  We construct a theoretical model for the dynamics of a microscale colloidal
particle, modeled as an interval, moving horizontally on a DNA-coated surface,
modelled as a line coated with springs that can stick to the interval.
Averaging over the fast DNA dynamics leads to an evolution equation for the
particle in isolation, which contains both friction and diffusion. The
DNA-induced friction coefficient depends on the physical properties of the DNA,
and substituting parameter values typical of a 1$\mu$m colloid coated densely
with weakly interacting DNA gives a coefficient about 100 times larger than the
corresponding coefficient of hydrodynamic friction. We use a mean-field
extension of the model to higher dimensions to estimate the friction tensor for
a disc rotating and translating horizontally along a line. When the DNA strands
are very stiff and short, the friction coefficient for the disc rolling
approaches zero while the friction for the disc sliding remains large.
Together, these results could have significant implications for the dynamics of
DNA-coated colloids or other ligand-receptor systems, implying that DNA-induced
friction between colloids can be stronger than hydrodynamic friction and should
be incorporated into simulations, and that it depends nontrivially on the type
of relative motion, possibly causing the particles to assemble into
out-of-equilibrium metastable states governed by the pathways with the least
friction.
