Subtree Prune Regraft & Horizontal Gene Transfer or Recombination

1. Subtree Prune Regraft&Horizontal Gene TransferorRecombination

2. We will discuss… • What are SPR moves? • Hein: Origins of SPR moves in biology. • Hill et. al.: A conjecture and software SPRIT. • Linz: Death of the conjecture. Replacement given and proved correct.

3. What is an SPR move?(Then some Biology!) • Pick a subtree (blue) • Cut off blue subtree. • Remove internal node where blue subtree was attached • Pick a new edge • Add an internal node to new edge • Attach blue subtree.

4. Another example Most of the biology-SPR literature assumes a rooted tree. Makes sense, right? That is ok. SPR still makes sense.

5. Introduction of SPR moves to Evolutionary Biology • First proposed by Hein in “Reconstructing Evolution of Sequences Subject to Recombination Using Parsimony”, Mathematical Biosciences, 1990. • Parsimony states that a history of sequences that minimizes the amount of evolution is a good approximation to the real evolutionary history.

6. “Parsimony is also applied to reconstruction of homologous sequences where recombinations or horizontals transfer can occur” – Hein. • Hein showed “The appropriate structure to represent the evolution of sequences with recombinations or horizontal transfers” is given by SPR moves.

7. Simply Put… • If two incongruent trees can be explained by a single reticulation event, then one tree can be constructed from the other by a single SPR move. • In general, if more than one reticulation event is needed to explain the difference between two trees.. • The number of SPR moves from one tree to the other is a lower bound on the number of reticulation events.

8. EVEN SIMPLER SPR moves between trees T1, T2 Reticulation events explaining T1, T2

9. From Hein • Two segments (thick parts), left and right, on separation chromosomes. • Then…recombination!!! • Left part of molecule related by tree with straight lines. • Right part of molecule described by following curved line.

11. Computational Aspects of SPR • The min-SPR problem: • Given two phylogenetic trees T1, T2 on taxa X compute minimum number of SPR moves from tree T1 to tree T2.

13. Computational Aspects of SPR • Calculating the minimum number of SPR moves between two trees is NP-hard. • What does that mean? • If you could quickly (polynomial time) solve the min-SPR problem, you could solve this list of extremely hard problems… • 3Sat, traveling salesman, vertex cover, dominating set, graph coloring, clique, independent set, Hamiltonian circuit, minimum degree spanning tree, maxcut, minimum edge-cost flow,…………… Many more! Oh, and you would be a millionaire. $$$$

14. Hill et. al. and SPRIT • Hill et. al. developed software SPRIT to solve the min-SPR problem. • They implemented a previously known exact method (slow). • They also proposed and implemented a new heuristic, which they conjectured gave them the correct SPR distance. • New method is a Divide & Conquer method.

15. Hill’s Conjecture • Let be phylogenetic trees on taxa X. • If can be decomposed into solvable clustersthen

16. Solvable Cluster Example

17. Hill Conjecture (Killed by Linz) But….

18. Hill’s Conjecture Disproven by Linz • Linz gave this counterexample to the Hill conjecture. • However, Linz proved that a slightly stronger clustering works, called sub-tree like clustering. • Hill has subsequently added this to the software SPRIT.

19. Even though the min-SPR problem is proven to be computationally hard, all hope is not lost…. • The running time of current algorithms & heuristics is exponential in the number of SPR events. • Good new: Typically the number of SPR events occurring in nature is low!

20. The End • Papers and link to SPRIT up on wiki.