Cooperativity, absolute interaction, and algebraic optimization
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Mathematical Biology
Cooperativity, absolute interaction, and algebraic optimization Nidhi Kaihnsa1 · Yue Ren2
· Mohab Safey El Din3 · Johannes W. R. Martini4
Received: 30 July 2019 / Revised: 4 May 2020 / Published online: 23 September 2020 © The Author(s) 2020
Abstract We consider a measure of cooperativity based on the minimal interaction required to generate an observed titration behavior. We describe the corresponding algebraic optimization problem and show how it can be solved using the nonlinear algebra tool SCIP. Moreover, we compute the minimal interactions and minimal molecules for several binding polynomials that describe the oxygen binding of various hemoglobins under different conditions. We compare their minimal interaction with the maximal slope of the Hill plot, and discuss similarities and discrepancies with a view towards the shapes of the binding curves. Mathematics Subject Classification 92C40 · 90C23
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Yue Ren [email protected] https://yueren.de Nidhi Kaihnsa [email protected] https://sites.google.com/view/kaihnsa/ Mohab Safey El Din [email protected] https://www-polsys.lip6.fr/~safey/ Johannes W. R. Martini [email protected]
1
Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA
2
Department of Mathematics, Swansea University, Swansea University Bay Campus, Swansea SA1 8EN, UK
3
Laboratoire d’Informatique de Paris 6, LIP6, Équipe PolSys, Sorbonne Université, CNRS, Paris, France
4
International Maize and Wheat Improvement Center, Texcoco, Mexico
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1 Introduction Interaction between components is a fundamental feature of biological systems. While a simple system of independent subunits is completely defined by its subunits, a complex system with interactions is more than the sum of its parts. A classical example of a small biological system with non-trivial interaction is hemoglobin with its four binding sites for oxygen (Bohr et al. 1904; Barcroft 1913; Hill 1913). The ligand oxygen binds to the four binding sites of the (target) molecule hemoglobin and the interaction can be seen on the overall (isotherm) binding curve, which relates the average ligand saturation inside the system to the ligand concentration outside the system at constant temperature (Hill 1985). While the binding curve of an independent system is linear, the binding curve of hemoglobin has a sigmoidal shape (Barcroft 1913; Hill 1913), see Fig. 1. The sigmoidal shape implies that the extremal states of full and zero saturation are more stable than the intermediate states of partial saturation. This phenomenon is commonly referred to as cooperativity, named after the intuition that bound ligands affect, either positively or negatively, the chances of new ligands binding to the still open sites (see Stefan and Le Novère 2013 for a review and interpretations). Cooperativity is ubiquitous in nature. It is an essential trait for transport molecules, and its importance in the formation of multi-protein complexes (Roy et al. 2017), in
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