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Abstract

We develop a theoretical foundation for a time-series analysis method suitable for revealing the spectrum of diffusion coefficients in mixed Brownian systems, where no prior knowledge of particle distinction is required. This method is directly relevant for particle tracking in biological systems, where diffusion processes are often non-uniform. We transform Brownian data onto the logarithmic domain, where the coefficients for individual modes of diffusion appear as distinct spectral peaks in the probability density. We refer to the method as the logarithmic measure of diffusion, or simply as the logarithmic measure. We provide a general protocol for deriving analytical expressions for the probability densities on the logarithmic domain. The protocol is applicable for any number of spatial dimensions with any number of diffusive states. The analytical form can be fitted to data to reveal multiple diffusive modes. We validate the theoretical distributions and benchmark the accuracy and sensitivity of the method by extracting multi-modal diffusion coefficients from 2D Brownian simulations of poly-disperse filament bundles. Bundling the filaments allows us to control the system non-uniformity and hence quantify the sensitivity of the method. By exploiting the anisotropy of the simulated filaments, we generalize the logarithmic measure to rotational diffusion. By fitting the analytical forms to simulation data, we confirm the method's theoretical foundation. An error analysis in the single-mode regime shows that the proposed method is comparable in accuracy to the standard mean squared displacement approach for evaluating diffusion coefficients. For the case of multi-modal diffusion, we compare the logarithmic measure against other more sophisticated methods, showing that both model selectivity and extraction accuracy are comparable for small data sets. Therefore we suggest that the logarithmic measure, as a method for multi-modal diffusion coefficient extraction, is ideally suited for small data sets, a condition often confronted in the experimental context. Finally, we critically discuss the proposed benefits of the method and its information content.