## Cooperativity and the Hill equation

In biology it often happens that a molecule of one substance has several binding sites for another molecule (call it the ligand). This is known as cooperative binding. A classical example is haemoglobin, which has four binding sites for oxygen. If the rate of binding of the ligand at one of the sites is affected (positively or negatively) by the fact that some of the other sites are occupied then this is known as allostery. Suppose for example that an enzyme has two binding sites for a ligand. Using a description of this process of Michaelis-Menten type for the two successive binding events leads to an equation where the right hand side is a quotient of two quadratic polynomials. A derivation of this can be found in Murray’s book ‘Mathematical Biology’. Assuming a reaction where both molecules combine with the enzyme simultaneously gives a different quotient of quadratic polynomials which is particularly simple, namely $\frac {Au^2}{u^2+B}$. This is not likely to be a genuine reaction mechanism but might arise in some way by telescoping two reactions to get a simpler model. It may be interpreted as what is called complete cooperativity: when one molecule of the ligand binds the probability of binding at the other site becomes much higher and so may be supposed to occur essentially immediately.

A generalization of this idea for $n$ substrate molecules is given by the Hill equation $\dot u=f(n)=\frac{Au^n}{u^n+B}$. It was introduced by the physiologist Archibald Hill on a paper on haemoglobin in 1910. (Hill studied mathematics before he turned to physiology. He received the 1922 Nobel Prize for Physiology or Medicine for his work on the production of heat in muscles.) There he first considers what kind of equations might result if several molecules of oxygen bind to one molecule of haemoglobin. He then suggests considering the equation which now bears his name independently of detailed reaction mechanisms. This is a common procedure in modern biology. The Hill equation is used primarily as a phenomenological ansatz. The idea of cooperativity is in the background but the link is not very direct. In particular, it can happen that the Hill equation is used with non-integer values of $n$. An important qualitative feature of the nonlinearity in the Hill equation is that it is sigmoid for $n>1$. In other words the first derivative increases for small values of the argument before decreasing for larger values. This is a feature which may be observed in experimental data. When it is seen it is possible to try to fit it to a description by the Hill equation using a Hill plot. This means fitting the linear relation $\log(\frac{f}{A-f})=n\log u-\log B$. In particular the slope of the graph gives the Hill coefficient $n$.

This equation has no relation to the linear ordinary differential equation called Hill’s equation which is named after the astronomer and mathematician George William Hill.

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