Abstract
Cellular signalling involves networks of small, interacting molecular populations and requires precise event timing, despite the presence of noise. The Poisson channel is an important communication model for such noisy interactions, providing a fundamental link between timing precision, information transfer and the rate of signalling. Constraints on this rate limit the Poisson channel capacity, which in turn bounds the precision with which molecular networks can solve estimation problems. We investigate these bounds as a function of signalling rate constraints, for problems in which information about a target molecular species, to be estimated, is encoded in the birth rate of a signalling species. Birth-following is a known heuristic encoder that asserts the maximum signalling rate until every target birth is recorded, then deactivates. Here we derive birth-following as a minimum time signalling (bang-bang) code, and prove that it outperforms these precision bounds for general target birth rate functions over arbitrary signalling network architectures. The simplest of these networks commonly models the dynamics of long-lived proteins. Birth-following is therefore an important reference strategy when the maximum signalling rate is high and the mean rate is no smaller than that of the target molecule. Discreteness is important in this regime. We examine the limit of this regime by removing the maximum signalling constraint. This leads to a Poisson channel with infinite capacity, which should allow completely precise timing. However, we find perfect estimation unrealisable unless the mean condition is maintained. The relationship between estimation and information is therefore not as simple or intuitive as in analogous problems on Gaussian channels. Higher Poisson capacities do not always imply better precision and realisable performance is more dependent on finding suitable encoders that capitalise on the information structure of the signalling problem of interest. There is a need for new information-theoretic metrics that can better account for both the gap between achievable and theoretical precision, and the idiosyncrasies of the Poisson channel.