From Qualitative to Quantitative Analyses of Circular Dichroism Spectra Using the Convex Constraint Algorithm

Abstract

Chirality is a geometric property of certain molecules, which results in that they interact differently with left- and right-circularly polarized light. The optical activity of dissimetric crystals was first observed almost two hundred years ago (Arago, 1811). However, the application of optical rotatory dispersion (ORD) and circular dichroism (CD) to study peptide and protein conformations started only in the last three decades. This was followed by the interesting ORD studies of synthetic polypeptides, performed in the fifties by Doty, Moffit and Yang. The pioneering discoveries of Crick and Kendrew (1957) on helices initiated several hundreds of valuable work on peptide and protein conformations. In 1965, Holzwarth and Doty measured, for the first time, the CD spectrum of an ot-helix. The CD spectrum of Pauling and Corey’s ß-pleated sheet was determined for (Lys)_n-type homopolymers, as well as for silk fibroin in solution. Since 1975, more attention has been focused on the CD of ß-turns, ß-bends, 3₁₀- bends, hairpins, loops, etc. (Chou, Fasman), as well as on the CD contribution of unordered or non- typical secondary structures. Theoretical attempts to give a general explanation for the origin of the various observed CD bands have also been performed (Moffit, Woody).

By assuming the additivity of the CD contributions originated in the different conformation structures (helices, ß-forms, etc.), a measured CD spectrum [f(λ)] is the weighted sum of the pure conformers’ CD spectra [g_i(λ)]:

$$ \begin{gathered} f\left( \lambda \right) = \sum {{p_i}*{g_i}\left( \lambda \right)} + noise \hfill \\ Where\,{p_i}\,is\,the\,weight\,of\,{g_i}\left( \lambda \right) \hfill \\ \end{gathered} $$

The fundamental question for a quantitative structure analysis using CD spectroscopy is the determination of the possible conformers and their associated g_i(λ) functions, and different methods of analysis have been developed in the past years to solve this problem. Traditionally, the determination of the Reference Spectra Set (the set of g_i((λ) curves) is based on model polypeptides and proteins, using a linear combination of CD spectra. The common weakness of these methods is their reduced flexibility, which makes them suitable only for qualitative or, in the best cases, semi-quantitative structure analysis. In contrast, a recent algorithm (Convex Constraint Deconvolution Algorithm - CCA) gives us, at least in principle, the possibility of reaching a quantitative level of accuracy in structure analysis studies by CD spectroscopy. In contrast to a linear combination based method, this procedure aims to determine simultaneously the weights pj and the pure component curves g_i(λ) by minimizing the following expression,

$$ {\left[ {\sum\limits_{{j = 1}}^N {f_j^m\left( \lambda \right) - \sum\limits_{{j = 1}}^N {f_j^c\left( \lambda \right)} } } \right]^2} = {\left[ {\sum\limits_{{j = 1}}^N {f_j^m\left( \lambda \right) - \sum\limits_{{j = 1}}^N {\sum\limits_{{i = 1}}^P {{p_{{ij}}}*{g_i}\left( \lambda \right)} } } } \right]^2} $$

which is possible since a set of measured f^m _j(λ) CD curves is available.

Assuming that all other CD determining factors (like temperature, solvent shift, concentration, number of chromophors, etc.) are constant, the calculated pure conformational CD curves (the set of g_i(λ) functions) and their weights p_ij are only related to the amide conformation, because this stays the unique variable considered.

The application of this algorithm was successfully performed on a series of proteins and peptides, and the obtained structural parameters were compared with the corresponding geometries determined by X-ray or NMR.