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A multidimensional approach to the analysis of chemical shift titration experiments in the frame of a multiple reaction scheme

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A multidimensional approach to the analysis of chemical shift titration experiments in the frame of a multiple reaction scheme
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  A multidimensional approach to the analysisof chemical shift titration experiments in theframe of a multiple reaction scheme Anthony D ’ Aléo, a, † Elise Dumont, a Olivier Maury a and Nicolas Giraud b * We present a method for  fi tting curves acquired by chemical shift titration experiments, in the frame of a three-step complex-ation mechanism. To that end, we have implemented a  fi tting procedure, based on a nonlinear least squares  fi tting method,that determines the best  fi tting curve using a  “ coarse grid search ”  approach and provides distributions for the different pa-rameters of the complexation model that are compatible with the experimental precision. The resulting analysis protocol is fi rst described and validated on a theoretical data set. We show its ability to converge to the true parameter values of the sim-ulated reaction scheme and to evaluate complexation constants together with multidimensional uncertainties. Then, we applythis protocol to the study of the supramolecular interactions, in aqueous solution, between a lanthanide complex and threedifferent model molecules, using NMR titration experiments. We show that within the uncertainty that can be evaluated fromthe parameter distributions generated during our analysis, the af  fi nities between the lanthanide derivative and each modelmolecule can be discriminated, and we propose values for the corresponding thermodynamic constants. Copyright © 2013John Wiley & Sons, Ltd.Keywords:  NMR  1 H; titration experiment; paramagnetic shift; fast exchange; thermodynamic constants determination; manyparameter curve  fi tting; multidimensional error bar Introduction NMR has proved to be a highly versatile tool, not only for probingever more complex molecular systems but also for monitoringthe dynamic equilibria that are developing when chemical spe-cies are interacting — or reacting — together. In the  fi eld of coordi-nation chemistry, this latter quality has made NMR spectroscopyan essential technique, notably to investigate the formation of supramolecular complexes. [1] Among the wide range of tech-niques that have been implemented to characterize the interac-tion network structuring supramolecular assemblies, [2] methodshave been developed to provide a thermodynamic insight intothe chemical equilibrium that is taking place when the speciesunder study are interacting with each other. [3] In this context, we have recently shown that  tris (dipicolinate) – lanthanide complexes ([Ln(DPA) 3 ] 3  (DPA=dipicolinate=pyridine-2,6-dicarboxylate); Fig. 1) can co-crystallize with proteins throughsupramolecular interactions, [4] which has highlighted the potentialof this class of lanthanide complexes for high-resolution structuredetermination of proteins. [5] The analysis of the binding mode be-tween this lanthanide derivative and different model proteins hasrevealed preferential af  fi nities for three cationic amino acids atphysiological pH: arginine, lysine and histidine (also noted R, K, H,accordingtotheconventionalone letter code).Todecipherthede-tail of the interaction mechanism with each of these residues, wehave chosen to introduce simple molecular models that mimic ar-ginine, histidine and lysine side chains, namely, ethylguanidinium(EtGua + ), imidazolium (Imz + ) and ethylammonium (EtNH 3+ ), respec-tively (Fig. 1; in the following, these molecules will be referred toammonium models, A M ). In this article, we aimed at determiningif there are any signi fi cant differences, at thermodynamic level,between the interactions that are created by the  tris -dipicolinateterbium complex with each ammonium model A M .Extending a preliminary analysis, [6] we have carried out, foreach A M cation, chemical shift titration experiments (Fig. 2). Theseexperiments take advantage of the strong pseudo-contact shiftinduced by the Tb(III) ion in the lanthanide complex, [7] whichallows to evidence the long range interactions occurring in thesecond sphere of this paramagnetic shift reagent. [8] Since eachA M cation is involved in fast equilibrium, when it is added to a so-lution of [Tb(DPA) 3 ] 3  , between the solvent and the second coor-dination sphere of the Terbium complex, an average shift can bemeasured for its proton spins. This shift depends on the quantityof A M that is introduced as well as on the af  fi nity constants of thesuccessive interaction reactions with the lanthanide complex.However, a fast and accurate determination of these thermo-dynamic constants in such system, which is a prerequisite for aproper comparison between the af  fi nities of the terbium com-plex for the different ammonium models, requires that somemethodological dif  fi culties should be overcome. First, in this * Correspondence to: Dr. Nicolas Giraud, Laboratoire de RMN en milieu orienté,ICMMO, UMR 8182 CNRS, Université Paris-Sud, bat 410, 91405 Orsay cedex,France. E-mail: nicolas.giraud@u-psud.fr  †  Present address: Aix-Marseille Universite, CNRS, CINaM UMR 7325, Campus deLuminy, Case 913, 13288 Marseille (France). E-mail: daleo@cinam.univ-mrs.fr  a Laboratoire de Chimie, UMR 5182 CNRS, ENS-Lyon, 46 allée d  ’   Italie, 69364 Lyoncedex 07, France b Laboratoire de RMN en milieu orienté, ICMMO, UMR 8182 CNRS, Université Paris-Sud, bat 410, 91405 Orsay cedex, France Magn. Reson. Chem.  2013 ,  51 , 641 – 648 Copyright © 2013 John Wiley & Sons, Ltd. Research Article Received: 27 April 2013 Revised: 28 June 2013 Accepted: 11 July 2013 Published online in Wiley Online Library: 16 August 2013 (wileyonlinelibrary.com) DOI 10.1002/mrc.3994  6  4 1   particular case where most of the chemical species can only existas mixtures, it is not possible to measure accurately the local shiftthat is experienced by an A M proton when it is either interactingwith the terbium complex or solvated. Second, there is a complexrelationship between (a) the experimental uncertainty, (b) the in-teraction model that is chosen to describe the shifts variationsand (c) the accuracy with which any thermodynamic or NMR pa-rameter can be evaluated from the  fi t of a model curve to theseexperimental data. The combination of these limitations withina crude  “ coarse grid search ”  approach results in general in a dra-matically lengthened computational time needed to constrain allthe parameters of the theoretical model. In addition, it is knownthat such a high number of parameters would also contributeto making the determination of the  K  k  ’ s all the less accurate: ina related area, Masiker  et al  . [9] have shown, using Monte Carlosimulations, that errors can propagate from chemical shift mea-surements to the determination of stability constants when theyhave used NMR chemical shift titration data to investigate thecomplexation of 12-crown-4 (12C4) with lithium or sodium cat-ions, in acetonitrile or methanol solvents.A rigorous comparison of the interaction strengths withinthese supramolecular assemblies thus implies that we evaluatewhat the accuracy on a given parameter is, when it has been de-termined over the analysis of NMR titration curves. With that aim,we have developed an improved analysis protocol that provides,in addition to the best  fi tting curve, statistical distributions mea-suring the deviation induced by the experimental uncertainty onthe determination of thermodynamic parameters. In other words,we propose to build multidimensional error bars that will comewith the determination of a given set of parameters. In the fol-lowing, we introduce the general scheme for a multidimensional fi tting of NMR titration curves, and we apply this protocol to theanalysis of the data that were recorded for each A M cation. Titration Curves Fitting Protocol Complexation model Similarly to our previous work, we have chosen to model the evo-lution of the proton shifts in A M upon addition to the terbiumcomplex, by a series of three successive equilibriums correspond-ing to the formation of the 1:1, 2:1 and 3:1 A M -[Tb(DPA) 3 ] 3  ad-ducts (noted A M Tb, A M2  Tb and A M3  Tb) with constants  K  1 ,  K  2  and K  3 , respectively (Scheme 1). [6] When A M is added to a solutionof [Tb(DPA) 3 ] 3  , it is involved in a rapid equilibrium between afree (solvated) state and the different adducts that result from aprogressive occupation of [Tb(DPA) 3 ] 3  interaction sites.Although a model that involves the formation of only 1:1 and2:1 adducts would yield an acceptable simulation of the NMRdata, the presence of the 3:1 adduct, that corresponds to thefavored formation of a neutral complex, is indicated by crystallo-graphic data, and has therefore been included in the equilibrium(adducts with a higher A M :Tb ratio are not considered). In thismodel, all the ion-pairs are supposed to be fully solvated inaqueous solution, and the contributions from counter-ions areneglected.Two additional relationships allow accounting for the conser-vation of matter:C 0  ¼  Tb ½  þ  A M Tb    þ  A M2 Tb    þ  A M3 Tb    (1) n eq C 0  ¼  A M    þ  A M Tb    þ  2 A M2 Tb    þ  3 A M3 Tb    (2)where  C  0  is the overall concentration in  Tb  and  n eq  is the numberof equivalents of A M that is added to the solution at each step of the titration experiment. Two proton sites were probed upon theinteraction process for each investigated ammonium model: weassume that each proton  H  i  ( i  =1, 2) undergoes a  δ  free i  shift when Figure 1.  Molecular structures of (a)  fi rst-shell  tris -dipicolinate lantha-nide complex and molecular ammonium models (b) ethylguanidinium(EtGua + ), (c) ethylammonium (EtNH 3+ ) and imidazolium (Imz + ). Figure 2.  NMR titration curves which were built from the measuredvariations in proton shifts with increasing A M equivalents (a)ethylguanidinium, (b) imidazolium and (c) ethylammonium ( n eq =[A M ] / [Cs 3 [Tb(DPA) 3 ]], in a solution of Cs 3 [Tb(DPA) 3 ]·7H 2 O (solvent: D 2 O; protonresonance frequency: 200MHz; T=298K;  C  0 =(a) 15.5mol.L  1 , (b)16.2mol.L  1 and (c) 16.2mol.L  1 ). Protons labelling refers to the assign-ment that is indicated on the ammonium structures displayed in Fig. 1. A. D ’ Aléo  et al  . wileyonlinelibrary.com/journal/mrc  Copyright © 2013 John Wiley & Sons, Ltd.  Magn. Reson. Chem.  2013 ,  51 , 641 – 648  6  4 2    A M is in solution and  δ  comp i  when it belongs to the second coordi-nation sphere of the terbium complex with a 1:1, 2:1 or 3:1 ratio.Under these conditions, in a fast exchange regime, we remindthat the experimental  1 H shifts  δ  exp i  ( i= 1, 2) can be expressed as: δ  exp i   ¼ δ  free i     A M    þ  δ  comp i     ∑ 3 k  ¼ 1 k    A Mk   Tb   n eq  C  0 ;  i   ¼  1 ; 2 ð Þ  (3)where [A M ] and [A M k  Tb ] ( k  =1, 2, 3) are the concentrations of theammonium model when it is free and in interaction with the ter-bium complex ( Tb ), respectively.The number of parameters of the interaction model that haveto be adjusted constitutes the main issue regarding the computa-tional time needed. Indeed, for any set of two titration curves, thequality of the  fi t depends on seven independent parameters:three adducts formation constants ( K  1 ,  K  2  and  K  3 ) and four chem-ical shifts ( δ  free 1  , δ  free 2  , δ  comp1  and δ  comp 2  ). Although each titration curve isa function of only two chemical shifts ( δ  free i   and  δ  comp i   ,  i  =1 or 2),the thermodynamic constants  K  k   ( k= 1, 2 or 3) contribute to both:the accuracy to which they can be determined may thus be en-hanced if both curves are  fi tted. As a result, we have chosen (1)to  fi t each curve separately, using an adapted  fi t procedure togenerate  K  k  ’ s distributions in a reasonable computational time,and then (2) to cross these distributions to improve theaccuracy on the determination of   K  k  ’ s. To that end, we haveimplemented the following protocol, which derives from thesrcinal nonlinear  fi t approach:1.  δ  free i  ( i  =1, 2) is evaluated from the last points of the NMRtitration curves, in the framework of the approximation[A M3  Tb] ≈ C  0  (i.e. for high  n eq  values).2. The best  fi t for each titration curve is determined over a “ coarse grid search ” : for each set of values of the parameters K  1 ,  K  2 ,  K  3  and  δ  comp i   taken at  fi xed intervals between  K  1min and  K  1max ,  K  2min and  K  2max ,  K  3min and  K  3max , and  δ  comp min i   and δ  comp max i   , respectively, the corresponding theoretical shifts δ  calc i   ( i  =1, 2) are calculated for each point of the studiedtitration curves, within the supramolecular association model. δ  calc i   is then compared with the experimental values  δ  exp i   , byevaluating a reliability factor that satis fi es the following rela-tionship: R i   ¼  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ∑ n δ  calc i   n ½    δ  expi   n ½    2 ∑ n δ  expi   n ½    2 v uuuut ;  i   ¼  1 ; 2 ð Þ  (4)where  n  corresponds to the  n th experimental point from each ti-tration curve, and  δ  calc i   and  δ  exp i   are the calculated and the exper-imental shifts respectively. We note that  R i   is a complex functionof the whole set of thermodynamic and NMR parameters:  R i  =f  ( K  1 ,  K  2 ,  K  3 , δ  comp i   ) . The set of parameters ( K  1 ,  K  2 ,  K  3 , δ  comp i   ) that leadsto the minimization of   R i   is selected as the best  fi t.3. Eachpoint (i.e. each shift value) fromthe best fi t foundat step2is then assumed to be at the center of a Gaussian distribution,withstandarddeviationtheexperimentalaccuracyofshiftmea-surements. A set of 1024 simulated curves is then calculated foreach proton site, with each point randomly generated fromthese normal distributions. The average value  |R i  | ref   ( i  =1, 2) of the reliability factor is then calculated between each simulatedcurve and the best  fi t, over this ensemble of simulations.4. The parameter space around the best values which were de-termined for  K  1 ,  K  2 ,  K  3  and  δ  comp i   at step 2 is then explored,again over a  “ coarse grid search ” : every set of parameters thatyields a simulated curve whose reliability factor satis fi es thecondition  “  R i  < |R i  | ref   ”  (calculated at step 3) is added to theparameter distributions.5. Theintersectionofthedistributionsthatwerecalculatedforeachtitration curve is  fi nally determined: it contains { K  1 ,  K  2 ,  K  3 ,  δ  comp1  , δ  comp2  } sets that are compatible with experimental uncertaintyfor the whole NMR data set, for a given ammonium model.We note that we did not implement any gradient method (orone of its derivatives) to converge toward the best  fi t more rapidlybecause on the one hand the ef  fi ciency of such method applied toa nonlinear  fi t depends heavily on the numerical accuracy withwhich the parameter space is explored (notably to avoid localminima).Ontheotherhand,weaimedatbuildingparameterdistri-butions,whichrequiresthatthecorrespondingparametersspaceisuniformly sampled (hence the  “ coarse grid search ”  strategy). Simulated set of NMR data Before the analysis of the experimental NMR data sets recordedon each A M , we have  fi rst generated a theoretical set of two titra-tion curves (with known thermodynamical and NMR parametervalues), and we have evaluated the ability of our  fi tting protocolto extract accurately these parameters. We have chosen  “ aver-age ”  values for adduct formation constants ( K  1th =10 1.7 ,  K  2th =10 1.0 , K  3th =10 1.0 ), and the A M proton shifts in solution ( δ  free th 1  = 4.5ppm, δ  free th 2  = 3.7ppm), or coordinated ( δ  free th 1  =   2.0ppm,  δ  free th 2  =   5.6ppm). Fig. 3 shows the evolution of the proton shifts and the con-centration pro fi les of the different chemical species that havebeen calculated over the simulation of an NMR titration experi-ment, within the interaction model described above.Experimental uncertainty has been accounted for by adding toeach calculated proton shift value a random deviation. We notethat the uncertainty of shift measurements in our data accountsfor the resolution of the NMR spectrometer (operating at a pro-ton resonance frequency of 200MHz), the loss of resolution due Scheme 1.  Schematic representation of the interaction between Cs 3 [Tb(DPA) 3 ] and A M in aqueous solution, that has been modelled according to ex-perimental  1 H NMR data. Water molecules that are involved in the  fi rst hydration shell of the terbium complex have been omitted because their con-centration does not affect the equilibrium. Tb indicates [Tb(DPA) 3 ] 3  . Analyzing NMR data with multidimensional error bars Magn. Reson. Chem.  2013 ,  51 , 641 – 648 Copyright © 2013 John Wiley & Sons, Ltd.  wileyonlinelibrary.com/journal/mrc  6  4  3    to signal broadenings resulting from paramagnetic relaxationand the accuracy of weighing of the different chemical species(which in fl uence the value of   n eq ). The number of points, the con-centration range over which the curves are simulated as well asthe order of magnitude of the random deviation are essentiallysimilar to those in our experiments. Direct determination of   δ free th 1  (  j  =1, 2) To reduce the number of parameters that need to be adjustedover the  fi t — and thus avoid a computational bottleneck  — we ex-amine in this part the possibility to determine  δ  free th 1  and  δ  comp i   di-rectly from each titration curve. We note that for high values of  n eq , the 3:1 adduct is predominant. When this approximation isapplied to Eqns (1) and (2), it allows to establish a simpli fi ed rela-tionship between [A M3  Tb], [A M ] and  C  0 :1bisA M3 Tb   ≈ C 0  (1bis)2bisA M   ≈ C 0    n eq – 3    (2bis)Under the same assumption, in the high  n eq  values region, Eqn3 becomes3bis δ  exp j   ≈ A M   δ  free  j   þ  3 A 3M Tb ½  δ  comp  j  n eq C  0  j   ¼  1 ; 2 ð Þ  (3bis)Finally, the combination of Eqs (1bis), (2bis) and (3bis) leads tothe simpli fi ed expression (which is available in principle for highvalues of   n eq ):3ter n eq    δ   j  exp ≈ n eq    δ   j  free þ  3    δ   j  comp   δ   j  free    j   ¼  1 ; 2 ð Þ  (3ter)The chemical shift values  δ  free  j   and  δ  comp  j   can hence in principlebe extracted from the slope and the  y  -intercept of the linear func-tions  n eq   δ  exp  j   = f   ( n eq ) (  j= 1, 2). In Fig. 4,  n eq   δ  th  j   has been cal-culated for the last points of the simulated shift curves (displayedin Fig. 3a) and is plotted against  n eq . First, we observe a good lin-earity for the simulated points. We note that the difference be-tween the shifts calculated from the slopes of these linearcurves ( δ  free 1  = 4.53ppm and  δ  free 2  = 3.69ppm) and the real valueswhich were entered as inputs to this simulation ( δ  free 1  =4.50ppmand  δ  free 2  = 3.70ppm) is the same order as the  “ experimental ”  un-certainty. Second, we remark that  δ  comp  j   could in theory be deter-mined over the same linear regression from the  y  -intercepts.To evaluate the accuracy on this determination of   δ  free  j   and δ  comp  j   , we have simulated an ensemble of 1024 theoretical titra-tion curves: for each  n eq  value,  δ   j  th (  j  =1,2) was calculated fromthe same set of values ( K  1th ,  K  2th ,  K  3th ,  δ  free th 1  ,  δ  c th 1  ,  δ  free th 2  ,  δ  c th 2  ), and af-fected by a random deviation to mimic experimental uncertaintyon the measurement of a chemical shift. For each curve, the lastfour points (corresponding to  n eq > 50) were analyzed within theapproximation mentioned above. Three different quantitiescould be evaluated for each proton site: (i) the average valuesof   δ  free  j   and  δ  comp  j   , (ii) the standard deviations for each ensemble δ  free  j   and δ  comp  j   and (iii) the average reliability factor calculated be-tween the ensemble of simulated curves, and the ideal curve thatis obtained using the exact values for  K  1 ,  K  2 ,  K  3 , δ  free  j   and δ  comp  j   , andassuming no  “ experimental ”  uncertainty ( i.e . adding no randomdeviation upon chemical shifts calculation). These results aresummarized in Table 1. First, we observe that the accuracy onthe determination of   δ  free  j   (that is measured by the standard devi-ation of the generated ensemble) is acceptable, regarding the ac-tual  “ experimental accuracy. ”  Second, we remark that thestandard deviation for  δ  comp  j   does not allow an accurate evalua-tion of this parameter through this approach because the quan-tity 3  (  δ  comp  j     δ  free  j   ) is highly sensitive to the inherentuncertainty of these data. The value of the average reliability Figure 3.  Semilogarithmic plots of (a) the theoretical evolutions  δ   j  th (  j= 1, 2) of two A M proton shifts that have been calculated for the additionof   n eq  equivalents of A M to a solution of Cs 3 [Tb(DPA) 3 ].7H 2 O (initial con-centration:  C  0 =15.5  10  3 mol.L  1 ) and (b) the evolution of the composi-tion of the solution. In panel a, plain lines correspond to  “ exact ” evolutions (i.e. proton shifts calculations without experimental deviation),and plain circles are the  “ experimental ”  points, which include a randomdeviation representing the experimental uncertainty on proton shiftsmeasurements (this deviation is generated from a Gaussian distribution,of standard deviation 0.04ppm). The concentrations in the differentchemical species, and the resulting shifts were calculated within theinteraction mechanism detailed in scheme 1. Figure 4.  A plot of   n eq   δ  th  j   against  n eq , that was calculated for bothsimulated shift evolutions from Figs. 3a, for  n eq  values where Eqns (1bis)and (2bis) hold. The  δ  free  j   values determined from the slopes of the linearregressions are also indicated (real values are given between brackets).The coef  fi cient of determination is higher than 0.9998 for both linearregressions. A. D ’ Aléo  et al  . wileyonlinelibrary.com/journal/mrc  Copyright © 2013 John Wiley & Sons, Ltd.  Magn. Reson. Chem.  2013 ,  51 , 641 – 648  6  4 4   factor that is obtained in this simulation will be used in thefollowing as a criterion to select data sets that are compatible withthe experimental accuracy, regarding their deviation to the best  fi t.To conclude for this part, we note that the analysis of the pointsfrom each titration curve that correspond to high values of   n eq leads to a good estimation of   δ  free  j   . This allows us to reduce thenumber of parameters ( K  1 ,  K  2 ,  K  3  and δ  comp  j   ) that have to be deter-mined using an iterative minimization of the reliability factor. Convergence of the nonlinear  fi t toward the true values of the theoretical model. We have performed a  “ coarse grid search ”  to determine the best fi t to each theoretical titration curve. We note that because thehigh A M concentration displaces each interaction equilibrium to-ward the formation of mainly the  tris  adduct [A M3  Tb] for high  n eq values, the last points of the titration curves do not have asigni fi cant impact on the value of the reliability factor. We havethus excluded these last points (that are used to determine δ  free  j   in the previous section) from the  fi tting procedure to shortencalculation time. Distributions of parameter sets { K  1 ,  K  2 ,  K  3 ,  δ  comp  j   , R  j  }  j=1,2  were built, by selecting sets allowing a correct  fi t of eachtitration curve (according to the  “  R  j  < |R  j  | av  ”  criterion). The best fi t was de fi ned for each distribution as the one correspondingto the smallest reliability factor evaluated in each case.Fig. 5 shows an overview of the  fi t convergence for each theo-retical curve, as a function of the overall thermodynamical con-stant (which measures the global stability of the supramolecularcomplex) and the shift  δ  comp  j   . We observe that within the uncer-tainty that is allowed by the  |R  j  | av  value, a great number of modelcurves are shown to  fi t each theoretical curve within an accept-able reliability factor. We also make the remark that althoughthe numerous parameters of the interaction model tend to com-pensate each other to lead, as expected, to a great number of  Figure 5.  The evolution of the reliability factor against the quantity (log 10 K  1 +log 10 K  2 +log 10 K  3 ) and δ  comp  j   (  j  =1,2) is shown for each theoretical titrationcurve. Each point corresponds to a set of values { K  1 ,  K  2 ,  K  3 ,  δ  comp  j   ,  R  j  }  j=1,2  that satis fi es the condition  R  j  < | R  j  | av  and is colored according to the value of itsreliability factor (red: highest value / blue lowest value, for each curve). This simulation was led by incrementing log 10 K  1  from 0.7 to 2.7 (by steps of 0.05),log 10 K  2  from 0.0 to 2.0 (0.05), log 10 K  3  from   2.0 to 2.0 (0.05),  δ  comp 1  from   4.8 to 1.56ppm (0.04ppm) and  δ  comp 2  from   8.44 to   1.68ppm (0.04ppm). Table 1.  In fl uence of the accuracy on chemical shift measurement on the determination of   δ  free  j   and  δ  comp  j  Curve number (  j  )  j= 1  j= 2 δ  free  j   (actual value) 4.49ppm (4.50ppm) 3.69ppm (3.70ppm)Standard deviation on  δ  free  j   0.055ppm 0.059ppm δ  comp  j   (actual value)   1.62ppm (  2.00ppm)   5.07ppm (  5.60ppm)Standard deviation on  δ  comp  j   1.59ppm 1.69ppm| R  j  | av  0.017 0.035 Figure 6.  A plot of the { K  1 ,  K  2 ,  K  3 } distributions that have been built, using a  “ coarse grid search, ”  for the model curves (a) and (b) (calculation details aregiven in Fig. 2). (c) The intersection between both  K  i  ’ s distributions. The actual value { K  1th ,  K  2th ,  K  3th } is indicated with dotted lines on each distribution. Thebest  fi t is indicated with plain lines for each distribution. Analyzing NMR data with multidimensional error bars Magn. Reson. Chem.  2013 ,  51 , 641 – 648 Copyright © 2013 John Wiley & Sons, Ltd.  wileyonlinelibrary.com/journal/mrc  6  4  5 
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