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Description of the Inept Pulse Sequence Using Product Operator Formalism Introduction

Nuclear Magnetic Resonance Spectroscopy (NMR) is a powerful analytical tool capable of providing atomicresolution structural data as well as information on molecular dynamics for an impressive array of molecules, ranging from small organic compounds to large biomolecular complexes. Though arguably the highest resolution, most versatile structural tool available, NMR suffers from a major drawback: lack of sensitivity. This is specially a problem when studying proteins, which need to be in high-enough quantity to be detected by NMR yet not so high that they aggregate or precipitate. While much of this was alleviated through the use of isotopic labeling of proteins, the development of higherfield magnets, and many other improvements, sensitivity remains the biggest challenge in NMR today. While on the one hand, there have been hardware developments, many improvements to sensitivity are achieved through the development of pulse sequences that rely on transfer of Boltzmann polarization from high gyromagnetic ratio nuclei, or from electrons, to low gyromagnetic ratio nuclei. Of the many methods of achieving this transfer, an experiment known as INEPT is one of the most widely used today and has been incorporated as an element in many modern pulse sequences. Insensitive nuclei enhanced by polarization transfer (INEPT) is a technique developed to enhance the NMR signals of nuclei with low gyromagnetic ratios. Published by Morris and Freeman in 1979, the seminal paper cited measurement of N and Si signals as a particularly useful application of INEPT because of their negative gyromagnetic ratios, which causes cancellation of signals in polarization transfer experiments utilizing cross-relaxation between nuclei. Although INEPT has indeed been implemented as a useful module for signal enhancement in pulse sequences, the true utility of the sequence has been demonstrated in multidimensional experiments, which provide information on connectivity between atoms in molecules. Such experiments are necessary in the structural characterization of proteins, as well as in analysis of complex organic molecules, in which the high degree of degeneracy lead to extensive overlap in one-dimensional spectra, rendering them un-interpretable. The mechanism of the INEPT experiment is to transfer the higher equilibrium magnetization of nuclei with high gyromagnetic ratios, which in most cases is H, to nuclei with low gyromagnetic ratios, like C and N, by means of the Jcoupling, which is a through-bond interaction mediated by the electrons connecting two nuclei. Morris & Freeman (1979) described this transfer of polarization using a vector model based on classical angular momentum theory. While the vector model provides an intuitive picture of what is happening to the nuclei in response to radiofrequency pulses (RF), it has a major limitation, which is that it is difficult to represent anti-phase coherence with vectors. As shown later in Figure 4E, the vector model does not provide an informative representation of anti-phase coherence. This observation highlights the superiority of product operators in explaining this phenomenon. While the details of anti-phase coherence will be presented below, this means that the vector model does not represent couplings well, in which applying an RF pulse to one nucleus directly affects the coupled nucleus. To describe these interactions accurately and completely, a quantum mechanical approach is used. This is accomplished using what is known by the NMR community as product operator formalism, developed by Richard Ernst in 1983 (Sorensen et al., 1983). Still widely used today to describe many pulse sequences, this formalism is grounded in quantum mechanics and describes the evolution of a population of nuclear spins in a static magnetic field under the influence of a series of RF pulses. In this report, we describe the INEPT pulse sequence using the product operator method. In addition, we introduce the more commonly used version of the pulse sequence, known formally as refocusedINEPT, which improves signal detection by converting the final signal from anti-phase to in-phase.