Deconvolution defined

Since the proline residue in peptides facilitates the cyclization, 3 sublibraries each containing 324 compounds were prepared with proline in each randomized position. Resolutions of 1.05 and 2.06 were observed for the CE separation of racemic DNP-glutamic acid using peptides with proline located on the first and second random position, while the peptide mixture with proline preceding the (i-alamine residue did not exhibit any enantioselectivity. Since the c(Arg-Lys-0-Pro-0-(i-Ala) library afforded the best separation, the next deconvolution was aimed at defining the best amino acid at position 3. A rigorous deconvolution process would have required the preparation of 18 libraries with each amino acid residue at this position. 

This led to the conclusion that these amino acids were essential for the resolution capability and only 6 new libraries of 18 compounds had to be synthesized with these amino acid residues to define the position 3. Surprisingly, the separation abilities of all six libraries were very similar. Therefore, tyrosine was chosen for continuing deconvolution, since it is convenient as its aromatic ring can easily be detected by UV spectrometry. The last step, defining position 5, required the synthesis and testing of 6 individual hexapeptides. 

Fig. 3-3. Comparison of the values of enantiomeric resolution of different DNP-D,L-amino acids at different deconvolution stages of a cyclic hexapeptide sublibrary. Resolution values in a cyclo(Arg-Lys-X-X-X-P-Ala) sublibrary, in the first line, are compared to those obtained in sublibraries with a progressively increasing number of defined positions. All the sublibraries were 30 mM in the running buffer while the completely defined cyclo(Arg-Lys-Tyr-P-Tyr-P-Ala) peptide is used at 10 mM concentration. Conditions cyclopeptide sublibrary in 20 mM sodium phosphate buffer, pH 7.0 capillary, 50 pm i.d., 65 cm total length, 57 cm to the window V = -20 kV, I = 40 electrokinetic injection, -10 kV, 3 s detection at 340 nm. (Reprinted with permission from ref. [75]. Copyright 1998, American Chemical Society.)...

It should be noted that, in two of these studies, " the perfusion parameter used to define the mismatch was not CBF or MTT, but instead the time it took for contrast concentration to reach peak concentration in each image voxel after contrast injection ( time to peak or TTP). TTP measurements are often used as rough approximations of MTT measurements because calculation of CBF and MTT are somewhat complex, requiring a mathematical process called deconvolution. The details of deconvolution are beyond the scope of this chapter, and the reader is referred to other sources for further explanation. In many clinical settings, maps of parameters like TTP that do not require deconvolution may be available much more quickly than those that do require deconvolution. TTP is less specific than MTT in detecting underperfused tissue because it does not distinguish between delayed contrast arrival time (such as that related to perfusion via collateral vessels) and truly prolonged intravascular transit time. 

The basic idea was to randomly acylate polyallylamine (MW = 50,000-65,000) all at once with eight different activated carboxylic acids. The relative amounts of acids used in the process was defined experimentally. Since the positions of attack could not be controlled, a huge family of diverse polymers (4) was formed. In separate runs the mixtures were treated with varying amounts of transition metal salts and tested in the hydrolysis reaction (1) —> (2) (Equation (1). The best catalyst performance was achieved in a particular case involving Fe3+, resulting in a rate acceleration of 1.5 x 105. The weakness of this otherwise brilliant approach has to do with the fact that the optimal system is composed of many different Fe3+ complexes, and that deconvolution and therefore identification of the actual catalyst is not possible. A similar method has been described in other types of reaction.30,31... 

In the range of linearity, Eq. (29) correctly represents the heat transfer within the calorimeter. It should be possible, then, by means of this equation to achieve the deconvolution of the thermogram, i.e., knowing g(l) (the thermogram) and the parameters in Eq. (29), to define f(t) (the input). This is evidently the final objective of the analysis of the calorimeter data, since the determination of the input f(t) not only yields the total amount of heat produced, but also defines completely the kinetics of the thermal phenomenon under investigation. 

Considerable effort has gone into solving the difficult problem of deconvolution and curve fitting to a theoretical decay that is often a sum of exponentials. Many methods have been examined (O Connor et al., 1979) methods of least squares, moments, Fourier transforms, Laplace transforms, phase-plane plot, modulating functions, and more recently maximum entropy. The most widely used method is based on nonlinear least squares. The basic principle of this method is to minimize a quantity that expresses the mismatch between data and fitted function. This quantity /2 is defined as the weighted sum of the squares of the deviations of the experimental response R(ti) from the calculated ones Rc(ti) ... 

A third problem with simulation of high resolution diffraction data is that there is no unique instrament function. In the analysis of powder diffraction data, the instalment function can be defined, giving a characteristic shape to all diffraction peaks. Deconvolution of these peaks is therefore possible and fitting techniques such as that of Rietveld can be used to fit overlapping diffraction peaks. No such procedure is possible in high resolution diffraction as the shape of the rocking curve profile depends dramatically on specimen thickness and perfection. Unless you know the answer first, you cannot know the peak shape. 

Next, custom software is used to interrogate the deconvoluted data set to identify the protein s mass and the intensity of the peak, determine any potential modification above a user-defined intensity threshold and, if there is a hit, calculate the mass and the relative conjugation of the fragment. In fact, the percent conjugation is used as a measure of relative affinities of the fragment hits. Since the library is mass encoded (all compounds in a well have a unique mass), the calculated mass of any hits are queried into a database to identify their structures. 

Given that the number of a and d protons are the same, an identical set of equations with Ad in place of Aa can also be used. Here we have used an average of Aa and Ad with the areas assessed after Lorentzian deconvolution in order to provide better separation of the 2.99 and 3.02 ppm lines. Mole fractions derived using this NMR method for reaction over the unpromoted copper catalyst are shown as a function of time in Figure 5. Clearly IDA is formed largely via HEG as intermediate since the concentration of the latter passes through a well-defined maximum. 

In spite of performance advantages in the use of nonlinear methods, it is instructive to start our deconvolution study by examining the linear methods they will give us insight into the process. The ensuing development will also define the applicability domain of linear methods and reveal their limitations. We shall see that in some circumstances a linear method is the method of choice. 

The two most common methods used to correct resolved peak profiles for the broadening imposed by the finite width of the X-ray beam in the diffractometer, are due to Jones (15) and Stokes (16). Both are essentially unfolding or deconvolution methods, but the Jones method defines specific functions for both the uncorrected and the instrumental broadening profile. If the uncorrected profile is Gaussian, then... 

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