5.10: Nuclear Magnetic Resonance (NMR) Theory and Solution NMR
Nuclear magnetic resonance (NMR) occurs when nuclei in an unmoving magnetic field is disturbed by an oscillating magnetic field; the nuclei generate an electromagnetic signal, whose frequency depends on the magnetic field applied. This happens near resonance, where the frequency of oscillation aligns with the frequency of the nuclei. The magnetic field strength, chemical environment, and isotope affect resonance. NMR spectroscopy is used to elucidate the structure of organic molecules, study crystals and non-crystals, and can be applied to medical diagnostic imaging. NMR has three basic steps: nuclear spins are aligned in a magnetic field, the nuclear spins are disturbed by a radio-frequency (RF) pulse, the NMR signal is detected during or after the RF pulse. NMR was described by Isidor Rabi in 1938 and he won the Nobel Prize in Physics in 1944. Felix Bloch and Edward Mills Purcell used NMR on lipids and solids and won the Nobel Prize in Physics in 1952. Yevgeny Zavoisky observed MNR in 1941, prior to Bloch and Purcell; however, he discarded the results as unreproducible. Alongside x-ray crystallography and cryogenic-electron microscopy, NMR is one of the three techniques that is used to elucidate the structure of proteins.
Nuclear Spin
Atomic nuclei of protons and neutrons have spin, an intrinsic quantum property; it is an angular momentum similar to the angular momentum of a revolving orb. The overall spin of nuclei is dictated by the spin quantum number, S. If the numbers of protons and neutrons in the nucleus are even, S=0, which means no overall spin. Electrons couple in non-degenerate atomic orbitals; even numbers of protons or neutrons, which are spin 1/2 fermions, also give no overall spin. 1
Protons and neutrons have lower energy when spins are aligned; the parallel spin alignment of particles does not violate the Pauli exclusion principle. The quark structure of protons and neutrons is responsible for the lower energy; the spin ground state for the nucleus of deuterium, which has one proton and one neutron, is one, not zero. Tritium, due to the Pauli exclusion principle, has two anti-parallel spin neutrons, and a proton spin of 1/2; the nuclear spin value for tritium is 1/2, comparable to 1 H. The nuclear magnetic resonance (NMR) absorption frequency for tritium is akin to 1 H. For non-radioactive nuclei, overall spin is non-zero; as an example, 27 Al nuclei have an overall spin of 5/2. 1
A non-zero spin is related to a non-zero magnetic dipole moment, µ: µ=gamma·S; gamma is the gyromagnetic ratio. This relates to the relation of the angular momentum and magnetic dipole moment of a rotating orb. They are vectors parallel to the revolution horizontal; the length increases directly with the rotating frequency. The magnetic moments’ interaction with magnetic fields permits the study of NMR values; these are associated with alterations between nuclear spin levels during RF incitement or due to Larmor precession of the magnetic moment after stimulation. Nuclei with even numbers of protons and neutrons have zero nuclear magnetic dipole moment and lack an NMR signal. 13 C, 31 P, 35 Cl, and 37 Cl generate an NMR signal, whereas 18 O does not. 4
In contrast to NMR, electron spin resonance (ESR) detects transitions in electronic spin, not nuclear spin. 17 A smaller number of molecules with unpaired electron spins show ESR (or electron paramagnetic resonance (EPR)) absorption than NMR; however, ESR has a higher signal per spin than NMR. 17 While NMR can be applied to proteins, EPR/ESR is mainly used to study metal complexes and organic radicals. 17
Magic Angle Spinning (MAS)
In NMR, magic-angle spinning (MAS) is an approach used in solid-state NMR and 1 H NMR. 2 The sample is spun at a frequency of 1 to 130 kiloHertz (kHz) at an angle of θ m =54.7356°, in regards to the magnetic field. 2 Wide lines become more narrow, which increases the resolution of the spectra. 2 Nuclear spin experiences three interactions interactions: dipolar, chemical shift anisotropy (CSA), and quadrupolar; the spectral lines generated are wide and difficult to analyze. 2 However, these interactions are dependent on position and MAS is used to average them. 2 Dipole interactions between nuclei are approximately zero at 54.7356°. 6 Nucleus-electron interactions are known as CSA, and are non-zero. 6 MAS can partially average the quadrupolar interactions. 6 In solution, these interactions are averaged out due to molecular movement; in solids, MAS causes narrow spectral lines, which can be used to determine CSA. 6
Spin Angular Momentum
Nuclear spin is a quantized intrinsic angular momentum. The values of S are limited and the x, y, and z-components are quantized; they are (half)-integer multiples of ħ, the reduced Planck constant. The (half)-integer quantum number related to spin alongside the z-axis or the magnetic field is the magnetic quantum number, m, which is +S to -S, in integer values. For all nuclei, there are a total of 2S+1 angular momentum states. The z-component of the magnetic moment is 4
\[μ_Z=γS_Z=γm\hbar.\]
Spin Energy in a Magnetic Field
Nuclei with a spin of 1/2, like 1 H, 13 C, or 19 F have two independent linear spin states, where m=1/2 or -1/2 for the z-portion; if a magnetic field is absent, the states possess the same energy. At thermal equilibrium, the number of nuclei per state is the same. When placed in a magnetic field, the energy varies due to interactions of the nuclear magnetic dipole moment and the magnetic field applied. The energy of the magnetic dipole moment, µ, in a magnetic field, B 0 , is
\[E=-µB_0=-µ_xB_{0x}-µ_yB_{0y}-µ_zB_{0z}.\]
The z-axis lies with B 0 , and the new equation is
\[E=-µ_zB_0=-γmℏB_0.\]
Therefore, in a magnetic field, energies vary. A spin state of 1/2 means the spin is aligned with the magnetic field, and is the lower energy state; conversely, a spin state of -1/2 means the spin is not aligned with the magnetic field, and is the higher energy state. A positive gamma indicates that m=1/2 is the lower energy state. The difference between the high and low states is
\[ΔE=γℏB_0.\]
The lower energy state is preferred at thermal equilibrium. If more spins are up than down, net spin magnetization aside B 0 occurs. 1
Larmor Precession & Radio-Frequency Pulses
The spin magnetization is proportional to the sum of spin vectors of atomic nuclei in equal magnetic sites and moves in a cone around the field, B. At non-equilibrium, precession of magnetization in B 0 arises with the Lamor frequency
\[ω_L=2\pi ν_L=-\gamma B_0.\]
The energy of the population does not change, as it is constant. The transverse magnetization formed by an oscillating field is detected in NMR when an RF field or pulse is applied. 4
Chemical Shielding (Shift)
Not all nuclei of the same element resonate at the same frequency. Electrons are charged and rotate and produce a magnetic field that is the reverse of the applied magnetic field. The electronic shielding reduces the magnetic field at the nucleus; the frequency needed to obtain resonance is decreased. This is known as chemical shift, and provides an explanation for NMR’s ability to ascertain chemical structures, which rely on electron density. Nuclei with higher degrees of shielding due to a higher electron density have their NMR frequency shifted up, which corresponds to a low chemical shift; in contrast, less shielding shifts NMR frequency down, which gives a high chemical shift. In solid state NMR, magic angle spinning averages these states to get frequency values at average shifts. In regular, NMR, molecular tumbling averages the CSA. 1
Relaxation
Relaxation occurs when nuclear spins revert to thermodynamic equilibrium; this is known as T 1 relaxation and indicates the average time it takes for nuclei to return to thermal equilibrium. Precessing nuclei eventually stop generating signals when they do not align; this is called T 2 relaxation. T 1 relaxation is normally longer than T 2 relaxation, because of small dipole-dipole interactions. 1
Solution NMR
Less than 2% of the 120,000 structures of solved proteins are of membrane proteins; very few of those structures were solved via NMR. 7 High quantities of protein must be used and they must be extracted from native environments using chemicals that maintain structural and functional integrity. 8 If NMR is used, crystallization is not needed, but, new problems arise. 8 Solution NMR can be used to determine the structures of macromolecules that go through fast rotational diffusion on timescales of less than 100 nanoseconds. 15 As a result, proteins must be placed in lipid micelles, small bicelles, and nanodiscs. 13 Proteins with a size of approximately 40 kiloDaltons or less can be obtained with solution NMR; this is due to dynamics and resonance line widths. 14
Heteronuclear NMR methods used for soluble proteins can be applied to membrane proteins. 18 For structural determination, proteins are 15 N- and 13 C-labeled by expression in minimal media with 15 N-ammonium sulfate and 13 C-glucose as nitrogen and carbon springs. 5 Protein side-chains are deuterated via 2 H, 13 C-glucose to obtain better spectral resolution. 7 This lowers 13 C relaxation, which causes lower resonance; however, less protons are available to measure 1 H-NOE’s, which give distant restraints. 11 2 H2O is used for culture growth and amide protons are replaced with the solvent during purification and preparation; if deuterium is used in the solvent, the NMR spectroscopy will reflect this. 7 The rates of amide proton exchange provide information on exposed versus buried moieties. 9
Correlation-Time Problem
The rotational correlation time of proteins is important for NMR. It determines extraction conditions, instrumental analysis, and data processing methods. It is hard to obtain high-resolution spectra of large or aggregated proteins with NMR due to slow reorientation. The correlation-time problem can be dealt with in two ways when it comes to membrane proteins. In solid state NMR, membrane proteins in lipid bilayers or sizable bicelles are paralyzed by environmental conditions; radio-frequency irradiation, magic angle sample spinning, or sample alignment are replacements for molecular motions for the line-narrowing mechanism. For small membrane proteins, micelles or small bicelles can be used to prepare the samples; this leads to fast reorientation, which is detectable by solution NMR. 4
Protein NMR
NMR of proteins is unlike magnetic resonance imaging (MRI), which obtains an image directly; protein NMR uses algorithms to create three-dimensional models of the sample of interest. 3 Protein NMR is conducted on thoroughly purified samples, which have a volume of approximately 500 µL and a concentration of approximately 0.2 mM. 16 Normally, proteins are obtained via recombinant DNA, which is performed by genetic engineering; this method permits isotopic labeling to track particular atoms. 13 Once the proteins is dissolved in a buffer, it is placed in a thin NMR tube for spectroscopy. 13
Large molecules, like proteins, have thousands of resonances, which will show overlaps of one-dimensional spectra. 10 As a result, multidimensional experiments are conducted; this reduces overlaps and focuses on certain nuclei in one section of the sample. 10 RF pulses permit the transfer of magnetization from chemical bonds and space (structure is irrelevant); this allows for the determination of chemical shifts and distance restraints, respectively. 10 Experiments of higher dimensions require more time than lower dimension experiments. 10
Homonuclear NMR
If the proteins of interest are unlabeled, correlation spectroscopy (COSY) is performed; two types of COSY are total correlation spectroscopy (TOCSY) and nuclear Overhauser effect spectroscopy (NOESY). 12 These two-dimensional NMR’s provide two-dimensional spectra. 12 Both axes are chemical shifts, in term of units. 12 These experiments build spin systems, a list of resonances of the chemical shift of protein’s protons. 12 To link the spin systems in the right pathway, NOESY must be used, which uses spin-lattice relaxation. 12 Magnetization is transferred via space in NOESY, which can be used to calculate distance relations. 12 NOESY can also determine chemical and conformational changes. 12 Peak overlap is an issue with homonuclear NMR; as a result, it is limited to small proteins. 12
References
- Abragam, Anatole, and Anatole Abragam. The principles of nuclear magnetism . No. 32. Oxford university press, 1961.
- Andrew, E. Raymond. "Magic angle spinning." Solid State NMR Studies of Biopolymers 2 (2010): 83.
- Clore, G. Marius, and Angela M. Gronenborn. "Determination of three-dimensional structures of proteins and nucleic acids in solution by nuclear magnetic resonance spectroscop." Critical reviews in biochemistry and molecular biology 24.5 (1989): 479-564.
- Feynman, Richard P., Robert B. Leighton, and Matthew Sands. "The feynman lectures on physics; vol. i." American Journal of Physics 33.9 (1965): 750-752.
- German, James. "The pattern of DNA synthesis in the chromosomes of human blood cells." The Journal of cell biology 20.1 (1964): 37-55.
- Hennel, Jacek W., and Jacek Klinowski. "Magic-angle spinning: a historical perspective." New techniques in solid-state nmr . Springer, Berlin, Heidelberg, 2005. 1-14.
- Johnson, Richard S., and Kenneth A. Walsh. "Mass spectrometric measurement of protein amide hydrogen exchange rates of apo‐and holo‐myoglobin." Protein Science 3.12 (1994): 2411-2418.
- Jørgensen, Thomas JD, et al. "Collisional activation by MALDI tandem time-of-flight mass spectrometry induces intramolecular migration of amide hydrogens in protonated peptides." Molecular & Cellular Proteomics 4.12 (2005): 1910-1919.
- Jørgensen, Thomas JD, et al. "Intramolecular migration of amide hydrogens in protonated peptides upon collisional activation." Journal of the American Chemical Society 127.8 (2005): 2785-2793.
- Liang, Binyong, and Lukas K. Tamm. "NMR as a Tool to Investigate Membrane Protein Structure, Dynamics and Function." Nature structural & molecular biology 23.6 (2016): 468.
- Marley, Jonathan, Min Lu, and Clay Bracken. "A method for efficient isotopic labeling of recombinant proteins." Journal of biomolecular NMR 20.1 (2001): 71-75.
- Opella, Stanley J., and Francesca M. Marassi. "Structure determination of membrane proteins by NMR spectroscopy." Chemical reviews 104.8 (2004): 3587-3606.
- Phinney, Karen W. "Expression of Stable Isotopically Labeled Proteins for Use as Internal Standards for Mass Spectrometric Quantitation of Clinical Protein Biomarkers." (2009).
- Polshakov, Vladimir I., Berry Birdsall, and James Feeney. "Effects of co-operative ligand binding on protein amide NH hydrogen exchange." Journal of molecular biology 356.4 (2006): 886-903.
- Rand, Kasper D., et al. "Electron capture dissociation proceeds with a low degree of intramolecular migration of peptide amide hydrogens." Journal of the American Chemical Society 130.4 (2008): 1341-1349.
- Wüthrich, Kurt. "Protein structure determination in solution by NMR spectroscopy." Journal of Biological Chemistry 265.36 (1990): 22059-22062.
- Zavoisky, E. "Spin-magnetic resonance in paramagnetics." J Phys USSR 9 (1945): 211-245.
- Zhang, Zhongqi, and David L. Smith. "Determination of amide hydrogen exchange by mass spectrometry: a new tool for protein structure elucidation." Protein Science 2.4 (1993): 522-531.