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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9 SDBSWeb:http://sdbs.db.aist.go.jpNationalInstituteofAdvancedIndustrialScienceandTechnology(Japan),AccessedMay2015