Decoding molecular distributional codes through collective instabilities

Biological information is often encoded in molecular variants that differ in just a few chemical traits, such as the number of phosphorylated sites or ubiquitin chain length, rather than in arbitrarily distinct species. These molecular distributions carry information about cellular state, yet reading them with conventional molecular circuits requires prohibitively many distinct sensors. In contrast, we show that collective physical instabilities can naturally integrate the information encoded in such distributions. Using an information-theoretic matching condition between an encoded distribution and a physical readout, we derive a geometric condition that any good decoder must satisfy, and establish that phase separation, percolation, and membrane curvature instabilities all approach it for biologically natural distributions while simple mass-action binding does not. Using mean-field theory and lattice Monte Carlo simulations, we find that phase separation reads the shape of a distribution beyond its mean, robustly capturing its variance and, more weakly, its skewness, whereas mass-action binding detects only the mean. Near phase boundaries the readout captures nearly all the information present in the molecular population. Finite valency, through the threshold for network formation, adds discriminatory power invisible to mean-field theory. These results suggest that cells can exploit collective physical instabilities as natural, compact, yet near-optimal sensors for decoding molecular distributional codes.