Genetic Linkage Analysis in the Presence of Germline Mosaicism Omer Weissbrod

Transcription

1 Technion - Computer Science Department - M.Sc. Thesis MSC Genetic Linkage Analysis in the Presence of Germline Mosaicism Omer Weissbrod

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3 Technion - Computer Science Department - M.Sc. Thesis MSC Genetic Linkage Analysis in the Presence of Germline Mosaicism Research Thesis Submitted in partial fulfillment of the requirements for the degree of Master of Science in Computer Science Omer Weissbrod Submitted to the Senate of the Technion Israel Institute of Technology Tishrei 77 Haifa October 0

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5 Technion - Computer Science Department - M.Sc. Thesis MSC The research thesis as done under the supervision of Prof. Dan Geiger in the Computer Science Department. This ork as published in Weissbrod, O. and Geiger, D. 0. Genetic Linkage Analysis in the Presence of Germline Mosaicism. Statistical Applications in Genetics and Molecular Biology, 0, 6. Mere gratitude is not enough to express my appreciation for my supervisor, Prof. Dan Geiger, hom has also been a mentor and a friend. Thanks to his constant guidance and profound insight, I can safely say that no student can hope for a more rapid and rearding entrance into the orld of academy. I ould also like to thank my friends in the Technion for making this such a great and fun orkplace. Last but not least, I ould like to thank my family for its immense support and advice. More than anything, I thank my beloved ife Michal, ho is alays a source of light. The generous financial support of the Technion is gratefully acknoledged.

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7 Technion - Computer Science Department - M.Sc. Thesis MSC Contents Abstract Abbreviations and Notations Introduction. Bayesian Netorks Likelihood Ratio Tests Genetic Terms and Definitions The Standard Model for Genetic Linkage Analysis Genetic Linkage Analysis Related Work Genetic Linkage Analysis in the Presence of Germline Mosaicism 7. Background Germline Mosaicism Detection of GM by Model Comparison Statistical Formulation of GM GM Statistical Model A Likelihood Ratio Test for Detecting GM Statistical Properties of the GM-LOD Statistic The Posterior Probability of GM Evaluation of the Test A Case Study Tools and Methods Results Proofs of Inequalities i

8 Technion - Computer Science Department - M.Sc. Thesis MSC Concluding Remarks 0 Bibliography ii

9 Technion - Computer Science Department - M.Sc. Thesis MSC List of Figures. A Bayesian netork representing a family ith to parents and one child. To adjacent genetic loci, denoted by i and i +, are shon A nuclear family affected ith a genetic trait, ith affected individuals shaded in black. Three markers have been measured for each individual. The affection status, marker data and phase of all measured markers is knon. A recombination event occurred in the meiosis of the paternal haplotype of individual II A nuclear family ith one untyped individual A nuclear family in hich the affection status of individuals II- and II- is unknon A nuclear family in hich the phase of the haplotypes of individuals II- and II- is not certain A three generations pedigree hose male founder I- has GM, ith affected individuals shaded in black. Individual I- is shaded in gray since his affection status is unknon. A haplotype ith four loci is shon for each individual, ith paternal haplotypes shon on the left and maternal on the right. Affected individuals carry the mutated gene m An extended model representation of a family hose father may have GM. The GM affects locus i but it does not affect locus i GM-LOD significance given a cutoff, hen the tested and trait locus are linked GM-LOD poer as a function of the number of GMs children and the number of generations in the pedigree GM-LOD poer using various values of the ratio ω/γ for pedigrees ith generations and 6 GMs children GM-LOD poer given a cutoff hen using phenotypic data only LOD poer in the presence of GM, using different genetic models iii

10 Technion - Computer Science Department - M.Sc. Thesis MSC The pedigree studied in Genzer-Nir et al. ith affected individuals shaded in black. A haplotype ith five markers is shos for each individual, ith the marker names shos on the top left. Individuals ith irrelevant haplotypes are denoted ith E u. The haplotype suspected of being linked to the trait is shaded in gray. The ten unaffected individuals ho carry the suspected haplotype are emphasized as bold numbers. The image is adapted from Genzer-Nir et al GM-LOD scores for the MDN pedigree. The horizontal line is the stringent cutoff of iv

11 Technion - Computer Science Department - M.Sc. Thesis MSC List of Tables. The probability that GMs passes a mutated allele to a child Cutoffs required to obtain a % GM-LOD genomeide significance level.. v

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13 Technion - Computer Science Department - M.Sc. Thesis MSC Abstract Genetic linkage analysis is a idely used statistical method for associating disease genes ith their location on the chromosome. This method has proved to be very successful in mapping genes involved in simple Mendelian diseases. Hoever, it is less poerful in mapping genes that do not follo Mendelian inheritance patterns. With the groing availability of genomic data, it becomes increasingly clear that some human genetic traits do not follo such patterns, and thus their associated genes may elude detection. This finding motivates the development of ne statistical tests that are more suitable to genetic mapping of such non-standard genetic traits. In this ork e consider a genetic condition called germline mosaicism that does not follo standard Mendelian inheritance patterns. Germline mosaicism violates some of the assumptions that underlie standard genetic linkage analysis. The standard method of genetic linkage analysis is therefore not suitable for analysis of traits caused by this condition. We develop a statistical model that is suitable for analysis of traits caused by this condition, and develop a statistical test to determine hether a genetic trait has been introduced by germline mosaicism. We also extend the method of genetic linkage analysis to incorporate germline mosaicism. Finally, e use our extended statistical model to provide solid statistical evidence for the location of the gene responsible for a syndrome recently discovered in Israel, called the MDN syndrome. This finding may prompt further genetic research to help identify exactly the causative gene of this syndrome.

14 Technion - Computer Science Department - M.Sc. Thesis MSC Abbreviations and Notations GM Germline Mosaicism LOD Logarithm of Odds M LE Maximum Likelihood Estimate M OI Mode of Inheritance df Degrees of Freedom GM s GM Suspect

15 Technion - Computer Science Department - M.Sc. Thesis MSC Chapter Introduction Genetic linkage analysis is a idely used statistical method for associating disease genes ith their location on the chromosome. The principal idea behind genetic linkage is that loci hich are located in close proximity on the chromosome tend to be passed together from parents to offspring. One can deduce the approximate location of a causative gene by finding a group of adjacent genetic markers that segregate healthy and affected individuals. Finding a causative gene sheds light on the biological mechanism that malfunctions in affected individuals and is sometimes the first step toards the development of a suitable remedy. Genetic linkage analysis has been very successful in mapping genes involved in simple Mendelian diseases. Well knon examples include Duchenne muscular dystrophy DMD, cystic fibrosis CF, and Huntington s disease HD Borecki and Rice, 00. Hoever, it is less poerful in mapping genes involved in traits that do not follo simple Mendelian inheritance patterns. Traits that do not follo such patterns ere once thought to rarely exist and received relatively little attention. With the groing availability of genomic data, it becomes increasingly clear that such human genetic traits are more frequent than previously thought Erickson and Leis, 99; Gropman and Adams, 007; Yaron and Orr-Urtreger, 00. Such traits violate some of the assumptions underlying classic genetic linkage analysis, and thus their associated genes may elude detection. When a genetic linkage analysis study fails, it may therefore still be possible to detect linkage by changing the assumptions underlying the model used for the analysis. To date, there has been little theoretical ork trying to formulate statistical tests for identifying biological phenomena that do not follo simple Mendelian inheritance in pedigrees. One common approach to dealing ith such traits is employing non-parametric linkage tests, hich do not assume a specific mode of inheritance. Hoever, these tests lack statistical poer in comparison to parametric tests that use an appropriate explicit model Strauch et al., 000.

16 Technion - Computer Science Department - M.Sc. Thesis MSC In this thesis e develop an extended statistical model for genetic linkage analysis to incorporate germline mosaicism GM, hich is a condition in hich some germ cells of an individual contain a mutation. Germline mosaicism has been found in a variety of inherited traits Barbosa et al., 008; Choi et al., 008; Fabrizi et al., 00; Khan et al., 00; Makri et al., 009; Pauli et al., 009. We use the extended statistical model to develop a parametric likelihood ratio test to evaluate hether a specific individual in a given pedigree has GM at a trait locus. Some theoretical aspects of GM have been studied before in the context of risk occurrence Edards, 989; Grimm et al., 990; Hartl, 97; Jeanpierre, 99; Murphy et al., 97, but no statistical test has been proposed previously for identification of GM in a pedigree. We demonstrate the effectiveness of the test by providing solid statistical evidence for GM in a pedigree affected ith MDN Genzer-Nir et al., 00, in hich GM has been hypothesized. A free computer package hich performs this test is available to donload at The rest of this chapter is organized as follos. Section. defines Bayesian Netorks. Section. defines likelihood ratio tests and provides some theoretical background regarding their use. Section. defines basic terms and definitions in Genetics that are used throughout this ork. Section. defines the standard model used in genetic linkage analysis. Finally, Section. defines genetic linkage analysis and provides some examples of its use in standard scenarios.. Bayesian Netorks A Bayesian Netork is a directed acyclic graph in hich each vertex corresponds to a discrete variable. The distribution for each variable V i is conditional upon the variables in the set of vertices from hich there are edges leading to V i in the graph, denoted as P Vi Jensen, 996; Pearl, 988; Dariche, 00. The distribution of each variable V i is therefore independent of any other variable V j / P Vi {V i } given an assignment for each V k P Vi. Due to this structure, the joint probability is given by PrV = v,..., V n = v n = V i PrV i = v i P Vi = p Vi, here p Vi is the joint assignment to the variables in P Vi. Bayesian netorks are ell suited to statistical inference tasks that are computationally intensive for to reasons. First, the independence assumptions underlying a Bayesian Netork allo probabilistic queries to be performed efficiently, compared ith the complexity required to perform queries on distributions in hich the probability of each joint assignment is specified. Several algorithms exist that exploit these independence assumptions in order to perform efficient inference computations on Bayesian netorks e.g. Dechter, 999. Second, inference computations on Bayesian netorks lend themselves ell to parallelization. Several methods of parallelizing probabilistic inference tasks have been studied

17 Technion - Computer Science Department - M.Sc. Thesis MSC Kozlov and Singh, 996; Pennock, 998; Madsen and Jensen, 999. Due to these characteristics, Bayesian netorks are often used for modeling and inference in the realm of genetic analysis Fishelson and Geiger, 00; Allen and Dariche, 008; Silberstein et al., 006; Sebastiani et al., 00.. Likelihood Ratio Tests A likelihood ratio test is a statistical hypothesis test used to determine hich of to given models is more likely to generate a set of given data Wasserman, 00. The test compares beteen to hypotheses, or statistical models. The first hypothesis is denoted as the null hypothesis and the second hypothesis is denoted as the alternative hypothesis. The alternative hypothesis has more free parameters than the null hypothesis, and thus the null hypothesis is said to be nested inside the alternative hypothesis. Since the alternative hypothesis has more parameters to optimize, it alays fits the given data better than the null hypothesis. The likelihood ratio test is used to determine hether the null hypothesis can be rejected in favor of the alternative hypothesis ith a given level of statistical significance. The likelihood ratio test is defined as follos. Denote by D a given dataset of observations and denote by H 0 and H a null and an alternative hypothesis, respectively. Further denote by θ a parameter used by the models, here θ is constrained to the set θ Θ 0 under hypothesis H 0 and to the set θ Θ under hypothesis H. Finally, assume that Θ 0 Θ, namely, that the hypothesis H 0 is nested inside H. The likelihood ratio is the ratio of the likelihood of the data under both hypotheses. It is given by Λ D = sup θ Θ PrD θ sup θ Θ0 PrD θ.. In order to determine hether the null hypothesis can be rejected, one must determine a cutoff R such that the null hypothesis is rejected hen Λ D R. Typically, the cutoff R is chosen such that the probability of exceeding R hen the null hypothesis holds and thus obtaining a false positive results is bounded by some value. The null distribution of the test is the distribution of the statistic Λ D given that H 0 holds. A cutoff value R hich guarantees a certain false positive rate can be determined analytically if the null distribution of the test is knon. If if is not knon, one may evaluate the null distribution of the test empirically via simulations. The number of degrees of freedom of a likelihood ratio test is the difference in the dimension of θ hen it is constrained to the set Θ and hen it is constrained to the set Θ 0. The larger this number, the higher the cutoff needed to ensure

18 Technion - Computer Science Department - M.Sc. Thesis MSC a fixed false positive rate. A useful theorem states that for many types of hypotheses, the null distribution of a likelihood ratio test follos a χ distribution, ith the number of degrees of freedom of the distribution corresponding to the number of degrees of freedom of the test Wilks, 98.. Genetic Terms and Definitions This section provides a concise description of basic terms and definitions from the realm of genetics that are used throughout this ork. The genetic material of individuals is organized in chromosomes. The human genome is composed of types of chromosomes, here each type of chromosome contains different genes. Every individual carries to different copies of each type of chromosome, one of hich as inherited from the father and one from the mother of this individual. To chromosomes that form such a pair are called homologous chromosomes. Each individual carries only 0% of the genetic contents of each parent. A meiosis is the process in hich a ne reproductive cell is formed. Reproductive cells carry only one copy of each type of chromosome, hile regular body cells carry to copies of each type of chromosome. During the meiosis, these chromosomes are formed by a random process that mixes the genetic material of the corresponding homologous chromosomes. Each chromosome in such a cell contains a combination of the genetic contents of the to homologous chromosomes that correspond to the type of this chromosome. The chromosomes in reproductive cells are composed of consecutive segments of genetic material. Each segment contains genetic material from one of the corresponding homologous chromosomes. The length of each such segment, as ell as the identity of the chromosome that corresponds to each segment, is chosen randomly. The event in hich to such segments sap during the meiosis is called a recombination. Several recombinations can occur in the formation of each chromosome during the meiosis. The folloing passage describes genetic terms that are mentioned in subsequent sections. A locus is a certain position across the genome of an individual. A marker is a locus hose genetic contents can be measured. Each marker can take one of several possible values, called alleles. Only markers ith more than one allele are of interest for genetic linkage analysis purposes. A SNP single-nucleotide polymorphism is a marker ith to alleles. In recent years, dense measurements of SNP markers have become idely affordable. The phase of the genetic marker of an individual is the information concerning here each of the measured alleles of the marker originated from. If one can determine the parental origin of each allele ith full certainty then the phase is fully knon. For example, if a father hose alleles at a certain marker are, and a mother hose alleles are, give birth 6

19 Technion - Computer Science Department - M.Sc. Thesis MSC to a child ith alleles,, one knos ith full certainty that the first measured allele originated from the father and the second measured allele originated from the mother. If, on the other hand, a father hose alleles are, and a mother hose alleles are, give birth to a child hose alleles are,, one can not kno hether each allele originated from the father or from the mother of this individual. A haplotype is a segment in the genome of an individual, that as received from a single chromosome of a single parent. The genotype of each individual can be decomposed into distinct haplotypes. Haplotypes can therefore be vieed as the basic unit of inheritance in human pedigrees. Contemporary genotyping technology cannot determine the phase of a genetic marker. It is therefore not alays possible to directly determine the decomposition of a genotype into haplotypes. Hoever, given a family ith several children, it is often possible to determine the correct decomposition of genotypes of all individuals into haplotypes. This decomposition makes use of the fact that one can compute the expected number of recombinations beteen every to markers and choose the most likely decomposition. A haplotyping algorithms finds the most likely decomposition of the genomes of all individuals in a pedigree into distinct haplotypes.. The Standard Model for Genetic Linkage Analysis The standard model for genetic linkage analysis provides a mathematical formulation that describes the flo of genetic material from individuals to offspring Elston and Steart, 97; Friedman et al., 000; Lander and Green, 987. Every pedigree can be described as a Bayesian Netork under this model Fishelson and Geiger, 00. For example, Figure. depicts a Bayesian netork model of a family of to parents and one child, denoted by a, b and c, respectively. The model defines to random variables for each locus of every individual in the pedigree, here the variables G i,k,p and G i,k,m denote the paternal and maternal allele of individual i at locus k, respectively. The model also defines selector variables also called meiosis variables denoted by S i,k,p and S i,k,m, hich indicate hether individual k received the paternal or the maternal allele of her respective parent at locus i. The variable S i,k,p is equal to 0 if the paternal allele of individual k at locus i is equal to the paternal allele of her father at locus i and is equal to otherise. The variable S i,k,m is defined similarly for the maternal allele. To loci on the same chromosome are passed together from parent to child if no recombination occurs beteen them during meiosis. The recombination frequency beteen loci i and i +, denoted as θ i,i+, lies in the interval [0, 0.], here higher values correspond to a higher probability of recombination beteen the to loci. Namely, θ i,i+ is equal to 0. hen the to loci segregate independently. In such a case, the to loci are said to be 7

20 Technion - Computer Science Department - M.Sc. Thesis MSC locus i locus i+ S i+,c,p G i,a,p G i+,a,p Father Father G i+,c,p G i,a,m G i+,a,m Child Child G i,b,p G i+,b,p Mother S i,c,p G i,c,p G i,c,m S i,c,m Mother G i+,c,m G i,b,m G i+,b,m S i+,c,m Figure.: A Bayesian netork representing a family ith to parents and one child. To adjacent genetic loci, denoted by i and i +, are shon. unlinked. Otherise, the to loci are linked. Figure. shos that the value of a selector variable at locus i + is related to the value of the corresponding variable at locus i. To such selector variables have a different value hen a recombination occurs beteen the to loci, an event hose probability is θ i,i+. Note that selector variables of siblings are independent of each other since every meiosis is an independent event. The recombination frequencies beteen genetic markers in the human genome are knon and are thus not parameters of the models. Similarly, the distribution of the alleles of each marker are also assumed to be knon. Thus, hen only measured markers are considered in the analysis, such as hen computing haplotypes, the model does not require the specification of any parameters. Hoever, typically one also considers the marker of a putative gene hich affects some genetic trait in the analysis. In this case, the distribution of the alleles of this marker and the recombination frequency beteen this marker and its 8

21 Technion - Computer Science Department - M.Sc. Thesis MSC to flanking markers are parameters that have to be specified.. Genetic Linkage Analysis Genetic linkage analysis is a statistical hypothesis test hich determines hether a certain genetic locus is linked to a genetic trait Ott, 999. Typically, the test is carried under the assumption that the genetic trait is caused by a single gene that resides in a certain unknon locus in the genome, denoted as the trait locus. The test compares the hypothesis that the tested locus and the trait locus are linked ith the null hypothesis that they are unlinked, using a likelihood ratio test. The test computes: [ Pr A, G L, γ, λ, θ ] LODA, G log 0.. Pr A, G U, γ, λ The quantities in Equation. are defined as follos. The variables A and G denote the phenotypic and genotypic data of the pedigree, respectively. Typically, the phenotypic data A corresponds to the affection status of individuals in the pedigree. When dealing ith discrete traits, the affection status of each individual can be either affected, unaffected or unknon. The genotypic data G corresponds to the genotypic data obtained from a set of genetic markers. It is composed of an unordered pair of the to alleles at each measured marker of each individual. The events L and U denote that the trait locus is linked and unlinked to the tested locus, respectively. The event L asserts that the recombination frequency beteen these to loci is smaller than 0., hile the event U asserts that it is equal to 0.. The parameter θ is the recombination frequency beteen the to loci in the event of linkage, and θ is the maximum likelihood estimate of θ in the range [0, 0.. Finally, the parameters γ and λ correspond to the prevalence and the penetrance parameters of the genetic trait, respectively. A value of LOD. is considered sufficient evidence for linkage Lander and Kruglyak, 99. The prevalence parameter γ corresponds to the probability that a chromosome of a randomly chosen individual carries a mutated allele. The penetrance parameter λ is the probability of the phenotype given the genotype of an individual. A genetic trait is said to have full penetrance if an individual ho carries the number of mutated alleles required to cause affection is affected ith 00% probability. Otherise, the trait has reduced penetrance. The penetrance parameter is represented by a tuple of three values in the range [0, ] hich correspond to the probability that an individual ho carries zero, one or to mutated alleles, respectively, is affected. For example, the penetrance parameter of a genetic trait that follos a dominant mode of inheritance MOI is given by 0,, if it fully penetrant and by 0, 0.7, 0.7 if carriers are affected ith 70% probability. The 9

22 Technion - Computer Science Department - M.Sc. Thesis MSC I II Figure.: A nuclear family affected ith a genetic trait, ith affected individuals shaded in black. Three markers have been measured for each individual. The affection status, marker data and phase of all measured markers is knon. A recombination event occurred in the meiosis of the paternal haplotype of individual II-. appropriate parameter for a fully penetrant trait ith a recessive MOI is given by 0, 0,. In order to compute the likelihood of a pedigree, one has to sum the joint probability of the visible data of the pedigree observed markers and affection statuses and the hidden data founder alleles and selector variables over all consistent assignments of the hidden data. A simple example of a nuclear family affected ith a genetic trait is given in Figure.. Under the assumptions of a fully penetrant dominant MOI and that the second marker is located on the trait locus, there is only one consistent assignment of hidden variables to this pedigree. this assignment given one mutated allele to individual I- and no mutated allele to individual II-. Additionally, the values of selector variables are fully knon, since it is knon here each allele originated from in each child. This is the only consistent assignment because there is a one to one correspondence beteen carries of the mutated allele and affected individuals due to the assumption of dominance and full penetrance. The more information is available to the analysis, the more likely it is that the LOD score exceeds the critical cutoff required to establish linkage. Less information is available hen the trait is not fully penetrant, hen the affection status of the measured markers of some individuals is unknon and hen the phases of some of the measured markers are not fully knon. Examples of all three conditions are no provided. If the genetic trait is not fully penetrant, then it is possible that an individual carries a mutated allele despite not being affected or the other ay around. For example, if the pedigree in Figure. is affected by a trait that is not fully penetrant, then it is possible that individual I- carries to mutated alleles or that individual II- carries one or to 0

23 Technion - Computer Science Department - M.Sc. Thesis MSC I II Figure.: A nuclear family ith one untyped individual. mutated alleles. In these cases, the measured markers do not determine hich individual carries a mutated allele and thus the poer to detect linkage is diminishes. If not all individuals have been genotyped then the poer to detect linkage diminished as ell. Figure. demonstrates that if individual II- is untyped then there is only a 0% probability that she received the same allele as individual II- from their father, and the likelihood ratio thus drops by 0%. The poer to detect linkage also drops hen the affection status of some of the individuals in the pedigree is unknon. Figure. demonstrates that if individuals II- and II- have an unknon affection status then there is only little evidence of linkage beteen the trait and the first left haplotype of individual I-. This is because only individuals I- and II- provide information regarding this haplotype. Finally, it is not alays possible to directly identify the phase of each meiosis. When to parents do not have distinct alleles, then it is not possible to fully determine here each allele of each child originated. In Figure., it is not knon here each allele of individuals II- and II- originated from. For example, one possibility is that individual II- received the alleles,, from individual I- and the alleles,, from individual II-, hich had a recombination event beteen the first and second markers. Another possibility is that individual II- received alleles,, from individual I- and alleles,, from individual II-. Assuming that the third marker locus is the trait locus, this affects the likelihood that individual II- received the mutated allele from individual I-. Is it not knon hich of the to haplotypes of individual I- is affected and the LOD score drops accordingly. In the special case in hich a genetic trait follos a fully penetrant dominant MOI and is caused by a gene located on a measured marker, each individual hose phase at the

24 Technion - Computer Science Department - M.Sc. Thesis MSC I II Figure.: A nuclear family in hich the affection status of individuals II- and II- is unknon. trait locus is fully knon contributes a value of 0. to the LOD score, provided that this individual has one affected parent ho carries one causative gene. This formula provides a rough upper bound on the contribution of each individual to the LOD score under ideal conditions, hen there is no uncertainty regarding the trait. The derivation of this formula is as follos. Denote a certain individual ho has one affected parent ith one causative gene by i and the set of ancestors of this individual by S. Further denote by A i and G i the phenotypic and genotypic data of individual i, respectively and denote by A S and G S the phenotypic and genotypic data of individuals in the set S, respectively. Since the trait is fully penetrant, it is knon hether individual i carries the causative gene. Under the null hypothesis of no linkage, the likelihood PrA i, G i A S, G S is /8. This holds because individual i has a probability of / of receiving the to alleles in the tested locus and a probability of / of receiving or not receiving the causative gene from the affected parent. Under the alternative hypothesis of linkage, the likelihood PrA i, G i A S, G S is /. This holds because individual i has a probability of / of receiving the to alleles in the tested locus and a probability of of receiving or not receiving the causative gene from the affected parent. This is because one knos hether individual i received the causative gene by observing the measured markers at the tested locus. Therefore, individual i causes the likelihood ratio to be multiplied by and thus contributes log 0 0. to the LOD score. A similar analysis shos that if a trait follos a recessive MOI, then under ideal conditions each individual contributes a value of 0.6 to the LOD score if both parents of this individual carry one causative gene. In general, the task of LOD scores calculation is computationally hard, since the assumptions given in the previous paragraph do not typically hold. In practice, genetic stud-

25 Technion - Computer Science Department - M.Sc. Thesis MSC I II Figure.: A nuclear family in hich the phase of the haplotypes of individuals II- and II- is not certain. ies often deal ith genetic traits that have reduced penetrance and ith large pedigrees that contain individual hose affection status is unknon or ho have not been genotyped. Moreover, the phase of some markers is often unknon, even if all other assumptions hold. Each of these elements contributes some amount of uncertainty to the studied pedigree. The computation of the LOD score becomes harder hen there is more uncertainty, since a likelihood computation requires summing over all consistent variable assignments in the probability space. In other ords, the computation of the LOD score is directly affected by the amount of uncertainty, so that the computation becomes harder hen the uncertainty is greater. Generally speaking, the LOD score computation is exponential in the number of alleles of each marker and in either the number of markers in the computation or in the number of individuals in the pedigree Elston and Steart, 97; Friedman et al., 000; Lander and Green, 987. Despite its computational complexity, several computer packages exist that perform LOD score computations Fishelson and Geiger, 00; Kruglyak et al., 996; O Connell and Weeks, 99; Gudbjartsson et al., 00; Abecasis et al., 00. These computer packages implement various optimizations that significantly speed up the LOD score computation. Namely, the program Superlink computes LOD scores by representing the pedigree as a Bayesian netork and computing the likelihood of the observed data in the netork. This provides the benefit of applying efficient inference algorithms from the realm of Bayesian netorks on the studied pedigree. Additionally, the program Superlink Online enables the LOD score computation of complex pedigrees by splitting the computational task into smaller subtasks and computing them concurrently on different computers Silberstein et al., 006.

26 Technion - Computer Science Department - M.Sc. Thesis MSC Chapter Related Work Model selection is the task of choosing a statistical model that is most likely to generate a given dataset of observations, out of a finite or infinite set of competing models. This is a fundamental problem in the fields of statistics and machine learning, hich has been studied for decades. Our ork is, at its core, a model selection problem here the standard genetic model is compared ith an extended model that accounts for germline mosaicism. Quite a fe studies have been published in the field of genetics, and of gene mapping in particular, that make use of model selection methodology as ell. A general revie of model selection in genetic mapping appeared in Sillanpää and Corander, 00. Many of the published studies of model selection in genetic mapping share some degree of similarity ith our ork, hich in a sense follos the footpath of these earlier studies. Here e survey these studies, restricting our focus to model selection in the field of gene mapping. The first model selection orks published in the field of genetic mapping focused on distinguishing beteen different modes of inheritance MOIs. One of the first such studies, dating back to 96, is Falconer, 96. This study attempted to discriminate beteen a discrete and a continuous trait in hich individuals hose phenotypic data exceed some critical cutoff are considered affected. More studies folloed in the same vein e.g. Morton et al., 970, 97; Reich et al., 97. An empirical study of model selection, tested by generating data sets and attempting to identify the model hich generated them, as published in Smith, 97. A key difference beteen these studies and our ork is that these studies used only phenotypic data for the model selection task, hile our ork uses both phenotypic and genotypic data for the analysis. Later, Greenberg 989 and Greenberg and Berger 99 studied the possibility of inferring the MOI of a genetic trait via comparison of LOD scores. These papers are similar to ours in the respect that they use the genotypic data of a pedigree in order to perform model selection. The papers concluded that if significant evidence for linkage is found then the correct model yields the highest

27 Technion - Computer Science Department - M.Sc. Thesis MSC LOD scores, though this may not alays be true if there is a strong bias in the analyzed data due to ascertainment. Most other model selection studies in the field of genetic mapping focused on choosing an appropriate genetic model in the absence of main effects, hen a genetic trait is affected by more than one gene. Namely, many authors studied model selection schemes for QTL Quantitative trait loci mapping e.g. Sen and Churchill, 00; Broman and Speed, 00; Yi et al., 00, 007; Manichaikul et al., 009. QTLs are continuous genetic traits that are affected by many different genetic and environmental factors. These orks differ from ours in several respects. First, these orks deal not only ith model selection but also ith model search. Our ork defines one ell-defined alternative model based on knon biological phenomena. Second, these orks mostly perform model selection using a Bayesian methodology, hile our ork uses a frequentist methodology that is more consistent ith the LOD test methodology. Third, these orks typically require large amounts of data that is not available in studies of human pedigrees. The poer to identify linkage in models ith one main effect is by definition higher than in models ith several secondary effects, since simpler models ith less parameters impose a less stringent selection criteria. Instead, these methods are used on experimental crosses in plants and animals, here the cost of each study is smaller and the amount of available data is much larger than in studies of human pedigrees. The statistical poer of these methods hen used on human pedigrees is bound to be smaller because of the limited availability of data. Other model-specification orks focused on epistatis, hich is a class of genetic traits determined by a small number of genes Cordell, 00. Unlike QTLs, epistatic traits are typically discrete and have ell-defined genetic models. As in QTLs, large amounts of data are required to identify linkage beteen a causative gene and an epistatic trait Cordell, 009. While many orks focus on various methods of modeling epistatic interactions beteen genes, the problem of distinguishing beteen epistatic and non-epistatic interactions received little attention. Greenberg 98 suggested to test for epistatis by testing for specific patterns via segregation analysis. Cordell 00 defined a statistical test to identify epistatis by formulating the models as logistic models and performing logistic regression. Both approaches use only phenotypic data to determine the correct model, unlike our method. Strauch et al. 00 defined several different models for epistatic interactions, and shoed that such models can be approximated ith a standard model that assumes only one causative gene. This stands in contrast to our ork, hich shos that modeling a trait caused by GM via a standard genetic model results in significantly reduced likelihood. Another body of research studied genetic traits affected by genes in the Mitochondrial genome. These orks are more similar to ours, in the sense that they study genetic traits hose underlying genes do not segregate according to Mendelian principles. Schork and

28 Technion - Computer Science Department - M.Sc. Thesis MSC Guo 99 defined pedigree models for genetic traits involving the Mitochondrial genome and suggested using likelihood based tests to identify Mitochondrial-caused genetic traits. Sun et al. 00 defined a class of statistical tests to detect Mitochondrial involvement in human diseases. These orks, like the ones previously surveyed, test for Mitochondrial inheritance via phenotypic data only. 6

29 Technion - Computer Science Department - M.Sc. Thesis MSC Chapter Genetic Linkage Analysis in the Presence of Germline Mosaicism This chapter describes germline mosaicism and ho it may be incorporated into the standard genetic model for genetic linkage analysis. The chapter is organized as follos. Section. defines germline mosaicism and explains the difficulty to identify GM in pedigrees. Section. provides a statistical genetic model that incorporates GM and a likelihood ratio test for detecting GM. It also demonstrates the statistical properties of this test and evaluates its effectiveness via computer simulations. Section. uses the statistical test to provide solid statistical evidence for GM in the pedigree reported by Genzer-Nir et al. Finally, Section. proves to inequalities that are defined in Section... Background This section provides background regarding germline mosaicism and demonstrates the difficulties of identifying GM by standard methods... Germline Mosaicism Germline mosaicism GM is a condition in hich part of the germ cells of an individual contain a mutation hich affects a genetic trait Zlotogora, 998. This mutation can affect the phenotype of every child ho as conceived from a mutated germ cell. The earlier the mutation occurred in the development of the individual, the larger the percentage of germ cells in the body hich carry this mutation. When an unaffected parent has several children affected ith a genetic trait, ithout the parent having a family history of the trait, germline mosaicism is an optional explanation. It is not possible to directly detect GM by 7

30 I III m m m m m II Figure.: A three generations pedigree hose male founder I- has GM, ith affected individuals shaded in black. Individual I- is shaded in gray since his affection status is unknon. A haplotype ith four loci is shon for each individual, ith paternal haplotypes shon on the left and maternal on the right. Affected individuals carry the mutated gene m. genotyping unless the mutation altered the allele of a marker that has been genotyped. It is therefore unlikely to directly detect GM even hen very dense marker maps are used. An example of a pedigree ith GM is given in Figure.. Assume that the first, second and fourth loci in each haplotype have been genotyped, hile the third locus has not been genotyped. This locus has to alleles, denoted by and m, here is a ildtype gene and m is a mutated gene hich causes affection. A certain percentage of the germ cells of individual I- carry a mutation at the third locus. The children of individual I- may therefore receive a mutated haplotype. Mutated haplotypes are shon in striped gray, hile non-mutated haplotypes are shon in solid gray. It is impossible to distinguish mutated haplotypes from non-mutated haplotypes by genotyping since the mutation only altered the gene at the third locus, hich has not been genotyped. Figure. demonstrates 8 Technion - Computer Science Department - M.Sc. Thesis MSC

31 Technion - Computer Science Department - M.Sc. Thesis MSC that germline mosaicism creates inheritance patterns that are highly unlikely under the assumptions of standard linkage analysis. For example, individuals II- and II- have both received the same markers from their parents, yet individual II- is affected hile individual II- is not. This is because individual II- received a mutated haplotype hile individual II- received a ildtype haplotype. It is impossible to distinguish beteen these to haplotypes, since the mutation has not altered any of the three measured markers. The same affection pattern holds for the offspring of individuals II- and II-. Namely, offspring of individual II- ho received the gray-shaded haplotype are affected hile offspring of individual II- are not affected, even if they carry the gray-shaded haplotype. This pattern is highly unlikely under the assumptions of standard genetic linkage analysis, since one subpedigree shos evidence of linkage beteen a certain haplotype and a genetic trait, hile the other subpedigree does not. Double recombination before and after the trait locus in the meiosis of every affected child is a possible explanation, but it is hardly an option in dense maps. Reduced penetrance is also a possible explanation, but this assumption becomes increasingly implausible as the trait segregates more clearly in large subpedigrees. Standard genetic linkage analysis is therefore unlikely to succeed in demonstrating linkage beteen the trait and the haplotype. In this ork e overcome this obstacle by incorporating GM into genetic linkage analysis... Detection of GM by Model Comparison Model comparison is typically performed in genetic linkage analysis by comparing the likelihood of the phenotypic data of a pedigree under several competing models. For each pair of models denoted by M and M, one computes the phenotypic likelihood ratio Pr A M / Pr A M, here A denotes the phenotypic data of the pedigree. This method is useful for differentiating beteen different modes of inheritance MOIs. For example, one may determine hether a genetic trait follos a dominant or recessive MOI by computing the likelihood ratio of the phenotypic data of a pedigree under these to competing models. In the pedigree given in Figure., the likelihood ratio of a dominant versus recessive fully penetrant MOI is 0.7 assuming a mutated allele prevalence of 0.%, indicating that the trait is over 00,000 more likely to follo a dominant MOI than a recessive MOI. Unlike differentiating recessive versus dominant models, differentiating a model hich assumes a genetic trait has been introduced by GM from a model hich assumes the converse via phenotypic data alone is a subtle and sometimes infeasible task. As an example, consider again the pedigree shon in Figure.. Denote by M a model hich assumes that the trait follos a fully penetrant dominant MOI and has been introduced into the 9

32 Technion - Computer Science Department - M.Sc. Thesis MSC pedigree from individual I- by inheritance and denote by M a model hich assumes the same MOI and that the trait has been introduced into the pedigree from individual I- by GM. Assuming that the paternal trait chromosome in half of the germ cells of individual I- carries a mutation, the log likelihood ratio of model M versus model M is.9, indicating that the trait is equally likely to be introduced into the pedigree by inheritance and by GM. More generally, consider an arbitrary pedigree ith a founder ho is suspected of having GM. Denote by M a model hich assumes that the GM suspect has received the mutated allele by inheritance and denote by M a model hich assumes that the GM suspect has GM hich causes half of the paternal trait chromosomes to carry a mutation. Under model M each child of the GM suspect receives a mutated allele ith probability 0., hile under model M each child receives a mutated allele ith probability 0.. The likelihood ratio of model M versus model M is approximately given by 0. n / 0. k 0.7 n k, here n denotes the number of children of the GM suspect and k denotes the number of such children ho are affected. The likelihood ratio depends only on k and n, regardless of the size of the studied pedigree, since the models M and M differ only in the origin of the mutated allele. Since human families are typically small, this ratio is not informative enough to determine hether the studied trait is caused by GM. The suggested resolution to the difficulty of identifying GM in pedigrees is to use both the phenotypic and the genotypic data of a pedigree. One can then possibly deduce that GM has occurred due to the irregular pattern of inheritance that emerges hen GM takes place; When the founder of a pedigree has GM, the pedigree can be divided into to subpedigrees, here one subpedigree shos high correlation beteen certain markers and the genetic trait, hile the second subpedigree does not. For example, in the pedigree shon in Figure., individuals II-, II-8 and their offspring sho high correlation beteen the trait and the haplotype shaded in gray. Other individuals in the pedigree do not sho such a correlation. This irregular pattern may indicate that GM has occurred. In this chapter e develop a statistical test that incorporates the likelihood of both the phenotypic and the genotypic data of a pedigree to identify GM.. Statistical Formulation of GM This section incorporates GM into the statistical model used for genetic linkage analysis. It develops a likelihood ratio test for identifying occurrences of GM in a pedigree and analyzes its statistical properties. The test is evaluated empirically via simulations. 0

33 Technion - Computer Science Department - M.Sc. Thesis MSC Table.: The probability that GMs passes a mutated allele to a child Standard Model GM Model.. GM Statistical Model GMs paternal, maternal alleles,, m m, m, m W = 0 S = W = 0 S = 0 0 W = S = 0 β β W = S = 0 0 The standard model described in Section. can be extended to account for GM. Consider a genetic trait determined by a gene that has to alleles, here denotes a ildtype allele and m denotes a mutated allele. Further consider a pedigree ith a founder ho may have GM at one of the to homologous chromosomes that carry the trait locus. Denote this founder as GM suspect GMs. To model GM in this founder, e add a ne random variable W and a ne parameter β to the standard model for genetic linkage analysis. The variable W is equal to either 0 or, here W = 0 indicates that GMs does not have GM and W = indicates that GMs has GM. The probability PrW = corresponds to the prior probability that GMs has GM. The probability β that GMs ill pass a mutated allele to a child is given by β = PrG = m W =, G =, S = 0, here G denotes the allele that the child received from GMs, G denotes the paternal allele of GMs and S = 0 indicates that the child received the paternal allele of GMs. In other ords, β is the probability that GMs ill pass a mutated allele instead of a ildtype allele to a child, given that GM occurred and given that the paternal segment of the mutated chromosome is passed. If at least one of the conditions is not satisfied either W = 0, G = m or S 0 then the probability that the child receives a mutated allele from GMs is unchanged from the standard model. Table. shos the probability that GMs passes a mutated allele to a child, given the trait alleles of GMs and given W and S. The left and right letter in each column of Table. corresponds to the paternal and maternal allele of GMs, respectively. Ros in Table. sho values of W and of S, here S = 0 and S = denote that the child receives the paternal and maternal allele of GMs, respectively. For example, if W=, S = and GMs alleles are, m, then the probability that GMs passes a mutated allele to a child is. Note that hen β = 0 the to models are identical. An illustration of the extended model is given in Figure.. Assume that locus i is the trait locus. If W =, S i,c,p = 0