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A new mathematical formula to link near equilibrium relaxation kinetics and conformational selection steps in enzymatic reactions


A new mathematical formula was derived for near equilibrium relaxation processes of enzyme reactions including the conformational selection (CS) modes. CS is one of the most accepted molecular recognition mechanisms, in which protein conformers (CS conformers) are in an equilibrium with varying degrees of ligand binding affinity so that ligands select a particular conformer among them to bind. Using computer simulation techniques, our previous study (Egawa and Callender in Math Biosci 313: 61–70, 2019) predicted that the rate constant for the near equilibrium relaxation processes (kNER) and the concentration-sum of substrate and product (CLt) of enzyme reactions uniquely related to the presence of CS steps in a manner that 1/kNER versus CLt plot transformed from linear to quadratic as the elementary rate constants of inter-conversions among the CS conformers were becoming smaller relative to the rate constants at other steps of the enzymatic reaction system. Thus our previous work could provide a potential tool to detect the presence of CS steps in an enzyme reaction simply by assays using only trace amount of enzyme samples, although logical basis to have the quadratic deformation in the 1/kNER versus CLt plot in response to the presence of CS steps could not be clarified. Employing mathematical approaches that were alternative to those in our previous study, this study succeeded in deriving a theoretical equation that fully explained why and how the CS modes caused the quadratic characters in the 1/kNER versus CLt plot.

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Scheme 1
Scheme 2
Fig. 1

adopted from Fig. 2 of ref [1]. Values of the rate constants k3-k16 were common in A and B, which are indicated at the top. The k19-k22 values were relatively fast (105 s−1) and slow (103 s−1) in A and B, respectively. CEt was set at 1 × 10–7 M. The green line in each panel is a result of linear least-squares fit to the data, for which the linear correlation coefficient \(\gamma\) was obtained (Color figure online)

Fig. 2


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Egawa, T. A new mathematical formula to link near equilibrium relaxation kinetics and conformational selection steps in enzymatic reactions. J Math Chem 59, 2270–2283 (2021).

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  • Enzyme reaction
  • Relaxation kinetics
  • Conformational selection
  • Kinetic differential equations

Mathematics Subject Classification

  • 92C45
  • 41A20
  • 34C20