The effects of voltage dependence and ion-binding reaction rates on a thermodynamically constrained mathematical model of the Na/Ca exchanger

Fig. 1. The consecutive ion exchange reaction cycle of the sodium–calcium exchanger has ten states (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{P}}_{1}$$\end{document}, …, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{P}}_{10}$$\end{document}). Here, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{E}$$\end{document} stands for the native enzyme, and the subscripts \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e$$\end{document} and…

Fig. 2. Conceptual diagram of the charge transfer occurring between each state transition in the proposed model. In the diagram, the upper direction represents the outside of the cell, and the lower direction represents the inside of the cell. Black circles indicate the charge transfer \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\delta }_{i}$$\end{document} due to ion binding and dissociation, whereas black squares represent the charge transfer \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\delta }_{i}{\prime}$$\end{document} within the NCX structural conformation. Note that an open circle represents the location before state transition, while a closed circle represents the location after state transition. In the proposed model, the amount of charge transfer for the parameters is the sum of these two charge transfers.

Fig. 3. A simplified two-state scheme characterizing NCX. (A) Consecutive ten-state ion exchange reaction cycle. (B) Simplified two-state diagram of (A). The ion binding and -unbinding reactions are included using the pseudo-first-order rate constants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma_{i}^{+/-}$$\end{document} (indicated by the dotted arrows).

Fig. 4. (A) Reproduction of the measured data (filled circles) in Figures 3 and 9 from Matsuoka et al.32 for the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\left[{Ca}^{2+}\right]}_{i}$$\end{document} dependence (top–left), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\left[{Na}^{+}\right]}_{i}$$\end{document} dependence (top–right), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\left[{Na}^{+}\right]}_{e}$$\end{document} dependence (middle–left), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek}…

Fig. 5. Convergence status of intracellular and extracellular Na+ and Ca2+ dissociation constants, NCX conformational change rate constants, and charge transfer values for each state transition in the model. The horizontal axis represents parameter convergence values, and the vertical axis represents optimization error. From top to bottom, it represents the dissociation constants of extracellular (intracellular) 1st, 2nd, and 3rd Na+ dissociations, extracellular (intracellular) Ca2+ dissociation, NCX conformational change rate constants related to intracellular and extracellular Na+ and Ca2+ transport, and the charge transfer between states.

Fig. 6. Correlations among parameters that did not converge. Both the horizontal and vertical axes represent optimal parameters. The black dots represent parameters with small, identified errors, and represent the values of the top 20 parameter sets. The dashed lines indicate correlation type, and the correlation coefficients are listed as text.

Fig. 7. Analysis of the steady-state ion concentration and voltage dependence data for NCX, as shown in Figures 4c and d of Hilgemann et al.41. (A) The intracellular [Ca2+] dependence, with the horizontal axis showing the intracellular [Ca2+] concentration and the vertical axis normalized to the maximum measured value at − 80 mV. (B) The intracellular [Na+] dependence, with the horizontal axis showing the intracellular [Na+] concentration and the vertical axis normalized to the maximum measured value at + 20 mV. Diamonds and squares indicate the experimental data measured at + 20 mV and − 80 mV, respectively, The solid lines represent the simulation results of the proposed model.