Intensity correlation-based calibration of FRET

Abstract

Dual-laser flow cytometric resonance energy transfer (FCET) is a statistically efficient and accurate way of determining proximity relationships for molecules of cells even under living conditions. In the framework of this algorithm, absolute fluorescence resonance energy transfer (FRET) efficiency is determined by the simultaneous measurement of donor-quenching and sensitized emission. A crucial point is the determination of the scaling factor α responsible for balancing the different sensitivities of the donor and acceptor signal channels. The determination of α is not simple, requiring preparation of special samples that are generally different from a double-labeled FRET sample, or by the use of sophisticated statistical estimation (least-squares) procedures. We present an alternative, free-from-spectral-constants approach for the determination of α and the absolute FRET efficiency, by an extension of the presented framework of the FCET algorithm with an analysis of the second moments (variances and covariances) of the detected intensity distributions. A quadratic equation for α is formulated with the intensity fluctuations, which is proved sufficiently robust to give accurate α-values on a cell-by-cell basis in a wide system of conditions using the same double-labeled sample from which the FRET efficiency itself is determined. This seemingly new approach is illustrated by FRET measurements between epitopes of the MHCI receptor on the cell surface of two cell lines, FT and LS174T. The figures show that whereas the common way of α determination fails at large dye-per-protein labeling ratios of mAbs, this presented-as-new approach has sufficient ability to give accurate results. Although introduced in a flow cytometer, the new approach can also be straightforwardly used with fluorescence microscopes.

Figure 1

Flow-chart and summary of the main formulae for determining FRET based on intensity correlations. As the starting point, single donor-only labeled samples are analyzed for determining the S₁, S₃ spectral spillage factors, the M_d intensity mean, and the d₀ intensity variance, then single acceptor-labeled samples are analyzed for determining the S₂ spectral spillage factor and the M_a intensity mean. In the knowledge of S₁, S₂, and S₃, the FRET-related quantity A′ is determined based on the double-labeled FRET sample: A′ is a single-valued, monotonously increasing function of the transfer efficiency E for the accurate calculation of which a scaling of A′ with the scaling factor α is necessary. After scaling of A′ with α, FRET efficiency E is calculated as E = A′/(α + A′), with the meaning A′/α = (R₀/R)⁶ (from Förster’s formula). Calculation of covariance-based α begins with determination of D, the difference between the variance of the hypothetical (immeasurable, indirectly determined) donor intensity when FRET is switched off (d), and the variance of donor intensity (measurable) when FRET is switched on (d′), both characterizing the double-labeled FRET sample. In the absence of any steric interaction between the dye-targeting labels (e.g., no competition) instead of d, the d₀ variance of the donor-only sample can be used for calculation of D. In the presence of steric influence, the functional dependence of the change in the intensity variance (Q′) on the change in mean intensity (Q) characterized with the m slope and b intercept parameters (Q′ = m · Q + b) can be used for estimating d and D on the FRET sample in the knowledge of d′, A′, and α. It can be noted that whereas in the former case D does not depend on α, in the latter case it does, with the consequence that a cubic equation should be used for α, instead of a quadratic one (see below). After determining D (or expressing it as the function of α), coefficients of p and q are calculated with I₁, and A′ on the double-labeled sample. In the knowledge of D, p, and q, a quadratic (or more complicated, depending on the nature of D, see also the Supporting Material) equation is solved for α. The fact that the D, p, and q coefficients are population averages of fluctuation (deviation from the mean) products implies two alternatives for solution of the equation for α: either it can be solved for α for each individual cell, i.e., on a cell-by-cell basis and histograms of α are created from which some characteristic expressing average behavior (mean, mode, median) is calculated (followed when D is independent of α); or the averages of the appropriate fluctuation product histograms (cell-by-cell distributions of D, p, q) are calculated first, then the equation is solved for an average α (followed when D depends on α). Finally, E is calculated with α and A′ on a cell-by-cell basis. The α-factor in the conventional way is determined from the relative absorption, labeling ratio, receptor number, and intensity (spectral α or α₀). FRET efficiency of E₀ is calculated with α₀ and A′ on a cell-by-cell basis. By taking a ratio, α/α₀ = (L_d/L_a)_effective can be introduced, expressing the reduction in quantum yield or binding efficiency of the fluorescently labeled ligand as the function of the degree of labeling of ligands.

Figure 2

Representative histograms of covariance-based FRET determination. For the application of the quadratic equation (Eq. 9) for α, first the second moments (or variances) d’ and d₀ are computed as mean values of the corresponding squared deviation (or fluctuation square) distributions (Panel B). In Panel B, d’ and d₀ are distributions of the squared deviations from the respective means (or fluctuation squares) computed from the distribution of the donor intensity of the double-labeled FRET sample (I₁) and the donor-only one (I₁,_d) of Panel A, respectively. Next, the p and q input constants for Eq. 9 are computed as averages of their respective distributions shown in Panels C, D. In Panel C, the distribution of p is that of the product of the deviations for I₁ (Panel A) and A’ (Panel F) from their respective means. In Panel D, the distribution of q is that of the squared deviation of the I₁·A’ product from its mean value. In Panel E shown is the distribution of α as computed by a cell-by-cell solution of Eq. 9 with the input distributions d’, p, q and with the average of the d₀ distribution as an input constant. As a summary, the distributions of d’, p, q, A’, and α (Panels B-F) are determined on the double-labeled FRET sample, distributions of I₁,d and d₀ (Panels A, B) on the donor-only sample. Finally, histogram of donor quenching efficiency Q (Panel G) is computed from the input histogram of I₁ and the average of I₁,_d as input constant according to Q=1-I₁/I₁,_d. Histogram of FRET efficiency E (Panel H), taking into account both donor quenching and sensitized acceptor emission, is computed from the distribution of A’ (Panel F) and the mean of the α distribution (Panel E) as input constant according to E=A’/(α+A’). During analysis, all intensities were divided by 1000, meaning that all moments are divided by 10⁶. The distributions (reminiscent to Weibull-type) are determined for the Alexa-Fluor 488-conjugated L368, Alexa-Fluor 546-conjugated W6/32 transfer pair labeling the β₂m and heavy-chain subunits of MHCI on the surface of FT T-lymphoblast cells. The details of the computations with the mean values of the shown distributions can be found in the Supporting Material.

Figure 3

Scatter-plots of pooled data expressing the general trends in the degree of correlations between the different FRET indices. Values of different FRET indices (indicated with dots, each representing a different sample) have been put on a common scale for the L368-W6/32 and W6/32-L368 (practically noncompeting) antibody pairs stained with either the Alexa-Fluor 488/Alexa-Fluor 546 or the Alexa-Fluor 546/Alexa-Fluor 647 (or Cy5) dye-pair, for the LS174T and FT cell lines. The different FRET indices are as follows: Q′, the relative change in the variance of donor intensity when the double-labeled sample is compared to the single donor-labeled one; Q, the relative change in the mean of donor intensity when the double-labeled sample is compared to the single donor-labeled one; E₀, FRET efficiency calculated by scaling A′ with the conventional α-factor (or spectral α, α₀); and E, FRET efficiency calculated by scaling A′ with the covariance-based α-factor. (A) Good correlation is between Q and Q′ (r(Q, Q′) = 79.3%). (D and F) The correlations of E are good with both Q and Q′ (r(E, Q) = 88.8%, r(E, Q′) = 89.4%). (B, D, and E) However, the correlations of E₀ with E, Q, and Q′ are bad (r(E₀, E) = 44.9%, r(E₀, Q) = 20.5%, and r(E₀, Q′) = 20.2%). Seemingly, in panel B, the correlations in these cases are ruined by points whose trend line is aligned approximately parallel with the horizontal and vertical axes, implying over- and underestimation of the real FRET efficiencies, due to the large acceptor and donor labeling ratios, respectively. For panels C and E, the opposite is true. This is supported by Fig. S6 in the Supporting Material directly showing the dependence of the FRET indices on the labeling ratios. Despite the good correlations of E with Q and Q′, the individual dispersion of these quantities, and also that of E₀, is substantial, due to the different types of dye-pairs and the different acceptor levels dictated by the different labeling ratios of the acceptor-targeting mAbs (L_a) as well as the different surface expression levels of MHCI (homo-association of MHCI has a contribution to the intramolecular FRET).