Reappraisal of Luo-Rudy Dynamic Cell Model

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1 Review Reappraisal of Luo-Rudy Dynamic Cell Model Acta Cardiol Sin 2010;26:69 80 Reappraisal of Luo-Rudy Dynamic Cell Model Yi-Hsin Chan, 1 Chia-Tung Wu, 1 Yung-Hsin Yeh 1 and Chi-Tai Kuo 1,2 The cardiac myocyte is unique in that it is responsible for the coupling of electrical impulse to mechanical function, a process with so-called excitation-contraction coupling. There are several ions, ion channels, and regulatory pathways participating in the generation of action potential (AP), and it is a complex process. In the cardiac myocyte, dynamic change of ionic concentrations, membrane voltage following time sequence and various regulatory pathways determine the ionic channel kinetics, which can be expressed as mathematical formalisms. The properties make it feasible to use the computational approach to analyze and elucidate the underlying mechanisms of the whole cardiac cell. Over the past decades, there have been several attempts to construct the cardiac cell models based on the approach. Among them, the Luo-Rudy dynamic (LRd) model has been one of the most famous models since it was first published by Luo and Rudy in In this paper, the theory of LRd cell model and its ability for investigating several cellular arrhythmogenic pathologies in cardiac cells will be reviewed. In conclusion, the LRd cell model based on mathematical approach is a powerful method which can help recognize and investigate the interactions between individual components of the cell and study their unique role underlying the whole-cell behavior. Key Words: Cardiac action potential model Ion channels Luo-Rudy dynamic cell model INTRODUCTION The cardiac myocyte is unique in that it is responsible for the coupling of electrical impulse to mechanical function, a process with so-called excitation-contraction coupling. 1 Among the ions participating in the complex processes of the cardiac myocyte, Ca 2+ may be the most important because it is directly involved in both electricity and mechanics. In the cardiac myocyte, dynamic change of ionic concentrations (Na +,K +,Ca 2+, and Cl - ), membrane voltage, time sequence and various regulatory pathways determine the ion channel kinetics, which can be expressed as mathematical formalisms. Received: February 23, 2010 Accepted: May 4, First Division of Cardiovascular Department, Chang Gung Memorial Hospital, Linkou Branch; 2 College of Medicine, Chang Gung University, Taoyuan, Taiwan. Address correspondence and reprint requests to: Dr. Chi-Tai Kuo, First Division of Cardiovascular Department, Chang Gung Memorial Hospital, Linkou, Taoyuan, Taiwan. Tel: , ext. 8162; Fax: ; chitai@adm.cgmh.org.tw There are several ion channels integrated in a single cell, and their interactions are intricate and nonlinear processes, making the single cardiac cell an interacting system with high synergism and integration. These properties make it feasible to use the computational approach to analyze and elucidate the underlying mechanisms of the whole cardiac cell. Over the past decades, there have been several attempts to construct cardiac cell models based on this approach. Among them, the Luo-Rudy dynamic (LRd) model has been one of the most famous models since it was first published by Luo and Rudy in The LRd model is a mathematical model of guinea pig ventricular myocyte, which is derived from individual channel-based approach by using Hodgkin-Huxley type formalism 3 initially. The LRd model contains several nonlinear ordinary differential equations (ODEs) regarding the dynamic change of intracellular ion concentrations and kinetic changes of the ion channels, which are used to derive macroscopic ionic currents through the channels and to construct the cell membrane AP. The model has gained popularity due to its elegant and feasi- 69 Acta Cardiol Sin 2010;26:69 80

2 Yi-HsinChanetal. ble design of formalism, and most important of all, its realistic simulation of the electrophysiological or pathological behaviors of cardiac cell in the real world. Specifically, the LRd model has kept evolving with advancing knowledge about the detailed molecular structure/function, the kinetic properties, and the modifications caused by genetic mutations of the cardiac ion channels in the past decades In this paper, the LRd cell model will be reviewed and its whole body will be built up step by step. The Hodgkin-Huxley AP model of nerve axon 3 will be the starting point, which is the basis of all modern cardiac and nerve AP models, followed by construction of the AP model by incorporating detailed ion (Na +,K + and Ca 2+ ) channel models into the LRd cell based on Markov chain formalism, 6-10 and finally the conjunction with two important intracellular regulatory pathways including the CaMKII 13 and the -adrenergic signaling cascades. 14 Furthermore, the ability of the integrated model in investigating several cellular arrhythmogenic pathologies caused by genetic mutations will be also illustrated. CONSTRUCTION OF CARDIAC ACTION POTENTIAL BASED ON MATHEMATICAL FORMALISM The nerve axon action potential (AP) model The Hodgkin-Huxley (HH) model describes the electrical characteristics of excitable cells like neurons or cardiac cells based on a mathematical approach. At the first, Hodgkin and Huxley used the voltage-clamp technique to study the changes of voltage-dependent conductance in excitable cells of the squid nerve axon. 15 The intracellular recording of the axon membrane potential (V m ) revealed that inward movement of Na + contributed to the initial rising phase, termed as depolarization, while outward flow of K + resulted in a falling phase, termed as repolarization. 16 A mathematical model was then developed to interpret the axon AP morphology that they had recorded. 3 For a single axonal cell, the relation of V m and the total transmembrane ionic current (I ion ) can be formulated as the following differential equations: where C m is the membrane capacitance ( F/cm 2 )provided by charge separation of the lipid bi-layer. I ion is the sum of three currents: I Na, which represents the depolarizing Na + current, I K, which accounts for the repolarizing K + current and I L, a small leakage current that is primarily carried by Cl - ion. The driving force for each ion current is generated by transmembrane ion concentration gradient, which is the difference between membrane potential V m and the equilibrium potential E for each ion (V m -E). Hodgkin and Huxley used several hypothetical gates to modulate the flow of ions through the channel. The conductance of the Na + channel is modeled by three identical fast gates m and one slow gate h, while the conductance of K + channel is modeled by four identical fast gates n during the AP. In voltage-gated ion channels, the channel gate is a function of both time and V m, and each gate can transition with first-order voltage-dependent kinetics between the opening and closed status, which is independent of the status of all other gates. The Na + and K + can only pass through the channel when all gates are open. The conductance of leakage ion I L is assumed constant and does not vary with time or V m. It is amazing that Hodgkin and Huxley derived such a relatively simple but precise mathematical model of the axonal AP even though the underlying mechanisms of time- and voltage-dependent gating in the ion channel still unknown at that time. It was several decades later that people are able to identify and record single-ion channels, clone specific ion channels and elucidate the detailed 3-dimensional conformation and function/structure of channels. Hodgkin and Huxley s originality and foresight of the ion channel model has remained the basis for modern neuronal and cardiac AP models. The Luo-Rudy dynamic (LRd) cardiac action potential model The first cardiac AP models were the Purkinje fiber, which formulated by McAllister et al. 17 (1975), and the mammalian ventricular myocyte (B-R model) formulated by Beeler and Reuter 18 (1977). These two models both describe the ion currents based on the Hodgkin- Huxley formalism. In 1991, Luo and Rudy had published the first Luo-Rudy model (LR-I) of the mammalian (Guinea pig) ventricular AP 2, which included the update experimental information that have been accumulated since the formulation of the B-R model in In Acta Cardiol Sin 2010;26:

3 Reappraisal of Luo-Rudy Dynamic Cell Model this model, the rate change of cardiac membrane potential V m with time is described by: where C m is the membrane capacitance ( F/cm 2 ), I st is the stimulus current, I i is the sum of total transmembrane ionic current ( A/cm 2 ) including: I Na, the fast inward Na + current; I K, the time-dependent K + current; I K1,the time-independent K + current; I KP, the plateau K + current; I b, the background current; and I Si, the slow inward current carried by Ca 2+. In this model, six ion currents were incorporated to construct the cardiac AP. The I Na, I K, I K1,andI KP currents were reformulated based on the Hodgkin-Huxley formalism. The I Si current was retained from the original B-R model to maintain the AP plateau. The LR-I model had gained its success in simulating the depolarization and repolarization phases of the AP and several electrophysiological phenomena like supernormal excitability, Wenckebach periodicity, and aperiodic patterns of the cardiac cell to periodic stimulation. 2 The LR-I cell did not account for dynamic changes of intracellular ion concentrations (Na +,K + and Ca + ) during the AP, especially the intracellular Ca 2+ [Ca 2+ ] i transients (CaT), which is the essential component of living cardiac myocyte expressing their contractile function. In 1994, Luo and Rudy had modified their LR-I to the Luo-Rudy dynamic model (LR-II), 4,5 where dynamic changes of ion concentrations during the AP were accounted for and formulated. Also, the L-type Ca 2+ channel current (I Ca(L) ) was reformulated based on the most advanced results from single-channel and single-cell experiments at that time. Several intracellular pumps and exchangers such the Na + /Ca 2+ exchanger (I NaCa ), the Na + /K + pump (I NaK ), the sacrolemma Ca 2+ pump (I p(ca) ) and the nonspecific Ca 2+ -activated current (I ns(ca) ) were incorporated to maintain and regulate the intracellular ion balance in the cardiac cell. The SR system was also reformulated to construct the detailed kinetics of intracellular Ca 2+ fluxes, including the following: separation of SR compartment into the network SR (NSR) and the junctional SR (JSR), calsequestrin (CASQ2) as the Ca 2+ buffer in thejsr,uptakeofca 2+ (I up ) through the SR Ca 2+ ATPase (SERCA) by the NSR, leakage of Ca 2+ (I leak ) from the NSR into intracellular space, translocation of Ca 2+ (I tr ) from the NSR to the JSR, and release of Ca 2+ fluxes (I rel ) via the ryanodine receptor 2 (RyR2) by the JSR. In addition, Ca 2+ buffers like calmodulin (CMDN) and troponin (TRPN) in the myoplasma were also included in the model. This model provides an accurate simulation of dynamic changes in ion concentrations and, in particular, intracellular CaT and fluxes during the AP. The properties of the model provided the basis of the excitation-contraction coupling process in cardiac cells. In addition, arrhythmogenic behaviors of the single cell including early afterdepolarization (EAD), delayed afterdepolarization (DAD), and triggered activity were also investigated in this model. 4,5 The LRd model provides a useful tool for basic and clinical investigation of electrophysiological processes and helps explain the underlying mechanisms that are not easily demonstrated in the experiments. Over the past decades, the LRd model has been updated to keep up with the accumulating information about ion channels and intracellular signaling pathway in cardiac cells, including the replacement of the time-dependent K + current I K with the rapid and slow components of the delayed rectifier K + current, I Kr and I Ks (LRd95), 19 formulation of I Ks for different cell types with endocardial, midmyocardial and epicardial cells (LRd99), 20,21 development of a novel theoretical model of the canine ventricular myocyte (Hund-Rudy dynamic (HRd) model), 13,22 formulation of the -adrenergic signaling cascade 14 and the Ca 2+ /calmodulindependent protein kinase (CaMKII) regulatory pathway, 13,22-24 and most important of all, reformulation of I Na, I Ks, I Kr, I Ca(L) and I rel based on a detailed Markov chain formalism (LRd00). 6-11,14,25 The Markov chain-based approach to model ion-channel kinetics The initial Luo-Rudy models (LR-I, LR-II and LRd99) construct the ionic currents based on Hodgkin- Huxley type approach. As more knowledge about the structure, function and kinetics of ion channels has been obtained, it is inadequate just to present the transitions of an ion channel between closed and open states based on the gating parameters (e.g. h, n, m) simply, and more specific kinetic status and information of channels should be defined. It is assumed that the proteins of ion 71 Acta Cardiol Sin 2010;26:69 80

4 Yi-HsinChanetal. channels have a few stable conformational states among which they can switch rapidly. There is no memory in each process of the model, and it is assumed that the probability of channel transition between states is only dependent on the present conformation of the channel but not on the history of previous states or behaviors in which the channel has remained in one state. The above phenomena indicate that the kinetics of the ion channels from a macroscopic view can be represented by the Markov chain based approach. 10,26-28 For example, there are three typical types of conformational states in ion channel kinetics: the opening state (O), closed state (C) and inactivated state (I). The relations between these three states are shown as Figure 1. The single ion channel can reside within any of these states, and there are different transition kinetics between any of the two states. The Markov chain models calculate occupancy of the channel in its several kinetic states depending on voltage, time and ligand binding. The ion (i.e. Na +,K +, Ca 2+,Cl - ) can pass through the channel only when the channel resides in its opening state. Therefore, the macroscopic ionic current I c through the ion channel can be formulated by the following equation: where for an arbitrary channel C, g C is the single channel maximal conductance, n is the number of channels per unit membrane area, O is the probability that a channel occupies the opening state, and (V m - E C ) is the driving force for specific ion. The equation specifically considers the fact that ionic current is contributed from a proportion of channels occupying in the opening state with a probability which is dependent on V m and time. This single-channel-based formulation of the current density can be integrated into a cell model. Due to the definite presentation of specific channel state, the model can be used to describe not only the macroscopic current contributing for the AP, but also the probability and transitions of each channel state, which gives a mechanistic and detailed link between the whole-cell AP and the structure/function of ion channels. Because the molecular interaction or structure abnormality caused by gene defects often depends on the conformational change of the channel, the Markov chain model is very useful to elucidate this behavior. Over the past decade, the major ion channel currents like I Na, I Ca(L), I Ks, I Kr,andI rel in the LRd cell model have been updated from the original Hodgkin-Huxley to the dedicated Markov chain formalism, and the detailed electrophysiological processes, mechanisms of anti-arrhythmic drugs, mutations of ion channels and related arrhythmogenic phenomena have been simulated successfully based on this approach. 6-11,14,29-31 The calcium transient and the major ionic currents during the time course of action potential The schematic diagram of the integrated LRd cell is shown in Figure 2. Details of the ion channel models and formulations can be found in the original articles. The constructed AP, CaT, and the major ionic currents contributing for the time course and morphology of the AP based on the model are illustrated in Figure 3. As the threshold of AP is reached, the I Na current provides a large (200~300 ua/uf) pulse within ~1 ms and results in the initial depolarizing upstroke of AP. At the same time, the I NaK exchanger activates immediately to extrude the intracellular Na + loading. Once the AP upstroke reaches around -25 mv, the I Ca(L) channel is activated and provides the depolarizing current to maintain the plateau of AP, which is counterbalanced by the outward repolarizing currents provided by I Ks and I Kr. The unique A Figure 1. (A) Schematic description of ion-channel transitions during the action potential (AP). (B) The three basic conformations with opening, inactivated, and closed states and their transition kinetics of the ionic channel can be represented by the Markov chain model (O, C, and I represent the probability of opening, closed, and inactivated states for each single channel;,,,,, and are the transition rates between each pair of states). More examples of Markov models of ionic currents are provided in Ref. 10. B Acta Cardiol Sin 2010;26:

5 Reappraisal of Luo-Rudy Dynamic Cell Model spike and dome morphology of I Ca(L) plays a important role in triggering the CICR process mediated by the activation of I rel. The activation of I Ca(L) and I rel generates the CaT and initiates myocyte contraction. The I Ca(L) is inactivated by both voltage and Ca 2+ -dependent processes. 32 During the plateau of AP, the I Ks and I Kr increase gradually to repolarize the V m to the resting level. The I NaCa also activates in the forward mode to extrude the Ca 2+ loading during this period. A large upstroke of I K1 is noted in the late repolarizing phase to stabilize the resting membrane potential (RMP). Note that the major ionic currents show different morphology and time course under slow (CL = 1000 ms, blue) and fast pacing rate (CL = 250 ms, red), contributing to the rate-dependent adaptation of AP duration (APD). A smaller I Na current is noted under fast pacing, which results in the lower AP peak and affects the subsequent activation of other channels (I Kr, I Ks,andI Ca(L) ) determining APD. Higher peak and plateau in I Ca(L) currents are noted under fast pacing, making more Ca 2+ entry into the cytoplasm and increase the Ca 2+ release from the JSR through the I rel receptor. The above synergistic process results in greater peak level of CaT under rapid pacing rate. The forward I NaCa are also highly activated to extrude excess intracellular Ca 2+ storage. The above processes indicate a high Ca 2+ loading status under fast pacing. More rapid rise and greater peak in I Ks current is noted to shorten the APD efficiently under fast pacing rate. The rapid rise in I Kr current is also noted under rapid pacing. However, no obvious difference in magnitude of I Kr peak is noted between slow or fast pacing, which makes I Kr secondary to I Ks in participating in the rate-dependent adaptation of APD in ventricular myocytes of guinea pig. Figure 2. Schematic diagram of the intergrated LRd cell model (modified from Refs. 4, 7, 8, 9, 10 and 11). The intracellular compartment is the sarcoplasmic reticulum (SR), which is divided into the network SR (NSR) and the junctional SR (JSR). The calsequestrin (CASQ2) acts as the Ca 2+ buffer in the JSR. Ca 2+ buffers like calmodulin (CMDN) and troponin (TRPN) are located in the myoplasm. The abbreviations representing ionic currents, pumps, and exchangers are defined as follows: I Na, fast Na + current; I Kr, rapid delayed rectifier K + current, I Ks, slow delayed rectifier K + current; I K1, time-independent K + current; I Kp, plateau K + current; I Ca(L),Ca 2+ current through L-type Ca 2+ channels; I Ca(T),Ca 2+ current through T-type Ca 2+ channels; I Na,b, background Na + current; I Ca,b, background Ca 2+ current; I NaK,Na + /K + -ATPase pump current; I NaCa,Na + /Ca 2+ exchange current; I NaCa(ss),Na + /Ca 2+ exchange current in the subspace; I p(ca), sarcolemmal Ca 2+ pump; SERCA, Sarco/Endoplasmic reticulum Ca 2+ -ATPase; I up,ca 2+ uptake from the myoplasm to NSR via SERCA; I leak,ca 2+ leakage from NSR to myoplasm; I tr,ca 2+ translocation from NSR to JSR; I rel,ca 2+ release by the ryanodine receptor 2 (RyR2) from JSR. Detail formulations of each ionic current are provided in Ref. 4, 7, 8, 9, 10 and 11. In addition, the -adrenergic regulatory pathway and the calcium/calmodulin-dependent protein kinase II (CaMKII) signaling cascade are also incorporated into the cell model. The binding of -agonist to the -receptor ( -AR) downstream affects the camp-dependent PKA, which regulate the kinetic behaviors of several ion channels via the phosphorylation process. The target lesions of PKA include I Ca(L), SERCA (I up ), I rel,i K1,I Ks and phospholambam (PLB) (Ref. 14). For the CaMKII signaling cascade, as activated by Ca 2+ /calmodulin, the CaMKII phosphorylates neighboring subunits (autophosphorylation) and substrates include I Ca(L), SERCA (I up ), I rel and phospholambam (PLB) (Ref. 13). 73 Acta Cardiol Sin 2010;26:69 80

6 Yi-HsinChanetal. Figure 3. The action potential (AP), intracellular calcium transient (CaT), and major ion channel currents (I Na,I Ks,I Kr,I K1,I Ca(L),I NaCa and I NaK )in the LRd cell model under different pacing rates (CL = 1000 ms, blue; CL = 250 ms, red). The -adrenergic signaling cascade and calcium/calmodulin-dependent protein kinase II (CaMKII) signaling cascade Increased heart rate is often concurrent with enhanced contraction to augment cardiac output under physiological demand, which can be mediated by adrenergic stimulation 33 or CaMKII regulatory pathway. 34,35 The -adrenergic signaling cascade enhances the contractility of the cardiac cell by elevating the CaT, which is primarily achieved via increasing the inward Ca 2+ flux through I Ca(L) channel. 33 Besides, the regulatory pathway also substrates several channels, including the I up, I rel, I NaK, I K1,andI Ks. 14 The above effects are formulated into the LRd cell (Figure 2) to simulate the electrophysiological process under the application of -agonist (i.e. isoproterenol (ISO)). 14 The -adrenergic signaling cascade in cardiac cell provides a basis for investigating many cardiac pathological conditions, since several arrhythmias are induced under high -adrenergic tone. Another important intracellular regulatory pathway is the CaMKII phosphorylation, which target on I Ca(L), I rel, I up, and phospholambam (PLB) 13 and mediate the modulation of frequency, amplitude, and duration of CaT. The above mechanisms link its effects in the promotion of positive frequency-contractility dependence and acceleration of relaxation, particularly during exercise or stress. The above processes are incorporated into the ventricular cell models of canine (Hund-Rudy dynamic (HRd) model) 13 and guinea pig (LRd) 24 (Figure 2) to simulate the CaMKII-mediated electrophysiological behaviors. The simulations reveal that gradual increase of pacing rate is accompanied by greater CaT amplitude and more rapid rate of relaxation, which is proportional to increased CaMKII activity. Another important property of CaMKII signaling cascade is that it mediates the process of APD alternans in cardiac myocytes, which is primarily driven by alternation of CaT. 36 Detailed clamp protocols are simulated to find the determinant of APD alternans. The CaT alternans persist after clamping the AP to the same morphology between each paced beat. However, the AP alternans disappear after clamping the CaT to either its small or large morphology, indicating that CaT alternans cause AP alternans. The SR Ca 2+ subsystem continues to oscillate during clamping with large CaT morphology, and the SR Ca 2+ release rate is higher during large depletion than during small depletion. The simulation reveals the alternans of V m,cat,i Ca(L) and several intracellular Acta Cardiol Sin 2010;26:

7 Reappraisal of Luo-Rudy Dynamic Cell Model Ca 2+ fluxes for two consecutive beats under rapid pacing rate with CaMKII application. The alternans are eliminated when CaMKII is inhibited, suggesting the key role of CaMKII in determining APD and CaT alternans. It has been shown that the mechanism of CaT alternans is caused by the refractoriness of the JSR Ca 2+ release process rather than the alternation of I Ca(L), since clamping of I Ca(L) only influences the amplitude of CaT alternation but fails to eliminate it. 37 Specifically, the two rate-limiting I up and I tr fluxes, conjugated with steep dependence of SR Ca 2+ release on SR Ca 2+ load, play a major role in the onset and offset of sustained alternans during rapid pacing. 24 Clinically, the APD alternans are reflected as the manifestation of T-wave alternans in surface ECG, which is associated with repolarization dispersion, susceptibility to lethal ventricular arrhythmias and sudden cardiac death. 38 Enhanced CaMKII activity is noted in several pathological conditions like cardiac hypertrophy, ischemia, and failure, which broadens the frequency range of CaT and APD alternans and elevates alternans amplitude, indicating its important role in lethal arrhythmia for these diseases. THE SIMULATIONS OF CELLULAR ARRHYTHMOGENIC BEHAVIORS IN THE LRD CELL MODEL The early afterdepolarization (EAD), delay afterdepolarization (DAD), and triggered activity The afterdepolarization is an abnormal depolarization of the cardiac cell that interrupt the normal AP, which occurs at phase 2/phase 3 (early) or phase 4 (delayed) during the whole APD. The early after-depolarizations (EADs) and delayed after-depolarizations (DADs) are single-cell phenomena which can cause abnormal electrophysiological behavior and cardiac arrhythmias. 39,40 The after-depolarizations may not reach threshold of AP initiation, but, if they do, they can induce another episode of after-depolarization or triggered activity, and thus self-perpetuate. All of these arrhythmogenic phenomena involve dynamic changes in intracellular Ca 2+. The long-qt syndrome (LQT) is a rare congenital heart disease with clinical manifestation of QT interval prolongation on the ECG, which is associated with syncope, seizures, or sudden cardiac death due to ventricular arrhythmias, possible torsade de pointes and ventricular fibrillation. 40 Several medications like class IA and III antiarrhythmic agents and pathological conditions like hypomagnesemia, hypokalemia, and cardiac hypertrophy also cause acquired QT prolongation (acquired LQTs). The arrhythmogenic mechansim of LQT is associated with the formation of EADs, which are mediated by the secondary reactivation of the I Ca(L) channel during the AP plateau. In the normal condition, the APD is tightly controlled by the balance of inward depolarizing currents with I Na and I Ca(L) and outward repolarizing currents with I Ks, I Kr,andI K1. Increase of I Na late currents, I Ca(L) plateau currents, or reduction of I Kr, I Ks and I K1 currents will cause APD prolongation, which in turn provides more chance for the I Ca(L) channels to recover from inactivation during phase 2 or 3 AP. Once sufficient recovery is achieved, the I Ca(L) channels reactivate and depolarize the cell membrane again. This is a positive feedback process and leads to the formation of EADs. The DADs are involved in spontaneous Ca 2+ release from the JSR compartment due to excess intracellular Ca 2+ loading, which includes several pathological or electrophysiological processes like rapid pacing stimulation, -adrenergic stimulation, drug intoxication (i.e. digitalis overdose), myocardial ischemic change, myocardial failure, genetic mutations like CASQ2 or RyR2 mutation, etc. The spontaneous Ca 2+ release from the JSR compartment then activates both the I ns(ca) current and I NaCa exchanger, which results in a net inward depolarizing current. The above arrhythmogenic processes are integrated into the LRd model, and the underlying mechanisms of the pathologies can be investigated. Two classic examples of cellular arrhythmogenesis will be illustrated in the following sections. 6,14 More simulations based on this approach can be found in the original articles. 6-11,14,30,31,41-43 The SCN5A mutation, LQT3 and Brugada syndrome Several distinct mutations in the SCN5A gene cause the so-called type 3 congenital long-qt syndrome (LQT3s). 44 The KPQ mutation 45 is one of the most severe LQT3s, which results in faster activation and recovery of the sodium channel from inactivation state and transient complete failure of the channel to remain inac- 75 Acta Cardiol Sin 2010;26:69 80

8 Yi-HsinChanetal. tivated, making the channel enter a bursting mode. The computational simulations 6 reveal a small but definite population of channels (< 1%) enter into the burst mode with a small transition rate. The lack of inactivated states causes frequent opening states in the burst mode, and the increased rate of recovery from inactivated state causes dispersed re-openings of channels in the background mode. The above combinations cause a significant late component of I Na during the plateau of AP and contribute to APD prolongation, which is reflected as the clinical manifestation of QT interval prolongation in ECG for the LQT3 syndrome. The prolonged APD may further lead to the generation of arrhythmogenic EADs during the AP plateau at slow pacing rates, which can explain why the arrhythmogenic episodes in congenital LQT3 most attack during bradycardiac conditions like relaxation or sleep. Another interesting example of SCN5A mutation is the 1795insD, 46 which can cause both clinical phenotypes of LQT3s and Brugada syndromes. Unlike the LQT3s, the Brugada syndrome results from decreased function of I Na, which is clinically manifested by ST segment elevation in the right precordial leads during tachycardia. The LQT3s and Brugada phenotypes and underlying mechanisms seem to be opposed to each other but can coexist paradoxically in the individual patient with 1795insD mutation, causing life-threatening arrhythmias and sudden cardiac death. The 1795insD mutant channel 8,10 has reduced channel availability and slow recovery from inactivation. Also, it contains a burst mode just like the KPQ model. In simulations of 1795insD mutation, three types of myocardial cells are considered to simulate the heterogeneity of ventricular myocardium: the epicardial (Epi-), endocardial (Endo-) and midmyocardial (M-) cells, which have different levels of I to (the transient outward current) and I Ks expression. 8 For the Epi-cells, especially from the right ventricular outflow tract (RVOT), have the highest levels of I to and I Ks, while the M-cells and Endo-cells have relatively low density of I to and I Ks. The difference of I to and I Ks density causes different AP morphology of three cell types 20,21 and contributes for myocardial transmural heterogeneicity. Under the condition of fast pacing rates with CL = 300 ms (Figure 4a), there is alternation between loss of dome and coved dome in the AP morphology for the 1795insD Epi-cells. The loss of dome is due to reduced peak I Na (Figure 4a: Panel I Na, arrow) and relatively large I to and I Ks in RVOT Epi-cells, contributing to premature repolarization and failure to activate the plateau currents of I Ca(L) effectively (Figure 4a: Panel I Ca(L), arrow). On the alternate beats, the Na + channel can recover more completely due to previous prolonged diastolic interval (DI) and thus results in a larger I Na.The larger I Na establishes a higher peak phase 0 V m and a shallow AP notch, which allows for subsequent activation of the I Ca(L) plateau currents and results in AP dome formation. The alternate beats do not happen in the M-cells or Endo-cells during rapid pacing due to relatively low density of repolarizing outward currents of I to and I Ks for the two cell types. As seen in the results, the different transmembrane potential of epicardial and other cells types at fast pacing generates an obvious transmural gradient - V m from the epicardial to M region during the AP plateau and repolarization phases. Because the potentials of surface ECG are proportional to the - V m, the transmural gradient causes ST segment elevation in the right precordial leads (i.e. V 1 and V 2 )of ECG, which is the closest distance to the RVOT epicardial cells. Furthermore, gradual reduction of I Na by accelerating the fast inactivation rate generates three levels of severity in Brugada syndrome: the saddleback, coved and triangular morphologies, 47 which are compatible with clinical patterns of Brugada patients. 48 The AP morphologies as previous described only happen at fast pacing rates. Under the condition of slow pacing with CL = 1000 ms (Figure 4b), the mutant channels have sufficient time to recover from inactivation state between beats and the condition of reduced I Na due to delay recovery does not happen, causing a minimal effect of the mutation on epicardial AP. However, the mutation affects the M-cell AP preferentially at slow rate. The long DI of slow pacing rates allows the channels to recover from the absorbing inactivated states and residue in the closed states, where they can enter the burst mode. Just like the KPQ mutation, the burst mode generates a small but persistent I Na current during the AP plateau, which causes the APD prolongation (Figure 4b: Panel I Na, arrow). In M-cells, the repolarizing effect of I Ks is smaller than in the other cell types and cannot counterbalance the late depolarizing I Na, which causes preferential APD prolongation of M-cell at slow rate. The APD of M-cell is only slightly prolonged at CL = Acta Cardiol Sin 2010;26:

9 Reappraisal of Luo-Rudy Dynamic Cell Model Figure 4. The comparison of M-cell (blue) and Epi-cell (red) of 1795insD mutation on the whole-cell AP, INa, and ICa(L) at different pacing rates. Under the condition of fast pacing rates with CL = 300 ms (Figure 4a), loss of dome in the AP morphology is noted in the 1795insD Epi-cell. The loss of dome is due to reduced peak INa (Figure 4a: Panel INa, arrow) and relatively large Ito and IKs (not shown here) in RVOT Epi-cells, which contribute to premature repolarization and failure to activate the plateau currents of ICa(L) effectively (Figure 4a: Panel ICa(L), arrow). The alternate beats do not happen in the M-cells during rapid pacing due to relatively low density of repolarizing outward currents of Ito and IKs (not shown here) for the M-cells. As a result, the different transmembrane potential of epicardial cells and M-cells at fast pacing generate an obvious transmural gradient -ÑVm from the epicardial to M region during the AP plateau and repolarization phases. Under the condition of slow pacing with CL = 1000 ms (Figure 4b), the mutant channels have sufficient time to recover from inactivated state between beats and the conditions of reduced INa due to delayed recovery do not happen, which cause a minimal effect in the AP of Epi-cell. However, the mutation affects the M-cell AP preferentially at slow rate. A small but persistent INa current was noted during the AP plateau, which causes the APD prolongation (Figure 4b: Panel INa, arrow). In the M cell, the repolarizing effect of IKs (not shown here) is smaller than in other cells types and cannot counterbalance the late depolaring INa, which causes preferential APD prolongation and EAD generation at slow rate of CL = 1000 ms. 77 Acta Cardiol Sin 2010;26:69-80

10 Yi-HsinChanetal. 300 ms; however, it is prolonged obviously and accompanied with EAD formation at CL = 1000 ms. The longest myocardial APD determines the QT interval on the ECG. Therefore, prolongation of the M-cell APD at slow rate is reflected by the QT interval prolongation of the LQT ECG phenotype. The calsequestrin mutation, DADs, triggered activity and catecholaminergic polymorphic ventricular tachycardia (CPVT) Catecholaminergic polymorphic ventricular tachycardia (CPVT) is a familial arrhythmogenic disease with the clinical presentation of repeat syncope episodes during exercise, emotional stress or other conditions of high -adrenergic tone. It has been shown that mutations in the RyR2 and CASQ2 genes are associated with the pathologies of CPVT. 49 To investigate the underlying mechanism by which the CASQ2 mutations result in the CPVT expression, the application of -adrenergic stimulation (ISO) and CASQ2 defects (CASQ2 D307H )wereincorporated into the LRd model to simulate the underlying pathologies. 14,50 The simulation reveals that DADs are generated under ISO application in the CASQ2 D307H mutant cell under fast pacing with CL = 500 ms. The CASQ2 D307H mutation decreases Ca 2+ buffering in the SR, which results in large free Ca 2+ in the JSR. The excess Ca 2+ loading in the JSR, accompanied by the application of ISO, further increases the intracellular Ca 2+ loading and causes spontaneous Ca 2+ release from the JSR due to exceeding its Ca 2+ storage capacity. The forward I NaCa then removes the subsequent rise of CaT and causes a net inward current, which depolarizes the membrane and results in DAD generation. It is shown that I Ca(L) does not play a role in the DAD formation. Figure 5 shows the conditions of CASQ2 D307H mutation and application of ISO with the elevation of RMP by reducing I K1 density by ~65% (CL = 500 ms). This elevation in RMP, as seen in several pathological conditions like hypokalemia, K + channel-blocking medications, ischemia, or myocardial failure, 51 allows for two DADs to trigger the spontaneous APs (Figure 5a, arrow), which are mediated by the activations of I Na and I Ca(L) (Figures Figure 5. The DAD can trigger a spontaneous AP (triggered activity). The simulation is performed under the condition of CASQ2 D307H mutation and application of ISO with additional blockade of I K1. Note that the last paced beat which is followed by two subsequent DADs that trigger two spontaneous APs ( arrow ). Owing to the elevated RMP by ~65% I K1 blockade, the depolarization is large enough to trigger activation of both I Ca(L) (Panel d) and I Na (Panel e), resulting in the AP generation. The third DAD is also noted, but cannot generate sufficient depolarizing current to trigger another spontaneous AP. Acta Cardiol Sin 2010;26:

11 Reappraisal of Luo-Rudy Dynamic Cell Model 5d and e). Note that a third DAD is also noted, but cannot generate sufficient depolarizing current to trigger another spontaneous AP. The above simulation reveals that the arrhythmogenic mechanisms underlying CASQ2 D307H mutation are related to store-overload-induced Ca2+ release (SOICR) and DAD generation due to excess free SR Ca 2+ loading following fast pacing and -adrenergic stimulation. 14 CONCLUSION Cardiac arrhythmia is a serious disease with multiple involvement at the molecular, cellular, and organ levels in the human body, and it is essential to understand the electro-pathological behaviors and underlying mechanisms for precise diagnosis and effective treatment. 52,53 Construction of a cardiac cell model with integration of individual sub-cellular components including ion channels, dynamic intracellular ionic handling, intracellular organelles and regulatory signaling pathways into the physiologically functioning system of the cardiac cell is necessary to facilitate insights into the interactions and mechanisms underlying electrophysiological behaviors. Among multiple cardiac models, the LRd cell model has demonstrated its potential to simulate the behavior and mechanism underlying different scales of organism: from the molecular level with structure, function and electrostatic properties of ion channel, to the sub-cellular level with intracellular microanatomy, ionic cycling and regulatory signaling cascade, and to the cellular level with an integrated cell model, cellular electrophysiology and pathologies. It is a powerful method which can help recognize and investigate the interactions between components of the cell and study their unique roles in the whole-cell behavior. With the accumulation of knowledge about the genomics, proteomics, and metabolomics and genetic engineering, accompanied by advance of computational technology, the approach of mathematical modeling will keep evolving and provide a cornerstone for investigating and integrating organic behaviors at the cellular level, and can be also expanded to model the electrophysiology, transport and biomechanics at the multiple-cellular level with tissue, organ, and even the whole body as well. REFERENCES 1. Bers DM. Cardiac excitation-contraction coupling. Nature 2002; 415: Luo CH, Rudy Y. A model of the ventricular cardiac action potential. Depolarization, repolarization, and their interaction. Circ Res 1991;68: Hodgkin AL, Huxley AF. A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 1952;117: Luo CH, Rudy Y. A dynamic model of the cardiac ventricular action potential. I. Simulations of ionic currents and concentration changes. Circ Res 1994a;74: Luo CH, Rudy Y. A dynamic model of the cardiac ventricular action potential. II. Afterdepolarizations, triggered activity, and potentiation. Circ Res 1994b;74: Clancy CE, Rudy Y. Linking a genetic defect to its cellular phenotype in a cardiac arrhythmia. Nature 1999;400: Clancy CE, Rudy Y. Cellular consequences of HERG mutations in the long QT syndrome: pre-cursors to sudden cardiac death. Cardiovasc Res 2001;50: Clancy CE, Rudy Y. Na + channel mutation that causes both Brugada and long-qt syndrome phenotypes: a simulation study of mechanism. Circulation 2002;105: Silva J, Rudy Y. Subunit interaction determines I Ks participation in cardiac repolarization and repolarization reserve. Circulation 2005;112: Rudy Y and Silva J. Computational biology in the study of cardiac ion channels and cell electrophysiology. Q Rev Biophys 2006; 39: Faber GM, Silva J, Livshitz L, Rudy Y. Kinetic properties of the cardiac L-type Ca 2+ channel and its role in myocyte electrophysiology: a theoretical investigation. Biophys J 2007;92: Silva JR, Pan H, Wu D, et al. Multiscale model linking ionchannel molecular dynamics and electrostatics to the cardiac action potential. Proc Natl Acad Sci USA 2009;106: Hund TJ, Rudy Y. Rate dependence and regulation of action potential and calcium transient in a canine cardiac ventricular cell model. Circulation 2004;110: Faber GM, Rudy Y. Calsequestrin mutation and catecholaminergic polymorphic ventricular tachycardia: a simulation study of cellular mechanism. Cardiovasc Res 2007;75: Hodgkin AL, Huxley AF, Katz B. Measurement of currentvoltage relations in the membrane of the giant axon of Loligo. J Physiol 1952;116: Hodgkin AL, Huxley AF. A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 1952;117: McAllister RE, Noble D, Tsien RW. Reconstruction of the electrical activity of cardiac Purkinje fibres. J Physiol 1975; 251: Acta Cardiol Sin 2010;26:69 80

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