Extracting Ramachandran torsional angle distributions from 2D NMR data using Tikhonov regularization /
Abstract
Solid state nuclear magnetic resonance (NMR) spectroscopy is a powerful technique for
studying structural and dynamical properties of disordered and partially ordered materials,
such as glasses, polymers, liquid crystals, and biological materials. In particular, twodimensional(
2D) NMR methods such as ^^C-^^C correlation spectroscopy under the magicangle-
spinning (MAS) conditions have been used to measure structural constraints on the
secondary structure of proteins and polypeptides. Amyloid fibrils implicated in a broad
class of diseases such as Alzheimer's are known to contain a particular repeating structural
motif, called a /5-sheet. However, the details of such structures are poorly understood, primarily
because the structural constraints extracted from the 2D NMR data in the form of
the so-called Ramachandran (backbone torsion) angle distributions, g{^,'4)), are strongly
model-dependent.
Inverse theory methods are used to extract Ramachandran angle distributions from a
set of 2D MAS and constant-time double-quantum-filtered dipolar recoupling (CTDQFD)
data. This is a vastly underdetermined problem, and the stability of the inverse mapping
is problematic. Tikhonov regularization is a well-known method of improving the stability
of the inverse; in this work it is extended to use a new regularization functional based
on the Laplacian rather than on the norm of the function itself. In this way, one makes
use of the inherently two-dimensional nature of the underlying Ramachandran maps. In
addition, a modification of the existing numerical procedure is performed, as appropriate for
an underdetermined inverse problem.
Stability of the algorithm with respect to the signal-to-noise (S/N) ratio is examined using
a simulated data set. The results show excellent convergence to the true angle distribution
function g{(j),ii) for the S/N ratio above 100.