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Phylogenetic Tree

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Title: Phylogenetic Tree


1
Phylogenetic Tree
2
Phylogenetic Tree What it is
  • Drawing evolutionary tree from characteristics of
    organisms or some measured distances between them
  • Represented as a tree where nodes are the
    organisms/objects and arcs are the proximity
    between the respective nodes
  • Based on how close the organisms are

3
Phylogenetic Tree Motivation
  • Pure curiosity biological science
  • One species can be studied for a related one
  • Drug test on monkeys for human
  • Rare species can be spared in a study
  • Drug design on evolution of micro-organism
    aids/flu vaccine/drug design depends on how do
    they evolve
  • Tracking pathogen sources
  • Genesis, archeology,,,

4
Phylogenetic Tree topology
  • Evolutionary distance is not same as elapsed
    time former is a crude approximation of the
    latter (if distance can be calculated at all)
  • Leaves are objects, internal nodes may or may not
    be objects (may represent hypothetical ancestors)
  • Mostly binary trees, sometimes not

5
Phylogenetic Tree source data types
  • Discrete characters
  • does it have long beaks?
  • Could be Boolean or multi-valued
  • Provided in matrix form (objects X characters)
  • Numerical distance matrix
  • Symmetric pairwise distances measured by some
    means, e.g., by aligning sequences
  • Continuous character character value is in
    numerical domain

6
Characters for phylogeny
  • Characters should be relevant in the context of
    phylogeny depends on the user scientist
  • Characters should be independent inherited
    without interference between the characters (eye
    color and hair color may not be a good
    combination in character set)
  • All characters must evolve from the same
    ancestor we presume that (1) it is tree, (2) it
    is a connected tree
  • Closest objects are called homologous max
    possible characters have same values or related
    values

7
Phylogeny using character state matrix
  • A state is a tuple with values for each
    character (value could be unassigned)
  • Internal node may be a state without any object
    assigned on it
  • Leaves are where the states correspond to objects
    with the respective assigned characters
  • P 178 a source character state matrix

8
Phylogeny using character state matrix Problems
  • Convergence evolution two non-homologous objects
    (most characters does not match, loosely
    speaking) happen to have same value on a
    character (needs a cycle in the graph)

9
Phylogeny using character state matrix Problems
  • In one case evolution suggests character value of
    c evolves from long to short, in another case
    the reverse confusion over the direction of
    evolution
  • Again, the tree property would be violated to
    accommodate this

10
Character domain types
  • Domain of character c could be
  • red lt - gt blue lt - gt yellow lt - gt green
  • C cannot evolve from blue to green without taking
    value yellow first
  • C is ordered
  • C can be directed and ordered, instead of
    undirected as above

11
Perfect phylogeny
  • Noise-free input
  • Each edge in phylogeny is a transition of the
    respective characters value
  • All nodes with the same value for a character
    must form a subtree (with the transition at its
    root)
  • Such a tree is perfect phylogeny

12
Perfect phylogeny problem
  • Given a character state matrix does there exist a
    perfect phylogeny over it
  • P 178 table does not have a perfect phylogeny
    (presume transitions always 0 -gt 1). Why?
  • P 180 table and its perfect phylogeny
  • What do you do when you do not have perfect
    phylogeny? Presume data is noisy and minimize
    errors in drawing perfect phylogeny

13
Perfect phylogeny problem
  • You can always try all possible trees over the
    objects and check whether each tree is perfect
    phylogeny or not
  • The total number of such trees is pi3n (2i-5)
    Exponential

14
Perfect phylogeny problem to check existence
(Boolean matrix)
15
Perfect phylogeny problem to check existence
(Boolean matrix)
  • Organize char state matrix colum-wise for each
    col i set of objects is Oi
  • Every pair of Oi and Ok should be
  • either Oi ? Ok
  • or Oi ? Ok
  • or Oi ? Ok null
  • Either one belongs to the other one or they do
    not overlap at all
  • If they overlap, no perfect phylogeny exist

16
Perfect phylogeny problem to check existence
(Boolean matrix)
  • In contrary, suppose Oi and Ok overlaps and a
    perfect phylogeny exists
  • say, i is the edge between (u, v) v and subtree
    has i1, but all other nodes have i0.
  • Suppose, three objects a, b, and c such that, a,
    b ? Oi, but c is not a,b in subtree of v and c
    is not there
  • But, suppose b, c ? Ok, and a is not b,c must
    belong to some other subtree separated by edge k
  • Contradiction

17
Perfect phylogeny problem to check existence
(Boolean matrix)
  • When no overlap exists
  • Contained sets go within same subtree, if Oi ?
    Ok, then i-subtree is subtree of k-subtree
  • Disjoint sets are separate subtrees
  • Proves if and only if of the condition for
    perfect phylogeny
  • Algorithm for checking Pairwise checking of
    object set may take O(m2) for m characters, but
    set overlap may check even more time

18
Perfect phylogeny problem Algorithm (Boolean
matrix)
  • Sort the columns by number of 1s (descending)
  • Scan each row to find which col number has the
    rightmost 1 for that box
  • Scan each column every box should agree
  • Complexity O(mn) count, O(m log m) sort, O(mn)
    index matrix creation, O(mn) checking over index
    matrix total O(mn) presuming n gt log m

19
Perfect phylogeny problem Algorithm (Boolean
matrix)
  • Exercise try the algorithm for tables 6.1 p 178
    and 6.2 p 180
  • Construction Algorithm (1) sort characters/col
    increasing order, (2) each object (3) each
    character (4) if edge for char exists put obj
    on the end, (5) else create an edge and put
    object at the end, (6 cosmetic step) if more
    objects in a leaf node create edges for each
    object
  • O(nm)
  • Exc. Try it on table 6.2 p180

20
Perfect phylogeny problem Algorithm (non-Boolean
matrix, but)
  • If two states per character but the order of
    transition not known, then presume an order
  • majority state 0, minority 1 (more ancestors are
    available)
  • Same Lemma must be applied after this
    presumption no overlapping set of objects

21
Phylogeny problem arbitrary domain size,
unordered characters
  • (Def) Triangulated graph no big hole cycle
    with gt3 vertices has a short-cut edge
  • Sub-trees of a tree form triangulated graph (as
    intersection graph?)
  • (Def) Intersection Graph over subsets subsets
    are nodes and edges between pairs of overlapping
    subsets

22
Phylogeny problem arbitrary domain size,
unordered characters
  • Fig 6.7, p187 intersection graph for Table 6.3
    p188 not triangulated, yet
  • (Def) c-Triangulated graph Connect edges of
    intersection graph G where nodes are of different
    characters, and if the graph becomes now
    triangulated, then G is c-triangulated
  • Fig 6.7 is c-triangulated

23
Phylogeny problem arbitrary domain size,
unordered characters
  • Iff a character state matrix translates to a
    c-triangulated graph then it admits perfect
    phylogeny
  • Creatingchecking c-triangulation is NP-hard
    (related to finding max-clique problem)

24
Phylogeny problem arbitrary domain size,
unordered characters 2 characters
  • For 2 characters, the intersection graph is
    bi-partite
  • Perfect phylogeny means (iff) the state
    intersection graph is acyclic

25
Phylogeny construction arbitrary domain size,
unordered characters 2 characters
  • Algorithm
  • (1) Construct intersection graph
  • (2) make nodes for edges (intersection of the
    objects in old nodes now goes to the new nodes)
  • (3) connect new nodes if they have overlapping
    objects
  • (4) spanning tree of the graph is phylogeny
  • (5 cosmetic step) objects huddled on a node
    should be put on separate leaves
  • Try on Table 6.4 p190, and check against Fig 6.8
    p189

26
When Perfect Phylogeny does not exist
  • Eliminate problematic characters which ones, an
    optimization problem min number of characters
    Compatibility criterion
  • Minimize convergence (character goes back to its
    previous value) Parsimony criterion
  • Both NP-complete problems

27
When Perfect Phylogeny does not exist Parsimony
  • Compatibility problem Does there exist a subset
    of characters such that Lemma 6.1
    (non-overlapping set of objects) is valid (or
    Perfect Phylogeny exists)?
  • Equivalent to K-clique problem does there exist
    a connected-subgraph with K or more nodes?

28
When Perfect Phylogeny does not exist Parsimony
  • Poly-transformation from Clique to compatibility
    problem nodes to character, 3 objects for each
    edge with specific character values
  • Every pair of NP-complete problems have two way
    poly-trans
  • Compatibility can also be poly-trans to Clique
    characters to nodes, non-overlapping (compatible)
    characters to edges

29
Phylogeny with Distance Matrix
30
Phylogeny with Distance Matrix
  • Input is a distance matrix (square, symmetric)
    between all pair of objects, instead of character
    state matrix
  • Output is phylogeny with leaves as objects and
    arcs have distances as labels

31
Phylogeny with Distance Matrix
  • Additive matrix when you can draw a tree where
    distance between every pair of leaves on the tree
    is the real distance on distance matrix
  • Matrices are unlikely to be additive in practice
  • For non-additive matrix, minimize deviation over
    the tree NP-hard problem

32
Phylogeny with Distance Matrix
  • Typically we have 2 matrices (1) upper bound on
    distances, and (2) that for lower bounds
  • Metric space
  • dijgt0, dii0, dijdji, for all i, j
  • dij lt dik dkj
  • Additive metric spaces follow 4 point condition
  • dijdkldikdjl gt dildjk

33
Phylogeny with Distance Matrix
  • Tree should have 3-degree internal nodes (Fig
    6.9, p194)
  • Arc xy to be split proportionately at c, to add a
    node z by arc cz, so that distances xz, zy are
    proper

34
Phylogeny with Distance Matrix
  • Mxz dxc dzc
  • Myz dyc dzc
  • Mxy dxc dyc
  • Three equations, three unknowns dxc, dyc, dzc to
    be solved for
  • The tree drawn is unique for 3 objects x, y and z

35
Phylogeny with Distance Matrix
  • Adding 4th object w is same as adding 3rd object
    z
  • Add between older objects x and y splitting xy at
    c2
  • If c2 coincides with c, ignore this and redo the
    same between zc
  • Object w may hang (from c2) between xz or yz, but
    will not have 2 different opportunities

36
Phylogeny with Distance Matrix
  • The property of uniqueness of the tree remain
    valid for any k objects for kgt4, for metric
    additive distance matrix
  • The algorithm may have to try all possible places
    to split an arc, but there will be a unique
    position, for metric additive space

37
Phylogeny Ultrametric tree
  • Excercise Get MST of a complete graph over table
    6.5 p195
  • Ultrametric tree construction
  • Input Distance matrices for High cut-off Mh, Low
    cut-off Ml (table 6.6 p 201)
  • Output Phylogeny where leaf-to-leaf distances
    are within the bounds provided by the 2 matrices
    (fig 6.16 p202)

38
Phylogeny Ultrametric tree
  • Algorithm
  • Compute MST T over Mh (algorithm?) provides
    basis for structure of the tree
  • Compute cut-off values between each edge on T
    using Ml provides basis for distances on the
    tree edges
  • Compute the ultrametric tree U and find distance
    on each arc using the cut-offs

39
Phylogeny Ultrametric tree
  • Step 2.1 input T, output is rooted tree R where
    internal nodes represent edges of T
  • Sort MST T by edge weights (from Mh)
    non-increasing
  • Pick up edges by the sort as root in each
    iteration
  • The path between the end nodes must go via the
    root the two nodes edge should be in two
    different subtrees
  • Next edge in the sort to be picked up that has
    the corresponding node (x) on the respective side
    of the previous root (xy)
  • Until no edge for a node (x) is left (all such xy
    is picked up), then the node x is on a leaf

40
Phylogeny Ultrametric tree
  • Step 2.2 (cut-off)
  • For each pair of nodes (x, y) look at the path in
    R
  • See which is the least common ancestor, say (ab)
    note each internal node represents an edge
  • Look up table Ml, if Ml_xy is more than current
    cut-off(ab) replace it with M_xy
  • In other words, the highest Ml value on any edge
    on the path from x to y in T should be its
    distance on the ultrametric tree
  • On example p201-202 root (ad) is updated for
    pairs of all nodes on the opposite sides EB(1),
    ED(1), AD(4), AB(3), CB(4), CD(3)

41
Phylogeny Ultrametric tree
  • Step 3 (ultrametric tree) Recompute R again same
    way as before
  • But, now put distance on internal nodes
  • Height of an internal node is its cut-off / 2
  • Note, computation of R starts with root downwards
  • Adjust distances between the nodes as heights are
    being calculated
  • Done

42
Phylogeny UPGMA
  • Intra-cluster mean distance
  • Inter-cluster distance
  • WPGMA distances from the root to every branch
    tip are equal

43
Phylogeny UPGMA
  • Intra-cluster mean distance

a b c d e a 0 17 21 31 23 b 17 0 30
34 21 c 21 30 0 28 39 d 31 34 28 0
43 e 23 21 39 43 0
44
Phylogeny UPGMA
a b c d e a 0 17 21 31 23 b 17 0 30
34 21 c 21 30 0 28 39 d 31 34 28 0
43 e 23 21 39 43 0
(a,b) c d e (a,b) 0 25.5 32.5 22 c 25.5
0 28 39 d 32.5 28 0 43 e 22 39 43 0
45
Phylogeny UPGMA
a b c d e a 0 17 21 31 23 b 17 0 30
34 21 c 21 30 0 28 39 d 31 34 28 0
43 e 23 21 39 43 0
(a,b) c d e (a,b) 0 25.5 32.5 22 c 25.5
0 28 39 d 32.5 28 0 43 e 22 39 43 0
Setting d ( a , u ) d ( b , u ) D 1 ( a , b )
/ 2
The branches joining a and b to u then have
lengths d ( a , u ) d ( b , u ) 17 / 2
8.5 Assuming, Ultra-metric space.
46
Metric Ultra-metric space
such that for all x , y , z ? M, one has d
( x , y ) 0 d ( x , y ) 0 d ( x , y
) d ( y , x ) (symmetry) d ( x , z ) max
d ( x , y ) , d ( y , z ) (strong triangle or
ultrametric inequality). For metric space,
d(x,z) d(x,y) d(y,z)
47
Comparing phylogenies
D 2 ( ( a , b ) , c ) ( D 1 ( a , c ) 1 D
1 ( b , c ) 1 ) / ( 1 1 ) ( 21 30 ) / 2
25.5 D 2 ( ( a , b ) , d ) ( D 1 ( a , d ) D
1 ( b , d ) ) / 2 ( 31 34 ) / 2 32.5 D 2
( ( a , b ) , e ) ( D 1 ( a , e ) D 1 ( b , e
) ) / 2 ( 23 21 ) / 2 22
48
Phylogeny UPGMA
(a,b) c d e (a,b) 0 25.5 32.5 22 c 25.5
0 28 39 d 32.5 28 0 43 e 22 39 43 0
We deduce the missing branch length d ( u , v )
d ( e , v ) - d ( a , u ) d ( e , v ) - d ( b
, u ) 11 - 8.5 2.5
49
Phylogeny UPGMA
calculated by proportional averaging D 3 ( ( (
a , b ) , e ) , c ) ( D 2 ( ( a , b ) , c ) 2
D 2 ( e , c ) 1 ) / ( 2 1 ) ( 25.5 2
39 1 ) / 3 30 Thanks to this proportional
average, the calculation of this new distance
accounts for the larger size of the ( a , b )
cluster (two elements) with respect to e (one
element). Similarly D 3 ( ( ( a , b ) , e ) , d
) ( D 2 ( ( a , b ) , d ) 2 D 2 ( e , d )
1 ) / ( 2 1 ) ( 32.5 2 43 1 ) / 3
36 Replace ltproportional averagegt with mean,
you get WPGMA
50
Phylogeny UPGMA
(a,b) c d e (a,b) 0 25.5 32.5 22 c 25.5
0 28 39 d 32.5 28 0 43 e 22 39 43 0
((a,b),e) c d ((a,b),e) 0 30 36 c 30 0
28 d 36 28 0
51
Phylogeny UPGMA
((a,b),e) c d ((a,b),e) 0 30 36 c 30 0
28 d 36 28 0
There is a single entry to update, keeping in
mind that the two elements c and d each have a
contribution of 1 in the average computation D
4 ( ( c , d ) , ( ( a , b ) , e ) ) ( D 3 ( c ,
( ( a , b ) , e ) ) 1 D 3 ( d , ( ( a , b ) ,
e ) ) 1 ) / ( 1 1 ) ( 30 1 36 1 ) / 2
33 Final step The final D 4 matrix
is ((a,b),e) (c,d) ((a,b),e) 0 33 (c,d) 33
0
52
Phylogeny UPGMA
53
Phylogeny UPGMA
Time Complexity O(n3) to O(n2 log n)
54
Phylogeny Neighbor-joining
  • Bottom up, as for UPGMA, but non-rooted
  • Distance matrix transformed to Q-matrix
    (negative)
  • Min Q-value used to connect cluster pairs
  • Cluster distance update formula (in
    distance-space not Q-space)
  • Iterate d ? Q ? cluster_join ? d_update ?
    iterate
  • Topology additive distances, node-pair distances
    are conserved
  • Assumption Balanced Minimum Evolution
  • Greedy optimization (underlying linear
    programming)
  • Fast (?), O(n3) complexity for n nodes
  • Correct optimized tree even if the source
    d-matrix is noisy
  • Wiki https//en.wikipedia.org/wiki/Neighbor_jo
    ining

55
Phylogeny General Comments
  • Morphological and molecular (sequences)
  • Distance matrix
  • Maximum parsimony
  • Maximum likelihood and Bayesian inference
  • Post-analysis of Tree-support evaluation
  • Shortcomings
  • convergent evolution, horizontal gene-transfer
  • hybrids, or non-binary tree, or phylogeny network
  • missing species/taxa
  • https//en.wikipedia.org/wiki/Computational_phylog
    enetics

56
Comparing phylogenies
  • Two trees are expected to be isomorphic
  • All nodes should be on the leaves, if not make it
    so
  • Pick up a node u and its sibling v on T1
  • Look for u in T2 and if its sibling is not v
    return False
  • If the sibling is v then merge uv into its parent
    (and remove subtree with u and v)
  • Continue bottom up until both T1 and T2 become
    single node trees, then return True
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