Phylogenetic Tree

1
Phylogenetic Tree
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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

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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,,,

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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

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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

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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

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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

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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)

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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

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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

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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

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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

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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

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Perfect phylogeny problem to check existence
(Boolean matrix)
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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

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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

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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

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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

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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

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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

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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

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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

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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)

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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

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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

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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

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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?

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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

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Phylogeny with Distance Matrix
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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

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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

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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

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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

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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

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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

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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)

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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

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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

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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)

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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

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Phylogeny UPGMA

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

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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
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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.
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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)
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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
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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
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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
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Phylogeny UPGMA
53
Phylogeny UPGMA
Time Complexity O(n3) to O(n2 log n)
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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

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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

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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