  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.
