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

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.

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.

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.

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

The Zeeman splitting is therefore

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)

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

A compass in a magnetic field

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

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

Larmor frequency and Zeeman splitting DE = h n0

Positive g  negative precession Negative g  positive precession

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

Generation of the NMR spectrum Fourier transform The NMR spectrum

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

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

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

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

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

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

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

Cholesterol proton spectrum chemical shift of TMS protons d = 0

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.

Examples of chemical equivalence

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)

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

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

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

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