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
 Johannes W. R. Martini,
 Martin Schlather,
 G. Matthias Ullmann
 … show all 3 hide
Abstract
The decoupled sites representation (DSR) for one type of ligand allows to regard complex overall titration curves as sum of classical HendersonHasselbalch (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 1dimensional 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 noninteracting sites for ligand L _{1}.
 Ackers, G.K., Shea, M.A., Smith, F.R. (1983) Free energy coupling within macromolecules: the chemical work of ligand binding at the individual sites in cooperative systems. J. Mol. Biol. 170: pp. 223242 CrossRef
 Bashford, D., Karplus, M. (1991) Multiplesite titration curves of proteins: an analysis of exact and approximate methods for their calculation. J. Phys. Chem. 95: pp. 95569561 CrossRef
 Becker, T., Ullmann, R.T., Ullmann, G.M. (2007) 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: pp. 29572968 CrossRef
 E. Bombarda, G.M. Ullmann, pHdependent 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)
 C.R. Cantor, P.R. Schimmel, Biophysical Chemistry. Part III. The Behavior of Biological Macromolecules, 1st edn. (W. H. Freeman, 1980)
 D. Cox, J. Little, D. O’Shea, Using Algebraic Geometry (2nd ed.). (Springer, New York, 2005
 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)
 GarciaMoreno, B.E. (1995) Probing structural and physical basis of protein energetics linked to protons and salt. Methods Enzymol. 259: pp. 512538 CrossRef
 Gnacadja, G. (2011) A method to calculate binding equilibrium concentrations in the allosteric ternary complex model that supports ligand depletion. Math. Biosci. 232: pp. 135141 CrossRef
 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)
 L.J. Henderson, The Fitness of the Environment. (Macmillan Company, New York, 1913)
 J. Martini, G. Ullmann (2012). A mathematical view on the decoupled sites representation. J. Math. Biol. 1–27 (2012). doi:10.1007/s002850120517x
 Medvedev, E., Stuchebrukhov, A. (2006) Kinetics of proton diffusion in the regimes of fast and slow exchange between the membrane surface and the bulk solution. J. Math. Biol 52: pp. 209234 CrossRef
 Onufriev, A., Case, D.A., Ullmann, G.M. (2001) A novel view of pH titration in biomolecules. Biochemistry 40: pp. 34133419 CrossRef
 Onufriev, A., Ullmann, G.M. (2004) Decomposing complex cooperative ligand binding into simple components: Connections between microscopic and macroscopic models. J. Phys. Chem. B 108: pp. 1115711169 CrossRef
 Schellman, J.A. (1975) Macromolecular binding. Biopolymers 14: pp. 9991018 CrossRef
 Tanford, C., Kirkwood, J.G. (1957) Theory of protein tiration curves. I. general equations for impenetrable spheres. J. Am. Chem. Soc. 79: pp. 53335339 CrossRef
 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
 Ullmann, R.T., Ullmann, G.M. (2011) Coupling of protonation, reduction and conformational change in azurin from Pseudomonas aeruginosa investigated with free energy measures of cooperativity. J. Phys. Chem. B 115: pp. 1034610359 CrossRef
 J. Wyman, S.J. Gill, Binding and Linkage: Functional Chemistry of Biological Macromolecules. (University Science Books, Mill Valley, California 1990)
 Title
 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
 Open Access
 Available under Open Access This content is freely available online to anyone, anywhere at any time.
 Journal

Journal of Mathematical Chemistry
Volume 51, Issue 2 , pp 672695
 Cover Date
 20130201
 DOI
 10.1007/s1091001201076
 Print ISSN
 02599791
 Online ISSN
 15728897
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

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

 Johannes W. R. Martini ^{(1)}
 Martin Schlather ^{(2)}
 G. Matthias Ullmann ^{(3)}
 Author Affiliations

 1. Institut für Mathematische Stochastik, GeorgAugust Universität Göttingen, Göttingen, Germany
 2. Institut für Mathematik, Universität Mannheim, Mannheim, Germany
 3. Bioinformatik/Strukturbiologie, Universität Bayreuth, Bayreuth, Germany