You ain't seen nothing, and yet: Future biochemical concentrations can be predicted with surprisingly high accuracy

Abstract

Accurate sensing of chemical concentrations is essential for numerous biological processes. The accuracy of this sensing, for small numbers of molecules, is limited by shot noise. Corresponding theoretical limits on sensing precision, as a function of sensing duration, have been well-studied in the context of quasi-static and randomly fluctuating concentrations. However, during development and in many other cases, concentration profiles are not random but exhibit predictable spatiotemporal patterns. We propose that leveraging prior knowledge of these structured profiles can improve and accelerate concentration sensing by utilizing information from current molecular binding events to predict future concentrations. By framing the constrained sensing problem as Bayesian inference over an allowed class of spatiotemporal profiles, we derive new theoretical limits on sensing accuracy. Our analysis reveals that maximum a posteriori (MAP) estimation can outperform the classical Berg-Purcell and maximum-likelihood (Poisson counting) limits, achieving a sensing precision of $\delta c/c = 1/\sqrt{a^2N}$, where $N$ is the number of binding events, and $a > 1$ in certain cases. Thus knowledge of the statistical structure of concentration profiles enhances sensing precision, providing a potential explanation for the rapid yet highly accurate cell fate decisions observed during development.