Lect18 Population Genetics

An Introduction to Bioinformatics Algorithms (Computational Molecular Biology)

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CSE182-L18 Population Genetics
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Perfect Phylogeny Assume an evolutionary model in which no recombination takes place, only  mutation. The evolutionary history is explained by a tree in which every mutation is on an  edge of the tree. All the species in one sub-tree contain a 0, and all species in the  other contain a 1. Such a tree is called a perfect phylogeny. How can one reconstruct such a tree?
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The 4-gamete condition A column i partitions the set of  species into two sets i 0 , and i 1   A column is homogeneous w.r.t a set  of species, if it has the same value for  all species. Otherwise, it is  heterogenous. EX: i is heterogenous w.r.t  {A,D,E} i A 0 B 0 C 0 D 1 E 1 F 1 i 0 i 1
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4 Gamete Condition 4 Gamete Condition There exists a perfect phylogeny if and only if for all pair of columns (i,j), either j is  not heterogenous w.r.t i 0 , or i 1 . Equivalent to There exists a perfect phylogeny if and only if for all pairs of columns (i,j), the  following 4 rows do not exist (0,0), (0,1), (1,0), (1,1)
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4-gamete condition: proof Depending on which edge the  mutation j occurs, either i 0 , or i 1   should be homogenous. (only if) Every perfect  phylogeny satisfies the 4-gamete  condition (if) If the 4-gamete condition is  satisfied, does a prefect  phylogeny exist? i 0 i 1 i
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An algorithm for constructing a  perfect phylogeny We will consider the case where 0 is the ancestral state, and 1 is the  mutated state. This will be fixed later. In any tree, each node (except the root) has a single parent. It is sufficient to construct a parent for every node. In each step, we add a column and refine some of the nodes  containing multiple children. Stop if all columns have been considered.
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Inclusion Property For any pair of columns i,j – i < j if and only if  i 1    j 1   Note that if i<j then the edge  containing i is an ancestor of the  edge containing i i j
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Example 1 2 3 4 5 A 1 1 0 0 0 B 0 0 1 0 0 C 1 1 0 1 0 D 0 0 1 0 1 E 1 0 0 0 0 r A B C D E Initially, there is a single clade r, and each node has r  as its parent
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Sort columns Sort columns according to the inclusion  property (note that the columns are already  sorted here). This can be achieved by considering the  columns as binary representations of numbers  (most significant bit in row 1) and sorting in  decreasing order 1 2 3 4 5 A 1 1 0 0 0 B 0 0 1 0 0 C 1 1 0 1 0 D 0 0 1 0 1 E 1 0 0 0 0
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Add first column In adding column i Check each edge and decide  which side you belong.
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Lect18 Population Genetics - CSE182-L18 Population Genetics...

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