Supplementary data
Roles of Subcellular Na+ Channel Distributions in the Mechanism of Cardiac Conduction Kunichika Tsumoto,†* Takashi Ashihara,‡ Ryo Haraguchi,§ Kazuo Nakazawa,§ and Yoshihisa Kurachi¶†* †
¶
The Center for Advanced Medical Engineering and Informatics, Osaka University, Division of Molecular and Cellular Pharmacology, Department of Pharmacology, Osaka University Graduate School of ‡ Medicine, Osaka, Japan; Department of Cardiovascular and Respiratory Medicine, Heart Rhythm § Center, Shiga University of Medical Science, Shiga, Japan; and Laboratory of Biomedical Science and Information Management, Research Institute, National Cerebral and Cardiovascular Center, Osaka, Japan
Supporting Material
Simulation Methods The following summarizes the methods we employed to conduct simulations of action potential (AP) propagation on a one-dimensional myofiber consisting of serially arranged ventricular cells. From the equivalent circuits in Fig. 1 B, we can derive a circuit equation at arbitrary time t using Ohm’s law and Kirchhoff’s law, leading to the simultaneous equation (Eq. S1): (S1) RI = V where R is the 320×320 resistance matrix, I is a current vector consisting of the transmembrane current from each node toward each membrane segment, and V is a voltage vector consisting of functions of the transmembrane potential of each membrane segment in the 64 cells. The current (I) and the voltage (V) vectors are denoted by Eqs. S2 and S3, respectively: I=
[i
1 m ,1
i
1 m,2
N −1 m ,5
i
i
i
N m ,1
i
N m,2
N m,3
N +1 m ,1
i
i
N m ,5
i
v
v
N ,4
v
N m,4
i ]t
i
64 m,4
(S2)
64 m ,5
and V=
[v v 1,1
1, 2
v
N −1, 5
v
N ,1
v
N ,2
N ,3
N ,5
v
N +1,1
v
64 , 4
v ]t 64 , 5
(S3)
where [ ]t represents the transpose operation. Each function, vN,k, for N = 1,…,64 and k = 1,…,5, except for v1,1 and v64,5, in the voltage vector is given by Eq. S4: N −1 N N v N ,1 = β V m ,5 − α V m ,1 + V m , 2 ,
v N , 2 = 2 V m ,1 − 3V m , 2 + V m ,3 , N
N
N
v N ,3 = V m , 2 − 2 V m ,3 + V m , 4 , N
N
N
(S4)
v N , 4 = V m , 3 − 3V m , 4 + 2 V m , 5 , N
N
N
v N ,5 = V m , 4 − α V m ,5 + β V m ,1 N
N
N +1
where α = 1+0.5Ri/Rg, β = 0.5Ri/Rg. When N = 1 and 64, the functions, v1,1 and v64,5, are 64 provided by v1,1 = − V 1m ,1 + V 1m , 2 and v64,5 = V 64 m , 4 − V m , 5 , respectively. The matrix, R, in
Eq. S1 is given by Eq. S5:
1
Ri / 2 Ri R=
0
Ri
0
0
0
0
Ri k 1 k 2 0 k3 k4 k4 k3 k3 k4 k4 k3 k2 0
0
k 1 Ri 0
Ri Ri k 1 k 2 0 k3 k4 k4 k3
0
0
k3 k4 k4 k3 k2 0
0
0
0
k 1 Ri Ri Ri
Ri / 2
(S5)
where k1 = –(Rd+2Rj), k2 = –2Rj, k3 = Rj+(Rd+Ri)/2, and k4 = Rj. The simultaneous equation (Eq. S1) can be solved, for the current I, with the membrane potentials, V mN , k , at time t, as an initial condition. The simultaneous equation in the non-cleft model in Fig. 1 C, on the other hand, can be derived by setting the cleft resistances, Rd and Rj, to zero in Eq. S1. Modulation of Intracellular Axial Current by Subcellular NaCh Distribution The amounts of the post-junctional INa and I gN , i.e., the intracellular axial current, both of which were modulated by the subcellular NaCh distribution, were associated with the formation of an AP upstroke of the cell. The reduction in intracellular axial current in the cleft model resulted from a decrease in the I gN comparing with the non-cleft model (Fig. S3 A). More importantly, the induction of the V Nj led to the simultaneous suprathreshold depolarization of the transmembrane potential in pre- and post-JMs at the cell junction under condition that the Gg was relatively high (Fig. S1 A-I). This simultaneous depolarization of the pre2
and post-JMs caused the decrease in the potential difference between the pre- and post-JMs of adjacent cells. Another reason of the reduction in the intracellular axial current might be the attenuation of post- and pre-junctional INa (24). As shown in Fig. S3 C, when the Gg was relatively high and NaChs were only localized in JM, the amplitudes of post- and pre-junctional INa in the cleft model were smaller than those in the non-cleft model (lower panel in Fig. S3 C). The reason for this difference was a marked decrease of the driving force of the pre- and post-junctional INa as discussed previously (24). The reduction in the driving force resulted in an attenuation of junctional INa even before the onset of the voltage-dependent inactivation of the NaChs. The attenuation of post-junctional INa in the cleft model was much greater than that in the non-cleft model because the extracellular cleft potential did not affect the transmembrane potentials of the JMs in the non-cleft model (Fig. S3 C). The attenuation in the post-junctional INa also resulted in the reduction in the intracellular axial current. In the presence of NaChs in the LM (Fig. S1 B), the influx of NaCh current from the LM into the cell could, however, boost the local currents and thereby enhanced the intracellular axial current. The enhancement of the intracellular axial current resulted in both the earlier upstroke and the augmented activation in the pre-JM of the cell. Since the earlier upstroke (phase-0) of transmembrane potential in pre-JM caused the increase in the potential difference between pre- and post-JMs at cell junctions (Fig. S1 B-I), the I gN increased as well. Therefore, the intracellular axial current in the cleft model in the presence of NaChs of LM was larger than that in the absence of NaChs of LM.
Effects of Alteration of Gap-junctional Conductance on Cleft Potential As the Gg was reduced, the I gN was decreased (Fig. S3 A). Thus, the amplitude of phase-0 of transmembrane potential in the post-JM becomes lower (compare Fig. S1 A and C). Since the reduction in the driving force in post-junctional INa was inhibited, the amplitude of the post-junctional INa was increased by reducing the Gg (see the bottom panel in Fig. S3 C) and then the intracellular axial current was enhanced. In this situation, a further increase of NaChs on the LM resulted in an additional increase in the intracellular axial current. Enhancing intracellular axial current causes an increase in the upstroke velocity of depolarization at phase-0 of the regional membrane voltage in the pre-JM. Thus, the depolarization peak of the regional membrane potential in the pre-JM was elevated by the cleft potential, and the depolarization peak of the regional 3
membrane voltage in the JM reached 86.6 mV (see Fig. S1 D-I). Therefore, the peak potential in the JM exceeded the reversal potential ~65 mV of the pre-junctional INa. As a consequence, the pre-junctional INa changed from inward to outward current (see Fig. S1 D-II). The negative cleft potential peak was markedly reduced by the outward NaCh current (see Fig. S3 B and Fig. S1 D-III). The reduction in the cleft-potential peak resulted in an insufficient elevation in the transmembrane potential in the post-JM, and thereby the NaChs in the post-JM could not be activated. Therefore, an increase of NaChs on LM resulted in a failure of AP propagation through the EF mechanism between some of the adjacent cells as the Gg was decreased. However, there was a slight current flow through the gap junction, resulting in the slower depolarization of the post-JM. As a result, AP propagation could be maintained even if the EF mechanism failed to act as the propagation mechanism. Nevertheless, because the propagation via gap-junctional mechanism became slow, the conduction velocity (CV) in the cleft model, where NaChs were present on LM, abruptly decreased with the reduction in Gg.
4
Supplemental Table
TABLE S1: Initial conditions and control parameter values Variable Definition
control value / initial value
Reference
Gg
Gap-junctional conductance
2.534 µS
(13)
Gj
Cleft conductance
4 MΩ
(24)
gkr
Maximum conductance of rapidly activating K+ current
0.03346×A mS/µF (28)
gks
Maximum conductance of slowly activating K+ current
0.0824×B mS/µF
(28)
gkp
Maximum conductance of plateau K+ current
0.01008 mS/µF
(28)
gto
Maximum conductance of the transient outward K+ current
0.11315 mS/µF
(28)
V m ,n
Transmembrane potentials of n-th segment of the N-th cell.
-87.0 mV
(28)
m
Activation gate of Na+ channel
0.0008
(30)
h
Fast inactivation gate of Na+ channel
0.993771
(30)
j
Slow inactivation gate of Na+ channel
0.995727
(30)
d
Voltage dependent activation variable for 3.210618×10-6 2+ L-type Ca channel
(29)
f1,fast
Fast component of the voltage dependent inactivation variable for L-type Ca2+ channel
1.0
(28)
f2,slow
Slow component of the voltage dependent inactivation variable for L-type Ca2+ 1.0 channel
(28)
N
5
Fast component of Ca2+ dependent inactivation variable for L-type Ca2+ channel
1.0
(28)
Slow component of Ca2+ dependent inactivation variable for L-type Ca2+ channel
1.0
(28)
b
Activation gate of T-type Ca2+ channel
0.000970231
(31)
g
Inactivation gate of T-type Ca2+ channel
0.994305
(31)
xr
Activation gate of rapidly activating K+ current
0.000124042
(31)
xs1
Fast activation gate of slowly activating K+ current
0.00445683
(31)
xs2
Slow activation gate of slowly activating K+ current
0.00445683
(31)
z
Activation gate of transient outward K+ current
0.0120892
(28)
y
Inactivation gate of transient outward K+ current
0.999978
(28)
[Na+]i
Intracellular Na+ concentration
12.1 mM
(31)
[K+]i
Intracellular K+ concentration
141.2 mM
(28)
[Ca2+]i
Intracellular Ca2+ concentration
0.00006 mM
(28)
2+
Ca2+ concentration of network sarcoplasmic reticulum
1.838 mM
(28)
2+
[Ca ]JSR
Ca2+ concentration of junctional sarcoplasmic reticulum
1.838 mM
(28)
[Na+]o
Extracellular Na+ concentration
140.0 mM
(31)
[K+]o
Extracellular K+ concentration
4.5 mM
(28)
[Ca2+]o
Extracellular Ca2+ concentration
1.8 mM
(28)
1
f CDI,fast
2
f CDI,slow
[Ca ]NSR
6
[K + ]o where A = , B =1+ 5.4
0.6 1.4
0.00038 1 + 2+ [Ca ]i
7
.
Supplemental Figure Legends FIGURE S1 Effects of cleft potentials on transmembrane potential in the cases of 100%gNa,JM with 0%gNa,LM (A and C), and of 100%gNa,JM with 50%gNa,LM (B and D). I (A through D) The phase-0 in transmembrane potentials in each segment from the 31st to the 33rd cell in the cleft model. II (A through D) The NaCh currents in each segment from the 31st to the 33rd cell. III (A through D) Cleft potentials. IV (A through D) The gap-junctional current (dashed lines), and the axial currents (solid lines). In III and IV, all data were obtained at the junctions between the 31st and the 32nd cells and between the 32nd and the 33rd cells. The gap-junctional conductance (Gg) in (A) and (B) was fixed at a normal value (2.534 µS), while the Gg in (C) and (D) was fixed at 3% of the normal value. %gNa,JM and %gNa,LM indicate the NaCh conductance of the junctional membrane (JM) and that of the lateral membrane (LM), respectively. In the strand model, we employed 0.25 µS for the cleft conductance (Gj). FIGURE S2 Conduction velocity (CV) versus gap-junctional conductance (Gg) when the NaCh distribution was altered while maintaining the total NaCh conductance per cell at the gNa, corresponding to ~1.23 µS. For comparison, the dashed line represents the CV in the case of 100%gNa,JM with 0%gNa,LM in the non-cleft model (red line, Fig. 5 A). FIGURE S3 Amplitudes of the gap junctional current Ig (A), cleft potential Vj (B), and NaCh current INa at junctional membrane (1st and 5th segments of the cell) (C) as a function of the gap-junctional conductance (Gg). All data were obtained at the junction between the 31st and 32nd cells. In the cleft model, we fixed the cleft conductance (Gj) at 0.25 µS.
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¶
The Center for Advanced Medical Engineering and Informatics, Osaka University, Division of Molecular and Cellular Pharmacology, Department of Pharmacology, Osaka University Graduate School of ‡ Medicine, Osaka, Japan; Department of Cardiovascular and Respiratory Medicine, Heart Rhythm § Center, Shiga University of Medical Science, Shiga, Japan; and Laboratory of Biomedical Science and Information Management, Research Institute, National Cerebral and Cardiovascular Center, Osaka, Japan
Supporting Material
Simulation Methods The following summarizes the methods we employed to conduct simulations of action potential (AP) propagation on a one-dimensional myofiber consisting of serially arranged ventricular cells. From the equivalent circuits in Fig. 1 B, we can derive a circuit equation at arbitrary time t using Ohm’s law and Kirchhoff’s law, leading to the simultaneous equation (Eq. S1): (S1) RI = V where R is the 320×320 resistance matrix, I is a current vector consisting of the transmembrane current from each node toward each membrane segment, and V is a voltage vector consisting of functions of the transmembrane potential of each membrane segment in the 64 cells. The current (I) and the voltage (V) vectors are denoted by Eqs. S2 and S3, respectively: I=
[i
1 m ,1
i
1 m,2
N −1 m ,5
i
i
i
N m ,1
i
N m,2
N m,3
N +1 m ,1
i
i
N m ,5
i
v
v
N ,4
v
N m,4
i ]t
i
64 m,4
(S2)
64 m ,5
and V=
[v v 1,1
1, 2
v
N −1, 5
v
N ,1
v
N ,2
N ,3
N ,5
v
N +1,1
v
64 , 4
v ]t 64 , 5
(S3)
where [ ]t represents the transpose operation. Each function, vN,k, for N = 1,…,64 and k = 1,…,5, except for v1,1 and v64,5, in the voltage vector is given by Eq. S4: N −1 N N v N ,1 = β V m ,5 − α V m ,1 + V m , 2 ,
v N , 2 = 2 V m ,1 − 3V m , 2 + V m ,3 , N
N
N
v N ,3 = V m , 2 − 2 V m ,3 + V m , 4 , N
N
N
(S4)
v N , 4 = V m , 3 − 3V m , 4 + 2 V m , 5 , N
N
N
v N ,5 = V m , 4 − α V m ,5 + β V m ,1 N
N
N +1
where α = 1+0.5Ri/Rg, β = 0.5Ri/Rg. When N = 1 and 64, the functions, v1,1 and v64,5, are 64 provided by v1,1 = − V 1m ,1 + V 1m , 2 and v64,5 = V 64 m , 4 − V m , 5 , respectively. The matrix, R, in
Eq. S1 is given by Eq. S5:
1
Ri / 2 Ri R=
0
Ri
0
0
0
0
Ri k 1 k 2 0 k3 k4 k4 k3 k3 k4 k4 k3 k2 0
0
k 1 Ri 0
Ri Ri k 1 k 2 0 k3 k4 k4 k3
0
0
k3 k4 k4 k3 k2 0
0
0
0
k 1 Ri Ri Ri
Ri / 2
(S5)
where k1 = –(Rd+2Rj), k2 = –2Rj, k3 = Rj+(Rd+Ri)/2, and k4 = Rj. The simultaneous equation (Eq. S1) can be solved, for the current I, with the membrane potentials, V mN , k , at time t, as an initial condition. The simultaneous equation in the non-cleft model in Fig. 1 C, on the other hand, can be derived by setting the cleft resistances, Rd and Rj, to zero in Eq. S1. Modulation of Intracellular Axial Current by Subcellular NaCh Distribution The amounts of the post-junctional INa and I gN , i.e., the intracellular axial current, both of which were modulated by the subcellular NaCh distribution, were associated with the formation of an AP upstroke of the cell. The reduction in intracellular axial current in the cleft model resulted from a decrease in the I gN comparing with the non-cleft model (Fig. S3 A). More importantly, the induction of the V Nj led to the simultaneous suprathreshold depolarization of the transmembrane potential in pre- and post-JMs at the cell junction under condition that the Gg was relatively high (Fig. S1 A-I). This simultaneous depolarization of the pre2
and post-JMs caused the decrease in the potential difference between the pre- and post-JMs of adjacent cells. Another reason of the reduction in the intracellular axial current might be the attenuation of post- and pre-junctional INa (24). As shown in Fig. S3 C, when the Gg was relatively high and NaChs were only localized in JM, the amplitudes of post- and pre-junctional INa in the cleft model were smaller than those in the non-cleft model (lower panel in Fig. S3 C). The reason for this difference was a marked decrease of the driving force of the pre- and post-junctional INa as discussed previously (24). The reduction in the driving force resulted in an attenuation of junctional INa even before the onset of the voltage-dependent inactivation of the NaChs. The attenuation of post-junctional INa in the cleft model was much greater than that in the non-cleft model because the extracellular cleft potential did not affect the transmembrane potentials of the JMs in the non-cleft model (Fig. S3 C). The attenuation in the post-junctional INa also resulted in the reduction in the intracellular axial current. In the presence of NaChs in the LM (Fig. S1 B), the influx of NaCh current from the LM into the cell could, however, boost the local currents and thereby enhanced the intracellular axial current. The enhancement of the intracellular axial current resulted in both the earlier upstroke and the augmented activation in the pre-JM of the cell. Since the earlier upstroke (phase-0) of transmembrane potential in pre-JM caused the increase in the potential difference between pre- and post-JMs at cell junctions (Fig. S1 B-I), the I gN increased as well. Therefore, the intracellular axial current in the cleft model in the presence of NaChs of LM was larger than that in the absence of NaChs of LM.
Effects of Alteration of Gap-junctional Conductance on Cleft Potential As the Gg was reduced, the I gN was decreased (Fig. S3 A). Thus, the amplitude of phase-0 of transmembrane potential in the post-JM becomes lower (compare Fig. S1 A and C). Since the reduction in the driving force in post-junctional INa was inhibited, the amplitude of the post-junctional INa was increased by reducing the Gg (see the bottom panel in Fig. S3 C) and then the intracellular axial current was enhanced. In this situation, a further increase of NaChs on the LM resulted in an additional increase in the intracellular axial current. Enhancing intracellular axial current causes an increase in the upstroke velocity of depolarization at phase-0 of the regional membrane voltage in the pre-JM. Thus, the depolarization peak of the regional membrane potential in the pre-JM was elevated by the cleft potential, and the depolarization peak of the regional 3
membrane voltage in the JM reached 86.6 mV (see Fig. S1 D-I). Therefore, the peak potential in the JM exceeded the reversal potential ~65 mV of the pre-junctional INa. As a consequence, the pre-junctional INa changed from inward to outward current (see Fig. S1 D-II). The negative cleft potential peak was markedly reduced by the outward NaCh current (see Fig. S3 B and Fig. S1 D-III). The reduction in the cleft-potential peak resulted in an insufficient elevation in the transmembrane potential in the post-JM, and thereby the NaChs in the post-JM could not be activated. Therefore, an increase of NaChs on LM resulted in a failure of AP propagation through the EF mechanism between some of the adjacent cells as the Gg was decreased. However, there was a slight current flow through the gap junction, resulting in the slower depolarization of the post-JM. As a result, AP propagation could be maintained even if the EF mechanism failed to act as the propagation mechanism. Nevertheless, because the propagation via gap-junctional mechanism became slow, the conduction velocity (CV) in the cleft model, where NaChs were present on LM, abruptly decreased with the reduction in Gg.
4
Supplemental Table
TABLE S1: Initial conditions and control parameter values Variable Definition
control value / initial value
Reference
Gg
Gap-junctional conductance
2.534 µS
(13)
Gj
Cleft conductance
4 MΩ
(24)
gkr
Maximum conductance of rapidly activating K+ current
0.03346×A mS/µF (28)
gks
Maximum conductance of slowly activating K+ current
0.0824×B mS/µF
(28)
gkp
Maximum conductance of plateau K+ current
0.01008 mS/µF
(28)
gto
Maximum conductance of the transient outward K+ current
0.11315 mS/µF
(28)
V m ,n
Transmembrane potentials of n-th segment of the N-th cell.
-87.0 mV
(28)
m
Activation gate of Na+ channel
0.0008
(30)
h
Fast inactivation gate of Na+ channel
0.993771
(30)
j
Slow inactivation gate of Na+ channel
0.995727
(30)
d
Voltage dependent activation variable for 3.210618×10-6 2+ L-type Ca channel
(29)
f1,fast
Fast component of the voltage dependent inactivation variable for L-type Ca2+ channel
1.0
(28)
f2,slow
Slow component of the voltage dependent inactivation variable for L-type Ca2+ 1.0 channel
(28)
N
5
Fast component of Ca2+ dependent inactivation variable for L-type Ca2+ channel
1.0
(28)
Slow component of Ca2+ dependent inactivation variable for L-type Ca2+ channel
1.0
(28)
b
Activation gate of T-type Ca2+ channel
0.000970231
(31)
g
Inactivation gate of T-type Ca2+ channel
0.994305
(31)
xr
Activation gate of rapidly activating K+ current
0.000124042
(31)
xs1
Fast activation gate of slowly activating K+ current
0.00445683
(31)
xs2
Slow activation gate of slowly activating K+ current
0.00445683
(31)
z
Activation gate of transient outward K+ current
0.0120892
(28)
y
Inactivation gate of transient outward K+ current
0.999978
(28)
[Na+]i
Intracellular Na+ concentration
12.1 mM
(31)
[K+]i
Intracellular K+ concentration
141.2 mM
(28)
[Ca2+]i
Intracellular Ca2+ concentration
0.00006 mM
(28)
2+
Ca2+ concentration of network sarcoplasmic reticulum
1.838 mM
(28)
2+
[Ca ]JSR
Ca2+ concentration of junctional sarcoplasmic reticulum
1.838 mM
(28)
[Na+]o
Extracellular Na+ concentration
140.0 mM
(31)
[K+]o
Extracellular K+ concentration
4.5 mM
(28)
[Ca2+]o
Extracellular Ca2+ concentration
1.8 mM
(28)
1
f CDI,fast
2
f CDI,slow
[Ca ]NSR
6
[K + ]o where A = , B =1+ 5.4
0.6 1.4
0.00038 1 + 2+ [Ca ]i
7
.
Supplemental Figure Legends FIGURE S1 Effects of cleft potentials on transmembrane potential in the cases of 100%gNa,JM with 0%gNa,LM (A and C), and of 100%gNa,JM with 50%gNa,LM (B and D). I (A through D) The phase-0 in transmembrane potentials in each segment from the 31st to the 33rd cell in the cleft model. II (A through D) The NaCh currents in each segment from the 31st to the 33rd cell. III (A through D) Cleft potentials. IV (A through D) The gap-junctional current (dashed lines), and the axial currents (solid lines). In III and IV, all data were obtained at the junctions between the 31st and the 32nd cells and between the 32nd and the 33rd cells. The gap-junctional conductance (Gg) in (A) and (B) was fixed at a normal value (2.534 µS), while the Gg in (C) and (D) was fixed at 3% of the normal value. %gNa,JM and %gNa,LM indicate the NaCh conductance of the junctional membrane (JM) and that of the lateral membrane (LM), respectively. In the strand model, we employed 0.25 µS for the cleft conductance (Gj). FIGURE S2 Conduction velocity (CV) versus gap-junctional conductance (Gg) when the NaCh distribution was altered while maintaining the total NaCh conductance per cell at the gNa, corresponding to ~1.23 µS. For comparison, the dashed line represents the CV in the case of 100%gNa,JM with 0%gNa,LM in the non-cleft model (red line, Fig. 5 A). FIGURE S3 Amplitudes of the gap junctional current Ig (A), cleft potential Vj (B), and NaCh current INa at junctional membrane (1st and 5th segments of the cell) (C) as a function of the gap-junctional conductance (Gg). All data were obtained at the junction between the 31st and 32nd cells. In the cleft model, we fixed the cleft conductance (Gj) at 0.25 µS.
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