Eukaryotic cells sense molecular gradients by measuring spatial concentration variation through the difference in the number of occupied receptors to which molecules can bind. They also secrete enzymes that degrade these molecules, and it is presently not well understood how this affects the local gradient perceived by cells. Numerical and analytical results show that these enzymes can substantially increase the signal-to-noise ratio of the receptor difference and allow cells to respond to a much broader range of molecular concentrations and gradients than they would without these enzymes.