May 12, 1998NMR Course. Lecture 14-15

Maltseva T. V.

Lecture 14-15

Part l:The NOESY and ROESY experimentsPart 2:The Back-Calculation of NOE spectra using the

Density Matrix~

~

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. James pp.169-183

Good review:* Macura,S., Emst,R.R..'Elucidation of cross relaxation in liquids by two-dimensional

NMR spectroscopy .' Molecular Physics ( 1980) pp.95-117

The extra, hut quite difficult without the knowledge of matrix analysis) to understanding the famous

book:

* 'Principle of Nuclear Magnetic Resonance in One and T wo Dimensions' by Ernst,R.,

Bodenhausen,G., Wokaun,A.Ch.9 pp.490-538"'

1

The of NOESY and ROESY(l)Laboratory-frame NOE spectroscopy (NOESY) is the 2D NMR experiment used to study the NOE inthe laboratory frame, and ROESY is the 2D NMR experiment used to study the NOE in the rotatingframe.

2D exchange spectroscopy

2DFT

r--

2D Fr

.I90 S(tl, tv

.--'

s( (J)\. 0>2)

~

,-~--: :-~-«1)I,fo'> (~.~

(90 S(tl. tv

~r--"'

90 90J

roJ

\.~,

itl 'tm t2 t,

, ,

/ (ö)i. roi) (rol,ro

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 cross-peak 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 rnatrixwith 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:

r

"

M( "l'm) = a( "l'm)M(O) = e-LrmM(O)

where a is the matrix of so -called mixing coefficients (1)

which are proportional to the measured 2D NOE intensities.

Incoherent magnetization transfer is driven by processes of random molecular motion. Among N

sites, the transfer can be described by the system ofN linear, coupled differential equantions:

~ = Lm( "l'm) with formal solution m( "l'm) = exp(L "l'm)m(O) (2)d"l'm

The vector m has elements nixiroi, 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 exchange or b~ cross-relaxation.

In the chemical exchange mechanism, the magnetization migrates together with the

spin, which changes its site i ~ j hut not its polarization:i(a) ~ j(a)

j(f3) ~ i({3)

i(a)j({3) ~ i({3)j(a)

In cross-relaxation, the spin changes its polarization a ~ {3 hut not its site:i(a) ~ i({3)

j({3) ~ j( a)

i(a)j({3) ~ i({3)j(a)BecaDse the net effect of each process is indistingDishable, the individDalcontribDtions 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 t wo possible mechanisms:

Kij, the chemical exchange rate and Rij, the cross-relaxation rate.

Lij =Kij-Rij (3)

Specialised 2D experiment (NOESY and ROESY) permit the individual rates to be determined.

The cross-peak intensities in NOESY and ROESY spectra depend on the respective cross-relaxation

and longitudinal relaxation rates hence the R is the matrix describing the complete dipole-dipole

relaxation network.

PI

(J2I

(J3I

0"12

P2

0"32

0'13

0'23

P3 (4)

The diagonal elements of the rate matrix are simply the longitudinal relaxation rates (Pi), while the

off-diagonal elements are the cross-relaxation rates (crij):Rii = Pi = 2(ni -l)(Wti+ W~)+ Ln)WH+ 2Wy+ W~)+ Rli (Sa)

j;ti

Rij = O'ij = ni(Wq -wg) (5b)

(3) The cross-relaxation rate CJij between t wo groups ofequivalent spins at sites i and j is given by:

O'ij'r = (wg)n,r -(w8)n,r

where ( w g ) and ( wg) represent two-spin transition probabilities for double quantum and zero

quantum transitions, respectively. The superscript n denotes the laboratory frame (NOESY) and therotating frame (ROESY) system of reference.

/""""'

TableRelaxation parameters for t wo groups of equivalent spins undergoing isotropic motion in Laboratory

Frame and Rotating Frame if q = 0.1 y4n2 ~ and J(mo) = 'l'c 2

/r~ 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 » 1

O"ij -q"t'c +2q"t'c

CONCLUSION:For the macromolecules in high magnetic field, OJo "t' c > > 1 ( so called II spin diffusion limit",

the cross-relaxation 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 Q):

a('Cm) = e-R'Cm :::: l-Rfm+~R2'C~ +l(-1)~JRn'C~+... (7)

For longer mixiIig periods, we may have to take into account in Eq.(7) the term quadratic in 'troandobtain:

aij('rm) = (-RijTm+~LRikRkjT~) (8)k

However, the dependence on longitudinal relaxation rates can be eliminated by nomlalizing the cross-

peaks to the corresponding diagonal peak in the same experiment

~

r--

.~

FARMER, MACURA, AND BROWN

!!!:.Q"r

f"

Flo. S. (A) Dipolar cross.rcJualion rates in Ihc 13boratory rrame (a") 3nd in the rouling rramc (-1:1$ arunclion ortbe isotropic COrTelalion time, r" all.1rmor rrcqueneic:s or 300 M Ht (-) and SOO M Ht (...).(D) The ratio a"o' U a runeliol\ orlhe isotropic eorTelatiol\ lime r,. .

-L/-

~... .~- 0.5

., 8AA

,,

,,

,,

, , .

...(.)oTc=O.112

...0.2

~~

~- -

0.1

o, \ 12 24 36 48 !m

l" s

----

C.)T :0.112oc

ro, -0.1

-0.2' .

UoTc « 1.A- u., UoTc » 1 A".--,

" I.' '.j

Figure 4. Dependence of the mixing coefficients a.~A=aSS and aAS=aS,( on the mixingtime Tm för cross rela.'tation in an AB spin system (25}. They indicnte ,the in-tensities of the t\vo auto- and of the t\\'O cross-penks, respectively. Three typicalcorrelation times TC have been assumed : "'o TC = 0,112 corresponds to a shortcorrelation time ( ~ extreme narrowing}, \vhile "'oTC = 11.2 represents a case of longcorrelation time (spin diffusion limit}. The critical case "'oTC= 1.12 leads tovanishing cross relaxation and disappearing cross-p~aks irrespective of the mixingtime Tm. The indicated time scale assumes a Larmor frequency "'0/2.. = 100 MHzand q = 3.33 x 10. $-2 (19}. ,T wo schematic representations of the resulting two-dimensional spectra are included.

r

"

-5""'-

p ART II(II) The back-calculation of nOe spectra using the density matrix.

The cross peaks in NOESY spectrum arise from cross relaxation via the dipole-dipole interactionsbetween protons. The NOESY spectrum can therefore be used to estimate lH-IH distance.The relaxation network for the macromolecule is described by a set of equations which are anextension of Solomon's equations describing the dipolar relaxation process in a two-spin system.The time course of the magnetisation during the mixing period of the 2D NOESY experiment isdescribed by the system of equations:

aM/at =- RM (1)

where M is the magnetization vector describing the deviation from thermal equilibriumM = M z -Mo and R is the matrix describing the complete dipole-dipole relaxation network.

PI 0'12 0'13 .--

0'21 P2 0'23 0'31 0'32 P3

r.

~The diagonal elements of the rate matrix are simply the longitudinal relaxation rates (Pi), while the

off-diagonal elements are the cross-relaxation rates (O'ij):R;j = P; = 2(n; -l)(W{; + W~) + ,L.nAwH + 2Wy + W~) + Rl; (2a)

1*1

(2b)Rij = O"ij = ni(W~ -wg)

nij is the number of equivalent spins in a group.

(3a)

(3c)

r

r

It gives the equation to the cross-relaxation rates:

r41i2 ( 6 'f c

JO"ij=~ ~-'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 sourceof the big errors! ! !

Inspection of Eqs.(2a,b) and (3a-c) 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:

6

M( 'rm) = a( 'rm)M(O) = e-R'rmM(O)

where a is the matrix of so -caJled mixing coefficients ( 4 )

which are proportional to the measured 2D NOE intensities.NOTE!!! The exponenti al of Eq.(4) can't be calculated directly by performing a term-by-term

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 ""'C m -::; O):

a('l'm) = e-R'l'm = l-R'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 areseveral protons in close vicinity to each other then a quick diffusion of magnetization occurs, leadingto "spin diffusion".

The extend of diffusion depends on 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 suchconditions ( linear regime ):

aij( 'l'm) = (Oij-Rij't"m) (6)for the cross peaks:

r4 h2 TC Tm (7)aii =

10,6.I)

where 'Y is the proton magnetogyric ratio and il is Planck's constant divided by 27t.

peaks(Interproton I H -I H distancescanbeestimated by measuring

Jthe intensitiesof cross peaks in the linear regime.

(l) Estimations of correlation times, ('tc), can be obtained from T2 and Tl measurements, accordingto the equation:

-}'2

'l'C = 2 (J)-1(3T~1r' ' which holds good for m'fC > > I

(2) If protons i, j, k, 1 have similar 'tc values and if rij is a known distance, then the unknowndistance fkl can be calculated by comparing the intensities Iij and Ikl in a single spectrum:

,I;Iij I

(""""

-- :klIkl ,rij)

In oligonucleotides, three reference distances can be utilised for this purpose, namely Cyt (H5-H6),(H2'-H2") and Thy (H6-CH3) 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 'tmand obtain:

~

I!~

,..-

rpHow 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 arepresentative case.Intensities were calculated according to Eq.5 for the protons in B-DNA assuming a correlation timeof 4 nsec ( at 500 MHz, (o'tc = 12.6) and mixing times of 10, 50, 100, 150 and 200 msec. Theresults are given in Tab. and Fig.

TableConvergence of serles expensio~ for 2D NOE i~te~sity calculation

a

calculated after N tenn.b Mixing timec Number of tenns in series expansion in Eq.5 required to achieve less than a 5% deviation.

l""""' .Intensities for short distances ($:3.0A) are typically overestimated by the single-term approximation. For longer distances the intensities are typicallyunderestimated.

8

The second term shows the diffusion of NOE from the i-spin a cross all spins to the j-spin. In thesimple example of three spins shown in Fig. a two-step pathway for the cross relaxation, spin i tospin k followed by spin k to spin 3, may under certain experimental conditions be more efficient thandirect cross relaxation between spin i and j.

(1/rik)6 ..G (1/rkj)6 P, H8

GJ.. : -\-.-6

(l/rij)

3. Calculating Intensities by CORMA.

A more expeditious approach to calculating intensities is to take advantage of linear algebra and thesimplifications which arise from working with the characteristic eigenvaJues and eigenvectors of amatrix.Since the rate matrix is symmetricaJ, one can express R as a product of matrices:

R=LALT

where A is a diagonal matrix of eigenvalues.

L is the unitary matrix of 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) = e-Rrm = e-LALTrm = l-LAL T 't"m+ ~LALTLALT't"~ (9)

~r-

L-l=LT::}LTL =E

a('l'm)=L e-AfmLT

Since multiplication is commutative for diagonal matrices, the ith diagonal element of e-A'l'm

is just e-Ai'l'm , where Åi is the ith eigenvalue of R.This calculation allows one to readily calculate all the cross-peak intensities for a proposed structuralmodel. Then comparison between calculated and measured intensities allows a determination as tothe 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 dipole-dipole relaxationrates and then distances. Rearrangement of Eq (5) gives the fundamentallogarithmic relationshipbetween the rates and mixing coefficients (se eq.10). The solution ofEq.(10) is typically performedby finding the eigenvalues of the mixing coefficient matrix and then performing the simple matrixmultiplication (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)

rNOTE!!! (1) Computer power is not the problem here ( minutes to hours formacromolecules ranging up to 1000 unique protons )

(2) Typical 2D NOE experiment does not yield all the information necessary forthe DIRECT calculation to work weIl (not weIl resolved cross and diagonal peaks)

9

~~,

5. A Hybrid NOR Volume Matrix/Restrained Molecular Dynamics Approach forstructural refinement (MORASS).

The hydrid matrix approach addresses the 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 thestructure. This process can be repeated until a satisfactory agreement between the calculated andobserved cross-peak volumes is obtained.Convergence is ffionitored using the following criterions:

~ [afhe-arXP ]2 ~laf!P-a~jel

RMS 1 = -L --'L,- h ~ R-factor = '}

vo N i a... e Lae)(pl' 00 '};I '}

~

Experimental Theoretical

@)C)

" /

- (§)C)

..

/

C>

.-"'

depeDdence on injtial struc~

[1

aexp

aexp

r

Schematic descriplion of merge matrix method.

A hybrid volume matrix ahyb is created by replacing the theoretical volume matrix elementsathe with the well-resolved experimental volume matrix elements aexp,

,10 -

,':/~ -~

/ ;.~:.;-;

'. .)';1...'~"!t;.. :.",,:~

experimental limitation

aexp

aexp

aexp

5. The MARDIGRAS Aigorithm.

The variation introduced in MARDIGRAS, and shown on the lower right in Fig,

utilizes~ rather than iterating through the cornputer time-consurning 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-