Chapter 4 – Making Light Work Harder in Biology  129

4.3.2  FÖRSTER RADIUS AND THE KAPPA-​SQUARED ORIENTATION FACTOR

The Förster radius R0 is given by a complex relation of photophysical factors:

(4.8)

R

Q

f

n N

f

A

0

2

0

4

4

0

1 6

0 529

≈

















∞

∞^∫

∫

.

/

κ

ε

λ

λ

λ

D

A

D

D

d

d

where

QD is the quantum yield of the donor in the absence of the acceptor

κ² is the dipole orientation factor

n is the refractive index of the medium (usually of water, ~1.33)

NA is the Avogadro’s number

the integral term in the numerator is for the spectral overlap integral such that fD is the

donor emission spectrum (with the integral in the denominator normalizing this)

εA is the wavelength-​dependent molar extinction coefficient or molar absorptivity of the

acceptor

Typical values of R0 for FRET pairs are in the range 3–​6 nm. The R⁶ dependence on ε results

in a highly sensitive response with distance changes. For example, for R < 5 nm, the FRET

efficiency ε is typically 0.5–​1, but for R > 5 nm, ε falls steeply toward zero. Thus, the technique

is very good for determining putative molecular interaction. The κ² factor is given by

(4.9)

κ

θ

θ

θ

2

2

3

=

−

(

)

cos

cos

cos

T

A

D

where angles θT, θA, and θD are relative orientation angles between the acceptor and donor,

defined in Figure 4.2e. The κ² factor can in theory vary from 0 (transition dipole moments

are perpendicular) to 4 (transition dipole moments are collinear), whereas parallel transi­

tion dipole moments generate a κ² of exactly 1. FRET donor–​acceptor fluorophore pairs that

rotate purely isotropically have an expected κ² of precisely 2/​3. However, care must be taken

not to simply assume this isotropic condition. The condition is only true if the rotational

correlation time for both the donor and acceptor fluorophores is significantly less than the

sampling time scale in a given experiment. Typical rotational correlation times scales are

~10⁻⁹ s, and so for fluorescence imaging experiments where the sampling times are 10⁻² to

10⁻³ s, the assumption is valid, though for fast nonimaging methods such as confocal fluor­

escence detection and fluorescence correlation spectroscopy (FCS) sampling may be over a

~10⁻⁶ time scale or faster and then the assumption may no longer be valid. An implication

of anisotropic fluorophore behavior is that an observed change in FRET efficiency could be

erroneously interpreted as a change in donor–​acceptor distance when in fact it might just be

a relative orientation change between their respective dipole moments.

4.3.3  SINGLE-​MOLECULE FRET

FRET is an enormously valuable tool for identifying putative molecular interactions between

biomolecules, as we discussed for the FLIM–​FRET technique previously (see Chapter 3). But

using light microscopy directly in nonscanning imaging methods enables powerful single-​

molecule FRET (smFRET) techniques to be applied to addressing several biological questions

in vitro (for a practical overview see Roy et al., 2008). The first smFRET biophysical inves­

tigation actually used near-​field excitation (Ha et al., 1996), discussed later in this chapter.

However, more frequently today, smFRET involves diffraction-​limited far-​field light micros­

copy with fluorescence detection of both the donor and accept or fluorophore in separate

color channels of high-​sensitivity fluorescence microscope (see Worked Case Example 4.2).

The majority of smFRET studies to date use organic dyes, for example, a common FRET pair