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May 12, 1998
NMR Course. Lecture 1415
Maltseva T. V.
Lecture
Part
~
1415
Part l:The
NOESY
and ROESY
2:The BackCalculation
of NOE
Density
Matrix
experiments
spectra using
the
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,
1996
The main reviews from the books:
*
'Method in Enzymology .V .176, Nuclear Magnetic Resonanc,Part A Spectral Thechniques
and Dynamics' by Norman J. Oppenheimer and Thomas L. Jamespp.169183
Good review:
*
Macura,S., Emst,R.R..'Elucidation of cross relaxation in liquids by twodimensional
NMR spectroscopy.' Molecular Physics ( 1980) pp.95117
The extra, hut quite difficult
without the knowledge of matrix analysis) to understanding the famous
book:
"'
*
'Principle of Nuclear Magnetic Resonancein One and T wo Dimensions' by Ernst,R.,
Bodenhausen,G., Wokaun,A.Ch.9 pp.490538
1
The
and
of NOESY
ROESY
(l)
Laboratoryframe NOE spectroscopy (NOESY) is the 2D NMR experiment used to study the NOE in
the laboratory frame, and ROESY is the 2D NMR experiment used to study the NOE in the rotating
frame.
2D exchange spectroscopy
2DFT
~
r90
90
90
S(tl.
~r"'
.I
'tm
,~: :~«1)I,fo'> (~.~
(
tv
90
S(tl, tv
.'
J
roJ
\.
tl
2D Fr
s( (J)\. 0>2)
,
~
i
/ (ö)i. roi) (rol,ro
t2
,
t,
002
(a)
(b)
(c)
Fig. l. Basic pulse sequences for 2D exchange spectroscopy: (a) in the laboratory frame.
(b) 2D exchange spectrum, and (c) in the rotating frame.
~
(2)
For the analysis of crosspeak evolution during the mixing time, it is convenient to represent the 2D
NOESY(ROESY)
spectrurn as rnatrix of peak volurnes which depends on the mixing time 'tm. For
'tm.=O, no rnagnetisation exchange takes place, and consequently, M(O) represents a diagonal rnatrix
with elernents proportional to equilibriurn populations of the individual spin sites. The volurne rnatrix
of the NOESY (ROESY) spectrurn recorded with a,fixed arbitrary mixing time M('tm) depends on the
equilibriurn populations M(O), on the mixing time 'tm, and on the dynamic rnatrix L:
M( "l'm)= a( "l'm)M(O)= eLrmM(O)
wherea is thematrixof socalledmixingcoefficients (1)
whichareproportional
tothemeasured
2DNOEintensities.
Incoherent magnetization transfer is driven by processesof random molecular motion. Among N
r
sites, the transfer can be describedby the system ofN linear, coupled differential equantions:
~
= Lm( "l'm) with formalsolution
m( "l'm)= exp(L "l'm)m(O)
d"l'm
The vector m has elements nixiroi,
"
(2)
where fl is the number of equivalent spins, X is the mole
fraction, and m is the deviation from thermal equilibrium of magnetisation at site I.
Magnetization
of spin
1/2 nuclei,
with
polarization
states
a., ~ can migrate
between
the sites
..
I and J
by t wo different mechanismz:b~ chemical exchangeor b~ crossrelaxation.
In the chemical
exchange
mechanism, the magnetization migrates together with the
spin, which changesits site i ~ j hut
i(a) ~ j(a)
not
its
polarization:
j(f3) ~ i({3)
i(a)j({3) ~ i({3)j(a)
In crossrelaxation,
the spin changes its polarization a ~ {3 hut
i(a) ~ i({3)
not
its
site:
j({3) ~ j( a)
i(a)j({3) ~ i({3)j(a)
BecaDse the net effect of each process is indistingDishable,
the individDal
contribDtions of each cannot be deteiinined from a single NOESY experiment.
The dynamic matrix L contains all the relevant data about the exchange rates in a given system.
Elements Li j of the dynamic matrix contain terms corresponding to the two possible mechanisms:
Kij,
the chemical exchangerate and Rij, the crossrelaxation rate.
Lij
=KijRij
(3)
Specialised2D experiment (NOESY and ROESY) permit the individual rates to be determined.
The crosspeak intensities in NOESY and ROESY spectra depend on the respective crossrelaxation
and longitudinal relaxation rates hence the R is the matrix describing the complete dipoledipole
relaxation network.
PI
0"12
0'13
(J2I
P2
0'23
(J3I
0"32
P3
(4)
The diagonal elements of the rate matrix are simply the longitudinal relaxation rates (Pi), while the
offdiagonal elements are the crossrelaxationrates (crij):
Rii = Pi = 2(ni l)(Wti+ W~)+ Ln)WH+ 2Wy+ W~)+ Rli (Sa)
j;ti
Rij = O'ij = ni(Wq
wg)
(3) The crossrelaxation
O'ij'r = (wg)n,r
(5b)
rate CJij between t wo groups ofequivalent
spins at sites i and j is given by:
(w8)n,r
where ( w g ) and ( wg)
represent twospin transition probabilities for double quantum and zero
quantum transitions, respectively. The superscript n denotes the laboratory frame (NOESY) and the
rotating frame (ROESY) system of reference.
Table
Relaxation parametersfor t wo groups of equivalent spins undergoing isotropic motion in Laboratory
Frame and Rotating Frame if q = 0.1 y4n2~
and J(mo) =
/r~
'l'c
2
l+(mo'l'c)
Parameter
Laboratory frame
Rotating frame
(wg)
qJ(o)
(1/4)qJ(o)
+ (3/4)qJ(2(J»)
(Wi)
6qJ(2(J»)
(9/4)qJ(o)
+ 3qJ((J») +(3/4)qJ(2(J»)
O"ij
6qJ(2(J») qJ(o)
2qJ(o) + 3qJ((J»)
/""""'
OJo"t'c »
O"ij
q"t'c
1
+2q"t'c
CONCLUSION:
For the macromolecules in high magnetic field, OJo "t'c > > 1 ( so called II spin diffusion limit",
the crossrelaxation rate in the rotating frame is twice as fast as in the laboratory frame and the rates
(4) Taylor range.
A simple approach to calculate the intensities or distances to recast the exponenti al into a series
expansion (when the mixing time is short enough 'T m 7
a('Cm)= eR'Cm::::lRfm+~R2'C~
+l(1)~JRn'C~+...
Q):
(7)
For longer mixiIig periods, we may have to take into account in Eq.(7) the term quadratic in 'troand
obtain:
aij('rm) = (RijTm+~LRikRkjT~)
k
(8)
However, the dependence on longitudinal relaxation rates can be eliminated by nomlalizing the crosspeaks to the corresponding diagonal peak in the same experiment
~
r
.~
FARMER,
!!!:.
Q"r
f"
Flo. S. (A) Dipolar cross.rcJualion rates in Ihc 13boratory rrame (a") 3nd in the rouling rramc (1:1$ a
runclion ortbe isotropic COrTelalion time, r" all.1rmor rrcqueneic:s or 300 M Ht () and SOOM Ht (...).
(D) The ratio a"o' U a runeliol\ orlhe isotropic eorTelatiol\ lime r,.
.
L/
MACURA,
AND
BROWN
~
0.5
~...
.,
.~
,
8AA
,
,
,
,
,
,
,
.
...(.)oTc=O.112
0.2
...
~

0.1

o
,
\ 12
24
36
48
"
ro,
s
l

0.1
C.)T
oc
0.2'
".,
!m
UoTc
:0.112
.
«
1.A
u.,
UoTc
»
1A
" I
.' '.j
Figure 4. Dependence of the mixing coefficients a.~A=aSS and aAS=aS,( on the mixing
time Tm för cross rela.'tation in an AB spin system (25}. They indicnte ,the intensities of the t\vo auto and of the t\\'O crosspenks, respectively.
Three typical
correlation times TC have been assumed : "'o TC= 0,112 corresponds to a short
correlation time ( ~ extreme narrowing}, \vhile "'oTC= 11.2 represents a case of long
correlation
time (spin diffusion
limit}.
The critical case "'oTC= 1.12 leads to
vanishing cross relaxation and disappearing crossp~aks irrespective of the mixing
time Tm. The indicated time scale assumes a Larmor frequency "'0/2.. = 100 MHz
and q = 3.33 x 10. $2 (19}. ,T wo schematic representations of the resulting twodimensional spectra are included.
r
"
5""'
p ART II
(II) The backcalculation
of nOe spectra using the density
matrix.
The cross peaks in NOESY spectrum arise from cross relaxation via the dipoledipole interactions
between protons. The NOESY spectrum can therefore be used to estimate lHIH distance.
The relaxation network for the macromolecule is described by a set of equations which are an
extension of Solomon's equations describing the dipolar relaxation process in a twospin system.
The time course of the magnetisation during the mixing period of the 2D NOESY experiment is
described by the system of equations:
aM/at = RM
(1)
where M is the magnetization vector describing the deviation from thermal equilibrium
M = M z Mo
and R is the matrix describing the complete dipoledipole relaxation network.
PI
0'12 0'13 .
0'21 0'32
0'31
P2 0'23
P3
r.
~
The diagonal elements of the rate matrix are simply the longitudinal relaxation rates (Pi), while the
offdiagonal elements are the crossrelaxationrates (O'ij):
R;j = P; = 2(n; l)(W{; + W~) + ,L.nAwH + 2Wy + W~) + Rl; (2a)
1*1
Rij = O"ij= ni(W~
wg)
nij is the number
of equivalent
(2b)
spins in a group.
(3a)
(3c)
r
It gives the equation to the crossrelaxation rates:
r41i2
O"ij=~
r
(
6 'f c
J
~'fc
The term R1i
represents external sources of relaxation such as paramagnetic impurities
or spin
labels. Usually, it is ignored in all mathematical approaches used in present but it could be the source
of the big errors! ! !
Inspection of Eqs.(2a,b) and (3ac) reveals the l/r6 distance dependence of the relaxation rates. How
to solve these equations which ffieans what someone should do to find the distance? Equation (1) has
the familiar solution:
M( 'rm) = a( 'rm)M(O)
= eR'rmM(O)
wherea is the matrixof socaJled mixingcoefficients
NOTE!!!
whichareproportional
to themeasured
2D NOEintensities.
The exponenti al of Eq.(4) can't be calculated
directly
6
(4 )
by performing
a termbyterm
The approach to calculating the intensities or distances from Eq.(4).
t. Taylor range.
A simple approach to calculate the intensities or distances to recast the exponenti al into a series
expansion (when the mixing time is short enough ""'Cm ::; O):
a('l'm) = eR'l'm = lR't"m+~R2'l'~
+l(1)~JRn'l'~+...
(5)
2. T wo proton approximation.
For large molecules which satisfy the condition OYCc>
1 cross relaxation is very efficient. If there are
several protons in close vicinity to each other then a quick diffusion of magnetization occurs, leading
to "spin diffusion".
The extend of diffusion dependson the length of mixing time ('tm) used in the NOESY experiment.
For short ('tm<50ms), the magnetization transfer is restricted to a single step and under such
conditions ( linear regime):
aij( 'l'm)= (OijRij't"m) (6)
for the cross peaks:
r4
h2
TC Tm
(7)
aii =
10,6.
I)
where 'Yis the proton magnetogyric ratio and il is Planck's constant divided by 27t.
(
peaks
Interproton
I H I H distancescanbeestimated
by measuring
J
the intensitiesof cross peaks in the linear regime.
(l) Estimations of correlation times, ('tc), can be obtained from T2 and Tl measurements,according
to the equation:
}'2
r
'
(""""
'l'C = 2 (J)1(3T~1
whichholdsgoodfor m'fC > > I
(2) If protons i, j, k, 1 have similar 'tc values and if rij is a known distance, then the unknown
distance fkl can be calculated by comparing the intensities Iij and Ikl in a single spectrum:
,I;
Iij
I
:kl
Ikl
,rij)
In oligonucleotides, three reference distances can be utilised for this purpose, namely Cyt (H5H6),
(H2'H2") and Thy (H6CH3) where Cyt and Thy refer to cytosine and thymine, respectively.
7
(3) For longer mixing periods, we may have to take into account in Eq.(5) the term quadratic in 'tm
and obtain:
The second term shows the diffusion of NOE from the ispin across all spins to the jspin. In the
simple example of three spins shown in Fig. a twostep pathway for the cross relaxation, spin i to
spin k followed by spin k to spin 3, may under certain experimental conditions be more efficient than
direct cross relaxation between spin i and j.
(1/rik)6 ..G
(1/rkj)6 P,
GJ..
H8
:\.(l/rij)
~
6
I!~
pr
,..
How to answer to the question Is it short enough mixing time or not to use Eqs (6) or (8)?
This question can be answered by running through the series expansion term by term for a
representativecase.
Intensities were calculated according to Eq.5 for the protons in BDNA assuming a correlation time
of 4 nsec ( at 500 MHz, (o'tc = 12.6) and mixing times of 10, 50, 100, 150 and 200 msec. The
results are given in Tab. and Fig.
Table
Convergenceof serlesexpensio~for 2D NOE i~te~sity calculation
a
calculated after N tenn.
b Mixing time
c Number of tenns in series expansion in Eq.5 required to achieve less than a 5% deviation.
l""""'
Intensities
for short
term
approximation.
underestimated.
.
distances
($:3.0A) are typically
overestimated
For
longer
distances
the
intensities
8
by the singleare
typically
3. Calculating Intensities by CORMA.
A more expeditious approach to calculating intensities is to take advantage of linear algebra and the
simplifications which arise from working with the characteristic eigenvaJues and eigenvectors of a
matrix.
Since the rate matrix is symmetricaJ, one can express R as a product of matrices:
R=LALT
whereA is adiagonal
matrixofeigenvalues.
L istheunitary
matrixof orthononnal
eigenvectors
(L I =L T)
The utiIity of making this transformation
is that, since
A is a diagonal matrix, the series
expansion for its exponential ( and consequently that of the mixing coefficient matrix) collapses:
a( 't"m)= eRrm = eLALTrm = lLAL T 't"m+ ~LALTLALT't"~
(9)
~
r
Ll=LT::}LTL
a('l'm)=L
=E
eAfmLT
Since multiplication is commutative for diagonal matrices, the ith
diagonal element of eA'l'm
is just eAi'l'm , where Åi is the ith eigenvalue of R.
This calculation allows one to readily calculate all the crosspeakintensities for a proposed structural
model. Then comparison between calculated and measuredintensities allows a determination as to
the validity of the model structure.
4. Direct Calculation of Distances.
The "ideal" way to calculate distances is to directl~ transform the scaled intensities ( mixing
coefficients) from the experimental 2D NOE spectra into their associated dipoledipole relaxation
rates and then distances. Rearrangement of Eq (5) gives the fundamentallogarithmic relationship
between the rates and mixing coefficients (se eq.10). The solution ofEq.(10) is typically performed
by finding the eigenvalues of the mixing coefficient matrix and then performing the simple matrix
multiplication (see.eq.ll)
{In [ a( 'l'mX
O
( 'l'm) a( )]} = R
(10)
R=
X (InXm)XT
(11)
a( O) refer to the diagonal matrix of intensities for
an experiment with mixing time zero ( 'l'm = 0)
r
NOTE!!!
(1) Computer power is not the problem here ( minutes to hours for
macromolecules ranging up to 1000 unique protons)
(2) Typical 2D NOE experiment does not yield all the information necessaryfor
the DIRECT calculation to work weIl (not weIl resolved cross and diagonal peaks)
9
~
~,
5. A Hybrid
NOR Volume Matrix/Restrained
structural refinement
Molecular Dynamics Approach
(MORASS).
for
The hydrid matrix approach addressesthe problem of incomplete experimental data.
T~e solution is to combine the information from the experimental NOESY volumes, aijxp
, and
calculated volumes, a~je ,derived from an initial or refined structure.
This hybrid volume matrix, ahyb, is then used to determine a rate matrix ( see Eq.ll),
and the
resulting distances are then utilised in restrained molecular dynamic simulation for refinement of the
structure. This process can be repeated until a satisfactory agreement between the calculated and
observed crosspeak volumes is obtained.
Convergence is ffionitored using the following criterions:
[
afhearXP
~
RMS
1=
vo
2
~laf!Pa~jel
'L, h
L
N
]
~
a... e
i
~
Rfactor
=
'}
Lae)(p
l'
;I
00
'}
'}
Theoretical
Experimental
C)
"
@)
/
~
~
C)
/
,':/
(§)

;.~:.;;
/
C>
'. .)';1
..
...'~"!t;..
:.",,:~
experimental limitation
1
."'
depeDdence on
injtial struc~
aexp
[
aexp aexp
aexp aexp
r
Schematic descriplion
of merge matrix method.
A hybrid volume matrix ahyb is created by replacing the theoretical volume matrix elements
athe with the wellresolved experimental volume matrix elementsaexp,
,10

5. The MARDIGRAS
The variation
utilizes
~
rather
introduced in MARDIGRAS,
Aigorithm.
and shown on the lower right in Fig,
than iterating through the cornputer timeconsurning
restrained MD
procedures af ter a single pass through the relaxation matrix ( as in MORASS).
r"'
{\
'"
l"""""
~
Fig. Schematic diagram of matrix analysis of relaxation for discerning geometry of
an aqueous structure (MARDIGRAS)
/1