Assembling programmable FRET-based photonic networks using designer DNA scaffolds

DNA demonstrates a remarkable capacity for creating designer nanostructures and devices. A growing number of these structures utilize Förster resonance energy transfer (FRET) as part of the device's functionality, readout or characterization, and, as device sophistication increases so do the concomitant FRET requirements. Here we create multi-dye FRET cascades and assess how well DNA can marshal organic dyes into nanoantennae that focus excitonic energy. We evaluate 36 increasingly complex designs including linear, bifurcated, Holliday junction, 8-arm star and dendrimers involving up to five different dyes engaging in four-consecutive FRET steps, while systematically varying fluorophore spacing by Förster distance (R0). Decreasing R0 while augmenting cross-sectional collection area with multiple donors significantly increases terminal exciton delivery efficiency within dendrimers compared with the first linear constructs. Förster modelling confirms that best results are obtained when there are multiple interacting FRET pathways rather than independent channels by which excitons travel from initial donor(s) to final acceptor.

for the 1.0R0 multi-dye stars where yield has been used as the fitting parameter(s). The level of "yield" required for these excellent fits are indicated by the percentages accompanying each line, with the rightmost percentage being that of the full structure and the others referring to partial structures that carry one less dye type and that are meant to capture the aggregate effect of all FRET-participating partial structures in the ensemble. The dyes not included in these specific assemblies are presumed to act as "free" dyes with no FRET contribution.   Notes: Estimates are  10%; fully formed structure: band present at the expected MW on the gel; partially formed structure: bands below fully formed, but not qualified as unformed; unformed structure: fluorescence present at <50 bp and consistent with ssDNA or unhybridized oligos.             Average scaled and normalized PL area of each fluorophore. PL area for each fluorophore was determined by decomposition of PL spectra collected from full Cy3-Cy5.5 construct. Across all data sets and constructs, the fluorophore contributions were then normalized to the direct excitation from molar equivalent Cy3 controls and scaled based on the construct (uni-, bi-directional, Holliday or 8-way junction). The resulting values for the full constructs were then all scaled by the maximum value (1.5R0 8-way junction) c Parenthetical values are average acceptor sensitized emission efficiencies d Average end-to-end efficiency through the three-step Cy3-Cy3.5-Cy5-Cy5.5 construct.

Controls (2 missing)
Cy3.5-Cy5 (c) Note: When multiple values are given, this is because a particular structure has arms that differ in length, generally by a single base pair. These differences were included in the simulations.

Supplementary
where Jij is the spectral overlap integral (estimated from the emission spectrum of donor i, the absorption spectrum of acceptor j, and the molar extinction coefficients), n is the refractive index of the medium, Qi is the fluorescence quantum yield (QY) of the donor i, and  2 is the dipole orientation factor that is usually taken to be 2/3 as is appropriate for the quasi-random dipole orientations found in these constructs (see below). 1 . The values for R0 obtained using (1) appear in Table 1. To obtain rough approximations to the nearest-neighbor dye separations we exploit the fact that each such dye pair is separated/supported by a DNA duplex that is much shorter than the persistence length and so can be regarded as straight/rigid. The distance between the dyes, or more precisely between the dye attachment points, can then be arranged by the DNA design as estimated by the cylinder model 2 in Å as: where Nij is the number of intervening bases, ij is either 0 or 180 o depending on whether the dyes are attached to the same DNA strand or to the complementary strand, d ~ 1 nm is the halfwidth of the DNA helix, and the factor of 34.3 comes from the 360° of rotation divided by the number of bases in a full turn. Dye attachment distances calculated in this way for the various DNA designs of this paper are shown in Supplementary Table 60.
The dyes were attached to the DNA both at terminal positions and internally. The former occurred at ends, nicks, or breaks in the DNA that were introduced no less than 9 bases apart (for reasons of stability), and that put the attachment points at the 3' (preferred) and/or 5' ends of a DNA strand. Where possible, the designs used a long scaffold strand (template) to which the dyes were attached via hybridization of shorter oligos, however, for the closest dye spacing's (i.e., 0.5R0) a staggered or concatenated DNA construction approach (no template) had to be used. To check for excessive distortions at junctions, all of the DNA designs (without dyes) were constructed within the program Nanoengineer 3 . Specific considerations regarding the attachment of the individual dyes are as follows: The Cy5 dye is the most tightly bound to the DNA scaffold because it is incorporated directly into a DNA strand as a phosphoramidite (with an unpaired adenine in the opposite strand) and is always an internal label that thus has two attachment points as shown in Supplementary Figure   32C. The Cy5 dyes can therefore be expected to sit parallel and very close to the DNA scaffold, and without much flexibility.
The Cy3 dye is also attached as phosphoramidite, but unlike the Cy5, it is usually situated at a terminal position as shown in Supplementary Figure 32C  and orientation (although there is evidence from molecular dynamics that it could stack onto the end of the duplex 4,5,6 ). We therefore assume the terminal Cy3 dyes will tend to be directed away from the end of the DNA, and counting the size of the elongated molecule, will add an extra 5-10Å to the spacing. The flexibility of the attachment means the dye's precise orientation will vary, thus justifying the random dipole approximation.
The Cy3.5 dye is also an internal label, but it is not a phosphoramidite and instead has the dye more loosely attached via a C6 linker following succinimidyl ester conjugation. Given its internal position, we assume it will tend to be radially directed and with a distance from the helical axis on the order of 10Å. Again, its flexibility in orientation supports the use of the random dipole approximation. The one exception to this is the 4-dye 2:1 dendrimer structure where Cy3.5 ester was replaced with a phosphoramidite ( Figure 3H).
The attachment of Cy5.5 is the same as for the Cy3.5, however, it sometimes is an internal label and sometimes a terminal label. In the former case, its positioning is assumed mainly radial, whereas for the latter a preference for an axial extension is assumed. The A488 dye is a terminal 86 label with a flexible C6 linker and an assumed axial preference. The A647 dye has the same attachment as the A488, but is internal and has an assumed radial preference.
The particular base sequences used for the designs of this paper are given in Supplementary Tables 6-42, with those for the "star" geometries adapted from Spillmann et al. 7  buffer. These sequences were also checked for self-complementarity and cross-complementarity using Operon's Oligo Analysis Tool, with a limit imposed of no more than 5 bases of nonspecific complementarity. internal Cy5 labels that occur at the branching junction. 40 base oligos were assembled to the 58 base center and doubly internally labeled with Cy3.5. Finally, at the ends were 18 base oligos with 3' and 5' labeled Cy3s.

Supplementary Note 2: Figures-of-merit/metrics
Viewing the fluorophore assemblies of this paper as light-harvesters, it is of central interest to assess their overall performance. To this end, we employ the following metrics or figures-ofmerit many of which can be estimated from experiment and all of which can be calculated from the Förster analysis. The latter also allows one to assess ideal performance, and to address questions of mechanism and of how to improve performance.
The terminal enhancement factor (TEF) is a relative measure of performance that compares the output of the terminal acceptor of each construct to that of a reference construct: For convenience we take the reference construct to be the unidirectional 1.5R0 photonic wire.
Having but a single arm and the largest dye spacing, this structure should have the smallest output and so the TEF is generally always an enhancement, i.e., greater than one. It is also important to note that the concentrations (and illumination, etc.) used in the test and reference experiments need to be the same. If not, then (3a) needs to be corrected as Obviously AG is the same as TEF with a particular definition for . As in (3b), (4) and (5) must be corrected if the test and reference experiments have different concentrations, illumination levels, etc.
We define exciton transfer efficiency as the conditional probability that an exciton will transfer from a given excited donor to a given acceptor, with the process of most interest being the endto-end transfer from a peripheral donor to a terminal acceptor. Like AG, this quantity is harder to estimate experimentally than TEF, AE or AG, but an empirical formula that has been used is where QA and QD are the QYs of the acceptor and donor, respectively 7,10,11 . In this expression, the denominator gives the number of excited donors (per second) while the numerator is the number of excited acceptors (per second) that did not become excited as a result of direct excitation. If D  and A  are the outputs of the peripheral donors and the terminal acceptor, respectively, then (6) will provide an approximation to the end-to-end efficiency with the proviso that direct excitations of intermediate dyes do not contribute significantly. It should be noted that the latter can be especially significant in all-organic systems that lack a QD's strong absorbance and its ability to be excited at "distant" wavelengths. On the other hand, if we are considering the constructs as light-harvesters, then any collection counts and the anywhere-toend efficiency of (6) is then the appropriate measure.
An efficiency analogous to that of (6) is also readily estimated from the Förster analysis as the average conditional probability that an exciton generated somewhere within the structure (including on the central dye itself) will result in excitons reaching the focus. The anywhere-toend efficiency so defined is computed as: It is apparent from the gel migration that the larger 8-arm stars do not migrate specific to what is expected just from DNA size given their non-linear shape. The latter manifest what appear to be multiple structures which may include cross-linked assemblies. We qualify these also as partial structures, although, they may still retain the ability to engage in efficient FRET.
This fact is exacerbated in the dendrimer structures and for those the FPLC method was more effective. We qualify our assembly efficiency values below as just estimates at this time and a more intensive study of their formation efficiency is underway for publication at a later date.  Figure 25) show the separation of the formed solution divided again into formed, partial, and unformed peaks. Since the absorption of each fraction is hard to determine, the dye molar extinction coefficient was used to calibrate the intensity of each peak. The FPLC separates according to size. Since the dendrimer structures are close to the same size, the primary peak runs close to the same at 11, 10 and 9.5 mL volume fractions for 2:1, 3:1 and 4:1 dendrimers, respectively. Peaks with all dyes present but not in the fully formed peak as stated above are considered partial formations and peaks where only one or two dyes are present are considered unformed. The larger size presents a peak with a lower fraction volume collection at 10 mL, as compared to 11 ml with the 4-dye 2:1 dendrimer. Given the similar nature of the peaks between the 4-dye and the 5-dye systems, the 2:1 dendrimer was used as a standard for calculating the relative concentration of the two 91 primary peaks that are seen. Tabulated results for all analysis are indicated in Supplementary   Tables 39 and 53.

Supplementary Note 4: spFRET analysis of 0.75R0 Cy3nCy5 structures
To better understand the ensemble FRET efficiencies of the linear, bifurcated, Holliday junction and 8-arm star Cy3nCy5 DNA structures, spFRET experiments were performed where each structure was labeled with Cy3 and Cy5, as described above for the 0.75R0 series. Figure 2D compares spFRET histograms for each of the structures. The FRET efficiency is calculated from the ratio IA/(IA+γID), where ID and IA are the photon burst signals from the donor and acceptor channels, respectively. The factor, γ, accounts for the photon detection efficiencies of the two channels, and fluorescence quantum yields of the donor and acceptor. For the experiments described here γ1. At initial incident laser powers <75 µW at the objective, we observed that the structures with more than one Cy3 donor showed increased relative intensity of the low FRET peak. This result may be attributed to an increased excitation generation rate as the number of Cy3 donors is increased within a structure, which in turn increases the probability for Cy5 photobleaching. To minimize the photobleaching, the laser power was reduced until the relative intensity of the low-FRET peak became insensitive to further power reduction. The high FRET peak observed for each construct (FRET efficiency  0.75 -0.80) indicates that at least a fraction of the ensemble undergoes efficient FRET. In addition, we observe a slight increase in efficiency of the high FRET peak as the number of arms increases from one to eight. A distinguishing feature of the 8-arm histogram is the significantly and larger relative intensity of the low FRET peak, which persists even at the lowest laser power used for excitation. The low FRET peak is asymmetric with a distribution that tails towards intermediate FRET efficiencies.
Taken together, these observations suggest that there are both high FRET and low FRET pathways within the more complex 8-arm structure, depending on which Cy3 is initially excited.
The relatively large ring-like opening at the center of the 8-arm structure (~30 Å, see Supplementary Figure 51) 8 creates Cy3-Cy5 separations significantly longer than the intended distance of 0.75R0. Therefore, FRET from a Cy3 donor located across the ring from the Cy5 acceptor would be less efficient. This explanation is the simplest that is consistent with the lower overall ensemble FRET efficiency of the 8-arm star structure.

Supplementary Note 5: Förster model
For a detailed understanding of the various structures under consideration it is useful to develop a model of the internal exciton dynamics. This is of most value when one wishes to predict the potential performance of different designs, to apprehend why a particular experimental realization of a design falls short of this ideal, and to explore ways of improving performance. We base such an analysis on a set of coupled rate equations that describe the various energy transfer processes that can occur in a well-mixed solution of the photo-active constructs. Because heterogeneous mixtures of constructs are considered, it is convenient to normalize these governing equations by the total concentration and the variables then become equivalent to probabilities. In particular, if the probability of the i th dye on the k th type of construct within the ensemble being excited at time t is Pik(t), then the system will be governed by the coupled ordinary differential equations: where Nk is the number of dyes in the k th construct, probability that an absorbed photon creates an exciton on a dye of type m (see below). We shall assume that there are S ≥1 different types of multi-dye FRET-active constructs in the heterogeneous ensemble that are labeled k = 1, . . . , S, with k = 1 being the target construct. In addition, we allow for the possibility of there being "constructs" that are not FRET-active (i.e., because the dyes are isolated or are all of the same type) and thus can be treated as separate "free" dyes; these are labeled k=S+1,…,S+M, and for them (8a) reduces to where Nk is the number of free dyes of type k -S. For the multi-dye constructs that obey (8a), according to Förster theory the couplings between the dyes are via point dipole-dipole interactions for which where mn R 0 is the Förster distance given in (1), and k j i r is the distance between them.
Since the emission rate from dye i on construct k is given by QikPik/ik where each of which can be solved separately for the remaining Wik. Lastly, to connect with experiment we need to relate the quantities in (11) to the PL areas m of (15) that represent the total emitted energy into the detector per second by dyes of type m. Specifically, we can write where  (k) is the molar concentration of construct k, (1) is the number of photons absorbed per second by a single target structure, L is the path length, and is a geometric factor expressing the fraction of emitted photons that make it to the detector). The first term in (12) is the contribution from the multi-dye FRET-active structures in the ensemble while the second term represents the "free" dye contribution. For an ideal assembly, Within Förster theory, the main geometrical factor affecting the photophysical response and efficiency is the relative positions of the fluorophores via (10). The orientation of the excited state fluorophore dipoles can also play a role through the dependence in (1) of R0 on the dipole orientation factor  2 . Given the flexible attachment of the dyes in our constructs we typically take  2 to have its ensemble-averaged random value of 2/3. However, given the fact that natural light-harvesting structures are known to control dipole orientation and that there is at least the possibility of such control in artificial systems (e.g., via double phosphoramidite linkages), we carried out a few simulations in which  2 took values other than 2/3.
A first point to be made about the effect of dipole orientation is that even if one had perfect control, the impact would be relatively small because of the 1/6 power dependence in (1).
Depending on the relative orientation of the dyes  2 can vary from zero to 4, and as shown in Supplementary Figure 30 at best (i.e., for parallel dipoles) R0 will be only about 35% larger than when randomly oriented.

95
For our DNA-organized FRET networks, Supplementary Figure 30 tends to exaggerate the size of the possible enhancements that could be derived from control of dipole orientation. A primary reason for this is that even if the dipoles were oriented perfectly parallel along a DNA duplex, this would not be true of the dyes in different DNA arms and so all inter-arm FRET processes would be non-optimal. To investigate things quantitatively we used simulation to study the impact of having oriented dipoles on the anywhere-to-end efficiencies of some of our FRET networks. In particular, in Figure 6E we compare simulations of an 8-arm, 4 dye star network ( Figure 1B) in which the dipoles are random ( 2 = 2/3) with a case when the dipoles are all parallel when the structure is flat. The main plot shows results with ideal formation, whereas the inset assumes "actual" formation as inferred by the curve-fittings discussed in this paper. In both cases, the effect of dipole orientation control on overall FRET efficiency is not large.
Moreover, the largest impact is seen when the dye spacing is near 1.0×R0 since this is where the Förster coupling is most sensitive to all parameters, and this represents another reason why dipole orientation is a secondary consideration in FRET network design: for maximum efficiency the main factor is dye spacing, and when this is reduced below 1.0×R0 the consequences of dipole orientation control are even further reduced.
The results of two other calculations regarding dipole orientation control appear in manuscript Figure 6F. In the main plot, the anywhere-to-end FRET efficiency of the fully formed 8-arm, 4-dye star with dye spacing of 1.0R0 is studied as a function of misalignment of the dipoles. In this simulation, the angle of the dipoles with respect to the DNA axes is varied, while the azimuthal angle is random so that as the dipoles incline away from the DNA axis they go out of parallel alignment and the efficiency falls. From this plot we see that one needs to keep the misalignment less than 20 o in order to preserve the efficiency gains. The second calculation in Fig. 6F examined more closely the random dipole case by studying two limits investigated in 12 . The first limit, referred to as dynamic averaging, follows the approach of elsewhere in this paper in which  2 = 2/3 as is appropriate if the dipole re-orientation time is fast compared to the lifetime. The other limit obtains when the dipole re-orientation time is slow compared to the lifetime, so that in a given construct the dipoles will be random but fixed in orientation during the measurement. The averaging that occurs is then over the ensemble and may be referred to as static averaging. The inset of Figure 6F compares these two limits with regards to the FRET 96 efficiency of linear photonic wires of varying length and with dye spacing of 0.75R0. Static averaging is seen to yield significantly lower efficiency; the relevance of this result to our studies (where we mostly assume dynamic averaging) is unknown.

Data Analysis/Simulations
In order to use the foregoing equations to analyze spectral data, we need values for the various parameters contained in these equations. Some of these parameters such as the quantum yields Qi and the R0 values (from (1)) are reasonably well established, while others such as the inter-dye distances k j i r are less so. In the case of a simple dye-pair it is well known that one can invert the procedure and use the FRET itself to deduce the separation distance 13 , but this spectroscopic ruler approach is unworkable for the complex structures considered in this paper.
Most intractable of all is the fact that our situations are usually heterogeneous due to the flexibility of the structures, possible self-quenching or photo-bleaching of dyes, inefficient assembly, etc. Given these realities, our goal has to be more modest, and we look merely for sufficient-but-not-necessary interpretations of the data through which we pursue not absolute agreement with data (which would perforce require uninformative curve fitting), but rather to address semi-quantitative questions such as are the systems describable by Förster theory, are the designs performing more or less as expected or is their evidence of problems, and most importantly what lessons can be learned about light-harvester design.
As already discussed, rough estimates of the nearest-neighbor dye spacing's can come from the DNA designs and the distances between the dye attachment points (see Eq. (15) and Supplementary Tables 62-64). However, for better accuracy one must also account for the small distances between the attachment points and the actual dyes (i.e., the locations of the pointdipoles of the Förster approximation) as determined by the dye linkage chemistries and the dye molecules themselves. And while there are sophisticated ways of gauging these dimensions, given the number and complexity of the situations considered in this paper, we instead employ a much less demanding approach. (The most accurate method at present for determining dye positions is one that combines molecular-dynamics (MD) simulations with single-pair FRET measurements 14 . MD simulations can also be used alone, and though less accurate, can still be quite informative 15 . Finally, for protein-based natural light-harvesters, X-ray diffraction has been invaluable.) In particular, we simply assert "reasonable" values for these linker/dye distances, and look for validation in the results obtained when these distances are held fixed across many other structures with the same dyes and linkage chemistries. For most of the dyes 98 considered, these attachments have significant flexibility and this aspect is represented both by taking the dipole orientation factor  2 to have its random value as already noted and by letting the linker orientation also be random over a defined range. For non-nearest-neighbor dyes, things are even more complicated because of the possibility that the interconnecting DNA scaffold can be flexible so that (15) no longer applies e.g., if there is an intervening Holliday junction. In our simulations we treat such flexible junctions by assuming they can take random angles over a specified range, and then capture the aggregate effect through ensemble averaging over many configurations. The most complex situations of this type are those of the 4-and 8arm stars that have central openings that force the central dye to be asymmetrically located as depicted in Supplementary Figure 31. These openings have diameters of approximately 15Å and 30Å, respectively, which we represent crudely as DNA rings (see Supplementary Figure 32A Our general approach to the simulations is to proceed from the simplest cases involving two dyes and work up to more complex structures, at each stage comparing the ensemble-averaged spectra derived from modeling with experimental data. Simulating the integrated sub-spectra m would be equivalent but we prefer fitting the spectra for it's more direct and intuitive connection with experiment. Three levels of simulation are considered and are ideal simulations, parameter adjustments, and low-yield simulations. For ideal simulations, perfect yield of the target structure is assumed and ideal parameter values including those given in Table 1 and Supplementary Tables 60-64 plus a set of dye/linker distances and a range of junction angles that, as noted above, are kept fixed to reflect the fact that our structures mostly involve the same dyes and linkage chemistries. The only specific fitting done in these simulations is of the multiplicative generation factor  that we adjust from the 99 values given in Supplementary Table 61, almost always by less than 20%, under the presumption   that this accounts for differences between the test and control experiments in sample   concentrations, illumination, etc. For parameter adjustments, if the discrepancies between the ideal simulations and the experimental spectra are relatively small, then we look to account for the differences with plausible adjustments of various parameters such as the dye/linker distances or the R0 values (that could be affected by local dielectric perturbation 7 ). These simulations continue to assume that the target structure is assembling properly, however, small reductions in yield can be considered as a way of representing, for example, the slight fall-off in assembly yield that would result if for example the initial concentrations were not precisely stoichiometric. Given the small contributions of the effects considered, identifying which one(s) is actually responsible is likely impossible within the present work because of its many uncertainties.
For low-yield simulations, should the parameter adjustments be found insufficient to produce agreement with experiment, we conclude that the assumption of perfect yield is flawed. To account for this we allow certain dyes to be inactive or quenched, or to simply be missing from the structure as a result of an incomplete assembly. As a result, the simulated ensemble will now be composed of the actual target structure plus various partial structures and leftover free dyes, all with specified concentrations, and with the aggregate photoemission described by (12).
While clearly justifiable in general, the fact that this approach introduces a large number of new parameters (i.e., the concentrations) means it can fit almost any data and so can easily turn into an exercise in curve-fitting with little physical meaning.
And unfortunately the electrophoresis/chromatography experiments do not provide a very substantial cross-check. For example, in a star construct that lacks a central dye the individual arms largely decoupled, and this implies that if the arms were physically separated the PL spectra would remain essentially unchanged whereas the gels would look completely different. Conversely, self-quenching of dyes in a structure would cause its spectrum to look entirely different while leaving the electrophoretic mobility unaffected. Given these realities, the quality of the individual fits to data are not especially meaningful, and we must instead judge the physical fidelity of our lowyield models from the reasonableness and simplicity of their basic assumptions and from the consistency and plausibility of the overall understanding.  Supplementary Table 60A. For the 4-arm and 8-arm designs our treatment is based on the idealized geometries of Supplementary Figures 32A,B in which a wide range of random arm angles is allowed and with the results found to be relatively insensitive to the exact choice. Lastly, we find that fixing the linker/dye distances to be 2Å for the Cy5 dye and 8Å for the Cy3 dye across all simulations works well and is also consistent with the known chemistry as discussed earlier.
As a first set of ideal simulations obtained using the parameter set just described, in Supplementary Figure 35a we compare simulation and experiment for 1-arm linear structures having all five dye-spacing's. As is evident from the figure, excellent agreement is obtained when the nominal dye spacing's are 1.5R0, 1.25R0, and 1.0R0, however, discrepancies appear when these spacing's are reduced to 0.85R0 and 0.75R0. Given the small size of the discrepancies, it is easy to adjust various simulation parameters and get improved fits, though as noted previously identifying which adjustment(s) is physical is problematic. For instance, one explanation could be that the inter-dye distance has somehow increased, with the needed additions being just 1Å and 5Å, respectively. Another possibility is that the R0 value has decreased by similar amounts. This could occur as a result of non-randomness in the dipole orientation factor  2 of these phosphoramidite-linked dyes, e.g., a 5Å reduction in R0 is produced if increases from 0.67 to 1.2. Finally, it could be that there has been a slight drop in the assembly yield of these shorter strand samples that is not detectable by gel electrophoresis (Supplementary   Tables 39 and 53) and that could result from slightly non-stoichiometric starting materials; under this assumption, the good fits shown in Supplementary Figure 35b are obtained when the respective yields are taken to be 93% and 82%. No matter which explanation is used, the simulations of Supplementary Figure 35a,b suggest that these constructs are well described by Förster theory.
Moving on to the multi-arm stars, we choose all parameters to be as in the 1-arm case plus for the 4-arm and 8-arm stars we employ the idealizations of Supplementary Figure 35a,b. The simulated spectra are plotted in Supplementary Figure 35c-e, and again we find good agreement with experiment for the larger dye-spacing's but growing discrepancies as these spacing's are reduced. That the disagreements between simulation and experiment are largest for the 4-arm and 8-arm stars and that electrophoresis also shows these cases to have issues with formation efficiency points to this as the cause. A fall-off in dye performance (e.g., by self-quenching) is also a possible interpretation, but this seems less likely as the inter-dye distances are relatively large (>35-40Å). Focusing then on assembly yield, the fact that the structures have just two dye types with a single nominal Cy3-Cy5 dye-spacing in any sample simplifies the interpretation in that, to first order, the ensemble can be treated as if it consisted just of fully formed structures and free Cy3 dye (since the individual arms function essentially independently and there is little direct excitation of any free Cy5 dyes). Excellent fits to the spectra can be obtained in this way (not shown except in the 1-arm case in Supplementary Figure 35b), and the required assembly yields of the putative fully formed structures are plotted in Supplementary Figure 36 (solid lines) as a function of the dye spacing as measured in DNA base pairs. Also shown in the figure are the yields as estimated by electrophoresis, with both the full (dash with symbols) and full+partial (solid with symbols) formation percentages plotted. The fact that the simulated yields are generally in much better agreement with the full+partial data can be interpreted to mean that the partial structures seen in electrophoresis contribute substantial FRET and so must contain both Cy3 and Cy5 dyes in fragmented assemblies. That the DNA strands containing the Cy5 dyes are much longer than those with the Cy3 dyes makes this conclusion unsurprising. More generally, the rough consistency with the electrophoresis suggests that the low-yield interpretation and its associated fitting parameter are physically meaningful, and especially for the 8-arm assembly where plausible adjustments of other parameters are manifestly insufficient to account for the differences between the ideal and experimental spectra. This is less clear for the other assemblies (and especially the 1-arm linear structure as discussed above) since their ideal spectra are much closer to the data and the interpretation of them as also arising from formation inefficiency ( Supplementary Figures 35b and 36), though consistent with the 8-arm treatment, could be incorrect.
To the extent that the above modeling provides an accurate representation of the photophysics of the two-dye star structures, we can now use that understanding to estimate and 9 refer to the arm that joins the ring at the point where the Cy5 is located so that it has the shortest Cy3-to-Cy5 distance and highest efficiency. The opposite extreme is path 5 that involves the arm that attaches at the opposite side of the ring and so has the longest Cy3-to-Cy5 distance and the lowest efficiency. Most interesting is the fact that these differences (in both geometries) are not as large as they would be if not for homoFRET. In particular, homoFRET produces an enhancement when the Cy3 is furthest away, and shows that the homoFRET pathway through multiple Cy3 dyes is making a substantial contribution.

Simulation Results, Photonic wire constructs
We next turn to the multi-dye star constructs considering 12 different DNA templates having  Figure 41a) where only minor discrepancies are seen, and for which even better agreement can easily be achieved with small parameter adjustments. An illustration appears in Supplementary Figure 42a where the yield has again been used as the fitting parameter (as in Supplementary Figure 35b) and only small adjustments are needed to obtain excellent fits. As additional dyes are included, the "ideal" emission is no longer as accurate, and especially when the Cy5.5 is included (Supplementary Figure 41c). In these cases the parameter adjustments needed start to become larger than seems consistent with the perfect yield assumption, and this conclusion us supported by the electrophoresis data. We therefore proceed with "low-yield" simulations, and using yield as the fitting parameter, we obtain the results shown in qualitatively track the gel results. The correspondence with the gel results suggests that this is due to a crowding effect in the 4-arm and 8-arm structures that impairs hybridization rather than to a self-quenching that has been reported to occur between Cy5s when these dyes are in close proximity. To first order, to have a functioning four-dye construct requires assembly of the three-dye construct, and the fact that the yield of the latter is poor shows that a weak Cy5 assembly is probably mostly responsible for the four-dye result as well. This is also supported by the fact that the DNA to which the Cy5 is attached is considerably shorter than that of the Cy5.5. Why the spectra-derived (but not gel-derived) yields are lowered also when the structure has one or two arms is less clear. It could be that with these short strands, the gel is not distinguishing a full one-arm structure from one without the Cy5 hybridized. As with the 1.0R0 situation, the foregoing suggests that (with appropriate caveats) the 0.5R0 spectra are also consistent with Förster theory and likely involve a greatly reduced formation efficiency of the Cy5 dye.
Assuming the above has indeed provided a plausible understanding of the photophysics of the multi-dye stars, we can now estimate their efficiency and gain parameters. In doing so, we focus on the complete four-dye structures that have a Cy5.5 at the center to which excitonic energy is being delivered. The results for the anywhere-to-end efficiency E1 as computed from (7a) are plotted in Supplementary Figure 45a with both actual and ideal results shown. The ideal results clearly display the expected behavior of the FRET efficiency rising strongly as the dye spacing is reduced. In addition, the fact that the ideal curves are relatively flat is further evidence of the independence of the arms in terms of energy transfer. That there is some rise in the ideal curves for the 1.0R0 and 0.5R0 cases shows, however, that there is a portion of the energy transfer that occurs on parallel paths (see below). The actual efficiencies are of course greatly reduced in the 1.0R0 and 0.5R0 cases due to yield issues. As we have seen these problems are especially severe in the 0.5R0 case and with multiple arms; in those cases, the efficiency is seen to fall even below that with the 1.0R0 spacing. The ideal end-to-efficiencies E2 are not shown in the plot but are roughly 10% less that E1 for the narrower dye spacing's. Interestingly, for the 1.5R0 constructs, E1 is in the range of 1.3-2.0% (as seen in Supplementary Figure 45a), whereas E2 is 0.1-0.3%, thus indicating that most of their E1 arises from direct excitation of the terminal dye.
To explore more carefully the effect of parallel paths, in Supplementary Figure 45b we plot the ideal E2 for the dye spacing's of 0.5R0 and 1.0R0 and compare simulations in which all FRET processes are permitted with ones in which the FRET is restricted to occurring only on the direct paths connected by DNA. As expected, there is no difference between these quantities when the constructs have one or two arms. However, in the case of 4-and 8-arm designs, a growing component along parallel paths (i.e., the difference between the curves) is seen.

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The antenna gain AG for the complete four-dye star structures, again calculated from (5) with the correction made as in (3b), is plotted in Supplementary Figure 46 with both ideal and actual curves shown. The ideal curve is close to the unity slope that would be expected if all arms operated independently; that the slope is slightly higher again reflects the contribution of parallel paths. Of course the actual AG is much lower, and again this is due to the low yield of these structures.

Simulation Results, Dendrimers
The  Table 1 and Supplementary Tables 60-64 and the same values for the linker/dye distances. The variation among arm angles is again treated as a random variable over a specific range with the results not especially sensitive to the choice of this range.
Starting with the dendrimers having potentially four dyes, we plot the ideal simulations along with the experimental data in Supplementary Figure 47a-c. The correspondence is even worse than before, with even the Cy3-Cy3.5 structures not seeming to show very close agreement.
Nevertheless, attempting to fit the Cy3-Cy3.5 dendrimer data using small parameter adjustments, we do find that small reductions in the yield do allow the experimental spectra to be fit quite nicely as is seen in Supplementary Figure 48a. So again it seems that the Cy3 and Cy3.5 dyes are assembling nearly perfectly in these structures. The other cases clearly require abandonment of the assumption of perfect assembly (just as was the case in other similar designs we have considered). How to do the low-yield simulations is less clear than before because the dendrimers lack the radial geometry of the stars and as a result there seems no obvious way of 110 representing the partial structures with just a few meaningful fitting parameters. Absent a better procedure we use the same approach as earlier of having the partial structures be full apart from missing entire layers of the inner dyes (Cy5 and/or Cy5.5). The results are plotted in Supplementary Figure 47b,c with the meaning of the percentage labels as before. Given the additional fitting parameters, the fits are of course excellent, but any real meaning must be looked for in the yield numbers. A plot of these yields appears in Supplementary Figure 47d, along with the experimental formation efficiencies derived from electrophoresis. That the yield on the full+partial structures is high in both simulation and experiment can again be taken as evidence of the assembly efficiency of the Cy3 and Cy3.5 dyes in all constructs including the partial structures observed in electrophoresis. The three-and four-dye yields are much lower, and without a strong correlation with the branching ratio, again suggest that the main effect is formation inefficiency with self-quenching likely playing no role. Although the nominal spacing in these structures and the third group of multi-dye stars was 0.5R0, the Cy5-Cy5.5 attachment distance in the dendrimer is ~35Å whereas in the stars it was ~20Å. This is another argument against a self-quenching mechanism. As with the multi-dye stars, to first order achieving a fourdye dendrimer requires the ability to assemble a three-dye dendrimer. That the yield of latter is greatly reduced and is similar in both magnitude and trend to the four-dye curve suggests that the Cy5 dye is again the source of the problem.
The ideal simulations for the 5-dye dendrimers appear in Supplementary Figure 49a Figures 41c and 42c); it should be noted that this is not a perfect analog in that its dye spacing's are somewhat different from the other dendrimer structures. The solid blue curve in Supplementary Figure 50a is for the actual structures as modeled above, while the long-dash red curve is for the ideal structure with perfect yield. Obviously the strong decline in the actual efficiency is due to the poor yield seen experimentally. In the ideal case, the efficiency rises with increasing branching ratio by about . Thus both intra-branch and inter-branch parallel paths contribute to the efficiency enhancement and these make the dendrimers inherently more efficient than the star constructs.
Lastly in Supplementary Figure 50c we plot the antenna gain of the actual and ideal 4-dye dendrimers, again including the linear 1-arm structure as the 1:1 branching ratio point. As expected, there is much potential for dramatic (exponential) increases in collection capacity with a dendrimer design, with the 4:1 dendrimer ideally producing a gain of nearly 400. Of course, as implemented by us the realized AG is far worse due to the yield issues, so much so that AG for the 4:1 dendrimer actually falls.

Design of DNA sequences
The DNA in these experiments were designed synthetic sequences that were purchased from Integrated DNA Technologies (Coralville, IA) with the exception of the Cy3.5, and internal labeled Cy5.5-functionalized strands which were purchased from Operon Biotechnologies, Inc.
(Huntsville, Alabama); see Supplementary Tables 1-37 where the final spacing is RDA, N is the number of separating bases, K is either 0 or 3.4 depending on whether the dye is on the same DNA helix or the opposite, and LD and LA are both 1.7, accounting for the 0.7 nm six carbon linker and 1 nm for half the width of the DNA helix.
The 34.3 factor comes from the 360° of rotation divided by 10.5, the number of bases in a full turn. This accounts for the linker to which the dye is attached to the DNA, the width of the DNA molecule itself and the radial position around the DNA helix. In Supplementary Table 67 an array of representative calculations is shown.

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The dyes were placed either on a 3' end (preferred), 5' end or an internal T* label (T* = amino C6-dT, see Figure S1). The Cy3nCy5 system implements an internal T labeling at the edge of the center junction for the Holliday junction and 8-arm star or at the center of the structure for the linear and bifurcated structures. The Cy3 dyes are all end-labeled. For the bifurcated, Holliday junction and 8-arm star the Cy3 is double labeled on the 3' and 5' ends. A variety of spacing's were investigated, including 0.75, 0.87, 1.0, 1.25 and 1.5R0. For the [Cy3Cy3.5Cy5]nCy5.5 systems, whenever possible the constructs were a template motif with the smaller dye labeled oligos assembled onto the long strand. The 0.5R0 system required the use of a staggered or concatenated DNA construction (no template) to afford the correct spacing for oligo assembly. For the Holliday junction and 8-arm star structures, the center templates had to be labeled with internal Cy5 in addition to the internal Cy5.5 label to account for the close spacing. The dendrimer structures relied on internal spacing for all dyes except the Cy3. Double labeling of the dyes was used in most cases to allow the needed dye numbers to be assembled.
The sequences were designed in such a way to maximize the base overlap so that there was a minimum of 9 bases before a nick or junction occurs. To further afford stability each portion of the structure was tested for melting temperature using Oligo Analyzer (https://www.idtdna.com/pages/scitools). A minimum melt temperature of 30°C at a salt concentration corresponding to 2.5X PBS buffer was required. The oligos were also checked for self-complementarity and cross-complementarity with non-desired sequences uses Operon's Oligo Analysis Tool (www.operon.com/tools/oligo-analysis-tool.aspx). Here a maximum of 5 bases of non-specific complementarity was set as the limit.
Given the added complexity, these structures are only measure at three different Forster distances, 0.5, 1.0 and 1.5R0. The 1.0 and 1.5R0 spacing structures are designed similar to the 2-dye system. To account for the additional space needed for the cascading FRET dyes, one side of the double stranded arm is extended. This occurs on the linear and bifurcated structures and on each arm of the Holliday and 8-arm star. The extended arms then act as a template for the smaller dye labeled oligos to be assembled onto. Given the close distances needed in the 0.5R0 114 spacing, the assembly proceeds differently. For the linear structure, 4 strands concatenate together to form the structure. Each of these strands contains one dye. The bifurcated structure also uses concatenated strands but requires the Cy5 to be double labeled and one of the oligos to be double labeled with Cy3 on the 3' end and Cy5.5 on the 5' end. All other oligos contain one dye either end-or internally -labeled as indicated in within individual sequences. The Holliday

Spectral Decomposition
For each construct, the basic data set consisted of its PL spectrum plus direct excitation spectra for molar equivalents of each of its constituent fluorophores (assembled on the DNA) and of the background. The first step in analysis of such data was to decompose the full PL spectrum into its individual fluorophore contributions, a task that was generally straightforward given their spectral separation (Supplementary Figure 53). The regression procedure used was much like that in our previous work 7, 10 , including the use of the Multipeak Fitting Tool in Igor Pro (v. 6.31) 18 , and was carried out starting with the primary donor and removing each successive contribution before moving on to the next dye. The quality of the fits, as judged by plots of the where fm()is the normalized emission spectrum of dye m.

Instrumental fluorescence response versus dye concentration.
In order to verify that the concentrations/volumes of dyes we were using did not suffer from inner filtering effects, we tested our Tecan Fluorometer for a linear response for excitation of each dye over a concentration range that spanned from 2 orders of magnitude above and below the ~1 μM working concentrations to be found within our structures. All responses using our typical range of instrumental settings and sample volumes were within a linear regime.

Experimental Setup for spFRET analysis of 0.75R0 Cy3nCy5 structures
Single-pair FRET (spFRET) experiments 19 were carried out using an Axiovert inverted microscope (Zeiss). Laser excitation was obtained using the 515 nm line of an argon ion laser that was coupled into a single mode optical fiber. The output from the fiber was then tightly focused into the sample solution using a 100X Neofluar objective (1.4 N.A., Zeiss). The DNA structures were placed in individual sample wells of an eight-well sample tray (Thermo-Scientific) at a concentration 30-50 pM in 2.5 X PBS. The freely diffusing single DNA structures were excited as they passed through the laser focus. To reduce photobleaching of the dyes an oxygen scavenging system was used consisting of 4 mg/ml of glucose, 2 mM trolox (Sigma Aldrich), 1 mg/ml glucose oxidase (Sigma Aldrich), and 0.04 mg/ml catalase (Sigma Aldrich) 20 .
To inhibit DNA melting the solutions were kept near 5C using a water-cooled metal jacket 116 placed around the sample tray. The fluorescence from the samples was focused onto a 75 micron pinhole to reject the out-of-focus emission. After the pinhole, the Cy3 and Cy5 fluorescence was separated using a dichroic filter (FF640-Di01, Semrock) and then detected using a single photon counting avalanche photodiode detector module (SPCM-ARQH-14, Excelitas) in each channel.
The fluorescence burst signal from each detector was collected using a counter/timer board (PCI-6602, National Instruments). The laser power was adjusted to give a maximum burst level of about 100 counts. For structures with a single Cy3 donor, the optical power was typically less than 75 μW before the objective. For other structures, the laser power was adjusted as explained below. The signals from each channel were processed into FRET histograms using custom designed software.