Mechanisms of Altered Excitation-Contraction Coupling in Canine...

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ISSN: 1524-4571 Copyright © 1999 American Heart Association. All rights reserved. Print ISSN: 0009-7330. Online TX 72514 Circulation Research is published by the American Heart Association. 7272 Greenville Avenue, Dallas, 1999;84;571-586 Circ. Res. Raimond L. Winslow, Jeremy Rice, Saleet Jafri, Eduardo Marbán and Brian O'Rourke Tachycardia-Induced Heart Failure, II : Model Studies Mechanisms of Altered Excitation-Contraction Coupling in Canine http://circres.ahajournals.org/cgi/content/full/84/5/571 located on the World Wide Web at: The online version of this article, along with updated information and services, is http://www.lww.com/reprints Reprints: Information about reprints can be found online at [email protected] 410-528-8550. E-mail: Kluwer Health, 351 West Camden Street, Baltimore, MD 21202-2436. Phone: 410-528-4050. Fax: Permissions: Permissions & Rights Desk, Lippincott Williams & Wilkins, a division of Wolters http://circres.ahajournals.org/subscriptions/ Subscriptions: Information about subscribing to Circulation Research is online at by on March 23, 2010 circres.ahajournals.org Downloaded from

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  • ISSN: 1524-4571 Copyright © 1999 American Heart Association. All rights reserved. Print ISSN: 0009-7330. Online

    TX 72514Circulation Research is published by the American Heart Association. 7272 Greenville Avenue, Dallas,

    1999;84;571-586 Circ. Res.Raimond L. Winslow, Jeremy Rice, Saleet Jafri, Eduardo Marbán and Brian O'Rourke

    Tachycardia-Induced Heart Failure, II : Model StudiesMechanisms of Altered Excitation-Contraction Coupling in Canine

    http://circres.ahajournals.org/cgi/content/full/84/5/571located on the World Wide Web at:

    The online version of this article, along with updated information and services, is

    http://www.lww.com/reprintsReprints: Information about reprints can be found online at

    [email protected]. E-mail: Kluwer Health, 351 West Camden Street, Baltimore, MD 21202-2436. Phone: 410-528-4050. Fax: Permissions: Permissions & Rights Desk, Lippincott Williams & Wilkins, a division of Wolters

    http://circres.ahajournals.org/subscriptions/Subscriptions: Information about subscribing to Circulation Research is online at

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  • Mechanisms of Altered Excitation-Contraction Coupling inCanine Tachycardia-Induced Heart Failure, II

    Model Studies

    Raimond L. Winslow, Jeremy Rice, Saleet Jafri, Eduardo Marba´n, Brian O’Rourke

    Abstract—Ca21 transients measured in failing human ventricular myocytes exhibit reduced amplitude, slowedrelaxation, and blunted frequency dependence. In the companion article (O’Rourke B, Kass DA, Tomaselli GF,Kääb S, Tunin R, Marba´n E. Mechanisms of altered excitation-contraction coupling in canine tachycardia-inducedheart, I: experimental studies.Circ Res. 1999;84:562–570), O’Rourke et al show that Ca21 transients recorded inmyocytes isolated from canine hearts subjected to the tachycardia pacing protocol exhibit similar responses.Analyses of protein levels in these failing hearts reveal that both SR Ca21 ATPase and phospholamban aredecreased on average by 28% and that Na1/Ca21 exchanger (NCX) protein is increased on average by 104%. Inthis article, we present a model of the canine midmyocardial ventricular action potential and Ca21 transient. Themodel is used to estimate the degree of functional upregulation and downregulation of NCX and SR Ca21 ATPasein heart failure using data obtained from 2 different experimental protocols. Model estimates of average SR Ca21

    ATPase functional downregulation obtained using these experimental protocols are 49% and 62%. Model estimatesof average NCX functional upregulation range are 38% and 75%. Simulation of voltage-clamp Ca21 transientsindicates that such changes are sufficient to account for the reduced amplitude, altered shape, and slowedrelaxation of Ca21 transients in the failing canine heart. Model analyses also suggest that altered expression of Ca21

    handling proteins plays a significant role in prolongation of action potential duration in failing canine myocytes.(Circ Res. 1999;84:571-586.)

    Key Words: excitation-contraction couplingn heart failuren midmyocardial ventricular action potentialn Ca21 transient

    Recent studies using the canine tachycardia pacing-induced model of heart failure1–8 demonstrate thatchanges in cellular electrophysiological and excitation-contraction (E-C) coupling processes are qualitatively similarto those observed in cells isolated from failing human heart.In human heart failure,IK1 current density measured athyperpolarized membrane potentials is reduced by'50%,9,10

    and density of the transient outward currentI to1 is reduced by'75% in subepicardial11 and'40% in midmyocardial ven-tricular cells9 and is unchanged in subendocardial ventricularcells.11 The magnitude ofIK1 is reduced by'40%, and that ofI to1 by '70% in failing canine midmyocardial cells.5 Expres-sion of proteins involved in E-C coupling is also altered inhuman heart failure. Sarcoplasmic reticulum (SR) Ca21 AT-Pase mRNA level,12–16 protein level,12,17,18and uptake rate19

    are reduced by'50% in end-stage heart failure. Na1/Ca21

    exchanger (NCX) mRNA levels are increased by'55% to79%,12,20and NCX protein levels increase 36% to 160%.12,20–22

    Less information is available with regard to NCX function inheart failure. However, Reinecke et al22 reported an 89%increase in sodium-gradient–stimulated45Ca21 uptake in humanheart sarcolemmal vesicles.

    As described in the preceding article by O’Rourke et al,23

    alterations of intracellular Ca21 handling in failing caninemidmyocardial ventricular myocytes parallel those observedin human. In particular, the time constant of Ca21 uptake inthe absence of Na1/Ca21 exchange is prolonged in failingcells (576683 versus 282630 ms in controls), suggesting afunctional downregulation of the SERCA2a. This observationis consistent with Western blot analyses indicating that SRCa21 ATPase protein levels are reduced in failing heart by28%. Additionally, in the presence of cyclopiazonic acid(CPA, a blocker of the SR Ca21 ATPase pump), the timeconstant of Ca21 extrusion is larger in normal than failingcells (8136269 versus 599648 ms). This observation isconsistent with Western blot analyses indicating a 104%

    Received April 27, 1998; accepted December 18, 1998.From the Departments of Biomedical Engineering (R.L.W., J.R., S.J.) and Computer Science (R.L.W.) and Center for Computational Medicine and

    Biology (R.L.W., J.R., S.J.), The Johns Hopkins University School of Medicine and Whiting School of Engineering, and Section of Molecular andCellular Cardiology (E.M., B.O.), Division of Cardiology, Department of Medicine, The Johns Hopkins University School of Medicine, Baltimore, Md.

    This manuscript was sent to Harry A. Fozzard, Consulting Editor, for review by expert referees, editorial decision, and final disposition.Correspondence to Raimond L. Winslow, PhD, The Johns Hopkins University School of Medicine, Department of Biomedical Engineering, 411

    Traylor Research Bldg, 720 Rutland Ave, Baltimore, MD 21205. E-mail [email protected]© 1999 American Heart Association, Inc.

    Circulation Researchis available at http://www.circresaha.org

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  • increase in the level of expression of the NCX in failing cells.Taken together, these results suggest that SR Ca21 uptake isimpaired and that Ca21 extrusion via the NCX is enhanced inmyocytes isolated from the failing canine heart in a way thatis similar qualitatively to that seen in human patients.

    In this article, we use the data of O’Rourke et al23 todevelop a computational model of the action potential and ofintracellular Ca21 handling in normal and failing canineventricular myocytes using biophysically detailed descrip-tions of both sarcolemmal currents and key components ofE-C coupling. With the limits of individual alterations fixedusing experimentally derived values, the model is used toquantify the extent to which each parameter (I to1, IK1, SR Ca21

    ATPase, and NCX) contributes to the overall change inelectrical and Ca21 dynamics in heart failure. The resultssupport the hypothesis that differences in expression ofsarcolemmal ion channels and Ca21 handling proteins mea-sured experimentally are sufficient to account for the alteredaction potential waveform and Ca21 transient of the failingcanine cardiomyocyte.

    Materials and MethodsNormal Canine Ventricular Cell ModelJafri et al24 have presented a model of Ca21 handling in the guineapig ventricular myocyte that incorporates the following: (1) sar-colemmal ion currents of the Luo-Rudy phase II ventricular cellmodel,25 (2) a state model of the L-type Ca21 current in whichCa21-mediated inactivation occurs via the mechanism of modeswitching,26 (3) calcium-induced calcium release from SR viaryanodine-sensitive calcium release (RyR) channels using a modeladapted from that of Keizer and Levine,27 and (4) a restrictedsubspace located between the junctional SR (JSR) and T tubules intowhich both L-type Ca21 and RyR channels empty. The model of thecanine midmyocardial ventricular cell used in this study is derivedfrom this guinea pig ventricular cell model. All dynamic equations,parameters, and initial conditions for this new model are given in theAppendix. The following modifications to the model of Jafri et al24have been made to better represent properties of canine midmyocar-dial ventricular cells.

    I to1Canine epicardial and midmyocardial ventricular cell action poten-tials exhibit a prominent notch in phase 1 of the action potential thatresults from the presence of 2 transient outward currents: a Ca21-independent 4-aminopyridine (4-AP)–sensitive current (I to1)5,28,29and a Ca21-dependent current (I to2).29,30 The Ca

    21-independent com-ponentI to1 is modeled on the basis of the formulation of Campbell etal31 for ferret ventricular cells. PeakI to1 conductance (Gto1) wasadjusted to yield a linear plot of peak current density in response to500-ms–duration voltage-clamp stimuli from a holding potential of–80 mV, with slope 0.3 pA/pF-mV andy-intercept 4.6 pA/pF. Thisagrees well with experimental measurements reported for canineI to1at 37°C by Liu et al28 (see their Figure10B: slope, 0.28 pA/pF-mV,andy-intercept, 5 pA/pF). Activation rate constants were scaled toyield a time to peak of'8 ms at a clamp potential of110 mV (seeFigure 5B of Tseng and Hoffman).29 Inactivation rate constants wereadjusted to yield a decay time constant of'20 ms.29 The Ca21-dependent chloride (Cl–) currentI to2 was not incorporated inthis model.

    I KrThe delayed rectifier currentIK in both canine and guinea pigventricular myocytes consists of rapid- and slow-activating compo-nents known asIKr and IKs, respectively. Models ofIKr and IKs inguinea pig ventricular cells have been developed.32 These modelshave been modified to approximate properties of corresponding

    currents measured in isolated canine midmyocardial ventricularcells. IKr is described using a closed-open–state model in whichforward (K12) and backward (K21) rate constants are exponentialfunctions of voltage (V) with the following form:

    (1) K ij~V!5eaij1bijV.

    Parameters of this model are fully constrained by knowledge of thetime constantt(V), defined as

    (2) t~V!51

    K12~V!1K21~V!,

    at 2 voltages and by knowledge of the steady-state activationfunction. Activation was fit using a Boltzmann function determinedby Liu and Antzelevitch33 (see their Figure 11). The time constant ofactivation at15 mV was set to 100 ms,33 and the time constant ofdeactivation at –60 mV was set to 3000 ms,34 thereby constrainingthe rate constantsK12(V) andK21(V). A fixed increment of 27 ms wasadded to Equation 2 to bound the time constant away from 0 atdepolarized potentials. The maximum conductanceGKr was adjustedto yield a tail current density of 0.2 pA/pF in response to avoltage-clamp step to125 mV for 3.0 seconds, followed by a stepto –35 mV for 1.0 seconds, as described by Gintant.35

    I KsThe slow-activating delayed rectifier currentIKs is present in epicar-dial, midmyocardial, and endocardial canine ventricular cells.IKs ismodeled as described in Zeng et al,32 with the exception that thesteady-state activation function is fit using a Boltzmann functiondetermined by Liu and Antzelevitch.33 The voltage-dependent timeconstant is also shifted by140 mV in the depolarizing direction tofit the experimental data of Liu and Antzelevitch33 (see their Figure13). Maximum conductance (#GKs) is adjusted to yield a tail currentdensity of 0.4 pA/pF in response to 3.0-second–duration voltage-clamp steps from the holding potential of –35 to125 mV, followedby a return to the holding potential34 (see Figure 5). The Ca21

    dependence ofIKs described in the Luo-Rudy phase II guinea pigmodel is not included, as there are no experimental data constrainingthis dependence in canine ventricular cells.

    I K1IK1 is fit using data measured at 22°C in isolated canine midmyocar-dial ventricular myocytes measured by Ka¨äb et al5 and scaled to37°C. These data indicate that maximum outwardIK1 density is'2.5pA/pF at –60 mV5 (see Reference5, Figure 4B). These data alsoshow thatIK1 density is nonnegligible at voltages within the plateaurange of the canine action potential. For example,IK1 density is 0.3pA/pF at 0 mV, a value comparable with the density ofIKr during theplateau phase of the action potential. The functional representation ofIK1 in the Luo-Rudy phase II model can therefore not be used, as itapproaches 0 at plateau membrane potentials. An alternative formu-lation better approximating the canine data is presented inthe Appendix.

    I Ca,LThe model of L-type Ca21 current used is identical to the mode-switching model presented in Jafri et al,24 with 3 exceptions. First,the voltage dependence of the activation transition ratesa(V) andb(V) and the inactivation variabley(V) are shifted by110 mV in thedepolarizing direction to position the peak L-type Ca21 current inresponse to voltage-clamp stimuli at15 mV, as measured experi-mentally.5 Second, the monotonic decreasing steady-state (voltage-dependent) inactivation functiony` is modified to have an asymp-totic value of 0.2 for large positive membrane potentialsV. Thismodification reproduces the slow component of Ca21 current ob-served under voltage-clamp stimuli in canine ventricular cells.5,37Finally, peak L-type Ca21 current density is adjusted to a value of 2.5pA/pF at a clamp voltage of15 mV.

    JupIn the model of Jafri et al,24 Ca21 uptake into network SR (NSR) ismodeled using a Hill function with coefficient of 2. Reverse pump

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  • rate is assumed to be 0, and Ca21 leak from NSR to cytoplasm isassumed to be proportional to the gradient of NSR and cytosolic Ca21

    concentrations. Recently, Shannon et al38 have proposed the hypoth-esis that SR Ca21 accumulation at rest is not limited by leak of Ca21

    from SR but rather is limited by a reverse component of SR Ca21

    ATPase pump current. They have proposed a new model of the SRCa21 ATPase pump that includes forward- and reverse-currentcomponents, each with its own binding constant and peak forwardand reverse rates (denotedVmaxf and Vmaxr, respectively).39 Theforward mode exhibits slight cooperativity, whereas the reversemode is noncooperative. The relative magnitudes of forward- andreverse-current components determine whether SR load increases, isconstant, or decreases during diastole. The model is presented inthe Appendix.

    Failing Canine Ventricular Cell ModelKääb et al5 have shown that in the canine tachycardia pacing-inducedmodel of heart failure,I to1 and IK1 are downregulated on average by66% and 32%, respectively, in terminal heart failure. Only thenumber of expressed channels is changed; the kinetic properties ofI to1 and gating behavior ofIK1 are unaltered. On the basis of thesedata, the effects of terminal heart failure are modeled by reducing thepeak conductance ofI to1 and IK1 by the factors indicated above.Downregulation of the SR Ca21 ATPase is modeled by simultaneousscaling of both the forward and reverse maximum pump ratesVmaxfandVmaxr by a scale factor,KSR. Upregulation of the NCX is modeledby increasing a scale factor,KNaCa.

    Numerical MethodsThe dynamical equations in the Appendix are solved on a SiliconGraphics workstation using the Merson modified Runge-Kuttafourth-order adaptive step algorithm (No. 25, Reference 52), with amaximum step size of 100 microseconds and maximum errortolerance of 10–6. The error from all variables is normalized to ensurethat each contributes equally to the calculation of global error, asdescribed in Jafri et al.24 Initial conditions listed in the Appendix areused in all calculations, unless noted otherwise. These initial condi-tions were computed in response to a periodic pulse train offrequency 1 Hz and were determined immediately before the 11thpulse. Action potentials are initiated using 0.1mAmF–1 currentinjection for 500 microseconds.

    The canine ventricular cell model is used to derive quantitativeestimates of the NCX scale factorKNaCa and the SR Ca

    21 ATPasescale factorKSR from experimental data by fitting model Ca

    21

    transient decay rates to those measured experimentally. To do this, aseries of 10 voltage-clamp stimuli (–97-mV holding potential, 3-mVstep potential, and 200-ms duration) are applied at a frequency of 1Hz. Ca21 transient decay rate is estimated from response to the finalvoltage-clamp stimulus to assure that model SR Ca21 concentrationshave reached equilibrium values.

    ResultsAction Potentials and Ca21 Transients: ModelVersus Experimental ResultsFigure 1 demonstrates the ability of the model to recon-struct action potentials and Ca21 transients of both normaland failing canine midmyocardial ventricular myocytes.The solid and dotted lines in Figure 1A show experimentalmeasurements of normal and failing action potentials,respectively. Model action potentials are shown in Figure1C. In this figure, the solid line shows a normal actionpotential. The dashed line shows an action potential whenI to1 is reduced by 66% of the normal values andIK1 by 32%of the normal values (the average percentage reductionsobserved in terminal heart failure.)5 The dotted line cor-responds to these same reductions ofI to1 andIK1, in additionto a 62% reduction of the SR Ca21 ATPase pump and a

    75% increase of the NCX. These values are model-basedestimates of the average percentage change in activity ofthese proteins determined using experimentally derivedlimits on their function, as described in the followingsections.

    The model data of Figure 1C show that downregulation ofI to1 andIK1 reduces the depth of the phase 1 notch. However,notch depth is larger in the experimental measurements fromthe failing myocyte (Figure 1A, dotted line) than is predictedby the model (Figure 1C, dashed line). This greater notchdepth is due to the presence of the Ca21-dependent transientoutward currentI to2, which is not included in the model. Themost significant change in model action potential duration(APD) occurs with upregulation of the NCX and downregu-lation of the SR Ca21 ATPase (Figure 1C, dotted line). These2 changes alone increase APDs at 90% repolarization(APD90) by '200 ms.

    Figure 1D illustrates model normal (solid line) and failing(dotted line) Ca21 transients. Amplitude of the Ca21 transientis reduced significantly in the heart failure model. Ca21

    transient shape is flattened, duration is prolonged, and relax-ation is slowed. These changes are similar qualitatively tothose seen in the experimental data of Figure 1B.

    Figures 1E and 1F show L-type Ca21 and Na1/Ca21

    exchange currents for normal (solid lines) and failing (dottedlines) model cells. The reduction in peak magnitude of theL-type Ca21 current seen in Figure 1E for the failing modelcell results from downregulation ofI to1, which reduces depthof the phase 1 notch and therefore driving force during onsetof the L-type Ca21 current. Figure 1E also shows that L-typeCa21 current is increased during the later plateau phase of theaction potential in failing model cells. The mechanism of thisincrease will be considered in subsequent sections. Figure 1Fshows that Na1/Ca21 exchange operates in reverse mode,generating a net outward current during most of the plateauphase of the action potential. The magnitude of this outwardcurrent decreases during the plateau phase, and in the failingcell model the current becomes significantly smaller than theinward L-type Ca21 current.

    These simulations demonstrate the ability of the model toreproduce both normal and failing canine myocyte actionpotentials and Ca21 transients. The following sections de-scribe application of the model to estimation of the degree offunctional change in the NCX and SR Ca21 ATPase in controland failing myocytes. The approach is as follows: (1) the timeconstant of Ca21 decay (tCa) measured with SR functionblocked using CPA data is used to estimate the modelNa1/Ca21 exchange scale factorKNaCa; (2) with KNaCa fixed atthis value, the model SR Ca21 ATPase scale factorKSRrequired to reproduce thetCa measured in physiologicalsolutions is determined; (3) the SR Ca21 ATPase reduction inheart failure is cross-checked independently by determiningthe model SR Ca21 ATPase scale factor required to reproducethe tCa measured under Na1-free conditions (0-Na data) withthe model Na1/Ca21 exchange set to 0; and (4) the modelNa1/Ca21 exchange scale factor is estimated independentlyfrom tCa in physiological solutions using the estimate of SRfunction determined in step 3.

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  • Figure 1. Model vs experimental action potentials and Ca21 transients. Each action potential and Ca21 transient is in response to a1-Hz pulse train, with responses measured in the steady state. A, Experimentally measured membrane potential as a function of time innormal (solid line) and failing (dotted line) canine myocytes. B, Experimentally measured cytosolic Ca21 concentration (mmol/L) as afunction of time for normal (solid line) and failing (dotted line) canine ventricular myocytes. C, Membrane potential as a function of timesimulated using the normal canine myocyte model (solid line), the myocyte model with Ito1 and IK1 downregulation (dashed line; down-regulation by 66% and 32%, respectively), and the heart failure model (dotted line; downregulation of Ito1 and IK1 as described previ-ously, KSR50.38 corresponding to 62% downregulation and KNaCa50.53 corresponding to 75% upregulation). D, Cytosolic Ca21 con-centration (mmol/L) as a function of time simulated using the normal (solid line) and heart failure (dotted line) model, with parameters asdescribed in panel A. E, L-type Ca21 current as a function of time for the normal (solid line) and failing (dotted line) cell models. F, Na1/Ca21 exchange current as a function of time for the normal (solid line) and failing (dotted line) cell models.

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  • Estimation of NCX and SR Ca21 ATPase Activityin Normal and Failing Myocytes:CPA ExperimentsIn the preceding article by O’Rourke et al, Ca21 transients inresponse to voltage-clamp stimuli were measured in thepresence and absence of CPA, a blocker of the SR Ca21

    ATPase pump. In the presence of CPA, Ca21 transient decayrate (tCa) following termination of a depolarizing voltage stepreflects the rate of extrusion of Ca21 from the cytosol by theNCX (extrusion by the sarcolemmal Ca21 ATPase is small).Estimates of the NCX pump current scale factorKNaCa maytherefore be obtained by setting the model value ofKSR to 0and varying KNaCa until model Ca21 transient decay ratesmatch those measured experimentally in the presence ofCPA. KNaCa may then be fixed at this value andKSR varieduntil model Ca21 transient decay rate matches that measuredexperimentally using physiological solutions. This procedurecan be applied to data obtained from both normal and failingcells to assess the extent of functional upregulation and

    downregulation of the NCX and SR Ca21 ATPase in heartfailure.

    To estimateKNaCa, model KSR was set to 0, 10 voltage-clamp steps (holding potential –97 mV, step potential 3 mV,and duration 200 ms) were applied at a frequency of 1 Hz toassure that Ca21 levels in each model Ca21 pool wereequilibrated, and modeltCa was measured by fitting anexponential function to the decay phase of the final Ca21

    transient. Figure 2A plots modeltCa (ordinate, ms) as afunction of KNaCa (abscissa) withKSR50.0 (open triangles).KNaCa50.30 yields atCa equal to the average value measuredexperimentally in normal myocytes in the presence of CPA(8136269 ms). One SD of experimental variability is ac-counted for byKNaCa values in the interval (0.21, 0.48). Thissame curve shows thatKNaCa50.53 produces atCa matchingthat measured in failing myocytes in the presence of CPA(599648 ms). One SD experimental variability is encom-passed byKNaCa values in the interval (0.48, 0.60). Assumingthe normal value ofKNaCa to be 0.30, these data suggest afunctional upregulation of the NCX in heart failure in therange of 60% to 100%, with average value'75%.

    Figure 2B plots modeltCa (ordinate, ms) as a function ofKSR (abscissa). The curve marked with open circles plots thisdependence whenKNaCa is constant at the normal valueestimated above (KNaCa50.30). The experimental value oftCameasured in normal myocytes using physiological solutionsis 219636 ms. The maximum forward and reverse SR Ca21

    ATPase pump ratesVmaxf and Vmaxr given in Table 4 of theAppendix have been selected to yield a similar time constantwhen KSR51.0. Measured variation about this value is ac-counted for byKSR values in the interval (0.85, 1.15).

    The experimental value oftCa measured in failing myo-cytes using physiological solutions is 292623 ms. Depen-dence of modeltCa on KSR when KNaCa is fixed at the valueestimated for failing canine myocytes (0.53) is shown by thecurve labeled with open squares in Figure 2B.KSR50.38yields a modeltCa equal to that observed experimentally. Theexperimental deviation oftCa is accounted for byKSR valuesin the interval (0.26, 0.51). Assuming the average value ofKSR in normal cells to be 1.0, these data suggest a functionaldownregulation of the SR Ca21 ATPase pump in heart failurein the range of 49% to 74%, with average value 62%.

    Estimation of NCX and SR Ca21 ATPase Activityin Normal and Failing Myocytes:0-Na ExperimentsTo provide a second, independent measure of altered Ca21

    handling protein expression in heart failure, O’Rourke et al23

    have measuredtCa in the presence and absence of Na1/Ca21

    exchange by removing Na1 ions from both intracellular andextracellular solutions. In the absence of Na1/Ca21 exchange,tCa reflects primarily the rate of Ca21 uptake from the cytosolby the SR Ca21 ATPase pump. Estimates of the SR Ca21

    ATPase pump rate scale factorKSR under 0-Na conditionsmay therefore be obtained by setting the model value ofKNaCato 0 and varyingKSR until simulated voltage-clamp Ca21

    transient decay rates match those measured experimentally.Once the model value ofKSR is constrained,KNaCacan then bedetermined by changing its value until model Ca21 transient

    Figure 2. A, Model voltage-clamp Ca21 transient relaxation timeconstant tCa as a function of NCX scale factor KNaCa at SR Ca21

    ATPase pump scale factors of KSR50.0 (corresponding to CPAblock), 0.51 (value estimated in 0-Na experiments for failingmyocytes), and 1.0 (normal value). ‚ indicates KSR50.0; M,KSR50.51; and E, KSR51.0. B, Model Ca21 transient relaxationtime constant tCa as a function of SR Ca21 ATPase pump scalefactor KSR for KNaCa values of 0.0 (0-Na conditions), 0.30 (valueestimated in normal myocytes in presence of CPA), and 0.53(value estimated for failing myocytes during CPA block). ‚ indi-cates KNaCa50.0; M, KNaCa50.53; and E, KNaCa50.30.

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  • decay rates match those measured experimentally usingphysiological solutions. This procedure can be applied to dataobtained from both normal and failing cells to assess theextent of functional upregulation and downregulation of theNCX and SR Ca21 ATPase in heart failure.

    To mimic 0-Na conditions,KNaCa was set equal to 0.KSRwas then varied, and the time constant for Ca21 reuptake intoSR was computed. ModeltCa values are plotted as a functionof KSR in Figure 2B (open triangles). Experimentally mea-sured values of this time constant are 282630 ms in normaland 576683 ms in failing canine ventricular cells studiedunder 0-Na conditions. AKSR value of 1.0 accounts fortCameasured experimentally in normal cells (282630 ms), andvalues in the interval (0.92, 1.07) account for the observed SDin these measurements. This estimate of the averageKSR valuein normal myocytes based on block of the NCX agrees withthat estimated using the CPA data. AKSR value of 0.51accounts for the averagetCa measured experimentally infailing cells (576683 ms), andKSR values in the interval(0.46, 0.59) account for the SD. Assuming the normalKSRvalue to be 1.0, these data suggest a functional downregula-tion of the SR Ca21 ATPase pump in failing myocytes in therange of 41% to 54%, with average value 49%. This estimateof SR Ca21 ATPase downregulation is qualitatively similar tothat obtained using CPA.

    Dependence of modeltCa on KNaCa whenKSR is fixed at thevalue estimated for normal canine myocytes (1.0) is shown bythe curve labeled with open circles in Figure 2A.KNaCa50.22yields a modeltCa equal to that observed experimentally usingphysiological solutions (219636 ms). Experimental deviationof tCa is accounted for byKNaCa values in the interval (0.13,0.43). Dependence of modeltCa on KNaCawhenKSR is fixed atthe value estimated for failing canine myocytes under 0-Naconditions (0.51) is shown by the curve labeled with opensquares in Figure 2A.KNaCa50.35 yields a modeltCa equal tothe average value observed experimentally using physiolog-ical solutions (292623 ms). Experimental deviation oftCa isaccounted for byKNaCa values in the interval (0.26, 0.46).Assuming the normal value ofKNaCa to be 0.22, these datasuggest a functional upregulation of the NCX in heart failurein the range of 18% to 109%, with average value 38%. Thisestimate of altered expression of NCX in heart failure hasgreater variability than that obtained previously using theCPA data but is consistent in that it also indicates increasedexpression.

    Parametric Dependence of Voltage-Clamp Ca21

    Transients on SR Ca21 ATPase and NCX LevelsThe above analyses provide estimates ofKSR and KNaCa innormal and failing myocytes. Results indicate functionaldownregulation of the SR Ca21 ATPase pump and upregula-tion of the NCX in heart failure. The parametric dependenceof model cytosolic Ca21 transients onKSR and KNaCa isexamined next.

    Model cytosolic Ca21 concentration (ordinate,mmol/L)versus time (abscissa, seconds) is shown in Figure 3A asKSRis varied. In these simulations,KNaCa is constant at the valueestimated using CPA data from normal cells (KNaCa50.30).KSR is varied from 1.0 to 0.0 in steps of 0.1. Ca21 transients

    are in response to a 1-Hz voltage-clamp stimulus (holdingpotential –97 mV, step potential 3 mV, and duration 200 ms).Response to the final stimulus of 10 stimulus cycles is shown,with the time origin translated to 0 seconds. These data showthat reduction of the model SR Ca21 ATPase pump, simulat-ing the effects of downregulation of this pump in heartfailure, reduces the amplitude of the early peak of the Ca21

    transient (marked by the arrow). This early peak disappears asKSR approaches 0. Figure 3B shows JSR Ca21 levels for eachof the responses in Figure 3A. Reduction of the early peak inthe data of Figure 3A coincides with depletion of JSR Ca21 atsmall values ofKSR. Thus, the early peak in the model Ca21

    transient is generated by Ca21 release from JSR, and the slowsecond peak, which is present even when JSR is depleted,results from influx of Ca21 through sarcolemmal L-type Ca21

    channels and reverse-mode Na1/Ca21 exchange. AsKSRdecreases, Ca21 levels in JSR decrease, and the Ca21 transientbecomes reduced in peak amplitude. The Ca21 transientexhibits a decrease, no change, or an increase of amplitudeduring the course of the voltage-clamp stimulus, dependingon the value ofKSR. Decay rate of the Ca21 transient decreases

    Figure 3. A, Cytosolic Ca21 concentration (mmol/L) as a func-tion of time in response to a 10-second–duration periodicsequence of voltage-clamp stimuli with frequency of 1 Hz. Hold-ing potential is –97 mV, and clamp potential is 13 mV, withduration of 200 ms. Only response to the final stimulus isshown. A family of responses is shown in which KNaCa is con-stant at 0.30 (value estimated in CPA experiments with normalcanine myocytes), and KSR is varied from 1.0 to 0.0 in steps of0.1. B, Model JSR Ca21 concentration in response to the stimuliin panel A.

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  • with decreasingKSR values, as shown in the data of Figure3A, as well as Figure 2B (open circles).

    Figure 4A shows model cytosolic Ca21 concentration(ordinate,mmol/L) versus time (abscissa, seconds) asKNaCa isvaried in steps of 0.5 from 0.5 to 2.5. A plot forKNaCa50.25is also shown.KSR is constant at the value estimated usingCPA data from normal myocytes (KSR51.0). Voltage clampsteps from –97 mV to13 mV with 200 ms duration areapplied at a rate of 1.0 Hz. The final Ca21 transient in asequence of 10 is displayed, with the time origin translated to0 seconds. There are 3 effects of increasedKNaCa. These are(1) increased rate of Ca21 extrusion and lower diastolic Ca21

    at the holding potential, (2) reduction in Ca21 transientamplitude in response to the13 mV voltage-clamp step, and(3) “flattening” of the Ca21 transient during the voltage-clampstep. The increased Ca21 extrusion at the holding potential isa direct consequence of increased NCX activity when theexchanger is operating in the forward mode at the –97-mVholding potential, as shown in Figure 4B. This figure alsoshows that the NCX operates in reverse mode at the13 mVclamp potential, thus generating Ca21 influx. The reduction inCa21 transient amplitude in response to the voltage step is aconsequence of the fact that total Ca21 extrusion at theholding potential is greater than total Ca21 influx at the steppotential. This produces a smaller Ca21 transient throughreductions in SR Ca21 loading and therefore a smaller Ca21

    release. The flattening of the Ca21 transient with increasedKNaCa is a direct consequence of increased Ca21 influx duringthe voltage step, as shown in Figure 4B. DecreasedKNaCavalues also produce smaller Ca21 transient decay rates, asseen by the data of Figure 4A, as well as Figure 2A (opencircles).

    Ca21 Transients in Response to Voltage-ClampStimuli: Model Versus Experimental ResultsFigure 5A shows model Ca21 transients in response to a 1-Hzvoltage-clamp pulse train. These transients were computedusing KSR and KNaCa parameter values determined from theexperimental series in the presence and absence of CPA. Thesolid line is the normal model Ca21 transient (KNaCa50.30 andKSR51.0). The peak Ca21 level (480 nmol/L) agrees well withthe value measured experimentally in normal myocytes(450675 nmol/L).23 The dotted line is the model Ca21

    transient computed using the averageKNaCa (0.53) andKSR(0.38) values for failing myocytes. The remaining 2 Ca21

    transients (dashed lines) correspond toKNaca and KSR valuesselected at61 SD from the average for failing myocytes. Theshort dashed line represents parameter choices producing ahigh degree of SR unloading (large NCX activity,KNaCa50.60; small SR Ca21 ATPase activity,KSR50.26). Thelong-dashed line represents parameter choices that minimizeSR unloading (small NCX activity,KNaCa50.48; large SRCa21 ATPase activity,KSR50.51). These data show that asKNaCa is increased from a normal value of 0.30 (taking onvalues of 0.48, 0.53, and 0.60) andKSR is decreased from thenormal value of 1.0 (taking on values 0.51, 0.38, and 0.26),Ca21 transient peak decreases monotonically from the normalvalue of 480 nmol/L, taking on values of 300, 266, and 230nmol/L. These values agree well with the average experimen-tal values measured in failing cells of 230640 nmol/L.23

    Figure 5B shows a Ca21 transient measured experimen-tally. The amplitude and waveform of the model predictionsin Figure 5A are in close agreement with these experimentaldata.

    Figure 5C shows a plot of the L-type Ca21 current duringthe Ca21 transients of Figure 5A. The parameter changes haverelatively little effect on peak current, but increases inKNaCaordecreases inKSR produce a monotonic increase in the latecomponent of the L-type Ca21 current. As shown in Figure5D, these same parameter changes also produce monotonicdecreases of the subspace Ca21 transient peak. Thus, theincrease in the late component of L-type Ca21 current seen inFigure 5C results from a decrease in Ca21-mediated inactiva-tion of this current due to reductions in magnitude of thesubspace Ca21 transient, which is in turn a consequence ofreduced SR Ca21 load. As can be appreciated by examiningthe magnitude of the change in L-type Ca21 current densitywith alterations in Ca21 handling, this late component of theL-type Ca21 current would be expected to play an importantrole in determining the action potential plateau. This suggeststhat in heart failure, alterations in the expression of Ca21

    handling proteins that decrease SR Ca21 load and reduce theamplitude of the Ca21 transient may contribute substantiallyto prolongation of APD by reducing Ca21-mediated inactiva-tion of the L-type Ca21 current.

    Figure 4. A, Cytosolic Ca21 concentration (mmol/L) as a function oftime in response to a 10-second–duration periodic sequence ofvoltage-clamp stimuli with frequency of 1 Hz. Holding potential is–97 mV, and clamp potential is 13 mV, with duration of 200 ms.Only response to the final stimulus is shown. A family of responsesis shown in which KNaCa is varied from 0.5 to 2.5 in steps of 0.5(response for KNaCa50.25 is also shown), while KSR is held constantat the value accounting for the Ca21 relaxation rate measured innormal myocytes under 0-Na conditions. B, Model NaCa exchangecurrent INaCa as a function of time in response to the stimulidescribed in Figure 3A.

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  • DiscussionIn this article, we present a model of the canine midmyocar-dial action potential and Ca21 transient. The model is used toestimate the magnitude of SR Ca21 ATPase pump rates andNCX current in normal and failing myocytes23 using 2methods. In the first method, model SR Ca21 ATPase currentis set to 0, and the NCX current is scaled to yield Ca21

    relaxation time constants in response to voltage-clamp stimulimatching those measured experimentally in normal andfailing myocytes in the presence of CPA, a blocker of the SRCa21 ATPase. The extent of functional upregulation of theNCX in heart failure estimated using this approach is in therange of 60% to 100%, with average value 75%. Havingconstrained the model NCX current, model SR Ca21 ATPasepump current is then estimated by matching the model Ca21

    relaxation rate to experimental data obtained in the absence ofCPA. Comparison of model SR Ca21 ATPase pump currentsestimated for normal and failing myocytes suggests a func-tional downregulation in heart failure in the range of 49% to74%, with average value 62%.

    In the second method, model NCX current is set to 0,and the SR Ca21 ATPase current is scaled to yield Ca21

    relaxation time constants matching those measured exper-imentally under 0-Na conditions. Functional downregula-tion of the SR Ca21 ATPase current in heart failureestimated using this approach is in the range of 41% to54%, with average value 49%. Having constrained themodel SR Ca21 ATPase current, NCX current is estimatedby matching the model Ca21 relaxation rate to experimen-tal data obtained in control intracellular and extracellularsodium concentrations. Functional NCX upregulation inheart failure estimated using this approach is in the rangeof 18% to 109%, with average value 38%.

    Analysis of protein levels in canine hearts subjected to thetachycardia pacing protocol reveal that both SR Ca21 ATPaseand phospholamban proteins are reduced on average by28%23 and that NCX protein is increased on average by104%.23 Both steady-state mRNA and expressed levels ofE-C coupling proteins in failing human ventricular cells have

    Figure 5. A, Cytosolic Ca21 concentra-tion (mmol/L) as a function of time inresponse to a 10-second–duration peri-odic sequence of voltage-clamp stimuliwith frequency of 1 Hz. Holding potentialis –97 mV, and clamp potential is 13mV, with duration of 200 ms. Onlyresponse to the final stimulus is shown.Results of 4 simulations are shown. Solidline indicates normal model Ca21 tran-sient (KNaCa50.30; KSR51.0). Dotted lineindicates the average failing model Ca21

    transient computed using values of KSRand KNaCa estimated in the presence/ab-sence of CPA, as described in the text(KSR50.38; KNaCa50.53). Short-dashedlines show failing model Ca21 transientscomputed using KSR50.26 andKNaCa50.60. Long-dashed lines show fail-ing model Ca21 transients computedusing KSR50.51 and KNaCa50.48. B,Experimentally measured cytosolic Ca21

    concentration (mmol/L) vs time inresponse to voltage-clamp stimuli (asdescribed in panel A) in normal (solidline) and failing (dotted line) canine ven-tricular myocytes. C, L-type Ca21 currentas a function of time for the stimuli andmodel parameters described in panel A.D, Subspace Ca21 concentration(mmol/L) as a function of time for thestimuli and model parameters describedin panel A.

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  • been measured. The majority of reports agree that there is a'50% reduction of: (1) mRNA encoding the SR Ca21

    ATPase pump,12–16 (2) expressed SR Ca21 ATPase proteinlevel,12,17,18and (3) direct SR Ca21 ATPase uptake rate duringheart failure.19 There is a 55% to 79% increase in Na-Caexchanger mRNA levels,12,20 a 36% to 160% increase inexpressed protein levels,12,20–22and an approximate factor of2 increase in Na1/Ca21 exchange activity in human heartfailure.22

    The model-based estimates of functional upregulation anddownregulation of the NCX and SR Ca21 ATPase pumpreported here are consistent with these reports. Model esti-mates of average SR Ca21 ATPase functional downregulationare 49% and 62%, depending on the estimation methods used.These values agree well with estimates of mRNA level,protein level, and SR Ca21 ATPase uptake rate measured inhuman heart failure, but suggest a slightly larger degree ofdownregulation than indicated by measurements of proteinlevel in canine tachycardia pacing-induced heart failure23

    (28%). Model estimates of average NCX upregulation are38% and 75%. These estimates agree well with measuredincreases in mRNA levels in human heart failure and arewithin the range of variability of measured NCX proteinlevels in human heart failure. However, the model estimatesare slightly lower than is suggested by the increased proteinlevels measured in the failing canine heart.23

    Ca21 transients measured in failing human and canineventricular myocytes exhibit reduced amplitude and slowedrelaxation.5,40–43 Model simulations of Ca21 transients inresponse to voltage-clamp stimuli reported here demonstratethat the altered expression of the NCX and SR Ca21 ATPasepump measured in failing canine myocytes is sufficient toaccount for these properties. Both changes contribute toreduced SR Ca21 load and release and therefore reducedamplitude of the early Ca21 transient peak (Figures 3A and4A). The shape of the Ca21 transient is also controlled by bothNCX and SR Ca21 ATPase levels. As the Ca21 ATPase pumpis downregulated (Figure 3A), the shape of the plateauportion of the voltage-clamp Ca21 transient changes fromnegative to 0, then to positive slope. This change in slope isproduced by a decrease in early Ca21 release from JSR, whichin turn increases the dependence of Ca21 transient shape onCa21 entry through the L-type Ca21 channel. Upregulation ofNCX also influences Ca21 transient shape, tending to flattenthe Ca21 transient plateau by increasing reverse-mode Ca21

    entry at depolarized potentials (Figure 4A). The interplaybetween both of these factors accounts for the flattened Ca21

    transient shape seen in failing myocytes (Figure 1D, model;Figure 1B, experimental data).

    Model Ca21 transients in response to voltage-clamp stimuliexhibit a “knob” at the early peak of the transient (see Figure3A, for example) that does not appear to be present in theexperimental data. This knob disappears as the SR Ca21 levelbecomes small (Figure 3A), indicating that the knob isdependent on SR Ca21 release. The knob is likely an artifactof model construction. All SR Ca21 release in this modeloccurs from a single functional unit, defined as a set of L-typeCa21 channels, RyR channels, and the subspace within whichthey interact. Stern has referred to such models as common

    pool models.44 The knob reflects a large, single Ca21 releaseevent from this single functional unit. In contrast, real cardiaccells have a large number of functional units in which there islocal control of calcium-induced calcium release. We haverecently implemented a local control model of Ca21 releaseconsisting of an ensemble of functional units, in which eachfunctional unit is defined as an L-type Ca21 channel interact-ing with a small set of RyR channels through a diadic space.Both L-type Ca21 channels and RyR channels are modeledstochastically using the channel models presented in Jafri etal.24 In such a model, the stochastic nature of RyR channelopenings produces a variable latency of Ca21 release in eachfunctional unit. The Ca21 transients computed using thismodel exhibit the property of gradedness and do not exhibitthe knob seen in Figure 3A due to temporal smearing of Ca21

    release times.A recent study has put forth the hypothesis that coupling

    between L-type Ca21 channels and RyR channels may bealtered in heart failure and that this altered coupling leads toa reduction in amplitude of the Ca21 transient.45 The resultspresented here cannot refute this hypothesis. Indeed, struc-turally detailed models of RyR channel and L-type Ca21

    channel interactions in the diadic space predict a strongdependence of these interactions on geometric factors.46–48

    However, the results reported here indicate that such anassumption is not necessary to account for reduced amplitudeof Ca21 transients in failing myocytes. Rather, these simula-tions indicate that the altered expression of Ca21 handlingproteins reported by several different groups in both failinghuman and canine myocytes could account for changes inCa21 transient amplitude and shape.

    The data of Figure 1 demonstrate that downregulation ofthe outward repolarizing currents IK1 and I to1, together withaltered expression of the NCX and SR Ca21 ATPase pump,can account for differences in both action potential and Ca21

    transient shape in heart failure. However, the data of Figure1C also indicate that downregulation ofIK1 andI to1, at least tothe extent measured on average in failing cells, has a smalleffect on APD. Instead, altered expression of Ca21 handlingproteins plays a significant role in APD prolongation.

    It is not surprising that downregulation of modelIK1 hasonly a modest impact on APD, asIK1 is primarily responsiblefor the terminal phase of repolarization. However, the findingthat reduction of modelI to1 has only a small effect on APDdiffers from the experimental results of Ka¨äb et al5 in dogmyocytes and of Beuckelmann et al9 in human cells. Theseexperiments were performed using EGTA as an intracellularCa21 buffer. This buffering minimizes the modulatory effectsof Ca21 and thus enhances the relative influence of outwardKcurrents on action potential characteristics. When effects ofEGTA buffering are simulated in the model described in thisarticle, block of I to1 has a greater influence on APD. Anexample is shown in Figure 6. The Ca21 buffering effects ofEGTA were modeled using the fast buffering approximationdeveloped by Wagner and Keizer,49 with EGTA5 10 mmol/Land the dissociation constantKm50.15mmol/L. Block of I to1by 95% increases APD90 by 73 ms, or'25% of the controlvalue. These results again emphasize the important modula-

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  • tory role of Ca21 on action potential characteristics in thecanine myocyte.

    It is also possible that 4-AP block ofK currents other thanI to1 occurred in the Ka¨äb et al5 experiments, but that sucheffects were not resolvable. Steady-state current-voltage re-lations were measured in the presence and absence of 4-AP toassess whether or not this was the case. Data are shown inFigure 10C of Ka¨äb et al5 and indicate that experimentalvariability in steady-state current at 0 mV (a potential nearthat of the action potential plateau) is roughly61.0 pA/pF.The sum of model outward currentsI to1, IK1, IKr, andIKs duringthe plateau is comparable with the magnitude of this variabil-ity in the experimental measurements ('1.0 pA/pF). Geneticapproaches for selective suppression ofI to150,51may turn out tobe more useful than pharmacological approaches in determin-ing the influence of this current on APD.

    The model predicts that one important mechanism of APDprolongation in heart failure is that shown in Figures 1 and 5.Under conditions of reduced SR Ca21 release, there is lessCa21-mediated inactivation of the L-type Ca21 current. Theresulting increase of inward current, as shown for voltage-clamp stimuli in Figure 5C and for action potentials in Figure1E, helps to maintain and prolong the plateau phase of theaction potential. Investigation into the relative contribution ofthe various Ca21-regulatory mechanisms and Ca21-dependentmembrane currents in determining the action potential shapeand duration is an important area for future experimental andmodeling studies.

    AppendixStandard Units (Unless Otherwise Noted) Used in the FollowingSet of Equations

    Quantity Units

    Membrane potential mV

    Membrane current mAmF21

    Membrane conductance mSmF21

    Ionic flux mMms21

    Concentration mM

    Time constant ms

    Rate constant ms21

    Membrane Currents

    Na1 Current INa

    (A.1) INa5G# Nam3hj~V2ENa!

    (A.2) ENa5RT

    FlnS @Na1#o@Na1# iD

    (A.3)dm

    dt5am~12m!2bmm

    (A.4)dh

    dt5ah~12h!2bhh

    (A.5)dj

    dt5a j~12j !2b j j

    (A.6) am50.32V147.13

    12e20.1~V147.13!

    (A.7) bm50.08e2V11

    For V$240 mV,

    (A.8) ah50.0

    (A.9) bh51

    0.13~11e~V110.66!/211.1!

    (A.10) a j50.0

    (A.11) b j50.3e22.535310

    27V

    11e20.1~V132!

    For V,240 mV,

    (A.12) ah50.135e~801V!/26.8

    (A.13) bh53.56e0.079V13.13105e0.35V

    (A.14) a j5~2127140e0.2444V23.474

    31025e20.04391V)V137.78

    11e0.311~V179.23!

    (A.15) b j50.1212e20.01052V

    11e20.1378~V140.14!

    Rapid-Activating Delayed Rectifier K1 Current IKr

    (A.16) IKr5G# Kr f~@K1#o! R~V! XKr~V2EK!

    Figure 6. Membrane potential as a function of time for the nor-mal model (solid line) and for the model with a 95% reduction inmagnitude of Ito1 (dotted line). Simulations are done in the pres-ence of EGTA (10 mmol/L; Km50.15 mmol/L), using the fastbuffering approximation of Wagner and Keizer.49 Response tofifth stimulus at a cycle length of 1200 ms is shown.

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  • (A.17) EK5RT

    FlnS @K1#o@K1# iD

    (A.18) R~V!51

    111.4945e0.0446V

    (A.19) f~@K1#o!5Î@K1#o/4

    (A.20)dXKrdt

    5K12~12XKr!2K21XKr

    (A.21) K125e25.49510.1691V

    (A.22) K215e27.67720.0128V

    Slow-Activating Delayed Rectifier K1 Current IKs

    (A.23) IKs5G# KsXKs2 ~V2EKs!

    (A.24) EKs5RT

    FlnS @K1#o10.01833@Na1#o@K1# i10.01833@Na1# i D

    (A.25)dXKsdt

    5~XKs` 2XKs!/tXKs

    (A.26) XKs` 5

    1

    11e2~V224.7!/13.6

    (A.27) tXKs51

    0.0000719~V210!

    12e20.148~V210!1

    0.000131~V210!

    e0.0687~V210!21

    Transient Outward K1 Current I to1

    (A.28) I to15G# to1Xto1Yto1~V2EK!

    (A.29)dXto1

    dt5aXto1~12Xto1!2bXto1Xto1

    (A.30)dYto1dt

    5aYto1~12Yto1!2bYto1Yto1

    (A.31) aXto150.04516e0.03577V

    (A.32) bXto150.0989e20.06237V

    (A.33) aYto150.005415e2~V133.5!/5

    110.051335e2~V133.5!/5

    (A.34) bYto150.005415e~V133.5!/5

    110.051335e~V133.5!/5

    Time-Independent K1 Current IK1

    (A.35) IK15G# K1K1`~V!S @K1#o@K1#o1KmK1D ~V2EK!

    (A.36) K1`~V!5

    1

    21e1.5F

    RT~V2EK!

    Plateau K1 Current IKp

    (A.37) IKp5G# KpKp~V!~V2EK!

    (A.38) Kp~V!51

    11e~7.4882V!/5.98

    NCX Current INaCa

    (A.39)

    INaCa5kNaCa5000

    Km,Na3 1@Na1#o

    3

    1

    Km,Ca1@Ca21#o

    1

    11ksate~h21!VF/RT

    ~ehVF/RT3@Na1# i3@Ca21#o2e

    ~h21!VF/RT@Na1#o3@Ca21# i!

    Na1-K1 Pump Current INaK

    (A.40) INaK5I#NaKfNaK1

    11SKm,Nai@Na1#iD1.5

    @K1#o@K1#o1Km,Ko

    (A.41) fNaK51

    110.1245e20.1VF/RT10.0365se2VF/RT

    (A.42) s51

    7~e

    @Na1#o67.3 21!

    Sarcolemmal Ca21 Pump Current Ip(Ca)

    (A.43) I p(Ca)5I#p(Ca)@Ca21# i

    Km,p(Ca)1@Ca21# i

    Ca21 Background CurrentICa,b

    (A.44) ICa,b5G# Ca,b~V2ECa!

    (A.45) ECa5RT

    2FlnS @Ca21#o@Ca21# iD

    Na1 Background CurrentINa,b

    (A.46) INa,b5G# Na,b~V2ENa!

    Membrane Potential

    (A.47)dV

    dt52~INa1ICa1ICa,K1IKr1IKs1I to11IK11IKp1INaCa

    1INaK1I p(Ca)1ICa,b1INa,b)

    Ca21 Handling Mechanisms

    L-Type Ca21 Current ICa

    (A.48) a50.4e~V12!/10

    (A.49) b50.05e2~V12!/13

    (A.50) a95aa

    (A.51) b95b

    b

    (A.52) g50.10375@Ca21#ss

    (A.53)dC0dt

    5bC11vCCa02~4a1g!C0

    (A.54)dC1dt

    54aC012bC21v

    bCCa1

    2~b13a1ga!C1

    (A.55)dC2dt

    53aC113bC31v

    b2CCa2

    2~2b12a1ga2!C2

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  • (A.56)dC3dt

    52aC214bC41v

    b3CCa3

    2~3b1a1ga3!C3

    (A.57)dC4dt

    5aC31gO1v

    b4CCa4

    2~4b1f1ga4!C4

    (A.58)dO

    dt5fC42gO

    (A.59)dCCa0

    dt5b9CCa11gC02~4a91v!CCa0

    (A.60)dCCa1

    dt54a9CCa012b9CCa21gaC1

    2Sb913a91vbDCCa1(A.61)

    dCCa2dt

    53a9CCa113b9CCa31ga2C2

    2S2b912a91vb2DCCa2(A.62)

    dCCa3dt

    52a9CCa214b9CCa41ga3C3

    2S3b91a91vb3DCCa3(A.63)

    dCCa4dt

    5a9CCa31g9OCa1ga4C4

    2S4b91f 91vb4DCCa4(A.64)

    dOCadt

    5f 9CCa42g9OCa

    (A.65) I#Ca5P#CaCsc

    4VF 2

    RT

    0.001e2VF/RT20.341@Ca21#oe2VF/RT21

    (A.66) ICa5I#Cay$O1OCa%

    (A.67) ICa,K5P9KCsc

    y$O1OCa%

    VF2

    RT

    @K1# ieVF/RT2@K1#o

    eVF/RT21

    (A.68) P9K5P# K

    11I#Ca

    ICahalf

    (A.69)d y

    dt5

    y`2y

    ty

    (A.70) y`50.8

    11e~V112.5!/510.2

    (A.71) ty5201600

    11e~V120!/9.5

    RyR Channel (Keizer and Levine)27

    (A.72)dPC1dt

    52ka1@Ca21#ss

    n PC11ka2PO1

    (A.73)dPO1dt

    5ka1@Ca21#ss

    n PC12ka2PO12kb

    1@Ca21#ssmPO1

    1kb2PO22kc

    1PO11kc2PC2

    (A.74)dPO2dt

    5kb1@Ca21#ss

    mPO12kb2PO2

    (A.75)dPC2dt

    5kc1PO12kc

    2PC2

    (A.76) Jrel5v1~PO11PO2!~@Ca21#JSR2@Ca

    21#ss!

    SERCA2a Pump (Shannon et al)38

    (A.77) fb5~@Ca21# i/Kfb!

    Nfb

    (A.78) r b5~@Ca21#NSR/Krb!

    Nrb

    (A.79) Jup5KSRvmaxf fb2vmaxrr b

    11fb1r b

    Intracellular Ca21 Fluxes

    (A.80) Jtr5@Ca21#NSR2@Ca

    21#JSRttr

    (A.81) Jxfer5@Ca21#ss2@Ca

    21# itxfer

    (A.82) Jtrpn5d@HTRPNCa#

    dt1

    d@LTRPNCa#

    dt

    (A.83)d@HTRPNCa#

    dt5khtrpn

    1 @Ca21# i~@HTRPN#tot2@HTRPNCa#!

    2khtrpn2 @HTRPNCa#

    (A.84)d@LTRPNCa#

    dt5kltrpn

    1 @Ca21# i~@LTRPN#tot2@LTRPNCa#!

    2kltrpn2 @LTRPNCa#

    Intracellular Ion Concentrations

    (A.85)d@Na1# i

    dt52~INa1INa,b13INaCa13INaK!

    AcapCscVmyoF

    (A.86)d@K1# i

    dt52~IKr1IKs1I to11IK11IKp1ICa,K

    22INaK)AcapCscVmyoF

    (A.87)d@Ca21#i

    dt5biHJxfer2Jup2Jtrpn2~ICa,b22INaCa1Ip(Ca)!AcapCsc2VmyoFJ

    (A.88) b i5H11 @CMDN#totKmCMDN~KmCMDN1@Ca21# i!2J21

    (A.89) bss5H11 @CMDN#totKmCMDN~KmCMDN1@Ca21#ss!2J21

    (A.90) bJSR5H11 @CSQN#totKmCSQN~KmCSQN1@Ca21#JSR!2J21

    (A.91)d@Ca21#ss

    dt5bssHJrelVJSRVss 2JxferVmyoVss 2~ICa!AcapCsc2VssF J

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  • (A.92)d@Ca21#JSR

    dt5bJSR$ Jtr2Jrel%

    (A.93)d@Ca21#NSR

    dt5Jup

    VmyoVNSR

    2JtrVJSRVNSR

    TablesTABLE 1. Cell Geometry Parameters

    Parameter Definition Value

    Csc Specific membrane capacity 1.00 mF cm22

    Acap Capacative membrane area 1.53431024cm2

    Vmyo Myoplasmic volume 25.8431026 mL

    VJSR JSR volume 0.1631026 mL

    VNSR NSR volume 2.1031026 mL

    VSS Subspace volume 1.231029 mL

    TABLE 2. Standard Ionic Concentrations

    Parameter Definition Value

    [K1]o Extracellular K1 concentration 4.0 mmol/L

    [Na1]o Extracellular Na1 concentration 138.0 mmol/L

    [Ca21]o Extracellular Ca21 concentration 2.0 mmol/L

    TABLE 3. Membrane Current Parameters

    Parameter Definition Value

    F Faraday constant 96.5coulomb mmol21

    T Absolute temperature 310 K

    R Ideal gas constant 8.314 Jmol21K21

    G# Kr Peak IKr conductance 0.0034 mSmF21

    G# Ks Peak IKs conductance 0.00271 mSmF21

    G# to1 Peak Ito1 conductance 0.23815 mSmF21

    G# K1 Peak IK1 conductance 2.8 mSmF21

    G# Kp Peak IKp conductance 0.002216 mSmF21

    G# Na Peak INa conductance 12.8 mSmF21

    kNaCa Scaling factor of Na1-Ca21 exchange 0.30 mAmF21

    KmNa Na1 half-saturation constant forNa1-Ca21 exchange

    87.5 mmol/L

    KmCa Ca21 half-saturation constant forNa1-Ca21 exchange

    1.38 mmol/L

    KmK1 Ca21 half-saturation constant for IK1 13.0 mmol/L

    ksat Na1-Ca21 exchange sat. factor atnegative potentials

    0.2

    h Controls voltage dependence ofNa1-Ca21 exchange

    0.35

    I#NaK Maximum Na1-K1 pump current 0.693 mAmF21

    Km,Nai Na1 half-saturation constant forNa1-K1 pump

    10.0 mmol/L

    Km,Ko K1 half-saturation constant forNa1-K1 pump

    1.5 mmol/L

    I#p(Ca) Maximum sarcolemmal Ca21 pumpcurrent

    0.05 mAmF21

    Km,p(Ca) Half-saturation constant forsarcolemmal Ca21 pump

    0.00005 mmol/L

    G# Ca,b Maximum background Ca21 currentconductance

    0.0003842 mSmF21

    G# Na,b Maximum background Na1 currentconductance

    0.0031 mSmF21

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  • TABLE 4. SR Parameters

    Parameter Definition Value

    v1 Maximum RyR channel Ca21 flux 1.8 ms21

    Kfb Forward half-saturation constant for Ca21 ATPase 0.16831023 mmol/L

    Krb Backward half-saturation constant for Ca21 ATPase 3.29 mmol/L

    KSR Scaling factor for Ca21 ATPase 1.0

    Nfb Forward cooperativity constant for Ca21 ATPase 1.2

    Nrb Reverse cooperativity constant for Ca21 ATPase 1.0

    vmaxf Ca21 ATPase forward rate parameter 0.81331024 mmol/L ms21

    vmaxr Ca21 ATPase reverse rate parameter 0.31831023 mmol/L ms21

    ttr Time constant for transfer from NSR to JSR 0.5747 ms

    txfer Time constant from subspace to myoplasm 26.7 ms

    ka1 RyR PC1 2 PO1 rate constant 12.15310

    9 mmol/L24 ms21

    ka2 RyR PO1 2 PC1 rate constant 0.576 ms

    21

    kb1 RyR PO1 2 PO2 rate constant 4.05310

    6 mmol/L23 ms21

    kb2 RyR PO2 2 PO1 rate constant 1.930 ms

    21

    kc1 RyR PO1 2 PC2 rate constant 0.100 ms

    21

    kc2 RyR PC2 2 PO1 rate constant 0.0008 ms

    21

    n RyR Ca21 cooperativity parameter PC1 2 PO1 4

    m RyR Ca21 cooperativity parameter PO1 2 PO2 3

    TABLE 5. L-Type Ca21 Channel Parameters

    Parameter Definition Value

    f Transition rate into open state 0.3 ms21

    g Transition rate out of open state 2.0 ms21

    f 9 Transition rate into open state for mode Ca 0.005 ms21

    g 9 Transition rate out of open state for mode Ca 7.0 ms21

    b Mode transition parameter 2.0

    a Mode transition parameter 2.0

    v Mode transition parameter 0.01 ms21

    P#Ca L-type Ca21 channel permeability to Ca21 3.125 1024 cm s21

    P#K L-type Ca21 channel permeability to K1 5.79 1027 cm s21

    ICahalf ICa level that reduces P9K by half 20.265 mAmF21

    TABLE 6. Buffering Parameters

    Parameter Definition Value

    [LTRPN]tot Total troponin low-affinity site concentration 70.031023 mmol/L

    [HTRPN]tot Total troponin high-affinity site concentration 140.031023 mmol/L

    khtrpn1 Ca21 on rate for troponin high-affinity sites 20.0 mmol/L21ms21

    khtrpn2 Ca21 off rate for troponin high-affinity sites 66.031026 ms21

    kltrpn1 Ca21 on rate for troponin low-affinity sites 40.0 mmol/L21 ms21

    kltrpn2 Ca21 on rate for troponin low-affinity sites 0.040 ms21

    [CMDN]tot Total myoplasmic calmodulin concentration 50.031023 mmol/L

    [CSQN]tot Total NSR calsequestrin concentration 15.0 mmol/L

    KmCMDN Ca21 half-saturation constant for calmodulin 2.3831023 mmol/L

    KmCSQN Ca21 half-saturation constant for calsequestrin 0.8 mmol/L

    584 E-C Coupling in Heart Failure II

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  • AcknowledgmentsThis research was supported by National Science Foundationgrant BIR91-17874, National Institutes of Health grant HL60133,NIH Specialized Center of Research on Sudden Cardiac Deathgrant P50 HL52307, Silicon Graphics Inc, and the WhitakerFoundation.

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    Variable Definition Initial Value

    t Time 0.00 ms

    V Membrane potential 295.87 mV

    m INa activation gate 2.467631024

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    C4 L-type Ca21 channel closed: mode normal 0.00

    O L-type Ca21 channel open: mode normal 0.00

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    CCa1 L-type Ca21 channel closed: mode Ca 0.00

    CCa2 L-type Ca21 channel closed: mode Ca 0.00

    CCa3 L-type Ca21 channel closed: mode Ca 0.00

    CCa4 L-type Ca21 channel closed: mode Ca 0.00

    OCa L-type Ca21 channel open: mode Ca 0.00

    y ICa inactivation gate 0.7959

    [LTRPNCa] Concentration of Ca21-bound low-affinity troponin sites 5.544331023 mmol/L

    [HTRPNCa] Concentration of Ca21-bound high-affinity troponin sites 136.6431023 mmol/L

    Winslow et al March 19, 1999 585

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