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Autocorrelation Due to Multicomponent Translational Diffusion

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 (f₁ and f₂ to fractions of the total particle number (f^P₁ and f^P₂) is given by

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

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:

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

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

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

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

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

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

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

If we substitute each component’s autocorrelation function for g₁ and g₂, Equation 9 becomes

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

We may generalize Equation 10 to M diffusing components, where

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

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

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

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^P₁ and f^P₂. These are defined as follows:

where f^P₁ + f^P₂ = 1.

To obtain the values of f^P₁ and f^P₂ from the values of f₁ and f₂, we can simplify the relationships by determining the relative values of each pair of fractions, as follows:

From these relationships, we can determine that

which, combined with the condition that f^P₁ + f^P₂ = 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