Lecture #5 – Relaxation
2/16/16
Clubb
Topics Covered
• Relaxation Time Constants (T1 and T2)
• Mechanisms of relaxation (dipole-dipole)
• Relaxation  protein/nucleic acid dynamics
Why do we care about NMR relaxation?
1) Various relaxation rates report on
molecular dynamics
2) Dipolar relaxation is the basis of the NOE:
crucial to structure determination by NMR
Reading: Claridge p 25-38
What do we mean by “relaxation”?
Typical one dimensional (1D) experiment
90o
(FID)
Delay to enable system
to relax back to equilibrium
90o
Return to equilibrium via
relaxation processes
Mz=Mo
equilibrium
Mz=0
Mz=Mo
equilibrium
1
Main Relaxation Parameters Measured in NMR
T1: Longitudinal (also called spin-lattice) relaxation. It is a measure of
rate of return of the magnetization to the Z-axis.
Importance:
1)
Dynamics
2)
Determines delay times between subsequent NMR experiments
90o
90o
(FID)
(FID)
~900ms
~100ms
Total recycle time
~1 s
T2: Transverse (also called spin-spin) relaxation. It is a measure of
The rate of decay of magnetization in the x-y plane.
Importance:
1)
Dynamics
2)
Determines line-width of peaks
3)
Can limit types of expts that can be performed
Linewidth  1/ ( x T2*)
FID
Note: formula describes reappearance
of Mz when it starts from X-Y plate
relaxation
Mz
Mz=Mo
Mz=0
Mz=Mo
2
•During T1 relaxation the spins lose energy to the lattice (surroundings)
as a result of spins “flipping” from the high energy (beta) state to lower
energy (alpha) state
•Spins “flip” during relaxation
process as a result of fluctuating
FID
magnetic fields that generate RF
at the appropriate frequency
T1 relaxation
Mz=Mo
Mz=Mo
Mz = 0
Physical origin of
oscillating magnetic fields
that cause relaxation:
Dipole-dipole (dipolar)
interactions
Each atom produces its own
magnetic field
Tumbling of protein
Tumbling of protein causes magnetic field from proximal
atoms to vary.
3
Other sources of fluctuating magnetic fields that cause relaxation
(not as important as dipolar relaxation in macromolecules)
Chemical shift anisotropy (CSA). Negligible for protons, but important for
nuclei with large chemical shift ranges such as carbon-13.
Quadrupolar (negligible in proteins). Atoms with I > ½ have non-spherical charge. Interaction
of resultant quadrupole moment with electric field gradients can cause relaxation.
Scalar relaxation (negligible in proteins): caused by time dependent fluctuations in
coupling constants
Paramagnetic (negligible for most proteins): Relaxation caused by very large
magnetic moment of unpaired electrons. Similar to nuclear dipole-dipole interaction.
Not significant in most proteins because they do not contain unpaired electrons.
T1 importance: Choose recycle times in multi-scan experiments to be
~5x T1 value of 1H atom
Total Recycle time
FID
t
Scan #1
t
t
#2
#3
#4
Delay between scans
to enable relaxation (t)
Summed FID
•Choose total recycle time to be
~5x T1 value of 1H atom
•1H T1 times in proteins are
typically ~200ms
FT
NMR Spectrum
4
Recall:
Recycle time = 5 * T1
Describes return of magnetization
to Z-axis when it starts from the
XY plane
At recycle time = 5 * T1
e(-t/T1) = 0.0067
Mz ~ M0
When Mz = Mo system
is at equilibrium
Inversion recovery experiment is used to
quantitatively measure T1
180o
90o
Perform a series of 1D experiments using different times for 
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Inversion recovery spectra at
various values of the delay

when Mt = 0
Note: formula describes reappearance
of Mz when it starts from –Z axis
Review:
p

Pulse of duration p
and strength B1
Pulses of different flip angles
dM(t)/dt = M(t) X B1
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Mz = -Mo
at very long  values
Mz = Mo
complete
recovery to Mo occurs
FT
Detector
At shorter values Mz does not
fully recover to Mo. Can give rise to
negative signal
Short
 values
FT
Mz = -Mo
Mz > -Mo
Very long
 values
Detector
FT
7
Recall:
Sign and phase
of signal depends
upon where magnetization
starts at the beginning
of detection.
Detector
T2 relaxation constant (spin-spin or transverse): Time
constant describing rate of decay in the transverse (XY) plane.
Larger T2 values slower
decay of magnetization
8
Rate of dephasing characterized by the
time constant T2 (transverse relaxation time or spin-spin relaxation time)
*
Linewidth at
half-maximal
intensity
 1/ ( x T2*)
Intrinsic time constant for
decay (transverse relaxation time,
also called spin-spin relaxation time)
Signal decay caused
by field inhomogeneity
Spin-Echo experiments to selectively measure intrinsic T2 value of a
nucleus. Experiment eliminates contribution from field inhomogeneity
Dephasing from
field inhomogeneity
Field inhomogeneity effects refocused.
However intrinsic dephasing caused by T2 is not.
Therefore the magnitude of the vector has decreased.
(note in relation to a simple one-pulse experiment the
spectrum would be of opposite sign. This can be
corrected by properly phasing the spectrum
9
Behavior of magnetization during the experiment
In real experiment you begin detection of
signal here. The FT of the resultant FID is a
regular 1D spectrum. However the
amplitude of the peaks are reduced as a
result of T2 relaxation
Behavior of signal
during experiment
Begin
detection
here
n=5
10
FT
Intensity α e –((time)/T2))
n=2
n=4
n=8
time
time = n x (2+180 pulse)
n=16
Relaxation and relationship to motion
( c ) molecular correlation time:
time required for molecule (protein) to rotate
through one radian
a
: viscosity
a: radius
k: Boltzman constant
T: temperature
11
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Overview: Relaxation  Motion
T2 (J()) and T1 (J())
Equations exist that relate
experimentally measured
relaxation parameters to J().
Relaxation Parameters
(measured as T1 and T2)
Spectral density function(J())
Fourier transform of autocorrelation
function. Describes frequencies of motions
that can promote relaxation
Autocorrelation function G()
Describes relationship between
protein dynamics and fluctuating
magnetic fields. Assumes motional
models (e.g. molecule tumbles as sphere
or as an elipsoid)
fluctuating magnetic fields
cause relaxation
Motions within macromolecule
create fluctuating fields
Spectral Density Function (J()): Power available from the lattice (molecular motions) to bring
about relaxation via transition probablities. It is a function of frequency (). All measurable relaxation
properties of a protein (T1, T2, NOE) can be expressed in terms of the spectral density function.
J(I  s) = J (~0)
• J() depends on how fast the macromolecule tumbles
in solution. It therefore depends on the size and shape of the
macromolecule, and the temperature and viscosity of the
solution.
J(I s)
On 500 MHz instrument
J(I s) = J (1GHz)
For spherically shaped
macromolecule
( isotropically tumbling)
J() = c/(1+2c2)
c ~100 MHz
c = 10 ns
13
Example: T1 dependence on J()
Relaxation depends
upon: c, r,  and o
T1 ~
Low frequency
motions
Mid-range
frequency motions
(on a 500MHz spectrometer
= 2 * 500 MHz)
High frequency motions
(on a 500MHz spectrometer
 = 2 * 1,000 MHz)
Representative Spectral Density curves for three molecules
of different sizes (assumes identical larmor frequencies)
Big molecules
Small molecules
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Later…Nitrogen-15 Relaxation time constants
• To gain insights into protein dynamics the T1 and T2 time
constants of backbone nitrogen-15 nuclei are typically measured.
• This nucleus is primarily relaxed by its directly bonded hydrogen-atom
• The T1 and T2 values report upon its
effective correlation time (eff), which
is a combination of the overall
molecular tumbling of the protein
and internal motions it experiences.
H
Bond
vector
eff -1 = c-1 + e-1
N
Protein
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