NOE

1
NOE

• Transferring magnetization through scalar
  coupling is a coherent process. This means
  that all of the spins are doing the same thing at
  the same time.
• Relaxation is an incoherent process, because it
  is caused by random fluxuations that are not
  coordinated.
• The nuclear Overhauser effect (NOE) is in
  incoherent process in which two nuclear spins
  cross-relax. Recall that a single spin can
  relax by T1 (longitudinal or spin-latice) or T2
  (transverse or spin-spin) mechanisms. Nuclear
  spins can also cross-relax through dipole-dipole
  interactions and other mechanisms. This cross
  relaxation causes changes in one spin through
  perturbations of the other spin.
• The NOE is dependent on many factors. The major
  factors are molecular tumbling frequency and
  internuclear distance. The intensity of the NOE
  is proportional to r-6 where r is the distance
  between the 2 spins.

2
Qualitative Description
2 spins I and S

• Two nuclear spins within about 5 Å will interact
  with each other through space. This interaction
  is called cross-relaxation, and it gives rise to
  the nuclear Overhauser effect (NOE).
• Two spins have 4 energy levels, and the
  transitions along the edges correspond to
  transitions of one or the other spin alone. W2
  and W0 are the cross-relaxation pathways, which
  depend on the tumbling of the molecule.

bb n4()
WI(2)
WS(2)
W2
() n2 ab
ba n3()
W0
WI(1)
WS(1)
aa n1()
3
dn1/dt -WS(1)n1-WI(1)n1W2n1
WS(1)n2WI(1)n3W2n4 etc for n2,3,4 using Iz
n1-n3n2-n4 Sz n1-n2n3-n4 2IzSz n1-n3-n2n4
One gets the master equation or Solomon
equation dIz/dt -(WI(1)WI(2)W2W0)Iz
(W2-W0)Sz (WI(1)-WI(2))2IzSz dSz/dt
-(WS(1)WS(2)W2W0)Sz (W2-W0)Iz
(WS(1)-WS(2))2IzSz d2IzSz/dt -(WI(1)WI(2)
WS(1)WS(2))2IzSz - (WS(1)-WS(2))Sz -
(WI(1)-WI(2))Iz (WI(1)WI(2)W2W0) auto
relaxation rate of Iz or rI (WS(1)WS(2)W2W0)
auto relaxation rate of Rz or rR (W2-W0) cross
relaxation rate sIS Terms with 2IzSz can be
neglected in many circumstances unless
(WI/S(1)-WI/S(2)) (D-CSA cross correlated
relaxation etc )
4
Spectral densities J(w)

• W0 ? gI2 gS2 rIS-6 tc / 1 (wI - wS)2tc2
• W2 ? gI2 gS2 rIS-6 tc / 1 (wI wS)2tc2
• WS ? gI2 gS2 rIS-6 tc / 1 wS2tc2
• WI ? gI2 gS2 rIS-6 tc / 1 wI2tc2
• Since the probability of a transition depends on
  the different
• frequencies that the system has (the spectral
  density), the
• W terms are proportional the J(w).
• Also, since we need two magnetic dipoles to have
  dipolar
• coupling, the NOE depends on the strength of
  the two
• dipoles involved. The strength of a dipole is
  proportional to
• rIS-3, and the Ws will depend on rIS-6
• for proteins only W0 is of importance W I,S,2 ltlt
• The relationship is to the inverse sixth power
  of rIS, which
• means that the NOE decays very fast as we pull
  the two
• nuclei away from each other.

5
d(Iz Iz0)/dt - rI (IzIz0) - sIS
(SzSz0) d(Sz Sz0)/dt - sIS (IzIz0) - rS
(SzSz0) Note that in general there is no
simple mono-exponential T1 behaviour !!
6
Steady State NOE Experiment
For a steady state with Sz saturation
Sz0 d(IzSS Iz0)/dt - rI (IzSSIz0) - sIS
(0Sz0) 0 IzSS sIS/rI Sz0 Iz0 for the NOE
enhancement h(IzSS-Iz0)/ Iz0 sIS/rI Sz0/Iz0
7
NOE difference
Ultrahigh quality NOE spectra The upper spectrum
shows the NOE enhancements observed when H 5 is
irradiated. The NOE spectrum has been recorded
using a new technique in which pulsed field
gradients are used the result is a spectrum of
exceptional quality. In the example shown here,
it is possible to detect the enhancement of H10
which comes from a three step transfer via H6 and
H9. One-dimensional NOE experiments using pulsed
field gradients, J. Magn. Reson., 1997, 125, 302.
8
Transient NOE experiment
Solve the Solomon equation With the initial
condition Iz(0)Iz0 Sz(0)-Sz0 For small mixing
times tm the linear approximation
applies d(Iz(t) Iz0)/dt -rI(Iz(t)Iz0) -
sIS(SzSz0) 2 sISSz0 Valid for tmrS and tmsIS
ltlt 1 (i.e. S is still inverted and very little
transfer from S) h(tm) (Iz(tm ) - Iz0)/ Iz0
2sIS tm The NOE enhancement is proportional to
sIS !
9
Longer mixing times
a system of coupled differential equations can be
solved by diagonalization or by numerical
integration Multi-exponential solution the
exponentials are the Eigenvalues of the
relaxation matrix
10
NOESY
The selective S inversion is replaced with a
t1evolution period Sz(0)cosWSt1Sz0,
Iz(0)cosWIt1Iz0 (using the initial rate
appx.) Sz(tm)sIStmIz0 rStmSz0 (a)
cosWIt1sIStmIz0 (b)
cosWSt1rStm-1Sz0 (c)
11
NOE vs. ROE
12
ROESY
90s
90
tm
t1
tm

• wSL ltlt wo, w tc ltlt 1
• The analysis of a 2D ROESY is pretty much the
  same than
• for a 2D NOESY, with the exception that all
  cross-peaks are
• the same sign (and opposite sign to peaks in
  the diagonal).
• Also, integration of volumes is not as
  accurate

13
Approaches to Identifying NOEs

• 15N- or 13C-dispersed (heteronuclear)

14
2D - 3D NOE
3D- NOESY-HSQC
15
4D NH-NH NOE
N1 H1
H2 N2
H1
H2
N1
N2