Monte Carlo simulations of isotropic <i>κ</i>²

<p><b>distribution. A.</b> Cartoon illustrating how the dipole orientation factor <i>κ</i>² is based on 2 angles, <i>θ</i> and <i>ω</i> (see Eq 6). <b>B.</b> Monte Carlo simulation of the <i>κ</i>² probability distribution assuming that <i>θ</i> and <i>ω</i> are randomly distributed (isotropic). Note that the mode of this distribution is 0 and the average is 2/3.</p

The donor fluorescence decay is an indicator of the distribution of FRET efficiency values in a population. A.

<p>Simulated distribution of FRET efficiencies for a narrow normally distributed <i>R_DA</i> population, <b>〈</b><i>E</i><b>〉</b> = 0.5. <b>B.</b> Simulated distribution of FRET efficiencies for a bimodal population with E = 0 & 1. <b>C.</b> Fluorescence decays from the populations depicted in panels A and B. For comparison the decay of donors in the absence of acceptors is also plotted (Black trace). Note that the decay of the population depicted in panel B (RED trace) was poorly fit by a single exponential decay model (dotted line), but was well fit using a double exponential decay model (dashed and dots line).</p

Large variance in separation (<i>R_DA</i>) produces bimodal FRET efficiency probability density histograms, multi-exponential decays, and alters the FRET efficiency from that pertaining to the mean separation.

<p><b>A.</b> A histogram of <i>R_DA</i> values from 5 Monte-Carlo simulations, each having a Gaussian distribution with its mode at 5.4 nm, but with standard deviations ranging from 1 to 200% of the mode. Note that points with negative separation were dropped from the distribution in the simulation. <b>B.</b> The distribution of FRET efficiency probabilities from the populations depicted in panel A. The <i>R</i>₀ value was set to 5.4 nm, the lifetime of the donor in the absence of acceptors was set to 3 ns, and <i>κ</i>² was set to 2/3 to simulate the dynamic random isotropic reorientational regime. green tinted area represents <i>E</i><0.05, and the yellow tinted area represents <i>E</i>>0.95. These points corresponded to the <i>R_DA</i> values in panel A with similar tints. <b>C.</b> Fluorescence decays from the populations depicted in panel A. D. The <i>R</i>₀ normalized dependence of 〈<i>E</i>〉 on both <i>R_DA</i> and its variance.</p

Long-lived acceptor dark states can produce bimodal FRET efficiency distributions, multi-exponential donor excited-state decays, and a decrease in measured average FRET efficiency compared with that for the mean separation.

<p><b>A.</b> Long-lived dark states (blinking and flickering) in donor and/or acceptor fluorophore populations may attenuate the apparent <i>R</i>₀ value for FRET, in the former by reducing the measured quantum yield, in the latter by reducing the measured extinction coefficient and thereby the spectral overlap integral. The attenuation factor () is plotted as a function of dark donor fraction (<i>f_Dd</i>) and dark acceptor fraction (<i>f_Ad</i>). <b>B.</b> Monte-Carlo simulations were used to model the distribution of FRET efficiencies from populations with 0 to 50% acceptor dark states. Separations were modeled using a Gaussian with a standard deviation equal to 1% of the mode. The true <i>R</i>₀ was fixed for all samples at a value of 5.4 nm, the lifetime of the donor in the absence of acceptors set to 3 ns, <i>κ²</i> set to 2/3, and it was assumed that dark states do not absorb in the region of donor emission. <b>C.</b> Fluorescence decays for the populations depicted in panel B. <b>D.</b> Dependence of measured 〈<i>E</i>〉 on the ratio of fixed (i.e., no distribution width) <i>R_DA</i> to true <i>R</i>₀ and the fraction <i>f_Ad</i> of acceptors in the dark state.</p

For the same separation distance, dynamic and static isotropic regimes have different FRET efficiency distributions and decays.

<p><b>A.</b> The FRET efficiency distributions from Gaussian populations with a mean <i>R_DA</i> value of 5.4 nm±1% in either the dynamic random isotropic reorientational regime (<i>κ</i>² = 2/3, GRAY peak) or the static random isotropic orientational regime (BLUE bimodal distribution). <b>B.</b> Fluorescence decays for the populations depicted in panel A. <b>C.</b> The dependence of 〈<i>E</i>〉 on <i>R_DA</i> in these dynamic (GRAY open circles) and static (BLUE squares) regimes. The blue area between these curves depicts the region between the dynamic and static regimes into which the FRET efficiencies for samples that have rotational correlation times similar to the inverse of the energy transfer rates will fall. <b>D</b> and <b>E.</b> FRET efficiency distributions (probability densities, <i>p(E)</i>) used to generate the dynamic (D) or static (E) average FRET efficiency curves displayed in panel C. Note that in the static random isotropic regime, samples tend to have FRET efficiencies either near 0% or centered at <i>F</i>/(1+<i>F</i>). <b>F.</b> FRET efficiency distributions, <i>p(E)</i>, for fixed separation ranging from 1 to 10 nm and a Förster separation of 5.4 nm (<i>F</i>-values ranging from 3.7×10⁴ to 3.7×10⁻²) calculated analytically.</p