  Many types of cells are able to accurately sense shallow gradients of
chemicals across their diameters, allowing the cells to move towards or away
from chemical sources. This chemotactic ability relies on the remarkable
capacity of cells to infer gradients from particles randomly arriving at
cell-surface receptors by diffusion. Whereas the physical limits of
concentration sensing by cells have been explored, there is no theory for the
physical limits of gradient sensing. Here, we derive such a theory, using as
models a perfectly absorbing sphere and a perfectly monitoring sphere, which,
respectively, infer gradients from the absorbed surface particle density or the
positions of freely diffusing particles inside a spherical volume. We find that
the perfectly absorbing sphere is superior to the perfectly monitoring sphere,
both for concentration and gradient sensing, since previously observed
particles are never remeasured. The superiority of the absorbing sphere helps
explain the presence at the surfaces of cells of signal degrading enzymes, such
as PDE for cAMP in Dictyostelium discoideum (Dicty) and BAR1 for mating factor
alpha in Saccharomyces cerevisiae (budding yeast). Quantitatively, our theory
compares favorably to recent measurements of Dicty moving up a cAMP gradient,
suggesting these cells operate near the physical limits of gradient detection.
