The Evolution Trees From: Computational Biology by R. C. T. Lee S. J. Shyu Department of Computer Science Ming Chuan University

S. J. Shyu Biological Assumption All species evolve from a common ancestor. Root: the suspected common ancestor Leaves: species (alive) Internal nodes: unknown species Length on edge ( a, b ): the time needed to evolve from a to b

S. J. Shyu Assumptions on Evolution Trees for Computer Scientists (I) Rooted evolution tree Degree of each internal node is 3, except the root. Unrooted evolution tree Degree of each internal node is 3.

S. J. Shyu Assumptions on Evolution Trees for Computer Scientists (II) The input is a distance matrix among all of the species. The distances satisfy the triangular inequality relationship. Depending upon different conditions, different ETs will be constructed to reflect the distances among species. Let d ( s i, s j ) ( dt ( s i, s j )) be the distance between s i and s i in the distance matrix (some evolution tree). Then dt ( i, j )  d ( i, j ). If the ET is rooted, then the distance from the root too all leaves are the same.

S. J. Shyu How many unrooted evolution trees are there? (I) NE ( n )=2 n -3 (by induction) Whenever a new species is added, #edgs+=2.

S. J. Shyu How many unrooted evolution trees are there? (II) Every edge is possible to be split to add a new species. Number of unrooted ETs for n species: TU ( n +1)=(2 n -3) TU ( n )or TU ( n )=(2 n -5) TU ( n -1)

S. J. Shyu How many rooted evolution trees are there? (I) Every edge in an unrooted ET is possible to be split to add a root to turn the ET into a rooted one.

S. J. Shyu How many rooted evolution trees are there? (II) Number of rooted ETs for n species: TR ( n )=(2 n -3) TU ( n ) TU ( n )= TU ( n +1) # of rooted ETs is much larger than that of unrooted ETs. It is desirable to consider unrooted Ets. Still, we cannot explain evolution by an unrooted tree.

S. J. Shyu Transforming unrooted ETs into rooted ones Add a species which is exceeding different from the species analyzed. This outlier species causes a long link that can be used to identify a root.

S. J. Shyu Distance matrix vs. evoluiton trees The input of an evolution tree problem is a distance matrix. We are asked to construct an evolution tree to perfectly reflect these distances. The goodness of an evolution tree is evaluated under some criterion.

S. J. Shyu Criteria of evoluiton trees Let d ( s i, s j ) ( dt ( s i, s j )) be the distance between s i and s j in the distance matrix (some evolution tree). 1. Minimax ETs max. of ( dt ( s i, s j )- d ( s i, s j )) is minimized 2. Minisum ETs total sum of all pairs of distances (  dt ( s i, s j )) is minimized 3. Minisize ETs total length of the tree is minimized

S. J. Shyu The complexity of ET problems MinimaxMinisumMinisize UnrootedNP-complete Unknown Rooted O(n2)O(n2) NP-complete

S. J. Shyu A Minimax rooted ET Algorithm Idea: Preserve the longest distance Suppose d ( s i, s j ) is the longest distance in the input matrix. dt ( s i, s j ) = d ( s i, s j ) Recursively apply for T i and T j Which species are in T i ( T i )?

S. J. Shyu Minimax rooted ET Algorithm

S. J. ShyuExample

5.4 The determination of weights when the ET structure is given What’s the minisize unrooted ET if the ET structure is given? As above for minisize unrooted ET As above for minimax unrooted ET How to determine the structure of ET is a problem. (# of possible ET is exponential to n.) By linear programming Open

S. J. ShyuExamples

5.5 UPGMA for rooted ETs A heuristic to determine a reasonably good structure of rooted ETs.

S. J. Shyu Example (UPGMA)

S. J. Shyu 5.6 The neighbor joining method for unrooted ETs A heuristic to determine a reasonably good structure of unrooted ETs.

S. J. Shyu Example (NJ) (I) w ( x, s 1 )=1/3( d ( s 1, s 2 )+ d ( s 1, s 3 )+ d ( s 1, s 4 )) =1/3(4+4+3)=3.67 (the mean of the distances from this species to all other species) w ( x, s i ) = average( s i ) = (  d ( s i, s j ))/( n -1) ijij 1 center

S. J. Shyu Example (NJ) (II) Geometrical center OC - NC = =2.34

S. J. Shyu Example (NJ) (III) OC - NC = (average( s 1 )+average( s 2 )) - (average( s 1 )+average( s 2 )+ d ( s 1,s 2 ))/2 =(average( s 1 )+average( s 2 )- d ( s 1,s 2 ))/2 d ( s 1,s 2 ) is preserved

S. J. Shyu 5.6 An Approximation Algorithm for an unrooted minisize ET unrooted minisize ET: no polynomial algorithm; never been proved to be NP-complete An approximation algorithm with size smaller than twice of the size of an optimal solution:

S. J. Shyu Example (I) Minimal spanning tree an unrooted minisize ET With error ratio  1

S. J. Shyu Proof of correctness (I) Evolution tree? Degree of each internal node=3 dt ( s i, s j )  d ( s i, s j ) dt ( s i, s j ) = d MST ( s i, s j ) d MST ( s i, s j )  dt ( s i, s j ) (triangular inequality)  1 Error ratio  1

S. J. Shyu Hamiltonian cycle and Euler cycle Hamiltonian cycle: a cycle visiting all of the nodes in G =( V, E ) exactly once Euler cycle: a cycle traversing each edge in G =( V, E ) exactly once

S. J. Shyu Proof of correctness (II)  1 Error ratio  1  1. |MST|  |TSP| P  Delete any edge in TSP  |P|<|TSP|  |MST|  |P|  2. |A|=|MST|  |TSP|  2 3. |TSP|  2|OPT| 4. Duplicate every edge of a tree  there is an Euler tour (Et)  2 5. T: optimal ET  |Et|  2|OPT|  6. CEt: the cycle of species of Et  |CEt|  |Et| ( dt ( s i, s j )  d ( s i, s j ))  2 7. |A|=|MST|  |TSP|  |CEt|  |Et|  2|OPT|

S. J. Shyu 5.6 The minimal spanning tree preservation approach for ET construction D : input distance matrix D t : distance matrix of the ET MST preservation: MST( D ) is an MST( D t )

S. J. Shyu Example (I)

S. J. Shyu Example (II)