Anomalous Transport Reshapes the Berg–Purcell Limit of Concentration Sensing Accuracy

Numerous biological processes require sensing chemical concentrations in the face of molecular shot noise. Classical limits(such as Berg-Purcell) on sensing accuracy as a function of measurement time $T$ and detector size $a$ have been derived for normally diffusing particles, yielding a $(Ta)^{-1/2}$ scaling. Yet in many biological processes, molecules exhibit anomalous transport. By treating the detector as a perfect volumetric counter, we establish fundamental bounds on concentration estimation accuracy for anomolously diffusing molecules. We focus on two representative mechanisms of anomalous diffusion: polymer-like diffusion and finite-speed Lévy walks, both motivated by the motion of DNA segments promoting the initiation of transcription. We show that the resulting accuracy scales differently with detector size and measurement time than classical results (e.g., Berg--Purcell). We confirm our theoretical results with numerical simulations.