This manuscript describes a simple manner to use data on probe orientation or alignment in membranes to estimate the distribution of the κ² FRET factor, using a small program described in more detail in Section 3. In this subsection we use the program to verify some results known from the literature, as well as to investigate other hypothetical scenarios of interest.

One important feature of our numerical procedure for estimation of κ² distributions of membrane-embedded fluorophores is that it allows the introduction of orientation heterogeneity, in the form of a normal distribution of cosθ_D and/or cosθ_A, as mentioned previously. This type of heterogeneity is expected in simple rotational models such as the “wobbling-in-cone” motions [21]. The standard deviations of the cosθ_D and cosθ_A distributions (σ_D and σ_A, respectively) are a measure of the extent of freedom of dipole orientation. Therefore, they serve a similar purpose to that of Dale et al.’s depolarization factors <d_D^x> and <d_A^x> [4]: σ and <d^x> are different ways to express the orientation distributions sampled by donor and acceptor. Whereas Dale et al.’s form of taking this orientation redistribution into account using depolarization factors was justified by their perceived possibility of estimation of <κ²> ranges from time-resolved fluorescence polarization measurements, σ is a technique-independent, more direct metric of the dipole orientation heterogeneity.

Intuitively, upon permitting sampling of different donor/acceptor relative orientations, κ² distributions and average values are necessarily affected. As discussed above, the limiting case of σ >> 1 leads to the expected isotropic case. Here we address the more physical possibility of moderate heterogeneity (σ ≈ 0.05–0.3) and its effect upon the orientation factor. Figures 4 and 5 illustrate the alterations produced on κ² distributions and average values upon widening the angular distributions of donor and acceptor dipoles for two distinct situations: identical donor and acceptor distributions (e.g., homo-FRET) centered around θ = 90° (Figure 4) and quite distinct donor and acceptor distributions centered around 60° and 120°, respectively (Figure 5). In these calculations, a larger number of pairs (at least 10⁶) should be considered, to allow a better sampling of the θ_D and θ_A distributions. In both cases, it can be seen that whereas moderate orientation heterogeneity produces visible alterations in κ² distribution, significant effects require a very large degree of heterogeneity. This is more apparent for the system where donor and acceptor have very distinct angular distributions (Figure 5), where, remarkably, very wide distributions of both probes affect κ² by a degree <10%.

In most applications of intramolecular FRET (involving a single acceptor fluorophore as quencher of each donor), distance measurements are carried out assuming a single fixed value of R₀, assuming a <κ²> value (usually 2/3) independent of donor-acceptor distance. However, this view has been challenged, as MD simulations revealed considerable correlation between κ² and donor-acceptor distance in a labeled protein [6], possibly related to constraints imposed by the linker used to attach the donor probe. Naturally, in systems where each donor may transfer its excitation energy to multiple acceptors at varying distances, correlation between donor-acceptor distance and their relative orientation will necessarily occur if the dynamic isotropic limit is not met. Consider for example that donors and acceptors are located in separate parallel planes (different depths in the membrane), and that both types of fluorophores have narrow orientation distribution. The range and relative weight of possible κ² values will certainly differ for a donor-acceptor pair in which the acceptor is located directly below or above the donor, relative to the case where their lateral separation is large. Therefore, the question is not whether there exists correlation, but to which extent κ² may vary as a function of distance, and how to best take this into account.

For the sake of illustration, we consider the same donor and acceptor orientations as in the previous subsection, but assuming fixed (no orientation heterogeneity) θ_D = θ_A = 90° and (θ_D, θ_A) = (60°, 120°). We also consider two separate possibilities in each case: (i) coplanar donor and acceptor spatial distributions; (ii) donor and acceptor planes separated by an arbitrary distance of h = 2.0 nm. In each of these scenarios, the size of the square systems l was varied up to 1000 nm. 10⁶ pairs were simulated for proper sampling of donor/acceptor distances. It is clear from the figure that location of both donor and acceptor dipoles in the same plane always lead to identical distributions (Figure 6A,E) and average values (Figure 6B,F). At variance, for donors and acceptors placed in parallel planes, there are very significant changes in both κ² distribution (Figure 6C,G) and <κ²> (Figure 6D,H), when the size of the simulated system (which constraints the maximal possible donor-acceptor distance) is <<100 nm.

In this case, definition of an average κ² becomes a problem. The best way to deal with this situation is to take notice of how FRET occurs from a given donor (with fluorescence lifetime in absence of acceptor equal to τ₀) to a distribution of M acceptors (where the distance between the selected donor and the ith-acceptor is denoted by R_i). The decay rate k is given by

where the different individual FRET rates are calculated according to

Finally, the Förster radius for each donor-acceptor pair is calculated using

where Φ₀ is the quantum yield of the donor in the absence of acceptor molecules, n is the refractive index of the medium, and the integral term is the overlap between the normalized donor emission (I(λ)) and the acceptor absorption (ɛ(λ)) spectra. The constant value in the equation above assumes nm units for both λ and R₀, as well as M⁻¹cm⁻¹ units for ɛ(λ). Combining these three equations, one concludes that the FRET rate is given by

where C is a constant term for all acceptors in the distribution. That is, the natural way to average κ² over different acceptors, with different distances to a given donor, is to use the inverse sixth power of the donor-acceptor distance as weight. In this way, acceptors located more closely contribute significantly more to the average (as they are also responsible for most of the quenching by FRET). For the purpose of this calculation, we adapted our program for averaging κ² in this way for FRET between a plane of donors and another of acceptors, and FRET in a bilayer geometry (with possibility of FRET between a plane of donors and two planes of acceptors—one in each bilayer leaflet). We then proceeded to calculate <κ²> for the two angular distributions presented above (θ_D = θ_A = 90°; and (θ_D, θ_A) = (60°, 120°)) assuming a bilayer geometry with h₁ = 0.0 and h₂ = 2.0 nm as the donor-acceptor interplanar distances. For coplanar fluorophores, an exclusion distance of R_e = 0.8 nm was also introduced, as donors and acceptors in the same plane cannot lie closer than a given minimum distance value. This would be similar to a typical experiment, with acceptor distribution in both bilayer leaflets. For a better sampling of all possible orientations for small R_i values, a value of l = 100 nm was used. Ten different simulations were carried out in each case, to assess whether proper convergence was obtained. For θ_D = θ_A = 90°, an average of 1.237 with standard deviation 0.010 was obtained, whereas for (θ_D, θ_A) = (60°, 120°) the corresponding values were 0.771 and 0.009. The relatively small standard deviations indicate that sampling was adequate. The average values were similar (very slightly lower) to the <κ²> calculated for the planar systems (1.250 and 0.776, respectively), and the small differences are due to the contribution of acceptors lying in the opposite leaflet. Because of the transverse distance of 2 nm, neglecting this leaflet does not lead to considerable error. In any case, for the following section that applies this methodology to computation of <κ²> for common experimental donor-acceptor pairs, whenever possible, information on donor-acceptor interplanar distances was taken into account.

Table 1 summarizes relevant information regarding membrane probes that are commonly involved in FRET experiments (as donors, acceptors or both), Förster pair combinations formed by these probes, examples of quantitative experimental FRET studies using these pairs, and information obtained from literature simulation or experimental studies regarding fluorophore location and transition dipole orientation of the probes. The latter information was used to calculate an average κ² value using the inverse sixth power of the donor-acceptor distance as weight, as described in the previous subsection. These averages are also shown in the Table. It can be seen that all these values but one (homo-FRET between Rhodamine B probes) lie within 15% or 0.1 of the isotropic limit of 2/3, even though the probe orientation distributions are quite varied. The most probable explanation for this is that all orientation data retrieved from the literature refer to liquid disordered bilayers, in which probes have generally broad angular distributions. Applications to FRET in liquid ordered or gel phases, where conformational freedom is largely reduced, would potentially lead to <κ²> further from the 2/3 value. This is illustrated in the t-PnA/DPH pair. For t-PnA, contrary to other probes, information regarding orientation distribution is also available in the gel phase (5°, 0.03 (gel) [22]). Using these input values instead of the fluid phase parameters, one would obtain <κ²> = 0.53 ± 0.01 instead of 0.58 ± 0.01 for both probes in the fluid phase. It should also be emphasized that the dipole orientation details shown in Table 1 refer to a specific probe compound labelled with the fluorophore under consideration, and other probes exist where the same fluorophore could be expected to show different transverse location and/or orientation distribution in the bilayer. For example, NBD data are based in a MD study of NBD-PC, whereas FRET studies also employ other NBD probes (NBD-PE, NBD-cholesterol).

The most common experimental observables in FRET are the decay of donor in the presence of acceptor, i_DA(t), and the FRET efficency, E. The decay law i_DA(t) for bilayer geometries and uniform probe distribution is available in the literature [23]

In this equation, c is the number of acceptors per unit area, γ is the incomplete gamma function, and the other symbols have the same meaning as in the previous equations. Although Equation 7 was originally derived for a plane of acceptors containing the donor (cis transfer), it is also valid if the donor molecule is separated from the acceptor plane by a distance R_e. From the time-resolved donor emission, the FRET efficiency, which is defined (and often experimentally measured) by

(where I_DA and I_D are the donor steady-state emission intensities in presence and absence of donor, respectively), can be calculated using

In the latter equation, i_D(t) = exp(-t/τ₀) is the donor time-resolved fluorescence decay in the absence of acceptor.

To appreciate the effect of <κ²> on the FRET efficiency, we first define R 0 ¯ as the value of the Förster radius calculated with κ² = 2/3, and then consider a form of Equation 7 in terms of dimensionless parameters ζ= t/τ₀ (reduced time), σ = π R 0 ¯ 2 c (number of acceptors in a circle of radius equal to R 0 ¯), α = R 0 ¯ / R e and β = (3/2)<κ²>:

This form of the decay law is suitable for calculation (by integration over ζ, done using Equation 9 after substitution of variables t and ζ) of universal curves E vs. σ, for fixed values of α and β. Representative examples are shown in Figure 7.

From Figure 7, it can be seen that variations in FRET efficiency upon changing the <κ²> value used to calculate R₀ are most important when the latter is smaller than or comparable to the exclusion distance, such as in panels (a) and (b). For example, for α = R 0 ¯ / R e = 0.50, the FRET efficiency obtained for <κ²> = 1.25 is >3 times larger than that obtained for for <κ²> = 0.30 over the whole σ < 5 range. However, it must be noted that larger values of α = R 0 ¯ / R e are often met in membranes, as long as R₀ has a reasonable (~2.5 nm or larger) value and there are acceptors in the same bilayer leaflet as the donors. In these cases, such as illustrated in panels (c) and (d), the effect of <κ²> on E, though still clearly detectable, is smaller, especially in the 0.50 < <κ²> < 0.80 range which is observed for most common FRET pairs (Table 1).

κ² is given by [5]

where Ψ_T is the angle between the transition moments of the donor and acceptor and Ψ_D and Ψ_A are the angles between the donor and acceptor transition moments and the vector uniting their centers, R⃗ (Figure 8). An equivalent expression, involving scalar products of &rrarr; = R⃗/||R⃗|| and unit vectors &urarr;_D and &urarr;_A (in the direction of the donor and acceptor dipoles, respectively), is

Two programs, written as macros in the Visual Basic Editor within a Microsoft Excel document, were developed for this article. The programs are available as supplemental files to this article, and may be edited by any user. Microsoft Excel Visual Basic was chosen as it may be easily used by most readers, and produces immediate numerical and graphic output.

Program 1 calculates κ² distributions and average values without weighting, for a single acceptor plane. The basic algorithm can be summarized as follows:

All calculations and distributions shown in Figures 1–6 were obtained with this program.

Program 2 follows a similar algorithm for calculation of κ² for the i^th donor-acceptor pair. However, it computes an average κ² value using R_i⁻⁶ as weight for each pair. It may be used for both one and two acceptor planes. The program reads θ_D,max, θ_A,max, σ_D, σ_A, l, N, and, for both planes, the respective donor-acceptor interplanar (h₁ and h₂) and exclusion (R_e1 and R_e2) distances. Two main cycles are carried out, one for each acceptor plane, with calculation of κ² distribution in each case. For both cycles, calculation of κ² for the i^th donor-acceptor pair is identical to that in the first program, with an additional check: if R_i < R_e1 (first cycle) or R_i < R_e2 (second cycle), then new donor and acceptor distance coordinates are allotted and a corresponding new R_i is calculated. Partial sums of κ²/R⁶ values are now computed, to obtain <κ²> = sum(κ²_i/R_i⁶)/sum(1/R_i⁶). As described in Section 3, this weighting scheme reflects the dependence of the FRET interaction with distance, and was used to compute the values of the right column of Table 1 (see footnote for typical input parameters).