Subtree Prune Regraft&

Horizontal Gene Transferor

Recombination

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.

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.

Another example

Most of the biology-SPR literature assumes a rooted tree. Makes sense, right?

That is ok. SPR still makes sense.

Introduction of SPR moves to Evolutionary BiologyFirst 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.

“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.

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.

EVEN SIMPLER

SPR movesbetweentrees T1, T2

Reticulation events explaining T1, T2

• 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.

From Hein

Computational Aspects of SPRThe min-SPR problem:Given two phylogenetic trees T1,

T2 on taxa X

compute minimum number of SPR moves from tree T1 to tree T2.

Computational Aspects of SPRCalculating 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. $$$$

Hill et. al. and SPRITHill 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.

Hill’s ConjectureLet be phylogenetic trees on

taxa X.If can be decomposed into

solvable clusters

then

Solvable Cluster Example

Hill Conjecture (Killed by Linz)

But….

Hill’s Conjecture Disproven by LinzLinz 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.

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!

The EndPapers and link to SPRIT up on

wiki.