Limits to biochemical signalling in a changing environment as an inference problem*

Cells must sense concentrations of external ligands as well as internal signalling molecules in order to adapt to environment changes and execute developmental programs. Berg and Purcell calculated an upper bound to the accuracy of concentration sensing by physical objects, due to the particle nature of molecules and finite sensor size. However, that bound assumed that the concentration to be sensed was constant. In realistic situations, concentrations may vary quickly over orders of magnitude. Here, we calculate a new bound to concentration sensing of a changing concentration by mapping the problem onto a field theory through Bayesian inference, which we solve using a Gaussian approximation. We find that the inverse square root dependency of the error as a function of concentration, ligand diffusivity, sensor size and time in the classical Berg and Purcell bound is replaced by a quartic root. The solution to the inference problem provides dynamical inference equations which can be mimicked by simple biochemical downstream networks, providing a plausible biological implementation of optimal inference.