{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"872b856f8b154776aec9a4bac48ba72bd7a57c7d41d9751d5bf1a260f122ce09","created":"2026-10-03T07:17:57Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"006e89bbec1d1298dcad18d41073611e94e57f15a384277fe9d28ac4a413a174","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"value","assisted_by":[],"external_id":"bio.cellular-sensing.energy-accuracy-bound","kind":"well-posed","literature_status":"partially-resolved","n":"1","parents":[],"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.","posed_since":"1977","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.","problem_ref":null,"references":"","settled_by":"A proof of the bound for general Markov readout networks together with an explicit network that saturates it.","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.","title":"Energy cost of sensing beyond the Berg-Purcell 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