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Modern Approaches toProtein structureDetermination(6 lectures)

Dr Matthew Crump

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Two types of angular momentum

• “Normal” or “extrinsic” angular momentum (due to rotational or orbital motion)

• “Intrinsic” or “spin angular momentum” (a property of fundamental particles -- cannot be visualized).

use your right hand to figure out the way the angular

momentum vector points

the direction of the spin angular momentum is indicated by an arrow.

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• The gyromagnetic ratio 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

Gyromagnetic ratio (1)

The gyromagnetic ratio is also

known as the magnetogyric

ratio

=IThis equation tells

us how much magnetism we get for a given spin.

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Quantum Angular Momentum

• 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 I z has 2I+1 values.

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ITOT = I(I +1)[ ]1/ 2

h

• In quantum mechanics, angular momentum is quantized.

• The total angular momentum of particles with spin takes the values of the form

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Iz = mh

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• the energy of the state with quantum number Iz is given by

Zeeman splitting

Energy

ground state;

no field

ground state; with field

Zeeman splitting h B/2π

Planck constant

gyromagnetic ratio

• Energy of interaction is given by E=-.B in a magnetic field B. The dot product tells us the energy depends on the size and relative orientation of B and .

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E z = −γhIzBz

• We take B to be along the Z axis, so the dot product becomes E=-zBz (I.e. xBz and yBz = 0

m=-1/2

m=+1/2

m=-1

m=+1

m= 0

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E z = −γhIzBz = −γh1

2Bz

m=-1/2

m=+1/2

I=1/2m=-1

m=+1

I=1

m= 0

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E z = −γhIzBz = γh1

2Bz

The Zeeman splitting is therefore

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hBz

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The gyromagnetic ratio determines how rapidly the Zeeman splitting increases when the magnetic field is increased.

Gryomagnetic ratio (2)

1H 15N 27Al

Note the ordering of the energy levels ( is negative

for 15N)

Note the ordering of the energy levels ( is positive

for 1H)

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Spins I and gyromagnetic ratios for some common nuclear isotopes:

Gyromagnetic ratio (3)

isotope naturalabundance

spin gyromagneticratio /rads–1T-1

1H 99.98% 1/2 267.5×106

2H 0.015% 1 41.1×106

10B 19.9% 3 28.7×106

12C 98.9% 0 -13C 1.1% 1/2 67.2×106

14N 99.6% 1 19.3×106

15N 0.37% 1/2 -27.1×106

16O 99.96% 0 -17O 0.04% 5/2 -36.3×106

19F 100% 1/2 251.8×106

23Na 100% 3/2 70.8×106

27Al 100% 5/2 69.8×106

31P 100% 1/2 108.4×106

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A compass in a magnetic field

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A nuclear spin precesses in a magnetic field

the circulating motion of the spin angular momentum is

called precession

Nuclear spins precess because:• they are magnetic•they have angular momentum

this arrow denotes the direction of the spin angular

momentum

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Precession frequency = Larmor frequency

0 = - Bz/2π

Larmor frequency in Hz (= cycles per second)

gyromagnetic ratio in rad s–

1 T–1

magnetic field inTesla (T)

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hBz = γh

2πBz = hv o

Compare with Zeeman Splitting

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Larmor frequency and Zeeman splitting

Zeeman splittingE = h 0

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Positive negative precession

Negative positive precession

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Precession frequencies for different isotopes

isotope naturalabundance

spin gyro-magnetic

ratio/rads–1T-1

Larmorfrequency(MHz)inafieldB0=11.7433T

1H 99.98% 1/2 267.5×106 -500.002H 0.015% 1 41.1×106 -76.7510B 19.9% 3 28.7×106 -53.7212C 98.9% 0 - -13C 1.1% 1/2 67.2×106 -125.7214N 99.6% 1 19.3×106 -36.1315N 0.37% 1/2 -27.1×106 +50.6816O 99.96% 0 - -17O 0.04% 5/2 -36.3×106 +67.7819F 100% 1/2 251.8×106 -470.47

23Na 100% 3/2 70.8×106 -132.2627Al 100% 5/2 69.8×106 -130.2931P 100% 1/2 108.4×106 -202.61

the Larmor frequency is proportional to the field

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Generation of the NMR spectrum

Fourier transform

The NMR spectrum

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The sense of the frequency axis

more rapid precession

increasing | |

less rapid precession

the sense of the precession is

ignored

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Chemical Shifts

The molecular environment distorts the magnetic field on a microscopic scale

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Mechanism of Chemical Shift

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.

The electrons in a molecule cause the localmagnetic fields to vary on a submolecular distancescale

2 steps…

1 2

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B jloc = Bo + B j

induced

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Proton Chemical Shifts

Chemical shifts correlate wellwith molecular structure and functional groups

“shielding” : magnetic field at nucleus reduced by

molecular environment

“deshielding” : magnetic field at nucleus enhanced by molecular environment

chemical shift

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This definition is used because it is field-independent

Definition of Chemical Shift

chemical shift

δj =|ν j |−|νref |

|νref |

chemical shift of site jLarmor frequency of site

j, ignoring the sign

Larmor frequency of spins in a reference compound,

ignoring the sign

By convention the spectrum is plotted with increasing from right to

left.

The result is usually quoted in units of ppm

(parts per million), where 1 ppm = 10-6

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A common reference compound: TMS

(Tetramethylsilane)

chemical shift

chemical shift of TMS protons

0

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Ethanol proton spectrum

chemical shift of TMS protons = 0

CH2 protons; = 3.7 ppm

OH proton; = 2.6 ppm

CH3 protons; = 1.2 ppm

chemical shift

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Cholesterol proton spectrum

chemical shift of TMS protons = 0

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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.

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Examples of chemical equivalence

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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)

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Chemical inequivalence in amino acids:

L-phenylalaninechiral centre

chemically inequivalent CH2 protons

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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

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J-couplings cause splittings

chemical shift

ethanol proton spectrum

multiplet structure caused by J-couplings

multiplet structures caused by homonuclear J-couplings

between protons

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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