A novel method to quantify contribution of channels and transporters to membrane potential dynamics

Abstract

The action potential, once triggered in ventricular or atrial myocytes, automatically proceeds on its time course or is generated spontaneously in sinoatrial node pacemaker cells. It is induced by complex interactions among such cellular components as ion channels, transporters, intracellular ion concentrations, and signaling molecules. We have developed what is, to our knowledge, a new method using a mathematical model to quantify the contribution of each cellular component to the automatic time courses of the action potential. In this method, an equilibrium value, which the membrane potential is approaching at a given moment, is calculated along the time course of the membrane potential. The calculation itself is based on the time-varying conductance and the reversal potentials of individual ion channels and electrogenic ion transporters. Since the equilibrium potential moves in advance of the membrane potential change, we refer to it as the lead potential, V(L). The contribution of an individual current was successfully quantified by comparing dV(L)/dt before and after fixing the time-dependent change of a component of interest, such as the variations in the open probability of a channel or the turnover rate of an ion transporter. In addition to the action potential, the lead-potential analysis should also be applicable in all types of membrane excitation in many different kinds of cells.

Figure 1

Equivalent electrical circuit of the cell membrane. (A) An equivalent circuit for the membrane electrical behavior expressed by Eq. 5. C_m, membrane capacitance; E_X, reversal potential for a channel X; I_Y, current through transporter Y; I_m, total membrane current. (B) A reduced circuit equivalent to Eq. 7.

Figure 2

Time-dependent changes of V_L (red) and V_m (black) in the SA node cell model (A) and ventricular myocyte model (B). The period for current injection was excluded from calculation of V_L in panel B (50–52 ms). The gray bars indicate the time spans analyzed in Figs. 3–5.

Figure 3

Time profile of r_c during slow diastolic depolarization in the SA node model. (A) (Top) Time-dependent changes of V_L (red) and V_m (black). These are the same curves as those shown in Fig. 2A. (Bottom) Time-dependent changes of r_c of the major current components calculated by Eq. 12. (Inset, right bottom) The numerators in Eq. 12 for the corresponding currents. (B) Changes of p(o) or amplitude of the corresponding currents, and [Ca²⁺]_i. It should be noted that the numerical calculation of r_c has poor precision when dV_L/dt approaches 0 because r_c is normalized with dV_L/dt (the denominator in Eq. 12). Therefore, we excluded the ranges in which dV_L/dt is close to zero from the analyses (also done in Figs. 4 and 5).

Figure 4

Time profile of r_c during repolarization of the SA node action potential. (A) (Top) Time-dependent changes of V_L (red) and V_m (black). These are the same curves as those shown in Fig. 2A. (Bottom) Time-dependent changes of r_c of the major membrane currents. (B) Changes of p(o) or amplitude of the corresponding currents, and [Ca²⁺]_i.

Figure 5

The time profile of r_c during repolarization of the ventricular action potential. (A) (Top) Time-dependent changes of V_L (red) and V_m (black). These are the same curves as those shown in Fig. 2B. (Bottom) Time-dependent changes of r_c of the major membrane currents. (B) Changes of p(o) or amplitude of the corresponding currents, and [Ca²⁺]_i.

Figure 6

Comparison of V_L determined by Eq. 8 (red solid) with the old V_L determined by Eq. 17 (blue solid). (A) Both V_L values meet with V_m (black dotted) at dV_m/dt = 0. The curves for the present V_L and V_m are the same as those shown in Fig. 2. The gray bars indicate the time spans analyzed in panel B. (B) Modified time courses of V_L after fixing of I_Kr (green dashed) or I_ha (gray dashed). The V_L (blue) curve is the same as the blue curve in panel A.