1. Incorporating Mutations
Previous we allowed for gene variants (alleles), but without a model of how they came into being
Rather than the coalescence of a single gene, next we consider successive generations of gene sets
Two things to consider
Variants of a gene (Alleles)
Variants in allele combinations (Sequences)
We begin by treating each independently Gn Gn Gn Gn Gn Gn Gn
Gn+1
Gn+2
Gn+3
Gn+4
4/17/2017
Comp 790– Genealogies to Sequences

2. Infinite Alleles Model
Assumes all that is knowable is if alleles are identical or different
No Spatial (i.e. sequence position) or quantitative information related to the observed differences
Only keeps track of how many of each allele type
Number of mutations that result in a variant is lost
Two event types, splits and mutations
Labels are arbitrary
(A)
(A,A)
(B)(A)
(B)(A)
(B)(A,A)
(B)(A)(C)
(B)(A)(C,C)
(B,B)(A)(C,C)
(B)(D)(A)(C,C)
(B)(D)(A)(C,C)
B D A C C
4/17/2017
Comp 790– Genealogies to Sequences

3. Comp 790– Genealogies to Sequences
Infinite Sites Model
Assumes mutations are rare events
Assumes DNA sequences are large
Multiple mutations at the same site are extremely rare
Infinite Sites Model assumes that multiple mutations never occur at the same sequence position
Thus, all genes are “Biallelic”
Lost haplotype
4/17/2017
Comp 790– Genealogies to Sequences

4. Comp 790– Genealogies to Sequences
SNP Panels
Observed Haplotypes and SNPs from previous example
Under the Infinite Sites Model the haplotype size equals number of historical mutations
While sequences can be lost, alleles cannot, in contrast to the Infinite Alleles Model
SNP Diversity Patterns (SDPs) can be repeated (eg. S1 and S2)
Since the assignment of 1s and 0s is arbitrary, a SNP and its complement share the same SDP
For N haplotypes, there are at most 2N-1 – 1 “possible” SDPs S1 S2 S3 S4 S5 H1 1 H2 H3 H4
4/17/2017
Comp 790– Genealogies to Sequences

5. A Different Kind of Tree
Unrooted “Perfect” Phylogeny
Nodes correspond to haplotypes (both visible and historical)
Edges correspond to SNPs
Removal of an edge creates a bipartition
Tree leaves correspond to mutations (allele variants) that are unique to a sequence, i.e. an SDP with only one minority allele instance, a singleton
4/17/2017
Comp 790– Genealogies to Sequences

6. Build a Phylogenetic Tree
Assume we only have direct access to observed haplotypes
Construct a pair-wise distance matrix between haplotypes using Hamming distances
Add smallest edge between all nodes which do not introduce a loop
If the smallest distance is greater than 1 add d-1 “hidden” nodes between the pair so that adjacent nodes have a hamming distance of 1
Augment the distance matrix with the new nodes and claim the introduced edges
Repeat finding the smallest distance, and augmenting until the graph is fully connected S1 S2 S3 S4 S5 H1 1 H2 H3 H4 H2 H3 H4 HA HB H1 1 3 2 4 H2 H3 H4 HA H1 1 3 2 4 H2 H3 H4 H1 1 3 4 2
4/17/2017
Comp 790– Genealogies to Sequences

7. Comp 790– Genealogies to Sequences
Four-Gamete Test
Under the assumption of the infinite sites model all SNP pairs exhibit the property no more that 3 out of the possible 4 allele combinations occur
Direct consequence of only one mutation per site
Showing that all SNP pair combinations satisfy the four gamete test is a necessary and sufficient condition for there to exist a perfect phylogeny tree S1 S2 S3 S4 S5 H1 1 H2 H3 H4
4/17/2017
Comp 790– Genealogies to Sequences

8. Comp 790– Genealogies to Sequences
Hard Questions
Which SDPs are compatible with any other SNP?
Given N distinct haplotype sequences resulting from an infinite sites model what is minimum number of SDPs?
Given N distinct haplotype sequences resulting from an infinite sites model what is maximum number of SDPs?
Singleton SNPs are compatible are compatible with any other SNP
N-1 edges are the fewest necessary to connect N haplotypes into a “linear” tree. How many singleton SNPs occur in such a tree? 2
2N-3 edges, the number of edges in an unrooted tree with N leaves
4/17/2017
Comp 790– Genealogies to Sequences

9. Comp 790– Continuous-Time Coalescence
Exercise
Consider the following SNP panel
Satisfies the four gamete test?
Construct the tree
Is the SDP 11001T possible? S1 S2 S3 S4 S5 H1 1 H2 H3 H4 H5
4/17/2017
Comp 790– Continuous-Time Coalescence