Neighbor-Joining (NJ) Algorithm

1. Neighbor-Joining (NJ) Algorithm

2. NJ Algorithm • Similar to FM (also removes molecular clock assumption) • but more sophisticated in how it selects clusters to join • Produces unrooted trees • Algorithm (similar to FM) • Add a leaf to the tree for each taxon • Initially make each taxon be its own cluster • Find the closest clusters (using more sophisticated criterion) • (place new node at distance given by a variant of 3-point formula) • Repeat previous step until all clusters are connected

3. NJ “closeness” Criterion • Suppose that you are given n taxa x1, x2, x3, …, xn, and suppose that you have some tree that fits the distance data x5 x4 x3 x1 z y x6 x2 observation: d(x1,x2) + d(xi,xj) < d(x1,xi) + d(x2,xj) (right side includes yz twice, left does not)

4. NJ “closeness” Criterion d(x1,x2) + d(xi,xj) < d(x1,xi) + d(x2,xj) • From previous slide d(x1,x2) + d(x3,x4) < d(x1,x3) + d(x2,x4) For a fixed i, say i = 3: d(x1,x2) + d(x3,x5) < d(x1,x3) + d(x2,x5) d(x1,x2) + d(x3,x6) < d(x1,x3) + d(x2,x6) … … … d(x1,x2) + d(x3,xn) < d(x1,x3) + d(x2,xn) ------------------------------------------------- Add d(x3,x1),d(x3,x2) , d(x3,x3), d(x2,x1), d(x2,x2) to both sides

5. NJ “closeness” Criterion • From previous slide, if x1 and x2 are neighbors • Let • Then in general, if xk and xl are neighbors • NJ uses this observation to determine “closeness” and computes the smallest value M(k, l) to determine a cluster • Unlike UPGMA and FM, NJ has a more global view of “closeness” when selecting neighbors

6. NJ new node Placement • If x1 and x2 are neighbors; where should new node y be by 3-point formula x4 x5 x3 … … … x1 -------------------------------------------------------------- y x2 add on right side d(x1,x1 ) + d(x1,x2) - d(x2,x1 ) - d(x2,x2 )

7. NJ mini summary • For each pair of nodes xk and xl compute the quantity • Actually, could compute • When xk and xl are replaced by new node y, place y at • From now on Si will always be divided implicitly by (n-2)

8. NJ Algorithm • From the distance matrix compute the criterion matrix • Find the smallest value in M(i, j) – cluster the corresponding pair • Connect taxa xi and xj with a new node y placed at distance • Remove xi and xj and replace with y; update the distance matrix using the 3-point formula • Repeat from beginning

9. Apply the NJ algorithm to the given distance matrix: First compute Si=sum-of-row / (n-2) S1= 11.75 S2=10.25 S3=12.75 S4=14.25 S5=11.25 S6= 12.25 Compute M(1,2) = d(1,2) – S1 – S2 = 8 – 22= -14 M(1,3) = d(1,3) – S1 – S3 = 3 – 24.5= -21.5 M(1,4) = d(1,4) – S1 – S4 = 14 – 26 = -12 M(1,5) = d(1,5) – S1 – S5 = 10 – 23 = -13 M(1,4) = d(1,4) – S1 – S4 = 12 – 24 = -12 and so on … Find min value, i.e. the pair to cluster

10. From previous slide we need to cluster x1 and x3 Add a new taxon x7 and place it at distance Recompute distances from x7 to all others using the 3-point formula x3 x1 2 1 x7 d(7,2) = ½(d(1,2) + d(3,2) – d(1,3)) = 7 d(7,4) = ½(d(1,4) + d(3,4) – d(1,3)) = 13 d(7,5) = ½(d(1,5) + d(3,5) – d(1,3)) = 9 d(7,6) = ½(d(1,6) + d(3,6) – d(1,3)) = 11

11. Apply the NJ algorithm to the new distance matrix: First compute Si=sum-of-row / (n-2) S2= S4= S5= S6= S7= Compute M(2,4) = d(2,4) – S2 – S4 = M(2,5) = d(2,5) – S2 – S5 = M(2,6) = d(2,6) – S2 – S6 = M(2,7) = d(2,7) – S2 – S7 = and so on … Find min value, i.e. the pair to cluster

12. From previous slide we need to cluster ? and ?? Add a new taxon x8 and place it at distance Recompute distances from x8 to all others using the 3-point formula x?? x? ? ? x8

13. NJ Summary • Distance-based algorithm that produces unrooted trees • Removes the assumption of molecular clock, but does not give information about the root (common ancestor) • Typically performs better than UPGMA and FM – uses a more global criterion to select pairs to cluster • To detect the root could introduce an extra taxon (outgroup) that is more distantly related to the given taxa