FRET or No FRET: A Quantitative Comparison

Pinhole

Fully opened for wide field imaging.

Amplifier gain and offset

Initial investigations with unlabeled streptavidin surfaces showed that in all our experiments, the background level was only dependent on the amplifier settings, but not on the laser power and detection gain (data not shown). This supports that the molecular backbone of our model system does not contribute to the total signal by autofluorescence. Therefore, there was no need to apply any compensation of a background signal by electronic background correction. The amplifier offset was set to 0. On the other hand, we found that the amplifier gain also increased noise. To avoid any complication in reconstructing ratiometric data from different image acquisition channels, we consistently set the gain to 1 (no amplification).

Filter set

Described in Table 3.

Laser power and detector gain

A precalibration of the microscope revealed that detector gains are linear within a certain working range, and therefore each channel can be tuned separately for maximum signal. For each set of experiments (variation of R_DA or R_SA), we used the donor-only sample (d) with maximum R_DA and R_SA to set the gain in the donor channel (D), and determined the minimum laser power necessary to acquire a strong signal (Dd) at maximum detector gain. The same process was repeated for the acceptor channel (A) using an acceptor-only sample (a) with maximum R_SA but minimum R_DA. We set the parameters of the FRET channel (F) by keeping the same laser power as for channel D and by adjusting the detector gain so that the signal measured from the R_DA = 1, R_SA = 1 sample containing both fluorophores (b) yielded a value around the middle of the dynamic range.

Once set, these parameters were used throughout the entire experiment.

Background subtraction

To eliminate residual background signals that originated from uncompensated dark current of the photo-multiplier tubes, but not from sample autofluorescence (see above), we imaged PLL-g-PEG-biotin surfaces coated with unlabeled streptavidin in all channel combination and subsequently subtracted the mean value from all fluorescence signals.

A system in two dimensions with multiple donors and multiple acceptors cannot be described by the single distance model

The single distance model describes the relationship between the distance r between one donor and one acceptor fluorophore and the transfer efficiency E (Lakowicz, 1999):

(1)

R₀ is the Förster distance, characteristic for the spectral overlap of the donor-acceptor pair (Lakowicz, 1999). In this model, it is assumed that one donor interacts with one acceptor. The model is applicable, for example, in the case where a donor and an acceptor dye molecule are coupled to two different domains of a molecule and variations in FRET efficiencies represent conformational changes (Suzuki et al., 1998; Mochizuki et al., 2001). However, for our situation where several donors and acceptors can interact, the single distance model cannot predict FRET efficiencies. Extensions of the model have been published by Wolber and Hudson (1979) and Dewey and Hammes (1980) for one donor with multiple acceptors. Yet, these more general models still do not describe the situation of multiple donors and multiple acceptors encountered with surface FRET. Here, an appropriate model should account for the following items:

1. Random distribution of donor and acceptor positions.

2. Random excitation of donors at random time points.

3. Already excited donors cannot be excited a second time. The energy is lost in the system.

4. Competition between excited donors to transfer energy to a nearby acceptor.

5. Saturation of the system when all acceptors around a donor are already excited and are not able to participate in the transfer process.

It seems difficult to find an analytical solution under all these conditions. However, the system can be elegantly simulated by an MC approach. The algorithm implements the events of fluorescence at the level of single fluorophores: A photon flux reaches the labeled surface. Whereas most of them are lost, those reaching a donor (and potentially participating in the process of its excitation) become “excitons”. In the MCS, each excited donor can then either transfer its energy to an acceptor or emit fluorescence, according to the rules listed above. The simulated efficiency is simply calculated as the number ratio between transfer incidences and the number of used excitons.

MCS stability is flux dependent

The exciton flux is an important parameter for the stability of the MC predictions. Two issues define the stability of our MC FRET simulations:

• How many excitons are necessary to guarantee robust statistics providing of FRET in a multi-donor, multi-acceptor system?

• What is the maximum flux of exciton such that sufficient donors and acceptors are still excitable at any time point to participate in the competition between donor fluorescence emission and FRET (see Appendix)?

We have performed systematic tests (data not shown) to determine the two parameters defining the exciton flux: N_ex, the number of excitons, and T_int, the integration time over which these excitons are randomly released over the simulated sample. It turns out that N_ex = 10⁴ excitons guarantee robust statistics, and that for a flux of J = 10 excitons/ns, the donor-acceptor system remains sufficiently unsaturated to ensure a largely undistorted stochastic decision between donor fluorescence emission and FRET. Interestingly, the maximum exciton flux guaranteed experimentally (laser power = 25 mW, at 488 nm, with a 100×/1.4 objective, surface = 10⁴ nm², extinction coefficient = 78,000 M⁻¹·cm⁻¹, fluorophore concentration = 4.01 pmol/cm², R_DA = 1, R_SA = 1) is in the range of 15 excitons/ns, in good agreements with the MCS flux. This flux is dependent on the cross section area of the donor. Implicitly, the more donors, the greater the probability for a photon to become an exciton. Therefore, the exciton flux is proportional to the number of donors, i.e., proportional to the fraction of labeled molecules on the surface R_SA, multiplied with the fraction of labeled molecules being donors R_DA/(R_DA + 1), hence R_SA·R_DA/(R_DA + 1). To be consistent with the experimental setup, the MCS adapts the simulated exciton flux J_Sim to R_DA or R_SA as J_Sim = 2·J·(R_SA·R_DA/(R_DA + 1)).

The efficiency increases when the ratio donor/acceptor (R_DA) decreases

Fig. 2 A shows the dependence of MC simulated FRET efficiencies on R_DA. With low R_DA, the surface is almost entirely composed of acceptors. In this configuration, an excited donor has a higher probability to transfer its energy to a neighboring acceptor than to emit energy as fluorescence. The second effect of a high number of acceptors is that the probability that two donors compete for the same acceptor is almost zero. In combination, the two effects yield a high efficiency.

In contrast, at high R_DA, a donor has mostly donors as neighbors, and they have to compete for a very low number of acceptors. The probability that a nearby acceptor is already being excited is then high, precluding the transfer of additional energy. The excited donor will emit fluorescence, leading to a decrease of efficiency.

To perform an experiment investigating the effect of changes in the mean distance between donors and acceptors, a good choice for R_DA is in the range 1–20. In this range, the efficiency goes almost linear with the concentration ratio and the steep slope predicts high sensitivity in determining donor-acceptor distances from FRET measurements (see Fig. 2 A). For R_DA > 20, the efficiency goes to zero, and for R_DA < 1, the efficiency reaches a plateau where changes in R_DA have little effect on the efficiency. Both ranges preclude a quantification of molecular distances. Note that the R_DA range of the plateau depends on the Förster distance R₀ (discussed in more detail below). Therefore, in experiments that aim at the detection of small efficiency variations, it might be useful to carefully select the dye pair so that the working range of R_DA is in the linear domain.

The efficiency increases when the fraction of labeled molecules (R_SA) increases

Fig. 2 B indicates that a decrease of the fluorophore concentration reduces efficiency. In these simulations, R_DA is set to 1, and the concentration of both kinds of fluorophores is varied to modulate the mean distance between donor and acceptor. Since the probability of transfer is directly related to r⁶, we expect a strong dependence of the efficiency on R_SA, as is confirmed by the MCS.

The efficiency increases when the Förster distance R₀ increases

Six simulations have been run with different Förster distances R₀ (2, 4, 5.55, 6.31, 8, and 10 nm). Both graphs, Fig. 2, A and B, show that also in a multi-donor, multi-acceptor system, FRET efficiency is highly dependent on R₀.

In Fig. 2 A, efficiency values calculated with the single-distance model (dashed lines) and those simulated at low R_DA (R_DA < 0.1) (solid lines) yield comparable results for all R₀. In this configuration, there is no competition between donors for the same acceptor, leading to a situation where the main parameter influencing the probabilities of transfer is the Förster distance. Interestingly, our multi-donor, multi-acceptor simulation even predicts systematically higher efficiencies than the single-distance model. This underlines the fact that with several acceptors per donor, the cumulative probability for having transfer versus fluorescence is higher than the probability for a single transfer (cf. Appendix).

When R_DA increases (Fig. 2 A), the competition between donors for the same acceptor increases and the efficiency drops to zero. The same happens with a decrease of R_SA (Fig. 2 B). Here, the reduction in efficiency is related to the increase in distance between the fluorophores.

FRET results depend on the method: a comparison of FRET efficiencies

We compare the methods E₁, E₄, E₆, E₇, and E₈ in Table 1. They include a ratio method using only one filter set and no correction for acceptor cross talk into the donor channel (E₁), three methods, which measure energy transfer directly with Fb and account for cross talk corrections with different schemes (E₄, E₆, E₇), and one method involving acceptor bleaching (E₈, discussed in a section below). Fig. 3 displays FRET efficiencies calculated with the five methods for changes in R_DA (A) and changes in R_SA (B). The data comprises two experiments using the dye pair Alexa 488-Alexa 546 (R₀ = 6.31 nm). All calculations rely on identical sets of input images, and the differences between the methods only relate to the differences in postprocessing. The methods can be examined in terms of 1), reproducibility between different experiments under identical conditions; 2), their ability to reflect changes in the parameters R_DA and R_SA consistently, and 3), their agreement with the MCS reference data (overlaid as black lines).

Fig. 3 shows that the stability of the curves is highly dependent on R_DA. In the range 0.1 < R_DA < 10, the results are stable and reproducible between experiments for methods E₄, E₆, and E₇. All three exhibit the expected decrease in efficiency with an increase of R_DA (Fig. 3 A) or a decrease of R_SA (Fig. 3 B, subject to R_DA = 1) in a consistent manner. Although the performances are nearly the same, it occurs that E₆ systematically provides results closest to the MCS reference curve. E₆ is the method in Table 1 with the most rigorous cross talk correction. We infer that, indeed, these cross talk terms remove essentially all artifacts from the calculated efficiency whereas E₄ and E₇ are still left with some biases. However, as shown later in this paper, the noise-induced uncertainty amounts to ±12% of FRET efficiency in this range of R_DA (“Uncertainty analysis” section). Therefore, the difference between E₄, E₆, and E₇ are statistically not significant, and our interpretation relies on the systematic shift of only two experiments per R_DA and R_SA settings.

In contrast, the efficiency E₁, calculated from the signal ratio of the donor in presence and in absence of acceptor, does not provide repeatable results. In some cases, it even delivers negative efficiencies. Negative efficiency values indicate that the fluorescence of the donor in the presence of the acceptor is enhanced instead of quenched. In our particular case of an experiment with equal donor and acceptor concentrations (R_DA = 1), three out of five images showed higher intensity in Db than in Dd. This demonstrates the weakness of indirect measurements of FRET. The method is only stable with absolutely repeatable detection of the donor signal before and after adding acceptor and thus, notably, between two different samples. Small changes in the fluorescence, whether noise- or sample-induced, can dramatically alter the efficiency and yield nonsensical negative values. This behavior is confirmed by the graph in Fig. 3 B when the concentration of fluorophore decreases. Similar concerns apply to method E₈, although the weakness of this method will mainly be observable with the results in Figs. 4 and 8. Because of the method-inherent weakness of E₁, it is discarded from the rest of the experimental performance analysis.

Outside the range 0.1 < R_DA < 10, the results obtained are unstable, independent of the method. Here, direct observation of Fb with appropriate compensation of cross talk alone does not guarantee accurate efficiency values. For example, for low R_DA, method E₆ predicts an increase of the efficiency, whereas the other methods suggest a decrease, notably based on identical raw data from the nine image channels. This cannot be explained by the differences in cross talk correction schemes. As will be shown below with the uncertainty analysis, image noise and any irreproducibility of fluorescence between experiments get amplified in an unfavorable manner outside 0.1 < R_DA < 10.

The spectral overlap influences FRET sensitivity

Our surface FRET system offers the possibility to exchange the dye pairs (see Material and Methods) and thus to alter the Förster distance. Results from the same set of experiments as discussed before, but for the dye pair Alexa 488-Alexa 633 (R₀ = 5.55 nm), are presented in Fig. 4. This new pair tests a donor-acceptor system with on the one hand less spectral overlap and on the other hand higher spectral separation such that cross talk between channels is reduced. A low spectral overlap implies lower probabilities for FRET, and thus a decrease of signal-to-noise ratio (SNR). It also implies that the cross talk ratios are calculated between channels where the cross talk is close to zero. The correction factors become very sensitive to image noise, as illustrated in Fig. 4 A by the substantially weaker reproducibility of the experiments as compared to Fig. 3 A. Only data in the range 0.1 < R_DA < 10 is presented (see above). As in Fig. 3 B, the two methods E₄ and E₆ appear to generate more consistent and stable FRET values than E₇ (Fig. 4 B).

Our comparisons of FRET pairs with different R₀ lead to the following findings: The instabilities induced by the choice of a well-separated dye pair prevail over the advantages of low cross talk corrections. Actually, Fig. 3 suggests that cross talk can be well corrected, even for a dye pair with a large Förster distance.

Despite the lower reproducibility of the experiments with shorter Förster distance pairs, the data in Fig. 4 B, as compared to Fig. 3 B, are in better agreement with the MCS reference. The effect is less obvious with the comparison between Figs. 4 A and 3 A, although the data in Fig. 3 A exhibit also a trend for systematically lower experimental efficiency in the range R_DA = 0.1–1 relative to the MCS predictions. This suggests that the model and experiments suffer a disagreement, which is more severe for long Förster distances. In our model, the Förster distance is a function of the spectral overlap and the geometric factor, χ², which takes into account the orientation of the donor dipole relative to the acceptor dipole (Lakowicz, 1999). The spectral overlap is characteristic for the spectral properties of the dye pair and is therefore a determined parameter. χ², however is a free parameter that is dependent on the system. Dale et al. (1979) calculated the average χ² to be ⅔ in the case where the dyes are freely rotating. We used this value in our initial MCS shown in Figs. 2, 3, and 4. However, the existence of a mismatch between MCS and experiment motivated us to modify our MCS and to introduce a random χ² for every donor-acceptor pair (see appendix, Eq. A2). The relative orientation of two dyes is calculated using three random angles, and the value of χ² can range from 0 to 4. This leads to different R₀, and thus variable FRET probabilities for every donor-acceptor pair. Fig. 5 shows the results of the modified MCS (dashed line) in comparison to the uncorrected MCS (solid line). The calculations have been made for the same dye pair as in Fig. 3. Lower efficiencies are obtained from an MCS with random χ² as compared to a fixed χ² = ⅔, due to the fact that the distribution of random χ² is skewed toward 0 (Fig. 5 B, inset), accompanied by a decrease of R₀.

Also in Fig. 5, we replot the experimental data, as calculated with method E₆. In comparison to Fig. 3, the randomization of χ² renders experiments and simulation in excellent agreement. This finding clearly reflects the stochastic nature of FRET and underlines the difficulties in representing the determinant statistical distributions by average characteristic parameters, as encountered in analytical predictions. An MCS approach has a fundamentally superior performance in predicting data under such conditions.

FRET indices as qualitative measures of surface FRET

Fig. 6 shows the results obtained for four FRET indices. They have been calculated according to Table 2 for the dye pair Alexa 488-Alexa 546 (A) and for the dye pair Alexa 488-Alexa 633 (B). R_DA was varied from 0.01 to 100 for the first dye pair and from 0.05 to 20 for the second dye pair to have more data in the center of the curve. The inset in Fig. 6 A displays the results of the first dye pair for this range and allows immediate visual comparison with the graph in Fig. 6 B. In contrast to efficiency, different indices cannot be compared on an absolute scale. Therefore we have arbitrarily normalized all index values such that the index value equals 1 for R_DA = 1. Two behaviors can be distinguished in the results in Fig. 6 A: FRET₁ and FRET₃ are close to the simulated curve for R_DA > 1 and monotonically increase when R_DA decreases in good qualitative agreement with the MCS. Interestingly, whereas both MCS and FRET efficiency values exhibit a plateau, the indices seem to amplify its sensitivity in the range 0.01–1. FRET₆ and FRET₇ perform in a similar manner for R_DA > 1, but exhibit a turning point at R_DA = 1, which makes them essentially useless, at least for the range R_DA < 1.

Results obtained with a dye pair with a shorter Förster distance (Fig. 6 B) confirm these findings, but like with the efficiencies, shorter R₀ tend to introduce more instability.

Photobleaching of the acceptor is a method to vary the concentration of acceptors locally

We have tested our system with acceptor photobleaching for the dye pair Alexa 488-Alexa 546. Sixteen regions of interest (ROIs) were defined and photobleaching was performed in these ROIs with 1, 5, 10, 15, 20, 25, 50, 75, 100, 200, 300, 500, 750, 1000, 1500, and 2000 cycles, as shown in Fig. 7. The laser (1 mW He-Ne, 543 nm, maximum power) bleached the acceptor only, as verified with the control experiment illustrated in the inset of Fig. 7 A. The donor signal (blue) is retained, whereas the acceptor signal (green) decays in the expected way. Fig. 7 A represents the efficiency map calculated with method E₆. The results are color-coded and clearly display a decrease of FRET with photobleaching of the acceptor (increasing number of cycles from the upper left to the lower right.). The mean value of the efficiency calculated in the ROI is represented in Fig. 7 B as a blue solid line. After 2000 cycles, E₆ ≈ 0, suggesting that this is sufficient to completely bleach the acceptor.

The assumption behind this experimental plan was that bleaching would provide an alternative to altering R_DA and R_SA for a modulation of the acceptor distance. To test this assumption, we have combined the results of Fig. 3 A (E(R_DA)) and Fig. 7 B (dashed line fitted to E(Bleaching cycle)) for E₆ to obtain an estimation for R_DA as a function of the bleaching cycles (inset of Fig. 7 B). This curve shows that 1000 cycles introduce a reduction of the acceptor concentration of a factor 10. For each of these concentrations, we have derived a mean distance for energy transfer between the donors and the acceptors. The mean distance for energy transfer is calculated by attributing every donor-acceptor distance with a weight that is proportional to the probability that a FRET event occurs , ∀ j acceptors in the influence area of donor i. Notice that such a distance definition is necessary in a multiple-donor, multiple-acceptor system. Combining these results with those in the inset, we obtained the relationship between the number of bleaching cycles and the mean distance between fluorophores illustrated by the black solid line in Fig. 7 B. The curve shows that the relationship is not linear but the mean distance between donor and acceptor increases exponentially. This is coherent with the fact that for a low number of bleaching cycles, few acceptors are bleached and every donor still has sufficient acceptor for energy transfer. After a certain number of cycles (∼200), the distance suddenly increases dramatically. The point is reached where the number of acceptors in the influence zone of the donor is so low that also longer donor-acceptor distances obtain significant weights. In agreement with our intuition, the curve goes to infinity when the number of bleaching cycles is high enough to destroy all acceptors. This data show that, in principle, it is possible to measure molecular distances also in a multiple-donor, multiple-acceptor system, but that the interpretation of the results is more demanding and much less obvious than with one pair where the single-distance model is applicable. For our system, a theoretical mean distance of 7.2 nm between the center of mass of the streptavidin molecules was predicted from its surface concentration. This predicted value is in good agreement with the mean distance for energy transfer of 6.8 nm shown in Fig. 7 B.

Incomplete photobleaching induces errors in the calculated efficiency

Acceptor photobleaching is also a frequently used approach to measure FRET, as discussed in Table 1. The corresponding efficiency is given by E₈ = 1 − Db/Db(ab), relying on the ratio of donor signal before and after complete bleaching of the acceptor. In our case of a homogeneously labeled surface, we chose a slightly different observation strategy. Only a part of the surface was bleached. Thus, the same image showed a region where both fluorophores were still present, providing a measure for Db, and an acceptor-bleached region, providing a measure for Db(ab). This protocol bears the advantage of circumventing problems of sequential observation, e.g., arising from global intensity changes due to focus drift between the acquisition of Db and Db(ab). In a less-controlled sample with inhomogeneous labeling, similar stability can be attained with sequential observation when a control region is coimaged, delivering two donor intensities, Db^c and Db^c(ab) that are unaffected by the acceptor bleaching. The modified method is insensitive to global variation of the intensity and may have the same characteristics as E₈ applied to our idealized model sample.

We have investigated the performance of this method in reporting FRET efficiency as a function of R_DA, R_SA, and R₀. Results are shown in Figs. 3 and 4. The reference value Db(ab) was taken after 2000 bleaching cycles, according to our findings in Fig. 7 B.

The results essentially agree with those obtained with the other methods, although in general the values seem to be lower. When R₀ decreases (Fig. 4), they exhibit large fluctuations, implying increased sensitivity to noise.

The method bears the advantage of using a single sample and a single filter set but strictly relies on complete photobleaching of the acceptor. In practice, such an approach is often problematic: First, to guarantee proper bleaching, one has to tune the laser power, bleaching wavelength, and bleaching time. Second, bleaching can have cross talk and thus affect the donor signal as well. Third, in live cell imaging, bleaching is known to cause phototoxicity and thus to severely affect the sample viability. It is therefore important to choose an acceptor that can be readily bleached. Our choice of Alexa 546 would obviously be not optimal for life experiments, since Alexa dyes are known to be very stable (as confirmed by the large number of cycles necessary for complete bleaching).

More critical for our performance analysis, however, are errors induced by incomplete bleaching. The method E₈ strictly relies on the assumption that the acceptor is entirely bleached. In the practice of, e.g., a live cell experiment, this can frequently not be guaranteed, as acceptor molecules are subjected to diffusion and other protein dynamic processes, and the assessment of the number of cycles necessary for complete bleaching is not straightforward. Fig. 8 shows the relative error estimated under incomplete acceptor photobleaching in comparison to method E₆. The results are presented as a function of the fraction of acceptor bleached (solid line) and as a function of the number of bleaching cycles (dashed line). If only 70% of the acceptors are bleached, the error in FRET efficiency is 50%. Even worse, the gradient in the error curve increases between 70% and 100% bleaching, which means that there is no tolerance at all for incomplete bleaching. Fig. 8 shows that despite a 100% photobleaching, the method E₈ still provides a 10% error. This error is mainly due to difference in the observation strategy and uncorrected cross talk.

Observation strategies relying on a physical exchange of samples are inferior to those recovering FRET from ratio and difference analysis of samples coimaged in different channels

Depending on the observation strategy, the methods described in Table 1 can be classified into two groups:

1. Methods E₁ and E₈ calculate the efficiency from the change of donor fluorescence in images taken from different samples (Db with and Dd or Db(ab) without acceptor). Fig. 3 shows that the efficiencies calculated with these methods can be negative. This problem is inherent to the chosen observation strategy. Any change in the absolute fluorescence intensity between the acquisition of the two images directly affects the FRET efficiency. Such changes are very likely. With method E₁, it is almost impossible to ensure twice the same donor distribution in an experiment, one in absence and one in presence of acceptor. With method E₈, mainly the bleed-through of the laser line used for acceptor bleaching and mobility of the donor bear the risk of altering the donor fluorescence in an uncontrolled manner. For the same reason, both methods are weak in analyzing FRET in dynamic systems.

2. In contrast, all other methods derive the efficiency from the signal obtained in the FRET channel in the presence of both fluorophores (Fb). They only differ in the applied cross talk correction factors, but none of them involves an exchange of the sample. To illustrate this, we examine, for example, method E₄. It relies on the measure of Fb, from which the cross talk in Db and Ab is subtracted. All three measures are taken from the same sample (b) coimaged in three different channels. The cross talk factors (FDd and AFd for sample d and FAa for sample a) are again calculated from signals comeasured on the same sample (either d or a). Importantly, the equations do not contain any ratio or subtraction that combines the signals of two different samples. Therefore, the only uncertainty of these ratio or subtraction terms arises from dynamic changes of one sample between the observation in two different channels. For many applications and microscope setups, including the one employed for this paper, sample variation during the switch of channels are negligible.

To illustrate the sensitivity of method E₁ to changes in the absolute level of fluorescence, we present in Table 4 A an example of intensities obtained for an experiment with R_DA = 1 and R_SA = 1 yielding negative E₁. In this particular case, Db is larger than Dd, most probably because of an uncontrolled increase of donor concentration between the sample d and b. In practice, it is often difficult to guarantee the same range of absolute donor fluorescence for two different samples. In our case, irreproducibilities can occur with different levels of donor protein adsorption and focus shifts. In live cell experiments, the problem gets even more prominent. Different cells will hardly ever express the same amount of protein, and changes in the experimental conditions, e.g., in temperature or pH, can have dramatic effects on the signal. We have measured the sample irreproducibility by taking five independent images per experiment. For the sample b, we found a standard deviation of 16%–38% of the channel mean intensity, indicating that even in our highly controlled surface FRET system, the fluorescence signal is subject to significant variation. These experimental difficulties affect methods E₂–E₇ much less for the reasons illustrated in the next two sections.

TABLE 4.

SNR analysis of efficiency calculation methods

		Aa	Da	Fa	Ad	Dd	Fd	Ab	Db	Fb
A. Data noise and irreproducibility for RDA = 1 and RSA = 1
Mean intensity		0.62	0.008	0.063	0.003	0.33	0.059	0.57	0.42	0.42
Std (mean 5 images)		–	–	–	–	–	–	0.09	0.15	0.14
% Irreproducibility		–	–	–	–	–	–	16	38	33
B. Relative influence factors for RDA = 1 and RSA = 1
E1		0	0	0	0	2.1	0	0	−2.1	0
E4		−0.24	0	0.24	−0.003	0.3	−0.3	−0.24	−1.5	1.7
E6		0.069	0.038	−0.11	0.003	0.023	−0.23	−0.14	−1.4	1.52
E7		0.17	0	−0.17	0	0.15	−0.15	−0.17	−0.5	0.71
C. Error propagation of method E6 for RSA = 1
RDA = 0.01	SNR	16.1	4.1	11.4	27.3	4.6	2.3	8.6	6.7	6.8
	|dE6(Ii)|	0.096	0.15	0.19	0.002	0.016	0.048	0.19	0.12	0.36
	dE6(Ii)2/	0.04	0.09	0.14	<0.01	<0.01	0.01	0.15	0.06	0.52
RDA=1	SNR	18.3	1.9	4.7	6.9	8.9	4.6	9.5	5	6.1
	|dE6(Ii)|	0.009	0.03	0.031	0.001	0.010	0.023	0.021	0.065	0.084
	dE6(Ii)2/	0.01	0.06	0.07	<0.01	0.01	0.04	0.03	0.3	0.50
RDA = 100	SNR	3.2	30.6	1.5	1.2	16.2	10.1	9.7	3.7	4
	|dE6(Ii)|	0.001	0.004	0.088	0.001	0.097	0.16	0.001	0.41	0.38
	dE6(Ii)2/)	<0.01	<0.01	0.02	<0.01	0.03	0.07	<0.01	0.47	0.41

Open in a new tab

Although tending toward the same instability in the presence of uncontrolled changes between Db and Db(ab), E₈ yields better results than E₁. This owes to the fact that in our model case, Db and Db(ab) are co-observed in two regions of the same, homogenously labeled surface. To a certain extend, this stability can also be rescued into more practical FRET imaging with E₈, when compensating global intensity changes with the ratio Db^c(ab)/Db^c of a control region. Nevertheless, the high chances for uncontrolled changes also in these measures and the clearly inferior performance of E₈ as compared to E₄, E₆, and E₇ (Figs. 3 and 4) even in our most idealistic case still support the use of a direct observation of FRET in presence of both donor and acceptor.

Influence factors indicate the effect of uncontrolled signal changes

To analyze the effect of uncontrolled signal changes, we have calculated for each of the methods the influence factor of every channel. The influence factor γ_i of a channel i denotes the change in FRET efficiency induced by a change in the intensity of this channel. In addition, we introduce the relative influence factor, ρ_i, as a measure of the relative change in efficiency induced by a relative change in the intensity of channel i. γ_i and ρ_i are calculated according to:

(2a)

(2b)

where is the nominal efficiency for a certain donor and acceptor configuration and I_i denotes the intensity of the ith channel, i = 1..9. In our case, is estimated by MCS. The relative influence factors ρ_i are listed in Table 4 B for methods E₁, E₄, E₆, and E₇ considering an experiment with R_DA = 1 and R_SA = 1.

For all methods except E₇, relative influence factors greater than 1 are obtained for at least one channel. This means that uncontrolled relative changes in the signal propagate adversely, amplifying the relative error of the FRET efficiency, as well. However, there is only a small difference in the magnitude of the relative influence factors between the method E₁, which we found unstable in presence of signal irreproducibility, and the clearly more stable methods E₄ and E₆. The maximum |ρ_i| of method E₁ is 2.1 in both Dd and Db, whereas E₄ and E₆ both have a maximum |ρ_i| in Fb of 1.7 and 1.5, respectively. Obviously, the instability in E₁ must be associated with the fact that irreproducibilities in Dd and Db propagate independent and uncompensated, whereas the channel contributions of E₄ and E₆ grant a compensation of irreproducibilities in Fb by other terms.

Methods with ratio and subtraction terms combining the signals of the same sample have compensating relative influence factors and thus are robust against image irreproducibility

It turns out that the fundamental difference between E₁ and the more robust methods E₄, E₆, and E₇ consists in the absence versus existence of cross-compensating influence factors. For example, an increase by x% of Db due to an uncontrolled increase in donor concentration of sample b relative to sample d yields a decrease of −2.1·x% in E₁. In contrast, the same increase yields a decrease of −1.5·x% in E₄, but at the same time, the signals Ab and Fb will increase, nearly canceling the effect of one another. Thus, uncontrolled variation of the sample has little effect on E₄ as long as the channels Ab, Db, and Fb are imaged under identical conditions. A similar cross-compensation of relative influence factors is found for the two other samples a and d in all methods listed in Table 4 B except E₁. Cross-compensation is indicated by bold numbers grouping the factors of the three channels A, D, and F for each sample a, d, and b. In each of the groups, e.g. {Aa, Da, Fa} the sum of the factors is almost zero, explaining the robustness of E₄, E₆, and E₇ against uncontrolled changes between the samples. The same is true for the groups {Ad, Dd, Fd} and {Ab, Db, Fb}.

The influence of image noise precludes robust analysis in extreme R_DA and R_SA

Fig. 9, A–D, illustrate the relative influence factors in the range 0.01 < R_DA < 100, R_SA = 1 for methods E₁, E₄, E₆, and E₇. Three domains can be observed in all panels:

Domain 1. The relative influence factors are in the range 5–10, implying that a change of 1% in the signal of one channel will yield a change of 5%–10% in the efficiency. Importantly, in the case the signal changes are associated with image noise, there is no cross-compensation between channels. Instead, noise-induced alterations and uncertainties of FRET add up according to the law of error propagation. Table 4 C presents example data for an error propagation in method E₆. Noise measurements and influence factors are listed in blocks for the three donor-acceptor ratios R_DA = 0.01, 1, and 100, all subject to R_SA = 1. The first row in each block contains the SNR of each channel. The SNR was determined experimentally by analyzing the variation in the signal of five images repeatedly taken from the same sample area. The SNR was then defined as the background subtracted mean of the five images divided by the mean of the pixelwise standard deviation of the signal. The second row specifies the magnitude in the FRET uncertainty |dE(I_i)| propagated from the noise in each channel according to Eq. 2a. Quadratic summation, yields the expected overall variance of the FRET efficiency due to image noise. Here, we assume that the noise distributions are mutually independent between the channels. The third row indicates the relative contribution of each channel to the overall FRET efficiency variance The channels with significantly higher contributions are highlighted as underlined numbers. For the first block with R_DA = 0.01, the propagated uncertainty due to noise amounts to 0.50, i.e., the FRET values E₆ displayed in Fig. 3 have a confidence interval (p = 66%) E₆ = 0.84 ± 0.50. On a relative scale, this corresponds to an uncertainty of ∼60%. Similar values are obtained for the methods E₄ and E₇ (data not shown). We infer from this that the observed instability in E₄, E₆, and E₇ of efficiency values for low R_DA originate in an unfavorable propagation of noise. Interestingly, the channels with the weakest SNR (Da, Dd, and Fd) contribute relatively little to the overall uncertainty, because of low influence factors. For R_DA = 0.01, the noise in Fb dominates the behavior of FRET despite a comparably high SNR of 6.8.

Domain 2. The influence factors are low (smaller than 5). This indicates that the efficiency calculated in this domain is much less susceptible to noise than in domain 1. Indeed, the data in block R_DA =1 in Table 4 C suggest an overall uncertainty of E₆ of 0.12, resulting in a confidence interval (E₆ = 0.6 ± 0.12, 20% relative uncertainty). This finding is supported by the small variation of FRET efficiencies in this domain in Fig. 3.

Domain 3. The influence factors increase dramatically when R_DA increases (e.g., Fig. 9 C, E₆). This renders the calculation of FRET efficiency instable. Comparable to domain 1, the uncertainty amounts to ±0.59, but owing to inherently low efficiencies in this domain, the relative uncertainty reaches now a level of up to ∼4000%.

We conclude from this analysis that the cumulated effect of noise propagation of each channel can predict the variation of FRET, calculated with E₆, including the nonsensical negative values for extreme R_DA found in Fig. 3. Similar conclusions can be drawn for the methods E₄ and E₇ (data not shown), whereas the instabilities of E₁ and E₈ originate in the unfavorable propagation of uncontrolled changes in donor concentration between the samples d and b, and focus shifts (see above).