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 [1
]. One fluorescence tool that enables researchers to observe protein interactions in living cells is Förster resonance energy transfer (FRET). FRET data has the potential to yield biochemical constants, which are critical for modeling biological systems, but measuring protein interactions from FRET data requires careful quantitative analysis.
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, Efr
) depends on the distance between the donor and acceptor (which must be between 1 and 10 nm for FRET to occur), their relative orientation, and the extent to which the donor's fluorescence emission peak overlaps the excitation peak of the acceptor [3
]. Optimal conditions for energy transfer occur when the distance between donor and acceptor is minimal, the molecules' electric dipoles are aligned, and the spectral overlap is significant.
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 [4
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 [7
]. Although they are straightforward to calculate, FRET indices often bear an indirect and nonlinear relationship to the underlying concentrations and strength of protein interactions [13
]. Alternatively, the ratio of donors to acceptors and the apparent FRET efficiency, which is the product of the intrinsic FRET efficiency and the fraction of donors (or acceptors) in complex, can be calculated [15
]. Such quantities can be measured, provided calibrations are carried out using constructs consisting of a donor linked to an acceptor [17
], but again do not relate directly to the dissocation constant (Kd
) of an interaction.
Another significant challenge is that both the Kd
and the FRET efficiency, Efr
, affect the observed signals, and so neither can be determined independently of the other based on data from a single sample. Efr
has been estimated from separate acceptor photobleaching [14
] and FLIM experiments [15
], but these approaches have drawbacks. Acceptor photo-bleaching is slow, irreversible, often fails to bleach all acceptors, and yields the intrinsic FRET efficiency only when all donors are in complex with acceptors [4
]. FLIM requires specialized equipment, is often slower than standard fluorescence imaging, and analysis of FLIM data is complicated by the multiexponential fluorescence decays of fluorescent proteins [20
]. There is however an alternative to directly measuring Efr
: Many values of Kd
will be consistent with data from a single cell or sample, but, by taking a set of data from samples that contain varying concentrations of donors and acceptors and analyzing it altogether, it is possible to find the unique values of Kd
that are consistent with the ensemble [16
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 Kd
along with the uncertainty of that inference is necessary for the standardization of quantitative FRET measurements, the design of informative experiments, and for providing in vivo
parameters for developing models of protein networks [22
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 [25
], we develop a Bayesian algorithm to infer both the interaction strength (in terms of the in vivo Kd
) and the FRET efficiency. Applying our algorithm to simulated data, we gain insight into how both experimental design factors such as measurement noise, number of measurements, fluorophore concentrations and ratios, and prior information impact our estimate and its uncertainty.