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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
 vector along only one of the three Cartesian
this
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
o
induced
Bloc

B

B
j
j

The magnetic field causes the
electrons to circulate
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 re f |
dj 
| n re f |
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
d0
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