Eventual Positivity of a Class of Double Star-like Sign Patterns

Identifying and classifying the potentially eventually positive sign patterns and the potentially eventually exponentially positive sign patterns of orders greater than 3 have been raised as two open problems since 2010. In this article, we investigate the potential eventual positivity of the class of double star-like sign patterns S(n,m,1) whose underlying graph G(S(n,m,1)) is obtained from the underlying graph G(S(n,m)) of the (n+m)-by-(n+m) double star sign patterns S(n,m) by adding an additional vertex adjacent to the two center vertices and removing the edge between the center vertices. We firstly establish some necessary conditions for a double star-like sign pattern to be potentially eventually positive, and then identify all the minimal potentially eventually positive double star-like sign patterns. Secondly, we classify all the potentially eventually positive sign patterns in the class of double star-like sign patterns S(n,m,1). Finally, as an application of our results about the potentially eventually positive double star-like sign patterns, we identify all the minimal potentially eventually exponentially positive sign patterns and characterize all the potentially eventually exponentially positive sign patterns in the class of double star-like sign patterns S(n,m,1)

Genes "Waiting" for recruitment by the adaptive immune system: The insights from amphioxus

in seeking evidence of the existence of adaptive immune system (AIS) in ancient chordate, cDNA clones of six libraries from a protochordate, the Chinese amphioxus, were sequenced. Although the key molecules such as TCR, MHC, Ig, and RAG in AIS have not been identified from our database, we demonstrated in this study the extensive molecular evidence for the presence of genes homologous to many genes that are involved in AIS directly or indirectly, including some of which may represent the putative precursors of vertebrate AIS-related genes. The comparative analyses of these genes in different model organisms revealed the different fates of these genes during evolution. Their gene expression pattern suggested that the primitive digestive system is the pivotal place of the origin and evolution of the AIS. Our studies support the general statement that AIS appears after the jawless/jawed vertebrate split. However our study further reveals the fact that AIS is in its twilight in amphioxus and the evolution of the molecules in amphioxus are waiting for recruitment by the emergence of AIS