In biochemistry and pharmacology, the Hill equation refers to two closely related equations that reflect the binding of ligands to macromolecules, as a function of the ligand concentration. A ligand is "a substance that forms a complex with a biomolecule to serve a biological purpose" (ligand definition), and a macromolecule is a very large molecule, such as a protein, with a complex structure of components (macromolecule definition).  Protein-ligand binding typically changes the structure of the target protein, thereby changing its function in a cell.
The distinction between the two Hill equations is whether they measure occupancy or response. The Hill equation reflects the occupancy of macromolecules: the fraction that is saturated or bound by the ligand. This equation is formally equivalent to the Langmuir isotherm. Conversely, the Hill equation proper reflects the cellular or tissue response to the ligand: the physiological output of the system, such as muscle contraction.
The Hill equation was originally formulated by Archibald Hill in 1910 to describe the sigmoidal O binding curve of haemoglobin.
The binding of a ligand to a macromolecule is often enhanced if there are already other ligands present on the same macromolecule (this is known as cooperative binding). The Hill equation is useful for determining the degree of cooperativity of the ligand(s) binding to the enzyme or receptor. The Hill coefficient provides a way to quantify the degree of interaction between ligand binding sites.
The Hill equation (for response) is important in the construction of dose-response curves.