134 4.4  Fluorescence Correlation Spectroscopy

This therefore can be rewritten as

(4.15)

I

t

P r P r

C r

C r t

V V

C

P r

Auto

′

′

′

( ) =

∫∫( )

′

( )〈

(

)

′

(

)〉

〈〉

( )

∫











δ

δ

,

,

0

d

d

dV

(

)

The displacement of a fluorophore as a function of time can be modeled easily for the case of

Brownian diffusion (see Equation 2.12), to generate an estimate for the number density auto­

correlation term 〈

(

)

(

)〉

′

δ

δ

τ

C r

C r

,

,

0

:

(4.16)

〈

(

)

′

(

)〉= 〈〉

(

)

−

−′

















′

′

′

δ

δ

π

C r

C r t

C

Dt

r

r

Dt









,

,

/

0

1

4

4

3 2

2

exp

An important result from this emerges at the zero time interval value for the autocorrel­

ation intensity function, which then approximates to 1/​V〈C〉, or 1/​〈N〉, where 〈N〉 is the mean

(time averaged) number of fluorophores in the confocal volume. The full form of the auto­

correlation function for one type of molecule diffusing in three spatial dimensions through a

roughly Gaussian confocal volume with anomalous diffusion can be modeled as Im:

(4.17)

I

t

I

I

t

t

m

m

m

′

′

′

( ) =

∞

( )+

( )

+ (

)

(

)

+ (

)

0

1

1

2

/

/

/

τ

τ α

α

Fitting experimental data IAuto with model Im yields estimates for parameters Im(0) (simply the

intensity due to the mean number of diffusing molecules inside the confocal volume), I(∞)

(which is often equated to zero), τ, and α. The parameter a is the anomalous diffusion coeffi­

cient. For diffusion in n spatial dimensions with effective diffusion coefficient D, the general

equation relating the mean squared displacement 〈R²〉 after a time t for a particle exhibiting

normal or Brownian diffusion is given by Equation 2.12, namely, 〈R²〉 =​ 2nDt. However, in the

more general case of anomalous diffusion, the relation is

(4.18)

〈

〉=

R

nDt

2

2

α

The anomalous diffusion coefficient varies in the range 0–​1 such that 1 represents free

Brownian diffusion. The microenvironment inside a cell is often crowded (certain parts of

the cell membrane have a protein crowding density up to ~40%), which results in hindered

mobility termed anomalous or subdiffusion. A “typical” mean value of a inside a cell is 0.7–​

0.8, but there is significant local variability across different regions of the cell.

The time parameter τ in Equation 4.17 is the mean “on” time for a detected pulse. This can

be approximated as the time taken to diffuse in the 2D focal plane, a mean squared distance,

which is equivalent to the lateral width w of the confocal volume (the full PSF width equiva­

lent to twice the Abbe limit of Equation 4.3, or ~400–​600 nm), indicating

(4.19)

τ

α

≈^

^



^

w

D

2

1

4

/

Thus, by using the value of τ determined from the autocorrelation fit to the experimental

data, the translational diffusion coefficient D can be calculated.

4.4.2  FCS ON MIXED MOLECULE SAMPLES

If more than one type of diffusing molecule is present (polydisperse diffusion), then the auto­

correlation function is the sum of the individual autocorrelation functions for the separate