1. Molecular Evolution and Phylogenetic Tree Reconstruction1 4 3 2 5
Molecular Evolution and Phylogenetic Tree Reconstruction

2. Phylogenetic Trees Nodes: species Edges: time of independent evolution
Edge length represents evolution time
AKA genetic distance
Not necessarily chronological time

3. Inferring Phylogenetic Trees
Trees can be inferred by several criteria:
Morphology of the organisms
Can lead to mistakes!
Sequence comparison
Example:
Mouse: ACAGTGACGCCCCAAACGT
Rat: ACAGTGACGCTACAAACGT
Baboon: CCTGTGACGTAACAAACGA
Chimp: CCTGTGACGTAGCAAACGA
Human: CCTGTGACGTAGCAAACGA

4. Inferring Phylogenetic Trees
Sequence-based methods
Deterministic (Parsimony)
Probabilistic (SEMPHY)
Distance-based methods
UPGMA
Neighbor-Joining
Can compute distances from sequences 4

5. Distance Between Two Sequences
Basic principles:
Degree of sequence difference is proportional to length of independent sequence evolution
Only use positions where alignment is certain – avoid areas with (too many) gaps 5

6. Distance Between Two Sequences
Given sequences xi, xj,
Define
dij = distance between the two sequences
One possible definition:
dij = fraction f of sites u where xi[u]  xj[u]
Better scores are derived by modeling evolution as a continuous change process

7. Outline Molecular Evolution Distance Methods Sequence Methods
UPGMA / Average Linkage
Neighbor-Joining
Sequence Methods
Deterministic (Parsimony)
Probabilistic (SEMPHY)

8. Molecular Evolution
Q: How can we model evolution on nucleotide level? (ignore gaps, focus on substitutions)
A: Consider what happens at a specific position for small time interval Δt
P(t) = vector of probabilities of {A,C,G,T} at time t
μAC = rate of transition from A to C per unit time
μA = μAC + μAG + μAT rate of transition out of A
pA(t+Δt) = pA(t) – pA(t) μA Δt + pC(t) μCA Δt + … 8

9. P(t+Δt) = P(t) + Q P(t) Δt
Molecular Evolution
In matrix/vector notation, we get
P(t+Δt) = P(t) + Q P(t) Δt
where Q is the substitution rate matrix 9

10. Molecular Evolution This is a differential equation: P’(t) = Q P(t)
A substitution rate matrix Q implies a probability distribution over {A,C,G,T} at each position, including stationary (equilibrium) frequencies πA, πC, πG, πT
Each Q is an evolutionary model (some work better than others) 10

11. Evolutionary Models
Jukes-Cantor
Kimura
Felsenstein
HKY 11

12. Estimating Distances
Solve the differential equation and compute expected evolutionary time given sequences
Jukes-Cantor
Kimura 12

13. Outline Molecular Evolution Distance Methods Sequence Methods
UPGMA / Average Linkage
Neighbor-Joining
Sequence Methods
Deterministic (Parsimony)
Probabilistic (SEMPHY) 13

14. A simple clustering method for building tree
UPGMA (unweighted pair group method using arithmetic averages)
Or the Average Linkage Method
Given two disjoint clusters Ci, Cj of sequences, 1
dij = ––––––––– {p Ci, q Cj}dpq
|Ci|  |Cj|
Claim that if Ck = Ci  Cj, then distance to another cluster Cl is:
dil |Ci| + djl |Cj|
dkl = ––––––––––––––
|Ci| + |Cj|

15. Algorithm: Average Linkage
Initialization:
Assign each xi into its own cluster Ci
Define one leaf per sequence, height 0
Iteration:
Find two clusters Ci, Cj s.t. dij is min
Let Ck = Ci  Cj
Define node connecting Ci, Cj, and place it at height dij/2
Delete Ci, Cj
Termination:
When two clusters i, j remain, place root at height dij/2 1 4 3 2 5 1 4 2 3 5

16. Average Linkage Examplew x y z 6 8 4 2 v w
xyz 6 8 vw
xyz 8 4 v w x yz 6 8 4 3 2 1 v w x y z

17. Ultrametric Distances and Molecular Clock
Definition:
A distance function d(.,.) is ultrametric if for any three distances dij  dik  dij, it is true that
dij  dik = dij
The Molecular Clock:
The evolutionary distance between species x and y is 2 the Earth time to reach the nearest common ancestor
That is, the molecular clock has constant rate in all species
The molecular clock results in ultrametric distances
years 1 4 2 3 5

18. Ultrametric Distances & Average Linkage1 4 2 3 5
Average Linkage is guaranteed to reconstruct correctly a binary tree with ultrametric distances
Proof: Exercise

19. Weakness of Average Linkage
Molecular clock: all species evolve at the same rate (Earth time)
However, certain species (e.g., mouse, rat) evolve much faster
Example where UPGMA messes up:
Correct tree
AL tree 3 2 4 1 4 2 3 1

20. Additive Distances 1
d1,4 12 4 8 3 13 7 9 5 11 10 6 2
Given a tree, a distance measure is additive if the distance between any pair of leaves is the sum of lengths of edges connecting them
Given a tree T & additive distances dij, can uniquely reconstruct edge lengths:
Find two neighboring leaves i, j, with common parent k
Place parent node k at distance dkm = ½ (dim + djm – dij) from any node m  i, j

21. d(x, y) + d(z, w) < d(x, z) + d(y, w) = d(x, w) + d(y, z)
Additive Distances z x w y
For any four leaves x, y, z, w, consider the three sums
d(x, y) + d(z, w)
d(x, z) + d(y, w)
d(x, w) + d(y, z)
One of them is smaller than the other two, which are equal
d(x, y) + d(z, w) < d(x, z) + d(y, w) = d(x, w) + d(y, z)

22. Reconstructing Additive Distances Given Tx T D y 5 4 v w x y z 10 17 16 15 14 9 3 z 3 4 7 w 6 v
If we know T and D, but do not know the length of each leaf, we can reconstruct those lengths

23. Reconstructing Additive Distances Given Tx T D y v w x y z 10 17 16 15 14 9 z w v

24. Reconstructing Additive Distances Given Tx v w x y z 10 17 16 15 14 9 T y z a w D1 v
dax = ½ (dvx + dwx – dvw) a x y z 11 10 9 15 14
day = ½ (dvy + dwy – dvw)
daz = ½ (dvz + dwz – dvw)

25. Reconstructing Additive Distances Given Tx a x y z 11 10 9 15 14 T y 5 4 b 3 z 3 a c 4 7 w D2 6
d(a, c) = 3
d(b, c) = d(a, b) – d(a, c) = 3
d(c, z) = d(a, z) – d(a, c) = 7
d(b, x) = d(a, x) – d(a, b) = 5
d(b, y) = d(a, y) – d(a, b) = 4
d(a, w) = d(z, w) – d(a, z) = 4
d(a, v) = d(z, v) – d(a, z) = 6
Correct!!! v a b z 6 10 D3 a c 3

26. Neighbor-Joining
Guaranteed to produce the correct tree if distance is additive
May produce a good tree even when distance is not additive
Step 1: Finding neighboring leaves
Define
Dij = (N – 2) dij – ki dik – kj djk
Claim: The above “magic trick” ensures that Dij is minimal iff i, j are neighbors 1 3
0.1
0.1
0.1
0.4
0.4 2 4

27. Algorithm: Neighbor-Joining
Initialization:
Define T to be the set of leaf nodes, one per sequence
Let L = T
Iteration:
Pick i, j s.t. Dij is minimal
Define a new node k, and set dkm = ½ (dim + djm – dij) for all m  L
Add k to T, with edges of lengths dik = ½ (dij + ri – rj), djk = dij – dik
where ri = (N – 2)-1 ki dik
Remove i, j from L;
Add k to L
Termination:
When L consists of two nodes, i, j, and the edge between them of length dij

28. Outline Molecular Evolution Distance Methods Sequence Methods
UPGMA / Average Linkage
Neighbor-Joining
Sequence Methods
Deterministic (Parsimony)
Probabilistic (SEMPHY) 28

29. Parsimony One of the most popular methods: Idea:
GIVEN multiple alignment
FIND tree & history of substitutions explaining alignment
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)

30. Example: Parsimony Cost of One Column
{A, B}
C++ A B A B A A
{A}
{B}
{A}
{A}

31. 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; node k = 2N – 1 (last leaf)
Iteration:
If k is a leaf, set Rk = { xk[u] } // Rk is simply the character of kth species
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 increment C if intersection is empty
Termination:
Minimal cost of tree for column u, = C

32. Example {B} {A,B} {A} {B} {A} {A,B} {A} A A A A B B A B {A} {A} {A}

33. Parsimony Traceback Traceback:
Choose an arbitrary nucleotide from R2N – 1 for the root
Having chosen nucleotide r for parent k,
If r  Ri choose r for daughter i
Else, choose arbitrary nucleotide from Ri
Easy to see that this traceback produces some assignment of cost C

34. inadmissible with Traceback
Example
Admissible with Traceback x B
Still optimal, but
inadmissible with Traceback A
{A, B} A B x
{A} B
{A, B} A B A B x B x A B A B A
{A}
{B}
{A}
{B} A B A B A x A x A B A B

35. Another Parsimony Algorithm
Let C(v) be cost for subtree rooted at node v
Let C(v,x) be cost for subtree rooted at v if we force v to have value x
Initialization:
For each leaf v
C(v) = 0
C(v,x) = 0 if x is input character that labels v; C(v,x) = ∞ otherwise
Iteration:
Let u, w be children of v
C(v,x) = min(C(u) + 1, C(u,x)) + min(C(v) + 1, C(v,x))
C(v) = min C(v,x)
Termination:
Minimal cost is C(root) 35

36. Probabilistic Methods
xroot t1 t2 x1 x2
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,
= pA¼(1 + 3e-4α) ¼(1 – e-4α) = (¼)3(1 + 3e-4α)(1 – e-4α)

37. Probabilistic Methods
xroot xu x2 xN x1
If we know all internal labels xu,
P(x1, x2, …, xN, xN+1, …, x2N-1 | T, t) = P(xroot)jrootP(xj | xparent(j), tj, parent(j))
Usually we don’t know the internal labels, therefore
P(x1, x2, …, xN | T, t) = xN+1 xN+2 … x2N-1 P(x1, x2, …, x2N-1 | T, t)

38. Felsenstein’s Likelihood Algorithm
To calculate P(x1, x2, …, xN | T, t)
Initialization:
Set k = 2N – 1
Iteration: 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,c P(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)
Let P(Lk | a) denote the prob.
of all the leaves below node k,
given that the residue at k is a

39. Felsenstein’s Likelihood Algorithm
Define:
and recursively compute: 39

40. Felsenstein’s Likelihood Algorithm
Now using u and U we can compute:
and 40

41. 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)

42. Current popular methods
HUNDREDS of programs available!
Some recommended programs:
Discrete—Parsimony-based
Rec-1-DCM3
Tandy Warnow and colleagues
Probabilistic
SEMPHY
Nir Friedman and colleagues