A Simple Derivation of Diffusion Fluorescence Correlation Spectroscopy Equations

Since its introduction in the 1970s, Fluorescence Correlation Spectroscopy (FCS) has become a standard biophysical and physical chemistry tool to investigate not only a diffusion but also a broad range of biochemical processes including binding kinetics and anomalous diffusion. Since the derivation of FCS equations for many biochemical processes shares many common steps with the diffusion FCS equation, it is important to understand the mathematical theory behind the diffusion FCS equation. However, because the derivation of FCS equations requires advanced Fourier Transform and inverse Fourier Transform theory, which most biologists and biochemists are not familiar with, it is often treated as a black box in practice. In this study, we provide a simple and straightforward step-by-step derivation of FCS equations for free diffusion based on calculus-level mathematics, so that FCS equations and its applications can be accessible to a broad audience. Additionally, we compare our derivation with the conventional Fourier Transform and inverse Fourier Transform theory based approach.