2.1.

FRET and FLIM

Upon the occurrence of FRET, donor fluorophores acquire an additional nonradiative decay pathway, which adds to the nonradiative rate constant and reduces lifetime. FLIM measurements can be carried out by exciting only donor fluorophores within the system. Fluorescence decay curves are collected on a per-pixel basis. For the study of interactions between proteins labeled with donor and acceptor fluorophores, the donor population consists of ones that are bound to acceptors and are undergoing FRET (FRETing) and free ones that are not undergoing FRET (nonFRETing). Therefore, in a FRET system with only one donor population and one acceptor population present, the total per-pixel fluorescence intensity decay measured is as follows:

2.2.

Global Fitting Algorithm

The global fitting algorithm developed by Pelet et al.³³ is utilized to fit our fluorescence decay data to the model expressed as Eq. (1) above. In summary, decay curves are scaled to a maximum of one before fitting. The prefactors are bounded such that $({\alpha }_D+{\alpha }_F)=1$, but are allowed to vary spatially. It is assumed that only two fluorophores exist in each pixel, nonFRETing and FRETing species; thus the fitted lifetimes are spatially invariant. The least-squares estimate ${\chi }^2$ is used as a merit function to optimize the values of all fit coefficients:

Image segmentation was carried out based on intensity thresholding whereby pixels of similar intensities were grouped together and their decay curves summed. A global fit is performed on this very limited set of decay curves (segment-sum) ³³ to obtain coefficients that were subsequently used as initial guesses in the final global fitting step (global-indiv). This method of using an educated initial guess of fit parameters was shown to greatly accelerate time to convergence.³³ It is possible to carry out segmentation because paxillin and FAT are cytosolic proteins that localize to FAs. Images obtained at the basal adhesion plane therefore capture the locations of FAs, which are more intense than the rest of the cell area and can be grouped together. Figure 1 shows intensity and segmented images of GPax-only and GPax/FATmCh cells.

Fig. 1

(a) Fluorescence intensity images of GPax (top) and GPax/FATmCh (bottom cells). (b) Images of the same cells in (a) segmented by intensity and arbitrarily colored to distinguish different intensity segments. Only pixels with LL1000 were selected for intensity segmentation. Scale bar is 10 μm.

Data fitting clearly depends on SNR level. Pixels with decay curves consisting of a minimum of 1000 photon counts are selected for fits unless otherwise specified. We express our lower limit (LL) in photon count symbolically as LL1000.

3.1.

Cell Preparation

BAECs were cultured in Dulbecco’s Modified Eagle’s Medium (DMEM) (Gibco/Invitrogen, Carlsbad, CA) supplemented with 10% fetal bovine serum, 1% penicillin/streptomycin, and 1% L-glutamine. Cells were seeded at an initial density of 160 000 cells/well in fibronectin-coated 35-mm glass-bottomed dishes (MatTek Corporation, Ashland, MA). Plasmids used for transfection were either GFP-paxillin (gift from K Yamada of the National Institutes of Health, Bethesda, MD) or FAT-mCherry. Mouse FAT from the dsRed-FAT plasmid (N. Mochizuki, National Cardiovascular Center Research Institute, Osaka, Japan) was amplified by polymerase chain reaction (PCR) with primer pairs ($5\prime$-CTAAGCAACCTGTCCAGCATCAGC-$3\prime$) and ($5\prime$-CGGATCCGGGTGTGGCCGTGTCTGCCCTAGC-$3\prime$). The resulting PCR fragment was ligated into a pcDNA4/HisMax© TOPO® vector (Invitrogen, Carlsbad, CA). mCherry from the pRSET-B mCherry plasmid (R. Tsien, University of California San Diego) was digested with EcoRI and BamHI and ligated into the pcDNA4-FAT vector at the $3\prime$ end of FAT. The FAT-mCherry plasmid was amplified using the QIAGEN MaxiPrep kit (QIAGEN, Valencia, CA). Plasmid notation (GPax, FATmCh) indicates the position of fluorophore within the fusion protein, where GFP is at the $N$ terminus of paxillin and mCherry is at the $C$ terminus of FAT. Cells were transfected at $\approx 80\%$ confluence with either GFP-paxillin plasmids only or cotransfected with both GFP-paxillin and FAT-mCherry plasmids using the FuGENE 6 transfection reagent from Roche Diagnostics (Indianapolis, IN). Note that plasmid notation indicates the position of the fluorophore within the fusion protein, where GFP is at the N terminus of paxillin and mCherry is at the C terminus of FAT. Lifetime imaging was carried out 24 h after transfection.

3.2.

Optical Setup

Details of the two-photon lifetime imaging microscope setup have been described previously.^18,33,37 In brief, a custom-built, two-photon microscope³⁷ was modified for obtaining lifetime images. This microscope is based on a modified inverted microscope, the Axiovert 110 by Zeiss (Göttingen, Germany), with a femtosecond laser source from the Mira family of mode-locked Ti:Sapphire oscillators by Coherent (80 MHz, 150 fs; Santa Clara, CA).^18,33 The laser, tuned at 890 nm, excites a subfemtoliter volume at the focal point of a $40\times$ Fluar lens objective (1.3 numerical aperture; Zeiss). Pixel images ($256\times 256$ pixels) spanning a field of view of $120\times 120{\mathit{\mu }\mathrm{m}}^2$ were imaged with a pixel residence time between 25 and 200 μs, depending on specimen brightness. The fluorescence signal from the cell is first selected with a short-pass Schott BG-39 filter and a short-pass 700-nm filter from Chroma Technology Corporation (Rockingham, VT), which minimizes contributions from scattered light and autofluorescence, before being sent to the top port of the microscope, where it is filtered for green wavelengths with a HQ500LP emission filter from Chroma. A photomultiplier tube (PMT) (R7400P; Hamamatsu Bridgewater, NJ) detects emitted photons, and this signal is sent to a time-correlated single-photon counting (TCSPC) card (SPC-730; Becker-Hickl, Berlin, Germany).

3.3.

FLIM Imaging

The instrument response (IR) of the system was measured using either fluorescein (flsn) in a well-slide or gallium arsenide (GaAs) dried onto a coverslip. Because flsn has a well-characterized decay of 4.0 ns,^38,39 the IR can be extracted as a Gaussian that best fits the fluorescein decay curve with a preset decay time of 4.0 ns. For GaAs, the second harmonic generation (SHG) signal from the GaAs surface is instantaneous, thus directly giving us the IR.

Cells selected for FLIM imaging were those in the middle range of fluorescence intensities (corresponding to cells containing fluorophores with concentrations in the micromolar range) as observed using epifluorescence with a 488-nm argon ion laser (MWK Industries, Austin, TX) or a 432-nm diode laser (Laserglow Technologies, Toronto, Ontario, Canada). This is a compromise between ensuring a bright-enough cell that has sufficient photon counts for decay curves with acceptable SNR and cells that are not grossly overexpressing the fusion proteins, which could introduce artifacts into the interactions we wish to measure. The selected cells typically have fluorescent proteins with concentration in the nanomolar to micromolar levels. FLIM imaging of transfected cells would take 7 or 8 minutes. All FLIM imaging was carried out at 890 nm. At this wavelength, mCherry is minimally excited, with fluorescence levels within the noise range. The microscope was carefully focused to the basal adhesion plane of the cell before we carried out FLIM imaging.

3.4.

Simulated Decay Curves

To test the conclusions derived from fitting of experimental data, sets of simulated decay curves, with Poisson noise added, were generated and fitted with the global fitting algorithm. Decay curves have either a single lifetime (${\tau }_D=2.6\mathrm{ns}$; single-exponential decay) or two lifetimes (${\tau }_D=2.6\mathrm{ns}$, ${\tau }_F=1.3\mathrm{ns}$; double-exponential decay). For sets of decay curves containing two lifetimes, the prefactors within each set were varied such that $0\le {\alpha }_F/({\alpha }_D+{\alpha }_F)\le 1$. Each decay curve is convolved with a Gaussian instrument response of 0.25 ns, with Poisson noise added to simulate the photon-counting process. The total photon counts within decay curves were allowed to vary between 100 and 3000, consistent with typical experimental data and depending on the fitting method being tested.

3.5.

Computation

The global fitting algorithm as developed by Pelet et al.³³ was carried out based on MATLAB (MathWorks, Natick, MA) and uses the MATLAB built-in fmincon optimization function. Convergence is defined as being reached when ${\chi }^2$ varies by less than ${10}^{-15}$.

4.1.

Instrument Response

One way to account for IR is to acquire SHG signal from a surface, GaAs in our case. Because SHG is instantaneous, this method directly gives us the IR. Another method would be to take FLIM data of a sample with a well-characterized lifetime, for example, flsn, which has a lifetime of 4.0 ns. Convolution of a Gaussian to a single-exponential decay of lifetime 4.0 ns and comparing this to the actual flsn data would recover the Gaussian that best depicts the IR. With either method, measurements of the IR ($I_{\mathrm{msr}}$) would typically be taken before carrying out cell measurements, and it is assumed that the IR does not vary throughout the imaging session. However, a number of factors in TCSPC microscopy may contribute to substantial variations in $I_{\mathrm{msr}}$, such as electronic circuit noise, or to laser instabilities. We found that the most difficult problem for some $\mathrm{Ti}:\mathrm{sapphire}$ lasers is the sudden, unpredictable changes in IR that are sometimes observed. We attribute these changes to the losing and regaining of mode-locking of the laser. Representative variation in IR due to periodic laser instability is shown in Figure 2(a), where IR width varies by about 100 ps within just 1 to 2 minutes.

Fig. 2

(a) Four representative variations instrument response (IR) functions were taken over a period of 2 min. A width variation of over 100 ps was seen. (b) Scaled IRs for two data sets. For the solid line plot of data set 1, the blue line is IRmsr(GaAs), the green line is IRext(GPax), and the red line is IRext(GPax / FATmCh). For the dashed line plot of data set 2, the blue line is IRext(flsn), the green line is IRext(GPax ), and the red line is IRext(GPax/FATmCh). The plot for data set 2 has been shifted to the right by 1.5 ns for clarity.

If the IR varies appreciably throughout the imaging session, it is inaccurate to use just one IR for all calculations. An adaptive approach that extracts IR directly from the image data is adopted for live-cell FRET imaging. Though it would be possible to interleave data collection from sample and reference specimens and collect IR data throughout the imaging session, this method is not very practical, as it significantly increases data acquisition time and perturbs the specimen, especially given that the IR profile changes unpredictably. In contrast, an adaptive approach allows continuous recovery of IRs over time throughout the experiment, mitigating the issue of sudden IR jumps observed with mode-locked lasers.

To overcome this problem, an adaptive solution was developed based on the assumption that the mean lifetime can be readily extracted and is invariant for a homogeneous population of cells. Our approach is based on many prior works, and we have extended them in FRET studies.^40–43 The IR is directly extracted from the cell FLIM data (${\mathrm{IR}}_{\mathrm{ext}}$). All decay curves from every pixel of the image are summed, and the IR is calculated by presetting mean lifetimes of $\tau =2.7\mathrm{ns}$ for GPax-only cells and $\tau =2.3\mathrm{ns}$ for GPax/FATmCh cells. The Gaussian with full-width half-maximum (FWHM) varied that, once convolved with a single-exponential decay having the preset lifetime, best fits the master decay curve is used as the IR for that particular image. Variations in preset lifetimes of $\pm 0.3\mathrm{ns}$ ns produced negligible variations in the FWHM of the Gaussian IR (data not shown), thus validating our method of direct IR extraction. Since this approach is most useful in kinetic or long-term FRET study of protein interactions in live cells when interleaving live-cell and reference compound imaging is impractical, the required mean lifetime values for this adaptive algorithm can be readily measured as control experiments prior to critical studies by referencing to known lifetime standards.

A plot of scaled IR showing the difference between ${\mathrm{IR}}_{\mathrm{msr}}$ and ${\mathrm{IR}}_{\mathrm{ext}}$ for two data sets is given in Fig. 2(b). The dashed line plot compares differences between ${\mathrm{IR}}_{\mathrm{msr}}$ from flsn and sample ${\mathrm{IR}}_{\mathrm{ext}}$ from GPax and GPax/FATmCh cells, whereas the solid line plot compares differences between ${\mathrm{IR}}_{\mathrm{msr}}$ from GaAs and sample ${\mathrm{IR}}_{\mathrm{ext}}$. Only very slight differences between different the IRs can be observed. However, these slight differences are sufficient to produce different fitted lifetime results, especially for double-exponential decays. This is shown in Table 1. When fitting single-exponential decays to GPax curves, both IR methods recover similar lifetimes. However, discrepancies become obvious when fitting double-exponential decays to data from double-transfected cells. Fits using both methods produce reasonable-looking fits (data not shown), and in all cases, the ${\chi }^2$ has been minimized. However, fitting with ${\mathrm{IR}}_{\mathrm{msr}}$, the expected two FRET lifetimes cannot be reliably recovered for either GaAs or flsn. ${\mathrm{IR}}_{\mathrm{msr}}$ GaAs recovered lifetimes of ${\tau }_D=2.16\pm 0.13\mathrm{ns}$ and ${\tau }_F=0.01\mathrm{ns}$, whereas ${\mathrm{IR}}_{\mathrm{msr}}$ Flsn recovered lifetimes of ${\tau }_D=2.30\pm 0.10\mathrm{ns}$ and ${\tau }_F=0.01\mathrm{ns}$. In both cases, the recovered ${\tau }_D$ is an average of the two lifetimes recovered using ${\mathrm{IR}}_{\mathrm{ext}}$. In contrast, expected single- and double-exponential lifetimes can be obtained using ${\mathrm{IR}}_{\mathrm{ext}}$.

Table 1

Lifetimes obtained from fits of GPax-only cells and GPax/FATmCh cells using either IRmsr (flsn or GaAs) or IRext.

Our results demonstrate that to obtain two relevant FRET lifetimes from double-exponential decay curves, the IR used in fits is important. Because the IR varies throughout the imaging session for our setup, an IR measured at the start of the experiment is rendered invalid for use with data generated later within the imaging session. Even a slight variation in IR could cause deviations in fit results. Therefore, a more robust adaptive method is used to obtain ${\mathrm{IR}}_{\mathrm{ext}}$ and obtain more consistent fits. ${\mathrm{IR}}_{\mathrm{ext}}$ is used for all subsequent fits for both FRETing and nonFRETing decay curves.

Theoretically, the decision to use GFP lifetime as an internal reference for single transfected cells is well-justified because GFP exhibits a single-exponential decay that is relatively insensitive to the cellular environment. For double-transfected cells, the mean lifetime is used as a reference because it can be more robustly measured compared to quantifying a double-exponential decay with noisy data. It also relies on the assumption that the cellular population studied is biochemically homogeneous within biological noise and that its mean lifetime is invariant with time. Clearly, these assumptions cannot always be met in some studies, and this approach needs to be used with caution.

Finally, this approach is unsuited to extracting very short lifetime components, such as from second harmonic generation. However, with our focus on FRET studies where the quenched donors normally still have fairly long lifetimes (for GFP, lifetime shortens from 2.6 ns to 1.8 ns), this is not a major concern. Furthermore, if strong SHG, Raman or Rayleigh scattering is present, their contribution may be removed by modeling an additional zero lifetime component convoluted with IR.

4.2.

Segmented and Global Fits

We examine the differences in lifetime fits observed between various fitting procedures and also between global fitting results from summed curves and individual curves. All fitting algorithms use the model represented by Eq. (1). For both GPax-only cells (eight cells) and GPax/FATmCh cells (five cells), fits were carried out with (i) a single-exponential decay, (ii) a double-exponential decay with ${\tau }_D$ fixed at 2.6 ns and (iii) both lifetimes freely varying. For each fitting procedure, results are compared between global fitting of individual decay curves carried out over the whole cell image (global-indiv) and global fitting of decay curves summed upon segmentation according to intensity thresholds (segment-sum) as previously described. LL1000 was chosen, which resulted in $\approx 8000$ decays fit for global-indiv and $\approx 6$ decays fit for segment-sum.

To verify the conclusions drawn from fitting of cell data, single- and double-exponential decay curves were simulated (five sets of curves each). The single-exponential decay curves had a lifetime of 2.6 ns, and the double-exponential decay curves had lifetimes of ${\tau }_D=2.6\mathrm{ns}$ and ${\tau }_F=1.3\mathrm{ns}$. The 2.6-ns lifetime was chosen because it is close to the nonFRETing lifetime of GFP, and a ${\tau }_D/{\tau }_F$ ratio of 0.5 was chosen, giving ${\tau }_F=1.3\mathrm{ns}$. The same fitting procedure as described above was carried out, with ${\tau }_D$ fixed at 2.6 ns for fit procedure (ii). The results are presented in Table 2.

Table 2

Lifetime fit results for cell and simulated data simulation data using various fitting procedures.

When cell data are fit with single-exponential decays, global-indiv and segment-sum extract similar ${\tau }_D$ for the two cell populations each, with segment-sum values being slightly higher. For GPax cells, segment-sum recovered a lifetime of $2.58\pm 0.07\mathrm{ns}$, whereas global-indiv recovered a lifetime of $2.47\pm 0.04\mathrm{ns}$. Summing multiple single-exponential decay curves produces another single-exponential decay curve with the same lifetime but a higher photon count and less noise (higher SNR). For GPax/FATmCh cells, the value of ${\tau }_D$ is underestimated. When a double-exponential decay is force-fit with a single-exponential decay, an average lifetime of the actual two lifetimes present is recovered: $2.37\pm 0.10\mathrm{ns}$ from segment-sum and $2.30\pm 0.19\mathrm{ns}$ from global-indiv.

The results from simulated decay curves verified our conclusions. For single-exponential decays, segment-sum accurately extracted the lifetime, whereas global-indiv extracted a lifetime within 5% of the known value of 2.6 ns. This shows that global-indiv single-exponential fitting on curves with a single lifetime is affected by the SNR of the decay curves, but that fit results are not greatly compromised. For double-exponential decays, both methods extracted lifetimes within 10% of the average of the two true lifetimes.

Fits were also carried out by fixing ${\tau }_D$ to 2.6 ns. GPax cells are not expected to have two lifetimes. Segment-sum indeed recovers a second lifetime of $2.56\pm 0.06\mathrm{ns}$, similar to the fixed ${\tau }_D$ value, indicating that it is possible to recover the single-decay lifetime even with a double-exponential fit from decay curves with very high SNR. Global-indiv is unable to recover the correct single lifetime because of the need for higher SNR data. It should be possible to recover the second lifetime present in GPax/FATmCh curves by fixing ${\tau }_D$ to the nonFRET lifetime value obtained from GPax cells. Fixing ${\tau }_D$ reduces the number of degrees of freedom given to the fitting algorithm, thus allowing the fits to converge faster. Segment-sum recovers a ${\tau }_F$ of $2.34\pm 0.12\mathrm{ns}$, whereas global-indiv recovered a lower ${\tau }_F$ of $1.99\pm 0.05$.

The results from simulated decay curves verified our conclusions for single-exponential decays. For double-exponential decays, both segment-sum and global-indiv are able to recover the correct ${\tau }_F$, to within 2%, though the segment-sum value is slightly more accurate. Though summing many double-exponential curves, each with its own fraction of the two lifetimes, averages over the prefactors in the fit equation, due to the very high resulting SNR in the summed decay, segment-sum is still well able to extract the known second lifetime. However, even though the extracted second lifetime obtained from segment-sum might be slightly more accurate, this method does not allow recovery of the prefactors. Therefore, global-indiv is the desired method for most applications. That the difference in lifetimes obtained from the two methods is larger for cell data compared to simulated data is an indication that experimental decay curves contain biological noise sources not accounted for when simulating decay curves only with Poisson noise.

When fits were carried out with both lifetimes free to vary, for GPax cells, segment-sum is again able to extract two lifetimes with similar values, $2.60\pm 0.05\mathrm{ns}$ and $2.56\pm 0.04\mathrm{ns}$, because of the high SNR. However global-indiv fails and in fact recovers two distinctly different lifetimes due to the low SNR. The recovered values, $2.76\pm 0.11\mathrm{ns}$ and $2.17\pm 0.08\mathrm{ns}$, lie on either side of the single lifetime present, and average to 2.47 ns, within 5% of 2.6 ns. For GPax/FATmCh decays, segment-sum also fails to recover the two expected lifetimes most likely because of the compound effect of averaging over the prefactors and because the two lifetimes present in the decay curves were not well-separated. Global-indiv, on the other hand, successfully recovered lifetimes of ${\tau }_D=2.56\pm 0.13\mathrm{ns}$ and ${\tau }_F=1.99\pm 0.08\mathrm{ns}$.

The results from simulated decay curves again verified experimental results for single-exponential decays. Segment-sum extracted two lifetimes within 1% of the known ${\tau }_D$, and global-indiv extracted two lifetimes well on either side of the known ${\tau }_D$. This result shows that a false FRET detection can actually be deduced from single-exponential decays if these data are fit with double-exponential fits. It underscores the importance of exercising some discretion when applying mathematical methodology to known or expected biological phenomena. For double-exponential simulated curves, both segment-sum and global-indiv are able to recover the two lifetimes present in the curves within 5% error.

It is unclear which method is best to accurately determine recovered lifetimes when carrying out double-exponential fits on experimental cell data. Both lifetimes could be allowed to vary freely. However, fitted lifetimes may vary for images obtained in a single imaging session, as shown in Table 3. One way to overcome this uncertainty is to assume that ${\tau }_D$ is the same in both GPax-only and GPax/FATmCh cells. This assumption says that the lifetime of GPax in cells is the same, regardless of whether it is the sole fusion protein species within the cell or whether it is coexpressed with a partner fusion protein. For cells of the same type that are cultured, transfected, and imaged under the same conditions, this assumption is reasonable. By doing this, ${\tau }_D$ can be fixed when double-exponential fits are carried out, thus recovering only ${\tau }_F$ and FR. With the use of this method, a comparison of the values of ${\tau }_F$ can be made across all images.

Table 3

Fit results for GPax and GPax/FATmCh cell data obtained during a single imaging session with two fitting procedures: (1) double-exponential fit with lifetimes free to vary and (2) double-exponential fit with τD fixed at 2.6 ns.

An important assumption underlying global analysis is the invariance of free and bound donor lifetimes within the cell. It is possible that different locations within the cell, delineated by varying fluorescence intensities, have microenvironments, such as protein aggregation state, protein phosphorylation state, and binding of cofactors, which may differentially affect the properties of fluorescent proteins. This would result in varying lifetimes, thus invalidating global analysis. Therefore, to test the validity of the global assumption on cell data, the double-exponential global analysis carried out on GPax-only cells as described above was further examined. Any variation between intensity regions would be apparent when looking at the FR distribution, even though the resultant lifetimes are artificial, because more fit parameters are allowed to vary. As can be seen in Fig. 3, however, FR distributes similarly, regardless of which intensity region is examined. This lack of intensity-dependent differences justifies the use of global analysis for our cell data.

Fig. 3

(a) GPax cell segmented by intensity at LL1000. (b) Histogram of FR distribution of cell in (a), color-coded by cell intensity segment. A double-exponential fit was carried out to obtain the FR. Scale bar is 10 μm.

As further validation, global analysis was carried out on pixels with similar intensities (and thus photon counts), but at different regions in the cell, either FAs or cytosol. Because FAs, as sites of cell attachment to the basal surface, transmit force and recruit a distinct set of proteins, these regions are morphologically and biologically different from the cytosol. It would be expected that if variations in fluorescent protein properties exist within the cell, comparing fit results from these two regions would extract such differences. Single-exponential fits carried out on these pixels retrieved ${\tau }_D=2.65\pm 0.03\mathrm{ns}$ for FA pixels, and ${\tau }_D=2.65\pm 0.04\mathrm{ns}$ for cytosolic pixels. There is indeed no difference between these lifetimes, which demonstrates the applicability of global analysis to cell data.

The choice of single exponential and biexponential fitting is partly motivated by the goal of finding the simplest models that can account for all the experimental observations. This choice is also partly justified by the limited number of photons in typical fluorescent images, and more complex models can rarely be distinguished, given Poisson statistics. We have demonstrated that paxillin-FAK binding is spatially invariant and that all experimental data can be accounted for by one free donor lifetime and one bound donor lifetime. In other protein systems, this spatially invariant condition may not be valid, and different bound and free lifetimes may be associated with different cellular locations. Furthermore, different initial conditions used in global fitting may converge to different lifetimes and preexponential fractions. The choice of initial conditions has been discussed extensively in literature and is not examined in detail here.^{18,22,32,33,44}

Finally, although it is not the main focus of this work, it should be noted that the data model and the initial condition choices significantly affect whether a global fitting algorithm converges as well as the speed of convergence. Strategies to improve global algorithm efficiency have been previously developed.^{22,30,32–34,44,45}

4.3.

Photon Counts Per Pixel: Lifetime Scaling

Typically, at least 100 counts per pixel are required to extract a lifetime with uncertainty of 20% from a fluorescence decay curve. A LL of 1000 counts per pixel is required to extract two lifetimes, provided the lifetimes are well-separated and the two prefactors are of comparable magnitude.^21,33 In cell samples, these parameters cannot be predicted and depend entirely on the biological sample. The binding interaction between fusion proteins dictates the distance between FRETing fluorophores and thus ${\tau }_F$. The expression rate of fusion proteins depends on the cell’s transcription and translation machinery and cannot be controlled externally. Thus the FRET ratio (prefactors) distribution varies at each pixel, depending on the amount of fusion protein present. As we attempt to select cells that are not overexpressing fusion proteins, this limits the total photon counts that can be collected within each pixel, where it might be difficult to collect even 1000 counts per pixel for a two-lifetime fit. Moreover, our fusion proteins localize to FAs, leading to a difference in pixel intensity between FA and cytosolic regions. Whereas FAs might contain pixels with more than 1000 counts, cytosolic regions might not.

To overcome these limitations, we examined how the fitted lifetimes and FRs deviate with varying LLs on photon counts for GPax/FATmCh cells. For GPax cells, this process is unnecessary because enough pixels per cell have counts higher than 1000. Besides, segment-sum adequately recovers the true ${\tau }_D$ from single-exponential decay curves, as previously shown. For GPax/FATmCh cells, ${\tau }_D$ was fixed at 2.6 ns, whereas ${\tau }_F$ and FR were recovered by carrying out fits at varying photon count LL values corresponding to the total number of integrated photons under a decay curve. Figure 4(a) shows representative double-exponential fits carried out with ${\tau }_D$ fixed at 2.6 ns for two sets of data corresponding to measured cell simulation. The open circles in Fig. 4(b) show the variation in fitted ${\tau }_F$ with LL. The fitted ${\tau }_F$ reaches a plateau at LL of $\approx 1500$. Fits carried out on simulated curves with a ${\tau }_F/{\tau }_D$ ratio of 0.8 and varying LL show similar trends (closed circles). This dependence on LL can also be seen by quantifying the percentage deviation from plateau value, it can be seen that the curves from both cells (open triangles) coincide, with the curve from simulated data (closed triangles) following a similar trend. Inclusion of decay curves with lower-intensity cell data (low LL) into fits leads to systematic underestimation of the second lifetime present in the double-exponential decays. Because the deviation with LL is systematic, it should be possible to carry out fits at lower LL to include as many pixels as possible from cell data, including dimmer pixels, and apply a correction factor to pixels with lower LL. To map out this systematic variation, fits were carried out on simulated curves at varying LL and ${\tau }_F/{\tau }_D$ ratios.

Fig. 4

Double-exponential fits were carried out with ${\tau }_{\mathit{D}}$ fixed at 2.6 ns for both cell and simulated data. (a) Lifetime decay data and the associated curve fits for five representative pixels of the LL100 data set (containing 1458 pixels in total) are shown. The normalized ${\chi }^2$ for fitting the whole data set is 0.26 without reweighting and is 0.67 with reweighting (see also Table 5). (b) Variation in fitted ${\tau }_F$ with varying LL. Left axis is fitted ${\tau }_F$ (circle) right axis is percentage difference in fitted ${\tau }_F$ compared to fitted ${\tau }_F$ at LL2000 (triangle). Open markers: two independent cell measurements; closed markers: simulated data. (c) Master curve for scaling ${\tau }_{\mathit{F}}$ fit to true ${\tau }_{\mathit{F}}$. Squares are percentage deviations from ${\tau }_{\mathit{F}}$ (LL2000) (500%dev2000 and 1000%dev2000), circles are percentage deviations from ${\tau }_{\mathit{F}}$ (LL1000) (500%dev1000), and triangles are percentage deviations from true ${\tau }_{\mathit{F}}$ (500%dev${\tau }_{\mathit{F}}$, 1000%dev${\tau }_{\mathit{F}}$, and 2000%dev${\tau }_{\mathit{F}}$). Dotted lines are fits of LL500 %dev points, dashed lines are fits of LL1000 %dev points, and solid line is a fit of LL2000 %dev points.

Trends in percentage deviation in fit ${\tau }_F$ with ${\tau }_F/{\tau }_D$ ratio are shown in Fig. 4(c), where the known ${\tau }_F$ is used when calculating ${\tau }_F/{\tau }_D$. Percentage deviation, such as that between LL500 and LL2000, is defined as follows:

Hereafter it is written as 500%dev2000. Squares are percentage deviation from ${\tau }_{F(\mathrm{LL}2000)}$ (500%dev2000 and 1000%dev2000), circles are percentage deviation from ${\tau }_{F(\mathrm{LL}2000)}$ (500%dev1000), and triangles are percentage deviation from known ${\tau }_F$ (500%dev${\tau }_F$, 1000%dev${\tau }_F$, and 2000%dev${\tau }_F$). Dotted lines are fits of LL500 %dev points, dashed lines are fits of LL1000 %dev points, and the solid line is a fit of LL2000 %dev points. The ${\tau }_F$ obtained from fits with LL2000 is very close to the true ${\tau }_F$, within 10% for lifetime ratios between 0.02 and 0.82. However, as previously mentioned, the lifetime is increasingly underestimated with higher${\tau }_F/{\tau }_D$ ratio. Although initially LL1000 was the minimum criteria for obtaining “good fits,” Fig. 4(c) shows that results obtained from LL1000 fits can in fact be scaled up to the true value. Thus Fig. 4(c) provides the ${\tau }_F$ master curve generated from linear regressions of simulation data, from which the value of the true ${\tau }_F$ can be determined for fits at any LL. As an example, for cell data, fits can be carried out at any two LL values, at LL500 to include most of the pixels within cells and at LL1000 to capture a smaller fraction of the pixels. The percentage deviation, 500%dev1000, can be calculated from the fitted ${\tau }_F$ obtained at these two LL values. Referring to the 500%dev1000 master curve, the true${\tau }_F/{\tau }_D$ ratio can be read from the $x$-axis.

This plot corroborates finding in literature that a minimum of LL1000 is sufficient to extract ${\tau }_F$ within 20% of the true value. However, we can actually scale the fitted lifetime to the true ${\tau }_F$, enabling us to calculate a more accurate value for FRET efficiency and thus interfluorophore distance, $r$. The ability to use pixel with lower photon counts further allows us to study regions where fluorescent protein expression is lower.

Though much effort has previously been put into analyzing the accuracy of fitted lifetime values obtained from FLIM data, not much work has been done to examine the variations in the exponential prefactors with various fitting protocols. In this study, we examined the effect of variations in ${\tau }_F/{\tau }_D$ and LL on prefactors, formulated as FR, as previously described.

Fits were carried out at ${\tau }_F/{\tau }_D=0.8$ and varying LL. A scaled histogram of the difference in fit compared to true FR (fitFR – truFR) is shown in Fig. 5(b). The histogram is similar for all LL values except LL100. This is very likely because at LL100, the SNR is below the threshold for obtaining reasonable fit parameters. The histogram is not quite symmetrical about 0 deviation, but is skewed slightly toward negative deviation. A plot of true FR against fit FR at LL1000 [Fig. 5(c)] shows the actual distribution of FR values via a linear regression. For comparison, a dashed line with zero $y$-intercept and slope of 1 is also plotted. A fit that produces results with a symmetrical histogram about 0 deviation would produce a true FR versus fit FR plot with data points scattered about this line. The open circles show the position where the points are most dense (histogram peak). When a line of best fit is drawn through the circles, the parameters recovered are a $y$-intercept of -0.12 and slope of 1.5. If a cell data fit were carried out at LL1000 and the ${\tau }_F/{\tau }_D$ ratio was found to be 0.8 by scaling from the ${\tau }_F$ master curve, the slope and $y$-intercept values would be used to shift fit FR values to true FR values. Figure 5(c) is a plot of slope and intercept for fits at a ${\tau }_F/{\tau }_D$ ratio of 0.8 and varying LL. It shows that the slope approaches 1 and the $y$-intercept approaches 0 with higher LL. This is an obvious trend, as decays with higher total counts have better SNR, leading to more accurate fits.

Fig. 5

(a) Scaled histogram of fit-truFR at a ${\tau }_{\mathit{F}}/{\tau }_{\mathit{D}}$ ratio of 0.8 and varying LL values. Color scheme is as follows: LL100 (dark blue), LL500 (green), LL750 (red), LL1000 (yellow), LL1500 (magenta), and LL2000 (light blue). (b) Plot of true FR versus fit FR at a ${\tau }_{\mathit{F}}/{\tau }_{\mathit{D}}$ ratio of 0.8 and LL1000. Red points: raw data. Plot is pseudocolored by data density; yellow: high density of data; green: low density of data, with a gradation for intermediate values. Solid line: line of best fit through data, dashed line: line with 0 $\mathit{y}$-intercept and slope of 1. (c) Slope (triangle) and intercept (circle) of true versus fitted FR plots at a ${\tau }_{\mathit{F}}/{\tau }_{\mathit{D}}$ ratio of 0.8 and with varying LL. A best-fit line is drawn through the data points based on a third-degree spline fit (using polyfit function of MATLAB) to generate a master curve.

A scaled histogram of fit-truFR at LL1000, with varying ${\tau }_F/{\tau }_D$ ratios, shows more pronounced differences between low lifetime ratios and higher lifetime ratios [Fig. 6(a)]. The histogram skews from more positive differences to more negative differences with increasing ${\tau }_F/{\tau }_D$ ratios. Besides that, relatively more pixels show higher differences with increasing ${\tau }_F/{\tau }_D$ ratios. We can again plot the variation in slope and $y$-intercept with varying ratios, and the plot shows that the slope approaches 1 and the $y$-intercept approaches 0 as ${\tau }_F/{\tau }_D$ decreases [Fig. 6(b)]. This suggests that FR can be extracted more accurately from FLIM data when the lifetimes are more widely separated.

Fig. 6

(a) Scaled histogram of fit-truFR at LL1000 and varying ${\tau }_{\mathit{F}}/{\tau }_{\mathit{D}}$ ratio. Color scheme is as follows: ${\tau }_{\mathit{F}}/{\tau }_{\mathit{D}}=0.25$ (magenta), ${\tau }_{\mathit{F}}/{\tau }_{\mathit{D}}=0.5$ (red), ${\tau }_{\mathit{F}}/{\tau }_{\mathit{D}}=0.7$ (green), ${\tau }_{\mathit{F}}/{\tau }_{\mathit{D}}=0.8$ (blue). (b) Slope and intercept of true versus fit FR plots at LL1000 and varying ${\tau }_{\mathit{F}}/{\tau }_{\mathit{D}}$ ratio. A best-fit spline is drawn through the data points.

With each variation in lifetime ratio and LL, it is possible to deterministically calculate the variation in fit FR. Therefore, it is possible to scale parameters obtained from low LL to true lifetime and FR, based on the slope and $y$-intercept values presented here. Conversely, it is possible to state the variance in true FR at any fit FR value.

It is interesting to examine the underlying reason for the systematic underestimation of the lifetime of the short component and the FRET ratio with lower photon counts. That is to say, for any given decay curve with high photon count, including decay curves with progressively lower photon counts in the fit, ${\tau }_F$ shifts for the whole set of curves to lower values and FR for that decay curve also falls (Table 4). The origin of this underestimation is due to fitting the decay curves using a least-squares function that is an unbiased estimator only for Gaussian noise, whereas the decay curves are dominated by Poisson noise. This behavior can be attributed to the presence of a noise floor. With the lifetime of the long component fixed and fitting to the noise floor at the longer time, the minimization of ${\chi }^2$ results in an overestimation of long lifetime component amplitude that is compensated by a reduction of the short component lifetime and the FRET ratio.

Table 4

Fit results for a single-decay curve with high photon counts when curves with increasingly lower LL values are included in the fit.

An alternative reweighting method to compensate for this bias in least-squares fitting has been developed by Selinger and co-workers and recently by Truton and co-workers.^46–48 It has been shown that using a modified ${\chi }^2$ weighted error function consisting of the values of the fitted curve (instead of the values of the data) in the final fitting step can effectively remove fitting bias. We have adopted this method to analyze some of our simulated data and compared the results with our reference master curve approach. The results are shown in Table 5, and we found that both methods allow recovery of the correct ${\tau }_F$, providing a cross-validation of our approach.

Table 5

Comparison of lifetimes recovered using weighting method with master curve lookup.