Stochastic Analysis of the Motion of DNA Nanomechanical Bipeds

Abstract

In this paper, we formulate and analyze a Markov process modeling the motion of DNA nanomechanical walking devices.We consider a molecular biped restricted to a well-defined one-dimensional track and study its asymptotic behavior.Our analysis allows for the biped legs to be of different molecular composition, and thus to contribute differently to the dynamics. Our main result is a functional central limit theorem for the biped with an explicit formula for the effective diffusivity coefficient in terms of the parameters of the model. A law of large numbers, a recurrence/transience characterization and large deviations estimates are also obtained.Our approach is applicable to a variety of other biological motors such as myosin and motor proteins on polymer filaments.

First-passage properties of molecular spiders

“…Analytical results regarding the asymptotic behaviors (limit theorems, transience, recurrence, and rate of escape) of spiders have been derived in Refs. [25,26]. In Refs.…”

Molecular spiders on a plane

“…An interesting new feature noticed in [16] is the emergence of a super-diffusive growth of the mean-square displacement, x 2 (t) ∼ t α with 1 < α < 2, which holds on a surprisingly long time span; eventually, the super-diffusive growth crosses over to the diffusive growth. Several rigorous results concerning the asymptotic behaviors (limit theorems, transience, recurrence, and rate of escape) of molecular spiders have been established in [18,19]. In Refs.…”

Spider walk in a random environment

“…The underlying random walk in [9] is a simple random walk on the integer lattice. The one-dimensional biped model of [9] was generalized by Ben-Ari et al in [12] to include asymmetric transition rates. For a general discussion on asymmetric spiders see, for instance, [69] and references therein.…”

“…The underlying random walk is a continuous-time process, which in general may be controlled, self-interacting or in random media, and thus it does not have to be Markovian. In [7] and [12] the underlying random walk is a usual nearest-neighbor random walk on the integer lattice Z, in [9] it is a 1-excited random walk (ERW) on Z introduced in [14], and in [37] it is a random walk in random environment (RWRE) on Z (see for instance [97] for a survey on RWRE). In [36] the authors considered a general graph G and a spider with leg relocation dynamics evolving according to the law of a nearest-neighbor symmetric random walk on G. Most of the work in [36] focuses on recurrence and transience criteria and on comparing the spider random walk to the simple nearest-neighbor random walk on the same tree.…”

“…In [12] the authors considered an asymmetric bipedal spider, sometimes referred to as a molecular biped in the following, with in general different transition mechanisms (underlying random walks) associated with each leg. The main results in [12] are a strong law of large numbers and a functional central limit theorem for the location of the molecular biped with explicit expressions for the asymptotic velocity and limiting variance.…”