Spatiotemporal constraints on concentrations improve cellular concentration sensing beyond classical limits

Numerous biological processes require sensing chemical concentrations. Limits on the accuracy of such sensing as a function of the sensing duration have been established in the presence of shot noise for quasi-static and randomly changing concentrations. However, in development, sensed concentration profiles are not random. We propose that this prior knowledge can be used to improve and speed up concentration sensing by using past molecular binding events to predict current concentration. By formulating the constrained sensing problem as Bayesian inference for a broad class of spatiotemporal profiles, we derive new limits on sensing accuracy. Our analysis shows that MAP estimation surpasses both the classical Berg-Purcell (Poisson) limit and maximum likelihood estimation, achieving sensing accuracy $\delta c/c = 1/\sqrt{aN}$, where sometimes $a > 1$. This demonstrates that knowing the statistical structure of concentration profiles can enhance sensing precision. This improved accuracy could help explain how cells make surprisingly fast, yet accurate, cell fate commitments during development.