Apparent superballistic dynamics in one-dimensional random walks with biased detachment

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, the 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. An MSD analysis of burnt-bridges ratchets similarly reveals superballistic behavior. Because superdiffusive MSDs are often used to infer biased, motorlike dynamics, these findings provide a cautionary tale for dynamical interpretation.

DNA Gold Nanoparticle Motors Demonstrate Processive Motion with Bursts of Speed Up to 50 nm Per Second

“…This value is on par with the speed of biological motor proteins. , Furthermore, analyses of 280 motors confirmed that most tracked particles (85%) were super diffusive with an average α = 1.37 (Figure g). A recent modeling study showed that in systems where motors progressively dissociate from the surface, selecting only the track-bound BBRs for analysis can lead to bias and an overestimate of α . However, in our system, we observed that a subset of motors tend to stall as well as dissociate over longer observation times.…”

“…A recent modeling study showed that in systems where motors progressively dissociate from the surface, selecting only the track-bound BBRs for analysis can lead to bias and an overestimate of α. 44 However, in our system, we observed that a subset of motors tend to stall as well as dissociate over longer observation times. Careful analysis of α confirmed that most S15).…”

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“…For each multivalency and span combination, we simulate N = 500 trajectories for walkers that remain processive and N = 20 000 trajectories for nonprocessive walkers. The N trajectories are then analyzed by an ensemble trajectory-averaged mean squared displacement (MSD ETA ) given by Δ r j 2 is the squared displacement of the walker in the j th trajectory at time-lag τ, T j is the duration of the j th trajectory, and Δ t is the simulation timestep . The position of a multivalent walker is defined as the mean position of all bound feet and is used to compute the walker’s displacement, Δ r j .…”

An Artificial Protein-Based Burnt-Bridges Molecular Motor Design

“…The mean squared displacement was calculated in two ways denoted by MSD TA and MSD ETA [58]. The trajectory-averaged MSD TA is computed independently for each trajectory and uses the mean of the squared displacements for each time lag τ : MSD TA,j (τ ) ≡ ∆r…”