Supporting Information - University of...
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SupportingInformation
Harnessing NMR relaxation interference effects to characterize
supramolecularassemblies
Gogulan Karunanithy, Arjen Cnossen, Henrik Muller, Martin D. Peeks, Nicholas H. Rees,
TimothyD.W.Claridge,HarryL.AndersonandAndrewJ.Baldwin
S.1Synthesisofporphyrins
Sixporphyrinderivativeswereconsideredhere:P1(R=t-Bu),l-P2(R=t-Bu),c-P6(R=THSandR=
C8H17),c-P6•T6(R=THS)andl-P6(R=THS).TheRgroupsareattachedatthemetapositionsofthearyl
rings (Main article Figure 1). c and l denote cyclic and linear assemblies respectively. Porphyrin
compoundswerepreparedusingpreviouslydescribedprotocols.1–3
FigureS1Fullstructuresforthesixporphyrinderivativesconsidered.
Electronic Supplementary Material (ESI) for ChemComm.This journal is © The Royal Society of Chemistry 2016
S.2RelaxationMeasurements
Relaxationmeasurements forαandβ coupled spinwhere carriedouton7.1T (300MHz), 14.1T
(600MHz),17.6T(750MHz)and22.3T(950MHz)narrowboresolution-stateNMRspectrometers.
Allspectrometerswereequippedwithroomtemperatureprobesandtheexperimentswerecarried
out at 298 K. The spectrometers were operating with Varian, GE Omega and Bruker operating
systems and so pulse sequences for all formats are available on request. The number of CPMG
elements(n)foreachsamplewasvariedsothatforthehighestvalueofntheintensityofthepeaks
droppedtoapproximately!!oftheirmaximumvalue.Atypicalexperimentemployedarecycledelay
of2s,acquisitiontime2s,16dummyscans,8transientsper1Dand16differentvaluesforn.Some
valuesofnarerepeatedforerroranalysisandtheorderinwhichthevaluesofnwerecarriedout
was randomized to minimise any systematic effects. All spectra were phased and Fourier
transformedusingNMRPipe4andexportedforanalysis.
S.3DataAnalysis
It is not possible to obtain reliable intensities through summing the signal due to the degree of
overlapbetween the twocomponentsof thedoublet (FigureS2B). Insteadwenote that thepeak
functiondescribingthecentralpositionoftheresonanceanditsshapewillnotvarywithn.Tothis
end,werepresentthedoubletsasthesumoftwoPseudo-VoigtfunctionsP(ω,ω0,Δ,f)inwhicheach
ischaracterisedbythreeparameters;apeakcentreω,apeakwidthΔandaGauss-to-Lorentzmixing
factorf.
z(ω ,ω 0,Δ) =ω −ω 0
Δ, L(ω ,ω 0,Δ) =
11+ z(ω ,ω 0,Δ)
2 , G(ω ,ω 0,Δ) = e− ln(2)z(ω ,ω0 ,Δ )
2
P(ω ,ω 0,Δ, f ) = ( fL(ω ,ω 0,Δ)+ (1− f )G(ω ,ω 0,Δ)
Thedecayofthetwopeaksisthenmodelledbymultiplyingeachfunctionwithasingleexponential
decayconstantsuchthatthespectrumatanygiventimeisgivenby:
S(ω ,t) = P(ω ,ω I−Sα,Δ I−Sα
, fI−Sα )e− tR2
I−Sα + P(ω ,ω I−Sβ,Δ I−Sβ
, fI−Sβ )e− tR2
I−Sβ
Theentiredatasetofspectraasafunctionoffrequencyandtimeisanalysedgloballytoobtainthe6
peakshapeparametersand2rateconstants.Thisprocedureis illustratedinFigureS2showingthe
comparisonbetweenrawandsimulateddatafortwotimepointsofoneexperiment(A).Thedecay
of thepeaksobtained from thismethoddiffers from themoreusualmethodof simply taking the
integral.Thedifferencebetweenthetworatesisillustratedin(B).Codetoprocessdatainthisway
waswritteninpythonandisavailableonrequest.
FigureS2A.Fitted(solid lines)andraw(dotted lines)spectraofan Ispindoubletattwodifferentrelaxationtimes.B.Theexpectedsignalintensitiesfromthefitting(solid)arenotinagreementwiththe intensities obtained from taking the peak height (dotted). The fitted intensities are themorereliableintensitymeasureasoverlapeffectsareremoved.
S.4RectangularPulsesforsemi-selectiverefocusing
In this study semi-selective refocusing is achieved through the use of low power rectangular
(constantamplitude)pulsesratherthanshapedpulses.Rectangularpulsesareabletoachievea180°
flipinmuchshortertimethanshapedpulsesandconsequentlyalargerrangeoftimepointsmaybe
interrogated before the transverse coherence decays. The difference in time for on resonance
inversion is compared in Figure S3A and B. The selective rectangular pulses will be ca. 6 times
shorterthanoptimisedinversionpulsessuchQ3orREBURP1(FigureS3B).
Thepulsesarecalibratedsuch that theon-resonancepulseundergoesa180° flipwhilst thescalar
coupledpartnerundergoesanullpulse(FigureS3C).Thisisachievedbysettingthepulsetime(pwsel)
oftheselective180°pulseaccordingto𝑝𝑤!"# = 3/2𝛥𝜈 where∆𝜈isthedifferenceinfrequency(in
Hz)between the spinof interest and its scalar coupledpartner. Thepulse isplacedon resonance
with the spinof interestand thepowermustbecalibrated so that the spinundergoesa180° flip
during a pulse of this length (Figure S3D, E and F). A soft rectangular pulse of this typewill have
residualeffectson thespectrumoutside the regionof interest,but itwilleffectivelydecouple the
spins of interest. Pulse times employed for the selective 180° pulses depend on the frequency
differenceof thesamples investigated,butwereapproximately1.7msat600MHz,1.4msat750
MHzand1.1msat950MHz.
FigureS3A.TheeffectsofapplyingpulsesofvariousshapesonasinglespininitiallyatequilibriumasafunctionofoffsetΩandB1fieldforcommonlyusedpulseshapes.TheREBURP1andQ3aremoreselective than SEDUCE or rectangular pulses. B. The offset between spins where on resonanceexperiencesinversion,and1%excitationisfeltatthespecifiedoffset,asafunctionofpulsedurationfor the variouspulse shapes. The selective rectangular pulseshave aduration ca. 5x shorter thanthose with more complex shapes making them highly amenable to measurements of this type,particularly when rapid relaxation is expected. C. The trajectory of magnetisation in the rotatingframeduringarectangularpulseasafunctionoffrequencyoffset.Theonresonancespinundergoesa180oflipwhereasthespinwhichisoffresonancebytheoptimalvalueeffectivelyexperiencesnopulse.D. A pulse sequence used to ensure that the selective pulse has been calibrated correctlyconsistsofaselective180opulse(showninred)anda‘hard’rectangular90opulse(black)executedat maximum power and minimum duration. As shown in E. and F. when the selective pulse iscalibratedcorrectlyandplacedonresonancewithoneofthespinsof interesttherewillbea180ophaseshiftbetweenthetwospinsastheoffresonancespineffectivelyexperiencesanullduringthefirstpulse.
S.5SummaryofRedfieldcalculationsofrelaxationratesforAXnspinsystems
Relaxation rates of the difference coherences can be calculated as follows using the approach of
Redfield. We consider here dipolar and CSA interactions between spins ½ and the correlations
betweenthetwo.Thestrengthsofthetwointeractionsaregivenrespectivelyby:
𝑑 = !!!!!!ℏ!!!!"
! and𝑐 = 𝛾!𝐵!Δ𝜎
Table 1 Summary ofthe spin state termscontained in thedipolar and chemicalshift anisotropyHamiltonians.
where 𝛾 is the gyromagnetic ratio of the relevant spin, 𝜇! is the magnetic constant, 𝑟!" is the
distancebetweenspinsIandS,𝛥𝜎isthechemicalshiftanisotropy(CSA)inppmandB0isthestatic
magneticfieldstrength.Therelaxationrateofagivencoherenceρi,suchasIzorIxduetoanother,
ρj,isobtainedfromRedfield’sequation:
Rρiρ j=∑k ,l
12Sl†Sk
< ρi | [Al†[Ak ,ρ j ]]>
< ρi | ρ j >Fl†FkJ(ω (Ak ))
ThesumhereextendstoallcombinationsofspinoperatorsintheappropriateHamiltonian,subject
tothesecularapproximationthatrequiresω(Ak)+ω(Al)=0whereω(Ak)=-ω(Akd),andFl=Fk*.Inthecase
ofdipolarandCSAinteractions,therelevanttermsaresummarisedintable1.ForanAXspinsystem,
asisconsideredhere,therewillbe12termsinthecombinedchemicalshiftanisotropyanddipolar
Hamiltonians, leadingtothesumextendingover144possiblecontributionstotherelaxationrate.
Ak Ald Fk Sk/l w(Ak)
IZSZ IzSz (1-3cos2(θ)) d 0
I+S- I-S+ ¼(1-3cos2(θ)) d ωI-ωS
I+S- I-S+ ¼(1-3cos2(θ)) d ωI-ωS
IzS+ IzS- 3/2sin(θ)cos(θ)exp(-iφ) d ωS
IzS- IzS+ 3/2sin(θ)cos(θ)exp(+iφ) d ωS
I+Sz I-Sz 3/2sin(θ)cos(θ)exp(-iφ) d ωI
I-Sz I+Sz 3/2sin(θ)cos(θ)exp(+iφ) d ωI
I+S+ I-S- ¾sin2(θ)exp(-2iφ) d ωI+ωS
I-S- I+S+ ¾sin2(θ)exp(2iφ) d ωI+ωS
IZ IZ 1/3(1-3cos2(θ)) c 0
I+ I- ¼sin(2θ)exp(-iφ) c ωI
I- I+ ¼sin(2θ)exp(+iφ) c ωI
Each individual term (Ak) is evaluatedwith respect to theHermitian conjugate (Ald) of all possible
terms.Anormalisedspectraldensityfunctionforisotropicrotationalmotionis:
J(ω ) = 25
τ c1+ω 2τ c
2
andtheensembleaveragedspatialtermsareevaluatedfrom
Fl†Fk =
14π
Fl†Fk sinθ dθ dφ
0
π
∫0
2π
∫
Ingeneral,theautorelaxationratewhereρi=ρjforanysinglequantumcoherencefromanAXnspin
systemwherethe‘A’isasinglequantumcoherenceandthe‘X’arethecompletesetofαandβspin
statesthatcanaccompanythecoherencewillbegivenbythefollowingresult:
R2S− ∏ Iα /β = 1+ 2 j2
jmax−1
⎛⎝⎜
⎞⎠⎟P20 (cosθDD )
⎛
⎝⎜⎞
⎠⎟2 jmaxRD
2 + RC2 + 4 jP2
0 (cosθCS )RDRC⎛
⎝⎜⎞
⎠⎟f + 2 jmaxRD
2 fADD
wherewemakeuseofthefollowing4definitions:
RD = d2 5
, RC = c3 5
, f = 4J(0)+ 3J(ω S ), fADD = 3J(ω I )+ J(ω S −ω I )+ 6J(ω S +ω I )
and P20 (x) = 1
23x2 −1( ) ,θDDistheanglebetweendipolarvectors(assumedconstanthere)andθCS
istheanglebetweentheprincipalaxisofthechemicalshifttensorandtheinternuclearvectorfor
thedipolarcoupling.Thenetspinofthecoupledspinsforthecoherenceofinterestisjandjmaxis
maximumpossibleangularmomentumfor thecoupledspins.Forexample, foracoherenceof the
formS-IαIαIβ,jmaxwouldbe3/2andjwouldbe½.Inthepresentmanuscript,themethodisappliedto
anAXspinsystem,andsoS-IαandS-Iβaretherelevantrelaxationrateswherejmax=1/2andj=+1/2or
-1/2. Using this equation, the method can be applied to any spin system of the type AmXn. The
relaxationrateswillneedtobefurthermodifiedformorecomplicatedspinsystemsandadditional
dynamicssuchasmethylgrouprotation.
S.6Globalanalysisofrelaxationdata
A set of R2 measurements for I-Sα, I-Sβ, IαS- and IβS- were obtained as described above for each
porphyrin sample as a function of magnetic field strength. The relaxation rates can be back
calculatedfromtheappropriaterelaxationequationsasafunctionofτcandΔσCSAoftheSspinand
σCSAoftheIspin,andtheanglebetweenthedipolarvectorandCSAprincipalaxis.Theseparameters
were globally minimised to obtain the fitted curves (Figure 3, Figure S4A). An additional term
correspondingtoR’D2fADDwasalsoincludedtoaccountfordipolarrelaxationduetoremoteprotons,
wherethedistanceinusedtocalculateRDisdefinedbytherelevantinter-protondistance.
Uncertainties in parameters were determined using a bootstrapping procedure. This involves
creatinganumberofsyntheticdatasetsbyrandomlyselectingdatapointsfromtheoriginalsample
withreplacement.Thesameoptimisationisthenrunonthesyntheticdatasettogivevaluesforthe
fittingparameters.Thisprocessisthenrepeatedca.1000timesenablingahistogramoffittedvalues
canbeplotted.Thestandarddeviationinthebootstrapvaluesforthefittingparameterscanbeused
as an estimate for the error (Figure S4B). In the analysis we obtain the cosine of the angle. The
specificanglestatedisthesolutionbetween90and180ofornumericalconvenience.
FigureS4A.Relaxationratesasafunctionofmagneticfieldstrength,andglobalfitsformonomericand dimeric porphyrins.B. The uncertainty distributions in parameters from a bootstrap analysis.Thestandarddeviationofthehistogramisanestimateoftheuncertaintyinthefittingparameter.
S.7NMRTranslationaldiffusionmeasurements
Thediffusioncoefficientsweremeasuredat11.74T(500MHz)onanAVIIspectrometer(Bruker)at
298 K. A convection-compensating double-stimulated echo experiment (DSTE) using bipolar
gradientswasused.Thediffusiontime∆was235.40ms,andthegradientpulselengthwas3ms.A
lineargradient incrementwasused.ThesamplecontainedP1, l-P6andc-P6•T6 inCDCl3.Diffusion
coefficients were determined by an exponential fit to the integrated area of resonances of each
componentusingtheStejskal-Tannerequation.5
𝑆! = 𝑆!𝑒!!!!!!!(!!!!)!!
Where Si is the measured signal intensity, S0 is the maximum signal intensity in the absence of
gradientdephasing,γisthegyromagneticratio,Δisthediffusiontime,δisthegradientpulselength
andGisthegradientfieldstrength.ArepresentativedecaycurveisshowninFigureS5B.
Figure S5 A. DOSY spectrum from which P1, l-P6 and c-P6•T6 can be identified. B. Example ofintensitydecreasewith increasedgradient fieldstrength(G) fromwhichadiffusioncoefficientcanbefittedforeachofthemolecules.
S.8Calculationofrotationalcorrelationtimesforlinearl-P6fromtranslationaldiffusion
ToobtainacorrelationtimeforthelinearmoleculeI-P6,itwasnecessarytoincludeacorrectionto
accountfor itsanisotropicdiffusion.Thediffusionconstantforamolecule isrelatedtothefriction
factor,F,throughtherelationship:
𝐷 = !!!!whereforarodoflengthLandradiusRthefrictionfactorisgivenby:𝐹 = 3𝜋𝜂𝐿(ln !
!−
0.3)!!. The radius of a rod-like molecule can be determined from a known length and its
translational diffusion coefficient from𝑅 = 𝐿(exp 0.3 + !!"#$!"
)!!. By equating the volume of a
spheretothevolumeoftherod,wecanobtaintheeffectivesphericalradiusoftherod,intermsof
theviscosity,theobserveddiffusioncoefficient(D)andthelengthoftherod.
rsphere = L34exp − 6
10− 6πηLD
kT⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟1/3
Fromtheeffectivesphericalradius,acorrelationtimecanbedetermined.TheviscosityofCDCl3at
298Kis0.542mPa.6
S.9Solid-stateNMRdeterminationof13CCSA
For themonomericporphyrin, itwaspossible to independentlymeasure theCSAusing solid-state
NMR spectroscopy as described in the main text. The intensity of the spinning side bands was
measuredasafunctionofspinningrate.Across-polarisation(CP)elementwasusedtoenhance13C
sensitivityandspectrawererecordedatmagicanglespinning(MAS)frequenciesof1kHz,3kHzand
10kHz.
Individual13CresonancesfromthecarbonnucleiattachedtoIandSprotonscouldberesolvedinthe13CMASspectra(MainarticleFigure4E)andwerecorrelatedtotheappropriate1Hresonanceusing
a 1H-13CHSQCspectrum.For theHerzfeld-Bergeranalysis the3kHzspinningspeedwaschosenas
theoverlapbetweenresonanceswasminimalwhileyetstillenablingresolutionofanumberofside
bandresonances.
Byfittingthesidebandpatternassociatedwithaparticularresonanceinasolid-stateNMRspectrum
usingaHerzfeld-Bergeranalysis7itispossibletoderivetheCSAofthatresonance.Solid-stateNMR
experimentswerecarriedoutonaP1sample(witht-Busidegroups)witha9.4T(400MHz)Bruker
AvanceIIIHDspectrometerequippedwitharoom-temperature4mmMASprobe.TheCSAsofthe
carbonnucleiattachedtotheIandSprotonswerefoundtobe124.0±6.3ppmand124.8±2.7ppm
respectively. These values compare favourably to those fromDFT calculations (section S.10). The
samemeasurementswerenotpossibledirectlyon1Hduetoresonanceoverlap.
S.10DFTCalculations
ThechemicalshifttensorwasestimatedusingDFTcalculations.UsingthesoftwareGaussian098,an
optimisedstructureisgenerated.Oncethegeometryisoptimised,theshieldingtensoriscalculated,
enablingisotropicandanisotropiccomponentstobedetermined.AllDFTcalculationswerecarried
outusingtheB3LYPdensityfunctionalwiththe6-31G(d)basisset.
Theexperimentalandcalculatedisotropicchemicalshiftsofadatabaseofsmallorganiccompounds
are shown together with values calculated for P1 (Figure S6). The correlation between the two
valuesisexcellentoverthisrange.
Calculationswere performedon the range of side groups used in theNMR study to confirm that
thesehavealimitedimpactonthemeasuredCSAs(TableS2).Thereisexcellentagreementbetween
the experimental 13C CSA measurements (section S.9) and those calculated using this method,
validating the calculation method. The agreement between the 1H CSA determined using the
relaxationmeasurements that are themain focus of this paper (Figure 4) and the calculations is
similarlyexcellent.
FigureS6A.CorrelationplotofcalculatedandexperimentalisotropicchemicalshiftsusingDFTwithB3LYP/6-31G(d).Bluedotsindicateexperimentalandcalculatedchemicalshiftsfor24smallorganicmolecules where experimental chemical shifts are taken form the Spectral Database for OrganicCompounds.9Reddots indicateexperimentalandcalculatedchemical shifts forP1 (R= t-Bu).B/C.EllipsoidrepresentationsofthecalculatedCSAtensorsoftheI(blue)andS(red)spins.Theellipsoidsarerotatedtoalignwiththeeigenvectorsofthechemicalshieldingtensorandtheiraxesarescaledto the correspondingeigenvalue.Eigenvaluesandeigenvectorsare calculated from the symmetriccomponent of the computed chemical shielding tensor and eigenvalues are subtracted from theisotropicshiftofTMScalculatedatthesameleveloftheory.Themostdeshieldedcomponentofthetensor for IandSspins isperpendicular to theplaneof theporphyrin.TheCSA fromtheDFTand
that measured experimentally from the relaxation interference measurements are in excellentagreement(Figure4).
R= I1HCSA/ppm S1HCSA/ppm I13CCSA/ppm S13CCSA/ppm
t-Bu** 16.5±0.1 12.9±0.4 131.317±0.138 126.6±0.2
t-Bu 15.3±0.1 12.1±0.1 132.495±0.035 128.4±0.1
THS 14.1±0.4 13.0±0.1 132.368±0.227 128.3±0.5
OC8H17 14.8±0.1 11.8±0.1 132.820±0.185 128.2±0.1
**Includes THS groups on the terminal acetylenes – in all other cases this was replaced with ahydrogenatom.
Table2ListofDFTderivedCSAsinboth1Hand13CinthepresenceofdifferentRgroups.
S.11CalculationofIntensityRatios
Assuming lorentzian lineshapes, the normalised intensity of a peak as a function of frequency is
givenby:
𝐼 = 𝑅!
𝑅!! + (𝜔 − 𝜔!)!
Where the peak is centered at frequency ω0. The maximum intensity of a peak is thefore
proportionalto !!!.The𝑅!containsacontributionfrombothintrinsicrelaxationandmagneticfield
inhomogeneity.Inthecaseofadoublet(e.g.𝐼!𝑆!and𝐼!𝑆!)theratioofthemaximumintensitiesis
givenby:!!!!!!
!!!!!!.
As shown in Figure S7, in the presence of magnetic field inhomogeneity the intensity difference
expectedduetorelaxationinterferenceislargelyobscuredforsmallmoleculesthathavecorrelation
timesonthepicosecondtimescale.Howeverforlargesupramolecularassembliesthattumbleonthe
nanosecondtimescale,asexemplifiedbytheporphyrinoligomers,significantdifferencesinintensity
canbedetectedbyinspectionofspectra,givenareasonablyhighCSA.Themagnitudeoftheeffectis
amplifiedwhenusingahigherstaticfieldstrength(FigureS7andmainarticleFigure3).Duetothe
dependence of the effect on correlation time, any factors that affect this e.g. a change in
temperature,solventviscosityorpresenceoflocalmotionwillalsoaffecttheratioofintensities.
Figure S7 Heat maps showing the ratio of signal intensities for a single scalar coupled doubletintensities !!
!! for a range of molecules/substituents at magnetic field strengths corresponding to
proton Lamor frequencies of 300, 600 and 950 MHz. The relaxation rate is calculated using theequations in the text assuming dipolar coupling to another proton at a distance of 2.5 Åwith anangle of 90o between the dipolar and CSA principal axes, as was the case for the porphyrinoligomers.1s-1 inhomogeneousbroadeninghasbeenaddedtoreflectatypicalexperimentalcase,andsoforabroadrangeofparameterspace,thetwocomponentsareofequal intensity (yellow).Specific CSA values for specific protons in functional groups are indicated (lines), including a CH3group, aromatic protons in a substituted benzene ring, twoporphyrins used in this study, and anamide proton, as frequently encounted in the context of proteins. The specific combinations ofcorrelation time and CSA, for specific molecules are expliticlty numbered. The magnitude of theeffectwillvarywiththemoleculebutthesevaluesserveasaqualitativeguideforwhentheeffectshouldbeanticipated. In general, increasing theeffective correlation timeabove thepointwherethe linewidth is limitedby inhomogeneousbroadeningwill render relaxation interferencedirectlyobservable in NMR spectra. This can be accomplished by, for example, reducing local internaldynamics, increasing theoverallmolecularweightof themolecule,decreasing the temperatureorincreasingtheviscosity.
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