Paterson function

As a result, we obtain the convolution of the density functirai p(r) with the same function inverted with respect of the origin of the reference frame p( r). Note that the minus sign appears due to different signs in the exponents for two complex conjugates in (5.28). The P(r) function is known as density autocorrelation function or the Paterson function when used in structural analysis. Thus, we may write the inverse and direct Fourier transforms as follows ... 

It means that the scattering (or diffraction) intensity and the autocorrelation function are reciprocal Fourier transforms similar to the reciprocal transforms of scattering amplitude F(q) and density p(r). It should be noted that in statistical physics one widely uses the density correlation function G(r) mentioned earlier (5.26) that is related to the structure factor S(q) exactly as the Paterson function is related to intensity of scattering 7(q). Below we prefer to use G(r). 

The Patterson function (Patterson, 1934) is a phaseless Fourier summation similar to that in Eq. 4 but employing as coefficients, thus it can be calculated directly from the experimentally measured amplitudes (Fp) without the need to determine the phase angle. In the case of macromolecules, (Fpn —Fp ) are used as coefficients in Eq. 4 to produce a Patterson map (hence the name difference Patterson). Such a map contains peaks of vectors between atoms (interatomic vectors). Thus in the case of a difference Patterson of macromolecules, it is a heavy-atom vector map. For example if a structure has an atom at position (0.25, 0.11, 0.32) and another atom at position (0.10, 0.35, 0.15), there will be a peak in the Paterson map at position (0.25-0.10, 0.11-0.35, 0.32-0.15), meaning a peak at (0.15, —0.24, 0.17). 

Cj) Caldirola Paterson Equation of State. Dunkle (Ref 10, p 183), stated that Cook (Ref 2c) found by working backward from experimental detonation rates to Corresponding values of covolume, that for all expls at very high pressures the Covolume is a function of the specific volume only. At these pressures all molecules have the same covolume per unit weight the dependence of a(T,v) on temperature is exceedingly small. The equation of state can be written pV = nRT +a(v)p... 

R. Paterson, Network Thermodynamics, in E.E. Bitter, Ed., Membrane Structure and Function, Vol. 2, Wiley, New York (1980). 

Moore B, Paterson M, Sturdee D (1987) Effect of oral hormone replacement therapy on liver function tests. Maturitas 9 7—15. 

In the enantioselective total synthesis of p-lactone enzyme inhibitor (-)-ebelactone A and B, I. Paterson and coworkers constructed seven stereocenters and a trisubstituted alkene plus a very sensitive p-lactone ring. The backbone of their strategy applied an aldol reaction / Ireland-Claisen rearrangement sequence and used minimal functional group manipulation. The Ireland-Claisen rearrangement was performed in the presence of an unprotected ketone moiety and set a precedent for this protocol. The diastereoselectivity was 96 4, indicating highly ( )-selective silylketene acetal formation. 

Spina D, Rigby PJ, Paterson JW, Goldie RG. a,-Adrenoceptor function and autoradiographic distribution in human asthmatic lung. BrJ Pharmacol 1989 97 701-708. 

Nguyen TV, Paterson IA, Juorio AV, Greenshow AJ, Boulton AA (1989) Trypt-amine receptors neurochemical and electrophysiological evidence for post-synaptic and functional binding sites. Brain Res 476 85-93... 

Solan, M., Raffaelh, D. G., Paterson, D. M., White, P. C. L., Pierce, G. J., 2006. Theme section introduction—Marine biodiversity and ecosystem function empirical approaches and future research need. Marine Ecology Progress Series, 311, 175-178. 

Aitken, R.J., Paterson, M., Fisher, H and Buckingham, D.W. (1995). Van Duin M. Redox regulation of tyrosine phosphorylation in human spermatozoa and its role in the control of human sperm function. J. Cell Sci. /0S 2017-2025... 

We have already collected some powerful tools for use in stereocontrolled aldol reactions, but we have not finished. We shall see now in Paterson s synthesis of (+)-discodermolide, how reagent control is used not to enhance the intrinsic substrate selectivity, but to overturn it. The aldol reaction is undoubtedly one of the most powerful ways of making carbon-carbon bonds and nature thinks so too. There are numerous natural products that are replete with 1,3 related oxygen functionality. Many of these are acetate or propionate-derived in nature. The methods detailed above developed from studies into the syntheses of these natural products. The manipulations of chiral ethyl ketones of this kind are of particular interest when it comes to natural products that are polypropionate-derived. 

Paterson s synthetic strategy is based on the coupling of three fragments of similar stereochemical and functional group complexity the C1-C6, C9-C16 and C17-C24 subunits. The absolute stereochemistry of these fragments was established via substrate-controlled, boron-mediated aldol reactions of chiral ethylketones. 

Effect of Temperature. Figures 2, 3, and 4 show that the fraction of the carbon that reacts according to the first phase, f, increases with temperature. Moseley and Paterson (29) pointed out that the fraction of carbon that reacts according to the first phase is a function of both temperature and pressure. Since the total pressures used in the present investigation were nearly constant at approximately 70 atm., total pressure had no significant effect on the value of /. Previous investigators... 

S. Mylchreest, C.R. Paterson et al. (1999). Novel mutations in the la-hydroxylase (P450cl) gene in three families with pseudovitamin D-deficiency rickets resulting in loss of functional enzyme activity in blood-derived macrophages. J. Bone Miner. Res. 14, 730-739. 

Berry MD, Juorio AV, Paterson lA. The functional role of monoamine oxidases A and B in the mammalian central nervous system. Prog Neurobiol 1994 42 375-391. 

---