A Bayesian method for inferring quantitative information from FRET data

Abstract

Background: Understanding biological networks requires identifying their elementary protein interactions and establishing the timing and strength of those interactions. Fluorescence microscopy and Förster resonance energy transfer (FRET) have the potential to reveal such information because they allow molecular interactions to be monitored in living cells, but it is unclear how best to analyze FRET data. Existing techniques differ in assumptions, manipulations of data and the quantities they derive. To address this variation, we have developed a versatile Bayesian analysis based on clear assumptions and systematic statistics.

Results: Our algorithm infers values of the FRET efficiency and dissociation constant, Kd, between a pair of fluorescently tagged proteins. It gives a posterior probability distribution for these parameters, conveying more extensive information than single-value estimates can. The width and shape of the distribution reflects the reliability of the estimate and we used simulated data to determine how measurement noise, data quantity and fluorophore concentrations affect the inference. We are able to show why varying concentrations of donors and acceptors is necessary for estimating Kd. We further demonstrate that the inference improves if additional knowledge is available, for example of the FRET efficiency, which could be obtained from separate fluorescence lifetime measurements.

Conclusions: We present a general, systematic approach for extracting quantitative information on molecular interactions from FRET data. Our method yields both an estimate of the dissociation constant and the uncertainty associated with that estimate. The information produced by our algorithm can help design optimal experiments and is fundamental for developing mathematical models of biochemical networks.

Figure 1

Overview of interaction and photophysical model. (A) In the underlying interaction, a donor-tagged protein binds an acceptor-tagged protein with dissociation constant K_d. (B) During a FRET experiment, photophysical events occur such as FRET, which occurs between donors and acceptors in complex with efficiency E_fr, and the excitation of and emissions from donors and acceptors, where the factor relating fluorescence observed in channel i to the concentration of species S is . We include all possible spectral overlap effects, allowing the donor and acceptor to be excited and emit photons in all spectral channels.

Figure 2

Analysis of typical FRET data recovers true values of K_d and E_fr. A typical three-cube FRET experiment is simulated from three virtual cells, each containing the indicated concentrations of donor- and acceptor- tagged proteins (A, left). The data is summarized in bar graphs with mean ± SD (A, right). Ten measurements/channel are made for each cell with 5% added measurement noise. For other parameters, see Methods. The data was analyzed in two ways to find the values of K_d and E_fr consistent with the data. We first calculated the energy for an array of parameter values; as the energy contour plot shows (B), the energy was minimal near the true parameter values. We also used a Monte Carlo algorithm to explore K_d-E_fr space, running 3 biased random walks starting from different initial positions (white *) and running for 12,000 steps. The paths of the walks (for clarity only the first 2,000 steps are shown) are superimposed on the contour plot (B) and all three converged to a region around the true value which coincides with the energy minimum. Histograms of the locations visited by all three walks, including only post-convergence steps (11,000 steps are included from each walk) act as approximate posterior distributions for K_d and E_fr (C). The dashed red lines indicate the true values of K_d and E_fr used to generate the data.

Figure 3

K_d estimates reflect data quality and quantity. We simulated and analyzed data with varying levels of measurement noise and numbers of measurements/cell/channel. The locations visited by MCMC walks (dots) for the three noise levels (A) and the three numbers of measurements/cell/channel (B) show that the highly probable region grows as measurement noise increases and as the number of measurements decreases. The 'True Value' (black and white spot) indicates values used to generate the data. Histograms of the locations visited by the walks (insets, histograms smoothed for clarity) approximate the corresponding posterior probability distributions for K_d, with true values indicated by black lines. The plots of error vs. measurement noise (C) and error vs. amount of data (D) illustrate that, in general, accuracy decreases with increasing noise and decreasing number of measurements although the mean of the error remains centered at zero. Even when the true value is unknown, the relative uncertainty of the parameter estimate (coefficient of variation of locations visited by a walk) is measurable; it also grows with increasing noise (E) and decreasing number of measurements/cell/channel (F). In (C-F), error bars are mean ± SD. 50 data sets were analyzed for each noise level or number of measurements, and each dataset was analyzed once with a random walk running for 20,000 steps, starting once the walk converged. Except when otherwise indicated, data had 10 measurements/cell/channel and 5% added Gaussian noise. For other parameters, see Methods.

Figure 4

Concentration variations determine shape of region of high probability. Using the Monte Carlo algorithm to analyze data from a single cell containing [D₀] = [A₀] = 1 μM produced an elongated region of highly probable values (A), indicated by the collection of dots showing the locations in (K_d, E_fr) visited by the biased random walk. These single-cell results are shown again in (B-E) (blue dots), with the results from analyzing other data sets superimposed. Analyzing three identical cells instead of one produced a narrower but still elongated region (B). However, when analyzing data from two cells (C and C inset) and three cells (D) with different concentrations of fluorophores, the resulting regions were contracted. Analyzing cells individually showed that the elongated highly probable regions for each all intersected near the true value (E). More extreme variation in concentrations led to an even smaller optimal region (F). In (A-F), there were 10 measurements/cell/channel, 3% added Gaussian noise, and 31,000 steps are shown for each walk. For other parameter values, see Methods.

Figure 5

Gaining insight into optimal experimental design. The approximate posterior probability distributions for K_d (A) have different shapes if the data analyzed was simulated from three cells containing equal concentrations of donors and acceptors which are much lower than (left), higher than (right) or about equal to K_d (centre). For the data analyzed in the left panel, for instance, the cells contained the concentrations [D₀] = [A₀] = 0.2·10^-3 μM, [D₀] = [A₀] = 1·10^-3 μM, and [D₀] = [A₀] = 5·10^-3 μM. Insets show amount of complex formed as a function of K_d for the indicated concentrations, demonstrating that where complex formation is insensitive to K_d corresponds to plateaus in the posterior probability distributions for K_d. In each plot (or inset), a vertical dashed line (or red circle) indicates 10^-6M, the true value of K_d. (A) had 36,000 steps/walk and 5% added noise. Exploring another aspect of fluorophore concentrations, increasing the ratio [D₀] : [A₀] increases the uncertainty in fitting K_d (B). As the ratio was increased (by keeping [D₀] constant for the three cells at 0.2·10^-6M, 1.0·10^-6M and 5.0 10^-6M while decreasing [A₀] according to the ratio), posterior probability distributions for K_d broadened (true value indicated by dashed vertical line). Insets show data used for fitting (bars marked 'E_fr = 0.4') from the donor channel (left) and FRET channel (right) contrasted with data from the same cells simulated with E_fr = 0, demonstrating that the relative contribution of FRET decreases as [D₀] : [A₀] increases. (B) had 50 measurements/cell/channel, 36,000 steps/walk and 3% added noise. Bars show mean ± SD. For other parameters, see Methods.

Figure 6

Prior information on E_fr improves estimate. Approximate posterior distributions for E_fr (upper panels) and K_d (lower panels) obtained from using the MCMC algorithm to analyze the same datasets in the absence (red) or presence (blue) of prior knowledge about E_fr show that prior knowledge helps to improve accuracy and reduce uncertainty of the estimate. True values are indicated by vertical dashed lines and the inset shows the prior distributions used (red is a uniform distribution and blue is a normal distribution centred at 0.45 with a variance of 0.15). The plots on the left show results from analyzing data where 10 measurements were made in each channel for each of 3 cells with [A₀] = [D₀] = [.2, 1, 5]·10^-6M. The plots on the right show results from data from the same cells, but with just 3 measurements/channel. 20,000 steps were recorded for each biased biased random walk, with 5% added noise. For other parameter values, see Methods.