Cardiac Arrhythmia After Myocardial Infarction

CARDIAC ARRHYTHMIA AFTER MYOCARDIAL INFARCTION:

INSIGHTS FROM A DYNAMIC CANINE VENTRICULAR MYOCYTE

MODEL

Thomas J. Hund

Submitted in partial fulfillment of the requirements For the degree of Doctor of Philosophy

Thesis Advisor: Dr. Yoram Rudy

Department of Biomedical Engineering CASE WESTERN RESERVE UNIVERSITY

May 2004

CASE WESTERN RESERVE UNIVERSITY

SCHOOL OF GRADUATE STUDIES

We hereby approve the dissertation of

______

candidate for the Ph.D. degree *.

(signed)______(chair of the committee)

______

(date) ______

*We also certify that written approval has been obtained for any proprietary material contained therein.

For Keila

iii

TABLE OF CONTENTS

List of Tables ...... 3

List of Figures...... 4

Acknowledgments...... 6

List of Abbreviations ...... 7

Glossary ...... 9

Chapter 1 - Introduction...... 13 Objective...... 14 Mathematical Modeling of Excitable Cells - A Brief History...... 14 Electrophysiological Remodeling...... 16 Ca2+/calmodulin-dependent protein kinase II ...... 19 Works Cited ...... 24

Chapter 2 - A mathematical model of the canine ventricular myocyte ...... 30 Introduction...... 31 Ca2+-Calmodulin-dependent Protein Kinase II...... 32 Subspace Compartment ...... 33 Ryanodine Receptor Ca2+ Release Channel...... 34 SR Ca2+-ATPase and phospholamban ...... 35 L-type Ca2+ Channel ...... 36 Two components of the delayed rectifier K+ current...... 38 Transient outward K+ current...... 39 Ca2+-dependent transient outward Cl - current...... 39 Chloride Homeostasis ...... 39 Na+-Ca2+ Exchanger...... 40 + The late Na current, INa,L ...... 40 Works Cited ...... 41

Chapter 3 - Rate dependence of action potential duration and calcium transient...... 47 Ionic mechanism of APD rate dependence (adaptation)...... 49 The CaTamp-Frequency Relationship ...... 52 Discussion...... 55 Works Cited ...... 60

Chapter 4 - Electrical remodeling in the epicardial border zone ...... 63 Introduction...... 64 Model Formulation ...... 65 Results...... 68 Discussion...... 76 Works Cited ...... 80

Chapter 5 - Discussion...... 83 Summary of Findings...... 84 Cell Modeling ...... 84 Myocardial Infarction ...... 86 CaMKII...... 86 Works Cited ...... 88

Appendix A - Conservation ...... 92 Introduction...... 94 Differential method...... 95 Algebraic method...... 96 Pacing protocols...... 97 Results...... 99 Discussion...... 105 Works Cited ...... 111

Appendix B - Model Equations ...... 113

Appendix C - Diseased Model Equations...... 136

Bibliography ...... 144

List of Tables

Table 1. Control and BZ electrophysiological properties...... 71

Table 2. Model calcium transient properties...... 71

Table 3. Abbreviations for conservation study...... 93

Table 4. Conservation initial conditions ...... 97

Table 5. Model abbreviations ...... 114

Table 6. Control ion concentrations at rest ...... 117

Table 7. BZ ion concentrations at rest ...... 137

List of Figures

Figure 1 The chain of events following a myocardial insult that creates a pro- arrhythmic substrate...... 17

Figure 2 Schematic illustrating the structure and functional states of Ca2+/calmodulin-dependent protein kinase II...... 20

Figure 3 Canine ventricular cell model schematic...... 32

Figure 4 Ratio of peak SR Ca2+ release flux to peak L-type Ca2+ channel flux vs. test potential...... 34

Figure 5 Validation of model ICa(L)...... 36

Figure 6 ICa(L) fast and slow voltage-dependent inactivation time constants fitted to canine ventricular data...... 37

Figure 7 Peak IKs and IKr tail currents compared to canine epicardial data...... 38

Figure 8 Ito1 I-V relationship compared to canine epicardial data...... 39

Figure 9 Steady-state action potentials and adaptation curve in the model and a canine epicardial myocyte...... 48

Figure 10 Ionic mechanism of APD adaptation...... 49

Figure 11 Effect of Ito1 on action potential and major ionic currents...... 50

Figure 12 APD adaptation curves with and without Ito1...... 51

Figure 13 Steady-state calcium transients measured in canine ventricular myocytes and simulated...... 52

Figure 14 The CaTamp-frequency relationship………………………………………53

Figure 15 Ionic mechanism of positive CaTamp-frequency relationship …………...54

Figure 16 Currents during the action potential in the HRd canine and LRd guinea pig cell models...... 58

Figure 17 Control and BZ model ICa(L) current-voltate (I-V) curves...... 65

Figure 18 Control and BZ INa steady-state fast inactivation gate, h∞...... 66

Figure 19 The action potential and ionic currents for the control model and BZ model...... 67

Figure 20 Adaptation curve for control and BZ models...... 68

Figure 21 Control and BZ experimental fura-2 transients compared to model Ca2+ transients at slow and fast pacing...... 69

Figure 22 Calcium handling during the AP in control and BZ during fast and slow pacing...... 72

Figure 23 Peak ICa(L) during application of voltage-pulse trains and recovery from inactivation in control and BZ models compared to experiment...... 73

Figure 24 CaT rate dependence for control model, BZ model, and control model with CaMKII suppressed...... 74

Figure 25 Effect of CaMKII suppression on AP and its rate dependence...... 75

Figure 26 Model CaT restitution compared to experimentally measured fura-2 restitution in control and BZ...... 76

Figure 27 Resting Vm as a function of time before onset of pacing in the differential method...... 99

Figure 28 Action potentials elicited with a voltage-pulse stimulus to -45 mV from resting steady state in the algebraic and differential methods...... 100

+ + Figure 29 Vm,dia, APD, [Na ]i, and [K ]i during application of a voltage-pulse train in the algebraic and differential methods...... 101

Figure 30 Vm,dia during application of a voltage-pulse train at long and short cycle lengths...... 102

+ + Figure 31 Vm,dia, APD, [Na ]i, and [K ]i vs. time during pacing with a current stimulus using the algebraic method and the differential method...... 103

Figure 32 Phase-space plots during pacing with a current stimulus for three different + initial values of [K ]i...... 105

Acknowledgments

First, I want to thank my committee, Dr. Yoram Rudy, Dr. Niels Otani, Dr. Igor

Efimov, Dr. Bob Harvey, and Dr. David Van Wagoner, for their guidance. I also thank

Yoram for his generous gift of resources and trust. Thanks to fellow Rudy labmates, past and present, for many engaging discussions and for their friendship. Jon Silva, Raja

Ghanem, and Charu Ramanathan deserve special recognition for donating their time and expertise to the entire Rudy group as computer administrators. Thanks also to Keith

Decker for proficient proofreading. Marlene Siegal has provided invaluable administrative assistance and an endless supply of peppermints. I am grateful for the extended CBRTC research community especially Dr. Niels Otani, Dr. Jianmin Cui, Dr.

Igor Efimov, and the members of their respective research groups.

Thanks and love to Keila, who has provided vital support, and to Mom, Dad,

Anita, Gina, Stephen, Aaron, Joanne, and Abe. Belleruth and Art had a very immediate role in this work as they provided shelter, food, and encouragement during the most difficult parts. Finally, thanks to members of our Cleveland community who have contributed dramatically to the quality of life, especially Steve Segar & Andy Getz, the

Schubert clan, Vanessa & Christian, the Tontis, Margaret & Kevin, the Leutenbergs, folks at WRUW, the CWRU GSS, students at Superior Elementary and Cleveland

State…

This work was supported by Grants R01-HL49054 and R37-HL33343 (to Yoram

Rudy) from the National Institutes of Health – National Heart, Lung and Blood Institute, by a Whitaker Foundation Development Award, and a departmental NIH training grant.

List of Abbreviations

APD Action potential duration measured at 90% repolarization APA Action potential amplitude, mV dV/dtmax Maximum rate of rise of the action potential upstroke, mV/ms Vm,rest Resting membrane potential, mV Vm,dia Maximum diastolic membrane potential, mV

CaT Calcium transient CaTamp Calcium transient amplitude W1/2 CaT width at half maximal amplitude, ms 2+ 2+ [Ca ]i,dia Diastolic intracellular Ca concentration, mmol/L

CaMKII Ca2+/calmodulin-dependent protein kinase II

CL Cycle length, ms

+ INa Fast Na current, µA/µF + INa,L Slowly inactivating late Na current, µA/µF

2+ 2+ ICa(L) Ca current through the L-type Ca channel, µA/µF

+ IKr Rapid delayed rectifier K current, µA/µF + IKs Slow delayed rectifier K current, µA/µF + Ito1 4AP-sensitive transient outward K current, µA/µF

2+ - Ito2 Ca -dependent transient outward Cl current, µA/µF

+ IK1 Time-independent K current, µA/µF + IKp Plateau K current, µA/µF + 2+ INaCa Na -Ca exchanger, µA/µF + + INaK Na -K pump, µA/µF 2+ Ip,Ca Sarcolemmal Ca pump, µA/µF 2+ ICa,b Background Ca current, µA/µF + - CTNaCl Na -Cl cotransporter, mmol/L per ms + - CTKCl K -Cl cotransporter, mmol/L per ms

SR Sarcoplasmic reticulum JSR Junctional SR NSR Network SR SS Ca2+ subspace

2+ Irel Ca release from JSR to myoplasm, mmol/L per ms 2+ Iup Ca uptake from myoplasm to NSR, mmol/L per ms 2+ Ileak Ca leak from JSR to myoplasm, mmol/L per ms

2+ Itr Ca transfer from NSR to JSR, mmol/L per ms

Istim Stimulus current, µA/µF

2+ CaMKbound Fraction of CaMKII binding sites bound to Ca /calmodulin CaMKactive Fraction of active CaMKII binding sites. CaMKtrap Fraction of autonomous CaMKII binding sites with trapped calmodulin. αCaMK, βCaMK Phosphorylation and dephosphorylation rates of CaMKII, respectively, ms-1

∆PCaMK CaMKII-dependent factor of substrate parameter P ∆P Maximal CaMKII-dependent change in substrate parameter P

PLB Phospholamban RyR Ryanodine receptor SR Ca2+ release channel LTCC L-type Ca2+ channel

G x Maximum conductance of channel x, mS/µF Km Half-saturation concentration, mmol/L

I x Maximum current carried through channel x, µA/µF Vm Transmembrane potential, mV

[]S o and [S ]i Extracellular and intracellular concentrations of ion S, respectively, mmol/L

2+ 2+ [Ca ]JSR Ca concentration in JSR, mmol/L 2+ 2+ [Ca ]NSR Ca concentration in NSR, mmol/L 2+ 2+ [Ca ]ss Ca concentration in subspace, mmol/L

BZ Canine epicardial border zone MI Myocardial infarction

Glossary

Action potential – Transient depolarization of the cell membrane that precedes contraction.

Action potential duration adaptation – Shortening of action potential duration as pacing rate increases.

Arrhythmia – Abnormal heart rhythm (e.g. too fast or too slow) that may lead to sudden cardiac death.

Autophosphorylation – Phosphorylation of a Ca2+/calmodulin-dependent protein kinase

II subunit by a neighboring active subunit, which increases subunit affinity for calmodulin 1000-fold. Autophosphorylation allows the kinase to remain active after intracellular calcium concentration falls to resting values.

Border zone – Thin rim of surviving tissue neighboring a myocardial infarct characterized by viable but abnormal cells and abnormal tissue structure. Arrhythmias are highly localized to the border zone.

Ca2+/calmodulin-dependent protein kinase II – Multifunctional and ubiquitous holoenzyme activated by Ca2+/calmodulin and believed to mediate a wide range of cellular functions from synaptic transmission in neurons to nitric oxide synthesis in

epithelial cells.

Calcium transient – Increase in intracellular Ca2+ during the cardiac action potential

crucial for cell contraction and heart pumping.

Calcium transient amplitude (CaTamp)-frequency relationship – Describes the change in calcium transient amplitude with a change in pacing frequency.

Calcium transient restitution – Dependence of the calcium transient on coupling interval.

Electrophysiological remodeling – Altered expression of ion channels prompted by an insult such as myocardial ischemia/infarction, rapid pacing, or heart failure.

Holoenzyme – Multiple subunit protein or combination of apoenzyme and cofactor.

Myocardial infarction – Formation of dead tissue in the heart wall following chronic

occlusion of a coronary artery.

Post-extrasystolic potentiation – Increased calcium transient amplitude with a pause

after rapid pacing.

Cardiac Arrhythmia after Myocardial Infarction:

Insights from a Dynamic Canine Ventricular Cell Model

Abstract

THOMAS JEFFREY HUND

Background: Electrophysiological remodeling of sarcolemmal ion channels in myocytes

neighboring a myocardial infarct is thought to create a substrate for life-threatening arrhythmias. Computational biology is a powerful tool in elucidating arrhythmogenic mechanisms at the cellular level, where complex interactions between ionic processes determine behavior. Methods and Results: A novel theoretical model of canine cardiac ventricular action potential (AP) and calcium cycling is used to investigate ionic mechanisms underlying Ca2+ transient (CaT) and AP duration (APD) rate dependence.

Remodeling changes measured in the 5-day old canine infarct epicardial border zone

(BZ) are integrated into the model to study ionic mechanisms underlying altered rate

dependence of AP and CaT in BZ myocytes. For the first time, Ca2+/calmodulin-

dependent protein kinase (CaMKII) regulatory pathway is integrated into a whole-cell

model, which is paced over the physiological and arrhythmic range of cycle lengths (CL).

Decreasing CL from 8,000 ms to 300 ms shortens APD due primarily to ICa(L) reduction, with additional contributions from Ito1, INaK, and late INa. CaT amplitude increases as CL

decreases from 8,000 ms to 500 ms. This positive force-frequency property depends on

CaMKII activity. We test the hypothesis that CaMKII dysfunction during infarction

contributes to abnormal rate-dependent properties of BZ myocytes. CaMKII suppression results in greater L-type calcium current reduction at fast pacing rates compared to control, loss of normal rate-dependent abbreviation of CaT, a negative relationship between CaT amplitude and frequency, and abnormal CaT restitution. Conclusions:

The model of CaMKII regulation serves as a paradigm for modeling other regulatory pathways and effects on cell function. This study identifies CaMKII dysfunction as a potential root component of the remodeling process, suggesting that pharmacological interventions targeting CaMKII may reduce heterogeneity in cardiac tissue postinfarction.

Chapter 1 - Introduction

13 14

Objective

In 2001, coronary heart disease was responsible for over 500,000 deaths in the

United States1. The immediate cause of death in the majority of these cases is thought to

have been ventricular fibrillation1. While a host of therapeutic strategies, including pharmaceuticals and medical devices, are available to help prevent fibrillation and sudden

cardiac death, clearly the need is great for a better understanding of what happens at the

cellular level and tissue level to predispose an ailing heart to life-threatening arrhythmias.

In an effort to address this need, the overall goal of our research is to gain mechanistic

insight into the initiation and maintenance of arrhythmias in the setting of myocardial

infarction (MI). To achieve this goal, we use computational biology, in close conjunction

with experimental data, to formulate hypotheses and to guide future experiments on the

mechanisms governing arrhythmogenesis during MI.

Mathematical Modeling of Excitable Cells - A Brief History

Mathematical models of excitable cells have undergone a remarkable evolution

since the pioneering work of Hodgkin and Huxley2 in the 1950’s. Hodgkin and Huxley

hypothesized the existence of transmembrane currents regulated by time-dependent and

voltage-dependent gates in the membranes of excitable cells. They also developed the

voltage-clamp experimental protocol that enabled critical experiments in giant squid

axons. Ultimately, their novel insight and technical proficiency led to the successful

reproduction of the squid nerve action potential waveform by solving a set of ordinary

differential equations with four state variables, known as the Hodgkin-Huxley equations.

In the early 1960's, two separate groups modified the Hodgkin-Huxley equations

to reproduce a cardiac-like action potential with its prominent plateau phase3,4. These early models described the time-varying behavior of the transmembrane potential, Vm, but assumed the concentrations of intracellular and extracellular ion concentrations to be fixed. DiFrancesco and Noble led a major advance in modeling complexity and utility by incorporating time-varying intracellular ion concentrations into a Purkinje cell model5.

The DiFrancesco-Noble model was the first of its kind to account for the fact that transmembrane currents are carried by ions and therefore affect the intracellular milieu.

This model also included ion pumps and exchangers responsible for maintaining homeostasis of intracellular ions and a crude Ca2+ release system. Many modern cardiac

cell models incorporate dynamic concentration changes derived from the Difrancesco-

Noble equations.

Cell models have evolved not only in their complexity but also in their diversity.

The cardiac action potential shows remarkable variability across regions of a single heart

and across species. Atrial, ventricular, nodal, and Purkinje cell models have been

developed for a wide range of species from rat to human. Since subsequent chapters deal

solely with electrophysiology and calcium handling of ventricular myocardial cells, the

rest of this survey focuses on the history of these cells. For a more thorough review of

ventricular cell modeling, including models of both the ventricular conduction system

(Purkinje fibers) and ventricular myocardium, see Noble and Rudy6. In 1977, Beeler and

Reuter published the first ventricular myocardial AP model using the Hodgkin-Huxley

formalism and voltage-clamp data from multicellular preparations7. The Beeler-Reuter

model would serve as the foundation for the Luo-Rudy phase 1 (LR1) ventricular AP

model, published almost fourteen years later8. Importantly, LR1 reformulated the fast

+ + + Na current INa, the delayed rectifier K current IK, and the inward rectifier K current IK1 based on experimental data from single-cell and single-channel recordings. A second phase of the Luo-Rudy model (LRd) incorporated dynamic concentration changes, Ca2+ release from the SR, and intracellular buffers9. Since then, LRd has continued to evolve

to account for new experimental findings10-12. Changes in ion channels during ischemia

have also been introduced into the model to study arrhythmogenesis13. LRd has also

been incorporated into multicellular models of cardiac tissue to study action potential

conduction14-17.

Today, dynamic ventricular cell models exist for guinea pig9,18-20, canine21-23, rat24, frog,25, ferret26, rabbit27, and human28,29. However, LRd remains one of the most

widely used cell models in the world.

Electrophysiological Remodeling

In response to chronic myocardial insult, the electrophysiological and structural

properties of both atrial and ventricular myocardium may change due to alterations in

gene expression, a process known as remodeling. Tachycardia, hypertrophy and heart

failure, and myocardial infarction are among the known triggers for remodeling (see

reviews30-32). In general, the insult triggers a signal (e.g. Ca2+ overload, cytokine, or

neurohormonal activation), altering gene expression, which in turn changes the properties of both sarcolemmal ion channels and gap junctions (Figure 1). It is possible for the remodeling pathway to be more complex. For example, chronic infarction leads not only to remodeling of ion channels (electrophysiological remodeling) and gap junctions

(structural remodeling) but also to remodeling of the size and shape of myocytes (cardiac

remodeling), promoting the progression of heart failure and further remodeling30.

Myocardial insult Infarction Tachycardia Electrophysiological Heart failure Remodeling Changes in ion channels

Altered Signal 2+ gene expression Fibrosis Ca overload Transcriptional Enhanced risk Neurohormonal/ Collagen Translational deposition of arrhythmia cytokine activation Post-translational

Structural Remodeling Changes in gap junction distribution

Figure 1 The chain of events following a myocardial insult that creates a pro-arrhythmic substrate.

The heart remodels as an adaptive response to myocardial insult. It has been hypothesized that during infarction, the heart remodels gap junctions to reduce infarct size33. Cardiac remodeling processes, such as cellular hypertrophy, serve to maintain

mechanical function of the heart. Even though remodeling may benefit the heart in the

short term, the end result is often maladaptive, creating a substrate that is highly prone to

arrhythmias. In a seminal study, Wiffjels and colleagues demonstrated clearly the

maladaptive properties of electrophysiological remodeling by intentionally inducing atrial

fibrillation (AF) in goats for extended periods of time34. The results of their research

revealed a relationship between the duration of AF and the ease of inducing AF after a

brief pause, suggesting that somehow the cardiac substrate had been altered in a

maladaptive way by the extended period of AF. Not only did this study clearly illustrate

electrophysiological remodeling in response to rapid pacing but also showed how

remodeling may facilitate the initiation of arrhythmias.

Ischemia and Infarction

The obstruction of a coronary artery leads to ischemia, which may cause a life-

threatening cardiac arrhythmia within minutes of artery occlusion. During this acute

phase of ischemia, the arrhythmia is caused by reentry made possible by slowed conduction and shortened action potential duration35. The presence of slow conduction

and shortened action potential duration (APD) is caused mainly by alterations in the electrophysiology of the cardiac myocyte, rather than changes in the tissue structure.

Cardiac arrhythmias may also occur hours after occlusion of a coronary artery. This

phase of ischemia, referred to as the subacute phase, is characterized by arrhythmias that

are generated by spontaneous depolarization of ischemic cells rather than by reentry35.

As in acute arrhythmias, the cell’s altered electrophysiology is the arrhythmogenic factor rather than altered tissue structure. A third class of arrhythmias occurs days to years after the original ischemic event once an infarct has formed and is in the process of healing35.

Studies have shown that the arrhythmias during the chronic phase are reentrant in nature36-38. Besides the important clinical relevance, the late stage of MI is interesting because of its complex and dynamic nature, with changes occurring in cell electrophysiology39-44 and tissue structure45-47.

MI has been intensively studied in a canine model where an infarct is created by

ligating the left anterior descending (LAD) coronary artery36,40-43,48,49. The procedure

results in a large infarct with a thin rim of surviving tissue located in the epicardial layer

overlaying the infarct. As will discussed in Chapter 4, the cells in this thin rim, also

called the epicardial border zone (BZ), are viable but unique in their electrophysiological

properties due to remodeling40-42,49-51. These cells also show reduced contractility due to abnormal calcium handling52. It is believed that electrophysiological remodeling of BZ myocytes creates a heterogeneous substrate favoring the initiation and maintenance of life-threatening arrhythmias. However, the cellular mechanism by which this occurs remains unknown. Identification of specific remodeling changes with pro-arrhythmic consequences may suggest novel pharmacological targets.

Ca2+/calmodulin-dependent protein kinase II

CaMKII structure

Ca2+/calmodulin-dependent protein kinase II (CaMKII) is a holoenzyme

belonging to a superfamily of serine/threonine Ca2+/calmodulin-dependent protein

kinases that includes myosin light chain kinase (MLCK), phosphorylase kinase, CaMKI,

CaMKIII, and CaMKIV (see reviews53-56). Because of its broad specificity, CaMKII has

also been referred to as the multifunctional Ca2+/calmodulin-dependent protein kinase55, however CaMKII is not the only multifunctional Ca2+/calmodulin-dependent protein

kinase. While myosin light chain kinase, phosphorylase kinase, and CaMKIII are dedicated to a single protein substrate54,55, CaMKI and CaMKIV share the

multifunctional nature of CaMKII.

Between 8 and 12 CaMKII subunits assemble to create an homomultimeric or

heteromultimeric structure, resembling a pinwheel. Each subunit is between 50 kDa and

60 kDa and consists of three functional domains: 1) N-terminal kinase domain with high

homology to other protein kinases, including the ATP-binding consensus sequence; 2)

Regulatory domain containing overlapping autoinhibitory and calmodulin-binding domains; 3) Association domain in the C-terminal region involved in assembly of subunits or association of holoenzyme with other proteins55 (Figure 2). Four different genes encode CaMKII isoforms α, β, δ, and γ, with every cell type containing at least one isoform54. The α and β isoforms are found only in nervous tissue while γ and δ are

distributed in most tissues53. The δ isoform is the dominant isoform found in cardiac

cells. A number of δ splice variants have been identified, including δB, localized in the

57 cell nucleus, and δC, found in the cytosol .

Autoinhibitory Protein kinase Regulatory Association domain domain domain ATP binding Ca2+/CaM Multimer formation binding Subunit structure Basal Active Autophosph’ed Functional 100% 100% states 0%

Ca2+/CaM P Ca2+

20-80% 20-80% 100% P CaM Holoenzyme Capped Autonomous Trapped

Figure 2 Schematic illustrating the structure and functional states of Ca2+/calmodulin-dependent protein kinase II.

CaMKII functional states

CaMKII can be in a number of functional states (Figure 2). In its basal state

without Ca2+/calmodulin bound, the kinase is inactive. Binding of Ca2+/calmodulin at the

C-terminal end of the regulatory domain relieves autoinhibition and yields a fully active

kinase. In its active state, a subunit may phosphorylate neighboring subunits

(autophosphorylation). Autophosphorylation at a specific threonine residue, Thr286, reduces the K0.5 of calmodulin binding to the kinase 1000-fold thereby trapping bound calmodulin56,58. The kinase remains fully active in the trapped state even after Ca2+ returns to resting values. Eventually calmodulin unbinds from the kinase; however, even in this autonomous state the kinase retains 20-80% of its maximal activity. Unbinding of calmodulin triggers a burst of Ca2+-independent autophosphorylation at specific residues

which reduces the affinity for calmodulin and caps kinase activity (even in the presence of Ca2+/calmodulin) to 20-80% of maximal activity. The kinase returns to its basal state

once complete dephosphorylation has occurred.

CaMKII cellular function

In neurons, CaMKII modulates cell excitability and synaptic transmission and

mediates some forms of cell memory (e.g. long-term potentiation). Neuronal targets for

CaMKII include tyrosine hydroxylase, synapsin I, nitric oxide synthase, and Ca2+- activated K+ channels53. Interestingly, several antidepressant and antipsychotic agents regulate CaMKII59. In smooth muscle cells, CaMKII regulates muscle contraction by

targeting caldesmon to relieve its inhibition of myosin ATPase and MLCK to decrease its

Ca2+/calmodulin sensitivity53. In epithelial cells, CaMKII affects secretion and volume

regulation via chloride channels and nonselective cation channels. CaMKII isoforms

regulate the expression of several genes, including c-fos (target for α-isoform),

53,57 interleukin-2(γ-isoform target), and atrial natriuretic factor (δB target) .

As will be discussed in Chapter 2, the δ isoform of CaMKII targets SERCA2a, phospholamban, the ryanodine receptor Ca2+ release channel, and the L-type Ca2+ channel in cardiac myocytes. This suggests an important role for CaMKII in intracellular calcium cycling. It has been hypothesized that CaMKII underlies rate-dependent changes in calcium transient relaxation60. Consistent with this hypothesis, CaMKII inhibitors

have a dramatic effect on calcium transient relaxation61 and excitation-contraction

coupling62. More recently, a role for CaMKII has been investigated in heart disease and

cardiac arrhythmia. The L-type Ca2+ current maintains the action potential plateau and is

responsible for generating potentially arrhythmogenic early after depolarizations

(EADs)63. The fact that CaMKII regulates the L-type Ca2+ channel has led some to

hypothesize that CaMKII over-activity may promote EAD formation64 and increase the

likelihood of life-threatening arrhythmias65. It has also been shown that mice

overexpressing the cytosolic splice variant of cardiac CaMKII, δC, develop hypertrophy

and heart failure66.

We propose a new paradigm for CaMKII function and role in heart disease. We

hypothesize that CaMKII mediates the normal rate-dependence of the calcium transient

amplitude and force generation. Furthermore, we propose that down-regulation of

CaMKII promotes abnormal CaT rate-dependence and compromised contractility during

MI. To test our hypotheses, we develop novel mathematical models of normal and

disease canine ventricular cells that incorporate the CaMKII regulatory pathway. Our

23 effort establishes a paradigm for inclusion of signal transduction pathways in whole cell computer models.

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Chapter 2 - A mathematical model of the canine ventricular

myocyte

Introduction

Dependence of action potential duration (APD) and Ca2+ transient (CaT) on

pacing rate is a fundamental property of cardiac myocytes that, when altered (e.g. in

electrophysiological remodeling), may promote life-threatening cardiac arrhythmias. The

canine is a common animal model used to investigate cardiac electrophysiology under

normal and diseased conditions (e.g., myocardial infarction and heart failure). We

develop here a detailed and physiologically-based mathematical canine ventricular cell

model (the Hund-Rudy dynamic, HRd, cell model) that simulates rate-dependent

phenomena associated with ion-channel kinetics, AP properties, and calcium handling.

An important rate-dependent regulatory pathway in ventricular myocytes is

mediated through the multifunctional Ca2+/calmodulin-dependent protein kinase

(CaMKII, see review1). Ca2+/calmodulin binding to a CaMKII subunit activates the enzyme. An active CaMKII subunit phosphorylates intracellular substrates including neighboring subunits (autophosphorylation), which makes CaMKII a functional sensor of

Ca2+ spike frequency2,3. In cardiomyocytes, CaMKII phosphorylates the L-type Ca2+ channel (LTCC), the ryanodine receptor Ca2+ release channel (RyR), the sarcoplasmic

reticulum Ca2+-ATPase (SR Ca2+-uptake pump) and phospholamban (PLB)4-11. The L-

2+ 2+ type Ca current (ICa(L)) is the primary SR Ca release trigger and shapes the action

2+ 2+ potential (AP) plateau, while Ca uptake (Iup) and release (Irel) from the SR govern Ca cycling. This suggests an important role for CaMKII in rate dependence of cardiac Ca2+ handling and electrophysiology. Here, we present a canine epicardial cell model that includes CaMKII and its participation in rate-dependent cellular processes. We use the model to gain new insights into ionic processes underlying AP and CaT rate dependence

and how CaMKII regulates these processes.

Complete HRd equations and variable definitions are provided in Appendix B.

Important model properties (schematic in Figure 3) are summarized below.

Ca2+-Calmodulin-dependent Protein Kinase II

Our dynamic CaMKII formulation is from Hanson et al.2 modified to respond

2+ dynamically to [Ca ]ss elevation during the CaT. We assume saturation of CaMKII by

−3 calmodulin and a half-maximal activation of the kinase (Km,CaM) at 1.5× 10 mmol/L free

Ca2+ (values between 0.7× 10−3 and 4.0× 10−3 mmol/L free Ca2+ have been cited in the literature12,13). We use phosphorylation and dephosphorylation rates an order of

INa,L ICa(L) INaCa Ip(Ca) ICa,b ICl,b INa

Ca2+ CTNaCl CaMKII Iup SER Calmodulin PLB

Irel JSR Itr NSR Ileak Calsequestrin Troponin Sarcoplasmic Reticulum

CTKCl Ito2

Ito1 IKr IKs IK1 IKp INaK IK(Na) IK(ATP)

Figure 3 Canine ventricular cell model schematic. Symbols are defined in text and Appendix B.

magnitude faster than those in the Hanson model to agree with the higher activity of

CaMKIIδ than neuronal isoforms14.

Model kinase subunits can be inactive, in a Ca2+/calmodulin-bound active state

(CaMKbound), or a “trapped” active state (CaMKtrap) in which calmodulin does not

2+ dissociate after return of [Ca ]ss to diastolic values. Activated subunits phosphorylate intracellular substrates and neighboring subunits (autophosphorylation), prompting a transition from CaMKbound to CaMKtrap. Trapped subunits are dephosphorylated at a

-1 constant rate βCaMK = 0.0007 ms , a moderate value compared to the pacing cycle length

(CL) range investigated in this study (CL between 300 and 8,000 ms).

Subspace Compartment

The junctional SR membrane is in close contact with the sarcolemmal membrane along the t-tubules, where LTCC and RyR clusters are localized15, creating a subspace

2+ 2+ where the Ca concentration ([Ca ]ss) rises faster and reaches a larger magnitude than

2+ [Ca ]i in the bulk myoplasm. We model the subspace as a compartment into which ICa(L)

2+ 2+ 2+ and RyR open, generating a local Ca concentration, [Ca ]ss, greater than [Ca ]i.

Anionic binding sites on sarcolemmal and SR membranes act as Ca2+ buffers16. CaMKII,

2+ - and Ito2 (Ca -dependent transient Cl outward current) are also located in the subspace pool.

Ryanodine Receptor Ca2+ Release

Channel 1.5 Model Our Irel formulation includes an ICa(L)- Experiment dependent activation gate, a Ca2+-dependent 1 Gain inactivation gate17,18, and modulation of 0.5 steady-state open probability by junctional Normalized to max.

2+ 2+ 19 0 SR [Ca ] and [Ca ]ss . The dependence of -20 0 20 Vtest (mV) 2+ steady-state activation (ro∞) on [Ca ]JSR, Figure 4 Ratio of peak SR Ca2+ release 19 flux to peak LTCC flux vs. test ∆ro∞,JSR, uses a Hill coefficient of 1.9 with potential (Vtest) in model (line in each panel) and experiment21 (circles in each 2+ an EC50 that depends on [Ca ]ss (EC50 = 2.6 panel). Model is clamped for 50 ms to - 40 mV holding potential followed by 50 2+ ms test pulse. at [Ca ]ss = 0.1 µM and EC50 = 10.6 at

2+ 19 [Ca ]ss = 1.0 µM ).

While it is generally accepted that the RyR is regulated by SR Ca2+ content19 and is also inactivated by cytosolic calcium17,18, the relative contribution of each process to

2+ termination and refractoriness of SR Ca release is unknown. Irel in our formulation terminates via a combination of inactivation and of SR activation-gate regulation20.

Voltage-dependent SR release gain (variable gain, shown for model and experiment21 in

Figure 4) is introduced through a multiplicative factor dependent on ICa(L) driving force.

Studies on canine SR vesicles show that phosphorylation of the RyR by CaMKII activates the channel6. While some studies have found a decrease in RyR activity in response to increased CaMKII activity22,23, drug studies24 and studies where the SR calcium content is tightly controlled25 provide strong evidence for a positive regulation of

RyR by CaMKII. Based on these data, we make the time constant of Irel inactivation (τri)

depend on CaMKII activity. Maximal CaMKII-dependent increase in τri of 10 ms yields a steady-state CaT amplitude (CaTamp) 95% greater for the control than for the model with CaMKII suppressed at rapid pacing (CL = 300 ms), which agrees with experiments using KN-93 to suppress CaMKII25.

SR Ca2+-ATPase and phospholamban

CaMKII phosphorylates the cardiac SR7, targeting the SR Ca2+ ATPase,

SERCA2a8, and phospholamban (PLB)9,11, which associates with SERCA2a to inhibit

2+ uptake. PLB phosphorylation relieves inhibition by shifting the K0.5 of Ca binding to the ATPase10. Toyofuku et al. have identified Ser38 as the site on SERCA2a phosphorylated by CaMKII8. They and others report increased maximum uptake rate

8,26 ( I up ) in response to CaMKII phosphorylation of SERCA2a , a finding which has been disputed10,27. Odermatt et al. argue that incubation of control cells in the presence of

10 EGTA destabilizes the cells, producing an apparent CaMKII-dependent change in I up .

Since then, Xu et al. have confirmed that CaMKII phosphorylation enhances I up in vitro28 and have reported phosphorylation of SERCA2a at Ser38 in vivo29, while a

different group has measured decreased I up in transgenic mice expressing a CaMKII inhibitory peptide30. Therefore, while controversy remains, mounting experimental evidence (in vitro and in vivo) supports the earlier findings that CaMKII phosphorylates

8,26 SERCA2a to increase I up . Based on these data, CaMKII phosphorylation of SR

uptake is modeled by making I up and K0.5 dependent on CaMKII activity. The maximal

8 CaMKII-dependent increase in I up is 75% , while the maximal decrease in K0.5 is 0.17

µM10.

V A test (mV) 1 -40 -20 0 20 40 B ICa(L)

F) 0

µ 0.8

A/ -1 µ 0.6 -2 -3 0.4 Inactivation Recovery from -4 to max. Normalized 0.2 inactivation -5

Current density ( density Current 0 ICa(L) -50 0 50 0 500 1000 -6 Vpre (mV) Interpulse interval (ms)

31 2+ Figure 5 (A) ICa(L) current-voltage (I-V) relationship compared to canine ventricular data . [Ca ]o = 31 2.0 mM. (B) (Left) ICa(L) voltage-dependent inactivation compared to experiment . Variable 1-s prepulse (Vpre) is followed by 10-ms holding interval at -50 mV and +80 mV test pulse. (Right) ICa(L) recovery from voltage-dependent inactivation compared to canine ventricular data35. 10-ms prepulse to +20 mV is followed by varying interpulse interval at -40 mV and +20 mV test pulse. Model Ca2+- dependent inactivation gates are clamped to one to isolate voltage-dependent inactivation.

L-type Ca2+ Channel

ICa(L) steady-state activation and current density yield a current-voltage relationship in agreement with canine ventricular data31 (Figure 5A). The formulation for

32 the time constant of activation, τd, is modified from Miyoshi et al . We assume two voltage-dependent inactivation gates with steady-state values (Figure 5B) and time constants fitted to canine ventricular data31 (Figure 6). Ca2+-dependent inactivation consists of a fast and a slow process33. Steady-state inactivation of the fast process

2+ 2+ 34 depends on [Ca ]ss and Ca entry through the LTCC (approximated as ICa(L) ), while

33 the slow process depends solely on ICa(L) . Recovery from voltage-dependent inactivation agrees with data from canine epicardial myocytes35 (Figure 5B).

Ca2+-dependent facilitation occurs via CaMKII phosphorylation of LTCC, which shifts the channel to a gating mode characterized by long openings4,5. Experiments indicate that CaMKII phosphorylation increases ICa(L) by between 40% and 50% at rapid

25,36 2+ pacing . We make the rapid Ca -dependent inactivation time constant (τfca)

25 dependent on CaMKII activity to produce a maximal increase of 40% in peak ICa(L) relative to the model with CaMKII suppressed at rapid pacing (CL = 500 ms).

37 ICa(L) is also facilitated by voltage-dependent pathways . A relatively slow (τ =

130 ms) voltage-dependent facilitation mediated by PKA has been observed in cardiac

LTCC38. More recently, a fast voltage-dependent facilitation that does not require PKA phosphorylation has been documented in HEK-293 cells expressing the α-subunit and β- subunit of the cardiac LTCC39. The facilitation process in this case has been attributed to intrinsic channel gating with τ = 10 ms39. We incorporate fast voltage-dependent

200

150 τf - Model 5 τf – Experiment

τs - Model 100 5 τs – Experiment

tau (ms) 50

0 -20 0 20 40 60

Figure 6 ICa(L) fast (thick line) and slow (thin lineV) voltage-dependent (mV) inactivation time constants fitted to canine ventricular data (circles). m

facilitation into the model by raising the activation gate, d, to a power which is a function

39 of voltage and time (τpow = 10 ms ). Without this modification, Ito1 significantly

40,41 increases peak ICa(L) (see previous canine models ) which disagrees with AP voltage- clamp experiments in canine ventricular myocytes42,43.

Two components of the delayed rectifier K+ current

In canine,

the delayed F)

µ 1.5

+ A/ rectifier K µ 1 current consists IKs IKr of a rapidly 0.5 activating 0

Tail current density ( Tail current -40 -20 0 20 40 60 80 component (I ) -40-200 20406080 Kr (mV) (mV) and a slowly

Figure 7 Peak IKs and IKr tail currents upon repolarization to -40 mV activating holding potential after 5-s test pulse compared to canine epicardial data44. component

44 (IKs) . The model IKs has fast (xs1, with time constant τxs1) and slow (xs2) activation

44 gates. Voltage dependence of τxs1 is fitted to canine data . The slow activation gate is

10 times slower than xs1 in agreement with the data. IKr has one activation gate, xr, whose

44 formulation is based on experimental data . The time constant of activation, τXr, is fit to

44 44 data from the canine . IKs and IKr conductances are chosen to agree with experiment

(Figure 7).

Transient outward K+ current

The 4AP-sensitive transient outward K+ F)

µ 40 A/ current, Ito1, incorporates activation and µ Ito1 30 inactivation kinetics of Dumaine et al45. To 20 account for the bixeponential recovery from 10 46 inactivation measured in canine , we add a second Current density ( 0 0 204060 inactivation gate with a slower time constant than Vtest (mV) the Dumaine formulation. Ito1 conductance is

Figure 8 Ito1 I-V relationship chosen to agree with canine epicardial myocyte compared to canine epicardial data47. data47 (Figure 8).

Ca2+-dependent transient outward Cl - current

Consistent with experimental findings in canine ventricular myocytes48, the Ca2+-

- dependent transient outward current (Ito2) is a ligand-gated channel selective for Cl with

2+ 48 a low Ca sensitivity (K0.5 = 0.1502 mM ). In our model, Ito2 is assumed to be in the

2+ subspace and is therefore a function of [Ca ]ss.

Chloride Homeostasis

- 49 Inclusion of Ito2 requires that the model be conservative for Cl ions . Our model

- + - 49 + - of Cl regulation includes the Na -dependent Cl cotransporter , CTNaCl, the K -Cl

50 - - 51 cotransporter , CTKCl, and a background Cl current, ICl,b. [Cl ]o = 100 mmol/L and

- resting [Cl ]i = 19 mmol/L.

Na+-Ca2+ Exchanger

+ 2+ 52 The Na -Ca exchanger (INaCa) is from Weber et al and includes allosteric

2+ 2+ interaction between intracellular Ca and the exchanger. We scale [Ca ]i by a factor of

2+ 53 1.5 to approximate higher [Ca ]i seen by the exchanger .

+ The late Na current, INa,L

Our formulation of the slowly-inactivating late sodium current INa,L, present in canine ventricular myocytes54, uses the steady-state activation curve and activation

55 kinetics from the fast sodium current INa in the LRd model . Voltage dependence of steady state inactivation is from Maltsev et al. with a voltage-independent inactivation time constant of 600 ms56. The conductance yields a current density of 0.3 µA/µF in response to a 295 ms pulse to 0 mV after a 2000 ms clamp to -130 mV to relieve inactivation (0.46 ± 0.068 pA/pF measured in canine epicardial myocytes54).

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38. Sculptoreanu A, Rotman E, Takahashi M, Scheuer T, Catterall WA. Voltage- dependent potentiation of the activity of cardiac L-type calcium channel α1 subunits due to phosphorylation by cAMP-dependent protein kinase. Proc Natl Acad Sci USA. 1993;90:10135-9.

39. Kamp TJ, Hu H, Marban E. Voltage-dependent facilitation of cardiac L-type Ca channels expressed in HEK-293 cells requires β-subunit. Am J Physiol Heart Circ Physiol. 2000;278:H126-36.

40. Greenstein JL, Wu R, Po S, Tomaselli GF, Winslow RL. Role of the calcium- independent transient outward current Ito1 in shaping action potential morphology and duration. Circ Res. 2000;87:1026-33.

41. Cabo C, Boyden P. Electrical remodeling of the epicardial border zone in the canine infarcted heart: a computational analysis. Am J Physiol Heart Circ Physiol. 2003;284:H372-H384.

42. Zygmunt AC, Robitelle DC, Eddlestone GT. Ito1 dictates behavior of ICl(Ca) during early repolarization of canine ventricle. Am J Physiol Heart Circ Physiol. 1997;273:H1096-H1106.

43. Banyasz T, Fulop L, Magyar J, Szentandrassy N, Varro A, Nanasi PP. Endocardial versus epicardial differences in L-type calcium current in canine ventricular myocytes studied by action potential voltage clamp. Cardiovasc Res. 2003;58:66-75.

44. Liu DW, Antzelevitch C. Characteristics of the delayed rectifier current (IKr and IKs) in canine ventricular epicardial, midmyocardial, and endocardial myocytes. A weaker IKs contributes to the longer action potential of the M cell. Circ Res. 1995;76:351-65.

45. Dumaine R, Towbin JA, Brugada P, Vatta M, Nesterenko DV, Nesterenko VV, Brugada J, Brugada R, Antzelevitch C. Ionic mechanisms responsible for the electrocardiographic phenotype of the Brugada syndrome are temperature dependent. Circ Res. 1999;85:803-9.

46. Liu DW, Gintant GA, Antzelevitch C. Ionic bases for electrophysiological distinctions among epicardial, midmyocardial, and endocardial myocytes from the free wall of the canine left ventricle. Circ Res. 1993;72:671-87.

47. Lue WM, Boyden PA. Abnormal electrical properties of myocytes from chronically infarcted canine heart. Alterations in Vmax and the transient outward current. Circulation. 1992;85:1175-88.

48. Collier ML, Levesque PC, Kenyon JL, Hume JR. Unitary Cl- channels activated by cytoplasmic Ca2+ in canine ventricular myocytes. Circ Res. 1996;78:936-44.

49. Kneller J, Ramirez RJ, Chartier D, Courtemanche M, Nattel S. Time-dependent transients in an ionically based mathematical model of the canine atrial action potential. Am J Physiol Heart Circ Physiol. 2002;282:H1437-H1451.

50. Piwnica-Worms D, Jacob R, Horres CR, Lieberman M. Potassium-chloride cotransport in cultured chick heart cells. Am J Physiol Cell Physiol. 1985;249:C337-44.

51. Baumgarten CM, Duncan SWN. Regulation of Cl- activity in ventricular muscle: - - + - Cl /HCO3 exchange and Na -dependent Cl cotransport. In: Dhalla NS, Pierce GN, Beamish RE, eds. Heart Function and Metabolism. Winnipeg, Cananda: Martinus Nijhoff; 1986:116-131.

52. Weber CR, Ginsburg KS, Philipson KD, Shannon TR, Bers DM. Allosteric regulation of Na/Ca exchange current by cytosolic Ca in intact cardiac myocytes. J Gen Physiol. 2001;117:119-31.

53. Weber CR, Piacentino V, Ginsburg KS, Houser S, Bers DM. Na+-Ca2+ exchange 2+ current and submembrane [Ca ]i during the cardiac action potential. Circ Res. 2002;90:182-189.

54. Zygmunt AC, Eddlestone GT, Thomas GP, Nesterenko VV, Antzelevitch C. Larger late sodium conductance in M cells contributes to electrical heterogeneity in canine ventricle. Am J Physiol Heart Circ Physiol. 2001;281:H689-97.

55. Luo CH, Rudy Y. A dynamic model of the cardiac ventricular action potential. I. Simulations of ionic currents and concentration changes. Circ Res. 1994;74:1071- 96.

56. Maltsev VA, Sabbah HN, Undrovinas AI. Late sodium current is a novel target for amiodarone: studies in failing human myocardium. J Mol Cell Cardiol. 2001;33:923-32.

Chapter 3 - Rate dependence of action potential duration and

calcium transient

47 48

The model is paced for 2000 s from rest (see Table 6 in

Appendix B for initial conditions) at a constant CL to steady state. CLs from 8,000 ms to

300 ms (frequency from 0.125 Hz to 3.3 Hz) are applied using a conservative current stimulus1 (Appendix A) with a duration of 1.0 ms and an amplitude of -80 µA/µF.

Steady-state APD (calculated at 90% repolarization) and CaTamp (calculated as peak

2+ 2+ systolic [Ca ]i - minimal diastolic [Ca ]i) are used to create the APD adaptation curve and CaTamp-frequency curve, respectively.

Figure 9A compares simulated APs to canine epicardial recordings2 after steady- state pacing at CLs between 8,000 ms and 300 ms. In model and experiment, the phase-1 notch depth and phase-2 dome amplitude decrease with CL. APD adaptation curves

(APD vs. CL) reveal close agreement between simulated and measured3 APD at every

A B Model AP

3 0 300 APD Experiment Model 40 mV 200 ms CL = 250 300 ms CL = 8,000 ms (ms) Experiment2 200 0 150

40 mV 0 2000 4000 6000 8000 200 ms Cycle length (ms)

Figure 9 (A) Steady-state AP simulated (top) and measured in a canine epicardial myocyte2 (bottom) for CLs of 8000, 4000, 2000, 1000, 500, and 300 ms. (B) Steady-state AP duration (APD) vs. CL (adaptation curve) in the model (line) and a canine epicardial myocyte3 (circles).

CL (Figure 9B). Decreasing CL from 8,000 ms to 300 ms shortens APD by 99 ms compared to 85 ms measured experimentally.

Ionic mechanism of APD rate dependence (adaptation)

To understand the mechanism underlying APD adaptation, important transmembrane currents are plotted during the AP (Figure 10A) for slow pacing (CL =

8,000 ms) and rapid pacing (CL = 300 ms). Rapid pacing reduces the sustained component of ICa(L) (Figure 10B) and decreases Ito1 due to its slow recovery kinetics

+ (Figure 10C). [Na ]i accumulation during rapid pacing increases the driving force for

INaK (Figure 10D), a repolarizing current that shortens APD. Late INa (INa,L, Figure 10E) inactivates and recovers from inactivation with a slow time course, resulting in

A CL = 8,000 ms C E 50 CL = 300 ms CL = 8,000 ms 0 20 -0.1 0 Vm

F) I to1 F) -0.2 µ µ

A/ 10 A/ (mV) µ µ ( -50 CL = 300 ms ( -150 INa +INa,L 0 -200 -100 100 ms 100 ms 100 ms

B 0 D F 0.3 INaK 0.2 IKs -2 F) F) 0.2 F) µ µ µ A/ A/ A/ 0.1 µ µ -4 µ ( ( I ( Ca(L) 0.1 -6 0 100 ms 0 100 ms 100 ms

Figure 10 Ionic mechanism of APD adaptation. (A) Vm, the transmembrane AP, (B) ICa(L), (C) Ito1 (arrow indicates peak current at each CL), (D) INaK, (E) INa+ INa,L (arrows indicate INa,L), (F) IKs. Values are shown at steady-state CLs of 8,000 ms (bold line) and 300 ms (thin line).

50 accumulation of inactivation and reduction ⎯ G =0.19mS/µF in this depolarizing current during rapid A ⎯to1 50 Gto1 =0mS/µF pacing. Due to rapid deactivation kinetics 0 Vm in canine, IKs does not accumulate with (mV) -50 rapid pacing and does not contribute -100 100 ms significantly to APD adaptation (Figure B 0 10F). -2

Interestingly, decreased Ito1, a F) µ A/

µ -4 repolarizing current, facilitates APD ( ICa(L) -6 shortening at fast rate contrary to its 100 ms expected opposite effect. The effect of Ito1 C 0.6 on APD is explored in Figure 11 where IKr 0.4 F) µ

steady-state AP at CL = 8,000 ms is shown A/ µ

( 0.2 with Ito1 present (normal maximum 0 100 ms conductance, Gto1 = 0.19 mS/µF) or absent D 0.5 (Gto1 = 0 mS/µF). At this slow rate, large INaCa F)

Ito1 produces rapid phase-1 repolarization µ

A/ 0 µ ( (Figure 11A, arrow) which increases the

-0.5 100 ms driving force for ICa(L) and enhances its voltage-dependent activation, resulting in Figure 11 (A) AP (arrow identifies rapid phase-1 repolarization) (B) ICa(L) (arrow indicates ICa(L) augmentation), (C) IKr, (D) augmentation of plateau ICa(L) (Figure 11B, INaCa, computed for steady-state pacing at CL = 8,000 ms. Values are computed with arrow). Phase-1 repolarization also Ito1 (thick line) and without Ito1 (thin line). increases IKr activation and decreases the

driving force for reverse-mode INaCa, reducing these repolarizing currents (Figure 11C and Figure 11D, respectively). By increasing ICa(L) and decreasing IKr and INaCa, Ito1 indirectly prolongs APD. During rapid pacing, Ito1 decreases due to its slow recovery from inactivation (Figure 10C) and phase-1 repolarization is slowed (Figure 10A).

Consequently, the indirect APD prolonging effect of Ito1 is suppressed, an effect that facilitates APD shortening as rate is increased. The greater APD rate adaptation when Ito1 is present in the model is clearly represented by the adaptation curves with and without

APD 250

(ms) 200

⎯ Gto1 =0.19mS/µF 150 ⎯ Gto1 =0mS/µF 0 2000 4000 6000 8000 Cycle Length (ms)

Figure 12 APD adaptation curves with Ito1 (thick line) and without Ito1 (thin line).

Ito1 (Figure 12).

In summary, the most important determinant of canine APD rate-adaptation is

ICa(L), an observation supported by the fact that eliminating the current in the model reduces adaptation (CL range of 8,000 to 300 ms) from 99 ms to 30 ms (71% decrease),

+ followed by Ito1 (Gto1 = 0 reduces adaptation by 40%), INaK (clamping [Na ]i during pacing reduces adaptation by 15%), and INa,L (eliminating IN,aL reduces adaptation by only

9%).

The CaTamp-Frequency Relationship

Steady-state CaT (Figure 13) measured in a canine epicardial myocyte4 (top) and simulated (bottom) agree with respect to amplitude and morphology at four different

4 frequencies. In simulation and experiment , CaTamp increases as pacing frequency increases from 0.25 Hz to 2.0 Hz (positive CaTamp-frequency relationship, Figure 14A).

Simulated CaMKII activity and excitation-contraction coupling (ECC) gain (Figure 14B) also increase at rapid pacing rates. PLB phosphorylation by CaMKII vs. pacing

Experiment 2 Hz (Sipido et al. Circulation, 2000)

0.25 Hz 0.5 Hz 1 Hz 450 nM

0.5 s 2 Hz Model

0.25 Hz 0.5 Hz 1 Hz 450 nM

0.5 s

Figure 13 Steady-state calcium transients (CaT) in canine ventricular myocytes4 (top) and simulated (bottom) for 0.25 Hz (CL = 4,000 ms), 0.5 Hz (CL = 2,000 ms), 1 Hz (CL = 1,000 ms), and 2 Hz (CL = 500 ms) pacing.

53 frequency in model and experiment5 show A CaT amplitude good agreement (Figure 14C). Experiment4 3 Steady-state [Ca2+] (Figure 15A), Model i 1.5 2+ [Ca ]ss (Figure 15B), minimal diastolic 1 CaMKII activity (Figure 15C), ICa(L) (Figure

15D), Iup (Figure 15E), and Irel (Figure 15F) 0.5 Normalized to 0.25 Hz 0123 Frequency (Hz) are computed during the AP for 0.125 Hz CaMKII Gain (CL = 8,000 ms) and 2.0 Hz (CL = 500 ms) B 1.5 25 to elucidate the ionic mechanism of gain 20 1 and CaTamp frequency dependence. During 15 slow and rapid pacing, a transient increase 0.5 10 2+ in [Ca ] (Figure 15B) produces a transient Normalized to 3.3 Hz ss 0 0123 Frequency (Hz) increase in CaMKII activity (Figure 15C).

However, the long diastolic interval during C PLB phosphorylation 1.5 slow pacing allows for complete Experiment5 Model0 dephosphorylation of “trapped” CaMKII 1 subunits, resulting in no accumulation from 0.5

beat-to-beat of total active CaMKII (Figure to baseline Relative 0 0123 15C, thick line). In contrast, active CaMKII Frequency (Hz)

subunits accumulate during rapid pacing Figure 14 (A) The CaTamp-frequency relationship. (B) Minimal CaMKII due to insufficient time for activity (normalized to 3.3 Hz) and excitation-contraction coupling gain. (C) PLB phosphorylation vs. pacing dephosphorlyation between beats (Figure frequency compared to experimental data5. 15C, thin line). In this way, CaMKII

54 detects Ca2+-transient frequency and contributes to frequency dependence of gain, which increases from 9 at 0.125 Hz to 18 at 2.0 Hz (Figure 14B) with a concomitant increase of

CaTamp (Figure 14A). A steep gain-frequency relationship is crucial to the positive

2+ CaTamp-frequency relationship because as pacing frequency increases, the primary Ca release trigger, ICa(L), decreases (Figure 15D, arrow). Greater Iup (Figure 15E) and Irel prolongation (Figure 15F, inset) underlie gain enhancement with increased pacing frequency.

A 0.125 Hz (CL = 8,000 ms) C E 2.0 Hz (CL = 500 ms) CaMKII 0.8 3 2+ I [Ca ] up i 10 0.6 2 M) µ ( 0.4 mol/L/ms) µ (

(% max activity) 1 0.2 0 0 0 100 ms 100 ms 100 ms

B D 0 F 25 CL = 500 ms 3 -1 CL = 8,000 ms 20 I -2 rel I 2

15 F) Ca(L)

µ -3 M) 2+ µ A/ ( 10 [Ca ] µ ss ( -4 1 CL = 500 ms (mmol/L/ms) 5 -5

0 -6 CL = 8,000 ms 0 100 ms 100 ms 100 ms

Figure 15 Ionic mechanism of positive CaTamp-frequency relationship. Quantities are shown during 2+ 2+ one AP. (A) [Ca ]i, (B) [Ca ]ss (arrows point to peak at each CL), (C) CaMKII activity, (D) ICa(L) (arrow indicates peak at each CL), (E) Iup, (F) Irel (declining phase expanded in inset) for steady- state pacing at CL = 8,000 ms (0.125 Hz, thick line) and CL = 500 ms (2.0 Hz, thin line).

Discussion

Summary of important mechanistic findings Major study findings are: 1) Canine APD adaptation is determined primarily by a reduction in ICa(L) at fast rates. 2) Ito1 contributes to APD adaptation indirectly by augmenting the phase-1 notch at slow rate, which increases the plateau component of

ICa(L) and decreases IKr and reverse-mode INaCa to prolong APD. 3) Increased INaK and reduced INa,L at fast rates also contribute to adaptation. 4) ECC gain increases with pacing frequency due to increased CaMKII activity, producing a positive CaTamp-frequency relationship for frequencies up to 2 Hz.

Comparison to existing models Several canine ventricular myocyte models exist in the literature6-8. The HRd model distinguishes itself from existing models in its ability to reproduce both the AP

(morphology and duration) and CaT over a wide CL range relevant for cardiac arrhythmias. Furthermore, HRd includes the following important physiological processes not found in existing canine models: 1) activation of CaMKII by the CaT; 2) dynamic

+ CaMKII regulation of Iup, ICa(L), and Irel; 3) the late Na current, INa,L; 4) Ito2 and dynamic chloride concentration changes; 5) a physiological but computationally efficient Irel formulation.

Ca2+ release based on “local control theory”9 has been integrated into a canine epicardial AP model10. In this paradigm, graded release occurs by statistical recruitment of individual Ca2+ release units. While local control models reproduce macroscopic release behavior based on events at the level of individual diads, their computational

56 demands discourage use in large scale (many cells and/or multiple cycles) simulations often required to model cardiac arrhythmia. Therefore, we sought to reproduce important features of local control models (variable gain and graded release) using a macroscopic and computationally-efficient approach. Our objective of developing a physiological yet tractable model also motivated our decision to retain the Hodgkin-Huxley formalism in representing transmembrane current kinetics.

The effect of CaMKII on Ca2+-handling Schouten11 first hypothesized and experiment12 later verified that CaMKII underlies rate-dependent changes in CaT duration. Through a computational approach, we show how CaMKII contributes to CaTamp rate dependence by facilitating ICa(L) and increasing ECC gain at fast pacing, consistent with experimental observations that

CaMKII enhances SR Ca2+ release efficacy13. We hope our results stimulate further experimental studies on the relationship between CaMKII and the CaTamp-frequency relationship.

Factors besides CaMKII determine the CaTamp-frequency relationship. As pacing rate increases, intracellular Na+ and Ca2+ accumulate due to a decrease in diastolic interval, during which the cell removes ions that entered during the previous beat. Ion accumulation at fast rates is CaMKII-independent and leads to a passive increase in SR load. A recent study suggests that intracellular Ca2+ buffer saturation at fast rates also contributes to positive force-frequency14. Consistent with these experimental findings,

2+ increasing Ca buffering capacity in our model blunts the positive CaTamp-frequency relationship (not shown).

The ionic mechanism of APD adaptation

IKr and IKs deactivation kinetics are important for APD adaptation. In guinea pig,

15 IKs activates and deactivates much more slowly than IKr . In canine, IKr and IKs

2 deactivation kinetics are reversed, with IKs deactivating faster than IKr , which implies different properties and ionic mechanism of APD rate-dependence in canine. Figure 16 compares rate-dependent changes in the AP (Figure 16A) and major ionic currents between the HRd canine model (Figure 16, left) and the LRd guinea-pig model16 (Figure

16, right). In the canine model, ICa(L) shows a reduction at fast rate not observed in the guinea pig model (Figure 16B). Guinea-pig adaptation is due primarily to accumulation

17 of the slow-deactivating IKs at fast rates (Figure 16C, right, arrow) . Canine IKs is small and does not accumulate between beats due to its faster deactivation (Figure 16C, left).

While guinea-pig IKs during the AP is almost twice in magnitude as IKr (Figure 16D), canine IKr is about 4 times larger than IKs (compare Figure 16C and Figure 16D). Our simulations suggest that IKs does not contribute significantly to canine APD adaptation due to its small amplitude and fast deactivation, consistent with recent canine

18,19 20 experiments . However, β-adrenergic stimulation enhances IKs possibly increasing

IKs importance in AP repolarization and adaptation in vivo.

The results also suggest an important role for Ito1 in determining APD. Consistent

21 22 with previous theoretical and experimental studies, we find that Ito1 creates a phase-1 notch that increases the driving force for ICa(L) and facilitates its activation, augmenting its early plateau phase. In addition, the phase-1 notch decreases the driving force for repolarizing currents IKr and reverse-mode INaCa. Together, these processes prolong APD.

New insight is obtained into the role of Ito1 in APD rate-adaptation: slow recovery from inactivation promotes Ito1 reduction, notch suppression, and less APD prolongation at fast

HRd model (canine) LRd model (guinea pig) CL = 2,000 ms A 50 CL = 300 ms 50

0 0 Vm Vm

(mV) -50 -50

-100 100 ms -100 100 ms B 0 0

F) -2 -2 µ A/ µ ( -4 -4 ICa(L) ICa(L) -6 100 ms -6 100 ms C 0.2 2 IKs IKs F) µ 0.1 1 A/ µ ( 0 0 100 ms 100 ms D 0.4 IKr 1 IKr F) µ

A/ 0.2 0.5 µ (

0 0 100 ms 100 ms

Figure 16 Currents during the action potential in the HRd canine (left panels) and LRd guinea pig16 (right panels) cell models. Steady-state values are shown at CL = 300 ms (thin line) and CL = 2,000 ms (thick line). (A) AP, (B) ICa(L), (C) IKs (arrow indicates IKs accumulation at fast rates in LRd), (D) IKr. rates, which enhances APD adaptation.

Our model includes Ito2, which does not play a major role in APD adaptation under control conditions (order of magnitude smaller than Ito1). However, it remains to

2+ be seen whether Ito2 will become more important under conditions of Ca overload or

23,24 during electrophysiological remodeling where Ito1 is downregulated . Future model studies will address these issues.

Limitations The HRd model is based wherever possible on recent experimental data and current understanding of cardiac electrophysiology and Ca2+-handling. However, the body of literature regarding CaMKII and its regulatory effects contains several controversies. There are disparate findings on whether CaMKII phosphorylates

SERCA2a directly to enhance SR Ca2+ uptake25 or whether CaMKII phosphorylates PLB alone26. Similarly, both increased13,27 and decreased28 RyR activity have been observed in response to increased CaMKII activity. These issues have yet to be resolved (see

Chapter 2). We hope that simulations presented here motivate experiments to fully characterize CaMKII regulatory effects on its substrates and their consequential cellular function.

In conclusion, we present a dynamic canine ventricular AP model that reproduces experimentally measured AP and CaT over a wide pacing frequency range. Given the broad frequency range during cardiac arrhythmias and growing appreciation of the interplay between calcium cycling and cellular electrophysiology as an important factor in arrhythmogenesis, this model should serve as a valuable theoretical research tool.

Works Cited

1. Hund TJ, Kucera JP, Otani NF, Rudy Y. Ionic charge conservation and long-term steady state in the Luo-Rudy dynamic model of the cardiac cell. Biophys J. 2001;81:3324-3331.

2. Liu DW, Antzelevitch C. Characteristics of the delayed rectifier current (IKr and IKs) in canine ventricular epicardial, midmyocardial, and endocardial myocytes. A weaker IKs contributes to the longer action potential of the M cell. Circ Res. 1995;76:351-65.

3. Liu DW, Gintant GA, Antzelevitch C. Ionic bases for electrophysiological distinctions among epicardial, midmyocardial, and endocardial myocytes from the free wall of the canine left ventricle. Circ Res. 1993;72:671-87.

4. Sipido KR, Volders PG, de Groot SH, Verdonck F, Van de Werf F, Wellens HJ, Vos MA. Enhanced Ca2+ release and Na/Ca exchange activity in hypertrophied canine ventricular myocytes: potential link between contractile adaptation and arrhythmogenesis. Circulation. 2000;102:2137-44.

5. Hagemann D, Kuschel M, Kuramochi T, Zhu W, Cheng H, Xiao RP. Frequency- encoding Thr17 phospholamban phosphorylation is independent of Ser16 phosphorylation in cardiac myocytes. J Biol Chem. 2000;275:22532-6.

6. Cabo C, Boyden P. Electrical remodeling of the epicardial border zone in the canine infarcted heart: a computational analysis. Am J Physiol Heart Circ Physiol. 2003;284:H372-H384.

7. Fox JJ, McHarg JL, Gilmour RF, Jr. Ionic mechanism of electrical alternans. Am J Physiol Heart Circ Physiol. 2002;282:H516-30.

8. Winslow RL, Rice J, Jafri S, Marban E, O'Rourke B. Mechanisms of altered excitation-contraction coupling in canine tachycardia-induced heart failure, II: model studies. Circ Res. 1999;84:571-86.

9. Stern M. Theory of excitation-contraction coupling in cardiac muscle. Biophys J. 1992;63:497-517.

10. Greenstein JL, Winslow RL. An integrative model of the cardiac ventricular myocyte incorporating local control of Ca2+ release. Biophys J. 2002;83:2918-45.

11. Schouten VJ. Interval dependence of force and twitch duration in rat heart explained by Ca2+ pump inactivation in sarcoplasmic reticulum. J Physiol (Lond). 1990;431:427-44.

12. Bassani RA, Mattiazzi A, Bers DM. CaMKII is responsible for activity-dependent acceleration of relaxation in rat ventricular myocytes. Am J Physiol Heart Circ Physiol. 1995;268:H703-12.

13. Li L, Satoh H, Ginsburg KS, Bers DM. The effect of Ca2+-calmodulin-dependent protein kinase II on cardiac excitation-contraction coupling in ferret ventricular myocytes. J Physiol (Lond). 1997;501:17-31.

14. Kuratomi S, Matsuoka S, Sarai N, Powell T, Noma A. Involvement of Ca2+ buffering and Na+/Ca2+ exchange in the positive staircase of contraction in guinea-pig ventricular myocytes. Pflugers Arch. 2003;446:347-355.

15. Sanguinetti MC, Jurkiewicz NK. Two components of cardiac delayed rectifier K+ current. Differential sensitivity to block by class III antiarrhythmic agents. J Gen Physiol. 1990;96:195-215.

16. Luo CH, Rudy Y. A dynamic model of the cardiac ventricular action potential. I. Simulations of ionic currents and concentration changes. Circ Res. 1994;74:1071- 96.

17. Viswanathan P, Shaw R, Rudy Y. Effects of IKr and IKs heterogeneity on action potential duration and its rate dependence: a simulation study. Circulation. 1999;99:2466-74.

18. Varro A, Balati B, Iost N, Takacs J, Virag L, Lathrop DA, Csaba L, Talosi L, Papp JG. The role of the delayed rectifier component IKs in dog ventricular muscle and Purkinje fibre repolarization. J Physiol (Lond). 2000;523:67-81.

19. Stengl M, Volders PG, Thomsen MB, Spatjens RL, Sipido KR, Vos MA. Accumulation of slowly activating delayed rectifier potassium current (IKs) in canine ventricular myocytes. J Physiol (Lond). 2003;551:777-786.

20. Sanguinetti MC, Jurkiewicz NK, Scott A, Siegl PK. Isoproterenol antagonizes prolongation of refractory period by the class III antiarrhythmic agent E-4031 in guinea pig myocytes. Mechanism of action. Circ Res. 1991;68:77-84.

21. Greenstein JL, Wu R, Po S, Tomaselli GF, Winslow RL. Role of the calcium- independent transient outward current Ito1 in shaping action potential morphology and duration. Circ Res. 2000;87:1026-33.

22. Banyasz T, Fulop L, Magyar J, Szentandrassy N, Varro A, Nanasi PP. Endocardial versus epicardial differences in L-type calcium current in canine ventricular myocytes studied by action potential voltage clamp. Cardiovasc Res. 2003;58:66-75.

23. Lue WM, Boyden PA. Abnormal electrical properties of myocytes from chronically infarcted canine heart. Alterations in Vmax and the transient outward current. Circulation. 1992;85:1175-88.

24. Netticadan T, Temsah RM, Kawabata K, Dhalla NS. Ca2+-overload inhibits the cardiac SR Ca2+-calmodulin protein kinase activity. Biochem Biophys Res Commun. 2002;293:727-32.

25. Toyofuku T, Curotto Kurzydlowski K, Narayanan N, MacLennan DH. Identification of Ser38 as the site in cardiac sarcoplasmic reticulum Ca2+-ATPase that is phosphorylated by Ca2+/calmodulin-dependent protein kinase. J Biol Chem. 1994;269:26492-6.

2+ 26. Odermatt A, Kurzydlowski K, MacLennan DH. The vmax of the Ca -ATPase of cardiac sarcoplasmic reticulum (SERCA2a) is not altered by Ca2+/calmodulin- dependent phosphorylation or by interaction with phospholamban. J Biol Chem. 1996;271:14206-13.

27. Witcher DR, Kovacs RJ, Schulman H, Cefali DC, Jones LR. Unique phosphorylation site on the cardiac ryanodine receptor regulates calcium channel activity. J Biol Chem. 1991;266:11144-52.

28. Lokuta AJ, Rogers TB, Lederer WJ, Valdivia HH. Modulation of cardiac ryanodine receptors of swine and rabbit by a phosphorylation-dephosphorylation mechanism. J Physiol (Lond). 1995;487:609-22.

Chapter 4 - Electrical remodeling in the epicardial border zone

63 64

Introduction

During the healing phase of myocardial infarction (MI), surviving tissue neighboring the infarct (the border zone, BZ) is altered in structure and electrophysiology. Clinically, there is increased likelihood of life-threatening reentrant tachyarrhythmias during this late stage of infarction1.

Arrhythmogenesis after MI has been studied extensively in the canine epicardial border zone formed 5 days after occlusion of the left anterior descending coronary artery

(see review2). Myocytes isolated from the BZ possess unique electrophysiological properties. While action potential (AP) duration (APD) measured at 90% repolarization is not significantly different from control, APD measured at 50% repolarization is reduced, producing a triangular AP3. In addition, the maximum rate of rise of the AP

4 upstroke, dV/dtmax, is significantly reduced . Underlying these AP changes are alterations in sarcolemmal ion channels, a process known as electrophysiological

5 remodeling. Specifically, densities of fast sodium current, INa , L-type calcium current,

6,7 ICa(L) , slow component, IKs, and fast component, IKr, of the delayed rectifier potassium

8 4 current , and transient outward potassium current, Ito1 , are reduced to varying degrees.

5,7,8 Furthermore, kinetic changes are observed in INa, ICa(L), IKr, and IKs . Molecular alterations responsible for these changes remain to be fully elucidated. However, it is clear that the remodeling process involves down-regulation of genes encoding major ion channels at the transcriptional and translational levels8-11.

BZ myocytes show alterations in calcium handling, as well. While control myocytes show an increase in calcium transient amplitude (CaTamp) during rapid pacing

(positive CaTamp-frequency staircase), BZ myocytes show a decrease in CaTamp (negative

12 CaTamp-frequency staircase) . BZ myocytes show a less pronounced increase in relaxation rate with pacing rate compared to control, enhanced postextrasystolic potentiation, and a slower time constant of CaT recovery12. The mechanism for these alterations remains unclear. Also unknown is why BZ myocytes show a greater reduction in peak ICa(L) at rapid pacing compared to control without any significant difference in

7 ICa(L) recovery from inactivation .

2+ CaMKII is a multifunctional enzyme that regulates the SR Ca ATPase, ICa(L),

13 and Irel in cardiac myocytes (see review ). In Chapter 3, we apply computational biology to show that CaMKII detects pacing frequency in the cardiac cell and promotes the positive CaTamp-frequency relationship observed in the canine. We hypothesize that dysfunction of CaMKII underlies the rate-dependent alterations in ICa(L) and CaT observed in BZ myocytes. To test our hypothesis, we use the canine ventricular

(epicardial) cell model presented in Chapter 2 (the HRd model), which accounts for activation of CaMKII and its regulation of intracellular substrates.

0 Model Formulation F)

µ -1

Details of the control HRd model are A/ µ -2 BZ provided in Chapter 2. Here, we describe -3 the implementation of remodeling changes Control -4 ICa(L) Current density ( that occur in epicardial BZ cells. BZ model -5 050 V (mV) equations are provided in Appendix C. m Figure 17 Control (thick line) and BZ Variable definitions are found in Table 5. (thin line) model ICa(L) current-voltate (I-V) 2+ curves. [Ca ]o = 2.0 mM.

2+ L-type Ca current, ICa(L)

Experiments show a 36% reduction in the peak of the ICa(L) I-V curve with no

7 significant difference in steady-state current after 250-ms test pulse . BZ ICa(L) also inactivates more rapidly than control. Based on these data, voltage-dependent and calcium-dependent inactivation time constants are reduced and the maximal conductance is reduced by 10% to yield a 33% reduction in peak ICa(L) (Figure 17).

+ Fast Na current, INa

The peak of the INa current- voltage (I-V) curve is 61% smaller Control

in BZ myocytes than in control and ∞ h steady-state inactivation is shifted BZ in a repolarizing direction relative to control5. To account for these data, we make the peak of the Vtest (mV) model BZ INa I-V curve 57% Figure 18 Control (thick line) and BZ (thin line) INa steady-state fast inactivation gate, h∞. smaller than control and shift the steady-state inactivation curve by 3.5 mV in the repolarizing direction (Figure 18).

Slow component and rapid component of the delayed rectifier K+ current

IKs and IKr densities are reduced by 77% and 67%, respectively, and IKr activation

8 is significantly faster in BZ myocytes compared to control . We reduce IKs and IKr densities (their maximum conductances G Ks and G Kr , respectively) and decrease the IKr activation time constant to account for these experimental observations. To yield the

measured APD in the BZ cell, G Ks and G Kr are reduced by 50% and 40%, respectively.

+ Transient outward K current, Ito1 4 Based on experimental findings that BZ myocytes do not contain Ito1 , Ito1 is completely blocked in BZ model.

2+ Homeostasis of [Ca ]i BZ myocytes show intracellular Ca2+ overload and accumulation of resting

2+ 6 2+ diastolic [Ca ]i . To achieve a comparable degree of Ca overload in the BZ model, we

2+ + + block the sarcolemmal Ca ATPase (Ip,ca) and decrease the maximum rate of the Na -K pump ( I NaK ) by 20%.

Ca2+/calmodulin-dependent protein kinase CaMKII CaMKII activity is suppressed completely to test the hypothesis that CaMKII dysfunction accounts for altered CaT rate-dependence in the BZ cell.

A 50 CL = 2,000 ms Control C E BZ 0 0.4 IKr 0 F) -2 F) µ µ 0.2 A/ BZ A/ (mV) I µ Ca(L) µ ( ( -50 -4 Vm 0 100 ms 100 ms 100 ms B D 0.3 F 0 0.4 IKs -50 0.2 F) F) F) µ µ BZ µ 0.2 A/ A/ -100 A/ µ µ µ ( ( I 0.1 ( -150 Na INaK 0 100 ms 0 100 ms 100 ms

Figure 19 (A) Vm, the transmembrane action potential (AP). (B-F) Ionic currents during the AP. Values are shown during steady-state pacing at CL = 2,000 ms for the control model (thick line) and BZ model (thin line). Arrows in panel B and C indicate peak BZ INa and ICa(L), respectively.

Results

AP rate dependence Figure 19 shows the epicardial action potential (AP) and remodeled currents in control (thick line) and BZ (thin line) cell models. Consistent with experiment3, remodeling produces a triangular AP (Figure 19A), which lacks the early repolarization notch and phase 2 dome characteristic of control. Peak INa during the AP is about 50% smaller than control (Figure 19B) resulting in a diminished dV/dtmax and AP amplitude

(APA) in the BZ myocyte. Comparison of dV/dtmax, APA, and Vm,rest in control and BZ models to experimentally measured values4 reveals close agreement (Table 1). Peak and sustained components of ICa(L) (Figure 19C) are diminished in BZ myocytes contributing to reduced plateau potential and triangular AP. Repolarizing currents INaK (Figure 19D),

240

220

200 Control 180 BZ APD (ms) 160 500 1000 1500 2000 Cycle Length (ms)

Figure 20 Action potential duration vs. cycle length (adaptation curve) for control (thick line) and BZ (thin line).

+ IKr (Figure 19E), IKs (Figure 19F), and the inward rectifier K current IK1 are also significantly reduced. Decreased depolarizing current (ICa(L)) and repolarizing currents counteract each other to produce AP duration (APD) comparable to control at CL = 2,000 ms. In Chapter 3, we show that ICa(L) is a major determinant of APD shortening at fast rate (APD adaptation) in canine, with contributions from Ito1, INaK, and the slowly

+ inactivating Na current INa,L (Figure 10). Consequently, reduction in ICa(L), Ito1, and INaK in the BZ cell is expected to alter APD adaptation. While APD at slow rate (CL = 2,000 ms) is comparable between BZ and control myocytes, at fast rate (CL = 500 ms) APD diverges significantly as the BZ myocyte has a greatly diminished capacity for APD adaptation (Figure 20).

Experiment Fura-2 Transients Model Calcium Transients (Licata et al. Cardiovasc Res, 1997)12

Control

5,000 ms CL 1,000 ms 2,000 ms CL 500 ms CL

BZ 0.2 µ M 1.0 800 ms 300 ms

Figure 21 Control (top) and BZ (bottom) experimental fura-2 transients12 compared to model Ca2+ transients at slow and fast pacing. The different time scales and compared CLs are attributable to temperature differences (experimental recordings are conducted at room temperature while the model is formulated for body temperature).

Calcium transient rate dependence Simulated CaT in control and BZ are compared to experimental recordings at slow and rapid pacing in Figure 21 (the different CLs in simulation and experiment reflect different temperature; experiments are at room temperature while simulations at body temperature). At slow pacing, CaTamp is only slightly smaller and the CaT width at half maximal amplitude (W1/2) is only slightly greater in BZ compared to control (Table

2). Decreasing CL from 2,000 ms to 500 ms in the simulation exaggerates these differences as CaTamp increases by 31% in control (positive CaT-frequency relationship) but decreases by 22% in the BZ cell (negative CaT-frequency relationship, Figure 21, right). In addition, W1/2 decreases by 35% (rate-dependent CaT abbreviation) in control compared to only 23% in the BZ cell. This behavior agrees qualitatively and quantitatively with experiments where CL is decreased from 5,000 ms to 1,000 ms at room temperature (CaTamp: 21% increase in control, 17% decrease in BZ; W1/2: 45% decrease in control, 27% decrease in BZ)12.

Figure 22 provides insight into the ionic mechanism underlying these rate-

dependent differences between BZ and control CaT (Figure 22A). Reduced I NaK (Figure

2+ 2+ 19D) and Ip,ca block in BZ lead to elevated diastolic [Ca ]i ([Ca ]i,dia, Figure 22A, left

2+ panel) and elevated [Ca ]JSR (Figure 22B, left panel) at slow pacing. Consistent with

2+ 12,14 canine experimental measurements, [Ca ]i,dia increases at rapid pacing in control and

BZ12. While control undergoes SR loading at fast rates (Figure 22B, compare diastolic segment of bold line in left and right panels, bold arrows), the BZ model experiences a

2+ decrease in [Ca ]JSR at fast rates (Figure 22B, compare thin line in left and right panels,

2+ 2+ thin arrows). Decreased BZ [Ca ]JSR at fast rates may be attributed to the fact that Ca uptake into the SR via Iup does not increase at fast rates in BZ as in control (Figure 22C),

which also accounts for weak rate-dependence of BZ W1/2. Negative CaTamp-frequency relationship in the BZ model is due to the fact that BZ Irel (Figure 22D, arrow indicates

BZ peak) is only 21% smaller than control at slow pacing compared to 39% at fast pacing.

Table 1. Control and BZ electrophysiological properties V * dV/dt † APA‡ Cell type m,rest max (mV) (mV/ms) (mV) Control model -86.5 283 125

Control experiment§ -88 ± 5 262 ± 59 121 ± 7

BZ model -85.8 178 109

BZ experiment§ -88 ± 4 162 ± 52 113 ± 6.5 *Resting membrane potential. †Maximum rate of rise of action potential upstroke. ‡Action potential amplitude §Lue WM and Boyden PA. Circulation. 1992;85:1175-1188.

Table 2. Model calcium transient properties

* 2+ † ‡ § Cell type CL (ms) [Ca ]i,dia (µM) CaTamp (µM) W1/2 (ms)

Control 2,000 0.095 0.398 252

BZ 2,000 0.124 0.347 274

Control 500 0.178 0.521 165

BZ 500 0.201 0.271 211 *Cycle length †Diastolic Ca2+ concentration ‡Ca2+ transient amplitude §CaT width at half maximum amplitude.

A CL = 2,000 ms Control CL = 500 ms Control 0.8 BZ 0.8 BZ 0.6 2+ 0.6 [Ca2+] [Ca ]i i M) M) µ 0.4 µ 0.4 ( (

0.2 2+ 0.2 [Ca ]i,dia 0 100 ms 0 100 ms B 2 2+ 2 2+ [Ca ]JSR [Ca ]JSR F) F) µ µ A/ A/

µ 1 µ 1 ( (

0 100 ms 0 100 ms C 3 3 I Iup up 2 2 M/ms) M/ms) µ µ ( 1 ( 1

0 100 ms 0 100 ms D 4 4

3 I 3 I BZ rel rel 2 2 BZ (mM/ms) 1 (mM/ms) 1

0 0 100 ms 100 ms

2+ Figure 22 (A) CaT (B) [Ca ]JSR (control and BZ diastolic values indicated by bold arrow and thin arrow), (C) Iup, (D) Irel (arrows indicate BZ peak) during the AP in control (thick line) and BZ (thin line) during steady-state pacing at CL of 2,000 ms (left) and 500 ms (right).

Differences in CaTamp-frequency relationship between BZ and control are also associated with altered ICa(L) rate dependence (Figure 23). Figure 23A shows peak ICa(L) in response to a voltage-pulse train applied at fast and slow rate in control and BZ model.

7 Consistent with experiment , BZ ICa(L) shows greater rate dependence (greater reduction at fast rate) than control (Figure 23A) despite no difference in recovery from inactivation

(Figure 23B).

A Peak I B I Recovery Ca(L) CL = 2,000 ms Ca(L) 1 1

Ca(L) Control 0.8 BZ Ca(L) 250 ms 0.9 +20 mV 0.6 -40 mV ... 0.4 350 ms CL = 500 ms +20 mV Normalized I 0.8 -40 mV

Normalized Peak I 0.2 IPI

0 0 10203040 05001000 Number of Beats Interpulse Interval, IPI (ms)

Figure 23 (A) Peak ICa(L) during application of voltage-pulse trains at cycle length of 2,000 ms and 500 ms in control (thick line) and BZ (thin line). Pacing protocol is shown in inset. Each simulation is preceded by 50 beats at CL of 8,000 ms. (B) ICa(L) recovery from inactivation in control (thin line) and BZ models (thick line) compared to experimental data in control (open circles) and BZ (filled circles)7. The control and BZ curves are superimposed. Pacing protocol is shown in inset. Model Ca2+-dependent inactivation gates are clamped to one to isolate voltage-dependent inactivation.

Altered rate-dependence of Iup, Irel, and ICa(L) in the BZ model is due to CaMKII suppression, which under control conditions enhances each of these processes at fast pacing rates (See Figure 15). Figure 24 shows rate dependence of CaTamp and excitation- contraction coupling (ECC) gain for the control model (thick line), BZ model (thin line), and control model with CaMKII suppressed (dashed line). Control CaTamp-frequency relationship is positive for frequencies ≤ 2 Hz while BZ CaTamp-frequency relationship is

74 negative for frequencies ≥ 1 Hz and relatively flat at lower frequencies (Figure 24A, top).

ECC gain increases at rapid pacing rates in control but not in BZ (Figure 24A, bottom).

Suppression of CaMKII produces the biphasic CaTamp-frequency relationship with negative slope (Figure 24A, top, dashed line) for frequencies ≥ 1 Hz and the flat gain- frequency relationship (Figure 24A, bottom) characteristic of the BZ myocyte. CaMKII suppression increases W1/2 to a value similar to that in the BZ cell (Figure 24B). It also

2+ decreases the SR Ca release trigger, ICa(L) (Figure 24C), and decreases gain by reducing

2+ SR loading via Iup (Figure 24D) and Ca release via Irel (Figure 24E). The greater reduction of peak ICa(L) in BZ than in control with CaMKII suppressed (Figure 24C) is due to ICa(L) remodeling in the BZ cell which reduces the current independently of

CaMKII effects (Figure 17). CaMKII suppression in control also eliminates differences

A B D CL = 500 ms CL = 500 ms 0.6 0.8 3 2+ CaTamp [Ca ]i 2.5 Iup 0.6 0.4 2 M) M) µ µ ( mol/ms) ( 0.4

µ 1.5 0.2 Control ( BZ 1 Con-CaMK 0.2 0 100 ms 0.5 100 ms C E CL = 500 ms CL = 500 ms 20 Gain 0 3

-1 BZ 15 2 Con-CaMK

F) -2

µ BZ A/ µ 10 ( -3 1 Con-CaMK (mmol/ms)

-4 Irel ICa(L) 5 0 0123 -5 Pacing frequency (Hz) 100 ms 100 ms

Figure 24 CaT rate dependence for control model (thick line), BZ model (thin line), and control model with CaMKII suppressed (dashed line). (A) CaTamp-frequency relationship (top) and gain- 2+ frequency relationship (bottom). (B) [Ca ]i (C) ICa(L) (D) Iup (E) Irel. Panels C-E are computed during steady-state AP at 2 Hz pacing (CL = 500 ms). Arrows indicate peak; Con-CaMK indicates control with CaMKII suppressed.

in rate-dependence of peak ICa(L) between control and BZ models (not shown).

CaMKII suppression has almost no effect on AP and its rate dependence (Figure

25), suggesting that while CaT rate-dependent abnormalties in BZ depend heavily on

CaMKII dysfunction, alterations in AP and APD adaptation (Figure 19 and Figure 20) are

CaMKII-independent and result from electrophysiological remodeling of ion channels.

40 CL = 500 ms CL = 2,000 ms 20 0 -20 Vm (mV) -40 -60 -80 100 ms 100 ms

Figure 25 Effect of CaMKII suppression on AP and its rate dependence. Values are shown for control (thick line), BZ (thin line), and control with CaMKII suppressed (dashed line). Steady-state AP is shown at CL = 500 ms (left) and CL = 2,000 ms (right). CaMKII suppression has negligible effect on AP.

Ca2+-transient restitution Figure 26 shows that CaT restitution is abnormal in BZ, consistent with experiment12. A pause after rapid pacing reveals that BZ CaT is slower to recover than control (Figure 26A) in both model (left) and experiment (right). Furthermore, BZ CaT experiences a greater increase following a pause after rapid pacing (post-extrasystolic potentiation) compared to control (Figure 26B). CaMKII suppression in the control

A Model Experiment 1 1.1 (Licata et al. Cardiovasc Res, 1997)12 Control BZ 1

amp 0.9

CaT 0.9 d 0.8 0.8 malize r 0.7 ... Control No 500 S1S2 Normalized0.7 Fura-2 ms interval BZ 0.6 0.6 400 600 800 1000 1000 2000 3000 S1S2 interval (ms) S1S2 Interval(ms) B Model Control Experiment 12 BZ 1.2 (Licata et al. Cardiovasc Res, 1997) amp 1.4 1.1

1 Normalized CaT Normalized0.9 Fura-2 Control BZ 1.3 1000 2000 3000 4000 5 101520 S1S2 interval (ms) S1S2 Interval (s)

Figure 26 Model CaT restitution (left) compared to experimentally measured fura-2 restitution12 (right) in control and BZ. (A) Recovery of CaT and fura-2 transient. The model is paced to steady state at CL = 500 ms after which an S2 stimulus is applied at a varying S1S2 interval (inset). CaTamp is normalized to its value at S1S2 = 7,000 ms. (B) Post-extrasystolic potentiation: Increased CaTamp following a long pause after rapid pacing (protocol is the same as in panel A; CaTamp is normalized to the steady-state value at CL = 500 ms). Note greater potentiation in BZ compared to control for both model and experiment. model eliminates differences in CaT restitution between control and BZ models (not shown).

Discussion

Myocardial infarction (MI) represents a major health risk in the industrial world.

It is therefore imperative that we understand cellular factors that predispose the heart to life-threatening arrhythmias post-infarction. An important step in this process is electrophysiological remodeling of ion channels and pumps that generate the cardiac AP,

CaT, and maintain intracellular homeostasis. In this chapter, we present a detailed model of remodeled cells isolated from the canine epicardial infarct BZ. We use the model to explore ionic mechanisms of altered AP and CaT rate dependence in BZ myocytes.

Importantly, we test the novel hypothesis that suppressed CaMKII activity in BZ cells may account for several seemingly unrelated experimental observations regarding rate dependence of ICa(L) and the CaT. Our model reproduces important AP and CaT properties of myocytes isolated from canine infarct BZ and shows that CaMKII suppression may give rise to altered rate dependence of CaT and ICa(L) in these cells.

In a previous study of electrophysiological remodeling post-infarction, Cabo et al. presented models of control and BZ canine epicardial myocytes15. The HRd model used here as the control model reproduces a number of physiological behaviors not accounted for by the control model used by Cabo, including: 1) physiological APD adaptation (as

CL decreases from 1000 ms to 300 ms, APD decreases by 10 ms in the Cabo model compared to 60 ms in experiment16 and 64 ms in HRd), 2) Ca2+ release formulation that

17 2+ 2+ 18 2+ accounts for variable gain and regulation of SR Ca release by [Ca ]JSR , 3) Ca subsarcolemmal space, and 4) dynamic CaMKII activity and its regulation of intracellular substrates. These properties are essential for physiological simulations of altered Ca2+ cycling in the BZ cell and its electrophysiological consequences.

Consistent with results of Cabo et al., we find that the BZ myocyte shows a diminished capacity for APD adaptation (Figure 20), which may lead to unidirectional

78 block and initiation of reentry in the BZ during rapid pacing19. Further studies are required to fully evaluate the importance of altered APD adaptation in initiating reentry.

With our BZ model, we test the hypothesis that alterations in Ca2+-handling and rate dependence of ICa(L) may be explained by a lesion in CaMKII. While CaMKII activity has not been characterized in BZ myocytes isolated from the 5-day old infarct,

CaMKII activity is reduced during heart failure induced by MI in rats20 and during heart failure induced by microemboliziation in dog left ventricle21. Ca2+-overload suppresses

CaMKII activity22 and may explain decreased CaMKII phosphorylation in MI20. BZ

2+ 6 myocytes experience elevated [Ca ]i under basal conditions , which may lead to CaMKII suppression in these cells. While our simulations suggest that CaMKII suppression

2+ produces a negative CaTamp-frequency relationship and may link Ca -overload conditions to the negative force-frequency relationship observed in MI, experiments are needed to validate these model predictions. It would be interesting to not only characterize CaMKII activity at 5 days postinfarction, but also at time points both earlier and later. We predict that CaMKII activity initially decreases over the first week or two after coronary occlusion but then begins to recover and eventually becomes hyperactivated as a compensatory mechanism. This may account for increased CaMKII activity in end-stage heart failure23. The HRd model may be used in the future to examine effects on Ca2+-handling of hyperactivity of CaMKII.

Our results regarding the effect of CaMKII suppression on CaT are consistent with experimental observations. Schouten24 first hypothesized and experiment25 later verified that CaMKII underlies rate-dependent changes in CaT width. Furthermore, the

CaMKII inhibitor KN-93 decreases CaTamp and increases W1/2 during rapid pacing and

79 increases CaT post-extrasystolic potentiation26, while CaMKII inhibitory peptides abolish

27 pacing-dependent facilitation of peak ICa(L) without slowing recovery from inactivation .

It is important to note that remodeling of other intracellular processes may affect

Ca2+ handling and contribute to abnormal CaT rate dependence in BZ myocytes. For example, decreased expression of SERCA2a has been observed in rat post-infarction28, which is expected to decrease CaTamp and increase W1/2 at all pacing frequencies. It is unknown whether SERCA2a is downregulated in canine 5 days postinfarction. However, our results suggest that observed alterations in Ca2+ handling do not require downregulation of SERCA2a and can be accounted for by CaMKII suppression.

Limitations Experimental data from the canine 5-day old BZ are used wherever possible to develop the BZ model. While data are readily available on remodeling of ion channels in this animal model of MI, less is known about what happens to cellular processes governing Ca2+ handling. For example, CaMKII activity during pacing has not been well characterized in either control or BZ canine myocyte. We hope that model predictions presented here will stimulate such experimental studies.

Works Cited

1. Wit AL, Janse MJ. The ventricular arrhythmias of ischemia and infarction: electrophysiological mechanisms.: Futura; 1992.

2. Pinto JM, Boyden PA. Electrical remodeling in ischemia and infarction. Cardiovasc Res. 1999;42:284-97.

3. Ursell PC, Gardner PI, Albala A, Fenoglio JJ, Jr., Wit AL. Structural and electrophysiological changes in the epicardial border zone of canine myocardial infarcts during infarct healing. Circ Res. 1985;56:436-51.

4. Lue WM, Boyden PA. Abnormal electrical properties of myocytes from chronically infarcted canine heart. Alterations in Vmax and the transient outward current. Circulation. 1992;85:1175-88.

5. Pu J, Boyden PA. Alterations of Na+ currents in myocytes from epicardial border zone of the infarcted heart. A possible ionic mechanism for reduced excitability and postrepolarization refractoriness. Circ Res. 1997;81:110-9.

6. Pu J, Robinson RB, Boyden PA. Abnormalities in Cai handling in myocytes that survive in the infarcted heart are not just due to alterations in repolarization. J Mol Cell Cardiol. 2000;32:1509-23.

7. Aggarwal R, Boyden PA. Diminished Ca2+ and Ba2+ currents in myocytes surviving in the epicardial border zone of the 5-day infarcted canine heart. Circ Res. 1995;77:1180-91.

8. Jiang M, Cabo C, Yao JA, Boyden PA, Tseng GN. Delayed rectifier K currents have reduced amplitudes and altered kinetics in myocytes from infarcted canine ventricle. Cardiovasc Res. 2000;48:34-43.

9. Yao JA, Jiang M, Fan JS, Zhou YY, Tseng GN. Heterogeneous changes in K currents in rat ventricles three days after myocardial infarction. Cardiovasc Res. 1999;44:132-45.

10. Gidh-Jain M, Huang B, Jain P, el-Sherif N. Differential expression of voltage- gated K+ channel genes in left ventricular remodeled myocardium after experimental myocardial infarction. Circ Res. 1996;79:669-75.

11. Huang B, Qin D, El-Sherif N. Early down-regulation of K+ channel genes and current in the postinfarction heart. J Cardiovasc Electrophysiol. 2000;11:1252- 1261.

12. Licata A, Aggarwal R, Robinson RB, Boyden P. Frequency dependent effects on Cai transients and cell shortening in myocytes that survive in the infarcted heart. Cardiovasc Res. 1997;33:341-50.

13. Maier LS, Bers DM. Calcium, calmodulin, and calcium-calmodulin kinase II: heartbeat to heartbeat and beyond. J Mol Cell Cardiol. 2002;34:919-39.

14. Laurita KR, Katra R, Wible B, Wan X, Koo MH. Transmural heterogeneity of calcium handling in canine. Circ Res. 2003;92:668-75.

15. Cabo C, Boyden P. Electrical remodeling of the epicardial border zone in the canine infarcted heart: a computational analysis. Am J Physiol Heart Circ Physiol. 2003;284:H372-H384.

16. Liu DW, Gintant GA, Antzelevitch C. Ionic bases for electrophysiological distinctions among epicardial, midmyocardial, and endocardial myocytes from the free wall of the canine left ventricle. Circ Res. 1993;72:671-87.

17. Wier W, Balke C. Ca2+ release mechanisms, Ca2+ sparks, and local control of excitation-contraction coupling in normal heart muscle. Circ Res. 1999;85:770-6.

18. Lukyanenko V, Gyorke I, Gyorke S. Regulation of calcium release by calcium inside the sarcoplasmic reticulum in ventricular myocytes. Pflugers Arch. 1996;432:1047-54.

19. Dillon SM, Allessie MA, Ursell PC, Wit AL. Influences of anisotropic tissue structure on reentrant circuits in the epicardial border zone of subacute canine infarcts. Circ Res. 1988;63:182-206.

20. Netticadan T, Temsah RM, Kawabata K, Dhalla NS. Sarcoplasmic reticulum Ca2+/Calmodulin-dependent protein kinase is altered in heart failure. Circ Res. 2000;86:596-605.

21. Mishra S, Sabbah HN, Jain JC, Gupta RC. Reduced Ca2+-calmodulin-dependent protein kinase activity and expression in LV myocardium of dogs with heart failure. Am J Physiol Heart Circ Physiol. 2003;284:H876-83.

22. Netticadan T, Temsah RM, Kawabata K, Dhalla NS. Ca2+-overload inhibits the cardiac SR Ca2+-calmodulin protein kinase activity. Biochem Biophys Res Commun. 2002;293:727-32.

23. Kirchhefer U, Schmitz W, Scholz H, Neumann J. Activity of cAMP-dependent protein kinase and Ca2+/calmodulin-dependent protein kinase in failing and nonfailing human hearts. Cardiovasc Res. 1999;42:254-61.

24. Schouten VJ. Interval dependence of force and twitch duration in rat heart explained by Ca2+ pump inactivation in sarcoplasmic reticulum. J Physiol (Lond). 1990;431:427-44.

25. Bassani RA, Mattiazzi A, Bers DM. CaMKII is responsible for activity-dependent acceleration of relaxation in rat ventricular myocytes. Am J Physiol Heart Circ Physiol. 1995;268:H703-12.

26. Li L, Satoh H, Ginsburg KS, Bers DM. The effect of Ca2+-calmodulin-dependent protein kinase II on cardiac excitation-contraction coupling in ferret ventricular myocytes. J Physiol (Lond). 1997;501:17-31.

27. Yuan W, Bers DM. Ca-dependent facilitation of cardiac Ca current is due to Ca- calmodulin-dependent protein kinase. Am J Physiol Heart Circ Physiol. 1994;267:H982-93.

28. Zhang XQ, Ng YC, Moore RL, Musch TI, Cheung JY. In situ SR function in postinfarction myocytes. J Appl Physiol. 1999;87:2143-50.

Chapter 5 - Discussion

83 84

Summary of Findings

We have developed a physiological mathematical model of the canine ventricular action potential and Ca2+ handling. Important aspects of our modeling effort include: 1) integration of dynamic CaMKII activity and its effects on intracellular substrates; 2) novel SR Ca2+ release formulation that accounts for variable gain and channel gating

2+ 2+ modulation by SR lumenal Ca ; 3) Ca subsarcolemmal space; 4) Ito2 and dynamic chloride. We incorporate electrophysiological remodeling changes known to occur after infarction to create a physiological canine BZ model. We find that down-regulation of several key ion channels results in abnormal APD adaptation in BZ myocytes.

Importantly, we find that CaMKII is important for normal CaT-frequency relationship.

We show that CaMKII dysfunction may account for the abnormal CaT rate dependence observed in BZ myocytes. In this final chapter, implications for our findings and areas of future study are discussed.

Cell Modeling

The frontiers for cardiac cell modeling are many and the evolution of ventricular modeling will continue for the foreseeable future. One rapidly expanding frontier is the development of cell models based on single-channel Markov formulations of transmembrane currents1-6. These Markov-based models have facilitated eloquent studies linking state-specific ion-channel genetic mutations to arrhythmic consequences in the context of the whole cell2,3,7,8. The potential is great for such models to illuminate the mechanism by which specific genetic mutations promote cardiac arrhythmia and sudden cardiac death. Transmembrane currents in the HRd model use the Hodgkin-Huxley

85 formalism to facilitate large-scale simulations needed to study the BZ. In the future,

Markov models of major currents may be incorporated into HRd to study genetic mutations and the effects of state-specific drugs. Incorporation of more detailed Markov- based models of CaMKII activity9 may be particularly useful in the study of single cell calcium-handling abnormalities after MI.

A second important frontier for cardiac cell modeling is physiological representation of intracellular calcium handling including the L-type Ca2+ channel and

SR Ca2+ release6,10. The L-type Ca2+ channel is an astoundingly complicated ion channel due in part to its regulation by calcium, voltage, and a host of intracellular enzymes. At the same time, SR Ca2+ release channels sense Ca2+ in a small dyadic space formed by the close proximity of the sarcolemmal and SR membranes11. The complex nature of the individual components of Ca2+ release and their interaction have motivated the development of highly detailed “local control”12 models that simulate thousands of individual dyadic spaces10. We have distilled the important aspects of these detailed

“local control” models into a simplified form that lends itself to simulations of a large number of cells and/or for a prolonged period of time. However, questions remain about important aspects of Ca2+ release such as termination and refractoriness13-17. As more data become available and our understanding of SR Ca2+ release becomes more complete, models of excitation-contraction coupling will evolve. While we present one viable paradigm, there is great opportunity for creative solutions to this problem6,10,18.

Integration of metabolic pathways and dynamic pH into whole cell models is a third area of active expansion in the field of cell modeling19. These efforts will ultimately lead to a dynamic representation of myocardial ischemia and hopefully a more complete

86 understanding of arrhythmogenesis in the minutes to days following artery occlusion.

Such models require chloride homeostasis due to the important chloride-bicarbonate exchanger. The HRd model includes dynamic chloride and therefore lends itself to incorporation of a metabolic model20 and eventually a dynamic representation of myocardial ischemia.

Finally, integration of key signaling pathways into whole cell models represents a fourth area with much potential for growth. The HRd model establishes a paradigm for such efforts by incorporating dynamic CaMKII activity. Recently, the β-adrenergic signaling pathway has also been integrated into a whole cell computer model21. Future modeling efforts will integrate other signaling pathways (for example, protein kinase C), and cross-talk between multiple pathways (for example, phosphorylation of PLB by PKA and CaMKII22) into whole cell models.

Myocardial Infarction

Development of single cell control and BZ models is only the first step in creating a complete model of the BZ after MI. The tools developed here must be incorporated into higher-dimensional models of cardiac tissue to examine the ionic mechanism of anisotropic propagation and initiation of reentry in the BZ23,24. These studies will provide valuable insight into the importance of electrophysiological remodeling and gap junction remodeling in creating a pro-arrhythmic substrate.

CaMKII

Model studies presented in this work support the hypothesis that CaMKII detects

87 pacing frequency in cardiac myocytes and mediates rate-dependent effects on the calcium transient. Furthermore, results show that CaMKII dysfunction during myocardial infarction promotes abnormal calcium handling and may be an important part of the remodeling process. Our model formulation is based on available experimental data and model behavior is consistent with a number of experimental findings, including the effect of CaMKII inhibitors on CaT relaxation25 and CaT amplitude26. However, the body of literature on CaMKII and its regulation of intracellular substrates is incomplete (see

Chapter 2 and Chapter 4 for complete discussion). More experiments are required to fully characterize CaMKII in cardiac myocytes and its role in CaT rate dependence.

Additional experiments are also needed to clarify the role of CaMKII in cardiac disease. While CaMKII activity is decreased in heart failure induced by MI in rats27 and canine28, CaMKII is up-regulated in animal models of cardiac hypertrophy29-33 and human heart failure34,35. One possible explanation for these disparate findings is that

CaMKII suppression due to Ca2+ overload36 is followed by a compensatory up-regulation of the kinase leading to hypertrophy and heart failure. The time course of alterations in

CaMKII activity following insult must be examined. Furthermore, modeling and experimental studies are needed to understand how CaMKII hyperactivity in heart failure promotes abnormal Ca2+ handling and cardiac arrhythmia37.

Works Cited

1. Zeng J. A Dynamic Model of a Cardiac Ventricular Action Potential. Biomedical Engineering Dissertation. Cleveland OH: Case Western Reserve University; 1997.

2. Clancy CE, Rudy Y. Linking a genetic defect to its cellular phenotype in a cardiac arrhythmia. Nature. 1999;400:566-9.

3. Clancy CE, Rudy Y. Cellular consequences of HERG mutations in the long QT syndrome: precursors to sudden cardiac death. Cardiovasc Res. 2001;50:301-13.

4. Jafri MS, Rice JJ, Winslow RL. Cardiac Ca2+ dynamics: the roles of ryanodine receptor adaptation and sarcoplasmic reticulum load. Biophys J. 1998;74:1149-68.

5. Winslow RL, Rice J, Jafri S, Marban E, O'Rourke B. Mechanisms of altered excitation-contraction coupling in canine tachycardia-induced heart failure, II: model studies. Circ Res. 1999;84:571-86.

6. Bondarenko VE, Bett GC, Rasmusson RL. A Model of Graded Calcium Release and L-type Ca2+ channel Inactivation in Cardiac Muscle. Am J Physiol Heart Circ Physiol. 2003:In Press.

7. Ficker E, Thomas D, Viswanathan PC, Dennis AT, Priori SG, Napolitano C, Memmi M, Wible BA, Kaufman ES, Iyengar S, Schwartz PJ, Rudy Y, Brown AM. Novel characteristics of a misprocessed mutant HERG channel linked to hereditary long QT syndrome. Am J Physiol Heart Circ Physiol. 2000;279:H1748-56.

8. Clancy CE, Rudy Y. Na+ channel mutation that causes both Brugada and long-QT syndrome phenotypes: a simulation study of mechanism. Circulation. 2002;105:1208-13.

9. Zhabotinsky AM. Bistability in the Ca2+/calmodulin-dependent protein kinase- phosphatase system. Biophys J. 2000;79:2211-21.

10. Greenstein JL, Winslow RL. An integrative model of the cardiac ventricular myocyte incorporating local control of Ca2+ release. Biophys J. 2002;83:2918-45.

11. Langer G, Peskoff A. Calcium concentration and movement in the diadic cleft space of the cardiac ventricular cell. Biophys J. 1996;70:1169-82.

12. Stern M. Theory of excitation-contraction coupling in cardiac muscle. Biophys J. 1992;63:497-517.

13. Sham J, Song L, Chen Y, Deng L, Stern M, Lakatta E, Cheng H. Termination of Ca2+ release by a local inactivation of ryanodine receptors in cardiac myocytes. Proc Natl Acad Sci USA. 1998;95:15096-101.

14. Shannon TR, Guo T, Bers DM. Ca2+ scraps: local depletions of free [Ca2+] in cardiac sarcoplasmic reticulum during contractions leave substantial Ca2+ reserve. Circ Res. 2003;93:40-45.

15. Lukyanenko V, Gyorke I, Gyorke S. Regulation of calcium release by calcium inside the sarcoplasmic reticulum in ventricular myocytes. Pflugers Arch. 1996;432:1047-54.

16. Gyorke I, Gyorke S. Regulation of the cardiac ryanodine receptor channel by luminal Ca2+ involves luminal Ca2+ sensing sites. Biophys J. 1998;75:2801-10.

17. Fabiato A. Time and calcium dependence of activation and inactivation of calcium-induced release of calcium from the sarcoplasmic reticulum of a skinned canine cardiac Purkinje cell. J Gen Physiol. 1985;85:247-89.

18. Shiferaw Y, Watanabe MA, Garfinkel A, Weiss JN, Karma A. Model of intracellular calcium cycling in ventricular myocytes. Biophys J. 2003;85:3666- 86.

19. Ch'en FF, Vaughan-Jones RD, Clarke K, Noble D. Modelling myocardial ischaemia and reperfusion. Prog Biophys Mol Biol. 1998;69:515-38.

20. Leem CH, Lagadic-Gossmann D, Vaughan-Jones RD. Characterization of intracellular pH regulation in the guinea-pig ventricular myocyte. J Physiol (Lond). 1999;517:159-80.

21. Saucerman JJ, Brunton LL, Michailova AP, McCulloch AD. Modeling beta- adrenergic control of cardiac myocyte contractility in silico. J Biol Chem. 2003;278:47997-8003.

22. Hagemann D, Kuschel M, Kuramochi T, Zhu W, Cheng H, Xiao RP. Frequency- encoding Thr17 phospholamban phosphorylation is independent of Ser16 phosphorylation in cardiac myocytes. J Biol Chem. 2000;275:22532-6.

23. Dillon SM, Allessie MA, Ursell PC, Wit AL. Influences of anisotropic tissue structure on reentrant circuits in the epicardial border zone of subacute canine infarcts. Circ Res. 1988;63:182-206.

24. Gough WB, Mehra R, Restivo M, Zeiler RH, el-Sherif N. Reentrant ventricular arrhythmias in the late myocardial infarction period in the dog. 13. Correlation of activation and refractory maps. Circ Res. 1985;57:432-42.

25. Bassani RA, Mattiazzi A, Bers DM. CaMKII is responsible for activity-dependent acceleration of relaxation in rat ventricular myocytes. Am J Physiol Heart Circ Physiol. 1995;268:H703-12.

27. Netticadan T, Temsah RM, Kawabata K, Dhalla NS. Sarcoplasmic reticulum Ca2+/Calmodulin-dependent protein kinase is altered in heart failure. Circ Res. 2000;86:596-605.

28. Mishra S, Sabbah HN, Jain JC, Gupta RC. Reduced Ca2+-calmodulin-dependent protein kinase activity and expression in LV myocardium of dogs with heart failure. Am J Physiol Heart Circ Physiol. 2003;284:H876-83.

29. Hagemann D, Bohlender J, Hoch B, Krause EG, Karczewski P. Expression of Ca2+/calmodulin-dependent protein kinase II δ-subunit isoforms in rats with hypertensive cardiac hypertrophy. Mol Cell Biochem. 2001;220:69-76.

30. Boknik P, Heinroth-Hoffmann I, Kirchhefer U, Knapp J, Linck B, Luss H, Muller T, Schmitz W, Brodde O, Neumann J. Enhanced protein phosphorylation in hypertensive hypertrophy. Cardiovasc Res. 2001;51:717-28.

31. Currie S, Smith GL. Calcium/calmodulin-dependent protein kinase II activity is increased in sarcoplasmic reticulum from coronary artery ligated rabbit hearts. FEBS Lett. 1999;459:244-8.

32. Maier LS, Zhang T, Chen L, DeSantiago J, Brown JH, Bers DM. Transgenic 2+ CaMKIIδC overexpression uniquely alters cardiac myocyte Ca handling: reduced SR Ca2+ load and activated SR Ca2+ release. Circ Res. 2003;92:904-11.

33. Zhang T, Maier LS, Dalton ND, Miyamoto S, Ross J, Jr., Bers DM, Brown JH. The δC isoform of CaMKII is activated in cardiac hypertrophy and induces dilated cardiomyopathy and heart failure. Circ Res. 2003;92:912-919.

34. Kirchhefer U, Schmitz W, Scholz H, Neumann J. Activity of cAMP-dependent protein kinase and Ca2+/calmodulin-dependent protein kinase in failing and nonfailing human hearts. Cardiovasc Res. 1999;42:254-61.

35. Hoch B, Meyer R, Hetzer R, Krause EG, Karczewski P. Identification and expression of δ-isoforms of the multifunctional Ca2+/calmodulin-dependent protein kinase in failing and nonfailing human myocardium. Circ Res. 1999;84:713-21.

36. Netticadan T, Temsah RM, Kawabata K, Dhalla NS. Ca2+-overload inhibits the cardiac SR Ca2+-calmodulin protein kinase activity. Biochem Biophys Res Commun. 2002;293:727-32.

37. Wu Y, Temple J, Zhang R, Dzhura I, Zhang W, Trimble R, Roden DM, Passier R, Olson EN, Colbran RJ, Anderson ME. Calmodulin kinase II and arrhythmias in a mouse model of cardiac hypertrophy. Circulation. 2002;106:1288-93.

Appendix A - Conservation

92 93

Table 3. Abbreviations for conservation study

2 Acap Capacitive area of membrane (cm )

Cm Membrane capacitance (µF)

F Faraday’s constant, 96,485 (C/mol) Total Ca2+ current through all ion channels in LRd model (µA/µF) ICa,t 1 ICa,t = ICa(L) + Ip(Ca) + ICa(T) + ICa,b Total K+ current through all ion channels in LRd model (µA/µF) IK,t 1 IK,t = IKr + IKs + IK1 + IKp + ICa,K Total Na+ current through all ion channels in LRd model (µA/µF) INa,t 1 INa,t = INa + INa,b + ICa,Na

2+ Ileak Ca leak from NSR (mM/ms)

+ 2+ INaCa Na -Ca exchanger (µA/µF)

+ + INaK Na -K pump (µA/µF)

2+ Irel Ca release from JSR (mM/ms)

2+ Itr Ca transfer from NSR to JSR (mM/ms)

2+ Iup Ca uptake into NSR (mM/ms)

JSR Junctional sarcoplasmic reticulum

NSR Network sarcoplasmic reticulum

Vmyo Volume of myoplasm (µL)

Vjsr Volume of JSR (µL)

Vnsr Volume of NSR (µL)

2+ 2+ [Ca ]i,tot Intracellular concentration of bound and free Ca (mM)

2+ 2+ [Ca ]jsr,tot JSR concentration of bound and free Ca (mM)

2+ 2+ [Ca ]nsr NSR concentration of Ca (mM)

+ + [K ]i Intracellular concentration of K (mM)

+ + [Na ]i Intracellular concentration of Na (mM)

∂t Time derivative

Introduction

The original modeling paradigm established by Hodgkin and Huxley2 uses the total current flowing through ion channels in the membrane to determine the

3-6 transmembrane potential (Vm) . In this paradigm, a differential equation is solved for each current and Vm. A later stage in the evolution of mathematical models has been the development of cell models that account for dynamic changes in intracellular ion concentrations7-9. These dynamic models build upon the original paradigm by calculating not only Vm but also intracellular ion concentrations from the transmembrane ionic currents.

While dynamic models have proven to be a useful and widely accepted tool for studying the electrophysiology and contractility of excitable cells, they face mounting concern regarding their behavior in response to prolonged periods of rapid pacing10-13.

Specifically, drift of intracellular ion concentrations and Vm, and the existence of an infinite number of steady states are often cited as problems with such models10,11. To address these concerns, models have been formulated based on the principle of charge conservation. In this formulation, the differential equation computing Vm from the transmembrane current (“differential” method) is replaced with an algebraic equation

12,14 relating Vm to intracellular ion concentrations (“algebraic” method) . It has been hypothesized that this approach produces a model that is stable with respect to drift in computed parameters (ion concentrations, Vm).

Here, we examine the phenomenon of drift in the Luo-Rudy dynamic (LRd) mathematical model of the mammalian ventricular cell during prolonged periods of pacing. We find that no drift occurs if ions carried by the stimulus current are accounted

for in the calculation of intracellular ion concentrations. When the stimulus charge is included in the formulation, the differential method and the algebraic method yield identical results.

Long-term pacing studies reproduce the physiological situation in the beating heart. During many cardiac arrhythmias, cells are subjected to periodic stimulation at a fast rate over extended periods of time. Therefore, simulation studies of cellular behavior in response to long periods of rapid pacing are essential for mechanistic understanding of arrhythmogenesis. The present study demonstrates that a dynamic model of the cardiac ventricular cell can be used for such studies and establishes the correct protocol for doing so.

Differential method

The LRd mathematical model of the mammalian ventricular myocyte1,8,15 is implemented in all simulations. Every time step, model equations provide the transmembrane currents through ion channels, pumps, and exchangers from which

+ + 2+ 2+ 2+ changes in [K ]i, [Na ]i, [Ca ]i,tot, [Ca ]jsr,tot, and [Ca ]nsr are determined as follows (IK,t,

+ + 2+ INa,t, ICa,t are sums of all currents in the LRd model through K , Na , Ca channels, respectively. See Table 3 for definitions and abbreviations).

+ Acap Cm ∂ t [K ]i = − [I K ,t − 2I NaK ] (1) Vmyo F

+ Acap Cm ∂ t [Na ]i = − [I Na,t + 3I NaK + 3I NaCa ] (2) Vmyo F

2+ Acap Cm Vnsr V jsr ∂ t [Ca ]i,tot = − [I Ca,t − 2I NaCa ] − [I up − I leak ] + I rel (3) 2Vmyo F Vmyo Vmyo

2+ ∂ t [Ca ] jsr,tot = I tr − I rel (4)

2+ V jsr ∂ t [Ca ]nsr = I up − I leak − I tr (5) Vnsr

In accordance with the original modeling paradigm established by Hodgkin and Huxley2, transmembrane currents also provide the change in Vm every time step as follows.

∂ tVm = −(I K ,t + I Na,t + I Ca,t + I NaK + I NaCa ) (6)

Eq. 6 is numerically integrated with the Forward Euler method and a time step of 5 µs.

Algebraic method

Vm can be calculated directly from intracellular ion concentrations based on a charge conservation principle12,14. To derive this formulation for the LRd model, Eqs. 1-

5 are combined as follows.

VmyoF + + 2+ 2V jsr 2+ 2Vnsr 2+ (∂t[K ]i + ∂t[Na ]i + 2⋅ ∂t[Ca ]i,tot + ⋅ ∂t[Ca ] jsr,tot + ⋅ ∂t[Ca ]nsr ) = AcapCm Vmyo Vmyo (7)

= −(IK ,t + I Na,t + ICa,t + I NaK + INaCa ) = ∂tVm

Eq. 7 is integrated to give the final form of the algebraic method

VmyoF + + 2+ 2V jsr 2+ 2Vnsr 2+ Vm = ([K ]i + [Na ]i + 2 ⋅[Ca ]i,tot + ⋅[Ca ] jsr,tot + ⋅[Ca ]nsr − Co ) (8) AcapCm Vmyo Vmyo where C0 is a constant of integration. Endresen et al. define C0 as the total extracellular

+ + 2+ 12 concentration of K , Na , and Ca . We instead determine C0 by substituting initial values for the dynamic model variables in Eq. 8. Initial conditions used in this study are provided in Table 4. Eq. 8 expresses the voltage difference across a capacitor with capacitance Cm and charge proportional to the total intracellular concentration of ions.

Table 4. Conservation initial conditions

+ [K ]i 138.9 mM

+ [Na ]i 11.5 mM

2+ -5 [Ca ]i 7.83 x 10 mM

+ [K ]o 4.5 mM

+ [Na ]o 132 mM

2+ [Ca ]o 1.8 mM

Vm -89 mV

2+ [Ca ]NSR 1.18 mM

2+ [Ca ]JSR 1.18 mM

Co 150.72 mM

In both the algebraic and differential methods, transmembrane currents are calculated for every time step, after which the changes in intracellular ion concentrations are determined using Eqs. 1-5. The differential method utilizes Eq. 6, a differential equation, to calculate Vm, while the algebraic method determines Vm from Eq. 8, an algebraic expression.

Pacing protocols

Two different pacing protocols are used: The first employs a voltage-pulse stimulus and the second a current stimulus. In the first protocol, the cell is paced from a resting steady state for 33 min. at basic cycle lengths (BCL) of 300 ms and 1000 ms. The voltage pulse depolarizes Vm to –45 mV for 0.5 ms. As will be explained below, this protocol violates conservation for both the differential and algebraic methods. At the end

of the voltage pulse, the differential method resumes calculation of Vm using the discrete form of Eq. 6:

Vm,n+1 = Vm,n − ∆t(I K ,t + I Na,t + I Ca,t + I NaK + I NaCa ) (9) where ∆t represents the time step and n is the time step index. Importantly, in this scheme Vm,0 is –45 mV for computing Vm,1 after cessation of the voltage pulse. The transmembrane current initiated by the voltage-pulse stimulus continues to depolarize Vm from the holding potential of –45 mV.

The voltage pulse violates conservation in the differential method because the depolarization from the maximum diastolic potential (Vm,dia) to –45 mV is not accompanied by a consistent change in intracellular ion concentrations. In the algebraic method, conservation is violated during the voltage pulse because Eq. 8 does not hold for the original value of C0. However, the violation is only temporary since Vm returns immediately to the value satisfying Eq. 8 for the original value of C0 once the voltage pulse is over.

In the second protocol, the cell is paced with a current stimulus 0.5 ms in duration and –150 µA/µF in amplitude. Unless stated, the stimulus current is assumed to carry K+

+ ions and is added to IK,t before calculation of [K ]i using Eq. 1. This protocol insures that conservation is not violated at any time in the differential or algebraic methods. Both methods remain conservative if the stimulus current is assumed to carry Na+ or Ca2+ ions, provided that the appropriate ion is accounted for in computing ion concentrations.

In dynamic model simulations as in experiments, a current stimulus is typically used to pace the cell. However, the algebraic method does not respond to a current stimulus unless an ion species is assumed as charge carrier. In order to compare the

differential and algebraic methods in response to pacing with a nonconservative stimulus, a voltage stimulus must therefore be used. In one simulation using the differential method, the stimulus current is added to the total current in Eq. 6 but is not assumed to carry a particular ion species for the purpose of computing ion concentrations. This is done to directly demonstrate that failure to account for ions carried by the stimulus current results in drift of model parameters.

Results

Before pacing, the model cell is left undisturbed for 15 minutes to attain a resting steady state. Figure 27 shows resting Vm as a function of time for three different initial

+ + values of [K ]i (indicated by [K ]i,0) using the differential method. Initial values of all other model variables

+ + except [K ]i are [K ]i,0 = 137.9 mM identical for all three -88.9 cases. Both the -89 [K+] = 138.9 mM differential and Vm i,0 (mV) algebraic methods -89.1 settle to the same [K+] = 139.9 mM -89.2 i,0 resting steady-state

+ 051015 for each [K ]i,0 (Vm = Time (min) –88.94 mV, -89.04 Figure 27 Resting Vm as a function of time before onset of pacing in the differential method. Resting steady states are shown for three different mV, -89.22 mV for + initial values of [K ]i: 137.9, 138.9, and 139.9 mM. All other initial conditions are the same for the three traces. + [K ]i,0 = 137.9 mM,

100

+ 138.9 mM, and 139.9 mM, respectively). A different value of [K ]i,0 results in a different steady state. However, once steady state is achieved (time constant to reach steady state is about 7 s), it does not drift for either the differential or algebraic methods.

Figure 28 shows the first action potential (AP) Algebraic Method 40 Differential Method generated with a voltage- 20 pulse stimulus from the 0 V -20 m +

[K ]i,0= 138.9 mM resting (mV) -40 state in Figure 27. The -60 -80 algebraic and differential 0 100 200 methods after one stimulus Time (ms)

Figure 28 APs elicited with a voltage-pulse stimulus to -45 mV produce APs that are + from resting steady state ([K ]i,0=138.9 mM) in the algebraic (solid line) and differential (dashed line) methods. Note that APs qualitatively and are superimposed. quantitatively very similar (error in AP duration of less than 0.001%). However, in

Figure 29 when the model is driven with a voltage pulse at a rapid rate (BCL = 300 ms) for a prolonged period of time, the algebraic and differential methods diverge with respect to several computed parameters, including Vm,dia, (Figure 29A), AP duration

+ + (APD, Figure 29B), [Na ]i (Figure 29C), and [K ]i (Figure 29D). Specifically, the computed parameters in the differential method display a linear drift that is absent in the algebraic method. Figure 30 shows that the drift of computed parameters depends on the pacing rate, with a less pronounced drift occurring when the cell is paced at a BCL of

1000 ms, consistent with previous statements in the literature11,13.

101

Figure 30 shows the time- Voltage stimulus A -87 V 300 ms BCL dependent behavior of computed m,dia parameters when the cell is paced with a -87.5 (mV) current stimulus rather than with a -88 voltage-pulse stimulus. Computed Algebraic Method -88.5 Differential Method parameters drift if the differential method B 108 APD is used with a current stimulus that does 106 not carry a particular ion species into the (ms) cell (D.M., no ion carrier in Figure 31). 104

Importantly, when the current stimulus is 102

+ C + assumed to carry K ions into the cell, 18 [Na ] i computed parameters do not drift and the 17.5 algebraic and differential methods yield 17 (mM)

16.5 identical results. The version of the LRd

16 model examined in this study contains a D + [K ] i formulation of INaCa that has been 134 modified from the original formulation to 132 (mM) saturate at positive voltages1,14. When the 130

128 simulation in Figure 31 is repeated with 0 102030Time (min)

8 0 2000 4000 6000 Beat the original formulation of INaCa , the number

+ Figure 29 (A) Vm,dia, (B) APD, (C) [Na ]i, and computed parameters again do not drift + (D) [K ]i during application of a voltage-pulse train at a BCL of 300 ms in the algebraic (solid (data not shown). line) and differential (dashed line) methods.

102

Figure 32 shows [K+] vs. V i m V -87 m,dia (phase-space plot) for nine -87.5 consecutive beats during pacing at a 300 ms BCL -88

BCL of 300 ms, after 33 minutes of (mV)

-88.5 pacing. When the model is paced 1000 ms BCL with a conservative stimulus (Figure -89 0102030 32A), the model’s trajectory in phase Time (min) space adheres to a single limit cycle Figure 30 Vm,dia during application of a voltage- pulse train at two different BCLs: 300 ms (dashed + line) and 1000 ms (solid line). (limit cycles for three different [K ]i,0 values are shown). Figure 32A, inset shows an enlarged limit cycle for the intermediate

+ + + value of [K ]i,0. As Vm depolarizes during the AP upstroke, [K ]i increases due to K ions carried into the cell by the stimulus current (stage 1 in Figure 32A, inset). Next, K+ ions leave the cell through K+ selective ion channels such as the rapidly activating and slowly activating delayed rectifier K+ currents, which contributes to the repolarization

+ phase of the AP (stage 2). During diastole, Vm repolarizes slowly and [K ]i increases as

+ + INaK restores cell homeostasis by transferring Na ions out of the cell and K ions into the cell (stage 3). When the next stimulus is applied, the trajectory has returned to the exact point from which it began the previous cycle (asterisk in Figure 32A, inset).

In a conservative system, the entire upstroke from Vm,dia to the AP peak is associated with a change in ion concentrations (stage 1 in Figure 32A, inset). However, this is not the case in the differential method when the cell is paced with a voltage-pulse stimulus (Figure 32B, shown on the same y-axis scale as the inset in Figure 32A). Notice that subsequent stimuli (1, 1’, 1’’ in Figure 32B) begin at a different point in phase space,

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reflecting a drift in computed parameters. As discussed in Methods, the depolarization of

Vm from Vm,dia to the holding potential of –45 mV is not associated with a parallel change in intracellular ion concentrations (stage 1 in Figure 32B, inset). During the voltage

+ pulse, Vm is held constant at –45 mV and [K ]i increases (stage 2 in Figure 32B, inset).

+ Upon cessation of the voltage pulse, the Na current depolarizes Vm from the already depolarized holding potential of –45 mV (stage 3 in Figure 32B, inset). Na+ ions enter

+ the cell and [K ]i continues to increase through INaK. Importantly, entry of ions is associated with a change in Vm for only a portion of the upstroke (from –45 mV to the

A V C + -82 m,dia 17 [Na ] i

-84 16.5

-86 16 (mV) (mM)

-88 15.5 Algebraic Method -90 Differential Method 15 D.M. (no ion carrier) B 116 APD D 140 + [K ] i 114 130

112

(ms) 120 (mM)

110 110 108 Time (min) 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Beat 0 2000 4000 6000 Number 0 2000 4000 6000

+ + Figure 31 (A) Vm,dia, (B) APD, (C) [Na ]i, and (D) [K ]i vs. time during pacing with a current stimulus using the algebraic method (solid line) and the differential method (dashed line, arrow). The stimulus carries K+ ions into the cell. An additional simulation is shown using the differential method and a current stimulus that does not carry a particular ion species (dashed-dotted line). BCL=300 ms.

104

+ peak Vm of 40 mV). Fewer Na ions entering the cell during the upstroke diminishes the

+ driving force for INaK. A reduced INaK results in less entry of K ions during and

+ following the AP, which is consistent with the steady downward drift of [K ]i between stimuli (Figure 29D and Figure 32B). Because INaK is a repolarizing current, reducing its driving force should depolarize Vm,dia and prolong APD. This prediction is consistent with the results of Figure 29A and Figure 29B.

The algebraic method, however, is immune to drift even when the cell is paced with a voltage-pulse stimulus (Figure 32C). Similar to the differential method, no change in ions is associated with the depolarization from Vm,dia to –45 mV (stage 1 in Figure

+ 32C, inset). Similar to the differential method, there is an increase in [K ]i during the voltage pulse (stage 2, Figure 32C, inset). During this stage, Eq. 8 does not hold and conservation is temporarily violated. However, in the algebraic method when the voltage pulse ends, Vm returns to the value determined by Eq. 8, which is also consistent with the new intracellular ion concentrations (stage 3, Figure 32C, inset). From this state, the fast

+ Na current activates and depolarizes the AP upstroke to the peak Vm value of +40 mV

(stage 4, Figure 32C, inset). It is apparent that in the algebraic method, the change in intracellular ion concentrations during the entire upstroke from Vm,dia to the AP peak is taken into account.

105

A Differential method Conservative current stimulus 134.5 133.602 134.25 1 133.6 134 + [K ]i 133.75 + 133.598 (mM) [K ] 133.5 i 2 (mM) 133.596 133.25 3* 133 133.594 132.75 -100 -50 0 -100 -50 0 Vm (mV) Vm (mV)

B Differential method 5mV C Algebraic method Voltage stimulus 15 3 Voltage stimulus nM 2 5mV 15 4 127.68 1 nM * 132.142 1 3 2 + 127.678 1' [K ] 1 132.14 + i 1'' * [K ]i 127.676 . (mM) . 132.138 (mM) . 127.674 132.136

127.672 132.134

-100 -50 0 -100 -50 0 (mV) Vm Vm (mV)

+ Figure 32 (A) [K ]i vs. Vm (phase-space plot) during pacing with a current stimulus (conservative + protocol) for three different initial values of [K ]i. BCL=300ms. Enlargement of limit cycle + corresponding to [K ]i,0=138.9 mM shown in inset. Asterisk signifies stimulus application. (B) Phase-space plot during voltage-pulse train using the differential method. Model trajectory near the stimulus is enlarged in inset. Note that upon cessation of the voltage pulse (end of stage 2), depolarization of Vm (stage 3) occurs from the -45 mV holding potential. (C) Same as (B), but using the algebraic method. Note that upon cessation of the voltage clamp (end of stage 2, inset), Vm returns to a slightly repolarized value (stage 3) before the upstroke (stage 4).

Discussion

Long term pacing is an essential protocol for the experimental and theoretical

investigation of the electrophysiological properties of cardiac cells. Importantly, during

many cardiac arrhythmias, cells are stimulated at a fast rate for prolonged periods of time.

Under such conditions, intracellular ion concentration changes are important

106

determinants of the cell electrophysiogical behavior. Simulation studies of these processes require the use of a dynamic cell model that accounts for ion concentration changes. It is essential to insure that the model does not suffer from nonphysiological behavior during long-term pacing. In this study, we examine the behavior of the LRd model during prolonged pacing protocols.

Important findings are: 1) The LRd cell model reaches a resting steady state with a time constant of about 7 s in both the differential and algebraic methods. 2) When pacing the LRd cell model, computed parameters do not drift if ions carried by the stimulus current are included when computing intracellular ion concentrations. 3) The differential and algebraic methods of calculating Vm produce identical results when the stimulus is accounted for in computing intracellular ion concentrations.

The algebraic method vs. the differential method Recently, mathematical cardiac cell models have been formulated that replace the

12,14 differential method of calculating Vm with the algebraic method . As shown in this study, the algebraic and differential methods compute identical results in a conservative system. Therefore, the choice of using the differential method over the algebraic method cannot be the fundamental reason for drift observed in such models. If charge is conserved in the system (including charge carried by the stimulus), then the choice of method has no impact on the behavior of the system.

The researcher interested in computer modeling must choose between the two different methods for calculating Vm. The differential method solves an additional differential equation, while the algebraic method contains an initial condition (C0) that

107

must be determined before running the simulation. Both methods produce non- physiological results when a nonconservative stimulus is used. In general, the algebraic method is more resistant to drift (Figure 32C). However, the differential method is just as robust when paced properly with a conservative stimulus (Figure 32A). Relative to the overall complexity of dynamic models, none of these distinctions is very significant in terms of implementation and therefore the choice of which method to use is left to the individual researcher’s discretion. Before a choice is made, however, it may be advisable to run the same simulation with both methods to ensure that conservation is not violated in the system.

Drift is caused by a nonconservative implementation of the stimulus It has been reported in the literature that dynamic cell models, which account for changes in intracellular ion concentrations, show a non-physiological drift in computed parameters11-13. This drift occurs during rapid pacing for prolonged periods of time11,13.

The results of the present study establish that such drift is due to a nonconservative implementation of the stimulus, and not to an intrinsic property of the LRd model. The drift disappears when ions carried by the stimulus current are accounted for in the computation of ion concentrations. In general, when using a dynamic model, any source of charge (such as the stimulus) must also be considered a source of ions. Failure to do so violates conservation and may produce a non-physiological behavior due to drift of model parameters. In this study, the problem is easily corrected by incorporating the stimulus current into the total K+ current in the LRd formulation. Assuming other ions in the system (Na+ and Ca2+) to be the stimulus charge carrier is also consistent with the conservation principle, as long as these ions are accounted for in the formulation. It is

108

worth mentioning that previous algebraic methods have been applied to automatic cells

(e.g., pacemaker cells) that require no external stimulus for excitation12,14. It would be expected therefore that the drift observed in the present study would not occur in such models because they do not require pacing. In addition, models that do not account for dynamic changes of intracellular ion concentrations2-6 are not susceptible to the stimulus- dependent drift discussed in this study.

Another problem with dynamic models cited in the literature is that limit cycles are not unique10,11. Figure 27 and Figure 32A confirm that steady state in the LRd model is not unique and depends upon initial conditions. However, the existence of an infinite number of steady states is only problematic when conservation is violated (Figure 29 and

Figure 32B). Figure 30 and Figure 32A illustrate that in a conservative system, the trajectory of the model in phase space does not stray from the limit cycle determined by the initial conditions.

While our results suggest that a nonconservative implementation of the stimulus current underlies the reported drift in computed parameters, other possible sources of drift exist. For example, we numerically integrate Eq. 6 with the Forward Euler method, which conserves the quantity of interest (C0 in Eq. 8). To understand the conservative nature of the Forward Euler method, we discretize Eq. 6 as follows.

Vn+1 = Vn − ∆t ⋅ ∑ I j (10) j where ∆t represents the time step and n is the time step index. Forward Euler applied to

Eqs. 1-5 gives the discrete form of the ion concentration differential equations as follows.

[X j ]n+1 = [X j ]n − ∆t ⋅ Cm ⋅κ j ⋅ I j,n (11)

109

where Xj is an ion species and κj is a constant related to the volume of distribution in the cell. We sum over ion species and combine Eqs. 10 and 11 to yield

1 1 Vn+1 − ∑ [X j ]n+1 = Vn − ∑ [X j ]n (12) j Cm ⋅κ j j Cm ⋅κ j

1 Eq. 12 shows that the quantity Vn − ∑ [X j ]n ≡ C0 is independent of n and is thus j Cm ⋅κ j conserved by the Forward Euler method. Numerical methods that do not conserve C0 from one time step to the next may result in drift. For example, a method that updates I

(the current) between Eq. 10 and Eq. 11 will not conserve C0, the result of which may be drift in computed parameters.

Simulating AP propagation in multicellular tissue models This study emphasizes the important contribution of an external stimulus current to changes in intracellular ion concentrations during prolonged pacing of the LRd single cell model. Microelectrodes that impale the cell for current stimulation are commonly filled with solution containing KCl. This guided our choice of K+ as the charge carrier of the stimulus current in the LRd model. In HRd, which includes dynamic Cl-, we assume the current to be a combination of K+ and Cl-. In the heart, cells are stimulated by electrotonic current that flows through gap junctions from depolarized neighboring cells.

+ Since K i concentration is much greater than that of other cations, it is safe to assume that

K+ is the charge carrier of this current (although anions such as Cl- may contribute as well). Similar to the case of external current stimulation, ions carried by the intercellular electrotonic current during AP propagation should be accounted for in the calculation of intracellular ion concentrations, in order to preserve conservation in the system. This is particularly true during fast, repetitive activation such as occurs experimentally when a

110

multicellular tissue preparation is subject to rapid pacing or in the in situ heart during tachyarrhythmias. For a given cell, the net ionic change (influx minus efflux) should be considered. Failure to do so may give rise to drift in computed parameters during propagation over prolonged periods of time.

It is important to note that this study examines drift in the LRd model, specifically. Non-physiological drift has been reported in other dynamic models11,13,16, which should be evaluated independently for any unique sources of drift.

111

Works Cited

+ 1. Faber GM, Rudy Y. Action potential and contractility changes in [Na ]i overloaded cardiac myocytes: A simulation study. Biophys J. 2000;78:2392-2404.

2. Hodgkin AL, Huxley AF. A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol. 1952;117:500-544.

3. Noble D. A modification of the Hodgkin-Huxley equations applicable to Purkinje fibre action and pacemaker potential. J Physiol (Lond) 1962;160:317-352.

4. McAllister RE, Noble D, Tsien RW. Reconstruction of the electrical activity of cardiac Purkinje fibres. J Physiol (Lond). 1975;251:1-59.

5. Beeler GW, Reuter H. Reconstruction of the action potential of ventricular myocardial fibres. J Physiol (Lond). 1977;268:177-210.

6. Luo CH, Rudy Y. A model of the ventricular cardiac action potential. Depolarization, repolarization, and their interaction. Circ Res. 1991;68:1501-26.

7. DiFrancesco D, Noble D. A model of cardiac electrical activity incorporating ionic pumps and concentration changes. Philos Trans R Soc Lond B Biol Sci. 1985;307:353-98.

8. Luo CH, Rudy Y. A dynamic model of the cardiac ventricular action potential. I. Simulations of ionic currents and concentration changes. Circ Res. 1994;74:1071- 96.

9. Winslow RL, Rice J, Jafri S, Marban E, O'Rourke B. Mechanisms of altered excitation-contraction coupling in canine tachycardia-induced heart failure, II: model studies. Circ Res. 1999;84:571-86.

10. Guan S, Lu Q, Huang K. A discussion about the DiFrancesco-Noble model. J Theor Biol. 1997;189:27-32.

11. Yehia AR, Jeandupeaux D, Alonso F, Guevara MR. Hysteresis and bistability in the direct transition from 1:1 to 2:1 rhythm in periodically driven single ventricular cells. Chaos. 1999;9:916-931.

12. Endresen LP, Hall K, Hoye JS, Myrheim J. A theory for the membrane potential of living cells. Eur Biophys J. 2000;29:90-103.

13. Rappel W-J. Filament instability and rotational anisotropy: A numerical study using detailed cardiac models. Chaos. 2001;11:71-80.

112

14. Varghese A, Sell GR. A conservation principle and its effect on the formulation of Na-Ca exchanger current in cardiac cells. J Theor Biol. 1997;189:33-40.

15. Zeng J, Laurita KR, Rosenbaum DS, Rudy Y. Two components of the delayed rectifier K+ current in ventricular myocytes of the guinea pig type. Theoretical formulation and their role in repolarization. Circ Res. 1995;77:140-52.

16. Michailova A, McCulloch A. Model study of ATP and ADP buffering, transport of Ca2+ and Mg2+, and regulation of ion pumps in ventricular myocyte. Biophys J. 2001;81:614-629.

Appendix B - Model Equations

113 114

Table 5. Model abbreviations + INa Fast Na current, µA/µF

+ INa,L Slowly inactivating late Na current, µA/µF

2+ 2+ ICa(L) Ca current through the L-type Ca channel, µA/µF + 2+ ICa,Na Na current through the L-type Ca channel, µA/µF + 2+ ICa,K K current through the L-type Ca channel, µA/µF

+ IKr Rapid delayed rectifier K current, µA/µF + IKs Slow delayed rectifier K current, µA/µF + Ito1 4AP-sensitive transient outward K current, µA/µF

2+ - Ito2 Ca -dependent transient outward Cl current, µA/µF

+ IK1 Time-independent K current, µA/µF + IKp Plateau K current, µA/µF + 2+ INaCa Na -Ca exchanger, µA/µF 2+ Allo Ca -dependent allosteric activation factor of INaCa vmax Maxmimal flux of INaCa, µA/µF ksat Saturation factor of INaCa at negative potentials η Position of energy barrier of INaCa + + INaK Na -K pump, µA/µF fNaK Voltage-dependent parameter of INaK + σ [Na ]o-dependent factor of fNaK 2+ Ip,Ca Sarcolemmal Ca pump, µA/µF 2+ ICa,b Background Ca current, µA/µF + - CTNaCl Na -Cl cotransporter, mmol/L per ms + - CTKCl K -Cl cotransporter, mmol/L per ms

SR Sarcoplasmic reticulum JSR Junctional SR NSR Network SR SS Ca2+ subspace

2+ Irel Ca release from JSR to myoplasm, mmol/L per ms τri Time constant of Irel inactivation, ms 2+ Iup Ca uptake from myoplasm to NSR, mmol/L per ms 2+ Ileak Ca leak from JSR to myoplasm, mmol/L per ms 2+ Itr Ca transfer from NSR to JSR, mmol/L per ms 2+ τtr Time constant of Ca transfer from NSR to JSR, ms

2+ Idiff Ca transfer from SS to myoplasm, mmol/L per ms 2+ τdiff Time constant of Ca transfer from SS to myoplasm, ms

115

2+ ICa,t Total transmembrane Ca current, ICa,t = ICa(L) + ICa,b + Ip,Ca - 2INaCa + INa,t Total transmembrane Na current, INa,t = INa + 3INaK + ICa,Na + 3INaCa + INa,L + IK,t Total transmembrane K current, IK,t = IKs + IKr + IK1 + ICa,K + Ito1 + IKp – 2INaK - ICl,t Total transmembrane Cl current, ICl,t = Ito2 + ICl,b Itot Total transmembrane current, Itot = ICa,t + INa,t + IK,t + ICl,t

Istim Stimulus current, µA/µF

2+ CaMKbound Fraction of CaMKII binding sites bound to Ca /calmodulin CaMKactive Fraction of active CaMKII binding sites. CaMKtrap Fraction of autonomous CaMKII binding sites with trapped calmodulin. CaMKo Fraction of active CaMKII binding sites at equilibrium. αCaMK, βCaMK Phosphorylation and dephosphorylation rates of CaMKII, respectively, ms-1

∆PCaMK CaMKII-dependent factor of substrate parameter P ∆P Maximal CaMKII-dependent change in substrate parameter P

APD Action potential duration measured at 90% repolarization CaT Calcium transient CaTamp Calcium transient amplitude

PLB Phospholamban RyR Ryanodine receptor SR Ca2+ release channel LTCC L-type Ca2+ channel

G x Maximum conductance of channel x, mS/µF Km Half-saturation concentration, mmol/L m, h, and j Activation gate, fast inactivation gate, and slow inactivation gate of INa, respectively mL and hL Activation gate and inactivation gate of INa,L, respectively d, f, and f2 Activation gate, fast voltage-dependent inactivation gate, and slow voltage-dependent inactivation gate of ICa(L), respectively pow Power applied to d 2+ 2+ fCa and fCa2 Fast Ca -dependent inactivation gate and slow Ca -dependent inactivation gate of ICa(L), respectively. Xs1 and Xs2 Fast activation gate and slow activation gate of IKs, respectively Xr Activation gate of IKr RKr Time-independent rectification gate of IKr K1 Inactivation gate of IK1

116

a, i, i2 Activation gate, fast inactivation gate, and slow inactivation gate of Ito1, respectively Rto1 Time-independent rectification gate of Ito1 2+ aa Ca -dependent activation gate of Ito2 ro and ri Activation gate and inactivation gate of Irel, respectively. 2+ ∆ro∞,JSR Modulation of ro∞ by [Ca ]JSR vg Variable gain factor for Irel y∞ Steady-state value of gate y -1 αy and βy Opening and closing rate constants of gate y, respectively, ms τy Time constant of gate y, ms csqn Calsequestrin, Ca2+ buffer in JSR trpn Troponin, Ca2+ buffer in myoplasm cmdn Calmodulin, Ca2+ buffer in myoplasm BSR Anionic SR binding sites for Ca2+ in SS BSL Anionic sarcolemmal binding sites for Ca2+ in SS

PS Membrane permeability to ion S, cm/s PS,A Permeability ratio of ion S to ion A γS Activity coefficient of ion S

I x Maximum current carried through channel x, µA/µF Vm Transmembrane potential, mV zs Valence of ion S Cm Total cellular membrane capacitance, 1 µF 2 ACap Capacitive membrane area, cm 2 AGeo Geometric membrane area, cm RCG Ratio of ACap to AGeo = 2

Ex Reversal potential of current x, mV

[]S o and [S ]i Extracellular and intracellular concentrations of ion S, respectively, mmol/L

2+ 2+ [Ca ]JSR Ca concentration in JSR, mmol/L 2+ 2+ [Ca ]NSR Ca concentration in NSR, mmol/L 2+ 2+ [Ca ]ss Ca concentration in subspace, mmol/L 2+ 2+ [Ca ]i,t Concentration of free and buffered intracellular Ca , mmol/L

∆[]S x Change in concentration of ion S in compartment x during one time step, mmol/L Vx Volume of compartment x, µL.

F Faraday constant, 96,487 C/mol R Gas constant, 8314 J/kmol/K T Temperature, 310 °K

117

Table 6. Control ion concentrations at rest

2+ −3 [Ca ]i 0.0814× 10 mmol/L - [Cl ]i 19.52 mmol/L + [K ]i 142.49 mmol/L + [Na ]i 10.02 mmol/L 2+ [Ca ]JSR 1.24 mmol/L 2+ [Ca ]NSR 1.24 mmol/L 2+ [Ca ]o 1.8 mmol/L - [Cl ]o 100 mmol/L + [K ]o 5.4 mmol/L + [Na ]o 140 mmol/L After model is undisturbed for 1000 s.

A. CaMKII 1 CaMKbound=⋅−⋅ CaMK o()1 CaMK trap KmCaM, 1+ 2+ []Ca ss

dCaMKtrap =⋅αβCaMKCaMK bound ⋅() CaMK bound + CaMK trap −⋅ CaMK CaMK trap dt

CaMKactive=+ CaMK bound CaMK trap

-1 -1 αCaMK = 0.05 ms ; βCaMK = 0.0007 ms

CaMKo = 0.05 ; Km,CaM = 0.0015 mmol/L

The CaMKII dependence of a substrate parameter, P, is defined by ∆PCaMK. We assume a half-saturation coefficient Km,CaMK of 0.15 to yield an almost-linear dependence on pacing frequency for frequencies up to 3 Hz1.

CaMKactive ∆=∆⋅PPCaMK KCaMKm, CaMK+ active

Km,CaMK = 0.15

118

∆P is the maximal CaMKII-dependent change for substrate parameter, P

for ICa(L),

∆=τ fca, CaMK 10 ms

for Irel,

∆=trel, CaMK 10 ms

for Iup,

∆=Iup, CaMK 0.75; ∆=KmPLB, 0.00017 mmol/L

B. ICa(L)

pow ICa() L=⋅df ca. f ca 2 ⋅⋅⋅ ffI 2 Ca

2 2 ()VFm −⋅15.0 γγCai⋅⋅[Ca ] ss exp( z Ca ⋅−⋅( V m 15.0)() F / RT) −⋅ Cao [ Ca ] o IPzCa() L =⋅⋅Ca Ca ⋅ RT exp()zVCa()() m −⋅ 15.0 FRT / − 1

1 d∞ = 1+−− exp() (Vm 4) / 6.74

exp( 0.052⋅+ (Vm 13)) τ d =+⋅0.59 0.8 1+⋅+ exp() 0.132 (Vm 13)

8 pow∞ =−9 1exp(+−+()Vm 65)/3.4

τ pow =10.0 ms

0.7 f∞ =+0.3 1.0++ exp() (Vm 17.12) / 7

0.77 f2∞ =+0.23 1.0++ exp() (Vm 17.12) / 7

119

1 τ f = 0.2411⋅− exp()⎣⎦⎣⎦⎡⎤⎡⎤ 0.045 ⋅−()()VVmm 9.6914 ⋅ 0.045 ⋅− 9.6914 + 0.0529

1 τ f 2 = 0.0423⋅− exp()⎣⎦⎣⎦⎡⎤⎡⎤ 0.059 ⋅−()()VVmm 18.5726 ⋅ 0.059 ⋅− 18.5726 + 0.0054

0.3 0.55 f =+ +0.15 Ca,∞ I []Ca2+ 1− Ca() L 1+ ss 0.05 0.003

1 f = Ca2,∞ I 1− Ca() L 0.01

CaMKactive ∆=∆⋅ττfca,, CaMK fca CaMK KCaMKmCaMK, + active

∆=τ fca, CaMK 10 ms

1 ττfca=∆ fca, CaMK +0.5 + 2+ ms 1.0+ [Ca ]ss / 0.003

350.0 τ fca2 =+125.0 ⎡⎤ 1+−− exp()⎣⎦ICa() L 0.175 / 0.04

KmCaMK, = 0.15

−4 PCa =×2.43 10 cm/s; γCai = 1; γCao = 0.341

C. IKs

⎛⎞ ⎜⎟0.6 G Ks =+0.0249 1 1.4 ⎜⎟13.810/[]+×−+52Ca ⎝⎠()i

120

RT ⎛⎞[]KPNa+++⋅ [ ] E =⋅ln oNaK, o Ks ⎜⎟++ FKPNa⎝⎠[]iNaK+⋅, [ ] i

1 XXss∞∞==2 1+−− exp() (Vm 10.5) / 24.7

1 τ = xs1 7.61×⋅+ 10−−54 (VV 44.6) 3.6 ×⋅− 10 ( 0.55) mm+ 1−− exp() 9.97(VVmm + 44.6) exp() 0.128 ⋅− ( 0.55) − 1

τ Xs21=⋅2 τ Xs

PNa, K = 0.01833

I KssSmKs=⋅⋅⋅−GXXVEKs 2 ()

D. IKr

+ []K o G Kr =⋅0.01385 5.4

+ RT ⎛⎞[]K o EKr =⋅ln ⎜⎟+ FK⎝⎠[]i

1 τ = Xr 0.6×⋅− 10−−34 (VV 1.7384) 3 ×⋅+ 10 ( 38.3608) mm+ 1− exp() − 0.136 ⋅− (VVmm 1.7384) exp() 0.1522 ⋅+ ( 38.3608) − 1

1 X r∞ = 1+−+ exp() (Vm 10.085) / 4.25

1 RKr = 1++ exp() (Vm 10) /15.4

I KrrKrmKr=⋅⋅⋅−GXRVEKr ()

121

E. Ito1

+ RT ⎛⎞[]K o Eto1 =⋅ln ⎜⎟+ FK⎝⎠[]i

RVto1 = exp() m / 300

25⋅− exp() (Vm 40) / 25 αa = 1+− exp() (Vm 40) / 25

25⋅−+ exp() (Vm 90) / 25 βa = 1+−+ exp() (Vm 90) / 25

0.03 αi = 1++ exp() (Vm 60) / 5

0.2⋅+ exp() (Vm 25) / 5 βi = 1exp(++()Vm 25)/5

0.0015 αi2 = 1++ exp() (Vm 60) / 5

0.1⋅+ exp() (Vm 25) / 5 βi2 = 1exp(++()Vm 25)/5

Gto1 = 0.19 mS/µF

3 Ito1211=⋅⋅⋅⋅⋅−GaiiRVEto1 to() m to

F. INaCa

I NaCa =⋅∆Allo E

122

1 Allo = 2 ⎛⎞KmCa, act 1+ ⎜⎟2+ ⎝⎠1.5⋅ [Ca ]i

⎛⎞++32⎛⎞VF + 3 2 + ⎛(1)η −⋅ VF ⎞ vNaCamax ⋅⋅⋅⋅−⋅⋅⋅⎜⎟[][]expio⎜⎟η []1.5[]exp NaCa o i ⎜ ⎟ ⎝⎠RT ⎝ RT ⎠ ∆=E ⎝⎠ ⎛⎞⎛⎞(1)η −⋅VF ⎜⎟1exp+⋅ksat ⎜⎟ ⎝⎠⎝⎠RT ⎛⎞⎛⎞1.5⋅ [Ca2+ ] KNaK[]+++33+⋅⋅+⋅ 1.5[] CaK 2 3 [ Ca 2 ]1 ⋅+ i ⎜⎟mCao,, i mNao i mNai , o ⎜⎟ ⎜⎟⎝⎠KmCai, ⎜⎟+ 3 ⎜⎟+++++33232⎛⎞[]Na +K ⋅⋅++⋅[]1Na⎜⎟i [][][]1.5[] Na Ca +⋅⋅ Na Ca ⎜⎟mCa, io⎜⎟K 3 i o o i ⎝⎠⎝⎠mNai,

vmax =µ4.5 A/µF; ksat = 0.27; η = 0.35

Km,Nai = 12.3 mmol/L; Km,Nao = 87.5 mmol/L

Km,Cai = 0.0036 mmol/L; Km,Cao = 1.3 mmol/L

−4 KmCa, act =×1.25 10 mmol/L

G. INa

Our formulation is the same as that in the Luo-Rudy model 2 with a reduced conductance.

G Na = 6.7 mS/µF

+ RT ⎛⎞[]Na o ENa =⋅ln ⎜⎟+ FNa⎝⎠[]i

123

0.32⋅+()Vm 47.13 αm = 1−−⋅+ exp() 0.1 (Vm 47.13)

⎛⎞Vm βm =⋅−0.08 exp⎜⎟ ⎝⎠11

If Vm ≥−40.0 mV,

αh = 0.0

1 βh = ⎛⎞⎛⎞Vm +10.66 0.13⋅+⎜⎟ 1 exp⎜⎟ ⎝⎠⎝⎠−11.1

α j = 0.0

−7 0.3⋅−() 2.535 × 10 ⋅Vm β j = exp()−⋅ 0.1()Vm + 32 + 1

else

⎛⎞(80.0 +Vm ) αh =⋅0.135 exp⎜⎟ ⎝⎠−6.8

5 βhm=⋅3.56 exp( 0.079 ⋅+×⋅VV) 3.1 10 exp( 0.35 ⋅ m)

55− ()−×⋅1.2714 10 exp() 0.2444 ⋅−×⋅−⋅⋅+VVVmmm 3.474 10 exp( 0.04391) () 37.78 α j = 1+⋅+ exp() 0.311()Vm 79.23

0.1212⋅− exp( 0.01052 ⋅Vm ) β j = 1+− exp() 0.1378 ⋅+()Vm 40.14

124

3 IGmhjVENa=Na ⋅ ⋅⋅⋅() m − Na

H. INa,L

+ RT ⎛⎞[]Na o ENa, L =⋅ln ⎜⎟+ FNa⎝⎠[]i

0.32⋅+()Vm 47.13 αmL, = 1−−⋅+ exp() 0.1()Vm 47.13

⎛⎞−Vm βmL, =⋅0.08 exp⎜⎟ ⎝⎠11.0

1 hL,∞ = 1++ exp()()Vm 91 / 6.1

τhL = 600 ms

G Na, L = 0.0065 mS/µF

3 IGmhVENa,, L=⋅⋅⋅−Na, L L L( m Na L )

I. Cl- currents

− RT ⎛⎞[]Cl o ECl =− ⋅ln ⎜⎟− FCl⎝⎠[]i

+ RT ⎛⎞[]K o EK =⋅ln ⎜⎟+ FK⎝⎠[]i

+ RT ⎛⎞[]Na o ENa =⋅ln ⎜⎟+ FNa⎝⎠[]i

+ - a. K -Cl cotransporter, CTKCl

()EEKCl− CTKCl =⋅ CT KCl (EEKCl−+ ) 87.8251

125

−6 CT KCl =×7.0756 10 mmol/L per ms

+ - b. Na -Cl cotransporter, CTNaCl

4 ()EENa− Cl CT=⋅ CT NaCl NaCl 4 4 ()EENa−+ Cl 87.8251

−6 CT NaCl =×9.8443 10 mmol/L per ms

−6 - −6 CT NaCl =×9.8443 10 mmol/ms yields a resting Cl uptake rate of 7.1× 10 mmol/ms, close to the value of 810× −6 mmol/ms measured experimentally3.

- c. Cl background current, IClb

−4 GCl, b =×µ2.25 10 A/µF

ICl, b=⋅−GVECl, b () m Cl

2+ d. Ca -dependent transient outward current, Ito2

2 [][]expCl−⋅−⋅⋅ Cl z V F / RT 2 VFm ⋅ io( Clm( )) IPzto2 =⋅⋅Cl Cl ⋅ RT 1exp−−⋅⋅()zVFRTCl m /()

1 aa∞ = Kmto,2 1+ 2+ []Ca r

τaa = 1 ms

−7 Pcl =×4.0 10 cm/s

126

Kmto,2= 0.1502 mM

Ito2 =⋅Iaato2

J. Irel

∆τ rel, CaMK = 10 ms ; KmCaMK, = 0.15

CaMKactive ∆=∆⋅ττrel,, CaMK rel CaMK KCaMKm, CaMK+ active

21.9+ []Ca JSR ∆=ro∞,JSR 1.9 2+ 21.9+ ⎛⎞49.28⋅ [Ca ]ss []Ca JSR + ⎜⎟2+ ⎝⎠[Ca ]ss + 0.0028

2 ICa() L ro∞∞=⋅∆22 ro ,JSR ICa() L +1.0

τ ro = 3 ms

1 cafac = ⎡⎤ 1++ exp()⎣⎦ICa() L 0.05 / 0.015

1 ri∞ = ⎡⎤2+ 1+−+⋅ exp()⎣⎦ [Ca ]ss 0.0004 0.002 cafac / 0.000025

350 − ∆τ rel, CaMK ττri =++∆3 rel, CaMK ms ⎡⎤2+ 1+−+⋅ exp()⎣⎦ [Ca ]ss 0.003 0.003 cafac / 0.0002

-1 Gvgrel =⋅3000 ms

1 vg = ⎡⎤ 1++ exp()⎣⎦I Ca() L 13 / 5

127

22++ Irel=⋅⋅⋅ Grel ro ri()[][] Ca JSR − Ca ss

K. SR fluxes

2 Formulations for Iup, Ileak, and Itr come from the LRd model . The transfer rate of

2+ Ca from the NSR to the JSR, τtr, is decreased from 180 ms to 120 ms to get faster refilling of releasable Ca2+ and less emptying of SR as reported experimentally4.

∆K m,PLB = 0.00017 mmol/L ; ∆=Iup, CaMK 0.75; KmCaMK, = 0.15

CaMKactive ∆=∆⋅KKmPLB,, mPLB KCaMKm, CaMK+ active

CaMKactive ∆=∆⋅IIup,, CaMK up CaMK KCaMKmCaMK, + active

2+ []Ca i IIup=∆() up, CaMK +1 ⋅ Iup ⋅ 2+ []CaimupmPLB+−∆ K,, K

I up = 0.004375 mmol/L per ms; Km,up = 0.00092 mmol/L

NSR =15 mmol/L

0.004375 ICa=⋅[]2+ leakNSR NSR

τ tr =120 ms

22++ [][]CaNSR− Ca JSR Itr = τ tr

L. Time-independent currents

Formulations for the time-independent currents with the exception of the Ca2+

128

2 background current, ICa,b, come from the LRd model . Conductances are reduced to account for the smaller conductances of the major time-dependent currents in the HRd model.

a. ICa,b

2 γγ⋅⋅[]expCa z ⋅⋅ V F / RT −⋅ [] Ca 2 VFm ⋅ Cai i( Ca m()) Cao o IPzCa,, b=⋅⋅ Ca b Ca ⋅ RT exp()zVFRTCa⋅⋅ m /() − 1

−7 PCa, b =×1.849 10 cm/s, ; γCai = 1; γCao = 0.341

b. INaK

+ exp() [Na ]o / 67.3− 1.0 σ = 7.0

1 fNaK = ⎛⎞−⋅0.1 VFmm ⎛⎞ − VF 1+⋅ 0.1245 exp⎜⎟ +⋅⋅ 0.0365σ exp ⎜⎟ ⎝⎠RT ⎝⎠ RT

KmNai, =10 mM ; KmKo, =1.5 mM

G NaK = 0.5625 mS/µF

⎛⎞+ fKNaK[] o IG=⋅NaK ⋅⎜⎟ NaK 2 ⎜⎟[]KK+ + ⎛⎞KmNai, ⎝⎠omKo, 1+ ⎜⎟+ ⎝⎠[]Na i

c. IKp

+ RT ⎛⎞[]K o EKp =⋅ln ⎜⎟+ FK⎝⎠[]i

129

1 K p = ⎛⎞7.488 −Vm 1exp+ ⎜⎟ ⎝⎠5.98

−3 G Kp = 2.76×10 mS/µF

I KppmKp=⋅⋅−GKVEKp ( )

d. IK1

+ RT ⎛⎞[]K o

EK1 =⋅ln ⎜⎟+ FK⎝⎠[]i 1.02 α K1 = 1+⋅−− exp() 0.2385()VEmK1 59.215

0.49124⋅ exp() 0.08032 ⋅−+(VEmK11 5.476) + exp( 0.06175 ⋅−−( VE mK 594.31)) β K1 = 1+− exp() 0.5143 ⋅−+()VEmK1 4.753

α K1 K1 = α K11+ β K

+ []K o G K1 =⋅0.5 mS/µF 5.4

I KmK11=⋅⋅−GKVEK1 ( 1)

e. Ip,Ca

G p,Ca = 0.0575 mS/µF

2+ []Ca i IGpCa, =⋅pCa, 2+ KCampCa, +[] i

130

M. Ca2+ buffers in myoplasm

2+ []Ca i Buffered [][]trpn=⋅ trpn 2+ []Caimtrpn+ K ,

2+ []Ca i Buffered [][]cmdn=⋅ cmdn 2+ []Caimcmdn+ K ,

[trpn ]= 0.07 mmol/L; [cmdn ]= 0.05 mmol/L;

Km,trpn = 0.0005 mmol/L ; Km,cmdn = 0.00238 mmol/L

N. Ca2+ buffer in JSR

2+ []Ca JSR Buffered [][]csqn=⋅ csqn 2+ []CaJSR+ K m, csqn

[csqn] = 10.0 mmol/L; Km,csqn = 0.8 mmol/L

O. Intracellular ion concentrations

a. Ca2+

dCa[]2+ ⎛⎞A V V i =−II + −2 ⋅ I ⋅Cap + II − ⋅NSR − I ⋅ ss ⎜⎟()Ca,, b p Ca Na , Ca() up leak Diff dt⎝⎠ Vmyo⋅2 F V myo V myo

222+++ [][][][][]Cait, =+ trpn cmdn + Ca i +∆ Ca i

2+ bcmdntrpnCa=+−++[][][]it,, K mtrpn K mcmdn ,

131

2+ cK=⋅−⋅+mcmdn,, K mtrpn[] Ca it ,,,( K mtrpn K mcmdn) +⋅+⋅ [] trpnK mcmdn , [ cmdnK ] mtrpn ,

2+ dK=−mtrpn,, ⋅ K mcmdn ⋅[] Ca it ,

219227⎛⎞⎛⎞bc−− b3 d b []Ca22+−=− b 3coscos c ⋅ 1 − i ⎜⎟⎜⎟21.5 332(3)3⎝⎠⎝⎠bc−

b. Na+

+ dNa[]i ACap =−I Na, t ⋅ +CT NaCl dt Vmyo ⋅ F

c. K+

+ dK[]i ACap =−I K ,t ⋅ +CT KCl dt Vmyo ⋅ F

d. Cl-

− dCl[]i ACap =⋅ICl, t +CT NaCl + CT KCl dt Vmyo ⋅ F

132

P. SR calcium concentrations

a. JSR

dCa[]2+ JSR =−I I dt tr rel

22++ b=−−[][][][] csqn csqn CaJSR −∆+ Ca JSR K m, csqn

22++ cK=⋅mcsqn, ()[][] csqnCa + JSR +∆ [ Ca ] JSR ;

bcb2 + 4 − []Ca2+ = JSR 2

b. NSR

2+ dCa[]NSR V JSR =−IIup leak −⋅ I tr dt VNSR

Q. Subspace calcium concentration

1 βss = BSR⋅⋅ KmBSR,, BSL K mBSL 1++22 22++ ()KCaKCamBSR,,++[] ss() mBSL [] ss

BSR = 0.047 mM; KmBSR, = 0.00087 mM

BSL =1.124 mM; KmBSL, = 0.0087 mM

22++ [][]Cassi− Ca I Diff = τ Diff

τ Diff = 0.2 ms

133

2+ dCa[]ss⎛⎞ACap V JSR =−⋅βss⎜⎟III Ca +⋅− rel Diff dt⎝⎠ Vss⋅2 F V ss

R. Conservative current stimulus

A conservative current stimulus5 (see Appendix A) is implemented during pacing protocols.

For duration of current stimulus,

I K ,,tKt=+⋅II0.5 stim

ICl,, t=+⋅II Cl t0.5 stim

S. Vm

dV− I mtot= dt Cm

T. Cell geometry

Length (L) = 0.01 cm; radius (r) = 0.0011 cm

26− Cell volume: VrLcell =⋅π ⋅=38 × 10 µL

24− 2 Geometric membrane area: ArrLgeo =⋅+⋅⋅=2ππ 2 0.767 × 10 cm

−4 2 Capacitive membrane area: ARAcap=⋅= CG geo 1.534 × 10 cm

−6 Myoplasm volume: VVmyo=⋅ cell 68% = 25.84 × 10 µL

−6 Mitochondria volume: VVmito=⋅ cell 26% = 9.88 × 10 µL

−6 SR volume: VVSR=⋅= cell 6% 2.28 × 10 µL

134

−6 NSR volume: VVNSR=⋅ cell 5.52% = 2.098 ×µ 10 L

−6 JSR volume: VVJSR=⋅ cell 0.48% = 0.182 × 10 µL

−6 Subspace volume: VVss=⋅ cell 2.0% = 0.76 × 10

U. Miscellaneous comments

2+ The T-type Ca current, ICa(T) is found in endocardial but not epicardial cardiac

6 ventricular myocytes . Therefore, the HRd model does not include ICa(T).

+ In the LRd and HRd models, INaCa removes Na from the cell at rest. In the LRd

+ + model, the background Na current INa,b allows for Na entry to maintain homeostasis of

+ [Na ]i at rest. INa,b is not included in the HRd model due to the presence of CTNaCl, which brings Na+ into the cell at rest.

ICa,K and ICa,Na are assumed to be insignificant and are eliminated from the HRd model.

135

Works Cited

1. Hagemann D, Kuschel M, Kuramochi T, Zhu W, Cheng H, Xiao RP. Frequency- encoding Thr17 phospholamban phosphorylation is independent of Ser16 phosphorylation in cardiac myocytes. J Biol Chem. 2000;275:22532-6.

2. Luo CH, Rudy Y. A dynamic model of the cardiac ventricular action potential. I. Simulations of ionic currents and concentration changes. Circ Res. 1994;74:1071- 96.

3. Baumgarten CM, Duncan SWN. Regulation of Cl- activity in ventricular muscle: - - + - Cl /HCO3 exchange and Na -dependent Cl cotransport. In: Dhalla NS, Pierce GN, Beamish RE, eds. Heart Function and Metabolism. Winnipeg, Cananda: Martinus Nijhoff; 1986:116-131.

4. Shannon TR, Guo T, Bers DM. Ca2+ scraps: local depletions of free [Ca2+] in cardiac sarcoplasmic reticulum during contractions leave substantial Ca2+ reserve. Circ Res. 2003;93:40-45.

5. Hund TJ, Kucera JP, Otani NF, Rudy Y. Ionic charge conservation and long-term steady state in the Luo-Rudy dynamic model of the cardiac cell. Biophys J. 2001;81:3324-3331.

6. Wang H-S, Cohen IS. Calcium channel heterogeneity in canine left ventricular myocytes. J Physiol (Lond). 2003;547:825-833.

Appendix C - Diseased Model Equations

136 137

Table 7. BZ ion concentrations at rest

2+ −3 [Ca ]i 0.0997× 10 mmol/L - [Cl ]i 19.43 mmol/L + [K ]i 140.84 mmol/L + [Na ]i 11.53 mmol/L 2+ [Ca ]JSR 1.467 mmol/L 2+ [Ca ]NSR 1.467 mmol/L 2+ [Ca ]o 1.8 mmol/L - [Cl ]o 100 mmol/L + [K ]o 5.4 mmol/L + [Na ]o 140 mmol/L After model is undisturbed for 1000 s.

This section contains BZ equations that differ from control HRd model. See

Appendix B for complete control model equations.

A. CaMKII

CaMKactive = 0.0

B. ICa(L)

pow ICa() L=⋅df ca. f ca 2 ⋅⋅⋅ ffI 2 Ca

2 2 ()VFm −⋅15.0 γγCai⋅⋅[Ca ] ss exp( z Ca ⋅−⋅( V m 15.0)() F / RT) −⋅ Cao [ Ca ] o IPzCa() L =⋅⋅Ca Ca ⋅ RT exp()zVCa()() m −⋅ 15.0 FRT / − 1

1 d∞ = 1+−− exp() (Vm 4) / 6.74

exp( 0.052⋅+ (Vm 13)) τ d =+⋅0.59 0.8 1+⋅+ exp() 0.132 (Vm 13)

8 pow∞ =−9 1exp(+−+()Vm 65)/3.4

τ pow =10.0 ms

138

0.7 f∞ =+0.3 1.0++ exp() (Vm 17.12) / 7

0.77 f2∞ =+0.23 1.0++ exp() (Vm 17.12) / 7

0.25 τ f = 0.2411⋅− exp()⎣⎦⎣⎦⎡⎤⎡⎤ 0.045 ⋅−()()VVmm 9.6914 ⋅ 0.045 ⋅− 9.6914 + 0.0529

1 τ f 2 = 0.0823⋅− exp()⎣⎦⎣⎦⎡⎤⎡⎤ 0.059 ⋅−()()VVmm 18.5726 ⋅ 0.059 ⋅− 18.5726 + 0.0054

0.3 0.55 f =+ +0.15 Ca,∞ I []Ca2+ 1− Ca() L 1+ ss 0.05 0.003

1 f = Ca2,∞ I 1− Ca() L 0.01

CaMKactive ∆=∆⋅ττfca,, CaMK fca CaMK KCaMKmCaMK, + active

∆=τ fca, CaMK 10 ms

1 ττfca=∆ fca, CaMK +0.25 + 2+ ms 1.0+ [Ca ]ss / 0.003

315.0 τ fca2 =+110.0 ⎡⎤ 1+−− exp()⎣⎦ICa() L 0.175 / 0.04

KmCaMK, = 0.15

−4 PCa =×2.16 10 cm/s; γCai = 1; γCao = 0.341

139

C. IKs

⎛⎞ ⎜⎟0.6 G Ks =+0.0112 1 1.4 ⎜⎟13.810/[]+×−+52Ca ⎝⎠()i

RT ⎛⎞[]KPNa+++⋅ [ ] E =⋅ln oNaK, o Ks ⎜⎟++ FKPNa⎝⎠[]iNaK+⋅, [ ] i

1 XXss∞∞==2 1+−− exp() (Vm 10.5) / 24.7

1 τ = xs1 7.61×⋅+ 10−−54 (VV 44.6) 3.6 ×⋅− 10 ( 0.55) mm+ 1−− exp() 9.97(VVmm + 44.6) exp() 0.128 ⋅− ( 0.55) − 1

τ Xs21=⋅2 τ Xs

PNa, K = 0.01833

I KssSmKs=⋅⋅⋅−GXXVEKs 2 ()

D. IKr

+ []K o G Kr =⋅0.00762 5.4

+ RT ⎛⎞[]K o EKr =⋅ln ⎜⎟+ FK⎝⎠[]i

1 τ = Xr 1.4×⋅− 10−−34 (VV 1.7384) 3 ×⋅+ 10 ( 38.3608) mm+ 1− exp() − 0.136 ⋅− (VVmm 1.7384) exp() 0.1522 ⋅+ ( 38.3608) − 1

140

1 X r∞ = 1+−+ exp() (Vm 10.085) / 4.25

1 RKr = 1++ exp() (Vm 10) /15.4

I KrrKrmKr=⋅⋅⋅−GXRVEKr ()

E. Ito1

Gto1 = 0.0 mS/µF

F. INa

G Na = 4.31 mS/µF

+ RT ⎛⎞[]Na o ENa =⋅ln ⎜⎟+ FNa⎝⎠[]i

0.32⋅+()Vm 47.13 αm = 1−−⋅+ exp() 0.1 (Vm 47.13)

⎛⎞Vm βm =⋅−0.08 exp⎜⎟ ⎝⎠11

If ()VVm−≥− shift 40.0 mV,

αh = 0.0

1 βh = ⎛⎞⎛⎞VVm−+ shift 10.66 0.13⋅+⎜⎟ 1 exp⎜⎟ ⎝⎠⎝⎠−11.1

α j = 0.0

−7 0.3⋅−() 2.535 × 10 ⋅(VVmshift − ) β j = exp()−⋅ 0.1()VVm − shift + 32 + 1

141

else

⎛⎞ (80.0 +−VVm shift ) α =⋅0.135 exp⎜⎟ h ⎜⎟−6.8 ⎝⎠

5 βh=⋅3.56 exp( 0.079 ⋅−(VV m shift)) +×⋅ 3.1 10 exp( 0.35 ⋅−() VV m shift )

5 α ja=−1.2714 × 10 ⋅ exp( 0.2444 ⋅(VV m − shift ))

−5 α jb=×⋅−3.474 10 exp( 0.04391 ⋅−(VV m shift ))

(VVmshift−+37.78) α jc = 1+⋅−+ exp() 0.311()VVm shift 79.23

α jjajbjc=−⋅()ααα

0.1212⋅− exp( 0.01052 ⋅−(VVm shift )) β j = 1+− exp() 0.1378 ⋅−+()VVm shift 40.14

Vshift =−3.5 mV

3 IGmhjVENa=Na ⋅ ⋅⋅⋅() m − Na

G. Time-independent currents

a. IK1

+ RT ⎛⎞[]K o

EK1 =⋅ln ⎜⎟+ FK⎝⎠[]i

142

1.02 α K1 = 1+⋅−− exp() 0.2385()VEmK1 59.215

0.49124⋅ exp() 0.08032 ⋅−+(VEmK11 5.476) + exp( 0.06175 ⋅−−( VE mK 594.31)) β K1 = 1+− exp() 0.5143 ⋅−+()VEmK1 4.753

α K1 K1 = α K11+ β K

+ []K o G K1 =⋅0.35 mS/µF 5.4

I KmK11=⋅⋅−GKVEK1 ( 1)

b. Ip,Ca

G pCa, = 0.0 mS/µF

c. INaK

+ exp() [Na ]o / 67.3− 1.0 σ = 7.0

1 fNaK = ⎛⎞−⋅0.1 VFmm ⎛⎞ − VF 1+⋅ 0.1245 exp⎜⎟ +⋅⋅ 0.0365σ exp ⎜⎟ ⎝⎠RT ⎝⎠ RT

KmNai, =10 mM ; KmKo, =1.5 mM

G NaK = 0.50625 mS/µF

143

⎛⎞+ fKNaK[] o IG=⋅NaK ⋅⎜⎟ NaK 2 ⎜⎟[]KK+ + ⎛⎞KmNai, ⎝⎠omKo, 1+ ⎜⎟+ ⎝⎠[]Na i

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