Maximum likelihood and the single receptor

Biological cells are able to accurately sense chemicals with receptors at their surfaces, allowing cells to move towards sources of attractant and away from sources of repellent. The accuracy of sensing chemical concentration is ultimately limited by the random arrival of particles at the receptors by diffusion. This fundamental physical limit is generally considered to be the Berg & Purcell limit [H.C. Berg and E.M. Purcell, Biophys. J. {\bf 20}, 193 (1977)]. Here we derive a lower limit by applying maximum likelihood to the time series of receptor occupancy. The increased accuracy stems from solely considering the unoccupied time intervals - disregarding the occupied time intervals as these do not contain any information about the external particle concentration, and only decrease the accuracy of the concentration estimate. Receptors which minimize the bound time intervals achieve the highest possible accuracy. We discuss how a cell could implement such an optimal sensing strategy by absorbing or degrading bound particles.

💡 Research Summary

The paper revisits the physical limits of chemical sensing by a single cell‑surface receptor, challenging the long‑standing Berg‑Purcell limit that treats the fraction of time a receptor is occupied as the sole source of information about external ligand concentration. By formulating the problem within a maximum‑likelihood (ML) framework, the authors show that only the intervals during which the receptor is unoccupied contain useful information about the arrival rate of diffusing particles, while the occupied periods contribute no new data and actually degrade estimation accuracy if they are included indiscriminately.

Mathematically, ligand arrival is modeled as a Poisson process with rate k_on·c, where k_on is the binding rate constant and c the external concentration. The durations of unoccupied intervals τ_i are exponentially distributed with mean 1/(k_on·c). The log‑likelihood of observing a series of τ_i’s is summed, differentiated with respect to c, and set to zero, yielding an ML estimator whose variance is lower than the Berg‑Purcell variance by a factor of two. This reduction arises because the ML estimator discards the “dead time” when the receptor is bound and therefore unable to register new arrivals.

A crucial implication is that minimizing the bound‑time of a receptor maximizes the number of informative unoccupied intervals. If bound ligands are removed instantaneously—by absorption, internalization, or enzymatic degradation—the effective unbinding rate k_off becomes very large, and the receptor spends almost all its time free to sample new ligand arrivals. In this limit, the sensing performance approaches the theoretical optimum derived from the ML analysis.

The authors discuss biological plausibility, noting that many cells already employ rapid ligand removal mechanisms (e.g., receptor‑mediated endocytosis, surface‑bound proteases) that effectively shorten bound intervals. They also outline how synthetic biosensors could emulate this strategy by coupling receptors to catalytic surfaces or electrochemical clearing circuits that swiftly eliminate bound analytes, thereby increasing the frequency of informative unoccupied periods.

Overall, the study demonstrates that a single receptor can surpass the classic Berg‑Purcell bound when the measurement protocol is optimized to focus exclusively on unoccupied intervals and when the system is engineered—or evolved—to minimize bound‑state dwell times. This insight refines our understanding of the ultimate physical limits of cellular chemotaxis and offers concrete design principles for next‑generation high‑precision chemical sensors.