Theory of fluorescence correlation spectroscopy at variable observation area for two-dimensional diffusion on a meshgrid

It has recently been proposed, with the help of numerical investigations, that fluorescence correlation spectroscopy at variable observation area can reveal the existence of a meshgrid of semi-permeable barriers hindering the two-dimensional diffusion of tagged particles, such as plasmic membrane constituents. We present a complete theory confirming and accounting for these findings. It enables a reliable, quantitative exploitation of experimental data from which the sub-wavelength mesh size can be extracted. Time scales at which fluorescence correlation spectroscopy must be performed experimentally are discussed in detail.

💡 Research Summary

The paper develops a comprehensive theoretical framework for using fluorescence correlation spectroscopy (FCS) with a variable observation area to probe two‑dimensional diffusion that is hindered by a semi‑permeable meshgrid of barriers, a situation commonly encountered in plasma membranes. The authors begin by modeling the membrane as a regular square lattice of spacing L, where each lattice line represents a barrier that allows particles to cross with a probability k (0 < k < 1). Inside each cell of the lattice particles undergo free Brownian motion with diffusion coefficient D.

From this physical picture they write a master equation for the probability density p(r,t) and, in the continuum limit, obtain a diffusion equation supplemented by mixed reflecting‑transmitting boundary conditions at the barriers. The observation region is defined by a Gaussian (or top‑hat) illumination profile of characteristic width w, which can be varied experimentally to change the effective observation area A ≈πw². The fluorescence intensity F(t) is proportional to the integral of p(r,t) over the illumination profile, and the normalized autocorrelation function G(τ;A) =⟨δF(t)δF(t+τ)⟩/⟨F⟩² is the central observable.

By expanding the solution in Fourier modes that respect the lattice periodicity, the authors derive an exact expression for G(τ;A) as a sum over lattice wavevectors q:

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