Abstract
The mean-squared displacement (MSD) is an averaged quantity widely used to assess anomalous diffusion. In many cases, such as molecular motors with finite processivity, dynamics of the system of interest produce trajectories of varying duration. Here we explore the effects of finite processivity on different measures of the MSD. We do so by investigating a deceptively simple dynamical system: a one-dimensional random walk (with equidistant jump lengths, symmetric move probabilities, and constant step duration) with an origin-directed detachment bias. By tuning the time dependence of the detachment bias, we find through analytical calculations and trajectory simulations that the system can exhibit a broad range of anomalous diffusion, extending beyond conventional diffusion to superdiffusion and even superballistic motion. We analytically determine that protocols with a time-increasing detachment lead to an ensemble-averaged velocity increasing in time, thereby providing the effective acceleration that is required to push the system above the ballistic threshold. MSD analysis of burnt-bridges ratchets similarly reveals superballistic behavior. Because superdiffusive MSDs are often used to infer biased, motor-like dynamics, these findings provide a cautionary tale for dynamical interpretation.
Highlights
The mean-squared displacement (MSD) is often used to assess anomalous diffusion in molecular systems
We demonstrate that this acceleration by detachment can manifest in a more realistic system by calculating MSD for burnt-bridges ratchets, which we find to exhibit apparently superballistic behavior
To demonstrate that the anomalous α shown above is observed in more realistic systems with finite processivity, we examine simulated trajectories of a burnt-bridges ratchet (BBR), reported previously [41]
Summary
The mean-squared displacement (MSD) is often used to assess anomalous diffusion in molecular systems. For processes that obey (1), such as a discrete-time random walk, the displacement distribution after N steps limits to a Gaussian distribution. The N individual steps of a discrete-time random walk are independent and identically distributed, with their sum governed by the central limit theorem [2]. Anomalous diffusion refers to systems that do not have a linear time dependence of the MSD. Anomalous diffusion is thought to emerge in stochastic systems whose displacement distributions are not Gaussian, and is intimately connected with the breakdown of the central limit theorem [2].
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