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The Biology Project > Biomath > Rational Functions > Applications > Hill Equation

Rational Function Applications

Hill Equation

Many biologically important molecules have multiple binding sites. For example hemoglobin, the oxygen carrying molecule in red blood cells, binds four molecules of molecular oxygen, each binding at its own distinct binding site. Hemoglobin is an example of a cooperative molecule, that is one where the binding of a ligand at one site alter that the affinity of other binding sites for their ligands. In the case of hemoglobin, the binding of a molecule of O2 at one site increases the affinity of the other sites for O2.

This property is critical for the function of hemoglobin, which picks up four molecules of O2 in the oxygen-rich environment of the lungs, then delivers it to the tissues that have a much lower concentration of oxygen. As each molecule of O2bind to hemoglobin, the affinity of the remaining binding site increases, making it easier for more O2 to bind . Conversely, as each molecule of O2 is released to a tissue, the affinity of the remaining sites for O2 decreases, making it easier for subsequent molecules of O2 to dissociate.


Scanning electron micrograph of blood. The doughnut-shaped cells are red blood cells. Photo credit: Bruce Wetzel. Courtesy of the National Cancer Institute.


The problem of how hemoglobin delivers oxygen throughout the body has been studied for the past 100 years. In 1910, biochemist Archibald Hill modeled this property of hemoglobin using the rational function,

where θ is the percentage of binding sites occupied, [L] is the concentration of ligand, n is the Hill coefficient, which represents the degree of cooperativity, and Kd is the dissociation constant. Recall that Kd is equal to the ligand concentration when half of the binding sites are filled.


A common application of the Hill equation is modeling cooperative enzymes. These enzymes are under allosteric control, that is the binding of a molecule at one site alters the affinity of the enzyme for its substrate and hence regulates the enzyme activity. In this case, the Hill equation is rewritten as the rational function,

where V is the reaction velocity, Vmax is the maximum reaction velocity, and [S] is the substrate concentration. The constant K is analogous to the Michaelis constant (Km) and n is the Hill coefficient indicating the degree of cooperativity.

Hill coefficient Cooperativity
n = 1 none
n > 1 positive
n < 1 negative

Positive cooperativity occurs when an enzyme has several sites to which a substrate can bind, and the binding of one substrates molecules increases the rate of binding of other substrates. Cooperativity can be recognized by plotting velocity against substrate concentration. An enzyme that displays positive cooperativity sill be sigmoidal (or S-shaped), while noncooperative enzymes display Michaelis-Menten kinetics and the plots are hyperbolic.


Use the Hill equation to answer the following questions:

Find the substrate concentration given the velocity of a reaction.

Determine what happens to the velocity of an enzyme-catalyzed reaction as the substrate concentration approaches infinity.

Determine how the velocity of a reaction catalyzed by a cooperative enzyme changes when the substrate concentration is halved.

Given a plot, determine if an enzyme exhibits cooperativity.

Determine if an enzyme exhibits cooperativity.


The Biology Project > Biomath > Rational Functions > Applications > Hill Equation

The Biology Project
Department of Biochemistry and Molecular Biophysics

The University of Arizona

April 2007
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