Rotating frame (angular velocity 𝝎𝝎𝒓𝒓𝒓𝒓):

𝑑𝑑𝝁𝝁(𝑡𝑡)𝑑𝑑𝑡𝑡

= 𝝎𝝎−𝝎𝝎𝒓𝒓𝒓𝒓 + 𝝎𝝎𝟏𝟏 × 𝝁𝝁 𝑡𝑡 = −𝛾𝛾 𝑩𝑩 +𝝎𝝎𝒓𝒓𝒓𝒓

𝛾𝛾 + 𝑩𝑩𝟏𝟏 × 𝝁𝝁 𝑡𝑡

𝜴𝜴 (offset) ∆𝑩𝑩 (residual vertical magnetic field in the rotating frame)

x

y

z 𝑩𝑩𝟎𝟎

𝑩𝑩𝟏𝟏

𝜔𝜔1

𝛼𝛼 𝑴𝑴

x

y

z 𝑩𝑩𝟎𝟎

𝜴𝜴

𝑴𝑴

1 2 3 4

Basic NMR experiment

Relaxation delay

90° pulse Free induction decay (FID)

Repeat 𝑛𝑛 times (to improve signal to noise ratio)

Free induction decay (FID) - Magnetization precesses in the x-y plane, this induces electrical current in a coil and is detected

90° pulse - Tips magnetization into the x-y plane

Relaxation delay - Allows magnetization to return to equilibrium along z

x

y

z 𝑩𝑩𝟎𝟎

𝑩𝑩𝟏𝟏

𝜔𝜔1

𝛼𝛼

Turn on 𝑩𝑩𝟏𝟏 for time 𝑡𝑡 such that 𝛼𝛼 = 𝜔𝜔1𝑡𝑡 = | − 𝛾𝛾𝐵𝐵1|𝑡𝑡 = 𝜋𝜋

2

𝑴𝑴

Rotating frame (angular velocity 𝝎𝝎𝒓𝒓𝒓𝒓)

90° pulse

x

y

z 𝑩𝑩𝟎𝟎

𝜴𝜴

Magnetization precesses in the x-y plane with frequency 𝜴𝜴 (and relaxes back to equilibrium)

𝑴𝑴

Free induction decay (FID)

1 2 3 4

Description and analysis of rotations (oscillations): Rotation with angular velocity 𝜴𝜴

x

y

𝑴𝑴

𝜴𝜴

x

y

𝑴𝑴 𝜴𝜴

𝑀𝑀𝑥𝑥

𝑀𝑀𝑦𝑦

time 𝑡𝑡

𝑀𝑀𝑥𝑥 = 𝑀𝑀0 𝑀𝑀𝑦𝑦 = 0

𝑀𝑀𝑥𝑥 = 𝑀𝑀0 cosΩ𝑡𝑡 𝑀𝑀𝑦𝑦 = 𝑀𝑀0 sinΩ𝑡𝑡

𝛼𝛼 = Ω𝑡𝑡

x

y

𝑴𝑴 𝜴𝜴

𝑀𝑀𝑥𝑥

𝑀𝑀𝑦𝑦

𝑀𝑀𝑥𝑥 = 𝑀𝑀0 cosΩ𝑡𝑡 𝑀𝑀𝑦𝑦 = 𝑀𝑀0 sinΩ𝑡𝑡

x

y

𝑴𝑴

−𝜴𝜴

𝑀𝑀𝑥𝑥

𝑀𝑀𝑦𝑦

𝑀𝑀𝑥𝑥 = 𝑀𝑀0 cos(−Ω𝑡𝑡) =𝑀𝑀0 cosΩ𝑡𝑡 𝑀𝑀𝑦𝑦 = 𝑀𝑀0 sin(−Ω𝑡𝑡) = −𝑀𝑀0 sinΩ𝑡𝑡

Need both 𝑀𝑀𝑥𝑥 and 𝑀𝑀𝑦𝑦 components to distinguish the direction (sign) of rotation

𝑀𝑀𝑥𝑥 and 𝑀𝑀𝑦𝑦 components formally treated like real and imaginary components of a complex number:

x

y

𝑴𝑴 𝜴𝜴

𝑀𝑀𝑥𝑥

𝑀𝑀𝑦𝑦

𝛼𝛼 = Ω𝑡𝑡

𝑀𝑀 = 𝑀𝑀𝑥𝑥 + 𝑖𝑖𝑀𝑀𝑦𝑦 = 𝑀𝑀0 cosΩ𝑡𝑡 + 𝑖𝑖𝑀𝑀0 sinΩ𝑡𝑡

𝑀𝑀 = 𝑀𝑀0 (cosΩ𝑡𝑡 + 𝑖𝑖 sinΩ𝑡𝑡) =𝑀𝑀0𝑒𝑒𝑖𝑖Ω𝑡𝑡

Return to equilibrium: relaxation • Longitudinal (spin-lattice) relaxation

𝑀𝑀𝑧𝑧

𝑀𝑀𝑧𝑧 = 𝑀𝑀0(1 − 𝑒𝑒−𝑅𝑅1𝑡𝑡) = 𝑀𝑀0(1 − 𝑒𝑒−𝑡𝑡𝑇𝑇1)

𝑀𝑀0

0 𝑡𝑡

Return to equilibrium: relaxation • Transverse (spin-spin) relaxation:

𝑀𝑀𝑥𝑥𝑦𝑦 = 𝑀𝑀0𝑒𝑒−𝑅𝑅2𝑡𝑡𝑒𝑒𝑖𝑖Ω𝑡𝑡 = 𝑀𝑀0𝑒𝑒− 𝑡𝑡𝑇𝑇2 𝑒𝑒𝑖𝑖Ω𝑡𝑡

𝑀𝑀𝑥𝑥

https://en.wikipedia.org/wiki/Relaxation_(NMR)

𝑡𝑡 0

Fourier transform S ω = ∫ 𝑠𝑠(𝑡𝑡)𝑒𝑒−𝑖𝑖𝑖𝑖𝑡𝑡∞−∞ 𝑑𝑑𝑡𝑡

Oscillation 𝑠𝑠 𝑡𝑡 = 𝑀𝑀0𝑒𝑒−𝑅𝑅2𝑡𝑡𝑒𝑒𝑖𝑖Ω0𝑡𝑡

𝑆𝑆 𝜔𝜔 = � 𝑀𝑀0𝑒𝑒−𝑅𝑅2𝑡𝑡𝑒𝑒𝑖𝑖Ω0𝑡𝑡 𝑒𝑒−𝑖𝑖𝑖𝑖𝑡𝑡∞

0

𝑑𝑑𝑡𝑡

Converting oscillating signal to a frequency spectrum

𝑖𝑖(Ω0−𝜔𝜔 − 𝑅𝑅2]𝑡𝑡 = y 𝑖𝑖(Ω0−𝜔𝜔 − 𝑅𝑅2]𝑑𝑑𝑡𝑡 = dy

= 𝑀𝑀0𝑖𝑖(Ω0−𝑖𝑖)−𝑅𝑅2

�𝑒𝑒𝑦𝑦𝑑𝑑𝑑𝑑 = 𝑀𝑀0𝑖𝑖(Ω0−𝑖𝑖)−𝑅𝑅2

[𝑒𝑒𝑖𝑖 Ω0−𝑖𝑖 𝑡𝑡𝑒𝑒−𝑅𝑅2𝑡𝑡]𝑡𝑡=0

= � 𝑀𝑀0𝑒𝑒[𝑖𝑖 Ω0−𝑖𝑖 −𝑅𝑅2]𝑡𝑡

∞

0

𝑑𝑑𝑡𝑡 = �𝑀𝑀0𝑒𝑒𝑦𝑦 𝑑𝑑𝑦𝑦𝑖𝑖 Ω0−𝑖𝑖 −𝑅𝑅2

∞ �𝑒𝑒𝑦𝑦𝑑𝑑𝑑𝑑 = 𝑒𝑒𝑦𝑦 + 𝐶𝐶

(0-1)

𝑆𝑆 𝜔𝜔 = −𝑀𝑀0𝑖𝑖 Ω0−𝑖𝑖 −𝑅𝑅2

= −𝑀𝑀0𝑖𝑖 Ω0−𝑖𝑖 −𝑅𝑅2

−𝑖𝑖 Ω0−𝑖𝑖 −𝑅𝑅2−𝑖𝑖 Ω0−𝑖𝑖 −𝑅𝑅2

𝐵𝐵 + 𝐴𝐴 −𝐵𝐵 + 𝐴𝐴 = −𝐵𝐵2 + 𝐴𝐴2

= 𝑀𝑀0𝑖𝑖 Ω0−𝑖𝑖 +𝑅𝑅2Ω0−𝑖𝑖 2+𝑅𝑅22

= 𝑀𝑀0[ 𝑅𝑅2Ω0−𝑖𝑖 2+𝑅𝑅22

+ 𝑖𝑖 Ω0−𝑖𝑖Ω0−𝑖𝑖 2+𝑅𝑅22

]

𝐴𝐴 𝜔𝜔 Absorption

𝐷𝐷(𝜔𝜔) Dispersion

Lorentzian line shape

𝜔𝜔 Ω0

∆𝜔𝜔 = 2𝑅𝑅2 (∆𝜈𝜈 = 𝑅𝑅2

𝜋𝜋 )

Full width at half height

2-dimensional (2D) NMR spectroscopy

Correlation between nuclear dipoles • Through-bond interactions (J-coupling) < 4 bonds

apart • Through-space interactions (nuclear Overhauser

effect, NOE) < 5 Å apart

Ω𝐴𝐴

Ω𝐵𝐵

Kumar A, Ernst RR, Wüthrich K. A two-dimensional nuclear Overhauser enhancement (2D NOE) experiment for the elucidation of complete proton-proton cross-relaxation networks in biological macromolecules. Biochem Biophys Res Commun. 1980 Jul 16;95(1):1-6.

The first 2D spectrum of a protein (BPTI)

2D NMR spectroscopy

Relaxation delay

90°y FID

𝑡𝑡1 𝑡𝑡2 𝑡𝑡𝑚𝑚𝑖𝑖𝑥𝑥

90°y 90°-y

x

y

z

𝑩𝑩𝟎𝟎

F B C D E A G

A

B

A: 𝑀𝑀𝑧𝑧 = 𝑀𝑀0 B: 𝑀𝑀𝑥𝑥 = 𝑀𝑀0

2D NMR spectroscopy

Relaxation delay

90°y FID

𝑡𝑡1 𝑡𝑡2 𝑡𝑡𝑚𝑚𝑖𝑖𝑥𝑥

90°y 90°-y

x

y

z

𝜴𝜴

F B C D E A G

B C

C: 𝑀𝑀𝑥𝑥 = 𝑀𝑀0 cosΩ𝑡𝑡1 𝑀𝑀𝑦𝑦 = 𝑀𝑀0 sinΩ𝑡𝑡1

x

𝜴𝜴

𝑀𝑀𝑥𝑥

𝑀𝑀𝑦𝑦

𝛼𝛼 = Ω𝑡𝑡1

C 𝑩𝑩𝟎𝟎 y

2D NMR spectroscopy

Relaxation delay

90°y FID

𝑡𝑡1 𝑡𝑡2 𝑡𝑡𝑚𝑚𝑖𝑖𝑥𝑥

90°y 90°-y

x

y

z

F B C D E A G

D

C

D: 𝑀𝑀𝑧𝑧 = −𝑀𝑀0 cosΩ𝑡𝑡1 𝑀𝑀𝑦𝑦 = 𝑀𝑀0 sinΩ𝑡𝑡1 (ignore 𝑀𝑀𝑦𝑦, it will get cancelled by phase cycling and/or gradients

𝑀𝑀𝑥𝑥 𝑀𝑀𝑦𝑦

𝑩𝑩𝟎𝟎

E: 𝑀𝑀𝑧𝑧 = −𝑀𝑀0 a cosΩ𝑡𝑡1 Relaxation during 𝑡𝑡𝑚𝑚𝑖𝑖𝑥𝑥

2D NMR spectroscopy

Relaxation delay

90°y FID

𝑡𝑡1 𝑡𝑡2 𝑡𝑡𝑚𝑚𝑖𝑖𝑥𝑥

90°y 90°-y

x

y

z

F B C D E A G

E

F

𝑩𝑩𝟎𝟎 E: 𝑀𝑀𝑧𝑧 = −𝑀𝑀0 a cosΩ𝑡𝑡1

F: 𝑀𝑀𝑥𝑥 = 𝑀𝑀0 a cosΩ𝑡𝑡1

G: 𝑀𝑀𝑥𝑥 = 𝑀𝑀0 a cosΩ𝑡𝑡1𝑒𝑒−𝑅𝑅2𝑡𝑡2 cosΩ𝑡𝑡2 𝑀𝑀𝑦𝑦 = 𝑀𝑀0 a cosΩ𝑡𝑡1𝑒𝑒−𝑅𝑅2𝑡𝑡2 sinΩ𝑡𝑡2

𝑀𝑀𝑥𝑥𝑦𝑦 = 𝑀𝑀0 a cosΩ𝑡𝑡1𝑒𝑒−𝑅𝑅2𝑡𝑡2𝑒𝑒𝑖𝑖Ω𝑡𝑡2

𝑀𝑀𝑥𝑥𝑦𝑦 = 𝑀𝑀0 a cosΩ𝑡𝑡1 𝑒𝑒−𝑅𝑅2𝑡𝑡2𝑒𝑒𝑖𝑖Ω𝑡𝑡2 Amplitude modulation

• Acquire data with increasing 𝑡𝑡1 • Fourier transform each FID (in 𝑡𝑡2) • Fourier transform the spectra in 𝑡𝑡1

Rule, G.S. and Hitchens, T.K. Fundamentals of Protein NMR Spectroscopy, 2006, Springer

250

250

2-dimensional (2D) NMR spectroscopy

Relaxation delay

90°y FID

𝑡𝑡1 𝑡𝑡2 𝑡𝑡𝑚𝑚𝑖𝑖𝑥𝑥

90°y 90°-y

F B C D E A G

𝑀𝑀𝑧𝑧 = −𝑀𝑀𝐴𝐴 cosΩ𝐴𝐴𝑡𝑡1 Nucleus A

D: 𝑀𝑀𝑧𝑧 = −𝑀𝑀𝐵𝐵 cosΩ𝐵𝐵𝑡𝑡1 Nucleus B

−𝑀𝑀𝐴𝐴 𝑎𝑎𝐴𝐴𝐴𝐴cosΩ𝐴𝐴𝑡𝑡1 E: −𝑀𝑀𝐵𝐵 𝑎𝑎𝐵𝐵𝐵𝐵cosΩ𝐵𝐵𝑡𝑡1

−𝑀𝑀𝐵𝐵 𝑎𝑎𝐵𝐵𝐴𝐴cosΩ𝐵𝐵𝑡𝑡1 −𝑀𝑀𝐴𝐴 𝑎𝑎𝐴𝐴𝐵𝐵cosΩ𝐴𝐴𝑡𝑡1

G: ( )𝑒𝑒−𝑅𝑅2𝑡𝑡2𝑒𝑒𝑖𝑖Ω𝐴𝐴𝑡𝑡2 ( )𝑒𝑒−𝑅𝑅2𝑡𝑡2𝑒𝑒𝑖𝑖Ω𝐵𝐵𝑡𝑡2

Ω𝐴𝐴

Ω𝐵𝐵

Ω𝐵𝐵

Ω𝐴𝐴

Rankin, Naomi & Preiss, David & Welsh, Paul & Burgess, Karl & Nelson, Scott & Lawlor, Debbie & Sattar, Naveed. (2014). The emergence of proton nuclear magnetic resonance metabolomics in the cardiovascular arena as viewed from a clinical perspective. Atherosclerosis. 237. 287–300.

Instrumentation

600 MHz Agilent VNMRS spectrometer: • four channels • z-axis gradients • HCN triple resonance probe 500 MHz Agilent VNMRS spectrometer: • three channels • three-axis gradients • HCN triple resonance probe • direct and indirect broadband

probes • accessories for MAS

University of Montana