Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

On the interaction of two different types of ligands binding to the same molecule part I: basics and the transfer of the decoupled sites representation to systems with n and one binding sites


The decoupled sites representation (DSR) for one type of ligand allows to regard complex overall titration curves as sum of classical Henderson-Hasselbalch (HH) titration curves. In this work we transfer this theoretical approach to molecules with different types of interacting ligands (e.g. protons and electrons), prove the existence of decoupled systems for n 1 and one binding sites for two different ligands, and point out some difficulties and limits of this transfer. A major difference to the DSR for one type of ligand is the loss of uniqueness of the decoupled system. However, all decoupled systems share a unique set of microstate probabilities and each decoupled system corresponds to a certain permutation of these microstate probabilities. Moreover, we show that the titration curve of a certain binding site in the original system can be regarded as linear combination of the titration curves of the individual sites of the decoupled system if the weights of the linear combination are substituted by functions in the activity of the second ligand. In the underlying model with only pairwise interaction, an important observation of our theoretical investigation is the following: Even though the binding sites of ligand L 1 may not interact directly, they can show secondary interaction due to the interaction with the second type of ligand. This means, if the activity of the second ligand is fixed and we regard the 1-dimensional titration curve of an individual binding site for ligand L 1 depending on its activity, we may observe a strong deviation from the classical HH shape in spite of non-interacting sites for ligand L 1.


  1. 1

    Ackers G.K., Shea M.A., Smith F.R.: Free energy coupling within macromolecules: the chemical work of ligand binding at the individual sites in co-operative systems. J. Mol. Biol. 170, 223–242 (1983)

  2. 2

    Bashford D., Karplus M.: Multiple-site titration curves of proteins: an analysis of exact and approximate methods for their calculation. J. Phys. Chem. 95(23), 9556–9561 (1991)

  3. 3

    Becker T., Ullmann R.T., Ullmann G.M.: Simulation of the electron transfer between the tetraheme subunit and the special pair of the photosynthetic reaction center using a microstate description. J. Phys. Chem. B 111(11), 2957–2968 (2007)

  4. 4

    E. Bombarda, G.M. Ullmann, pH-dependent pKa values in proteins- A theoretical analysis of protonation energies with practical consequences for enzymatic reactions. J. Phys. Chem. B 114(5), 1994–2003. PMID: 20088566 (2010)

  5. 5

    C.R. Cantor, P.R. Schimmel, Biophysical Chemistry. Part III. The Behavior of Biological Macromolecules, 1st edn. (W. H. Freeman, 1980)

  6. 6

    D. Cox, J. Little, D. O’Shea, Using Algebraic Geometry (2nd ed.). (Springer, New York, 2005

  7. 7

    D.A. Cox, J. Little, D. O’Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3rd ed. (Springer, Secaucus, NJ, USA, 2008)

  8. 8

    Garcia-Moreno B.E.: Probing structural and physical basis of protein energetics linked to protons and salt. Methods Enzymol. 259, 512–538 (1995)

  9. 9

    Gnacadja G.: A method to calculate binding equilibrium concentrations in the allosteric ternary complex model that supports ligand depletion. Math. Biosci. 232(2), 135–141 (2011)

  10. 10

    K. Hasselbalch, Die Berechnung der Wasserstoffzahl des Blutes aus der freien und gebundenen Kohlensäure desselben, und die Sauerstoffbindung des Blutes als Funktion der Wasserstoffzahl. (Julius Springer, Berlin, 1916)

  11. 11

    L.J. Henderson, The Fitness of the Environment. (Macmillan Company, New York, 1913)

  12. 12

    J. Martini, G. Ullmann (2012). A mathematical view on the decoupled sites representation. J. Math. Biol. 1–27 (2012). doi:10.1007/s00285-012-0517-x

  13. 13

    Medvedev E., Stuchebrukhov A.: Kinetics of proton diffusion in the regimes of fast and slow exchange between the membrane surface and the bulk solution. J. Math. Biol 52, 209–234 (2006). doi:10.1007/s00285-005-0354-2

  14. 14

    Onufriev A., Case D.A., Ullmann G.M.: A novel view of pH titration in biomolecules. Biochemistry 40(12), 3413–3419 (2001)

  15. 15

    Onufriev A., Ullmann G.M.: Decomposing complex cooperative ligand binding into simple components: Connections between microscopic and macroscopic models. J. Phys. Chem. B 108(30), 11157–11169 (2004)

  16. 16

    Schellman J.A.: Macromolecular binding. Biopolymers 14(5), 999–1018 (1975)

  17. 17

    Tanford C., Kirkwood J.G.: Theory of protein tiration curves. I. general equations for impenetrable spheres. J. Am. Chem. Soc. 79(20), 5333–5339 (1957)

  18. 18

    M.S. Till, T. Essigke, T. Becker, G.M. Ullmann, Simulating the proton transfer in gramicidin a by a sequential dynamical monte carlo method. J. Phys. Chem. B 112(42), 13401–13410 (2008). PMID: 18826179

  19. 19

    Ullmann R.T., Ullmann G.M.: Coupling of protonation, reduction and conformational change in azurin from Pseudomonas aeruginosa investigated with free energy measures of cooperativity. J. Phys. Chem. B 115, 10346–10359 (2011)

  20. 20

    J. Wyman, S.J. Gill, Binding and Linkage: Functional Chemistry of Biological Macromolecules. (University Science Books, Mill Valley, California 1990)

Download references

Author information

Correspondence to Johannes W. R. Martini.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Reprints and Permissions

About this article

Cite this article

Martini, J.W.R., Schlather, M. & Ullmann, G.M. On the interaction of two different types of ligands binding to the same molecule part I: basics and the transfer of the decoupled sites representation to systems with n and one binding sites. J Math Chem 51, 672–695 (2013). https://doi.org/10.1007/s10910-012-0107-6

Download citation


  • Decoupled sites representation
  • Protonation
  • Electron binding
  • Different ligands
  • Binding polynomial
  • Interaction energy
  • Binding energy
  • Transport
  • Transfer
  • Photosynthesis
  • Receptor