Limits to concentration sensing in inhomogeneous environments

Information vital to the life of cells is often encoded by small changes in the concentration of crucial molecules, as with transcription factors in the control of gene expression. At low concentrations, the stochasticity of diffusion sets a bound, the Berg-Purcell limit, on the precision of these measurements. Recent experiments indicate that transcription factors form regions of elevated concentration in the neighborhood of active genes. These inhomogeneities could reflect either intrinsic phase-separation or attraction to scaffold domains with many binding sites, effectively forming an external potential. Do these effects change the limits to concentration sensing beyond just increasing the mean concentration? We use statistical mechanics to treat the case of an external potential, and find that the noise floor is controlled by the spectrum of a linear operator that includes a term describing the non-uniformity. Surprisingly, we find that this noise floor is unchanged in two different models of the external potential. In the case of condensates, the internal concentration is buffered and there is essentially no local response to overall changes in concentration, making sensing more difficult. The noise floor could be lowered if the cluster could act as a single large sensor, and we explore ways in which this might work.