Förster distance ($R_0$)

The Förster distance is the separation giving FRET efficiency $E=1/2$ for the specified pair and conditions. It is a property of the optical pair and its environment, rather than a universal length. In the usual pair model,where ${\kappa }^2$ is the FRET dipole orientation factor, $Q_D$ the FRET donor quantum yield, $J$ the FRET spectral overlap integral, and $n$ the medium's refractive index. The omitted proportionality constant depends on the unit convention for $J$.^[ref]

If FRET efficiency is known and those assumptions hold, inversion givesThe derivative explains why errors become troublesome near $E=0$ or $E=1$. An illustrative pair with $R_0=4$ nanometres and another with $R_0=6$ nanometres both give $E=1/2$ at their different Förster distances: knowing FRET efficiency alone does not determine $r$.

A heterogeneous population adds another ambiguity. In a synthetic example with equal weights per FRET donor excitation, efficiencies $0.1$ and $0.9$ average to $0.5$, but their separations are $9^{1/6}{R}_0$ and $9^{-1/6}{R}_0$. Their average separation is approximately $1.068R_0$, whereas inverting the average FRET efficiency gives $R_0$. A measured emission ratio has additional detection weights; it need not equal even this excitation-weighted average. The transferable lesson is that an average readout does not uniquely specify a distribution of molecular separations.

FRET spectral overlap quantifies overlap between the FRET donor's emission and the FRET acceptor's absorption as functions of wavelength $\lambda$. With normalized donor emission $f_D$ and acceptor molar absorption coefficient ${\epsilon }_A$, one convention isMatching two emission colors is insufficient: the receiving absorption matters.^[ref]

The FRET dipole orientation factor describes the geometry of the two transition electric dipoles relative to their separation. Let ${\hat{\mathbf{\mu }}}_D$, ${\hat{\mathbf{\mu }}}_A$ and $\hat{\mathbf{r}}$ be their unit vectors. ThenFor a separation along $z$, parallel electric dipoles along $z$ give ${\kappa }^2=4$, parallel ones along $x$ give $1$, and one along $x$ with the other along $y$ gives $0$. Thus rotating a pair can change FRET efficiency without changing its separation.

The familiar value ${\kappa }^2=2/3$ requires suitable rapid, independent orientation averaging during the FRET donor lifetime. Random orientations across a population of slowly rotating fluorescent proteins do not establish that dynamic condition. ^[ref]

The FRET donor quantum yield is the emitted fluorescence photon fraction per absorbed excitation photon, measured without a FRET acceptor. In the constant-rate model used for FRET efficiency, $Q_D=k_{\mathrm{r}}/(k_{\mathrm{r}}+k_{\mathrm{nr}})$. Changing the FRET donor quantum yield changes the Förster distance, even if label positions remain fixed.