Department of Physiology, McGill University, 3655 Promenade Sir William Osler, Montreal, Quebec H3G 1Y6, Canada

Centre for Systems Biology at Edinburgh, University of Edinburgh, Edinburgh EH9 3JD, UK

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, _{d}
_{d}

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.

Background

Proteins work together continuously in the cells of all living things, generating cascades of reactions that are vital for life. To fully understand each individual protein's task requires discovering the timing, location, and strength of its interactions. To acquire this detailed information, fluorescence microscopy methods are ideal because they can provide dynamic, single-cell data at high spatial resolution

FRET is a physical process where a molecule in an excited energetic state (the donor) transfers energy to a nearby ground-state molecule (the acceptor). The chance that an excited donor will transfer its energy to an acceptor (known as the FRET efficiency, _{fr}

When FRET is used to study protein interactions in living cells, the proteins under investigation are fused to fluorescent tags (often variants of the green fluorescent protein) that act as the donors and acceptors. When the proteins interact, they bring the fluorescent tags together so that FRET may occur. FRET increases the number of photons emitted by acceptors and reduces both the number of photons emitted by donors and the donor's fluorescence lifetime. To observe these effects, the most common techniques for collecting FRET data include fluorescence lifetime imaging (FLIM) and using a fluorescence microscope or spectrofluorometer to record fluorescence intensity after exposing samples to light that mainly excites either donors or acceptors

For FRET data to reveal information about the underlying protein interactions, complicating factors must be dealt with. One confounding issue is spectral contamination, which arises from the requirement that the donor and acceptor must have overlapping spectra for FRET to occur. Due to the overlap, excitation light intended to excite one fluorophore may also excite the other (spectral crosstalk), and, conversely, one fluorophore may emit photons in the emission range of the other (spectral bleed-through). Several FRET analysis methods address this issue by calculating a FRET index, which is the FRET signal corrected for contamination from spectral overlap and normalised by donor or acceptor concentrations _{d}

Another significant challenge is that both the _{d }
_{fr}
_{fr }
_{fr}
_{d }
_{fr }
_{d }
_{fr }

Given these different approaches, it is not always obvious which one should be applied in different situations and there is no consensus on the statistical analysis, with each method processing the data differently and most giving no procedure to test the reliability of any estimates. A general method for inferring the _{d }

Here we propose a systematic analysis method that explicitly includes models of the photophysics and underlying chemical interactions and of measurement noise. Building on a spectral model for FRET _{d}

Results and Discussion

Algorithm

Overview of the mathematical model

FRET enables the study of molecular interactions in diverse settings. To design a widely applicable analysis technique, we consider a general system containing proteins (or other molecules) that form complexes and are labelled with fluorophores that act as FRET donors and acceptors (Figure

Overview of interaction and photophysical model

**Overview of interaction and photophysical model**. (A) In the underlying interaction, a donor-tagged protein binds an acceptor-tagged protein with dissociation constant _{d}_{fr}

Our model relies on a few other assumptions. First, we assume that donors and acceptors may be free or form bimolecular, donor-acceptor complexes (i.e. [_{fr }

For our general mathematical description of the FRET system, we use a previously described spectral mixing model

where **C **is a vector of the concentrations of fluorophores (**C **= ([^{T }
**I _{obs }
**is a vector of the fluorescence intensities observed for each spectral channel, and

The observed fluorescence intensities could represent data obtained from fluorescence microscopy, spectrophotometry or a flow cytometer. Microscopy images would require standard image processing steps for quantification, such as subtracting background fluorescence and defining regions of interest such as areas within the cytosol or nucleus for localized fluorescence

Photophysics in the mathematical model

The matrix

The spectral mixing framework can represent any number of spectral channels by increasing the dimensions of the matrix, but we focus on the particular case of three, which describes the most common 'three-cube' FRET experiments

where

Here the subscript _{
i
}is the illumination intensity for the excitation wavelength of channel ^{(S) }is the quantum yield of species (

To illustrate Eq (2), consider the expression it yields for

Continuing the matrix multiplication yields analogous expressions for

Molecular interactions in the mathematical model

To include _{d }
_{0 }] = [_{0}] = [_{0}], [_{0}], and _{d}

with [_{0}] - [_{0}] - [

Bayesian inference

Our aim is to infer the values of _{d }
_{fr }
_{d}
_{fr}
_{d}
_{fr}

The likelihood function, _{d}
_{fr}
_{d }
_{fr }

The observed intensities for each channel are _{D}
_{A}
_{F}
_{D}
_{A }
_{F }
_{0 }and _{0 }that maximize the likelihood (see section on marginalization of _{0 }and _{0 }in the Methods). Combining the likelihood and prior probability distribution allows us to calculate the posterior probability distribution and determine the most probable values for our parameters of interest.

Illustration of method

To illustrate our method, we simulated data from three cells (Figure _{0}]) and acceptor ([_{0}]). In the first cell, [_{0}] = [_{0}] = 0.2 _{0}] = [_{0}] = 1 _{0}] = [_{0}] = 5

Analysis of typical FRET data recovers true values of K_{d }and E_{fr}

**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

To obtain the posterior probability distribution that corresponds to this data for two parameters of interest, _{d }
_{fr}
_{d }
_{fr}
_{d }
_{fr}
_{d }
_{fr }
_{d }
_{fr }

Inferring other quantities from FRET data

To further illustrate the versatility of our method, we show that we can also use our model and Bayesian framework to estimate other parameters. Two instrument-independent parameters that have been a focus of interest are the apparent FRET efficiency, _{d }
_{da}
_{0}] with [_{0}] = _{da}
_{0}] and _{fr}
_{d }r_{da}
_{0}]. We can analytically maximize the posterior probability of _{da }
_{d }
_{0}] by setting the derivative of the probability with respect to [_{0}] equal to zero and solving for this optimal [_{0}] in terms of _{da }
_{d}
_{da }
_{d }

Testing

Algorithm reflects data quality

An algorithm for estimating parameter values should report not just an estimate of the most probable parameter values but also the reliability of that estimate. To evaluate this aspect of our algorithm, we measured both the error and the uncertainty of its output. We defined the error as the discrepancy between the mean of the posterior probability distribution and the true value. We calculated error as _{d }
_{fr}

We first tested our algorithm in the presence of varying levels of measurement noise. As the examples in Figure _{d }
_{fr}
_{d }
_{d }
_{fr }

K_{d }estimates reflect data quality and quantity

**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

Next, we verified that gathering more data would improve the quality of the estimates of _{d }
_{fr}
_{d }
_{d }
_{fr }

Overall, our algorithm reliably reflects the quality of the data analyzed in the uncertainty it gives of its estimate and accurately infers the values of _{d }
_{fr }
_{d }

The total amount of donor, _{0}]_{0}]_{
d
}and _{
fr
}

_{d }and E_{fr }requires variations in _{0}]_{0}]

A challenge in analyzing FRET data is that both _{fr }
_{d }
_{fr }
_{d }
_{fr }
_{d}

To demonstrate the problem, we analyzed a set of simulated data from a single sample with [_{0}] = [_{0}] = 1 _{d}
_{fr}

Concentration variations determine shape of region of high probability

**Concentration variations determine shape of region of high probability**. Using the Monte Carlo algorithm to analyze data from a single cell containing [_{0}] = [_{0}] = 1 _{d}_{fr}

The difficulty in determining _{d }
_{fr }
_{0}] = [_{0}] = 1 _{d }
_{d }
_{d }

_{0}] = [_{0}] ≫ _{d }
_{0}] = [_{0}] ≪ _{d}

Plots of complex formation vs. _{d }
_{d }
_{d }
_{0}] = [_{0}] ≈ _{d}
_{d }
_{0}] = [_{0}] ≪ _{d }
_{0}] = [_{0}] ≫ _{d }
_{d }
_{d }
_{d }

Gaining insight into optimal experimental design

**Gaining insight into optimal experimental design**. The approximate posterior probability distributions for _{d }_{d }_{0}] = [_{0}] = 0.2·10^{-3 }_{0}] = [_{0}] = 1·10^{-3 }_{0}] = [_{0}] = 5·10^{-3 }_{d }_{d }_{d}^{-6}_{d}_{0}] : [_{0}] increases the uncertainty in fitting _{d }(B). As the ratio was increased (by keeping [_{0}] constant for the three cells at 0.2·10^{-6}^{-6}^{-6}_{0}] according to the ratio), posterior probability distributions for _{d }_{fr }_{fr }_{0}] : [_{0}] increases. (B) had 50 measurements/cell/channel, 36,000 steps/walk and 3% added noise. Bars show mean ± SD. For other parameters, see Methods.

By finding the posterior probability distribution of _{d}
_{d }
_{d}
_{d }
_{d }
_{0}] = [_{0}] ≪ _{d }
_{d }
^{-6}
_{0}] = [_{0}] ≫ _{d }
_{d }
^{-6}

This analysis illustrates how posterior probability distributions for the parameters of interest can be more informative than single values. For example, information from posterior distributions can be useful for improving experimental design. If one were to obtain plateaued distributions like the two shown in Figure _{d }

The ratio [_{0}]:[_{0}] affects inference

We also used our algorithm to explore the effects of varying the ratio, [_{0}] : [_{0}], on the algorithm's ability to infer _{d }
_{0}] : [_{0}] deviates from unity _{d }

The insets in Figure _{0}] : [_{0}] = 1 and drops to 0.7% for [_{0}] : [_{0}] = 10 (FRET is included by changing the FRET efficiency from 0 to 0.4). As the ratio increased, the relative shortage of acceptors meant that fewer complexes could form, making the contributions from FRET smaller and causing the uncertainty of the estimate to grow. In the absence of measurement noise, however, our algorithm can estimate _{d }
_{d }

Knowing how spectral overlap and the ratio of donors to acceptors affect the inference of protein interaction strength helps make it possible to design informative experiments. For example, we find that in each channel, the difference generated by FRET depends on spectral overlap. In the donor channel, where FRET decreases the signal, the relative change is _{0}] = [_{0}], the change in the donor channel is proportional to

Prior information improves estimate

A further advantage of our method is its flexibility. The Bayesian framework makes it possible to incorporate additional details we know about the system as prior information, allowing us to more accurately represent the system being analyzed and potentially improve our estimate of the parameters of interest. To exploit this feature, we tested whether including additional information about the FRET efficiency would improve our inference of _{fr }
_{d}
_{fr }

Figure _{fr }
_{d }
_{fr}
_{fr }
_{fr }
_{d }
_{fr }
_{d}

Prior information on E_{fr }improves estimate

**Prior information on E _{fr }improves estimate**. Approximate posterior distributions for

Summary of testing

We have shown, using typical, simulated FRET data, that our algorithm accurately recovers the parameters of interest. It responded consistently and intuitively to changes in the amount of measurement noise present in the data and the quantity of data. We also used the posterior probability distributions we obtain for the parameters to gain insight into how the magnitude and variation of the donor and acceptor concentrations affect the ability of the algorithm to infer _{d}
_{d }
_{d }
_{fr }

Conclusions

Fluorescence microscopy and FRET open a window onto the cell, allowing us to observe protein interactions as the cell functions as a complete system. However, for protein interaction information from FRET data to be integrated into models and improve our understanding of biological systems, it must be reliably quantified, including the uncertainty in the estimates produced

For this purpose, we have presented an algorithm for inferring the most probable values of the absolute or relative _{fr }
_{d}

In the examples described here, we make a few assumptions, but these are not necessarily part of our methodology. First, we use molar extinction coefficients that must be measured separately or taken from the literature and assume that the literature measurements are valid in the cellular environment. Second, we assume Gaussian measurement noise, but our model could be straightforwardly adapted to include log-normal or other types of noise. Finally, we have not taken into consideration photo-bleaching, incomplete labelling or dark states, but our model could readily be extended to include these factors.

We have used simulated data to illustrate principles that apply to real data, showing how, in practice, it is best to infer the dissociation constant from FRET data. Three-cube FRET experiments are a well-established experimental technique _{d}
_{d}

We have demonstrated that to infer the values of _{d }
_{fr}

While a number of

Methods

Calibration of fluorescence constants,

The _{d}

To infer the values of the constants

Determining the ^{(D) }and ^{(A)}, may not always be available at these wavelengths, but they can be estimated from literature values of the molar extinction coefficients (usually measured at the fluorophore's excitation peak) and the excitation spectra of the donor and acceptor

When the relationship between brightness and concentration cannot be determined absolutely, relative values for

The three-cube measurements on samples containing only donors or only acceptors would correspond to Eqs (8) and (9). Three-cube data obtained from samples containing donor-acceptor constructs would correspond to the following equations:

where we have replaced

where ^{th }

We are interested in the values of the _{0}], [_{0}], and [_{k }
_{k }
^{th }
_{
k
}as parameters to be inferred in the procedure we describe below. Assuming a prior probability that only specifies positive values for the

which is valid for the three cube experiments for each sample (donor, acceptor, or construct). For the donor-only sample, _{i }

For a given data set, we can infer _{calib}. Alternatively, one can use a numerical solver to find the most probable values by solving for the roots of the system of equations consisting of the derivatives of Eq (12) with respect to each of the six variables. Using this relative calibration procedure, the final _{d }
_{d }

Data simulation

We designed our simulated data to mimic the key features of experimental data, which could come from various sources, such as fluorescence reader measurements of solutions of purified proteins or images of cells from fluorescence microscopy that have been processed and quantified. To simulate data, we wrote a function in Matlab (The Mathworks, Natick, MA) that takes as input _{fr}
_{d}
**A _{0}
**,

For each pair of concentrations, [_{0}] and [_{0}], we calculate [_{D}
_{A}
_{F }

In our examples, we use _{fr }
_{d }

The constants corresponding to the donor, acceptor, and complex fluorescing in their respective channels

Incorporating prior information

The Bayesian framework makes it possible to incorporate prior information about any of the parameters, including uncertainty in _{fr }
_{fr }
_{E}

Markov chain Monte Carlo (MCMC) estimation

To sample _{d }
_{fr }

We use the Metropolis-Hastings algorithm ^{j}
^{
j-1}) is the posterior probability of the current location. The walk is run for a sufficiently long time (about 10,000 steps) to generate independent samples from the posterior distribution and the step size chosen to maintain an acceptance rate of 40 - 60%

Once the walk has converged with the energy fluctuating around a minimum value, we record the steps taken and use the histogram of these sampled values as our estimate of the posterior probability distribution. From this estimated distribution, we obtain the mean and standard deviation of each parameter being inferred. The algorithm is summarized below.

Marginalization of _{0 }and _{0}

We have defined the likelihood in Eq 7 and assume that the measurement noise in each channel is independent. Because we have a Gaussian model, the values of the measurement noise parameters, _{D}
_{A }
_{F }
_{d }
_{fr}

Although _{d}
_{fr}
_{0 }and _{0 }are important parameters, we cannot measure them directly. For our purposes, we are interested in the values of _{d }
_{fr}
_{0 }and _{0}. Rather than fit _{0 }and _{0}, we marginalize or integrate them out:

To indicate that we have no knowledge about the values of _{0 }and _{0}, we set the prior, _{0}, _{0}), to a constant for positive _{0 }and _{0 }(and 0 otherwise).

As a result,

Because this expression is difficult to integrate analytically, we consider the 'energy', _{d}
_{fr}
_{0}, A_{0})), and approximate

where ∇∇_{0 }and _{0}.

Therefore,

This approximation of the likelihood results in a Gaussian integrand, which we can then integrate analytically

In summary,

where |_{d }
_{fr }
_{d }
_{fr}

Algorithm summary

Our algorithm for sampling from the posterior probability distribution of (_{d}
_{fr}
_{fr }

1. Perform calibration to obtain values (absolute or relative) for the

2. Define prior probability distributions for parameters to be inferred based on initial information that is known, if any.

3. Run Markov chain Monte Carlo algorithm:

Choose initial step

For j = 2 to n,

(a) Choose proposal step

(b) Compute the posterior probability,

i. Find _{0}, _{0}} that minimise the energy of the posterior probability using nonlinear optimisation (the Nelder-Mead simplex algorithm implemented in Matlab's fminsearch function (The Mathworks, Natick, MA)).

ii. Compute

iii. Using Eq (18), compute the likelihood,

iv. Using Eq (13), compute the prior,

v. Compute the posterior probability distribution, _{j}

(c) Check whether to accept the move to

• If _{j }
_{
j-1}, accept.

• Otherwise, accept with probability

Steps (a)-(c) are repeated until _{d }
_{fr}

4. Repeat step 3, varying initial (_{d}
_{fr}

Availability

We have made our data simulation and analysis software available at

Authors' contributions

PSS conceived of the study. CL and PSS developed the methodology. CL implemented and tested the algorithm. Both authors wrote the paper and read and approved the final manuscript.

Acknowledgements

We thank Marko Laine (Finnish Meteorological Institute) for providing his MCMC code and helpful feedback. We also wish to acknowledge Stephane Laporte, May Simaan and Jay Nadeau (McGill University) for useful discussions and Andrea Weisse, Bruno Martins, Christos Josephides, Clive Bowsher (University of Edinburgh), Nathan Scales (McGill University), and Vahid Shahrezaei (Imperial College London) for helpful comments on the manuscript. CL and PSS are supported by the Scottish Universities Life Sciences Alliance.