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APL, a powerful research tool in
Magnetic Resonance Spectroscopy
Claude Chachaty
e-mail : claude.chachaty@wanadoo.fr
Keywords : NMR, ESR, spin, hyperfine coupling, spectra, simulations, automated fitting.
I-Introduction
In spite of its outstanding scientific potential, APL is up to now ignored or scarcely
exploited by research workers. During 15 years as the head of the Magnetic Resonance
Laboratory of the Nuclear Research Center at Saclay, the author has extensively used APL in
his works [1-3] and continues to promote its scientific applications.
The Magnetic Resonance Spectroscopy (MRS) includes two main branches, the Nuclear
Magnetic Resonance (NMR) and the Electron Spin Resonance (ESR) also called Electron
Paramagnetic Resonance (EPR). The NMR is a priviledged method for the identification and
conformational analysis of organic and biological molecules and is well known for its medical
application, the Magnetic Resonance Imaging. The ESR/EPR which is the main subject of
this topics, is the specific method for studying paramagnetic molecules i.e. molecules
possessing at least one unpaired electron, namely the free radicals resulting from the breaking
of a chemical bond, triplet fundamental (e.g. the oxygen of air) or lowest excited states and
some metal coordination complexes. Most of these species are very reactive and are initiators
or intermediates in a large number of chemical and biological processes : oxidation,
combustion, polymerization, radiation damaging, photosynthesis etc… An important
application common to the NMR and ESR is the molecular dynamics which provides
thorough information on some physical properties of condensed matter.
II-Principles
Electrons and most of the nuclei possess a spin angular momentum, denoted S and I , respectively, as well as magnetic moments µ = g β S = γ η ge and gI are the spectroscopic factors, the latter being specific of each nucleus, γ the relevant gyromagnetic ratios, β e and n the Bohr and nuclear magnetons and η the Planck's constant divided by 2π . In a magnetic field B0 , the spins and magnetic moments undergo a precession of angular frequency ω = γ B about B0 . For a spin quantum number S or I, multiple of ½, the spins and magnetic moments take 2S+1 or 2I+1 orientations defined by the projections M = S ,S -1,.1- S,-S or M To each magnetic quantum number MS or MI corresponds an energy level. A magnetic resonance experiment consists in submitting a small sample (0.1-1 ml) placed in a very strong and homogeneous magnetic field B0 to a rotating radiofrequency (NMR) or
microwave (ESR) magnetic field B
1 perpendicular to B0 ( B 0 ). The resonance phenomenon corresponds to a transition between adjacent energy levels which occurs when the angular frequency of B B and involves the absorption of a photon of Claude Chachaty energy hν = gβB 0 , ν0 being the spectrometer frequency and h the Planck's constant. For technical reasons, the resonance is obtained by varying ν0 at constant field (NMR) or B0 at
constant frequency (ESR) and the ESR spectra are usually recorded as the first derivative.
The nuclear and electron spins are seldom isolated and generally experience local magnetic
fields due to other spins. The 2S+1 fundamental energy levels of a spin S interacting with a
spin I are splitted into 2I+1 sublevels and the resulting (2S+1)(2I+1) levels are :
E←(B0×(ge×be×MS)°.-gn×bn×MI)+a×MS°.×MI
where a is the hyperfine coupling constant expressed in energy units. The allowed ESR
transitions between these levels follow the selection rule ∆M = ±1, ∆M =
expression is easily extended to any number of spins of any quantum number and is an usual approximation when its first term is much larger than the second one. Figure 1 shows a simple application of these principles to a S=1/2, I=1/2 system, the H• atom, the smallest and one of the most reactive free radical. MS MI
I -1/2 1/2
II -1/2 -1/2
III 1/2 -1/2
IV 1/2 1/2
Magnetic field (gauss)
Figure 1 : Energy levels and ESR resonance lines of the hydrogen atom. Hyperfine coupling
constant a = 1.42 GHz or 508 gauss (1 gauss = 0.1 mT), spectrometer frequency ν0 = 9.24
GHz. Allowed transitions : I⇔IV and II⇔III.
III-Applications to the ESR spectroscopy.
The interpretation of the ESR spectra in terms of identification of the paramagnetic species
we are dealing with, of the hyperfine coupling parameters and sometimes of dynamical
behaviour is generally not feasible without the help of computer simulations. A visual
comparison of the experimental spectum with the simulated one tell us if the starting
assumptions made about this species are likely or not. The hresol function listed below is a
simplified version for the simulation of high resolution ESR spectra of radicals in solution.
APL and Magnetic Resonance hresol;ŒIO;A;Y;i ŒIO„1 ª 'Central field (mT) :' ª Bc„Œ 'Nuclear spin quantum numbers :' ª NS„½SN„,Œ L0:'Hyperfine coupling constants (mT)' ª …(NS¬½HFC„,Œ)/L0 SW„Bc+˜0.5+¯1.2 1.2×SN+.× HFC Spectral window centered on Bc 'Number of points€:' ª dH„--/SW÷NPTS„Œ C0„SW[1],SW[1]+dH×¼NPTS ª DIM1„1+2×SN DIM2„½V„(DIM1½¨HFC)×DIM1½¨MI„SN,¨SN-¼¨2×SN ª i„0 ª HR„Bc L1:HR„,HR°.+¹V[i„i+1] ª …(i<DIM2)/L1 ª INT„+/HR°.=HR„HR["HR] MASK„MASK,1†MASK„(¯1‡HR)¬1‡HR ª HR„MASK/HR ª INT„MASK/INT 'Linewidth at half-height : ' Y2„Y×Y„HR°.-C0 ª SP„INT+.ׯ2×Y÷A×A„Y2+RHL×RHL„0.5׌ PLOT CY„C0,[1.5]SP„SP÷+/+ SP Spectrum first derivative The figure 2 shows the spectrum of the benzyl ( C H − 2 ) radical generated by this Figure 2. Simulated ESR spectrum (first derivative) of the C H − 2 radical in fluid solution. The electron spin is coupled to 3 pairs of equivalent protons and a single proton. The interactions between the spins and the magnetic field and between the spins are of the 0 . g. and S. A. I , respectively, where g and A are symmetric second rank tensors
whose components are the sum of an isotropic term (g factor and hyperfine coupling constant)
and an anisotropic one. In fluids, the latter is averaged to zero by fast molecular motions but
is partially or not averaged in anisotropic systems as solids and liquid crystals.
The functions for fitting the spectra of spin S=1/2 species (free radicals, copper and
vanadyl ions) in anisotropic media proceed by the following steps :
1 - Parameters
Invariant : spectrometer frequency, spectral width and nuclear spin quantum numbers.
Claude Chachaty Adjustable : principal values of A, g and σ (linewidth) tensors, width of Gaussian line
broadening and rate of rotational motion if any.
2
– Angular dependence of the transitions probabilities P (θ , ϕ , M
σ (θ,ϕ, M , the angles θ and φ defining the orientation of B 0 in the frame of the g tensor.

3
– Calculation of resonance fields B θ ,ϕ, M
4 – For each transition, summation of spectra over all orientations :
S (B M I )= 1 P (θ ,φ , M (B B (θ ,φ , M );σ (θ ,φ , M ) )sin θ d θ d φ
where B is the scanning magnetic field, F the form function and N the normalization factor.
For a Lorentzian form function F(x) = 1/(1+x2), the relevant APL expression is :
(d1 d2 d3)„½¨B th phi
U„B°.-,Br ª F„÷RU×1+U×U„U÷RU„(d1½1)°.×,sigma
S„S÷+/S„F+.×,P×(1±th)°.×d3½1

5
– The overall spectrum obtained by summing S(B, MI ) over MI is convoluted by a
Gaussian and derived numerically.
6 – If the agreement with the experimental spectrum is not satisfactory, return to 1 to reajust
the parameters. This step may be automated by means of an optimization function based on
the Levenberg-Marquardt's algorithm [4] to minimize the variance between the experimental
and computed spectra.
Figure 3 shows an example of an automated fitting using the method outlined above.
.Figure 3. Experimental (solid line) and computed (•••••) spectra of an ESR spin-probe, a
nitroxide radical, in a model phospholipid membrane before (a) and after (b) addition of
cholesterol. This membrane is constituded by phospholipid bilayers separated by water and
behaves as a liquid crystal.The broadening and increased asymmetry of the lines from (a) to
(b) are significant of an increase of the membrane rigidity and molecular ordering upon
cholesterol addition, which may be quantitavely estimated [5].
IV-Conclusion

APL and Magnetic Resonance The theory of magnetic resonance is for a large part founded on matrix algebra, one of the
strong points of APL, making quite easy the programming of spectral simulations and fitting
of experimental data. For this reason the author has chosen APL rather than other
programming languages currently used by the scientific community (Fortran, Basic, C,
Pascal) in spite of its small diffusion and of some problems of portability.
The Magnetic Resonance software is written in APL2 (IBM) and APL+WIN (APL2000).
Descriptions of the workspaces are given in the sites www.garpe.org and
ftp://ierc.scs.uiuc.edu/pub/SoftwareDatabase.
References
[1 ] C. Chachaty et G. Langlet
Logiciels d'étude conformationnelle par RMN, de molécules flexibles.
Journal de Chimie-Physique et de Physico-Chimie Biologique, 82, 613 (1985).
[2] C. Chachaty
Simulations de spectres de résonance magnétique appliquées à la dynamique de
molécules en milieux anisotropes.
ibid., 82, 621 (1985).
[3] C. Chachaty and E. Soulié
Determination of electron spin resonance parameters by automated fitting of the spectra.
Journal de Physique III France, 5, 1927 (1995).
[4] D.W. Marquardt
An algorithm for least-squares estimation of nonlinear parameters.
Journal of the Society of Industrial Applied Mathematics, 11, 431 (1963).
[5] C. Wolf and C. Chachaty
Compared effects of cholesterol and 7-dehydrocholesterol on sphingomyelin-
glycerophospholipid bilayers studied by ESR.
Biophysical Chemistry, 84, 269 (2000).

Claude Chachaty

Source: http://www.afapl.asso.fr/MDR2002.pdf

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