FRET efficiency ( $E$ )

FRET efficiency is the probability that an excitation of the FRET donor ends in transfer to the FRET acceptor. Write

k_{D} = k_{r} + k_{nr}

for the total decay rate without a FRET acceptor, including radiative fluorescence and other decay, and

k_{T}

for the transfer rate. For constant competing rates,

k_{T} = k_{D} (\frac{R _{0}}{r})^{6}, E = \frac{k _{T}}{k _{D} + k _{T}} = \int_{0}^{\infty} k_{T} e^{- (k_{D} + k_{T}) t} d t .

(1)

The integral counts transfer events among all excitations in a population with exponential decay. Substitution gives the separation law above. This short rate argument assumes one FRET donor, one FRET acceptor, weak coupling, an available FRET acceptor, and fixed geometry during the decay; it is not a model for arbitrary interacting labels.^[ref]

Table of contents

Multiple-acceptor FRET efficiency ( $E_{n} = n E / [1 + (n - 1) E]$ )

 0  0

For n identical independent available FRET acceptors, transfer rates add. With single-pair FRET efficiency E, the donor exit probability is

E_{n} = n E / [1 + (n - 1) E]

. It is not nE; competition with the donor's other decay routes must be retained.

Optimal Förster distance for two-state contrast ( $R_{0, opt} = A B$ )

 0  0

For FRET efficiency

E (R) = 1/ [1 + (R / R_{0})^{6}]

and distinct positive reporter distances A and B, the absolute contrast is maximized at the geometric mean Förster distance

R_{0} = A B

. Substitution of

x = R_{0}^{6}

makes the derivative have the sign of

A^{6} B^{6} - x^{2}

. The maximum is

∣ B^{3} - A^{3} ∣/ (B^{3} + A^{3})

and the two efficiencies then sum to one. If

A = B

, every choice gives zero contrast and there is no unique optimum.

Förster distance ( $R_{0}$ )

 0  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,

R_{0}^{6} \propto κ^{2} Q_{D} J n^{- 4},

(1)

where

κ^{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 gives

r = R_{0} (E^{- 1} - 1)^{1/6}, \frac{d l o g r}{d E} = - \frac{1}{6 E ( 1 - E )} .

(2)

The derivative explains why errors become troublesome near

E = 0

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

1.068 R_{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 ( $J$ )

 0  0

FRET spectral overlap quantifies overlap between the FRET donor's emission and the FRET acceptor's absorption as functions of wavelength

λ

. With normalized donor emission

f_{D}

and acceptor molar absorption coefficient

ε_{A}

, one convention is

\int_{0}^{\infty} f_{D} (λ) d λ = 1, J = \int_{0}^{\infty} f_{D} (λ) ε_{A} (λ) λ^{4} d λ .

(1)

Matching two emission colors is insufficient: the receiving absorption matters.^[ref]

FRET dipole orientation factor ( $κ^{2}$ )

 0  0

The FRET dipole orientation factor describes the geometry of the two transition electric dipoles relative to their separation. Let

μ_{D}

μ_{A}

and

r

be their unit vectors. Then

κ^{2} = [μ_{D} \cdot μ_{A} - 3 (μ_{D} \cdot r) (μ_{A} \cdot r)]^{2}, 0 \leq κ^{2} \leq 4.

(1)

For a separation along

z

, parallel electric dipoles along

z

give

κ^{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

κ^{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]

FRET donor quantum yield ( $Q_{D}$ )

 0  0

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_{r} / (k_{r} + k_{nr})

. Changing the FRET donor quantum yield changes the Förster distance, even if label positions remain fixed.

FRET donor lifetime ( $τ_{D}, τ_{D A}$ )

 0  0

The FRET donor lifetime is the mean time spent in the excited state. For the constant-rate model of FRET efficiency,

τ_{D} = \frac{1}{k _{D}}, τ_{D A} = \frac{1}{k _{D} + k _{T}}, E = 1 - \frac{τ _{D A}}{τ _{D}} .

(1)

After a sufficiently brief excitation pulse, the FRET donor's fluorescence follows exponential decay,

I (t) = I (0) e^{- t / τ}

. With the illustrative values

τ_{D} = 4

nanoseconds and

τ_{D A} = 2

nanoseconds, the inferred FRET efficiency is

1/2

. Multiplying

I (0)

changes brightness without changing this decay time.

The lifetime comparison requires the same nontransfer decay rates in the reference and paired samples. Changes in the FRET donor's environment or a different nonradiative loss can also shorten the FRET donor lifetime. Heterogeneous FRET efficiencies or slowly varying FRET dipole orientation factors need an appropriate decay model; substituting any fitted average lifetime into the single-pair formula can mislead. ^[ref]