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Accepted · first revision · record

Title: Energy cost of sensing beyond the Berg-Purcell limit
Plain: 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: 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.
Settled by: A proof of the bound for general Markov readout networks together with an explicit network that saturates it.
Kind: well-posed
Answer type: value
References: (empty)
Literature status: partially-resolved
Status 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.
Posed since: 1977