A natural classification of the basic helix–loop–helix class of transcription factors

A natural (evolutionary) classification is provided for 242 basic helix–loop–helix (bHLH) motif-containing proteins. Phylogenetic analyses of amino acid sequences describe the patterns of evolutionary change within the motif and delimit evolutionary lineages. These evolutionary lineages represent well known functional groups of proteins and can be further arranged into five groups based on binding to DNA at the hexanucleotide E-box, the amino acid patterns in other components of the motif, and the presence/absence of a leucine zipper. The hypothesized ancestral amino acid sequence for the bHLH transcription factor family is given together with the ancestral sequences of the subgroups. It is suggested that bHLH proteins containing a leucine zipper are not a natural, monophyletic group.

On the classification of binary shifts of finite commutant index

We provide a complete classification up to conjugacy of the binary shifts of finite commutant index on the hyperfinite II1, factor. There is a natural correspondence between the conjugacy classes of these shifts and polynomials over GF(2) satisfying a certain duality condition.