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Allele Recoding Tutorial #6: Computational Difficulties and Savings in Multipoint Likelihood Computations

This tutorial explores the computational difficulties in multipoint likelihood computations due to highly polymorphic markers and provides possible savings through allele recoding. It discusses the implementation of allele recoding in Vitesse and explains the concept of transmitted and non-transmitted sets of alleles. Fuzzy inheritance and set-recoding techniques are also covered.

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Allele Recoding Tutorial #6: Computational Difficulties and Savings in Multipoint Likelihood Computations

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  1. Allele Recoding Tutorial #6 by Ma’ayan Fishelson

  2. Computational Difficulties • Algorithms that perform multipoint likelihood computations sum over all the possible ordered genotypes for each person. • When highly polymorphic markers are analyzed, any person who is untyped at these markers will have a very large number of possible genotypes. • The computations become increasingly difficult with regards to both memory and time requirements.

  3. Possible Savings • All alleles that do not appear in the genotype of any typed person can be lumped together in a single new allele. • What would be the gene frequency of this allele ? • Perform genotype elimination: • Ignore genotypes incompatible with phenotypes. • Delete genotypes of parents which are incompatible with children’s genotypes. • Vitesse does even better !

  4. Allele-Recoding in Vitesse • In order to reduce the computational complexity, a set-recoding scheme which recodes each person’s genotypes has been implemented in Vitesse (a program for Genetic Linkage Analysis). • The number of genotypes that need to be summed over is decreased.

  5. k 1/1 1/2 Key Idea • Consider an untyped individual k who has only two typed descendants, and a locus with 4 alleles. • Q: • Which alleles need to be specified • individually ? • Can we distinguish between alleles 3 & 4 ? • (These alleles don’t appear in any of the • genotypes of k’s typed descendants). • A: • There’s no need to distinguish between alleles 3 & 4. • There’s also no need to consider allele 2 separately, • because person k couldn’t have passed it to her 2nd • typed descendant.

  6. Transmitted & Non-Transmitted Sets • For each untyped person, sets of alleles with identical roles as recombination indicators are combined into a single representative allele.

  7. Transmitted & Non-Transmitted Sets(cont.) • There are 2 sets of alleles: ‘transmitted’ and ‘non-transmitted’. • In terms of recombination indicators, a person’s non-transmitted alleles are indistinguishable from one another.

  8. Some Notations… • Assume there are n alleles at a marker. An untyped person will initially have an ordered genotype list consisting of n x n orderedpairs of alleles formed by {1,…n}X{1,…,n}. • The set of maternal alleles (Ma) – the set of alleles inherited from the mother. • The set of paternal alleles (Pa) is defined in a similar way. • The set of paternal transmitted alleles – Pt. • The set of paternal non-transmitted alleles – Pn • In a similar way for the maternal alleles.

  9. Transmitted Alleles An allele is transmitted if : • The allele appears in the ordered genotype list of a typed descendent D of P, as inheritable from. • There is some path from P to D containing only untyped descendants in the pedigree, so D is the “nearest” typed descendant on that path.

  10. Non-Transmitted Alleles The remaining alleles are defined to be non-transmitted.

  11. Example 2 1 5 4 6 3 1/2 7 8 • Assume a locus • with 4 alleles. 10 11 9 3/3 2/2

  12. Example (2) 2/3 2/3 {2/2, 2/3, 3/3} 8 7 Pt = {2,3} Pn = Ø 11 9 10 3/3 2/2 Pt = Ø Pn = {2,3}

  13. Example (3) 2 1 Pt = {1,2,3} Pn = {4} 4 5 6 3 1/2 8 7 Pt = {2,3} Pn = Ø 10 11 9 3/3 2/2 Pt = Ø Pn = {2,3}

  14. Clarifications: • The term ‘transmitted’ doesn’t mean that transmission necessarily occurred. • A person’s transmitted / non-transmitted set may be empty. • An untyped parent may have more (but not fewer) transmitted alleles than its untyped offspring. • An untyped parent will have fewer non-transmitted alleles than its untyped offspring, unless the offspring’s non-transmitted set is empty.

  15. Fuzzy Inheritance • Regular Inheritance case: if a parent has the genotype A|B and its child has allele C, then C is inherited from the parent if A=C or A=B. • Set-Recoding: A,B,C are sets. C is inherited from the parent if or . • Set-recoding of typed loci: (A|B = {A}|{B})

  16. 2 {6}/{4,5} 1 {1}/{2,3} 3 {1,2,3}/{4,5} Example – fuzzy inheritance

  17. 2/3 3/2 2/4 2/2 1/1 1/2 3/4 1/2 1/2 2/3 Another Example… • A locus with 5 alleles.

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