Fig 1.
Photon arrival-time histograms are composed of the sum of two exponential distributions.
(A) Photon arrival histogram composed of two exponential distributions, with a short-lifetime fraction fS, a long-lifetime fraction fL, and a background fraction fB = (1 − fS − fL) (B) Inferred posterior distribution generated from data in Fig 1A.
Fig 2.
(A) Control dyes having known long (Coumarin 153) and short (Erythrosin B) lifetimes were mixed at a fixed ratio. From the measured master curve of photon arrival times, a variable number of photons are randomly sampled, generating histograms with a variable number of photons. (B) Bias in the estimated short-life photon fraction, fS, decreases with increasing photon number. Data points represent the average of the posterior mean (squares) or mode (circles) for 300 independent samplings for each photon count. Error bars are s.e.m. (C) Black circles: measured sample standard deviations from data in Fig 2B averaged across the 300 independent samplings. The sample standard deviation decreases approximately as . Power law fit to a × xb for all but the four lowest values of nphoton shown in gray, with a = 0.04 ± 0.01 and b = −0.48 ± 0.04 (95% confidence interval).
Fig 3.
(A) Samples of dyes with short-lifetime (Erythrosin B) and long-lifetime (Coumarin 153) were prepared, and fluorescence lifetime measurements were collected for each dye separately, leading to separate master photon histograms. Test histograms were constructed by randomly sampling a fixed number of photons, with varying fractions being drawn from the master lists of short-lifetime and long-lifetime photons. These histograms were then analyzed in order to estimate the fraction of short-lifetime photons, fS. (B) The estimated short-lifetime fraction, fS, varies linearly with the constructed short-lifetime fraction for three different total photon numbers, with a small offset. Squares: estimate from posterior mode. Dots: estimate from posterior mean. Dashed lines: linear fits with slopes 0.9933 ± 0.0026, 1.0085 ± 0.0024, and 1.0106 ± 0.0031, and offsets of 0.002000 ± 0.0484 × 10−4, 0.004400 ± 0.0814 × 10−4, and 0.009500 ± 0.1992 × 10−4 for low, medium, and high intensities respectively (95% confidence interval). Intensities correspond to data collected at ≈1.5 × 105, 1.2 × 106, and 4.8 × 106 counts per second, for low, medium, and high intensity respectively. Inset: Data from main figure shown on a log-log scale. (C) Changes in the estimated short-lifetime fraction track the known changes in the short-lifetime fraction. Squares: estimate from posterior mode. Dots: estimate from posterior mean. Dashed lines: Linear fits with slopes of 0.9573 ± 0.1895, 1.0013 ± 0.1445, and 0.9580 ± 0.1717, and offsets of (0.2332 ± 0.3712) × 10−5, (0.3215 ± 0.3088) × 10−5, and (0.4617 ± 0.4286) × 10−5 for low, medium, and high intensities respectively (95% confidence interval). (D) Sample standard deviations decrease with increasing photon number as Squares: Posterior standard deviation Dashed line: power law fit to all intensities with exponent −0.4764 ± 0.0471 (95% confidence interval).
Fig 4.
(A) FLIM images depicting the long-lifetime fraction from measurements of mTurquoise2 in a U2OS cell. Photons were pooled from pixels grouped using boxcar windowing into groups of either 3 × 3, 7 × 7, or 11 × 11 pixels and analyzed using either the Bayesian analysis presented here, or least-squares fitting. (B) Histograms showing the probability density of the long-lifetime fraction from images in (A). The probability density functions from Bayesian analysis were found to have mean values of 0.648 ± 0.069, 0.642 ± 0.037, and 0.640 ± 0.026 (mean ± s.d.) for 3 × 3, 7 × 7, and 11 × 11 binning respectively, while the means values were found to be 0.558 ± 0.137, 0.648 ± 0.050, and 0.652 ± 0.034 (mean ± s.d.) for 3 × 3, 7 × 7, and 11 × 11 binning respectively using least-squares-fitting.