Inference rules for supernetwork construction Katharina Huber, School of Computing Sciences, University of East Anglia.

An ultimate goal gene2( ) gene1( )

But, … gene2( ) gene1( ) ?

Or, even worse gene2( ) gene1( ) ?

We could, … gene2( ) gene1( ) ?

or, … gene2( ) gene1( ) ?

Not very satisfactory!

So far,.. Z-closure supernetwork (Huson et al, 2004) Q-imputation (Holland et al, 2007), Attractive but produce many splits Filtering approaches

Weak compatibility (Bandelt and Dress, 1992) A1A1 A3A3 One of intersections marked by a dot is empty! A2A2 A2A2 A3A3 A1A1

Weak compatibility (Bandelt and Dress, 1992) A1A1 A3A3 A2A2 A2A2 A3A3 A1A1

Y- inference rule

M - inference rule (Meacham, 1972)

Repeat until inference process stabilizes Collection of partial splits apply inference rule and add (if underlying condition is violated stop) remove partial splits that can get extended A|B extends C|D if either A  C and B  D or A  D and B  C.

Theorem (Gruenewald, Huber, Wu) Suppose  is an irreducible collection of partial splits and  is either the Y- or M- or M/Y-rule. Then any two closures of   obtained via are the same. Irreducible: no split in  extends another split in . Closure: if the underlying condition(s) is (are) never violated, the set of partial splits generated when inference process stabilizes, and  otherwise.

Circular collections of partial splits S 1 =123|4567 S 2 =23|45671 S 3 =345|6712 S1S S2S2 S3S3 A collection  of partial split is said to be displayed by a cycle if every split in  can get extended to a full split such that the resulting split system is circular.

Theorem (Gruenewald, Huber, Wu) Suppose  is an irreducible collection of partial splits. Then  is displayed by a cycle C if and only if the closure of   via M/Y is displayed by C. In that case the closure of  via Y and the closure of  via M is also displayed by C.

Rivera et al’s ring of life Rivera et al, most probable phylogenetic trees from a study of 10 bacterial genomes from Rivera et al, 2004  in its early stages life was more like a network than a tree. How much does this result depend on the fact that trees were all on the same taxa set?

The ring of life Rivera et al, 2004 M/Y-inference rules Z-closure supernetwork