If you want to understand how drugs work/behave (or just how 2 molecules interact with each other) then you need to understand one of the oldest and most useful theories in biochemistry: the Hill Equation. The Hill Equation mathematically models how Ligands(L)/Drugs interaction with their Receptors(R)/targets and generally assumes that the amount of complex that forms (RL) is proportional to some biological response . Here the __ K_{d}__ represents the “dissociation constant” (which is a measure of the strength of the interaction between R and L) and

__denote concentrations/dosages (with a t subscript indicating total concentrations).__

*brackets*

The Langmuir-Hill Equation was co-discovered by the bio-chemisty Archibald Hill in 1910 and by the surface-chemist Irving Langmuir in 1916. Generally, this equation describes the shape of dose-response curves one obtains while titrating their drug or ligand of interest (bottom figure).

There are two parts to the Langmuir-Hill Equation. First, we have the **total concentration of the receptor/target ([ R]_{t})**. This term represents the maximum complex that can form and is generally proportional to the magnitude of the biological response you obtain (more complex = more downstream effect). Second, we have a

**fractional term which describes the shape of the dose-response curve in response to increasing concentrations of ligand/drug**. If you data is normalized to 100% then this fractional term is usually sufficient to describe your data.

An interesting feature of this fractional term is that it defines the dose of ligand or drug which elicits a 50% maximal response or EC_{50} as being equal to the dissociation constant (*K _{d}*). This simple equation underlies all structure-based drug design approachs which endeavor to improve the clinical potency of a drug (EC

_{50}) by altering the chemistry so that it binding constant(

*K*) improves.

_{d}On limitation of the Hill Equation is that is only works when the receptor concentration is much lower than the dissociation constant (i.e. [R]t << *K _{d}*). While this is true in most

*in vivo*and cell-based experiments its sometimes breaks down in many biophysical experiment that require large amounts of protein. In these cases, a more general, quadratic-equation must be used in place of the Hill-Equation and is described in a previous post: Understanding Ligand-Receptor Dose-Response Curves.

**REFERENCES: **

- Hill, A.V. The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves.
*Proceedings of the Phsiological Society.***1910**, 40, iv-vii. - Langmuir, I. The Constituion and Fundamental Properties of Solids and Liquids.
*J. Am. Chem. Soc.***1916**, 2221-2295. - Lineweaver, H.; Burke, D. The determination of enzyme dissociation constants.
*J. Am. Chem. Soc.***1934**,56, 658-666. - Straus, O.H.; Goldstein, A.; Plachte, W. Zone Behavior of Enzymes.
*J. Gen. Physiol.***1943**, 26, 559-585. - Cha, S. Kinetic Behavior at High Enzyme Concentrations.
*J. Biol. Chem.***1970**, 245, 4814-4818. - Gesztelyi, R.; Zsuga, J.; Kemeny-Beke, A.; Varga, B.; Juhasz, B.; Tosaki, A. The Hill equation and the origin of quantitative pharmacology.
*Archive for History of Exact Sciences***2012**, 66, 427-438. - Goutelle, S.; Maurin, M.; Rougier, F.; Barbaut, X.; Bourguignon, L.; Ducher, M.; Maire, P. The Hill equation: a review of its capabilities in pharmacological modelling.
*Fund. Clin. Pharmacol.***2008**, 22, 633-648. - Lauffenburger, D.A. Receptors: Models for Binding, Trafficking and Signalling, Oxford University Press
**1993**. - Segel, I.H. Enzyme Kinetics: Behavior and Analysis of Rapid Equilibrium and Steady-State Enzyme Systems, Wiley-Interscience,
**1993**

This work by Eugene Douglass and Chad Miller is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.