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## PowerPoint Slideshow about 'Phylogenetic Tree' - katina

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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

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

- 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

- 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 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

- 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

- 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: 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

- 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

- 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

- 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 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)

- 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 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 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)

- 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 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…)

- 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

- (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 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 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

- 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

- 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

- 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

- 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: 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

- 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 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 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 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 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 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 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

- 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 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 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 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 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

- 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

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