Molecular spiders on a plane

Abstract

Synthetic bio-molecular spiders with "legs" made of single-stranded segments of DNA can move on a surface covered by single-stranded segments of DNA called substrates when the substrate DNA is complementary to the leg DNA. If the motion of a spider does not affect the substrates, the spider behaves asymptotically as a random walk. We study the diffusion coefficient and the number of visited sites for spiders moving on the square lattice with a substrate in each lattice site. The spider's legs hop to nearest-neighbor sites with the constraint that the distance between any two legs cannot exceed a maximal span. We establish analytic results for bipedal spiders, and investigate multileg spiders numerically. In experimental realizations legs usually convert substrates into products (visited sites). The binding of legs to products is weaker, so the hopping rate from the substrates is smaller. This makes the problem non-Markovian and we investigate it numerically. We demonstrate the emergence of a counter-intuitive behavior -the more spiders are slowed down on unvisited sites, the more motile they become.

Interacting quantum walkers: two-body bosonic and fermionic bound states

“…A classical analogue consists of multi-pedal molecular devices whose legs perform random walks, known as molecular spiders [47]. Their behavior has been investigated theoretically, both on a one-dimensional [48] and a two-dimensional [49] substrate. The quantum version of the problem yields a special kind of N -fermion bound state, which can be analyzed by techniques from integrable systems.…”

Dynamics of the chemical master equation, a strip of chains of equations ind-dimensional space

“…To investigate the validity of our main assumption Eq. (14), one needs to undertake further work with more thorough mathematics. Because nowadays the HJE is becoming popular in solving the CME, as well as for the solution of the diffusion process in case of weak noise, this issue (the validity of the HJE ansatz) is becoming more important.…”

“…(12) is crucial for the ansatz Eq. (14). In [13] we considered an example, when the exponential ansatz p ∼ exp[N u(x)] is invalid for non-smooth functions of transition rates.…”

“…In [10] an implicit expression was obtained for the dynamics of the population distribution variance, following the HJE method of [11], [12], [13]. To obtain an exact complete solution of CME in dimensions D ≥ 2 is a very hard problem and only a few solutions are known, see [14].…”

Cooperative effects enhance the transport properties of molecular spider teams

“…Semenov et al [16] showed that spiders experience a rather extended time period of superdiffusion given that the cleavage rate r is small. More complex spiders in quasi-one [19] and in two dimensions [20] have also been studied. Moreover, there have also been recent studies focussing on mathematical aspects like recurrence, transience and ergodicity [21,22], as well as random environments [23,24].…”