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2001 EMBO Course: Structure Determination by NMR Spectroscopy

[Lawrence McIntosh - Practical 9]

Analysis of 15N Relaxation Data:

Global and Internal Motions of Proteins

TENSOR2

Patrice Dosset, Dominique Marion & Martin Blackledge

Institut de Biologie Structurale J.P. EBEL CEA-CNRS

Laboratoire de Resonance Magnetique nucleaire

Grenoble

The following notes are condensed from the original versions (courtesy of MartinBlackledge):

http://www.ibs.fr/ext/labos/LRMN/softs/tensor/TENSORV2_DOC/theory.html

http://www.ibs.fr/ext/labos/LRMN/softs/tensor/TENSORV2_DOC/practice.html

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A practical analysis of 15N relaxation data to obtain global andinternal mobility values using TENSOR2

(A) SET-UP

• set-up directories and start the program with tensor2

• Input a pdb structural co-ordinate fileFile -> Open PDB File -> c2anisocoord.pdb

Choose the style for displaying the molecule with Style and BackgroundActivate the Select icon to allow the identification and displaying of selected atoms.

• Input relaxation data fileFile -> Open Data File -> Data600_index_Helix

View and choose the residues to be used for global motional analysis withVisualization -> Definition

(a . indicates that data is available, while residues flagged as '1' in the input fileare shown in colored boxes)

activate 2nd vue icon to color map these residues on the molecular structure

include(or exclude) residues by highlighting with the mouse and choosingthe select (or unselect) icon

(B) DETERMINE GLOBAL CORRELATION TIME USING THE R2/R1 RATIO

• Set the spectrometer frequency to 600 MHz for 1H (Setup -> Spectro)

• Select Isotropy on the lower menu and click Fit

τc will be given in the text window and can be seen using Visualization -> R1/R2 Tc

• For error analysis, set Nb Cycles to 100 (use more for better statistics) and clickStart Monte-Carlo

The resulting τc and its error are given in the text window and the quality of the fit ofthe relaxation data to the isotropic model can be viewed withVisualization -> Monte-Carlo

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(C) DETERMINE THE DIFFUSION TENSOR FOR ANISOTROPIC TUMBLING

• Select Anisotropy on the lower menu and click Fit

The principle components of the diffusion tensor for the 2 axially symmetric models (Dpar, Dper) and for the fully anisotropic model (Dxx, Dyy, Dzz) are given in the text window

Visualization -> Diffusion Axis will show the orientation of the fully anisotropic tensor on the molecular structure (for viewing, you can rotate the molecule and tensor with the left mouse button, and you can change the Euler rotation of the tensor with Control -> Rotation)

Visualization -> Orientation will also display the molecule oriented along the various diffusion axes

• For error analysis, set Nb Cycles to 50 (use more for better statistics) and clickStart Monte-Carlo

The resulting values of the diffusion tensors and their errors, as well as the statisticalevaluation of the fit of the relaxation data to the various diffusion models, are given inthe text window

The quality of these fits can be seen under Visualization -> Monte Carlo(Note that the statistical analysis is highly sensitive to the magnitude of theexperimental errors in R1, R2, and the NOE)

The dispersion of direction of the ansiotropic diffusion tensor due to experimentalerror can displayed on the molecular structure usingVisualization -> Diffusion Axis or Orientation

A comparison of the calculated (yellow squares) and observed data (with error bars) in given in Visualization -> R1/R2 Back (fully anisotropic)

R1/R2 Alpha (axial symmetric models)

(D) COMPARE THE DIFFUSION AND INERTIA TENSORS

• Highlight the Inertia Tensor icon to calculate the inertia tensor for the input structure (therelative moment of inertia are given in the text window)

Use Visualization -> Diffusion Axis or Orientation to display the orientation of the inertia tensor on the structure along with the diffusion tensor

Highlight Inert./Diffus. to determine the angles between the principle components of the diffusion and inertia tensors.

• Note: The pdb file can be saved in the diffusion or inertial frames under FileThe results of the above τc and D calculations are written to the filesresiso.* and resaniso.*

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(E) MODEL FREE ANALYSIS OF INTERNAL DYNAMICS

• Choose the residues to analyze under Visualization -> Definition(all the input data is selected by default, and can be modified with select or unselect)

• Choose either Internal Mobility D-iso or Internal Mobility D-anisoset Nb cylces = 50 (more for better statistics) and click Start Mobility

The previously fit isotropic τc or fully anisotropic diffusion tensor will be used by default (but an arbitrary τc can be set under Control -> Tc)

The input value of τc or the fully anisotropic diffusion tensor and the resulting modelfree fits are displayed in the text window, along with the fit values. These results canbe saved by Visualization -> Mobility Values -> File -> Save As

The results can be viewed under Visualization -> Mobility -> S2 teKexModel (simplest of 1-5 used to fit the data; 6=none)All (the above four together)S2/te (dispersion of S2 versus te for the fitting)

Note: In the windows for S2, te, and Kex, use the left mouse button to move a vertical cursor along the residue axis. The position of the residue at the cursor can is mapped on the structure seen the 2nd Vue option)

(and printed under File -> Print)

The resulting S2 values can also be mapped on the structure viaVisualization -> View Molecule(use the S2 scroll bar to choose the upper limit on S2 for the color display)

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Relaxation Data AnalysisThe 15N heteronuclear relaxation rates R1 and R2 and the heteronuclear {1H}15N NOEdepend on the spectral density function J(ω) in the following manner :

R1 = d2[J(ωH - ωN) + 3J(ωN) + 6J(ωH+ ωN)] + c2J(ωN)

R2 = (d2/2)[4J(0) + J(ωH - ωN) + 3J(ωN) + 6J(ωH + ωN) + 6J(ωH)] + (c2/6)[4J(0) + J(ωN)]

NOE = 1 + [(γA/γN)d2{6J(ωH + ωN) - J(ωH - ωN)}/R1]

where d2 = (1/10) γH2γN

2(h/2π)2(<rNH-3>)2

and c2 = (2/15) ωN2 (σ|| - σ⊥ )2

h is Planck's constant, γHand γNthe gyromagnetic ratios of 1H and 15N, ωH and ωN their Larmorfrequencies, and rNH the internuclear distance (assumed to average to 1.02 Å). σ|| - σ⊥ are theparallel and perpendicular components of the axially symmetric 15N chemical shift tensor(approximated to -170ppm) which in a first approximation is assumed to be coaxial withrespect to the dipolar interaction.

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Format of Input Files.

Robust methods for measuring R1 and R2 and the heteronuclear {1H}15N NOE and theirassociated errors are described by (Farrow et al., 1994).

Relaxation Data:

Example :

0 2 1.1407 0.043841 12.6434 0.28419 0.768764 0.0145671 3 1.2425 0.033048 13.3264 0.20920 0.80534 0.0233111 4 1.2274 0.029092 13.0570 0.16854 0.57453 0.0456321 5 1.2125 0.055488 12.8067 0.20193 0.764352 0.067832.........

The 1st column (optional) is a flag describing whether the residue should be taken intoaccount in the fit or not. (if 0 the residue is not taken included in the error function).The 2nd column specifies the number of the resiude in the primary sequence.Columns 3 and 4 represent R1 and associated uncertainty (in s-1).Columns 5 and 6 represent R2 and associated uncertainty (in s-1).Columns 7 and 8 represent the heteronulcear nOe and associated uncertainty (no units) -these are not necessary (and can be left out) if no analysis of internal mobility is to beperformed.

Coordinate File :

The coordinate file is a standard *.pdb file with protons

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(A) Relaxation Data Analysis for Global Motions

Isotropic tumbling

In the case of isotropic tumbling of the molecule, and rapid internal mobility (τi < τc)which is negligible in terms of relaxation effects, the auto-correlation function isdescribed by a single decaying exponential and the spectral density function is given by:

where τc is the overall correlation time of the molecule.

Estimation of c

It has been proposed that the (R2i /R1i) ratio of the most rigid vectors in the moleculecan provide an initial estimate of τc This is valid because:

as τi −> 0

and the ratio (R2i /R1i,) becomes independent of τi and S2.

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As long as we are careful to exclude residues whose internal mobility is negligible, theoverall correlation time of the molecule can be fitted to the ratio R2/R1 which in this motionalregime, is determined by τc. It is also necessary that these residues do not experiencechemical exchange which could increase R2eff. A reasonable criteria is to exclude residueswith:

(i) a heteronuclear {1H}15N NOE < 0.65, indicative of fast internal mobility.

(ii) values of [(<R2> - R2i)/<R2>) - (<R1> - R1i)/<R1>)] > 1.5 x the std. deviation of thisdifference. Note that this allows for for variations in the R2/R1 ratios due to anisotropictumbling, while excluding those residues with significant Rex contributions to R2eff.

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Fit of relaxation data using an isotropic tensor:In this case all directions are assumed equal and the contribution to rotationaldiffusion is independant of vector orientation. This is simply a fit of the average R2/R1ratio to τc for those residues chosen in the selection proceedure performed above.

Error calculation and confidence limit testing:Specify the number of simulations for the Monte-Carlo procedure and click on Monte-Carlo. This calculation not only estimates the uncertainty in the overall correlation timedetermined from the selected residues, but also provides a measure of the confidence inthe predicted value. The comparison with the simulated χ2 distribution allows you todecide whether isotropic tumbling is in fact applicable to your molecular system.

In the case shown below such a model is rejected using 95% confidence limits, as theexperimental χ2 (whose value is shown as the yellow line) is much higher than thatsimulated from the optimal correlation time (95% of simulated χ2values are lower thanthe orange line), assuming the experimental errors describe the random fluctuation of themeasured relaxation rates.

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Fitting of the anisotropic rotational diffusion tensor:In the case of anisotropic tumbling, the relaxation rates are dependant on the orientationof the relaxation mechanisms, relative to the rotational diffusion tensor of the molecule.

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The spectral density function describing anisotropic rotational diffusion (Woessner 1962)is given by:

where

A1=3y2z2, A2=3x2z2, A3=3x2y2

A4,5=0.25{3(x4+y4+z4)-1}±(1/12){δx((3x4+6y2z2-1)+δy((3y4+6x2z2-1)+δz((3z4+6y2x2-1)}

τ1,2,3=(4Dxx+Dyy+Dzz)-1, (Dxx+4Dyy+Dzz)-1, (Dxx+Dyy+4Dzz)

-1,

τ4,5=(6Diso ± 6(Diso2-L2)1/2),

Diso=(Dxx+Dyy+Dzz)/3,

L2=(DxxDyy+DxxDzz+DyyDzz)/3

δm=(Dmm-Diso)/(Diso2-L2) 1/2.

m=(x,y,z) and (x,y,z) are the direction cosines of the N-H vector in the principal axisframe of the diffusion tensor.

6 parameters (Dxx,Dyy,Dzz, α, β, γ) are optimized, describing the orientation and amplitudeof the principle components of the diffusion tensor in the chosen molecular frame. Thediffusion parameters are extracted by minimizing:

where σ is the uncertainty in the experimental R2/R1 ratio.

Axial symmetry:

A commonly used simplification is to assume axial symmetry and fit the relaxation datato 4 parameters (D//, D⊥, θ, φ). This implies that the protein can be adequately described asa prolate (D// / D⊥ > 1) or oblate (D// / D⊥ < 1) ellipsoid. In either case, the expression forJ(ω) reduces from 5 to 3 coefficients Ai and correlation times τi, as given in the followingdiagram.

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In fact, as shown below, two physically reasonable minima can exist in the fit using theaxially symmetric rotational diffusion tensor, whose unique axes are related to the real axesof the asymmetric tensor by the following approximations (Blackledge et al JACS 1998): inthe case of an asymmetric tensor with components (Dxx,Dyy,Dzz) the components of the axiallysymmetric tensor (D⊥ and D//) would be approximated to ((Dxx+Dyy)/2, Dzz), and ((Dyy+Dzz)/2,Dxx) in the three minima. The relative importance of these minima with respect to the targetfunction is indicative of the nature of the diffusion tensor.

Below the target function in reduced parametric space is shown - χ2is plotted with respect tothe polar angles θ and φ, and the ratio D// to D⊥.

The assignment of D// to D⊥ to the summed or the isolated term depends on the geometryof the molecule. In the case of three components which are significantly different (e.g.DxxDyyDzz) as for the cytochrome c2 : one finds two minima of similar significance (χ2 =44.3, 44.6 for the helical dataset). The orientation of the tensor in the oblate and prolatemodels is illustrated below (Blackledge et al 1998, Cordier et al 1998).

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Clearly the presence of these two similarly significant minima is indicative of a rotationaldiffusion tensor with significant rhombicity. Blackledge et al. have tested many proteins forwhich relaxation data and structural coordinates are available and find that this is far from arare occurrence, but that often only one of the two minima is reported in published analysesof rotational diffusion using relaxation data. This is perhaps not surprising is we look at the χ2

space illustrated above; depending on the values of θ and φ, equally valid minima can besteep-sided and therefore difficult to find using a grid-search type of minimisation algorithm,or broad and consequently much easier to localise. (The values of θ and φ, are of coursedependent only on the reference frame, usually taken from the database coordinates, andtherefore completely arbitrary.)

Structure determination:Note that for this analysis, we start with a known structure and apply a "rigid body" rotationby the best-fit Euler angles α, β, and γ (full anistropy) or θ and φ (axial symmetry) into theco-ordinate system of the diffusion tensor. The R2/R1 value of each individual amide isderived from the directional cosine of its N-H vector within this co-ordinate system.

Conversely, if the diffusion tensor can be estimated, then R2/R1 ratios can be used as arestraint to help determine the global structure of a protein (i.e. the orientation of each rigidN-H vector with respect to the overall diffusion tensor of the protein).

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Model selection (i.e. axial symmetry versus full anisotropy) is based on Monte-Carlo simulations to characterize the random variation in the fit, to provide probabilitystatistics and estimate uncertainty. An F characteristic is used to judge the statisticalsignificance of introducing an additional parameter to model (1). F is defined as:

for the comparison of models fitting N variables with m and n parameters. In the case wherethe reduction of χ2is less than the α=0.20 critical value for random statistical improvement wereject the more complex model and propose the model (1) parameterization.

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(B) Relaxation Data Analysis for Internal Mobility Measurement of heteronuclear relaxation rates have been widely used to investigate thebackbone dynamics of proteins. Commonly, 15N R1 and R2 autorelaxation rates and theheteronuclear 1H-15N NOE are measured for each available residue in the protein. Dataare normally interpreted using the abstract modelfree dynamics formalism (Lipari &Szabo,1982) or by simple matrix inversion to provide direct samples of the spectraldensity function (Peng & Wagner,1992,1995) of NH vectors along the peptide chain.

In the modelfree approach mobility is characterized using an order parameter S2, whichmay be interpreted as the amplitude of the motion and a correlation time τi- thecharacteristic time constant of this motion. The physical nature of the mobility is notconstrained, but the internal and global motion are assumed independent and the overalland internal autocorrelation functions are assumed to have an exponential nature. Theinterpretation has been further extended to take account of two uncoupled internalmotions and hence three independent terms in the time correlation function (Clore et al.,1992).

This approach has the advantage of being quantitative, intuitive and relatively easy to testfor statistical significance (Mandel et al., 1995). Although non-specific and valid only inrestricted timescales with respect to overall motion, this simple analysis has gained inpopularity with the routine investigation of 15N labelled proteins to become a standardmeasure in the solution study of proteins using heteronuclear nmr spectroscopy.

Lipari-Szabo Approach:Internal and global motion are assumed to be independent and the autocorrelation function isdescribed by the two exponential processes - internal motion is defined by two parametersdescribing the amplitude (S2 - called the square of the order parameter) and a characteristiccorrelation time of the motion i.

,

The plateau value of the correlation functions S2 ( ) describes the spatial restrictionof the X-H vector, and τithe internal correlation time of this vector in the molecular frame.

Limiting cases - S2 approaches 0; relaxation described only by internal motion.

S2 approaches 1; relaxation described only by global motion.

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In the case of isotropic overall motion Cg(τ) is given by ;

Similarly the spectral density function can be described using this formalism -

where , τe is the effective internal correlation time and τc is the isotropicglobal correlation time. In the limit τe << τc and S2 1

The spectral density function applicable in the relaxation rate equations can thus be defined(in its simplest form) for all J(ω) terms by two parameters describing the internal motion (S2,τi) and by one term (τc) describing the overall motion.

This spectral density function is then a non-linear function with respect to (S2, τi)It is necessary to fit the relaxation data by non-linear least squares optimisation of a targetfunction:

where Rx represents the measured relaxation rates R1,R2 and nOe.

In the case of isoptropic overall tumbling, the parameter τc is common to all points in themolecule - this formalism allows for three measured variables to characterise the internalmobility using two parameters.

In the case of anisotropic tumbling, the same arguments apply with the common orientationand principle components of the diffusion tensor introduced into the first term of the generalspectral density function:

with

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Exchange term (Rex): For specific sites in the molecule it may be necessary tosupplement these two parameters {(S2, τi} with a chemical shift exchange contribution Rex, tothe measured transverse relaxation rate R2, due to rapid (µs-ms) interconversion of sitesexperiencing varying chemical environments.

Conformational Exchange - (τexch - µs)

R2 (eff) = R2(dip) + R2(csa) + R2(exch)

Transverse relaxation can have a component due to exchange between twochemical shift sites

In practice, Rex reflects contributions to R2 that cannot be fit by the simple Lipari-Szabo model. More extensive measurements (e.g. CPMG-type experiments ormeasurements at multiple fields) are likely necessary to define Rex accurately.

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Extended Lipari-Szabo Model: It is sometimes necessary introduce additionalmovement with an intermediate correlation time to explain the relaxation data:

the movement is parametrised by τfand τsandtwo order parameters and for the

fast and slow motions respectively. τf << τs << τc. .

The correlation function drops rapidly to a plateau ( ) then more slowly to a secondplateau (S2).

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The full spectral density function for isotropic tumbling becomes:

where and are the effective correlation times of the

two motions ( ).

With , the equation reduces to :

The internal motion is now parametrised by Ss2, Sf

2 and τs.

Now we see why we have to be very careful with this type of analysis — there areN (measurables) and N (parameters)!

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Available models for internal dynamics:

Following the expression for J(ω) given above, five models of internal mobility are possible,assuming D (rotational diffusion tensor) or τc (isotropic tumbling) is known.

1 - we consider that τi is very fast ( <20ps):

2 - is relaxation active (classical Lipari Szabo):

3 - J(ω) as for M1 with a chemical exchange contribution (Rex).

4 - J(ω) as for M2 with a chemical exchange contribution (Rex).

5 - Extended Lipari Szabo model, including a very fast and a slower internal motion

( and with ).

Dynamic Parameters

Model 1

Model 2 ,

Model 3 ,

Model 4 , ,

Model 5

, , =

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Sets of relaxation data are fitted to the different dynamic models, using the optimized globalcorrelation function by minimizing the function:

where i represents the relaxation parameters used in the calculation. , andare the experimental and calculated relaxation rates/nOe and the estimated experimental

uncertainty of respectively.

The program will test the models (1-5) shown above against the experimental{R1,R2,nOe} for each residue selected. Once the data can be reproduced satisfactorilyfrom one of the models (within a 95% confidence limit, as determined from Monte-Carlosimulations) the program selects this model and then goes onto the next residue. If one ofthe models (2 or 3) does not improve the fit enough to satisfy the F-test (α=0.2}, then thepreviously failed, but marginal, model 1 is accepted.

If none of the models can reproduce the data satisfactorily the program will assign model 6 -this means that neither models 4 or 5 {3 parameters for 3 measureables} was sufficientlyclose to a zero χ2 for the model to be acceptable. The mobility parameters for the best ofthese models are nevertheless given in the final table.

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References.

General:

Farrow et al., Biochemistry, 1994, 33, 5984-6003.

Lipari, G. & Szabo. AJ. Am. Chem. Soc. . 1982a ,104, 4546-4558.

Lipari, G. & Szabo. A. J. Am. Chem. Soc. 1982b ,104, 4559-4570.

Peng, J. & Wagner, G., Biochemistry, 1992,31, 8571-8586.

Peng, J. & Wagner, G., Biochemistry, 1995,34, 16733-16752.

Clore, G. M., Szabo, A., Bax, A., Kay, L. E., Driscoll, P. C. & Gronenborn, A. M., J. Am.Chem. Soc., 1990,112, 4989-4991.

Mandel, A.M., Akke, M. & Palmer, A.G. J. Mol. Biol. 1995 246, 144-163.

Schurr, J.M., Babcock, H.P. & Fujimoto, B.S.. J.Magn. Reson. Ser B, 1994,105, 211-224.

Woessner, D. E. J.Chem.Phys. 1962, 37, 647-654.

Tensor:

Precision and Uncertainty in the characterization of rotational diffusion from heteronuclearrelaxation data. Blackledge, M.; Cordier, F.; Dosset, P.; Marion, D. J.Am.Chem.Soc. 1998,120, 4538-4539.

Solution structure, rotational diffusion anisotropy and local backbone dynamics ofRhodobacter capsulatus cytochrome c2. Cordier, F.; Caffrey, M.; Brutscher, B; Cusanovich,M.; Marion, D.; Blackledge, M. J.Mol.Biol. 1998, 281, 341-361.

Efficient analysis of macromolecular rotational diffusion from heteronuclear relaxation data.Dosset, P.; Hus, J-C; Blackledge, M.; Marion, D. J.Biomol.NMR. 2000, 16: 23-28.

Rotational diffusion anisotropy and local backbone dynamics of carbon monoxide boundRhodobacter capsulatus cytochrome c'. Tsan, P.; Hus, J-C; Caffrey, M. ; Marion. D. &Blackledge, M. J.Am.Chem.Soc. 2000, 121,2311-2312.