We then apply this method to functionally related clusters previously studied in Saccharomyces cerevisiae.

35 22 The fourth chapter is an analysis of transcriptional variation of gene expression levels in yeast individual amino acid biosynthetic pathways. We look at the number of QTL and the extent of pleiotropy in each pathway. We also estimate genetic correlations and heritabilities for genes in individual amino acid biosynthetic pathways. In this dissertation, we treated RNA abundance as a phenotypic trait for two experimental crosses: one from eucalyptus and the other from yeast. Both crosses are modeled as a backcross in the QTL experiment design. We analyzed a subset of Kirst and colleagues (2005) eucalyptus data. We used their cdna arrays consisting of 2,610 gene expression levels measured on 88 samples of a eucalyptus pseudo-backcross: E. grandis x F1 hybrid (see Figure 1.1). The average marker spacing is about 1 marker for every 10 centimorgans. The final version of the yeast data set we used was a subset of data set published in Brem and Kruglyak (2005). This data set comes from a cross between a wild (BY) and laboratory (RM) strains of Saccharomyces cerevisiae. The expressions of 6,195 genes were measured in 112 haploid samples using a platform of custom open reading frames (Yvert et al. 2003). In chapter 2, only 86 yeast samples were used which was the number of available samples at that time. In chapters 3 and 4, 112 yeast samples are used in the analysis. The marker density of this cross is on average one marker for every 3kb.

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