Mathematical models of the action
potential in the periphery and center of the rabbit sinoatrial (SA)
node have been developed on the basis of published experimental data.
Simulated action potentials are consistent with those recorded
experimentally: the model-generated peripheral action potential has a
more negative takeoff potential, faster upstroke, more positive peak
value, prominent phase 1 repolarization, greater amplitude, shorter
duration, and more negative maximum diastolic potential than the
model-generated central action potential. In addition, the model
peripheral cell shows faster pacemaking. The models behave
qualitatively the same as tissue from the periphery and center of the
SA node in response to block of tetrodotoxin-sensitive Na+
current, L- and T-type Ca2+ currents,
4-aminopyridine-sensitive transient outward current, rapid and slow
delayed rectifying K+ currents, and
hyperpolarization-activated current. A one-dimensional model of a
string of SA node tissue, incorporating regional heterogeneity, coupled
to a string of atrial tissue has been constructed to simulate the
behavior of the intact SA node. In the one-dimensional model, the
spontaneous action potential initiated in the center propagates to the
periphery at ~0.06 m/s and then into the atrial muscle at 0.62 m/s.
 |
INTRODUCTION |
THE RHYTHMIC BEATING
of the heart is the result of action potentials initiated in the
pacemaker of the heart, the sinoatrial (SA) node. Mathematical models
of the electrical activity of the SA node of the rabbit (the species
for which most data have been obtained) have been produced. The first
models were produced by Yanagihara et al. (61) and Noble
and Noble (49), and subsequent models were developed from
the earlier models (13, 18, 59).
All the above models are of a typical SA node action potential.
However, the SA node, functionally, anatomically, and
electrophysiologically, is not homogeneous. In the rabbit the SA node
measures ~8 mm × ~10 mm (5). In the vertical
direction it is bounded by the superior and inferior venae cavae, and
in the horizontal direction it is bounded by the crista terminalis (a
thick bundle of atrial muscle) and the interatrial septum. The action
potential is initiated in a small part of the SA node, the leading
pacemaker site. Normally, the leading pacemaker site is approximately
midway between the two venae cavae and 1-2 mm from the crista
terminalis (3). This region is referred to as the center
of the SA node. From the leading pacemaker site in the center, the
action potential propagates to the periphery of the SA node and then
onto the atrial muscle of the crista terminalis. Conduction toward the
interatrial septum is blocked (3). The periphery of the SA
node (the region of the SA node close to the crista terminalis) is
referred to by some authors as perinode or transitional tissue.
Although the principal function of the periphery of the SA node is to
conduct the action potential from the leading pacemaker site in the
center to the atrial muscle, the periphery does show pacemaker
activity. In response to a variety of interventions, for example,
autonomic nerve stimulation, the leading pacemaker site shifts from the center, and in many cases it shifts toward the periphery
(53); the pacemaker activity of the periphery of the SA
node is, therefore, important physiologically. Most work on regional
differences in the SA node has been carried out on tissue around the
leading pacemaker site midway between the venae cavae and has focused on peripheral-central differences (little is known about the
tissue from the more superior and inferior regions and also toward the interatrial septum). There are important anatomic differences; for
example, in the center the cells are smaller and have fewer and more
poorly organized myofilaments than in the periphery (3). There are electrophysiological differences; these have been studied in
the intact SA node or in small balls of tissue from different regions
of the SA node. In the periphery the takeoff potential is more
negative, the action potential upstroke velocity is higher, the action
potential is shorter, the maximum diastolic potential (also resting
potential in quiescent tissue) is more negative, and the intrinsic
pacemaker activity is paradoxically faster than in the center
(33). Ion channel block has different effects in the
different regions: block of tetrodotoxin-sensitive Na+
current (iNa), 4-aminopyridine (4-AP)-sensitive
transient outward current (ito), or
hyperpolarization-activated current (if) has a
greater effect in the periphery, whereas block of L-type
Ca2+ current (iCa,L) or rapid
delayed rectifying K+ current (iK,r)
has a greater effect in the center (6, 32, 33, 45). These differences in the response to
ion channel block suggest regional differences in ionic currents.
Single cells have not been isolated from different regions of the SA
node to confirm this. However, we isolate single cells from the whole of the SA node and then distinguish between cells on the basis of cell
capacitance (Cm), a measure of cell size, which
is known to vary between the periphery and the center (see above)
(27, 28, 36, 39).
The action potential characteristics vary with Cm in a manner consistent with the regional
differences (see above) (27). For example, in large cells
with a high Cm (presumably from the periphery)
the upstroke velocity is high, whereas in small cells with a low
Cm (presumably from the center) the upstroke velocity is low (27). We have measured the density of some
ionic currents; whereas the density of iCa,L is
not significantly different in cells of different size, the densities
of iNa, ito,
iK,r, iK,s, and
if are greater in larger cells (27, 28, 36, 39;
iK,r and iK,s data from
unpublished observations).
Models incorporating regional differences within the SA node have been
developed (49, 60). However, the models were
based on speculation because of the absence of data on regional
differences in ionic currents. The aim of the present study was to
develop, on the basis of the evidence reviewed above, biophysically
detailed models of action potentials in the periphery and center of the rabbit SA node.
Glossary
| 4-AP |
4-Aminopyridine
|
| AM |
Atrial muscle
|
| APD |
Action potential duration
|
| CL |
Spontaneous cycle length
|
| Cm |
Cell capacitance
|
| Cma(x),
Cms(x) |
Capacitance of atrial muscle cell or SA node cell in
one-dimensional model of intact SA node at distance x from
center of SA node
|
| dL, dT |
Activation variables for iCa,L and
iCa,T
|
| dNaCa |
Denominator constant for iNaCa
|
| dV/dtmax |
Maximum upstroke velocity of action potential
|
| Da, Ds |
Diffusion coefficient between atrial muscle cells or
SA node cells in one-dimensional model of the intact SA node
|
| EK,s |
Reversal potential for iK,s
|
| ENa, ECa,
EK |
Equilibrium potentials for Na+, Ca2+, and
K+
|
| ECa,L, ECa,T |
Reversal potentials for iCa,L and
iCa,T
|
| F |
Faraday's constant
|
| FK,r |
Fraction of activation of iK,r that occurs
slowly
|
| FNa |
Fraction of inactivation of iNa that occurs
slowly
|
| fL, fT |
Inactivation variables for iCa,L and
iCa,T
|
| gp, gc |
Conductance of a current in peripheral or central SA node cell
models
|
| ga(x),
gs(x) |
Conductance of a current in atrial muscle cell or SA node cell in
one-dimensional model of intact SA node at distance x from center of SA node
|
| gNa |
Conductance of iNa
|
| gCa,L, gCa,T |
Conductance of iCa,L and
iCa,T
|
| gto, gsus |
Conductance of ito and
isus
|
| gK,r, gK,s |
Conductance of iK,r and
iK,s
|
| gf,Na, gf,K |
Conductance of Na+ and K+ components of
if
|
| gb,Na, gb,Ca,
gb,K |
Conductance of ib,Na,
ib,Ca, and ib,K
|
| h1, h2 |
Fast and slow inactivation variables for
iNa
|
| h |
Net fractional availability of iNa
|
| iNa |
TTX-sensitive Na+ current
|
| iCa,L, iCa,T |
L- and T-type Ca2+ currents
|
| ito, isus |
Transient and sustained components of 4-AP-sensitive current
|
| iK,r, iK,s |
Rapid and slow delayed rectifying K+ currents
|
| iK |
Sum of iK,r and iK,s
|
| if |
Hyperpolarization-activated current
|
| if,Na, if,K |
Na+ and K+ components of
if
|
| ib,Na, ib,Ca,
ib,K |
Background Na+, Ca2+, and K+
currents
|
| iNaCa |
Na+/Ca2+ exchanger current
|
| ip |
Na+-K+ pump current
|
 |
Maximum ip
|
| itot |
Total ionic current in a cell
|
| itota(x),
itots(x) |
Total ionic current in atrial muscle cell or SA node cell in
one-dimensional model of intact SA node at distance x from
center of SA node
|
| ist |
Sustained current
|
| iK,ACh |
ACh-activated K+ current
|
| iK,ATP |
ATP-sensitive K+ current
|
| kNaCa |
Scaling factor for iNaCa
|
| Km,Na, Km,K |
Dissociation constants for Na+ and K+
activation of ip
|
| L |
Length of string of SA node and atrial tissue in one-dimensional model
of intact SA node
|
| Ls |
Length of string of SA node tissue in one-dimensional model of intact
SA node
|
| m |
Activation variable for iNa
|
| MDP |
Maximum diastolic potential
|
n |
Steady-state value of n
|
| pa |
General activation variable for iK,r
|
| pa,f, pa,s |
Fast and slow activation variables for
iK,r
|
| pi |
Inactivation variable for iK,r
|
| Q10 |
Fractional change in a variable with a 10°C increase in
temperature
|
| r |
Activation variable for ito
|
| R |
Universal gas constant
|
| q |
Inactivation variable for ito
|
| SA node, SAN |
Sinoatrial node
|
| t |
Time
|
| T |
Absolute temperature
|
| TOP |
Takeoff potential
|
| V |
Membrane potential
|
| Va, Vs |
Membrane potential of atrial muscle cell or SA node cell in
one-dimensional model of intact SA node
|
| Va(x),
Vs(x) |
Membrane potential of atrial muscle cell or SA node cell in
one-dimensional model of intact SA node at distance x from
center of SA node
|
| x |
Distance from center of SA node in one-dimensional model of
intact SA node
|
| xs |
Activation variable for iK,s
|
| y |
Activation variable of if
|
| z |
Valency of ion
|
| [Na+]i,
[Ca2+]i |
Intracellular Na+, Ca2+, and
K+
|
| [K+]i |
concentrations
|
| [Na+]o,
[Ca2+]o |
Extracellular Na+, Ca2+, and
K+
|
| [K+]o |
concentrations
|
 n |
Voltage-dependent opening rate constant of n
|
 n |
Voltage-dependent closing rate constant of n
|
 NaCa |
Position of Erying rate theory energy barrier controlling voltage
dependence of iNaCa
|
 n |
Time constant of n
|
 |
Space constant
|
| |
MODEL DEVELOPMENT |
Mathematical models of the action potential in peripheral and
central cells of the rabbit SA node at 37°C were developed using experimental data from rabbit SA node preparations. New formulations for a number of ionic currents were developed on the basis of newly
published data from rabbit SA node cells: iNa,
iCa,L, iCa,T, ito, 4-AP-sensitive sustained outward current
(isus), iK,r,
iK,s, and if. Full
details are given below. The models also include formulations for
background currents (ib,Na,
ib,Ca, and ib,K), ip, and iNaCa; these
formulations are similar to those in other models (13,
17, 25). The membrane potential is calculated using Eq. 2 (Table 1). The
Glossary defines all abbreviations used. Formulations for
ionic currents are shown in Tables
2-9. All parameter values are listed in Table
10. Differences in current densities
between the peripheral and central SA node cell models are listed in
Table 11. Initial values of variables
used to run the models are listed in Table
12.
Model of iNa
The iNa was thought to be absent in SA
node cells, and most previous models of the SA node action potential do
not include iNa. However, recent experimental
results show that iNa is present and
physiologically important (27, 33). Demir et
al. (13) introduced iNa in their
model of the rabbit SA node action potential; the formulation for
iNa was based on the experimental data of Colatsky (11) from rabbit Purkinje fibers. However, on the
basis of the formulation for iNa from Demir et
al., the time dependence of iNa is different
from that seen experimentally in rabbit SA node cells
(27). The classic formulation for the Na+
current assumes that the Na+ conductance is controlled by
m3h, in which m is the
activation variable and h is the inactivation variable
(26). Most previous formulations for cardiac
iNa have used the same term (17).
However, recent voltage-clamp experiments on rabbit SA node cells
(27, 43) and other cardiac cells have shown
that the time course of recovery from inactivation can be best fitted
by two exponentials; therefore, there are two components of
inactivation; hence, two inactivation variables are needed. In the
present formulation, three variables are used to govern the kinetics of
iNa: m and h1
and h2, a fast and a slow inactivation variable.
The equations for iNa are listed in Table 2.
Activation and inactivation curves.
Activation curves (corresponding to the steady-state value of
m3) are shown in Fig.
1A. The filled squares show
the activation curve based on data from Baruscotti et al.
(2) from young rabbit SA node cells at room temperature,
and the filled triangles represent data from Muramatsu et al.
(43) from cultured rabbit SA node cells at 22-24°C
(fits to the experimental data rather than the original data are
shown). The solid line is the model-generated activation curve, which
fits well with the data of Baruscotti et al. The dashed line is the
activation curve from a model of a rabbit atrial cell
(41). The general inactivation variable h is
the weighted sum of h1 and
h2 (Eq. 7, Table 2). FNa
is the fraction of inactivation that occurs slowly and is dependent on the membrane potential; h1 and
h2 change with different time constants but have
the same steady-state value. Inactivation curves (corresponding to the
steady-state value of h) are also shown in Fig.
1A. The open squares show the inactivation curve based on
data from Baruscotti et al. from young rabbit SA node cells, and the
open triangles represent data from Muramatsu et al. from cultured
rabbit SA node cells (fits to the experimental data rather than the
original data are shown). There is a substantial difference between the data from the two groups. It is known that ion channels can change in
culture, and this perhaps explains the difference. The model-generated inactivation curve (solid line) is closer to the data of Baruscotti et
al. from young rabbit SA node cells. The dashed line is the inactivation curve from the model of a rabbit atrial cell
(41).
View larger version (24K):
[in this window]
[in a new window]
|
Fig. 1.
TTX-sensitive Na+ current
(iNa). A: activation
(m3, filled symbols) and inactivation
(h, open symbols) curves. B: time
constant of activation (m). C:
time constant of fast inactivation
(h1). D: time constant
of slow inactivation (h2).
E: fraction of iNa inactivation that
occurs slowly (FNa). F: simulated
iNa during 10-ms voltage-clamp pulses to  55 to
+40 mV (in 5-mV increments) from a holding potential of 60 mV
(top) and current-voltage relationships for
iNa (bottom). For the current-voltage
relationships, iNa was measured as peak inward
current. G: density of iNa (measured
from the peak inward current during a pulse to 5 mV) plotted against
cell capacitance (Cm).  , Data
from Honjo et al. (27) from rabbit sinoatrial (SA) node
cells; dotted line, regression line;  , values used in
the peripheral (Cm = 65 pF) and central
(Cm = 20 pF) SA node cell models.
|
|
Kinetics.
Because of the rapid activation of iNa, study of
iNa activation is difficult, and there are no
data from rabbit SA node cells on the time constant of activation
(m). Data from the study of Brown et al.
(8) on rat ventricular cells were used. Inasmuch as the
experiments of Brown et al. were carried out at room temperature (22°C), a Q10 of 1.7 (41) was used to
correct the data for 37°C. The m is plotted as a
function of membrane potential in Fig. 1B, in which the
circles show the temperature-corrected experimental data, and the solid
line was generated by the model. In Fig. 1C, the time
constant of fast inactivation
(h1) is plotted as a function of
membrane potential. The formulation for
h1 was based on data from
Muramatsu et al. (43) from cultured rabbit SA node cells
(circles in Fig. 1C), Honjo et al. (27) from
rabbit SA node cells (triangles), and Brown et al. from rat ventricular cells (squares). A Q10 of 1.7 (41) was used to
correct the experimental data (collected at 22°C) for 37°C (the
temperature-corrected data are shown in Fig. 1C). In Fig.
1C, the solid line was generated by the model and the dashed
line is from the model of a rabbit atrial cell (41). In
Fig. 1D, the time constant of slow inactivation (h2) is plotted as a function of
membrane potential. The formulation for
h2 was based on data from
Muramatsu et al. from cultured rabbit SA node cells (circles in Fig.
1D) and Brown et al. from rat ventricular cells (squares). A
Q10 of 1.7 (41) was used to correct the
experimental data (collected at 22°C) for 37°C (the
temperature-corrected data are shown in Fig. 1D). In Fig.
1D, the solid line was generated by the model and the dashed
line is from the model of a rabbit atrial cell (41). In
Fig. 1E, the fraction of slow inactivation (FNa)
is plotted as a function of membrane potential. The formulation for FNa was based on data from Muramatsu et al. from cultured
rabbit SA node cells. In Fig. 1E, the circles show
experimental data, and the solid line was generated by the model. Over
a wide range of membrane potentials, FNa is ~10% of
total inactivation of iNa. Inasmuch as the
density of iNa is large in a peripheral SA node cell, ~10% of iNa that inactivates slowly
will contribute a substantial inward current during the early period of
the action potential.
Simulated current.
Figure 1F shows simulated iNa from
the peripheral SA node cell model during depolarizing voltage-clamp
pulses as well as the current-voltage relationship of
iNa from the model (solid line and filled
squares). The open circles show the experimental data of Honjo et al.
(27) from a rabbit SA node cell with a
Cm of 54.5 pF.
Models of iCa,L and iCa,T
The equations for iCa,L are listed in
Table 3. Activation and inactivation curves [corresponding to the
steady-state values of the activation variable
(dL) and the inactivation variable (fL)] are shown in Fig.
2A. The squares and circles
show data from Hagiwara et al. (22) and Fermini and Nathan
(19), respectively, from rabbit SA node cells at
36-37°C, and the solid lines were generated by the model. For
the time constants of activation and inactivation, we followed Demir et
al. (13) and used the data of Nilius (46)
from guinea pig SA node cells after temperature correction (from 25 to
37°C with a Q10 of 2.3). Figure 2C shows simulated iCa,L from the peripheral SA node cell
model during depolarizing voltage-clamp pulses. Figure 2D
shows current-voltage relationships for iCa,L
(circles); the open circles show data from Hagiwara et al. from a
rabbit SA node cell, and the filled circles (and solid line) show data
from the peripheral SA node cell model.
View larger version (18K):
[in this window]
[in a new window]
|
Fig. 2.
L- and T-type Ca2+ currents
(iCa,L and iCa,T).
A: activation (dL, filled symbols)
and inactivation (fL, open symbols) curves
for iCa,L. B: activation
(dT, filled symbols) and inactivation
(fT, open symbols) curves for
iCa,T; error bars, SE. C: simulated
iCa,L (300-ms voltage-clamp pulses to 30 to
+40 mV in 10-mV increments from a holding potential of 40 mV) and
iCa,T (300-ms voltage-clamp pulses to 70 to
+10 mV in 10-mV increments from a holding potential of 80 mV).
D: current-voltage relationships for
iCa,L (circles) and iCa,T
(squares); iCa,L and
iCa,T were measured as peak inward current.
E: density of iCa,L (measured from
the peak inward current during a pulse to 0 mV) plotted against
Cm. , Data from Honjo et al.
(27) from rabbit SA node cells; , values
used in the peripheral (Cm = 65 pF) and
central (Cm = 20 pF) SA node cell models.
|
|
The equations for iCa,T are listed in
Table 4. Activation and inactivation curves [corresponding to the
steady-state values of the activation variable
(dT) and the inactivation variable (fT)] are shown in Fig. 2B. The
squares, circles, and triangles show data from Hagiwara et al.
(22), Fermini and Nathan (19), and Lei et al.
(38), respectively, from rabbit SA node cells at
~37°C. In the model the activation and inactivation curves (solid
lines) were computed using equations (Eqs. 33 and 38, Table 4) formulated by Lei (35) and Lei et
al. (38) based on their experimental data from rabbit SA
node cells at 37°C. In the model we used equations (Eqs.
30-32 and 35-37, Table 4) formulated by Hagiwara et al. (22) for the time constants of activation
and inactivation (dT and
fT) in rabbit SA node cells. Figure 2C shows simulated
iCa,T during depolarizing voltage-clamp pulses.
Figure 2D shows current-voltage relationships for
iCa,T (squares); the open squares show data from
Hagiwara et al. from a rabbit SA node cell, and the filled squares (and
solid line) show data from the peripheral SA node cell model.
Model of 4-AP-Sensitive Current
Previous models of the SA node action potential did not
incorporate ito. However,
ito is now known to be present in the rabbit SA
node and to play an important role (6, 28,
39). The ito is known to be blocked
by 4-AP. In rabbit SA node cells, 4-AP blocks a transient outward
current as well as a sustained outward current. It is unclear whether
the transient and sustained components represent two phases of one
current or two separate currents (28). We chose to treat
the two components as separate mathematical entities:
ito and isus. Honjo et
al. (28) found no difference in the activation curves for
ito and isus, and
therefore in the model we used the same activation variable
(r) for ito and
isus. Of course, the inactivation variable
(q) only governs ito. The equations
for ito and isus are
listed in Table 5.
Activation and inactivation curves.
Activation curves (corresponding to the steady-state value of the
activation variable, r) are shown in Fig.
3A. The filled triangles,
filled squares, and filled diamonds show data from Honjo et al.
(28) from rabbit SA node cells with capacitances of 63.4, 34.5, and 20.3 pF at 25°C; the activation curves are for the sum of
ito and isus. The filled
hexagons show data from Lei et al. (39) from rabbit SA
node cells at 35°C; the activation curve is for
ito only. The filled circles show data from
Giles and van Ginneken (21) from rabbit crista terminalis
cells at ~20.5°C; because of the method used, the activation curve
is for the sum of ito and
isus (if the latter was present). The solid line
shows the activation curve generated by the model. Inactivation curves
for ito only (corresponding to the steady-state
value of q) are also shown in Fig. 3A.
Inactivation curves are shown from Honjo et al. (28) from
rabbit SA node cells (open triangles, open squares, open diamonds: data
from cells with capacitances of 63.4, 47.1, and 23.6 pF, respectively),
Lei et al. from rabbit SA node cells (open hexagons), Giles and van
Ginneken from rabbit crista terminalis cells (open circles), and the
present model (solid line). The model-generated inactivation curve is
close to the data of Lei et al. from rabbit SA node cells at 35°C.
View larger version (22K):
[in this window]
[in a new window]
|
Fig. 3.
Transient and sustained components of 4-aminopyridine
(4-AP)-sensitive currents (ito and
isus). A: activation
(r, filled symbols) and inactivation
(q, open symbols) curves. B: time
constant of activation (r). C:
time constant of inactivation (q).
D: simulated 4-AP-sensitive current during 200-ms
voltage-clamp pulses to 70 to +60 mV (in 10-mV increments) from a
holding potential of 80 mV (top, voltage-clamp protocol;
bottom, current). E: current-voltage relationship
for ito (measured as peak outward current at the
start of the pulse and the current at the end of the pulse; currents
have been normalized to the maximum current at +60 mV); error bars, SE.
F and G: densities of ito
(F, measured as the difference between the peak
4-AP-sensitive outward current during a 200-ms pulse to +50 mV from a
holding potential of 80 mV and the current at the end of the pulse)
and isus (G, measured as the
4-AP-sensitive current at the end of a 200-ms pulse to +50 mV from a
holding potential of 80 mV) plotted against
Cm. , Data from Lei et al.
(39) from rabbit SA node cells; dotted lines, regression
lines; , values used in the peripheral
(Cm = 65 pF) and central
(Cm = 20 pF) SA node cell models.
|
|
Kinetics.
The time constant of activation of ito and
isus (r) is shown in
Fig. 3B. The circles show data from Giles and van Ginneken
(21) from rabbit crista terminalis cells. Inasmuch as the
experimental data were collected at 24°C, a Q10 of 2.18 (41) was used to correct the data for 37°C (the
temperature-corrected data are shown in Fig. 3B). The solid
line was generated by the model. Figure 3C shows the time
constant of inactivation of ito (q). The squares and triangles show data from
Honjo et al. (28) from rabbit SA node cells for the fast
and slow components of inactivation [data collected at 25°C
corrected for 37°C with a Q10 of 2.18 (41)]. The circles show data from Giles and van Ginneken
from rabbit crista terminalis cells [data collected at ~20.5°C
corrected for 37°C with a Q10 of 2.18 (41)]. The solid line was generated by the model.
Simulated current.
Figure 3D shows simulated 4-AP-sensitive current
(ito + isus) from
the peripheral SA node cell model during depolarizing voltage-clamp pulses. The currents are similar to 4-AP-sensitive currents in rabbit
SA node cells (28, 39). Current-voltage
relationships for ito are shown in Fig.
3E. The open circles show data from Lei et al.
(39) from rabbit SA node cells, and the filled circles (and solid line) show data from the peripheral SA node cell model.
Model of iK,r
Recent experiments have shown that iK in
rabbit SA node cells (29, 37,
51) can be separated into two kinetically different components, iK,r and
iK,s. A formulation for the delayed rectifying K+ current with two components, iK,r
and iK,s, was constructed; the equations are
listed in Tables 6 and 7. Equations for iK,r take the general form suggested by Shibasaki (56) with an
activation variable (pa) and inactivation
variable (pi).
Activation and inactivation curves.
Activation and deactivation of iK,r in rabbit SA
node cells have double-exponential time courses (37,
51). To model this, we have used two activation variables:
a fast activation variable (pa,f) and a slow
activation variable (pa,s). The general
activation variable (pa) is the weighted sum of
the fast and slow activation variables (Eq. 48, Table 6). In
rabbit SA node cells, experimental data have shown no distinct
dependence of the fraction of inactivation that occurs slowly
(FK,r) on membrane potential, and the ratio of the slow to
the fast component of activation of iK,r is 2:3 (37). In the model, FK,r is assumed to be
constant with a value of 0.4. Figure
4A shows activation curves
corresponding to the steady-state value of pa
(we assume that pa,f and
pa,s are the same and equal to
pa). The triangles represent data from Lei and Brown (37) from rabbit SA node cells at 37°C (fit to the
experimental data rather than the original data is shown). The solid
line was generated by the present model and is close to the data of Lei and Brown. In a model of a guinea pig ventricular cell, Noble et el.
(50) assumed that the steady-state values of
pa,f and pa,s are
different, and in the right-hand part of Fig. 4A the long
dashed and short dashed lines show the dependence of
pa,f and pa,s,
respectively, from the model of Noble et al. on membrane potential.
View larger version (26K):
[in this window]
[in a new window]
|
Fig. 4.
Rapid and slow delayed rectifying currents
(iK,r and iK,s).
A: activation (pa, filled symbols)
and inactivation (pi, open symbols) curves
for iK,r; error bars, SE. B: fast
time constant of activation of iK,r
(pa,f).
C: slow time constant of activation of
iK,r
(pa,s);
error bars, SE. D: activation curve for
iK,s. E: time constant of activation
of iK,s. F and G:
simulated iK,r (F) and
iK,s (G) during 1-s pulses applied to
potentials between 50 and +40 mV (in 10-mV increments) from a holding
potential of 60 mV. H and I: current-voltage
relationships for iK,r (H) and
iK,s (I). Currents were measured at
the end of the voltage-clamp pulses. J and K:
densities of iK,r (J, measured
as the peak 3 µM E-4031-sensitive tail current after a 1-s pulse to
10 mV from a holding potential of 50 mV) and
iK,s (K, measured as the peak 3 µM E-4031-insensitive tail current after a 1-s pulse to +40 mV from a
holding potential of 50 mV) plotted against
Cm. , Data from M. Lei
(unpublished observations) from rabbit SA node cells; dotted lines,
regression lines; , values used in the peripheral
(Cm = 65 pF) and central
(Cm = 20 pF) SA node cell models.
|
|
Inactivation variable.
Figure 4A shows inactivation curves (corresponding to the
steady-state value of pi). In the model the
inactivation curve (solid line) was computed using an equation
(Eq. 56, Table 6) formulated by Ito and Ono
(29, 51) based on their data (circles) from rabbit SA node cells at 33°C. The dashed line is from the model of a
guinea pig ventricular cell (50). There appears to be a nearly 18-mV shift in the inactivation curve in rabbit SA node cells
compared with that in guinea pig ventricular cells.
Kinetics.
The time constants of the fast and slow activation variables
(pa,f and
pa,s) are bell-shaped functions of membrane potentials, as shown in Fig. 4, B and
C. In the model the time constants (solid lines) were
computed using equations (Eqs. 52 and 53, Table
6) formulated by Ono and Ito (51) based on their
experimental data (circles) for rabbit SA node cells. The dashed lines
in Fig. 4, B and C, show the time constants used for the guinea pig ventricular cell model of Noble et al.
(50). It appears that
pa,f and
pa,s
are larger in guinea pig ventricular cells than in rabbit SA node
cells. No experimental data are available on the relationship between
the time constant of inactivation
(pi) and potential in the rabbit
SA node. Here we assume that pi
is independent of the membrane potential and has a fixed value of 0.002 s, as suggested by Ono and Ito.
The iK,r has been consistently observed to
reverse direction at the K+ equilibrium potential
(EK) (51); therefore, only
K+ is assumed to carry iK,r, and the
reversal potential is equal to EK.
Simulated current.
Figure 4F shows simulated iK,r from
the peripheral SA node cell model during depolarizing voltage-clamp
pulses. Figure 4H shows the current-voltage relationship of
iK,r (iK,r measured at
the end of the pulse). The open circles show data from Ito and Ono
(29) from rabbit SA node cells, and the filled circles (and solid line) show data from the peripheral SA node cell model. The
current-voltage relationship from the model is roughly similar to the
experimental data.
Model of iK,s
The slow sigmoidal activation of
iK,s is modeled by squaring a gating variable
(xs). Figure 4D shows the activation
curve (corresponding to the steady-state value of
xs2). The circles show data from Lei and
Brown (37) from rabbit SA node cells at 37°C, and the
solid line is the activation curve generated by the model. There are
limited experimental data available for the time constant of the
activation (xs) of
iK,s in rabbit SA node cells [triangle in Fig.
4E (37)]. Instead, we used equations
(Eqs. 61, 62, and 64, Table 7) formulated by Heath and Terrar (24) based on their data (circles) from
guinea pig ventricular cells at 35-36°C, as shown in Fig.
4E; in Fig. 4E the solid line was generated using
the model. The iK,s has been observed to reverse
at voltages positive to EK, which suggests that
the iK,s channel is permeable to an ion in
addition to K+ (23). We assumed that
iK,s is permeable to a small extent to Na+. The reversal potential of iK,s
is shown in Table 7. Figure 4G shows simulated
iK,s from the peripheral SA node cell model during depolarizing voltage-clamp pulses. Figure 4I shows a
current-voltage relationship for iK,s from the
peripheral SA node cell model.
Model of if
The if is a mixed current and is carried
by Na+ and K+. In our formulation,
if has two components,
if,K and if,Na. The
equations are listed in Table 8. Figure
5A shows activation curves
(corresponding to the steady-state value of y). In Fig.
5A, the triangles and squares show data from Liu et al.
(42) from rabbit SA node cells at 35-36°C, the
circles show data from van Ginneken and Giles (57) from
rabbit SA node cells at 30-33°C, and the solid line was
generated by the model. Figure 5B shows the time constant of
activation of if (y).
In Fig. 5B, the triangles and circles show data from Liu et
al. and van Ginneken and Giles, respectively, from rabbit SA node
cells, and the solid line was generated by the model. Figure
5C shows simulated if from the peripheral SA node cell model during hyperpolarizing voltage-clamp pulses. Figure 5D shows current-voltage relationships for
if. The open triangles show data from Honjo et
al. from a rabbit SA node cell at 36°C with a
Cm of 44.7 pF, and the open circles show data
from Honjo et al. from a smaller rabbit SA node cell at 36°C with a
Cm of 20.2 pF. The filled circles (and solid
line) show data from the peripheral SA node cell model
(Cm = 65 pF), and the filled squares (and
solid line) show data from the central SA node cell model
(Cm = 20 pF). The threshold of
if in the model is 40 mV, which is in the
range of that seen experimentally in rabbit SA node cells
(14-16, 57) but more positive than that seen by Honjo et al.
View larger version (19K):
[in this window]
[in a new window]
|
Fig. 5.
Hyperpolarization-activated current
(if). A: activation curves.
B: time constant of activation
(y). C: simulated
if during 300-ms pulses to potentials from 50
to 120 mV (in 10-mV increments) from a holding potential of 40 mV
(top, voltage-clamp protocol; bottom,
if). D: current-voltage relationships
for if; if was measured
as the increase in inward current from the beginning to the end of the
pulses. E: density of if (measured
during pulse to 110 mV) plotted against Cm.
, data from Honjo et al. (27) from rabbit
SA node cells; dotted lines, regression lines; , values
used in the peripheral (Cm = 65 pF) and
central (Cm = 20 pF) SA node cell models.
|
|
Regional Differences in Ionic Current Densities
The variation in electrical activity from the periphery to the
center of the SA node (as described in the introduction) could be the
result of a gradual decrease in the electrotonic influence of the
surrounding atrial muscle (with more hyperpolarized diastolic potentials) from the periphery to the center. However, experiments using ligated or dissected small pieces of tissue from different regions of the rabbit SA node (in which the large mass of surrounding atrial muscle is removed) also show marked differences in the electrophysiological characteristics of the tissue from the periphery and center of the SA node (33). Immunocytochemical studies
have shown that atrial cells are intermingled with SA node cells in the
periphery of the SA node in several species (52). It has been conjectured (58) that the electrophysiological
properties of pacemaker cells in the SA node are roughly uniform
throughout the SA node, and the apparent regional differences in
electrical activity in the intact SA node are the result of a
progressive decrease in the percentage of intermingling atrial cells
toward the center, giving rise to a progressive decrease in their
hyperpolarizing influence from the periphery toward the center.
However, we have little evidence of such intermingling in the rabbit SA
node (12). Furthermore, in single SA node cells from the
rabbit, we record a range of electrical activities similar to that
observed in the different regions of the intact SA node
(27). Therefore, we believe that the properties of rabbit
SA node cells are not uniform, as suggested by others
(58). The size of cells varies from the periphery to the
center of the rabbit SA node: cells are large in the periphery and
small in the center (in the center, the cells are <8 µm diameter and
25-30 µm long) (3). We have shown that the
electrical activity of rabbit SA node cells is correlated with cell
size (as measured by Cm), and, as expected,
large cells have properties characteristic of the periphery of the SA
node and small cells have properties characteristic of the center (see the introduction). The densities of a number of ionic currents are
dependent on cell size (see the introduction and below). In construction of the models, we assumed that a peripheral cell is large
in size, with a Cm of 65 pF, and a central cell
is small in size, with a Cm of 20 pF. These
values of Cm represent the approximate maximum
and minimum values measured experimentally for rabbit SA node cells
(27, 28, 36, 39).
Ionic current densities for the peripheral and central SA node
cell models are listed in Table 11. For some but not all currents, the
relationship between current density and Cm has
been determined for rabbit SA node cells, and the final panels of Figs.
1-5 show the densities of various currents plotted against
Cm. The open circles are experimental data (27, 28, 39; iK,r and iK,s
data from unpublished observations), and the filled squares are the chosen values for the peripheral and central SA node cell models. The
experimentally measured densities of iNa,
ito, isus,
iK,r, iK,s, and
if are significantly correlated with
Cm and are larger in cells with higher
Cm (Figs. 1 and 3-5) (27, 28, 36, 39;
iK,r and iK,s data from
unpublished observations). Values chosen for the densities of these
currents for the peripheral and central SA node cell models are within
the experimental range and are greater in the peripheral SA node cell
model. Although the density of iCa,L measured
experimentally is not significantly correlated with
Cm (27), it was necessary to
increase the density of iCa,L in the peripheral
SA node cell model compared with that in the central SA node cell model
(Fig. 2E). Despite this, the chosen values for the density
of iCa,L in the peripheral and central SA node
cell models are reasonably consistent with the experimental values
(Fig. 2E).
During the development of the models, there were no experimental data
available on the relationship between the densities of
iK,r and iK,s and
Cm for rabbit SA node cells. Some data were available: Lei and Brown (37) had reported that the ratio
of the amplitude of iK,r to
iK,s tails after a 1-s pulse to +40 mV from a
holding potential of 40 mV was 1:0.3-0.4, and, therefore, in the
peripheral and central SA node cell models, the ratio was set at 1:0.3.
Although there were no direct measurements of the density of
iK,r in rabbit SA node cells with different
values of Cm, various indirect lines of evidence
suggested that the density of iK,r is greater in
peripheral than in central SA node cells. First, although complete
block of iK,r by 1 µM E-4031 abolishes pacemaker activity in tissue from the periphery and center of the
rabbit SA node, partial block of iK,r by 0.1 µM E-4031 abolishes pacemaker activity in central but not peripheral
tissue (32). One explanation for this different response
to partial block of iK,r is that the density of
iK,r is greater in the periphery of the SA node
(32). Second, an immunocytochemical study examined the
distribution of ERG (channel protein responsible for
iK,r) in the ferret SA node (7). If
the anatomy of the SA node in the ferret is similar to that of the
rabbit, the data from the study support the possibility that the
density of iK,r is greater in the periphery of
the SA node: the labeling of ERG was little in the intercaval region
(where the center of the SA node is located in the rabbit at least) and
substantial close to the crista terminalis (where the periphery of the
SA node is located in the rabbit at least). Finally, in the models of
peripheral and central SA node action potentials, it was necessary to
increase the density of iK,r in the peripheral
cell model compared with that in the central cell model to obtain some
of the characteristic differences between the two regions (see Fig.
10). After the development of the models, experimental measurements of
the densities of iK,r and
iK,s became available (Fig. 4, J and
K, open circles) from our laboratory in rabbit SA node cells
(unpublished observations). Although the densities of
iK,r and iK,s in the
peripheral and central SA node models were set before the experimental
data became available, they are within the experimental range obtained
from rabbit SA node cells.
In the case of currents for which there are no data concerning current
density and Cm, the current density was assumed
to be the same in peripheral and central SA node cells.
Intracellular Ionic Concentrations
Previous SA node models have included computation of
concentrations of intracellular Na+ and Ca2+
(13, 18, 49, 59).
In the present models, all intracellular ion concentrations are assumed
to be constant (see Table 10 for values). There are various reasons for
this: 1) little is known about intracellular Na+
in the SA node, and there is only one report about intracellular Ca2+ in the rabbit SA node (40); 2)
nothing is known about differences in intracellular Na+ and
Ca2+ handling between the periphery and center of the SA
node; 3) intracellular Na+ changes only slowly
over several minutes, and over a few beats intracellular
Na+ is approximately constant; 4) buffering
intracellular Ca2+ with
1,2-bis(2-aminophenoxy)ethane-N,N,N',N'-tetraacetic acid-AM or reducing the amplitude of the intracellular Ca2+
transient with ryanodine does not abolish spontaneous activity in
rabbit SA node cells (40), although it can produce a
decrease in the rate of spontaneous action potentials
(40); and 5) much of our data concerning the
dependence of electrical activity and the density of ionic currents on
Cm was obtained from rabbit SA node cells in
which intracellular Ca2+ was buffered with EGTA
(27, 28). Because of points 1 and 2, inclusion of intracellular Na+ and
Ca2+ handling in the models would be based on speculation
to a large degree, and, because of points 3-5,
inclusion of intracellular Na+ and Ca2+
handling in the models is not essential for reasonably accurate calculation of electrical activity (over a few beats at least). Inclusion of intracellular Ca2+ handling may be important
if inward iNaCa triggered by the intracellular Ca2+ transient plays a significant role in electrical
activity: the slowing of spontaneous activity by
1,2-bis(2-aminophenoxy)ethane-N,N,N',N'-tetraacetic acid-AM
or ryanodine could indicate such a role, but even in these cases the
slowing may be the result of actions on other currents (40); the role of inward iNaCa
triggered by the intracellular Ca2+ transient in the SA
node is, therefore, unclear. Despite these arguments that the inclusion
of intracellular Na+ and Ca2+ handling is not
crucial, future models will have to address this shortcoming.
Comparison of the Models of Peripheral and Central SA Node Action
Potentials With Action Potentials Measured Experimentally From the
Rabbit SA Node
After construction of the models as described above, the
configuration of the simulated action potentials was compared with the
configuration of action potentials measured experimentally from rabbit
SA node cells (with different values of Cm) at
35°C (27) and from small balls of tissue from the
periphery and center of the rabbit SA node at 32°C (33).
The experimental data from rabbit SA node cells with various values of
Cm at 35°C (27) are similar to
the experimental data from small balls of tissue from the periphery and
center of the rabbit SA node at 32°C (33), although the
action potential is shorter and the rate of spontaneous activity is
faster in the single cells. We assume that these differences are the
result of the higher temperature at which the experiments on single
cells were conducted; for this reason, when model and experimental data
were compared, more importance was placed on the data from single
cells. In addition to this, the response of the simulated peripheral
and central action potentials to block of various currents was
calculated and compared with the experimentally measured response of
peripheral and central action potentials to various blockers. If there
were differences, we adjusted the densities of appropriate currents
until the simulated and experimental results were more similar.
Although the current densities in the models were adjusted in this
manner, the final current densities had to be consistent with current
densities measured experimentally (the scope for change was, therefore,
limited). The models finally developed do not produce action
potentials that are exact matches of any one experimental recording;
instead, the action potentials have characteristics within the range
observed experimentally and behaviors to ion channel block within the
range observed experimentally.
One-Dimensional Model of the SA Node and Surrounding Atrial
Muscle
On the basis of the models developed for peripheral and central
SA node cells, a one-dimensional, partial differential equation, multicellular model for the SA node and atrium was developed. The
equations for the one-dimensional model are listed in Table 13. In the model the multicellular SA
node and atrium are modeled as a string of tissue with a length
(L) of 12.6 mm; of this the string of SA node tissue has a
length (Ls) of 3 mm [similar to the distance
from the center of the SA node to the atrial muscle in the rabbit
(3)], and the string of atrial tissue has a length of 9.6 mm. It could be argued that a two-dimensional model of the SA node and
atrium as used by others would be more appropriate. However, the
one-dimensional model is computationally more efficient, and,
furthermore, in a radially symmetrical, two-dimensional model of a
circular SA node within a concentric atrium, the behavior of a string
of tissue extending from the center of the SA node to atrial muscle
will be no different from the one-dimensional model described here,
because there will be no net current flow laterally (i.e., at a tangent
to the propagating wavefront of the action potential) in the one- or
the two-dimensional model. Within the string of SA node tissue, we
assume that Cm changes from 20 pF
(Cm in the central SA node cell model) to 65 pF
(Cm in the peripheral SA node cell model)
exponentially (Eq. 80, Table 13) and ionic current
conductances are functions of Cm (Eq.
81, Table 13). In the model, single atrial cells are represented by the Earm-Hilgemann-Noble equations (48). Electrotonic
interactions within the tissue are modeled by the diffusive
interactions of membrane potentials (Eqs. 82 and 83, Table 13). We used nonflux boundary conditions for both
ends of the model (Eqs. 79 and 84, Table 13).
Ds and Da scale the
conduction velocity of the action potential in the SA node and the
atrial muscle. The conduction velocity for near-planar waves is
~0.001-0.1 m/s in the SA node and ~0.3-0.8 m/s in the atrium (20). We set Ds at 0.6 cm2/s, which gives a conduction velocity of ~0.06 m/s in
the SA node, and Da at 1.25 cm2/s,
which gives a conduction velocity of 0.62 m/s in the atrial muscle.
Coupling at the junction of the SA node and atrial muscle is by the
diffusion coefficient (Ds).
Computational Methods
The models were coded in C and Fortran languages. The programs
were run on Indigo 2, Silicon Graphics machines with an IRIS 6.0 operating system. A fourth-order Runge-Kutta-Merson numerical integration method (55) was used to solve the ordinary
differential equations. The time step was 0.1 ms, which gives a stable
solution of the equations and maintains the accuracy of the computation of membrane current and potential. One-dimensional partial differential equations were solved by an explicit Euler method with a three-node approximation of the Laplacian operator (55), with a time
step of 0.1 ms and a space step of 0.1 mm for SA node tissue and 0.32 mm for atrial muscle. The chosen time and space steps are sufficiently small for a stable and accurate solution.
| |
RESULTS |
Peripheral and Central SA Node Action Potentials
Figure 6 shows the action potentials generated using the models of
a peripheral cell (Cm = 65 pF) and a
central cell (Cm = 20 pF) at 37°C from
the SA node of the rabbit at fast (Fig. 6A) and slow (Fig.
6B) time bases. For comparison, Fig.
6 also shows action potentials recorded
experimentally from rabbit SA node preparations: Figure 6C
shows action potentials at a fast time base recorded from small balls
of tissue from the periphery and center of the SA node at 32°C, and
Fig. 6D shows action potentials at a slow time base recorded
from single cells with Cm of 57.5 and 22.0 pF at
35°C. The simulated action potentials are similar to those recorded
experimentally. The action potential of the peripheral model has a more
negative takeoff potential, a more rapid upstroke, a more positive peak
value, a greater amplitude, a shorter duration, and a more negative
maximum diastolic potential than the action potential of the central
model. Furthermore, the spontaneous activity of the peripheral model is
faster than that of the central model. All these are characteristic
differences seen experimentally between small balls of tissue from the
periphery and center of the rabbit SA node (31,
33) or large and small rabbit SA node cells
(27) (Fig. 6, C and D). The action
potential from the peripheral model has an early rapid phase of
repolarization (phase 1) after the action potential upstroke. Such an
early rapid phase of repolarization after the action potential upstroke
can be observed frequently in the periphery of the intact SA node (but
not in the center) and in small balls of tissue from the periphery (but
not from the center) (6, 34, 45)
(Fig. 6C).
View larger version (25K):
[in this window]
[in a new window]
|
Fig. 6.
Simulated peripheral and central SA node action
potentials. A: simulated action potentials at a fast time
base. B: simulated action potentials at a slow time base.
C: action potentials recorded from small balls of peripheral
and central rabbit SA node tissue (balls B and D)
at a fast time base (unpublished observations). D: action
potentials recorded from rabbit SA node cells with
Cm of 57.5 and 22.0 pF at a slow time base
(unpublished observations). E-H, takeoff potential
(TOP; E), maximum upstroke velocity
(dV/dtmax, F), maximum
diastolic potential (MDP, G) and cycle length (CL,
H) plotted against Cm.
, Data from Honjo et al. (27) from rabbit
SA node cells; dotted lines, regression lines; , values
computed from the peripheral (Cm = 65 pF)
and central (Cm = 20 pF) SA node cell
models.
|
|
Figure 6, E-H, compares the characteristics of the
simulated action potentials with the average characteristics of action potentials recorded experimentally from rabbit SA node cells by Honjo
et al. (27) at 35°C. The open circles in Fig. 6,
E-H, show experimental measurements of the takeoff
potential, maximum upstroke velocity, maximum diastolic potential, and
cycle length (time between successive spontaneous action potentials) of
rabbit SA node cells plotted against Cm. In all
cases, there are significant correlations of the variables with
Cm (27). In Fig. 6,
E-H, the filled squares show
TOP
ABSTRACT
INTRODUCTION
