Physical limits on chemical sensing in bounded domains

Cells respond to chemical signals, and the precision with which they can sense these signals is fundamentally limited by the stochastic nature of diffusion and ligand binding. Berg and Purcell famously investigated how well a small sensor in an infinite ligand bath can determine the ligand concentration, and a number of subsequent analyses have refined and built upon their classical estimates. Not all concentration-sensing problems, however, occur in such an infinite geometry. At different scales, subcellular sensors and cells in tissues are both often confronted with signals whose diffusion is affected by confining boundaries. It is thus valuable to understand how basic limits on chemosensation depend on the sensor's size and on its position in the domain in which ligand diffuses. Here we compute how sensor size and proximity to reflecting boundaries affect the diffusion-limited precision of chemosensation for various geometries in one and three dimensions. We derive analytical expressions for the sensing limit in these geometries. Among our conclusions is the surprising result that, in certain circumstances, smaller sensors can be more effective than larger sensors. This effect arises from a trade-off between spatial averaging and time averaging that we analyze in detail. We also find that proximity to confining boundaries can degrade a sensor's precision significantly compared to the precision of the same sensor far from any boundaries.