Abstract
The decoupled sites representation (DSR) is a theoretical instrument which allows to regard complex pH titration curves of biomolecules with several interacting proton binding sites as composition of isolated, non-interacting sites, each with a standard Henderson–Hasselbalch titration curve. In this work, we present the mathematical framework in which the DSR is embedded and give mathematical proofs for several statements in the periphery of the DSR. These proofs also identify exceptions. To apply the DSR to any molecule, it is necessary to extend the set of binding energies from \({\mathbb{R}}\) to a stripe within \({\mathbb{C}}\). An important observation in this context is that even positive interaction energies (repulsion) between the binding sites will not guarantee real binding energies in the decoupled system, at least if the molecule has more than four proton binding sites. Moreover, we show that for a given overall titration curve it is not only possible to find a corresponding system with an interaction energy of zero but with any arbitrary fix interaction energy. This result also effects practical work as it shows that for any given titration curve, there is an infinite number of corresponding hypothetical molecules. Furthermore, this implies that—using a common definition of cooperative binding on the level of interaction energies—a meaningful measure of cooperativity between the binding sites cannot be defined solely on the basis of the overall titration. Consequently, all measures of cooperativity based on the overall binding curve do not measure the type of cooperativity commonly defined on the basis of interaction energies. Understanding the DSR mathematically provides the basis of transferring the DSR to biomolecules with different types of interacting ligands, such as protons and electrons, which play an important role within electron transport chains like in photosynthesis.
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References
Ackers GK, Shea MA, Smith FR (1983) 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
Bashford D, Karplus M (1991) Multiple-site titration curves of proteins: an analysis of exact and approximate methods for their calculation. J Phys Chem 95(23): 9556–9561
Becker T, Ullmann RT, Ullmann GM (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(11): 2957–2968
Ben-Naim A (2001) Cooperativity and Regulation in Biochemical Processes. Kluwer Academic/Plenum Publishers, New York
Berg J, Tymoczko J, Stryer L (2007) Biochemistry. W.H. Freeman, New York
Bombarda E, Ullmann GM (2010) 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
Cantor CR, Schimmel PR (1980) Biophysical Chemistry. Part III. The Behavior of Biological Macromolecules, 1st edn. W. H. Freeman, New York
Cox D, Little J, O’Shea D (2005) Using Algebraic Geometry, 2nd edn. Springer, Berlin
Cox DA, Little J, O’Shea D (2008) Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3rd edn. Springer, New York
Garcia-Moreno BE (1995) Probing structural and physical basis of protein energetics linked to protons and salt. Methods Enzymol 259: 512–538
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: 209–234. doi:10.1007/s00285-005-0354-2
Onufriev A, Case DA, Ullmann GM (2001) A novel view of pH titration in biomolecules. Biochemistry 40(12): 3413–3419
Onufriev A, Ullmann GM (2004) Decomposing complex cooperative ligand binding into simple components: Connections between microscopic and macroscopic models. J Phys Chem B 108(30): 11157–11169
Schellman JA (1975) Macromolecular binding. Biopolymers 14: 999–1018
Tanford C, Kirkwood JG (1957) Theory of protein tiration curves. I. General equations for impenetrable spheres. J Am Chem Soc 79(20): 5333–5339
Till MS, Essigke T,Becker T, Ullmann GM (2008) Simulating the proton transfer in gramicidin a by a sequential dynamical monte carlo method. J Phys Chem B 112(42):13401–13410. PMID: 18826179
Ullmann RT, Ullmann GM (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: 10346–10359
Wyman J, Gill SJ (1990) Binding and Linkage: Functional Chemistry of Biological Macromolecules. University Science Books, Mill Valley
Acknowledgments
First of all, we thank Martin Schlather for comments on the original manuscript. Moreover, we thank Tobias Dorsch for discussing ideas.
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GMU was supported by the Deutsche Forschungsgemeinschaft through SFB 840 research project B2.
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Martini, J.W.R., Ullmann, G.M. A mathematical view on the decoupled sites representation. J. Math. Biol. 66, 477–503 (2013). https://doi.org/10.1007/s00285-012-0517-x
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DOI: https://doi.org/10.1007/s00285-012-0517-x
Keywords
- Decoupled sites representation
- Protonation
- Binding polynomial
- Interaction energy
- Binding energy
- Ligand binding