Vol. 283, Issue 5, H2074-H2101, November 2002
Dynamical description of sinoatrial node pacemaking: improved
mathematical model for primary pacemaker cell
Yasutaka
Kurata1,
Ichiro
Hisatome2,
Sunao
Imanishi1, and
Toshishige
Shibamoto1
1 Department of Physiology, Kanazawa Medical
University, Ishikawa 920-0293; and 2 First Department of
Internal Medicine, Tottori University School of Medicine, Yonago
683-0826, Japan
 |
ABSTRACT |
We developed an improved mathematical
model for a single primary pacemaker cell of the rabbit sinoatrial
node. Original features of our model include 1)
incorporation of the sustained inward current
(Ist) recently identified in primary pacemaker
cells, 2) reformulation of voltage- and
Ca2+-dependent inactivation of the L-type Ca2+
channel current (ICa,L), 3) new
expressions for activation kinetics of the rapidly activating delayed
rectifier K+ channel current (IKr),
and 4) incorporation of the subsarcolemmal space as a
diffusion barrier for Ca2+. We compared the simulated
dynamics of our model with those of previous models, as well as with
experimental data, and examined whether the models could accurately
simulate the effects of modulating sarcolemmal ionic currents or
intracellular Ca2+ dynamics on pacemaker activity. Our
model represents significant improvements over the previous models,
because it can 1) simulate whole cell voltage-clamp data for
ICa,L, IKr, and
Ist; 2) reproduce the waveshapes of
spontaneous action potentials and ionic currents during action
potential clamp recordings; and 3) mimic the effects of
channel blockers or Ca2+ buffers on pacemaker activity more
accurately than the previous models.
rabbit sinoatrial node; nonlinear dynamical system; computer
simulation; bifurcation diagram
 |
INTRODUCTION |
PACEMAKER ACTIVITY
of the sinoatrial (SA) node is well known as the initiation of the
spontaneous heart beat. A large body of information on ionic current
systems underlying the SA node pacemaker activity has been obtained
from recent single cell patch-clamp experiments. On the basis of single
cell patch-clamp data, several mathematical models describing the
pacemaker activity of a single rabbit SA node cell have been developed
in the past decade.
In 1991, Wilders et al. (124) first developed a single SA
node cell model based on single cell patch-clamp data to reproduce the
electrical behavior of a single rabbit SA node cell quantitatively. Later, Demir et al. (17) proposed a more detailed model
for transitional-type cells (rather than for primary pacemaker cells) including intracellular Ca2+ buffering (by calmodulin,
troponin, and calsequestrin) and new formulations for Ca2+
handling by the sarcoplasmic reticulum (SR) based on microanatomic data. Dokos et al. (24) developed a different SA node
model incorporating new formulations of the
Na+/Ca2+ exchanger current
(INaCa) and muscarinic K+ channel
current (IK,ACh) as a background K+
current. However, these previous models have several difficulties, as
follows. First, the formulations of voltage-dependent inactivation and
recovery of the L-type Ca2+ channel current
(ICa,L) were not based on experimental data. Second, the Ca2+-dependent inactivation of
ICa,L, known as an essential property of the
L-type Ca2+ channel in the rabbit SA node (10,
67), was not formulated [or not directly dependent on the
intracellular Ca2+ concentration
([Ca2+]i), but a function of the inactivation
gating variable fL in Wilders et al.
(124)]. Although Dokos et al. (24)
incorporated a second inactivation gating variable to represent the
[Ca2+]i-dependent inactivation process, they
assumed an extraordinarily large maximum ICa,L
conductance (gCa,L) and a very high affinity for
Ca2+ binding to the inactivation site of L-type
Ca2+ channels; such a large gCa,L
with a high affinity for Ca2+ binding is rather unlikely
and has not been demonstrated. Third, these models do not include the
slow component of the delayed rectifier K+ current
(IKs), 4-aminopyridine (4-AP)-sensitive
currents, or sustained inward current (Ist),
whereas these currents are known to be present in primary pacemaker
cells. Fourth, intracellular Ca2+ transients in their
models (2.5~10 µM at the peak) were much higher than those in
atrial or ventricular myocytes (~1 µM) (e.g., Refs.
106 and 110; see also Refs. 68
and 71), probably too high for SA node cells. Finally, SR
volumes or Ca2+ concentrations in the SR are comparable to
or higher than those experimentally determined for ventricular myocytes
(see Refs. 17, 63, and 107),
probably too large for SA node cells. Recently, Zhang et al.
(130) developed separate models for central and peripheral
SA node cells based on recent experiments in which the regional
differences in action potential (AP) parameters, ionic current
densities, and pharmacological responses (i.e., electrophysiological
effects of various current blockers) between central and peripheral SA
node cells have been studied. Their central and peripheral models
provide a theoretical basis for regional differences in AP parameters
and are superior to the previous models in that they incorporate novel
current systems such as IKs and 4-AP-sensitive
currents. However, their models have at least two apparent
difficulties. First, intracellular ion concentrations were assumed to
be constant in their models (the net ion flux of Na+,
K+, or Ca2+ during an AP cycle is not zero),
whereas intracellular Ca2+ dynamics and changes in
intracellular Na+ ([Na+]i) and
K+ concentrations ([K+]i) are
known to exert substantial effects on pacemaker activity (e.g., see
Refs. 67, 98, and 122). Second,
their models lack some sarcolemmal currents, such as
Ist and IK,ACh, known to
play important roles in regulating the pacemaker activity of rabbit SA
node cells. Thus the previous SA node models all have several drawbacks
or serious disadvantages; a satisfactory single cell model is not available.
The aim of this study was to develop an improved mathematical model of
a single "primary pacemaker cell" of the rabbit SA node that is
more suitable than the previous models for investigating the dynamical
mechanisms of pacemaker generation. In addition to the drawbacks stated
above, the previous single cell models exhibited different bifurcation
structures, i.e., different ways of abolishing automaticity (see Ref.
30) from those of real SA node cells during inhibition of
ICa,L or the rapid component of the delayed
rectifier K+ current (IKr) not
suitable for a study on the mechanisms of pacemaker generation (for
details, see RESULTS AND DISCUSSION). Thus an improved
model cell should have the same bifurcation structures as real SA node
cells have as well as the capability of reproducing experimental data
more accurately than the previous models. Such a model would also serve
as a base model for developing more sophisticated models. On the basis
of recent experimental findings, we were able to update the previous
models in several ways: 1) Ist, not included in the previous models, has been incorporated; 2)
voltage- and Ca2+-dependent inactivation kinetics of
ICa,L have been reformulated; 3)
expressions for activation kinetics of IKr have
renewed; 4) revised kinetic formulas for 4-AP-sensitive
currents have been incorporated; 5) voltage- and
concentration-dependent kinetics of the Na+-K+
pump current (INaK) have been reformulated; and
6) the subsarcolemmal space as a diffusion barrier for
intracellular Ca2+ has been incorporated.
To validate our model, we first compared the simulated dynamics of the
model (spontaneous APs and ionic currents during pacemaker activity)
with experimental data from the rabbit SA node as well as with those of
the previous SA node models. We further validated our model by
simulating the effects of modulations of sarcolemmal ionic currents or
intracellular Ca2+ dynamics on pacemaker activity. The
experimental findings for verification include 1) whole cell
voltage-clamp data for ICa,L, IKr, and Ist;
2) waveshapes of spontaneous APs and ionic currents (such as
ICa,L and IKr) as
observed during AP-clamp recordings; 3) modulations and
cessation of pacemaker activity by applications of
ICa,L or IKr blockers;
4) modifications of AP waveforms (changes in AP parameters)
by blocking the T-type Ca2+ channel current
(ICa,T), hyperpolarization-activated current (Ih), or 4-AP-sensitive currents; 5)
effects of blocking SR Ca2+ release (by ryanodine) on
pacemaker frequency; and 6) negative chronotropic effects of
Ca2+ buffers. Our model could reproduce these experimental
data more accurately than the previous SA node models. In this study,
we particularly focused on the bifurcation structures during
applications of ICa,L or
IKr blockers and also the differential effects
of BAPTA and EGTA on pacemaker frequency. During inhibition of
ICa,L or IKr, our model
exhibited essentially the same bifurcation structures as observed in
real SA node cells, whereas previous models did not; only our model
could simulate the differential responses of SA node cells to BAPTA and
EGTA. Detailed comparisons of our model with the previous models as
well as experimental data are addressed in THEORY AND
METHODS (on modeling) and RESULTS AND DISCUSSION (on
simulated results).
Glossary
General
| 4-AP |
4-Aminopyridine
|
| AP |
Action potential
|
| APA |
AP amplitude
|
| APD50 |
AP duration at 50% repolarization
|
| CL |
Cycle length
|
| F |
Faraday constant
|
| I |
Current
|
| MDP |
Maximum diastolic potential
|
| POP |
Peak overshoot potential
|
| R |
Universal gas constant
|
| SA |
Sinoatrial
|
| SR |
Sarcoplasmic reticulum
|
| t |
Time
|
| T |
Absolute temperature
|
| V |
Membrane potential (in mV)
|
Ionic Channel Currents
| Ib,Ca |
Background Ca2+ current
|
| Ib,K |
Background K+ current
|
| Ib,Na |
Background Na+ current
|
| ICa,L |
L-type Ca2+ channel current
|
| ICa,T |
T-type Ca2+ channel current
|
| Ih |
Hyperpolarization-activated current
|
| IK |
Delayed rectifier K+ current
|
| IK,ACh |
Muscarinic K+ channel current
|
| IKr |
Rapid component of the delayed rectifier K+ current
|
| IKs |
Slow component of the delayed rectifier K+ current
|
| INa |
Fast Na+ channel current
|
| INaCa |
Na+/Ca2+ exchanger current
|
| INaen |
Electroneutral Na+ influx current
|
| INaK |
Na+-K+ pump current
|
| INaKmax |
Maximum INaK
|
| Ip,Ca |
Sarcolemmal Ca2+ pump current
|
| Ist |
Sustained inward current
|
| Isus |
Sustained component of the 4-AP-sensitive current
|
| Ito |
Transient component of the 4-AP-sensitive current
|
Cell Geometry
| Cm |
Cell membrane capacitance
|
| Vcell |
Cell volume
|
| Vi |
Myoplasmic volume available for Ca2+ diffusion
|
| Vrel |
Volume of junctional SR (Ca2+ release store)
|
| Vsub |
Subspace volume
|
| Vup |
Volume of network SR (Ca2+ uptake store)
|
Ionic Concentrations
| Ca2+i |
Myoplasmic Ca2+ concentration
|
| [Ca2+]o |
Extracellular Ca2+ concentration
|
| [Ca2+]rel |
Ca2+ concentration in the junctional SR
|
| [Ca2+]sub |
Subspace Ca2+ concentration
|
| [Ca2+]up |
Ca2+ concentration in the network SR
|
| [K+]i |
Intracellular K+ concentration
|
| [K+]o |
Extracellular K+ concentration
|
| [Mg2+]i |
Intracellular Mg2+ concentration
|
| [Na+]i |
Intracellular Na+ concentration
|
| [Na+]o |
Extracellular Na+ concentration
|
Equilibrium (Reversal) Potentials
| ECa,L |
Apparent reversal potential of ICa,L
|
| ECa,T |
Apparent reversal potential of ICa,T
|
| EK |
Equilibrium (Nernst) potential for K+
|
| EKr |
Reversal potential of IKr
|
| EKs |
Reversal potential of IKs
|
| Emh |
Reversal potential of INa
|
| ENa |
Equilibrium (Nernst) potential for Na+
|
| Est |
Apparent reversal potential of Ist
|
Sarcolemmal Ionic Currents
| L |
Activation gating variable for ICa,L
|
dL, |
Steady-state dL
|
| dT |
Activation gating variable for ICa,T
|
dT, |
Steady-state dT
|
| fCa |
Ca2+-dependent inactivation gating variable for
ICa,L
|
fCa, |
Steady-state fCa
|
| fL |
Voltage-dependent inactivation gating variable for
ICa,L
|
fL, |
Steady-state fL
|
| fT |
Inactivation gating variable for ICa,T
|
fT, |
Steady-state fT
|
| FNa |
Fraction of slow inactivation of INa
|
| gb,Na |
Background Na+ conductance
|
| gCa,L |
Maximum ICa,L conductance
|
| gCa,T |
Maximum ICa,T conductance
|
| gh |
Maximum Ih conductance
|
| gh,K |
K+ current component of gh
|
| gh,Na |
Na+ current component of gh
|
| gK,ACh |
Scaling factor for IK,ACh
|
| gKr |
Maximum IKr conductance
|
| gKs |
Maximum IKs conductance
|
| gNa |
Maximum INa conductance
|
| gst |
Maximum Ist conductance
|
| gsus |
Maximum Isus conductance
|
| gto |
Maximum Ito conductance
|
| h |
Inactivation gating variable for INa
|
h |
Steady-state h
|
| hF |
Fast inactivation gating variable for INa
|
| hS |
Slow inactivation gating variable for INa
|
| jCa,sm |
Net Ca2+ flux through the sarcolemmal membrane
|
| kNaCa |
Scaling factor for INaCa
|
| K1ni |
Dissociation constant for intracellular Na+ binding to
first site on INaCa transporter
|
| K2ni |
Dissociation constant for intracellular Na+ binding to
second site on INaCa transporter
|
| K3ni |
Dissociation constant for intracellular Na+ binding to
third site on INaCa transporter
|
| K1no |
Dissociation constant for extracellular Na+ binding to
first site on INaCa transporter
|
| K2no |
Dissociation constant for extracellular Na+ binding to
second site on INaCa transporter
|
| K3no |
Dissociation constant for extracellular Na+ binding to
third site on INaCa transporter
|
| Kci |
Dissociation constant for intracellular Ca2+ binding to
INaCa transporter
|
| Kco |
Dissociation constant for extracellular Ca2+ binding to
INaCa transporter
|
| Kcni |
Dissociation constant for intracellular Na+ and
Ca2+ simultaneous binding to INaCa
transporter
|
| KmfCa |
Dissociation constant for Ca2+-dependent inactivation of
ICa,L
|
| KmKp |
Half-maximal [K+]o for
INaK
|
| KmNap |
Half-maximal [Na+]i for
INaK
|
| m |
Activation gating variable for INa
|
m |
Steady-state m
|
| n |
Activation gating variable for IKs
|
n |
Steady-state n
|
| pa |
Activation gating variable for IKr
|
pa, |
Steady-state pa
|
| paF |
Fast activation gating variable for IKr
|
| paS |
Slow activation gating variable for IKr
|
| pi |
Inactivation gating variable for IKr
|
pi, |
Steady-state pi
|
| q |
Inactivation gating variable for Ito
|
q |
Steady-state q
|
| qa |
Activation gating variable for Ist
|
qa, |
Steady-state qa
|
| qi |
Inactivation gating variable for Ist
|
qi, |
Steady-state qi
|
| Qci |
Fractional charge movement during intracellular Ca2+
occlusion reaction of INaCa transporter
|
| Qco |
Fractional charge movement during extracellular Ca2+
occlusion reaction of INaCa transporter
|
| Qn |
Fractional charge movement during Na+ occlusion reactions
of INaCa transporter
|
| r |
Activation gating variable for Ito and
Isus
|
r |
Steady-state r
|
| x |
Gating variable
|
x |
Steady-state value of x
|
| y |
Activation gating variable for Ih
|
y |
Steady-state value of y
|
dL |
Opening rate constant of dL
|
fCa |
Ca2+ dissociation rate constant for
ICa,L
|
qa |
Opening rate constant of qa
|
qi |
Opening rate constant of qi
|
n |
Opening rate constant of n
|
dL |
Closing rate constant of dL
|
fCa |
Ca2+ association rate constant for
ICa,L
|
n |
Closing rate constant of n
|
qa |
Closing rate constant of qa
|
qi |
Closing rate constant of qi
|
dL |
Time constant of dL
|
dT |
Time constant of dT
|
fCa |
Time constant of fCa
|
fL |
Time constant of fL
|
fT |
Time constant of fT
|
hF |
Time constant of hF
|
hS |
Time constant of hS
|
m |
Time constant of m
|
n |
Time constant of n
|
paF |
Time constant of paF
|
paS |
Time constant of paS
|
q |
Time constant of q
|
qa |
Time constant of qa
|
qi |
Time constant of qi
|
r |
Time constant of r
|
x |
Time constant for a gating variable x
|
y |
Time constant of y
|
Ca2+ Diffusion
| jCa,dif |
Ca2+ diffusion flux from subspace to myoplasm
|
dif,Ca |
Time constant of Ca2+ diffusion from the subspace to
myoplasm
|
SR Function
| jrel |
Ca2+ release flux from the junctional SR to subspace
|
| jtr |
Ca2+ transfer flux from the network to junctional SR
|
| jup |
Ca2+ uptake flux from the myoplasm to network SR
|
| Krel |
Half-maximal [Ca2+]sub for Ca2+
release from the junctional SR
|
| Kup |
Half-maximal [Ca2+]i for Ca2+
uptake by the Ca2+ pump in the network SR
|
| Prel |
Rate constant for Ca2+ release from the junctional SR
|
| Pup |
Rate constant for Ca2+ uptake by the Ca2+ pump
in the network SR
|
tr |
Time constant for Ca2+ transfer from the network to
junctional SR
|
Ca2+ Buffering
| CQtot |
Total calsequestrin concentration
|
| [CM]tot |
Total calmodulin concentration
|
| fCMi |
Fractional occupancy of calmodulin by Ca2+ in myoplasm
|
| fCMs |
Fractional occupancy of calmodulin by Ca2+ in subspace
|
| fCQ |
Fractional occupancy of calsequestrin by Ca2+
|
| fTC |
Fractional occupancy of the troponin-Ca site by Ca2+
|
| fTMC |
Fractional occupancy of the troponin-Mg site by Ca2+
|
| fTMM |
Fractional occupancy of the troponin-Mg site by Mg2+
|
| kbBAPTA |
Ca2+ dissociation constant for BAPTA
|
| kbCM |
Ca2+ dissociation constant for calmodulin
|
| kbCQ |
Ca2+ dissociation constant for calsequestrin
|
| kbEGTA |
Ca2+ dissociation constant for EGTA
|
| kbTC |
Ca2+ dissociation constant for the troponin-Ca site
|
| kbTMC |
Ca2+ dissociation constant for the troponin-Mg site
|
| kbTMM |
Mg2+ dissociation constant for the troponin-Mg site
|
| kfBAPTA |
Ca2+ association constant for BAPTA
|
| kfCM |
Ca2+ association constant for calmodulin
|
| kfCQ |
Ca2+ association constant for calsequestrin
|
| kfEGTA |
Ca2+ association constant for EGTA
|
| kfTC |
Ca2+ association constant for troponin
|
| kfTMC |
Ca2+ association constant for the troponin-Mg site
|
| kfTMM |
Mg2+ association constant for the troponin-Mg site
|
| [TC]tot |
Total concentration of the troponin-Ca site
|
| [TMC]tot |
Total concentration of the troponin-Mg site
|
 |
THEORY AND METHODS |
Model Development
We formulated a mathematical model describing the dynamic
properties of a single primary pacemaker cell of the rabbit SA node under space-clamp conditions (at 37°C). Our model is an extension of
previous SA node models (17, 24, 124, 130) that utilized classical Hodgkin-Huxley formalism, including variations in
intracellular ion concentrations, Ca2+ handling by the SR,
and Ca2+ buffering. The standard model for normal pacemaker
activity is described as a nonlinear dynamical system of 27 simultaneous, first-order, ordinary differential equations. A complete
list of the equations and standard parameter values is presented in the
APPENDIX.
Geometrical considerations.
We assumed our model cell to be a 70-µm-long by 8-µm-diameter
cylinder (i.e., a "spindle-shaped" cell with a length of 70 µm
and a mean width of 8 µm) and set the cell volume and membrane capacitance to 3.5 pl and 32 pF, respectively. The cell volume of 3.5 pl was nearly identical to that used by Demir et al.
(17) but smaller than the value of 5 pl estimated by
Denyer and Brown (19) for spindle-shaped isolated SA node
cells and used for the models of Wilders et al. (124) and
Dokos et al. (24). Primary pacemaker cells are thought to
be smaller than transitional or peripheral cells (see Refs.
44 and 130); thus we chose the smaller value
of 3.5 pl for our primary pacemaker cell model. The membrane
capacitance of 32 pF is derived from Wilders et al. (124)
and Dokos et al. (24), smaller than the value of 55 pF used by Demir et al. (17) for their transitional cell
model and larger than the value of 20 pF for the central model of Zhang et al. (130). Microanatomic data such as SR volumes used
in this study are from Demir et al. (17). The effective
intracellular volume of a SA node cell wherein free Ca2+ is
available to enter into reaction was set to 46% of the cell volume
(1.6 pl), because there are various intracellular structures and
organelles, as summarized by Demir et al. (17).
There is now evidence that there exists a small restricted
subsarcolemmal domain where Ca2+ concentrations may
transiently reach higher levels than in the bulk myoplasm (29,
47, 62, 86, 88, 122). Therefore, we assumed the subsarcolemmal
space as a barrier for Ca2+ diffusion to the myoplasm. The
subspace volume Vsub was tentatively set to 1% of the cell
volume (0.035 pl), assuming that the subspace is limited to 20 nm below
the sarcolemmal membrane (see Fig.
1A). The time constant of the
Ca2+ diffusion from subspace to myoplasm
(
dif,Ca) was set to 40 µs; this value corresponded to
a diffusion coefficient of
1 × 10
9
cm2/ms if the mean distance between the openings of the
L-type Ca2+ channel or SR Ca2+ release channel
and the myoplasmic compartment is assumed to be 200 nm (see Refs.
29, 61, and 113). A
dif,Ca value of 40 µs was chosen to yield a peak
[Ca2+]sub of ~1.8 µM, as reported in the
modeling studies of Glukhovsky et al. (29) and Snyder et
al. (113). The subspace of our model corresponds to the
"fuzzy" space, a functionally restricted intracellular space
accessible to the Na+/Ca2+ exchanger as well as
to the L-type Ca2+ channel and Ca2+-gated
Ca2+ channel in the SR (see Refs. 51 and
62). Note that it is different from the "diadic" space
between the L-type Ca2+ channel and the ryanodine receptor
(Ca2+-gated Ca2+ channel), where
Ca2+ concentrations increase to several tens of
micromolars; in ventricular myocytes, the ryanodine receptor
and SR Ca2+ release are closely linked to the L-type
Ca2+ channel and its Ca2+-dependent
inactivation in the diadic space (1, 5, 50, 93, 106, 114,
115). In SA node cells, however, such a microdomain or
cross-signaling between the L-type Ca2+ channel and SR
Ca2+ release channel has not been demonstrated, being
possibly absent because the SR is poorly developed in SA node cells
compared with ventricular or atrial myocytes (see Refs. 17
and 47). Thus we did not incorporate the diadic space into
our model.

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|
Fig. 1.
Schematic diagrams depicting a cross section of the model cell
(A) and intracellular fluid compartments for
Ca2+ (B). R and L
represent the radius of the cell (4 µm) and the thickness (depth) of
the subsarcolemmal space (20 nm), respectively. Volumes of the cell and
each compartment are given in the text. The Ca2+ release
from junctional SR (JSR) and Ca2+ uptake to the network SR
(NSR) were assumed to be strictly into the restricted subspace and from
the bulk myoplasm, respectively. Definitions of the symbols for the
Ca2+ concentration in each compartment and Ca2+
flux between compartments are given in the Glossary.
|
|
Descriptions of membrane currents.
The mathematical expressions used for the membrane current system are
essentially the same as those formulated previously (17, 24, 124, 130) with modifications according to newly reported experimental data. The complete model for the normal pacemaking includes 13 membrane current components. The differential equation for the membrane potential (V) is
|
(1)
|
where ICa,L and
ICa,T represent the L-type and T-type
Ca2+ channel currents, respectively. The rapid and slow
components of the delayed rectifier K+ current are denoted
as IKr and IKs,
respectively. The membrane current system also includes the transient
(Ito) and sustained (Isus) components of 4-AP-sensitive currents,
hyperpolarization-activated current (Ih),
sustained inward current (Ist), Na+
channel current (INa), background
Na+ current (Ib,Na), muscarinic
K+ channel current (IK,ACh),
Na+-K+ pump current
(INaK), and Na+/Ca2+
exchanger current (INaCa) charging the membrane
capacitance (Cm). For parameter adjustments, the
regional differences of current densities in the SA node were taken
into account (44, 130).
Model equations for channel gating behaviors are essentially the same
as those of the previous Hodgkin-Huxley type models. A gating variable,
x, can be computed as a solution of the first-order differential equation of the form
|
(2)
|
where x
and
x are the steady-state x value and
relaxation time constant, respectively, as functions of V. Relaxation time constants have been appropriately scaled for the temperature of 37°C with the use of a Q10 of 1.6~3.0.
The functions x
(V) and
x(V) for individual gating
variables are provided in the APPENDIX (Fig. 16).
L-type Ca2+ channel current.
The kinetics of ICa,L are described with
activation (dL), voltage-dependent inactivation
(fL), and Ca2+-dependent
inactivation (fCa) gating variables. The
inactivation of ICa,L consists of two
exponential terms: rapid and slow components. The rapid component is
mediated by the intracellular Ca2+-dependent inactivation
with a time constant ranging from 10 to 30 ms for rabbit SA node cells,
whereas the slower component reflects the voltage-dependent
inactivation with a time constant of 30~70 ms (5, 34, 72, 81,
102). To describe the inactivation process, we adopted a simple
model in which the Ca2+-dependent inactivation process is
independent of the voltage-dependent one (see Refs. 35 and
105); the inactivation process was described by the two
Hodgkin-Huxley type gating variables fL and
fCa.
The voltage dependences of ICa,L activation and
inactivation (steady-state probabilities and time constants for
dL and fL) are shown in
Fig. 2, top. For the
steady-state activation and inactivation curves
(dL,
and fL,
), we
used the formulas of Demir et al. (17) based on the data
from Fermini and Nathan (27). Expressions of the
activation time constant
dL were
adopted from Demir et al. (17), who modified the
formulation of Nilius (84). To formulate the time constant
of the voltage-dependent inactivation/recovery
(
fL), the previous models
employed different equations with different voltage dependences;
however, these equations did not fit the recovery time course of
ICa,L experimentally observed in single rabbit
cardiac myocytes (see Fig. 2). Therefore, we originally formulated the
inactivation time constant
fL
from the data of Nakayama et al. (81), Hagiwara et al.
(37), and Kawano and Hiraoka (54). According to Demir et al. (17), a Q10 factor of 2.3 was
applied to scale the gating time constant data for a temperature of
37°C. The inactivation/recovery time constant data were fitted to a
function similar to that used by Lindblad et al. (68) for
a rabbit atrial model and by Nygren et al. (88) for a
human atrial model, using a least-square minimization procedure.
Formulas for the Ca2+-dependent inactivation
fCa are similar to those used by DiFrancesco and
Noble (22). We included the subsarcolemmal domain and
modeled the Ca2+-dependent inactivation as a function of
the subspace Ca2+ concentration
([Ca2+]sub). The half-maximum
Ca2+ concentration for the Ca2+-dependent
inactivation was set to 0.35 µM, the same value as used by
Courtemanche et al. (15). To reproduce the
Ca2+-dependent inactivation with a time constant of ~10
ms during voltage-clamp pulses, the rate constant of Ca2+
binding was set to 60 ms
1 · mM
1.

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Fig. 2.
Kinetics of ICa,L. Top: voltage
dependence of steady-state probabilities (dL and
fL) and time constants for
ICa,L activation
( dL) and inactivation
( fL). The equations used for the
present model are shown as thick lines (K). For comparison, those used
for previous models are also shown as thin lines: W, Wilders et al.
(124); De, Demir et al. (17); Do, Dokos et
al. (24); and Z, Zhang et al. (130). The
experimental data for fL are from
Kawano and Hiraoka (54) ( ), Hagiwara et
al. (37) ( ), and Nakayama et al.
(81) ( ). Bottom: computed
voltage-clamp records for ICa,L
(left) and the peak
ICa,L-V relationship
(right). Currents were evoked by 100-ms step pulses to test
potentials ranging from 30 to +50 mV (in 10-mV increments). The
holding potential was 40 mV. Simulating the whole cell
perforated-patch recording, [Ca2+]i was not
fixed (intracellular Ca2+ was not buffered by EGTA or
BAPTA), whereas [Na+]i and
[K+]i were fixed at 10 and 140 mM,
respectively. The experimental data for the peak
ICa,L-V relationship (peak currents
normalized to the maximum value at 0 mV) are from Honjo et al.
(44) ( ), Hagiwara et al. (37)
( ), Vinogradova et al. (122)
( ), and Verheijck et al. (120)
( ). See the Glossary for definitions of the
abbreviations.
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Maximum ICa,L has been formulated as a fully
selective Ca2+ current, with its reversal potential
(ECa,L) fixed at +45 mV and the maximum
conductance gCa,L set to 0.58 nS/pF at 2 mM
[Ca2+]o. As Dokos et al. (24)
have pointed out, ICa,L has a relatively low
ECa,L albeit its high Ca2+
selectivity. Wilders et al. (124) adopted the
constant-field formulation to describe the conductance property of
ICa,L with a low ECa,L;
as suggested by Dokos et al. (24), however, their formulas
introduce large unnecessary fluxes of Na+ and
K+, leading to an overestimation of
INaK. The ECa,L has been
reported to shift with changing [Ca2+]o in
accordance with an ideal Nernstian Ca2+ electrode, although
displaced negatively below ECa (see Ref. 11); thus Dokos et al. (24)
formulated ICa,L as a fully selective Ca2+ current with ECa,L displaced at
a constant 75 mV negative to ECa. The negative
displacement of ECa,L may be attributable to the
existence of a high-concentration Ca2+ domain at the
intracellular surface of the L-type Ca2+ channel pore.
Nevertheless, there are no available data on the [Ca2+]i dependence of
ECa,L. It has been suggested that once the
L-type Ca2+ channel is open, the Ca2+
concentration at the inner mouth does not change; the Ca2+
concentration in the vicinity of the inner mouth of the channel may be
nearly constant (see Refs. 50, 111, and
128). In our model, therefore,
ECa,L was set to a constant value of +45 mV, as
in the model of Demir et al. (17) utilizing a fully
Ca2+-selective formulation of ICa,L
with the ECa,L fixed at +46.4 mV. It might be
difficult to express ECa,L (or permeation
kinetics) as a simple elementary function of ion concentrations,
because the mechanisms of ion permeation in the L-type Ca2+
channel have been shown to involve complex ion-ion or ion-channel interactions (e.g., Refs. 41, 76, and
116).
The model-generated ICa,L during voltage-clamp
pulses and the peak ICa,L-V
relationship are depicted in Fig. 2, bottom. Our model can
simulate the Ca2+-dependent inactivation of
ICa,L experimentally observed in rabbit SA node
cells (10, 67); inactivation time courses of the simulated currents are very similar to the experimental data reported by Hagiwara
et al. (37), Honjo et al. (44), Verheijck et
al. (120), and Vinogradova et al. (122). The
simulated peak ICa,L-V relation is
comparable to the experimental data as shown by the symbols. The
maximum amplitude of ICa,L measured in the
simulated voltage-clamp experiment was 9.44 pA/pF, attained at 0 mV;
this value is in good agreement with experimental data (44, 81, 119, 120, 122).
T-type Ca2+ channel current.
Contributions of ICa,T to pacemaker
depolarization in the previous models are different: Wilders et al.
(124) and Dokos et al. (24) assumed small
contribution of ICa,T according to the experimental report of Hagiwara et al. (37), whereas Demir
et al. (17) assumed much larger
ICa,T based on the data from Doerr et al.
(23). With the use of the ICa,T
expressions of Demir et al. (17), our standard model could
reproduce the relatively large effects of blocking
ICa,T on CL as observed experimentally (23, 37, 103); in contrast, the spontaneous AP of our
standard model cell was little affected by incorporating the
ICa,T expressions of Wilders et al.
(124), with CL increasing only by 2.1% on eliminating ICa,T. Therefore, we adopted the expressions of
Demir et al. (17) rather than those of Wilders et al.
(124) or Dokos et al. (24) for the
steady-state probabilities and time constants of the activation gating
variable dT and inactivation gating variable
fT. The conductance property of
ICa,T was formulated using a linear voltage
relation, with the reversal potential ECa,T
fixed at +45 mV and the maximum conductance
gCa,T set to 0.458 nS/pF, according to Demir et
al. (17).
Delayed rectifier K+ currents.
Recent studies in rabbit, guinea pig, and human cardiac myocytes have
identified two types of delayed rectifier K+ currents:
1) the rapidly activating component
IKr, exhibiting strong inward rectification, and
2) the slowly activating component IKs, exhibiting only weak rectification
(2, 34, 64, 66, 100, 101, 123). Whereas
IK in the rabbit SA node appears to predominantly reflect only IKr-type behavior
(2, 108), the IKs-type component
has also been identified in isolated rabbit SA node cells (34,
48, 64). For the standard pacemaker model, therefore, we assumed
both IKr- and IKs-type
components of IK.
The voltage dependences of IKr activation and
inactivation (steady-state probabilities and time constants for the
activation gating variable pa and inactivation
gating variable pi) are shown in Fig.
3, top, along with those for
the previous models. To describe the gating kinetics of
IKr, the previous models (17, 24,
124) used the equations provided by Shibasaki
(108). However, Ono and Ito (90) recently
reported the complete quantitative data on the activation kinetics of
IKr in single rabbit SA node cells and
mathematically described the activation kinetics with two gating
variables, paF and paS.
According to their report, therefore, we described the general
activation variable pa as a weighted sum of the
fast (paF) and slow (paS)
activation variables and used their original expressions for
IKr activation kinetics
(pa,
,
paF, and
paS). A modified version of the
formulation of Ono and Ito (90) for
IKr activation was also utilized by Zhang et al.
(130). We also employed the expression of Ono and Ito
(90) for steady-state inactivation
(pi,
). No detailed experimental data are
available on the time constant of the voltage-dependent IKr inactivation
(
pi); thus we adopted the
expression of Shibasaki (108) for
pi.

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Fig. 3.
Kinetics of IKr. Top: voltage
dependence of steady-state probabilities for IKr
activation (pa) and inactivation
(pi) and activation time constants
( pa). Thick lines represent the
present model (K); those for previous models (W, De, Do, and Z) are
also shown as thin lines for comparison. and
, Experimental data from Ono and Ito (90).
Bottom: computed voltage-clamp records for
IKr (left) and amplitudes of
IKr measured at the end of test pulses as a
function of the test potential (right). Currents were
elicited by 1-s step pulses from a holding potential of 60 mV to test
potentials ranging from 50 to +40 mV (in 10-mV increments).
, Experimental data from Ono and Ito (90).
See the Glossary for definitions of the abbreviations.
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The conductance of IKr was chosen as to allow a
maximum diastolic potential (MDP) between
60 and
55 mV
(approximately equal to
58 mV) to be achieved. The standard
gKr value of our model is smaller than that
determined by Ono and Ito (90); this difference in
gKr may reflect the regional difference in
IKr density between the center and periphery of
the SA node (see Ref. 56). We also included a term to
describe the "square root" activation of IKr single channel conductance by [K+]o,
as reported by Shibasaki (108). Dokos et al.
(24) pointed out that the reversal potential of
IKr (EKr) is positive to
EK by 10~19 mV, thus assumed that
IKr channels are slightly permeable to
Na+. According to recently published reports (e.g., Refs.
90 and 119), however, EKr is nearly
equal to EK; therefore, the
IKr channel was assumed to be highly selective
to K+. The model-generated IKr
during the voltage-clamp pulses simulating the experiment of Ono and
Ito (90) are shown in Fig. 3, bottom.
The kinetics of IKs were described by the
formulation of Zhang et al. (130). The steady-state
activation curve for IKs used in this study is
from Lei and Brown (64). There are limited experimental
data available for the time constant of the voltage-dependent activation of IKs in rabbit SA node cells. Thus
the time constant of IKs activation was
described using the expressions of Zhang et al. (130),
i.e., the equations of Heath and Terrar (40) based on
their data from guinea pig ventricular myocytes.
4-AP-sensitive K+ currents.
Recent studies for rabbit SA node cells have identified the transient
and sustained outward currents sensitive to 4-AP (46, 65,
119). The models of Zhang et al. (130) incorporated
these currents, whereas most previous SA node models did not.
Therefore, we incorporated the transient (Ito)
and sustained (Isus) components of
4-AP-sensitive currents into our model. According to Zhang et al.
(130), we treated the two 4-AP-sensitive components as separated currents and used the same activation variable r
for both Ito and Isus.
The steady-state activation and inactivation curves
(r
and q
) were
based on the experimental data from Honjo et al. (46) and
Lei et al. (65). The time constant of the voltage-dependent activation (
r) was
formulated from the data of Giles and van Ginneken (28)
for rabbit crista terminalis cells; the inactivation time constant
(
q) was from Giles and van Ginneken
(28) and Honjo et al. (46). To correct the data collected at 20.5~25°C for 37°C, Zhang et al.
(130) used a Q10 of 2.18. However, the use of
Q10 = 2.18 yielded a pronounced phase 1 notch in APs.
Thus we slightly accelerated the gating kinetics by using a
Q10 of 3.0. The corrected time constants were comparable to
those reported by Honjo et al. (46) and Uese et al.
(117).
The values of the scaling parameters (conductances) for
Ito and Isus were
determined by exploring the change in peak overshoot potential (POP)
and variation of [K+]i . The experimentally
measured densities of Ito and
Isus were significantly correlated with
Cm, i.e., cell size, and are larger in cells
with a higher Cm (46, 65); this
probably reflects the regional differences of the current densities
(see Ref. 7). We selected relatively small conductance
values for our primary pacemaker cell model with relatively small
Cm, because the current densities in the central
SA node region would be relatively small (see Refs. 64 and
130). Whereas Zhang et al. (130) set the Isus conductance to a very small value of 3.3 pS/pF for their central model, we used a larger value of 20 pS/pF to
accentuate the prolongation of AP duration during the blockade of the
4-AP-sensitive currents, i.e., to reproduce the experimental data of
Boyett et al. (7).
Hyperpolarization-activated current.
It is difficult to quantify Ih and its
participation in the diastolic depolarization, in part due to the large
variation in its threshold of activation ranging from
30 to
70 mV,
as well as the variation in current density (18, 21, 73, 81,
118). In previous models, the kinetic data from van Ginneken and
Giles (118) were used to formulate the gating kinetics of
Ih; however, the contributions of
Ih to pacemaker depolarization in the model cells are quite different (refer to Table 3). For our standard model,
we adopted the formulation of Wilders et al. (124) to reproduce the relatively large effects of blocking
Ih on CL as observed in experiments. The data of
van Ginneken and Giles (118) were collected at
30~33°C; following Demir et al. (17), activation time
constants were corrected for 37°C by the use of a
Q10 = 2.3. According to van Ginneken and Giles
(118), the maximum conductance and reversal potential of
Ih were assumed to be 0.375 nS/pF and
24 mV,
respectively. These values were achieved by setting
gh,K = 7.4 nS and
gh,Na = 4.6 nS for a model cell, as used in
Wilders et al. (124).
Sustained inward current.
Guo et al. (32) reported a novel pacemaker current
activated within the range of pacemaker depolarization in the rabbit SA
node (see also Ref. 33). This current, named the sustained inward current (Ist), is carried by
Na+ under physiological conditions, blocked by both organic
and inorganic Ca2+ channel blockers as well as by external
Ca2+ and Mg2+, and enhanced by isoprenaline or
a Ca2+ agonist, BAY K 8644. These biophysical and
pharmacological characteristics are compatible with those of the
monovalent cation conductance of the L-type Ca2+ channel;
thus Guo et al. (32) concluded that
Ist is generated by a novel subtype of
the L-type Ca2+ channel. Ist has
also been recorded in SA node cells of other animal species, such as
guinea pigs and rats (31, 77, 78, 109). Because
Ist was observed only in spontaneously beating SA node cells but was absent in quiescent cells,
Ist may play an essential role in the generation
of intrinsic cardiac automaticity (32, 78). Therefore, we
incorporated Ist in our standard SA node model
for a primary pacemaker cell, whereas previous SA node models did not
include Ist.
The voltage dependences of Ist activation and
inactivation (steady-state probabilities and time constants for the
activation gating variable qa and inactivation
gating variable qi) are shown in Fig.
4, top. Because there is no
detailed report on the gating kinetics (time constant) of
Ist in the rabbit SA node, we modified the
formulation of Shinagawa et al. (109) for the rat SA node Ist. From the data of Guo et al.
(32), the half-activation voltage and slope factor for the
activation curve were set to
57.0 and 5.0 mV, respectively. The
inactivation and recovery of the rabbit Ist, as
reported in Guo et al. (32), appear to be slower than those of the rat Ist reported by Shinagawa et
al. (109). Thus the time constants of inactivation and
recovery of the rabbit Ist were assumed to be
6.65 times larger than those of the rat Ist. The
maximum conductance and reversal potential of
Ist were set to 15.0 pS/pF and +37.4 mV,
respectively, according to Guo et al. (32). As shown in
Fig. 4, bottom, our equations for Ist could reproduce the kinetic properties of Ist
reported by Guo et al. (32).

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Fig. 4.
Kinetics of Ist. Top: voltage
dependence of steady-state probabilities (qa and
qi) and time constants for
Ist activation
( qa) and inactivation
( qi). Thick lines, present model;
thin lines, qa and
qi formulated by Shinagawa et al.
(109) for the rat SA node Ist.
and , Experimental values approximated
from the data of Guo et al. (32). Bottom:
computed voltage-clamp records for Ist
(left) and the peak Ist-V
relation (right). Currents were evoked by 500-ms
depolarizing test pulses ranging from 70 to +50 mV (in 10-mV
increments). The holding potential was 80 mV. See the Glossary
for definitions of the abbreviations.
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Na+ channel current.
Although most of the previous SA node models incorporated the fast
Na+ channel current (INa), the
contribution of INa to model cell dynamics is
relatively small: in the models of Wilders et al. (124)
and Dokos et al. (24), eliminating
INa yielded only a 0.7~0.9% increase in CL,
whereas it yielded a 8.4% increase in the model of Demir et al.
(17) for transitional cells. In a preliminary study, we
incorporated INa into our model to assess the
possible contribution of INa to the pacemaker
activity of our model cell. The kinetics of INa
were reformulated from the recent experimental data: 1)
steady-state activation and inactivation curves are based on the data
from Muramatsu et al. (80) and Baruscotti et al.
(3); 2) for the time constant of activation, we
used the formulation of Zhang et al. (130); and
3) the inactivation of INa was
described by a weighted sum of two gating variables, hF and hS, with the
fraction of slow inactivation (FNa) being expressed as a function of V (see Ref. 130).
The formulation for the inactivation time constant is based on data
from Muramatsu et al. (80). According to Lindblad et al.
(68), a Q10 of 1.7 was used to correct the
experimental data for 37°C. Incorporating INa
with a conductance of 1.8 pS/pF, the mean experimental value for
transitional or peripheral cells (44), decreased CL only by 5.6% (from 307.4 to 290.2 ms), indicating that the contribution of
INa to pacemaking is relatively small in our
model cell as well.
INa appears to be completely absent or almost
negligible in primary pacemaker cells located in the central region of
the SA node (57, 59, 82, 118, 130). Baruscotti et al.
(3, 4) reported that both the density and frequency of
occurrence of INa in the SA node decrease with
development, with INa in adult rabbit SA node
cells being very small or negligible. Thus INa would not play an important role in normal pacemaking of primary pacemaker cells. The central model of Zhang et al. (130)
for the leading pacemaker cell did not include
INa. In our simulations for primary pacemaker
(central SA node) cells, therefore, INa was
assumed to be zero (negligible or completely inactivated).
Na+-dependent background current
and muscarinic K+ channel current.
Our model includes two background current components: 1)
Na+-dependent background current
(Ib,Na), reported by Hagiwara et al.
(38), in which Ib,Na was measured
as 0.73 ± 0.21 pA/pF at
50 mV; and 2) muscarinic
K+ channel current (IK,ACh),
reported by Ito et al. (49), who attributed the entire
background K+ component in rabbit SA node cells to the
spontaneous opening of muscarinic K+ channels in the
absence of an ACh agonist and characterized
IK,ACh as exhibiting inward rectification with
an amplitude of 0.33 ± 0.28 pA/pF at
50 mV under physiological
conditions. Following their experimental data, we modeled
Ib,Na with an ohmic I-V
relationship and defined IK,ACh as a background
K+ current component exhibiting inward rectification. The
Ib,Na conductance gb,Na
was set to 5.4 pS/pF, yielding a current density of 0.65 pA/pF at
50
mV. The gb,Na value of 5.4 pS/pF is comparable to that in the previous SA node models (2.9~7.5 pS/pF). To model the
inward rectifying behavior for IK,ACh, we chose
the formula of Dokos et al. (24) adopted from Egan and
Noble (26). The standard value of
IK,ACh amplitude used in this study was 0.23 pA/pF at
50 mV, smaller than the value of 0.46 pA/pF in Dokos et al.
(24).
The background Ca2+ current (Ib,Ca)
has been frequently incorporated in mathematical models of the SA node
to balance the Ca2+ extrusion via
Na+/Ca2+ exchange during diastole: the models
of Wilders et al. (124), Demir et al. (17),
and Zhang et al. (130) included
Ib,Ca with a conductance of 0.66~1.25 pS/pF.
In our model, however, incorporating an Ib,Ca of
0.66~1.25 pS/pF attenuated pacemaker activity (decreased APA to <70
mV) and induced Ca2+ overload (diastolic
[Ca2+]i > 0.3 µM) via unnecessary
Ca2+ influx, unfavorable for improving the model. Although
the sarcolemmal Ca2+ pump current
(Ip,Ca) may balance
Ib,Ca, the contribution of
Ip,Ca to Ca2+ efflux through the SA
node cell membrane is unknown. Ib,Ca has not yet
been recorded directly in SA node cells; the report of Hagiwara et al.
(38) suggests that the rabbit SA node membrane is
impermeable to Ca2+. In our modeling, therefore,
Ib,Ca (as well as Ip,Ca)
was assumed to be negligible.
Na+-K+
pump current.
In previous SA node models, the formulations of
INaK were not based on experimental data from SA
node cells. The novel INaK formulation used in
this study is based on the recent experimental work of Sakai et al.
(99) for rabbit SA node cells. Parameter values were
determined from the voltage- and concentration-dependent data provided
by Sakai et al. (99): Km values for
Na+ and K+ binding were set to 14.0 and 1.4 mM,
respectively. The voltage dependence of INaK in
our model is steeper than that in previous models (see Fig. 17),
consistent with experimental data from the rabbit SA node
(99). The magnitude of INaK is
linked to the influx of Na+ through
Ib,Na (and Ist) and
efflux of K+ through K+ channels. According to
the data of Sakai et al. (99), we set the maximum
attainable current (INaKmax) to 3.6 pA/pF, a value high enough to maintain [Na+]i
of <10 mM and [K+]i of ~140 mM during long
runs of the normal pacemaker activity.
Na+/Ca2+
exchange current.
Most existing models of SA node activity have relied on the
hypothetical INaCa formulation utilized by
DiFrancesco and Noble (22), describing the
Na+/Ca2+ exchanger as a simultaneous transfer
of the ions with an exponential voltage dependency. In contrast, Dokos
et al. (24) utilized a more accurate model of consecutive
translocation, i.e., the "E4" model originally formulated by
Matsuoka and Hilgemann (74), which correctly reproduces
the saturation characteristics of INaCa at large
[Ca2+]i and for negative potential-low
[Ca2+]i conditions. To describe
INaCa kinetics, we adopted the formulation of
Dokos et al. (24), a modified version of the
Matsuoka-Hilgemann "E4" model. The scaling parameter
kNaCa was set to 125 pA/pF, the same value as
determined by Dokos et al. (24) from the study of Hagiwara
and Irisawa (36), who reported an
INaCa density of 1 pA/pF at
40 mV and
[Ca2+]i = 0.5 µM. A
kNaCa value of 125 pA/pF yielded an
INaCa of 0.83 pA/pF at
40 mV and
[Ca2+]i = 0.5 µM, giving a diastolic
free Ca2+ level in the presumed physiological range of
<0.3 µM.
SR functions.
The SR was modeled as consisting of two compartments: the
Ca2+ uptake store (network SR) and release store
(junctional SR), as shown in Fig. 1B. Owing to the lack of
available data for updating the kinetic formulation of Ca2+
uptake and release by the SR in SA node cells, we utilized simple formulas similar to those of DiFrancesco and Noble (22),
which have been incorporated in previous SA node models (24, 85, 124). The model of Demir et al. (17) utilized more
complex SR kinetics based largely on the modeling study by Hilgemann
and Noble (42), including the internal SR Ca2+
buffering by calsequestrin. Their formulation is, however, overly complex, given the complete lack of associated data in SA node cells;
in their model, Ca2+ concentrations in the SR are
unreasonably high compared with those in other models, much higher than
experimental values for ventricular myocytes (see Refs. 63
and 107). We therefore opted for simpler kinetics, which are able to
reproduce the essential features of both the uptake and release
processes. According to Demir et al. (17), the fractional
volumes of Ca2+ uptake and release stores in our model cell
were taken to be 1.16 and 0.12%, respectively, of the cell volume,
corresponding to one-third of the values given for ventricular myocytes.
Because no experimental data are available to estimate the kinetic
parameters for SR Ca2+ uptake and release in SA node cells,
the parameter values were determined from previous modeling studies.
The formula for the Ca2+ release mechanism was adopted from
Dokos et al. (24), with the kinetics of Ca2+
release to subspace expressed as a function of
[Ca2+]sub (not
[Ca2+]i). The values of
Krel used in previous models ranged from 0.8 to
2.0 µM; we chose a medium value of 1.2 µM. The
Prel value was determined so as to make the peak
[Ca2+]sub and
[Ca2+]i nearly maximum with a
Krel of 1.2 µM. The formulation for
Ca2+ uptake was adopted from Luo and Rudy
(71). The values of
tr used in previous SA
node models ranged from 6.64 to 400 ms. In our modeling, the
tr value was set to a medium value of 60 ms, although
some reports have suggested a smaller
tr or single
compartment for SR (e.g., Refs. 29 and 113).
Ca2+ buffering.
Our model includes the dynamics of three Ca2+ buffers:
calmodulin, troponin, and calsequestrin. The amounts of calmodulin and troponin available for Ca2+ binding, and the rate constants
for Ca2+ binding to and release from these buffers, were
adopted from Demir et al. (17) and Lindblad et al.
(68), who used data from Robertson et al.
(95). The rate constants for Ca2+ binding to
calmodulin and troponin were scaled with the use of a Q10
of 1.8, according to Lindblad et al. (68). The
concentration of calsequestrin within the SR release compartment was
set to a relatively low value of 10 mM, the value used by Luo and Rudy (71), Courtemanche et al. (15), and Priebe
and Beuckelmann (91). The on and off rates for
Ca2+ binding to calsequestrin were based on the study of
Cannell and Allen (12), adjusted to 37°C via a
Q10 of 1.6 (see Ref. 68).
We also examined the effects of exogenous Ca2+ buffers,
BAPTA and EGTA, on pacemaker activity. The rate constants for
Ca2+ binding to and release from these buffers were adopted
from Sham (106), Smith et al. (112), and You
et al. (128).
Ion concentration homeostasis.
Our model also includes material balance expressions to define the
temporal variations in [Na+]i,
[K+]i, and [Ca2+]i.
[Na+]i and [K+]i
were assumed to be homogeneous, because subspace [Na+]
(or [K+]) was nearly equal to myoplasmic
[Na+] (or [K+]) unless the diffusion time
constant for Na+ (or K+) was more than 10 times
larger than that for Ca2+. As suggested by Nygren et al.
(88), there may be electroneutral Na+ influx
(and K+ efflux as well) via electroneutral transport
mechanisms such as Na+/H+ exchange and
Na+-K+-2Cl
cotransport; thus it
is difficult to estimate [Na+]i (and
[K+]i) accurately from model simulation. In
our simulations, net electroneutral Na+ and K+
fluxes were assumed to be zero; in the free running model with the
standard parameter values listed in the APPENDIX, the
intracellular free Na+ and K+ levels averaged
9.44 and 140.0 mM, respectively, during a long period of normal
pacemaking. [Na+]o,
[K+]o, and [Ca2+]o
were set equal to 140, 5.4, and 2 mM, respectively.
Numerical Integration (Dynamic Simulation)
Dynamic behavior of the model cell was determined by solving the
simultaneous nonlinear ordinary differential equations numerically. We
employed a fourth-order adaptive-step Runge-Kutta algorithm, which
includes an automatic step-size adjustment based on an error estimate,
or a variable time-step numerical differentiation approach selected for
its suitability to stiff systems. The maximum relative error tolerance
for our integration methods was set to 1 × 10
6.
The APA as a voltage difference between the local potential minimum and
maximum, as well as the CL, were determined for each calculation of a
cycle. Numerical integration was continued until the differences in
both APA and CL between the newly calculated cycle and the preceding
one became <1 × 10
3 of the preceding APA and CL
values. When periodic behavior was irregular or unstable, model
dynamics were computed for 30~60 s; all potential extrema and CL
values were then plotted in the diagram. The value of current
conductance is expressed as a ratio to the control value unless
otherwise stated.
Numerical computations were performed on Power Macintosh G4 computers
(Apple Computers; Cupertino, CA) using Matlab 5.2 (MathWorks; Natick, MA).
 |
RESULTS AND DISCUSSION |
Dynamic Properties of Simulated Pacemaker Activity
Spontaneous APs and sarcolemmal ionic currents.
Figure 5, left, shows
spontaneous APs and temporal behavior of sarcolemmal ionic currents
simulated by the present model with the standard parameter values
listed in the APPENDIX. The MDP, POP, and APA are
58.6,
+16.6, and 75.2 mV, respectively; the CL and AP duration at 50%
repolarization (APD50) are 307.5 and 107.0 ms,
respectively. The simulated AP parameters of our model cell are listed
in Table 1, along with those of previous
SA node models as well as corresponding experimental data for
comparison (see also Fig. 6). The AP
parameters of our model appear to be in reasonable agreement with the
mean experimental values recently determined for central SA node
(primary pacemaker) cells and spindle- or spider-shaped cells. The
previous models developed by Wilders et al. (124), Demir
et al. (17), and Dokos et al. (24) are all
likely to reflect the activity of a typical transitional cell. Compared
with these previous models for transitional cells, our model for
primary pacemaker cells exhibited 1) more positive MDP, 2) relatively small APA, and 3) long
APD50. These AP characteristics of our model are comparable
to those of primary pacemaker cells in the central region of the rabbit
SA node (8, 44, 55, 57), reflecting the regional
difference in AP parameters.

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Fig. 5.
Model-generated behavior of spontaneous APs, underlying
transmembrane ionic currents (left) and intracellular
Ca2+ dynamics (right) in the steady state. The
differential equations were numerically solved for 11 s with an
initial condition appropriate to a zero current potential ( 24.3 mV)
and a 1-ms stimulus of 1 pA/pF for triggering an AP. Model cell
behavior during the last 1 s (from MDP) of the simulated
free-running activity is depicted. Note the differences in the ordinate
scales for individual currents. The intracellular Ca2+
dynamics shown on the right include the changes in
Ca2+ concentrations (in the subspace, myoplasm, and SR),
associated percent changes in the occupancies of intracellular
Ca2+ buffers, and Ca2+ fluxes produced by the
SR during Ca2+ uptake, release, and transfer. Changes in
[Na+]i and [K+]i
during pacemaker activity are also shown at the bottom. See
the Glossary for definitions of the abbreviations.
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Fig. 6.
Spontaneous APs and underlying transmembrane
time-dependent currents in various SA node models developed by Wilders
et al. (124) (A), Demir et al.
(17) (B), Dokos et al. (24)
(C), Zhang et al. (130) (D; central
model), and 5) the present study (E). The
differential equations for previous models were numerically solved for
11 s with the initial conditions provided in the original
articles. Model cell behaviors during the last 600 ms (from MDP) are
depicted. Spontaneous APs recorded in the central SA node (leading
pacemaker site) and the currents experimentally determined by the AP
clamp method are also shown (F) for validation of the
models: AP, Miyamae and Goto (79);
ICa,L, Zaza et al. (129);
IKr, Ono and Ito (90);
ICa,T, Doerr et al. (23); and
Ih, Zaza et al. (129). See the
Glossary for definitions of the abbreviations.
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As shown in Fig. 6, our model-generated ICa,L
waveform during spontaneous APs was very similar to that recorded by
Zaza et al. (129) and Doerr et al. (23)
during the "AP clamp" of a single rabbit SA node cell. The
model-generated IKr waveform was also very
similar to that measured as an E-4031-sensitive current during AP clamp
(90, 129). Furthermore, the time course of the simulated
ICa,T was similar to that of
ICa,T measured as a Ni2+-sensitive
current by Doerr et al. (23); the
Ih during pacemaker depolarization was similar
to the Cs+-sensitive current recorded by Zaza et al.
(129) and also to the Ih computed
by Maruoka et al. (73). Thus the simulated changes in ICa,L, IKr,
ICa,T, and Ih during
pacemaking are comparable to the experimentally observed changes in
these currents during spontaneous APs in rabbit SA node cells.
Simulated ionic currents, as well as an AP waveform, in our model were
compared with those in the previous models (Figs. 6 and
7). The time courses of
ICa,L and IKr during
pacemaker activity in the models were different. In the models of
Wilders et al. (124), Demir et al. (17), and
Dokos et al. (24), the fast inhibition of
ICa,L was followed by a secondary increase
(inward "hump"); in contrast, the inward hump was absent in our
simulated ICa,L. Consistent with our model
simulation, the secondary increase in ICa,L was
not observed in the AP-clamp experiments (23, 129). The
rapid inactivation of IKr on AP upstroke (during
phase 0) observed in the AP-clamp experiments (90, 129)
was reproduced only by our model and the model of Zhang et al.
(130). Thus our model is apparently superior to previous
models in generating the ICa,L and
IKr waveforms experimentally observed in single rabbit SA node cells. The amplitudes of Ib,Na
and IK,ACh (as Ib,K) during pacemaker activity of our model are comparable to those of
previous models (see Fig. 7). On the other hand, the time courses of
INaCa in the model cells were quite different,
reflecting the different intracellular Ca2+ dynamics. In
our model, INaCa during phase 4 was relatively
large, suggesting the large contribution of
INaCa to pacemaker depolarization.

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Fig. 7.
Time-independent background and transporter currents
during spontaneous pacemaking in various SA node models formulated by
Wilders et al. (124) (A), Demir et al.
(17) (B), Dokos et al. (24)
(C), Zhang et al. (130) (D; central
model), and the present study (E). See the Glossary
for definitions of the abbreviations.
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Intracellular Ca2+ dynamics.
Intracellular Ca2+ dynamics (concentration changes, SR
Ca2+ uptake and release, and Ca2+ buffering)
during normal pacemaking are shown in Fig. 5, right. The
free Ca2+ concentrations in subspace
([Ca2+]sub) and myoplasm
([Ca2+]i) during the spontaneous APs were
0.18~1.81 and 0.25~0.68 µM, respectively. Although there are no
available data on intracellular Ca2+ concentrations in the
rabbit SA node cell, these values are comparable to those measured in
other pacemaker cells (47, 53).
As shown in Fig. 8, the simulated
intracellular Ca2+ dynamics in our model cell were quite
different from those in previous models, because we assumed a
subsarcolemmal space and relatively small SR volumes. The myoplasmic
Ca2+ transient in our model, the peak value of which is
0.68 µM, was much smaller than that in previous SA node models but is
comparable to that experimentally or theoretically determined for
atrial or ventricular myocytes (e.g., Refs. 68,
71, and 110). In contrast, peak
[Ca2+]i values in previous SA node models
were >2.5 µM, unreasonably larger than those in atrial or
ventricular models. The Ca2+ concentrations in the SR
([Ca2+]rel and
[Ca2+]up) of our model are similar to those
of previous models except for the model of Demir et al.
(17), in which the SR Ca2+ concentrations
(>10 mM) are much higher than experimental values for ventricular
myocytes (see Refs. 63 and 107). The
Ca2+ release from the SR in our model is much smaller than
that in the model of Wilders et al. (124) or Demir et al.
(17), whereas it is comparable to that in the model of
Dokos et al. (24), suggesting a relatively small
contribution of SR Ca2+ release to intracellular
Ca2+ transients in primary pacemaker cells.

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Fig. 8.
Intracellular Ca2+ dynamics during spontaneous
pacemaking in various SA node models formulated by Wilders et al.
(124) (A), Demir et al. (17)
(B), Dokos et al. (24) (C), and the
present study (D). Note the differences in the ordinate
scales for individual models. See the Glossary for
definitions of the abbreviations.
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Contribution of Ist to pacemaker activity.
Figure 9 shows the simulated effects of
block of Ist on pacemaker activity of our model
cell. When the maximum Ist conductance (gst) was reduced from the control value to
zero, pacemaking slowed, with CL increasing from 307.5 to 473.2 ms, but
did not cease. The effects of eliminating Ist on
MDP and POP were relatively small, with MDP hyperpolarized from
58.6
to
63.6 mV. These alterations in the AP waveform by the removal of
Ist could partly be compensated for by reducing
the maximum IKr conductance
(gKr): the Ist-removed model cell with gKr reduced to 75% of the
control value produced spontaneous APs with MDP =
58.9 mV and
CL = 379.8 ms (see Fig. 9, bottom right). Thus
Ist is not essential for generating pacemaker activity.

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Fig. 9.
Contribution of Ist to pacemaker
activity of the SA node. Left: dynamics of the model cell
during inhibition of Ist. Changes in MDP/POP
(top) and CL (bottom) are shown as functions of
the maximum Ist conductance
(gst). The gst value,
shown as a ratio to the control value of 15 pS/pF, was reduced at an
interval of 0.01. Right: simulated spontaneous APs of the
standard (control) system (top),
Ist-removed system (middle), and
Ist-removed system with the
IKr conductance (gKr)
reduced to 75% of the control (bottom). See the
Glossary for definitions of the abbreviations.
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Effects of Modulating Sarcolemmal Ionic Currents on Pacemaker
Activity
We simulated the effects of inhibition of sarcolemmal ionic
currents on pacemaker activity and then compared the simulated behaviors of our model cell with experimental findings as well as with
those of previous models. We used two different systems, the
Ist-incorporated (standard) system and the
Ist-removed (gKr = 0.75) system, because Ist is not always present
in spontaneously beating SA node cells (see Ref. 120).
Simulated blockade of ICa,L.
We first examined the effects of blocking ICa,L
on pacemaker activity of the model cell by decreasing the maximum
ICa,L conductance (gCa,L). Figure
10A shows the dynamic
behaviors of the Ist-incorporated [Ist(+)] and
Ist-removed [Ist(
)]
systems during gCa,L decrease, depicting MDP/POP
and CL as functions of gCa,L (i.e., bifurcation diagrams for the bifurcation parameter gCa,L).
As gCa,L diminished, APA was reduced and CL
increased; POP gradually decreased with reducing
gCa,L, whereas MDP was relatively stable.
Blocking ICa,L by 76.6% (reducing
gCa,L to 23.4% of the control) abolished
spontaneous activity of the standard system, with V settling
at
29.5 mV; the complete block of
ICa,L yielded a resting potential of
32.6 mV.
In the Ist-removed system, pacemaker activity
abruptly ceased at gCa,L = 0.298 (V =
37.6 mV) via irregular (chaotic) dynamics between gCa,L = 0.439 and 0.304; the
complete block of ICa,L yielded a resting
potential of
42.0 mV. Note that the bifurcation structure as a way to
quiescence during inhibition of ICa,L depends on
whether the model cell includes Ist or not: APA
of our standard model gradually and continuously decreased during the
gCa,L decline, whereas the irregular (chaotic)
dynamics occurred when Ist was removed or
concomitantly blocked.

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Fig. 10.
Effects of block of ICa,L on
pacemaker activity of the SA node. A: dynamics during
ICa,L inhibition of our standard model system
including Ist [Ist(+)]
and the Ist-removed system with
gKr reduced to 75% of the control
[Ist( )]. Changes in MDP/POP (top)
and CL (bottom) are shown as functions of the maximum
ICa,L conductance
(gCa,L). The gCa,L value,
shown as a ratio to the control value of 0.58 nS/pF, was reduced at an
interval of 0.001. Note the difference in the scales of CL.
B: experimentally observed behaviors of SA node cells in the
presence of a Ca2+ antagonist (ICa,L
blocker) and simulated dynamics of our model cell with blocked
ICa,L. The experimental data are from Kohlhardt
et al. (58) (top) and Vinogradova et al.
(122) (bottom). Simulated spontaneous APs of
the standard (top) or Ist-removed
(bottom) system with reduced gCa,L
are shown at the bottom of each experimental record. See the
Glossary for definitions of the abbreviations.
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Dynamic behaviors of real SA node cells, as well as the model cell,
during inhibition of ICa,L are shown in Fig.
10B. In experiments using Ca2+ antagonists
(L-type Ca2+ channel blockers), such as nifedipine and
verapamil, two ways of abolishing pacemaker activity have been
observed: one is characterized by a gradual and continuous decline in
APA (with a marked decrease in POP) to quiescence as well as an
increase in CL (57, 58) and the other is characterized by
irregular (chaotic) dynamics, called "skipped-beat runs," as
observed in the rabbit SA node (122) or cat subsidary
pacemaker cells (97). Our model could simulate both of the
two distinct bifurcation structures experimentally observed during
applications of Ca2+ antagonists. In the experiment for
small balls of rabbit SA node tissues, nifedipine (2 µM) abolished
spontaneous APs in the center of the SA node (i.e., the leading
pacemaker site); the resting potential at which SA node cells settled
after block of ICa,L was
40 ± 5 mV
(57). The resting potential in this experiment was ~10
mV more negative than the prediction of our standard model (approximately
29.5 to
32.6 mV), whereas it was close to the value
in the Ist-removed system (approximately
37.6
to
42.0 mV). This inconsistency may be due in part to a concomitant
block of Ist by the Ca2+ channel
blocker, which is known to inhibit Ist (see
Refs. 32 and 78), as well as to the absence
of Ist in some preparations; the complete block
of both ICa,L and Ist
in the standard model yielded a resting potential of
43.7 mV.
The effects of blocking ICa,L (decreasing
gCa,L) on spontaneous APs of previous SA node
models are shown in Fig. 11 for
comparison. In all of the models, blocking ICa,L
caused cessation of pacemaker activity; resting potentials of the
quiescent model cells at critical gCa,L values
and with complete block of ICa,L are compared in Table 2. The simulated behaviors of
previous models during gCa,L decrease are
inconsistent with experimental observations (30, 57, 58,
122) in the following respects: 1) the model of Dokos et al. (24) did not exhibit the gradual decline in APA
(POP) observed in experiments, with pacemaker activity being abruptly abolished via irregular dynamics; 2) in the central model of
Zhang et al. (130), pacemaker activity abruptly ceased
with no irregular dynamics; 3) in the models of Demir et al.
(17) and Zhang et al. (130), CL monotonously
decreased with reducing gCa,L, whereas pacemaking slowed during applications of Ca2+ antagonists;
and 4) the resting potentials predicted by the models of
Wilders et al. (124), Demir et al. (17), and
Zhang et al. (130) were 5~15 mV more positive than those
experimentally determined by Kodama et al. (57) and
predicted by our Ist-removed system. Thus our
model is superior to previous models in predicting changes in AP
parameters, bifurcation structures, and resting potentials during
applications of Ca2+ antagonists.

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Fig. 11.
Effects of block of ICa,L on
pacemaker activities of various SA node models developed by Wilders et
al. (124) (A), Demir et al. (17)
(B), Dokos et al. (24) (C), Zhang et
al. (130) (D; central model), and the present
study (E; standard model). Changes in MDP/POP
(top) and CL (bottom) during
gCa,L decrease are depicted. The
gCa,L value, shown as a ratio to the control
value for each model, was reduced at an interval of 0.001. Note the
differences in the scales of CL. See the Glossary for
definitions of the abbreviations.
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Simulated blockade of IKr.
Figure 12 illustrates the simulated
effects of blocking IKr [decreasing the maximum
IKr conductance (gKr)]
on pacemaker activity of the standard and
Ist-removed model systems along with
experimental data showing the behaviors of central SA node cells during
applications of a selective IKr blocker, E-4031
(56). In the simulations, reducing
gKr markedly depolarized MDP, whereas POP was
little changed until gKr diminished to ~0.5;
MDP depolarization during the gKr decline was
more prominent than that during the gCa,L decline (compare Figs. 10 and 12). With reducing
gKr, CL in the standard system first increased
and then decreased, being fairly stable; in contrast, CL in the
Ist-removed system monotonously decreased.
Whether Ist is incorporated or not, the
bifurcation structure of the model system during
gKr reduction was essentially the same: APA
gradually and continuously declined to zero. Pacemaker activities of
the standard and Ist-removed systems ceased when gKr decreased by 64.0 and 64.9%, respectively,
with V settling at
13.8 and
13.5 mV,
respectively. The complete block of IKr yielded a resting potential of
7.6 mV in the standard system and
8.3 mV in the Ist-removed system.

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Fig. 12.
Effects of block of IKr on
pacemaker activity of the SA node. Left: simulated dynamics
during IKr inhibition of our standard model
[Ist(+)] and
Ist-removed model
[Ist( )]. Changes in MDP/POP (top)
and CL (bottom) are shown as functions of
gKr. The gKr value, shown
as a ratio to the control value, was reduced at an interval of 0.001. Note that the control gKr value for the
Ist-removed system is 0.75 times that for the
standard system. Right: experimentally observed behaviors of
SA node cells during applications of a selective
IKr blocker, E-4031, from Kodama et al.
(56). Note the gradual and continuous decline in APA with
accentuated depolarization of MDP on the way to cessation of pacemaker
activity. See the Glossary for definitions of the
abbreviations.
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Selective IKr blockers such as E-4031 have been
experimentally found to cause a gradual and continuous decline in APA
with marked depolarization in MDP and then quiescence (see Fig. 12, right). These dynamic changes during applications of
IKr blockers are in good agreement with the
predictions of our model cell. Our model predicted only a slight
increase or even decrease in CL during the gKr
decline. In most experiments, however, CL was increased by applications
of IKr blockers (e.g., see Refs.
56, 64, and 119), although 0.1 µM E-4031 did not significantly change CL in Ono and Ito
(90). This inconsistency may result from the
state-dependent kinetics of IKr block.
Alternatively, IKr blockers may also modulate
other ionic currents directly or secondary via changing intracellular
ion concentrations. In recent experiments, block of
IKr by 1 µM E-4031 caused a cessation of spontaneous activity; after block of IKr, the
V of rabbit SA node cells or tissues settled at
32 ± 2 mV (56),
37.4 ± 2.9 mV (90), or
24.5 ± 1.8 mV (119). These experimentally observed resting potentials after block of IKr were
10~20 mV more negative than the predictions of our model. In other
reports, however, the resting or zero current potential after block of
IKr was relatively positive: 1)
Yanagihara and Irisawa (126) reported a cessation of
pacemaker activity with a resting potential of
10 mV on application of Ba2+ to block IK and
2) in the report of Verheijck et al. (119), 10 µM E-4031 stabilized V at
19.6 ± 1.8 mV and
yielded a zero current crossing of approximately
10.0 mV in the
I-V curve. These experimental data are in
reasonable agreement with the predictions of our model. The relatively
deep resting potentials in some experiments may be due to 1)
the regional difference between the central (primary pacemaker) cell
and the transitional or peripheral (latent pacemaker) cell (e.g., in
the density of background currents); 2) the difference in
recording methods or other experimental conditions (when
[Na+]i and [K+]i
were fixed at constant values of 10 and 140 mM, respectively, as in the
perforated-patch recording for single SA node cells, the resting
potential of our standard system was
18.5 mV at
gKr = 0.11 and
17.5 mV with the complete
block of IKr); 3) a
Ca2+-dependent increase of background Cl
or
K+ conductance in the region of
ICa,L window current (e.g., see Ref.
92); or 4) direct modifications of ionic
currents other than IKr by
IKr blockers.
The effects of block of IKr (decreasing
gKr) on spontaneous APs of previous SA node
models are shown in Fig. 13 for
comparison. During the decrease in gKr,
pacemaker activity of previous models abruptly ceased or attenuated
with MDP only slightly depolarized; in the model of Dokos et al.
(24), irregular dynamics appeared before the cessation of
pacemaker activity. In experiments using IKr
blockers such as E-4031, however, application of an
IKr blocker always induced the gradual and
continuous decline in APA to quiescence with a marked depolarization in
MDP and no irregular dynamics, as simulated by our model (see Fig. 12).
Abrupt cessation of pacemaker activity or irregular dynamics as
predicted by previous models has never been observed during
applications of IKr blockers. Thus the
bifurcation structures during IKr inhibition of
real SA node cells are different from those of previous models, whereas
they are essentially the same as those of our model. Block of
IKr caused a cessation of pacemaker activity in
all of the models; the resting potentials at critical
gKr values and with the complete block of
IKr for SA node models are compared in Table 2,
along with the experimental data. Resting potentials in the models of
Wilders et al. (124), Demir et al. (17), and
Dokos et al. (24) were more positive than those
predicted by our model as well as those observed in the
experiments. Taken together, our model appears to be superior to
previous models in predicting the behavior of real SA node cells during
applications of IKr blockers.

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Fig. 13.
Effects of block of IKr on
pacemaker activities of various SA node models formulated by Wilders et
al. (124) (A), Demir et al. (17)
(B), Dokos et al. (24) (C), Zhang et
al. (130) (D; central model), and the present
study (E; standard model). Changes in MDP/POP
(top) and CL (bottom) during
gKr decrease are depicted. The
gKr value, shown as a ratio to the control value
for each model, was reduced at an interval of 0.001. See the
Glossary for definitions of the abbreviations.
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Block of ICa,T.
The effects of block of ICa,T on pacemaker
frequency have been largely different in previous experimental reports:
Doerr et al. (23) found that the block of
ICa,T by 40 µM Ni2+ exerted a
pronounced negative chronotropic effect on pacemaker activity by nearly
doubling the basal CL (see also Refs. 84 and
103). In contrast, Wilders et al. (124)
reported that the Ni2+-induced block of
ICa,T had a negligible influence on pacemaker frequency. Because of this inconsistency in experimental results, the
simulated effects of eliminating ICa,T on CL of
the previous models were also quite different: the increase in CL was
relatively small in the models of Wilders et al. (124) and
Dokos et al. (24), whereas it was relatively large in the
others (see Table 3). Incorporating the
ICa,T expressions of Demir et al.
(17) based on the AP-clamp experiment of Doerr et al.
(23), our model predicted a 17.0~22.4% increase in CL
by the complete block of ICa,T (Fig.
14, top); this relatively
large effect is comparable to the experimental data from Hagiwara et
al. (37) and Satoh (103). Owing to the large
variation in experimental results, however, it is difficult to validate
a model from the contribution of ICa,T to
diastolic depolarization.

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Fig. 14.
Effects of block of ICa,T
(top), Ih (middle), or
4-AP-sensitive currents (bottom) on spontaneous APs of our
model cell. Both the standard system including
Ist (left) and the
Ist-removed system with
gKr reduced to 75% of the control
(middle) were tested. Control APs are shown as thin lines;
dashed lines indicate zero potentials. Experimentally observed
responses of rabbit SA node cells to application of 40 µM
Ni2+ (to block ICa,T), 2 mM
Cs+ (to block Ih), or 5 mM 4-AP are
also shown for comparison (right). The data are from
Hagiwara et al. (37) (top), Nikmaram et al.
(83) (middle), and Boyett et al.
(7) (bottom). See the Glossary for
definitions of the abbreviations.
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The discrepancy in the apparent contribution of
ICa,T to pacemaker activity may in part result
from the heterogeneity of SA node cells (regional difference) in the
density of Ist as well as of
INa or ICa,T itself. In
most experiments, Ni2+ was used as the selective
ICa,T blocker. However, Ni2+ has
been found to block Ist as well:
Ni2+ at 40 µM, which blocks ICa,T
completely, also decreases the amplitude of Ist
to 46~64% of the control; Ni2+ at 1 mM completely
abolishes Ist (32, 78). As shown in
Fig. 9, the pacemaker activity of our model cell was dramatically
slowed by reducing the Ist conductance: the
complete block of Ist increased CL by 53.0%
(from 307.5 to 470.5 ms). When a SA node cell possesses Ist sensitive to Ni2+, therefore,
the effect of block of ICa,T would be
overestimated because of a concomitant block of
Ist by Ni2+. Thus the differences in
the Ni2+-induced CL prolongation between experimental
studies may reflect the difference in Ist
density. The Ni2+-sensitive current recorded by Doerr et
al. (23) during pacemaker depolarization, considerably
larger (~20 pA/cell) than ICa,T in our model
(~7 pA/cell), may also include Ist.
Block of Ih.
We also simulated the effect of Cs+-induced
Ih block on pacemaker activity by setting
Ih = 0. As listed in Table 3, the
contributions of Ih to pacemaker depolarization
of previous models were different: the increase in CL on the removal of
Ih was relatively large in the models of Wilders
et al. (124) and Dokos et al. (24), whereas it was relatively small in the others. Incorporating the
Ih expressions of Wilders et al.
(124), our model predicted a 8.4~25.3% increase in CL
by the complete block of Ih (Fig. 14,
middle). The effect of block of Ih on
CL was much greater in the Ist-removed system than in the standard system.
The simulated effect of elimination of Ih in our
standard model for the primary pacemaker cell (8.4% increase in CL) is
in good agreement with experimental data from central SA node tissues: Nikmaram et al. (83) reported that in central SA node
tissues, block of Ih by 2 mM Cs+
caused a 5~10% increase in CL (see also Ref. 130). In
other previous reports, the effects of block of ih on CL
were greater than in the report of Nikmaram et al. (83):
Denyer and Brown (18) and van Ginneken and Giles
(118) reported increases in CL of 19~30% and 57%,
respectively (see Table 3). Nikmaram et al. (83) also
reported a regional difference in the effect of Cs+ on CL:
the decrease in the spontaneous rate induced by 2 mM Cs+
was largest in the periphery (~19%) and least in the center
(~7%). Because Honjo et al. (44) have shown that the
density of Ih in the rabbit SA node was
significantly greater in larger cells from the periphery than in
smaller cells from the center, the different effects of block of
Ih on CL may reflect the regional difference in
Ih density. Furthermore, the degree of CL
prolongation on block of Ih strongly depends on
basic CL, MDP, and thus on densities of ionic currents other than
Ih itself; values of MDP and CL would vary
depending on both the experimental condition and area of the SA node
from which cells were isolated. As already mentioned, the effect of
elimination of Ih on CL was much greater in the
Ist-removed system with longer CL than in the
standard system with shorter CL. When Ist was
eliminated from the standard system (MDP =
63.64 mV), the
complete block of Ih increased CL by 43.9%
(from 470.5 to 677.0 ms). Thus the large differences in the effect of
block of Ih on CL (different contributions of Ih to pacemaker depolarization) reported
previously (18, 20, 83, 87, 118) would arise from the
regional variations in MDP and basic CL as well as in the density or
activation threshold of Ih.
Block of 4-AP-sensitive currents.
The effects of block of 4-AP-sensitive currents
(Ito and Isus) on AP
configurations are shown in Fig. 14, bottom. Complete block
of 4-AP-sensitive currents in the standard and
Ist-removed systems prolonged APD50
by 26.7 and 22.3%, respectively, and also caused positive shifts in
POP (from +16.6 to +32.9 mV in the standard system). On elimination of
both Ito and Isus, the CL
of the standard system increased by 5.1%, whereas that of the
Ist-removed system decreased by 7.2%. The
simulated increase in APD50 (22.3~26.7%) is in good
agreement with the experimental data of Boyett et al. (7):
block of 4-AP-sensitive currents by 5 mM 4-AP caused an AP duration
prolongation of 25 ± 5% in small balls from central SA node
tissues. They also reported a 5 ± 2% decrease in CL by 5 mM
4-AP, consistent with the behavior of our
Ist-removed system.
Effects of Modulating Intracellular
Ca2+ Dynamics on Pacemaker Activity
Effects of block of SR Ca2+ release.
SR Ca2+ release is known to affect intracellular
Ca2+ transients, Ca2+-dependent inactivation of
ICa,L, and activation of
INaCa, possibly playing an important role in
regulating pacemaker activity (39, 67, 104). We therefore
assessed the contributions of SR Ca2+ release to pacemaker
activities of the model cells by reducing the SR volume to 0.1% of the
control. As shown in Table 4, the influences of block of SR Ca2+ release on pacemaker
frequencies of the model cells were different but relatively small. In
our standard model for primary pacemaker cells, elimination of SR
Ca2+ release decreased the peak
[Ca2+]sub only by 22.4% (from 1.81 to 1.41 µM) and increased CL by only 3.4% (from 307.5 to 317.6 ms). Thus, in
our primary pacemaker model, SR Ca2+ release plays only a
minor role in generating subsarcolemmal Ca2+ transients and
regulating pacemaker activity (via modifications of the
Ca2+-dependent inactivation of ICa,L
and activation of INaCa). This simulated result
agrees well with the reports of Janvier and Boyett (52)
and Miyamae and Goto (79), in which application of
ryanodine to abolish SR Ca2+ release resulted in little or
no slowing of pacemaker activity in SA node cells. The
Ca2+-dependent inactivation of L-type Ca2+
channels in rat ventricular or atrial myocytes is mediated primarily by
Ca2+ release from the SR to the microdomain (5,
60). In our SA node model, however, the
Ca2+-dependent inactivation of L-type Ca2+
channels is chiefly mediated by Ca2+ influx through the
L-type Ca2+ channel itself. Consistent with our model
behavior, ryanodine little slowed the fast decay of
ICa,L in SA node cells (67), suggesting that SR Ca2+ release does not contribute to the
Ca2+-dependent inactivation of L-type Ca2+
channels in the SA node. Morphological studies indicate that the SR is
poorly formed in typical SA node cells and is therefore unlikely to
play a major role in modulation of pacemaker activity (see Ref.
17). As suggested by Janvier and Boyett (52),
the contributions of SR Ca2+ release to intracellular
Ca2+ transients and to pacemaker activity (via inactivation
of ICa,L or activation of
INaCa) would be relatively small in SA node
cells because of the poor development of the SR.
In some experimental reports (6, 39, 67, 104), the block
of SR Ca2+ release by ryanodine has been shown to exert a
negative chronotropic effect on rabbit SA node cells, suggesting the
important roles of SR Ca2+ release, intracellular
Ca2+ transients, and subsequent activation of
INaCa in regulating pacemaker activity of the SA
node. This inconsistency may be due to the regional difference in the
development of the SR between the central and peripheral SA node, i.e.,
poor development of the SR in primary pacemaker cells (see Refs.
47 and 96). Alternatively, ryanodine may
directly block sarcolemmal ionic currents, such as
ICa,T and INaCa, or
secondarily modify Ca2+-dependent currents including
ICa,L, IKr, and
Ih via affecting intracellular Ca2+
dynamics (see Refs. 6, 39, 67,
94, and 104). Recent experiments have shown
that the ICa,T-triggered focal Ca2+
release from the junctional SR to subspace stimulates
INaCa, thereby playing an important role in
regulating pacemaker activity (6, 47). Our model cannot
simulate such local events (i.e., Ca2+ sparks and waves)
due to the spatial heterogeneity of intracellular Ca2+
distribution; further compartmentalization for intracellular Ca2+ or the use of partial differential equations is
required for simulating these phenomena involved in pacemaker regulation.
Effects of Ca2+ buffers.
To buffer intracellular Ca2+, Ca2+ buffers such
as EGTA and BAPTA are routinely used for whole cell patch-clamp
experiments. These Ca2+ buffers to reduce intracellular
Ca2+ transients have been shown to affect the pacemaker
activity of SA node cells (67, 122). We therefore examined
the effects of the Ca2+ buffers BAPTA and EGTA on the
pacemaker activity of SA node models. Wilders et al. (124)
simulated EGTA buffering by fixing [Ca2+]i to
80 nM. However, it is known that the slow buffer EGTA cannot efficiently buffer free Ca2+ in the subsarcolemmal space:
although both BAPTA and EGTA can efficiently suppress global
(myoplasmic) Ca2+ transients, only the fast buffer BAPTA is
efficient in buffering local (subsarcolemmal) Ca2+
transients (see Refs. 5, 106,
122, and 128). Therefore, we calculated
intracellular Ca2+ dynamics in the presence of BAPTA or
EGTA to simulate the differential effects of Ca2+ buffers
on [Ca2+]sub and on spontaneous APs. As shown
in Fig. 15 and Table
5, the fast buffer BAPTA at 10 mM
remarkably reduced both the subsarcolemmal and myoplasmic
Ca2+ transients and dramatically slowed pacemaker activity
with CL increasing by 43.4% in our standard model. This negative
chronotropic effect is a consequence of the reduction in
INaCa during the late phase of pacemaker
depolarization. In contrast to BAPTA, the slow buffer EGTA at 10 mM
could not sufficiently inhibit the subsarcolemmal Ca2+
transient, whereas it almost completely suppressed the myoplasmic Ca2+ transient to ~0.1 µM; pacemaker frequency was
little affected by 10 mM EGTA. These differential effects of BAPTA and
EGTA on pacemaker frequency have been experimentally observed
(122): only BAPTA significantly reduced the rate of
spontaneous APs (by 54%), whereas EGTA did not (see Fig. 15). In
contrast to our model, previous models have failed to simulate the
differential responses of SA node cells to Ca2+ buffers:
both BAPTA and EGTA dramatically suppressed intracellular Ca2+ transients, exerting negative (or positive)
chronotropic effects (see Table 5). Thus only our model could reproduce
the differential responses of SA node cells to BAPTA and EGTA. Our
model incorporates two compartments for intracellular Ca2+,
including the subsarcolenmal space as a diffusion barrier for Ca2+, whereas previous models assumed only one
intracellular compartment. Subspace Ca2+ dynamics are
indispensable for simulating the differential effects of BAPTA and EGTA
on SA node dynamics. A recent experimental report suggests local high
Ca2+ gradients in the subsarcolemmal microdomain in SA node
cells (122).

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Fig. 15.
A: effects of Ca2+ buffers, BAPTA
and EGTA (10 mM), on spontaneous APs (top) and intracellular
Ca2+ concentrations (middle and
bottom) in our standard model. Simulating the whole cell
patch-clamp recording for internally perfused SA node cells,
[Na+]i and [K+]i
were fixed at 10 and 140 mM, respectively. The equations were
numerically integrated for 11 s with an initial condition of a
steady state at 60 mV, and membrane potential behavior and
intracellular Ca2+ dynamics during the last 1 s (from
MDP) were then depicted. Control APs and Ca2+ transients
are shown as thin lines. B: experimentally observed
responses of rabbit SA node cells to BAPTA and EGTA, the data from
Vinogradova et al. (122), are also shown for comparison.
See the Glossary for definitions of the abbreviations.
|
|
Achievements and Limitations of the Present Model
On the basis of recently published experimental data, we were able
to develop an improved SA node model incorporating 1) the novel pacemaker current Ist not included in
previous models, 2) new formulations for voltage- and
Ca2+-dependent inactivation kinetics of the L-type
Ca2+ channel, 3) new expressions for activation
kinetics of IKr, 4) revised kinetic
formulas for 4-AP-sensitive currents (Ito and Isus), 5) new formulations for
voltage- and concentration-dependent kinetics of
INaK, and 6) the subsarcolemmal space
as a diffusion barrier for Ca2+. The present model provides
well-integrated explanations of the electrophysiological behavior of
primary pacemaker cells in the rabbit SA node, representing significant
improvements over earlier models. As described above, our model can
1) simulate whole cell voltage-clamp data for
ICa,L, IKr, and
Ist; 2) reproduce the waveshapes of
spontaneous APs and sarcolemmal ionic currents
(ICa,L, IKr, ICa,T, and Ih) observed
during AP-clamp recordings; 3) reproduce the bifurcation
structures seen during applications of ICa,L or IKr blockers; 4) mimic the
reported effects of block of ICa,T, Ih, or 4-AP-sensitive currents on pacemaker
activity; and 5) simulate the differential effects of BAPTA
and EGTA on pacemaker frequency more accurately than previous SA node models.
Despite several improvements over previous models, the present model
still has some limitations. Quantitatively, our model predictions
(e.g., simulated changes in pacemaker frequency or resting potentials
during inhibition of IKr) exhibited some
inconsistencies with experimental data. These discrepancies may be due
to 1) incomplete experimental data on the kinetics (or
density) of ionic currents as well as on intracellular Ca2+
dynamics (and SR functions) in the SA node; 2) the large
heterogeneity of SA node cells (e.g., regional differences in current
densities); 3) poor selectivity of the agents used
experimentally to block ionic currents or state-dependent kinetics of
channel blockade; 4) the existence of some ionic current
components not included in our model (e.g., Ca2+-activated
Cl
current; see Ref. 92); 5) the
spatial heterogeneity of intracellular distributions of
Ca2+ and Na+ (see Ref. 13);
6) the lack of intracellular or intramembrane regulatory
systems in our model; and 7) inappropriate kinetic formulations based on the Hodgkin-Huxley formalism for ionic channel currents such as ICa,L and
IKr, which are essential to pacemaker generation
(see Refs. 14, 43, 50,
88, 115, and 127). These points
are of great importance in future modeling to develop more
sophisticated models.
It should be noted that one limitation in estimating the contribution
of a current to pacemaking, as well as in measuring a current by AP
clamp, is the need of a pharmacological block of the current (see Ref.
72). Bindings of channel blockers used for experiments are
known to be nonspecific and voltage (state) dependent (5, 70,
119, 129). Therefore, the currents recorded during AP clamp may
be different from the real pure current, rather reflecting the total
current blocked by an agent. As Verheijck et al. (119)
suggested, changes in AP parameters induced by a drug are most likely
the result of a combination of direct and indirect effects on various
ionic currents. An agent used to block a current may directly affect
other ionic currents, or, alternatively, secondarily affect ion channel
properties via changes in intracellular Ca2+ concentrations
and/or modifications of second messengers (see Refs. 9 and
129). Thus we should be cautious when simulated current
waveforms or blocking effects are compared with experimental data.
There are numerous factors involved in the regulation of ion channel
and SR functions in intact cardiac myocytes: intracellular second
messengers (e.g., Ca2+, cyclic nucleotides, and protein
kinases), as well as intramembrane modulators (e.g., receptors, G
proteins, and enzymes), have been shown to modify various ionic current
systems (for reviews, see Refs. 5, 9,
16, 25, and 91). Therefore,
incorporating the dynamics of these modulating factors would be
indispensable for predicting SA node behavior more accurately and might
at least in part solve the discrepancies between model predictions and experimental data. Tentative models including putative intracellular Ca2+-dependent changes in ionic currents or modifications
of channel functions by receptors and second messengers have recently
been published (9, 16, 25, 75). In this study, however, we did not incorporate these modulating factors, because the aim of this
study was not to develop a complete model but to develop an improved
model for investigating the dynamical mechanisms of pacemaker
generation. The present model could also serve as a base model for the
development of a complete model incorporating the intramembrane
modulators and intracellular second messengers and further the genetic
regulation of ion channels and transporters.
Despite many limitations, our model and simulations can provide the
guidance for future developments of more sophisticated single SA node
cell models, multicellular (two or three dimensional) models of the
intact SA node, and whole heart models.
 |
APPENDIX |
The mathematical expressions used in our SA node model are given
below. The units used are millivolts, picoamps, nanosiemens, milliseconds, nanofarads, millimolars, and liters. The temperature assumed for the computations was 37°C; the experimental data of gating kinetics obtained at <37°C were corrected for temperature with a Q10 = 1.6~3.0 (see THEORY AND
METHODS). The functions x
(V) and
x(V) for individual gating
variables and steady-state I-V relations for
P
individual current systems are plotted in Figs.
A1 and
A2, respectively; the model constants
(standard parameter values) are given in Table
A1.

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Fig. 16.
Computed gating characteristics of each ion channel.
Steady-state probabilities (top) and time constants
(middle and bottom) for activation and
inactivation are shown as functions of the membrane potential. Curves
are drawn according to the equations listed in the
APPENDIX. See the Glossary for definitions of
the abbreviations.
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|

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Fig. 17.
Voltage dependence of steady-state time-dependent
and -independent currents computed with
[Na+]i, [K+]i, and
[Ca2+]i (equal to
[Ca2+]sub) fixed at 10, 140, and 0.0001 mM,
respectively. See the Glossary for definitions of the
abbreviations. Ib,sum represents the sum of the
background and transporter currents (i.e.,
Ib,sum = Ib,Na + IK,Ach + INaK + INaCa). The total sarcolemmal membrane
current is denoted as Itotal.
|
|
Sarcolemmal Ionic Currents
L-type Ca2+ channel current.
T-type Ca2+ channel current.
Rapidly activating delayed rectifier
K+ current.
Slowly activating delayed rectifier
K+ current.
4-AP-sensitive currents.
Hyperpolarization-activated current.
Sustained inward current.
Na+ channel current.
Na+-dependent background current.
Background muscarinic K+ channel
current.
Na+-K+
pump current.
Na+/Ca2+
exchange current.
Intracellular Ca2+ Dynamics
Ca2+ diffusion flux.
Ca2+ handling by the SR.
State Variables and Differential Equations
Membrane potential.
Gating variables.
Intracellular ion concentrations.
Ca2+ buffering.
 |
ACKNOWLEDGEMENTS |
This work was supported in part by Ministry for Education,
Science, Sports and Culture of Japan Grant-in-Aid for Scientific Research (c) 11670717 (to S. Imanishi) and by Kanazawa Medical University Grant for Project Research P99-6 (to Y. Kurata).
 |
FOOTNOTES |
Address for reprint requests and other correspondence: Y. Kurata, Dept. of Physiology, Kanazawa Medical University, 1-1 Daigaku, Uchinada-machi, Kahoku-gun, Ishikawa 920-0293, Japan (E-mail: yasu{at}kanazawa-med.ac.jp).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
10.1152/ajpheart.00900.2001
Received 16 October 2001; accepted in final form 27 June 2002.
 |
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