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Phylogeny Tree ReconstructionPowerPoint Presentation

Phylogeny Tree Reconstruction

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Inferring Phylogenies

Trees can be inferred by several criteria:

- Morphology of the organisms
- Sequence comparison
Example:

Orc: ACAGTGACGCCCCAAACGT

Elf: ACAGTGACGCTACAAACGT

Dwarf: CCTGTGACGTAACAAACGA

Hobbit: CCTGTGACGTAGCAAACGA

Human: CCTGTGACGTAGCAAACGA

Modeling Evolution

During infinitesimal time t, there is not enough time for two substitutions to happen on the same nucleotide

So we can estimate P(x | y, t), for x, y {A, C, G, T}

Then let

P(A|A, t) …… P(A|T, t)

S(t) = … …

… …

P(T|A, t) …… P(T|T, t)

x

x

t

y

Modeling Evolution

A

C

Reasonable assumption: multiplicative

(implying a stationary Markov process)

S(t+t’) = S(t)S(t’)

That is, P(x | y, t+t’) = z P(x | z, t) P(z | y, t’)

Jukes-Cantor: constant rate of evolution

1 - 3

For short time , S() = I+R = 1 - 3

1 - 3

1 - 3

T

G

Modeling Evolution

Jukes-Cantor:

For longer times,

r(t) s(t) s(t) s(t)

S(t) = s(t) r(t) s(t) s(t)

s(t) s(t) r(t) s(t)

s(t) s(t) s(t) r(t)

Where we can derive:

r(t) = ¼ (1 + 3 e-4t)

s(t) = ¼ (1 – e-4t)

S(t+) = S(t)S() = S(t)(I + R)

Therefore,

(S(t+) – S(t))/ = S(t) R

At the limit of 0,

S’(t) = S(t) R

Equivalently,

r’ = -3r + 3s

s’ = -s + r

Those diff. equations lead to:

r(t) = ¼ (1 + 3 e-4t)

s(t) = ¼ (1 – e-4t)

Modeling Evolution

Kimura:

Transitions: A/G, C/T

Transversions: A/T, A/C, G/T, C/G

Transitions (rate ) are much more likely than transversions (rate )

r(t)s(t)u(t)s(t)

S(t) = s(t)r(t)s(t)u(t)

u(t)s(t)r(t)s(t)

s(t)u(t)s(t)r(t)

Where s(t) = ¼ (1 – e-4t)

u(t) = ¼ (1 + e-4t – e-2(+)t)

r(t) = 1 – 2s(t) – u(t)

Parsimony

- One of the most popular methods
Idea:

Find the tree that explains the observed sequences with a minimal number of substitutions

Two computational subproblems:

- Find the parsimony cost of a given tree (easy)
- Search through all tree topologies (hard)

Parsimony Scoring

Given a tree, and an alignment column u

Label internal nodes to minimize the number of required substitutions

Initialization:

Set cost C = 0; k = 2N – 1

Iteration:

If k is a leaf, set Rk = { xk[u] }

If k is not a leaf,

Let i, j be the daughter nodes;

Set Rk = Ri Rj if intersection is nonempty

Set Rk = Ri Rj, and C += 1, if intersection is empty

Termination:

Minimal cost of tree for column u, = C

Number of labeled unrooted tree topologies

- How many possibilities are there for leaf 4?
For the 4th leaf, there are 3 possibilities

2

1

4

3

Number of labeled unrooted tree topologies

- How many possibilities are there for leaf 5?
For the 5th leaf, there are 5 possibilities

2

1

4

5

3

Number of labeled unrooted tree topologies

- How many possibilities are there for leaf 6?
For the 6th leaf, there are 7 possibilities

2

1

4

5

3

Number of labeled unrooted tree topologies

- How many possibilities are there for leaf n?
For the nth leaf, there are 2n – 5 possibilities

2

1

4

5

3

Number of labeled unrooted tree topologies

- #unrooted trees for n taxa: (2n-5)*(2n-7)*...*3*1 = (2n-5)! / [2n-3*(n-3)!]
- #rooted trees for n taxa: (2n-3)*(2n-5)*(2n-7)*...*3 = (2n-3)! / [2n-2*(n-2)!]

2

1

N = 10

#unrooted: 2,027,025

#rooted: 34,459,425

N = 30

#unrooted: 8.7x1036

#rooted: 4.95x1038

4

5

3

Probabilistic Methods

A more refined measure of evolution along a tree than parsimony

P(x1, x2, xroot | t1, t2) = P(xroot) P(x1 | t1, xroot) P(x2 | t2, xroot)

If we use Jukes-Cantor, for example, and x1 = xroot = A, x2 = C, t1 = t2 = 1,

= pA¼(1 + 3e-4α) ¼(1 – e-4α) = (¼)3(1 + 3e-4α)(1 – e-4α)

xroot

t1

t2

x1

x2

Computing the Likelihood of a Tree

xk

- Define P(Lk | a): probability of subtree rooted at xk, given that xk = a
- Then, P(Lk | a) = (bP(Li | b) P(b | a, tki))(cP(Lj | c) P(c | a, tki))

tkj

tki

xj

xi

Felsenstein’s Likelihood Algorithm

To calculate P(x1, x2, …, xN | T, t)

Initialization:

Set k = 2N – 1

Recursion: Compute P(Lk | a) for all a

If k is a leaf node:

Set P(Lk | a) = 1(a = xk)

If k is not a leaf node:

1. Compute P(Li | b), P(Lj | b) for all b, for daughter nodes i, j

2. Set P(Lk | a) = b, cP(b | a, ti)P(Li | b) P(c | a, tj) P(Lj | c)

Termination:

Likelihood at this column = P(x1, x2, …, xN | T, t) = aP(L2N-1 | a)P(a)

Probabilistic Methods

Given M (ungapped) alignment columns of N sequences,

- Define likelihood of a tree:
L(T, t) = P(Data | T, t) = m=1…M P(x1m, …, xnm, T, t)

Maximum Likelihood Reconstruction:

- Given data X = (xij), find a topology T and length vector t that maximize likelihood L(T, t)

ML Reconstruction

- Task 1
- Edge length estimation
- Given a topology, find the edge lengths that maximize the likelihood
- Accomplished by iterative methods such as Expectation Maximization (EM) or using Newton’s Raphson optimization
- Each iteration requires computations that take on the order of the number of taxa times the number of sequence positions
- Guaranteed to find local maxima but, in practice, usually find the global maximum

- Edge length estimation

ML Reconstruction

- Task 2
- Find a tree topology that maximizes the likelihood
- More challenging
- Naïve, exhaustive search of tree space is infeasible
- Effectiveness of heuristic paradigms, like simulated annealing, genetic algorithms or other search methods is hampered by re-estimating edge lengths afresh for different trees

- Problem is tackled by iterative procedures that greedily construct the desired tree

Structural EM Approach

- Based on development of Structural EM method in learning Bayesian networks
- Similar to standard EM procedure, with exception that we optimize not only edge lengths but also the topology during each EM iteration
- Overview to Algorithm
- Choose a tree (T1, t1) using, say, Neighbor-Joining
- Improve the tree in successive iterations
- In the l’th iteration, we start with the bifurcating tree (Tl, tl) and construct a new bifurcating tree (Tl+1, tl+1). Basically we use (Tl, tl) to define a measure Q(T,t:Tl, tl) of expected log likelihood of trees and then to find a bifurcating tree that maximizes this expected log likelihood

Definition of Terms

- Σ – fixed, finite alphabet of characterse.g. Σ = {A,C,G,T} for DNA sequences
- T – bifurcating tree, in which each internal node is adjacent to exactly three other nodes
- N – number of sequences/nodes in the tree
- Xi – random variable representing the node i in the tree T, where i = 1,2,..,N
- (i,j) Є T – nodes i and j in T have an edge between them
- t – vector comprising of a non-negative duration or edge length ti,j for each edge (i,j) Є T
- The pair (T,t) constitutes a phylogenetic tree

Definition of Terms

- D – complete data set
- pa->b(t)
- Probability of a character to tranform from ‘a’ into ‘b’ in duration t
- Equal to Σc pa->c(t) pc->b(t’) where c ЄΣ

- pa - prior distribution of character ‘a’
- Si(a)
- frequency count of occurrences of letters
- It is the number of times we see the character ‘a’ at node i in D
- Si(a) = Σm 1{Xi[m] = a}

- Si,j(a,b)
- frequency count of co-occurrences of letters
- It is the number of times we see the character ‘a’ at node i and the character ‘b’ at node j in D
- Si,j(a,b) = Σm 1{Xi[m] = a, Xj[m] = b}

Structural EM Iterations

- Three steps
- E-Step: Compute expected frequency counts for all links (i,j) and for all character states a,b ЄΣ
- M-Step: Optimize link lengths by computing for each link (i,j), its best length
- M-Step II:
- Construct a topology T*l+1 that maximizes Wl+1(T), by finding maximizing spanning tree
- Construct a bifurcating topologyTl+1 such that L(T*l+1, tl+1) = L(Tl+1,tl+1)

E-Step

- For each iteration calculate expected frequency count for all links (i,j) using the formula,E[Si,j(a,b) |D, Tl, tl] = Σm P(Xi[m] = a, Xj[m] = b | x[1…N][m], Tl,tl)
- Here x[1…N][m]= the set of characters observed at all nodes at position ‘m’ in the data
- (Tl,tl) is the current topology
- Computation of these expected counts takes O(M.N2|Σ|3) and storing them requires O(N2|Σ|2 ) space
- We can improve the speed by O(|Σ|) by calculating approximate counts instead S*i,j(a,b) =Σm P(Xi[m] = a| x[1..N][m], Tl,tl) P(Xj[m] = b | x[1..N][m], Tl,tl)

M-Steps

- M-Step I
- After each E-Step, we calculate each link (i,j)’s best length using,
tl+1i,j= argmaxt Llocal(E[Si,j(a, b) | D, Tl, tl], t), where

Llocal(E[Si,j(a, b) | D, Tl, tl], t) = Σa,b E[Si,j(a,b)] [log pa->b(t) – log pb]

- After each E-Step, we calculate each link (i,j)’s best length using,
- M-Step II
- Construct a topology T*l+1 that maximizes Wl+1(T), by finding maximizing spanning tree
- Construct a bifurcating topologyTl+1 such that L(T*l+1, tl+1) = L(Tl+1,tl+1)

Empirical Evaluation

- The Algorithm has been written in C++ and been implemented in a program called SEMPHY
- For protein sequences
- Compared results to MOLPHY, which is a leading ML application for phylogeny reconstruction based on protein sequences
- Used the basic Structural EM algorithm, using a Neighbor-joining tree as a starting point
- Evaluated performance on synthetic data sets, generated by constructing an original phylogeny and then sampling from its distributions
- The synthetic data sets were broken into a test set and training set
- Ran both programs on the data

Empirical Evaluation

- For protein sequences:
- Results of basic Structural EM
- Quality of both methods deteriorated when number of taxa increases and when the number of training position decreases. In this case, both methods did poorly
- SEMPHY outperforms MOLPHY in terms of quality of solutions
- MOLPHY’s running time grows much faster than SEMPHY’S running time, which is only quadratic in number of taxa
- Both MOLPHY’s and SEMPHY’s running times grows linearly in number of training positions

- Results of basic Structural EM

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