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Appendices

FCS Classroom

Autocorrelation Due to Multicomponent Translational Diffusion

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

equation

and

equation

where

equation

When the diffusing components have unequal brightness/particle, the correction needed to convert fractional contributions of each component to the correlation function (f1 and f2 to fractions of the total particle number (fP1 and fP2) is given by

equation where q2/q1 is the ratio of the brightness/particle of component 2 and 1.

Detailed Derivation

Combining Autocorrelation Functionsback to top

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

equation
, (1)

where N1 and N2 are the number of particles of components 1 and 2, respectively, and q1 and q2 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:

equation
. (2)

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

equation
. (3)

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

equation
. (4)

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

equation
. (5)

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

equation
. (6)

Simple Case: All Components are Equally Bright (q1 = q2)back to top

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

equation
. (7)

If we define the total number of diffusing particles as Np = <N1> + <N2> and the fractions of the two components as

equation
, (8)

where f1 + f2 = 1, we can rewrite Equation 7 as

equation
. (9)

If we substitute each componentís autocorrelation function for g1 and g2, Equation 9 becomes

equation
, (10)

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

We may generalize Equation 10 to M diffusing components, where

equation
, (11)

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

equation
. (12)

Complex Case: Components have Different Brightnesses (q1 ≠ q2)back to top

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 (f1 and f2 in Equation 10) to fractions of the total particle number.

If we define the ratio of quantum efficiencies as α = q2/q1, Equation 6 becomes

equation
. (13)

If we use the form of Equation 9 to determine the values of f1 and f2 in Equation 13, we get that

equation
. (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 fP1 and fP2. These are defined as follows:

equation
. (15)

where fP1 + fP2 = 1.

To obtain the values of fP1 and fP2 from the values of f1 and f2, we can simplify the relationships by determining the relative values of each pair of fractions, as follows:

equation
. (16)

From these relationships, we can determine that

equation
. (17)

which, combined with the condition that fP1 + fP2 = 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