CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Distance Matrix Methods Anders Gorm Pedersen Molecular Evolution Group Center for Biological Sequence Analysis Technical University of Denmark

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Distance Matrix Methods 1. Construct multiple alignment of sequences 2. Construct table listing all pairwise differences (distance matrix) 3. Construct tree from pairwise distances Gorilla : ACGTCGTA Human : ACGTTCCT Chimpanzee: ACGTTTCG GoHuCh Go-44 Hu-2 Ch- Go Hu Ch

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Finding optimal branch lengths S1S1 S2S2 S3S3 S4S4 S1S1 -D 12 D 13 D 14 S2S2 -D 23 D 24 S3S3 -D 34 S4S4 - Observed distance S1 S3 S2 S4 a b c d e Distance along tree (patristic distance) D 12  d 12 = a + b + c D 13  d 13 = a + d D 14  d 14 = a + b + e D 23  d 23 = d + b + c D 24  d 24 = c + e D 34  d 34 = d + b + e Goal:

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Exercise (handout) Construct distance matrix (count different positions)Construct distance matrix (count different positions) Reconstruct tree and find best set of branch lengthsReconstruct tree and find best set of branch lengths

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Optimal Branch Lengths for a Given Tree: Least Squares Fit between given tree and observed distances can be expressed as “sum of squared differences”:Fit between given tree and observed distances can be expressed as “sum of squared differences”: Q =  (D ij - d ij ) 2 Q =  (D ij - d ij ) 2 Find branch lengths that minimize Q - this is the optimal set of branch lengths for this tree.Find branch lengths that minimize Q - this is the optimal set of branch lengths for this tree. S1 S3 S2 S4 a b c d e Distance along tree D 12  d 12 = a + b + c D 13  d 13 = a + d D 14  d 14 = a + b + e D 23  d 23 = d + b + c D 24  d 24 = c + e D 34  d 34 = d + b + e Goal: j>i

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Optimal Branch Lengths: Least Squares Longer distances associated with larger errorsLonger distances associated with larger errors Squared deviation may be weighted so longer branches contribute less to Q:Squared deviation may be weighted so longer branches contribute less to Q: Q =  (D ij - d ij ) 2 Q =  (D ij - d ij ) 2 Power (n) is typically 1 or 2 S1 S3 S2 S4 a b c d e Distance along tree D 12  d 12 = a + b + c D 13  d 13 = a + d D 14  d 14 = a + b + e D 23  d 23 = d + b + c D 24  d 24 = c + e D 34  d 34 = d + b + e Goal: D ij n j>i

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Finding Optimal Branch Lengths ABC A-D AB DACDAC B-DBCDBC C- Observed distance A B C v1v1 v3v3 v2v2 Distance along tree D AB  d AB = v 1 + v 2 D AC  d AC = v 1 + v 3 D BC  d BC = v 2 + v 3 Goal:

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Finding Optimal Branch Lengths

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Finding Optimal Branch Lengths

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Finding Optimal Branch Lengths

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Finding Optimal Branch Lengths

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Finding Optimal Branch Lengths

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Finding Optimal Branch Lengths

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Finding Optimal Branch Lengths

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Finding Optimal Branch Lengths System of n linear equations with n unknowns Can be solved using substitution method or matrix-based methods

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Least Squares Optimality Criterion Search through all (or many) tree topologiesSearch through all (or many) tree topologies For each investigated tree, find best branch lengths using least squares criterion (solve N equations with N unknowns)For each investigated tree, find best branch lengths using least squares criterion (solve N equations with N unknowns) Among all investigated trees, the best tree is the one with the smallest sum of squared errors.Among all investigated trees, the best tree is the one with the smallest sum of squared errors. Least squares criterion used both for finding branch lengths on individual trees, and for finding best tree.Least squares criterion used both for finding branch lengths on individual trees, and for finding best tree.

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Minimum Evolution Optimality Criterion Search through all (or many) tree topologiesSearch through all (or many) tree topologies For each investigated tree, find best branch lengths using least squares criterion (solve N equations with N unknowns)For each investigated tree, find best branch lengths using least squares criterion (solve N equations with N unknowns) Among all investigated trees, the best tree is the one with the smallest sum of branch lengths (the shortest tree).Among all investigated trees, the best tree is the one with the smallest sum of branch lengths (the shortest tree). Least squares criterion used for finding branch lengths on individual trees, minimum tree length used for finding best tree.Least squares criterion used for finding branch lengths on individual trees, minimum tree length used for finding best tree.

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Superimposed Substitutions Actual number ofActual number of evolutionary events:5 Observed number ofObserved number of differences:2 Distance is (almost) always underestimatedDistance is (almost) always underestimated ACGGTGC C T GCGGTGA

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Model-based correction for superimposed substitutions Goal: try to infer the real number of evolutionary events (the real distance) based onGoal: try to infer the real number of evolutionary events (the real distance) based on 1. Observed data (sequence alignment) 2. A model of how evolution occurs

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Jukes and Cantor Model Four nucleotides assumed to be equally frequent (f=0.25)Four nucleotides assumed to be equally frequent (f=0.25) All 12 substitution rates assumed to be equalAll 12 substitution rates assumed to be equal Under this model the corrected distance is:Under this model the corrected distance is: D JC = x ln( x D OBS ) For instance:For instance: D OBS =0.43 => D JC =0.64 ACGT A -3     C    G    T   

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Clustering Algorithms Starting point: Distance matrixStarting point: Distance matrix Cluster the two nearest nodes:Cluster the two nearest nodes: –Tree: connect pair of nodes to common ancestral node, compute branch lengths from ancestral node to both descendants –Distance matrix: replace the two joined nodes with the new (ancestral) node. Compute new distance matrix, by finding distance from new node to all other nodes Repeat until all nodes are linked in treeRepeat until all nodes are linked in tree Results in only one tree, there is no measure of tree-goodness.Results in only one tree, there is no measure of tree-goodness.

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Neighbor Joining Algorithm For each tip compute u i =  j D ij /(n-2)For each tip compute u i =  j D ij /(n-2) (essentially the average distance to all other tips, except the denominator is n-2 instead of n-1) Find the pair of tips, i and j, where D ij -u i -u j is smallestFind the pair of tips, i and j, where D ij -u i -u j is smallest Connect the tips i and j, forming a new ancestral node. The branch lengths from the ancestral node to i and j are:Connect the tips i and j, forming a new ancestral node. The branch lengths from the ancestral node to i and j are: v i = 0.5 D ij (u i -u j ) v j = 0.5 D ij (u j -u i ) Update the distance matrix: Compute distance between new node and each remaining tip as follows:Update the distance matrix: Compute distance between new node and each remaining tip as follows: D ij,k = (D ik +D jk -D ij )/2 Replace tips i and j by the new node which is now treated as a tipReplace tips i and j by the new node which is now treated as a tip Repeat until only two nodes remain.Repeat until only two nodes remain.

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Neighbor Joining Algorithm ABCD A B-1218 C-14 D-

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Neighbor Joining Algorithm ABCD A B-1218 C-14 D- iuiui A( )/2=32.5 B( )/2=23.5 C( )/2=23.5 D( )/2=29.5

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Neighbor Joining Algorithm ABCD A B-1218 C-14 D- iuiui A( )/2=32.5 B( )/2=23.5 C( )/2=23.5 D( )/2=29.5 ABCD A B- C--39 D- D ij -u i -u j

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Neighbor Joining Algorithm ABCD A B-1218 C-14 D- iuiui A( )/2=32.5 B( )/2=23.5 C( )/2=23.5 D( )/2=29.5 ABCD A B- C--39 D- D ij -u i -u j

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Neighbor Joining Algorithm ABCD A B-1218 C-14 D- iuiui A( )/2=32.5 B( )/2=23.5 C( )/2=23.5 D( )/2=29.5 ABCD A B- C--39 D- D ij -u i -u j C D X

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Neighbor Joining Algorithm ABCD A B-1218 C-14 D- iuiui A( )/2=32.5 B( )/2=23.5 C( )/2=23.5 D( )/2=29.5 ABCD A B- C--39 D- D ij -u i -u j C D v C = 0.5 x x ( ) = 4 v D = 0.5 x x ( ) = X

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Neighbor Joining Algorithm ABCDX A B-1218 C-14 D- X- C D 4 10 X

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Neighbor Joining Algorithm ABCDX A B-1218 C-14 D- X- C D 4 10 X D XA = (D CA + D DA - D CD )/2 = ( )/2 = 17 D XB = (D CB + D DB - D CD )/2 = ( )/2 = 8

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Neighbor Joining Algorithm ABCDX A B C-14 D- X- C D 4 10 X D XA = (D CA + D DA - D CD )/2 = ( )/2 = 17 D XB = (D CB + D DB - D CD )/2 = ( )/2 = 8

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Neighbor Joining Algorithm ABX A-17 B-8 X- C D 4 10 X D XA = (D CA + D DA - D CD )/2 = ( )/2 = 17 D XB = (D CB + D DB - D CD )/2 = ( )/2 = 8

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Neighbor Joining Algorithm ABX A-17 B-8 X- C D 4 10 X iuiui A(17+17)/1 = 34 B(17+8)/1 = 25 X

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Neighbor Joining Algorithm ABX A-17 B-8 X- C D 4 10 X ABX A B- X- D ij -u i -u j iuiui A(17+17)/1 = 34 B(17+8)/1 = 25 X

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Neighbor Joining Algorithm ABX A-17 B-8 X- C D 4 10 X D ij -u i -u j ABX A B- X- iuiui A(17+17)/1 = 34 B(17+8)/1 = 25 X

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Neighbor Joining Algorithm ABX A-17 B-8 X- C D 4 10 X D ij -u i -u j ABX A B- X- iuiui A(17+17)/1 = 34 B(17+8)/1 = 25 X v A = 0.5 x x (34-25) = 13 v D = 0.5 x x (25-34) = 4 A B Y 4 13

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Neighbor Joining Algorithm ABXY A-17 B-8 X- Y C D 4 10 X A B Y 4 13

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Neighbor Joining Algorithm ABXY A-17 B-8 X-4 Y C D 4 10 X A B Y 4 13 D YX = (D AX + D BX - D AB )/2 = ( )/2 = 4

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Neighbor Joining Algorithm XY X-4 Y- C D 4 10 X A B Y 4 13 D YX = (D AX + D BX - D AB )/2 = ( )/2 = 4

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Neighbor Joining Algorithm XY X-4 Y- C D 4 10 A B D YX = (D AX + D BX - D AB )/2 = ( )/2 = 4

CENTER FOR BIOLOGICAL SEQUENCE ANALYSIS Neighbor Joining Algorithm C D A B ABCD A B-1218 C-14 D