Making role assignment feasible: A polynomial-time algorithm for computing ecological colorings.

In this paper, we study the problem of ecologically coloring a graph. Intuitively, an ecological coloring of a graph is a role assignment to the nodes of the graph, such that two nodes surrounded by the same set of roles must be assigned the same role (Borgatti and Everett, 1992). We prove that, for any simple undirected graph $G$ with $n_G$ distinct neighborhoods and for any integer $k$ with $1 \leq k \leq n_G$, $G$ admits an ecological coloring which uses exactly $k$ roles, and that this coloring can be computed in polynomial time. Our result strongly contrasts with the NP-completeness result of the regular coloring problem, where it is required that two nodes with the same role must be surrounded by the same set of roles (Fiala and Paulusma, 2005). Hence, we conclude that not only the ecological coloring is easier to understand as a model of social relationships (Borgatti and Everett, 1994), but it is also feasible from a computational complexity point of view

Importance of the N Terminus of Rous Sarcoma Virus Protease for Structure and Enzymatic Function

All retrovirus proteases (PRs) are homodimers, and dimerization is essential for enzymatic function. The dimer is held together largely by a short four-stranded antiparallel beta sheet composed of the four or five N-terminal amino acid residues and a similar stretch of residues from the C terminus. We have found that the enzymatic and structural properties of Rous sarcoma virus (RSV) PR are exquisitely sensitive to mutations at the N terminus. Deletion of one or three residues, addition of one residue, or substitution of alanine for the N-terminal leucine reduced enzymatic activity on peptide and protein substrates 100- to 1,000-fold. The purified mutant proteins remained monomeric up to a concentration of about 2 mg/ml, as determined by dynamic light scattering. At higher concentrations, dimerization was observed, but the dimer lacked or was deficient in enzymatic activity and thus was inferred to be structurally distinct from a wild-type dimer. The mutant protein lacking three N-terminal residues (ΔLAM), a form of PR occurring naturally in virions, was examined by nuclear magnetic resonance spectroscopy and found to be folded at concentrations where it was monomeric. This result stands in contrast to the report that a similarly engineered monomeric PR of human immunodeficiency virus type 1 is unstructured. Heteronuclear single quantum coherence spectra of the mutant at concentrations where either monomers or dimers prevail were nearly identical. However, these spectra differed from that of the dimeric wild-type RSV PR. These results imply that the chemical environment of many of the amide protons differed and thus that the three-dimensional structure of the ΔLAM PR mutant is different from that of the wild-type PR. The structure of this mutant protein may serve as a model for the structure of the PR domain of the Gag polyprotein and may thus give clues to the initiation of proteolytic maturation in retroviruses