FCSXpert Solutions: Fluorescence Correlation Spectroscopy Simplified!.
## FCS Classroom

#### Autocorrelation Due to Multicomponent Translational Diffusion

_{2}/q_{1} is the ratio of the brightness/particle of component 2 and 1.
### Detailed Derivation

#### Combining Autocorrelation Functionsback to top

#### Simple Case: All Components are Equally Bright (q_{1} = q_{2})back to top

#### Complex Case: Components have Different Brightnesses (q_{1} ≠ q_{2})back to top

For multicomponent, three-dimensional diffusion, the autocorrelation functions are given by

and

where

When the diffusing components have unequal brightness/particle, the correction needed to convert
fractional contributions of each component to the correlation function (f_{1} and f_{2}
to fractions of the total particle number (f^{P}_{1} and f^{P}_{2}) is given by

We next consider the case of multi-component diffusion. Suppose that a sample contains two independently diffusing species making up the intensity, I, such that

, (1)

where N_{1} and N_{2} are the number of particles of components 1 and 2, respectively, and q_{1} and q_{2} are the proportionality
constants relating intensity to particle number for each component. In physical terms, q is a combination of absorbance efficiency,
quantum (emission) efficiency, and number of dyes per particle.

We can calculate the autocorrelation function of the intensity due both components as follows:

. (2)

We can also define the autocorrelation function for each component separately as follows:

. (3)

Because the two components move independently, the cross terms of Equation 2 are zero and Equation 2 becomes

. (4)

If we substitute Equations 3 into Equation 4, we get:

. (5)

Substituting the definitions of I from Equations 1 into Equation 5, we get

. (6)

For this derivation, we will make the assumption that q1 = q2 = q, meaning that both components have equal intensities/particle, q. Under this assumption, Equation 6 becomes

. (7)

If we define the total number of diffusing particles as N_{p} = <N_{1}> + <N_{2}> and the fractions of the two components as

, (8)

where f_{1} + f_{2} = 1, we can rewrite Equation 7 as

. (9)

If we substitute each component’s autocorrelation function for g_{1} and g_{2}, Equation 9 becomes

, (10)

which is the autocorrelation function for two-component isotropic diffusion.

We may generalize Equation 10 to M diffusing components, where

, (11)

so that the equation for multi-component isotropic diffusion is written as:

. (12)

Here we consider the case where each component has a different intensity proportionality coefficient
and calculate the correction needed to convert fractional contributions of each component to the correlation
function (f_{1} and f_{2} in Equation 10) to fractions of the total particle number.

If we define the ratio of quantum efficiencies as α = q_{2}/q_{1}, Equation 6 becomes

. (13)

If we use the form of Equation 9 to determine the values of f_{1} and f_{2} in Equation 13, we get that

. (14)

The physically-relevant parameter we are interested in is the fraction of the total particle number that is
associated with each component, which we will call f^{P}_{1} and f^{P}_{2}. These are defined as follows:

. (15)

where f^{P}_{1} + f^{P}_{2} = 1.

To obtain the values of f^{P}_{1} and f^{P}_{2} from the values of f_{1} and f_{2}, we can simplify the
relationships by determining the relative values of each pair of fractions, as follows:

. (16)

From these relationships, we can determine that

. (17)

which, combined with the condition that f^{P}_{1} + f^{P}_{2} = 1, gives us a convenient
way of calculating the fraction of total particle number associated with each diffusing component
from the fractional contributions to the correlation functions.

For examples of applying Equation 17, please see Interpreting Fitted Fractions in FCS in our Analysis Notes