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Lecture_7_phylogenetics

# T tree of a single edge of length d12 4 return t 5 if

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Unformatted text preview: ng a Tree with 3 Leaves •  Tree reconstruc?on for any 3x3 matrix is straightorward •  We have 3 leaves i, j, k and a center vertex c Observe: dic + djc = Dij dic + dkc = Dik djc + dkc = Djk Reconstruc?ng a Tree with 3 Leaves dic + djc = Dij + dic + dkc = Dik 2dic + djc + dkc = Dij + Dik 2dic + Djk = Dij + Dik dic = (Dij + Dik – Djk)/2 Similarly, djc = (Dij + Djk – Dik)/2 dkc = (Dki + Dkj – Dij)/2 Addi?ve Distance Matrices Matrix D is ADDITIVE if there exists a tree T with dij(T) = Dij NON- ADDITIVE otherwise Trees with > 3 Leaves •  An tree with n leaves has 2n- 3 edges •  This means ﬁrng a given tree to a distance matrix D requires solving a system of “n choose 2” equa?ons with 2n- 3 variables •  This is not always possible to solve for n > 3 Distance Based Phylogeny Problem •  Goal: Reconstruct an evolu?onary tree from a distance matrix •  Input: n x n distance matrix Dij •  Output: weighted tree T with n leaves ﬁrng D •  If D is addi?ve, this problem has a solu?on and there is a simple algorithm to solve it 7 9/16/13 Finding Neighboring Leaves Using Neighboring Leaves to Find a Tree •  Find neighboring leaves i and j with parent k •  Remove the rows and columns of i and j •  Add a new row...
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