Comparative introduction to mathematical theory of the Fluorescence Correlation Spectroscopy for a diffusion process: Classical approach vs simplified approach
Abstract: Since its introduction in the 1970s, Fluorescence Correlation Spectroscopy (FCS) has become a standard physical chemistry tool to investigate diffusion processes in living cells complementing other fluorescence microscopy techniques including but not limited to Fluorescence Recovery after Photobleaching (FRAP) and Single Particle Tracking (SPT). For diffusion FCS analysis, an autocorrelation curve of fluorescence fluctuation data is compared with a theoretical autocorrelation function or FCS equation for a diffusion coefficient. Additionally, FCS is now being applied not only to a diffusion process but also to 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 derivation steps with the diffusion FCS equation, it is important to understand mathematical theory behind the diffusion FCS equation. However, because the derivation of FCS equations requires advanced Fourier Transform and inverse Fourier Transform involving intermediate to advance level mathematical techniques, which most biologists and bioengineers are not familiar with, it is often treated as a black box in the classroom. In this work, we first provide a simple and straightforward derivation of FCS equation for free diffusion based on calculus-level mathematics without the knowledge of Fourier transform and inverse Fourier transform, so that FCS equations and its applications are accessible to a broad audience. Additionally, we present the classical Fourier transform based derivation of the diffusion FCS equation and compare it with our new simplified approach.
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