Lecture 12

NMR Course. Lecture 12
May 10,1999
Maltseva T. V.
Lecture
Practical
multinuclear,
12
introduction
to theory
and
multidimensional
nuclear
implementation
of
magnetic
resonance
experiment.
1""""-
Book: II Protein NMR speectroscopy" by Cavanagh, J., Fairbrother, W. J., Palmer III, A. G.,
Skelton, N. J. Academic Press. Inc., San Diego, N.Y., Boston, London, Sydney, Tokyo, Toronto,
f"
..;
1996
The main reyiews from the books:
* 'Method in Enzymology.V.239, Nuclear Magnetic Resonanc,Part C' by Thomas L.james
and Norman j. Oppenheimer. p.3- 79
Good reyiews:
* Kessler,H.,Gehrke,M.,Griesinger,C.'Two-dimensional NMR spectrosopy:background
and oyerview of the experiments' Angew. Chem. Int. Ed.Engl. (1988) pp.490-536
* Sorensen,O.W., Eich,G.W., Leyitt. M.H., Bodenhausen,G., Ernst,R.R.'Product
operator formalism for the description of NMR pulse experiments' Progress in NMR
spectroscopy, (1983) Y. 16,pp.163-192
I'""'
This book is quite difficult :
* 'Principle of Nuclear Magnetic Resonancein One and T wo Dimensions' by Emst,R
Bodenhausen,G., Wokaun,A. pp 1-69 and pp.3-79
ry
1
Practical
introduction
multidimensional
to theory and implementation
nuclear magnetic resonance
(I) Pulse Sequences.
of multinuclear
experiment.
,
The easiest way to understand complex NMR experiments is to be able to identify the small number
of building blocks from which all experiment are constructed. Once the input and output
characteristics of the building blocks are known, a coherence flow network can be constructed to
illustrate graphically the flow of coherence through an experiment. For this purposes we will
introduce a small number of product operator rules. Although many of these experiments correlate
tour or more different spins, the majority of interactions can be understood in terms of pairwise, ~
sl2ininteraction. We will focus on the NMR experiments for through bond, sequential assignments.
Part
,,-.
(I):
Product
Operators
In the context of an 2-spin systern, we distinguish one-spin and two-spin operators. One-spin
operators are associated with a rnagnetisation.
We willIabeIled t wo spins by I and S. A subscript to I and S refers to the rotating frame axis along
which the rnagnetisation is oriented. Thus Ix, refer to the rnagnetisation of the I spin oriented along
the x axis.
r--
For the two-s12ininteraction there are 15 combination of the magnetization:
Iz, Sz (longitudinal)
Ix, Sx, ly, Sy (transverse)
21zSz (J -ordered spin state )
""'""
I'""""'
,.".
21xSz , 21ySz , 2Sxlz ,2Sylz
(antiphase)
21xSx , 21xSy , 21ySx ,2lySy
(multiple quantum)
(The coefficient of 2 arises as a normallsation factor)
When only a single spin is involved , they call the inphasemagnetisation.
x -and y- magnetisation are narned as transverse
z -aligned along the magnetic field, is called longitudinal .
Only three operators are neededto describemost NMR experiments:
1) radio frequency (rf) pulse
2) chemical shift
3) coupling
Both the rf pulse and chemical shift operators describe simple rotations in the three- dimensional
space.
The notation of product operatorsis:
a
operator
b
where the operator transforrns a into b. In our case a and b are 15 combinations presented above
the operator we will discuss below. You should know that "hats" indicate an operator.
2
The directionof rotationfrom rf pulsesor chemicalshift betterdeterminethrough" right-handrule"
RULE:
The thumb of the right hand points along the positive axis, the direction
of the rotation
is the direction
the fingers will close to make a fist.
(I)
Radio
frequency
pulse
lz
x
(\
y
z
Ix
"
~
If3-~If3COS(l/J)
+
Irsin(l/J)
a,~, y are x,y ,z axis; <I>is the pulse flip angle
A
A
4>1x
4>1y
Iz
~Izcos(4» -Iysin(4»
Iz
lz
~Ix
180° Ix
Iz
~lx
Ix
90° ix
Iz
~-Iz
~-Iy
180° ix
1800Iy
~-lz
+ Ixsin(4»
9001 y
90°j y
t/>lx
Ix
~Izcos(4»
Ix
Ix
~-Ix
(II)
Chemical
~Ix
shift
The chemical shift has the effect of an "rf' pulse along the z axis.
r'
r
Ix
QI tiz
-~IxCOS(QIt)
ly
A
!2Itlz
-~ lyCOS(!2It)-Ixsin(QIt)
+ Iysin(QIt)
!2 is the chemical
Except
of strongly
coopled
state (where
JIS>~cr)
shifts act on combination
of I and S as if the other
Example:
2IzSx~2(lzCOS(l/»
polses and chemical
spins were absent.
-lysin(l/»)Sx = 2IzSxcos(l/» -2IySxsin(l/»
.Q.Itiz
-~2(IxCOS(.Q.It)+Iysin(.Q.It)Sz=
2Ix SzCOS(.Q.I
t) + 2IySzsin(.Q.I
t)
Conclusion
Chemical
shift
and rf polses are operations
dimensional
rotation.
Theyare
not
able
to
starting on spin I to spin S .
2IxSz
3
interpreted
transfer
as
magnetisation
three-
(II) Conpling
The coupling operator is the essentia!feature of multidimensiona! experiments that provide correlation
through coherent transfer of magnetisation from spin I to spin S.
la
~
(a,f3
7rJ1St2izsz
--~
.
laCOS(1rJlst) + 2IpSzsm(1rJlst)
= x,y)
I""""'
Ix
1rJ1St2iz§z
.
--~
Ixcos(1T:Jlst) + 2IySzsm(1T:Jlst)
2IySz
Example:
{"l
1IJJst2iz§z
--~2Iy
Sz
1IJJst2iz§z
z
2IySz
.--~-Ix
Conpling operator can
antiphase
r
.
-Ixsm(1T:Jlst)
if =} COS(1T:J1St)
= O,=} 1T:J1St
= !E.then =} t = ~
2
2J1S
Ix
RULE:
1rJlst2iz§zz
--~2IySzCOS(1T:Jlst)
conrse inphase
coherence to go to antiphase,
or
coherence to go inphase.
Part (2): Building
Blocks
and Coherence
Flow Networks.
There are surprisingly few ways to transfer magnetisation ( coherence)from one spin to another ( for
t wo spin system).
All of the method can be placed into two broad categories:
I) Coherent ( All multinuclear, multidimensional experimentsrely on one or more coherenttransfer
stepsmediated by scalar coupling)
2) Incoherent transfer ( including the dipolar interactions going rise to NOEs and ROEs as weIl as
chemically exchanging nuclei)
For the former one, the main (coherent) building blocks are:
a) HMQC ( heteronuclearmultiple-quantum correlation spectroscopy)
b) INEPT (Intensive nucleus enhancementby polarisation transfer)
c) TOCSY (Total correlated spectroscopy)
d) COSY (Correlated spectroscopy)
We discuss how these building blocks work by using the product operator rules given above.
4
.
An important component of many of the "building blocks" are variations of the "spin-echo
experiment,
tt
so we will describe these first.
(I) "Spin-echo"The
spin echo is simply a clelay followecl by 180° pulse followecl by a seconcl
clelay.We will consiclerin present only the casewith two equal clelays.In part "a" we have Ix (At this
step it is not important how we get this magnetisation ).
(A),(B):
Dec SE
180x
Ix
I
't
-'t
a
I
b
(C): JIS SE
I
t
c
d
180
D
,
-.:
b
c
1-
r'-
s
L---O---,
a
(D):
d
Dec CS
I
't
s
0
180
't
1a
I
b
c
d
Variation of the spin-echo building block. (A) and (B) Building block DEC SE which refocuses I
chemical shift and decouples S. (C) Building block JIS SE which refocuses I chemical shift hut
~
J
allows coupling to occur with S. (D) Building block Dec CS allows chemical on I to occur while
decoupling S.
5
(A)
In first case we assume no scalar
a
r"
b
c
conpling
the spin I spin S.
d
The following
steps show the product operator transfornlations
(Let us go together through all step):
A
QI 'l"lz
.
a-+b: Ix
-~lxCOS(QI'l")+lysm(QI'l")
from points "a" to "d":
b ~
c:
Sin(.aI
C~
d:
~
IxCOS(.aI
nI 'l'lz
~[ IxcoS(nI
'f) + [Iycos(1800)
'l') + lysin(nI
+ Lsin(1800)]
'l') ]cos(nI
= Ix[ cos2(nI 'l') + sin2(nI 'l') ] + ly[ cos(nI 'l')sin(nI
'l') -[ lycos(nI
'l') -cos(nI
"a"."b"
'l')] ~
'I"",,
y Sin(.aI
'f)
'l') =
Ix
z
"c"."d"
--
Iysin(c!I)
'f) -I
'l') ]sin(nI
z
"b"-"c"
--
xY-'-
'l') -Ixsin(nI
'l')sin(nI
If to repeat the same, but using the vector representation:
z
z
'f) =:} IxCOS(.aI
--
-Iysin(~)
y
ar
y
~
Ix
x
~
180
o
.
The net result:
WE END UP WITH
WHAT
WE START
WITH!!!
RULE: Whenever a 180° pulse is placed syrnmetrically between t wo time intervals, no chemical
shift takes place. (ECHO)
(B) In next case, consider the same spin -echo experiment in the presence of scalar
coupling between I and S spin: (We neglect the chemical shift)
The following steps show the product operator transformations from points "a" to "d":
"
a~
b
: Ix
7r.l1s't2iz§z
b ~ c: ~
r--
c ~
d:
.
~IxCOS(7r.l1S't)+2IySzsm(7r.l1s't)
Ixcos(7r.l1s't) + Sz[Iycos(180O) + Izsin(180O) ] x sin(7r.l1S't)=> Ixcos(7r.l1s't) -2IySzsin(1rJ1s't)
1r.Ils'Z"2i.s.
~[ Ix cos( 1r.I1S'Z")
+ 2Iy S. sin( 1r.I1S'Z")
] COS(1r.I1S'Z")
-[ 2Iy S. COS(1r.I1S'Z")
-Ix
sin( 1r.I1S'Z")
] sin( 1r.I1S'Z")
~
=> Ix[ cos2(QI 'Z")+ sin2(QI 'Z")] + [ 2Iy S. -2IyS.]sin(1r.I1S'Z")COS(1r.I1S'Z") => Ix
The
Det
from
S
This
Ix
resDit:
In addition
is a decoupled
Dec SE
to refocusing
chernical
shift,
spin-echo:
~ Ix
a
b
in teml of a CFN Diagram :
6
this
simple
experiment
decouples
spin
I
( C) Let us consider two additional related experiments.
First, we include a 180° pulse on S spin simultaneous to the 180° pulse on I spin
(C):JISSE
~O
IzSz
I
't
IZ~
'--
0180
x
IySz
ly
s
zQy
1-
I
a
b
c
IX~
IX~
d
( presenceof scalar conpling betweenI andS spin)
Note:
That since Ix and Sx do not interfere with unlike spins( Sy~Sy)
a 180° S
and I pulses
can be applied together or in tandem, in either order.
"
Once again, we start with Ix magnetisation and neglect chemical shift which is refocnsed by a 180°1
pulse.
n
a ~
b ~
b
:
1r:.lls'r2IzSz
Ix
c:
180°
180°
c ~
d:
ix
-I.
~ Ixcos(/IJls'r)
~ Ix cos( 1r:.llS'r) + Sz [ Iycos(180°
A
Sx
~ Ixcos(7rJ1s'r)
7rJls't'2jz§z
=} Ixcos(7rJls2't')
The
)and
2sin(
S. cos(180°
+ 2IySzsin(7rJ1S't')
(7rJ1S't') ] + +2IySz[
( 'P ) = cos(2'P
) + Iz sin(180°
-2Iysin(7rJ1S'r)[
~[IxCOS(7rJls't')
=} Ix [ COS2(7rJ1S't') -sin2
Iicos2 ( 'P) -sin2
+ 2IySzsm(1r:.llS'r)
) -sysin(180°
) ]=> Ixcos(7rJ1s'r)
+ Sin(7rJ1S't')COS(7rJ1S't')
Det
Jr
JrJls2't"= -=>
2
resDit:
In ternl
] =
The
result
of
this
building
block
is
't"=
the
same
as the
single
on both S and I in spin echo
+ 2IySzsin(1CJIs2T)
1
Then Ix
1tllSr2iz§z
~2IySz
{=>
Ix
JlsSE
~2IySz
4JIs
1rJJs'r2iz§z
2IySz
~
+ 2IySzsin(7rJ1S2't')
IxCOS(1CJIs2T)
or
]Sin(7rJ1S't')
'P) = sin(2'P)11
operation, 7r.lIs(2r)2Iz §z
Unlike the previous example, simultaneous by a 180° pulses
experiment leave
the t wo spin coDpled. Tum back to the final resu1t:
r-
Sz sin( 1r:.llS'r)
+ 2IyS.sin(7rJ1S't")
]COS(7rJ1S't') + +[ 2IySz COS(7rJ1S't') -Ixsin(7rJ1S't')
Sin(7rJ1S't')COS(7rJ1S't')
'P)cos(
) ] x sin( 1r:.llS'r) => Ix cos( 1r:.llS'r) -2Iy
-Ix
~
JIsSE
~
2IySz
~ -Ix
of a CFN .
I
a
d
CFN
s
represents changes in state both from in-phase
'7
to antiphase
and antiphase
to in-phase.
~
(D)
Finally, consider the experiment shown in Fig.
(D): Deccs
I
0
't
180
s
't
1-
I
a
b
C
d
As ahove, we start the analysis with
chemical
a-tb: Ix
Ix magnetization
at point "a", hut this time we cannot
neglect
shift! ! !
QI 't"lz
.
IxCOS(QI 't") + IySm(QI 't")
-~
7rJls't"2Iz§z ~[IxCOS(QI 't") + Iysin(QI 't")] cos(7rJlS't") + [2IySzCOS(QI't") -2IxSzsin(QI
't")] sin( 7rJ1S't")
~
b ~
C:~[
[ S.cos(180°
r
C~
-
IxCOS(.0.1 'r) + Iysin(.0.1
) -Sysin(180°
A A
1I:JIs'r2I.S.
d:
)J ~
'r) JCOS(1I:JIS'r) + [ 2IyCOS(.0.1 'r) -2lxsin(.0.1
[ IxCOS(.0.1 'r) + Iysin(.0.1
'r) JCOS(1I:JIS'r) -[
21yS. COS(.0.1 'r) -2IxS.sin(.0.1
~[ Ix COS(.0.1 'r) + ly sin(.0.1 'r) J cos2 ( 1I:JIS'r) + [ 21y S. COS(.0.1 'r) -21x
{[ 21yS. COS(.0.1'r) -21~
'r) ]sin(1I:JIs'r)
S. sin( .0.1'r) ]sin(
'r) ]sin(1I:JIs'r)
1I:JIS'r) cos( 1I:JIS'r) -
S. sin(.0.1 'r) ] sin( 1I:JIS'r)COS(1I:JIS'r) --
=
-[ Ix COS(.0.1'r) + ly sm(.0.1 'r) Jsin2 ( 1I:JIS'r)
-[ Ix COS(.0.1'r) + ly sin(.0.1 'r) J[ cos2( 1I:JIS'r) + sin2 ( 1I:JIS'r)] =
~
IxCOS(QI
'f)
+ Iysin(QI
'f)
A
~
Ixcos2(.QI'f) + IyCOS(.QI
'f) sin (.QI 'f) + IyCOS(.QI
'f)sin(.QI 'f)-Iysin2(.QI'f) =.J
=> Ix [ cos2(.QI'f) -sin2 (.QI'f) ] + 21y[sin(.QI'f) COS(.QI
'f) ] = => Ix COS(.QI
2 'f) + lysin(.QI2 'f)
The
Det result:
The experiment decouples I from S while
letting I evolve under chemical shift for the
period 2't.
This experiments are referred as a decoupled chemical shift (DecCS) and can be represented as:
Ix
DecCS ~
Ix COS(.Q.I 2 -r) + ly sin(.Q.I
2 -r)
~
In term of a CFN
l'"'
evvvvvvvvvvvvvve
a
d
RULES
l. Only products having a single transverse component give rise to detected NMR signal (e.g., Ix, Sy,
2IxSz, and 2IzSy).
2. Products having more than one transversecomponent correspond to multiple quantum coherence,and
are not detected (e.g., 2IxSx, 2IySx, or for a three-spin system, .4IxSxXx, 4IxSzXx).
3. Products having one transverse term and one or several z terms represent antiphase signal. In other
words, the I doublet appears in the spectrum with one line up and the other line down, having an
overall integral of zero.
8