Convexity constraint

A series of 14 linear and cyclic peptides have been studied 135 by CD, NMR, and MD simulations. Quantitative NOE measurements and MD were used to determine the fractions of type I and II (3-turns in each peptide. The CD spectra for these same peptides were analyzed by convex constraint analysis 136 to derive a set of CD curves for the two types of 13-turns (Figure 6). These basis spectra gave fractional (3-turn contents in good agreement with NMR measurements. Two basis spectra (components 1 and 4) were obtained for the type I... 

Figure 6 The Four Component CD Curves from Convex Constraint Analysis of CD Spectra of Cyclic (S-Tum Models11351"1 ...

Recently, attempts have been made to derive CD spectra of various p turns using deconvolution procedures (see below). Employing an approach termed convex constraint analysis, Perczel, Fasman, and others [58-61] have derived CD spectra for both types I and H turns. Three basis spectra were extracted for a data base of fourteen peptides which displayed either types I or II p turns. One basis spectrum for type I turns resembled a class C spectrum and the other a variation of a class B. The type II turn is said to display a classic class B spectrum [59-61],... 

In optimization, it is well-known that non-hnear constraint problems are generally "hard" to solve [61] particularly for a priori non-convex constraints. To overcome this difficulty, the equations of problem (P) are formulated as problem (P ), defined as ... 

In addition, Quesada and Grossmann (1995b) showed that the nonlinear convex constraint... 

Levy and Perdew[24] recently derived a very severe low density convexity constraint on Exc. Define the convex functional A[n] by... 

CCA, Convex constraint analysis. No X-ray data are required. The two sets of values for the unordered conformation are due to two different classifications of secondary structure. 

Lee and Grossmann (2000) have derived the convex hull relaxation of problem (GDP). The basic idea is as follows. Consider a disjunction keK that has convex constraints. 

Except for constraint (90), this is a linear programming problem in the variables Xj p which can be solved by the simplex method provided a restricted basis entry rule is used which enforces (90). However, in the case when each fj is strictly convex and each is convex, constraint (90) can be... 

Convex programming refers to a minimization problem with a convex objective and a convex constraint set. For example, if/and g are convex functions, then... 

Peptide CD structure analysis differs firom that of proteins as peptides are usually a combination of a particular secondary structure and random coil (i.e. unordered) residues. Proteins seldom if ever have random coils, so the CD of the proteins in the basis set of the protein anal3rsis program has no random coil component. Further, the analysis of the CD of a peptide is not under-determined, and so does not require a flexible method like Variable Selection. A Convex Constraint Analysis (22) that extracts the component spectra has been developed into a peptide structure analysis program by Greenfield (14). 

Perczel, A. Park, K. Fasman, G.D. Analysis of the circular dichroism spectrum of proteins using the convex constraint algorithm a practical guide. Anal. Biochem. 1992, 203, 83-93. 

Perczel A, Holldsi M, Tusnady G and Fasman GD (1991) Convex constraint analysis a natural deconvolution of circular dichroism curves of proteins. Protein Engineering 4 669-679. 

Figure 4 (A) Solvent-induced conformational transition of PrP27-30 scrapie amyloid (prion) protein in the solid state on quartz glass. Films were exposed for 20 to 30 min to SDS, sodium dodecyl sulfate, TFE, trifluorethanol HFIP, hexafluorisopropanol TFA, trifluoro-acetlc acid FA, 99% formic acid, f is the spectrum with no treatment at 23°C. (B) Data on PrP27-30 obtained under various conditions, deconvoluted into five basis curves using the convex-constraint algorithm. (—)/S-tum type I or III and/or 3,o-type helix (.) alpha...

Perczel et al. [91P2] also base their convex constraint analysis on CD data of both 18 proteins published by Chang et al. [78C1] and 15 proteins used by Hennessey and Johnson [81H1] (cf. Tables VIII and IX in Yang et al. [86Y1]). The two sets of CD parameters give somewhat different results of the CCA... 

A program for the convex constraint analysis is available from the authors upon request. 

VS variable selection procedure [87M1] applied to the Hennessey-Johnson method [81H1] the fp values are the combined values of parallel and antiparallel P-sheets, and negative values are counted as zero. CCA convex constraint analysis [91P2] this method has a fifth or more components for additional chiral contributions other than the four conformations and Efi plus this fifth or more weights equals one. See text. SC self-consistent method [93S12],... 

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