  Living cells often need to extract information from biochemical signals that
are noisy. We study how accurately cells can measure chemical concentrations
with signaling networks that are linear. For stationary signals of long
duration, they can reach, but not beat, the Berg-Purcell limit, which relies on
uniformly averaging in time the fluctuations in the input signal. For short
times or nonstationary signals, however, they can beat the Berg-Purcell limit,
by non-uniformly time-averaging the input. We derive the optimal weighting
function for time averaging and use it to provide the fundamental limit of
measuring chemical concentrations with linear signaling networks.
