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Abstract: We study basic properties of monomial ideals with linear quotients. It is shown that if the monomial ideal I has linear quotients, then the squarefree part of I and each component of I as well as have linear quotients, where is the graded maximal ideal of the polynomial ring. As an analogy to the Rearrangement Lemma of Björner and Wachs we also show that for a monomial ideal with linear quotients the admissible order of the generators can be chosen degree increasingly. [Copyright &y& Elsevier]

Abstract: We show that an arbitrary monomial ideal I is pretty clean if and only if its polarization is clean. This yields a new characterization of pretty clean monomial ideals in terms of the arithmetic degree, and it also implies that a multicomplex is shellable if and only the simplicial complex corresponding to its polarization is (non-pure) shellable. We also discuss Stanley decompositions in relation to prime filtrations. [Copyright &y& Elsevier]

Abstract: The multi-domain scaffolding protein NHERF1 modulates the assembly and intracellular trafficking of various transmembrane receptors and ion-transport proteins. The two PDZ (postsynaptic density 95/disk large/zonula occluden 1) domains of NHERF1 possess very different ligand-binding capabilities: PDZ1 recognizes a variety of membrane proteins with high affinity, while PDZ2 only binds limited number of target proteins. Here using NMR, we have determined the structural and dynamic mechanisms that differentiate the binding affinities of the two PDZ domains, for the type 1 PDZ-binding motif (QD T R L ) in the carboxyl terminus of cystic fibrosis transmembrane regulator. Similar to PDZ2, we have identified a helix–loop–helix subdomain coupled to the canonical PDZ1 domain. The extended PDZ1 domain is highly flexible with correlated backbone motions on fast and slow timescales, while the extended PDZ2 domain is relatively rigid. The malleability of the extended PDZ1 structure facilitates the transmission of conformational changes at the ligand-binding site to the remote helix–loop–helix extension. By contrast, ligand binding has only modest effects on the conformation and dynamics of the extended PDZ2 domain. The study shows that ligand-induced structural and dynamic changes coupled with sequence variation at the putative PDZ binding site dictate ligand selectivity and binding affinity of the two PDZ domains of NHERF1. [Copyright &y& Elsevier]