Conformation of self-assembled porphyrin dimersin liposome vesicles by phase-modulation 2Dfluorescence spectroscopyGeoffrey A. Lotta,1,2, Alejandro Perdomo-Ortizb,1, James K. Utterbacka, Julia R. Widomc, Alán Aspuru-Guzikb, andAndrew H. Marcusc,3

aDepartment of Physics, Oregon Center for Optics, University of Oregon, Eugene, OR 97403; bDepartment of Chemistry and Chemical Biology, HarvardUniversity, Cambridge, MA 02138; and cDepartment of Chemistry, Oregon Center for Optics, Institute of Molecular Biology, University of Oregon,Eugene, OR 97403

Edited* by Michael D. Fayer, Stanford University, Stanford, CA, and approved July 12, 2011 (received for review November 18, 2010)

By applying a phase-modulation fluorescence approach to 2Delectronic spectroscopy, we studied the conformation-dependentexciton coupling of a porphyrin dimer embedded in a phospholipidbilayer membrane. Our measurements specify the relative angleand separation between interacting electronic transition dipolemoments and thus provide a detailed characterization of dimerconformation. Phase-modulation 2D fluorescence spectroscopy(PM-2D FS) produces 2D spectra with distinct optical features, simi-lar to those obtained using 2D photon-echo spectroscopy. Specifi-cally, we studied magnesium meso tetraphenylporphyrin dimers,which form in the amphiphilic regions of 1,2-distearoyl-sn-glycero-3-phosphocholine liposomes. Comparison between experimentaland simulated spectra show that although a wide range of dimerconformations can be inferred by either the linear absorptionspectrum or the 2D spectrum alone, consideration of both types ofspectra constrain the possible structures to a “T-shaped” geometry.These experiments establish the PM-2D FS method as an effectiveapproach to elucidate chromophore dimer conformation.

fluorescence-detected 2D photon echo ∣ nonlinear spectroscopy ∣supramolecular conformation ∣ excitonically coupled dimer

The ability to determine three-dimensional structures ofmacromolecules and macromolecular complexes plays a cen-

tral role in the fields of molecular biology and material science.Methods to extract structural information from experimentalobservations such as X-ray crystallography, NMR, and opticalspectroscopy are routinely applied to a diverse array of problems,ranging from investigations of biological structure-function rela-tionships to the chemical basis of molecular recognition.

In recent years, two-dimensional optical methods have becomewell established to reveal incisive information about noncrystal-line macromolecular systems—information that is not readilyobtainable by conventional linear spectroscopic techniques. Two-dimensional optical spectroscopy probes the nanometer-scalecouplings between vibrational or electronic transition dipole mo-ments of neighboring chemical groups, similar to the way NMRdetects the angstrom-scale couplings between adjacent nuclearspins in molecules (1). For example, 2D IR spectroscopy probesthe couplings between local molecular vibrational modes and hasbeen used to study the structure and dynamics of mixtures of mo-lecular liquids (2), aqueous solutions of proteins (3), and DNA(4). Similarly, 2D electronic spectroscopy (2D ES) probes corre-lations of electronic transitions and has been used to study themechanisms of energy transfer in multichromophore complexes.Such experiments have investigated the details of femtosecondenergy transfer in photosynthetic protein–pigment arrays (5–8),conjugated polymers (9), and semiconductors (10, 11).

Following the examples established by 2D NMR and 2D IR,2D ES holds promise as a general approach for the structuralanalysis of noncrystalline macromolecular systems, albeit for thenanometer length scales over which electronic couplings occur. It

is well known that disubstitution of an organic compound withstrongly interacting chromophores can lead to coupling of theelectronic states and splitting of the energy levels (12–14). Thearrangement of transition dipoles affects both the splitting andthe transition intensities, which can be detected spectroscopically.Nevertheless, weak electronic couplings relative to the monomerlinewidth often limits conformational analysis by linear spectro-scopic methods alone. Two-dimensional ES has the advantagethat spectroscopic signals are spread out along a second energyaxis and can thus provide the information needed to distinguishbetween different model-dependent interpretations. Several the-oretical studies have examined the 2D ES of molecular dimers(15–19), and the exciton-coupled spectra of multichromophorelight harvesting complexes have been experimentally resolvedand analyzed (20–22).

Because of its high information content, 2D ES presents pre-viously undescribed possibilities to extract quantum informationfrom molecular systems and to determine model Hamiltonianparameters (23). For example, experiments by Hayes and Engelextracted such information for the Fenna–Matthews–Olsen lightharvesting complex (24). Recently, it was demonstrated by Brinkset al. that single molecule coherences can be prepared usingphased optical pulses and detected using fluorescence (25).The latter experiments exploit the inherent sensitivity of fluores-cence and demonstrate the feasibility to control molecular quan-tum processes at the single molecule level. Fluorescence-basedstrategies to 2D ES, such as presented in the current work, couldprovide a route to extract high purity quantum informationfrom single molecules. It may also be a means to study molecularsystems in the ultraviolet regime where background noise due tosolvent-induced scattering limits ultrafast experiments.

Here we demonstrate a phase-modulation approach to 2D ESthat sensitively detects fluorescence to resolve the excitoncoupling in dimers of magnesium meso tetraphenylporphyrin(MgTPP), which are embedded in 1,2-distearoyl-sn-glycero-3-phosphocholine (DSPC) liposomal vesicles. MgTPP is a nonpolarmolecule that preferentially enters the low dielectric amphi-philic regions of the phospholipid bilayer. At intermediate con-centration, MgTPP forms dimers as evidenced by changes inthe linear and 2D absorption spectra. Quantitative comparison

Author contributions: A.H.M. designed research; G.A.L., A.P.-O., J.K.U., J.R.W., and A.H.M.performed research; A.P.-O. and A.A.-G. contributed new reagents/analytic tools; G.A.L.,A.P.-O., J.K.U., J.R.W., and A.H.M. analyzed data; and G.A.L., A.P.-O., A.A.-G., and A.H.M.wrote the paper.

The authors declare no conflict of interest.

*This Direct Submission article had a prearranged editor.1G.A.L. and A.P.-O. contributed equally to this work.2Present address: Boise Technology, Inc., 5465 East Terra Linda Way, Nampa, ID 83687.3To whom correspondence should be addressed. E-mail: ahmarcus@uoregon.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1017308108/-/DCSupplemental.

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between our measurements and simulated spectra for a broaddistribution of selected conformations, screened by a global op-timization procedure, shows that the information contained inlinear spectra alone is not sufficient to determine a unique struc-ture. In contrast, the additional information provided by 2Dspectra constrains a narrow distribution of conformations, whichare specified by the relative separation and orientations of theMgTPP macrocycles.

In our approach, called phase-modulation 2D fluorescencespectroscopy (PM-2D FS), a collinear sequence of four laserpulses is used to excite electronic population (26). The ensuingnonlinear signal is detected by sweeping the relative phases of theexcitation pulses at approximately kilohertz frequencies and byusing lock-in amplification to monitor the spontaneous fluores-cence. This technique enables phase-selective detection of fluor-escence at sufficiently high frequencies to effectively reducelaboratory 1∕f noise. Because the PM-2D FS observable dependson nonlinear populations that generate fluorescence, a differentcombination of nonlinear coherence terms must be consideredthan those of standard photon-echo 2D ES (referred to hereafteras 2D PE). In 2D PE experiments, the signal—a third-orderpolarization generated from three noncollinear laser pulses—isdetected in transmission. The 2D PE signal depends on the super-position of well-known nonlinear absorption and emission pro-cesses, called ground-state bleach (GSB), stimulated emission(SE), and excited-state absorption (ESA) (27). Analogous excita-tion pathways contribute to PM-2D FS. However, the relativesigns and weights of contributing terms depend on the fluores-cence quantum efficiencies of the excited-state populations.Equivalence between the two methods occurs only when all ex-cited-state populations fluoresce with 100% efficiency (28). Thus,self-quenching of doubly excited exciton population can give riseto differences between the spectra obtained from the two meth-ods—differences that may depend, in themselves, on dimer con-formation. For the conformations realized in the current study,we find that the PM-2D FS and 2D PE methods produce spectrawith characteristic features distinctively different from oneanother.

Results and DiscussionMonomers of MgTPP have two equivalent perpendicular transi-tion dipole moments contained within the plane of the porphyrinmacrocycle (see Fig. 1B, Inset). These define the molecular-framedirections of degenerate Qx and Qy transitions between groundjgi and lowest lying excited electronic states, jxi and jyi. The col-lective state of two monomers is specified by the tensor productjiji ½i;j ∈ fg;x;yg�, where the first index is the state of monomer 1and the second that of monomer 2. When two MgTPP monomersare brought close together, their states can couple through reso-nant dipole–dipole interactions Vkl ½k;l ∈ fjijig� with signs andmagnitudes that depend on the dimer conformation. We adoptthe convention that a conformation is specified by the monomercenter-to-center vector ~R, which is oriented relative to molecule 1according to polar and azimuthal angles θ and ϕ, and the relativeorientation of molecule 2 is given by the Euler angles α and β (seeFig. 1A and details provided in SI Text). The effect of the inter-action is to create an exciton-coupled nine-level system, withstates labeled jXni, comprised of a single ground state (n ¼ 1),four singly excited states (n ¼ 2–5), and four doubly excited states(n ¼ 6–9). Transitions between states are mediated by the collec-tive dipole moment, miu1 þmiu2, which also depends on thestructure of the complex.

In Fig. 1B are shown vertically displaced linear absorptionspectra of MgTPP samples prepared in toluene, and 70∶1 and7∶1 1,2-distearoyl-sn-glycero-3-phosphocholine ðDSPCÞ∶MgTPPliposomes. For the 70∶1 sample, the line shape and position of thelowest energy Qð0;0Þ feature, centered at 606 nm, underwent aslight redshift relative to the toluene sample at 602 nm. For

the elevated concentration 7∶1 sample, the line shape broadened,suggesting the presence of a dipole–dipole interaction and exci-ton splitting between closely associated monomer subunits.

In principle, it is possible to model the linear absorption spec-trum in terms of the structural parameters ~R, α and β that deter-mine the couplings Vkl and the collective dipole moments, andwhich ultimately determine the energies and intensities of theground-state accessible transitions. To test the sensitivity of the

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Fig. 1. (A) Energy level diagram of two chemically identical three-levelmolecules, each with degenerate transition dipole moments directed alongthe x and y axes of the molecular frames. (Inset) A random configuration oftwo MgTPP monomers whose relative conformation is defined by the mole-cular center-to-center vector ~R and the angles θ, ϕ, α, and β. Electronic inter-actions result in an exciton-coupled nine-level system, with a single groundstate, four nondegenerate singly excited states, and four doubly excitedstates. Multipulse excitation can excite transitions between ground, singlyexcited, and doubly excited state manifolds. (B) Absorption spectra of theMgTPP samples studied in this work. Spectra are vertically displaced forclarity. The samples correspond to MgTPP in toluene (Bottom), aqueous lipo-some suspension with 70∶1 DSPC∶MgTPP (Middle), and 7∶1 DSPC∶MgTPP(Top). The dashed vertical line represents the lowest energy monomer transi-tion energy used in our calculations. (Insets) Molecular formulas for MgTPPand lipid DSPC. (C) Overlay of the 7∶1 DSPC∶MgTPP absorbance and the laserpulse spectrum. The laser spectrum (solid black curve) has been fit to a Gaus-sian (dashed gray curve) with center frequency 15;501 cm−1 (606 nm), andFWHM ¼ 327.0 cm−1 (12 nm). The linear absorbance (solid black curve) iscompared to the simulated spectrum (dashed black curve), which is basedon the “T-shaped” conformation (Inset). Also shown are the positions ofthe underlying exciton transitions (discussed in text).

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linear absorption spectrum to different conformational models,we numerically generated approximately 1,000 representativeconformations and simulated their linear spectra (details pro-vided in SI Text). By comparing experimental and simulated data,we established that a wide distribution of approximately 100 con-formations can reasonably explain the linear absorption spec-trum. Nevertheless, only a very small conformational subspacecould be found to agree with the experimental 2D spectra (pre-sented below), and which is also consistent with the linear spec-trum. In Fig. 1C is shown the simulated linear spectrum (Top,third column) and the four underlying component transitionsof the optimized “T-shaped” conformation. The linear spectrumcorresponding to this conformation is composed of two intensespectral features at 16,283 and 16;619 cm−1, one weak featureat 16;718 cm−1, and one effectively dark feature at 16;382 cm−1

(see SI Text for intensity values). The relatively unrestrictive con-straint imposed on dimer conformation by the linear spectrum isa consequence of the many possible arrangements and weightsthat can be assigned to the four overlapping Gaussian featureswith broad spectral width.

The PM-2D FS method uses four collinear laser pulses toresonantly excite electronic population, which depends on theoverlap between the lowest energy electronic transition [theQð0;0Þ feature] and the laser pulse spectrum (as shown inFig. 1C). We assigned the nonlinear coherence terms GSB, SE,and ESA to time-ordered sequences of laser-induced transitionsthat produce population on the manifold of singly excited states(n ¼ 2–5) and the manifold of doubly excited states (n ¼ 6–9).The theoretically derived expressions for PM-2D FS were foundto differ from those of 2D PE (details provided in SI Text). Thisis because ESA pathways that result in population on the doublyexcited states have a tendency to self-quench by, for example,exciton–exciton annihilation or other nonradiative relaxationpathways, so that these terms do not fully contribute to thePM-2D FS signal. In 2D PE experiments, signal contributions toESA pathways interfere with opposite sign relative to the GSBand SE pathways; i.e., S2D PE ¼ GSBþ SE − ESA. In PM-2DFS experiments, quenching of doubly excited state populationleads to interference among GSB, SE, and surviving ESA path-ways with variable relative sign; i.e., SPM-2D FS ¼ GSBþ SEþð1 − ΓÞESA, where 0 ≤ Γ ≤ 2 is the mean number of fluorescentphotons emitted from doubly excited states relative to the aver-age number of photons emitted from singly excited states. In ouranalysis of PM-2D FS spectra (described below), we treated Γ asa fitting parameter to obtain the value that best describes our

experimental data. As we show below, the difference betweensignal origins of the two methods can result in 2D spectra withmarkedly different appearances, depending on the specific dimerconformation.

In Fig. 2 are shown complex-valued experimental PM-2DFS data for the 7∶1 lipid∶MgTPP sample (Top), the 70∶1 lipid∶MgTPP (Middle), and the toluene sample (Bottom). Rephasingand nonrephasing data, shown respectively in Fig. 2A and B, wereprocessed from independently detected signals according to theirunique phase-matching conditions. The two types of spectraprovide complementary structural information, because each de-pends on a different set of nonlinear coherence terms. Both re-phasing and nonrephasing 2D spectra corresponding to the 7∶1liposome sample exhibit well resolved peaks and cross-peaks withapparent splitting approximately 340 cm−1. This is in contrast tothe 2D spectra obtained from control measurements on the 70∶1liposome and toluene samples, which as expected exhibit onlythe isolated monomer feature due to the absence of electroniccouplings in these samples. The 2D spectra of the 7∶1 liposomesample are asymmetrically shaped, with the most prominent fea-tures a high energy diagonal peak and a coupling peak directlybelow it. We note that the general appearance of the 7∶1 lipo-some PM-2D FS spectra is similar to previous model predictionsfor an exciton-coupled molecular dimer (15–18, 29). We nextshow that the information contained in these spectra can be usedto identify a small subspace of dimer conformations.

By extending the procedure to simulate linear spectra (de-scribed above), we numerically simulated 2D spectra for a broaddistribution of conformations (details provided in SI Text). Weperformed a least-squares regression analysis that compared si-mulated and experimental spectra to obtain an optimized confor-mation consistent with both the 2D and the linear datasets. In ouroptimization procedure, we treated the fluorescence efficiency Γof doubly excited excitons as a parameter to find the value thatbest represents the experimental data. In Fig. 3, we directly com-pare our experimental and simulated PM-2D FS spectra for theoptimized conformation. The values obtained for the parametersof this conformation are θ ¼ 117.4°, ϕ ¼ 225.2°, α ¼ 135.2°,β ¼ 137.2°, R ¼ 4.2 Å, and Γ ¼ 0.31, with associated trust inter-vals: −16° < Δθ < 4°, −11° < Δϕ < 11°, −11° < Δα < 11°,−2° < Δβ < 2°, −0.05 Å < ΔR < 0.05 Å, and −0.1 < ΔΓ ¼ 0.1(details provided in SI Text). For both rephasing and nonrephas-ing spectra, the agreement between experiment and theory isvery good, with an intense diagonal peak and a weaker couplingpeak (below the diagonal) clearly reproduced in the simulation. A

Fig. 2. Comparison between rephasing (A) and nonrephasing (B) experimental 2D spectra corresponding to the MgTPP samples of Fig. 1B. Complex-valuedspectra are represented as 2D contour plots, with absolute value (Left), real (Center) and imaginary (Right) parts. The color scale of each plot is linearand normalized to its maximum intensity feature. Positive and negative contours are shown in black and white, respectively, and are drawn at 0.8, 0.6,0.4, 0.2, and 0.

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notable feature of the experimental 2D spectra is the asymmetricline shape. A possible explanation for these asymmetries is theexistence of distinct interactions between the various excitonstates and the membrane environment. The discrepancy betweenexperimental and simulated 2D line shapes is an indication of ashortfall in the model Hamiltonian, which could be addressed infuture experiments that focus on system–bath interactions.

In Fig. 4, we show the results of our calculations for threerepresentative conformations. We compare simulated PM-2D FS

spectra (with Γ ¼ 0.31 optimized to the data, first column), 2DPE spectra (with Γ ¼ 2, second column), and linear spectra (thirdcolumn). It is evident that dimers with different conformationscan produce very similar linear spectra. However, these samestructures can be readily distinguished by the combined behaviorsof both linear and 2D spectra. We note that for both PM-2D FSand 2D PE methods, the 2D spectrum depends on dimer confor-mation. However, we found that the qualitative appearance ofsimulated PM-2D FS spectra appear to vary over a greater range

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Fig. 4. Comparison between simulated 2D and linear spectra for three selected dimer conformations. Each simulated linear spectrum (gray dashed curve) iscompared to the experimental line shape for the 7∶1 DSPC∶MgTPP sample. The laser spectrum is shown fit to a Gaussian (dashed gray curve) with centerfrequency 15;501 cm−1 (606 nm), and FWHM approximately 327.0 cm−1 (12 nm). Also shown are the positions of the underlying exciton transitions. Eachof the three conformations produce a linear spectrum in agreement with experiment, whereas only the first (optimized) conformation produces simulatedspectra that agree with PM-2D FS data (with Γ ¼ 0.31). Two-dimensional PE spectra (with Γ ¼ 2) are shown for comparison. Conformations are shown in thefourth column. The squares indicate the size of the MgTPP molecules, with monomer 1 in blue and monomer 2 in red with their respectiveQx andQy transitiondipoles indicated. (Top) (Optimized) conformation with θ ¼ 117.4°, ϕ ¼ 225.2°, α ¼ 135.2°, β ¼ 137.2°, and R ¼ 4.2 Å. (Middle) Conformation with θ ¼ 44.3°,ϕ ¼ 26.0°, α ¼ 29.2°, β ¼ 138.6°, and R ¼ 3.7 Å. (Bottom) Conformation with θ ¼ 82.4°, ϕ ¼ 18.7°, α ¼ 47.9°, β ¼ 124.0°, and R ¼ 7.6 Å. Color scale and contoursare the same as in Fig. 2.

Fig. 3. Comparison between rephasing (A) and nonrephasing (B) experimental (Left) and simulated 2D spectra (Right). Absolute value spectra (Top), real part(Middle) and imaginary part (Bottom). The simulated spectra are based on the optimized T-shaped conformation depicted in Fig. 4 (Top, fourth column) anddiscussed in the text. Color scale and contours have the same values as in Fig. 2.

16524 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1017308108 Lott et al.

and to exhibit a higher sensitivity to structural parameters in com-parison to simulated 2D PE spectra.

Our confidence in the conformational assignment we havemade is quantified by the numerical value of the regression ana-lysis target parameter χ2tot ¼ χ2linear þ χ22D ¼ 7.39þ 9.87 ¼ 17.26,which includes contributions from both linear and 2D spectra.By starting with this conformation and incrementally scanningthe structural parameters θ, ϕ, α, and β, we observed that χ2tot in-creased, indicating that the favored conformation is a local mini-mum when both linear and 2D spectra are included in the analysis(see SI Text). Similarly, we found that the value Γ ¼ 0.31 corre-sponds to a local minimum (see SI Text). If only one of the twotypes of spectra is included, the restrictions placed on the dimerconformation are significantly relaxed. As shown in Fig. 4, con-formations that depart from the optimized structure do not simul-taneously produce 2D and linear spectra that agree well withexperiment.

We found that the average conformation for the MgTPP dimeris a T-shaped structure with mean separation between Mg centersR ¼ 4.2 Å. Close packing considerations alone would suggest themost stable structure should maximize π–π stacking interactions.However, entropic contributions to the free energy due to fluc-tuations of the amphiphilic interior of the phospholipid bilayermust also be taken into account. It is possible that the averageconformation observed is the result of the system undergoingrapid exchange among a broad distribution of energetically equi-valent structures. In such a dynamic situation, the significance ofthe observed conformation would be unclear. However, at roomtemperature the DSPC membrane is in its gel phase (30), andstatic disorder on molecular scales is expected to play a promi-nent role. It is possible that the observed dimer conformation—an anisotropic structure—is strongly influenced by the shapesand sizes of free volume pockets that form spontaneously insidethe amphiphilic membrane domain. Future PM-2D FS experi-ments that probe the dependence of dimer conformation ontemperature and membrane composition could address this issuedirectly.

We have shown that PM-2D FS can uniquely determine theconformation of a porphyrin dimer embedded in a noncrystallinemembrane environment at room temperature. The appearanceof the PM-2D FS spectra is generally very different from thatproduced by simulation of the 2D PE method. This effect isdue to partial self-quenching of optical coherence terms thatgenerate population on the manifold of doubly excited states. Inthe current study on MgTPP chromophores in DSPC liposomes,we find that PM-2D FS spectra are quite sensitive to dimer con-formation (20–22).

The PM-2D FS method might be widely applied to problemsof biological and material significance. Spectroscopic studies ofmacromolecular conformation, based on exciton-coupled labels,could be practically employed to extract detailed structural infor-mation. Experiments that combine PM-2D FS with circulardichroism should enable experiments that distinguish betweenenantiomers of chiral structures. PM-2D FS opens previouslyundescribed possibilities to study exciton coupling under low light

conditions, in part due to its high sensitivity. This feature mayfacilitate future 2D experiments on single molecules, or UV-absorbing chromophores.

MethodsLiposome Sample Preparation. Samples with 7∶1 and 70∶1 DSPC∶MgTPP num-ber ratio were prepared according to the procedure described by MacMillanand Molinski (31). An additional control sample was prepared by dissolvingMgTPP in spectroscopic grade toluene. Details are provided in SI Text.

Linear Absorption Spectra. All samples were loaded into quartz cuvettes with3-mm optical path lengths. Concentrations were adjusted so that the opticaldensity was approximately 0.15 at 602 nm. Absorption spectra for each sam-ple were measured using a Cary 3E spectrophotometer (Varian, resolution<0.7 nm), over the wavelength range 520–640 nm. Each spectrum showedthe vibronic progression of the lowest lying electronic singlet transition withQð0;0Þ centered at approximately 602 nm in the toluene sample, and Qð0;0Þapproximately 606 nm in the 70∶1 lipid sample. The current work focused onthe electronic coupling betweenmonomerQð0;0Þ transition dipolemoments.

PM-2D FS. The PM-2D FS method was described in detail elsewhere (26).Samples were excited by a sequence of four collinear optical pulses withadjustable interpulse delays (see SI Text). The phases of the pulse electricfields were continuously swept at distinct frequencies using acoustoopticBragg cells, and separate reference waveforms were constructed from theresultant intensities of pulses 1 and 2, and of pulses 3 and 4. The referencesignals oscillated at the difference frequencies of the acoustooptic Braggcells, which were set to 5 kHz for pulses 1 and 2, and 8 kHz for pulses 3and 4. The reference signals are sent to a waveformmixer to construct “sum”and “difference” side-band references (3 and 13 kHz, respectively). Theseside-band references were used to phase-synchronously detect the fluores-cence, which isolates the nonrephasing and rephasing population terms,respectively. All measurements were carried out at room temperature. Thesignals were measured as the delays between pulses 1 and 2, and betweenpulses 3 and 4 were independently scanned. Fourier transformation ofthe time-domain interferograms yielded the complex-valued rephasing andnonrephasing 2D spectra. Further details are provided in SI Text.

Computational Modeling. A nonlinear global optimization with 13 variableswas performedwith the aid of the package KNITRO (32). Five variables definethe structural arrangements of the dimer; seven variables are associated withthe transition intensities, broadening, and line shapes for the linear and 2Dspectra, and the remaining variable Γ accounts for the quantum yield ofthe doubly excited manifold relative to the singly excited manifold. To suc-cessfully obtain good simulation/experimental agreement, we designed anonlinear least-squares optimization that included in its target functionthe six experimental 2D datasets (real, imaginary, and absolute value rephas-ing and nonrephasing spectra) and also a contribution from deviationsbetween the experimental and simulated linear spectra. Further detailsabout the construction of the target function are given in SI Text.

ACKNOWLEDGMENTS. A.H.M. thanks Professor Jeffrey A. Cina of theUniversity of Oregon and Professor Tadeusz F. Molinski of the Universityof California at San Diego for useful discussions. This material is based onwork supported by grants from the Office of Naval Research (GrantN00014-11-1-0193 to A.H.M.) and from the National Science Foundation,Chemistry of Life Processes Program (CHE-1105272). A.P.-O. and A.A.-G. weresupported as part of the Center for Excitonics, an Energy Frontier ResearchCenter funded by the US Department of Energy, Office of Basic Sciences(DE-SC0001088).

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16526 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1017308108 Lott et al.

Supporting InformationLott et al. 10.1073/pnas.1017308108SI Text1. Liposome Sample Preparation. Samples were prepared accordingto the procedure described by MacMillan and Molinski (1). Mag-nesium meso tetraphenylporphyrin (MgTPP) was purchasedfrom Strem Chemicals (Boston) and used without further purifi-cation. One and a half milligrams of MgTPP were dissolved in20 mL of toluene, transferred to a 50-mL spherical flask, and thesolvent was evaporated. In a separate flask, 12.8 mg of the phos-pholipid 1,2-distearoyl-sn-glycero-3-phosphocholine (DSPC, Sig-ma Aldrich) were dissolved in 20 mL of dichloromethane. Thecontents of the two flasks were combined to create a solution with7∶1 DSPC∶MgTPP number ratio. The organic solvent was re-moved, and 30 mL of nanopure water were added to the flask.The sample was alternately heated to 70 °C and agitated by ultra-sonication for a period of 15–30 min until an aqueous lipid/por-phyrin emulsion was fully formed. The mixture was prefilteredtwice through glass wool, and then extruded through a 100- to1,000-nm pore nylon membrane (Avestin) to create a suspensionof liposome vesicles. A second sample with 70∶1 DSPC∶MgTPPwas prepared using the same procedure. It was confirmed usingfluorescence microscopy that the MgTPP was localized to themembrane phase. An additional control sample was preparedby dissolving MgTPP in spectroscopic grade toluene.

2. Phase-Modulation 2D Fluorescence Spectroscopy. The phase-mod-ulation 2D fluorescence spectroscopy (PM-2D FS) method wasdescribed in detail elsewhere (2). Samples were excited by a se-quence of four collinear optical pulses with adjustable interpulsedelays (see Fig. S1). The pulse sequence was produced using ahigh repetition regenerative amplifier (Coherent, RegA 9050,250 kHz, pulse energy approximately 10 μJ), which was pumpedby a Ti∶sapphire seed oscillator (Coherent, Mira, 76 MHz, pulseenergy approximately 9 nJ, pulse width approximately 35 fs) and ahigh power continuous wave ND∶YVO4 laser (Coherent VerdiV-18, 532 nm). The amplified pulses were sent to two identicaloptical parametric amplifiers (Coherent, OPA 9400), with outputpulse energies approximately 70 nJ. The relative phase of pulses 1and 2, and pulses 3 and 4 were independently swept at distinctfrequencies (5 kHz and 8 kHz, respectively) using acoustoopticBragg cells. Electronic references were detected from the pulsepairs and sent to a waveform mixer to generate “sum” and “dif-ference” sideband signals (13 kHz and 3 kHz, respectively). Thesereference waveforms were used to phase-synchronously detectthe nonlinear fluorescence, which separately determined the non-rephasing and rephasing signals. The signal phase was calibratedto zero at the origin of the interferograms, i.e., when all interpulsedelays were set to zero. The measured pulse spectrum at thesample was Gaussian with FWHM approximately 327 cm−1 (ap-proximately 12 nm, shown in Fig. 1C). Separate dispersion com-pensation optics were used for each OPA, and the temporal pulsewidth determined by autocorrelation was approximately 60 fs forpulses 1 and 2, and approximately 80 fs for pulses 3 and 4. Thesample cuvette was a flow cell (Starna Cells, 583.3/Q/3/Z15, pathlength 3 mm, 0.1-mL volume), which was fitted to a peristalticpump (flow rate approximately 1 mL∕min, approximately 6 mLreservoir volume). The excitation beam was focused into the sam-ple using a 5-cm focal length lens. Fluorescence from the samplewas collected using a 3-cm lens, spectrally filtered (620-nm long-pass, Omega Optical) and detected using an avalanche photodiode (Pacific Silicon Sensor). All measurements were carriedout at room temperature. The signals were measured as thedelays between pulses 1 and 2 and between pulses 3 and 4 were

independently scanned. Fourier transformation of the time-do-main interferograms yielded the rephasing and nonrephasing2D optical spectra.

3. Exciton-Coupled Dimer of Three-Level Molecules. Monomers ofMgTPP have two equivalent perpendicular transition dipole mo-ments contained within the plane of the macrocycle (see Fig. 1B,Inset). These define the directions of degenerate Qx and Qy tran-sitions between the ground and lowest lying excited electronicstates (3–6). Both transition moments contribute to the collectiveexciton interactions in a molecular complex, as illustrated inFig. 1A.

To specify dimer conformations, we adopt a molecular-framecoordinate system similar to that described in refs. 4 and 5. Foreach monomer, a right-handed coordinate system is taken withthe x and y axes lying parallel to the Qx and Qy transition direc-tions, and the z axis perpendicular to the porphyrin plane. Weadopt the convention that a conformation is specified by themonomer center-to-center vector ~R, which is oriented relativeto molecule 1 according to polar and azimuthal angles θ andϕ. The relative orientation of molecule 2 is given by the Eulerangles α and β. Because of the degeneracy of the Qx and Qy tran-sitions, all of the results are independent of the third Euler angle,γ, which we set to zero from this point on (5).

For the Hamiltonian of a dimer of chemically identical three-level molecules in which system-bath effects are neglected, onedefines the tensor product states jiji, where i;j ¼ g;x;y respectivelylabel the states on monomer 1 and 2, and fjijig is the dimerHilbert space basis. Notice xðyÞ is short-hand notation for theexcited electronic state associated with the Qx (Qy) transitionon each monomer.

Within this localized basis description, one can write themolecular Hamiltonian for the dimer

H˜

¼ H˜

ð1Þ þH˜

ð2Þ þ V˜

¼ H˜

0 þ V˜; [S1]

where ~Hð2Þ (H˜

ð2Þ) is the Hamiltonian associated with monomer 1

(monomer 2). Within the point-dipole approximation, the elec-tronic coupling term can be expressed as V

˜¼ 1

4πεR3 ~μ˜

1 ·

ð1 − 3~R ~RR2 Þ · ~μ

˜2; with ~R the monomer center-to-center vector, ~μ1

ð ~μ2Þ the dipole operator for monomer 1 (monomer 2), and εthe dielectric constant.

We simplify our notation by denoting the nine basis statesfjliig, with jl1i ¼ jggi, jl2i ¼ jxgi, jl3i ¼ jygi, jl4i ¼ jgxi,jl5i ¼ jgyi, jl6i ¼ jxxi, jl7i ¼ jxyi, jl8i ¼ jyxi, jl9i ¼ jyyi. In thisbasis, the total Hamiltonian can be written as a nine-by-nine matrixof the form (5)

H∼≈

0

ε1 V 23 V 24 V 25

V 32 ε1 V 34 V 35

V 42 V 43 ε1 V 45

V 52 V 53 V 54 ε12ε1

2ε12ε1

2ε1

0BBBBBBBBBBBB@

1CCCCCCCCCCCCA

: [S2]

Here we have assumed all the diagonal contributions in the termsassociated withH∼ 0; i.e., we have assumed that hlijV∼ jlii ¼ 0 for all li.

Lott et al. www.pnas.org/cgi/doi/10.1073/pnas.1017308108 1 of 9

To set the reference energy scale, we set εðiÞg ¼ 0 withH∼

ðiÞjgi ¼ εðiÞg jgi, and therefore H∼ 0jggi ¼ ðεð1Þg þ εð2Þg Þjggi ¼ 0jggi.The value of ε1 used in our simulations was 16;500.7 cm−1, whichcorresponds to the monomer excitation energy associated witheither of the degenerate Qx or Qy transitions for the 70∶1 sample(see Fig. 1 in main text). Then H∼ 0jlki ¼ εkjlki with εk ¼ ε1 for any

of the states containing one excitation (k ¼ 2–5) and εk ¼ 2ε1 forthe states containing two excitations (k ¼ 6–9). Diagonalizationof the Hamiltonian is straightforward because it involves onlythe 4 × 4 block associated with the singly excited state manifold.Note that the eigenenergies of the singly excited state manifoldcorrespond to the exciton transitions underlying the region of in-terest in the experimental and simulated linear spectra. The posi-tions of these eigenenergies depend on the structural parametersof the dimer through the dependence on the couplings:

V∼ ij ¼1

4πεR3ð ~μ∼1Þij ·

�1 − 3

~R ~RR2

�· ð ~μ∼2Þij ¼

jμj24πεR3

κ2ij: [S3]

Here the orientation factor κ2ij is related to the directions of thetransition dipole moments and the vector connecting their centersaccording to κ2ij ¼ ðμ

∼1Þij · ðμ∼2Þij − ½3ðμ∼1Þij · R�½R · ðμ

∼2Þij�, where

R ¼ ðsin θ cosϕ; sin θ sinϕ; cos θÞ is the monomer center-to-centerunit vector, and ðμ

∼nÞij ¼ hlijμ∼njlji∕jμj is the normalized transition

dipole moment operator. The relationship between the squareof the monomer transition dipole moment and its absorption coef-ficient α, is given by (7)

jμj2 ¼ 3εℏcπNA

Z∞

−∞dν

αðνÞν

: [S4]

In Eq. S4, ε is the dielectric constant of the medium, ℏ isPlanck’s constant divided by 2π, c is the speed of light, andNA is Avogadro’s number. The factor ∫ ∞

−∞dναðνÞ∕ν is the opticallinewidth of the Qð0;0Þ transition, measured in wavenumbers,and divided by its peak value. We estimated this number bynumerical integration of the line shape to be 44.3 M−1 cm−1.

4. Theoretical Comparison Between PM-2DFS and 2D Photon-EchoSpectroscopy (2D PE) Signals. The PM-2D FS and 2D PE methodsare conceptually similar, yet important distinguishing factorscan result in their nonequivalence. The 2D PE signal can be in-terpreted as the third-order polarization of the sample, which isthe source of the detected signal field. In contrast, PM-2D FS isa technique based on fluorescence detection (2). The signal maybe considered proportional to the fourth-order excited statepopulation. We thus compare the signals of the two methodsbased on interpretation of 2D PE signals using third-orderperturbation theory, and PM-2D FS signals using fourth-orderperturbation theory.

We consider the semiclassical light-matter interaction Hamil-tonian,

H∼ sc ¼ H∼ 0 þH∼ intðtÞ; H∼ intðtÞ ¼ − ~μ∼· ~EðtÞ: [S5]

In PM-2DFS experiments, the electric field for P sequentialcollinear pulses polarized in the x direction can be described by~EðtÞ ¼ ∑p

j EjðtÞx, whereEjðtÞ ¼ λjAjðt − tjÞ cos½ωjðt − tjÞ þ ϕj�; [S6]

with λj the electric field maximum intensity, Ajðt − tjÞ ¼ e− 4 ln 2τ2fwhm

ðt−tjÞ2

the pulse envelope, and ωj is the laser frequency of the jth pulse.

Analogously, in 2D PE experiments the pulses are described byEjðtÞ ¼ λjAjðt − tjÞ cos½ωjðt − tjÞ − ~kj · ~r�. Using the density matrixformalism, the evolution of the system is described by the Liou-ville–von Neumann equation

iℏ∂ρ∼ðtÞ∂t

¼ ½H∼ intðtÞ; ρ∼ðtÞ�; [S7]

where we have used the “hat” notation to indicate that thecorresponding operators are in the interaction picture, i.e.,

O∼ ðtÞ≡ eiH∼ 0ðt−t0ÞO∼ e

−iH∼0ðt−t0Þ. A formal solution to Eq. S7 is

ρ∼ðtÞ ¼ ρ

∼ðt0Þ þ∑

∞

n¼1

ρ∼ðnÞðtÞ; [S8]

with

ρ∼ðnÞðtÞ≡ ð−1Þn

�iℏ

�nZ

t

t0

dτn

Zτn

t0

dτn−1…Z

τ2

t0

dτ1½H∼ intðτnÞ;

× ½H∼ intðτn−1Þ;½…;½H∼ intðτ1Þ;ρ∼ðt0Þ�…���: [S9]

The expectation of any observable, hO∼ ðtÞi≡ trfO∼ ðtÞρ∼ðtÞg, can be

expressed as hO∼ ðtÞi ¼ ∑∞n¼0hO∼

nðtÞi with hO∼nðtÞi≡ trfO∼ ðtÞρ

∼ðnÞðtÞg.

As previously mentioned, the 2D PE signal is associated withthe third-order polarization and therefore requires

Pð3ÞðtÞ≡ trfμ∼ðtÞρ∼ð3ÞðtÞg; [S10]

whereas the PM-2D FS signal is associated with the fourth-orderexcited state population

Að4ÞðtÞ≡ trfA∼ðtÞρ∼ð4ÞðtÞg; [S11]

with A∼ ¼ ∑vjνihνj the projector into all the states fjνig of theexcited state manifold.

We focus our discussion to the case of the nine-level model ofthe exciton-coupled dimer (see Fig. 1A in the text). Two-dimen-sional PE signals have been derived and studied for this model (8,9). In Fig. S2, we show the double-sided Feynman diagrams(DSFDs) contributing to the nonrephasing and rephasing signals,collected in the phase-matched directions KI ≡ k1 − k2 þ k3 andKII ≡ −k1 þ k2 þ k3, respectively. Neglecting dissipation for themoment, and assuming the rotating wave approximation in theimpulsive limit (8), one obtains the following expressions for eachof the nonrephasing terms:

R�1a ∝ ∑

e;e0½μegμgeμe0gμge0 �e1e2e3e4e−iωegτe−iωe0g t; [S12]

R2a ∝ ∑e;e0

½μegμge0μe0gμge�e1e2e3e4e−iωegτe−iωee0Te−iωegt ; [S13]

R�3b ∝ ∑

e;e0;f

½μegμge0μe0fμf e�e1e2e3e4e−iωegτe−iωee0Te−iωf e0 t : [S14]

Similarly, the rephasing terms are

R4a ∝ ∑e;e0

½μgeμegμge0μge0 �e1e2e3e4e−iωgeτe−iωe0gt : [S15]

Lott et al. www.pnas.org/cgi/doi/10.1073/pnas.1017308108 2 of 9

R3a ∝ ∑e;e0

½μgeμe0gμegμge0 �e1e2e3e4e−iωgeτe−iωe0eTe−iωe0g t: [S16]

R�2b ∝ ∑

e;e0;f

½μgeμe0gμf e0μef �e1e2e3e4e−iωgeτe−iωe0eTe−iωf e t: [S17]

Here, e;e0 ∈ fX2;X3;X4;X5g is the singly excited state manifoldafter diagonalization of the 4 × 4 block of the Hamiltonian inEq. S2, f ∈ fX6;X7;X8;X9g is the doubly excited state manifold,and ½μabμcdμjkμlm�e1e2e3e4 denotes the three-dimensional orienta-tionally averaged product hðμab · e1Þðμcd · e2Þðμjk · e3Þðμlm · e4Þi,where ei denotes the polarization of the ith pulse (9).

The detailed derivation of these expressions and their relationto the PM-2D FS terms will be published elsewhere. In Fig. S2, wepresent the corresponding PM-2D FS nonrephasing and rephas-ing DSFDs obtained from the fourth-order perturbation expan-sion (Eq. S11). For our current purpose, we provide here theconnection to the 2D PE expressions presented in formulasS12–S17. For example, it can be shown that for the case of thenonrephasing contributions, the following relations between 2DPE and PM-2D FS hold: R�

1a ¼ Q�5a ≡GSB1, R2a ¼ Q2a ≡ SE1,

R�3a ¼ Q�

3b ≡ ESA1, and also Q�3b ¼ Q7b. For the rephasing sig-

nals, we have R4a ¼ Q4a ≡GSB2, R3a ¼ Q3a ≡ SE2, R�2b ¼ Q�

2b≡ESA2, and Q�

2b ¼ Q�8b.

Although most of the 2D PE and PM-2D FS contributions areequal, there are two key differences that make their signalsunique:

1. Because PM-2D FS is a fluorescence-detection technique, it isimportant to consider the nature of the resulting excited stateof the system after the interaction with the four ultrafastpulses. As a consequence, even though mathematically Q�

3b ¼Q7bðQ�

2b ¼ Q�8bÞ, they do not contribute equally because the

terms Q�3bðQ�

2bÞ end in the singly excited manifold fjeigwhereas the terms Q7bðQ�

8bÞ end in the doubly excited statesfjf ig. Because the quantum yield of singly and doubly excitedstates are different in general, we must account for this factwhen simulating the signals. We introduced a multiplicativefactor Γ in front of the diagrams ending in a doubly excitedpopulation (see Q7b and Q�

8b in Fig. S2) to capture the relativequantum yield of this doubly excited state compared to thesingly excited states. Because of the abundance of nonradia-tive relaxation pathways for highly excited states, one expectsthe relative quantum yield of the doubly excited states to besignificantly smaller than the singly excited states. In a fullyideal coherent case, where two photons are emitted via thepathway jf i → jei → jgi, then Γ ¼ 2. In general, 0 ≤ Γ ≤ 2.For the dimer studied in the current work, the value ofΓ ¼ 0.31 was obtained from the global optimization that com-pared simulated and experimental spectra. A visual illustra-tion of these differences can be found in Fig. 4 of the maintext, where we compare for three different conformationsPM-2D FS spectra (Γ ¼ 0.31) to the corresponding 2D PEspectra (Γ ¼ 2). Table S2 shows the sensitivity of the optimiza-tion target function to the parameter Γ around the optimalvalue of 0.31.

2. The ground-state bleach, stimulated emission, and excited-state absorption (ESA) terms add up differently for 2D PEand PM-2D FS. This is a consequence of the third-order versusfourth-order perturbation approach, respectively. This is themain reason for the different appearances of PM-2D FS versus2D PE spectra.

The nonrephasing and rephasing 2D PE signals are written

S2D PENRP ðτ;T;tÞ ∝ R�

1a þ R2a − R�3b ∝ GSB1 þ SE1 − ESA1; [S18]

S2D PERP ðτ;T;tÞ ∝ R4a þ R3a − R�

2b ∝ GSB2 þ SE2 − ESA2: [S19]

Taking account of the differences between the two methodsmentioned above, and making use of Fig. S2, the nonrephasingand rephasing PM-2D FS signals are written

SPM-2D FSNRP ðτ;T;tÞ ∝ −ðQ�

5a þQ2a þQ�3b − ΓQ7BÞ

∝ −½GSB1 þ SE1 þ ð1 − ΓÞESA1�; [S20]

SPM-2D FSRP ðτ;T;tÞ ∝ −ðQ4a þQ3a þQ�

2b − ΓQ�2bÞ

∝ −½GSB2 þ SE2 þ ð1 − ΓÞESA2�: [S21]

Although the signal expressions corresponding to the two tech-niques are closely related, the variable sign contribution of theESA terms in the PM-2D FS expressions (formulas S20 andS21), in comparison to the well-known negative sign ESA contri-bution in 2D PE spectroscopy (formulas S18 and S19), can lead toconsiderably different appearances of the 2D spectra. The differ-ences in sign assignments of these terms arises from the commu-tator expansions of Eq. S11.

In the current work, we have considered the case where thepopulation time T ¼ 0 fs. To account for optical dephasing,inhomogeneous broadening, and other dissipative processes,we multiplied each term given by formulas S18–S21 by a phenom-enological line-broadening function, which is assumed to beGaussian in both coherence times, τ and t. That is, the rephasingsignals were multiplied by the factors e−τ

2∕σ2RP and e−t2∕κ2RP . Simi-

larly, we have used factors that contain the parameters σNRP andκNRP to describe the broadening of the nonrephasing signals.Fourier transformation of these equations to the ωτ and ωt do-mains provide the real, imaginary, and absolute value 2D spectrapresented in Fig. 3 of the text, with very good agreement to ex-periment. We note that although the intensities and positions of2D optical features are well accounted for by the molecular dimerHamiltonian, the observed spectral line shapes deviate markedlyfrom this simple model. The asymmetric line shapes could be dueto a number of factors, including differences in the system-bathcoupling and population times of the various excited states, aswell as the effects of laser pulse overlap. Understanding the ori-gins of the line shape asymmetries is important to future studies.

5. Computational Modeling. The search for the porphyrin-dimerconformation consistent with both linear and 2D experimentaldata involved a constraint-nonlinear-global optimization with13 variables. Optimizations performed separately on the linearand 2D spectra did not provide solutions consistent with bothsets of experimental data. We therefore employed a joint targetoptimization function, which involved a least-squares regressionoptimization using both sets of data—i.e., χ2tot ¼ χ2lin þ χ22D, whichis described in the next section.

Construction of target function for linear spectra. The Qð0;0Þ transi-tion of the monomer in the lipid bilayer membrane has energy16;500.7 cm−1 (see 70∶1 lipid∶MgTPP linear spectra shown inFig. 1B of the text). The Qð0;0Þ feature contains contributionsfrom both degenerate Qx and Qy transitions. Formation of theelectronically coupled dimer results in four new transitions, whicharise from the couplings between the states on each monomer.The energies of the resulting exciton transitions are given bythe eigenvalues obtained from diagonalization of the 4 × 4 blockof the Hamiltonian matrix (formula S2). The relative intensitiesof the exciton transitions are computed from the eigenvectors,which determine the transition dipole moments (5). All of thetransitions are broadened and modeled as Gaussians centeredat their respective eigenvalues, with equal line widths σlin. Thevalue of σlin was treated as an optimization parameter. The trialfunction used to reproduce the linear spectra can be written

Lott et al. www.pnas.org/cgi/doi/10.1073/pnas.1017308108 3 of 9

triallinðθ;ϕ;α;β;R;a0;η;σlinÞ

¼ a0 þ η

�∑4

i¼1

aiðθ;ϕ;α;β;RÞe−½ν−νiðθ;ϕ;α;β;R�2∕σ2lin�: [S22]

In Eq. S22, a0 accounts for background absorption, η is a multi-plicative factor that uniformly adjusts the intensities ai, and νiare the eigenenergies of the transitions. All of the optimizationparameters are determined by a least-squares regression analysiswhen compared to experimental data. We isolated the experi-mental data inside the region-of-interest frequency window16;300–16;810 cm−1, which is centered around the uncoupledmonomer transition energy (ε1 ¼ 16;500.7 cm−1). We denote theleast-squares sum as targetlin, and the contribution to the totaloptimization function is defined as χ2lin ¼ 105 targetlin. For exam-ple, the value of χ2lin corresponding to the best fit to both linearand 2D spectra is 7.39. The values of the eigenenergies for the op-timized conformation are ν1 ¼ 16;283 cm−1, ν2 ¼ 16;382 cm−1,ν3 ¼ 16;619 cm−1, and ν4 ¼ 16;718 cm−1, with respective relativeintensities a1 ¼ 0.867, a2 ¼ 1.94 × 10−13, a3 ¼ 1.00, anda4 ¼ 0.133.

Construction of the target function for the 2D spectra. The simula-tions of the 2D spectra involves the five geometrical parametersθ, ϕ, α, β, and R; the line-broadening parameters σRP, σNRP, κRP,and κNRP discussed above; and the doubly excited state manifoldfluorescence efficiency parameter Γ. For the least-squares analy-sis of 2D spectra we used the experimental data in the frequencywindow ωτ ∈ ½3.04 rad fs−1;3.15 rad fs−1� and ωt ∈ ½3.04 rad fs−1;3.15 rad fs−1�, where the most intense diagonal peaks and cross-peaks were located. The least-squares sum χ22D includes the sixsets of 2D experimental data, i.e., the real, imaginary, and abso-lute value spectra for rephasing and nonrephasing signals. Forexample, the value of χ22D for the best fit to both linear and 2Dspectra is 9.87.

Importance of the combined target function. Finding a single con-formation that agrees well with the linear and 2D data proved tobe a restrictive task, suggesting a definitive structural determina-tion. For example, the optimization of either χ2lin or χ

22D by them-

selves did not result in solutions that were consistent with theother type of spectra. A single solution was only possible whenthe combined target function χ2tot ¼ χ2lin þ χ22D was used. Asshown in Fig. 4 of the text, it was possible to find examplesfor which χ2lin was smaller than the value obtained for the optimalconformation. Yet in these cases the 2D spectra departed signif-icantly from the experimental data. Similarly, the optimization ofonly the target function χ22D could lead to misleading results. InTable S1, we list values for the target function and its linear and2D components for several values of the structural angles, whichwere scanned relative to the optimized conformation. We notethat Table S1 contains some negative values for either χ2lin orχ22D, indicating that a departure from the χ2tot minimum can yieldimproved agreement with one type of spectra at the expense ofagreement with the other. The results presented in Table S1 sug-gests that the sensitivity of the search to structural parametersallows for a quantitative estimate of dimer conformation.

6. Error Analysis and Propagation of Uncertainties in PM-2D FS Signals.In this section we calculate trust intervals for the structural para-meter values we have obtained for the MgTPP dimers embeddedin DSPC liposomes. We discuss here the uncertainties in ourresults, which arise from two different sources: (i) the qualityof the optimization search performed with the KNITRO packageand (ii) the uncertainty in the reference experimental data used toconstruct the target function χ2tot.

To determine the quality of the KNITRO search, e.g., the ab-sence of convergence to local minima, we performed a fine-resolution parameter scan to verify the extent to which the valuesobtained by the program indeed correspond to a global minimumof the target function, i.e., the best minimum from the multistartsearch. In Fig. S3, we plot the relative deviation Δχ2tot∕χ2tot ¼ðχ2 − χ2tot;refÞ∕χ2tot;ref from the reference value of χ2tot;ref , whichcan be interpreted as a relative error when moving away fromthe optimal conformation. Fig. S3 shows that the structure foundis the minimum, to within�1° for the each of the angles,�0.05 Åfor the R distance, and�0.01 units in Γ. The few missing points inthe scans for α and ϕ were removed because these converged to ahigher local minima above the predominant branch where themajority of points appear to lie. For all of the scans, one para-meter was varied while the remaining parameters that enteredthe calculation of the 2D spectra were held constant. The lackof convergence we refer to here is due to the additional optimi-zation required to relax the parameters needed for the linearspectra (i.e., a0, faig, η and σlin in Eq. S22). Because the few datapoints that converged above the predominant branch do not sug-gest an alternative minimum, it was not necessary to convergethese points since enough were present to clearly show the beha-vior upon approaching the minimum.

The scans in Fig. S3 also serve to assess the degree of sensi-tivity. For example, it is clear that the scans are more sensitive tothe parameters β, R, and θ, when compared to other degrees offreedom such as α, ϕ, and Γ. As a consequence, under a certainfixed relative error, one expects that the uncertainty will be smal-ler for β and θ while slightly larger for α and ϕ.

Having established that our search routine is almost exact, wenext address the error propagation due to uncertainties in theexperimental measurements. In the following, we base our discus-sion on χ22D motivated by the assumption that Δχ2tot∕χ2tot≈Δχ22D∕χ22D, i.e., that these relative errors are comparable. We thususe our estimate of Δχ22D∕χ22D to read out the trust intervalsdirectly from the scans shown in Fig. S3. This relative error wasestimated to be approximately 1%, and it is indicated with thered-shaded rectangles in Fig. S3.

We next explain the assumptions we have made to obtainthe 1% estimate using standard error propagation analysis (10).The 2D target function is defined according to

χ22D ¼ ∑ωiτ ;ω

jt

fAbs½NRPsimðωiτ;ω

jtÞ� −Abs½NRPexpðωi

τ;ωjtÞ�g2

þ fRe½NRPsimðωiτ;ω

jtÞ� −Re½NRPexpðωi

τ;ωjtÞ�g2

þ fIm½NRPsimðωiτ;ω

jtÞ� − Im½NRPexpðωi

τ;ωjtÞ�g2

þ fAbs½RPsimðωiτ;ω

jtÞ� −Abs½RPexpðωi

τ;ωjtÞ�g2

þ fRe½RPsimðωiτ;ω

jtÞ� −Re½RPexpðωi

τ;ωjtÞ�g2

þ fIm½RPsimðωiτ;ω

jtÞ� − Im½RPexpðωi

τ;ωjtÞ�g2: [S23]

In Eq. S23, the subscripts “sim” and “exp” indicate simulatedand experimental spectra, respectively. The indices “i” and “j”indicate the 2D frequency coordinate. For the error propagationanalysis, we include every data point from each of the six Fourier-transformed experimental signals [AbsðNRPexpÞ, AbsðRPexpÞ,ReðNRPexpÞ, ReðRPexpÞ, ImðNRPexpÞ, and ImðRPexpÞ] to definea variable with its own uncertainty. For simplicity, we define

Abs½NRPexpðωiτ;ω

jtÞ�≡ f ij1; Re½NRPexpðωi

τ;ωjtÞ�≡ f ij2;

Im½NRPexpðωiτ;ω

jtÞ�≡ f ij3; Abs½RPexpðωi

τ;ωjtÞ�≡ f ij4;

Re½RPexpðωiτ;ω

jtÞ�≡ f ij5; and Im½RPexpðωi

τ;ωjtÞ�≡ f ij6:

Lott et al. www.pnas.org/cgi/doi/10.1073/pnas.1017308108 4 of 9

The sum in Eq. S23 is performed over the discrete frequencyvalues inside the interval ωτ;ωt ∈ ð3.04;3.15Þ rad fs−1. Becausethere are N ¼ 101 points per frequency axis inside this interval,the number of terms in the summation containsN2 ¼ 10;201 vari-ables of the form f ijk for each value of k. Because we are dealingwith k ¼ 1–6, the number of independent variables in the errorpropagation analysis is 61,206. We define z≡ χ22Dðff ijkgÞ ¼χ22DðfgngÞ, where gn ¼ f ijk, with n running from 1–61,206 denotingall possible combinations of i, j, and k. Under the assumptionthat all variables are independent, we estimate the uncertaintyof z by (10)

Δz ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi∑61;206

n¼1

�∂z∂gn

Δgn

�2

vuut : [S24]

In terms of the gn variable, Eq. S23 for χ22D can be rewritten as

z ¼ ∑61;206

n¼1

ðgsimn − gnÞ2: [S25]

The partial derivative can be calculated according to∂z∕∂gn ¼ −2ðgsimn − gnÞ2. Once the uncertainties Δgn are calcu-lated, the error in Eq. S24 can be easily determined.

As previously stated, each of the gn corresponds to a data pointfrom any of the 2D spectra involved in the calculation of χ22D.To estimate the value of uncertainty associated with each of thesevariables, we divide the 61,206 variables into two groups; the firsthalf (n ¼ 1–30;603) associated with the absolute value, real andimaginary parts of the rephasing data, and the remaining half(n ¼ 30;604–61;206) associated with that of the nonrephasingdata. To simplify these calculations, we find a single uncertaintyvalue representative for each of the two types of spectra.We denote these as ΔgRP and ΔgNRP for the rephasing and non-rephasing data, respectively. Calculations of these uncertaintiesare illustrated in Fig. S4. The uncertainty is estimated from fourdifferent experimental runs performed on a ZnTPP monomer indimethylformamide solution, which were processed using anidentical procedure to the MgTPP samples studied here. The2D absolute value rephasing and nonrephasing spectra of onedata run are shown in Fig. S4 A and B, respectively. In Fig. S4 Cand D are shown overlays of the absolute value rephasing andnonrephasing signals, sRPðNRPÞ

ω , for each of the four data runsalong the diagonal profile, with ωτ ¼ ωt ¼ ω. Fig. S4 E and F

show the average signal sRPðNRPÞω ≡ hsRPðNRPÞ

ω isets along the diag-onal profile, where h⋯isets indicates the average performed overindividual datasets. We similarly calculate the variance at eachvalue of ω according to σ2RPðNRPÞðωÞ ¼ h½sRPðNRPÞ

ω − sRPðNRPÞω �2isets,

which are shown in Fig. S4 G and H.The representative uncertainties, ΔgRP and ΔgNRP, are esti-

mated as the frequency average of the standard deviations alongthe diagonal profiles, i.e., ΔgRPðNRPÞ ¼ hσRPðNRPÞðωÞiω. The aver-age over frequency was done to include most of the significantdata, taking approximately twice the full-width at half-maximumfrom the main peak for both the rephasing and nonrephasing pro-files—i.e., over the interval ω ∈ ð3.07;3.20Þ rad fs−1. By using theresulting values forΔgRP ¼ 0.0086 andΔgNRP ¼ 0.016 in Eq. S24,we find that Δz∕zref ¼ Δχ22D∕χ22D ≈ Δχ2tot∕χ2tot ¼ 0.0096∼ 1%. Thevalue of χ22D ¼ 9.87 used for this estimate corresponds to the re-ference value obtained for the optimal conformation. Having es-tablished that the expected error is approximately 1%, wedetermine the trust intervals directly from the parameter scanplots shown in Fig. S3, as indicated by red-shaded rectangles.These intervals correspond to −16° < Δθ < 4°, −11° < Δϕ <11°, −11° < Δα < 11°, −2° < Δβ < 2°, −0.05 Å < ΔR < 0.05 Å,and −0.1 < ΔΓ ¼ 0.1, where Δx≡ x − xref , and xref is taken fromthe optimized outcomes.

We conclude this section by commenting on the uncertainty ofthe variable R. In addition to the uncertainties discussed above,an accurate estimate of ΔR must also account for its dependenceon the calculated value of the monomer square transition dipolemoment jμj2. Uncertainty in the estimation of jμj2 (Eq. S4) willappear in the electronic couplings (Eq. S3) as a rescaling of theend-to-end distance R. For example, too small an estimation ofjμj2 will result in an apparent value of R that is also too small.Although we have attempted to make our estimate of jμj2 asaccurate as possible, we cannot discount the possibility that asystematic error is present. We note that the values we have ob-tained for the angles θ, ϕ, α, and β constrain the conformationsignificantly. We therefore propose that further refinements inthe conformation could be achieved through quantum chemicalcalculations. For example, semiempirical calculations on theMgTPP dimer, in which only the distance R is varied, could beused to obtain its value where the energy minimum occurs. Giventhe degree of molecular detail provided by quantum chemicalcalculations, it should in principle be possible to capture theeffects of steric interactions between bulky phenyl groups. Suchan approach might be useful to further refine the values of thestructural parameters within their trust intervals.

1. MacMillan JB, Molinski TF (2004) Long-range stereo-relay: Relative and absoluteconfiguration of 1,n-glycols from circular dichroism of liposomal porphyrin esters.J Am Chem Soc 126:9944–9945.

2. Tekavec PF, Lott GA, Marcus AH (2007) Fluorescence-detected two-dimensionalelectronic coherence spectroscopy by acousto-optic phase modulation. J Chem Phys127:214307.

3. GoutermanM (1979) The Porphyrins, ed Dolphin D (Academic Press, New York), Vol III,pp 1–156.

4. Stomphorst RG, Koehorst RBM, van der Zwan G, Benthem B, Schaafsma TJ (1999)Excitonic interactions in covalently linked porphyrin dimers with rotational freedom.J Porphyr Phthalocyanines 3:346–354.

5. Koolhaas MHC, van der Zwan G, van Mourik F, van Grondelle R (1997) Spectroscopyand structure of bacteriochlorophyll dimers. I. Structural consequences of non-conservative circular dichroism spectra. Biophys J 72:1828–1841.

6. Won Y, Friesner RA, Johnson MR, Sessler JL (1989) Exciton interactions in syntheticporphyrin dimers. Photosynth Res 22:201–210.

7. Hardwick JL (2003) Absorption and emission of electromagnetic radiation. Handbookof Molecular Physics and Quantum Chemistry, ed Wilson S (Wiley, New York).

8. Mukamel S (1995) Nonlinear Optical Spectroscopy (Oxford University Press, Oxford,UK).

9. ChoM (2009) Two-Dimensional Optical Spectroscopy (CRC Press, Boca Raton, FL), 1st Ed.10. Taylor JR (1997) An Introduction to Error Analysis: The Study of Uncertainties in

Physical Measurements (University Science Books, Sausalito, CA), 2nd Ed.

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T

250 kHz OPA

driver 1

APD

mono 2

sampleτ

twaveformmixer

‘sum’ lock-in

‘dif’ lock-in

computer

fluorescence

250 kHz OPA

mono 1

driver 2

ref 1

ref 2

MZI 1

MZI 2

BS

BS

BS

BS

BS

B

sample

A pulse 4 pulse 3 pulse 2 pulse 1

T τ, φ21t, φ43

Fig. S1. (A) Collinear sequence of optical pulses used in PM-2D FS experiments. The coherence, population, and measurement periods (τ, T , and t) are in-dicated, as well as the relative phase of pulses 1 and 2 (ϕ21), and pulses 3 and 4 (ϕ43). (B) Schematic of the PM-2D FS apparatus, described in the text and in ref. 2.The phases of the pulse electric fields are swept using acoustooptic Bragg cells, which are placed in the arms of two Mach–Zehnder interferometers (MZI 1 andMZI 2). The excitation pulses are made to be collinear before entering the sample. Reference waveforms are constructed from the pulse pairs from eachinterferometer. The reference signals oscillate at the difference frequencies of the acoustooptic Bragg cells (5 kHz and 8 kHz for refs. 1 and 2, respectively).The reference signals are sent to a waveform mixer to construct “sum” and “difference” side band signals (3 kHz and 13 kHz). These reference side bands areused to phase-synchronously detect the fluorescence, which isolates the nonrephasing and rephasing population terms, respectively.

Fig. S2. DSFDs representing the light-matter interactions contributing to the rephasing and nonrephasing signals measured experimentally. The four-levelmodel used to describe the coupled dimers of MgTPP are shown in Fig. 1A of the text. The collective dipole moment allows transitions from the ground state tothe first-excited manifold, and from the latter to the final doubly excited state. The sign associated with each diagram is determined by the number of arrows(dipole interactions) on the right vertical line of each ladder diagram (“bra” side). An even (odd) number of interactions picks up a positive (negative) sign forthe term under consideration. Therefore, the nonrephasing and rephasing 2D PE signals are S2D PE

NRP ðτ;T ;tÞ ∝ R�1a þ R2a − R�

3b and S2D PERP ðτ;T ;tÞ ∝ R4a þ R3a − R�

2b,

respectively, whereas the corresponding PM-2D FS signals are SPM-2D FSNRP ðτ;T ;tÞ ∝ −ðQ�

5a þQ2a þQ�3b − ΓQ7BÞ and SPM-2D FS

RP ðτ;T ;tÞ ∝ −ðQ4a þQ3a þQ�2b − ΓQ�

2bÞ. Theparameter Γ accounts for the different fluorescence quantum yields between doubly and singly excited state manifolds.

Lott et al. www.pnas.org/cgi/doi/10.1073/pnas.1017308108 6 of 9

-30 -20 -10 10 20 30

0.5

1.0

1.5

2.0

∆θ (°)-30 -20 -10 10 20 30

0.05

0.10

0.15

0.20

0.25

∆φ (°)

-30 -20 -10 10 20 30

0.05

0.10

0.15

0.20

0.25

∆α (°)-30 -20 -10 10 20 30

2

4

6

8

∆β (°)

-1.5 -1.0 -0.5 0.5 1.0 1.5

5

10

15

20

25

∆R (Å)-0.3 -0.2 -0.1 0.1 0.2 0.3

0.02

0.04

0.06

0.08

∆Γ

A B

C D

E F

∆χ2 / χ2tottot ∆χ2 / χ2

tottot

∆χ2 / χ2tottot ∆χ2 / χ2

tottot

∆χ2 / χ2tottot

∆χ2 / χ2tottot

20 15 10 5 5 10

0.02

0.06

0.10

4 2 2 4

0.02

0.06

0.10

0.10 0.05 0.05 0.10

0.02

0.04

0.06

Fig. S3. Relative deviation of the target function, Δχ2tot∕χ2tot, from the optimized reference value, χ2tot;ref, as a function of structural parameter uncertainties.Cross-sections of the target function are shown for the uncertainties (A) Δθ, (B) Δϕ, (C) Δα, (D) Δβ, (E) ΔR, and (F) ΔΓ, where Δx ≡ x − xref, and xref is the valuecorresponding to the optimized conformation. The optimized conformation corresponds to a minimum of the multidimensional parameter surface. As in-dicated by the red shaded rectangles, trust intervals are directly read out from these plots, based on the approximately 1% relative error associated with theexperimental data quality. The trust interval regions are expanded and shown (Insets) for the parameters Δθ, Δβ, and ΔR. The resulting intervals are−16° < Δθ < 4°, −11° < Δϕ < 11°, −11° < Δα < 11°, −2° < Δβ < 2°, −0.05 Å < ΔR < 0.05 Å, and −0.1 < ΔΓ ¼ 0.1.

Lott et al. www.pnas.org/cgi/doi/10.1073/pnas.1017308108 7 of 9

3.0 3.1 3.2 3.3

0.1

0.2

0.3

0.4

3.0 3.1 3.2 3.3

0.1

0.2

0.3

0.4

3.0 3.1 3.2 3.3

0.1

0.2

0.3

0.4

3.0 3.1 3.2 3.3

0.1

0.2

0.3

0.4

Rephasing Nonrephasing

3.0 3.1 3.2 3.3

0.5e-4

1.0e-4

1.5e-4

2.0e-4

3.0 3.1 3.2 3.3

2.0e-4

4.0e-4

6.0e-4

ω (rad / fs) ω (rad / fs)

ampl

itude

(a.

u.)

ampl

itude

(a.

u.)

varia

nce

(a. u

.)

A B

C D

E F

four datasets:

diagonalprofile

four datasets:

diagonalprofile

avergare offour data sets

diagonalprofile

average of four datasets

diagonalprofile

G H

3.0 3.1 3.2 3.3 3.0 3.1 3.2 3.3

3.0

3.1

3.2

3.3

3.0

3.1

3.2

3.3

ωτ (rad / fs) ωτ (rad / fs)ω

t (ra

d / f

s)

Fig. S4. Experimental data runs performed on ZnTPPmonomer in dimethylformamide solution, which were used for error propagation analysis. InA and B areshown, respectively, the 2D absolute value rephasing and nonrephasing spectra of a single representative dataset. In C and D are shown overlays of theabsolute value rephasing and nonrephasing signals for each of the four data runs along the diagonal profile. E and F show the average of the four datasetsalong the diagonal profile. In G and H are shown the corresponding variances along the diagonal profile. By integrating the standard deviation of the dataover the interval ω ∈ ð3.07;3.20Þ rad fs−1, we obtain the relative uncertainties ΔgRP ¼ 0.0086 and ΔgNRP ¼ 0.016 (defined in SI Text). These values are input toEq. S24 to estimate the relative target function uncertainty Δχ2tot∕χ2tot ¼ 0.0096∼ 1%, which in turn establishes the trust intervals of the structural parametersrelative to the optimized outcome.

Lott et al. www.pnas.org/cgi/doi/10.1073/pnas.1017308108 8 of 9

Table S1. Linear least-squares target function χ 2tot ¼ χ 2lin þ χ 22D dependence on structural angles

Δθ Δϕ Δα Δβ

deg Δχ2lin Δχ22D Δχ2tot Δχ2lin Δχ22D Δχ2tot Δχ2lin Δχ22D Δχ2tot Δχ2lin Δχ22D Δχ2tot−30 8.124 5.447 13.57 3.580 0.807 4.388 3.580 0.807 4.388 141.2 −0.353 140.9−25 2.311 2.143 4.454 1.933 0.495 2.429 1.934 0.495 2.429 101.2 −0.340 100.9−20 0.327 0.715 1.041 0.965 0.282 1.248 0.965 0.282 1.248 64.68 −0.306 64.38−15 −0.018 0.150 0.132 1.077 0.144 1.221 1.077 0.144 1.221 34.84 −0.255 34.58−10 0.080 −0.057 0.022 0.606 0.059 0.665 0.606 0.059 0.665 14.05 −0.190 13.86−5 0.381 −0.100 0.285 0.557 0.014 0.571 0.557 0.014 0.571 3.031 −0.107 2.9240 0 0 0 0 0 0 0 0 0 0 0 05 0.382 0.317 0.698 0.557 0.014 0.571 0.557 0.014 0.571 1.034 0.147 1.18210 1.333 1.145 2.478 0.606 0.059 0.665 0.606 0.059 0.665 3.220 0.359 3.58015 3.600 3.172 6.772 1.077 0.144 1.221 1.077 0.144 1.221 5.531 0.680 6.21020 11.91 7.672 19.58 0.965 0.282 1.248 0.965 0.282 1.248 7.352 1.187 8.54025 27.24 16.55 43.79 1.933 0.495 2.429 1.934 0.495 2.430 7.875 2.007 9.88230 52.35 32.65 85.00 3.580 0.807 4.388 3.580 0.807 4.388 7.00 3.325 10.33

Target function values are given relative to the reference values: χ2lin ¼ 7.39, χ22D ¼ 9.87, and χ2tot ¼ 17.26, which correspond to the conformation with

structural parameters θ ¼ 117.4°, ϕ ¼ 225.2°, α ¼ 135.2°, β ¼ 137.2°, R ¼ 4.2 Å, and Γ ¼ 0.31, and line-broadening parameters σRP ¼ 108.1 fs, σNRP ¼ 96.2 fs,κRP ¼ 98.1 fs, and κNRP ¼ 102.9 fs.

Table S2. Linear least-squares target function χ 22D dependence on fluorescence efficiency Γ of the doubly excited statemanifold

Γ 0 0.2 0.31 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0Δχ22D 0.857 0.104 0 0.083 0.814 2.319 4.580 7.520 10.87 13.97 17.25 21.20

Values are given relative to the optimized conformation with χ22D ¼ 9.87 and Γ ¼ 0.31.

Lott et al. www.pnas.org/cgi/doi/10.1073/pnas.1017308108 9 of 9