Evolutionary Tree Reconstruction. Distance-based methods. Multiple Sequence Alignment. Distance-based methods. How distance matrices are obtained

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1 Eoltionar ree Reconstrction ien obserations of similarities or differences beteen k species, find the hpothesis (tree) that best eplains the data ith respect to some criterion: Maimm parsimon (character data) Minimm eoltion (distance data) Maimm Likelihood (character data) ssmptions: election dominates Mtations are rare No mltiple sbstittions Parsimon: haracter data Find the tree that reqires the feest changes to eplain the data ssmptions: Netral mtation dominates Mltiple sbstittions occr Minimm Eoltion: Distance data Find the tree that s the pairise distances beteen taa Maimm Likelihood: haracter data Find the most likel tree Distance-based methods Ho to obtain a distance matri orrecting for mltiple sbstittions onditions for obtaining an eact fit dditie distances Minimm Eoltion: finding the tree ith the reed algorithms UPM Ho distance matrices are obtained ien seqences from k taa onstrct a mltiple seqence alignment Determine pairise distance from each pair of taa sing the M orrect for mltiple sbstittions Mltiple eqence lignment ~~~~LEKQELLKQWEVLKQNIPHLRLFLIIE ~~~MLEKQELLKQWEVLKQNIPHLRLFLILE ~~~MLERQELLKQWEVLKQNIPHLRLFLIIE ~~~~~~~~~~ELLKQWEVLKQNIPHLLFLIIE he distance beteen taon i and taon j is the distance of the pairise alignment indced b the M. Distance-based methods Ho to obtain a distance matri orrecting for mltiple sbstittions onditions for obtaining an eact fit dditie distances Minimm Eoltion: finding the tree ith the reed algorithms UPM 1

2 bstittion patterns bstittion patterns ingle sbstittion: 1 change, 1 difference Mltiple sbstittion: changes, 1 difference oincidental sbstittion: bstittion patterns bstittion patterns oincidental sbstittion: changes, 1 difference Parallel sbstittion: oincidental sbstittion: changes, 1 difference Parallel sbstittion: changes, no difference bstittion patterns bstittion patterns onergent sbstittion: onergent sbstittion: 3 changes, no difference ack sbstittion:......

3 bstittion patterns Mltiple sbstittions onergent sbstittion: 3 changes, no difference ack sbstittion: changes, no difference eqence difference atration ime since diergence orrecting for mltiple sbstittions Distance-based methods eqence difference Epected difference ime since diergence orrection Obsered difference Ho to obtain a distance matri orrecting for mltiple sbstittions Fitting distances to a tree onditions for obtaining an eact fit dditie distances Minimm Eoltion: finding the tree ith the reed algorithms UPM Match distance matri to branch lengths arp ebrafish almon rot arp ebrafish almon 0 6 rot 0 Obsered distances arp ebrafish almon rot arp ebrafish almon 0 6 rot 0 Obsered distances + = 3 carp ebrafish trot salmon + + = = = = 8 + = 6 carp ebrafish trot salmon 3

4 arp ebrafish almon rot arp ebrafish almon 0 6 rot 0 an eer matri be fitted to a tree? trot carp ebrafish salmon an eer matri be fitted to a tree? = 7 + = + + = = = = 3 + = = 6 dditie Matrices: matri can be fitted to a tree, if and onl if the eqations + = + + = = = = 3 + = hae a soltion. matri is additie if and onl if it satisfies the for point condition. For point condition: +D <= ma(+d,d+) +D <= ma(+d,d+) D+ <= ma(+d,+d) D D 0 he for-point condition: a test for additiit +D <= ma(+d,d+) +D <= ma(+d,d+) D+ <= ma(+d,+d) + D + D D + D D D 4

5 +D <= ma(+d,d+) +D <= ma(+d,d+) D D 0 +D <= ma(+d,d+) +D <= ma(+d,d+) D D 0 D+ <= ma(+d,+d) D+ <= ma(+d,+d) Does this matri satisf the for point condition? Does this matri satisf the for point condition? We kno the matri arp almon ebrafish rot arp almon ebrafish 0 8 rot 0 +D <= ma(+d,d+) arp almon ebrafish rot arp almon ebrafish 0 8 rot 0 We don t kno the tree topolog +D <= ma(+d,d+) D+ <= ma(+d,+d) he for-point condition also gies the topolog: + D + D D D + he matri is additie he for point condition holds: +D <= ma(+d,d+) +D <= ma(+d,d+) D+ <= ma(+d,+d) he eqations + = + + = + + = D + + = + + = D + = D Eqialent statements hae a soltion. he topolog and branch lengths are niqel determined. 5

6 Distance-based methods Ho to obtain a distance matri orrecting for mltiple sbstittions onditions for obtaining an eact fit dditie distances Minimm Eoltion: finding the tree ith the Heristics UPM Ultrametric distances onsider a rooted tree ith constant mtation rate on all branches (moleclar clock) Note: 1. ame distance from the root to eer leaf. D[,] < D[,] = D[,] 3. + < ++ = ++ hree point condition We kno the matri We don t kno the tree topolog + < ++ = ++ For eer triple, {,,} in ma(,) ma(,) ma(,) Is the matri ltrametric? Eqialent statements matri is ltrametric satisfies the three point condition fits a rooted tree ith eqal distances from the root to all leaes mtation rates are the same in all lineages. hree point condition an eample For eer triple, {,,} in ma(,) ma(,) ma(,) 6

7 hree point condition an eample hree point condition For eer triple, {,,} in ma(,) ma(,) ma(,) For eer triple, {,,} in ma(,) ma(,) ma(,) hree point condition an eample For eer triple, {,,} in ma(,) ma(,) ma(,) hree point condition another eample ll ltrametric matrices fit rooted trees bt not all rooted trees are ltrametric. If the matri is not ltrametric, the closest pair ma not be neighbors mmar mmar, cont d matri is additie if it satisfies the for point condition. tree defines a tree metric, [i,j]; i.e., the pairise distances beteen all pairs of leaes. ll tree metrics are additie. If a matri, O[i,j], is additie there eists a niqe tree topolog ith branch lengths sch that [i,j] = O[i,j]. his tree can be obtained in polnomial time. In real life, obsered distance matri, O[i,j] is neer additie. matri is ltrametric if it satisfies the three point condition. ll ltrametric matrices fit rooted trees. Not all rooted tree metrics are ltrametric. n ltrametric tree satisfies the moleclar clock hpothesis. ll distances from the root to a leaf are the same. Its branch lengths are proportional to time. For k > 3, ll ltrametric matrices are additie t, an additie matri is not necessaril ltrametric. 7

8 Distance-based methods Ho to obtain a distance matri orrecting for mltiple sbstittions onditions for obtaining an eact fit dditie distances Minimm Eoltion: finding the tree ith the reed algorithms UPM Ho to reconstrct a tree hen O[i,j] is not additie. Ehastie search: onsider all trees and select the tree that is the closest fit; i.e., the difference beteen O[i,j] and [i,j] is smallest Measres of fit: norms L α O, α = ( O[ i, j] [ i, j] i< j α 1 α O, = ma{ O[i,j] [i,j] } ), α = 1,,... Minimm Eoltion ien an obsered distance matri, O[i,j] For eer topolog,, ith k leaes Let D*() be the matri that minimies = O = O[ i, j] [ i, j] i< j L, Let core() be the sm of the branch lengths of D*() he best tree is the tree ith minimm score. Distance-based methods Ho to obtain a distance matri orrecting for mltiple sbstittions onditions for obtaining an eact fit dditie distances Minimm Eoltion: finding the tree ith the reed algorithms UPM Uneighted Pair rop Method ith rithmetic Means UPM 8

SUBJECT:ENGINEERING MATHEMATICS-I SUBJECT CODE :SMT1101 UNIT III FUNCTIONS OF SEVERAL VARIABLES. Jacobians

SUBJECT:ENGINEERING MATHEMATICS-I SUBJECT CODE :SMT0 UNIT III FUNCTIONS OF SEVERAL VARIABLES Jacobians Changing ariable is something e come across er oten in Integration There are man reasons or changing