Conformal dimension and canonical splittings of hyperbolic groups

We prove a general criterion for a metric space to have conformal dimension one. The conditions are stated in terms of the existence of enough local cut points in the space. We then apply this criterion to the boundaries of hyperbolic groups and show an interesting relationship between conformal dimension and some canonical splittings of the group.     

Global conformations of proteins as predicted from the modeling of their CZE mobility data

Estimations of protein global conformations in well-specified physicochemical microenvironments are obtained through global structural parameters defined from polypeptide-scale analyses. For this purpose protein electrophoretic mobility data must be interpreted through a physicochemical CZE model to obtain estimates of protein equivalent hydrodynamic radius, effective and total charge numbers, hydration, actual ionizing pK and pH-near molecule. The electrical permittivity of protein domain is also required. In this framework, the solvent drag on proteins is obtained via the characteristic friction power coefficient associated with the number of amino acid residues defining the global chain conformation in solution. Also, the packing dimension related to the spatial distribution of amino acid residues within the protein domain is evaluated and discussed. These scaling coefficients together with the effective and total charge number fractions of proteins provide relevant interpretations of protein global conformations mainly from collapsed globule to hybrid chain regimes. Also, protein transport properties may be estimated within this framework. In this regard, the central role played by the friction power coefficient in the evaluation of these properties is highlighted.     

Determination of the microenvironment-pH and charge and size characteristics of amino acids through their electrophoretic mobilities determined by CZE

Effective electrophoretic mobility data of 20 amino acids reported in the literature are analyzed and interpreted through simple physicochemical models, which are able to provide estimates of coupled quantities like hydrodynamic shape factor, equivalent hydrodynamic radius (size), net charge, actual pK values of ionizing groups, partial charges of ionizing groups, hydration number, and pH near molecule (microenvironment-pH of the BGE). It is concluded that the modeling of the electrophoretic mobility of these analytes requires a careful consideration of hydrodynamic shape coupled to hydration. In the low range of pH studied here, distinctive hydrodynamic behaviors of amino acids are found. For instance, amino acids with basic polar and ionizing side chain remain with prolate shape for pH values varying from 1.99 to 3.2. It is evident that as the pH increases from low values, amino acids get higher hydrations as a consequence each analyte total charge also increases. This result is consistent with the monotonic increase of the hydrodynamic radius, which accounts for both the analyte and the quite immobilized water molecules defining the electrophoretic kinematical unit. It is also found that the actual or effective pK value of the alpha-carboxylic ionizing group of amino acids increases when the pH is changed from 1.99 to 3.2. Several limitations concerning the simple modeling of the electrophoretic mobility of amino acids are presented for further research.     

A monopole mass filter with a hyperbolic V-shaped electrode

In this article we compare the classical monopole mass filter of von Zahn and the monopole mass filter with a hyperbolic V-shaped electrode. The experimental results and those of computer simulation for both mass spectrometers are presented. We show that the replacement of a conventional 90 degrees V-shaped electrode by an electrode with a hyperbolic profile substantially improves the peak shape of any given mass, and increases the mass resolution by a factor of 3-4 and the abundance sensitivity by a factor of 100. The potential of high analytical performance combined with electroforming techniques for electrode manufacture indicate future practical uses of such instruments. Copyright 1999 John Wiley & Sons, Ltd.     

A theoretical and NMR lanthanide‐induced shift (LIS) investigation of the conformations of lactams.

Molecular mechanics (MM) with MMFF94 and MMX force fields and ab initio (RHF/6‐31G*,RHF/6‐311G**, and B3LYP/6‐311G**) calculations are used with lanthanide‐induced shift (LIS) to investigate the conformations of N‐methyl‐2‐pyrrolidone 1 , N‐methyl‐2‐piperidone 2 , ε‐caprolactam 3, γ‐valerolactam (1,5‐dimethyl‐2‐pyrrolidone) 4, 2 ‐ azetidinone 5 , 4‐methyl azetidinone 6 , 4‐phenyl azetidinone 7 , and N‐methyl‐4‐phenyl azetidinone 8 . The Yb(fod)₃ paramagnetic induced shifts of all the ¹H and ¹³C nuclei are measured and the corresponding diamagnetic complexation shifts obtained by the addition of Lu(fod)₃. The complexation model (two‐, three‐, or four‐site) used depends on the relative rates of the processes involved. The amide inversion is the same order as that of the 5‐ and 6‐membered lactam rings and much faster than the lanthanide complexation and the inversion of the 7‐membered ring. Both MM and ab initio calculations give an envelope conformation for 1 with C‐4 out of the ring plane in agreement with the LIS analysis. For the piperidone ring of 2 , the half‐chair is calculated as the most stable form. The LIS analysis confirms this but cannot exclude a small amount (<2%) of the boat conformation. For 3 , the LIS analysis gives a minimum for 90:10% chair to boat conformation, and 4 exists in two envelope conformations with the C₅‐Me ps‐eq and ps‐ax in an eq/ax ratio of 94:6%. In 2‐azetidinone 5 , the ab initio calculations gave both ring and nitrogen planar, but the MMFF94 calculations give a butterfly ring and pyramidal nitrogen. The LIS analysis for 5 gave good agreement (Rcryst 0.46%) for the MMFF94 geometry with endo NH but the planar ab initio geometries worse agreement (Rcryst = 1.1%). For 4‐methyl‐2‐azetidinone 6 , the MMFF94 geometry gave good agreement (Rcryst 0.96%) with two butterfly conformations with axial and equatorial methyl groups in 1:1 ratio. All the planar geometries gave worse agreement (Rcryst >1.5%). In 4‐phenyl azetidinone 7 , the MMFF94 geometry with 60% of the axial conformer gave Rcryst 1.2% but the other geometries Rcryst >1.5%. In contrast the N‐methyl‐4‐phenyl‐2‐azetidinone 8 gave good agreement for all the geometries. The butterfly conformation gave Rcryst 1.1% for 80% of the axial conformer and the planar geometries Rcryst 0.98%. The LIS results confirm the ab initio and MM optimised geometries, but the conformer energies at times differ from the calculated values. They also differ considerably from the corresponding values for the lactones studied previously, and possible reasons for this are discussed.     

Exploring the evaluation of net charge, hydrodynamic size and shape of peptides through experimental electrophoretic mobilities obtained from CZE

This work explores the validity of simple CZE models to analyze the electrophoretic mobilities of 102 peptides reported in literature. These models are based mainly on fundamental physicochemical theories providing analytical expressions amenable to relatively simple numerical analysis. Thus, the Linderstr?m-Lang capillary electrophoresis model (LLCEM) and its perturbed version (PLLCEM), proposed and applied previously to the CZE of globular proteins, are adapted and used here for peptides. Also the effects of pK-shifts on net charge, hydration and hydrodynamic size and shape of peptides are analyzed and discussed. Emphasis is placed on the fact that these parameters are physically coupled, and thus a variation in the net charge may produce an appreciable change in the hydrodynamic size of peptides. Within the framework of CZE, peptides may be assumed as having a hydrodynamic volume associated with either spherical or spheroidal particles. The effects on peptide net charge and hydrodynamic size, of electrostatic interaction between a pair of charged groups in the chain and electrical permitivitty around the peptide domain are studied. The predictions of the PLLCEM and LLCEM are in good agreement with results reported previously in the literature. Several limitations concerning these models and some needs for further research are also described.