Developing a numerical inverse-theory-based extraction of orientation-dependent relaxation rates from partially- relaxed spectra

Second-rank tensor interactions, such as quadrupolar interactions between the spin- 1 deuterium nuclei and the electric field gradients created by chemical bonds, are affected by rapid random molecular motions that modulate the orientation of the molecule with respect to the external magnetic field. In biological and model membrane systems, where a distribution of dynamically averaged anisotropies (quadrupolar splittings, chemical shift anisotropies, etc.) is present and where, in addition, various parts of the sample may undergo a partial magnetic alignment, the numerical analysis of the resulting Nuclear Magnetic Resonance (NMR) spectra is a mathematically ill-posed problem. However, numerical methods (de-Pakeing, Tikhonov regularization) exist that allow for a simultaneous determination of both the anisotropy and orientational distributions. An additional complication arises when relaxation is taken into account. This work presents a method of obtaining the orientation dependence of the relaxation rates that can be used for the analysis of the molecular motions on a broad range of time scales. An arbitrary set of exponential decay rates is described by a three-term truncated Legendre polynomial expansion in the orientation dependence, as appropriate for a second-rank tensor interaction, and a linear approximation to the individual decay rates is made. Thus a severe numerical instability caused by the presence of noise in the experimental data is avoided. At the same time, enough flexibility in the inversion algorithm is retained to achieve a meaningful mapping from raw experimental data to a set of intermediate, model-free

Extracting Ramachandran torsional angle distributions from 2D NMR data using Tikhonov regularization /

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.

Student Perceptions of Discretion in Discipline: Seeking Resolution and Restoration in a Punitive Culture

This study examined high school student perceptions of discretion utilized by educators in high school disciplinary proceedings. Using a sample of 6 high school students who had experienced differing levels of formal discipline, the study investigated the discretionary factors that influence an educator's decision making. The study was a generic qualitative study where the primary source of data collection was open-ended interviews to ensure the integrity of the research as a study of student voices and perceptions. Journaling was also employed to record observations and to identify researcher assumptions. The data were analyzed employing aspects of a grounded theory approach. The findings were coded to reveal 5 areas high school students identified in relation to discipline and discretion: punitive discipline versus problem resolution, effective processes, educator discretion, student discretion, and the student-educator relationship. The final discussion highlights the need for a community vision for high school discipline in order to channel discretion and to uphold students' best interests. Restorative justice is proposed as a feasible vision for high school discipline, whereby participants' responses are measured against a restorative paradigm.