A mathematical

model of a bullfrog

cardiac pacemaker

cell

R. L. RASMUSSON, J. W. CLARK, W. R. GILES, E. F. SHIBATA, AND D. L. CAMPBELL Department of Electrical and Computer Engineering, Rice University, Houston, Texas 77251-1892; Departments of Medical Physiology and Medicine, University of Calgary Medical School, Calgary, Alberta T2N 4N1, Canada; Departments of Cell Biology and Pharmacology, Duke University Medical Center, Durham, North Carolina 27710; and Department of Physiology and Biophysics, University of Iowa School of Medicine, Iowa City, Iowa 52242

RASMUSSON, R. L., J. W. CLARK, W. R. GILES, E. F. SHIBATA, AND D. L. CAMPBELL. A mathematical model of a bullfrog cardiac pacemaker ceZZ. Am. J. Physiol. 259 (Heart Circ. Physiol. 28): H352-H369, 1990.-Previous models of cardiac cellular electrophysiology have been based largely on voltageclamp measurements obtained from multicellular preparations and often combined data from different regions of the heart and a variety of species. We have developed a model of cardiac pacemaking based on a comprehensive set of voltage-clamp measurements obtained from single cells isolated from one specific tissue type, the bullfrog sinus venosus (SV). Consequently, sarcolemmal current densities and kinetics are not influenced by secondary phenomena associated with multicellular preparations, allowing us to realistically simulate processes thought to be important in pacemaking, including the Na+-K+ pump and Na’-Ca2’ exchanger. The membrane is surrounded extracellularly by a diffusion-limited space and intracellularly by a limited myoplasmic volume containing Ca”‘-binding proteins (calmodulin, troponin). The model makes several predictions regarding mechanisms involved in pacing. 1) Primary pacemaking cannot be attributed to any single current but arises from both the lack of a background K’ current and a complex interaction between Ca”‘, delayedrectifier K+, and background leak currents. 2) Ca2’ current displays complex behavior and is important during repolarization. 3) Because of Ca2’ buffering by myoplasmic proteins, the Na’-Ca2’ exchanger current is small and has little influence on action potential repolarization but may modulate the maximum diastolic potential. 4) The Na+-K+ pump current does not play an active role in repolarization but is of sufficient size to modulate the rate of diastolic depolarization. 5) K’ accumulation and Ca2+ depletion may occur in the extracellular spaces but play no role in either the diastolic depolarization or repolarization of a single action potential. This model illustrates the importance of basing simulations on quantitative measurements of ionic currents in myocytes and of including both electrogenic transporter mechanisms and Ca2’ buffering by myoplasmic Ca2+ -binding proteins.

MOST MODELS of membrane excitability that have been used in cardiac electrophysiology are of the HodgkinHuxley (HH) type (52). The HH formalism was first applied to Purkinje fibers by Noble (81). This model was later extensively revised by McAllister, Noble, and Tsien (MNT) WA variations of which were used in attempts to simulate the electrophysiological responses of ventric-

ular (5) and sinoatrial (SA) cells (8,9,84, 102). However, later experimental findings necessitated significant changes in the MNT model in order to incorporate mathematical descriptions of the ATP-dependent Na’K+ pump, the Na+-Ca2+ exchanger (e.g., Ref. 39), and a lumped fluid compartment for describing intra- and extracellular concentration changes in Na’, K+, and Ca” (e.g., Ref. 29). A comprehensive Purkinje fiber model incorporating these additional features has been developed by DiFrancesco and Noble (DN) (33). These models of cardiac cellular electrophysiology have been very useful. However, it is important to note that they are based largely on voltage-clamp data obtained from multicellular preparations that contain small restricted extracellular spaces (ECS), significant diffusion barriers, and electrical connections with other cells (25,59,60,73). Although these factors are physiologically important in intact cardiac tissue, their presence severely limits experimental attempts to obtain reproducible quantitative data on transmembrane currents. As a consequence, quantitative interpretation of multicellular voltage-clamp data is extremely difficult, if not impossible. Reliable data on sarcolemmal current density, as well as net current changes that can be separated from secondary phenomena, are an essential prerequisite to accurate simulations of numerous processes hypothesized to be of physiological importance in cardiac electrical activity. Such processes include the Na+-K’ pump, Na+-Ca2+ exchanger, extracellular potassium concentration ([K+],) accumulation, and extracellular calcium concentration ( [Ca”‘],) depletion. Within recent years both voltage-clamp methods and experimental preparations have been significantly improved. The two most significant advances have been the development of patch-clamp technology (cf. Ref. 49) and the development of enzymatic dispersion techniques that yield viable single isolated cardiac cells (cf. Ref. 90). The use of single isolated cardiac myocytes eliminates or greatly minimizes many of the problems associated with multicellular preparations and therefore allows for much more accurate separation and measurement of sarcolemma1 current densities. The electrophysiological properties of a wide variety of enzymatically isolated cardiac cell types have now been studied (for reviews see Refs. 13, 23, 56). Work in our laboratory over the past few years has been devoted to studying single myocytes en-

H352

the American

cardiac myocyte; ering; ion pumps

action

potential;

intracellular

0363-6135/90

$1.50

calcium

Copyright

buff-

0 1990

Physiological

Society

Downloaded from www.physiology.org/journal/ajpheart by ${individualUser.givenNames} ${individualUser.surname} (130.070.008.131) on January 17, 2019.

SINUS

VENOSUS

PACEMAKER

zymatically isolated from the bullfrog sinus venosus (SV) and atrium. Single SV cells have a very simple microanatomy and exhibit relatively few ionic currents (45, 96, 97). Nevertheless, a number of important features of their electrophysiology cannot be understood by simple extrapolation or intuition. Moreover, it is apparent that very small current changes, which are often difficult if not impossible to measure experimentally, are very important in cardiac pacemaker activity. Modeling can thus serve as an important theoretical adjunct to our understanding of the pacemaking process in that it may be used to predict behavior of a variety of processes which may be experimentally inaccessible but are hypothesized to be important (e.g., dynamic changes in channel currents, changes in internal concentrations of Na’, K+, or Ca’+, buffer occupancies). The main objective of this work was to formulate a mathematical model of the electrical activity of a single pacemaker cell from the SV of the frog heart that is based on voltage-clamp data recorded from individual SV cells. The model consists of two parts: 1) a cell membrane consisting of membrane capacitance, gated ionic channel currents, and pump and exchanger currents; and 2) a lumped fluid compartment that describes changes in the concentrations of Na+, K+, and Ca”+. The mathematical characterization of the transmembrane and background currents is largely based on experimental data from our laboratory (44, 96, 97). A HH-type formalism is used to describe the gating characteristics of the individual transmembrane currents, model-generated current-voltage (I-V) characteristics are compared with experimentally determined I-V characteristics from conventional voltage-clamp studies, and spontaneous nonpropagated action potentials are compared with experimental data from single SV cells. The singlecell model is then extended by considering the effects of placing the cell back into a physiological (in situ) situation where ionic diffusion between the ECS and bathing solution is characterized by a slow time constant (cf. Refs. 24, 84, 85). Our SV model (as well as our atria1 model described in the next paper) incorporates two new and important features that have not been used in previous cardiac models. First, it combines current densities and kinetics obtained from single cells with multicellular morphological data so as to more realistically estimate the effects of accumulation and depletion of extracellular ions on the pacemaker depolarization. Previous modeling studies based on multicellular data required that both measured current densities and kinetics be corrected for the effects of a restricted ECS. Thus they represented a recapitulation of the underlying assumptions used in the original data analysis. The assumptions employed in such ECS corrections are prone to error, which in turn can lead to biasing of model results. By utilizing current measurements obtained without the effects of a restricted ECS, we have minimized the bias in our initial estimation of its effects. Second, this model attempts to describe, in a physiologically realistic manner, the interactions between sarcolemmal electrical activity and cellular Ca2+ homeostatic mechanisms by incorporating the effects of

H353

MODEL

physiologically important myoplasmic Ca2+-binding proteins. The significance of such Ca2+-binding proteins will be elaborated on in our atria1 model described in the accompanying paper (92). This SV model successfully reproduces experimentally observed pacemaker depolarization and action potential behavior. It predicts that cleft accumulation and depletion of cations is not important in the time course of SV pacing on a beat-to-beat basis but that coupled transport mechanisms may contribute significantly to net membrane current during diastolic depolarization. DESCRIPTION

OF MODEL:

GENERAL

ASPECTS

The lumped electrical equivalent circuit for the sarcolemma of a single SV pacemaker cell is shown in Fig. 1. This membrane model contains both the channelmediated currents responsible for generation of the action potential and the pump, exchanger, and background currents. The currents are designated as follows: 1k, the time- and voltage-dependent “delayed-rectifier” potassium current; I caythe time- and voltage-dependent calcium current; &, a linear background current, composed of a background sodium current 1Na,Band a background calcium current 1ca,R(1Na,Bis assumed to be much larger the electrogenic sodium-potassium than ICa B); INaK, pump current; I Naca,the electrogenic Na+-Ca2+ exchanger current; and I ca,p, the ATP-dependent Ca2+ pump current. C, is the membrane capacitance. It should be noted that, consistent with experimental data (44, 96, 97), our model frog SV cell does not have 1) a voltage-dependent tetrodotoxin (TTX) -sensitive sodium current INa, 2) an inwardly rectifying background potassium current $, 3) a calcium-activated potassium current, 4) a calciumactivated nonspecific current, or 5) the hyperpolarization-activated current 1F The absence or extremely small size of Ik, in primary cardiac pacemaker tissue, both amphibian and mammalian, is now generally recognized. However, even though 1f has been consistently recorded in man-made strips (32, 102), trabeculae from bullfrog SV (10, 103, 104), and single cells derived from the SA node (e.g., Refs. 45, 56, 65), it has never been observed in our single spontaneously active frog SV cells (96) (see DISCUSSION). Thus 1f cannot be considered as a universal “pacemaking current” important to cardiac pacemaking cells in general (see DISCUSSION and Ref. 45). Under space-clamp conditions, the differential equation describing membrane potential (V) is dV dt

--

-

-tICa

+

IK

+

IB

+

INaK

+

INaCa

+

ICaP) 9

(1j

cm

In our description of the HH-type gating variables for Ica and Ik we have adopted the original nomenclature of the classical MNT model of the Purkinje fiber (67). Accordingly, d and f are, respectively, the activation and inactivation gating variables for Ica, and n is the activation gating variable for Ik (see Tables 1 and 2). The potential and time dependence of d, f, and n are given by solutions of first-order differential equations of the general form

~(V,t) = [z(V) - ~WJ)l dt

4 V)

(2)

Downloaded from www.physiology.org/journal/ajpheart by ${individualUser.givenNames} ${individualUser.surname} (130.070.008.131) on January 17, 2019.

H354

SINUS

VENOSUS

Cleft

PACEMAKER

MODEL

space ?

I Na-K

I

INa-Ca

1 /Capump~

exchanger

Pump

--1

r ------! I Na+

‘K

1

FIG. 1. Diagram of equivalent circuit of sarcolemma of frog sinus venosus (SV) cell. Membrane capacitance (C,) is shunted by a background current (1B), time- and voltage-dependent calcium and potassium currents ( Ica, 1k), electrogenic pumping mechanisms (Na’-K’ and Ca2’ pumps), and an electrogenic Na’-Ca2’ exchanger. Ek, Ec,, ENa, equilibrium potentials for potassium, calcium, and sodium. This equivalent circuit is referred to as the membrane model, and it, together with the lumped fluid compartmental model of Fig. 2, comprise our overall model of SV pacemaker cell.

ka

-1 I

--------___

Ca++

-I

i 1

1

A Inside

value of where z = d, f, or n, z(V) is the steady-state gating variable z at potential V, and 7,( V) is the z gating variable time constant at V. The individual rate functions a,( V) and ,&( V), used in the definitions of z( V)[= a/(cr + ,&)I and 7,( V)[= l/(a + ,@I, are also given in Tables 1 and 2. The mathematical expressions for the pump, exchanger, and channel currents require the specification of the concentrations of Na’, K+, and Ca2+ both inside and outside the cell. In the case of the single SV cell, the extracellular environment is the bathing medium. In situ conditions for the single cell require that the nature of the extracellular cleft spaces within the muscle bundle, or trabeculum, be taken into account. The extracellular space of a single cell is modeled in terms of the lumped three-compartment system shown in Fig. 2. Inclusion of the “cleft space” depends on whether the goal is to mimic action potential data from isolated SV myocytes in the bathing medium or to have the model predict the behavior of the cells under in situ conditions where the cleft acts as an important restrictive diffusion space. Based on the analysis of the general compartmental system, differential equations have been derived describing the time-dependent concentration changes for Na+, Ca’+, and K+ in the lumped intra- and extracellular (cleft) media (see Table 3). Of special interest in the intracellular medium of the compartmental model (Fig. 2) are the differential equations describing the binding of intracellular calcium to specific myoplasmic proteins. This Ca2+ buffering system is very important for the regulation and limitation of free intracellular Ca2’ concentration ( [ Ca”‘]J transients. Mathematical expressions for the Na+-K+ pump mechanism, the Na+-Ca2+ exchanger, and the calcium pump are given in Table 4. Computational

Details

The complete model of the electrophysiological behavior of the single SV cell in its physiological environment is described mathematically by a system of 14 first-order differential equations. Four of these differential equa-

tions pertain to the membrane portion of the model, and the remaining 10 are associated with the fluid compartment model that includes internal Ca2’ buffering by myoplasmic proteins. A fourth-order Runge-Kutta method was used in conjunction with a variable-step-size algorithm to accomplish numerical integration. The variable-step-size algorithm was adopted to moderate computation time consistent with accurate numerical integration (69). Changes in each gating variable, fractional occupancy of each of the protein Ca2+-binding sites, and the transmembrane potential (V) were all monitored by the variable-step-size integration routine. The amount by which each of these variables was allowed to change per time step was limited according to the following inequality statements At*dV/dt

< 0.15 mV

At=d(occupancy)/dt

At d( gating variable)/d l

< 0.02 t c 0.02

(3) (4) (5)

The accuracy of numerical integration was ensured by checking sensitivity to the step size and setting a maximum limit on the step size of 0.1 ms. This simulation program was implemented in FORTRAN 77 using double-precision arithmetic. Various versions of the program have been run on a VAX 11-750 computer employing a VMS operating system, a VAX 11-750 under UNIX, a Pyramid computer under UNIX, Hewlett-Packard 9000 using UNIX, a Cyber 205 using NOS, and a Texas Instruments Personal Computer using MSDOS. A complete model simulation of a two action potential sequence runs on a DEC VAX 750 (VMS) in -15 min. Detailed Description and Initial Results

of Model Components

Our complete SV model consists of two major components, a membrane model and a lumped fluid compartment model. Each in turn consist of several subcomponents. This section will describe these components in

Downloaded from www.physiology.org/journal/ajpheart by ${individualUser.givenNames} ${individualUser.surname} (130.070.008.131) on January 17, 2019.

SINUS

PACEMAKER

H355

MODEL

1. Inward and background currents

TABLE Calcium

VENOSUS

current Ica I (:a= df&a 0.0274V([Ca]i[eXp(O.O78V) exp(0.078V)

&a = . f =

&

7;‘(fm

-

$(d,

- d)

[Ca],]j

- 1

-f)

d, = l+exp(:s)

rT1 = 0.0197 exp(-

[0.0337(

V + 10)12) + 0.02

0.8

r= exp

50 - v 20 1

(-

1 + 1

fee= ”

1 +exp(v+8T’06)+r Total Sodium

background

current

I Na,B = 0.000115(

V - 54)

For explanation

of abbreviations

TABLE

background

current

(IN,,B)

(IB = IN~,B + L,B) Calcium

background

current

(Ica,B)

I Ca,B = O.OOOOOO3( V - 54) see text.

2. Outward currents Delayed rectifier IK = n21K

(Ik) TK = O.OllS(V 95

J-l =

V - EK - 78

1 + exp fin = 0.000286

- 23~ + r’)

- 95

25 exp[-0.0381(

V + 26.5)]

a, =

0.0000144( 1 - exp[-O.l28(V

V + 26.5) + 26.5)]

EK = 59 log cKl

WI i

For explanation

I

potassium

NaK = o’145

Calcium

I

pump

(INak)

Sodium-calcium

(jK],[:j6.621)?

[Cali +

exchanger

0.001

)

(INaca)

[Na]~[Ca],exp(O.O195V) 1 + O.OOOl([Ca]i[Na]~

For explanation

(s)

(1Ca.P)

([Cali

o*oo675

I NaCa - 0.000004{

pump

((Na;y?5.46r

Ca,P =

see text.

3. Pump and exchanger currents

TABLE Sodium

of abbreviations

of abbreviations

- [Na]z[Ca];exp(-0.0195V)j + [Ca],[Na]“) see text.

detail and indicate the contribution of each to the modelgenerated waveforms of transmembrane voltage and current. Membrane model. An essential property of any mathematical model of a primary pacemaking cell is its ability

to reproduce the spontaneous oscillations in transmembrane potential recorded under physiological conditions. Figure 3A shows that realistic spontaneous pacemaker and action potential waveforms can be reproduced by our SV model utilizing the equations and nominal parameter values given in Tables 1-5. The waveforms of the simulated channel-mediated currents ( Ica, Ik, and &) are shown in Fig. 34 and the Na+-K’ and calcium pump currents and Na’-Ca2’ exchanger current (INak, Ica,P, are shown in Fig. 3C. Table 6 compares the and IN&a) model-generated action potential waveform of Fig. 3A with several parameters of action potential waveforms obtained from experimental recordings from 19 different single SV cells (44, 96). Note from Table 6 that the parameters of the model-generated waveform are very similar to the average experimental values. Because of the small but significant variability in the experimental measurement of each parameter, more detailed comparison or parameter matching is not justified at present. A typical spontaneous pacemaker and action potential action is shown in Fig. 4 along with a model-generated

Downloaded from www.physiology.org/journal/ajpheart by ${individualUser.givenNames} ${individualUser.surname} (130.070.008.131) on January 17, 2019.

H356

SINUS

VENOSUS

PACEMAKER

MODEL

Bulk

Cleft

Available

Medium

Space

Intracellular /

CNal b

-

T

‘)

Space

1 El

’

FIG. 2. Lumped fluid compartment model of a SV pacemaker cell in its simulated physiological environment. This model consists of 3 well-stirred fluid compartments containing Na+, K+, and Ca”’ in different concentrations. Compartments are I) an intracellular fluid compartment that also contains protein binding sites for Ca”, 2) a small volumerestricted diffusion space (called a cleft space) that approximates the extracellular fluid space “seen” by the single SV cell, and 3) a very large volume referred to as the bulk extracellular bathing medium. Tp, time constant for diffusion between bulk medium and cleft space, is 7p.

Calmodulin

P

*

‘Ca

9

WI b

Troponin

[Cal b

1

C

’ Ca(P)

CNal WI.

[Cal i

4. Compartmental ion flux equations

TABLE Extracellular

concentrations

[Nab - [WC +

PJa,=1 .

WI

2lNaK

7P

.

[Ca]b

.

+ INaB

-

IK

flc

-

[Cal,

2INaCa

+

+

ka

+ kaP

+

kaB

2JYY

of Na+ and K’

-3INaK

=

3INaCa

7P

concentrations

[Na3;

+

J%

WIb - WI c+

[Ca,=1 Internal

3INaK

7P =

C

of Na+, K+, and Ca2’

-

3INaCa

-

. WI i =

INaB

2INaK

mi

Internal

calcium

-

OTC = 39[Ca]i(l

Oc)

- OTC)

-

0.238Oc

- 0.1960Tc

0 TMgC = loo[ca];(l

-

OTMaC - OTM,,)

-

0.00330~~,c

0 TMgM = O*l[Mg]i(l

-

OTM~C

-

0*3330TMgM

& = 0.000045& =

For explanation

IK

concentrations

Oc = lOO[Ca];(l

[Cal;

mi

2INaCa

-

ka

-

OTM~M)

+ 0.00008426Tc + 0.0001684~~~,c -

ka,P

-

ka,B

2ViF of abbreviations

--

OB Vi

see text.

F, Faraday

constant.

potential. To obtain the model-generated waveform, the nominal parameter set of Table 5 was employed with the magnitudes of the component membrane currents appropriately scaled to yield acceptable fits to the experimental data. Our SV model is therefore able 1) to generate spontaneous pacemaker and action potential waveforms that reproduce the characteristic behavior which is experimentally observed and 2) to reproduce such electrical activity by only changing current densities within the experimentally observed range of values and without alteration of either their experimentally measured kinetics or steady-state gating variables. These two properties provide an initial indication of the success of this model

J

in reconstructing our experimental data. The majority of the model parameters used to characterize specific time- and voltage-dependent membrane currents were derived from voltage-clamp measurements conducted on single cells in our laboratory. It is therefore essential that the overall model be able to reproduce the results of various voltage-clamp experiments. In the remainder of this section we will describe the results of applying simulated voltage-clamp protocols to the isolated single SV cell model for each of the transmembrane ionic currents as well as the pump and exchanger currents. Calcium buffering by myoplasmic Ca2+-binding proteins will then be discussed in conjunction with a detailed description of the portion of the model used for the compartmental flux analysis and in vivo simulations. BACKGROUND CURRENTS. SV pacemaker cells, unlike atria1 or ventricular cells, appear not to have an inwardly rectifying K+ current (I kl, using the nomenclature of the DN model). Instead, the I-V relationship for the background currents in this type of cell is approximately linear (44, 97). As shown in Table 1 the total inward background current (1B) is modeled as a linear combination of a sodium (& B) and a calcium (Ica B) current, with INaB being very much larger than IcaB. The experimental ‘data describing the total background current (ITB) in SV cells (Fig. 5) were obtained by blocking IQ and measuring the instantaneous change in current in response to voltage-clamp steps. Thus this experimentally measured I TB consists of a number of components in addition to 113,including I&k, &Q, and INaca. In addition, at membrane voltages greater than -20 mV, & may also contribute. An analogous voltage-clamp protocol was used in our sinus model, yielding the modelgenerated curve in Fig. 5. At potentials negative to -20 mV, the two I-V curves are very similar in magnitude and slope. However, at more positive potentials the relationships differ due to activation of Ik in the experimental recordings Of 1~. CALCIUM CURRENT. In cardiac pacemaker tissue, the last part of diastolic depolarization, as well as the rapid upstroke of the action potential, is strongly influenced

Downloaded from www.physiology.org/journal/ajpheart by ${individualUser.givenNames} ${individualUser.surname} (130.070.008.131) on January 17, 2019.

SINUS

VENOSUS

PACEMAKER

TABLE

H357

MODEL

5. Model constants and initial conditions F= 96,500 C/m01 V, = 0.0004 nl WI c = 2.6 mM [Na]; = 7.5 mM [Na]b = 111 mM [C 1 = 2.25 mM [M:j:i = 2.5 mM

Vi = 0.0025 ms TP = 10,000 [K]; = 130 mM WI = 2.5 mM Wa! = 111 mM [Cali = 0.5 pM [Calb = 2.25 mM C, = 0.075 nF

0

C

> E

d=O n = 0.05 oc = 0.2 0 TMgM = 0.9

f=l V= -75 0 TC = 0.1 oTMrM = 0.04

-50

For explanation

of abbreviations

see text.

6. Comparison of indexes from experimental and model-generated action potentials TABLE

Shibata

Max diastolic potential, Overshoot, mV Duration, ms Upstroke velocity, V/s Frequency, Hz

0
E

101B

0

0

1

2

i

i

i

3

FIG. 11. Model-generated output waveforms for single SV pacemaker cell in its simulated physiological environment (i.e., cleft space medium included). Nominal set of parameter values given in Tables I5 are used in simulation. A: spontaneous action potential waveforms; B: time course of changes in internal free Ca”+ concentration ([Cali); C: fluctuations in fractional calcium occupancy binding sites for calmodulin (Oc), troponin-Ca (OTC), and troponin-Mg (OrrM&; D: time course of changes in cleft space concentrations of potassium ([K’],) and calcium ([ Ca”‘],. Time courses of [ Na+]i, [ K+]i, and [ Na+], (not shown) are very nearly constant during action potential.

Downloaded from www.physiology.org/journal/ajpheart by ${individualUser.givenNames} ${individualUser.surname} (130.070.008.131) on January 17, 2019.

H364

SINUS

VENOSUS

PACEMAKER

(0 TM& binds Ca2+ much more slowly.

Thus both calmodulin and troponin-Ca “blunt out” the rapid [Ca2+]; transient that would otherwise be produced by loa. Furthermore, the time course of Ca2+ binding to troponinCa suggests that this site controls phasic tension generation. In contrast, troponin-Mg is largely occupied during the entire pacemaking cycle (Fig. 11); nonetheless, this site is capable of binding a significant amount of Ca2+ and approaches saturation during each action potential. This suggests that troponin-Mg is importantly involved in longer-term regulation of [Ca2+];. These Ca2+-binding simulations suggest that these single SV myocytes possess a significant intracellular Ca2+ buffering capacity. A more detailed description of the effects and interactions between these myoplasmic Ca”+-binding proteins is given in the accompanying paper describing our atria1 model (92), where these effects become very important for both modulation of the Na+-Ca2+ exchanger current lNaca and the excitation-contraction coupling process. The time course of change in extracellular cleft concentrations of [K’lc and [ Ca2+], are shown in Fig. 1lD. The fluctuation of [K’lc is -0.5 mM and that of [Ca”‘], is -0.3 mM. The model waveforms shown in Fig. 11 were obtained under steady-state conditions so that concentration changes due to ion movements from the bulk medium would have stabilized. Pizzaro, Cleeman, and Morad (88) have shown that in frog ventricular trabeculum there is a calcium depletion effect produced during the action potential as well as during voltage-clamp depolarizations that activate Ica. DISCUSSION

Summary of Results Electrophysiological models of the type presented here serve several important functions. At their most basic level they represent a concise summary of current data, analytic techniques, and quantitative interpretation of several biophysical systems that are usually studied in isolation but interact and function as part of a larger and more complex physiological system. Instead of being merely a simplified catalogue of the behavior of each individual system, a realistic model based on experimental data, due to its predictive capabilities, can serve as an extremely valuable experimental tool. For example, modeling can allow the theoretical examination of both the potentially complex interactions that may occur among various systems and the behavior of individual systems and proposed mechanisms under physiological conditions. Such behavior may be difficult (even impossible) to examine experimentally. Important interactions among systems may be significantly altered or obscured by perturbations introduced by experimental techniques. Thus, when modeling is coupled closely to a reliable and consistent set of experimental data obtained from both the same species and same cell type, either the failure or the success of the model to realistically reproduce observed physiological phenomena may provide significant insight into the hypothesized function, or description of various processes believed to be important in the generation of normal electrical activity. In other words, mod-

MODEL

eling provides both an important check on the assumptions underlying experimental design and analysis and serves as a means for quantitatively testing and realistically bracketing the influence of any given system within the framework of presently available data. Although a number of models of cardiac pacemaking have been published recently (5, 8, 84, 102), many of their features were not based on data obtained from the same preparation. The various anatomically distinct regions of the heart exhibit quite pronounced differences in their normal electrical activity, indicating significant differences in both the presence or absence of individual current components and their relative densities. In an attempt to avoid such problems in our SV model we have used voltage-clamp data obtained from measurements of the total background current, the calcium current lca, and the delayed-rectifier potassium current 1k from a single cell type isolated from a distinct anatomic region, and have made reasonable estimates of the maximal capacities of the Na+-K+ pump and Na+-Ca”+ exchanger currents from a very similar tissue, single cells from the bullfrog atrium. As indicated in RESULTS, the parameters used in simulating 1k were based entirely on quantitative voltage-clamp analysis of this current in single SV cells, and the kinetics and magnitude of lca were also matched as closely as possible to voltage-clamp data. Pacemaker potential. Comparison of our SV model with our bullfrog atria1 model (92) indicates that it is the lack of a stabilizing inwardly rectifying potassium current Irk, that promotes spontaneous pacing. It is also important to note that spontaneous pacing in bullfrog SV cells occurs in the absence of the so-called pacemaker current 1f (45, 96, 97) and that in various mammalian cardiac pacemaker preparations pharmacological block of 1f slows (by -IO-20%) but does not block pacing (reviewed in Ref. 83). These results suggest that 1f, when present, acts as a modulator of pacemaker activity rather than as its primary source (11, 44, 83). The term “the pacemaker current” therefore may be somewhat misleading, since no one current is responsible for pacemaking. Instead, our SV model demonstrates a pacemaker potential that results from an interaction of several current systems. The earliest phase of the diastolic depolarization (i.e., that immediately following the maximum diastolic potential) arises from an interaction between the inward background current 1B, the inward Na+-Ca2+ exchanger current I Naca,and the deactivation of the outward delayed-rectifier potassium current 1k. The inward calcium current Ic,a becomes activated during the last one-third of the diastolic depolarization, where it makes a significant contribution to this phase. In contrast to the dynamic behavior displayed by INaca, 1k, and Ica, the Na’-K+ pump current I NaK remains relatively constant during the diastolic depolarization. However, this current is of a similar magnitude to 1B and can therefore also significantly influence the diastolic depolarization. The possible influence of the Na+-EC+ pump current IN& and the Na+-Ca2+ exchanger current INaca in generation of the spontaneous pacemaker potential in bullfrog SV cells can be better appreciated from the model results presented in Fig. 12. This figure compares waveforms

Downloaded from www.physiology.org/journal/ajpheart by ${individualUser.givenNames} ${individualUser.surname} (130.070.008.131) on January 17, 2019.

SINUS

VENOSUS

PACEMAKER

generated by our SV model in the absence of both lNak and Ih:aCa (“ simplified model”) to those generated by the “complete model” (i.e., both I&k and &&, present). The removal of I Nak and INaca produces tW0 major effects On the pacemaker potential. 1) A more negative maximum diastolic potential is reached in the simplified model. This is due mainly to the removal of the depolarization produced by the inward INaca flowing during this phase. 2) The rate of the diastolic depolarization is greater, and thus the diastolic interval is shortened, in the simplified model. This is due to the removal of the hyperpolarization produced by INak. These simulations suggest that even though both I Naca and IN& are Sldl Currents, they can play important roles in pacemaking by helping to determine both the magnitude of the maximum diastolic potential and the rate of the spontaneous diastolic depolarization. Action potential. The upstroke of the SV action potential is produced by activation of IcaT which occurs during the latter one-third of the diastolic depolarization. In addition, our model suggests that the incomplete inactivation displayed by the Ica gating variable f leads to “reactivation” of Ica during the later phases of the action potential. This reactivation produces a maintained inward component of Ica that helps to sustain the action potential plateau. This predicted behavior of the macroscopic Ica during the action potential is comparable to that experimentally observed by Mazzanti and DeFelice

1

I

\

1

/

MODEL

H365

(66) for single calcium channels in embryonic chick heart cell aggregates and for the whole cell macroscopic 1oa of both embryonic chick myocytes (1) and rabbit SA node cells (35). With regard to the mechanism(s) underlying repolarization of the action potential, all three of the hyperpolarizing currents IK,, IN&, and Ik have been proposed as being the cause of repolarization in cardiac tissue (21, 26, 55, 58, 72). Our SV model may therefore help in clarifying the mechanism(s) of repolarization in cardiac pacemaker tissue. First, it is clear, both from our model and experimental results, that repolarization can occur in the absence of 1~ Second, our model does repolarize in the absence of the Na’-K’ pump current IN& (Fig. 12). These results suggest th .at neither Ik, nor IN& is the primary current underlying repolarization. Rather, our SV model indicates that repola .rization occurs as the result of a complex interaction between the delayedrectifier Ik and the maintained (reactivated) inward component Of Ica. It is important to note that the maintained component of net inward current in our SV model is due to Ica. In this respect our bullfrog SV model differs significantly from the recent model of the rabbit SA node proposed by Noble and Noble (84). In their SA node model, Noble and Noble (84) used the DN equations (33) and consequently included a mathematical formulation for Ca”+induced Ca”+ release from the SR. There are genuine differences in both the ionic currents and the EC coupling mechanisms in rabbit vs. bullfrog. There is also one other important fundamental difference between these two models: the Noble and Noble (84) model does not include any provision for buffering of [ Ca2+]; by myoplasmic Ca”+ -binding proteins, the effects of which have been postulated to significantly modulate the magnitude of the [Ca2+]i transient resulting from release of Ca from the SR (17). For both these reasons the time course of the [ Ca2+]; transient and the size and time cou:se of the Na+-Ca”+ >xchanger current INaca differ significantly in our SV model compared with those predieted by the Noble and Noble model (see also Ref. 11). In particular, while the Noble and Noble model does contain a sustained component of inward calcium current during the plateau, it also predicts that a very significant portion of the inward current flowing during the early portion of the plateau is due to Na+-Ca2+ exchange. Although the contribution of Na+-Ca2+ exchange during the plateau decreases on repolarization, INaca always remains a net inward (i.e., depolarizing) current in the Noble and Noble model. In contrast, in our SV model INaca is an outward current during the plateau and a larger inward current during the peak diastolic potential.

1

2

1

3

Set 12. Effects of removing both Na’-K’ pump and Na+-Ca”+ exchanger currents. Waveforms for complete model and a simplified model where both I ~~~~ and IN&a are absent are compared. A : 2 simulated spontaneous action potential waveforms, the more rapid being generated by simplified model in absence of pump and exchanger currents. B: pump and exchanger currents of complete model that are responsible for differences seen in action potential waveforms.

Importance of Including Compartmental Descriptions of Extracellular and Intracellular Spaces in Modeling SV Electrical Activity

FIG.

A wide variety of experimental findings [summarized by Noble (83)] strongly indicates that in syncytial structures, such as heart or smooth muscle, there are significant changes in the concentrations of extracellular K+ and intracellular Na+ and Ca2+ during normal physiolog-

Downloaded from www.physiology.org/journal/ajpheart by ${individualUser.givenNames} ${individualUser.surname} (130.070.008.131) on January 17, 2019.

H366

SINUS

VENOSUS

PACEMAKER

ical activity. The DN model (33) of the Purkinje fiber included mathematical descriptions of these concentration changes in the restricted extracellular clefts and evaluated the possible contribution of electrogenic ion pumps (e.g., the Na’-K+ pump) and cotransport processes (the electrogenic Na+-Ca2+ exchanger) on electrophysiological activity. A similar, although less elaborate, model of extracellular K+ accumulation and depletion had been developed by Coulombe and Coraboeuf (29). As described previously, in frog heart there are both small ECS and caveolae that could act as unstirred layers adjacent to cells. The results shown in Fig. 11 indicate that the presence or absence of restricted ECS do not cause significant changes in pacemaking rate or repolarization. There are two reasons for this: 1) 1k is relatively small, and 2) the ECS in the SV are estimated to be very large compared with those in the Purkinje fiber (12% for the SV compared with much less than 1% of the total intracellular volume for the Purkinje fiber). These results suggest that changes in [K’lc are not a significant factor in modulation of spontaneous SV pacemaker activity at the normal physiological rate. In addition, our model indicates that no significant changes in [Na’]i take place. These SV pacemaker cells do not exhibit a time- and voltage-dependent Na+ current. Therefore the only Na+ influx that occurs in these cells is through the small time-independent background sodium current lNa B and the Na’-Ca2’ exchanger current lNaca operating ‘in reverse mode (i.e., as an outward current during the plateau). Thus our model also suggests that changes in [Na’]; are not significantly involved in spontaneous SV pacemaker activity. Both [Ca”], and [Ca2+]; have been hypothesized to change significantly during each beat as a result of Ica. In the absence of intracellular Ca2+-binding proteins or other sequestration mechanisms the predicted change in [Ca2+]; produced by Ica is very large-indeed (>lOO-fold increase above resting levels; cf. Ref. 15). This large change in [Ca2+]; could have a number of very important electrophysiological co nsequences, since it would both significantly change the driving force for Ica and strongly activate the Na+-Ca2+ exchanger. In the models of Fischmeister and Vassort (39), DN (33), and Noble and Noble (84) this is the case, because no provision is made in these models for intracellular Ca2+ buffering. However, Ca2+ -binding proteins are included, when myoplasmic the change in [Ca2+1; is much smaller and therefore the Na+-Ca”+ exchanger generates much less peak current (cf. 6, 15). This appears to be a very important point, since we discovered that repolarization of the SV model would not proceed to normal levels in the presence of lNaca unless significant Ca2+ buffering by myoplasmic proteins was included. Limitations of Our Model of Cardiac Pacemaker Activity in Bullfrog SV In the development of any model simplifying assumptions and formulations must always be included. It is important that these underlying assumptions and simplifications be clearly stated and understood when evaluating the predictions of the model. Our SV model has

MODEL

predictive value, but it also has limitations. Nonetheless, because it is based on a consistent set of experimental data, our model in its present form does provide some significant insights and directions for future experimentation. Some of the more intriguing and important limitations and questions are as follows. Calcium current. The kinetics of the delayed-rectifier 1k have been well characterized over the entire range of physiological potentials and over time periods that allowed it to reach steady-state conditions (i.e., times much longer than the duration of the action potential). In contrast, for the reasons previously discussed, neither the kinetics of activation or deactivation of Ica could be reliably recorded in these single SV cells. More importantly, significant uncertainty remains concerning the mechanism, and hence the correct mathematical formulation, of inactivation of loa in most excitable cells. The formulation for both the kinetics and steady-state inactivation relation of Ica used in our model was based on experiments using voltage-clamp pulses of 5200 ms. Thus, in contrast to 1k, the behavior of Ica is formulated in our model by extrapolating the available experimental Its. data over a much longer time frame than was originally recorded. We have recognized that the incomplete inactivation displayed by the f gating variable was crucial in maintaining the plateau of the action potential. However, because of the extrapolation of Ica kinetics to longer time periods, we also discovered that repolarization could not proceed properly without shifting the f inactivation relationship 10 mV in the depolarized direction. Clearly, our model makes interesting predictions regarding calcium channel activity during the action potential and indicates a need for quantitative evaluation of calcium current activity over longer periods of time than those applied to date. Calcium buffering. The Ca”+ buffering by both calmodulin and troponin in our SV model is a conservative estimate of the total Ca2+-binding capacity that exists in these cells. The extent to wh .ich our mathematical formulation of Ca”+ buffering is act urate depends both on how well measurements made from reconstituted systems of contractile proteins can be extrapolated to intact tissues, such as skeletal muscle or heart, and the accuracy of the conversion factors we used to arrive at the concentrations of the proteins and their distribution (cf. Ref. 15). Our present formulation of Ca”+ buffering by the contractile proteins and calmodulin should therefore be considered as only a first approximation. Nonetheless, it is important to point out that all previous models of cardiac electrical activity have failed to incorporate the effects of myoplasmic Ca”+ -binding proteins. The minimal values chosen for our SV model were designed to demonstrate that myoplasmic Ca2+-binding proteins are important physiological modulators of calcium transients and Na+-Ca2+ exchanger current (6, 15). Sodium-calcium exchanger current. Since the size and the time course of this current are modulated significantly by the extent of Ca”+ buffering from various Ca2+binding sites, the same quantitative reservations apply to our calculations of I NaCaas have previously been stated for the Ca”+ buffering calculations. In addition, whether

Downloaded from www.physiology.org/journal/ajpheart by ${individualUser.givenNames} ${individualUser.surname} (130.070.008.131) on January 17, 2019.

SINUS

VENOSUS

the DN (33) formulation is an appropriate description of lNaca in bullfrog SV cells remains to be determined. The present limitations of our SV model therefore illustrate several very important areas that deserve careful future experimental analysis, including studies on basic intracellular biochemical mechanisms as well as electrophysiological and biophysical studies. In conclusion, our SV model is in agreement with the previous experimental findings of Shibata and Giles (97) and can thus be used to explore the feasibility of various working hypotheses. This is particularly important in cardiac pacemaker tissue, in which important effects can arise from very small net current changes, that can often not be resolved experimentally or can be recorded only with considerable difficulty. The existence of a mathematical model of the kind described here should therefore contribute to both the design and interpretation of further studies of cardiac pacemaker tissue, including the actions of autonomic transmitters, changes in intercellular resistance and cell-to-cell coupling on pacemaker activity, and interactions with electroneutral ionic transport mechanisms (e.g., Na-K-Xl cotransport, Na-H exchange). NOTE

ADDED

IN

PACEMAKER

5.

6.

7. 8.

9.

10.

11.

12.

13.

14.

PROOF

Recently, a modification of the SA node model described in Ref. 103 has been used to study phase resetting [M. Guevara and H. J. Jongsma. Am. J. Physiol. 258 (Heart Circ. Physiol. 27): H734-H747, 19901 and pharmacological agents that cause bradycardia (T. Doerr and W. Trautwein. Nuunyn-Schmiedeberg’s Arch. Pharmacol. 341: 331-341, 1990).

15.

16.

17. A special note of appreciation is due C. Richard Murphey and John M. Shumaker, Rice University, who contributed many helpful comments and suggestions in the revision of this paper. This work was supported by National Heart, Lung, and Blood Grants HL-27454 and HL-36475 and by grants from the Canadian Medical Research Council, the Canadian Heart Foundation, the Alberta Heritage Foundation for Medical Research (AHFMR), and Control Data Corporation. J. W. Clark held an AHFMR Visiting Professor Award. W. R. Giles held an Established Investigator’s Award from the American Heart Association and is presently an AHFMR Medical Scientist. E. F. Shibata was supported by a National Institutes of Health Postdoctoral Fellowship. D. L. Campbell was a Pre- and postdoctoral Fellow of the Canadian Heart Foundation. Much of this work was included in a thesis submitted by R. L. Rasmusson for the degree of Master of Science in Electrical Engineering, Rice University, Houston, TX. Address for reprint requests: W. R. Giles, Depts. of Medical Physiology and Medicine, University of Calgary Medical School, PO Box 1892, Calgary, Alberta T2N 4N1, Canada. Received

24 May

1988; accepted

in final

form

9 February

1990.

REFERENCES 1. ANDERSON, A. J., AND L. PATMORE. Calcium currents evoked in embryonic chick myocytes during depolarization with an action potential voltage waveform (Abstract). J. Physiol. Lond. 407: 126P, 1988. 2. BARR, L., M. DEWEY, AND W. BERGER. Propagation of action potentials and the structure of the nexus in cardiac muscle. J. Gen. Physiol. 48: 797-823, 1965. 3. BAYERD~RFFER, E., L. ECKHARDT, W. HAASE, AND I. SCHULZ. Electrogenic calcium transport in plasma membrane of rat pancreatic acinar cells. J. Membr. BioZ. 84: 45-60, 1985. 4. BAYLOR, S. M., W. K. CHANDLER, AND M. W. MARSHALL. Sarcoplasmic reticulum calcium release in frog skeletal muscle

18.

19.

19a.

19b.

20.

21.

22. 23.

24.

25.

26.

MODEL

H367

fibres estimated from arsenazo III calcium transients. J. Physiol. Lond. 344:625-666,1983. BEELER, G. W., AND H. REUTER. Reconstruction of the action potential of ventricular myocardial fibres. J. Physiol. Lond. 268: 177-210, 1977. BEUCKELMANN, D. J., AND W. G. WIER. Sodium-calcium exchange in guinea-pig cardiac cells: exchange current and changes in intracellular Ca2+. J. Physiol. Lond. 414: 499-520, 1989. BORLE, A. B. Control, modulation, and regulation of cell calcium. Rev. Physiol. Biochem. Pharmacol. 90: 14-153, 1981. BRISTOW, D., AND J. W. CLARK. A mathematical model of primary pacemaking cell in S-A node of the heart. Am. J. Physiol. 243 (Heart Circ. PhysioZ. 12): H207-H218, 1982. BRISTOW, D. G., AND J. W. CLARK. A mathematical model of the vagally driven primary pacemaker. Am. J. Physiol. 244 (Heart Circ. Physiol. 13): H150-H161, 1983. BROWN, H. F., W. GILES, AND S. NOBLE. Membrane currents underlying activity in frog sinus venosus. J. Physiol. Lond. 271: 783-816, 1977. BROWN, H. F., J. KIMURA, D. NOBLE, S. NOBLE, AND A. TAUPIGNON. The ionic currents underlying pacemaker activity in rabbit sino-atria1 node: experimental results and computer simulations. Proc. R. Sot. Lond. B Biol. Sci. 222: 329-347, 1984. CAMPBELL, D. L. Calcium Current and Calcium Homeostasis in B&frog AtriaL Myocytes (PhD thesis). Galveston: Univ. of Texas, 1986. CAMPBELL, D. L., AND W. R. GILES. Calcium in cardiac muscle. In: Cakium and the Heart, edited by G. Langer. New York: Raven, 1989, p. 27-83. CAMPBELL, D. L., W. R. GILES, J. R. HUME, D. NOBLE, AND E. F. SHIBATA. Reversal potential of the calcium current in bull-frog atria1 myocytes. J. Physiol. Lond. 403: 276-286, 1988. CAMPBELL, D. L., W. GILES, K. ROBINSON, AND E. F. SHIBATA. Studies of the sodium-calcium exchanger in bull-frog atria1 myocytes. J. Physiol. Lond. 403: 317-340, 1988. CAMPBELL, D. L., R. L. RASMUSSON, AND H. C. STRAUSS. Theoretical study of the voltage and concentration dependence of the anomalous mole fraction effect in single calcium channels. Biophys. J. 54: 945-954, 1988. CANNELL, M. B., AND D. G. ALLEN. Model of calcium movements during activation in the sarcomere of frog skeletal muscle. Biophys. J. 45: 913-925,1984. CARONI, P., M. ZURINI, A. CLARK, AND E. CARAFOLI. Further characterization and reconstitution of the purified Ca-pumping ATPase of heart sarcolemma. J. BioL. Chem. 258: 7305-7310, 1983. CHAD, J. E., AND R. ECKERT. Calcium “domains” associated with individual channels may account for anomalous voltage-relations of Ca-dependent responses. Biophys. J. 45: 993-999, 1984. CHAMPIGNY, G., P. BOIS, AND J. LENFANT. Characterization of the ionic mechanism responsible for the hyperpolarization-activated current in frog sinus venosus. Pfluegers Arch. 410: 159-164, 1987. CHAMPIGNY, G., AND J. LENFANT. Block and activation of the hyperpolarization-activated inward current by Ba and Cs in frog sinus venosus. Pfluegers Arch. 407: 684-690, 1986. CHAPMAN, J. B., E. A. JOHNSON, AND J. M. KOOTSEY. Electrical and biochemical properties of an enzyme model of the sodium pump. J. Membr. Biol. 74: 139-153, 1983. CHAPMAN, J. B., J. M. KOOTSEY, AND E. A. JOHNSON. A kinetic model for determining the consequences of electrogenic active transport in cardiac muscle. J. Theor. Biol. 80: 405-424, 1979. CHAPMAN, R. A. Excitation-contraction coupling in cardiac muscle. Prog. Biophys. Mol. Biol. 35: l-52, 1979. CLAPHAM, D. E. A brief review of single channel measurements from isolated heart cells. In: The Heart Cell in Culture. edited by A. Pinson. Boca Raton, FL: CRC, 1987, p. 159-167. CLEEMAN, L., AND L. GAUGHAN. Measurement of intracellular 42K diffusion in frog ventricular strips. Pfluegers Arch. 401: lOl103, 1984. CLEEMAN, L., G. PIZARRO, AND M. MORAD. Optical measurements of extracellular calcium depletion during a single heartbeat. Science Wash. DC 226: 174-177, 1984. COHEN, I. S., N. B. DAYTNER, G. A. GINTANT, AND R. P. KLINE. Time-dependent outward currents in the heart. In: The Heart and

Downloaded from www.physiology.org/journal/ajpheart by ${individualUser.givenNames} ${individualUser.surname} (130.070.008.131) on January 17, 2019.

H368

27.

28.

29.

30. 31.

32. 33.

34.

35.

36.

37.

38.

39.

40.

41. 42. 43.

44.

45.

46.

47.

48. 49.

50.

SINUS

VENOSUS

Cardiovascular System: Scientific Foundations, edited by H. A. Fozzard, E. Haber, R. B. Jennings, and A. M. Katz. New York: Raven, 1986. COHEN, I. S., N. B. DAYTNER, G. A. GINTANT, N. K. MULRINE, AND P. PENNEFATHER. Properties of an electrogenic sodiumpotassium pump in isolated canine Purkinje myocytes. J. PhysioZ. Lond. 383: 251-267,1987. COHEN, I. S., R. P. KLINE, P. PENNEFATHER, AND N. K. MULRINE. Models of the Na/K pump in cardiac muscle predict the wrong intracellular Na+ activity. Proc. R. Sot. Lond. B BioZ. Sci. 231: 371-382,1987. COULOMB, A., AND E. CORABOEUF. Simulation of potassium accumulation in clefts of Purkinje fibres: effect on membrane electrical activity. J. Theor. BioZ. 104: 211-229, 1983. CRANK. J. The Mathematics of Diffusion. Oxford, UK: Clarendon, 1975. DE WEER, P., D. C. GADSBY, AND R. F. RAKOWSKI. Voltage dependence of the Na-K pump. Annu. Rev. Physiol. 50: 225-241, 1988. DIFRANCESCO, D. A new interpretation of the pacemaker current in calf Purkinje fibres. J. Physiol. Lond. 314: 359-376, 1981. DIFRANCESCO, D., AND D. NOBLE. A model of cardiac electrical activity incorporating ionic pumps and concentration changes. Philos. Trans. R. Sot. B BioZ. Sci. 307: 353-398, 1985. DIPOLO, R., AND L. BEAUG~. The calcium pump and sodiumcalcium exchange in squid axons. Annu. Rev. Physiol. 45: 313324, 1983. DOERR, T., R. DENGER, AND W. TRAUTWEIN. Calcium currents in single SA nodal cells of the rabbit heart studied with action potential clamp. Pfluegers Arch. 413: 599-603, 1989. EISNER, D. A., AND W. J. LEDERER. Na-Ca exchange: stoichiometry and electrogenicity. Am. J. Physiol. 248 (Cell Physiol. 17): C198-C202,1985. FABIATO, A. Calcium release in skinned cardiac cells: variations with species, tissues, and development. Federation Proc. 41: 22382244,1982. FISCHMEISTER, R., AND M. HORACKOVA. Variation of intracellustudy of ionic lar Ca” following Ca2+ entry in heart. A theoretical diffusion inside a cylindrical cell. Biophys. J. 41: 341-348, 1983. FISCHMEISTER, R., AND G. VASSORT. The electrogenic Na+/Ca2+ exchange and the cardiac electrical activity. 1. Simulation on Purkinje fibre action potential. J. Physiol. Paris 77: 705-709, 1981. FOGELSON, A. L., AND R. S. ZUCKER. Presynaptic calcium diffusion from various arrays of single channels. Biophys. J. 48: 10031017,1985. GABELLA, G. Inpocketings of the cell membrane (caveolae) in the rat myocardium. J. Ultrastruct. Res. 65: 135-147, 1978. GADSBY, D. C. The Na/K pump of cardiac cells. Annu. Rev. Biophys. Bioeng. 13: 373-398, 1984. GADSBY, D. C., J. KIMURA, AND A. NOMA. Voltage dependence of Na/K pump current in isolated heart cells. Nature Lond. 315: 63-65,1985. GILES, W. R., AND E. F. SHIBATA. Voltage clamp of bullfrog cardiac pacemaker cells: a quantitative analysis of potassium currents. J. Physiol. Lond. 368: 265-292, 1985. GILES, W., A. C. G. VAN GINNEKEN, AND E. F. SHIBATA. Ionic currents underlying pacemaker activity: a summary of voltage clamp data from single cells. In: Cardiac MuscZe: the Regulation of Excitation and Contraction, edited by R. Nathan. New York: Academic, 1986. GORMAN, A. L. F., AND M. V. THOMAS. Intracellular calcium accumulation during depolarization in a molluscan neurone. J. Physiol. Lond. 308: 259-285, 1980. HAAS, H. G., R. MEYER, H. EINWACHTER, AND W. STOCKEM. Intracellular coupling in frog heart muscle, electrophysiological and morphological aspects. Pfluegers Arch. 399: 321-335, 1983. HAGIWARA, S., AND L. BYERLY. Calcium channel. Annu. Rev. Neurosci. 4: 69-125, 1981. HAMILL, 0. P., M. E. NEHER, B. SAKMANN, AND F. J. SIGWORTH. Improved patch-clamp techniques for high-resolution current recording from cells and cell-free membrane patches. Pfluegers Arch. 391: 85-100,198l. HASUO. H.. AND K. KOKETSU. Potential denendencv of the elec-

PACEMAKER

51. 52.

53. 54.

55.

56. 57. 58.

59.

60.

61.

62. 63.

64.

65.

66.

67.

68.

69.

70.

71.

72. 73.

74.

MODEL

trogenic Na-pump current in bullfrog atria1 muscles. Jpn. J. Physiol. 35: 89-100, 1985. HESS, P., AND R. W. TSIEN. Mechanism of ion permeation through calcium channels. Nature Lond. 309: 453-456, 1984. HODGKIN, A. L., AND A. F. HUXLEY. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. Lond. 117: 500-544, 1952. HODGKIN, A. L., AND H. KEYNES. Movements of labelled calcium in giant squid axons. J. Physiol. Lond. 138: 253-281, 1957. HOLROYDE, M. J., S. P. ROBERTSON, J. D. JOHNSON, R. J. SOLARO, AND J. D. POTTER. The calcium and magnesium binding sites on cardiac troponin and their role in the regulation of myofibrillar adenosine triphosphatase. J. Biol. Chem. 255: 1166811693,1980. HUME, J. R., W. GILES, K. ROBINSON, E. F. SHIBATA, R. D. NATHAN, K. KANAI, AND R. RASM~JSSON. A time- and voltagedependent K+ current in single current cardiac cells from bullfrog atrium. J. Gen. Physiol. 88: 777-798, 1986. IRISAWA, H. Electrophysiology of single cardiac cells. Jpn. J. Physiol. 34: 375-388, 1984. JACK, J. J. B., D. NOBLE, AND R. W. TSIEN. Electric Current FZow in Excitable Cells. Oxford, UK: Clarendon, 1975. JOHNSON, E. A., J. B., CHAPMAN, AND J. M. KOOTSEY. Some electrophysiological consequences of electrogenic sodium and potassium transport in cardiac muscle: a theoretical study. J. Theor. Biol. 87: 737-756, 1980. KLINE, R. P., AND I. S. COHEN. Extracellular [K’] fluctuations in voltage-clamped canine cardiac Purkinje fibers. Biophys. J. 46: 663-668,1984. KLINE, R. P., AND M. MORAD. Potassium efflux and accumulation in heart muscle. Evidence from potassium electrode experiments. Biophys. J. 16: 367-372, 1976. KORT, A. A., AND E. G. LAKATTA. Calcium-dependent mechanical oscillations occur spontaneously in unstimulated mammalian cardiac tissues. Circ. Res. 54: 396-404, 1984. KUSHMERICK, M. J., AND R. J. PODOLSKY. Ionic mobility in muscle cells. Science Wash. DC 166: 1297-1298, 1969. LAKATTA, E. G., M. C. CAPOGROSSI, A. A. KORT, AND M. D. STERN. Spontaneous myocardial Ca oscillations: an overview with emphasis on ryanodine and caffeine. Federation Proc. 44: 29772983,1985. LEVIN, K. R., AND E. PAGE. Quantitative studies on plasmalemma1 folds and caveolae of rabbit ventricular myocardial cells. Circ. Res. 46: 244-255, 1980. MAYLIE, J., AND M. MORAD. Ionic currents responsible for the generation of the pacemaker current in the rabbit S-A node. J. Physiol. Lond. 355: 215-235, 1984. MAZZANTI, M., AND L. J. DEFELICE. Regulation of the Naconducting Ca channel during the cardiac action potential. Biophys. J. 51: 115-121,1987. MCALLISTER, R. E., D. NOBLE, AND R. W. TSIEN. Reconstruction of the electrical activity of cardiac Purkinje fibers. J. Physiol. Lond. 251: l-59,1975. MEYER, R., W. STOCKEM, M. SCMITZ, AND H. G. HAAS. Histochemical demonstration of an ATP-dependent Ca2+-pump in bullfrog myocardial cells. 2. Naturforsch. 37: 489-501, 1982. MOORE, J. W., AND F. RAMON. On numerical integration of the Hodgkin and Huxley equations for a membrane action potential. J. Theor. Biol. 45: 249-273, 1974. MOORE, L. E., R. B. CLARK, E. F. SHIBATA, AND W. R. GILES. Comparison of steady-state electrophysiological properties of isolated cells from bullfrog atrium and sinus venosus. J. Membr. Biol. 89: 131-138, 1986. MOORE, L. E., A. SCHMID, AND G. ISENBERG. Linear electrical properties of isolated cardiac cells. J. Membr. Biol. 81: 29-40, 1984. MORAD, M. Physiological implications of K accumulation in heart muscle. Federation Proc. 39: 1533-1539, 1980. MORAD, M., AND Y. GOLDMAN. Excitation-contraction coupling in heart muscle: membrane control of development of tension. Prog. Biophys. Mol. Biol. 27: 259-313, 1973. MORAD, M., Y. E. GOLDMAN, D. R. TRENTHAM. Rapid photochemical inactivation of Ca2+-antagonists shows that Ca2’ entry directly activates contraction in frog heart. Nature Lond. 304: 635-638.1983.

Downloaded from www.physiology.org/journal/ajpheart by ${individualUser.givenNames} ${individualUser.surname} (130.070.008.131) on January 17, 2019.

SINUS

VENOSUS

75. MORCOS,

PACEMAKER

N. C. Isolation of frog heart sarcolemma possessing -ATPase and Ca pump activities. Biochim. Biophys. [C a’+-Mg2+] Acta 643: 55-62, 1981. 76. MULLINS, L. J. A mechanism for Na/Ca transport. J. Gen. Physiol. 70: 681-695, 1977. 77. MULLINS, L. J. Ion Transport In Heart. New York: Raven, 1981. 78. NAKAYAMA, T., AND H. IRISAWA. Transient outward current carried by potassium and sodium in quiescent atrioventricular node cells of rabbits. Circ. Res. 57: 65-73, 1985. 79. NAKAYAMA, T., Y. KURACHI, A. NOMA, AND H. IRASAWA. Action potential and membrane currents of single pacemaker cells of the rabbit heart. Pfluegers Arch. 402: 248-257, 1984. 80. NASI, E., AND D. TILLOTSON. The rate of diffusion of Ca2+ and Ba2+ in a nerve cell body. Biophys. J. 47: 735-738, 1985. 81. NOBLE, D. A modification of the Hodgkin Huxley equations applicable to Purkinje fiber action and pacemaker potentials. J. PhysioZ. Lond. 160: 317-352, 1962. 82. NOBLE, D. Conductance mechanisms in excitable cells. In: Biomembranes 3, edited by F. Kreuzer and J. F. G. Slegers. New York: Plenum, 1972, p. 427-447. 83. NOBLE, D. The surprising heart: a review of recent progress in cardiac electrophysiology. J. Physiol. Lond. 353: l-50, 1984. 84. NOBLE, D., AND S. NOBLE. A model of S-A node electrical activity using a modification of the DiFrancesco-Noble (1984) equations. Proc. R. Sot. Lond. B Biol. Sci. 222: 295-304, 1984. 85. NOBLE, S. J. Potassium accumulation and depletion in frog atria1 muscle. J. Physiol. Lond. 258: 579-613, 1976. 86. PAGE, E., AND L. P. MCALLISTER. Quantitative electron microscopic description of heart muscle cells. Am. J. CardioZ. 31: 172-

181, 1973. 87. PAGE, S. R., AND R. NEIDERGERKE.

Structures of physiological interest in the frog heart ventricle. J. CeZZ Sci. 11: 179-203, 1972. 88. PIZZARO, G., L. CLEEMAN, AND M. MORAD. Optical measurement of voltage-dependent Ca” influx in frog heart. Proc. N&l. Acad. Sci. USA 82: 1864-1868,1985. 89. POTTER, J. D., AND H. G. ZOT. The role of actin in modulating Ca2’ binding to troponin (Abstract). Biophys. J. 37: 43A, 1982. 90. POWELL, T. Methods for the isolation and preparation of single adult myocytes. In: Biology of Isolated Adult Cardiac Myocytes,

H369

MODEL

edited by W. A. Clark, R. S. Decker, and T. K. Borge. New York: Elsevier, 1988. 91. RASMUSSON, R. L. A Model of Frog Atria1 and Sinus Electrical Activity (Master’s thesis). Houston, TX: Rice University, 1986. 92. RASMUSSON, R., J. W. CLARK, W. R. GILES, K. ROBINSON, R. B. CLARK, E. F. SHIBATA, AND D. L. CAMPBELL. A mathematical model of electrophysiological activity in a bullfrog atria1 cell. Am. J. Physiol. 259 (Heart Circ. Physiol. 28): H370-H389, 1990. 93. ROBERTSON, S., D. JOHNSON, AND J. POTTER. The time-course of Ca2’ exchange with calmodulin, troponin, parvalbumin, and myosin in response to transient increases in Ca2+. Biophys. J. 34:

559-569,198l. 94. SAFFORD, R. E., AND J. B. BASSINGTHWAIGHTE. sion in transient

and steady

136,1977. 95. SCHATZMAN,

H. J. The and of the sarcoplasmic

states

in muscle.

Calcium diffuBiophys. J. 20: 113-

calcium pump of the surface membrane reticulum. Annu. Reu. Physiol. 51: 473-

485, 1989. 96. SHIBATA, E. F. Ionic

Currents in Isolated Cardiac Pacemaker Cells (PhD thesis). Galveston: Univ. of Texas, 1984. 97. SHIBATA, E. F., AND W. R. GILES. Ionic currents that generate the spontaneous depolarization in individual cardiac pacemaker cells. Proc. N&Z. Acad. Sci. USA 82: 7796-7800, 1985. 98. SHIBATA, E. F., Y. MOMOSE, AND W. R. GILES. An electrogenic Na+/K+ pump current in individual bullfrog atria1 myocytes (Abstract). Biophys. J. 45: 136A, 1984. 99. SORDAHL, L. A. Effects of magnesium, ruthenium red, and the antibiotic ionophore A23187 on initial rates of calcium uptake and release by heart mitochondria. Arch. Biochem. Biophys. 167:

104-115, 1974. 100. TSIEN, R. W. Calcium 101.

channels in excitable cell membranes. Annu. Rev. Physiol. 45: 341-358, 1983. WIER, W. G., M. B. CANNELL, J. R. BERLIN, E. MARBAN, AND W. J. LEDERER. Cellular and subcellular heterogeneity of [Cal; in single heart cells received by fura-2. Science Wash. DC 235: 325-

328, 1987. 102. YANAGIHARA, sino-atria1 exneriments.

K., A. NOMA, AND H. IRISAWA. Reconstruction of node pacemaker potential based on the voltage clamp Jpn. J. Phvsiol. 30: 841-857. 1980.

Downloaded from www.physiology.org/journal/ajpheart by ${individualUser.givenNames} ${individualUser.surname} (130.070.008.131) on January 17, 2019.