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Phylogenetic Tree. 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

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phylogenetic tree what it is
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
phylogenetic tree motivation
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,,,
phylogenetic tree topology
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
phylogenetic tree source data types
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
characters for phylogeny
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
phylogeny using character state matrix
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
phylogeny using character state matrix problems
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)
phylogeny using character state matrix problems9
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
character domain types
Character domain types
  • Domain of character c could be:

red < - > blue < - > yellow < - > 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
perfect phylogeny
Perfect phylogeny
  • Problem-free source
  • Each edge in phylogeny is a transition of the respective character’s value
  • All nodes with the same value for a character must form a subutree (with the transition at its root)
  • Such a tree is “perfect phylogeny”
perfect phylogeny problem
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 -> 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
perfect phylogeny problem13
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 Pi[i=3 to n] (2i-5): Exponential
perfect phylogeny problem to check existence boolean matrix
Perfect phylogeny problem: to check existence (Boolean matrix)
  • Organize char state matrix columnwise: 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 another one or they do not overlap at all
  • If they overlap, no perfect phylogeny exist
perfect phylogeny problem to check existence boolean matrix15
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 i=1, but all other nodes have i=0.
  • 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
perfect phylogeny problem to check existence boolean matrix16
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
  • Provesif and only if of the condition for perfect phylogeny
  • Algorithm for checking: Pairwise checking of object set may take O(m^2) for m characters, but set overlap may check even more time
perfect phylogeny problem algorithm boolean matrix
Perfect phylogeny problem: Algorithm (Boolean matrix)
  • Sort the columns by number of 1’s (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 > log m
perfect phylogeny problem algorithm boolean matrix18
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
perfect phylogeny problem algorithm non boolean matrix but
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
phylogeny problem arbitrary domain size unordered characters
Phylogeny problem: arbitrary domain size, unordered characters
  • (Def) Triangulated graph: [no big hole] cycle with >3 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
phylogeny problem arbitrary domain size unordered characters21
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
phylogeny problem arbitrary domain size unordered characters22
Phylogeny problem: arbitrary domain size, unordered characters
  • Iff a character state matrix translates to a c-triangulated graph then it admits perfect phylogeny
  • Creating+checking c-triangulation is NP-hard (related to finding max-clique problem)
phylogeny problem arbitrary domain size unordered characters 2 characters
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
phylogeny construction arbitrary domain size unordered characters 2 characters
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
when perfect phylogeny does not exist
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
when perfect phylogeny does not exist parsimony
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?
when perfect phylogeny does not exist parsimony27
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
phylogeny with distance matrix
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
phylogeny with distance matrix29
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
phylogeny with distance matrix30
Phylogeny with Distance Matrix
  • Typically we have 2 matrices: (1) upper bound on distances, and (2) for lower bounds
  • Metric space:
    • dij>0, dii=0, dij=dji, for all I, j
    • dij =< dik + dkj
  • Additive metric spaces follow 4 point condition:

dij+dkl=dik+djl >= dil+djk

phylogeny with distance matrix31
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
phylogeny with distance matrix32
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
phylogeny with distance matrix33
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
phylogeny with distance matrix34
Phylogeny with Distance Matrix
  • The property of uniqueness of the tree remain valid for any k objects for k>4, 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
phylogeny ultrametric tree
Phylogeny: Ultrametric tree
  • Exc: 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)
phylogeny ultrametric tree36
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
phylogeny ultrametric tree37
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
phylogeny ultrametric tree38
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)
phylogeny ultrametric tree39
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
comparing phylogenies
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 (an dremove subtree with u and v)
  • Continue bottom up until both T1 and T2 become single node trees, then return True