Characterization of supercommuting matrices

For a given n-by-n matrix A, we consider the set of matrices which commute with A and all of whose principal submatrices commute with the corresponding principal submatrices of A. The properties of this set are examined, with particular attention to its dimension.     

The evaluation of Cauchy principal value integrals involving unknown poles

A method is described for the evaluation of a Cauchy principal value integral of the formf ₀ ^p f(t)dt, wheref is analytic in the interval [0,p] except at a simple pole at an unknown point in (0,p), with an unknown residue. The method is based on the trapezoidal rule.     

N-α-Benzyloxyacetyl derivatives of (S)-4-benzyl-5,5-dimethyloxazolidin-2-one for the asymmetric synthesis of differentially protected α,β-dihydroxyaldehydes

α-Dibenzylamino- and α-benzyloxy- derivatives of N-acetyl-(S)-4-benzyl-5,5-dimethyloxazolidin-2-one readily undergo highly stereoselective boron mediated syn-aldol reactions with a range of aromatic and aliphatic aldehydes, generating the syn-aldol products in good to excellent yields as single diastereoisomers after purification. In the α-dibenzylamino series, deprotection of the functionalised aldol fragments to the corresponding α-amino-β-hydroxy methyl ester or α-amino-β-hydroxyaldehyde proved problematic, with a range of N- and O-protecting groups giving mixtures of products arising from endocyclic and exocyclic cleavage pathways. However, in the α-benzyloxy series, O-silyl protection of the aldol products, and subsequent DIBAL reduction gives stereoselectively the corresponding N-1′-hydroxyalkyloxazolidin-2-ones, which undergo base promoted fragmentation to the desired highly functionalised and differentially protected α,β-dihydroxyaldehydes in good yields and without loss of stereochemical integrity.     

PROTEIN STRUCI'URE MODELLING OF THE BACTERIAL LIGHT-HARVESTING COMPLEX

Abstract Protein structure modelling offers a method of obtaining 3-dimensional information that can be tested and used to plan mutagenesis experiments when a crystallographically determined structure is not available. At its simplest a model may consist of little more than a secondary structure prediction coupled with a determination of the likely regions of transmembrane/membrane surface/globular configuration. These methods can yield an interesting topology map of the protein, which places the residues in their likely positions with respect to, for example, the membrane interface. If it is a member of a large family of related proteins then aligned protein sequences can be used to predict the residues that have an important function as these. will be largely conserved in the alignments. Using all these methods a model can be constructed (using for example, the Nicholson Molecular Modelling Kit) to visualize the proposed structure in three dimensions following the premise of good design, that is, avoiding obvious steric clashes, packing of helices in a realistic manner, observing the correct H-bond lengths, etc . In this latter exercise the review of Chothia ( Annu. Rev. Biochem . 53 , 537–572, 1984) of the principles of protein structure is particularly helpful as it clearly sets out how proteins pack and their preferred configuration. There is a wealth of information about individual amino acid conformational preferences and observed frequencies of occurrence in known protein structures, which can help decide how the residues in the model can be oriented.
In this article we have collated the various protein models of the bacterial light-harvesting complexes and present our own model, which is a synthesis of the available biophysical data and theoretical predictions, and show its performance in explaining recent results of site-directed mutants of the LHI and LH2 light-harvesting complexes of Rhodobacter sphaeroides .