wpgma input: distance matrix d ij; initially each element is a cluster. n r - size of cluster r find...

WPGMA

• Input: Distance matrix Dij; Initially each

element is a cluster. nr- size of cluster r• Find min element Drs in D; merge clusters

r,s• Delete elts. r,s, add new elt. t with

Dit=Dti=nr/(nr+ns)•Dir+ ns/(nr+ns) • Dis

• Repeat

The distance table

dog bear raccon weasel seal sea lion

cat chimp

dog 0 32 48 51 50 48 98 148bear 0 26 34 29 33 84 136raccon 0 42 44 44 92 152weasel 0 44 38 86 142seal 0 24 89 142sea lion 0 90 142cat 0 148chimp 0

The distance table

dog bear raccon weasel seal sea lion

cat chimp

dog 0 32 48 51 50 48 98 148bear 0 26 34 29 33 84 136raccon 0 42 44 44 92 152weasel 0 44 38 86 142seal 0 24 89 142sea lion 0 90 142cat 0 148chimp 0

Starting tree

seal sea lion

We call the father node of seal and sea lion “ss”.

12 12

Distance between these two taxa was 24, so each branch has a length of 12.

ss

Removing the seal and sea-lion rows and columns,and adding the ss row and columns

dog bear raccon weasel ss cat chimp

dog 0 32 48 51 ? 98 148bear 0 26 34 ? 84 136raccon 0 42 ? 92 152weasel 0 ? 86 142ss 0 89 142cat 0 148chimp 0

Computing dog-ss distance

dog bear raccon weasel seal sea lion

cat chimp

dog 0 32 48 51 50 48 98 148

),())()(

)((),()

)()(

)(()),(( kjD

jnin

jnkiD

jnin

inkijD

Here, i=seal, j=sea lion, k = dog.

n(i)=n(j)=1.

D(ss,dog) = 0.5D(sea lion,dog) + 0.5D(seal,dog) = 49.

The new table. Starting second iteration…

dog bear raccon weasel ss cat chimp

dog 0 32 48 51 49 98 148bear 0 26 34 31 84 136raccon 0 42 44 92 152weasel 0 41 86 142ss 0 89 142cat 0 148chimp 0

Starting tree

We call the father node of seal and sea lion “ss”.

Distance between bear and raccoon was 26, so each branch has a length of 13.

seal sea lion

12 12

ss

bear raccoon

13 13

br

Computing br-ss distance

dog bear raccon weasel ss cat chimp

ss 49 31 44 41 0 89.5 142

Here, i=raccoon, j=bear, k = ss.

n(i)=n(j)=1. D(br,ss) = 0.5D(bear,ss)+0.5D(raccoon,ss)=37.5.

),())()(

)((),()

)()(

)(()),(( kjD

jnin

jnkiD

jnin

inkijD

The new table. Starting second iteration…

dog br weasel ss cat chimp

dog 0 40 51 49 98 148br 0 38 37.5 88 144weasel 0 41 86 142ss 0 89 142cat 0 148chimp 0

Starting tree

Distance between br and ss was 37.5, so each branch has a length of 18.75. But this is the distance from brss to the leaves. The distance brss to ss is 18.75-12=6.75. The distance between brss to br is 18.75-13=5.75

seal sea lion

12 12

ss

bear raccoon

6.75

13

brss

br

5.75

13

Computing dog-brss distance

dog br weasel ss cat chimp

dog 0 40 51 49 98 148

Here, i = br, j = ss, k = dog.

n(i)=n(j)=2. D( brss , dog ) = 0.5D( br , dog ) + 0.5D( ss , dog )=44.5.

),())()(

)((),()

)()(

)(()),(( kjD

jnin

jnkiD

jnin

inkijD

The new table. Starting second iteration…

dog brss weasel cat chimp

dog 0 44.5 51 98 148brss 0 39.5 88.75 143weasel 0 86 142cat 0 148chimp 0

Starting tree

Distance between brss and w was 39.5, so wbrss is mapped to the line 19.75. The distance to brss, is thus, 1

seal sea lion

0

ss

bear raccoon

brss

br 1312

19.7518.75

weasel

wbrss

Computing dog-wbrss distance

dog brss weasel cat chimp

dog 0 44.5 51 98 148

Here, i = brss, j = weasel, k = dog.

n(i)=4, n(j)=1. D( wbrss , dog ) = 0.8D( brss , dog ) + 0.2D( weasel , dog )=

44.5*8/10+51*2/10 = (356+102)/10=45.8

),())()(

)((),()

)()(

)(()),(( kjD

jnin

jnkiD

jnin

inkijD

The new table. Starting second iteration…

dog wbrss cat chimp

dog 0 45.8 98 148wbrss 0 88.2 142.8cat 0 148chimp 0

Starting tree

Distance between wbrss and dog was 45.8, so dwbrss is mapped to the line 22.9 The distance to wbrss, is thus, 3.15

seal sea lion

0

ss

bear raccoon

brss

br 1312

22.9

18.75

weasel

dwbrss

19.75

dogl

wbrss

The new table. Starting second iteration…

dwrbss cat chimp

dwrbss 0 89.833 143.66cat 0 148chimp 0

Starting tree

Distance between dwbrss and cat was 89.833, so cdwbrss is mapped to the line 44.9165The distance to dwbrss, is thus, 22.0165

seal sea lion

0

ss

bear raccoon

brss

br 1312

44.9165

18.75

weasel

cdwbrss

19.75

dog

wbrss22.9

cat

dwbrss

The new table. Starting second iteration…

cdwrbss chimp

cdwrbss 0 144.2857chimp 0

Starting tree

Distance between cdwbrss and chimp was 144.2857, so THE ROOT is mapped to the line 72.14285The distance to dwbrss, is thus, 27.22635

seal sea lion

0

ss

bear raccoon

brss

br 1312

72.14

18.75

weasel

dwbrss

19.75

dog

wbrss22.9

cat

cdwbrss44.9165

chimp

Neighbor Joining Algorithm Saitou & Nei, 87

• Input: Distance matrix Dij; Initially each element is a cluster.

• Find min element Drs in D; merge clusters r,s

• Delete elts. r,s, add new elt. t with Dit=Dti=(Dir+ Dis – Drs)/2

• Repeat• Present the hierarchy as a tree with similar

elements near each other