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...
View
Full Document
 Fall '13
 AsaBenHur
 DNA, Distance, Orangutan

Click to edit the document details