On the determinant of the QQ-walk matrix of rooted product with a path

Let $G$ be an $n$-vertex graph and $Q(G)$ be its signless Laplacian matrix. The $Q$-walk matrix of $G$, denoted by $W_Q(G)$, is $[e,Q(G)e,\ldots,Q^{n-1}(G)e]$, where $e$ is the all-one vector. Let $G\circ P_m$ be the graph obtained from $G$ and $n$ copies of the path $P_m$ by identifying the $i$-th vertex of $G$ with an endvertex of the $i$-th copy of $P_m$ for each $i$. We prove that, $$\det W_Q(G\circ P_m)=\pm (\det Q(G))^{m-1}(\det W_Q(G))^m$$ holds for any $m\ge 2$. This gives a signless Laplacian counterpart of the following recently established identity [17]: $$\det W_A(G\circ P_m)=\pm (\det A(G))^{\lfloor\frac{m}{2}\rfloor}(\det W_A(G))^m,$$ where $A(G)$ is the adjacency matrix of $G$ and $W_A(G)=[e,A(G)e,\ldots,A^{n-1}(G)e]$. We also propose a conjecture to unify the above two equalities.Comment: 16 pages, 1 figur

The evolution of BIR domain and its containing proteins

AbstractBIR domain and its containing proteins play critical roles in cell apoptosis and cell division. Here several lines of novelty were revealed based on a comprehensive evolutionary analysis of BIR domains in 11 representative organisms. First, the type II BIR domains in Survivin and Bruce showed more conservation compared with the type I BIR domains in the inhibitors of apoptosis proteins (IAPs). Second, cIAP was derived from a XIAP duplicate and emerged just after the divergence of invertebrates and vertebrates. Third, the three BIR domains of NAIP displayed significantly elevated evolutionary rates compared with the BIR domains in other IAPs

Phylogenetic and evolutionary analysis of the septin protein family in metazoan

AbstractSeptins, a conserved family of cytoskeletal GTP-binding proteins, were presented in diverse eukaryotes. Here, a comprehensive phylogenetic and evolutionary analysis for septin proteins in metazoan was carried out. First, we demonstrated that all septin proteins in metazoan could be clustered into four subgroups, and the representative homologue of every subgroup was presented in the non-vertebrate chordate Ciona intestinalis, indicating that the emergence of the four septin subgroups should have occurred prior to divergence of vertebrates and invertebrates, and the expansion of the septin gene number in vertebrates was mainly by the duplication of pre-existing genes rather than by the appearance of new septin subgroup. Second, the direct orthologues of most human septins existed in zebrafish, which suggested that human septin gene repertoire was mainly formed by as far as before the split between fishes and land vertebrates. Third, we found that the evolutionary rate within septin family in mammalian lineage varies significantly, human SEPT1, SEPT 10, SEPT 12, and SEPT 14 displayed a relative elevated evolutionary rate compared with other septin members. Our data will provide new insights for the further function study of this protein family

The ancient function of RB-E2F Pathway: insights from its evolutionary history

<p>Abstract</p> <p>Background</p> <p>The RB-E2F pathway is conserved in most eukaryotic lineages, including animals and plants. E2F and RB family proteins perform crucial functions in cycle controlling, differentiation, development and apoptosis. However, there are two kinds of E2Fs (repressive E2Fs and active E2Fs) and three RB family members in human. Till now, the detail evolutionary history of these protein families and how RB-E2F pathway evolved in different organisms remain poorly explored.</p> <p>Results</p> <p>We performed a comprehensive evolutionary analysis of E2F, RB and DP (dimerization partners of E2Fs) protein family in representative eukaryotic organisms. Several interesting facts were revealed. First, orthologues of RB, E2F, and DP family are present in several representative unicellular organisms and all multicellular organisms we checked. Second, ancestral E2F, RB genes duplicated before placozoans and bilaterians diverged, thus E2F family was divided into E2F4/5 subgroup (including repressive E2Fs: E2F4 and E2F5) and E2F1/2/3 subgroup (including active E2Fs: E2F1, E2F2 and E2F3), RB family was divided into RB1 subgroup (including RB1) and RBL subgroup (including RBL1 and RBL2). Third, E2F4 and E2F5 share more sequence similarity with the predicted E2F ancestral sequence than E2F1, E2F2 and E2F3; E2F4 and E2F5 also possess lower evolutionary rates and higher purification selection pressures than E2F1, E2F2 and E2F3. Fourth, for RB family, the RBL subgroup proteins possess lower evolutionary rates and higher purification selection pressures compared with RB subgroup proteins in vertebrates,</p> <p>Conclusions</p> <p>Protein evolutionary rates and purification selection pressures are usually linked with protein functions. We speculated that function conducted by E2F4/5 subgroup and RBL subgroup proteins might mainly represent the ancient function of RB-E2F pathway, and the E2F1/2/3 subgroup proteins and RB1 protein might contribute more to functional diversification in RB-E2F pathway. Our results will enhance the current understanding of RB-E2F pathway and will also be useful to further functional studies in human and other model organisms.</p> <p>Reviewers</p> <p>This article was reviewed by Dr. Pierre Pontarotti, Dr. Arcady Mushegian and Dr. Zhenguo Lin (nominated by Dr. Neil Smalheiser).</p