Förster Resonance Energy Transfer (FRET)

What is FRET?

FRET is a distance-dependent energy transfer between two fluorescent molecules — a donor and an acceptor. When they are close (1–10 nm), excited donor molecules transfer energy non-radiatively to the acceptor via dipole-dipole coupling.

FRET is called a "molecular ruler" because the transfer efficiency depends sharply on the donor-acceptor distance.

Transfer Efficiency

$E = \frac{1}{1 + \left(\frac{r}{R_0}\right)^6}$

• r = donor-acceptor distance (nm)
• R_0 = Förster radius — the distance at which E = 0.5 (typically 2–10 nm)

The sixth-power dependence makes FRET highly sensitive to distance near R_0:

Distance from Efficiency

Rearranging the efficiency equation:

$r = R_0 \left(\frac{1}{E} - 1\right)^{1/6}$

FRET Rate

The rate of energy transfer from donor to acceptor is:

$k_T = \frac{1}{\tau_D} \left(\frac{R_0}{r}\right)^6$

where τ_D is the donor fluorescence lifetime without acceptor (typically 1–10 ns). The FRET efficiency can also be expressed as:

$E = \frac{k_T}{k_T + 1/\tau_D}$

Biological Applications

• Protein conformational changes (intramolecular FRET)
• Protein-protein interactions
• DNA hybridization assays
• Live-cell imaging of signaling events

Your Task

Implement three functions:

1. fret_efficiency(r_nm, R0_nm) — FRET transfer efficiency (0 to 1)
2. fret_distance_nm(E, R0_nm) — donor-acceptor distance from measured efficiency
3. fret_rate_s(r_nm, R0_nm, tau_D_s) — FRET rate constant in s⁻¹