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Pulse/Fourier Transform NMR Chemical Exchange Summary

Biophysical Chemistry: NMR Spectroscopy

Spin Dynamics & Chemical Exchange Lieven Buts

Vrije Universiteit Brussel

25th November 2011

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary

Outline

1

Pulse/Fourier Transform NMR Thermal Equilibrium Effect of RF Pulses The Fourier Transform

2

Chemical Exchange Symmetric Exchange Between Two Sites Asymmetric Two-Site Exchange Applications

3

Summary

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary 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

Chemical Exchange Symmetric Exchange Between Two Sites Asymmetric Two-Site Exchange Applications

3

Summary

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Quantum Description of a Spin-1/2

Pulse/Fourier Transform NMR Chemical Exchange Summary 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

• f the spin remains constant throughout this motion. De

frequency of the precession is the Larmor frequency ν = γ(1−σ)B0

2π

• r ω = γ(1 − σ)B0.

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary 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

• ther 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

• r the motions of the molecule which it is part of.

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary Thermal Equilibrium Effect of RF Pulses The Fourier Transform

Populations (1)

The ratio between the populations of the two energy levels (nα and nβ) is determined by the energy difference ∆E and the temperature T: nβ nα = e− ∆E

kT

from which we find that nα − nβ nα + nβ = ∆E 2kT The Boltzmann constant (k = kB = 1.38066 × 10−23 J

K) functions

a conversion factor from temperature to thermal energy.

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary 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 × 10−21J . The energy difference between the two stationary states of a spin I = 1/2 is very small, even for 1H (which has the largest gyromagnetic ratio of all practically available nuclei) in a strong external field: γ = 26.73 × 107T−1s−1; B0 = 9.4T; ∆E = γB0 = 2.65 × 10−25J This implies that the difference between the two populations is very small: nα − nβ nα + nβ = ∆E 2kT = 3.2 × 10−5 In other words, about one low-energy spin out of every 105 has no counterpart in the high-energy orientation.

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary 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 |cα|2) and a complementary fraction "β character" (proportional to |cβ|2 = 1 − |cα|2) to the ensemble (the collection of all spins). The populations nα and nβ of the two energy levels are the avarage values |cα|2 and |cβ|2 over all spins in the ensemble.

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary 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.

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary 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.

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary 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 nα: nα − nβ nα + nβ = ∆neq 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 ∆neq: M0 = 1 2γ∆neq

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary 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.

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary 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

Chemical Exchange Symmetric Exchange Between Two Sites Asymmetric Two-Site Exchange Applications

3

Summary

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Design of a Modern NMR Spectrometer (1)

Design of a Modern NMR Spectrometer (2)

Pulse/Fourier Transform NMR Chemical Exchange Summary 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.)

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

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 B1 rotates in the x,y plane, with a frequency ωRF and a phase φRF determined by the operator: B1,x = B1 cos(ωRFt + φRF) B1,y = B1 sin(ωRFt + φRF)

Pulse/Fourier Transform NMR Chemical Exchange Summary 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 (ωRF):

• e′

x = cos(Φ(t))

ex + sin(Φ(t)) ey

• e′

y = cos(Φ(t))

ey − sin(Φ(t)) ex

• e′

z =

ez Φ(t) = ωRFt + φRF

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary 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 ω0 of the spins has to be replaced by the offset frequency Ω0: Ω0 = ω0 − ωRF The pulse frequency ωRF is generally chosen to lie in the middle

• f the natural frequency range of the spins in the sample.

Therefore, offset frequencies can be both positive and negative.

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

The Effect of Resonance

The offset frequency in the rotating frame corresponds to Larmor precession around a reduced magnetic field ∆B: ∆B = Ω γ = ω0 − ωRF γ When ωRF = ω0, ∆B = 0 and the effective magnetic field Beff is completety determined by B1 along the x axis. This is a geometric representation of the resonance principle.

Pulse/Fourier Transform NMR Chemical Exchange Summary 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 βp around the x axis is determined by the intensity of the pulse (B1) and by its duration (tp): βp ∼ γB1tp Since all individual spins are coherently rotated by this same angle, the bulk magnetisation also rotates by this angle.

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary 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.

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary 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 tp corresponds to a flip angle βp

• f 180 degrees. Once this value is known, the required duration

for any desired flip angle can be easily calculated.

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary 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

• bservable current in the detector coil.
• 10
• 5

5 10 2 4 6 8 10 12 14 Mx t

• 10
• 5

5 10 2 4 6 8 10 12 14 My t

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary 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

Chemical Exchange Symmetric Exchange Between Two Sites Asymmetric Two-Site Exchange Applications

3

Summary

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary 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:

• 10
• 5

5 10 2 4 6 8 10 12 14 Mx t

+

• 10
• 5

5 10 2 4 6 8 10 12 14 Mx t

=

• 15
• 10
• 5

5 10 15 2 4 6 8 10 12 14 Mx t

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Fourier Analysis

Pulse/Fourier Transform NMR Chemical Exchange Summary 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
• 5

5 10 2 4 6 8 10 12 14 Mx t

• 10
• 5

5 10 2 4 6 8 10 12 14 My t

Mx = M0 sin(Ω0t) exp(− t T2 ) My = −M0 cos(Ω0t) exp(− t T2 ) in which T2 is a characteristic time constant.

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

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 S(Ω) = A λ λ2 + (Ω − Ω0)2 where A is the amplitude of the signal is, and λ = 1

T2 .

Pulse/Fourier Transform NMR Chemical Exchange Summary 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

Chemical Exchange Symmetric Exchange Between Two Sites Asymmetric Two-Site Exchange Applications

3

Summary

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary 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

• f 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.

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary 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 νA and νB corresponding to the A and B states.

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary Symmetric Exchange Between Two Sites Asymmetric Two-Site Exchange Applications

Transition from Slow to Fast Exchange

−20 20 x k (s−1)

νA+νB 2

νA νB

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Intermediate Exchange: k = 100 Hz

Pulse/Fourier Transform NMR Chemical Exchange Summary 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 = νA − νB, is

satisfied, the two signal remain centered around νA and νB. 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.

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Intermediate Exchange: k = 1000 Hz

Pulse/Fourier Transform NMR Chemical Exchange Summary 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: νpeak = νaverage = νA + νB 2 The line broadening contribution ∆ν in this regime ∆ν = π(δν)2

2k

, and therefore decreases as k increases.

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Intermediate Exchange: k = 20000 Hz

Phase Differences during Slow Exchange

20 40 60 80 100 120 140 160 180 200 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

100 200 300 400 500 600 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

100 200 300 400 500 600 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

Pulse/Fourier Transform NMR Chemical Exchange Summary 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

Chemical Exchange Symmetric Exchange Between Two Sites Asymmetric Two-Site Exchange Applications

3

Summary

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

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 (kA and kB) are no longer equal. The equilibrium mixture will contain more of the lower-energy conformer. If pA and pB = 1 − pA are the fractional populations of the two forms, at equilibrium the relation pAkA = pBkB holds.

Pulse/Fourier Transform NMR Chemical Exchange Summary Symmetric Exchange Between Two Sites Asymmetric Two-Site Exchange Applications

Transition from Slow to Fast Exchange (1)

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary 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 νA and νB 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 ∆νA = kA

π and ∆νB = kB π ) is initially observed, followed by a

merging into a single broad line, which subsequently starts becoming sharper again (with a term ∆ν = 4πpApB(δν)2

kA+kB

). The combination line is no longer exactly at the average freqeuncy, but is shifted towards the frequency of the more abundnt conformation: νpeak = pAνA + pBνB

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary 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

Chemical Exchange Symmetric Exchange Between Two Sites Asymmetric Two-Site Exchange Applications

3

Summary

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary 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(− Eact NAkBT ) where NA is Avogadro’s number, kB is the Boltzmann constant, and Eact is the activation energy of the process.

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary Symmetric Exchange Between Two Sites Asymmetric Two-Site Exchange Applications

Measuring Rate Constants

For N,N’-dimethylformamide the rate constant could be determined experimentally

• ver a wide range of
• temperatures. Fitting the

Arrhenius equation resulted in values of Eact = 90.1kJmol−1 and A = 1.16 × 1014s−1.

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

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.

Pulse/Fourier Transform NMR Chemical Exchange Summary 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

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary

Summary (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.

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary

Summary (2)

An RF pulse with a frequency close to the resonance frequency of the spins can rotate the bulk magnetisation

• ver 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

• scillation of the transverse magnetisation gives rise to an
• bservable 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.

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary

Summary (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.

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary

Summary (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.

Lieven Buts Biophysical Chemistry: NMR Spectroscopy

Pulse/Fourier Transform NMR Chemical Exchange Summary

Summary (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

• f 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.

Lieven Buts Biophysical Chemistry: NMR Spectroscopy