Comparing skew Schur functions: a quasisymmetric perspective

Reiner, Shaw and van Willigenburg showed that if two skew Schur functions sA and sB are equal, then the skew shapes $A$ and $B$ must have the same "row overlap partitions." Here we show that these row overlap equalities are also implied by a much weaker condition than Schur equality: that sA and sB have the same support when expanded in the fundamental quasisymmetric basis F. Surprisingly, there is significant evidence supporting a conjecture that the converse is also true. In fact, we work in terms of inequalities, showing that if the F-support of sA contains that of sB, then the row overlap partitions of A are dominated by those of B, and again conjecture that the converse also holds. Our evidence in favor of these conjectures includes their consistency with a complete determination of all F-support containment relations for F-multiplicity-free skew Schur functions. We conclude with a consideration of how some other quasisymmetric bases fit into our framework.

Structure-Function Relations in the PI-PLC/GDPD Enzyme Superfamily

Phosphatidylinositol-specific phospholipases C (PI-PLC) are known to participate in many eukaryotic signal transduction pathways and act as virulence factors in lower organisms. Glycerophosphoryl diester phosphodiesterase (GDPD) enzymes are involved in phosphate homeostasis and phospholipid catabolism for energy production. Streptomyces antibioticus phosphatidylinositol-specific phospholipase C (SaPLC1) is a 38 kDa enzyme that displays characteristics of both enzyme superfamilies, representing an evolutionary link between these divergent enzyme classes. SaPLC1 also boasts a unique catalytic mechanism that involves a trans 1,6-cyclic inositol phosphate intermediate instead of the typical cis 1,2-cyclic inositol phosphate. The mechanism by which this occurs is still unclear. To attack this problem, we established a wide mutagenesis scan of the active site and measured activities of alanine mutants. A chemical rescue assay was developed to verify that the activity loss was due to the removal of the functional role of the mutated residue. 31P-NMR was employed in characterizing and quantifying intermediates in mutants that slowed the reaction sufficiently. We found that the H37A and H76A mutations support the hypothesis that these structurally conserved residues are also conserved in terms of their catalytic roles. H37 was found to be the general base (GB), while H76 plays the role of general acid (GA). K131 was identified as a semi-conserved key positive charge donor found at the entrance of the active site. By elucidating the SaPLC1 mechanism in relation to its active site architecture, we have increased our understanding of the structure-function relations that support catalysis in the PI-PLC/GDPD superfamily. These findings provide groundwork for in vivo studies of SaPLC1 function and its possible role in novel signaling or metabolism in Streptomyces.