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Phylogenetic Reconstruction: Parsimony. Anders Gorm Pedersen [email protected] Trees: terminology. Trees: terminology. Trees: terminology. “Reptilia” is a non-monophyletic group. Trees: representations. Three different representations of the same tree-topology. Trees: rooted vs. unrooted.

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Phylogenetic reconstruction parsimony l.jpg

Phylogenetic Reconstruction: Parsimony

Anders Gorm Pedersen

[email protected]




Trees terminology4 l.jpg
Trees: terminology

“Reptilia” is a non-monophyletic group


Trees representations l.jpg
Trees: representations

Three different representations of the same tree-topology


Trees rooted vs unrooted l.jpg
Trees: rooted vs. unrooted

  • A rooted tree has a single node (the root) that represents a point in time that is earlier than any other node in the tree.

  • A rooted tree has directionality (nodes can be ordered in terms of “earlier” or “later”).

  • In the rooted tree, distance between two nodes is represented along the time-axis only (the second axis just helps spread out the leafs)

Early

Late


Trees rooted vs unrooted7 l.jpg
Trees: rooted vs. unrooted

  • A rooted tree has a single node (the root) that represents a point in time that is earlier than any other node in the tree.

  • A rooted tree has directionality (nodes can be ordered in terms of “earlier” or “later”).

  • In the rooted tree, distance between two nodes is represented along the time-axis only (the second axis just helps spread out the leafs)

Early

Late


Trees rooted vs unrooted8 l.jpg
Trees: rooted vs. unrooted

  • A rooted tree has a single node (the root) that represents a point in time that is earlier than any other node in the tree.

  • A rooted tree has directionality (nodes can be ordered in terms of “earlier” or “later”).

  • In the rooted tree, distance between two nodes is represented along the time-axis only (the second axis just helps spread out the leafs)

Early

Late


Trees rooted vs unrooted9 l.jpg
Trees: rooted vs. unrooted

  • In unrooted trees there is no directionality: we do not know if a node is earlier or later than another node

  • Distance along branches directly represents node distance


Trees rooted vs unrooted10 l.jpg
Trees: rooted vs. unrooted

  • In unrooted trees there is no directionality: we do not know if a node is earlier or later than another node

  • Distance along branches directly represents node distance



Cladistics group organisms based on shared derived characters synapomorphies l.jpg
Cladistics: group organisms based on shared, derived characters (“synapomorphies”)


Homology limb structure l.jpg
Homology: limb structure characters (“synapomorphies”)

Homology: any similarity between characters that is due to their shared ancestry


Homology vs homoplasy l.jpg
Homology characters (“synapomorphies”)vs. Homoplasy

X

X

X

X

Homology: similar traits inherited from a common ancestor

Homoplasy: similar traits are

not directly caused by common ancestry (convergent evolution).


Homoplasy wings l.jpg
Homoplasy: wings characters (“synapomorphies”)


Molecular phylogeny l.jpg

A characters (“synapomorphies”) A G C G T T G G G C A A

B A G C G T T T G G C A A

C A G C T T T G T G C A A

D A G C T T T T T G C A A

1 2 3

DNA and protein sequences

Homologous characters inferred from alignment.

Other molecular data: absence/presence of restriction sites, DNA hybridization data, antibody cross-reactivity, etc. (but losing importance due to cheap, efficient sequencing).

Molecular phylogeny


Morphology vs molecular data l.jpg
Morphology vs. molecular data characters (“synapomorphies”)

African white-backed vulture

(old world vulture)

Andean condor

(new world vulture)

New and old world vultures seem to be closely related based on morphology.

Molecular data indicates that old world vultures are related to birds of prey (falcons, hawks, etc.) while new world vultures are more closely related to storks

Similar features presumably the result of convergent evolution


Phylogenetic reconstruction l.jpg
Phylogenetic reconstruction characters (“synapomorphies”)


Phylogenetic reconstruction19 l.jpg
Phylogenetic reconstruction characters (“synapomorphies”)


Parsimony criterion choose simplest hypothesis l.jpg
Parsimony criterion: characters (“synapomorphies”)choose simplest hypothesis


Parsimonious reconstruction l.jpg
Parsimonious reconstruction characters (“synapomorphies”)

A

G..

B

G..

C

T..

D

T..


Parsimonious reconstruction22 l.jpg
Parsimonious reconstruction characters (“synapomorphies”)

A

G..

B

G..

C

T..

D

T..

G..

T..

T..


Parsimonious reconstruction23 l.jpg
Parsimonious reconstruction characters (“synapomorphies”)

A

G..

B

G..

C

T..

D

T..

G..

T..

T..


Alternative tree homoplasy l.jpg
Alternative tree: homoplasy characters (“synapomorphies”)

A

G..

B

G..

C

T..

D

T..

G..

T..

T..

A

G..

C

T..

D

T..

B

G..


Alternative tree homoplasy25 l.jpg
Alternative tree: homoplasy characters (“synapomorphies”)

A

G..

B

G..

C

T..

D

T..

G..

T..

T..

A

G..

C

T..

D

T..

B

G..

T..

T..

T..


Alternative tree homoplasy26 l.jpg
Alternative tree: homoplasy characters (“synapomorphies”)

A

G..

B

G..

C

T..

D

T..

G..

T..

T..

A

G..

D

T..

C

T..

B

G..

T..

T..

T..


One character assumption of no homoplasy is equivalent to finding shortest tree l.jpg
One character: Assumption of no homoplasy is equivalent to finding shortest tree

A

G...

B

G...

C

T...

D

T...

G..

T..

T..

A

G..

C

T..

D

T..

B

G..

T..

T..

T..


Phylogenetic reconstruction28 l.jpg
Phylogenetic reconstruction finding shortest tree

A

..G

B

..G

C

..T

D

..T

..G

..T

..T


Phylogenetic reconstruction29 l.jpg
Phylogenetic reconstruction finding shortest tree

A

G.G

B

G.G

C

T.T

D

T.T

G.G

T.T

T.T


Phylogenetic reconstruction conflicts l.jpg
Phylogenetic reconstruction: conflicts finding shortest tree

A

B

C

D

A

.G.

C

.G.

B

.T.

D

.T.

.G.

.T.

.T.


Phylogenetic reconstruction conflicts31 l.jpg
Phylogenetic reconstruction: conflicts finding shortest tree

A

.G.

B

.T.

C

.G.

D

.T.

.T.

.T.

.T.

A

C

B

D


Phylogenetic reconstruction conflicts32 l.jpg
Phylogenetic reconstruction: conflicts finding shortest tree

A

B

C

D

A

G.G

C

T.T

B

G.G

D

T.T

T.T

T.T

T.T


Several characters choose shortest tree equivalent to fewer assumptions of homoplasy l.jpg
Several characters: choose shortest tree finding shortest tree(equivalent to fewer assumptions of homoplasy)

A

GGG

B

GTG

C

TGT

D

TTT

GTG

Total length of tree: 4

Total length of tree: 5

TTT

TTT

A

GGG

C

TGT

B

GTG

D

TTT

TGT

TTT

TTT


Maximum parsimony l.jpg
Maximum Parsimony finding shortest tree

  • Maximum parsimony: the best tree is the shortest tree (the tree requiring the smallest number of mutational events)

  • This corresponds to the tree that implies the least amount of homoplasy (convergent evolution, reversals)

  • How do we find the best tree for a given data set?


Maximum parsimony first approach l.jpg
Maximum Parsimony: first approach finding shortest tree

  • Construct list of all possible trees for data set

  • For each tree: determine length, add to list of lengths

  • When finished: select shortest tree from list

  • If several trees have the same length, then they are equally good (equally parsimonious)


Maximum parsimony problems l.jpg
Maximum Parsimony: problems finding shortest tree

  • We need algorithm for constructing list of all possible trees

  • We need algorithm for determining length of given tree

  • Should all mutational events have same cost?


Constructing list of all possible unrooted trees l.jpg
Constructing list of all possible unrooted trees finding shortest tree

  • Construct unrooted tree from first three taxa. There is only one way of doing this

  • Starting from (1), construct the three possible derived trees by adding taxon 4 to each internal branch

  • From each of the trees constructed in step (2), construct the five possible derived trees by adding taxon 5 to each internal branch.

  • Continue until all taxa have been added in all possible locations


Maximum parsimony problems38 l.jpg
Maximum Parsimony: problems finding shortest tree

  • We need algorithm for constructing list of all possible trees 

  • We need algorithm for determining length of given tree

  • Should all mutational events have same cost?


Algorithm for determining length of given tree fitch l.jpg
Algorithm for determining length of given tree: Fitch finding shortest tree

C

A

A

G

C

What is the length of this tree? (How many mutational steps are required?)


Algorithm for determining length of given tree fitch40 l.jpg
Algorithm for determining length of given tree: Fitch finding shortest tree

  • Root the tree at an arbitrary internal node (or internal branch)

  • Visit an internal node x for which no state set has been defined, but where the state sets of x’s immediate descendants (y,z) have been defined.

  • If the state sets of y,z have common states, then assign these to x.

    If there are no common states, then assign the union of y,z to x, and increase tree length by one.

  • Repeat until all internal nodes have been visited. Note length of current tree.



Algorithm for determining length of given tree fitch42 l.jpg
Algorithm for determining length of given tree: Fitch finding shortest tree

C

A

A

G

C


Algorithm for determining length of given tree fitch43 l.jpg
Algorithm for determining length of given tree: Fitch finding shortest tree

A

C

C

A

G



Algorithm for determining length of given tree fitch45 l.jpg
Algorithm for determining length of given tree: Fitch finding shortest tree

C

A

C

A

G

Length so far = 0


Algorithm for determining length of given tree fitch46 l.jpg
Algorithm for determining length of given tree: Fitch finding shortest tree

C

A

C

A

G

{C, A}

Length so far = 1


Algorithm for determining length of given tree fitch47 l.jpg
Algorithm for determining length of given tree: Fitch finding shortest tree

C

A

C

A

G

{A, G}

{C, A}

Length so far = 2


Algorithm for determining length of given tree fitch48 l.jpg
Algorithm for determining length of given tree: Fitch finding shortest tree

C

A

C

A

G

{A, G}

{A, C}

{A, C, G}

Length so far = 3


Algorithm for determining length of given tree fitch49 l.jpg
Algorithm for determining length of given tree: Fitch finding shortest tree

C

A

C

A

G

{A, G}

{A, C}

{A, C, G}

Length so far = 3

{A, C}


Algorithm for determining length of given tree fitch50 l.jpg
Algorithm for determining length of given tree: Fitch finding shortest tree

C

A

C

A

G

{A, G}

{A, C}

{A, C, G}

Length of tree = 3

{A, C}


Algorithm for determining length of given tree fitch51 l.jpg
Algorithm for determining length of given tree: Fitch finding shortest tree

C

A

C

A

G

A

A

A

Length of tree = 3

A

One possible reconstruction (several others exist)


Maximum parsimony problems52 l.jpg
Maximum Parsimony: problems finding shortest tree

  • We need algorithm for constructing list of all possible trees 

  • We need algorithm for determining length of given tree 

  • Should all mutational events have same cost?


Mutational events need not have the same cost l.jpg
Mutational events need not have the same cost finding shortest tree

C

A

C

A

G

A C G T

A 0 3 1 3

C 3 0 3 1

G 1 3 0 3

T 3 1 3 0

Sankoff algorithm


Maximum parsimony problems54 l.jpg
Maximum Parsimony: problems finding shortest tree

  • We need algorithm for constructing list of all possible trees 

  • We need algorithm for determining length of given tree 

  • Should all mutational events have same cost? 


How many branches are there on an unrooted tree with x tips l.jpg
How many branches are there on an unrooted tree with x tips? finding shortest tree

  • There is only one way of con-structing the first tree. This tree has 3 tips and 3 branches

  • Each time an extra taxon is added, two branches are created.

  • A tree with x tips will therefore have the following number of branches:

    nbranches = 3+(x-3)*2

    = 3+2x-6

    = 2x-3

A

B

C

A

B

D

C


How many unrooted trees are there l.jpg
How many unrooted trees are there? finding shortest tree

  • A tree with x tips has 2x-3 branches

  • For each tree with x=n-1 tips, we can therefore construct 2(n-1)-3 derived trees (with n tips).

  • The number of unrooted trees with n tips is therefore:

    (2i-3) = 1 x 3 x 5 x 7 x ...

n-1

i=2



Branch and bound shortcut to perfection step 1 construction of initial tree by sequential addition l.jpg
Branch and bound: shortcut to perfection finding shortest treeStep 1: Construction of initial tree by sequential addition


Branch and bound shortcut to perfection step 2 search treespace discard suboptimal parts l.jpg
Branch and bound: shortcut to perfection finding shortest treeStep 2: search treespace, discard suboptimal parts


Heuristic search l.jpg
Heuristic search finding shortest tree

  • Construct initial tree (e.g., sequential addition); determine length

  • Construct set of “neighboring trees” by making small rearrangements of initial tree; determine lengths

  • If any of the neighboring trees are better than the initial tree, then select it/them and use as starting point for new round of rearrangements. (Possibly several neighbors are equally good)

  • Repeat steps 2+3 until you have found a tree that is better than all of its neighbors.

  • This tree is a “local optimum” (not necessarily a global optimum!)


Heuristic search hill climbing l.jpg
Heuristic search: hill-climbing finding shortest tree


Heuristic search local vs global optimum l.jpg
Heuristic search: local finding shortest treevs. global optimum


Types of rearrangement i nearest neighbor interchange nni l.jpg
Types of rearrangement I: finding shortest treenearest neighbor interchange (NNI)

Original tree

Two neighbors per internal branch:

tree with n tips has 2(n-3) neighbors

(For example, a tree with 20 tips has 34 neighbbors)


Types of rearrangement ii subtree pruning and regrafting spr l.jpg
Types of rearrangement II: finding shortest treesubtree pruning and regrafting (SPR)


Types of rearrangement iii tree bisection and reconnection tbr l.jpg
Types of rearrangement III: finding shortest treetree bisection and reconnection (TBR)










Sankoff minimal length of possible subtrees having nucleotide a at internal node l.jpg
Sankoff: minimal length of possible subtrees having nucleotide “A” at internal node

C

A

SA = mini [costAi + Sleft, i] +

minj [costAj + Sright, j]

costAA = 0, costAC = costAG = costAT = 1


Sankoff minimal length of possible subtrees having nucleotide a at internal node75 l.jpg
Sankoff: minimal length of possible subtrees having nucleotide “A” at internal node

C

0

SA = mini[costAi+ Sleft, i] +

minj [costAj + Sright, j]


Sankoff minimal length of possible subtrees having nucleotide a at internal node76 l.jpg
Sankoff: minimal length of possible subtrees having nucleotide “A” at internal node

C

0

SA = mini[costAi + Sleft, i] +

minj [costAj + Sright, j]


Sankoff minimal length of possible subtrees having nucleotide a at internal node77 l.jpg
Sankoff: minimal length of possible subtrees having nucleotide “A” at internal node

C

0

SA = mini[costAi + Sleft, i] +

minj [costAj + Sright, j]


Sankoff minimal length of possible subtrees having nucleotide a at internal node78 l.jpg
Sankoff: minimal length of possible subtrees having nucleotide “A” at internal node

C

1

SA = mini [costAi + Sleft, i] +

minj [costAj + Sright, j]


Sankoff minimal length of possible subtrees having nucleotide a at internal node79 l.jpg
Sankoff: minimal length of possible subtrees having nucleotide “A” at internal node

C

1

SA = mini [costAi + Sleft, i] +

minj [costAj + Sright, j]


Sankoff minimal length of possible subtrees having nucleotide a at internal node80 l.jpg
Sankoff: minimal length of possible subtrees having nucleotide “A” at internal node

C

1

SA = mini [costAi + Sleft, i] +

minj [costAj + Sright, j]


Sankoff minimal length of possible subtrees having nucleotide a at internal node81 l.jpg
Sankoff: minimal length of possible subtrees having nucleotide “A” at internal node

C

1

SA = mini [costAi + Sleft, i] +

minj [costAj + Sright, j]


Sankoff minimal length of possible subtrees having nucleotide a at internal node82 l.jpg
Sankoff: minimal length of possible subtrees having nucleotide “A” at internal node

A

0

SA = 1 + minj [costAj + Sright, j]


Sankoff minimal length of possible subtrees having nucleotide a at internal node83 l.jpg
Sankoff: minimal length of possible subtrees having nucleotide “A” at internal node

A

1

SA = 1 + minj [costAj + Sright, j]


Sankoff minimal length of possible subtrees having nucleotide a at internal node84 l.jpg
Sankoff: minimal length of possible subtrees having nucleotide “A” at internal node

A

1

SA = 1 + minj [costAj + Sright, j]


Sankoff minimal length of possible subtrees having nucleotide a at internal node85 l.jpg
Sankoff: minimal length of possible subtrees having nucleotide “A” at internal node

A

1

SA = 1 + minj [costAj + Sright, j]


Sankoff minimal length of possible subtrees having nucleotide a at internal node86 l.jpg
Sankoff: minimal length of possible subtrees having nucleotide “A” at internal node

A

0

SA = 1 + minj [costAj + Sright, j]


Sankoff minimal length of possible subtrees having nucleotide a at internal node87 l.jpg
Sankoff: minimal length of possible subtrees having nucleotide “A” at internal node

C

A

0

1

SA = mini [costAi + Sleft, i] +

minj [costAj + Sright, j]

SA = 1 + 0 = 1


Sankoff minimal length of possible subtrees having nucleotide a at internal node88 l.jpg
Sankoff: minimal length of possible subtrees having nucleotide “A” at internal node

C

A

C

A

G


Sankoff minimal length of possible subtrees having nucleotide c at internal node l.jpg
Sankoff: minimal length of possible subtrees having nucleotide “C” at internal node

C

A

1

0

SC = 0 + 1 = 1


Sankoff minimal length of possible subtrees having nucleotide g at internal node l.jpg
Sankoff: minimal length of possible subtrees having nucleotide “G” at internal node

C

A

1

1

SG = 1 + 1 = 2


Sankoff minimal length of possible subtrees having nucleotide t at internal node l.jpg
Sankoff: minimal length of possible subtrees having nucleotide “T” at internal node

C

A

1

1

ST = 1 + 1 = 2





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