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CSCI1950‐Z Computa3onal Methods for Biology Lecture 7
Ben Raphael February 9, 2009
hHp://cs.brown.edu/courses/csci1950‐z/
Outline
Probabilis3c Models of Phylogeny
- 1. Models of nucleo3de change
- 2. Compu3ng likelihood of trees
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Distances from Sequences
Chimpanzee: CCTGCCAGTTAGCAAACGG Ancestor: CCCGCGACTTAACAAACGC Human: CCTGCGAGTTAACAAACGA
Hamming distance = 3. DS = differences per site = 3/20 12 total muta3ons.
T G G
coincident back parallel convergent mul3ple
Jukes‐Cantor Model
A C G T A ‐3α α α α C α ‐3α α α G α α ‐3α α T α α α ‐3α
Q =
P(t) = eQt pxy(t) = Pr[Xt = x | X0 = y] = ¼ + ¾ e‐4αt if x = y
¼ ‐ ¼ e‐4αt if x ≠ y
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Observed Differences vs. Jukes‐Cantor
= αt
Other Models
Kimura 2 parameter model Other models
- HKY model for DNA
- Other models for protein sequences (20 x 20
matrices)
A C G T A ‐2β‐ α β α β C β ‐2β‐ α β α G α β ‐2β‐ α β T β α β ‐2β‐ α
In biology, not all subs3tu3ons are equally likely. {A, G} {C, T} purines pyrimidines
transi3on transversion
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Probabilis3c Model
Given a character matrix M, what is the “most likely” tree T that generated M? Pr[ x | y, t] = probability that y mutates to x in 3me t
y x t
Rat: ACAGTGACGCCCCAAACGT Mouse: ACAGTGACGCTACAAACGT Gorilla: CCTGTGACGTAACAAACGA Chimpanzee: CCTGTGACGTAGCAAACGA Human: CCTGTGACGTAGCAAACGA
Likelihood
Given data D and a model (hypothesis) H for genera3on of D, the likelihood of H is the quan3ty L[H; D] = Pr[ D | H ]. Note: likelihood of H is not the probability of H.
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Probabilis3c Model
Given a character matrix M, what is the “most likely” tree T that generated M?
Assume:
- 1. Characters evolve independently.
- 2. Constant rate of muta3on on each branch.
- 3. State of a node depends only on parent and branch length:
i.e. Pr[ x | y, t] depends only on y and t. (Markov process)
Pr[ x | y, t] = probability that y mutates to x in 3me t
y x t
Maximum Likelihood
Given a character matrix M, a tree T with branch lengths t* = t1, …, t2n‐2: L(T, t*) = Pr[ M | T, t*] is called the likelihood. Maximum likelihood: Find argmaxT, t L(T, t*) First: How to compute L(T, t*)?
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Probabilis3c Model
Given a tree (T, t*) with leaves labeled by characters in M, Pr[ M | T, t*] is the probability of a labeling of ancestral nodes.
Assume: 1. Characters evolve independently: Pr[ M | T, t*] = Πi Pr[ Mi | T, t*] so consider each character separately 2. Constant rate of muta3on on each branch. 3. State of a vertex depends only on parent and branch length: i.e. Pr[ x | y, t] depends only on y and t. (Markov process)
Pr[ x | y, t] = probability that y mutates to x in 3me t
y x t
Example
t1 t3 t2 t5 t4 t6 A T C G y z x
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Probabilis3c Model
Two species Pr[ x1, x2, a | T, t1, t2] = qa Pr[x1 | a, t1] Pr[x2 | a, t2] Pr[ x | y, t] = probability that y mutates to x in 3me t
T = tree topology x1 , x2 : characters for each species a : character for ancestor a x1 x2 t1 t2 y x t qa = Pr[ ancestor has character a]
Probabilis3c Model
Two species Pr[ x1, x2 | T, t1, t2] = Σa qa Pr[x1 | a, t1] Pr[x2 | a, t2]
T = tree topology x1 , x2 : characters for each species a : character for ancestor a x1 x2 t1 t2 qa = Pr[ ancestor has character a]
Pr[ x1, x2, a | T, t1, t2] = qa Pr[x1 | a, t1] Pr[x2 | a, t2]
Follows from Law of Total Probability: P(X) = Σ P(X| Yi) P(Yi).
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Probabilis3c Model
n species: x1, x2, …, xn Let α(i) = ancestor of node i. Let an+1, an+2, …, a2n‐1 = characters on internal nodes, where nodes are number from internal ver3ces up to root.
qa2n−1
2n−2
Pr[ai|aα(i), ti]
n
Pr[xi|aα(i), ti]
Pr[x1, ..., xn|T, t1, ..., t2n−2] =
Follows from Law of Total Probability: P(X) = Σ P(X| Yi) P(Yi).
Felsenstein’s Algorithm
Let Pr[Tk | a] = probability of leaf nodes “below” node k, given ak = a. Compute via dynamic programming
Pr[Tk|a] =
Pr[b|a, ti]Pr[Ti|b]
Pr[c|a, tj]Pr[Tj|c]
Ini3al condi3ons. For k = 1, …, n (leaf nodes) Pr[Tk | a] = 1, if a = xk
0, otherwise.
a b c
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Compu3ng the Likelihood
Let Pr[Tk | a] = probability of leaf nodes “below” node k, given ak = a.
Pr[x1, . . . , xn|T, t∗] =
Pr[T2n−1|a]qa
Note: Root is node 2n‐1
Maximum Likelihood when T unknown
Must search over all trees T. Complexity unknown un3l recently:
– Felsenstein book (2004): “There has also been no proof that the problem is NP‐hard (as there has been for many
– Shamir notes (2000): “[Maximum likelihood] not proven to be NP‐complete.”
- ML is NP‐hard (B. Chor and T. Tuller, RECOMB 2005).
Pr[x1, . . . , xn|T, t∗] =
Pr[T2n−1|a]qa
Find T, t* that maximize:
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Unknown branch lengths
- T fixed, branch lengths t* are unknown.
- Use local op3miza3on rou3ne: e.g. Newton’s
method or Expecta3on Maximiza3on
Finding Ancestral States
Let Pr[Tk | a] = probability of best assignment of ancestral states to nodes “below” node k, given ak = a. Traceback as before with Sankoff’s algorithm.
Pr[Tk|a] =
b
Pr[b|a, ti]Pr[Ti|b] max
c
Pr[c|a, tj]Pr[Tj|c]
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- Max. Parsimony vs. Max. Likelihood
- Set δij = ‐log P(j | i) in weighted parsimony
(Sankoff algorithm)
- Weighted parsimony produces “maximum
probability” assignments, ignoring branch lengths
Searching over Tree Space
- How to find T with maximum likelihood?
- How to find T with maximum parsimony?
