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A mathematical view on the decoupled sites representation.

Johannes W R Martini1, G Matthias Ullmann

  • 1Institut für Mathematische Stochastik, Georg-August Universität Göttingen, Göttingen, Germany. jmartin2@uni-goettingen.de

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Summary

The decoupled sites representation (DSR) mathematically models complex biomolecule pH titration curves. This study proves DSR exceptions and shows cooperativity cannot be solely determined from overall binding curves.

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Area of Science:

  • Biochemistry
  • Theoretical Chemistry
  • Physical Chemistry

Background:

  • Complex pH titration curves of biomolecules arise from interacting proton binding sites.
  • The decoupled sites representation (DSR) simplifies these curves by treating sites as isolated.
  • Standard Henderson-Hasselbalch curves describe isolated sites.

Purpose of the Study:

  • To present the mathematical framework for the decoupled sites representation (DSR).
  • To provide mathematical proofs for statements regarding DSR, including exceptions.
  • To investigate the implications of DSR for understanding cooperativity in biomolecules.

Main Methods:

  • Mathematical framework development for DSR.
  • Rigorous mathematical proofs for DSR-related statements and exceptions.
  • Analysis of binding energies and interaction energies within the DSR model.

Main Results:

  • Mathematical proofs identify exceptions to DSR application.
  • Even with positive interaction energies, decoupled systems may not yield real binding energies for >4 sites.
  • A given titration curve corresponds to infinite hypothetical molecules with varying interaction energies.

Conclusions:

  • Cooperativity cannot be solely defined by overall binding curves using interaction energy definitions.
  • Measures of cooperativity based on overall binding curves differ from those based on interaction energies.
  • Mathematical understanding of DSR facilitates its extension to other interacting ligands like electrons.