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Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2012, Article ID 183978, 10 pagesdoi:10.1155/2012/183978

Research Article

Simulation of Arrhythmogenic Effect of Rogue RyRs in FailingHeart by Using a Coupled Model

Luyao Lu,1 Ling Xia,2 and Xiuwei Zhu1

1 Department of Biomedical Engineering, Wenzhou Medical College, Wenzhou 325035, China2 Department of Biomedical Engineering, Zhejiang University, Hangzhou 310027, China

Correspondence should be addressed to Ling Xia, [email protected]

Received 21 June 2012; Accepted 22 August 2012

Academic Editor: Feng Liu

Copyright © 2012 Luyao Lu et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Cardiac cells with heart failure are usually characterized by impairment of Ca2+ handling with smaller SR Ca2+ store and highrisk of triggered activities. In this study, we developed a coupled model by integrating the spatiotemporal Ca2+ reaction-diffusionsystem into the cellular electrophysiological model. With the coupled model, the subcellular Ca2+ dynamics and global cellularelectrophysiology could be simultaneously traced. The proposed coupled model was then applied to study the effects of rogueRyRs on Ca2+ cycling and membrane potential in failing heart. The simulation results suggested that, in the presence of rogueRyRs, Ca2+ dynamics is unstable and Ca2+ waves are prone to be initiated spontaneously. These release events would elevate themembrane potential substantially which might induce delayed afterdepolarizations or triggered action potentials. Moreover, thevariation of membrane potential depolarization is indicated to be dependent on the distribution density of rogue RyR channels.This study provides a new possible arrhythmogenic mechanism for heart failure from subcellular to cellular level.

1. Introduction

Calcium is considered to be the key ion in mediatingthe process of cardiac excitation-contraction coupling (E-C coupling). Since the discovery of Ca2+ sparks in 1993[1], Ca2+ sparks have been widely accepted to be thestereotyped elementary Ca2+ release events in the intactmyocyte. Sparks arise via clusters of ryanodine receptors(RyRs) localized in the junctional SR (jSR) which is in closeapposition to transverse tubules (TTs) [2]. In a diastolicmyocyte, spontaneous Ca2+ sparks occur randomly at verylow frequency, even in the absence of Ca2+ influx. Duringa single muscle twitch, Ca2+ influx via sarcolemmal L-type Ca2+ channels will trigger synchronously occurrence ofthousands of sparks, summation of which in space and timecauses a global steep rise of Ca2+ concentration named Ca2+

transient. However under some pathological conditions,successive recruitment of Ca2+ sparks tends to evolve intoCa2+ waves propagating across the myocytes which mighttrigger ventricular arrhythmias [3].

With the improvement of optical methods and innova-tive techniques, microscopic Ca2+ signals at the subcellularlevel have been extensively investigated and characterized. In

addition to Ca2+ sparks via clustered RyRs, nonspark Ca2+

release events, named Ca quarks, activated by low-intensityphotolysis of Ca2+-caged compounds [4] or by inward Na+

current, INa [5], could elicit spatially homogeneous but smallCa2+ transient. These quarks are likely to be mediated via oneor a few RyR channels called rogue RyRs [6]. Differing fromRyR clusters that underly sparks, rogue RyRs are thoughtto be uncoupled with each other and behave in ways morelike the characteristic of single RyR channels [7]. Althoughdetection of these small rogue RyR channels is difficult byconventional instruments, some researchers have suggestedthat, besides sparks, the nonspark pathway via rogue RyRsexplains a part of SR Ca2+ leak [8, 9]. Quantitatively, withthe optical superresolution technique, Baddeley et al. haveindicated that there are greater numbers of rogue RyR groupsthan large RyR clusters [10]. An experimental study thatCa2+ waves are inhibited without affecting Ca2+ sparks byruthenium red suggests a nonspark producing RyR channelswhich are important to propagation of Ca2+ wave [11].Direct visualization of small local release events has beenmade possible by recent technical innovations. Brochet etal. claimed that they have directly visualized quark-like or“quarky” Ca2+ release events which might depend on the

2 Computational and Mathematical Methods in Medicine

opening of rogue RyRs (or small cluster of RyRs) in rabbitventricular myocytes [12].

SR Ca2+ leak consists of two components: RyR-dependent leak and RyR-independent leak [8]. The formeris thought to be comprised of spark-mediated leak (visibleleak) and non-spark-mediated leak (invisible leak). ElevatedSR Ca2+ leak would contribute to delayed afterdepolariza-tions (DADs) and consequently arrhythmia in heart failure(HF) [13]. Besides spark-mediated leak, additional Ca2+ leakvia rogue RyRs may be an important factor in disturbingCa2+ dynamics and triggering Ca2+ waves [11, 14]. However,how do these abnormal Ca2+ release events affect cellularelectrophysiological properties? The precise relationshipsbetween property of rogue RyRs and Ca2+ handling as well ascellular electrophysiology in failing heart are not completelyclear.

In this paper, we developed a coupled mathematicalmodel including Ca2+ cycling processes from subcellularto cellular level and electrophysiology of the ventricularmyocyte. The proposed coupled model was then applied tostudy the effects of Ca2+ release via rogue RyRs on subcellularspatiotemporal Ca2+ cycling and on the possible membranepotential changes in failing heart.

2. Methods

Subcellular Ca2+ release events and cellular Ca2+ cycling aswell as corresponding membrane potential were simulatedsynchronously by a coupled model. The model consists oftwo parts: a two-dimensional (2D) spatial Ca2+ reaction-diffusion model and an electrophysiological model of theventricular myocyte.

2.1. A Subcellular Ca2+ Reaction-Diffusion Model. The shapeof the cardiac myocyte in the model is represented as acircular cylinder 100 μm in length and 10 μm in radius.However, because of quasi-isotropic diffusion of Ca2+ onthe transverse section [15], a 2D model was used in oursimulation work (Figure 1), where x axis denotes the cell’slongitudinal direction and y axis is along the Z-line. It couldstill describe most of the key properties of Ca2+ waves, butneeds much less computation work than a 3D model. The 2Dspatiotemporal Ca2+ reaction-diffusion model is describedbased on a reaction-diffusion system proposed by Izu et al.[16]. Figure 1 shows the subcellular structural representationof RyRs network. The x-axis denotes the cell’s longitudinaldirection and the y-axis is along the Z-line. The blue dotsrepresent RyR clusters which account for Ca2+ sparks. Thesmall red dots are the rogue RyR channels which raise Ca2+

quarks. Rogue RyRs are distributed in a stochastic manner.Nrogue is referred to the distributing density of rogue RyRswith the unit of rogue RyR/μm2.

The free Ca2+ concentration [Ca2+]i in the reaction-diffusion is described by a differential equation as follows:

∂[Ca2+

]i

∂t= Dx

∂2[Ca2+

]i

∂x2+ Dy

∂2[Ca2+

]i

∂y2+ Jdye

+ Jbuffers + Jpump + Jleak + Jsub-rel,

(1)

ly

lxZ

y

x

-line

Figure 1: Geometry of RyRs distribution. The blue dots representCa2+ release units consisting of clusters of RyRs. lx = 2.0μm andly = 1.0 μm. The small red dots are rogue RyRs which scatter overthe plane randomly. In this figure, Nrogue is equivalent to 1.0 rogueRyR/μm2.

where Dx and Dy are the diffusion coefficients; Jdye and Jbuffers

are due to fluorescent indicator dye and endogenous Ca2+

buffer, respectively; Jpump is pumping rate of SR Ca2+ ATPase;Jleak is defined as a RyR-independent leak flux which is smalland invisible and persists in the presence of RyR inhibition[8]; Jsub-rel is summation of Ca2+ release fluxes in the 2Dsubcellular model which consists of two types of Ca flows asfollows:

Jsub-rel =∑

i, j

Jcluster

(xi, yj

)+∑

m,n

Jrogue(xm, yn

),

Jcluster

(xi, yj

)= Vcluster

([Ca2+]

SR −[Ca2+]

i,(xi,yj )

),

(2)

where Jcluster(xi, yj) is Ca2+ release flux via a cluster of RyRslocated on (xi, yj), Vcluster is maximal Jcluster conductanceequivalent to 1.97 × 10−8 ms−1, and Jrogue(xm, yn) is Ca2+

release flux via a rogue RyR channel located on (xm, yn),equivalent to 3.3166× 10−9 pmol/ms.

Firings of the two types of RyR channels are consideredto be stochastic processes and treated by the Monte Carlosimulation in our work. To evaluate the effects of SRluminal Ca2+ concentration ([Ca2+]SR) on SR Ca2+ release,we integrate a new parameter kCaSR into the probabilityof firing of Ca2+ sparks or quarks (Pj , j = cluster for RyRclusters, and j = rogue for rogue RyRs) as follows:

kCaSR = kmax

1 + (DSR/[Ca2+]SR)nSR,

Pj = Pmax

1 +(Kmj/[Ca2+]i

)njkCaSR,

(3)

where kmax = 2.0, the Hill coefficient nSR = 4.5, Pmax =0.3/event/ms, ncluster = 1.6, and nrogue = 1.0 for the lesscoupled gating of rogue RyRs than RyR clusters. DSR isluminal Ca2+ sensitivity parameter of Ca2+ release events,and Kmj is cytoplasmic Ca2+ sensitivity parameter of RyRclusters or rogue RyR channels. In our simulation work,

Computational and Mathematical Methods in Medicine 3

Kmrogue was always set to be of the same value as Kmcluster; thusKm was used to represent the value of Kmcluster and Kmrogue.

In this study, the simulation of subcellular Ca2+ handlingwas performed on the longitudinal section of a cardiacmyocyte with the size of 100μm × 20μm along the cellularlongitudinal direction (x-axis) and Z-line (y-axis), respec-tively. The number of RyR clusters was 49 × 19 along x andy axes, respectively, and the total number of rogue RyRs wasNrogue×2000μm2. The diffusion partial differential equationwas approximated by the finite difference method (FDM)with a time-step size of 0.01 ms and a mesh size of 0.1 μm.

Because of the stochasticity of opening of RyR clustersand rogue RyRs, the properties of Ca2+ signalling weredescribed by statistical results by carrying out repeatedMonte Carlo simulations. All averaged data were expressedas mean± SEM. One-way analysis of variance (ANOVA) wasused for comparison and P < 0.05 was taken to indicatestatistical significance.

2.2. A Cellular Electrophysiological Model. The electrophysi-ological behavior of a myocardial cell is modelled based ona cardiac action potential model proposed by Ten Tusscherand Panfilov [17]. The voltage across the cell membrane canbe described with the following differential equation:

dVm

dt= − 1

Cm(INa + ICaL + Ito + IKr + IKs + IK1 + INaK

+ INaCa + IpCa + IpK + IbCa + IbNa + Istim

),

(4)

where Cm is the membrane capacitance, Istim is a stimuluscurrent, and Ix denotes all kinds of ionic currents across thesarcolemma.

However, different from the Ca2+ dynamical system byTen Tusscher et al., global SR Ca2+ release current Jrel at thecellular level is calculated by the summation of local Ca2+

release fluxes in the 2D subcellular model:

Jrel = krelJsub-rel

= krel

⎛

⎝∑

i, j

Jcluster

(xi, yj

)+∑

m,n

Jrogue(xm, yn

)⎞

⎠,(5)

where krel is a constant multiplier equivalent to 22.25 in ourcoupled model.

2.3. Heart Failure Model. Changes of Ca2+ cycling as well asother ionic currents have been observed in failing heart; thuswe modified the parameters of our coupled model to mimicabnormal Ca2+ dynamics and electrophysiological propertiesin heart failure from subcellular to cellular levels.

2.3.1. Ca2+ Handling

(a) SR Ca2+ Release Channels. In HF, RyR channels wouldbecome unstable due to phosphorylation of protein kinase A(PKA) [18] or Ca2+/calmodulin-dependent-protein-kinase-I- (CaMKI-) induced hyperphosphorylation [19] and beoversensitive to cytoplasmic Ca2+ and SR luminal Ca2+ [20].

In our simulation study, Km was set to be 7.5 μM and DSR was2.5 mM under the condition of heart failure, while 15 μM and3.25 mM, respectively, under control condition.

(b) SR Ca2+ Pump. Pumping activity of SR Ca2+ ATPase infailing heart is reduced as shown in experimental studies[21]. A 45% reduction in Jpump of a failing myocyte isincorporated into our HF model.

(c) SR Ca2+ Leak. Spontaneous openings of RyR clustersand rogue RyRs at diastole are the main contributors to SRCa2+ leak as the form of Ca2+ sparks and quarks. Becauseof instability of RyR channels, RyR-mediated Ca2+ leak fromSR increased in the resting HF myocyte. However, RyR-independent leak was unaltered in our HF model.

2.3.2. Ionic Current across the Sarcolemma

(a) Inward Rectifier Potassium Current: IK1. In heart failure,IK1 was shown to be reduced in many studies [22, 23]. In ourHF model the current density of IK1 was assumed to decreaseby 20%.

(b) Slowly Activated Delayed Rectifier Potassium Current: IKs.IKs is the slowly activated component of delayed rectifierpotassium current. In the failing canine hearts IKs has beenshown to be downregulated by nearly a half [24]. Therefore,maximal IKs conduction was changed to 50% of the valueused in nonfailing myocytes.

(c) Transient Outward Potassium Current: Ito. According toan experimental result, the current density of Ito in HFdeclined to 64% of the value in control cardiac cells [25],so that in our simulations Ito was reduced to 64% in failingmyocytes.

(d) Fast Na Current: INa. It has been reported that the peakdensity of INa decreased significantly in heart failure [26].Therefore, the maximal INa conductance GNa was set tobe 8.902 nS/pF in the failing myocytes, while equivalent to14.838 nS/pF in the nonfailing myocytes.

(e) Na-Ca Changer Current: INaCa. The activity and/or geneexpression of Na/Ca changer was found to be increased obvi-ously in many experiments [27, 28]. Thus we upregulatedINaCa by 65% in the failing myocytes.

(f) Na-K Pump Current: INaK . As shown in the experimentalstudy, the concentration of Na/K ATPase in the failing heartwas reduced by 42% [29], so that reduction of INaK by thesame proportion was incorporated in our HF model.

(g) Ca Background Current: IbCa. Inward IbCa was consideredto balance Ca2+ extrusion via Na/Ca exchanger and sar-colemmal Ca2+ pump at resting potential. In our HF modelthe conductance of IbCa was increased due to the increase ofINaCa.


Table 1: Parameters in nonfailing versus failing myocyte models.

Parameters Definition Nonfailing Failing

Vmax Maximal SR Ca2+ pumping rate 0.006375 mM/ms 0.0035 mM/ms

GKs Maximal IKs conductance 0.392 nS/pF 0.196 nS/pF

GK1 Maximal IK1 conductance 5.405 nS/pF 4.324 nS/pF

Gto Maximal Ito conductance 0.294 nS/pF 0.185 nS/pF

GNa Maximal INa conductance 14.838 nS/pF 8.902 nS/pF

PNaK Maximal INaK 2.724 pA/pF 1.57 pA/pF

GbCa Maximal IbCa conductance 0.000592 nS/pF 0.0009045 nS/pF

kNaCa Maximal INaCa 1000 pA/pF 1650 pA/pF

Km Ca2+ sensitivity parameters of RyR clusters or rogue RyR when they take the same value 15 μM 7.5 μM

DSR luminal Ca2+ sensitivity parameter of Ca2+ release events 3.25 mM 2.5 mM

The different values of parameters in nonfailing andfailing myocyte models are shown in Table 1.

3. Results

3.1. Ca2+ Cycling and Vm in HF. With the proposed coupledmodel, firstly we simulated the action potential and calciumcycling by applying a stimulus with a frequency of 1 Hz,duration of 1 ms, and an amplitude of 7 pA. Figure 2 showsthe simulation results of membrane potential, cytoplasmicCa2+ concentration, Ca2+ concentration in SR lumina, andthe Na+/Ca2+ exchanger current after 10th stimulus. Whileblue curves in Figure 2 are obtained under the physiologicalconditions, red curves are under the pathological conditions,that is, heart failure. Compared with that in nonfailingmyocytes, the plateau of action potential (AP) shows alarger amplitude and longer duration, causing a significantincrease in AP duration (∼45% longer than in normalcondition) in heart failure. Meanwhile, decrease of themaximal conductivity of IK1 in heart failure makes the restingpotential elevate 2∼3 mV. However, the amplitude of APovershot is smaller in heart failure, which is due to thereduction of fast inward current INa. Then at the early stage ofrapid repolarization, a weakened notch is observed in heartfailure AP, which is caused by a decrease of Ito. Moreover,the prolonged plateau is mainly due to decease of maximalconductivity of IKs.

For the calcium handing in heart failure, it is mainlycharacterized by a significant impair of global Ca2+ tran-sient and a much slower decay of calcium concentration.Moreover, the SR Ca2+ store is smaller in heart failure,that is, [Ca2+]SR is ∼15% lower in resting cells, and therestoring rate of SR calcium is slower than that on controlcondition. Due to the changes of AP morphology andcalcium transient curves together with increase of the activityof Na/Ca exchanger, the curve of INaCa in heart failurediffers significantly from that under normal condition. Thiscan be seen in Figure 2(d); that is, in heart failure, boththe inward and outward currents of INaCa are increased.However, it takes a longer time to reach the peak ofinward current, and the amplitude of inward INaCa inresting stage is also larger compared with that under controlconditions.

3.2. Dependence of DAD on Rogue RyR. In heart failuremyocytes, the RyR channels become very unstable and aremore likely to open with the same values of [Ca2+]i and[Ca2+]SR as those under normal conditions. However, thecalcium release current through a RyR cluster decreases asthe SR calcium store is partly unloaded, which is observed asreduction of amplitude and area of calcium sparks. Indeed,the simulation results by using our coupled model showthat, although more spontaneous calcium sparks occur infailing myocytes, propagating Ca2+ waves are seldom foundwhen there is no rogue RyRs on the 2D subcellular spacewithout any stimulus. These spontaneous calcium sparkswould slightly elevate global [Ca2+]i on the cellular level (Δ[Ca2+]i = (1.16 ± 0.06) × 10−4 mM, n = 10) and depolarizetransmembrane potential with a tiny amplitude (ΔVm =4.21 ± 0.23 mV, n = 10) as shown in Figure 3(a).

However, how do Ca2+ dynamics and electrophysiologi-cal properties of a failing myocyte change in the presence ofrogue RyR channels? We integrate rogue RyR channels intothe 2D RyR grid and vary their distribution density Nrogue

to investigate the precise effect of rogue RyR channels onCa2+ handling and membrane potential. When the densityof rogue RyRs is relatively low, for example, Nrogue = 0.25rogue RyR/μm2, spontaneously occurring calcium sparks arefrequently observed in the subcellular region of HF myocytesduring the resting state. Occasionally Ca2+ waves are formed,albeit at a small area, by recruiting several adjacent Ca2+

sparks. Those small Ca2+ waves could not propagate acrossthe whole myocyte, but self-abort during a short time.Similar to the condition without rogue RyRs, global [Ca2+]iand membrane potential are not affected severely by thoseCa2+ release events under the condition with low densityof rogue RyRs as shown in Figure 3(a) (Δ[Ca2+]i = (1.61 ±0.08) × 10−4 mM, ΔVm = 5.95± 0.29 mV, n = 10).

Amplification and increase rate of [Ca2+]i and Vm shouldbe two groups of principal parameters to evaluate the effectsof rogue RyRs on Ca2+ dynamics and electrophysiologicalproperties. Besides Δ[Ca2+]i and ΔVm, two new parametersTpeakCa and TpeakVm are used in our simulation. TpeakCa

represents the mean time to reach the peak of [Ca2+]ifrom the end of resting stage, and TpeakVm is the time toreach the peak of Vm. Increase rate of Ca2+ concentrationand depolarization velocity could be estimated indirectly


0 500 1000−100

−50

0

50

Vm

(mV

)

Time (ms)

(a)

0

0.2

0.4

0.6

0.8

1

0 500 1000

Time (ms)

×10−3

[Ca2+

] i(m

M)

(b)

2

2.5

3

3.5

0 500 1000

Time (ms)

[Ca2+

] SR

(mM

)

(c)

0

0.5

−1.5

−1

−0.5

0 500 1000

Time (ms)

I NaC

a(p

A/p

F)

(d)

Figure 2: Simulation results of membrane potential (Vm) (a), cytoplasmic Ca2+ concentration ([Ca2+]i) (b), SR luminal Ca2+ concentration([Ca2+]SR) (c), and INaCa (d) during a single twitch. The red curve represents the time course of those parameters in failing myocytes, whilethe blue curve is the time course of those parameters in control myocytes.

by the two parameters, which is shown in Figure 3(b).The simulated TpeakCa decreases significantly when Nrogue isupregulated from 0 to 0.25 rogue RyR/μm2 (TpeakCa = 418 ±19.7 ms and 325 ± 15.2 ms (n = 10), resp., n = 10,P < 0.05).However, decrease of TpeakVm is slight when Nrogue is from 0to 0.25 rogue RyR/μm2 (TpeakVm = 406 ± 38.5 ms and 340 ±25.1 ms, resp., n = 10,P > 0.05). In Figure 3(c) three bluecurves when Time >1000 ms represent repeated simulationresults of Vm without any stimulus when Nrogue = 0.25 rogueRyR/μm2. The results indicate a smooth variation without asignificant peak in the membrane potential morphology.

WhenNrogue is increased to 0.5 rogue RyR/μm2, similar toabove, small spontaneous calcium waves cannot propagate in

a long distance and quickly decay as well. The depolarizationof membrane potential caused by calcium release events hasbigger amplitude compared with that when Nrogue = 0.25rogue RyR/μm2 (P < 0.05), but is also relatively weak withan average of 7.48± 0.25 mV. Again, the membrane potentialmorphology is smooth.

However, as Nrogue is further increased, specifically whenNrogue ≥ 0.75 rogue RyR/μm2, large calcium waves couldbe initiated spontaneously in the 2D subcellular region.Moreover, our Monte-Carlo simulation results show thatthese large calcium release events cause a significant largercalcium transient at the whole cell level and subsequentlydepolarize the membrane potential to a larger extent when


0 0.5 1 1.51

2

3

4

Nrogue (rogue RyR/µm2)

×10−4

0

5

10

15

20

ΔVm

(mV

)

Δ[C

a2+] i

(mM

)

0 0.5 1 1.5


(a)

T pea

kVm

(ms)

150

200

250

300

350

400

450

150

200

250

300

350

400

450

Tpe

akC

a(m

s)

0 0.5 1 1.5


0 0.5 1 1.5


(b)

0 500 1000 1500 2000

0

20

40

Time (ms)

1100 1200 1300 1400 1500

Nrogue = 0.25 rogue RyR/µm2

Nrogue = 1 rogue RyR/µm2

Nrogue = 1.75 rogue RyR/µm2

−20

−40

−60

−80

−60

−70

−80

Vm

(mV

)

(c)

Figure 3: (a) Effects of distribution density of rogue RyRs on global change in [Ca2+]i (Δ[Ca2+]i, left image) and depolarization of membranepotential (ΔVm, right image ) without any external stimulus in heart failure. (b) Dependence of TpeakCa and TpeakVm on Nrogue. TpeakCa

represents the time to reach the peak of global [Ca2+]i from the end of resting stage, and TpeakVm is the time to reach the peak of Vm.For each Nrogue, we simulated 10 times under the same conditions to get statistical results due to the randomness of opening of RyR channels.The values denoted by arrows in (a) and (b) are recorded when INa = 0 pA. (c) Three groups of simulation results of Vm when Nrogue = 0.25,1.0, and 1.75 rogue RyR/μm2 (blue, red, and black curves, resp.). The inset is the enlarged view of depolarization of membrane potentialduring the period of 1100 ms ∼ 1500 ms. INa is also set to be 0 pA when Nrogue = 1.75 rogue RyR/μm2.


0

0

50

1

0.5

0

1

1

2

3

Lin

e sc

an

0 1000 3000 4000

Time (ms)

−100

−1

−2

−50

Vm

(mV

)

×10−3

[Ca2+

] i(m

M)

[Ca2+

] SR

(mM

)I N

aCa

(pA

/pF)

0 1000 3000 4000

0 1000 2000 3000 4000

0 1000 2000 3000 4000

Time (ms)

2000

Time (ms)

2000

Time (ms)

Figure 4: Occurrence of DAD following three action potentials infailing myocyte when Nrogue = 1.0 rogue RyR/μm2. Five figures fromtop to bottom represent membrane potential, cytoplasmic [Ca2+]i,SR luminal [Ca2+]SR, INaCa, and line-scan image along longitudinaldirection of the cell, respectively, in failing ventricular myocyte.

Nrogue is increased by a step of 0.25 rogue RyR/μm2 (P <0.05) as shown in Figure 3. Figure 4 shows typical simulationresults when Nrogue = 1.0 rogue RyR/μm2. After 3 actionpotentials paced by the cycle length of 1000 ms, a relativelylarge Ca2+ transient is observed, as well as a big inwardINaCa and consequently a DAD without external stimulus inthe heart failure myocytes. The linescan image in Figure 4indicates the underlying microcosmic Ca2+ cycling on thesubcellular level.

Besides, as Nrogue is increased to a larger value, Ca2+

transient elicited by spontaneous Ca2+ release and thedepolarization of membrane potential enlarge further (P <0.05), while the values of TpeakCa and TpeakVm decreasesignificantly (P < 0.05) (as shown in Figure 3). All theseresults together suggest close relationship between rogueRyRs and DADs.

3.3. Triggered Action Potential. As demonstrated above, thespontaneous Ca2+ release from SR causes a Ca2+ transientin cytoplasm and subsequently depolarizes the membranepotential even in the resting stage without any stimulus.Furthermore, the amplitude of calcium transient and the

0

2

4

0

0

1

0.5

0

1

100

Time (ms)

0 1000 2000 3000 4000

0 1000 2000 3000 4000

0 1000 2000 3000 4000

0 1000 3000 4000

Vm

(mV

)

×10−3

[Ca2+

] i(m

M)

[Ca2+

] SR

(mM

)I N

aCa

(pA

/pF)

−100

−1

−2

Lin

e sc

an

Time (ms)

2000

Time (ms)

Time (ms)

Figure 5: Triggered action potential in failing myocyte when Nrogue

= 1.75 rogue RyR/μm2. Five figures from top to bottom representmembrane potential, cytoplasmic [Ca2+]i, SR luminal [Ca2+]SR,INaCa, and line-scan image along longitudinal direction of themyocyte, respectively, in failing ventricular cell.

degree of depolarization are positively correlated to thedistribution density of rogue RyR channels. Therefore, oncethe density of rogue RyRs is large enough, the membranepotential may be depolarized to reach the threshold that willtrigger an action potential.

Figure 5 shows the time course of simulated membranepotentials: cytoplasmic Ca2+ concentration, SR Ca2+ store,the Na/Ca exchange current, as well as the Ca2+ dynamics atsubcellular level. From the line-scan image we can see that,along the cellular longitudinal direction, many spontaneousCa2+ waves are initiated nearly at the same time. These Ca2+

waves could propagate and diffuse to finally form a largewave. These Ca2+ releases rise the [Ca2+]i quickly and drivea strong inward component of INaCa, which causes the depo-larization of the membrane potential. When the membranepotential is depolarized to reach the threshold for activationof fast Na+ channel, the large INa is produced very fast andinduces an upstroke of the membrane potential. Then, theL-type Ca2+ channels will be subsequently activated and aflux of extracellular calcium ions burst into cytoplasm viathe L-type calcium channels. This inflow of Ca2+ together


with the previously released Ca2+ from SR can activate theremaining available RyR channels and elicit an even largerCa2+ transient in the cytoplasm. The other ionic channelsat the membrane are successively opened and determine themorphology of AP together. Particularly, for the INaCa, itswitches to a weak outward current during the plateau stage,but turns to a strong inward current in the repolarizationstage of AP, by which the Ca2+ is ejected to the extracellularspace.

When Nrogue = 1.5 rogue RyR/μm2, triggered APs areobserved in 11 simulations (totally 22 Monte Carlo sim-ulations), that is, the probability of triggered AP is 50%.Moreover, when Nrogue = 1.75 rogue RyR/μm2, triggered APsare found in 18 out of 21 Monte Carlo simulations, thatis, the probability of triggered AP rises to 87.5%. On thecontrary, when Nrogue < 1.5 rogue RyR/μm2, no triggered APis seen in our simulations.

To quantitatively investigate the effect of high denserogue RyRs on Ca2+ handling, we also recorded the variationsof [Ca2+]i and membrane potential (Δ[Ca2+]i and ΔVm)as well as TpeakCa and TpeakVm . However, ΔVm elicited byintensive Ca2+ release events is often large enough to activateINa. Consequently, the voltage upstroke induced by INa wouldoverlap the original ΔVm. Then, Δ [Ca2+]i by Ca2+ releaseevents is also overlapped by the following inward-flowingICaL during a triggered AP. Under those conditions, themeasurement of Δ[Ca2+]i and ΔVm as well as TpeakCa andTpeakVm becomes very difficult. Thus, in our simulation,when Nrogue � 1.5 rogue RyR/μm2, we set the INa to bezero after removing the external stimulus. By doing this,triggered AP will not be formed even when a DAD makes themembrane potential more positive than the threshold for INa.Therefore, we can calculate these parameters easily. Actually,our simulation results are shown in Figure 3 marked byarrows, from which we can conclude that, as Nrogue isincreased from 1.5 to 1.75 rogue RyR/μm2, the amplitude of[Ca2+]i enlarges (P < 0.05), whereas the time to reach thepeak does not change obviously (P > 0.05). However, theamplitude of a DAD increases significantly (P < 0.05), whilethe time to observe the DAD decreases (P < 0.05).

4. Discussion

4.1. Mechanism of Ca Handling in HF. Heart failure, asyndrome caused by significant impairments in cardiacfunction, has become one of the biggest human killers with apoor prognosis [30]. Ca2+ handling of cardiac cells in heartfailure is always characterized by reduction in the amplitudeas well as by slowed decay of Ca2+ transient [31]. The primaryreason for decrease in the amplitude of Ca2+ transient isthe partly unloaded Ca2+ store in SR. Three factors mainlyaccounting for the smaller store are (1) increased Ca2+ leak inthe resting myocyte, (2) decreased activity of SR Ca2+ pump,and (3) increase in expression and/or activity of Na+-Ca2+

exchanger.Despite the increased activity of Na/Ca exchanger, at

the early stage of [Ca2+]i decay, the net current of Na/Caexchanger might be outward current (i.e., Ca2+ influx) or

weak inward current, due to longer AP plateau and higherplateau potential in failing myocyte. Therefore, slowed decayof Ca2+ transient is mainly due to decreased SR Ca2+ pump,which removes major amount of Ca2+ at the early stageof [Ca2+]i decay. Only when the membrane potential isrepolarized to a relatively negative voltage and [Ca2+]i isstill high that INaCa turns to a strong inward current andaccelerates decay of Ca2+ transient at the late stage of [Ca2+]idecay.

4.2. Arrhythmogenic Effect of Rogue RyRs. Besides pumpfailure, patients with severe heart failure are at high riskof sudden cardiac death generally triggered by a lethalarrhythmia [23, 32]. DADs are thought to be the primarymechanism underlying arrhythmia in failing heart [33].In our simulation work, although the probability of firingof RyR clusters increases in resting failing cardiocytes, thespontaneous Ca2+ sparks could not elicit enough amplitudeof Ca2+ transient to induce an obvious DAD in the absentof rogue RyRs. The existence of rogue RyR channels is ofimportance in initiation and propagation of spontaneousCa2+ waves in ventricular myocytes with heart failure [14].

In this work, we propose a coupled mathematic modelby integrating the spatiotemporal Ca2+ reaction-diffusionsystem into the cellular electrophysiological model. RogueRyR channels are then incorporated into the coupled modelto simulate subcellular Ca2+ dynamics and global cellularelectrophysiology simultaneously under heart failure con-ditions. Our simulation results show that, in the presenceof rogue RyRs, Ca2+ dynamics is more unstable and Ca2+

waves are more likely to be initiated than the conditionwithout rogue RyRs. Different from sporadic sparks ina resting myocyte without Ca2+ waves, a number of SRCa2+ release events occur intensively during the process ofspontaneous occurrence of Ca2+ waves. These release eventscould elevate the amplitude of Ca2+ transient effectivelyand thus induce Ca2+-dependent inward current (mainlyvia Na/Ca exchanger) which depolarizes the sarcolemmaand lends to a DAD, or a triggered AP sometimes. Fora given level of Ca2+ release in failing myocytes, inwarddepolarizing current becomes larger due to increased activityof Na/Ca exchanger. And increased membrane resistanceowing to reduction of IK1 enables the same inward currentto produce greater depolarization. Once a DAD elevatesmembrane potential to the threshold for activation of INa,a triggered AP is then formed. DADs and triggered AP arethe primary triggered activities accounting for arrhythmiasin heart failure.

4.3. Dependence of Vm on Density of Rogue RyRs. With-out rogue RyR channels or with low distribution density,occurrence of spontaneous Ca2+ sparks and/or quarks isindependent in time and space, which is unlikely to evolveinto propagating Ca2+ waves with partially unloaded SRCa2+ store. [Ca2+]i variation elicited by these Ca2+ releaseevents is slight. In our simulation, when Nrogue = 0 or0.25 rogue RyR/μm2, the amplitude of membrane potentialdepolarization is very small, only several mV, and the average


TpeakVm is big and extensive with large SEM. The morphologyof membrane potential is very smooth without a distinctpeak, so that this type of depolarization could not be referredto a genuine DAD.

However, as more rogue RyRs are distributed over the 2Dplane, ΔVm increases gradually, while the value of TpeakVm

decreases but becomes more intensive. When the valueof Nrogue is elevated to 1.5 rogue RyR/μm2 or bigger, thelarge membrane potential depolarization tends to evoke atriggered AP, and the probability of occurrence of triggeredAPs increases with the larger Nrogue. The reason is that largernumber of rogue RyRs would increase the amplitude and rateof DADs by initiating more Ca2+ waves which occur moresynchronously. Therefore, depolarization of Vm is indicatedto be dependent on the distribution density of rogue RyRchannels.

4.4. Limitations and Further Work. Because the rogue RyRremains to be a hypothetical channel rather than a deter-minate concept, experimental parameters of rogue RyRsare lacked. Some assumptions were made regarding thedensity, distribution, and kinetic of rogue RyR channels. Inour work, a constant was used to represent Ca2+ releaseflux via a rogue RyR channel, while Ca2+ release fluxvia a cluster of RyRs was dependent on global luminalCa2+ concentration ([Ca2+]SR) and local Ca2+ concentration([Ca2+]i(xi,yi)). Besides, different Nrogue values were usedto evaluate the effect of rogue RyR on Ca2+ cycling andmembrane potential in failing heart.

Numerous key regulatory proteins, such as protein kinaseA(PKA), Calstalin, CaMKII, and phosphatase, are bound toRyR, thus forming the junctional complex. RyR channelswould be regulated via different signaling pathways [34]. Forexample, PKA phosphorylation dissociates FKBP12.6 fromthe RyR and thus makes the RyR channel unstable in failinghearts [35]. These regulating processes could not be mimicedby our present model. Besides, defective Ca2+ handlingalso occurs in many cardiac diseases, such as myocardialinfarction, atrial fibrillation, and various arrhythmogenicparadigms. The coupled model is planned to be improved,and more relevant parameters should be added to investigatethe potential mechanisms of Ca2+ dynamics in various kindsof cardiac diseases.

5. Summary

By integrating the spatiotemporal Ca2+ reaction-diffusionmodel into the cellular electrophysiological model, appear-ance of subcellular Ca2+ release events and evolutionof waves together with dynamics of ionic concentrationand membrane potential on the cellular level could beenmonitored simultaneously. By using the coupled model weinvestigate the effects of rogue RyRs on Ca2+ handling fromsubcellular to cellular level as well as electrophysiologicalproperties in failing heart. The simulation results suggestthat rogue RyR with tiny Ca2+ release flux should be animportant factor in triggering arrhythmia in failing cardiaccells. Our work suggests the importance of rogue RyRs in

initiation of Ca2+ release events (especially Ca2+ waves) andconsequently DADs or triggered APs. Our study indicates thearrhythmogenic effect of rogue RyRs and helps to elucidate apossible arrhythmia mechanism in failing heart.

Acknowledgments

This project is supported by the 973 National Basic Research& Development Program (2007CB512100) and the NationalNatural Science Foundation of China (81171421).

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