Defining reversible binding rates in 1D systems dependent on diffusion, density, and excluded volume

Binding reactions in effectively one-dimensional systems, such as proteins diffusing along DNA or other filaments, pose a fundamental coarse-graining challenge because stochastic trajectories are recurrent in one dimension and, therefore, do not admit a unique, separation-independent macroscopic association rate. As a result, continuum rate equations are not exact in 1D even for initially homogeneous systems. Here, we develop a practical framework for mapping stochastic 1D reaction-diffusion dynamics onto effective kinetic models. We consider reversible protein dimerization (A+A or A+B) and reversible target binding (A+B), with and without excluded volume. Using mean-first-passage arguments and particle-based simulations, we define a density-dependent association rate and a corresponding single-rate approximation and quantify when each provides an accurate description of the underlying stochastic dynamics. We implement 1D reaction-diffusion in the NERDSS software using a free-propagator reweighting algorithm and validate it against known pairwise and many-body limits. Our results show that ordinary rate equations with a single effective rate can accurately reproduce 1D reaction kinetics when the dimensionless parameter governing the ratio of intrinsic to diffusion-limited reactivity is small, with excellent agreement in the strongly rate-limited regime and increasing deviations as diffusion control strengthens. We further show that excluded volume in 1D can appreciably alter both kinetics and equilibrium populations, even at modest particle densities, by reducing accessible length and introducing blockade effects. Together, these results provide quantitative guidance for selecting between spatial simulations, density-dependent rate models, and single-rate continuum descriptions of reversible 1D binding reactions.