Theory for the optimal detection of time-varying signals in cellular sensing systems.

Living cells often need to measure chemical concentrations that vary in time. To this end, they deploy many resources, e.g. receptors, downstream signaling molecules, time and energy. Here, we present a theory for the optimal design of a large class of sensing systems that need to detect time-varying signals, a receptor driving a push-pull network. The theory is based on the concept of the dynamic input-output relation, which describes the mapping between the current ligand concentration and the average receptor occupancy over the past integration time. This concept is used to develop the idea that the cell employs its push-pull network to estimate the receptor occupancy and then uses this estimate to infer the current ligand concentration by inverting the dynamic input-output relation. The theory reveals that the sensing error can be decomposed into two terms: the sampling error in the estimate of the receptor occupancy and the dynamical error that arises because the average ligand concentration over the past integration time may not reflect the current ligand concentration. The theory generalizes the design principle of optimal resource allocation previously identified for static signals, which states that in an optimally designed sensing system the three fundamental resource classes of receptors and their integration time, readout molecules, and energy are equally limiting so that no resource is wasted. However, in contrast to static signals, receptors and power cannot be traded freely against time to reach a desired sensing precision: there exists an optimal integration time that maximizes the sensing precision, which depends on the number of receptors, the receptor correlation time, and the correlation time and variance of the input signal. Applying our theory to the chemotaxis system of Escherichia coli indicates that this bacterium has evolved to optimally sense shallow gradients.