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Doug RaifordLesson 9
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3 Approaches Distance Parsimony Maximum Likelihood
Have already seen a distance method
04/22/23 2Phylogenetics Part II
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What’s wrong with UPGMA?Let’s revisit the exampleCan this be? Doesn’t the derived tree
imply that B is equidistant from C and D
04/22/23 Phylogenetics Part II 3
A B C D
A B C D
A 0 7 6 7
B 0 4 5
C 0 3
D 0
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UPGMA averaged the two and put them both (branches for C and D) at 1.5
What if don’t have equal rates of evolution after a divergence
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A B C D
A B C D
A 0 7 6 7
B 0 4 5
C 0 3
D 0
4
.5.5
1 22.5
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Differing rates of evolution can sometimes cause problems with UPGMA
Especially if very similar (small distances)
04/22/23 Phylogenetics Part II 5
A B C
A 0 4 3
B 0 3
C 0A B C
1
2 11
This tree Yields this matrix Yields this tree
BCA
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Also called minimum evolution method
Definition of parsimony:1 a : the quality of being careful with money or resources : thrift b : the quality or state of being stingy
2 : economy in the use of means to an end; especially : economy of explanation in conformity with Occam's razor
Ockham's razor: the simplest explanation is usually the best
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Looks at each column of an MSA and attempts to find a tree that describes
Builds a consensus tree
atgccgca-actgccgcaggagatcaggactttcatgaatatcatcatgcgtggga-ttcagacctccatacgtgccccaggagatctggactttcacc---tggatcatgcgaccgtacctact-atgg-t-cgtgccgcaggagatcaggactttca-gt--g-aatcatctgg-cgc--c-aat--tcgt-ac-tgccccaggagatctggactttcaaa---ca-atcatgcgcc-g-tc-tataattccgtacgtgccgcaggagatcaggactttcag-t--a-tatcatctgtc-ggc--tag
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What do we mean when we say “attempts to find a tree that describes”
Attempts to fit all possible trees in each column and choose best
How determine all possible trees? How determine which one has the best fit? Assume that majority nucleotide represents
ancestor
AGCTAACTAACTAACT
One possible tree
A A A G
A
00
A or a G
A or a G0 if A
0 if A
0 if A 1 if A
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Total mutations that explain this
tree = 1
Pretty darn good
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When there are two organisms there is only one possible tree
A B
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What about when there are threeThird could go…
A B04/22/23 10Phylogenetics Part II
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For each of the previous 3 trees, could add 4th to any of its branches (or could form a new root)
Each of the possible trees had 4 branches so could add to one of 4 locations (or splice in at top)
So total number of trees with 4 leaves: 3*5=15
04/22/23 Phylogenetics Part II 11A B
If this were the tree
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Ni is number of trees given i taxa
Bi is the number of branches in a tree given i taxa
Bi=Bi-1+2, also i x 2-2 Ni=Ni-1*(Bi-1+1)
plus 1 due to possible new root
N2= 1 B2=2
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TaxaBranch
esTrees
2 2 1
3 4 3
4 6 15
5 8 105
6 10 945
7 12 10,395
8 14 135,135
9 16 2,027,025
10 1834,459,42
5
11 20654,729,0
75
Defined by a recurrence relation
so …
That’s right, as usual, exponential
Defined by a recurrence relation
so …
That’s right, as usual, exponential
What does this growth rate look like?
What does this growth rate look like?
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Rooted vs. un-rootedWherever the root is, un-kink it
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Always bifurcated Can never have 3 branches “from” a
single node What are the odds?
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A
B C
D
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Three possible trees
04/22/23 Phylogenetics Part II 15
A
B C
D
A
D C
B
A
C B
D
Are there any other combinations?
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For each of the three trees (having 4 taxa) could add a branch to any of the 5 branches
3*5=15 trees
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A
B C
D
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Outgroup Include an organism that is known to be
further away from all taxa than they are from each other
04/22/23 17Phylogenetics Part II
A
B C
D
If outgroup goes here…
outgroup A B C D
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Ni is number of trees given i taxa
Bi is the number of branches in a tree given i taxa
Bi=Bi-1+2, also i x 2-3 Ni=Ni-1*(Bi-1)
No need for a “plus 1” for a possible new root because there are no roots
N2= 1 B2=2
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TaxaBranch
esTrees
3 3 1
4 5 3
5 7 15
6 9 105
7 11 945
8 13 10,395
9 15 135,135
10 17 2,027,025
11 1934,459,42
5
12 21654,729,0
75
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Noticed that for un-rooted trees: Bi=2i-3 (for i 2)
Also noticed Ni=Ni-1*Bi-1
And reduced to (2n-5)(2n-7)(2n-9)…(3)(1)
where n is number of taxa Shorthand: (2n-5)!!
For rooted Ni=Ni-1*(Bi-1+1)
Reduced to (2n-3)!!
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Ni=Bi-1*Ni-1
=(2(i-1)-3)Ni-1
=(2i-5)Ni-1
=(2i-5)(2i-7)Ni-2
Till the N term gets to 3
Double factorial: each successive number
reduced by two
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Radical reduction in the number
Still only bought one additional taxa
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TaxaUn-rooted
treesRooted trees
3 1 3
4 3 15
5 15 105
6 105 945
7 945 10,395
8 10,395 135,135
9 135,135 2,027,025
10 2,027,025 34,459,425
11 34,459,425 654,729,075
12 654,729,07513,749,310,5
75
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Even brighter mathematicians
04/22/23 21Phylogenetics Part II
Can you see why?
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Not really a candidate for dynamic programming Don’t repeat a bunch of
sub-problems over and over Each sub-problem is a tree,
and they are all unique
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Still exponen
tial
Still exponen
tial
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Discard large subsets of possible solutions
Use heuristics or predictions
04/22/23 Phylogenetics Part II 23
Don’t bother
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Calculate a reasonable upper bound using a fast algorithm like UPGMA (hierarchical clustering)
Incrementally grow potential treesAny branch that any that go over
threshold stop investigating
04/22/23 Phylogenetics Part II 24
A
B C
DXX
X
Don’t bother, over threshold
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Some columns all same Add no meaning All trees minimum
Columns that are all different Also add no meaning
Must have minimum 2 nt’s (or aa’s) that are the same
Useful in one respect If all the same infer makeup of
ancestor
04/22/23 Phylogenetics Part II 25
AGCTAACTAACTACCT
A A A A
A
00
A
A00
0 0
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Each column yields a tree If all agree done If some different use
majority rule If sample too small
perform bootstrapping randomly draw sequences
from MSA Generate more trees labeled branches with the
percentage of bootstrap trees in which they appear
Used as a measure of support (repeatability)
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Still have maximum likelihoodAlso, some inferential stuff, but
that’s all in the next lecture
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04/22/23 28Phylogenetics Part III