1. Introduction

Förster resonance energy transfer (FRET) microscopy, owing to its spatial range similar to intracellular protein interaction scale and high sensitivity to intermolecular distance, has become a common method for monitoring biochemical activities in living cells [1–8]. As an instrument-independent metric, FRET efficiency (E) is well defined as the ratio of the energy of the donor that is transferred to the acceptor, and its quantification is necessary for data comparison in academic exchange.

In particular, 3-cube FRET [9–11] is the most widely used approach for live cell quantitative FRET measurement owing to its nondestructiveness and high speed. In addition, there is a method called FRET hybrid assay, which can comprehensively characterize the interaction of biomolecules by simultaneously obtaining spatial distances, relative affinity constants, stoichiometry, and other parameters [12]. However, before applying quantitative FRET imaging, this method must obtain the spectrum crosstalk coefficients (a, b, c, d) and the system calibration parameters (G, k, $\frac {\varepsilon _A}{\varepsilon _D}$). Conventionally, the constants a, b, c, d can be measured from samples containing pure donors or acceptors. Moreover, G, k, $\frac {\varepsilon _A}{\varepsilon _D}$ can be measured from a standard plasmid featuring a 1:1 donor-to-acceptor ratio. This implies that traditional methods require at least three transfected cells with different plasmids to measure spectral crosstalk coefficients and calibration parameters. In addition, guaranteeing that the measured parameters have been obtained under the same experimental conditions is difficult, which reduces the reliability of the results.

In 2022, we proposed a protocol, named SCC-FRET (single-cell-based calibration FRET), to simultaneously determine $\beta$, $\delta$, G and k using single-cell imaging [13]. The spectral crosstalk coefficients a, b, c and d are replaced with $\beta = a-bd$ and $\delta = d-ac$ in FRET imaging. The cultured cells are expressed as a standard FRET plasmid with known FRET efficiency ($E_D^0$) and donor-acceptor concentration ratio ($R_C^0$). The measurement values ($E_D^1$ and $R_C^1$) containing unknown calibration parameters ($\beta$, $\delta$, G, and k) are established using E-FRET theory with three-channel fluorescence intensity obtained from the region of interest (ROI) in the cell image. By minimizing the residual under the constraints established by a given parameter and $F(\beta,\delta, {G}, {k})=|E_D^1-E_D^0|+|R_C^1-R_C^0|$ the calibration parameters of the FRET system can be obtained. SCC-FRET can simultaneously measure crosstalk correction parameters and system calibration parameters under the same experimental conditions while reducing the complexity of calibration steps and amount of sample types in a more convenient way than prior methods [9,10,14–16].

However, we found that the SCC-FRET method shows two drawbacks: (1) $\frac {\varepsilon _A}{\varepsilon _D}$, which is a key calibration parameter to calculate the acceptor-centric FRET efficiency $E_A$ [12] is not estimated; and (2) the calculation results strongly dependent on the range of $\beta$ and $\delta$. In SCC-FRET, to obtain correct results, the range of $\beta$ and $\delta$ has to be limited to $\pm$10% of theoretical value.

To make the SCC-FRET more robust and universal, we here in propose an improved single-cell-based calibration for FRET system (Im-SCC-FRET). According to the principle of least squares, the target function is changed from the sum of absolute values to the sum of squares, and the acceptor-donor extinction coefficient ratio ($\frac {\varepsilon _A}{\varepsilon _D}$) is added. The initial estimations $\beta$ and $\delta$ are changed to the integral rather than the maximal value of spectral overlap of fluorophore and filter; the quantum response of the camera is also considered. Moreover, the calibration parameter with the smallest error is fitted using the sequence quadratic programming (SQP) algorithm. Compared to SCC-FRET, the experiment results demonstrate that Im-SCC-FRET exhibits greater completeness, accuracy, and robustness.

2. Methods and materials

2.1 Im-SCC-FRET theory

In Im-SCC-FRET theory, the conversion of donor-centric FRET ($E_D$), and acceptor-centric FRET ($E_A$) efficiencies, and donor-acceptor concentration ratio ($R_C$) in 3-cube FRET are expressed as follows [17,18]:

Based on the least-squares optimization theory, the calibration parameters of FRET systems, that is, $\beta$, $\delta$, G, k, and $\frac {\varepsilon _A}{\varepsilon _D}$, can be determined by the following equations:

Equation (4) differs significantly from SCC-FRET in that we apply least squares, that is, we express the total error in terms of a sum of squares, and add the acceptor-to-donor extinction coefficient ratio ($\frac {\varepsilon _A}{\varepsilon _D}$) as a factor. The SQP algorithm is used for calculations that present good convergence, high computational efficiency, and strong boundary search ability; SQP is a medium-scale algorithm that meets the task objectives.

The SQP algorithm requires a set of initial values that satisfy all constraints. Thus, we need to estimate the initial value of the calibration parameters. The initial estimates of $\delta _D$ and $\beta _A$ can be calculated from spectral information:

According to Eq. (8), the spectral information in FPbase [18] is used to calculate the initial estimates of the parameters $\delta$ and $\beta$. Figure 1 shows the Cerulean excitation and emission spectrum as well as the donor and acceptor emission filters; the donor emission filter is 480/40 bp and the acceptor emission filter is 535/30 bp. Figure 1 also shows the Venus excitation and emission spectrum together with the donor and acceptor excitation filters; in this case, the donor excitation filter is 436/20 bp and the acceptor emission filter is 500/20 bp. Trapezoidal approximation of overlapping areas yields: $e_{D^{em}A_{filter}^{em}}^s=13.5$, $e_{D^{em}D_{filter}^{em}}^s=31.78$, $e_{A^{ex}D_{filter}^{ex}}^s=0.78$, $e_{A^{ex}A_{filter}^{ex}}^s=12.9$.

 

Fig. 1. Spectra of the FRET system. (a) Absorption and fluorescence emission spectra of Cerulean and filter sets. (b) Absorption and fluorescence emission spectra of Venus and filter sets.

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The quantum response of the camera can be queried on the official website of Zeiss. According to this response, $\varphi =1.071$.

The relative intensity ratio of the light source X-Cite120Q at the center wavelength of the donor excitation filter (446 nm) and acceptor excitation filter (510 nm), denoted as $\phi$ is $\sim$2.500. Substituting these values into Eq. (8) yields $\delta _D=\varphi \cdot \frac {e_{D^{em}A_{filter}^{em}}^s}{e_{D^{em}D_{filter}^{em}}^s}=0.4554$, $\beta _A=\phi \cdot \frac {e_{A^{ex}D_{filter}^{ex}}^s}{e_{A^{ex}A_{filter}^{ex}}^s}=0.151$.

We found that values of $\beta _A$ and $\delta _D$ have a slight impact on the quantitative FRET calculation that the relative errors of $E_D$, $E_A$ and $R_C$ change $\sim 1{\% }$ when $\beta _A$ or $\delta _D$ changes $20{\% }$. Taking into accounts some experimental factors, such as the fluorescent proteins’ maturity and expression level, $1{\% }$ of variations in FRET measurement are acceptable. For more details, please refer to the Supplement 1.

We also used the interior-point algorithm to fit a set of G, k, and $\frac {\varepsilon _A}{\varepsilon _D}$ values with initial values of $\beta _A$ and $\delta _D$. The initial value of each calibration parameter was set to meet the constraints of SQP and realize Im-SCC-FRET.

2.2 Cell culture, transfection, and plasmids

Cell culture, transfection, and plasmids were described in our previous publication [13].

2.3 Living-cell fluorescence imaging

Living-cell fluorescence imaging was described in our previous publication [13].

3. Results and discussion

As shown in Fig. 2, to verify that our method can accurately obtain the system calibration parameters from single-cell data, we selected ten ROIs in a single cell expressing C5V ($E_D^0=E_A^0=0.43$, $R_C^0=1.00$), read out the gray values of each ROI for three channels, and performed SCC-FRET and Im-SCC-FRET calculations to obtain the calibration parameters ($\beta$, $\delta$, G, k, and $\frac {\varepsilon _A}{\varepsilon _D}$) for different ranges of $\beta$ and $\delta$. The results are shown in Fig. 3.

 

Fig. 2. Representatives of $I_{AA}$ (a), $I_{DA}$(b) and $I_{DD}$(c) images of cells expressing C5V, and ten ROIs of a single cell.

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Fig. 3. SCC-FRET and Im-SCC-FRET fitting results with different fitting ranges of $\beta$ and $\delta$. (a) $\beta$, (b) $\delta$, (c) $G$, (d) $k$, (e) $\epsilon$.

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Figure 3 shows that as the ranges of $\beta$ and $\delta$ change, the results of SCC-FRET vary significantly. Moreover, the coefficients $\beta$ and $\delta$ increase with the increase of the upper limit of the value range, and the calibration parameter G decreases accordingly. The results yielded by Im-SCC-FRET remain mostly unchanged. This shows that SCC-FRET has strict requirements for the fitting range, and when the fitting range is expanded to 0-1, it does not produce stable fitting results. Therefore, Im-SCC-FRET significantly improves this issue.

Furthermore, the parameters calculated by SCC-FRET and Im-SCC-FRET within the range 0-1 are used for quantitative FRET analysis. Table 1 lists the results of standard plasmids C17V ($E_D^0=E_A^0=0.38$, $R_C^0=1$) [20] and C32V ($E_D^0=E_A^0=0.32$, $R_C^0=1$) [21].

Table 1. Calculations of SCC-FRET and Im-SCC-FRET for a fitting range 0-1

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When the ranges of $\beta$ and $\delta$ are set to the theoretical maximum range 0-1, the quantitative FRET results ($E_D$, $E_A$, and $R_C$) of Im-SCC-FRET are significantly closer to the reported values than those of SCC-FRET. This shows that Im-SCC-FRET can produce more accurate results even for the maximum fitting range. In combination with the Im-SCC-FRET results presented above, we demonstrated that stability is achieved when the fitting range changes, indicating Im-SCC-FRET is a robust method.

4. Conclusion

We developed the Im-SCC-FRET method, which relaxes the restrictions on the range of the calibration parameters and exhibits stronger robustness and higher accuracy compared with SCC-FRET. In addition, the calibration parameter $\frac {\varepsilon _A}{\varepsilon _D}$, which is significant for FRET hybrid assay, can be estimated using the Im-SCC-FRET method, which makes this method more complete. The Im-SCC-FRET method can effectively calibrate the commonly used 3-cube FRET microscopy and reduce the complexity of sample preparation, thereby opening a promising avenue for developing an intelligent FRET correction system. It will also contribute to using quantitative FRET techniques to characterize biochemical scenes within living cells and study or whitewash molecular mechanisms.