Biophysical Chemistry: NMR Spectroscopy
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1 Nuclear Magnetism Vrije Universiteit Brussel 21st October 2011
2 Outline 1 Overview and Context 2 3
3 Outline 1 Overview and Context 2 3
4 Context Proteins (and other biological macromolecules) Functional characterisation (binding studies, enzymology, in vivo studies) Function and dysfunction Structural characterisation (information about larger complexes, high-resolution structures of the components) High-resolution NMR (HNMR) X-ray crystallography (diffraction)
5 Prerequisites and References This part of the course assumes basic familiarity with the theory of electromagnetism and organic chemistry. The following books are used as reference material: (Oxford Chemistry Primers #32), P.J. Hore, Oxford Science Publications, ISBN Spin Dynamics: Basics of (2nd edition), M.H. Levitt, Wiley, ISBN Understanding NMR Spectroscopy, J. Keeler, Wiley, ISBN
6 Outline 1 Overview and Context 2 3
7 The Electric Field Coulomb s law describes the force between two static charges q and q 0 : F = 1 4πɛ 0 qq 0 r 2 1 r The deflection of an electron between two charged plates is a classical application of this idea: and leads to the concept of the electric field emanating from one charge and influencing the other: E = F q 0 = 1 4πɛ 0 q r 2 1 r
8 Magnetism The magnetic field is introduced to describe interactions between moving charges: F = q v B
9 Magnetic Dipoles (1) A magnetic dipole produces a magnetic field with a characteristic pattern of field lines, and can be describe by the following equations: B µ,x = µ 0 µ (3 sin(θ) cos(θ)) 4π r3 B µ,y = 0 B µ,z = µ 0 µ 4π r 3 (3 cos2 (θ) 1)
10 Magnetic Dipoles (2) In certain positions the magnetic field vector has special properties: parallel with the dipole moment on the z axis antiparallel to the dipole moment on the x axis perpendicular to the dipole moment on a line making an angle θ = 54.7 (for which 3 cos 2 (θ) 1 = 0) with the z axis.
11 Magnetic Dipoles (3) The energy of a magnetic dipole in an external magnetic field is determined by their strengths and relative orientation: E = µ B = µ B cos(θ)
12 Induction and EM Waves Electric currents give rise to magnetic fields, and changing magnetic fields induce currents in conductors. An alternating current produces electromagnetic waves, in which the electric and magnetic fields evolve in a coupled way, and both become functions of position and time: E = E( r, t); B = B( r, t); B E The most complete description of all EM phenomena is provided by the Maxwell equations.
13 Outline 1 Overview and Context 2 3
14 The Quantum Mechanical Atom The classical "solar system" model with particles following a well-defined trajectory is replaced by a probabilistic description with an inherent uncertainty principle.
15 Molecular Orbitals
16 Outline Overview and Context 1 Overview and Context 2 3
17 Nuclear Spin Overview and Context Elementary particles, such as electrons, neutrons and protons, have been found to possess an intrinsic angular momentum, known as spin. Spin is a fundamental property of particles, just like their mass and charge, and cannot be intepreted in terms of an actual physical rotation. The spin angular momentum is a vector quantity I with a magnitude of I(I + 1), where I is the spin quantum number of the particle. For electrons, neutrons and protons, I = 1 2. In atomic nuclei the spins of the component protons and neutrons partially or completely compensate each other, leaving the nucleus with a relatively small spin quantum number I of 0, 1 2, 1, 3 2, 2,...
18 Nuclear Magnetism The intrinsic angular momentum I inevitably gives rise to a magnetic dipole moment µ: µ = γ I in which the gyromagnetic ratio γ is a characteristic constant for each type of nucleus. Because the nuclei of different isotopes have different numbers of neutrons, they will have different spin quantum numbers and magnetogyric ratios. In NMR, isotopes are generally referred to as nuclides.
19 Biologically Relevant Nuclides Nuclide I γ/10 7 radt 1 s 1 Abundance/% 1 H H C C N N O O O
20 Quantisation The angular momentum, and therefore the dipole moment, are further quantised in a single direction, which is chosen to lie along the z axis by convention. The quantisation rule states that the z component of I can only adopt values of the form I z = m. m is the magnetic quantum number, which can adopt values between I and I, in integer steps: m = I, I 1, I 2,..., I + 1, I = h 2π, where h = J.s is the Planck constant.
21 Effect of an External Magnetic Field In the absence of any significant external magnetic field, the direction of quantisation (the z axis) is arbitrary, and all magnetic substates have the same energy. In the presence of a strong external magnetic field ( B 0 with magnitude B 0 ), the direction of quantisation aligns with the field, and each substate acquires a different energy determined by its magnetic quantum number: E = m γb 0 This gives rise to 2I energy differences, all equal to E = γb 0
22 The Simplest Case: I = 1/2 When I = 1 2 there are two possible energy levels with m = (generally denoted α) and with m = 1 2 (β). α and β are two special, stationary states of a spin-1/2. In general, a spin-1/2 exists as a quantum mechanical superposition of the two stationary states. Its state is described by the general wave function Ψ, which is a linear combination of the wave functions of the stationary states: with Ψ = c α α + c β β c α, c β C
23 Interactions with EM Waves A spin in an external field can absorb or emit electromagentic waves when two conditions are satisfied: the magnetic quantum numbers of the nuclear states before and after the interaction can differ by only one unit (this is the selection rule): m = ±1 the energy of the photons, determined by their frequency ν or wavelength λ, must match the energy difference betwdeen the two states: E = hν = hc λ = γb 0
24 Outline Overview and Context 1 Overview and Context 2 3
25 NMR in the EM Spectrum (1)
26 Basic NMR Instrumentation Nuclear magnetic resonance was first observed using relatively simple experimental setups: The first experiments were done on simple pure compounds, such as water and ethanol (shown here):
27 NMR in the EM Spectrum (2) Gamma rays X rays Visible UV light (Hz) IR Micro waves Radio waves NMR (MHz) B = 9.4 T 0 B = 21.2 T 0 1 H F P C H 15 N (ppm) B = 9.4 T 0 B = 21.2 T 0 4 khz 9 khz
28 Continuous Wave versus Puls/FT There are two obvious ways of recording an NMR spectrum. One possibility is to irradiate the sample with an RF source of constant amplitude and frequency, while varying the intensity of the external magnetic field. The other is to generate a constant magnetic field, while varying the frequency of the RF source. Since in both cases the sample is continuously exposed to RF radiation, this approach is known as continuous wave NMR spectrosocpy. As we shall see, it is also possible to apply a short and powerful RF pulse to the sample, which simultaneously excites all nuclei in the sample, after which the different resonance frequencies can be deduced using a Fourier analysis. This pulse/ft approach has essentially completely displaced continuous wave methods because of its enormous practical advantages.
29 (1) Electrons and nuclei possess an intrinsic angular momentum I, which is subject to quantisation rules involving a spin quantum number I and a magnetic quantum number m. Some nuclei, including 12 C, have a spin quantum number I = 0 and are magnetically inert. Many biologically important nuclides, including 1 H, 13 C and 15 N, have I = 1 2. Unpaired electrons are also in this category of "spins-1/2". Other nuclei with I > 1 2 can also be studied by NMR, but will be ignored here. Any spin-1/2 behaves like a magnetic dipole with a magnetic moment µ = γ I, where γ is a characteristic gyromagnetic ratio.
30 (2) For a spin-1/2 (I = 1 2 ) the magnetic quantum number m can adopt two different values (+ 1 2 and 1 2 ), corresponding to two distinct energy states of the spin in an external magnetic field B 0. A group of spins-1/2 can absorb electromagnetic radiation when the frequency ν of the photons matches the energy difference between the two magnetic states according to the relationship E = hν = γb 0
31 (3) An NMR spectrometer has hardware capable of generating a strong magnetic field B 0 as well as RF radiation of a defined frequency ν. It is capable of detecting resonance when the combination of these two parameters causes absorption of the RF energy by the nuclei in the sample. A complicated sample will contain nuclei of the same type experiencing different chemical environments, resulting in slightly different resonance frequencies and a spectrum with distinct lines.
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