A Bayesian method for single molecule, fluorescence burst analysis
Spotlight summary: A modern microscopist will probably agree with the assertion that "Fluorescence imaging is an indispensable instrument in the treatment of nearly every recondite question in modern biology" (with apologies to Lord Kelvin who reckoned that the Fourier's theorem is an indispensable instrument in the treatment of nearly every recondite question in modern physics). Two features of fluorescence imaging are responsible for its wide-ranging utility: (a) it can report the behavior of single molecules, and (b) the linear relationship between the recorded intensity and the number of photons emitted by the fluorophore permits rigorous quantitative analysis of fluorescence data. The current paper by Barber et al. exemplifies the above assertion by developing an excellent approach for identification and quantitative analysis of lifetime of single fluorophores from highly noisy data. Fluorescence lifetime (the time delay after which an excited fluorophore emits a photon) is a sensitive reporter of the microenvironment around the fluorophore—such as changes in pH and presence of an acceptor molecule (a molecule that can accept the photon from the fluorophore) within distances measuring nm. As we approach the single molecule specificity, the diminishing signal-to-noise ratio (SNR) limits the accuracy of measurement. The poor SNR becomes a problem especially when the photons are divided over several temporal bins for measuring the fluorescence lifetime. The current paper develops a statistical analysis approach based on Bayes' theorem that uses the information contained in the arrival times of individual photons.
A reliable approach of ascertaining fluorescence lifetime within a microenvironment (selected by focusing the excitation light with high numerical aperture) is to measure the arrival times of photons emitted by a population of fluorophores with reference to a pulsed excitation. This approach is called time-correlated single-photon counting (TCSPC). The histogram of the arrival times follows a characteristic exponential decay whose time constant is the measure of the lifetime. It is necessary to measure the arrival statistics from a large number of fluorophores having the same characteristic lifetime to obtain a histogram that can be easily fit to an exponential curve. However, the biological environment changes rapidly, and it is usually desirable to reduce the excited specimen volume to the minimum possible size. Thus, there is a fundamental tradeoff in measuring the lifetime of single molecules—if we measure for a short time over small volume, the accuracy of the measurement is limited by the poor SNR, but if we measure for a long time over larger volume, the accuracy of the measurement is limited by the averaging effects. The spotlighted paper shows a robust way of balancing the tradeoff. Instead of fitting an exponential curve to the histogram of arrival times, Barber et al. calculate the Bayesian (conditional or posterior) probability of the burst of photons having certain lifetime given the measured arrival times. They even go one step further, by calculating the conditional probability of a burst of photons being present within a noisy background in the first place. This is a promising way of thinking about the problem of measuring fluorescence lifetime and, as illustrated with simulated and experimental results in the paper, allows the estimation of lifetime from the recordings with the SNR as poor as 0.11. Bayes' theorem provides a fundamental means of estimating parameters of certain physical phenomenon given the analytical model of the phenomenon and the noisy measurements. The authors employ the Poissonian model for the noise and the monoexponential decay model for the photons emitted by the fluorophores to measure the presence of the burst and the lifetime of the burst in probabilistic terms. Equation (1) of the paper forms the basis of the analysis and ties together the model of the signal (photon emission statistics with a characteristic lifetime), the model of noise (Poissonian arrival of the photons), and the probability of the burst being present when certain photon arrival times are measured.
The results of the paper are valuable as they demonstrate that by moving from the deterministic paradigm to the probabilistic paradigm, one can robustly analyze very noisy fluorescence measurements. Contrary to the approach of thresholding and fitting the fluorescence data to a given model, the probabilistic approach will degrade gracefully in its performance as the SNR worsens. It will be interesting to see if similar probabilistic analysis is extended to the other approaches of measuring lifetime—such as phase modulation of the fluorescence detected in response to the excitation that varies sinusoidally over time.
ToC Category: Optical Biophysics
|OCIS Codes:||(000.3860) General : Mathematical methods in physics|
|(180.0180) Microscopy : Microscopy|
|(180.1790) Microscopy : Confocal microscopy|
|(180.2520) Microscopy : Fluorescence microscopy|
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