A Thermodynamically Consistent Reaction--Diffusion Model of Spatial Proofreading in One and Two Dimensions

Spatial proofreading is a mechanism that can enhance molecular discrimination by exploiting nonequilibrium diffusive transport between spatially separated source and readout regions. Here, we introduce a thermodynamically consistent reaction--diffusion model in which a gradient of active substrates is generated self-consistently by a reversible kinase--phosphatase switch coupled to nucleotide chemostats. The resulting chemical-potential gradient explicitly controls the nonequilibrium driving and allows the discrimination potential to be related to microscopic chemical rates and diffusive transport. A simple one-dimensional analysis shows that the ability of an enzyme to discriminate between wrong and right substrates is governed by a subtle balance between substrate-gradient confinement, controlled by phosphatase activity, the diffusive crossing time across the source--readout domain, and selective complex dissociation. We then extend the model to two-dimensional domains, showing that source, i.e. kinase, localization modulates spatial specificity by shaping the effective diffusive paths to the readout boundary. Finally, stochastic simulations reveal that molecular fluctuations generate intermittent wrong-readout events, which can be characterized through an event-weighted measure of specificity. Interestingly, we find that fluctuations promote frequent transitions to long-residence states in which discrimination is better than predicted by the deterministic estimate. Overall, our work highlights the importance of thermodynamically consistent descriptions of spatial proofreading and clarifies how energy input, transport, and spatial organization jointly shape biochemical discrimination.