Modeling Depolarization Delay, Sodium Currents, and Electrical Potentials in Cardiac Transverse Tubules

Figures

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  FIGURE 1. Delay of membrane depolarization in murine and human T-tubules. (A) Schematic representation of the model. T-tubule is subdivided in segments consisting of intraluminal resistance, membrane capacitance, and membrane resistance. Cytosolic resistance is assumed negligible and T-tubular mouth is assumed perfectly clamped. Panels (B–E) depict simulations of different T-tubules: healthy mouse (B), healthy human (C), mouse post-myocardial infarction (MI) (D), and human post-myocardial infarction (E). Dotted lines indicate the opening threshold for voltage-gated calcium channels (around –40 mV). Insets in panels (B–E) represent the time of depolarization to –40 mV vs. position along the T-tubule. T-tubular geometries are schematically depicted in blue and detailed in Table 1. Membrane potentials of every tenth (B,D) and second (C,E) node are depicted. Note the different axes in panels (C) and (E).

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  FIGURE 2. Varying parameters in the murine and human T-tubule models. In the models of mouse healthy (red solid line), mouse post-myocardial infarction (MI) (red dashed line), human healthy (black solid line), and human post-MI (black dashed line) T-tubules, we varied tubule length (A), membrane capacitance (B), resistivity of the extracellular space (C), and membrane conductance (D). The time delay after which the deepest membrane segment reached –40 mV is plotted against the respective varied parameter. Vertical lines represent reference values from the respective models.

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  FIGURE 3. Delay of membrane depolarization in murine T-tubules with constrictions. In the healthy murine T-tubule (Figure 1B), we introduced five 10× constrictions of luminal diameter (A) or an overall 10× constriction (B). Tubule geometries are schematically given in blue and detailed in Table 1. Membrane potentials of each (A) and every tenth (B) node are shown. Insets represent the time of depolarization to –40 mV versus position along the T-tubule.

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  FIGURE 4. Modeling sodium current in a healthy murine T-tubule. A voltage-gated sodium current [formulated according to Luo and Rudy (1991) and Livshitz and Rudy (2009)] with a conductance of 23 mS/μF (Luo and Rudy, 1991) was introduced into the healthy mouse T-tubule model in parallel to membrane resistance (Figure 1A). (A) Schematic representation of the morphology of a healthy murine T-tubule (Table 1). Membrane potentials (B), extracellular potentials (C), and simulated sodium current density upon a voltage-clamp step of the tubule mouth from –85 to –20 mV (D) are given. Note the decrease of peak sodium current and delayed activation in deeper segments of the tubules (D). This correlates with changes in the biophysical properties of the sodium current: product of activation gates (m3, E) and inactivation gates (hj, F) shows faster activation and inactivation in deeper T-tubular segments, respectively; and peak open probability (defined as m3hj) very slightly increases in deeper T-tubular segments (G). Data are shown for every tenth node.

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  FIGURE 5. Modeling sodium current in a murine T-tubule with five constrictions. (A) Schematic representation of the morphology of the tubule (Table 1). Membrane potentials (B), extracellular potentials (C), and simulated sodium current densities upon a voltage-clamp step of the tubule mouth from –85 to –20 mV (D) are given. The peak sodium current decreases in deeper tubular segments, while activation is delayed and inactivation is faster (D). This correlates with a lower driving force (B), and changes in the activation (m3, E) and inactivation gates (hj, F). Peak open probability (defined as m3hj) slightly increases in proximal T-tubular segments and then decreases in deeper T-tubular segments (G). Curve colors represent luminal diameter from largest (black) to smallest (light blue). Data are shown for each node.

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  FIGURE 6. Modeling sodium current in an overall constricted murine T-tubule. (A) Schematic representation of the morphology of the tubule (Table 1). Membrane potentials (B), extracellular potentials (C), and simulated sodium current densities upon a voltage-clamp step of the tubule mouth from –85 to –20 mV (D) are given. The sodium current decreases in deeper segments, while activation is delayed and inactivation is faster in deeper segments of the tubules (D). This correlates with a lower driving force (B), and changes in the activation gates (m3, E) and inactivation gates (hj, F). Peak open probability (defined as m3hj) slightly increases in proximal T-tubular segments and then decreases in deeper T-tubular segments, G). Data are shown for every tenth node.

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  FIGURE 7. Varying sodium current parameters in the healthy murine T-tubular model with and without constrictions. The Luo–Rudy–Livshitz sodium current (INa) model [blue (Luo and Rudy, 1991; Livshitz and Rudy, 2009)] is compared to the ten Tusscher et al. (2004) INa model (orange). For the tubule without constrictions (left), the tubule with five constrictions (middle), and the overall constricted tubule (right), the plots show the delay after which –40 mV is reached in the innermost segment (A–C; in panel A, both curves overlap), the most negative extracellular potential Ve (D–F), and the quantification of self-attenuation (G–I) versus the maximal sodium current conductance gNa. gNa was systematically varied using the following values: 2.3, 4.6, 6.9, 9.2, 11.5, 13.8, 16.1, 18.4, 20.7, 23, 34.5, and 46 mS/μF. Self-attenuation was quantified as the smallest peak INa in the tubule (in absolute value) normalized by the largest peak INa.