  We consider a single-species diffusion-limited annihilation reaction with
reactants confined to a two-dimensional surface with one arbitrarily large
dimension and the other comparable in size to interparticle distances. This
situation could describe reactants which undergo both longitudinal and
transverse diffusion on long filamentous molecules (such as microtubules), or
molecules that undergo truly one-dimensional translational diffusion (e.g. a
transcription factor on DNA) but simultaneously exhibit diffusive behavior in a
second dimension corresponding to a rotational or conformational degree of
freedom. We combine simple analytical arguments and Monte Carlo simulations to
show that the reaction rate law exhibits a crossover from one-dimensional to
two-dimensional diffusion as a function of particle concentration and the size
of the smaller dimension. In the case of a reversible binding reaction, the
diffusion-limited reaction rate is given by the Smoluchowski expression, but
the crossover is revealed in the statistics of particle collision histories.
The results can also be applied to a particle-antiparticle annihilation
reaction.
