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Modern Approaches to Protein structure Determination (6 lectures) Dr Matthew Crump 1 Two types of angular momentum • “Normal” or “extrinsic” angular momentum (due to rotational or orbital motion) use your right hand to figure out the way the angular momentum vector points • “Intrinsic” or “spin angular momentum” (a property of fundamental particles -cannot be visualized). the direction of the spin angular momentum is indicated by an arrow. 2 Gyromagnetic ratio (1) • The gyromagnetic ratio g determines the ratio of the nuclear magnetic moment to the nuclear spin. • It is a fundamental property of each nuclear isotope • Fundamental symmetry theorems predict that spin and magnetic moment are co-linear m The gyromagnetic ratio is also known as the magnetogyric ratio m =gI This equation tells us how much magnetism we get for a given spin. 3 Quantum Angular Momentum • In quantum mechanics, angular momentum is quantized. • The total angular momentum of particles with spin takes the values of the form ITOT I(I +1) 1/ 2 • If we specify an I value, quantum mechanics restricts us as well to specifying the projection of this vector along only one of the three Cartesian components of I. By convention the z-axis is chosen and Iz is given by Iz m • where m is a second quantum number which can take values m=-I,-I+1,-I+2,..,I. Therefore Iz has 2I+1 values. 4 Zeeman splitting • Energy of interaction is given by E=-m.B in a magnetic field B. The dot product tells us the energy depends on the size and relative orientation of B and m. • We take B to be along the Z axis, so the dot product becomes E=-mzBz (I.e. mxBz and myBz = 0 • the energy of the state with quantum number Iz is given by E z g Iz Bz Energy gyromagnetic ratio Planck constant m=-1/2 m=-1 m= 0 m=+1/2 ground state; no field ground state; with field m=+1 Zeeman splitting h g B/2π 5 1 E z g Iz Bz g Bz 2 I=1/2 I=1 m=-1/2 m=-1 m= 0 m=+1/2 m=+1 1 E z g Iz Bz g Bz 2 The Zeeman splitting is therefore g Bz 6 Gryomagnetic ratio (2) The gyromagnetic ratio g determines how rapidly the Zeeman splitting increases when the magnetic field is increased. 1H Note the ordering of the energy levels (g is positive for 1H) 15N 27Al Note the ordering of the energy levels (g is negative for 15N) 7 Gyromagnetic ratio (3) Spins I and gyromagnetic ratios g for some common nuclear isotopes: isotope natural abundance spin gyromagnetic ratio g/rad s–1 T-1 1 H 99.98% 1/2 267.5 106 2 H 0.015% 1 41.1 106 10 19.9% 3 28.7 106 12 98.9% 0 - 13 C 1.1% 1/2 67.2 106 14 N 99.6% 1 19.3 106 15 0.37% 1/2 -27.1 106 16 99.96% 0 - 17 0.04% 5/2 -36.3 106 F 100% 1/2 251.8 106 Na 100% 3/2 70.8 106 100% 5/2 69.8 106 100% 1/2 108.4 106 B C N O O 19 23 27 Al 31 P 8 A compass in a magnetic field 9 A nuclear spin precesses in a magnetic field the circulating motion of the spin angular momentum is called precession this arrow denotes the direction of the spin angular momentum Nuclear spins precess because: • they are magnetic •they have angular momentum 10 Precession frequency = Larmor frequency n0 = - g Bz/2π magnetic field in Tesla (T) Larmor frequency in Hz (= cycles per second) gyromagnetic ratio in rad s–1 T– 1 Compare with Zeeman Splitting h o g Bz g Bz hv 2 11 Larmor frequency and Zeeman splitting Zeeman splitting DE = h n0 12 Positive g negative precession Negative g positive precession 13 Precession frequencies for different isotopes isotope natural abundance spin gyromagnetic ratio g/rad s–1 T-1 Larmor frequency (MHz) in a field B0 = 11.7433 T 1 H 99.98% 1/2 267.5 106 -500.00 2 H 0.015% 1 41.1 106 -76.75 10 19.9% 3 28.7 106 -53.72 12 98.9% 0 - - 13 C 1.1% 1/2 67.2 106 -125.72 14 N 99.6% 1 19.3 106 -36.13 15 0.37% 1/2 -27.1 106 +50.68 16 99.96% 0 - - 17 0.04% 5/2 -36.3 106 +67.78 F 100% 1/2 251.8 106 -470.47 Na 100% 3/2 70.8 106 -132.26 100% 5/2 69.8 106 -130.29 100% 1/2 108.4 106 -202.61 B C N O O 19 23 27 Al 31 P the Larmor frequency is proportional to the field 14 Generation of the NMR spectrum Fourier transform The NMR spectrum 15 The sense of the frequency axis less rapid precession more rapid precession increasing | n | the sense of the precession is ignored 16 Chemical Shifts The molecular environment distorts the magnetic field on a microscopic scale 17 Mechanism of Chemical Shift The electrons in a molecule cause the local magnetic fields to vary on a submolecular distance scale 2 steps… 2 1 B loc j B B The magnetic field causes the electrons to circulate o induced j The circulating electrons generate an additional magnetic field which is sensed by the nuclei.This is called the induced field. It is proportional to the applied field. 18 Proton Chemical Shifts chemical shift d “deshielding” : magnetic field at nucleus enhanced by molecular environment “shielding” : magnetic field at nucleus reduced by molecular environment Chemical shifts correlate well with molecular structure and functional groups 19 Definition of Chemical Shift chemical shift of site j Larmor frequency of site j, ignoring the sign Larmor frequency of spins in a reference compound, ignoring the sign | n j | | n ref | dj | n ref | chemical shift d By convention the spectrum is plotted with d increasing from right to left. The result is usually quoted in units of ppm (parts per million), where 1 ppm = 106 This definition is used because it is field-independent 20 A common reference compound: TMS (Tetramethylsilane) chemical shift of TMS protons chemical shift d d0 21 Ethanol proton spectrum CH3 protons; d = 1.2 ppm OH proton; d = 2.6 ppm CH2 protons; d = 3.7 ppm chemical shift d chemical shift of TMS protons d = 0 22 Cholesterol proton spectrum chemical shift of TMS protons d = 0 23 Chemical equivalence Two spins are chemically equivalent if • there is a molecular symmetry operation that exchanges their positions, or • there is a dynamic process between two or more energetically equivalent conformations, in which the positions of the two nuclei are exchanged. Chemically equivalent spins have the same chemical shift. 24 Examples of chemical equivalence 25 An example of chemical inequivalence chiral centre the rotation around the CC bond exchanges the protons but the onformations are not equivalent (different energies and different chemical shifts) 26 Chemical inequivalence in amino acids: L-phenylalanine chiral centre chemically inequivalent CH2 protons 27 Spin-spin couplings Direct DD coupling (averages to zero in ordinary liquids) Indirect DD coupling or J– coupling (doesn’t average to zero in ordinary liquids) electrons 28 J-couplings cause splittings ethanol proton spectrum chemical shift d multiplet structures caused by multiplet structure caused by homonuclear J-couplings J-couplings between protons 29 J-multiplets J-coupling to N magnetically equivalent spins-1/2 splits the spectrum into N+1 multiplet components 1 coupling partner: doublet 2 coupling partners: triplet 3 coupling partners: quartet 30