Finite-range viscoelastic subdiffusion in disordered systems with inclusion of inertial effects

Abstract

This work justifies further paradigmatic importance of the model of viscoelastic subdiffusion in random environments for the observed subdiffusion in cellular biological systems. Recently, we showed (2018, PCCP , 20 , 24140) that this model displays several remarkable features, which makes it an attractive paradigm to explain the physical nature of subdiffusion occurring in biological cells. In particular, it combines viscoelasticity with distinct non-ergodic features. We extend this basic model to make it suitable for physical phenomena such as subdiffusion of lipids in disordered biological membranes upon including the inertial effects. For lipids, the inertial effects occur in the range of picoseconds, and a power-law decaying viscoelastic memory extends over the range of several nanoseconds. Thus, in the absence of disorder, diffusion would become normal on a time scale beyond this memory range. However, both experimentally and in some molecular-dynamical simulations, the time range of lipid subdiffusion extends far beyond the viscoelastic memory range. We study three 1d models of correlated quenched Gaussian disorder to explain the puzzle: singular short-range (exponentially correlated), smooth short-range (Gaussian-correlated), and smooth long-range (power-law correlated) disorder. For a moderate disorder strength, transient viscoelastic subdiffusion changes into the subdiffusion caused by the randomness of the environment. It is characterized by a time-dependent power-law exponent of subdiffusion α ( t ), which can show nonmonotonous behavior, in agreement with some recent molecular-dynamical simulations. Moreover, the spatial distribution of test particles in this disorder-dominated regime is shown to be a non-Gaussian, exponential power distribution with index χ = 1.45–2.3, which also correlates well with molecular-dynamical findings and experiments. Furthermore, this subdiffusion is nonergodic with single-trajectory averages showing a broad scatter, in agreement with experimental observations for viscoelastic subdiffusion of various particles in living cells.

Keywords

• anomalous diffusion
• viscoelasticity
• random environment
• non-Gaussian diffusion
• non-ergodic features