1. Modern Approaches to Protein structure Determination (6 lectures)
Dr Matthew Crump

2. 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.

3. This equation tells us how much magnetism we get for a given spin.
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
m =gI
The gyromagnetic ratio is also known as the magnetogyric ratio
This equation tells us how much magnetism we get for a given spin.

4. Quantum Angular Momentum
In quantum mechanics, angular momentum is quantized.
The total angular momentum of particles with spin takes the values of the form
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
where m is a second quantum number which can take values m=-I,-I+1,-I+2,..,I. Therefore Iz has 2I+1 values.

5. 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
Planck constant
gyromagnetic ratio
Energy
m=-1/2
m=-1
m= 0
m=+1/2
m=+1
ground state; no field
Zeeman splitting h g B/2π
ground state; with field

6. The Zeeman splitting is therefore

7. Gryomagnetic ratio (2)
The gyromagnetic ratio g determines how rapidly the Zeeman splitting increases when the magnetic field is increased. 1H
15N
27Al
Note the ordering of the energy levels (g is positive for 1H)
Note the ordering of the energy levels (g is negative for 15N)

8. Gyromagnetic ratio (3)
Spins I and gyromagnetic ratios g for some common nuclear isotopes:

9. A compass in a magnetic field

10. 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

11. 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

12. Larmor frequency and Zeeman splitting
DE = h n0

13. Positive g  negative precession
Negative g  positive precession

14. Precession frequencies for different isotopes
the Larmor frequency is proportional to the field

15. Generation of the NMR spectrum
Fourier
transform
The NMR spectrum

16. The sense of the frequency axis
less rapid precession
more rapid precession
increasing | n |
the sense of the precession is ignored

17. Chemical Shifts The molecular environment
distorts the magnetic field on a microscopic scale

18. Mechanism of Chemical Shift
The electrons in a molecule cause the local
magnetic fields to vary on a submolecular distance
scale
2 steps… 1 2
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.
The magnetic field causes the electrons to circulate

19. 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

20. Definition of Chemical Shift
Larmor frequency of site j, ignoring the sign
chemical shift of site j
Larmor frequency of spins in a reference compound, ignoring the sign
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 = 10-6
This definition is used because it is field-independent

21. A common reference compound: TMS (Tetramethylsilane)
chemical shift of TMS protons
chemical shift d
d = 0

22. 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

23. Cholesterol proton spectrum
chemical shift of TMS protons d = 0

24. 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.

25. Examples of chemical equivalence

26. An example of chemical inequivalence
chiral centre
the rotation around the C-C bond exchanges the protons but the onformations are not equivalent (different energies and different chemical shifts)

27. Chemical inequivalence in amino acids: L-phenylalanine
chiral centre
chemically inequivalent CH2 protons

28. Direct DD coupling (averages to zero in ordinary liquids)
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

29. J-couplings cause splittings
ethanol proton spectrum
chemical shift d
multiplet structures caused by homonuclear J-couplings between protons
multiplet structure caused by J-couplings

30. 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