Confined subdiffusion in three dimensions
Provides a theoretical framework for confined subdiffusion in 3D, relevant for biophysics and single-particle tracking, but the approach is incremental.
The authors solved the 3D Fick's and fractional diffusion equations for reflecting boundaries, deriving an analytical expression for the ensemble-averaged time-averaged MSD via a Mittag-Leffler function. Their model successfully simulated mRNA diffusion in E. coli, matching experimental data.
The three-dimensional (3D) Fick's diffusion equation and fractional diffusion equation are solved for different reflecting boundaries. We use the continuous time random walk model (CTRW) to investigate the time averaged mean square displacement (MSD) of 3D single particle trajectory. Theoretical results show the ensemble average of the time averaged MSD can be expressed analytically by a Mittag-Leffler function. Our new expression is in agreement with previous formulas in two limiting cases which are $ < \overline{δ^2} > \simΔ$ in short lag time and $ < \overline{δ^2}> \simΔ^{1-α}$ in long lag time. We also simulate the experimental data of mRNA diffusion in living E. coli using 3D CTRW model under confined and crowded conditions. The simulated results are well consistent with experimental results. The calculations of power spectral density (PSD) indicate further the subdiffsive behavior of individual trajectory.