## Data Analysis

1. FRET efficiency: FRET efficiency (E) is the ratio of the rate of energy transfer to the acceptor, divided by the total rate of relaxation from the excited donor including: fluorescence, quenching, internal conversion, and intersystem crossing (2). FRET efficiency is strongly dependent on the distance between the donor and the acceptor. A shorter distance allows a faster rate of energy transfer and a higher FRET efficiency. The relationship between E and the donor-to-acceptor distance (R) in a dual-fluorophore system follows Eq. 1.

R0 is the distance between the donor and the acceptor at which FRET efficiency is 0.5 (also known as the Förster distance). R0 is usually considered a constant that is governed by the properties of the FRET pair and experimental conditions.

R06 = 8.785 ■ 1023 ■ ®d ■ k2 ■ n-2 ■ J(v)6 Ä6 (2)

Od is the quantum yield of the donor. k2 is the orientation factor for dipole coupling. When both the donor and the acceptor can rotate freely during the excited state lifetime of the donor, k2 has an average value of two-thirds (17). n is the refractive index of the media. J(v) is the overlap integral of the fluorescence spectrum of the donor and the absorption spectrum of the acceptor.

2. The (ratio)A method in dual-fluorophore systems. In this chapter we use the (ratio)A method to determine FRET efficiency and donor-to-acceptor distance (2,6). First, the (ratio)A method in a dual-labeled system is reviewed. To obtain (ratio)A, the donor is excited and emissions from both the donor and the acceptor are collected. When the donor is excited, emissions in the acceptor region contain three components: (1) the "red tail" emission of the donor that tails to the acceptor region; (2) emission from the directly excited acceptor; and (3) acceptor emission from energy transfer. Therefore, only component (2) and (3) are actually emitted from the acceptor. In a second excitation, the acceptor is directly excited. Components (2) and (3) of the acceptor fluorescence from the first excitation, divided by the acceptor fluorescence in the second excitation is defined as (ratio)A. Therefore, by definition,

(ratio)A = {F(^m, XDa) - aFD(kem, XDex)}/F(Xem, XAa) (3)

is the FRET spectrum when the donor is excited at XDex (the acceptor may also have some absorption at this wavelength). F(Xem, XDex) is the fluorescence spectrum of a donor singly labeled sample (no acceptor). a is a coefficient to fit the donor singly labeled spectrum FD(Xem, XDem) to subtract the "red tail" of the donor from the FRET spectrum FD(Xem, XDem).

F(Xem, XAex) is the spectrum of the directly excited acceptor (exciting acceptor at XAex). Therefore, the value of (ratio) A can be determined from the collected fluorescence spectra. As previously discussed, after subtracting the donor contribution, emission of the FRET spectrum at the acceptor region is from two sources: the directly excited acceptor and the acceptor emission through energy transfer. Fluorescence from the directly excited acceptor is proportional to ceA(kDex)^A, and fluorescence from energy transfer is proportional to EceA(XDex)^A. Therefore,

(ratio)A = [ceA(^Dex)0A + EceD(^Dex)0A]/[ceA(^Aex)0A] =

{[ea(^deX)]/[ea(^aex)]} + E{[ed(^deX)]/[ea(^aex)]} (4)

c is concentration of the sample. eA(XDex) is the extinction coefficient of the acceptor fluorophore at wavelength at which the donor is excited. eD(XDex) is the extinction coefficient of the donor fluorophore at wavelength at which the donor is excited. eA(^Aex) is the extinction coefficient of the acceptor fluorophore at wavelength at which the acceptor is excited. Oa is the quantum yield of the acceptor fluorophore.

On the right side of Eq. 4, all the extinction coefficients can be measured using absorption spectroscopy. Therefore, E can be calculated from Eq. 4.

3. The (ratio)A method in the tri-fluorophore-labeled DNAzyme, the TMR-Cy5 pair: the tri-fluorophore-labeled DNAzyme (see Fig. 1) is used as an example to discuss the calculation of (ratio)A, FRET efficiency (E) and fluorophore-to-fluorophore distance (R) for tri-fluorophore FRET systems. There are three fluorophores in the system: FAM (excitation at 490 nm and emission at 520 nm), TMR (excitation at 560 nm and emission at 580 nm), and Cy5 (excitation at 647 nm and emission at 660 nm). Considering the three fluorophores two by two, three FRET pairs are formed. Each pair is considered separately. The TMR-Cy5 pair is discussed first. When either TMR or Cy5 is excited, FAM is not excited. No energy is transferred from TMR or Cy5 to FAM. Therefore, FAM can be ignored while considering the TMR-Cy5 pair (see Fig. 2A), and the calculation for this pair is identical to that in dual fluorophore systems. When TMR (donor) is excited at 560 nm, the FRET spectrum is shown in Fig. 3A (black solid line). Two peaks corresponding to TMR and Cy5 emission are observed. Eliminate the "red tail" emission of TMR by fitting a TMR singly labeled spectrum (black dashed line) to the FRET spectrum. The resulting difference spectrum is shown as the gray dashed line. Only a Cy5 peak at 660 nm is observed. The ratio of this peak, divided by the peak when Cy5 is directly excited at 647 nm (gray solid line) is the (ratio)A for the TMR-Cy5 pair (ratio) atC). Based on Eq. 4, the FRET efficiency of the TMR-Cy5 pair (Etc) can be calculated from Eq. 5.

(ratio) A = [eCy5(560)/eCy5(647)J + |ETC[eTMR(560)/eCy5(647)]j (5)

The superscripts describe the name of the fluorophores and the numbers in parentheses indicate the corresponding wavelengths.

4. The FAM-TMR pair: when FAM (donor) is excited at 490 nm, the FRET spectrum is shown in Fig. 3B (black solid line). Three emission peaks corresponding to FAM (520 nm), TMR (580 nm, only shown as a shoulder), and Cy5 (660 nm) are observed. Eliminate the "red tail" emission of FAM by fitting a FAM singly labeled spectrum (black dashed line) to the FRET spectrum. The resulting difference spectrum is shown as the gray dashed line. Two peaks from TMR and Cy5 are left. The spectrum when TMR is excited at 560 nm is the gray solid line. Therefore, the ratio of the two peaks at 580 nm (TMR emission) is the (ratio) A for the FAM-TMR pair ((ratio)AFT). According to Eq. 4, the FRET efficiency of the FAM-TMR pair (EFT) can be calculated from Eq. 6.

(ratio) aft = [eTMR(490)/eTMR(560)] + |EKr[eFAM(490)/eTMR(560)]j (6) Besides energy transfer from FAM to TRM, energy can also transfer from FAM to Cy5 and from TMR to Cy5. Therefore, when considering the FAM-TMR pair, Cy5 acts as a quencher to decrease the quantum yield of both FAM (donor) and TMR (acceptor) (see Fig. 2B). However, in the (ratio) A method, quantum yield of acceptors does not appear in any of the equations and does not affect the results. Quantum yield of donors appears only in Eq. 2, and affects only R0 but not FRET efficiencies. Therefore, the presence of Cy5 does not affect FRET efficiency cal-

Fig. 2. Schematics of energy transfer for each fluorescence resonance energy transfer pair in the tri-fluorophore-labeled DNAzyme. (A) When the TMR-Cy5 pair is considered, FAM can be ignored. (B) When the FAM-TMR pair is considered, Cy5 acts as a quencher to quench fluorescence from both FAM and TMR. (C) When the FAM-Cy5 pair is considered, there are multiple sources for energy to transfer to Cy5. TMR acts as a quencher to quench FAM fluorescence.

Fig. 2. Schematics of energy transfer for each fluorescence resonance energy transfer pair in the tri-fluorophore-labeled DNAzyme. (A) When the TMR-Cy5 pair is considered, FAM can be ignored. (B) When the FAM-TMR pair is considered, Cy5 acts as a quencher to quench fluorescence from both FAM and TMR. (C) When the FAM-Cy5 pair is considered, there are multiple sources for energy to transfer to Cy5. TMR acts as a quencher to quench FAM fluorescence.

Fig. 3. Fluorescence spectra decomposition for the tri-fluorophore-labeled DNAzyme to acquire (ratio) A values for each FRET pair. Decomposition for the TMR-Cy5 pair (A), the FAM-TMR pair (B), and the FAM-Cy5 pair (C). The inset in (C) shows the zoomed-in figure of the spectra fitting to subtract TMR emission.

Fig. 3. Fluorescence spectra decomposition for the tri-fluorophore-labeled DNAzyme to acquire (ratio) A values for each FRET pair. Decomposition for the TMR-Cy5 pair (A), the FAM-TMR pair (B), and the FAM-Cy5 pair (C). The inset in (C) shows the zoomed-in figure of the spectra fitting to subtract TMR emission.

### No culations for the FAM-TMR pair.

5. The FAM-Cy5 pair: when compared to the previous two pairs, more factors contribute to the emission at the acceptor (Cy5) region: directly excited Cy5, energy transfer to Cy5 from FAM, from TMR, and from FAM via TMR. To make the calculation even more complicated, there are two "red tail" emissions to be subtracted: from FAM and from TMR. In Fig. 3B, the "red tail" emission of FAM is already subtracted and the difference spectrum (gray dashed line) is moved to Fig. 3C as the black solid line. Again, eliminate the TMR "red tail" emission by fitting the spectrum with a TMR singly labeled spectrum (black dashed line). The resulting difference spectrum is the gray dashed line with only a Cy5 peak left. The ratio of this peak divided by the peak when Cy5 is directly excited (gray solid line) is the (ratio)A for the FAM-Cy5 pair ((ratio)AFC). Therefore, combined with Eq. 5 and Eq. 6, FRET efficiency of the TMR-Cy5 pair (ETC) can be solved from Eq. 7.

(ratio) afc = [eCy5(490)/eCy5(647)J + |EFC[eFAM(490)/ecy5(647)]j +

|ETC[eTMfl(490)/ecy5(647)]j + {EFTETC[£FAM(490)/£C'5(647)]} (7)

On the right side of Eq. 7, the first part corresponds to directly excited Cy5; the second part corresponds to energy transfer to Cy5 from FAM; the third part corresponds to energy transfer to Cy5 from directly excited TMR; and the fourth part corresponds to energy transfer to Cy5 from FAM via TMR.

6. Quenching of donors: As previously discussed (see Subheading 3.3.4.), quantum yields of acceptors do not affect calculations in the (ratio)A method. However, there are two ways for the quantum yield of donors (®d) to affect FRET results. The first is static quenching. A fraction of donor fluorophores may be quenched by forming ground-state complexes with quenchers to make the donor completely non-fluorescent. Incomplete labeling of donors can also be categorized in this class. For those donors that are not quenched, their fluorescence properties remain the same as if no quenchers were present. The fluorescence lifetime of the donor does not change in the presence of such quenchers, although fluorescence intensity drops. Equation 4 is derived by assuming 100% labeling of fluorescent donors. To correct for static quenching or missing of donors, a parameter d+ is introduced. d+ is the fraction of acceptor that has fluorescent donors to pair with (2). Therefore Eq. 4 is rewritten as Eq. 8.

(ratio) A = [c£A(^Bex)®A + Ecd+£D(XDex)^A]/[c£A(XAex)^A] =

{[tA(lDex)/tA(Kx)] + d+e[£d(^dex)/[£a(^aex)]} (8)

The second way in which quantum yield of donors affects FRET results is dynamic quenching. Some quenchers, such as oxygen or heavy metal ions can decrease quantum yield of donors by processes like collisions. The fluorescence lifetime decreases in the presence of this type of quencher. Although dynamic quenching does not affect the calculation of FRET efficiency from Eq. 4, R0 is affected based on Eq. 2. Therefore, corrections on R0 should be made to acquire correct R values from Eq. 1 (see Note 3). Examples of corrections on dynamic quenching are given in the section below (see Subheading 3.3.7.).

7. Calculation of R in the tri-fluorophore system (see Note 4). in Eq. 2 is defined as the quantum yield of the donor in the absence of the acceptor. In the tri-fluorophore-labeled DNAzyme, when the FAM-TMR pair is considered, the quantum yield of the donor (FAM) in the absence of the acceptor (TMR) is less than that of free FAM, owing to energy transfer to Cy5. This energy transfer here is considered as a dynamic quenching. The decrease of quantum yield of FAM because of energy transfer to Cy5 (A®FC) is expressed as Eq. 9:

R0FC is the Förster distance of the FAM-Cy5 pair. RFC is the distance between FAM and Cy5.

According to Eq. 2, R0 of the FAM-TMR (R0FT) pair is changed to R0FT, and

(RoFT')6 = (R0FT)6 ■ {1 - [(R0FC)6/[(R0FC)6 + (RFC)6]]} (10)

According to Eq. 1,

RFT is the distance between FAM and TMR.

Similarly, when the FAM-Cy5 pair is considered, TMR acts as a quencher for FAM. The decrease of quantum yield because of energy transfer from FAM to TMR is

According to Eq. 2, R0 for the FAM-Cy5 pair (R0FC) is changed to R0FC', and

(R0FC')6 = (R0FC)6 ■ {1 - [(R0FT)6/[(R0FT)6 + (RFT)6]]} (13)

According to Eq. 1,

From Eqs. 10, 11, 13, and 14, RFT and RFC are solved as

8. Systems containing more than three different fluorophores: Mathematically, there is no significant difference in results when applying the tri-fluorophore (ratio)A method to systems with even higher numbers of fluorophores. For example, in a system with four different fluorophores (see Fig. 4), there are six FRET pairs (see Note 5 for the choice of fluorophores). The same spectra decomposition method can be used to acquire (ratio)A and FRET efficiency for each pair. The distance between each pair can be calculated in a similar manner with the results listed next.

R12 = (R0X2 ■ [(1 - E\2 - EVi - E^/E^r R13 = (R0X3 ■ [(1 - E\2 - EVi - E14)/E13]1/6 R14 = (R0)14 ■ [(1 - EX2 - EVi - E14)/E14]1/6

Fig. 4. Schematics of a tetrafluorophore system. Fluorophores 1, 2, 3, 4 have increasing absorption wavelength maximum. Fluorophore 1 transfers energy to fluorophore 2, 3 and 4. Fluorophore 2 transfers energy to fluorophore 3 and 4. Fluorophore 3 transfers energy to fluorophore 4. Considering the four fluorophores two-by-two, there are six FRET pairs.

R34 = (Ro)34 ■ [(1 - EJ/^F R12 is the distance between fluorophore 1 and fluorophore 2 in Fig. 4. (R0)12 is the Förster distance of fluorophore 1 and fluorophore 2. The same notations are used for other pairs.

In practice, with multi-step energy transfer, errors accumulate in data acquisition and processing. For the first donor (fluorophore 1 in Fig. 4) and the last acceptor (fluorophore 4 in Fig. 4), the spectra overlap would be very small, giving a very small (Ro)14 value based on Eq. 2. If this pair is positioned at a distance exceeding 2(R0)14, the distance information on this pair is lost. Despite this, FRET efficiency and distance information can still be obtained from other pairs with efficient energy transfer (see Note 5).

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