  We combine stochastic thermodynamics, large deviation theory, and information
theory to derive fundamental limits on the accuracy with which single cell
receptors can estimate external concentrations. As expected, if estimation is
performed by an ideal observer of the entire trajectory of receptor states,
then no energy consuming non-equilibrium receptor that can be divided into
bound and unbound states can outperform an equilibrium two-state receptor.
However, when estimation is performed by a simple observer that measures the
fraction of time the receptor is bound, we derive a fundamental limit on the
accuracy of general nonequilibrium receptors as a function of energy
consumption. We further derive and exploit explicit formulas to numerically
estimate a Pareto-optimal tradeoff between accuracy and energy. We find this
tradeoff can be achieved by nonuniform ring receptors with a number of states
that necessarily increases with energy. Our results yield a novel thermodynamic
uncertainty relation for the time a physical system spends in a pool of states,
and generalize the classic 1977 Berg-Purcell limit on cellular sensing along
multiple dimensions.
