Diffusion-limited rates on low-dimensional manifolds with extreme aspect ratios

Abstract: 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.