Archive for History of Exact Sciences

, Volume 66, Issue 4, pp 427–438 | Cite as

The Hill equation and the origin of quantitative pharmacology

  • Rudolf Gesztelyi
  • Judit Zsuga
  • Adam Kemeny-Beke
  • Balazs Varga
  • Bela Juhasz
  • Arpad Tosaki


This review addresses the 100-year-old Hill equation (published in January 22, 1910), the first formula relating the result of a reversible association (e.g., concentration of a complex, magnitude of an effect) to the variable concentration of one of the associating substances (the other being present in a constant and relatively low concentration). In addition, the Hill equation was the first (and is the simplest) quantitative receptor model in pharmacology. Although the Hill equation is an empirical receptor model (its parameters have only physico-chemical meaning for a simple ligand binding reaction), it requires only minor a priori knowledge about the mechanism of action for the investigated agonist to reliably fit concentration-response curve data and to yield useful results (in contrast to most of the advanced receptor models). Thus, the Hill equation has remained an important tool for physiological and pharmacological investigations including drug discovery, moreover it serves as a theoretical basis for the development of new pharmacological models.


Hill Equation Receptor Model Richards Equation Hill Model Nonparametric Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Black J.W., Leff P. (1983) Operational models of pharmacological agonism. Proceedings of the Royal Society B: Biological Sciences 220: 141–162CrossRefGoogle Scholar
  2. Clark A.J. (1926) The antagonism of acetylcholine by atropine. Journal of Physiology (London) 61: 547–556Google Scholar
  3. Colquhoun D. (1998) Binding, gating, affinity and efficacy: The interpretation of structure-activity relationships for agonists and of the effects of mutating receptors. British Journal of Pharmacology 125: 924–947CrossRefGoogle Scholar
  4. Colquhoun D. (2006) The quantitative analysis of drug-receptor interactions: A short history. Trends in Pharmacological Sciences 27: 149–157CrossRefGoogle Scholar
  5. Danhof M., de Jongh J., De Lange E.C., Della Pasqua O., Ploeger B.A., Voskuyl R.A. (2007) Mechanism-based pharmacokinetic-pharmacodynamic modeling: Biophase distribution, receptor theory, and dynamical systems analysis. Annual Review of Pharmacology and Toxicology 47: 357–400CrossRefGoogle Scholar
  6. Del Castillo J., Katz B. (1957) Interaction at end-plate receptors between different choline derivatives. Proceedings of the Royal Society B: Biological Sciences 146: 369–381CrossRefGoogle Scholar
  7. Furchgott R.F., Bursztyn P. (1967) Comparison of dissociation constants and of relative efficacies of selected agonists acting on parasympathetic receptors. Annals of the New York Academy of Sciences 144: 882–899CrossRefGoogle Scholar
  8. Giraldo J., Vivas N.M., Vila E., Badia A. (2002) Assessing the (a)symmetry of concentration-effect curves: Empirical versus mechanistic models. Pharmacol Ther 95: 21–45CrossRefGoogle Scholar
  9. Goutelle S., Maurin M., Rougier F., Barbaut X., Bourguignon L., Ducher M., Maire P. (2008) The Hill equation: A review of its capabilities in pharmacological modelling. Fundamental and Clinical Pharmacology 22: 633–648CrossRefGoogle Scholar
  10. Hill A.V. (1909) The mode of action of nicotine and curari, determined by the form of the contraction curve and the method of temperature coefficients. Journal of Physiology (London) 39: 361–373Google Scholar
  11. A. V. Hill. 1910. The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. Journal of Physiology (London) 40: Proceedings iv–vii.Google Scholar
  12. Keller F., Giehl M., Czock D., Zellner D. (2002) PK-PD curve-fitting problems with the Hill equation? Try one of the 1-exp functions derived from Hodgkin, Douglas or Gompertz. International Journal of Clinical Pharmacology and Therapeutics 40: 23–29Google Scholar
  13. Kenakin T. (2006) Data-driven analysis in drug discovery. Journal of Receptors and Signal Transduction Research 26: 299–327CrossRefGoogle Scholar
  14. Kenakin T. (2004) Principles: Receptor theory in pharmacology. Trends in Pharmacological Sciences 25: 186–192CrossRefGoogle Scholar
  15. Kenakin T. (2009) Quantifying biological activity in chemical terms: A pharmacology primer to describe drug effect. ACS Chemical Biology 4: 249–260CrossRefGoogle Scholar
  16. Langmuir I. (1918) The adsorption of gases on plane surfaces of glass, mica and platinum. Journal of the American Chemical Society 40: 1361CrossRefGoogle Scholar
  17. Maehle A.H. (2004) Receptive substances”: John Newport Langley (1852–1925) and his path to a receptor theory of drug action. Medical History 48: 153–174CrossRefGoogle Scholar
  18. Maehle A.H. (2009) A binding question: The evolution of the receptor concept. Endeavour 33: 135–140CrossRefGoogle Scholar
  19. Mager D.E., Wyska E., Jusko W.J. (2003) Diversity of mechanism-based pharmacodynamic models. Drug Metabolism and Disposition 31: 510–518CrossRefGoogle Scholar
  20. Michaelis L., Menten M.L. (1913) Die Kinetik der Intertinwerkung. Biochemische Zeitschrift 49: 333–369Google Scholar
  21. Motulsky, H.J., and A. Christopoulos. 2003. Fitting models to biological data using linear and nonlinear regression. A practical guide to curve fitting. Oxford: Oxford Press (Corrected online version:
  22. Neubig R.R., Spedding M., Kenakin T., Christopoulos A. (2003) International union of pharmacology committee on receptor nomenclature and drug classification. XXXVIII. update on terms and symbols in quantitative pharmacology. Pharmacological Reviews 55: 597–606Google Scholar
  23. Pelner L. (1972) Corpora non agunt nisi fixata. Maxim behind all of Ehrlich’s great discoveries. New York State Journal of Medicine 72: 620–624Google Scholar
  24. Rang, H.P. 2006. The receptor concept: Pharmacology’s big idea. British Journal of Pharmacology 147: S9–16.Google Scholar
  25. Richards F.J. (1959) A flexible growth function for empirical use. Journal of Experimental Botany 10: 290–300CrossRefGoogle Scholar
  26. Ruffolo R.R. Jr. (1982) Review important concepts of receptor theory. Journal of Autonomic Pharmacology 2: 277–295CrossRefGoogle Scholar
  27. Scheindlin S. (2001) A brief history of pharmacology. Modern Drug Discovery 4: 87–88Google Scholar
  28. Stephenson R.P. (1956) A modification of receptor theory. British Journal of Pharmacology 11: 379–393Google Scholar
  29. Van der Graaf P.H., Danhof M. (1997) Analysis of drug-receptor interactions in vivo: A new approach in pharmacokinetic-pharmacodynamic modelling. International Journal of Clinical Pharmacology and Therapeutics 35: 442–446Google Scholar
  30. Van der Graaf P.H., Schoemaker R.C. (1999) Analysis of asymmetry of agonist concentration-effect curves. Journal of Pharmacological and Toxicological Methods 41: 107–115CrossRefGoogle Scholar
  31. Van der Graaf P.H., Stam W.B. (1999) Analysis of receptor inactivation experiments with the operational model of agonism yields correlated estimates of agonist affinity and efficacy. Journal of Pharmacological and Toxicological Methods 41: 117–125CrossRefGoogle Scholar
  32. Weiss J.N. (1997) The Hill equation revisited: Uses and misuses. FASEB Journal 11: 835–841Google Scholar
  33. Zimmer, H.G. (1996) Carl Ludwig: the man, his time, his influence. Pflugers Archiv 432: R9–22.Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Rudolf Gesztelyi
    • 1
  • Judit Zsuga
    • 2
  • Adam Kemeny-Beke
    • 3
  • Balazs Varga
    • 1
  • Bela Juhasz
    • 1
  • Arpad Tosaki
    • 1
  1. 1.Department of Pharmacology, Faculty of PharmacyUniversity of DebrecenDebrecenHungary
  2. 2.Department of Neurology, Faculty of MedicineUniversity of DebrecenDebrecenHungary
  3. 3.Department of Ophthalmology, Faculty of MedicineUniversity of DebrecenDebrecenHungary

Personalised recommendations