ABSTRACT
A well-known conjecture of Alspach says that every 2 k -regular Cayley graph of a finite abelian group can be decomposed into Hamiltonian cycles. We consider an analogous question for infinite abelian groups. In this setting one natural analogue of a Hamiltonian cycle is a spanning double-ray. However, a naive generalisation of Alspach's conjecture fails to hold in this setting due to the existence of 2 k -regular Cayley graphs with finite cuts F , where ⣠F ⣠and k differ in parity, which necessarily preclude the existence of a decomposition into spanning double-rays. We show that every 4-regular Cayley graph of an infinite abelian group all of whose finite cuts are even can be decomposed into spanning double-rays, and so characterise when such decompositions exist. We also characterise when such graphs can be decomposed either into Hamiltonian circles, a more topological generalisation of a Hamiltonian cycle in infinite graphs, or into a Hamiltonian circle and a spanning double-ray.
ABSTRACT
The impact of the incorporation of a non-natural amino acid (NNAA) on protein structure, dynamics, and ligand binding has not been studied rigorously so far. NNAAs are regularly used to modify proteins post-translationally in vivo and in vitro through click chemistry. Herein, structural characterisation of the impact of the incorporation of azidohomoalanine (AZH) into the model protein domain PDZ3 is examined by means of NMR spectroscopy and X-ray crystallography. The structure and dynamics of the apo state of AZH-modified PDZ3 remain mostly unperturbed. Furthermore, the binding of two PDZ3 binding peptides are unchanged upon incorporation of AZH. The interface of the AZH-modified PDZ3 and an azulene-linked peptide for vibrational energy transfer studies has been mapped by means of chemical shift perturbations and NOEs between the unlabelled azulene-linked peptide and the isotopically labelled protein. Co-crystallisation and soaking failed for the peptide-bound holo complex. NMR spectroscopy, however, allowed determination of the protein-ligand interface. Although the incorporation of AZH was minimally invasive for PDZ3, structural analysis of NNAA-modified proteins through the methodology presented herein should be performed to ensure structural integrity of the studied target.
Subject(s)
Alanine/analogs & derivatives , Disks Large Homolog 4 Protein/chemistry , Ligands , Alanine/chemistry , Amino Acid Sequence , Circular Dichroism , Crystallography, X-Ray , Disks Large Homolog 4 Protein/genetics , Disks Large Homolog 4 Protein/metabolism , Isotope Labeling , Magnetic Resonance Spectroscopy , Mutagenesis , PDZ Domains/genetics , PDZ Domains/physiology , Peptides/chemistry , Peptides/metabolism , Protein Binding , Protein Structure, Tertiary , Recombinant Proteins/biosynthesis , Recombinant Proteins/chemistryABSTRACT
An exchange system is presented, which allows ultrafast experiments with high excitation rates (1 kHz) on samples with reaction cycles in the range of a few seconds and small sample volumes of about 0.3 ml. The exchange is accomplished using a commercially available cuvette by the combination of a special type of magnetic stirring with transverse translational motion of the sample cuvette.
