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

Abstract

Specialized mathematical models have been proposed to quantitatively assess the ion transporter performance in heart-related physiological experiments. The Na/Ca exchanger imports three Na⁺ ions into the cell and exports one Ca²⁺ ion; therefore, it plays a vital role in cardiomyocyte ion homeostasis. The detailed characteristics of the voltage and ion concentration dependencies of the exchanger current were reported by Matsuoka and Hilgemann in 1992, whereas existing mathematical models can only reproduce a limited range of experimental data. This study primarily focuses on the development of a new mathematical model by introducing charge movements to all consecutive state transition processes under thermodynamic constraints, which accomplish voltage dependencies and current generation for each ion binding and dissociation process. The proposed model includes 22 charge movement, dissociation constant, and rate constant parameters, which were optimized to fit the voltage and ion concentration dependencies of the steady-state transporter currents. Through this process, most parameters converged effectively. Using the optimal parameters, our model successfully replicated the experimental steady-state current data as well as the transient current data. This underscores the accuracy and reliability of our model for reflecting the complex dynamics of cardiomyocyte electrophysiology.

Keywords: Charge Transfer; Mathematical Model; Na/Ca Exchanger; Thermodynamics; Voltage Dependence.

Fig. 1

The consecutive ion exchange reaction cycle of the sodium–calcium exchanger has ten states (, …, ). Here, stands for the native enzyme, and the subscripts and refer to conformations in which cation-binding sites are exposed to the extracellular medium and the cytosol, respectively. The assumption of independent and distinct ion-binding sites results in each ion having an independent and distinct dissociation constant, or , which represents the dissociation constant for an interaction between a single ion and a single binding site. One of the rate constants with a subscript + indicates forward cycling, which is assumed to be clockwise.

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 due to ion binding and dissociation, whereas black squares represent the charge transfer 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 (indicated by the dotted arrows).

Fig. 4

(A) Reproduction of the measured data (filled circles) in Figures 3 and 9 from Matsuoka et al. for the dependence (top–left), dependence (top–right), dependence (middle–left), dependence (middle-right) of the ion exchange current, and for the voltage dependence of the ion exchange current. The horizontal axis for the ion concentration dependence data represents the ion concentration, and the horizontal axis for the voltage dependence data represents the membrane potential. In all cases, the vertical axis represents the current. Red lines represent the simulation results of the proposed model. (B) Reproduction of the measured data (black line) of the time-dependent ion exchange current in Figures 2 (a1–a3) from Hilgemann et al.. The horizontal axis represents time, and the vertical axis represents current. The red line represents the simulation results of the proposed model.

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.. (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.