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Am J Physiol Heart Circ Physiol 275: H301-H321, 1998;
0363-6135/98 $5.00
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Vol. 275, Issue 1, H301-H321, July 1998

Ionic mechanisms underlying human atrial action potential properties: insights from a mathematical model

Marc Courtemanche1,2, Rafael J. Ramirez1, and Stanley Nattel1,3,4

1 Research Center, Montreal Heart Institute, Montreal, Quebec H1T 1C8; Départements de 2 Physiologie and 4 Médecine, Université de Montréal, Montreal, Quebec H3C 3J7; and 3 Department of Pharmacology, McGill University, Montreal, Quebec H3G 1Y6, Canada

    ABSTRACT
Top
Abstract
Introduction
Results
Discussion
Appendix
References

The mechanisms underlying many important properties of the human atrial action potential (AP) are poorly understood. Using specific formulations of the K+, Na+, and Ca2+ currents based on data recorded from human atrial myocytes, along with representations of pump, exchange, and background currents, we developed a mathematical model of the AP. The model AP resembles APs recorded from human atrial samples and responds to rate changes, L-type Ca2+ current blockade, Na+/Ca2+ exchanger inhibition, and variations in transient outward current amplitude in a fashion similar to experimental recordings. Rate-dependent adaptation of AP duration, an important determinant of susceptibility to atrial fibrillation, was attributable to incomplete L-type Ca2+ current recovery from inactivation and incomplete delayed rectifier current deactivation at rapid rates. Experimental observations of variable AP morphology could be accounted for by changes in transient outward current density, as suggested experimentally. We conclude that this mathematical model of the human atrial AP reproduces a variety of observed AP behaviors and provides insights into the mechanisms of clinically important AP properties.

action potential morphology; action potential rate dependence; transient outward current; L-type calcium current; sodium/calcium exchanger

    INTRODUCTION
Top
Abstract
Introduction
Results
Discussion
Appendix
References

THE HUMAN ATRIAL ACTION POTENTIAL (AP) and ionic currents that underlie its morphology are of critical importance to our understanding of the electrical properties of atrial tissues in normal and pathological conditions. AP shape and underlying ionic current densities are frequency dependent and may be modulated by specific pharmacological agents (33). Their properties are known to change during the course of atrial arrhythmias or as a result of other cardiac pathologies (32, 39). In addition, APs have shown significant variability in morphology, ranging from triangular APs with no sustained plateaus to long APs with definite spike-and-dome morphology, even when recorded from cells isolated from the same atrial sample (4, 33, 54). There is a need to integrate the information obtained from measurements of single currents to understand complex current interactions during the AP and their role in controlling AP morphology.

Until recently, limited data were available on the various currents underlying the AP. New findings have revealed important properties and significant interspecies differences. For example, the transient outward current (Ito), present in human and rabbit atrial cells (8, 16, 25, 47), has been shown to recover from inactivation at least two orders of magnitude faster in humans than in rabbits (20, 47). A novel delayed rectifier current (IKur) has been identified in human atrial cells (1, 18, 41, 48, 53) and is different from the sustained current (Isus) observed in rabbit atrial cells (12). IKur also differs from delayed rectifiers in other species that show kinetically similar characteristics (3, 6, 29). Additional data have been obtained on the classic delayed rectifier currents (IKr and IKs) (54, 55), the time-independent inward rectifier current (IK1) (30, 31), the fast Na+ current (INa) (42), the L-type Ca2+ current (ICa,L) (35) and its inactivation by voltage and intracellular Ca2+ (26, 49), and the Na+/Ca2+ exchanger current (INaCa) (4).

Models of atrial cells based on animal data only have been published (14, 36, 40, 60). Although these models have provided valuable insights into the mechanisms underlying AP generation in these species, the existence of significant interspecies differences and the amount of human data available indicate a need for an AP model based specifically on direct measurements of human atrial currents. To address this issue, we developed a mathematical model of the AP based on ionic current data obtained directly in human atrial cells. When human data were insufficient to characterize a given atrial current fully, we supplemented with animal data and referred to existing published models of atrial and ventricular APs (14, 36, 37, 40, 60). The most recent published models of mammalian cardiac cells are those of Luo and Rudy (37) (LR2 model) and Linblad et al. (36) (LMCG model), based on measurements from guinea pig ventricular cells and rabbit atrial cells, respectively. Our model builds mostly on the work of Luo and Rudy to develop a working model of the human atrial AP.

Our primary goal was to develop a useful model of the AP from which we could gain insights into experimental observations made on human atrial cells and tissues and make predictions regarding the behavior of these cells under previously untested conditions. Throughout our description of the model, we clearly indicate the sources of data used to derive model parameters, how the parameters were obtained, and whether they were derived directly from data or modified to improve our representation of the AP. We then use the model to investigate the mechanism of experimentally measured AP rate dependence, the changes in AP morphology in the presence of pharmaceutical blockers of the ICa,L and the Na+/Ca2+ exchanger, and the variability in experimentally observed AP morphology.

Glossary

V Transmembrane potential
R Gas constant
T Temperature
F Faraday constant
Cm Membrane capacitance
AP Human atrial action potential
Vmax Maximal AP upstroke velocity (dV/dtmax)
APA AP amplitude (peak AP voltage minus diastolic voltage at onset of AP)
APO AP overshoot (peak AP voltage >0 mV)
APD AP duration
APD50 APD measured from AP onset to 50% of total APA repolarization
APD90 APD measured from AP onset to 90% of total APA repolarization
AF Atrial fibrillation
I-V Current-voltage
Q10 Temperature adjustment factor, k(T ) = k(T0)Q(T-T0)/1010

SR

Sarcoplasmic reticulum

JSR

Junctional SR, SR release compartment

NSR

Network SR, SR uptake compartment

Vcell

Cell volume

Vi

Intracellular volume

Vup

SR uptake compartment volume

Vrel

SR release compartment volume

[X ]o

Extracellular concentration of ion X

[X ]i

Intracellular concentration of ion X

Cmdn

Calmodulin, sarcoplasmic Ca2+ buffer

Trpn

Troponin, sarcoplasmic Ca2+ buffer

Csqn

Calsequestrin, JSR Ca2+ buffer

[Ca2+]rel

Ca2+ concentration in release compartment

[Ca2+]up

Ca2+ concentration in uptake compartment

[Ca2+]Cmdn

Ca2+-bound calmodulin concentration

[Ca2+]Trpn

Ca2+-bound troponin concentration

[Ca2+]Csqn

Ca2+-bound calsequestrin concentration

EX

Equilibrium potential for ion X

Iion

Total ionic current

Ist

Stimulus current

alpha x

Forward rate constant for gating variable x

beta x

Backward rate constant for gating variable x

tau x

Time constant for gating variable x

xinfinity

Steady-state relation for gating variable x

INa

Fast inward Na+ current

gNa

Maximal INa conductance

m

Activation gating variable for INa

h

Fast inactivation gating variable for INa

j

Slow inactivation gating variable for INa

IK1

Inward rectifier K+ current

gK1

Maximal IK1 conductance

Ito

Transient outward K+ current

gto

Maximal Ito conductance

KQ10

Q10-based temperature adjustment factor

oa

Activation gating variable for Ito

oi

Inactivation gating variable for Ito

IKur

Ultrarapid delayed rectifier K+ current

gKur

Maximal IKur conductance

ua

Activation gating variable for IKur

ui

Inactivation gating variable for IKur

IKr

Rapid delayed rectifier K+ current

gKr

Maximal IKr conductance

xr

Activation gating variable for IKur

IKs

Slow delayed rectifier K+ current

gKs

Maximal IKs conductance

xs

Activation gating variable for IKs

ICa,L

L-type inward Ca2+ current

gCa,L

Maximal ICa,L conductance

d

Activation gating variable for ICa,L

f

Voltage-dependent inactivation gating variable for ICa,L

fCa

Ca2+-dependent inactivation gating variable for ICa,L

Ip,Ca

Sarcoplasmic Ca2+ pump current

Ip,Ca(max)

Maximal Ip,Ca

INaK

Na+-K+ pump current

INaK(max)

Maximal INaK

fNaK

Voltage-dependence parameter for INaK

sigma

[Na+]o-dependence parameter for INaK

Km,Na(i)

[Na+]i half-saturation constant for INaK

Km,K(o)

[K+]o half-saturation constant for INaK

INaCa

Na+/Ca2+ exchanger current

INaCa(max)

INaCa scaling factor

Km,Na

[Na+]o saturation constant for INaCa

Km,Ca

[Ca2+]o saturation constant for INaCa

ksat

Saturation factor for INaCa

gamma

Voltage-dependence parameter for INaCa

Ib,Na

Background Na+ current

gb,Na

Maximal Ib,Na conductance

Ib,Ca

Background Ca2+ current

gb,Ca

Maximal Ib,Ca conductance

Irel

Ca2+ release current from the JSR

krel

Maximal Ca2+ release rate for Irel

u

Activation gating variable for Irel

v

Ca2+ flux-dependent inactivation gating variable for Irel

w

Voltage-dependent inactivation gating variable for Irel

Fn

Sarcoplasmic Ca2+ flux signal for Irel

Iup

Ca2+ uptake current into the NSR

Iup(max)

Maximal Ca2+ uptake rate for Iup

[Ca2+]up(max)

Maximal Ca2+ concentration in NSR

Itr

Ca2+ transfer current from NSR to JSR

tau tr

Ca2+ transfer time constant

Iup,leak

Ca2+ leak current from the NSR

[Cmdn]max

Total calmodulin concentration in myoplasm

[Trpn]max

Total troponin concentration in myoplasm

[Csqn]max

Total calsequestrin concentration in JSR

Km,Cmdn

Ca2+ half-saturation constant for calmodulin

Km,Trpn

Ca2+ half-saturation constant for troponin

Km,Csqn

Ca2+ half-saturation constant for calsequestrin

Vrest

Resting membrane potential

    MODEL DESCRIPTION

General

We model the cell membrane as a capacitor connected in parallel with variable resistances and batteries representing the ionic channels and driving forces. The time derivative of the membrane potential V (with the assumption of an equipotential cell) is given by
 <FR><NU>d<IT>V</IT></NU><DE>d<IT>t</IT></DE></FR> = <FR><NU>−(<IT>I</IT><SUB>ion</SUB> + <IT>I</IT><SUB>st</SUB>)</NU><DE><IT>C</IT><SUB>m</SUB></DE></FR> (1)
where Iion and Ist are the total ionic current and stimulus current flowing across the membrane and Cm is the total membrane capacitance. The total ionic current is given by
<IT>I</IT><SUB>ion</SUB> = <IT>I</IT><SUB>Na</SUB> + <IT>I</IT><SUB>K1</SUB> + <IT>I</IT><SUB>to</SUB> + <IT>I</IT><SUB>Kur</SUB> + <IT>I</IT><SUB>Kr</SUB> + <IT>I</IT><SUB>Ks</SUB> + <IT>I</IT><SUB>Ca,L</SUB>
 + <IT>I</IT><SUB>p,Ca</SUB> + <IT>I</IT><SUB>NaK</SUB> + <IT>I</IT><SUB>NaCa</SUB> + <IT>I</IT><SUB>b,Na</SUB> + <IT>I</IT><SUB>b,Ca</SUB> (2)
The ionic and pump currents, along with the handling of intracellular Ca2+ concentration ([Ca2+]i) by the sarcoplasmic reticulum (SR) system, are described in Membrane Currents. The model keeps track of the [Ca2+]i as well as intracellular concentrations of Na+ ([Na+]i) and K+ ([K+]i). No extracellular cleft spaces are included in the model (extracellular ion concentrations are fixed). Figure 1 shows a schematic representation of the currents, pumps, and exchangers included in the model. Cm is taken to be 100 pF. The length and diameter of the cell are set to 100 and 16 µm, respectively. Unless otherwise noted, physical units are as follows: time (t) is in milliseconds, V is in millivolts, Cm is in picofarads, current density is in picoamperes per picofarad, conductance is in nanosiemens per picofarad, and concentrations are in millimoles per liter. Physical parameters for the model cell are given in Table 1. Cell compartment volumes are identical to those used by Luo and Rudy (37).


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Fig. 1.   Schematic representation of currents, pumps, and exchangers included in model. Cell includes 3 intracellular compartments: myoplasm, sarcoplasmic reticulum (SR) release compartment [junctional SR (JSR)], and SR uptake compartment [network SR (NSR)]. See Glossary for definitions.

                              
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Table 1.   Model constants

A modified Euler method is used to integrate Eq. 1 with a fixed time step Delta t = 0.005 ms (see APPENDIX, Numerical Integration). All simulations were performed on a local workstation (SGI Indigo 2XZ R4400) using double-precision arithmetic. A description of the model current formulations and their general mathematical representations are given below. A detailed listing of all equations and parameters can be found in the APPENDIX.

Membrane Currents

Fast Na+ current. The model implements INa as a modification of the widely used Ebihara-Johnson model (15) proposed by Luo and Rudy (37). The current is given by
<IT>I</IT><SUB>Na</SUB> = <IT>g</IT><SUB>Na</SUB><IT>m</IT><SUP>3</SUP><IT>hj</IT>(<IT>V</IT> − <IT>E</IT><SUB>Na</SUB>),  <IT>g</IT><SUB>Na</SUB> = 7.8 (3)
Using the formulation of Luo and Rudy, we find that the model steady-state inactivation and its time constants are shifted 20 mV positive compared with the data of Sakakibara et al. (42). However, the formulation for steady-state inactivation is consistent with more recent human atrial steady-state inactivation data from Schneider et al. (46) (see DISCUSSION for a detailed evaluation of available human atrial data). Maximum Na+ conductance (gNa) was adjusted to produce an appropriate value for the AP amplitude (APA) and maximal upstroke velocity (Vmax). We use gNa = 7.8 nS/pF, which is ~1.3 times the temperature-adjusted value (with the assumption of Q10 = 3) reported by Sakakibara et al. Figure 2 shows the steady-state activation and inactivation curves, as well as time constants, used in the model. Data from Sakakibara et al., shifted by the amounts described above and temperature corrected, are overlaid on the model curves. For activation time constants, temperature-adjusted data from Schneider et al. are shown. The peak current-voltage (I-V ) relationship for INa is displayed in Fig. 3. No experimental I-V curve was included, because available data (19, 42, 46), as a result of equal intra- and extracellular Na+ concentrations, display a nonphysiological reversal potential for INa close to 0 mV.


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Fig. 2.   Model representation of parameters describing fast Na+ current. Steady-state activation and inactivation curves (A) and gating variable time constants (B) are shown. Experimental data from Sakakibara et al. (42) (minfinity , hinfinity jinfinity , tau h, tau j) and Schneider et al. (46) (tau m), voltage shifted (+20 mV) and temperature corrected (Q10 = 3), are included for comparison. Experimental steady-state relationships are lines fitted by Sakakibara et al. to their experimental data.


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Fig. 3.   Current-voltage (I-V) relationships of major currents in model with corresponding experimental data. Model curves (solid lines, dashed line without symbols for INa) represent peak current measured during 3,000-ms voltage steps to various potentials from a holding potential of -80 mV. Experimental data sets (dashed lines with symbols) are scaled so that maximum current for a given experimental I-V curve matches current amplitude of corresponding model I-V curve at that potential. Data sources and scaling factors are as follows: Li and Nattel (35) for ICa,L scaled by 0.38, Feng et al. (19) for IKur scaled by 0.81 and Ito scaled by 1.2, Wang et al. (55) for IKs scaled by 3.0 and IKr scaled by 0.38. Refer to original studies for exact experimental voltage-clamp protocols used. Experimental I-V curves for IKur and Ito were shifted 10 mV negative before they were scaled, matching voltage shift introduced in model to account for known effects of divalent cations used experimentally to inhibit ICa,L (13, 28). Experimental I-V data for IK (55) were transformed to pA/pF with use of Cm = 100 pF before scaling was performed. No experimental data are shown for INa.

Inward rectifier K+ current. The model implements IK1 on the basis of available current data and resting membrane resistance measurements. The current is given by
<IT>I</IT><SUB>K1</SUB> = <FR><NU><IT>g</IT><SUB>K1</SUB>(<IT>V</IT> − <IT>E</IT><SUB>K</SUB>)</NU><DE>1 + exp [0.07(<IT>V</IT> + 80)]</DE></FR> ,  <IT>g</IT><SUB>K1</SUB> = 0.09 (4)
We assume no temperature dependence of the inward rectifier current on the basis of preliminary experimental data. The current is assumed to reverse at EK, with a voltage-dependent conductance and no additional dependence on outward K+ concentration ([K+]o). Koumi et al. (30, 31) reported measurements of IK1 from human atrial cells. When incorporated in the model, their large conductance value of 0.25 nS/pF at EK yielded a low input membrane resistance (~40 MOmega for a 100-pF cell). We decreased IK1 conductance to compensate for this using gK1 = 0.09 nS/pF. When the latter is used, input resistance calculated for a voltage step from -80 to -90 mV is ~174 MOmega , closer to experimentally recorded values (e.g., ~150 MOmega in Ref. 52). However, there is a large range of measured values for the input resistance of human atrial cells that may reflect differences in the isolation procedure and its effect on isolation-sensitive currents such as IK1. The I-V relationship of IK1 is shown in Fig. 4.


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Fig. 4.   I-V relationships for IK1 and INaK obtained from model. Sum of instantaneous time-independent membrane currents (Iv) is also shown. For INaK, the amplitude of which depends on Na+ and K+ concentrations, resting concentrations from Table 2 are used.

Transient outward and ultrarapid rectifier K+ currents. The model implements Ito and IKur on the basis of data from our laboratory (20, 53). The currents are given by
<IT>I</IT><SUB>to</SUB> = <IT>g</IT><SUB>to</SUB><IT>o</IT><SUP>3</SUP><SUB>a</SUB><IT>o</IT><SUB>i</SUB>(<IT>V</IT> − <IT>E</IT><SUB>K</SUB>),  <IT>g</IT><SUB>to</SUB> = 0.1652 (5)
<IT>I</IT><SUB>Kur</SUB> = <IT>g</IT><SUB>Kur</SUB><IT>u</IT><SUP>3</SUP><SUB>a</SUB><IT>u</IT><SUB>i</SUB>(<IT>V</IT> − <IT>E</IT><SUB>K</SUB>),
<IT>g</IT><SUB>Kur</SUB> = 0.005 + <FR><NU>0.05</NU><DE>1 + exp <FENCE><FR><NU>(<IT>V</IT> − 15)</NU><DE>−13</DE></FR></FENCE></DE></FR> (6)
Both currents share similar activation kinetics but differ in their inactivation properties. Ito has fairly rapid inactivation kinetics; IKur displays only partial slow inactivation. Currents were modeled at room temperature, and Q10 = 2.2 was used to adjust their kinetics to 37°C (53). Current amplitudes were adjusted to match AP morphology. Figure 5 shows steady-state activation, inactivation, and corresponding time constants for IKur, along with experimental data from our laboratory (53). Figure 5 shows similar results for Ito. All model fits shown in Fig. 5 are shifted 10 mV negative for computations to account for the known positive shift (13, 28) induced by divalent cations such as Cd2+, which was used experimentally to block ICa,L. The peak I-V relationship of both currents, along with scaled experimental I-V curves from Feng et al. (19), are shown in Fig. 3. Figure 6 displays traces for IKur and Ito in response to voltage-clamp steps from a holding potential of -80 mV. The traces of Ito + IKur show typical rapid inactivation of Ito followed by a sustained outward current carried by IKur. The combined model current traces are overlaid with scaled experimental recordings from Wang et al. (53).


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Fig. 5.   Model representation of parameters describing ultrarapid delayed rectifier current (A and B) and transient outward current (C and D). Steady-state activation and inactivation curves (A and C) and gating variable time constants (B and D) are shown. Experimental data from Wang et al. (53), to which model relations are fitted, are included for comparison.


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Fig. 6.   IKur and Ito generated by model in response to voltage-clamp steps. Results are shown for combined currents (A), Ito alone (B), and IKur alone (C). From resting state, model is held at -80 mV for 1,000 ms, followed by potential steps to -20, 0, and +20 mV. Experimental recordings of IKur + Ito, from Wang et al. (55), are included for comparison (dashed lines in A). Experimental recordings were scaled using Cm = 100 pF to match peak amplitude of model trace at +20 mV.

Rapid and slow delayed rectifier K+ currents. The model implements the two components of the classic delayed rectifier, IKr and IKs, on the basis of data from our laboratory (54, 55). The currents are given by
<IT>I</IT><SUB>Kr</SUB> = <FR><NU><IT>g</IT><SUB>Kr</SUB><IT>x</IT><SUB>r</SUB>(<IT>V</IT> − <IT>E</IT><SUB>K</SUB>)</NU><DE>1 + exp <FENCE><FR><NU><IT>V</IT> + 15</NU><DE>22.4</DE></FR></FENCE></DE></FR> ,  <IT>g</IT><SUB>Kr</SUB> = 0.0294 (7)
<IT>I</IT><SUB>Ks</SUB> = <IT>g</IT><SUB>Ks</SUB><IT>x</IT><SUP>2</SUP><SUB>s</SUB>(<IT>V</IT> − <IT>E</IT><SUB>K</SUB>),  <IT>g</IT><SUB>Ks</SUB> = 0.129 (8)
Both currents have only a single activation gate, although IKs activation uses a squared (x2) activation gate, as suggested in previous studies (61). Partial inactivation of IKr is observed in some experiments (55) but is not included in this model. Figure 7 shows the steady-state activation curves and time constants for both delayed rectifier currents as well as experimentally measured time constants. The steady-state activation parameters are taken directly from fits to experimental data presented by Wang et al. (55). Some experimental evidence (27, 43, 45) suggests that activation and deactivation of IKs may be multiexponential. To account for these measurements and obtain results consistent with the overall experimental literature, the model computations use time constants for IKs activation that are one-half the fitted values shown in Fig. 7. In addition, observations of slow deactivation of the delayed rectifier at negative potentials (27, 43) were considered by using asymmetric relationships to represent the dependence of tau x(r) and tau x(s) on V. Current amplitudes were adjusted to match AP morphology. The I-V relationships of both currents, along with scaled experimental I-V curves from Wang et al., are shown in Fig. 3. Figure 8 displays traces for IKr and IKs in response to 3,000-ms voltage-clamp steps from a holding potential of -80 mV. The model traces are overlaid with scaled experimental recordings from Wang et al. The combined model IK trace shows rapid and slow activation phases linked to activation of IKr and IKs, respectively.


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Fig. 7.   Model representation of parameters describing rapid and slow delayed rectifier currents. A: steady-state activation curves; B: gating variable time constants. Experimental data from Wang et al. (55), to which model relations are fitted, are included for comparison (open symbols in B).


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Fig. 8.   IKr and IKs generated by model in response to voltage-clamp steps. Results are shown for combined currents (A), IKs alone (B), and IKr alone (C). From resting state, model is held at -80 mV for 1,000 ms, followed by potential steps to -20, 0, and +20 mV. Experimental recordings of IKr and IKs, from Wang et al. (55), are included for comparison (dashed lines in B and C). With use of Cm = 100 pF to obtain units of pA/pF, experimental recordings were further scaled by a factor of 3.5 (IKs) and 0.47 (IKr) to match peak amplitude of model currents at +20 mV.

L-type Ca2+ current. The model implements the slow inward Ca2+ current on the basis of previous models and data from human atrial cells (35, 37, 40, 49). The current is given by
<IT>I</IT><SUB>Ca,L</SUB> = <IT>g</IT><SUB>Ca,L</SUB><IT>d f f</IT><SUB>Ca</SUB>(<IT>V</IT> − 65.0),  <IT>g</IT><SUB>Ca,L</SUB> = 0.1238 (9)
Our formulation follows that of Luo and Rudy (37) and includes voltage-dependent activation and inactivation gates, as well as a Ca2+-dependent inactivation gate. The steady-state activation relation was fit to human atrial data from Li and Nattel (35) (Fig. 9) and hence differs slightly from that of Luo and Rudy and Rasmusson et al. (40) by providing a better fit to the slower onset of activation observed in the experimental data. Because of the absence of activation time constant measurements from human atrial myocytes, we retained the activation time constant formulation reported previously (36, 37, 40). Steady-state voltage-dependent inactivation data from Li and Nattel were accurately fit by the relation of Rasmusson et al. (40) (Fig. 9). Hence, we retained their formulation in our model. Our formulation of steady-state Ca2+-dependent inactivation is slightly modified from that of Luo and Rudy to better match the Ca2+ transient and reproduce the extent of Ca2+-dependent inactivation of ICa,L observed experimentally. A fixed intrinsic time dependence (tau f,Ca = 2 ms) is incorporated into the fCa gate, as reported by Friedman et al. (21, 22) (see SR Ca2+ storage and release), to better reproduce the time course of Ca2+-dependent inactivation of ICa,L.


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Fig. 9.   Model representation of parameters describing ICa,L. A: steady-state activation curves; B: gating variable time constants. Experimental data from Li and Nattel (35), to which steady-state relationships are fitted, are included for comparison (open symbols in A).

In experiments on human atrial cells, Sun et al. (49) showed two fast Ca2+-dependent inactivation time constants for ICa,L, on the order of 10-40 and 300-400 ms, respectively, at room temperature. In addition, when monovalent cations or Sr2+ was used as charge carrier, a very slow voltage-dependent process was apparent with inactivation time constants in the range of 600-800 ms at room temperature. Our ability to reproduce these results is dependent on our formulation of the current, which implements two inactivation mechanisms for ICa,L. We have selected the time constant for voltage-dependent inactivation in our model to match qualitatively the measurements of Sun et al. The voltage dependence of the time constant uses the form proposed by Luo and Rudy (37), but the magnitude of the time constant is scaled to better agree with the data of Sun et al. The time constant at 0 mV is ~240 ms in the model, which is within the experimentally measured range with the assumption of a threefold temperature correction to 37°C. Ca2+-dependent inactivation is mediated directly through the fCa gate. We rely on the Ca2+ transient to drive Ca2+-dependent inactivation, with its time constants being set indirectly through the dynamics of Ca2+ release and reuptake. This provides rapid inactivation of ICa,L on a time scale on the order of 10 ms. The model does not account for the existence of a slower Ca2+-dependent inactivation, as identified by Sun et al. (see DISCUSSION). The I-V relationships for ICa,L, along with a scaled experimental I-V curve from Li and Nattel (35), are shown in Fig. 3. Figure 10A displays model traces for ICa,L in response to voltage-clamp steps from a holding potential of -80 mV. The current traces exhibit rapid activation, followed by two distinct inactivation time scales. Rapid inactivation mediated by intracellular Ca2+ inactivates a large fraction of the current soon after the onset of the voltage step and is followed by a slower voltage-dependent inactivation. The model response after a voltage-clamp step to 0 mV is compared with a scaled averaged experimental recording from Sun et al. in Fig. 10B.


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Fig. 10.   A: ICa,L generated by model in response to voltage-clamp steps. From resting state, model is held at -80 mV for 1,000 ms, followed by potential steps to -40 to +20 mV. B: model response at a voltage-clamp potential of +20 mV, overlaid with experimental data from Sun et al. (49). Experimental data represent an average experimental trace from 7 voltage-clamp current recordings to +20 mV. Peak amplitude of normalized recording was scaled to match model peak amplitude at +20 mV.

Na+-K+ pump. Our formulation of the Na+-K+ pump follows the model of Luo and Rudy (37). The amplitude of the pump current [INaK(max)] is adjusted to maintain stable intracellular ion concentrations at rest (37, 59). The pump current is given by
<IT>I</IT><SUB>NaK</SUB> = <IT>I</IT><SUB>NaK(max)</SUB><IT>f</IT><SUB>NaK</SUB> <FR><NU>1</NU><DE>1 + {<IT>K</IT><SUB>m,Na(i)</SUB>/[Na<SUP>+</SUP>]<SUB>i</SUB>}<SUP>1.5</SUP></DE></FR>
· <FR><NU>[K<SUP>+</SUP>]<SUB>o</SUB></NU><DE>[K<SUP>+</SUP>]<SUB>o</SUB> + <IT>K</IT><SUB>m,K(o)</SUB></DE></FR> ,  <IT>I</IT><SUB>NaK(max)</SUB> = 0.6  (10)
The voltage dependence of INaK at resting [Na+]i ([K+]o is fixed at 5.4 mM in the model) is shown in Fig. 4.

Na+/Ca2+ exchanger. The exchanger current follows the revised formulation of Luo and Rudy (37). The exchanger current is given by

<IT>I</IT><SUB>NaCa</SUB> = <FR><NU><IT>I</IT><SUB>NaCa(max)</SUB>{exp [&ggr;<IT>VF</IT> /(<IT>RT</IT>)][Na<SUP>+</SUP>]<SUP>3</SUP><SUB>i</SUB>[Ca<SUP>2+</SUP>]<SUB>o</SUB> − exp [(&ggr; − 1)<IT>VF</IT> /(<IT>RT</IT>)][Na<SUP>+</SUP>]<SUP>3</SUP><SUB>o</SUB>[Ca<SUP>2+</SUP>]<SUB>i</SUB>}</NU><DE>(<IT>K</IT><SUP>3</SUP><SUB>m,Na</SUB> + [Na<SUP>+</SUP>]<SUP>3</SUP><SUB>o</SUB>)(<IT>K</IT><SUB>m,Ca</SUB> + [Ca<SUP>2+</SUP>]<SUB>o</SUB>) · {1 + <IT>k</IT><SUB>sat</SUB> exp [(&ggr; − 1)<IT>VF</IT> /(<IT>RT</IT>)]}</DE></FR> , <IT>I</IT><SUB>NaCa(max)</SUB> = 1600.0 (11)

As with INaK, the amplitude is adjusted to maintain stable physiological intracellular ion concentrations at rest (Table 2). Figure 11 shows the voltage dependence of the exchanger current at resting [Na+]i for two values of [Ca2+]i: at rest ([Ca2+]i = 0.1 µM) and near peak Ca2+ release concentration ([Ca2+]i = 1 µM). The downward shift in the curve is associated with a shift from Ca2+ entry to Ca2+ extrusion mode at depolarized potentials as Ca2+ concentration increases.

                              
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Table 2.   State variables of the AP model at rest


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Fig. 11.   I-V relationships for INaCa obtained from model. Current amplitude depends on Na+ and Ca2+ concentrations. Resting concentration shown in Table 2 is used for [Na+]i. Two different Ca2+ concentrations are used to illustrate shift from reverse to forward exchanger modes at plateau potentials in response to an increase in intracellular Ca2+.

Background Ca2+ and Na+ currents. The Ca2+ leak current is adjusted to maintain a stable [Ca2+]i at rest. The Na+ leak current is adjusted along with INaK to maintain stable [Na+]i and [K+]i at rest. Both approaches have been used extensively in previous AP models (37, 59). The background currents are given by
<IT>I</IT><SUB>b,Ca</SUB> = <IT>g</IT><SUB>b,Ca</SUB>(<IT>V</IT> − <IT>E</IT><SUB>Ca</SUB>),  <IT>g</IT><SUB>b,Ca</SUB> = 0.00113 (12)
<IT>I</IT><SUB>b,Na</SUB> = <IT>g</IT><SUB>b,Na</SUB>(<IT>V</IT> − <IT>E</IT><SUB>Na</SUB>),  <IT>g</IT><SUB>b,Na</SUB> = 0.000674 (13)

Ca2+ pump current. A sarcolemmal Ca2+ pump is included in the model to maintain [Ca2+]i at physiological levels. The pump current formulation (37, 40) is given by
<IT>I</IT><SUB>p,Ca</SUB> = <IT>I</IT><SUB>p,Ca(max)</SUB> <FR><NU>[Ca<SUP>2+</SUP>]<SUB>i</SUB></NU><DE>0.0005 + [Ca<SUP>2+</SUP>]<SUB>i</SUB></DE></FR> ,
<IT>I</IT><SUB>p,Ca(max)</SUB> = 0.275  (14)
Figure 4 shows the total time-independent I-V relationship for the model, which includes the background currents, IK1, and the pump and exchanger currents.

SR Ca2+ storage and release. SR Ca2+ uptake and release are implemented using a two-compartment model, as reported by Luo and Rudy (37). Intracellular Ca2+ is taken up into an SR uptake compartment [network SR (NSR), 5.52% of cell volume] coupled to a release compartment [junctional SR (JSR), 0.48% of cell volume]. Uptake is [Ca2+]i dependent and includes a leak from the uptake compartment back into the intracellular space. The main currents (in mM/ms) involved in SR Ca2+ handling are given by
<IT>I</IT><SUB>rel</SUB> = <IT>k</IT><SUB>rel</SUB><IT>u</IT><SUP>2</SUP><IT>vw</IT>([Ca<SUP>2+</SUP>]<SUB>rel</SUB> − [Ca<SUP>2+</SUP>]<SUB>i</SUB>),  <IT>k</IT><SUB>rel</SUB> = 30 (15)
<IT>I</IT><SUB>tr</SUB> = <FR><NU>[Ca<SUP>2+</SUP>]<SUB>up</SUB> − [Ca<SUP>2+</SUP>]<SUB>rel</SUB></NU><DE>&tgr;<SUB>tr</SUB></DE></FR> ,  &tgr;<SUB>tr</SUB> = 180 (16)
<IT>I</IT><SUB>up</SUB> = <FR><NU><IT>I</IT><SUB>up(max)</SUB></NU><DE>1 + (<IT>K</IT><SUB>up</SUB>/[Ca<SUP>2+</SUP>]<SUB>i</SUB>)</DE></FR> ,  <IT>I</IT><SUB>up(max)</SUB> = 0.005 (17)

Triggering of Ca2+ release from the JSR is modified from the original formulation of Luo and Rudy (37) following the work of Friedman et al. (21, 22). Release from the JSR is triggered by Ca2+ flux into the cell with emphasis placed on flux through the closely coupled ICa,L and JSR release channels. Release is controlled by two Ca2+ flux-dependent activation and inactivation gates, u and v. Steady-state activation and inactivation curves for u and v, as well as their respective time constants, are shown in Fig. 12. This trigger mechanism results in a transient activation of the SR Ca2+ release current in response to a suprathreshold Ca2+ flux. Friedman (21) showed that, for AP simulations, this formulation is equivalent to using a threshold on [Ca2+]i entry, as done by Luo and Rudy (37). The formulation used here has the advantage of being a continuous analytic representation, thereby avoiding possible discontinuities and artifacts in the cellular response near the threshold for Ca2+ release (21). The voltage-dependent inactivation gate w contributes to the decrease in SR release current amplitude at positive potentials, when membrane voltage approaches the ICa,L reversal potential. The steady-state voltage-dependent inactivation curve for w and its time constant are also shown in Fig. 12. Detailed expressions for the flux computations and gating variables are given in the APPENDIX. In a recent experimental study on human atrial myocytes, Hatem et al. (26) recorded Ca2+ transients in response to 150-ms voltage-clamp pulses to 0 mV applied at frequencies of 0.1 and 1 Hz. Figure 13 shows a comparison of experimental data from Hatem et al. and model-generated Ca2+ transients elicited using the same voltage-clamp protocol. A detailed comparison of model and experimental Ca2+ transient morphologies can be found in the DISCUSSION.


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Fig. 12.   Model representation of parameters describing Ca2+-induced Ca2+ release. Steady-state activation and inactivation curves (A) and corresponding time constants (B) for Ca2+ flux-dependent gating variables u and v are shown. C: steady-state inactivation curve and corresponding inactivation time constant for voltage-dependent gating variable w.


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Fig. 13.   Comparison of model-generated (solid lines) and experimentally measured (dashed lines) Ca2+ transients in response to 150-ms voltage-clamp pulses to 0 mV applied at a frequency of 0.1 and 1 Hz. Experimental data are based on published measurements of Hatem et al. (26) in human atrial myocytes.

Myoplasmic and SR Ca2+ buffers. Ca2+ buffering within the cytoplasm is mediated by troponin and calmodulin. Ca2+ buffering in the release compartment is mediated by calsequestrin. In particular, calsequestrin is essential in providing sufficient Ca2+ stores for release within the limited JSR volume. Buffers are considered at equilibrium throughout our simulations, and incoming Ca2+ in the myoplasm and JSR compartments is instantaneously equilibrated into free and buffer-bound fractions (see APPENDIX for computational methods). Buffer concentrations and binding constants are given in Table 1 and follow the models of Luo and Rudy (37) and Rasmusson et al. (40).

    RESULTS
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Abstract
Introduction
Results
Discussion
Appendix
References

Resting Properties

The model has a stable resting potential near -81 mV. All intracellular concentrations are stable at rest with [Ca2+]i = 0.1 µM, [Na+]i = 11.2 mM, and [K+]i = 139.0 mM. As stated earlier, the resting input resistance measured as the current change from -80 to -90 mV is ~174 MOmega . The first 100-ms period of Fig. 14 shows the balance of membrane currents in diastole during stimulation at 1,000 ms. During the diastolic phase the steady state involves a balance between the pump and exchanger currents, the background currents, and IK1.


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Fig. 14.   Model action potential during stimulation at 1 Hz. Membrane potential and corresponding ionic currents are shown. Model was stimulated from rest at a period of 1,000 ms. Model output from 12th action potential is depicted.

Model AP

Figure 14 shows a model AP generated during stimulation at 1,000 ms in response to 2-ms pulses of 2-nA amplitude (twice diastolic threshold). The time course of membrane currents during the AP is also shown in Fig. 14. AP duration (APD) at 50 and 90% repolarization (APD50 and APD90), APA, AP overshoot (APO), and Vmax are reported in Table 3. The AP generated with the control parameters exhibits a spike-and-dome morphology commonly observed in human atrial AP recordings. This morphology is similar to experimentally recorded spike-and-dome APs reported by Benardeau et al. (4) and Wang et al. (53). Intracellular Ca2+ dynamics during the AP are shown in Fig. 15.

                              
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Table 3.   AP characteristics and their rate dependence


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Fig. 15.   Model action potential and intracellular Ca2+ dynamics during stimulation at 1 Hz. Membrane potential, intracellular Ca2+ flux (trigger for Ca2+ release), intracellular, uptake, and release compartment Ca2+ concentrations, and uptake, transfer, and release Ca2+ fluxes are shown. Model was stimulated from rest at a frequency of 1 Hz. Model output from 12th action potential is depicted.

AP Rate Dependence

Rate dependence of APD and AP refractoriness is an essential property of atrial cells that is central to our understanding of excitability and propagation patterns in the atria, particularly during reentrant arrhythmias. Table 3 summarizes the main properties of the model APs at various pacing periods. Figure 16 displays model APs, after 12 s of pacing, for various stimulation periods ranging from 1,000 to 300 ms. There is no significant increase in APD at >1,000 ms. This is in agreement with data from Fermini et al. (20) that showed no change in AP morphology and APD on changing the stimulation period abruptly from 10 to 1 s. Larger relative rate-dependent changes in APD are measured with APD50 than with APD90, as in experimental preparations (20, 32). Figure 16 also shows APs obtained in a tissue preparation at stimulation periods of 1,000, 600, and 300 ms (56). The experimentally observed changes in APD and AP morphology with stimulation rate are qualitatively similar to those observed in the model.


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Fig. 16.   Rate dependence of action potential duration and morphology. Top left: model action potentials recorded after 12 s of pacing at various basic cycle lengths (BCLs). Action potential duration decreases as BCL decreases from 1 to 0.3 s. Top right: experimental recordings of action potentials obtained at pacing periods of 1, 0.6, and 0.3 s (data from Ref. 56). Bottom panels: currents obtained during model action potentials at BCLs of 1 and 0.3 s. Currents during 2 consecutive action potentials are shown at BCL of 0.3 s to facilitate comparison to currents at BCL of 1 s.

The mechanism underlying APD changes may be investigated by examining the detailed current traces for APs at stimulation periods of 1,000 and 300 ms depicted in Fig. 16. The traces reveal that an increase in stimulation rate conspicuously reduces ICa,L. We inspected the time course of gating variables for ICa,L, IKr, and IKs to quantify their role in rate-dependent AP changes. We first consider a comparison of the slow gating variables f, fCa, xr, and x2s at the onset of the AP for stimulation periods of 1,000 and 300 ms. These slow variables are most susceptible to alteration at fast rates. For IKr and IKs, we found that incomplete deactivation at 300 ms resulted in an increase of 0.05 for xr and 0.004 for x2s compared with their values at 1,000 ms. This compares with reductions of 0.2 for f and 0.16 for fCa. The maximal conductances of ICa,L and IKs are comparable, whereas the maximal conductance of IKr is four times smaller. Hence, the changes in the f and fCa gate result in larger changes in current at the faster rate, pointing to ICa,L as an important determinant of AP rate dependence. However, the absolute magnitude of current changes with rate must be considered relative to the amplitude of other currents activating at a similar stage during the AP. For this reason, a small rate-dependent change in IK amplitude may have significant effects during late repolarization, when IK is the dominant repolarizing current.

To better ascertain the relative importance of IK and ICa,L in AP rate adaptation, we use a simulation protocol where the rate-induced changes in a specific current are identified and implemented individually to determine its specific contributions to AP rate dependence. We begin with a simulation of pacing at 1,000 ms that is interrupted just before the onset of a stimulus. Before continuation of the simulation and application of the scheduled stimulus, the values of the gating variables for the current under study are substituted with their values just before a stimulus during pacing at 300 ms. When the simulation is continued and the scheduled stimulus is applied, the resulting AP displays the rate-adaptation properties associated with the current under study, the state of which has been reset to its value during rapid pacing. We have carried out this procedure to examine the rate-dependent effect of ICa,L (via the d and f gating variables), IKr (via the xr gating variable), IKs (via the xs gating variable), and [Ca2+]i (via [Ca2+]i, [Ca2+]up, [Ca2+]rel, and the u, v, and w gating variables) on the AP.

Figure 17 displays the APs after 12 s of pacing at 1,000 and 300 ms, along with the APs arising from isolating the rate-adaptation effects of ICa,L, IK (IKr + IKs), and ICa,L + IK. We note that when the effect of either IKr or IKs alone was examined, the observed change in APD90 was ~50% of the total reduction seen with both currents combined (IK in Fig. 17). Also, the effect on APD90 of [Ca2+]i and ICa,L combined was similar to that of ICa,L alone, which is why the results for [Ca2+]i were not included in Fig. 17. The limited role of [Ca2+]i at fast pacing rates may reflect the opposing effects of a higher resting [Ca2+]i and a lower Ca2+ transient amplitude on ICa,L inactivation. Our first observation is that the effect of the rate-dependent decrease in available ICa,L alone is to lower the plateau potential and to significantly accelerate early repolarization. The lower plateau level reduces IK activation, slows terminal repolarization, and results in no net change in total APD. Second, we observe that the effect of the rate-dependent increase in available IK alone is to lower the plateau potential and to compensate for the effect of the reduced plateau level on IK activation, leaving the rate of terminal repolarization unchanged. The effect of IK rate adaptation alone explains less than one-half of the total rate-dependent reduction in APD90. Third, the effect of ICa,L and IK combined is synergistic, lowering the plateau potential and accelerating late repolarization relative to the effect of ICa,L alone. Changes in these two currents together are sufficient to account for the total rate-dependent AP shortening observed on reducing the pacing period from 1,000 to 300 ms. Hence, we conclude that ICa,L and IK act together to produce the observed rate-dependent AP shortening.


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Fig. 17.   Role of ICa,L and IK in producing rate-dependent shortening of action potential on reducing pacing period from 1,000 to 300 ms. Steady-state action potentials during pacing at 1,000 and 300 ms are shown. In addition, action potentials resulting from rate adaptation of ICa,L alone, IK (IKr + IKs) alone, and ICa,L + IK are shown. See RESULTS for a detailed description of numerical simulation procedure. Combination of ICa,L and IK rate adaptation is sufficient to explain full extent of rate-dependent action potential shortening on reduction of pacing period from 1,000 to 300 ms.

ICa,L and Rate Dependence

Li and Nattel (35) showed that rate dependence can be abolished in isolated human atrial myocytes by blocking ICa,L. To compare these experimental data with the behavior of the model, we evaluated the effects of a strong decrease in L-type Ca2+ channel conductance on model AP rate dependence. Figure 18A shows model APs after 12 s of pacing with stimulation at 5,000 and 300 ms with a 90% reduction in ICa,L conductance. Figure 18B shows corresponding experimental results from Li and Nattel in which nifedipine was used to block ICa,L. In the model and experiments, block of ICa,L abolishes AP rate dependence. Although we have shown that ICa,L and IK play a role in rate adaptation, it appears that when ICa,L is strongly inhibited, as in Fig. 18, the plateau level is lowered to the point where IK activation is greatly reduced and can no longer contribute to rate adaptation.


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Fig. 18.   Role of ICa,L in rate dependence of action potential. A: model action potentials recorded after 12 s of pacing at various BCLs with 90% decrease in ICa,L conductance. No rate dependence is observed. B: experimentally recorded action potentials during stimulation at BCL = 5 and 0.3 s in presence of 10 µM nifedipine to block ICa,L. No rate dependence is observed.

Block of INaCa and Role in APD

Recently, Benardeau et al. (4) investigated the role of Na+/Ca2+ exchange in AP morphology and APD. In particular, they examined the effect of INaCa block by Li+ on AP morphology. Their results revealed a shortening of APD and transient hyperpolarization after the AP in the presence of Li+. Figure 19 shows control and INaCa-blocked (90% decrease in maximum exchanger activity compared with control) model APs with stimulation at 1 s. Model APs display both of the changes observed in experimental preparations (see Fig. 7A in Ref. 4), i.e., AP shortening and hyperpolarization. Inspection of the current records shows that the decrease in APD may be related to two mechanisms: 1) an increase in Ca2+-dependent ICa,L inactivation caused by an increase in resting [Ca2+]i and Ca2+ transient amplitude and 2) a reduction in depolarizing exchanger current during the late phase of the AP. The observed membrane hyperpolarization is due to a reduced inward exchanger current at rest, causing a negative shift of the resting membrane potential.


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Fig. 19.   Model action potentials during stimulation at 1 Hz under control conditions and with a 90% decrease in INaCa maximal amplitude. Block of INaCa is meant to mimic experimental conditions where Li+ is used to block exchanger. Action potential shortening and slight hyperpolarization are observed, as in experimental recordings (see Fig. 7A in Ref. 4).