PHYSICAL REVIEW E 101, 062101 (2020) Effect of polyvalency on tethered molecular walkers on independent one-dimensional tracks David Arredondo 1, 2 and Darko Stefanovic 3, 2, 1 1 Nanoscience and Microsystems Engineering, University of New Mexico, Albuquerque, New Mexico 87131, USA 2 Center for Biomedical Engineering, University of New Mexico, Albuquerque, New Mexico 87131, USA 3 Department of Computer Science, University of New Mexico, Albuquerque, New Mexico 87131, USA (Received 5 September 2019; revised manuscript received 20 March 2020; accepted 30 April 2020; published 1 June 2020) We study the motion of random walkers with residence time bias between first and subsequent visits to a site, as a model for synthetic molecular walkers composed of coupled DNAzyme legs known as molecular spiders. The mechanism of the transient superdiffusion has been explained via the emergence of a boundary between the new and the previously visited sites, and the tendency of the multilegged walker to cling to this boundary, provided residence time for a first visit to a site is longer than for subsequent visits. Using both kinetic Monte Carlo simulation and an analytical approach, we model a system that consists of unipedal walkers, each on its own one-dimensional track, connected by a tether, i.e., a kinematic constraint that no two walkers can be more than a certain distance apart. Even though a single unipedal walker does not at all exhibit directional, superdiffusive motion, we find that a team of unipedal walkers on parallel tracks, connected by a flexible tether, does enjoy a superdiffusive transient. Furthermore, unipedal walker teams exhibit a greater expected number of steps per boundary period and are able to diffuse more quickly than bipedal walker teams, which leads to longer periods of superdiffusion. DOI: 10.1103/PhysRevE.101.062101 I. INTRODUCTION Cargo transport is ubiquitous in biological systems, and is often carried out by molecular walkers that are able to perform very specific actions despite the chaotic nature of the environment in which they act. Cyclic patterns of movement by biological molecular motors are typically the result of a catalyzed conformational change that converts chemical energy into directional movement [1–5]. These mechanisms are highly complex and still not fully understood and therefore pose a problem when we try to design and synthesize novel molecular walkers in the laboratory. Simpler walkers that do not rely on complex conformations can be constructed from DNA, which move stochastically and can be biased in a given direction through rational design of their environment and the track on which they move. The track can be a two-dimensional (2D) surface from which DNA strands protrude, and the walker can be one or more DNA legs that move from site to site via branch migration or detachment and rehybridiza- tion. Some models use additional DNA strands in solution to block and unblock sites adjacent to the walker to achieve directional movement [6,7]. Thubagere et al. [8] employed a diffusive walker that simply wanders about its track to pick up and deliver cargo to marked locations. We are interested in modeling designs that employ restriction endonucleases [9–13] or DNAzymes [14–19] to permanently modify sites via cleavage. Inspired by Rank et al. [20], our abstract model explores the effect of polyvalency (leg number) for teams of tethered walkers on adjacent parallel one-dimensional (1D) tracks, which modify but do not destroy sites, in order to explore the relationship between geometric constraints and probabilistic biases that define walker motion. DNAzyme cleavage of track sites by the walker results in decreased residence time at sites already visited. The DNAzyme is composed of two recognition arms separated by a catalytic core. Once the recognition arms bind to their complements, the catalytic core cleaves the bound strand at its single RNA base facing the catalytic core. Rates of association to and dissociation from recognition sites of the DNAzyme are a function of environmental factors such as ion concentration, choice of ion, and pH [21]. These factors influence the kinetics of the cleaving mechanism. Cha et al. [17] show how this parameter may be optimized for maximum processive motion in the case of a single DNAzyme that walks along a 1D track via branch migration of its recognition arms. Experimental data suggest the ratio of dwell time at cleaved versus uncleaved sites could be in the range (0.05,0.1) [22]. In numerical studies, the difference in residence times for legs at visited and unvisited sites has been explored extensively as it relates to first passage properties and directional movement of bipedal walkers in various environments [23–26]. The effect of increasing dwell time at unvisited sites relative to visited sites is to promote superdiffusive motion, especially in the 1D case. Two general classes of DNAzyme-based walkers exist: those that destroy sites as they are visited, and those that do not. Walkers that destroy sites as they are visited are known as “burnt-bridge” walkers. This walking mechanism results in a self-avoiding random walk and can produce continuous processive motion in one dimension so long as the walker remains attached to the track. Korosec et al. [27] simu- late the behavior of multilegged (polyvalent) burnt-bridge walkers as the width of the track grows, as a function of leg number (polyvalency) and span. In short, they find that 2470-0045/2020/101(6)/062101(10) 062101-1 ©2020 American Physical Society