Relating FRET Data to a Biochemical Quantity of Interest
Before trying to understand the linear range of FRET measurements, it is important to understand precisely how the FRET signal is related to specific biochemical entities within the cell, such that FRET data can be compared to appropriate quantities in a mathematical model. In general, the transition of the inactive FRET probe P to an active state P*, where FRET is more likely to occur, can be represented by the scheme in . The enzyme F catalyzes the forward reaction and the enzyme R catalyzes the reverse reaction, both with a standard Michaelis-Menten mechanism. The fraction of donor fluorophores that are capable of transferring energy by FRET (ϕF), is directly proportional to the concentration of active FRET probes, P* (the quantity ϕF is related but not exactly equal to the traditional FRET efficiency E; this will be important and discussed in below analyses). Here, we consider how P* is related to biochemical quantities.
Kinetic Scheme for FRET Probe Activation.
First, consider the scenario where the enzyme F is saturated (
is the Michaelis constant for the forward enzyme and PTOT
is the total probe concentration) and the enzyme R is in its linear range (
is the Michaelis constant for the reverse enzyme). Then, the steady-state level of P*
is given by
is the activity of the forward enzyme and Ra
is the activity of the reverse enzyme. Under these conditions, we see that the concentration of active probe molecules is linearly proportional to the forward enzyme activity. However, consider the steady-state levels of P*
when R is saturated and F is in its linear range,
or when both enzymes are in their linear range,
Equations 2 and 3 yield non-linear relationships between the forward enzyme activity Fa
. Considering a more complex scenario where both enzymes follow Michaelis-Menten kinetics (not saturated nor in the linear range of operation) we have
which again gives a non-linear relationship between Fa
. Considering non-steady-state or non-Michaelean conditions, which is particularly important given that intracellular reactions are rarely at steady-state or obey Michaelis-Menten kinetics, only further complicates the relationship between P*
and forward enzyme activity. Therefore, we conclude that when FRET data are used for quantitative modeling, they should be compared to an explicitly modeled downstream substrate 
, and not the activity state of the forward enzyme. For example, if the FRET probe is responsive to a kinase activity, then the amount of active FRET probes are proportional to the level of a phosphorylated substrate (which should be explicitly modeled), but not the kinase activity itself.
The most common way of quantifying FRET is by a technique called ratiometric imaging. The standard protocol in such an experiment is as follows 
. Cells are exposed to excitation light for the donor channel, and then fluorescence emission is divided into donor and acceptor channels. The output from ratiometric imaging, R
, is the intensity in the acceptor channel, IAdon
, divided by the intensity in the donor channel, IDdon
Below we derive an expression for R in terms of properties of intrinsic donor and acceptor properties, the excitation and emission channel characteristics, and of greatest importance, the fraction of donor molecules capable of transferring energy by FRET (ϕF), which is directly indicative of the number of active probes.
The intensity in the donor channel is the sum of donor emission (IDD
) and acceptor emission crosstalk (IDA
In most circumstances, it is reasonably easy to exclude acceptor emission from the donor channel, and therefore we assume that IDA
is negligible compared to IDD
. The donor emission can be represented in terms of the total number of excited donor molecules (ND*
), the fraction of donor molecules actually transferring energy by FRET (the fraction of donor molecules capable of transferring energy by FRET multiplied by the FRET efficiency E of such molecules: EϕF
), and the fraction of the donor emission captured by the donor channel (fDD
), which has units of photons per molecule.
Note that fDD is proportional to the integral of the emission spectra between the emission filter wavelengths.
The intensity in the acceptor channel, similar to that of the donor channel, is the sum of acceptor emission (IAA
) and donor emission crosstalk into the acceptor channel (IAD
As above, the individual emission intensities can be represented in terms of the number of excited molecules, FRET fraction, and emission spectrum coverage, giving
Here, the acceptor may be excited either by FRET from the donor (with the number of molecules denoted by
) or by direct excitation at the donor wavelength (with the number of molecules denoted by
), and fAA
are the fractions of the acceptor and donor emission captured by the acceptor channel, respectively (again units of photons per molecule).
Given these expressions for the acceptor and donor emission intensities, the ratio R becomes
Providing that the amount of direct acceptor excitation is negligible, R simplifies to
The behavior of Eq. 11 is shown in , for the conditions where (i) the apparent fraction of FRETing molecules (EϕF) ranges from 0 to a maximum of approximately 60% and (ii) the donor and acceptor emission coverages are of approximately the same magnitude (1~fAA/fDD). Although condition (ii) may seem restrictive, in practice deviation from this condition can be easily adjusted at the level of digital intensities by changing detector gains for the acceptor and donor channels. A striking feature of is the non-linearity of R as a function of ϕF for all values of fAD/fDD (which quantifies the relative amount of crosstalk from the donor into the acceptor channel). Even with no crosstalk from the donor into the acceptor channel, ratiometric FRET should give a non-linear signal-response relationship.
To test this prediction of a non-linear relationship between ϕF
and the observed ratio R
we analyzed a Raichu-RhoA FRET probe 
that has mTFP1 
as the donor and mVenus as the acceptor 
. This combination is superior to the previously used CFP/YFP pair as mTFP1 has a mono-exponential fluorescence decay, and is more photostable and brighter than ECFP (or its derivative mCerulean). The mTFP1/mVenus donor/acceptor combination has been shown previously to undergo significant FRET 
. The FRET output of this probe increases when it is bound to GTP, and decreases when it is bound to GDP. We expressed the Raichu-RhoA FRET probe in E. coli
and purified it. To create active and inactive populations of the probe, we incubated the recombinant Raichu-RhoA protein in vitro
with either GTPγS or GDPβS, which stably bind to small G-proteins such as those found in the Raichu-RhoA probe sensing unit, and therefore holds it either in the “on” (GTP) or “off” (GDP) state. We then mixed the GTP and GDP bound forms of the probe in various proportions and analyzed the ratiometric FRET with a fluorescence plate reader (). To our surprise, the resulting relationship was linear across the entire spectrum of GTP bound fractions. However, the ratio R only ranges between approximately 1.8 and 2.1, which is quite small compared to the range calculated in . Given this small ratio range, the inherently non-linear relationship predicted by Eq. 11 would appear effectively linear. Therefore, we tested a system where a greater range of ratios may be explored. We expressed and purified both mTFP1 alone (non-FRETing protein) and an mTFP1-mVenus tandem fusion (FRETing protein), mixed these two proteins together in various proportions, and then measured the resulting ratiometric FRET again with a fluorescence plate reader (). With this system, we could explore a much wider range of ratio changes. The results confirm that the ratios depend non-linearly on the fraction of molecules undergoing FRET, in a manner consistent with the predictions of . We verified that adding matched amounts of pure mVenus, rather than mTFP1-mVenus, did not change the observed ratios, showing that the observed ratio changes were a result of intra-molecular rather than inter-molecular FRET (data not shown).
An alternative method of quantifying FRET via ratiometric methods is to divide the donor emission intensity by the emission intensity of direct acceptor excitation (IAacc
. In this method, a reduction of the donor intensity is indicative of increased FRET, and this FRET indicator is normalized by the directly-excited acceptor intensity, which is an indicator of the total number of probes. In this formulation, one avoids introducing an inherent non-linearity into the denominator of the ratio. We denote this alternative ratio as Ralt
In the common situation where we can excite the acceptor without exciting the donor, and following the above nomenclature and assumptions, we arrive at
which shows that the ratio Ralt
should be linearly proportional to the fraction of probes capable of FRET.
To test this hypothesis, we again mixed various dilutions of mTFP1-mVenus and mTFP1. But in contrast to the above experiment, as the concentration of mTFP1-mVenus was reduced, equal concentrations of both mTFP1 and mVenus were added to retain a constant total mVenus concentration (monomer plus tandem). The results confirmed that indeed there is a linear relationship between Ralt
0.99). Note, however, that this same linearity result does not apply to Ralt−1
, when one divides the emission intensity of direct acceptor excitation with the donor emission intensity.
Fluorescence Lifetime Imaging
Another way of measuring FRET is by observing the lifetime of a population of excited donor molecules, which is commonly called fluorescence lifetime imaging microscopy (FLIM). When a donor molecule undergoes FRET rather than standard fluorescence emission, the average fluorescence lifetime decreases 
. In particular, the FRET efficiency E
is related to the fluorescence lifetime by
are the fluorescence lifetimes in the absence and presence of acceptor, respectively 
. Now, consider a situation where there are two populations of excited donor molecules, one that is capable of FRET (FY*
) and one that is not (FN*
), and FRET occurs with rate constant ket
while fluorescence occurs with rate constant kf
(). Given (i) standard first-order kinetics for these processes and (ii) that the resultant measured lifetime of the mixture is a fractional weighted average of the individual components, it can be shown that
are the overall FRET efficiency and fluorescence lifetime of such a mixture of molecules, respectively. Condition (ii) will be the case so long as measurement error of the pure and FRETting lifetimes is the approximately the same, and the measurement method does not bias the estimate toward one population or another. Rearrangement of Eq. 15 gives
which shows that the fraction of molecules capable of undergoing FRET, ϕF
, should be linearly proportional to the measured fluorescence lifetime τmix
. Moreover, both the slope and y-intercept of this line should be the related to donor fluorescence lifetime.
To test these hypotheses we again prepared mixtures of mTFP1 and the mTFP1-mVenus tandem fusion in various proportions, and then measured their fluorescence lifetimes (). As predicted, we see a linear correlation between the fraction of mTFP1-mVenus molecules and the fluorescence lifetime (R2
0.98), with a y-intercept that agrees with that measured for pure mTFP1 (2.98 ns vs. 2.94+/−0.07 ns). Based on the calculated slope (−0.78 ns—see ) and the measured value of the mTFP1 lifetime (2.94 ns), we calculate the FRET efficiency of mTFP1-mVenus tandem molecules as 27% based on Eq. 16 (0.78/2.94). This corresponds closely to the measured FRET efficiency of 25% based on standard definition in Eq. 14 ((2.94 ns-2.2 ns)/2.94 ns), further justifying Eq. 16 and the calculated value of the slope.
Relating Measured FRET to the Fraction of Active Probes
When comparing the measured FRET responses to a mathematical model simulation, it is useful to extract the absolute fraction of probes that are in the active state (ϕa=P*/PTOT
). This is possible, so long as one uses either τmix
to quantify FRET (which both vary linearly with the fraction of probes capable of FRET), and can force all of the probes in a cell either to the active state or inactive state. If the probe is for a kinase, for example, then one can use saturating doses of phosphatase or kinase inhibitors to accomplish this. Then, the FRET measurements of these completely active and inactive states may be measured; denote these Fact
, respectively. Here, FRET measurements may refer to either τmix
Then, because the measurement scales linearly with the number of probes capable of FRET, the fraction of active probes can be written in terms of known quantities as follows
denotes either τmix
. Thus, linear FRET measurements combined with simple controls allows for direct estimation of the fraction of active probes in the cell.
Our theoretical analysis supported by experimental data yields important guidelines for using FRET probe data with quantitative modeling. We have shown that only in rare circumstances will the fraction of molecules capable of FRET be linearly related to an upstream enzymatic activity. Therefore, FRET data should only be directly compared to model variables analogous to the active FRET probe state, and not upstream enzyme activities. For instance, FRET data from the probe for ERK kinase, EKAR 
, should only be compared to the phosphorylation levels of an ERK substrate (corresponding to P*
), and not the levels of active, doubly-phosphorylated ERK (proportional to enzyme activity). Of relevance to this study, FRET data from a probe for a small GTPase such as RhoA should only be compared to levels of RhoGTP (P*
) or an explicitly modeled FRET probe, and not directly to GEF or GAP activities. Furthermore, we show that FRET measurements obtained via the standard ratiometric method have an inherently non-linear signal-response relationship and should therefore be avoided if possible. Although we found the dynamic range of the Raichu-RhoA FRET probe was not great enough to observe this inherent non-linearity, some probes have a greater dynamic range, and further probe improvements will push experiments into a regime where ratiometric measurements would become troublesome. Importantly, however, we find that measuring FRET either by (1) ratiometric methods where the donor emission intensity is divided by the emission intensity of direct acceptor excitation (Ralt
) or (2) FLIM (τmix
), results in a linear signal-response relationship between the measurement (Ralt
) and the fraction of probes capable of undergoing FRET. Quantifying FRET via Ralt
removes the inherent non-linearity built into the denominator of the typically-used ratio R
. As ratiometric methods involve less expensive, simpler microscope equipment than does FLIM, it may be preferable to use such methods, so long as data are quantified via Ralt
, and not with other commonly used forms of the ratio. Moreover, it has been shown in one case that ratiometric methods have a slightly better signal-to-noise ratio than lifetime imaging methods 
, albeit the difference is small, a non-linear form of the ratiometric method was employed, and it is not clear whether it holds true for different probes and fluorescent protein pairs. Also, some donors, such as ECFP, have multi-exponential lifetimes, and therefore are not suitable for FLIM. On the other hand, FLIM has several technical advantages. First, slight photobleaching of the donor will not affect FLIM measurements, whereas it would introduce significant artifacts into ratiometric methods, showing up as increased FRET. Second, photobleaching of the acceptor plays a minimal role in FLIM, but is likely to occur in ratiometric imaging due to direct acceptor excitation, and can introduce artifacts showing up as decreased FRET. This is particularly important given the well-known weak photostability of the YFP derivatives, which are commonly-used as acceptors. Lastly, dark acceptors, such as REACh 
, may be used in FLIM, greatly increasing the ability to multiplex FRET measurements for observing multiple activities in the same cell in real time. Thus, FLIM may be preferable when one is interested in analysis of networks over long time scales (or with high frequency measurements), which is usually the case in the context of mathematical modeling.