FIG. 4.

FRET-lines of dynamic molecules. (a) Pure states are characterized by a single lifetime, and the corresponding lifetime distributions show a single peak. In the presence of dynamics, pure states are mixed at different ratios. The lifetime distributions show two peaks weighted by the species fractions x⁽¹⁾ and x⁽²⁾ = 1 − x⁽¹⁾. The pure states are defined by lifetimes ${\mathit{\tau }}_{\mathit{D}\left(\mathit{A}\right)}^{(1)}$ = 0.8 ns and ${\mathit{\tau }}_{\mathit{D}\left(\mathit{A}\right)}^{(2)}$ = 3.2 ns. Species fractions are color coded from red ($\left.{\mathit{x}}^{(1)}=1\right)$ to blue ($\left.{\mathit{x}}^{(1)}=0\right)$. (b) The corresponding fluorescence decays of the lifetime distributions are shown in (a). For pure states, the decays are single exponentials, while mixed states have two-component lifetimes. (c) The dependency between the species fractions x⁽¹⁾ and x⁽²⁾ is given by x⁽¹⁾ + x⁽²⁾ = 1. (d) In a plot of the FRET efficiency E vs the intensity-weighted average fluorescence lifetime ${⟨{\mathit{\tau }}_{\mathit{D}\left(\mathit{A}\right)}⟩}_{\mathit{F}}$, pure states define the static FRET-line (grayscale diagonal line). Mixed states are displaced from the static FRET-line and fall onto a curved line connecting the pure states, described by Eq. (27). The dynamic FRET-line is color-coded by the contribution of species 1. The arrow indicates the maximum possible dynamic shift ds from the static FRET-line. (e) Exemplary dynamic FRET-lines for limiting states with FRET efficiencies E₁ = 0.1/E₂ = 0.9 (orange, ds = 0.28), E₁ = 0.3/E₂ = 0.7 (purple, ds = 0.06), and E₁ = 0.5/E₂ = 0.95 (cyan, ds = 0.17) are shown in a plot of the FRET efficiency vs the normalized intensity-weighted average fluorescence lifetime ${⟨{\mathit{\tau }}_{\mathit{D}\left(\mathit{A}\right)}⟩}_{\mathit{F}}/{\mathit{\tau }}_{\mathit{D}\left(0\right)}$. (f) Contour plot of the dynamic shift, ds, as a function of the FRET efficiencies of the limiting states, E₁ and E₂. The dynamic shifts for the examples given in E are shown as circles. In (d) and (e), the static FRET-lines are given by Eq. (22). Dynamic FRET-lines were calculated according to Eq. (27).