FRET Efficiency and Donor–Acceptor Distance Calculator (Förster Equation)

FRET three-way calculator: efficiency E from distance R and Förster radius R₀, distance R from efficiency E, or efficiency from donor fluorescence quenching; covers 2–14 nm.

Calculation mode

Förster radius R₀ (nm)

Donor–acceptor distance R (nm)

FRET efficiency E (%)

Donor fluorescence F_D (donor-only sample)

Donor fluorescence F_DA (donor–acceptor sample)

What is FRET

Fluorescence (Förster) Resonance Energy Transfer (FRET) is the physical process by which an excited-state donor fluorophore transfers its excitation energy non-radiatively to a nearby acceptor fluorophore through dipole–dipole coupling, without photon emission. The coupling efficiency falls off as the sixth power of the separation, making FRET exquisitely sensitive to distance changes in the 2–10 nm range. This has earned it the nickname ‘molecular ruler,’ and it is widely used for real-time monitoring of protein conformational changes, protein–protein interactions, nucleic acid structure, and intracellular signal transduction.

Core equations

The FRET efficiency E (fraction of energy transferred to the acceptor, ranging from 0 to 1) and the donor–acceptor chromophore separation R are related by the Förster equation (Förster T, Ann. Phys. 1948;2:55-75; Stryer L, Annu. Rev. Biochem. 1978;47:819-846, PMID: 354506):

E = 1 / (1 + (R / R₀)^6)

Back-calculating the distance:

R = R₀ × ((1 - E) / E)^(1/6)

Here R₀ is the Förster radius: it is defined as the separation at which E = 0.5 (50% transfer), and is an intrinsic property of each donor–acceptor dye pair.

The third mode derives efficiency directly from donor fluorescence quenching, without needing R₀:

E = 1 - F_DA / F_D

F_D is the fluorescence intensity of a donor-only sample (same batch, no acceptor label); F_DA is the donor-channel fluorescence intensity of the doubly labelled donor–acceptor sample.

Förster radius R₀

R₀ is determined by four factors:

Spectral overlap integral J: the overlap between the donor emission spectrum and the acceptor molar extinction spectrum in wavelength; larger overlap gives larger R₀
Donor quantum yield Φ_D: higher quantum yield gives larger R₀
Orientation factor κ²: describes the relative orientation of the donor and acceptor transition dipoles; the default value is 2/3, which applies when both fluorophores rotate freely and rapidly (isotropic average); when fluorophores are rigidly fixed, κ² can range from 0 to 4, shifting R₀ by up to about 35%
Refractive index n: typically 1.33–1.40 for aqueous solutions

This tool does not calculate R₀. Look up the R₀ for your specific dye pair in the dye manufacturer’s datasheet or dedicated literature. Typical ranges: Cy3/Cy5 ≈ 5.4 nm, ECFP/EYFP ≈ 4.9 nm, most organic dye pairs fall between 3–8 nm. R₀ for a given pair can shift with solvent, temperature, or protein microenvironment, so literature values should be treated as initial estimates only.

Applicable range

Sensitive window: 0.5 R₀ to 2 R₀ (for Cy3/Cy5, approximately 2.7–10.8 nm): outside this range, E is nearly insensitive to changes in R — when R < 0.5 R₀ the efficiency is ≈ 1, and when R > 2 R₀ it is ≈ 0, so back-calculating R from E becomes highly error-prone and quantitative analysis is unreliable
Point-dipole approximation: the Förster equation treats chromophores as point dipoles; the theory breaks down when R is smaller than about 1 nm (approaching the physical size of the chromophore itself)
Single-pair assumption: the equations apply to one donor–acceptor pair; they are not directly applicable when each donor is coupled to multiple acceptors or when multiple FRET pairs are present
Artefacts in the F_DA/F_D method: direct excitation of the acceptor and acceptor autofluorescence both contaminate the donor-channel F_DA reading; these contributions must be subtracted using appropriate controls; FLIM avoids this problem entirely

Common mistakes

Unmatched donor concentrations in the fluorescence method: if the donor concentration in the two samples differs by 10%, the calculated E carries a corresponding absolute error that is amplified at high efficiencies (E > 80%)
Forcing quantitative analysis at R > 2 R₀ (low efficiency): E is very small and instrumental noise is comparable to the signal; only qualitative judgements (‘FRET present or absent’) are valid; reliable distance inference is not possible
Treating R₀ as universal across dye pairs: the 5.4 nm value for Cy3/Cy5 does not apply to Alexa 488/Alexa 647; always use the R₀ specific to your experimental system

Verification

The values in the table below follow directly from the equations and can be checked by hand or mental arithmetic:

Input	Calculation	Expected result
R₀ = 5.4 nm, R = 5.4 nm	(5.4/5.4)^6 = 1; E = 1/(1+1)	E = 50.00 % (R = R₀ by definition)
R₀ = 5.4 nm, R = 2.7 nm	(0.5)^6 = 0.015625; E = 1/1.015625	E = 98.46 %
R₀ = 5.4 nm, R = 10.8 nm	2^6 = 64; E = 1/65	E = 1.54 %
E = 50 %, R₀ = 5.4 nm (back-calculation)	((1−0.5)/0.5)^(1/6) = 1; R = 5.4×1	R = 5.400 nm (inverse of first row)
F_D = 1000, F_DA = 600	1 − 600/1000 = 0.4	E = 40.00 %

FAQ

What is R₀ and where do I find it?

The Förster radius R₀ is the donor–acceptor separation at which FRET efficiency is exactly 50%. It depends on the donor quantum yield, the spectral overlap integral, and the orientation factor κ² of the specific dye pair. R₀ is an intrinsic property of each dye pair and cannot be transferred between pairs; typical organic dye pairs fall in the 3–8 nm range. Look up R₀ in the dye manufacturer's datasheet or the relevant literature. This tool does not calculate R₀.

What is the difference between calculating E from distance and from fluorescence, and which should I use?

The fluorescence-intensity method (E = 1 − F_DA/F_D) is a direct experimental measurement that reflects the true ensemble efficiency, but it requires the donor concentration to be strictly identical in both samples — any concentration mismatch propagates directly into the efficiency error. Calculating from R and R₀ is a forward prediction (estimating signal from a known structure) and depends on the accuracy of R₀. If you need to avoid concentration-matching problems, fluorescence lifetime imaging (FLIM, E = 1 − τ_DA/τ_D) is the recommended approach for intracellular FRET measurements, but requires a time-resolved instrument.

Does the calculated R equal the surface-to-surface distance between two proteins?

No. R is the straight-line distance between the dipole centres of the two chromophores. Fluorophores are attached to proteins via flexible linkers, so the FRET-measured R is typically 0.5–2 nm larger than the protein surface-to-surface distance. In addition, when a molecule exists in multiple conformations, the R back-calculated from the ensemble-average efficiency is not equal to the mean of the individual inter-chromophore distances (Jensen's inequality); single-molecule FRET can resolve distinct conformational sub-populations.