Inference rules for supernetwork construction

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## Inference rules for supernetwork construction

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**Inference rules for supernetwork construction**Katharina Huber, School of Computing Sciences, University of East Anglia.**gene2( )**gene1( ) gene2( ) gene1( ) gene2( ) gene1( ) gene2( ) gene1( ) gene2( ) gene1( ) An ultimate goal**gene1( )**But, … ? gene1( ) gene2( ) gene1( ) gene1( ) gene2( ) gene2( ) gene2( )**gene1( )**gene1( ) gene1( ) gene1( ) gene1( ) Or, even worse ? gene2( ) gene2( ) gene2( ) gene2( )**gene1( )**gene1( ) gene1( ) gene1( ) gene1( ) We could, … ? gene2( ) gene2( ) gene2( ) gene2( )**gene1( )**gene1( ) gene1( ) gene1( ) gene1( ) or, … ? gene2( ) gene2( ) gene2( ) gene2( )**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)**A1 A1 A2 A2 A3 A3 One of intersections marked by a dot is empty!**Weak compatibility(Bandelt and Dress, 1992)**A1 A1 A2 A2 A3 A3**Repeat until inference process stabilizes**apply inference rule and add (if underlying condition is violated stop) Collection of partial splits Collection of partial splits 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.**S1**7 1 S2 6 2 5 S3 3 4 Circular collections of partial splits S1=123|4567 S2=23|45671 S3=345|6712 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 lifeRivera et al, 2004**5 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?**Z-closure supernetwork**The ring of lifeRivera et al, 2004 M/Y-inference rules