page_title: Hill equation (biochemistry)
section_title: Hill plot
hierarchical_section_title: Hill equation (biochemistry) / Proportion of ligand-bound receptors / Hill plot
caption_attribution_description: English: Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.
context_page_description: 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", and a macromolecule is a very large molecule, such as a protein, with a complex structure of components. Protein-ligand binding is an example of this kind of binding, which 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–Langmuir 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–Langmuir equation was originally formulated by Archibald Hill in 1910 to describe the sigmoidal O₂ binding curve of haemoglobin.
context_section_description: The Hill plot is the rearrangement of the Hill–Langmuir Equation into a straight line.
  Taking the reciprocal of both sides of the Hill–Langmuir equation, rearranging, and inverting again yields: . Taking the logarithm of both sides of the equation leads to an alternative formulation of the Hill-Langmuir equation:
  .
  This last form of the Hill–Langmuir equation is advantageous because a plot of  versus  yields a linear plot, which is called a Hill plot. Because the slope of a Hill plot is equal to the Hill coefficient for the biochemical interaction, the slope is denoted by . A slope greater than one thus indicates positively cooperative binding between the receptor and the ligand, while a slope less than one indicates negatives cooperative binding.
  Transformations of equations into linear forms such as this were very useful before the ubiquity of computers, as they allowed researchers to determine parameters by fitting lines to data. However, these transformations affect how errors are propagated, which may result in undue weight to error in data points near 0 or 1. This impacts the parameters of linear regression lines fitted to the data. Furthermore, the advent of computers in experiments allows for more robust analysis with nonlinear regression.
caption: A Hill plot, where the x-axis is the logarithm of the ligand concentration and the y-axis is the transformed receptor occupancy. X represents L and Y represents theta.
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