Energy cost of sensing beyond the Berg-Purcell limit
In plain words
In 1977 Berg and Purcell estimated how precisely a cell can measure a chemical concentration in a fixed time. Spending energy in the readout chemistry can do better than simple counting, and the exact trade-off between energy and precision is not settled.
Precise statement
For a receptor or receptor array of size a, concentration c, diffusion constant D and integration time T, the Berg-Purcell estimate is $(dc/c)^2 \sim 1/(D a c T)$, and maximum-likelihood readout lowers it by a factor of 2. Find the tight lower bound on $(dc/c)^2$ as a function of free energy dissipated per measurement by an arbitrary downstream reaction network, including receptor and readout noise, and identify the network architecture that reaches it.
What would settle it
A proof of the bound for general Markov readout networks together with an explicit network that saturates it.
Status in the literature
Unverified note
Bounds for specific readout networks were derived around 2012 to 2014; a general tight bound for arbitrary networks is not established as of 2026, to this survey's knowledge.