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.