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

The least squares method c e a b d ABCDE A0 B0 C0 D0 E Expected matrix

The least squares method c e a b d ABCDE A00.23 B0 C0 D0 E Expected matrix

The least squares method c e a b d ABCDE A B C D E 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 = A == B A -1 = | A | = 1 1*(-5)- 3* = X = A -1 B = = =

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 bear raccoon weasel seal sea lion cat monkey 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 bear raccoon weasel seal sea lion cat monkey 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 bear raccoon weasel SS 0 cat monkey

Clustering algorithms: UPGMA dogbearraccoonweaselsealsea lioncatmonkey dog bear raccoon weasel seal sea lion cat monkey 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 bear raccoon weasel SS 0 cat monkey

Clustering algorithms: UPGMA dogbearraccoonweaselsealsea lioncatmonkey dog bear raccoon weasel seal sea lion cat monkey dogbearraccoonweaselSScatmonkey dog bear raccoon weasel SS 0 cat monkey 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 bear raccoon weasel seal sea lion cat monkey dogbearraccoonweaselSScatmonkey dog bear raccoon weasel SS 0 cat monkey 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 bear raccoon weasel seal sea lion cat monkey dogbearraccoonweaselSScatmonkey dog bear raccoon weasel SS cat monkey 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 bear raccoon weasel SS cat monkey

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 bear raccoon weasel SS cat monkey dogBRweaselSScatmonkey dog BR weasel SS cat monkey 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 BR weasel SS cat monkey

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

Clustering algorithms: UPGMA dogBRweaselSScatmonkey dog BR weasel SS cat monkey dogBRSSweaselcatmonkey dog BRSS weasel cat monkey 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 BRSS weasel cat monkey

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 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 BRSS weasel cat monkey dogBRSSWcatmonkey dog BRSSW 0 cat monkey = (4* *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 BRSS weasel cat monkey dogBRSSWcatmonkey dog BRSSW cat monkey = (4* *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 BRSS weasel cat monkey dogBRSSWcatmonkey dog BRSSW cat monkey

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 BRSSW cat monkey

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 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 BRSSW cat monkey BRSSWDcatmonkey BRSSWD 0 cat 0148 monkey 1480 = (5* *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 BRSSW cat monkey BRSSWDcatmonkey BRSSWD cat monkey 1480 = (5* *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 BRSSW cat monkey BRSSWDcatmonkey BRSSWD cat monkey = (5* *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 cat monkey

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 weasel dog 22.9 cat

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 cat monkey BRSSWDmonkey BRSSWD 0 monkey 0 = (6* *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 cat monkey BRSSWDmonkey BRSSWD monkey = (6* *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 weaseldog 22.9 cat monkey

Clustering algorithms: Neighbour-joining 1.Calculate S x = (  D x )/(n-2) dogbearraccoonweaselsealsea lioncatmonkey dog bear raccoon weasel seal sea lion cat monkey

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 bear raccoon weasel seal sea lion cat monkey dogbearraccoonweaselsealsea lioncatmonkey dog bear raccoon weasel seal sea lion cat monkey = =

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 bear raccoon weasel seal sea lion cat monkey dogbearraccoonweaselsealsea lioncatmonkey dog bear raccoon weasel seal sea lion cat monkey

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 bear raccoon weasel seal sea lion cat monkey branch length cat-cm = 148/2 + ( )/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 bear raccoon weasel seal sea lion cat monkey branch length cat-cm = 148/2 + ( )/2 = branch length monkey-cm = 148/2 + ( )/2 =

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 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 bear raccoon weasel seal sea lion cat monkey dogbearraccoonweaselsealsea lioncm dog bear raccoon weasel seal sea lion 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 ( )/2 = 49 ( )/2 = 49

Clustering algorithms: Neighbour-joining dogbearraccoonweaselsealsea lioncatmonkey dog bear raccoon weasel seal sea lion cat monkey dogbearraccoonweaselsealsea lioncm dog bear raccoon weasel seal sea lion cm 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 ( )/2 = 49 ( )/2 = 49

Clustering algorithms: Neighbour-joining dogbearraccoonweaselsealsea lioncm dog bear raccoon weasel seal sea lion cm Calculate S x = (  D x )/(n-2)

Clustering algorithms: Neighbour-joining dogbearraccoonweaselsealsea lioncm dog bear raccoon weasel seal sea lion cm 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 bear raccoon weasel seal sea lion cm

Clustering algorithms: Neighbour-joining dogbearraccoonweaselsealsea lioncm dog bear raccoon weasel seal sea lion cm 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 + ( )/2 = branch length sealion-ss = 24/2 + ( )/2 = 11.65

Clustering algorithms: Neighbour-joining cat sea lion seal monkey weasel bear raccoon dog cm 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 bear raccoon weasel seal sea lion cm 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 dog bear raccoon weasel ss cm

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

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

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

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

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