Distance matrix methods calculate a measure of distance between each pair of species, then find a tree that predicts the observed set of distances.

Branch lengths and times in distance matrix methods, branch lengths reflect the expected amount of evolution in different branches of the tree. branch length = r i t i rate of evolution elapsed time

The least squares method ABCDE A0D ab D ac D ad D ae BD ab 0D bc D bd D be CD ac D bc 0D cd D ce DD ad D bd D cd 0D de ED ae D be D ce D de 0 Observed matrix minimise the difference between the observed matrix of distances and the matrix of distances predicted by the tree.

The least squares method ABCDE A0d ab d ac d ad d ae Bd ab 0d bc d bd d be Cd ac d bc 0d cd d ce D d ad d bd d cd 0d de Ed ae d be d ce d de 0 Expected matrix c e a b d 0.08 0.05 0.10 0.07 0.06 0.05 0.03

The least squares method c e a b d 0.08 0.05 0.10 0.07 0.06 0.05 ABCDE A0 B0 C0 D0 E0 0.03 Expected matrix

The least squares method c e a b d 0.08 0.05 0.10 0.07 0.06 0.05 ABCDE A00.23 B0 C0 D0 E0 0.08+0.05+0.10 0.03 Expected matrix

The least squares method c e a b d 0.08 0.05 0.10 0.07 0.06 0.05 ABCDE A00.230.160.200.17 B0.230 0.170.24 C0.160.2300.150.11 D0.200.170.1500.21 E0.170.240.110.210 0.03 Expected matrix

The least squares method Q =  w ij (D ij – d ij ) 2 i=1j=1 n n observed distance between species i and j expected distance between species i and j Q is a measure for the discrepancy between the observed and the expected matrix.

The least squares method Q =  w ij (D ij – d ij ) 2 i=1j=1 n n weight (1, 1/D 2, 1/D) distances can be weighed or not.

The least squares method c e a b d v1 v7 v2 v4 v5 v3 v6 x ij,k = 1 if branch k is on the path between species j and k = 0 if branch k is not on the path between species j and k X ij, k is a handy variable

The least squares method c e a b d v1 v7 v2 v4 v5 v3 v6 X a-b,1 = 1

The least squares method c e a b d v1 v7 v2 v4 v5 v3 v6 X a-b,1 = 1 X a-b,7 = 1

The least squares method c e a b d v1 v7 v2 v4 v5 v3 v6 X a-b,1 = 1 X a-b,7 = 1 X a-b,3 = 0

The least squares method Q =  w ij (D ij – d ij ) 2 i=1j=1 n n d ij =  x ij,k v k k rewrite d ij, the expected values

The least squares method Q =  w ij (D ij –  x ij,k v k ) 2 i=1j=1 n n k

The least squares method Q =  w ij (D ij –  x ij,k v k ) 2 i=1j=1 n n k = -2  w ij x ij, k (D ij –  x ij,k v k ) i=1j=1 n n dQ dv k k differentiate Q and equate the derivative to zero

The least squares method = -2  x ij, k (D ij –  x ij,k v k ) = 0 i=1j=1 n n dQ dv k k for the unweighted case

The least squares method = -2  x ij, 1 (D ij –  x ij,k v k ) = 0 i=1j:j≠1 n n dQ dv 1 k x AB,1 (D AB -  x AB  k v k ) + x AC,1 (D AC -  x AC  k v k ) + x AD,1 (D AD -  x AD  k v k ) + x AB,1 (D AE -  x AE  k v k ) + x BC,1 (D BC -  x BC  k v k ) + x BD,1 (D BD -  x BD  k v k )+ x BE,1 (D BE -  x BE  k v k ) + x CD,1 (D CD -  x CD  k v k ) + x CE,1 (D CE -  x CE  k v k ) + x DE,1 (D DE -  x DE  k v k ) = 0 i=1 i=2 i=3 i=4 j=2j=3j=4j=5 j=3j=4j=5 j=4j=5 written in full

The least squares method c e a b d v1 v7 v2 v4 v5 v3 v6 X ij,1 ABCDE A-1111 B-000 C-00 D-0 E-

The least squares method = -2  x ij, 1 (D ij –  x ij,k v k ) = 0 i=1j=1 n n dQ dv 1 k 1 (D AB -  x AB  k v k ) + 1 (D AC -  x AC  k v k )+ 1 (D AD -  x AD  k v k )+ 1 (D AE -  x AE  k v k ) + 0 (D BC -  x BC  k v k ) + 0 (D BD -  x BD  k v k )+ 0 (D BE -  x BE  k v k ) + 0 (D CD -  x CD  k v k ) + 0 (D CE -  x CE  k v k ) + 0 (D DE -  x DE  k v k ) = 0 X ij,1 ABCDE A-1111 B-000 C-00 D-0 E- many terms are zero

The least squares method = -2  x ij, 1 (D ij –  x ij,k v k ) = 0 i=1j=1 n n dQ dv 1 k (D AB -  x AB,k v k ) + (D AC -  x AC  k v k ) + (D AD -  x AD  k v k ) + (D AE -  x AE  k v k ) = 0 c e a b d v1 v7 v2 v4 v5 v3 v6 =1v 1 + 1v 2 + 0v 3 + 0v 4 + 0*v 5 + 0v 6 + 1*v 7 non-zero terms expanded

The least squares method = -2  x ij, 1 (D ij –  x ij,k v k ) = 0 i=1j=1 n n dQ dv 1 k (D AB -  x AB  k v k ) + (D AC -  x AC  k v k ) + (D AD -  x AD  k v k ) + (D AE -  x AE  k v k ) = 0 c e a b d v1 v7 v2 v4 v5 v3 v6 =1v 1 + 0v 2 + 1v 3 + 0v 4 + 0*v 5 + 1v 6 + 0*v 7

The least squares method = -2  x ij, 1 (D ij –  x ij,k v k ) = 0 i=1j=1 n n dQ dv 1 k (D AB -  x AB  k v k ) + (D AC -  x AC  k v k ) + (D AD -  x AD  k v k ) + (D AE -  x AE  k v k ) = 0 D AB + D AC + D AD + D AE – 4v 1 – v 2 – v 3 – v 4 – v 5 – 2v 6 – 2v 7 = 0 D AB + D AC + D AD + D AE = 4v 1 + v 2 + v 3 + v 4 + v 5 + 2v 6 + 2v 7 rearranging to

The least squares method = -2  x ij, 1 (D ij –  x ij,k v k ) = 0 i=1j=1 n n dQ dv 1 k (D AB -  x AB  k v k ) + (D AC -  x AC  k v k ) + (D AD -  x AD  k v k ) + (D AE -  x AE  k v k ) = 0 D AB + D AC + D AD + D AE – 4v 1 – v 2 – v 3 – v 4 – v 5 – 2v 6 – 2v 7 = 0 D AB + D AC + D AD + D AE = 4v 1 + v 2 + v 3 + v 4 + v 5 + 2v 6 + 2v 7 equation for v1

The least squares method D AB + D AC + D AD + D AE = 4v 1 + v 2 + v 3 + v 4 + v 5 + 2v 6 + 2v 7 D AB + D BC + D BD + D BE = v 1 + 4v 2 + v 3 + v 4 + v 5 + 2v 6 + 3v 7 equation for v1 equation for v2 mutatis mutandis for v2

The least squares method D AB + D AC + D AD + D AE = 4v 1 + v 2 + v 3 + v 4 + v 5 + 2v 6 + 2v 7 D AB + D BC + D BD + D BE = v 1 + 4v 2 + v 3 + v 4 + v 5 + 2v 6 + 3v 7 D AC + D BC + D CD + D DE = v 1 + v 2 + 4v 3 + v 4 + v 5 + 3v 6 + 2v 7 D AD + D BD + D CD + D DE = v 1 + v 2 + v 3 + 4v 4 + v 5 + 2v 6 + 3v 7 D AE + D BE + D CE + D DE = v 1 + v 2 + v 3 + v 4 + 4v 5 + 3v 6 + 2v 7 D AC + D AE + D CE + D BE + D CD + D DE = 2v 1 + 2v 2 + 3v 3 + 2v 4 + 3v 5 + 6v 6 + 4v 7 D AB + D AD + D BC + D CD + D BE + D DE = 2v 1 + 3v 2 + 2v 3 + 3v 4 + 2v 5 + 4v 6 + 6v 7 equation for v1 equation for v2 v3 v4 v5 v6 v7 and all other branches

The least squares method solving linear equations with matrices x + 2y = 4 3x - 5y = 1 1 2 3 -5 4141 A == B A -1 = -5 -2 -3 1 1 | A | = 1 1*(-5)- 3*2 -5 -2 -3 1 = - -5 -2 -3 1 1 11 X = A -1 B = - -5 -2 -3 1 1 11 4141 = - 1 11 -22 -11 2121 =

Clustering algorithms clustering methods have no criterion but apply algorithms to come up with trees

Clustering algorithms: UPGMA an ultrametric tree UPGMA assumes that evolutionary rates are the same in all lineages Unweighted Pair Group Method with Arithmetic mean

Clustering algorithms: UPGMA dogbearraccoonweaselsealsea lioncatmonkey dog 0324851504898148 bear 3202634293384136 raccoon 482604244 92152 weasel 5134420443886142 seal 502944 02489142 sea lion 4833443824090142 cat 9884928689900148 monkey 148136152142 1480 1.Find species i and j with the smallest distance. 2.Calculate branch length between i and j.

Clustering algorithms: UPGMA 1.Find species i and j with the smallest distance. 2.Calculate branch length between i and j. sea lionseal 12

Clustering algorithms: UPGMA dogbearraccoonweaselsealsea lioncatmonkey dog 0324851504898148 bear 3202634293384136 raccoon 482604244 92152 weasel 5134420443886142 seal 502944 02489142 sea lion 4833443824090142 cat 9884928689900148 monkey 148136152142 1480 1.Find species i and j with the smallest distance. 2.Calculate branch length between i and j. 3.Lump i and j into a new group. dogbearraccoonweaselSScatmonkey dog 032485198148 bear 320263484136 raccoon 482604292152 weasel 513442086142 SS 0 cat 988492860148 monkey 1481361521421480

Clustering algorithms: UPGMA dogbearraccoonweaselsealsea lioncatmonkey dog 0324851504898148 bear 3202634293384136 raccoon 482604244 92152 weasel 5134420443886142 seal 502944 02489142 sea lion 4833443824090142 cat 9884928689900148 monkey 148136152142 1480 1.Find species i and j with the smallest distance. 2.Calculate branch length between i and. 3.Lump i and j into a new group. 4.Compute distance between new group and all other groups (weigh for number of species in groups). dogbearraccoonweaselSScatmonkey dog 032485198148 bear 320263484136 raccoon 482604292152 weasel 513442086142 SS 0 cat 988492860148 monkey 1481361521421480

Clustering algorithms: UPGMA dogbearraccoonweaselsealsea lioncatmonkey dog 0324851504898148 bear 3202634293384136 raccoon 482604244 92152 weasel 5134420443886142 seal 502944 02489142 sea lion 4833443824090142 cat 9884928689900148 monkey 148136152142 1480 dogbearraccoonweaselSScatmonkey dog 03248514998148 bear 320263484136 raccoon 482604292152 weasel 513442086142 SS 0 cat 988492860148 monkey 1481361521421480 1.Find species i and j with the smallest distance. 2.Calculate branch length between i and. 3.Lump i and j into a new group. 4.Compute distance between new group and all other groups (weigh for number of species in groups).

Clustering algorithms: UPGMA dogbearraccoonweaselsealsea lioncatmonkey dog 0324851504898148 bear 3202634293384136 raccoon 482604244 92152 weasel 5134420443886142 seal 502944 02489142 sea lion 4833443824090142 cat 9884928689900148 monkey 148136152142 1480 dogbearraccoonweaselSScatmonkey dog 03248514998148 bear 32026343184136 raccoon 482604292152 weasel 513442086142 SS 0 cat 988492860148 monkey 1481361521421480 1.Find species i and j with the smallest distance. 2.Calculate branch length between i and. 3.Lump i and j into a new group. 4.Compute distance between new group and all other groups (weigh for number of species in groups).

Clustering algorithms: UPGMA dogbearraccoonweaselsealsea lioncatmonkey dog 0324851504898148 bear 3202634293384136 raccoon 482604244 92152 weasel 5134420443886142 seal 502944 02489142 sea lion 4833443824090142 cat 9884928689900148 monkey 148136152142 1480 dogbearraccoonweaselSScatmonkey dog 03248514998148 bear 32026343184136 raccoon 48260424492152 weasel 51344204186142 SS 49314441089.5142 cat 9884928689.50148 monkey 148136152142 1480 1.Find species i and j with the smallest distance. 2.Calculate branch length between i and. 3.Lump i and j into a new group. 4.Compute distance between new group and all other groups (weigh for number of species in groups).

Clustering algorithms: UPGMA 1.Find species i and j with the smallest distance. 2.Calculate branch length between i and j. dogbearraccoonweaselSScatmonkey dog 03248514998148 bear 32026343184136 raccoon 48260424492152 weasel 51344204186142 SS 49314441089.5142 cat 9884928689.50148 monkey 148136152142 1480

Clustering algorithms: UPGMA 1.Find species i and j with the smallest distance. 2.Calculate branch length between i and j. sea lionseal 12 raccoon bear 13

Clustering algorithms: UPGMA dogbearraccoonweaselSScatmonkey dog 03248514998148 bear 32026343184136 raccoon 48260424492152 weasel 51344204186142 SS 49314441089.5142 cat 9884928689.50148 monkey 148136152142 1480 dogBRweaselSScatmonkey dog 040514998148 BR 4003837.588144 weasel 513804186142 SS 4937.541089.5142 cat 98888689.50148 monkey 148144142 1480 1.Find species i and j with the smallest distance. 2.Calculate branch length between i and. 3.Lump i and j into a new group. 4.Compute distance between new group and all other groups (weigh for number of species in groups).

Clustering algorithms: UPGMA 1.Find species i and j with the smallest distance. 2.Calculate branch length between i and j. dogBRweaselSScatmonkey dog 040514998148 BR 4003837.588144 weasel 513804186142 SS 4937.541089.5142 cat 98888689.50148 monkey 148144142 1480

Clustering algorithms: UPGMA 1.Find species i and j with the smallest distance. 2.Calculate branch length between i and j. sea lionseal 12 raccoon bear 13 18.75 6.75 5.75

Clustering algorithms: UPGMA dogBRweaselSScatmonkey dog 040514998148 BR 4003837.588144 weasel 513804186142 SS 4937.541089.5142 cat 98888689.50148 monkey 148144142 1480 dogBRSSweaselcatmonkey dog 044.55198148 BRSS 44.5039.588.75143 weasel 5139.5086142 cat 9888.75860148 monkey 1481431421480 1.Find species i and j with the smallest distance. 2.Calculate branch length between i and. 3.Lump i and j into a new group. 4.Compute distance between new group and all other groups (weigh for number of species in groups).

Clustering algorithms: UPGMA 1.Find species i and j with the smallest distance. 2.Calculate branch length between i and j. dogBRSSweaselcatmonkey dog 044.55198148 BRSS 44.5039.588.75143 weasel 5139.5086142 cat 9888.75860148 monkey 1481431421480

Clustering algorithms: UPGMA 1.Find species i and j with the smallest distance. 2.Calculate branch length between i and j. sea lionseal 12 raccoon bear 13 19.75 6.75 5.75 weasel

Clustering algorithms: UPGMA 1.Find species i and j with the smallest distance. 2.Calculate branch length between i and j. Lump i and j into a new group. 3.Lump i and j into a new group. 4.Compute distance between new group and all other groups (weigh for number of species in groups). dogBRSSweaselcatmonkey dog 044.55198148 BRSS 44.5039.588.75143 weasel 5139.5086142 cat 9888.75860148 monkey 1481431421480 dogBRSSWcatmonkey dog 098148 BRSSW 0 cat 980148 monkey 148 0 = (4*44.5 + 1*51)/5 4 species in BRSS 1 species in weasel

Clustering algorithms: UPGMA 1.Find species i and j with the smallest distance. 2.Calculate branch length between i and j. Lump i and j into a new group. 3.Lump i and j into a new group. 4.Compute distance between new group and all other groups (weigh for number of species in groups). dogBRSSweaselcatmonkey dog 044.55198148 BRSS 44.5039.588.75143 weasel 5139.5086142 cat 9888.75860148 monkey 1481431421480 dogBRSSWcatmonkey dog 045.898148 BRSSW 45.80 cat 980148 monkey 148 0 = (4*44.5 + 1*51)/5 4 species in BRSS 1 species in weasel

Clustering algorithms: UPGMA 1.Find species i and j with the smallest distance. 2.Calculate branch length between i and j. Lump i and j into a new group. 3.Lump i and j into a new group. 4.Compute distance between new group and all other groups (weigh for number of species in groups). dogBRSSweaselcatmonkey dog 044.55198148 BRSS 44.5039.588.75143 weasel 5139.5086142 cat 9888.75860148 monkey 1481431421480 dogBRSSWcatmonkey dog 045.898148 BRSSW 45.8088.2142.8 cat 9888.20148 monkey 148142.81480

Clustering algorithms: UPGMA 1.Find species i and j with the smallest distance. 2.Calculate branch length between i and j. Lump i and j into a new group. dogBRSSWcatmonkey dog 045.898148 BRSSW 45.8088.2142.8 cat 9888.20148 monkey 148142.81480

Clustering algorithms: UPGMA 1.Find species i and j with the smallest distance. 2.Calculate branch length between i and j. sea lionseal 12 raccoon bear 13 19.75 6.75 5.75 weasel dog 22.9

Clustering algorithms: UPGMA 1.Find species i and j with the smallest distance. 2.Calculate branch length between i and j. Lump i and j into a new group. 3.Lump i and j into a new group. 4.Compute distance between new group and all other groups (weigh for number of species in groups). dogBRSSWcatmonkey dog 045.898148 BRSSW 45.8088.2142.8 cat 9888.20148 monkey 148142.81480 BRSSWDcatmonkey BRSSWD 0 cat 0148 monkey 1480 = (5*88.2 + 1*98)/6 1 species in dog 5 species in BRSSW

Clustering algorithms: UPGMA 1.Find species i and j with the smallest distance. 2.Calculate branch length between i and j. Lump i and j into a new group. 3.Lump i and j into a new group. 4.Compute distance between new group and all other groups (weigh for number of species in groups). dogBRSSWcatmonkey dog 045.898148 BRSSW 45.8088.2142.8 cat 9888.20148 monkey 148142.81480 BRSSWDcatmonkey BRSSWD 089.833 cat 89.8330148 monkey 1480 = (5*88.2 + 1*98)/6 1 species in dog 5 species in BRSSW

Clustering algorithms: UPGMA 1.Find species i and j with the smallest distance. 2.Calculate branch length between i and j. Lump i and j into a new group. 3.Lump i and j into a new group. 4.Compute distance between new group and all other groups (weigh for number of species in groups). dogBRSSWcatmonkey dog 045.898148 BRSSW 45.8088.2142.8 cat 9888.20148 monkey 148142.81480 BRSSWDcatmonkey BRSSWD 089.833143.66 cat 89.8330148 monkey 143.661480 = (5*88.2 + 1*98)/6 1 species in dog 5 species in BRSSW

Clustering algorithms: UPGMA 1.Find species i and j with the smallest distance. 2.Calculate branch length between i and j. Lump i and j into a new group. BRSSWDcatmonkey BRSSWD 089.833143.66 cat 89.8330148 monkey 143.661480

Clustering algorithms: UPGMA 1.Find species i and j with the smallest distance. 2.Calculate branch length between i and j. sea lionseal 12 raccoon bear 13 19.75 6.75 5.75 weasel dog 22.9 cat 44.9166 22.0166

Clustering algorithms: UPGMA 1.Find species i and j with the smallest distance. 2.Calculate branch length between i and j. Lump i and j into a new group. 3.Lump i and j into a new group. 4.Compute distance between new group and all other groups (weigh for number of species in groups). BRSSWDcatmonkey BRSSWD 089.833143.66 cat 89.8330148 monkey 143.661480 BRSSWDmonkey BRSSWD 0 monkey 0 = (6*143.66 + 1*148)/7 1 species in cat 6 species in BRSSWD

Clustering algorithms: UPGMA 1.Find species i and j with the smallest distance. 2.Calculate branch length between i and j. Lump i and j into a new group. 3.Lump i and j into a new group. 4.Compute distance between new group and all other groups (weigh for number of species in groups). BRSSWDcatmonkey BRSSWD 089.833143.66 cat 89.8330148 monkey 143.661480 BRSSWDmonkey BRSSWD 0144.2857 monkey 144.28570 = (6*143.66 + 1*148)/7 1 species in cat 6 species in BRSSWD

Clustering algorithms: UPGMA 1.Find species i and j with the smallest distance. 2.Calculate branch length between i and j. sea lionseal 12 raccoonbear 13 19.75 6.75 5.75 weaseldog 22.9 cat 44.9166 22.0166 monkey 72.142827.22619

Clustering algorithms: Neighbour-joining 1.Calculate S x = (  D x )/(n-2) dogbearraccoonweaselsealsea lioncatmonkey dog 0324851504898148 bear 3202634293384136 raccoon 482604244 92152 weasel 5134420443886142 seal 502944 02489142 sea lion 4833443824090142 cat 9884928689900148 monkey 148136152142 1480 79.2 62.3 74.7 72.8 70.3 69.8 114.5 168.3 79.262.374.772.870.369.8114.5168.3

Clustering algorithms: Neighbour-joining 1.Calculate S x = (  D x )/(n-2) 2.Calculate M ij = D ij -S i -S j and select pair with smallest M ij dogbearraccoonweaselsealsea lioncatmonkey dog 0324851504898148 bear 3202634293384136 raccoon 482604244 92152 weasel 5134420443886142 seal 502944 02489142 sea lion 4833443824090142 cat 9884928689900148 monkey 148136152142 1480 79.262.374.772.870.369.8114.5168.3 dogbearraccoonweaselsealsea lioncatmonkey dog -109.50 bear raccoon weasel seal sea lion cat monkey 32 - 79.2 - 62.3 = -109.5 32 - 79.2 - 62.3 = -109.5

Clustering algorithms: Neighbour-joining 1.Calculate S x = (  D x )/(n-2) 2.Calculate M ij = D ij -S i -S j and select pair with smallest M ij dogbearraccoonweaselsealsea lioncatmonkey dog 0324851504898148 bear 3202634293384136 raccoon 482604244 92152 weasel 5134420443886142 seal 502944 02489142 sea lion 4833443824090142 cat 9884928689900148 monkey 148136152142 1480 79.2 62.3 74.7 72.8 70.3 69.8 114.5 168.3 79.262.374.772.870.369.8114.5168.3 dogbearraccoonweaselsealsea lioncatmonkey dog -109.50-105.83-101.00-99.50-101.00-95.67-99.50 bear -109.50-111.00-101.17-103.67-99.17-92.83-94.67 raccoon -105.83-111.00-105.50-101.00-100.50-97.17-91.00 weasel -101.00-101.17-105.50-99.17-104.67-101.33-99.17 seal -99.50-103.67-101.00-99.17-116.17-95.83-96.67 sea lion -101.00-99.17-100.50-104.67-116.17-94.33-96.17 cat -95.67-92.83-97.17-101.33-95.83-94.33-134.83 monkey -99.50-94.67-91.00-99.17-96.67-96.17-134.83

Clustering algorithms: Neighbour-joining 1.Calculate S x = (  D x )/(n-2) 2.Calculate M ij = D ij -S i -S j and select pair with smallest M ij 3.Create a node that joins this pair and calculate branch lengths as (D ij /2)+(S i -S j )/2 dogbearraccoonweaselsealsea lioncatmonkey dog 0324851504898148 bear 3202634293384136 raccoon 482604244 92152 weasel 5134420443886142 seal 502944 02489142 sea lion 4833443824090142 cat 9884928689900148 monkey 148136152142 1480 79.2 62.3 74.7 72.8 70.3 69.8 114.5 168.3 79.262.374.772.870.369.8114.5168.3 branch length cat-cm = 148/2 + (114.5-168.5)/2 = 47.08

Clustering algorithms: Neighbour-joining 1.Calculate S x = (  D x )/(n-2) 2.Calculate M ij = D ij -S i -S j and select pair with smallest M ij 3.Create a node that joins this pair and calculate branch lengths as (D ij /2)+(S i -S j )/2 dogbearraccoonweaselsealsea lioncatmonkey dog 0324851504898148 bear 3202634293384136 raccoon 482604244 92152 weasel 5134420443886142 seal 502944 02489142 sea lion 4833443824090142 cat 9884928689900148 monkey 148136152142 1480 79.2 62.3 74.7 72.8 70.3 69.8 114.5 168.3 79.262.374.772.870.369.8114.5168.3 branch length cat-cm = 148/2 + (114.5-168.5)/2 = 47.08 branch length monkey-cm = 148/2 + (168.5-114.5)/2 = 110.92

Clustering algorithms: Neighbour-joining cat sea lion seal monkey weasel bear raccoon dog 1.Calculate S x = (  D x )/(n-2) 2.Calculate M ij = D ij -S i -S j and select pair with smallest M ij 3.Create a node that joins this pair and calculate branch lengths as (D ij /2)+(S i -S j )/2 4.Join the two species and make all other taxa in form of a star.

Clustering algorithms: Neighbour-joining cat sea lion seal monkey weasel bear raccoon dog cm 47.08 100.92 1.Calculate S x = (  D x )/(n-2) 2.Calculate M ij = D ij -S i -S j and select pair with smallest M ij 3.Create a node that joins this pair and calculate branch lengths as (D ij /2)+(S i -S j )/2 4.Join the two species and make all other taxa in form of a star.

Clustering algorithms: Neighbour-joining dogbearraccoonweaselsealsea lioncatmonkey dog 0324851504898148 bear 3202634293384136 raccoon 482604244 92152 weasel 5134420443886142 seal 502944 02489142 sea lion 4833443824090142 cat 9884928689900148 monkey 148136152142 1480 dogbearraccoonweaselsealsea lioncm dog03248515048 49 bear32026342933 raccoon482604244 weasel51344204438 seal502944 024 sea lion48334438240 cm 1.Calculate S x = (  D x )/(n-2) 2.Calculate M ij = D ij -S i -S j and select pair with smallest M ij 3.Create a node that joins this pair and calculate branch lengths as (D ij /2)+(S i -S j )/2 4.Join the two species and make all other taxa in form of a star. 5.Create a new matrix. Calculate the distances between the new node and other taxa as D xij =(D ix +D jx -D ij )/2 (98+148-148)/2 = 49 (98+148-148)/2 = 49

Clustering algorithms: Neighbour-joining dogbearraccoonweaselsealsea lioncatmonkey dog 0324851504898148 bear 3202634293384136 raccoon 482604244 92152 weasel 5134420443886142 seal 502944 02489142 sea lion 4833443824090142 cat 9884928689900148 monkey 148136152142 1480 dogbearraccoonweaselsealsea lioncm dog03248515048 49 bear32026342933 36 raccoon482604244 48 weasel51344204438 40 seal502944 024 41.5 sea lion48334438240 42 cm 4936484041.5420 1.Calculate S x = (  D x )/(n-2) 2.Calculate M ij = D ij -S i -S j and select pair with smallest M ij 3.Create a node that joins this pair and calculate branch lengths as (D ij /2)+(S i -S j )/2 4.Join the two species and make all other taxa in form of a star. 5.Create a new matrix. Calculate the distances between the new node and other taxa as D xij =(D ix +D jx -D ij )/2 (98+148-148)/2 = 49 (98+148-148)/2 = 49

Clustering algorithms: Neighbour-joining dogbearraccoonweaselsealsea lioncm dog 03248515048 49 bear 32026342933 36 raccoon 482604244 48 weasel 51344204438 40 seal 502944 024 41.5 sea lion 48334438240 42 cm 4936484041.5420 55.6 38 50.4 49.8 46.5 45.8 51.3 55.63850.449.846.545.851.3 1.Calculate S x = (  D x )/(n-2)

Clustering algorithms: Neighbour-joining dogbearraccoonweaselsealsea lioncm dog 03248515048 49 bear 32026342933 36 raccoon 482604244 48 weasel 51344204438 40 seal 502944 024 41.5 sea lion 48334438240 42 cm 4936484041.5420 55.6 38 50.4 49.8 46.5 45.8 51.3 55.63850.449.846.545.851.3 1.Calculate S x = (  D x )/(n-2) 2.Calculate M ij = D ij -S i -S j and select pair with smallest M ij dogbearraccoonweaselsealsea lioncm dog -61.60-58.00-54.40-52.10-53.40-57.90 bear -61.60 -62.40-53.80-55.50-50.80-53.30 raccoon -58.00-62.40 -58.20-52.90-52.20-53.70 weasel -54.40-53.80-58.20 -52.30-57.60-61.10 seal -52.10-55.50-52.90-52.30 -68.30-56.30 sea lion -53.40-50.80-52.20-57.60-68.30 -55.10 cm -57.90-53.30-53.70-61.10-56.30-55.10

Clustering algorithms: Neighbour-joining dogbearraccoonweaselsealsea lioncm dog 03248515048 49 bear 32026342933 36 raccoon 482604244 48 weasel 51344204438 40 seal 502944 024 41.5 sea lion 48334438240 42 cm 4936484041.5420 55.6 38 50.4 49.8 46.5 45.8 51.3 55.63850.449.846.545.851.3 1.Calculate S x = (  D x )/(n-2) 2.Calculate M ij = D ij -S i -S j and select pair with smallest M ij 3.Create a node that joins this pair and calculate branch lengths as (D ij /2)+(S i -S j )/2 branch length seal-ss = 24/2 + (46.5-45.8)/2 = 12.35 branch length sealion-ss = 24/2 + (45.8-46.5)/2 = 11.65

Clustering algorithms: Neighbour-joining cat sea lion seal monkey weasel bear raccoon dog cm 47.08 100.92 ss 1.Calculate S x = (  D x )/(n-2) 2.Calculate M ij = D ij -S i -S j and select pair with smallest M ij 3.Create a node that joins this pair and calculate branch lengths as (D ij /2)+(S i -S j )/2 4.Join the two species and make all other taxa in form of a star.

Clustering algorithms: Neighbour-joining dogbearraccoonweaselsealsea lioncm dog 03248515048 49 bear 32026342933 36 raccoon 482604244 48 weasel 51344204438 40 seal 502944 024 41.5 sea lion 48334438240 42 cm 4936484041.5420 1.Calculate S x = (  D x )/(n-2) 2.Calculate M ij = D ij -S i -S j and select pair with smallest M ij 3.Create a node that joins this pair and calculate branch lengths as (D ij /2)+(S i -S j )/2 4.Join the two species and make all other taxa in form of a star. 5.Create a new matrix. Calculate the distances between the new node and other taxa as D xij =(D ix +D jx -D ij )/2 dogbearraccoonweaselsscm dog0324851 3749 bear3202634 19 36 raccoon4826042 32 48 weasel5134420 29 40 ss 37193229029.75 cm 4936484029.750

Clustering algorithms: Neighbour-joining cat sea lion seal monkey weasel bear raccoon dog cm 47.08 100.92 ss br Round 3 bear+raccoon

Clustering algorithms: Neighbour-joining cat sea lion seal monkey weasel bear raccoon dog cm 47.08 100.92 ss br brd Round 4 (bear+raccoon)+dog

Clustering algorithms: Neighbour-joining cat sea lion seal monkey weasel bear raccoon dog cm 47.08 100.92 ss br brd cmw Round 5 (cat+monkey)+weasel

Clustering algorithms: Neighbour-joining cat sea lion seal monkey weasel bear raccoon dog cm 47.08 100.92 ss br bdr cmw bdrss Round 6 (seal+sealion)+(bear+raccoon+dog)

Clustering algorithms: Neighbour-joining cat sea lion seal monkey weasel bear raccoon dog cm 47.08 100.92 ss br bdr cmw bdrss

Clustering algorithms: Neighbour-joining cat sea lion seal monkey weasel bear raccoon dog sea lionsealraccoonbearweaseldogcatmonkey UPGMA

Distance matrix methods calculate a measure of distance between each pair of species, then find a tree that predicts the observed set of distances.

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Distance matrix methods calculate a measure of distance between each pair of species, then find a tree that predicts the observed set of distances.

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Presentation on theme: "Distance matrix methods calculate a measure of distance between each pair of species, then find a tree that predicts the observed set of distances."— Presentation transcript:

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