exchange experiments involving the transfer of heteronuclear longitudinal magnetization of product operator terms such as izsz or sz were introduced by montelione et al .
to quantitate transitions between different conformations of biomolecules for which isolated resonances are observed ( montelione and wagner 1989 ) .
these experiments have played an important role in nmr dynamic studies of biological systems ever since , because they provide access to functionally important dynamic processes occurring on time scales ranging from ~0.5 to ~50  s. in addition , they offer the possibility to resolve these processes on a per - residue basis and without perturbing the system from thermodynamic equilibrium , a feature that makes nmr spectroscopy an indispensable instrument for the analysis of macromolecular dynamics ( mittermaier and kay 2009 ) .
longitudinal exchange experiments are based on the quantification of magnetization transfer between exchanging conformers during a mixing time . in the standard
( frequency - labeled ) setup , four peaks are observed for each spin , i.e. two direct correlation ( diagonal ) peaks and two exchange cross peaks ( fig .  1 ) .
in order to measure the interconversion rates between the exchanging conformers , a series of spectra is recorded for a range of mixing times , and the magnitudes of the four peaks are quantified by fitting the appropriate theoretical curves to the experimental data ( fig . 
several variants of this setup have been described , including 2d , 3d and pseudo-4d trosy selected / detected versions ( hwang and kay 2005 ; sahu et al .
2002 ) as well as zero - quantum / double quantum coherence versions to resolve chemical shift degeneracy ( robson et al . 2010).fig
.  1a scheme of a 2d longitudinal exchange experiment ( montelione and wagner 1989 ) .
after frequency - labeling in an indirect dimension , t1 , magnetization associated with state a ( b ) ( shown in red and black , respectively ) either remains a ( b ) or is converted to b ( a ) during a variable mixing time ( mix ) .
subsequent frequency - labeling in the direct dimension ( t2 ) gives rise to two direct correlation ( aa and bb ) and two exchange cross peaks ( ab , ba ) , respectively .
b if the order of chemical shift evolution ( t1 ) and mixing times is interchanged , exchange cross peaks are not resolved but remain part of the two direct correlation peaks a and b , which corresponds to the collapse of aa with ba and bb with ab , respectivelyfig
.  2a time dependence of normalized direct correlation peaks aa and bb ( red and blacksolidlines , respectively ) and exchange cross peaks ab and ba ( red and blackdashedlines ) , simulated using ( 1 ) and ( 2 ) . parameters : kab  =  0.8  s , kba  =  1.2  s , r1a  =  2.0  s , r1b 
=  2.0  s. b as in a , but without normalization by maa(0 ) and mbb(0 ) ( but maa(0 )  +  mbb(0 ) set to 1 ) .
c time dependence of normalized direct correlation peaks a and b [ ( 3 ) , red and blackdashedlines ] , as well as aa and bb [ ( 1 ) , red and blacksolidlines ] , using the same parameters as in a. the curves for a and b are identical if r1a  =  r1b . by taking the difference between a and aa ( b and bb )
curves it is possible to reconstruct the correctly normalized exchange curves ab ( ba ) shown in a. d as in c but without normalization by ma(0 ) , maa(0 ) and mb(0 ) , mbb(0 ) , respectively . here
, the differences between a and aa ( b and bb ) are equal a scheme of a 2d longitudinal exchange experiment ( montelione and wagner 1989 ) .
after frequency - labeling in an indirect dimension , t1 , magnetization associated with state a ( b ) ( shown in red and black , respectively ) either remains a ( b ) or is converted to b ( a ) during a variable mixing time ( mix ) .
subsequent frequency - labeling in the direct dimension ( t2 ) gives rise to two direct correlation ( aa and bb ) and two exchange cross peaks ( ab , ba ) , respectively .
b if the order of chemical shift evolution ( t1 ) and mixing times is interchanged , exchange cross peaks are not resolved but remain part of the two direct correlation peaks a and b , which corresponds to the collapse of aa with ba and bb with ab , respectively a time dependence of normalized direct correlation peaks aa and bb ( red and blacksolidlines , respectively ) and exchange cross peaks ab and ba ( red and blackdashedlines ) , simulated using ( 1 ) and ( 2 ) . parameters :
kab  =  0.8  s , kba  =  1.2  s , r1a  =  2.0  s , r1b 
=  2.0  s. b as in a , but without normalization by maa(0 ) and mbb(0 ) ( but maa(0 )  +  mbb(0 ) set to 1 ) .
c time dependence of normalized direct correlation peaks a and b [ ( 3 ) , red and blackdashedlines ] , as well as aa and bb [ ( 1 ) , red and blacksolidlines ] , using the same parameters as in a. the curves for a and b are identical if r1a  =  r1b . by taking the difference between a and aa ( b and bb ) curves it
is possible to reconstruct the correctly normalized exchange curves ab ( ba ) shown in a. d as in c but without normalization by ma(0 ) , maa(0 ) and mb(0 ) , mbb(0 ) , respectively . here , the differences between a and aa ( b and bb ) are equal the apparent simplicity of the analysis of longitudinal exchange experiments is , however , deceiving , as there are several pitfalls that have to be accounted for in order to avoid systematic errors .
first , different relaxation properties of the two species result in different transfer efficiencies during inept step(s ) ( tollinger et al .
second , and more importantly , the analysis of kinetic properties from this experiment may suffer from substantial systematic errors if the line - widths of the two species are different ( fig . 
unless precautions are taken to ensure precise determination of peak volumes , errors due to systematic over- or underestimation of the magnitudes of exchange cross peaks can have a significant impact on the fitting results.fig . 
a peaks associated with state a during t2 detection ( aa direct correlation and ba exchange cross peaks ) have identical line widths ( lwa ) in the direct dimension , which can be different from that of bb and ab ( lwb ) . because quantification of ab ( ba ) exchange cross peaks requires normalization by aa ( bb ) direct correlation peaks at zero mixing time
, exact peak volumes rather than intensities or partial peak volumes have to be used for analysis of the data .
b simulation of the effect of approximating peak volumes by intensities for normalized exchange curves , using ( 2 ) .
parameters : kab  =  1.0  s , kba  =  1.0  s , r1a  =  2.0  s , r1b  = 2.0  s. assuming lorentzian line shapes ( where int  =  vol / lw ) , normalized exchange curves based on intensity measurements are simulated for lwa / lwb  =  0.7 , 0.8 , 0.9 , and 1.0 .
ab and ba exchange curves are shown in red and black , respectively . only if lwa  =  lwb , or if exact ( direct correlation and exchange cross peak ) volumes are used , ab and ba exchange curves are identical , as is expected for kab  =  kba ( superimposed red and blackdashedlines ) effect of differential line - broadening in 2d frequency - labeled experiments .
a peaks associated with state a during t2 detection ( aa direct correlation and ba exchange cross peaks ) have identical line widths ( lwa ) in the direct dimension , which can be different from that of bb and ab ( lwb ) . because quantification of ab ( ba ) exchange cross peaks requires normalization by aa ( bb ) direct correlation peaks at zero mixing time
, exact peak volumes rather than intensities or partial peak volumes have to be used for analysis of the data .
b simulation of the effect of approximating peak volumes by intensities for normalized exchange curves , using ( 2 ) .
parameters : kab  =  1.0  s , kba  =  1.0  s , r1a 
= 2.0  s. assuming lorentzian line shapes ( where int  =  vol / lw ) , normalized exchange curves based on intensity measurements are simulated for lwa / lwb  =  0.7 , 0.8 , 0.9 , and 1.0 .
ab and ba exchange curves are shown in red and black , respectively . only if lwa  =  lwb , or if exact ( direct correlation and exchange cross peak ) volumes are used , ab and ba exchange curves are identical , as is expected for kab  =  kba ( superimposed red and blackdashedlines ) because determining peak volumes by integration requires fitting to a functional form , this is particularly challenging in cases where non - canonical line - shapes ( resulting from conformational exchange processes occurring on the mss time scale ) are encountered , which are not known a priori .
moreover , it can sometimes be impractical to precisely measure the volumes of small exchange cross peaks with low signal - to - noise ratio , as typically encountered for short mixing times , in particular in the vicinity of strong direct correlation peaks . to circumvent integration , peak volumes
are often approximated by partial peak volumes that are obtained by summation of the intensities of data points within a defined interval .
partial peak volumes are , however , dependent on the choice of the summation interval with respect to the line - width ( rischel 1995 ) , thereby complicating the normalization of peaks with different line - widths . to improve the precision in the determination of longitudinal exchange kinetics , a quadratic approximation was proposed for the analysis of the data by employing a composite ratio of exchange and direct correlation peak intensities .
this approach does , however , require that equilibrium constants can be determined from other experiments ( miloushev et al .
we present a simple and robust alternative to the standard setup of multidimensional longitudinal exchange experiments for biomolecules .
our approach is based on collecting two complementary data sets , one where exchange cross peaks are resolved , and a second one where they are not .
this strategy is reminiscent of ( one - dimensional ) approaches comparing selective and non - selective inversion recovery measurements , as pioneered by hoffman and forsen ( 1966a , b ) . combining the two data sets
eliminates systematic errors that can be caused by differential line - broadening and enables reliable quantification of kinetic and relaxation parameters . as an example
, we have used this experimental setup to characterize conformational exchange in a selectively c labeled sample of a bistable dna oligomer .
a sample of bistable dna ( tcgtaccggaaggtacgaaccttccg ) ( puffer et al . 2009 ) was synthesized and purified as described ( kloiber et al .
2011 ) , introducing isolated methyl ( chd2 ) labels into thymidines to avoid complications due to cross - correlated relaxation in the methyl group .
the nmr sample contained 1.0  mm dna , 50  mm sodium arsenate buffer , ph6.5 , and 100% d2o , all nmr spectra were recorded at 33c .
2-dimensional longitudinal exchange experiments were recorded using pulse sequences that are based on the scheme described by farrow et al . , but employing the exchange of two - spin heteronuclear order , izsz ( farrow et al .
c frequency labeling preceded the mixing time , while in the second experiment the order of indirect evolution ( c frequency labeling ) and mixing times was interchanged .
deuterium decoupling was employed during c ( t1 ) and h ( t2 ) detection .
nmr spectra were recorded on a varian inova spectrometer operating at a larmor frequency of 500  mhz .
series of 10 2d spectra ( with and without frequency labeling ) were collected , with mixing times ranging from 10 to 800  ms .
spectra were recorded as complex data matrices composed of 512    38 points , with maximum acquisition times of 127 and 64  ms in the c and h dimensions , respectively .
32 scans per fid were obtained with a recycle delay of 2  s , resulting in a total experimental time of 18.5  h for each series .
1995 ) and in - house written fitting scripts . the t1 ( t2 ) time domain data were apodized using a gaussian function , linear predicted to 128 data points ( t1 ) , and zero filled to 4,096    256 points .
partial peak volumes were obtained by adding the intensities in boxes of 7    7 , 5    5 or 3    3 data points centered on the peak maximum . a least - squares fitting procedure
was employed to extract kab , kba and r1a , r1b by fitting the expressions given in ( 1 ) and ( 3 ) to the measured partial peak volumes , along with maa(0 ) , mbb(0 ) and ma(0 ) , mb(0 ) values , respectively . for comparison ,
the two direct correlation peaks along with the single resolved exchange cross peak for each spin label were used to extract kab , kba and r1a , r1b by fitting the appropriate expressions given in ( 1 ) and ( 2 ) to the measured partial peak volumes ( 3    3 data points ) . in all cases ,
experimental uncertainties were estimated via a monte carlo approach in which 200 synthetic data sets were generated from the best - fit values of kab , kba and r1a , r1b by adding random errors based on the signal - to - noise ratio to the best - fit curves .
errors quoted in the paper are the standard deviations in fitted parameters that were obtained in this procedure .
a sample of bistable dna ( tcgtaccggaaggtacgaaccttccg ) ( puffer et al . 2009 ) was synthesized and purified as described ( kloiber et al .
2011 ) , introducing isolated methyl ( chd2 ) labels into thymidines to avoid complications due to cross - correlated relaxation in the methyl group .
the nmr sample contained 1.0  mm dna , 50  mm sodium arsenate buffer , ph6.5 , and 100% d2o , all nmr spectra were recorded at 33c .
2-dimensional longitudinal exchange experiments were recorded using pulse sequences that are based on the scheme described by farrow et al . , but employing the exchange of two - spin heteronuclear order , izsz ( farrow et al . 1995 ) . in one experiment
c frequency labeling preceded the mixing time , while in the second experiment the order of indirect evolution ( c frequency labeling ) and mixing times was interchanged .
deuterium decoupling was employed during c ( t1 ) and h ( t2 ) detection .
nmr spectra were recorded on a varian inova spectrometer operating at a larmor frequency of 500  mhz .
series of 10 2d spectra ( with and without frequency labeling ) were collected , with mixing times ranging from 10 to 800  ms .
spectra were recorded as complex data matrices composed of 512    38 points , with maximum acquisition times of 127 and 64  ms in the c and h dimensions , respectively .
32 scans per fid were obtained with a recycle delay of 2  s , resulting in a total experimental time of 18.5  h for each series .
data were processed and analyzed using nmrpipe and nmrdraw software ( delaglio et al . 1995 ) and in - house written fitting scripts . the t1 ( t2 ) time domain data were apodized using a gaussian function , linear predicted to 128 data points ( t1 ) , and zero filled to 4,096    256 points .
partial peak volumes were obtained by adding the intensities in boxes of 7    7 , 5    5 or 3    3 data points centered on the peak maximum . a least - squares fitting procedure
was employed to extract kab , kba and r1a , r1b by fitting the expressions given in ( 1 ) and ( 3 ) to the measured partial peak volumes , along with maa(0 ) , mbb(0 ) and ma(0 ) , mb(0 ) values , respectively . for comparison ,
the two direct correlation peaks along with the single resolved exchange cross peak for each spin label were used to extract kab , kba and r1a , r1b by fitting the appropriate expressions given in ( 1 ) and ( 2 ) to the measured partial peak volumes ( 3    3 data points ) . in all cases ,
experimental uncertainties were estimated via a monte carlo approach in which 200 synthetic data sets were generated from the best - fit values of kab , kba and r1a , r1b by adding random errors based on the signal - to - noise ratio to the best - fit curves .
errors quoted in the paper are the standard deviations in fitted parameters that were obtained in this procedure .
the first one is analogous to the two - dimensional experiment described by montelione et al . , where heteronuclear chemical shift evolution precedes the mixing time and four resonances , two direct correlation and two exchange cross peaks , all frequency - labeled in both dimensions , are observed for each spin ( montelione and wagner 1989 ) . in the second experiment ,
the order of chemical shift evolution and mixing times is interchanged , so that exchange cross peaks are not resolved but remain part of the corresponding direct correlation peaks ( fig .  1 ) .
analysis of the time dependence of only the direct correlation peaks in the two experiments with and without frequency - labeling eliminates systematic errors that arise from differential line broadening effects , as described below . in the frequency - labeled experiment
, the time dependence of the direct correlation peaks of a two - site exchanging system is given by ( farrow et al .
2001)1\documentclass[12pt]{minimal } 
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				 \begin{document}$$ \begin{aligned } { { { \text{m}}_{\text{bb } } \left ( t \right ) } \mathord{\left/ { \vphantom { { { \text{m}}_{\text{bb } } \left ( t \right ) } { { \text{m}}_{\text{bb } } \left ( 0 \right ) } } } \right . \kern-\nulldelimiterspace } { { \text{m}}_{\text{bb } } \left ( 0 \right ) } } & = \frac{1}{{\left ( { \lambda_{1 } - \lambda_{2 } } \right)}}\left [ { \left ( { a_{11 } + \lambda_{1 } } \right ) \cdot e^{{\lambda_{1 } t } } - \left ( { a_{11 } + \lambda_{2 } } \right ) \cdot e^{{\lambda_{2 } t } } } \right ] \\ { { { \text{m}}_{\text{aa } } \left ( t \right ) } \mathord{\left/ { \vphantom { { { \text{m}}_{\text{aa } } \left ( t \right ) } { { \text{m}}_{\text{aa } } \left ( 0 \right ) } } } \right .
\kern-\nulldelimiterspace } { { \text{m}}_{\text{aa } } \left ( 0 \right ) } } & = \frac{1}{{\left ( { \lambda_{1 } - \lambda_{2 } } \right)}}\left [ { \left ( { a_{22 } + \lambda_{1 } } \right ) \cdot e^{{\lambda_{1 } t } } - \left ( { a_{22 } + \lambda_{2 } } \right ) \cdot e^{{\lambda_{2 } t } } } \right ] \\ \end{aligned }
$ $ \end{document}where maa and mbb are the magnitudes of the direct correlation peaks corresponding to sites a and b , which are normalized by the amount of magnetization associated with states a and b ( the magnitudes of the direct correlation peaks at the start of the mixing period , maa(0 ) and mbb(0 ) , respectively ) .
the eigenvalues of the spin density matrix which includes magnetization transfer effects due to chemical exchange , 1/2 , are given by \documentclass[12pt]{minimal } 
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				 \begin{document}$$ \lambda _
{ 1/ 2 } = 1/ 2 [   - ( { a _ { 1 1 } + { \text{a } } _ { 2 2 } ) \pm [ ( { a _ { 1 1 } - a _ { 2 2 } } ) ^ { 2 } + 4a _ { 1 2 } a _ { 2 1 } ] ^ { 1/ 2 } } ] $ $ \end{document } , where elements aij are defined as \documentclass[12pt]{minimal } 
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				 \begin{document}$$ a _ { 1 1 } = k_{\text{ab } } + r _ { 1}^{\text{a } } , a _ { 2 2 } = k_{\text{ba } } + r _ { 1}^{\text{b } } , a _ { 1 2 } = - k_{\text{ba } } $ $ \end{document } and \documentclass[12pt]{minimal } 
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				 \begin{document}$$ a _ { 2 1 } = - k_{\text{ab } } $ $ \end{document}. parameters kab and kba denote the microscopic kinetic rate constants that describe the interconversion between states a and b , while r1a and r1b are the longitudinal relaxation rates of magnetization in sites a and b. likewise , the normalized magnitudes of the associated exchange cross peaks in the frequency - labeled experiment are given by ( farrow et al .
2001)2\documentclass[12pt]{minimal } 
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				 \begin{document}$$ \begin{aligned } { { { \text{m}}_{\text{ab } } \left ( t \right ) } \mathord{\left/ { \vphantom { { { \text{m}}_{\text{ab } } \left ( t \right ) } { { \text{m}}_{\text{aa } } \left ( 0 \right ) } } } \right . \kern-\nulldelimiterspace } { { \text{m}}_{\text{aa } } \left ( 0 \right ) } } & = \frac{1}{{\left ( { \lambda_{1 } - \lambda_{2 } } \right)}}\left [ { - a_{21 } \cdot e^{{\lambda_{1 } t } } + a_{21 } \cdot e^{{\lambda_{2 } t } } } \right ] \\ { { { \text{m}}_{\text{ba } } \left ( t \right ) } \mathord{\left/ { \vphantom { { { \text{m}}_{\text{ba } } \left ( t \right ) } { { \text{m}}_{\text{bb } } \left ( 0 \right ) } } } \right .
\kern-\nulldelimiterspace } { { \text{m}}_{\text{bb } } \left ( 0 \right ) } } & = \frac{1}{{\left ( { \lambda_{1 } - \lambda_{2 } } \right)}}\left [ { - a_{12 } \cdot e^{{\lambda_{1 } t } } + a_{12 } \cdot e^{{\lambda_{2 } t } } } \right ] . \\ \end{aligned } $ $ \end{document } in the experiment without frequency labeling only two direct correlation peaks are observed , one of them with a magnitude , ma , that corresponds to the sum of aa and ab , while the magnitude of the second peak , mb , results from bb and ab .
the normalized dependencies of ma and mb are then given by3\documentclass[12pt]{minimal } 
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				 \begin{document}$$ \begin{aligned } { { { \text{m}}_{\text{a } } \left ( t \right ) } \mathord{\left/ { \vphantom { { { \text{m}}_{\text{a } } \left ( t \right ) } { { \text{m}}_{\text{a } } \left ( 0 \right ) } } } \right . \kern-\nulldelimiterspace } { { \text{m}}_{\text{a } } \left ( 0 \right ) } } & = \frac{1}{{\left ( { \lambda_{1 } - \lambda_{2 } } \right)}}\left [ { \left ( { a_{22 } - a_{21 } + \lambda_{1 } } \right ) \cdot e^{{\lambda_{1 } t } } - \left ( { a_{22 } - a_{21 } + \lambda_{2 } } \right ) \cdot e^{{\lambda_{2 } t } } } \right ] \\ { { { \text{m}}_{\text{b } } \left ( t \right ) } \mathord{\left/ { \vphantom { { { \text{m}}_{\text{b } } \left ( t \right ) } { { \text{m}}_{\text{b } } \left ( 0 \right ) } } } \right . \kern-\nulldelimiterspace } { { \text{m}}_{\text{b } } \left ( 0 \right ) } } & = \frac{1}{{\left ( { \lambda_{1 } - \lambda_{2 } } \right)}}\left [ { \left ( { a_{11 } - a_{12 } + \lambda_{1 } } \right ) \cdot e^{{\lambda_{1 } t } } - \left ( { a_{11 } - a_{12 } + \lambda_{2 } } \right ) \cdot e^{{\lambda_{2 } t } } } \right ] \\ \end{aligned } $ $ \end{document}where ma(0 ) and mb(0 ) are the magnitudes of the peaks a and b at zero mixing time . using the standard frequency - labeled strategy to extract r1a , r1b and kab , kba involves simultaneous least - squares fitting of ( 1 ) and ( 2 ) to exchange cross peaks and direct correlation peaks ( fig .  2 ) .
it is obvious from ( 2 ) , however , that this requires the normalization of the two exchange cross peaks mab(t ) , mba(t ) by the direct correlation peaks maa(0 ) , mbb(0 ) , respectively . because mab and maa ( as well as mba and mbb ) have different line widths in the direct ( t2 ) dimension in most cases ( see fig . 
3 ) , special care has to be taken to ensure precise determination of peak volumes in order to avoid errors due to systematic over- or underestimation of exchange cross peak magnitudes , and concomitant over- or underestimation of the corresponding kinetic rate constants .
this complication can be avoided by using the four direct correlation peaks in experiments with and without frequency - labeling , maa , mbb and ma , mb , respectively , for the analysis of longitudinal exchange . in this case
all peaks are normalized by their respective magnitudes at the start of the mixing period , maa(0 ) , mbb(0 ) and ma(0 ) , mb(0 ) , with inherently identical line widths .
thus , longitudinal relaxation and kinetic rate constants ( r1a , r1b and kab , kba ) can be extracted from the four direct correlation peaks by simultaneous fitting of ( 1 ) and ( 3 ) to the experimental data , yielding values that are not distorted by differential line - broadening effects .
consequently , the results are independent of the peak integration method , i.e. intensities , partial peak volumes or ( exact ) peak volumes can be used .
we have applied this simple strategy to a bistable dna 26-mer that exists in two different conformational ( hairpin ) sub - states a and b , which interconvert slowly on the nmr chemical shift time scale ( puffer et al .
both states are populated to a considerable extent at ambient temperature and in aqueous buffer ( ph 6.5 ) , and discrete sets of resonances are observed for a and b ( fig . 
selective c labeling ( chd2-methyl ) in thymidines enables the measurement of longitudinal exchange ( 2-spin order ) between folds a and b in time series of 2-dimensional experiments performed with and without c frequency - labeling , respectively .
two thymidines , t4 and t22 , were chosen as probes for the interconversion process between the two sub - states because the t22 spin label clearly displays differential line - broadening in folds a and b , consistent with the presence of an a - t / a - t step ( a10-t23/a11-t24 ) in fold b ( lingbeck et al .
the ab exchange cross peaks of t4 and t22 overlap at 500  mhz and 33c in the chd2-methyl labeled sample , preventing straightforward analysis of the interconversion process between folds a and b by the standard ( frequency - labeled ) approach.fig . 
a thymidine methyl group region of a 2-dimensional hc correlation spectrum without ( top ) and with ( bottom ) frequency - labeling in ( t1 ) ( mixing time 120  ms ) .
duplex dna , which does not participate in the exchange process , is indicated by an asterisk .
b normalized partial peak volumes of a , b , aa and bb direct correlation peaks of spin labels t4 and t22 , together with best - fit curves ( dashedlines for a , b , solidlines for aa , bb ) .
all four curves ( ( 1 ) and ( 3 ) ) were fitted simultaneously to the experimental data for each spin label to yield kinetic rate constants kab  =  0.51    0.08  s ( t4 ) , kab  =  0.42    0.07  s ( t22 ) , kba  =  0.41    0.07  s ( t4 ) , kba  =  0.49    0.06  s ( t22 ) , and longitudinal relaxation rate constants r1a  =  1.21    0.04  s ( t4 ) , r1a  =  0.67    0.05  s ( t22 ) , r1b  =  0.90    0.04  s ( t4 ) and r1b  =  1.59    0.04  s ( t22 ) .
uncertainties were derived from a montecarlo approach experimental longitudinal exchange data recorded for ( t4 , t22)-methyl c labeled bistable dna .
a thymidine methyl group region of a 2-dimensional hc correlation spectrum without ( top ) and with ( bottom ) frequency - labeling in ( t1 ) ( mixing time 120  ms ) .
duplex dna , which does not participate in the exchange process , is indicated by an asterisk .
b normalized partial peak volumes of a , b , aa and bb direct correlation peaks of spin labels t4 and t22 , together with best - fit curves ( dashedlines for a , b , solidlines for aa , bb ) .
all four curves ( ( 1 ) and ( 3 ) ) were fitted simultaneously to the experimental data for each spin label to yield kinetic rate constants kab  =  0.51    0.08  s ( t4 ) , kab  =  0.42    0.07  s ( t22 ) , kba  =  0.41    0.07  s ( t4 ) , kba  =  0.49    0.06  s ( t22 ) , and longitudinal relaxation rate constants r1a  =  1.21    0.04  s ( t4 ) , r1a  =  0.67    0.05  s ( t22 ) , r1b  =  0.90    0.04  s ( t4 ) and r1b  =  1.59    0.04  s ( t22 ) .
uncertainties were derived from a montecarlo approach figure  4 shows the experimental data obtained for the aa , bb , a and b direct correlation peaks in experiments with and without frequency - labeling , together with best - fit curves using ( 1 ) and ( 3 ) .
kinetic rate constants of kab  =  0.51    0.08  s ( t4 ) and kab  =  0.42    0.07  s ( t22 ) , as well as kba  =  0.41    0.07  s ( t4 ) and kba  =  0.49    0.06  s ( t22 ) were obtained , with uncertainties derived from a montecarlo approach . despite the considerable difference in line - broadening between
folds a and b for t22 , the extracted kinetic rates for this spin label are in good agreement with those of t4 , which is consistent with a collective two - state conformational transition between the two folds .
as expected , the extracted parameters do not depend on the peak integration method : using partial peak volumes by adding the intensities in 7    7 grids centered on the peak maximum yields values that are identical ( within experimental uncertainties ) to the ones obtained using 5    5 or 3    3 grids , or by using peak intensities . for comparison ,
the frequency - labeled data set was used to determine kinetic rate constants by analyzing the single ( ba ) exchange cross peak that is resolved for each spin label along with its two direct correlation peaks . in this case , only values for one rate constant ( kba ) could be determined reliably . only for the t4 spin
label the value of kba ( 0.58    0.10  s ) is in good agreement with the value obtained by analyzing aa , bb , a and b direct correlation peaks , while for t22 the magnitudes of normalized ba exchange cross peaks are overestimated due to considerable line - broadening of fold b , which results in an increase of kba ( 1.17    0.21  s ) .
it is evident from ( 1 ) and ( 3 ) and fig .  2 that the bi - exponential decays of a and b direct correlation peaks differ from those of the aa and bb direct correlation peaks by an amount that corresponds to the pertinent exchange peak.1 thus
, subtraction of normalized aa intensities from a ( and bb intensities from b ) can , in principle , be employed to formally reconstruct the ( correctly ! ) normalized magnitudes of the corresponding exchange cross peaks ab ( ba ) that are not affected by differential line - broadening , as they are computed from two peaks with identical line - widths , and irrespective of cross peak overlap .
these exchange cross peak intensities can then be analyzed , together with aa and bb direct correlation peaks , as in the conventional frequency - labeled approach using ( 1 ) and ( 2 ) to obtain longitudinal exchange and kinetic rate constants .
we opt not to do so , however , since analysis of the four direct correlation peaks directly is robust and straightforward .
we have presented an alternative way of analyzing longitudinal exchange in two - state exchanging systems by recording pairs of two- ( or more ) dimensional spectra with and without frequency - labeling in an indirect dimension .
analysis of the time - dependence of the four ( 2  +  2 ) direct correlation peaks that are observed in the two experiments enables reliable and robust quantification of the interconversion rates between the two species , along with their longitudinal relaxation rates .
this approach partly resolves the issue of cross peak overlap and eliminates errors caused by differential line - broadening of the two species , thereby expanding the applicability of longitudinal exchange experiments to systems with additional conformational exchange on the mss time scale . while our approach increases the experimental time by a factor of two , it significantly reduces the number of sources for systematic errors .
we recommend to generally use this strategy in cases where ( 1 ) resonances show different line broadening , or ( 2 ) exchange cross peaks are overlapped or close to intense direct correlation peaks ( which may preclude the determination of their exact volumes ) .
 longitudinal exchange experiments facilitate the quantification of the rates of interconversion between the exchanging species , along with their longitudinal relaxation rates , by analyzing the time - dependence of direct correlation and exchange cross peaks . here 
 we present a simple and robust alternative to this strategy , which is based on the combination of two complementary experiments , one with and one without resolving exchange cross peaks . 
 we show that by combining the two data sets systematic errors that are caused by differential line - broadening of the exchanging species are avoided and reliable quantification of kinetic and relaxation parameters in the presence of additional conformational exchange on the mss time scale is possible . 
 the strategy is applied to a bistable dna oligomer that displays different line - broadening in the two exchanging species . 