Basics of Linkage Analysis Idea of Linkage Analysis Types of Linkage Analysis Parametric Linkage Analysis Nonparametric Linkage Analysis Conclusions

Basics of Linkage Analysis Idea of Linkage Analysis Types of Linkage Analysis Parametric Linkage Analysis Nonparametric Linkage Analysis Conclusions

Gene mapping problem Lähde: Morgan Genetics Tutorial. http://morgan.rutgers.edu/morganwebframes/level1/page2/karyotype.html

Linkage Analysis One of the two main approaches in gene mapping. Uses pedigree data.

Genetic linkage and linkage analysis Two loci are linked if they appear closeby in the same chromosome. The task of linkage analysis is to find markers that are linked to the hypothetical disease locus Complex diseases in focus  usually need to search for one gene at a time Requires mathematical modelling of meiosis

Meiosis and crossover Number of crossover sites is thought to follow Poisson distribution. Their locations are generally random and independent of each other.

  L( |data ) DIS Marker The simple idea Recombination fraction Find that maximises Obtain measure for degree of evidence in favour of linkage (LOD score)

Markers and inheritance 1 2 4 3 2 1 3 4 2 3 1 2 Father Mother 2 3 1 4 3 1 Child Polymorphic loci whose locations are known Point mutations (SNP) or lengths of repetitive sequences Inherited together with the chromosomal segments

Markers and information Two individuals share same allele label  they share the allele IBS (identical by state) Two individuals share an allele with same grandparental origin  they share an allele IBD (identical by descent) IBS sharing can easily be deduced from genotypes. IBD sharing provides more information. One can try to deduce IBD sharing based on family structure and inheritance.

Markers and information The children share allele 1 IBS. 1,2 2,3 They also share it IBD. 1,2 1,3

Markers and information The children share allele 1 IBS. 1,2 1,3 They do not share alleles IBD. 1,2 1,3

Markers and information The children share allele 1 IBS. 1,1 2,3 They either share or do not share it IBD. 1,2 1,3

Building blocks of linkage analysis Marker maps Chr. 1 Pedigree structures 1 2 1 12 1 2 1 1 2 1 1 5 2 14 1 3 2 1 2 2 Chr. 2 1 3 4 5 1 2 3 2 4 2 1 3 4 4 7 1 4 3 4 4 2 3 Genotypes Chr. 22 Phenotypes 2 1 1 3 2 2 2 3 3 4

Building blocks of linkage analysis Information about disease model (in parametric analysis)  (aa), probability of a homozygote being affected  (Aa), probability of a heterozygote being affected  (AA), probability of a non-carrier being affected (phenocopy rate) Information about environmental variables

13 0 0 5 0 0.0 0.0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 1 2 0.99 0.01 1 0.001000 0.999000 0.999000 3 5 # M0 0.172 0.036 0.176 0.283 0.333 3 5 # M1 0.100 0.345 0.310 0.164 0.081 (---clip---) 3 5 # M10 0.169 0.432 0.147 0.130 0.122 3 5 # M11 0.397 0.204 0.151 0.043 0.205 0 0 0.10 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 1 0.1 0.45 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 1 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 5 1 2 1 1 3 1 2 1 4 5 1 6 3 2 3 3 5 4 2 5 4 3 2 5 3 2 4 5 1 6 0 0 2 2 4 4 3 4 2 5 9 7 1 7 1 3 2 5 5 3 4 6 2 5 2 1 2 1 1 7 0 0 1 1 5 1 2 2 5 2 6 6 5 1 1 3 3 2 5 5 3 4 5 5 4 4 2 1 1 8 1 2 2 1 3 3 2 2 4 5 6 6 3 5 1 4 5 4 1 5 4 3 2 5 3 2 4 5 1 9 3 4 1 1 3 4 2 2 5 2 6 6 5 5 4 1 4 5 5 3 3 6 2 5 2 3 2 1 1 10 3 4 2 1 3 4 2 5 4 3 1 6 3 2 1 3 5 5 2 3 4 2 4 5 4 3 5 1 1 11 5 6 1 1 1 4 1 4 5 5 6 7 2 7 3 3 5 5 2 3 4 6 2 5 3 1 4 1 1 12 5 6 2 2 3 4 2 4 4 5 1 7 3 7 3 3 5 5 2 3 4 6 5 5 2 1 5 1 1 13 5 6 1 2 3 4 1 4 5 5 6 7 2 7 3 3 5 5 2 3 4 6 2 5 3 1 5 1 1 14 7 8 1 1 5 3 2 2 2 5 6 6 5 5 1 4 3 4 5 5 3 3 5 5 4 2 2 5 1 15 7 8 2 1 1 3 2 2 2 5 6 6 5 5 1 4 3 4 5 5 3 3 5 2 4 3 2 4 1 16 0 0 1 1 5 5 2 4 5 6 3 1 1 1 3 4 4 4 3 7 1 3 5 4 3 2 1 4 1 17 16 12 1 2 5 4 2 4 5 5 3 7 1 7 3 3 4 5 7 3 3 6 4 5 2 1 4 5 1 18 16 12 2 1 5 3 4 2 6 4 1 1 1 3 3 3 4 5 3 2 1 6 4 5 2 1 4 1

Example of linkage analysis results for one chromosome

Basics of Linkage Analysis Idea of Linkage Analysis Types of Linkage Analysis Parametric Linkage Analysis Nonparametric Linkage Analysis Conclusions

Types of linkage analysis Parametric vs. non-parametric Dichotomous vs. continuous phenotypes Elston-Stewart vs. Lander-Green vs. heuristic Two-point vs. multipoint Genome scan vs. candidate gene

Basics of Linkage Analysis Idea of Linkage Analysis Types of Linkage Analysis Parametric Linkage Analysis Nonparametric Linkage Analysis Conclusions

Maximum likelihood estimation A common approach in statistical estimation Define hypotheses Generate likelihood function Estimate Test hypotheses Draw statistical conclusions

Hypotheses in linkage analysis  = 0.5 the disease locus is not linked to the marker(s) HA:   0.5 the disease locus is linked to the marker(s)

Likelihood function for a single nuclear family Lj = gF P(gF) P(yF | gF) gM P(gM)P(yM | gM) gOi P(gOi | gF, gM) P(yOi | gO) G = genotype probabilities y = phenotype probabilities The parameter  is incorporated here

Several independent families The likelihood functions of multiple indpendent families are combined: L =  Lj or logL =  log Lj

Testing of hypotheses Compute values of likelihood function under null and alternative hypotheses. Their relationship is expressed by LOD score (essentially derived from the likelihood ratio test statistic.

On significance levels P-value gives a probability that a null hypothesis is rejected even though it was true. A LOD-score threshold of 3 corresponds to a single-test p-value of approximately 0.0001. In genome-wide gene mapping study, one conducts several (partially dependent) statistical tests. Applying the aforementioned threhold, the global p-value of 400 mutually independent test would be 1 - (1-0.0001)400 = 0.039  0.05. What if one focuses on individual candidate regions?

An example of ML estimation Single marker, dominant disease All genotypes known 1,3 2,4 2,3 1,4 1,2 1,4 1,2 3,4 1,2

Paternal haplotype combinations Haplotype combinations of children, assuming unlinked loci

1,3 2,4 2,3 1,4 1,2 3,4 1 + D 3 1-  4 1 2 3 + 1 D ½

Recombination fraction 0.56 0.5 LOD score 0.0 0.0 0.14 0.5 Recombination fraction LOD>3 taken as evidence of linkage.

Basics of Linkage Analysis Idea of Linkage Analysis Types of Linkage Analysis Parametric Linkage Analysis Nonparametric Linkage Analysis Conclusions

Idea of nonparametric linkage analysis No assumption is made on disease model. The tests measure IBD sharing of alleles among affected relatives. ASP (Affected-Sib-Pair test) is the simplest form of NPL Requires nuclear families of two affected children Extendable to arbitrary pedigrees, missing data, and arbitrary group of affected relatives

Example analysis for one marker 3 4 3 4 1 2 1 2 2 4 1 3 1 3 2 3 1 3 2 3 3 4 3 4

Idea of ASP test Collect large number of families with two affected offspring and deduce IBD status for each pair of offspring. Let us mark the number of sib-pairs with IBD status zero by n0. Respectively, n1 ja n2 are observed counts of the sib-pairs that share 1 or 2 alleles IBD. Compare the counts against the expected distribution by computing the value of the 2 test statistic.

Test statistic for ASP where e0 = 0.25n, e1 = 0.5n ja e2 = 0.25n. 2-test with 2 degrees of freedom. homozygous parents are a problem. lots of variants and implementations.

1 1 1 1 1 1 1 1 1 1 1 7

Idea of nonparametric linkage analysis Compare to the 2 cumulative distribution function (with 2 degrees of freedom): P=0.0012 The sample is too small for the 2 test to be reliable.

Basics of Linkage Analysis Idea of Linkage Analysis Types of Linkage Analysis Parametric Linkage Analysis Nonparametric Linkage Analysis Conclusions

Conclusions Linkage analysis is a pedigree-based approach to gene mapping. Parametric vs. nonparametric methods. Hypothesis-driven vs. explorative analysis. Meta-analysis becoming increasingly popular.