On the computation of zone and double zone diagrams

Classical objects in computational geometry are defined by explicit relations. Several years ago the pioneering works of T. Asano, J. Matousek and T. Tokuyama introduced "implicit computational geometry", in which the geometric objects are defined by implicit relations involving sets. An important member in this family is called "a zone diagram". The implicit nature of zone diagrams implies, as already observed in the original works, that their computation is a challenging task. In a continuous setting this task has been addressed (briefly) only by these authors in the Euclidean plane with point sites. We discuss the possibility to compute zone diagrams in a wide class of spaces and also shed new light on their computation in the original setting. The class of spaces, which is introduced here, includes, in particular, Euclidean spheres and finite dimensional strictly convex normed spaces. Sites of a general form are allowed and it is shown that a generalization of the iterative method suggested by Asano, Matousek and Tokuyama converges to a double zone diagram, another implicit geometric object whose existence is known in general. Occasionally a zone diagram can be obtained from this procedure. The actual (approximate) computation of the iterations is based on a simple algorithm which enables the approximate computation of Voronoi diagrams in a general setting. Our analysis also yields a few byproducts of independent interest, such as certain topological properties of Voronoi cells (e.g., that in the considered setting their boundaries cannot be "fat").Comment: Very slight improvements (mainly correction of a few typos); add DOI; Ref [51] points to a freely available computer application which implements the algorithms; to appear in Discrete & Computational Geometry (available online

Tryptophan and Non-Tryptophan Fluorescence of the Eye Lens Proteins Provides Diagnostics of Cataract at the Molecular Level

The chemical nature of the non-tryptophan (non-Trp) fluorescence of porcine and human eye lens proteins was identified by Mass Spectrometry (MS) and Fluorescence Steady-State and Lifetime spectroscopy as post-translational modifications (PTM) of Trp and Arg amino acid residues. Fluorescence intensity profiles measured along the optical axis of human eye lenses with age-related nuclear cataract showed increasing concentration of fluorescent PTM towards the lens centre in accord with the increased optical density in the lens nucleolus. Significant differences between fluorescence lifetimes of “free” Trp derivatives hydroxytryptophan (OH-Trp), N-formylkynurenine (NFK), kynurenine (Kyn), hydroxykynurenine (OH-Kyn) and their residues were observed. Notably, the lifetime constants of these residues in a model peptide were considerably greater than those of their “free” counterparts. Fluorescence of Trp, its derivatives and argpyrimidine (ArgP) can be excited at the red edge of the Trp absorption band which allows normalisation of the emission spectra of these PTMs to the fluorescence intensity of Trp, to determine semi-quantitatively their concentration. We show that the cumulative fraction of OH-Trp, NFK and ArgP emission dominates the total fluorescence spectrum in both emulsified post-surgical human cataract protein samples, as well as in whole lenses and that this correlates strongly with cataract grade and age

Hydroimidazolone Modification of the Conserved Arg12 in Small Heat Shock Proteins: Studies on the Structure and Chaperone Function Using Mutant Mimics

Methylglyoxal (MGO) is an α-dicarbonyl compound present ubiquitously in the human body. MGO reacts with arginine residues in proteins and forms adducts such as hydroimidazolone and argpyrimidine in vivo. Previously, we showed that MGO-mediated modification of αA-crystallin increased its chaperone function. We identified MGO-modified arginine residues in αA-crystallin and found that replacing such arginine residues with alanine residues mimicked the effects of MGO on the chaperone function. Arginine 12 (R12) is a conserved amino acid residue in Hsp27 as well as αA- and αB-crystallin. When treated with MGO at or near physiological concentrations (2–10 µM), R12 was modified to hydroimidazolone in all three small heat shock proteins. In this study, we determined the effect of arginine substitution with alanine at position 12 (R12A to mimic MGO modification) on the structure and chaperone function of these proteins. Among the three proteins, the R12A mutation improved the chaperone function of only αA-crystallin. This enhancement in the chaperone function was accompanied by subtle changes in the tertiary structure, which increased the thermodynamic stability of αA-crystallin. This mutation induced the exposure of additional client protein binding sites on αA-crystallin. Altogether, our data suggest that MGO-modification of the conserved R12 in αA-crystallin to hydroimidazolone may play an important role in reducing protein aggregation in the lens during aging and cataract formation

Beyond Genetic Factors in Familial Amyloidotic Polyneuropathy: Protein Glycation and the Loss of Fibrinogen's Chaperone Activity

Familial amyloidotic polyneuropathy (FAP) is a systemic conformational disease characterized by extracellular amyloid fibril formation from plasma transthyretin (TTR). This is a crippling, fatal disease for which liver transplantation is the only effective therapy. More than 80 TTR point mutations are associated with amyloidotic diseases and the most widely accepted disease model relates TTR tetramer instability with TTR point mutations. However, this model fails to explain two observations. First, native TTR also forms amyloid in systemic senile amyloidosis, a geriatric disease. Second, age at disease onset varies by decades for patients bearing the same mutation and some mutation carrier individuals are asymptomatic throughout their lives. Hence, mutations only accelerate the process and non-genetic factors must play a key role in the molecular mechanisms of disease. One of these factors is protein glycation, previously associated with conformational diseases like Alzheimer's and Parkinson's. The glycation hypothesis in FAP is supported by our previous discovery of methylglyoxal-derived glycation of amyloid fibrils in FAP patients. Here we show that plasma proteins are differentially glycated by methylglyoxal in FAP patients and that fibrinogen is the main glycation target. Moreover, we also found that fibrinogen interacts with TTR in plasma. Fibrinogen has chaperone activity which is compromised upon glycation by methylglyoxal. Hence, we propose that methylglyoxal glycation hampers the chaperone activity of fibrinogen, rendering TTR more prone to aggregation, amyloid formation and ultimately, disease