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Nuclear Magnetic Resonance Spectroscopy is a method widely used to determine structural information of proteins of an atomic level resolution as well as interactions with binding partners, reaction kinetics and molecular dynamics (Montelione, Zheng et al. 2000; Cavanagh 2007; Kay 2011). As of August 2012, 6302 out of 33444 (18.8%) of all structures in the Protein Data Bank (PDB) have been produced by way of NMR.

NMR spectroscopy exploits the physical phenomenon of nuclear magnetic resonance i.e. that the resonant frequency of a nucleus is influenced by its chemical environment and that one nucleus can influence other nuclei through chemical bonds. Since the first observation of NMR in solids and liquids in the 1940s, subsequent advances such as the pulse-FT approach, multipulse and multidimensional NMR techniques, high field spectrophotometers and TROSY experiments have greatly increased NMRs scope of applications (Kay, Clore et al. 1990; Pervushin, Riek et al. 1997; Tugarinov, Sprangers et al. 2004; Claridge 2009)

4.1.3.1 General Principles

NMR arises due to an intrinsic property of nuclei known as then nuclear spin angular momentum or ‘spin’, this can be described by the nuclear spin quantum number (I) and its magnetic quantum number (m). Nuclei with an even mass number and atomic number have zero spin (I = 0) and are deemed to be NMR inactive e.g 2H, 12C, 14N. The magnetic moment (m) of a positively charged nucleus can adopt 2I+1 possible orientations in the presence of an external magnetic field (B0). Hence, a proton, with I = ½, can adopt two possible energy states in the presence of an external magnetic field (B0) (Fig. 4.4).

143 Figure 4.4: Two possible orientations of the magnetic moment (µ) of a proton in the presence of an external magnetic field (B0). The two states are termed the α-state (m = + ½ ) and the β-state (m = -½)

and correspond to nuclear spins which are parallel or antiparallel to B0 respectively. The α-state has a lower energy level than the β-state.

144 The two states are termed the α-state (m = + ½ ) and the β-state (m = -½) and correspond to nuclear spins which are parallel or antiparallel to B0 respectively. The α-state has a lower energy level than the β-state. Nuclei adopting such spins have a magnetic dipole and behave as bar magnets within a magnetic field.

Application of electromagnetic radiation with an appropriate frequency will allow transition between the α- and β- states and is termed resonance. In the case of NMR, the appropriate frequency like in the radio frequency range (MHz)

4.1.3.2.1 Lamor precession

Nuclei with a non-zero spin have an associated magnetic moment and will experience torque in the presence of an external magnetic field. The resultant torque of the magnetic moment will cause it to precess about the field axis and a frequency known as the Lamor frequency which is directly proportional to the gyromagnetic ratio and external magnetic field strength (Fig 4.5). The transition between α- and β- states occurs at a frequency equal to the Lamor frequency.

In summary, for a nucleus precessing in a magnetic field, the frequency of resonance is its Lamor frequency.

145 Figure 4.5: Lamor precession. Nuclei with a non-zero spin have an associated magnetic moment (µ)

146

4.1.3.2.2 Boltzmann distribution

At thermal equilibrium, a system of energy levels will have more particles adopting a lower energy level than higher energy levels. The population difference between the two possible energy levels can be described by the Boltzmann distribution:

𝑁

𝑁

= 𝑒

∆𝐸/𝑘𝑇

Where N is the number of nuclei in each state, k is the Boltzmann constant and T is the temperature in K.

Population differences under typical experimental conditions can be small (i.e approximately. 1 in 105 nuclei will realign upon application of the external magnetic field). This means that NMR spectroscopy can be a relatively insensitive technique. However, the slight differences in population results in a bulk magnetization along the z-axis and results in a net magnetization vector which can be visualized as the vector sum of all individual magnetic moments.

4.1.3.2 Relaxation

The process by which the bulk magnetization vector reaches its equilibrium level is known as relaxation. Relaxation typically takes place over a relatively long time-frame (0.001 – 1 second) and means that magnetization persists for a sufficiently long time to be manipulated and collected. However, this also imposes a time limit upon the acquisition of the signal. Relaxation occurs due to the presence of localised magnetic fields intrinsic to the sample oscillating at the appropriate Lamor frequency.

Relaxation can be separated into two processes, longitudinal relaxation and transverse relaxation.

147

4.1.3.2.1 Longitudinal Relaxation (T1)

Longitudinal (or spin-lattice) relaxation refers to the process by which the energy gained by the nuclei is lost to the lattice structure of the surroundings. Neighbouring spins have their own individual magnetic moments and tumble randomly due to thermal motion. The resultant transient magnetic fields at the Lamor precession frequency will cause the associated magnetic moment to precess. Effects from the neighbouring spins thus cause the z-component of the bulk magnetization vector to decay towards its equilibrium state and re-establish the normal Gaussian population distribution.

Longitudinal relaxation is characterised by the time constant T1 according to the equation: