A methodology for the constructive approximation of nonlinear operators defined on noncompact sets

We consider the constructive approximation of a non-linear operator that is known on a bounded but not necessarily compact set. Our main result can be regarded as an extension of the classical Stone-Weierstrass Theorem and also shows that the approximation is stable to small

disturbances.

This problem arises in the modelling of real dynamical systems where an input-output mapping is known only on some bounded subset of the input space. In such cases it is desirable to construct a model of the real system with a complete input-output map that preserves, in some approximate sense, the known mapping. The model is normally constructed from an algebra of elementary continuous functions.

We will assume that the input space is a separable Hilbert space. To solve the problem we introduce a special weak topology and show that uniform continuity of the given operator in the weak topology provides an alternative compactness condition that is sufficient to justify the desired approximation.     

Toward Best Approximation of Nonlinear Systems: A Case of Models with Memory

The problem of nonlinear dynamical system modeling, considered in this paper, is motivated by restrictions arising in real-world tasks. The restrictions are that first, a system input cannot be entirely observed for one trial. Second, the system model must be subjected to the causality principle. Third, the input is corrupted by noise so that no relationship between the reference input and noise is known. Fourth, the model should have some degrees of freedom so that the associated accuracy can be regulated by a variation of these freedom degrees. We propose and justify new procedures for the nonlinear system modeling that are initialized by these motivations. The models are nonlinear and given by so called r-degree operators that can be reduced to a matrix form presentation. To satisfy the restrictions above, the matrices have special structures that we call the lower p-band matrices. The degree r of the models is the required degree of freedom. The rigorous analysis of errors associated with the presented techniques is given. Numerical experiments with real data demonstrate the efficiency of the proposed approach.     

ESI‐MS/MS of expanded porphyrins: a look into their structure and aromaticity

Electrospray mass spectrometry/mass spectrometry was used to investigate the gas‐phase properties of protonated expanded porphyrins, in order to correlate those with their structure and conformation. We have selected five expanded meso‐pentafluorophenyl porphyrins, respectively, a pair of oxidized/reduced fused pentaphyrins (22 and 24 π electrons), a pair of oxidized/reduced regular hexaphyrins (26 and 28 π electrons) and a regular doubly N‐fused hexaphyrin (28 π electrons). The gas‐phase behavior of the protonated species of oxidized and reduced expanded porphyrins is different. The oxidized species (aromatic Hückel systems) fragment more extensively, mainly by the loss of two HF molecules. The reduced species (Möbius aromatic or Möbius‐like aromatic systems) fragment less than their oxidized counterparts because of their increased flexibility. The protonated regular doubly fused hexaphyrin (non‐aromatic Hückel system) shows the least fragmentation even at higher collision energies. In general, cyclization through losses of HF molecules decreases from the aromatic Hückel systems to Möbius aromatic or Möbius‐like aromatic systems to non‐aromatic Hückel systems and is related to an increase in conformational distortion. Copyright © 2016 John Wiley & Sons, Ltd.     

Steric trigger as a mechanism for CB1 cannabinoid receptor activation

To determine the moiety that behaves as the steric trigger to activate the CB(1) cannabinoid receptor, conformational properties of the nonclassical cannabinoid CP55244, one of the most potent CB(1) receptor agonists, were characterized by conformational analysis, rotational barrier calculations, and molecular dynamics (MD) simulations. It was shown from the present MD simulations that the torsion angles phi1 and phi4 of the C3 side chain showed the most dramatic change when compared with the ground-state receptor-bound conformation, indicating that rotation around these torsion angles is responsible for releasing the ligand strain energy. Multiple stages would be involved in the ligand conformational change. As a molecular mechanism for the ligand-induced CB(1) receptor conformational change, we propose that the C3 side chain serves as the steric trigger, while the ACD-ring moiety of CP55244 serves as the plug. Steric clash with helices within the binding pocket would induce microconformational adaptation within the protein. This mechanism would suggest that rotational flexibility in a ligand may be as important a determinant of agonist activity as the pharmacophoric elements that can be identified.     

Effects of mutation on the amyloidogenic propensity of apolipoprotein C-II(60-70) peptide

Using experimental and computational methods we identified the effects of mutation on the structure and dynamics of the amyloidogenic peptide apoC-II(60-70), in monomeric and oligomeric states. Methionine (Met60) substitutions to hydrophilic Gln, hydrophobic Val, and methionine sulfoxide residues were investigated and the results compared with observations of fibril formation by the wild-type, Met60Gln, Met60Val, and oxidised Met60 (oxi-Met) apoC-II(60-70) peptides. ThT fluorescence measurements showed fibril formation by all peptides, however with different kinetics. The wild-type and Met60Val peptides formed fibrils fastest, while oxi-Met and Met60Gln peptides exhibited significantly longer lag phases. Molecular dynamics simulations showed that the mutated monomers exhibited structural features consistent with fibril-forming propensity, such as β-hairpin conformation and a hydrophobic core. However, important differences to the wild-type were also noted, such as increased structural flexibility (oxi-Met and Met60Gln systems) and a broader distribution of the aromatic angle orientation, which could contribute to the different fibrillation kinetics observed in these peptides. Our results also showed that the critical nucleus size for fibril formation by apoC-II(60-70) may not be very large, since tetrameric oligomers in anti-parallel configuration were very stable within the 100 ns of simulations. The single-point mutations Met60Val and Met60Gln had no significant effect on the structural stability of the tetramer. The rate of fibril formation by apoC-II(60-70) peptides was generally much faster compared to longer apoC-II(56-76) peptides. Also, the effects of amino acid modifications on the kinetics of peptide fibril formation differ from the effects observed for apoC-II(56-76) and full-length apoC-II, suggesting that additional mechanisms are involved in fibril formation by mature apoC-II.