Chapter 3 – Making Light Work in Biology  67

single fluorophore, from the measured emission intensity parallel or perpendicular (I2) to the

incident linear E-​field polarization vector after a time t:

(3.9)

r t

I

t

I

t

I

t

I

t

( ) =

( ) −

( )

( )+

( )

1

2

2

2

2

In fluorescence anisotropy measurements, the detection system will often respond differ­

ently to the polarization of the emitted light. To correct for this, a G-​factor is normally used,

which is the ratio of the vertical polarization detector sensitivity to the horizontal polariza­

tion detector sensitivity. Thus, in Equation 3.9, the parameter I2(t) is replaced by GI2(t).

Note that another measure of anisotropy is sometimes still cited in the literature as a par­

ameter confusingly called the polarization, P, such that P =​ 3r/​(2 +​ r) =​ (I1 − I2)/​(I1 +​ I2), to be

compared with Equation 3.9. The anisotropy decays with time as the fluorophore orientation

rotates, such that for freely rotating fluorophores

(3.10)

r t

r

t

( ) =

−













0^exp

R

τ

where r0 is called the “initial anisotropy” (also known as the “fundamental anisotropy”), which

in turn is related to the relative angle θ between the incident E-​field polarization and the tran­

sition dipole moment by

(3.11)

r

cos

0

2

3

5

=

−

(

)

θ

1

This indicates a range for r0 of −0.2 (perpendicular dipole interaction) to +​0.4 (parallel dipole

interaction). Anisotropy can be calculated from the measured fluorescence intensity, for

example, from a population of fluorophores such as in in vitro bulk fluorescence polariza­

tion measurements. The rotational correlation time is inversely proportion to the rotational

diffusion coefficient DR such that

(3.12)

τR

R

=

1

6D

The rotational diffusion coefficient is given by the Stokes–​Einstein relation (see Chapter 2),

replacing the drag term for the equivalent rotational drag coefficient. Similarly, the mean

squared angular displacement 〈θ²〉 observed after a time t relates to DR in an analogous way

as for lateral diffusion:

(3.13)

〈

〉=

θ²

2D t

R

For a perfect sphere of radius r rotating in a medium of viscosity η at absolute temperature T,

the rotational correlation time can be calculated exactly as

(3.14)

τ

π

η

R

B

= ⁴

3

3r

k T

Molecules that integrate into phospholipid bilayers, such as integrated membrane proteins

and membrane-​targeting organic dyes, often orientate stably parallel to the hydrophobic

tail groups of the phospholipids such that their rotation is confined to that axis with the

frictional drag approximated as that of a rotating cylinder about its long axis using the