Background on NMR

1
Background on NMR exchange spectroscopy
S. Macura, W. M. Westler, and J. L. Markley,
Two-Dimensional Exchange Spectroscopy of
Proteins, Methods Enzymol. 239 106-144 (1994).
2
Consider two spins I and S with equilibrium
magnetization I? and S? and longitudinal
magnetization Iz and Sz dIz / dt - ?I (Iz
- I?) - ? (Sz - S?) dSz / dt - ?S (Sz -
S?) - ? (Iz - I?) ? represents longitudinal
relaxation (leakage to the lattice) ? represents
cross relaxation (heat transfer from
one spin to another)
3
The relaxation matrix R, which contains diagonal
elements Rii (auto-relaxation) and off-diagonal
elements Rij (cross-relaxation), describes the
time course of the magnetization vector m, where
m has elements m1, m2, , mn for the n spins in
the system. dm / dt - R m multispin Solomon
equation
4
A NOESY experiment with a mixing time ?m yields a
spectrum in which the integrated peak areas can
be thought of as the elements of the matirx a a
(?m ) a (0) exp (-R ?m ) where R is the
relaxation matrix. S. Macura R. R. Ernst
(1980) Molec. Phys. 41, 95-117.
5
In order to extract information about distances
from cross-relaxation data, one must take into
consideration the relationship between cross
relaxation rates given by the relaxation matrix
and the transition probabilities for simultaneous
parallel (double-quantum, W2ij) and anti parallel
(zero-quantum , W0ij) spin flips Rij W2ij -
W0ij
6
The transitions are stimulated by motions of the
vectors between cross-relaxing spins
characterized by the correlation time ?cij For
macromolecules, Rij - q ?cij where q 5.69
? 104 rij -6 s-2 with rij in nm
7
For short mixing times this expands to aij(?m)
/ aii(0) Rij (?m) 1/2 ? Rik Rkj (?m)2
1/6 ? ? Rik Rkj
(?m)2 . . . . . where aij(?m) is the peak
volume at mixing time , and summations are over
all spins in the relaxation network. In the
isolated spin pair approximation, this reduces to
rij rfix (Rfix / Rij)1/6 rfix (afix /
aij)1/6 This allows distances to be estimated
from NOESY peak intensities.
8
Assumed geometry
2.5 Å
5.5 Å
2.5 Å
3.5 Å
9
(assumed correlation time ?c 4.8 ns)
0
0.08
Cross peak intensity
0 ms
350 ms
Mixing time ?m