Biochemical assays can identify potential protein–protein interactions in cell extracts, but studying these interactions in living cells is a more challenging task. The most promising approach for the measurement of molecular interaction dynamics exploits the energy transfer between fluorophores over short distances (Fluorescence Resonance Energy Transfer; FRET) (1). This transfer occurs only if both molecules are in close proximity to one another (typically less than 10 nm) (2).
Several strategies are used to obtain this measurement which are mainly based on fluorescence intensity (3), anisotropy (4), and lifetime (5) measurements. The latter is rather accurate, especially when using a Time Correlated Single Photon Counting (TCSPC) system (6) and an instrumental setup with a narrow Instrumental Response Function (IRF) (7). This spectroscopy technique can either be used for the measurement of fluorescence lifetime of selected areas or for the acquisition of complete images; in which case it is referred to as FLIM (Fluorescence lifetime imaging microscopy). TCSPC is an indirect method that allows determination of the fluorescence lifetime species by fitting measured photon decay curves using the following equation:
With “b” the background level, “IRF”, IRF of the acquisition system, “i” an index associated with each exponential, “a” the proportion of each component and “τ” the corresponding lifetime.
A mean fluorescence lifetime defined as in Eq. (2) is used to facilitate comparison between experiments (8),
With “a” the proportion and “τ” the lifetime of each fluorescent component.
In general, fluorescence lifetime analysis is performed based on the Least squares method relying on the iterative minimization of the χ2 parameter (9, 10). Parameters that according to Marquardt (11) describe the difference between the model and the measured data are:
With “i” the time channel, “V” the photon number, “b” the background, “IRF”, IRF of our system, “j” an index associated with each exponential, “a” the proportion of each component and “τ” the corresponding lifetime.
Using a least squares based fit procedure implies the choice of the fluorescence species used in Eq. (1). This choice is usually achieved based on the fit curve residual distribution.
To accurately calculate the fluorescence lifetime, the choice of “n” in Eq. (1) is essential as demonstrated in Figure 1. On one hand, a missing exponential term leads to an overestimation of fluorescence lifetime (Fig. 1A). On the other hand, an overestimation of the model's degree of freedom, i.e. exponential terms results in instability of the fitting algorithm (12), and so, in higher dispersion of estimated lifetimes (Fig. 1B). The Occam's or Ockham's razor (13), also referred to as parsimony law, states a preference for simple theories. Consequently, the fit model that describes the photon decay curve with the lowest number of exponentials is preferred.
The choice of the optimal fit model based on the residual statistics has a strong theoretical foundation in literature of both statistics and information theory fields (14, 15). However, the information that is crucial to apply this theory is not easily accessible from the FLIM analysis software. Moreover, it cannot be directly used for the generation of lifetime images.
In this article, we describe an easy-to-use analysis procedure based on the χ2 variation that allows for best model choice on a pixel-by-pixel basis. It uses information available in all FLIM analysis systems without complex modification of the fitting algorithm. We demonstrate its robustness throughout the analysis of series of simulated photons decay curves. We then show the improvement gained when applied to FRET investigation in living cells.
MATERIALS AND METHODS
The memb-eGFP-mCherry was constructed using the memb-mCherry (16) which is derived from the PM-eGFP (17). We inserted, upstream of the mCherry coding sequence using standard molecular biological techniques, a hydrophilic linker (18) and the necessary restriction enzyme cutting sites for the in-frame subcloning of the eGFP fragment derived from the pEGFP-1 (Clontech). Plasmid clones were propagated in XL1 blue (Stratagene) and checked by sequencing.
For Rab6 and Rab11 plasmid constructs see (19) and (20), respectively.
Cell Culture and Transfection
A HeLa stable cell line expressing eGFP was generated after transfection of 1 μg of pEGFP-1, using 3 μl of transfectin (Bio-Rad). Clones were selected by repeated flow cytometry sorting.
HeLa cells were grown in plastic flasks at 37°C in 5% CO2 in Dulbecco's modified Eagle's medium (GIBCO/BRL) supplemented with 10% Fetal Calf Serum, 4 mM L-glutamine and 5 mM sodium pyruvate. Cells were plated on 32 mm diameter glass coverslips 12 h prior to transfection. One hour before transfection, culture medium was removed and replaced with culture medium containing 100 units/ml penicillin/streptomycin. Transient transfections of Rab plasmids were performed by calcium phosphate method on native HeLa cells (21).
FuGENE HD was used for the transfection of memb-eGFP-mCherry on HeLa cells stably expressing eGFP. Observations were conducted 24 h after transfection. For FLIM imaging, culture medium was replaced by L15 medium (Invitrogen) for pH stabilization during experiments.
Two-Photon Fluorescence Lifetime Microscopy
The Time Correlated Single Photon Counting (TCSPC) FLIM system was built around a Leica SP2 confocal Microscope (Leica Microsystems), a pulsed laser source (Ti: Sapphire with a 5 W Verdi pump laser, Mira900-F, Coherent Inc.), a detector with high temporal resolution (MCP PMT model R3809U-52, Hamamatsu) and a dedicated photon-counting and timing electronic card (SPC 730 TCSPC card, Becker & Hickl). For more details about the system implementation and characterization, see (7).
Monte Carlo Simulation of Photon Decay Curves
To assess the curve fitting estimation methods, a number of data set or photon histograms with controlled/known parameters were generated using a Monte Carlo approach.
In brief, the method consisted of drawing in a random number generator as many discrete time values as the desired number of photons in the final data set. The density probability function of the random generator corresponded to the theoretical decay curve of the simulated data set. To generate a data set that mimics the instrument data acquisition characteristics, we added uniformly distributed false photons to recapitulate instrument noise. We also took into account that the tail of photon decay curves could exceed laser pulse period and consequently some photons might be collected in the wrong time channels (data wrapping). This was solved by implementing a modulo function that wrapped around the generated data set.
The controlled parameters were:
n the number of time channels (1024 bins) in the photon histogram,
th the time length of a histogram bin, r the pulse repetition period (12 ns),
tp the start of the pulse,
N the total number of photons,
IRF the instrument response function,
m the number of exponential decay functions fi such that fi(t) = ηi exp(−(t−tp)/τi) when t ≥ tp and fi(t) = 0 otherwise, with ηi the proportion and τi the decay parameter,
Nf the number of false photons due to instrument noise.
Let's call D, the decay curve such that D(t) = IRF ⊗ ∑fi(t) where ⊗ denotes convolution.
The algorithm for data sets generation was the following:
1Set all histogram bins to zero and variable count to zero,
2Within a range from 0 to 1, draw a value x from a random number generator with an uniform probability,
3Find the tx value so that the normalised integral of D from 0 to tx equals x,
4Calculate the bin number b ← integer[(tx mod r)/th], where mod denotes modulo,
5If b < n then add a photon in bin b of the photon histogram and increment count,
6Repeat steps 2 to 5 until the counter equals N-Nf.
7Within a range from 0 to 1023, draw an integer number k from a random number generator with a uniform probability,
8Add a photon in bin k of the photon histogram and increment count,
9Repeat steps 7 to 8 until count equals N.
Photon Decay Curve Analysis
Lifetime images were analyzed by fitting data with a mono and bi-exponential function [see Eq. (1)] using the SPC Image software (Becker & Hickl). First, for each point of the curve, fi(aj,τj) was calculated according to Eq. (4).
The software then iteratively modifies aj and τj in order to minimize the χ2 parameter [Eq. (3)].
Following analysis, fluorescence lifetimes, proportions and χ2 values are exported in ASCII format and processed using the MFS homemade ImageJ plug-in (22), available as a freeware.
Our aim was to optimize the fitting method, based on the Occam's razor or parsimony law, which implies that an additional exponential in Eq. (1) is only needed if it generates a significantly improved solution. Although the natural tendency of a fitting algorithm is to increase the number of variables in response to the system's fluctuation, it is detrimental to the overall understanding of the system. Therefore, implementing a fitting quality criterion is essential and would render both data analysis and biological interpretation, accurate and relevant, respectively. Here, we propose to ground this fitting quality criterion to the χ2 variation as a function of the incremental number of exponential terms.
We deliberately limited our demonstration to the choice between a mono and bi-exponential model since it is representative of most interaction texture studies in living cells. It could however be easily extended to a higher number of exponentials terms.
Δχ2 Definition and Threshold Determination
Although the χ2 value is dependent on both the relevance of the exponential model and the total photon number, a threshold value may not be directly generated from these parameters to choose the best model. However, the difference between the χ2 after mono-exponential and bi-exponential fit is only derived from the number of species included in the model, as illustrated in Figure 2A. For instance, inaccuracy linked to proper IRF estimation and convolution is the same regardless of the number of species in the model. Thus, the difference between χ2 after mono-exponential and bi-exponential fit results from the relevant usage of each exponential model.
We next proposed to use the variation termed Δχ2 in Eq. (5) as a fitting quality criterion in order to ascribe the most adapted exponential model.
With “χ” and “χ” (2) being χ2 value obtained respectively after a mono-exponential or a bi-exponential fit applied to the photon decay curve.
To test the relevance of this parameter as a fitting quality criterion, we performed Monte Carlo simulations of photon decay curves with different total photon numbers and exponential component proportions (see MATERIALS AND METHODS section). For each of these curves, the Δχ2 was calculated. We then tested different values of Δχ2 as a discriminating criterion to determine the optimal fit model. Results from these simulations are summarized in Table 1.
Table 1. Frequency of occurrence when the wrong model is selected; depending on the number of photons and Δχ2 threshold [Color table can be viewed in the online issue, which is available at www.interscience.wiley.com.]
Simulations were performed on a mix of τ1 = 0.6 ns and τ2 = 2.4 ns with different proportions (0, 25, 50, 75, and 100%) and a noise ratio of 100. 30 simulations per condion.
To accurately quantify the fluorescence lifetime, the following requirements should be fulfilled: a minimum photon number per curve and an appropriate Δχ2 threshold. When setting the Δχ2 threshold to 20%, errors could be found in less than 2% and 12% of the cases for acquisition respectively collecting at least 105 and 104 photons per curve. When considering the usual amount of photons collected during FLIM acquisitions on living cells, the above Δχ2 threshold value leads to both optimal accuracy and reproducibility.
Robustness of Δχ2 as a Discriminating Factor
To further investigate the relevance of the Δχ2 threshold, we tested it on a series of photon decay curves, generated via Monte Carlo simulations, with variable lifetimes, proportions and signal-to-noise ratios values.
Firstly, we performed an extensive study by mimicking different behaviors of fluorophores solutions through the mixture of simulated fluorescent lifetimes of τ1 = 0.6 ns and τ2 = 2.4 ns as shown in Figure 2. These values were chosen because they are representative of the ones obtained in classical FRET studies, using the most common FRET pairs (i.e. CFP-YFP, GFP-mRFP, and their variants). The first step was to assess whether any advantages could be drawn from a Δχ2 study over a regular χ2 study. Data from Figure 2A clearly showed that no χ2 value could be found to classify experiments between mono-exponential and bi-exponential curves. However, all mono-exponential curves exhibited a Δχ2 below 10%, while bi-exponential curves resulted in a Δχ2 greater than 40%. Thus the previously chosen fitting quality criterion, Δχ2, set to 20% was hereby validated since it provided the optimal exponential model choice. Confirmation came from Figure 2B where fluorescence lifetimes obtained after a model choice based on a Δχ2 = 20% were consistent with expected theoretical lifetimes. In addition, the calculated mean lifetime benefited from both, mono-exponential fitting stability, and bi-exponential accuracy.
Secondly, we determined the robustness of the Δχ2 criterion on simulated photon decay curves with variable lifetimes and signal to noise ratios.
50%/50% mixes of different lifetime (Figs. 3A1 and 3A2)
50%/50% of τ1 = 0.6 ns and τ2 = 2.4 ns with different signal to noise ratio (Figs. 3B1 and 3B2).
In all cases, Δχ2 enabled the optimal model choice while keeping well above the 20% threshold value. The robustness of the fitting quality criterion was validated, and applied to live cell FRET measurements in classical fitting configurations encountered in FLIM experiments.
The different characterization steps enabled us to propose an algorithm, Liχ, based on Δχ2 value, which is applicable to lifetime images. It is integrated in MFS, a Java plug-in for ImageJ.
Liχ is divided into four steps:
The photons/pixel threshold should be set to 103 to exclude background pixels.
The photons/pixel threshold should be set to 104. Based on both simulations and theory, only a mono-exponential fitting is reliable (17).
Then, for each remaining pixel, the Δχ2 is calculated [Eq. (5)]: If Δχ2 is less than 20%, a mono-exponential fitting is chosen. If Δχ2 is more than 20%, a bi-exponential is more reliable.
Finally, fluorescence lifetime values from previous steps are gathered into a single lifetime image.
Extending Liχ to more than two exponentials is easily feasible. In order to do so, one needs to fix a new threshold value for each additional exponential term and to calculate the Δχ2 between “n” and “n + 1” exponentials. Then, every image calculated from each cycle should be gathered into one.
Application to Enhance Live Cell Lifetime Imaging
Upon Liχ validation with Monte Carlo simulated photon decay curves, we tested the algorithm by analyzing HeLa cells stably expressing eGFP and transiently transfected with a membrane targeted construct expressing eGFP and mCherry in tandem (see MATERIALS AND METHODS section for details). This plasmid was used as positive FRET reference (Fig. 4).
This experiment enabled the acquisition of images exhibiting an heterogeneous photon count, with mono-exponential areas, characterized by signal in the green channel only (Fig. 4A, cell No. 1), and bi-exponential areas characterized by signal from both green and red channels (Fig. 4C, cell No. 2 and No. 3). When considering the fitting residuals (Fig. 4B), the random distribution after mono-exponential fit obtained in cell No. 1 and No. 2 confirmed the relevance of such a model choice while the non-random distribution obtained from cell No. 3 suggested a lack of exponential species in the model. The automated Liχ based fit model choice presented in Figure 4D revealed an analyzed image that consistently took into account the above observation: a mono-exponential fit was applied to cell No. 1 and No. 2, and a bi-exponential fit to cell No. 3. Most important, these results were achieved without the need for a fastidious pixel by pixel manual inspection of the fitting residuals. The resulting contrasted fluorescence lifetime variation image is presented in Figure 4E.
Finally, we applied Liχ to the analysis of HeLa cells co-expressing Rab6A-CFP, Rab11A-YFP and a non-tagged Rab6IP1 (21). These proteins are localized in the dispersed Golgi apparatus upon nocodazol treatment (Fig. 5). Interactions between Rab6A and Rab11 triggered by Rab6IP1, which occur only in some vacuoles, provide a good example of interactions heterogeneity in living cells.
A classical mono-model fitting (mono or bi-exponential model applied to the whole image) yielded fluorescence lifetime images shown in Figures 5A and 5B, resulting in errors in interaction interpretations. However, using our algorithm, we generated a fluorescence lifetime image (Fig. 5C), containing areas with one or two fluorescent species. The lifetime distribution given for each pixel (Fig. 5D) underlined the mis-interpretation resulting from a mono-model analysis. Indeed, the mono-exponential analysis showed a single population, where the bi-exponential model exhibited a bimodal distribution but with wrong proportions. Figures 5E and 5F present photon decay curves (blue) with associated mono-exponential (Figs. 5E1 and 5F1) or bi-exponential (Fig. 5E2 and 5F2) fitting curves (red) and fitting residuals (black). Photon decay curves presented in Figure 5E were extracted from a bi-exponential area in accordance with both Liχ and fitting residuals analysis (red arrow). Obviously, there was a degree of freedom missing in Eq. (1) after a mono-exponential fit (Fig. 5E1) when compared to a bi-exponential fit (Fig. 5E2). In contrast, photon decay curves, gathered from a mono-exponential area according to Δχ2 (green arrow), were perfectly adjusted with a mono-exponential model (Fig. 5F1), while a bi-exponential fit improved neither the χ2 value nor the fitting residual (Fig. 5F2).
Fluorescence lifetime imaging is a powerful tool for localizing interactions in living cells. However, interaction texture studies become increasingly interesting in the case of heterogeneity, which implies different behavior of photon decay curves. FLIM analysis software, such as SPCImage, only proposes a mono-model analysis, which dramatically decreases the lifetime contrast and leads to erroneous interpretation of FRET measurements. Our algorithm, Liχ, which is based on the calculation of the Δχ2 parameter, greatly improves the information that can be extracted from lifetime images.
Optimal conditions were reached, for both simulated and measured photon decay curves, when the Δχ2 threshold was set to 20% and the number of photons was greater than 103 photons per curve. In this context, the sole obstacle to information extraction is the potential application of a bi-exponential analysis to photon decay curves with poor statistics. This became quite obvious upon analysis of individual lifetimes of either mix with low contribution of the longest lifetime (less than 25% of the long lifetime component in Fig. 1), or mixtures of simulated fluorescence species with respective lifetimes of 1.6 ns and 1.4 ns. A mono-exponential equation offered the best result as a bi-exponential fit was not able to extract the appropriate lifetime and proportion and only induced a higher instability of the fitting algorithm.
Such an example is provided by cell No. 2 (Fig. 4). Indeed, the presence of both eGFP (stable cell line) and memb-eGFP-mCherry (signal in the red channel) implied the presence of two fluorescent species, and thus suggested a bi-exponential behavior. Considering the mean photon number per curves (∼2.104), and in view of the fitting residual and Liχ based choice, a mono-exponential fit was more appropriate in this context. In addition, a bi-exponential fit did not lead to a proper estimation of individual lifetime and proportions, and resulted in higher instability of the fitting algorithm. When a higher number of photons was collected (Fig. 4, cell No. 3, ∼4.104 photons per curve), a bi-exponential fit was needed to calculate the proper mean lifetime and allowed the determination of proportion of interacting molecules (∼60%), as well as lifetimes from both eGFP fluorescent species (∼1.6 and ∼2.5 ns).
As shown in Figure 5, the use of Liχ also provided great improvement in the mean lifetime estimation. We observed a significant difference between mean lifetimes calculated from mono-exponential or bi-exponential fits which led to different interaction interpretations. Moreover, the examination of the residual shape of fitted pixel, revealed and emphasized that different models should be applied. For instance, the pixel pointed out by the red arrow (Fig. 5) required a bi-exponential fit whereas the one indicated by the green arrow suffered a mono-exponential fit. For these two pixels, mono and bi-exponential fits resulted respectively in a fluorescence lifetime of 2.22 ns and 1.76 ns for the red arrow and of 2.50 ns and 2.34 ns for the green arrow, corresponding to an error of 0.46 ns and 0.16 ns, respectively.
In summary, this easy-to-use procedure, based on the application of the Δχ2 fitting quality criterion within the Liχ algorithm provides an accurate fit model decision on a pixel by pixel basis, ensuring a robust interpretation of interacting populations, which would not be possible with a mono-model approach. This validated approach opens a new way towards interaction texture studies in heterogeneous biological samples, which mostly need quality fluorescence lifetime imaging analysis.
We kindly thank François Waharte and Nicolas Dross for providing biological applications and Jim Smith for his generous gift of the pCS-memb-mCherry plasmid.