Functional divergence after gene duplication and sequence-structure relationship: a case-study of G-protein alpha subunits

In this study, I use animal G-protein alpha subunit family as an example to illustrate a comprehensive analytical pipeline for detecting different types of functional divergence of protein families, which is phylogeny-dependent, combined with ancestral sequence inference and available protein structure information. In particular, I focus on (i) Type-I functional divergence, or site-specific rate shift, as typically exemplified by amino acid residue highly conserved in a subset of homologous genes but highly variable in a different subset of homologous genes, and (ii) Type-II functional divergence, or the shift of cluster-specific amino acid property, as exemplified by a radical shift of amino acid property between duplicate genes, which is otherwise evolutionally conserved. We utilized the software DIVERGE2 to carry out these analyses. In the case of G-protein alpha subunit gene family, I have tested the significance of functional divergence between subfamily Gq and Gs, and predicted the candidate amino acid residues related to either Type-I or Type-II functional divergence. Then, the inferred ancestral sequences and current amino acid configuration of these candidate sites were combined with phylogenies to explore the trends of functional divergence. Finally, these predicted residues are mapped to the protein structures to test whether these residues may have structures or solvent accessibility preferences

Root multiplicities for Nichols Algebras of diagonal type

In this thesis we chase the root multiplicities for Nichols algebras of diagonal type. Based on an inequality for the number of Lyndon words and an identity for the shuffle map, we illustrate when the multiplicity of a root is smaller than in the tensor algebra of a braided vector space of diagonal type, and determine the dimension of the kernel of the shuffle map considered as an operator acting on the free algebra. Moreover, we give an complete expression for the multiplicities of a class of roots for Nichols algebras of diagonal type of rank two