Biophysical Chemistry: NMR Spectroscopy

Spin Dynamics & Vrije Universiteit Brussel 25th November 2011

Outline 1 Pulse/Fourier Transform NMR Thermal Equilibrium Effect of RF Pulses The Fourier Transform 2 Symmetric Exchange Between Two Sites Asymmetric Two-Site Exchange Applications 3

Thermal Equilibrium Effect of RF Pulses The Fourier Transform Outline 1 Pulse/Fourier Transform NMR Thermal Equilibrium Effect of RF Pulses The Fourier Transform 2 Symmetric Exchange Between Two Sites Asymmetric Two-Site Exchange Applications 3

Quantum Description of a Spin-1/2

Thermal Equilibrium Effect of RF Pulses The Fourier Transform Larmor Precession (1) The interaction between an individual spin and a uniform external magnetic field leads to precession of the spin around the direction of the external field: The angle θ between the direction of the field and the direction of the spin remains constant throughout this motion. De frequency of the precession is the Larmor frequency ν = γ(1 σ)b 0 2π or ω = γ(1 σ)b 0.

Thermal Equilibrium Effect of RF Pulses The Fourier Transform Larmor Precession (2) The interaction between the spin and the external field is far stronger than all other interactions between the nucleus and other particles in its environment. Therefore, as a first approximation, the nucleus behaves like an isolated gyroscope which rotates independently, with no regard for its surroundings or the motions of the molecule which it is part of.

Thermal Equilibrium Effect of RF Pulses The Fourier Transform Populations (1) The ratio between the populations of the two energy levels (n α and n β ) is determined by the energy difference E and the temperature T: n β = e E kt n α from which we find that n α n β n α + n β = E 2kT The Boltzmann constant (k = k B = 1.38066 10 23 J K ) functions a conversion factor from temperature to thermal energy.

Thermal Equilibrium Effect of RF Pulses The Fourier Transform Populations (2) At a temperature of T = 300K the average thermal energy is kt = 4.14 10 21 J. The energy difference between the two stationary states of a spin I = 1/2 is very small, even for 1 H (which has the largest gyromagnetic ratio of all practically available nuclei) in a strong external field: γ = 26.73 10 7 T 1 s 1 ; B 0 = 9.4T; E = γb 0 = 2.65 10 25 J This implies that the difference between the two populations is very small: n α n β = E = 3.2 10 5 n α + n β 2kT In other words, about one low-energy spin out of every 10 5 has no counterpart in the high-energy orientation.

Thermal Equilibrium Effect of RF Pulses The Fourier Transform Populations at Thermal Equilibrium Each individual spin contributes a certain fraction of "α character" (proportional to c α 2 ) and a complementary fraction "β character" (proportional to c β 2 = 1 c α 2 ) to the ensemble (the collection of all spins). The populations n α and n β of the two energy levels are the avarage values c α 2 and c β 2 over all spins in the ensemble.

Thermal Equilibrium Effect of RF Pulses The Fourier Transform Thermal Coupling Very infrequently, the nucleus does interact with a surrounding particle, which can lead to a change of its orientation with respect to the external field, as expressed by the angle θ. The energy that drives these interactions comes from the thermal energy of the atoms, that is associated with their random motions. The minuscule energy changes of the nuclear spins are associated with equally minuscule temperature changes of the system. Because of the energy difference between the α and β states there is a small preference for random flips that move the spin state towards the lower energy level. As a result, a thermal equilibrium between the α and β populations is slowly established. This equilibrium is described by the Boltzmann distribution.

Thermal Equilibrium Effect of RF Pulses The Fourier Transform Bulk Magnetisation at Equilibrium (1) The individual dipole moments of all spins can be added together to find the total or bulk magnetisation of the sample.men. In the x and y directions, the spins are oriented completely randomly: which results in a net magnetisation of zero in these directions.

Thermal Equilibrium Effect of RF Pulses The Fourier Transform Bulk Magnetisation at Equilibrium (2) In the z direction there is a small preference for the low energy state, as reflected by the slightly larger population n α : n α n β = n eq n α + n β N = E 2kT Because of this, a small net magnetisation remains in the direction of the positiove z axis. The magnitude of this remainder is proportional to the population difference n eq : M 0 = 1 2 γ n eq

Thermal Equilibrium Effect of RF Pulses The Fourier Transform Larmor Precession at Equilibrium At equilibrium, the x and y components of the spin dipoles remain randomly distributed throughout the precessional motion, and theirsum remains zero. The distribution of the α and β components is also unaffected by the precessional motion, and therefore the z component of the total magnetisation also remains constant. The bulk magnetisation vector therefore remains constant as the individual spins precess around the z axis.

Design of a Modern NMR Spectrometer (1)

Design of a Modern NMR Spectrometer (2)

Thermal Equilibrium Effect of RF Pulses The Fourier Transform Bulk Magnetisation at Equilibrium (3) At thermal equilibrium the spins are almost equally distributed in all directions, with a small preference for the low-energy state: (For the purpose of the illustration, the population difference has been greatly exaggerated.)

Design of a Modern NMR Spectrometer (3)

Effect of an RF Pulse A well-tuned RF pulse coherently rotates all spins about the x axis. The net effect is that the bulk magnetisation as a whole undergoes the same rotation:

Design of a Modern NMR Spectrometer (4)

Return to Equilibrium (Relaxation) When the excitation by the RF pulse ends, the system returns to its equilibirium state. The oscillating variation of the net magnetisation in the (x, y) plane is the source of the obervable signal:

The Magnetic Field of an RF Pulse Physics tells us that only the magnetic component of the RF radition coming from the excitation coil affects the spins. Because of the position of the coil with respect to the sample this magnetic component B 1 rotates in the x,y plane, with a frequency ω RF and a phase φ RF determined by the operator: B 1,x = B 1 cos(ω RF t + φ RF ) B 1,y = B 1 sin(ω RF t + φ RF )

Thermal Equilibrium Effect of RF Pulses The Fourier Transform The Rotating Frame In order to simplify the description of the precession of spins around a field that is itself rotating, we introduce a new frame of reference that rotates around the z axis at the frequence of the RF wave (ω RF ): e x = cos(φ(t)) e x + sin(φ(t)) e y e y = cos(φ(t)) e y sin(φ(t)) e x e z = e z Φ(t) = ω RF t + φ RF

Thermal Equilibrium Effect of RF Pulses The Fourier Transform Implications In the rotating frame the magnetic field of the RF pulse appears to be fixed on the x axis. The precession frequency ω 0 of the spins has to be replaced by the offset frequency Ω 0 : Ω 0 = ω 0 ω RF The pulse frequency ω RF is generally chosen to lie in the middle of the natural frequency range of the spins in the sample. Therefore, offset frequencies can be both positive and negative.

The Effect of Resonance The offset frequency in the rotating frame corresponds to Larmor precession around a reduced magnetic field B: B = Ω γ = ω 0 ω RF γ When ω RF = ω 0, B = 0 and the effective magnetic field B eff is completety determined by B 1 along the x axis. This is a geometric representation of the resonance principle.

Thermal Equilibrium Effect of RF Pulses The Fourier Transform Coherent Excitation (1) At resonance the spins therefore precess around an effective field along the x axis. The angle of rotation β p around the x axis is determined by the intensity of the pulse (B 1 ) and by its duration (t p ): β p γb 1 t p Since all individual spins are coherently rotated by this same angle, the bulk magnetisation also rotates by this angle.

Thermal Equilibrium Effect of RF Pulses The Fourier Transform Coherent Excitation (2) A 90-degree pulse converts the equilibrium population difference (on the z axis) completely into a coherent orientation in the x, y plane. The magnitude of the measurable transverse signal is therefore determined by the original population difference.

Thermal Equilibrium Effect of RF Pulses The Fourier Transform Pulse Length Calibration By executing a series of test pulses of increasing duration, one can determine which duration t p corresponds to a flip angle β p of 180 degrees. Once this value is known, the required duration for any desired flip angle can be easily calculated.

Thermal Equilibrium Effect of RF Pulses The Fourier Transform Larmor Precession after Excitation When the RF pulse ends, the spins resume their precession around the external field. Since they are all rotating at the same Larmor frequency, the direction of their preferred orientation, and therefore the bulk magnetisation vector, also rotate at the Larmor frequency. This rotational change of the bulk magnetisation in the x,y plane is equivalent to a variable magnetic field and induces an observable current in the detector coil. 10 10 5 5 Mx 0 My 0-5 -5-10 -10 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 t t

Thermal Equilibrium Effect of RF Pulses The Fourier Transform Mixed Ensembles If there are different ensembles of spins with distinct Larmor frequencies mixed together in the sample, all spins are excited simultaneously by the RF pulse. Subsequently each subgroup precesses at its own Larmor frequency, and the total observed signal is the sum of the contributions of all subgroups at different frequencies: Mx 10 5 0-5 -10 0 2 4 6 8 10 12 14 t + Mx 10 5 0-5 -10 0 2 4 6 8 10 12 14 t = Mx 15 10 5 0-5 -10-15 0 2 4 6 8 10 12 14 t

Fourier Analysis

Thermal Equilibrium Effect of RF Pulses The Fourier Transform Relaxation Due to a number of relaxation mechanisms, the bulk magnetisation ultimately returns to its equilibrium value, and the oscillating signal gradually fades away. 10 10 5 5 Mx 0 My 0-5 -5-10 0 2 4 6 8 10 12 14 t -10 M x = M 0 sin(ω 0 t) exp( t T 2 ) M y = M 0 cos(ω 0 t) exp( t T 2 ) in which T 2 is a characteristic time constant. 0 2 4 6 8 10 12 14 t

The Lorentzian Curve The Fourier transform of such an oscillating and exponentially fading signal is called a Lorentzian curve and can be desribed analytically as λ S(Ω) = A λ 2 + (Ω Ω 0 ) 2 where A is the amplitude of the signal is, and λ = 1 T 2.

Symmetric Exchange Between Two Sites Asymmetric Two-Site Exchange Applications Outline 1 Pulse/Fourier Transform NMR Thermal Equilibrium Effect of RF Pulses The Fourier Transform 2 Symmetric Exchange Between Two Sites Asymmetric Two-Site Exchange Applications 3

Symmetric Exchange Between Two Sites Asymmetric Two-Site Exchange Applications Isomerisation of a Partial Double Bond The bond between the two nitrogen atoms in the nitroso group of N,N -dimethylformamide has a partial double bond character. The cis and trans forms both occur and have identical energies, but there is a significant energy barrier for the transition of one conformer to the other.

Symmetric Exchange Between Two Sites Asymmetric Two-Site Exchange Applications Slow Exchange When the exchange between the two states is very slow (or, more accurately, very rare) each individual molecule is eiher in state A or in state B during the whole course of the NMR measurement, without switching. The sample can then be considered as a mixture of two distinct, unchanging molecular species, and the spectrum will simply consist of two independent signals at the respective frequencies ν A and ν B corresponding to the A and B states.

Symmetric Exchange Between Two Sites Asymmetric Two-Site Exchange Applications Transition from Slow to Fast Exchange ν A x 0 ν B 20 20 k (s 1 ) ν A +ν B 2

Intermediate Exchange: k = 100 Hz

Symmetric Exchange Between Two Sites Asymmetric Two-Site Exchange Applications Slow-Intermediate Exchange In the slow intermediate exchange regime some molecules will undergo a small number of conformational changes during the course of the experiment. As long as the condition k < δν 2, where δν = ν A ν B, is satisfied, the two signal remain centered around ν A and ν B. However, the lines gradually broaden by an amount ν = k π = 1 πτ, until they finally coalesce into one very wide, very weak signal. τ = 1 k is the average lifetime of each state.

Intermediate Exchange: k = 1000 Hz

Symmetric Exchange Between Two Sites Asymmetric Two-Site Exchange Applications Fast-Intermediate Exchange In the fast-intermediate exchange regime, where k > δν/2, the merged signal starts to get sharper again, and is centered around the average position of the two frequencies: ν peak = ν average = ν A + ν B 2 The line broadening contribution ν in this regime ν = π(δν)2 2k, and therefore decreases as k increases.

Intermediate Exchange: k = 20000 Hz

Phase Differences during Slow Exchange 200 180 160 140 120 100 80 60 40 20 0 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

Phase Differences during Intermediate Exchange 600 500 400 300 200 100 0 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

Phase Differences during Fast Exchange 600 500 400 300 200 100 0 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

Asymmetric Exchange In the case of asymmetric exchange there is an energy difference between the A and B states, and the rate constants in both directions (k A and k B ) are no longer equal. The equilibrium mixture will contain more of the lower-energy conformer. If p A and p B = 1 p A are the fractional populations of the two forms, at equilibrium the relation p A k A = p B k B holds.

Symmetric Exchange Between Two Sites Asymmetric Two-Site Exchange Applications Transition from Slow to Fast Exchange (1)

Symmetric Exchange Between Two Sites Asymmetric Two-Site Exchange Applications Transition from Slow to Fast Exchange (2) In slow exchange, two "normal" peaks at frequencies ν A and ν B are observed, with relative intensities that are proportional to the populations of states A and B in the mixture. In the intermediate regime, a broadening of the two lines (with ν A = k A π and ν B = k B π ) is initially observed, followed by a merging into a single broad line, which subsequently starts becoming sharper again (with a term ν = 4πp Ap B (δν) 2 k A +k B ). The combination line is no longer exactly at the average freqeuncy, but is shifted towards the frequency of the more abundnt conformation: ν peak = p A ν A + p B ν B

Symmetric Exchange Between Two Sites Asymmetric Two-Site Exchange Applications Energy Profile The effect of temperature on the reaction rate is expressed by the Arrhenius equation: k(t) = A exp( E act N A k B T ) where N A is Avogadro s number, k B is the Boltzmann constant, and E act is the activation energy of the process.

Symmetric Exchange Between Two Sites Asymmetric Two-Site Exchange Applications Measuring Rate Constants For N,N -dimethylformamide the rate constant could be determined experimentally over a wide range of temperatures. Fitting the Arrhenius equation resulted in values of E act = 90.1kJmol 1 and A = 1.16 10 14 s 1.

Rotation of Tyrosine Side Chains A tyrosine side chain can in priciple rotate freely around the single bond between Cα and Cβ. In the tightly packed hydrophobic core of a protein this motion can however be limited, in which case the signals of symmetrically positioned hydrogen atoms can be distinguished.

Symmetric Exchange Between Two Sites Asymmetric Two-Site Exchange Applications Effect of Exchange on Scalar Coupling Very pure ethanol Ethanol with a catalytic amount of acid

(1) A realistic NMR sample contains vast numbers of individual spins, each in its own quantum superposition state and precessing at the Larmor frequency around the direction of the external field. The total magnetisation of all spins can be represented by a bulk magnetisaion vector, which obeys a few relatively simple rules. At thermal equilibrium the bulk magnetisatiom points towards the positive z axis, and has a magnitude determined by the population difference between the two energy levels of the spins.

(2) An RF pulse with a frequency close to the resonance frequency of the spins can rotate the bulk magnetisation over any desired angle around the x or y axis. The most commonly used flip angles are 90 and 180 degrees. Once displaced from equilibrium, the bulk magnetisation itself precesses in x,y plane at the Larmor frequency. This oscillation of the transverse magnetisation gives rise to an observable signal in the detector coil. The initial amplitude of the signal is proportional to the equilibrium magnetisation, and thus to the population difference between the two energy states.

(3) A number of relaxation mechanism cause the bulk magnetisation to slowly return to its equilibrium value along then z axis. As a result, the observed signl becomes progressively weaker, ans becomes a free induction decay (FID). The Fourier transform of an FID signal is a Lorentzian curve around the resonance frequency, with a line width determined by the rate of relaxation.

(4) When a nucleus can change between two states with distinct Larmor frequencies (due to chemical reactions or conformational changes), the appearance of the spectrum is determined by the rate of the exchange process. In the very slow exchange regime, two separate signals at the two distinct Larmor frequencies are observed. In the very fast exchange regime, a single signal at the average frequency is observed.

(5) Going from very slow to very fast transitions the signal passes through a transition region. At first, the two separate signals, each at its own frequency, become wider and wider, until they flow together and start getting sharper again around their average frequency. In the intermediate regime, an analysis of the spectra can provide the rate constant of the transition process. Outside of this regime, the only conclusion that can be drawn is whether the process is occuring too fast or too slow for analysis by NMR methods.