Maximum likelihood estimation of FRET efficiency and its implications for distortions in pixelwise calculation of FRET in microscopy

Abstract

Ratiometric determination of the efficiency of fluorescence or F€ orster resonance energy transfer (FRET) is one of the most widespread methods for the characterization of protein clustering and conformation. Low photon numbers, often present in pixel-by-pixel determination of FRET efficiency in digital microscopy, result in large uncertainties in the derived FRET parameter. Here, we propose a method based on maximum likelihood estimation (MLE) of FRET efficiency using photon counting detectors to overcome this limitation. Intensities measured in the donor, FRET, and acceptor channels were all assumed to follow Poisson statistics as a result of detector shot noise. The joint probability of photon numbers detected in the donor, FRET, and acceptor channels was derived using an equation describing the relationship between the three measured intensities. The FRET efficiency generating the measured photon numbers with the largest likelihood was determined iteratively providing a single FRET value for all pixels in the calculation. Since as few as 100 pixels are sufficient to provide a maximum likelihood estimate for FRET, biological variability in FRET values can be revealed by performing the analysis for regions of interests in an image. Since the algorithm provides the probability of a combination of donor, FRET, and acceptor intensities observed in each individual pixel given a certain FRET efficiency, outlier pixels with low probabilities could be excluded from the analysis. Simulations carried out with low photon numbers in the presence and absence of outlier pixels revealed that the proposed approach can reliably and reproducibly estimate FRET efficiency. In addition, systematic evaluation of the simulation results showed that the distribution of pixel-by-pixel FRET efficiencies is skewed, and the mean of these FRET values is a biased and unreliable estimate of the FRET efficiency. In the absence of outlier pixels, FRET calculated from summed donor, FRET, and acceptor intensities proved to be as reliable as MLE. We conclude that MLE of FRET outperforms calculations using summed and pixel-bypixel intensities in biologically relevant situations involving low photon numbers and outlier pixels.

rFRET: A comprehensive, Matlab‐based program for analyzing intensity‐based ratiometric microscopic FRET experiments

“…RiFRET (6), pixFRET (9)), only the pixel-by-pixel calculation approach is implemented. As demonstrated previously (13,14) the pixelwise calculation method is the most sensitive for distortions introduced by outlier pixels. This drawback is the most prominent at low signal-to-noise ratios.…”

“…MLE, taking the Poissonian nature of photon detection into consideration explicitly, has been shown to be a powerful approach for estimating both overspill parameters and the FRET efficiency (13). In most other applications, for the evaluation of ratiometric microscopic FRET experiments (e.g.…”

“…Once the overspill parameters and a have been determined, the user can choose from three different methods to calculate the FRET efficiency: (a) pixel-by-pixel; (b) FRET calculated from summed intensities; and (c) maximum likelihood estimation (13). After defining the input images recorded in the donor, FRET and acceptor channels the mask image must be chosen and background correction must be performed.…”

“…If the signal-tonoise ratio is low, the spread of calculated parameters as a result of measurement error related to the statistical nature of photon detection may be significant. In these cases, the parameter distribution is skewed, and consequently the mean is an unreliable estimate for the central tendency (13). Therefore, it is important to look at the distribution of parameters to appreciate these factors.…”

“…Although simple histogram-based thresholding is available in some applications, this approach has severe limitations if the signal is weak relative to the background. Only pixelwise calculation of overspill factors and the FRET efficiency is usually available, although it has been shown that regression-based and maximum likelihood estimation (MLE) of these parameters as well as the use of summed fluorescence intensities is superior to pixelwise calculation if the signal-to-background ratio is low (13,14). The capability of the available programs to calculate parameter a is limited although many such methods have been published (7,8,(15)(16)(17).…”