Molecular spiders with memory

Abstract

Synthetic biomolecular spiders with "legs" made of single-stranded segments of DNA can move on a surface which is also covered by single-stranded segments of DNA complementary to the leg DNA. In experimental realizations, when a leg detaches from a segment of the surface for the first time it alters that segment, and legs subsequently bind to these altered segments more weakly. Inspired by these experiments, we investigate spiders moving along a one-dimensional substrate, whose legs leave newly visited sites at a slower rate than revisited sites. For a random walk (one-leg spider), the slowdown does not affect the long time behavior. For a bipedal spider, however, the slowdown generates an effective bias toward unvisited sites, and the spider behaves similarly to the excited walk. Surprisingly, the slowing down of the spider at new sites increases the diffusion coefficient and accelerates the growth of the number of visited sites.

FIG. 1

The scaled distribution (27) of the number of visited sites by the exited walk for several values of bias.

FIG. 2

The amplitude A in the growth law $\langle N\rangle =A\sqrt{t}$ as a function of r. For the bipedal L = 2 spider with maximal distance between the legs s = 2, simulations are in excellent agreement with the theoretical prediction (43). The same is also valid for the L = 3 spider with S = 3, where the theoretical prediction is given by (49).

FIG. 3

Simulation results for the diffusion coefficient as a function of rate r for the bipedal L = 2 spider with maximal distance between the legs S = 2, and for the L = 3 spider with S = 3. The dashed lines are the approximation (44).