Ca2! Alternans in a Cardiac Myocyte Model that Uses Moment Equationsto Represent Heterogeneous Junctional SR Ca2!

Marco A. Huertas,†§* Gregory D. Smith,†6 and Sandor Gyorke‡6†Department of Applied Science, College of William and Mary, Williamsburg, Virginia; ‡Davis Heart and Lung Research Institute, Department ofPhysiology and Cell Biology, Ohio State University, Columbus, Ohio; and §Neuroscience Center of Excellence, Louisiana State UniversityHealth Sciences Center, New Orleans, Louisiana

ABSTRACT Multiscale whole-cell models that accurately represent local control of Ca2!-induced Ca2! release in cardiacmyocytes can reproduce high-gain Ca2! release that is graded with changes in membrane potential. Using a recently introducedformalism that represents heterogeneous local Ca2! using moment equations, we present a model of cardiac myocyte Ca2!

cycling that exhibits alternating sarcoplasmic reticulum (SR) Ca2! release when periodically stimulated by depolarizing voltagepulses. The model predicts that the distribution of junctional SR [Ca2!] across a large population of Ca2! release units is distincton alternating cycles. Load-release and release-uptake functions computed from this model give insight into howCa2! fluxes andstimulation frequency combine to determine the presence or absence of Ca2! alternans. Our results show that the conditions forthe onset of Ca2! alternans cannot be explained solely by the steepness of the load-release function, but that changes in therelease-uptake process also play an important role. We analyze the effect of the junctional SR refilling time constant on Ca2!

alternans and conclude that physiologically realistic models of defective Ca2! cycling must represent the dynamics of heteroge-neous junctional SR [Ca2!] without assuming rapid equilibration of junctional and network SR [Ca2!].

INTRODUCTION

The phenomenon of cardiac alternans—mechanical or elec-trical oscillations that alternate in magnitude on a beat-to-beat basis—has received considerable attention in the pastyears, because it is closely related to an increase in the riskof cardiac arrhythmias and is a good marker for suddencardiac death. Because up to 50% of deaths related to heartfailure can be attributed to these arrhythmias, understandingthe underlying mechanisms that lead to cardiac alternans isof grave relevance.

Many cardiac arrythmias are linked to spatially discordantrepolarization alternans, which themselves are causally con-nected with changes in action potential duration (APD) (1).These variations in APD are due to irregularities in the mech-anisms of Ca2! cycling in the cell, given the bidirectionalcoupling between membrane potential and intracellular[Ca2!] (2–4). In general, the coupling frommembrane poten-tial to intracellular Ca2! is a positive one, since the voltage-dependent activity of L-type (dihydropyridine receptor(DHPR)) Ca2! channels tends to increase the intracellularCa2! by triggering Ca2! release from the sarcoplasmic retic-ulum (SR) via Ca2!-induced Ca2! release (CICR). On theother hand, the coupling from intracellular [Ca2!] tomembrane potential can be either positive or negative (2,3)depending on the net effect of Ca2! on regulation of theactivity of the DHPR (Ca2! influx), the Na!-Ca2! exchanger(NCX) (Ca2! efflux), and other Ca2!-regulated currents. Asaction potential duration can be either prolonged or shortened,

depending on the magnitude of SR Ca2! release, beat-to-beatalternations in the size of SR Ca2! release may underlie themechanisms leading to repolarization alternans in cardiacmyocytes. It is interesting to note that recent experimentalevidence indicates that Ca2! release alternans can occurindependently of action potential duration alternans (2,5).

What are the mechanisms underlying SR Ca2! releasealternans? One possible cause is cellular Ca2! cycling prop-erties that limit the amount of released Ca2! that can beresequestered back into the SR before the next voltage pulseor action potential occurs, leading to alternations in the SRcontent and subsequent SR release. Other proposed mecha-nisms include the generation of Ca2! waves, the time courseof ryanodine receptor (RyR) inactivation and refractoriness,and dysfunctions in the release mechanism of the RyR(reviewed in Laurita and Rosenbaum (3)). Both modelingand experimental studies indicate that two important aspectsof cellular Ca2! dynamics favor the occurrence of Ca2!

release alternans: 1), a strong dependence of Ca2!-releaseamplitude on the SR content before the voltage pulse; and 2),a strong relationship between the cytoplasmic [Ca2!] and theamount of Ca2! extruded from the cell in a given cycle.

Some prior computational studies have formulated min-imal dynamic models that represent these features of Ca2!

cycling using discrete-time maps that can be analyzed usingbifurcation theory (2,4,6,7). In such models, the load-releasefunction specifies how the amount of SR release depends onSR content, and the release-uptake function specifies thesubsequent process of Ca2! uptake into the SR. These speci-fied functional relationships are sometimes fitted to experi-mental data and usually assumed to be either linear (2) orsigmoidal (6); however, they are not derived from biophysical

Submitted December 23, 2009, and accepted for publication April 12, 2010.6Gregory D. Smith and Sandor Gyorke contributed equally to this work

*Correspondence: mhuer1@lsuhsc.edu

Editor: David A. Eisner

! 2010 by the Biophysical Society0006-3495/10/07/0377/11 $2.00 doi: 10.1016/j.bpj.2010.04.032

Biophysical Journal Volume 99 July 2010 377–387 377

properties of Ca2! currents and pumps. On the other hand,detailed compartmental models that take into account theCa2! dynamics of large numbers of junctional SR (jSR)compartments have been proposed (8,9). Thesemodels utilizeaminimal description ofCa2! release throughRyRs, focusingmainly on a nonlinear relationship between SR content andrelease (8) or describing it through a Hill equation dependentonly on the local (diadic subspace) [Ca2!] (9), without takinginto account the dynamics of stochastic gating of RyRs orthe L-type Ca2! channel (DHPR).

This study investigates features of Ca2! alternans froma computational perspective with an emphasis on a novelwhole-cell modeling approach that has been applied success-fully to the simulation of local control mechanisms in exci-tation-contraction coupling (10). The modeling approachutilizes a probability density formulation and an associatedmoment-closure technique to represent the stochastic activityof Ca2! release units (CaRUs) composed of a single L-typeCa2! channel (DHPR) and RyR megachannel, with Ca2!-dependent dynamics that depend on the local [Ca2!] ina large number of dyadic-subspace and jSR compartments.It is important to note that the moment-based formalismaccounts for the heterogeneous distribution of local [Ca2!]across the population of CaRUs, a feature that sets thismodeling approach apart from prior work that assumes thatlocal Ca2! concentrations are an instantaneous function ofCaRU state (8,11–13).

MODEL FORMULATION

The whole-cell model of Ca2! cycling used here is consistentwith (and derived from) a multicompartment local-controlformulation that represents the dynamics of bulk myoplas-mic [Ca2!] (cmyo), network SR [Ca2!] (cnsr), N diadicsubspace Ca2! concentrations (cnds ), and N jSR domainCa2! concentrations (cnjsr) using a system of N ! 2 concen-tration balance equations (14,10). These take the formlxdcx=dt "

Pi J

ix, where x is an index over compartments,

lx " Vx/Vmyo is the effective volume ratio for the compart-ment using the myoplasm as a reference, cx is the [Ca2!]in compartment x, and each Jix is a Ca2! flux (see Fig. S1and Eq. S1, Eq. S2, Eq. S3, Eq. S4, Eq. S5, Eq. S6, andEq. S7 in the Supporting Material). These equations arecoupled to N Markov chains, each of which represents thestochastic gating of a single CaRU composed of a (two-state)L-type (DHPR) Ca2! channel and a (six-state) RyR mega-channel. Note that these channels do not gate independentlyof one another, because they are coupled via changes in cdsoccurring in the restricted diadic subspace. Thus, a singleCaRU is described by a 12-state transition-state diagram

CC1 # CC2 # CC3 # CC4 # CC5 # COq q q q q qOC1 # OC2 # OC3 # OC4 # OC5 # OO

;

(1)

where the first character (C or O) indicates the state of theDHPR and the second character (Ci or O) refers to the stateof the RyR megachannel (see Supporting Material for rateconstants). The transitions XCi/XCj, where j " i ! 1 aremediated by cds. The transitions XC5/XO depend on cjsr,so that depletion of luminal Ca2! decreases the open proba-bility of the RyR megachannel. The transitions CX/OX arevoltage-dependent, whereas the reverse reactions XCi)XCj,XC5)XO, and CX)OX are independent of both voltageand Ca2!.Under the physiologically realistic assumption of a large

number of CaRUs (N z 20,000), the ordinary differentialequations (ODEs) for the jSR Ca2! concentrations (Eq. S3)can be replaced by a set of probability density functions,

ri!cjsr; t

"dcjsr " Pr

#cjsr < ~cjsr#t$< cjsr ! dcjsr and ~S#t$" i

$;

(2)

where i is an index over the M " 12 CaRU states, ~S is thestate of a randomly sampled CaRU, and ~cjsr is the associatedjSR [Ca2!]. These densities satisfy a system of advection-reaction equations (Eq. S18) (10,14–16).

The moment-based description of jSR [Ca2!] begins bydefining the qth moment of ri(cjsr, t) as

miq#t$ "

Z !cjsr

"qri!cjsr; t

"dcjsr: (3)

The first three moments (q" 0, 1, 2) have simple interpre-tations: mi

0 is the probability that a randomly sampled CaRUis in state i; mi

1 is proportional to the expected value of thejSR [Ca2!] conditioned on CaRU state (Ei%~cjsr " mi

1=mi0

%),

and mi2 is related to the conditional variance of the jSR

[Ca2!] through Vari%~cjsr " mi2=m

i1 & #mi

1=mi0$

2i

. Using thedefinition in Eq. 3 and the evolution equation for the proba-bility density ri(cjsr, t), one can derive an infinite system ofODEs for the time evolution of these moments. The systemis truncated to include equations for mi

0, mi1, and mi

2. Theequations are closed by expressing mi

3 as an algebraic func-tion of the lower moments, mi

3 " 4#mi0;m

i1;m

i2$, that would

be strictly correct if the probability density functions werescaled b-distributions.

In summary, the whole-cell model of Ca2! cycling that isthe focus of this article utilizes concentration balanceequations for myoplasmic and network SR [Ca2!] and amoment-based description of heterogeneous diadic subspaceand jSR [Ca2!] that assumes rapid equilibration of diadicsubspace Ca2!, but accounts for slow dynamics of jSR Ca2!

associated with a large population of CaRUs. The moment-based whole-cell modeling approach has been validated asan alternative to Monte Carlo simulation and shown to becapable of reproducing important electrophysiological prop-erties of cardiac myocytes (10). Parameters consistent withprior modeling and experiments are presented in Table S1,Table S2, and Table S3.

Biophysical Journal 99(2) 377–387

378 Huertas et al.

RESULTS

To initiate this investigation of Ca2! alternans, parametersconsistent with prior modeling and experiment (17–20)were adjusted to qualitatively fit experimental data frompostinfarction models of ventricular fibrillation in thedog (21), resulting in the standard parameters used here(Table S1, Table S2, and Table S3). Although parametersleading to Ca2! alternans are not unique, the standard param-eters of the model retain the high-gain graded Ca2! release(see Fig. S2 and Fig. S3) that was the focus of prior work(10). In addition, the standard parameters result in simulatedCa2! cycling that is consistent with the following experi-mental observations from postinfarction models of ventric-ular fibrillation in the dog: 1), the presence of alternatingCa2! responses during periodic stimulation of 100-ms volt-age pulses; 2), myoplasmic Ca2! transients with 200- to500-ms duration; 3), an onset stimulation frequency forCa2! alternans of ~1 Hz; and 4), the absence of Ca2! alter-nans at stimulation frequencies >5 Hz.

Alternating Ca2! responses in the moment-basedwhole-cell model

Fig. 1 A is an example of alternating SR Ca2! release (i.e.,Ca2! alternans) when the moment-based whole-cell modelis periodically stimulated by 100-ms depolarizing voltagepulses at 1 Hz. The thick solid line shows the time courseof the total myoplasmic [Ca2!] starting at its steady-stateconcentration (dotted line) and progressing through severalstimulation cycles. The total myoplasmic and SR [Ca2!](ctotmyo and ctotsr , respectively) are given by the expressions

ctotmyo " cmyo ! ldsE%~cds'1 ! lds

and ctotsr "lnsrcnsr ! ljsrE

&~cjsr

%

lnsr ! ljsr;

(4)

where lds and ljsr are volume ratios and E%~cds' and E%~cjsr' areexpected values, e.g., E%~cjsr "

Pi m

i1

%, where i is an index of

CaRU states and mi1 is the first moment of the jSR [Ca2!] prob-

ability densityri(cjsr, t) (seeEq. 3, Eq.S32, and associated text).Note that in Fig. 1 A, the onset of alternans is apparent only

after the fourth voltage pulse, before which a gradual increasein Ca2! transient amplitude is observed. Fig. 1 B shows thecorresponding dynamics of total SR [Ca2!] (heavy solidline). Before the onset of alternans, ctotsr increases in a stepwisefashion to ~1100 mM during the first four pulses. After theonset of alternans, larger SR Ca2! depletion events are associ-ated with the larger increases in myoplasmic Ca2!. DuringCa2! release events, the average jSR [Ca2!] (E%~cjsr'; Fig. 1 B,dashed line) is often far more depleted than the network SR[Ca2!] (cnsr; Fig. 1B, thin solid line), whereas between pulses,junctional and network SR [Ca2!] equilibrate. Because ctotsr isa weighted average of E%~cjsr' and cnsr (Eq. 4), the thick solidline is between the thin solid and dashed lines.

The circles in Fig. 1,C andD, show that the moment-basedwhole-cell model exhibits Ca2! alternans when periodicallystimulated at 2–5 Hz. These traces and Fig. 1, C andD, corre-spond reasonably well to experiments in postinfarctionmodels of ventricular fibrillation in the dog (cf. Figs. 1 Dand 2A ofBelevych et al. (21)).When two symbols are plottedin Fig. 1 for a particular frequency, these represent the twodifferent maximum myoplasmic Ca2! concentrations (orminimum SRCa2! concentrations) observed during the alter-nating response. When one symbol is plotted, Ca2! alternanswere not observed. The bubble in these plots indicates therange of stimulation frequencies leading to Ca2! alternans.

Distribution of jSR [Ca2!] during alternans

Although the joint distributions defined in Eq. 2 are notcalculated in the moment-based simulation, they can be

A

Rn

12

3

4

0

0.5

1

c myo

tot

(mM

)

Ln L

n+1

Un

B

0 5 10Time (s)

400

600

800

1000

1200

c nsr, E

[cjs

r], c srto

t (mM

)

C

D0

0.5

1

1.5

max

(cm

yoto

t) (m

M)

10!1

100

101

Stimulus Frequency (Hz)

400

600

800

min

(csrto

t ) (m

M)

FIGURE 1 Representative Ca2! alternans exhibited by

the moment-based model when stimulated by periodic

100-ms voltage pulses from &80 to 0 mV. (A) Total myo-

plasmic [Ca2!] (Eq. 4) during 1 Hz stimulation (solid line)and in the absence of stimulation (dotted line). (B) NetworkSR [Ca2!] (thin solid line), expected jSR [Ca2!] (E%~cjsr ';dashed line), and total SR [Ca2!] (ctotnsr ; heavy solid line)during 1 Hz stimulation and in the absence of stimulation(dotted line). The quantities Rn, Ln andUn in A and B define

the Ca2! release, SR load, and SR uptake, respectively, for

the nth stimulus cycle. (C and D) Frequency dependence ofCa2! alternans. The maximum total myoplasmic [Ca2!]

(C) and corresponding minimum total SR [Ca2!] (D)during stimulation at various frequencies.

Biophysical Journal 99(2) 377–387

Ca2! Alternans in a Cardiac Myocyte Model 379

constructed post hoc from the computed moments mi0, m

i1,

and mi2 as scaled b-distributions with integrated area of mi

0

and mean and variance consistent with Ei%~cjsr' andVari%~cjsr' (Eq. S21 and Eq. S22) (10). Fig. 2 A shows theoverall distribution of jSR [Ca2!] irrespective of CaRU state,given by rT#cjsr; t$ "

PMi"1r

i#cjsr; t$ at two times duringa 0.5-Hz stimulus cycle that does not result in Ca2! alter-nans. The jSR [Ca2!] distribution (I) corresponds to thebeginning of the 100-ms depolarizing voltage pulse (cf.Fig. 2 C). It has little dispersion, indicating that most jSRcompartments have similar [Ca2!]. In contrast, distributionII corresponds to the phase of the stimulus cycle when ctotmyoreaches its maximum (cf. Fig. 2 C) and is quite dispersed,indicating a wide range of jSR depletion. The value ofEi%~cjsr' for these distributions is indicated by the blacktriangles.

Fig. 2 B shows jSR distributions obtained using 1-Hzstimulation that results in Ca2! alternans. The timing of thesesnapshots is identical to that for Fig. 2 A, but here there arefour distributions, because the Ca2! responses are differenton consecutive cycles. In Fig. 2 B, distributions i and iiiare both quite focused, indicating that the jSR compartmentshave similar [Ca2!] at the onset of the voltage pulse on alter-nating cycles. However, distributions ii and iv differ signifi-cantly from each other, indicating that jSR [Ca2!] is distincton alternating cycles at the time the maximum myoplasmic[Ca2!] is observed. Fig. 2 D shows that distribution ii corre-sponds to the large Ca2! transient in which nearly all the jSRcompartments have [Ca2!] <850 mM, whereas distributioniv corresponds to the small Ca2! transient in which a broader

distribution of jSR [Ca2!] is observed, with a large fractionof jSR compartments only slightly depleted relative to thenetwork SR. Fig. S4 shows these jSR [Ca2!] distributionsdependent on the RyR megachannel being closed/open.

Discrete-time map of alternating Ca2! responses

Prior computational studies have formulated minimaldynamic models of Ca2! cycling and used discrete-timemaps to analyze the bifurcations that give rise to alternatingSR Ca2! release (4,6,8,22). The moment-based whole-cellmodel used here provides an opportunity to explore howthese relations depend on specific Ca2! fluxes and heteroge-neous jSR [Ca2!].

Fig. 3 B defines the SR load (Ln) as the total SR [Ca2!](ctotnsr in Eq. 4) at the onset of the nth voltage pulse. Fig. 3 Ashows the discrete-time map, H, that relates the size of Lnwith Ln!1 of the subsequent cycle, Ln!1 " H(Ln). The solidline in Fig. 3 B is computed by integrating the model equa-tions over one cycle for various initial SR loads, underthe assumption that at the beginning of the nth cycle,E%~cjsr#tn$ " cnsr#tn$' and Var%~cjsr#tn$ " 0' . This is justifiedby the observation that before a voltage pulse, the junctionaland network SR [Ca2!] are often approximately in equilib-rium (Fig. 1 D, dashed and solid lines). The equilibriumpoint of the discrete-time map satisfies L* " H(L*) andcorresponds to a balance of SR fluxes over the stimuluscycle. In Fig. 3 A, the fixed point is unstable (jH0(L*)j < 1)and consecutive values of Ln are alternately larger andsmaller than L*. The black circles and dotted lines obtained

C

(I)

(II)

(I)

(II)

0

0.5

1

c myo

(mM

)

0 1 2 3 4Time (s)

400

600

800

1000

1200

c nsr, E

[cjs

r], c srto

t (mM

)

D

BA

(i)

(ii)

(iii)

(iv)

0

0.5

1

c myo

(mM

)

0 0.5 1 1.5 2Time (s)

400

600

800

1000

1200

c nsr, E

[cjs

r], c srto

t (mM

)

FIGURE 2 (A and B) Distribution of

jSR [Ca2!] irrespective of CaRU state

in the absence (A) and presence (B) ofCa2! alternans, at 0.5 Hz and 1 Hz stim-ulation, respectively, as computed from

the moments of jSR [Ca2!] (mi0, mi

1,

and mi2) in the whole-cell model. The

probability densities labeled I and i

and iii correspond to the onset of the

voltage pulse; those labeled II and ii

and iv correspond to the phase of thestimulus cycle when the total myoplas-

mic [Ca2!] reaches its maximum value.

Triangles indicate the expected jSR

[Ca2!] (E%~cjsr '). (C and D) Upper tracesshow myoplasmic Ca2! for the nonal-

ternating and alternating cases, respec-

tively, with labels I and II and i–iv

corresponding to the probability densityregions in A and B. Lower traces showthe network SR [Ca2!] (dashed line),the expected jSR [Ca2!] (E%~cjsr ', dashedline), and the total SR [Ca2!] (ctotnsr , solidline).

Biophysical Journal 99(2) 377–387

380 Huertas et al.

from the moment-closure simulation (Fig. 3) cobweb thediscrete-time map, H. Thus, we conclude that the map accu-rately represents the dynamics of the alternating response.

Load-release and release-uptake functions

A related approach to analyzing Ca2! alternans (2,7) beginsby defining the release (Rn) as the difference between themaximum total myoplasmic [Ca2!] (ctotmyo; Eq. 4) observedin the nth stimulus cycle and that observed at the beginningof the depolarizing voltage pulse (Fig. 4 A). In a similar way,the uptake on the nth cycle (Un) is defined as the differencebetween the SR load of the subsequent cycle (Ln!1) and theminimum SR [Ca2!] of cycle n (Fig. 4 B). These quantitiesare used to define a load-release (Rn " r(Ln)) and a release-

uptake function (Ln!1 " u(Rn)). When composed, thesefunctions yield the discrete-time map of the previous section,

Ln! 1 " u%Rn' " u%r#Ln$' " H#Ln$: (5)

In agreement with prior experimental (23,24) andmodeling(6) studies, the calculated load-release curve is a monotonicincreasing function of the SR load (see Fig. S5 A). However,the release-uptake curve is not monotonic (see Fig. S5 B).When release events are small, increasing release leads toincreased resequestration by SERCA pumps. Large releaseevents can lead to accelerated extrusion of Ca2! by NCXand decreased resequestration of Ca2! into the network SR.

Fig. 3 B combines the load-release function (solid curve)and the inverse of the release-uptake function, Rn "u&1(Ln!1) (dashed curve) to construct a two-step discrete-time map (Eq. 5) analogous to maps that have been used toanalyze minimal models of alternating Ca2! responses (2,7).The black and gray circles show the SR load and releasefrom the moment-based simulation of Fig. 1. Similar to theone-step map (Fig. 3 A), these values are consistent with cob-webbing the computed load-release and release-uptakecurves. Because the stability condition jH0j " ju0r0j < 1is not satisfied, the intersection of the load-release andrelease-uptake curves correctly predicts an unstable equilib-rium and alternating Ca2! response.

Stimulus frequency shifts the release-uptakefunction

The left columns of Fig. 4, A and B, show load-release andrelease-uptake curves similar to those in Fig. 3 B, but withthe stimulation frequency increased to 1.33 and 2 Hz, respec-tively. At these stimulation frequencies, the load-release andrelease-uptake curves (solid and dashed lines, respectively)correctly predict the Ca2! alternans observed in the moment-based simulations (circles). Note that the load-release functiondoes not change significantly as the stimulation frequency ischanged (see also Fig. S7 A). Conversely, the release-uptakecurve, Rn " u&1(Ln!1), changes shape and shifts to the right,reflecting the frequency dependence of the steady-state SRload determined by the balance of Ca2! influx during thevoltage pulse and Ca2! extrusion via NCX during the inter-pulse interval. The loss of stability that occurs between 0.5and 1 Hz and results in Ca2! alternans (see Figs. 1 and 2) isdue to the changing slopes at the intersection of the load-releaseand release-uptake curves as the latter curve moves rightward.

When the stimulation frequency is increased to 4 Hz, theload-release and release-uptake curves predict the presenceof alternans, though none are observed in simulation(Fig. 4 C). In this case, the load-release and release-uptakecurves do not accurately predict the model response, becausethe assumptions made for their computation are no longervalid, i.e., E%~cjsr#tn$ scnsr#tn$' and Var%~cjsr#tn$ s0' . Notethe increased dispersion of the distribution of jSR [Ca2!]during high-frequency stimulation.

1

2

3

4

SR uptake

SR release

A

700 800 900 1000 1100 1200L

n (mM)

700

800

900

1000

1100

1200L n+

1 (mM

)

Load!ReleaseCurve

Rn = r(L

n)

Release!UptakeCurve

Rn = u!1(L

n+1)

B

12

3

4

700 800 900 1000 1100 1200Load: L

n, L

n+1 (mM)

0

0.2

0.4

0.6

0.8

1

1.2

Rel

ease

: Rn (m

M)

FIGURE 3 (A) Return map of the moment-based whole-cell model that

relates the total SR load in a given cycle, Ln, to that of the following cycle

Ln!1. Numbered circles correspond to peaks in Fig. 1 A. The solid curve is

obtained by integrating the model equations over the 1-Hz stimulus cycleunder the assumption that jSR compartments are in equilibrium with the

network SR at the beginning of the voltage pulse (see text). (B) Correspond-ing load-release (solid line) and release-uptake (dashed line) functions.

Black and gray circles show values obtained directly from the simulationshown in Fig. 1, where the quantities Rn, Ln, and Ln!1 are defined.

Biophysical Journal 99(2) 377–387

Ca2! Alternans in a Cardiac Myocyte Model 381

Ca2! fluxes and load-release/release-uptakefunctions

Fig. 5 shows how two important parameters of the moment-based whole-cell model influence the load-release andrelease-uptake functions at 1 Hz (solid and dashed lines,

respectively; for additional examples, see Fig. S6, A–C).

The four black circles in each panel correspond to the alter-

nating Ca2! response observed with standard parameters (cf.

Fig. 2 B). The open and gray circles of Fig. 5, A and B, and

those in Fig. S6, A–C, show that Ca2! alternans can be

A

0.8 1 1.2L

n (mM)

0

0.5

1

1.5

Rn (µ

M) 0

0.5

1

1.5

Cm

yoto

tal (µ

M)

0 2 4 6 8 10Time (s)

0.2

0.4

0.6

0.81

1.2

Csrto

tal (

mM

)

B

0.8 1 1.2L

n (mM)

0

0.5

1

1.5

Rn (µ

M) 0

0.5

1

1.5

Cm

yoto

tal (µ

M)

0 2 4 6 8 10Time (s)

0.2

0.4

0.6

0.81

1.2

Csrto

tal (

mM

)

C

0.4 0.6 0.8 1 1.2L

n (µM)

0

0.5

1

1.5

Rn (µ

M) 0

0.5

1

1.5

Cm

yoto

tal (µ

M)

0 1 2 3 4 5Time (s)

0.2

0.4

0.6

0.8

1

1.2

Csrto

tal (

mM

)

FIGURE 4 Periodic responses of the moment-based whole-cell model at successively higher stimulation frequencies (1.33, 2, and 4 Hz in A–C, respec-tively). The leftmost panels show the load-release (solid lines) and release-uptake (dashed lines) curves and values obtained from on-going moment-based

simulation (dark and light circles). Middle panels show the total myoplasmic (upper trace) and SR (lower trace) [Ca2!] as a function of time. Rightmostcolumn shows distributions of jSR [Ca2!].

Biophysical Journal 99(2) 377–387

382 Huertas et al.

eliminated by either increasing or decreasing each of fiveparameters studied: the SERCA pump rate (vserca), theNCX pump rate (I0ncx), the cytosolic Ca2! activation rateconstant of the RyR megachannel (r!ryr), the rate constantfor luminal Ca2! regulation of the RyR megachannel(r!ryr;(), and the network-to-junctional-SR Ca2! transfer rate(vrefill).

Increasing either vserca or vrefill shifts both curves right-ward into the region corresponding to high SR loads. Theload-release curve also becomes steeper as the SERCApump rate increases, reminiscent of experimental observa-tions during b-adrenergic stimulation (25,26). Desensitizingthe RyR megachannel through changes in either cytosolic orluminal Ca2! regulation results in a similar shift of bothcurves to higher SR loads. Increasing I0ncxalso shifts therelease-uptake curve to higher SR loads, but has little effect

on the load-release curve. In general, the steepness of theload-release function (viewed in isolation from the release-uptake curve) cannot be used to predict the presence orabsence of Ca2! alternans.

Fig. 6, A and B, also shows how the alternating Ca2!

response exhibited by the model with standard parameterscan be eliminated by increasing or decreasing both vsercaand I0ncx. The maximum bulk myoplasmic [Ca2!] (circles)is an increasing function of vserca and a decreasing functionof I0ncx; consequently, these two parameters can be playedoff one another. For example, the range of vserca leading toalternans shifts to greater values when I0ncx is increased (notshown). The black circles of Fig. 4 C show that differentnetwork-to-junctional-SR transfer rates (vrefill) can alsochange the dynamics of alternating Ca2! responses.

Variance and slow dynamics of jSR [Ca2!]influence alternans

To investigate the functional significance of the variance andslow dynamics of jSR [Ca2!] in alternating Ca2! responses,the standard model was compared to a simplified model thatassumes rapid refilling of jSR compartments (Eq. S36).In this no-variance/fast-jSR model simplification, balanceof release (Jnryr) and refill (Jnrefill) fluxes enslaves jSR [Ca2!]to the diadic subspace and network SR [Ca2!] (the relation-ship depends on the CaRU state). The triangles in Fig. 6 Bshow that the no-variance/fast-jSR model exhibits alter-nating Ca2! responses at 1 Hz (note bubble when I0ncx is50% of the standard value). On the other hand, for mostparameter values surveyed in Fig. 4, the response of theno-variance/fast-jSR model is dramatically different fromthat of the standard model (compare triangles and circles).The observation that this model simplification often doesnot exhibit Ca2! alternans for the same parameter valuesas the standard model holds over a wide range of stimulationfrequencies (Fig. S8). The calculated load-release andrelease-uptake curves of the no-variance/fast-jSR model arealso quite distinct from the standard model (see Fig. S7 B).The lack of agreement between these two models under-scores the importance of accounting for heterogeneous jSR[Ca2!] in whole-cell simulations of Ca2! alternans.

DISCUSSION

Ca2! alternans, beat-to-beat variations in Ca2! transients, arerecognized as an important factor in the development ofcardiac arrhythmias. Ca2! alternans are generally thought toarise from inherent instability of myocyte Ca2! handlingthat can be probed using discrete-time maps and bifurcationtheory. For example, Ca2! alternans are associated with anunstable equilibrium of the map relating SR Ca2! loads onsubsequent cycles (Fig. 3 A). Alternating and nonalternatingresponses can be further dissected by considering how theamount of SR release depends on SR content (the load-release

A

vsercastandard

0.25X3.5X

600 800 1000 1200 1400 1600 1800L

n,L

n+1 (mM)

0

0.5

1

1.5R

n (mM

)

B

Iostandard

0.25X2X

700 800 900 1000 1100 1200L

n,L

n+1 (mM)

0

0.5

1

1.5

Rn (m

M)

FIGURE 5 Effect of various model parameters on the load-release (solidlines) and release-uptake (dashed lines) curves computed from the moment-

based whole-cell model. Solid circles indicate period-1 or -2 oscillatory

responses during 1 Hz stimulation. (A) SERCA pump rate (vserca) at standardvalue (see Table S1, Table S2, and Table S3), 0.25) (decreased), and 3.5)(increased). (B) NCX pump rate (I0ncx) at standard value, 0.6), and 1.2).

Biophysical Journal 99(2) 377–387

Ca2! Alternans in a Cardiac Myocyte Model 383

function) and how the process of Ca2! uptake into the SRdepends on the myoplasmic Ca2! transient (release-uptake function; see Fig. 3 B).

In this study, we used a recently introduced moment-basedmodeling formalism to explore the dynamics ofCa2! alternansin a whole-cell model that represents heterogeneous diadicsubspace and jSR [Ca2!] and the consequences of this hetero-geneity on Ca2! cycling (10,14). Although themoment-basedmodeling approach is not explicitly spatial, the formalismtakes into account the translocation ofCa2! between the diadicsubspace and the bulk myoplasm, as well as the network andjunctional SR. The collective behavior includes coupling ofCaRUs that occurs when diadic subspace Ca2! from an acti-vated CaRU diffuses through the bulk myoplasm into thediadic subspaces of other CaRUs. The description of theCaRU presented here is based on our previous work (10,14)and does not include a mechanism for Ca2! or voltage inacti-vation of DHPRs or refractory RyR states. However, theluminal dependence of RyR open probability introducesa refractory time, because the jSR needs to replenish beforethe RyR megachannel is capable of opening again.

The moment-based approach to representing heteroge-neous jSR [Ca2!] requires far less computer time than a tradi-tional Monte Carlo approach. This computational efficiencygreatly facilitates our parameter studies, exploring the effectson Ca2! alternans of both stimulation frequency and manip-ulations of key Ca2! handling processes, including SERCA-mediated SR Ca2! uptake, sensitivity of RyR-mediatedrelease, NCX activity, and the network-to-jSR Ca2! transferrate. We find that Ca2! alternans occur within specific win-dows of stimulation frequency and values for the above-mentioned Ca2! handling parameters. We also found thatthe steepness of the load-release function is not by itselfa good predictor of Ca2! alternans. Rather, the loss ofstability of period-1 oscillations and the transition to Ca2!

alternans arises from the interrelation of SR Ca2! load-release and release-uptake functions (Fig. 5). Althoughboth curves can be manipulated by parameter changes thatinfluence the dynamic interplay between cellular Ca2!

fluxes, the release-uptake function is more sensitive to stim-ulation frequency (Fig. S7 A) and NCX activity (Fig. 5 B).

Junctional SR [Ca2!] heterogeneityand Ca2! alternans

The moment-based modeling approach encourages detailedanalysis of heterogeneous jSR [Ca2!] during Ca2! alternans.Stimulation at 0.5 Hz results in normal Ca2! cycling (a stableperiod-1 oscillation) with Ca2! release heterogeneity mani-festing itself as a broad u-shape distribution of jSR [Ca2!](Fig. 2 A). At 1–2 Hz, alternating Ca2! responses (stableperiod-2 oscillations) were associated with two distinct jSR[Ca2!] distribution patterns on subsequent cycles: 1), abroad, nearly normal u-shape distribution associated withthe small Ca2! transient; and 2), an n-shape distribution

A

standard modelno!variance/fast!jSR

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5times standard v

serca

0

0.5

1

1.5

2c m

yom

ax (m

M)

B

0 0.5 1 1.5 2 2.5 3times standard Io

ncx

0

1

2

3

4

c myo

max

(mM

)

C

0.5 1 5 10 50times standard v

refill

0

0.5

1

1.5

c myo

max

(mM

)

FIGURE 6 Maximum of the myoplasmic [Ca2!] transient during stableoscillations is plotted as a function of SERCA pump rate, vserca (A), NCXpump rate, I0ncx (B), and network-to-junctional-SR Ca2! transfer rate, vrefill(C). One, two, or four distinct maxima for a given parameter value indicate

period-1, -2, and -4 cycles, respectively. Results from the standard modelthat uses moment-based simulation to represent heterogeneous jSR [Ca2!]

(circles) are compared to those from the no-variance fast-jSR model simpli-

fications (triangles).

Biophysical Journal 99(2) 377–387

384 Huertas et al.

skewed toward lower jSR [Ca2!] associated with the largeCa2! transient (Figs. 2 B and 4, A and B). At 0.5–2 Hz,the jSR [Ca2!] distribution is focused near the network SR[Ca2!] at the beginning of the stimulus pulse, and distributedmore broadly and with a lower mean at the end of the stim-ulus. During 4-Hz stimulation, the period-1 oscillation isagain stable and the jSR [Ca2!] distribution pattern beforethe stimulus also becomes broadly distributed (Fig. 4 C).

These results are intuitively straightforward.During normalCa2! transients the broad u-shaped jSR [Ca2!] distributionat the end of the stimulus pulse reflects the nonuniform prob-ability ofCaRU recruitment and stochastic RyR-megachannelopen dwell times, leading to varying extents of jSR depletion.During Ca2! alternans, activation of CaRUs in the cycle withlarger SR Ca2! release leads to excessive depletion of jSR[Ca2!] and diminished CaRU activation in the subsequentcycle. Conversely, the cycle with smaller SR Ca2! releaseresults in less extensive jSR [Ca2!] depletion and greaterCaRU activation on the next cycle. Differences in activationof CaRUs during the large versus small Ca2! transient area consequence of both the fidelity of triggered release aswell as the extent of cross-activation of CaRUs coupled viathe bulk myoplasm (Fig. S4). These observations are consis-tent with a recent analysis of period-doubling bifurcations ina two-dimensional array of coupled stochastically excitableelements (27) and traditional Monte Carlo simulations ofspatially explicit local control models (9,28,29).

Myocyte Ca2! handling parametersand Ca2! alternans

The computational efficiency of the moment-based whole-cell model permitted us to investigate the relationshipbetween alternans and parameter values for key Ca2!

handling systems including SERCA-mediated SR Ca2!

uptake, Ca2! regulation of the RyR megachannel, and NCX-mediated Ca2! removal. Slowed SR Ca2! uptake by SERCAis generally considered to be conducive to alternans, whereasaccelerated uptake is believed to stabilize Ca2! cycling(2,3,5,7,30). Reports on the consequences of alterations inRyR function have been conflicting; some studies showalternans after partial inhibition of RyRs (7,31), and otherslink alternans to enhanced RyR activity (21,32). The impactof changes of NCX on Ca2! alternans has, to our knowledge,not been explored. This study demonstrates that alternansoccurs within a certain window of parameter values for thesedifferent Ca2! transport systems (Figs. 5 and 6). Thisbehavior is analogous to the frequency dependence of alter-nans observed in whole-cell models using traditional MonteCarlo simulation (9,28,29), minimal formulations of Ca2!

cycling (2,4,6–8,22), and to that observed here (Fig. 1, Cand D), in which a band of intermediate stimulation frequen-cies leads to alternans. This biphasic dependence of Ca2!

alterans on Ca2!-handling parameter values could accountfor some of the apparent discrepancies in reports regarding

the functional consequence of changes in certain Ca2! trans-port systems. For example, differences in the set point forcytosolic and luminal Ca2! regulation of RyRs could explainwhy alternans was caused by RyR inhibition in some studiesand linked to enhanced RyR activity in others (Fig. S6, Aand B).

Dynamic interactions between load-releaseand release-uptake functions

Our analysis of Ca2! alternans closely follows the approachused to probe minimal models of Ca2! cycling for period-doubling bifurcations (2,4,6,7). To our knowledge, thisis the first time such techniques have been methodicallyapplied to a whole-cell model that accurately represents het-erogeneous jSR [Ca2!] (Fig. 3). At low stimulation frequen-cies, load-release and release-uptake functions calculatedfrom the moment-closure model under the assumption thatjunctional and network SR Ca2! are equilibrated at thebeginning of the voltage pulse are consistent with the simu-lations of Ca2! alternans that do not make this assumption(Fig. 4, A and B). However, at higher stimulation frequen-cies, load-release and release-uptake functions calculated inthis fashion do not correctly predict the onset of Ca2! alter-nans, because the jSR does not equilibrate with the networkSR during the interpulse interval (Fig. 4 C).

A steep relationship between SR Ca2! content and Ca2!

release is often considered to be a crucial factor accountingfor the generation of Ca2! alternans (2,5,9). Our moment-based whole-cell simulations show that the load-releaserelationship by itself is an inadequate predictor of Ca2! alter-nans. For example, conditions such as increasing pacingfrequency or reducing NCX-mediated Ca2! extrusion causealternans without modifying the load-release function(Fig. 5 B and Fig. S7 A). In addition, changes in RyR param-eters that led to a steepened load-release function dimin-ished, rather than promoted, Ca2! alternans (Fig. 5 B andFig. S6, A and B). The reason alternans cannot be understoodentirely by the slope of the load-release function is, ofcourse, the dynamic linkage between SR Ca2! release anduptake. Just as SR load determines release, the amplitudeof release effects SR Ca2! uptake and load on subsequentcycles by activating various amounts of cellular Ca2! extru-sion (Fig. 3 D). Consequently, release and uptake mutuallydetermine the onset of Ca2! alternans (Fig. 3, A and B). Forexample, both increased stimulation frequency (Fig. S7 A)and reduced NCX activity (Fig. 5 B) shift the release-uptakefunction to higher SR loads and thereby change the intersec-tion of this curve with the load-release function. The slope ofthese curves at the fixed point also changes and, as discussedabove and in previous studies (7), this aspect of the load-release and release-uptake functions determines the stabilityof the period-1 oscillation. In a similar way, lowering RyRCa2! sensitivity increases the steepness of the load-releasecurve and shifts this curve to the right, but it also drastically

Biophysical Journal 99(2) 377–387

Ca2! Alternans in a Cardiac Myocyte Model 385

alters the slope of the release-uptake function, thereby stabi-lizing Ca2! cycling (Fig. S6 A).

Limitations of the model

The compartmental structure of the whole-cell model and thefunctional form of the Ca2! fluxes are consistent with priorwork and do not warrant extensive discussion (10,14). Onthe other hand, the 12-state Ca2! release unit used in thismodeling study (Eq. 1) is quite minimal. Following proce-dures in our previous work (10), we used a six-state RyRmegachannel model with essentially all-or-none gating(33–35) and a two-state model of the L-type Ca2! channelthat includes voltage-dependent activation but, for sim-plicity, not voltage- and Ca2!-dependent inactivation.Neglecting these features of stimulated Ca2! influx is amodel limitation that will be overcome in future work. Notethat use of more complex and realistic stochastic models ofthe L-type Ca2! channel or RyR cluster will result in aCaRU model with far more than 12 states, which reducesthe efficiency of moment-based simulations compared toMonte Carlo simulations. Therefore, an important avenueof future research is the automated reduction of CaRUmodels for multiscale simulation of local and global Ca2!

responses in myocytes and other cell types (36).Because the dynamics of each CaRU is responsive to the

associated diadic subspace and jSR [Ca2!], the moment-based modeling formalism is properly described as astochastic local-control whole-cell model. However, thespecifics of the formalism presented here assume that the dia-dic subspace [Ca2!] associated with each CaRU rapidlyequilibrates with the corresponding jSR [Ca2!] and the bulkmyoplasmic [Ca2!] (10). This rapid equilibrium assumptionis reasonable given the small effective volume of the diadicsubspace and does not represent a significant model limita-tion. On the other hand, the failure of the no-variance/fast-jSR model simplification to recapitulate Ca2! alternansexhibited by the full model (Fig. 6 and Fig. S7) demonstratesthat an assumption of rapid equilibration of jSR [Ca2!] isdebilitating.

The most important caveat to this modeling approach isthat one must consider the trade-off between 1), the compu-tational efficiency of using moment equations to representheterogeneous local Ca2!; and 2), the limitations of usinga mathematical formalism that is not explicitly spatial. Theformalism takes into account heterogeneous diadic subspaceand jSR [Ca2!], the (fast) translocation of Ca2! betweenthe diadic subspace and the bulk myoplasm, and the (slow)translocation of Ca2! between network and junctional SR.On the other hand, the CaRUs influence each other onlythrough their contribution to increases in bulk myoplasmic[Ca2!] and decreases in network SR [Ca2!]. This globalcoupling of local Ca2! signals represents an intriguingbalance between computational efficiency and physiologicalrealism. However, it is exact only in the limit of fast diffusion

of myoplasmic and network SR Ca2! and cannot be used tostudy the possible role of subcellular Ca2! waves in thegenesis of Ca2! alternans.

The importance of accounting for heterogeneousjSR Ca2!

Our calculations show that it is important to account forheterogenous jSR [Ca2!] in whole-cell simulations of Ca2!

alternans. This heterogeneity can be simulated using themoment-based approach chosen here (10), the original pop-ulation density formulation (14), or other computationallyless efficient simulation techniques that involve integrationof ODEs coupled to Markov chains (37). Comparison ofthe moment-based standard model with the no-variance/fast-jSR simplification suggests that—whatever modelingapproach may be preferred in a given context—it is essentialfor physiological realism to represent the slow dynamics ofjSR [Ca2!], as well as CaRU state-dependent heterogeneity(Fig. 6). That is, if the dynamics of jSR refilling is indeed onthe order of 10–200 ms, as suggested by prior modeling andexperiment (17–20), then multiscale modeling approachesthat assume rapid equilibration of jSR [Ca2!] cannot beexpected to correctly reproduce alternating Ca2! responses,despite the fact that such an approach has been successfullyemployed in previous studies of cardiac CICR (11–13).

Our standard parameter set corresponds to a jSR refillingtime constant of trefill " lTjsr=v

Trefill " 32 ms (see Table S1).

In this case, the no-variance/fast-jSR model simplificationdoes not even qualitatively agree with the standard modeluntil jSR refilling has been accelerated 50-fold (Fig. 4 C).When vrefill is chosen so that trefill< 1 ms, the jSR in the stan-dard model is essentially in a CaRU state-dependent quasi-static equilibrium with the bulk myoplasmic and SR [Ca2!],and the no-variance/fast-jSR model simplification agreeswith the standard model (no alternans at a stimulationfrequency of 1 Hz). This suggests that slow jSR dynamicsis an aspect of local Ca2! signaling that has consequencesfor global Ca2! responses that may be difficult to predict.In particular, Fig. 6 A shows that it may not be possible tocompensate for an assumption of fast jSR refilling bydecreasing SERCA activity, regardless of how intuitivethis suggestion may seem.

SUPPORTING MATERIAL

Eight figures, three tables, and further details of the model formulation

are available at http://www.biophysj.org/biophysj/supplemental/S0006-

3495(10)00530-8.

G.D.S. and S.G. jointly mentored M.A.H. Some of these results previously

appeared in abstract form (38).

This material is based upon work supported by the National Science Foun-

dation (grants 0133132 and 0443843 to G.D.S.) and the National Institutesof Health (grants HL074045 and HL063043 to S.G.). G.D.S. gratefully

acknowledges a research leave during academic year 2007–2008 supported

Biophysical Journal 99(2) 377–387

386 Huertas et al.

by the College of William and Mary and a long-term visitor position at the

Mathematical Biosciences Institute at Ohio State University.

REFERENCES

1. Myles, R. C., F. L. Burton, ., G. L. Smith. 2008. The link betweenrepolarisation alternans and ventricular arrhythmia: does the cellularphenomenon extend to the clinical problem? J. Mol. Cell. Cardiol.45:1–10.

2. Weiss, J. N., A. Karma, ., Z. Qu. 2006. From pulsus to pulseless: thesaga of cardiac alternans. Circ. Res. 98:1244–1253.

3. Laurita, K. R., and D. S. Rosenbaum. 2008. Cellular mechanisms ofarrhythmogenic cardiac alternans. Prog. Biophys. Mol. Biol. 97:332–347.

4. Shiferaw, Y., D. Sato, and A. Karma. 2005. Coupled dynamics ofvoltage and calcium in paced cardiac cells. Phys. Rev. E Stat. Nonlin.Soft Matter Phys. 71:021903.

5. Eisner, D. A., Y. Li, and S. C. O’Neill. 2006. Alternans of intracellularcalcium: mechanism and significance. Heart Rhythm. 3:743–745.

6. Qu, Z., Y. Shiferaw, and J. N. Weiss. 2007. Nonlinear dynamics ofcardiac excitation-contraction coupling: an iterated map study. Phys.Rev. E Stat. Nonlin. Soft Matter Phys. 75:011927.

7. Xie, L. H., D. Sato, ., J. N. Weiss. 2008. Intracellular Ca alternans:coordinated regulation by sarcoplasmic reticulum release, uptake, andleak. Biophys. J. 95:3100–3110.

8. Shiferaw, Y., M. A. Watanabe, ., A. Karma. 2003. Model of intracel-lular calcium cycling in ventricular myocytes. Biophys. J. 85:3666–3686.

9. Tao, T., S. C. O’Neill,., H. Zhang. 2008. Alternans of cardiac calciumcycling in a cluster of ryanodine receptors: a simulation study. Am. J.Physiol. Heart Circ. Physiol. 295:H598–H609.

10. Williams, G. S. B., M. A. Huertas, ., G. D. Smith. 2008. Momentclosure for local control models of calcium-induced calcium release incardiac myocytes. Biophys. J. 95:1689–1703.

11. Hinch, R., J. L. Greenstein,., R. L. Winslow. 2004. A simplified localcontrol model of calcium-induced calcium release in cardiac ventricularmyocytes. Biophys. J. 87:3723–3736.

12. Greenstein, J. L., R. Hinch, and R. L. Winslow. 2006. Mechanisms ofexcitation-contraction coupling in an integrative model of the cardiacventricular myocyte. Biophys. J. 90:77–91.

13. Hinch, R., J. L. Greenstein, and R. L. Winslow. 2006. Multi-scalemodels of local control of calcium induced calcium release. Prog. Bio-phys. Mol. Biol. 90:136–150.

14. Williams, G. S. B., M. A. Huertas,., G. D. Smith. 2007. A probabilitydensity approach to modeling local control of calcium-induced calciumrelease in cardiac myocytes. Biophys. J. 92:2311–2328.

15. Mazzag, B., C. J. Tignanelli, and G. D. Smith. 2005. The effect ofresidual Ca2! on the stochastic gating of Ca2!-regulated Ca2! channelmodels. J. Theor. Biol. 235:121–150.

16. Huertas, M. A., and G. D. Smith. 2007. The dynamics of luminal deple-tion and the stochastic gating of Ca2!-activated Ca2! channels andrelease sites. J. Theor. Biol. 246:332–354.

17. Luo, C. H., and Y. Rudy. 1994. A dynamic model of the cardiac ventric-ular action potential. I. Simulations of ionic currents and concentrationchanges. Circ. Res. 74:1071–1096.

18. Brochet, D. X., D. Yang, ., H. Cheng. 2005. Ca2! blinks: rapidnanoscopic store calcium signaling. Proc. Natl. Acad. Sci. USA.102:3099–3104.

19. Sobie, E. A., L. S. Song, and W. J. Lederer. 2005. Local recovery ofCa2! release in rat ventricular myocytes. J. Physiol. 565:441–447.

20. Jafri, M. S., J. J. Rice, and R. L. Winslow. 1998. Cardiac Ca2!

dynamics: the roles of ryanodine receptor adaptation and sarcoplasmicreticulum load. Biophys. J. 74:1149–1168.

21. Belevych, A. E., D. Terentyev, ., S. Gyorke. 2009. Redox modifica-tion of ryanodine receptors underlies calcium alternans in a caninemodel of sudden cardiac death. Cardiovasc. Res. 84:387–395.

22. Shiferaw, Y., Z. Qu, ., J. N. Weiss. 2006. Nonlinear dynamics ofpaced cardiac cells. Ann. N. Y. Acad. Sci. 1080:376–394.

23. Bassani, J. W., W. Yuan, and D. M. Bers. 1995. Fractional SR Carelease is regulated by trigger Ca and SR Ca content in cardiac myo-cytes. Am. J. Physiol. 268:C1313–C1319.

24. Shannon, T. R., K. S. Ginsburg, and D. M. Bers. 2000. Potentiation offractional sarcoplasmic reticulum calcium release by total and free intra-sarcoplasmic reticulum calcium concentration. Biophys. J. 78:334–343.

25. Callewaert, G., L. Cleemann, and M. Morad. 1988. Epinephrineenhances Ca2! current-regulated Ca2! release and Ca2! reuptake inrat ventricular myocytes. Proc. Natl. Acad. Sci. USA. 85:2009–2013.

26. Hussain, M., and C. H. Orchard. 1997. Sarcoplasmic reticulum Ca2!

content, L-type Ca2! current and the Ca2! transient in rat myocytesduring b-adrenergic stimulation. J. Physiol. 505:385–402.

27. Cui, X., R. J. Rovetti, ., Z. Qu. 2009. Period-doubling bifurcation inan array of coupled stochastically excitable elements subjected to globalperiodic forcing. Phys. Rev. Lett. 103:044102.

28. Restrepo, J. G., J. N. Weiss, and A. Karma. 2008. Calsequestrin-medi-ated mechanism for cellular calcium transient alternans. Biophys. J.95:3767–3789.

29. Restrepo, J. G., and A. Karma. 2009. Spatiotemporal intracellularcalcium dynamics during cardiac alternans. Chaos. 19:037115.

30. Kameyama, M., Y. Hirayama, ., T. Takano. 2003. Possible contribu-tion of the sarcoplasmic reticulum Ca2! pump function to electrical andmechanical alternans. J. Electrocardiol. 36:125–135.

31. Dıaz, M. E., D. A. Eisner, and S. C. O’Neill. 2002. Depressed ryanodinereceptor activity increases variability and duration of the systolic Ca2!

transient in rat ventricular myocytes. Circ. Res. 91:585–593.

32. Lehnart, S. E., C. Terrenoire, ., A. R. Marks. 2006. Stabilization ofcardiac ryanodine receptor prevents intracellular calcium leak andarrhythmias. Proc. Natl. Acad. Sci. USA. 103:7906–7910.

33. Stern, M. D. 1992. Theory of excitation-contraction coupling in cardiacmuscle. Biophys. J. 63:497–517.

34. Rice, J. J., M. S. Jafri, and R. L. Winslow. 1999. Modeling gain andgradedness of Ca2! release in the functional unit of the cardiac diadicspace. Biophys. J. 77:1871–1884.

35. Sobie, E. A., K. W. Dilly, ., M. S. Jafri. 2002. Termination of cardiacCa2! sparks: an investigative mathematical model of calcium-inducedcalcium release. Biophys. J. 83:59–78.

36. Hao, Y., P. Kemper, and G. D. Smith. 2009. Reduction of calciumrelease site models via fast/slow analysis and iterative aggregation/disaggregation. Chaos. 19:037107.

37. Greenstein, J. L., and R. L. Winslow. 2002. An integrative model of thecardiac ventricular myocyte incorporating local control of Ca2! release.Biophys. J. 83:2918–2945.

38. Huertas, M. A., G. D. Smith, and S. Gyorke. 2009. Analysis of calciumalternans in a cardiac myocyte model that uses moment equations torepresent heterogenous junctional SR calcium 2009 Biophysical SocietyMeeting Abstracts. Biophy. J. (Supplement 1), Abstract, 1414-Pos.

Biophysical Journal 99(2) 377–387

Ca2! Alternans in a Cardiac Myocyte Model 387

Biophysical Journal, Volume 99 Supporting Material

Ca2+ alternans in a cardiac myocyte model that uses moment equations to represent heterogeneous

junctional SR Ca2+

Marco A. Huertas, Gregory D. Smith, and Sándor Györke

Biophysical Journal

Supplemental Material

Ca2+ alternans in a cardiac myocyte model that uses moment equations torepresent heterogeneous junctional SR Ca2+

Marco A. Huertas, Gregory D. Smith, and Sandor Gyorke

Model Formulation

The whole cell model of Ca2+ cycling used in this study employs a moment-based descriptionof heterogenous diadic subspace and junctional SR (jSR) [Ca2+], a mathematical techniquethat has been validated in our previous work by comparison with traditional Monte Carlosimulations (10). Here we review the most important aspects of the model formulation.

Traditional Monte Carlo formulation

The Monte Carlo formulation of the whole cell model of Ca2+ cycling simulates the dynam-ics of bulk myoplasmic [Ca2+] (cmyo), network SR [Ca2+] (cnsr), N diadic subspace Ca2+

concentrations (cnds) and N jSR domain Ca2+ concentrations (cn

jsr) using a system of N + 2concentration balance equations. These ordinary di!erential equations (ODEs) are coupledto N Markov chains, each of which represents the stochastic gating of a CaRU consistingof one L-type Ca2+ channel (DHPR) and one RyR “megachannel” (described below). Theconcentration balance equations take the form,

dcmyo

dt=Jleak + JT

efflux ! Jncx ! Jserca + Jin (S1)

dcnsr

dt=

1

!nsr

!

Jserca ! JTrefill ! Jleak

"

(S2)

dcnjsr

dt=

1

!jsr

!

Jnrefill ! Jn

ryr

"

(S3)

where 1 " n " N , !nsr and !jsr are volume ratios defined in Table S1, and the fluxes appearwith signs consistent with Fig. S1. The flux through the RyR megachannel associated withthe n-th CaRU (Jn

ryr) is given by

Jnryr = "n

ryr

vTryr

N(cn

jsr ! cnds) (S4)

where "nryr is a stochastic variable that takes the value 1 or 0 depending on whether the n-th

RyR megachannel is open or closed, and cnds is the associated diadic subspace concentration

defined below (Eq. S8). Similarly, di!usion from the network SR to each jSR compartmentis given by

Jnrefill =

vTrefill

N(cnsr ! cn

jsr). (S5)

while the flux out of the n-th diadic subspaces into the myoplasm is given by

Jnefflux =

vTefflux

N(cn

ds ! cmyo). (S6)

The total flux is obtained from the sum of these individual contributions and is given by

JT! =

N#

n=1

Jn! (S7)

where the subscript # is ryr, refill, or efflux. See Description of Ca2+ Fluxes below fordetails of the remaining four fluxes that appear in Eqs. S1–S3 and Fig. S1: Jn

dhpr (influx

into the diadic subspaces via L-type Ca2+ channels that is a function of the random variable!n

dhpr), Jin (voltage-independent Ca2+ influx), Jncx (Na+-Ca2+ exchange), Jserca (SR Ca2+-ATPase), and Jleak (the network SR leak).

Diadic subspace Ca2+ concentration

Note that Eqs. S1–S3 do not include a concentration balance equation for diadic subspace[Ca2+], because in previous studies we observed rapid equilibrium of the diadic subspace[Ca2+] with the [Ca2+] in bulk myoplasm and the associated juctional SR (14). Using thisfact, the [Ca2+] in each of the N diadic subspaces is well-approximated by balancing fluxesin and out of that compartment. That is, setting

Jnryr(c

njsr, c

nds) + Jn

dhpr(cnds) = Jn

efflux(cnds, c

nmyo)

the rapidly equilibrated diadic subspace [Ca2+] (cnds) can be written as a linear function of

the associated jSR Ca2+ concentrations,

cnds = cn

ds,0 + cnds,1 cn

jsr, (S8)

where the coe!cients cnds,0 and cn

ds,1 are functions of the bulk myoplasmic and network SR[Ca2+],

cnds,0 =

!ndhprJ

0dhpr + veffluxcmyo

!nryrvryr + vefflux ! !n

dhprJ1dhpr

(S9)

cnds,1 =

!nryrvryr

!nryrvryr + vefflux ! !n

dhprJ1dhpr

. (S10)

In these expressions, the quantities !ndhpr and !n

ryr indicate whether the n-th L-type Ca2+

channel and RyR megachannel are open or closed vryr = vTryr/N , vefflux = vT

efflux/N , andJ0

dhpr and J1dhpr are the following functions of plasma membrane voltage defined by

Jndhpr = !n

dhpr

!

J0dhpr + cn

dsJ1dhpr

"

(S11)

where the functional form of J0dhpr and J1

dhpr are given by Eqs. S33 and S34.

Twelve-state CaRU model

As mentioned before, each of the N CaRUs considered here consists of one voltage-dependentL-type Ca2+ channel and one RyR megachannel. Upon membrane depolarization the L-typechannel opens, allowing an increase in [Ca2+ ] in the dyadic subspace (cds). This increasein Ca2+ concentration activates the RyR megachannel which in turn releases Ca2+ from thejSR. This process is known as Ca2+induced Ca2+ release (CICR).

The RyR megachannel model used follows (10) and is consistent with several studiesindicating that the gating of the RyR cluster associated with each CaRU is essentially all-or-none (34–36), and is described by a six-state transition state diagram

4k+ryrcds 3"k+

ryrcds 2"2k+ryrcds "3k+

ryrcds k+ryr,!cjsr

C1 ! C2 ! C3 ! C4 ! C5 ! O#3k"

ryr 2#2k"

ryr 3#k"

ryr 4k"

ryr k"

ryr,!

. (S12)

The opening of the RyR assumes the sequential binding of diadic subspace Ca2+ ions—to achieve highly cooperative Ca2+-dependent opening of the RyR megachannel—and anexplicit jSR Ca2+-dependent transition—so that depletion of luminal Ca2+ decreases theopen probability of the RyR megachannel.

Also as in ref. (10), the two-state model of the L-type Ca2+ channel (DHPR) is describedby the transition state diagram

k+dhpr (V )

C ! Ok!

dhpr

, (S13)

where k+dhpr is the voltage-dependent activation rate (40) given by

k+dhpr = k+

dhpr

e(V !V !dhpr)/!dhpr

1 + e(V !V !dhpr)/!dhpr

, (S14)

and k!

dhpr is the constant de-activation rate that sets the mean open time (0.2 ms) andmaximum open probability (0.1) of the channel. For simplicity, this two-state DHPR modeldoes not include voltage- and Ca2+-dependent inactivation of L-type Ca2+ channels.

Because both channels are coupled through cds one must describe the dynamics of eachCaRU using a single transition state diagram. Each CaRU can then be in one of twelvepossible states and transitions between these states are mediated by changes in voltage andin Ca2+ concentration in the diadic subspace and jSR. The resulting twelve-state transitionstate diagram takes the form,

CC1 ! CC2 ! CC3 ! CC4 ! CC5 ! CO"# "# "# "# "# "#

OC1 ! OC2 ! OC3 ! OC4 ! OC5 ! OO(S15)

where horizontal and vertical transitions are governed by Eqs. S12 and S13, respectively.Each state is identified by the juxtaposition of the states of the L-type channel and the RyRmegachannel. The first character (C or O) indicates the state of the DHPR while the secondcharacter (C1, C2, C3, C4, C5, or O) refers to the state of the RyR megachannel. Note thatthe 12-by-12 generator matrix that collects the rate constants of the CaRU model (Eq. S15)can be written compactly in the form,

Q = K"(V ) + cds Kds + cjsr Kjsr (S16)

where the elements of K"(V ) are the Ca2+-independent transitions (both voltage-dependentand voltage-independent with units of time!1), and the elements of Kds and Kjsr are theassociation rate constants for the transitions mediated by diadic subspace (cds) and jSR (cjsr)[Ca2+], respectively.

Probability density formulation and moment closure technique

Underlying the moment closure technique is a univariate probability density function (PDF)denoted by !i(cjsr, t) that describes the distribution of [Ca2+] across a large population of NjSR compartments, that is,

!i(cjsr, t) dcjsr = Pr{cjsr < cjsr(t) < cjsr + dcjsr and S(t) = i}. (S17)

In this expression, i is an index over CaRU state, S is the state of a randomly sampled CaRU,and cjsr is the associated jSR [Ca2+]. These densities satisfy a system of advection-reactionequations of the form (10,14–16),

!"i

!t= !

!

!cjsr[f i

jsr "i] + [!Q]i, (S18)

where 1 " i " M , M=12 is the number of states in the CaRU model, Q is the M # Mgenerator matrix (Eq. S16), the row-vector !(cjsr, t) = ("1, "2, · · · , "M) collects the time-dependent probability densities and [!Q]i is the ith element of the vector-matrix product!Q. In this expression the advection rates f i

jsr are consistent with Eq. S3, that is,

f ijsr =

1

#Tjsr

!

vTrefill(cnsr ! cjsr) ! $i

ryrvTryr(cjsr ! c i

ds)"

, (S19)

where #Tjsr is the volume ratio associated with all jSR compartments in aggregate (see Ta-

ble S1), and c ids is given by Eqs. S8–S10 with the replacement of i for n. In the resulting

expressions for c ids,0 and c i

ds,1 used in Eqs. S18 and S19, $idhpr and $i

ryr indicate whether theL-type Ca2+ channel and RyR megachannel are open or closed in the ith CaRU state.

Moments of junctional SR [Ca2+]

The moment-based description of jSR [Ca2+] begins by writing

µiq(t) =

#

(cjsr)q "i(cjsr, t) dcjsr (S20)

where the index i indicates the CaRU state, "i(cjsr, t) is the probability density function(Eq. S17), and the non-negative integer q indicates the moment degree, which appears asan exponent in the term (cjsr)q. The first three moments (q = 0, 1, 2) have simple inter-pretations. The zeroth moment µi

0 corresponds to the probability that a randomly sampledCaRU is in state i, that is,

µi0(t) =

#

"i(cjsr, t)dcjsr = Pr{S(t) = i},

where by conservation of probability we have$

i µi0 = 1. The first moment µi

1 is related tothe expected value of the jSR [Ca2+] conditioned on CaRU state through

Ei[cjsr] =

µi1

µi0

. (S21)

The second moment µi2 is related to the conditional variance of the jSR [Ca2+] by

Vari[cjsr] =

µi2

µi0

!

%

µi1

µi0

&2

. (S22)

Moment equations for junctional SR [Ca2+]

As discussed in our prior work (10), the definition of µi0, µi

1, and µi2 (Eq. S20) can be combined

with the evolution equation for the probability density !i(cjsr, t) (Eq. S18) to derive a systemof ODEs for the time-evolution of the moments of jSR [Ca2+],

dµiq

dt=

qµiq!1

"Tjsr

!

vTrefill cnsr + #i

ryr vTryr c

ids,0

"

+qµi

q

"Tjsr

!

#iryr vT

ryr c ids,1 ! vT

refill ! #iryr vT

ryr

"

+M

#

j=1

µjq

!

Kj,i! + c j

ds,0 Kj,ids

"

+M

#

j=1

µjq+1

!

c jds,1 Kj,i

ds + Kj,ijsr

"

(S23)

where q = 0, 1, 2, c jds,0 and c j

ds,1 are defined as in Eq. S19, and the superscripts in Kj,i! , Kj,i

ds ,

and Kj,ijsr indicate the transition rate or bimolecular rate constant in the jth row and ith

column of these matrices defined in Eq. S16. When moment µi3 is required in the equation

for µi2, we use an algebraic relation

µi3 = $

!

µi0, µ

i1, µ

i2

"

(S24)

that would be strictly correct if the probability density functions were scaled beta distribu-tions. This truncation and closure of the moment equations (Eq. S23) is further motivated inref. (10) and has been validated by comparing results thus obtained with traditional MonteCarlo simulation.

Whole cell model fluxes and the moments of junctional SR [Ca2+]

To put it all together, the Ca2+ fluxes that appear in the concentration balance equationsfor the bulk myoplasmic and network SR (Eqs. S1 and S2) are expressed in terms of themoments µi

q. For example, the total flux leaving the network SR and moving into the jSRcompartments is given by

JTrefill =

M#

i=1

$

vTrefill (cnsr ! cjsr) !i(cjsr, t) dcjsr (S25)

= vTrefill

M#

i=1

!

cnsrµi0 ! µi

1

"

(S26)

= vTrefill

%

cnsr !M

#

i=1

µi1

&

. (S27)

Similarly, the flux into the myoplasm from the diadic subspaces is

JTefflux = vT

efflux

%

M#

i=1

!

c ids,0 µi

0 + c ids,1 µi

1

"

! cmyo

&

(S28)

and the flux through the L-type channels and RyR megachannels are

JTdhpr =

M!

i=1

!idhpr

"

J0dhprµ

i0 + J1

dhpr

#

cids,0 µi

0 + cids,1 µi

1

$%

(S29)

and

JTryr =

M!

i=1

vTryr!

iryr

#

µi1 ! ci

ds,0 µi0 ! ci

ds,1 µi1

$

. (S30)

Note that the average diadic subspace and jSR [Ca2+] can also be written in terms of themoments,

cavgds = E[cds] =

M!

i=1

"iE

i[cids,0 + ci

ds,1 cjsr] =M

!

i=1

#

cids,0 µi

0 + cids,1 µi

1

$

(S31)

cavgjsr = E[cjsr] =

M!

i=1

"iE

i[cjsr] =M

!

i=1

µi1, (S32)

and JTefflux = vT

efflux [cavgds ! cmyo] and JT

refill = vTrefill

"

cnsr ! cavgjsr

%

when expressed in termsusing these quantities.

Description of Ca2+ fluxes

Fluxes that appear in Eqs. S1 and S2, but only involve the bulk myoplasm and network SR,are modeled as in our prior work (10).

For example, the Na+-Ca2+ exchanger takes the form Jncx = !AmIncx/F where (20, 40,41)

Incx = Ioncx

[Na+]3myocexte!ncxFV/RT ! [Na+]3extcmyoe(!ncx!1)FV/RT

#

K3ncx,n + [Na+]3ext

$

(Kncx,c + cext) (1 + ksatncxe

(!ncx!1)FV/RT ),

Am = Cm#myo/Vmyo, cext is the extracellular Ca2+ concentration, and [Na+]myo and [Na+]ext

are the intracellular and extracellular sodium concentrations, respectively (for parameterssee Tables S1–S3). The SERCA-type Ca-ATPase flux includes both forward and reversemodes (42) and is given by

Jserca = vserca(cmyo/Kfs)

!fs ! (cnsr/Krs)!rs

1 + (cmyo/Kfs)!fs + (cnsr/Krs)

!rs.

The leakage Ca2+ flux that appears in Eqs. S1 and S2 is given by

Jleak = vleak(cnsr ! cmyo).

Following (41), Eq. S1 includes a constant background Ca2+ influx that takes the formJin = !AmIin/zF where Iin = gin(V ! ECa) and ECa = (RT/2F ) ln(cext/cmyo).

The e!ective volume ratios $nsr and $jsr that appear in Eqs. S2 and S3 are defined withrespect to the physical volume (Vmyo) and include a constant-fraction Ca2+ bu!er capacity

for the myoplasm (!myo). For example, the e!ective volume ratio associated with the networkSR is

"nsr =Vnsr

Vmyo

=Vnsr/!nsr

Vmyo/!myo

with e!ective volumes defined by Vnsr = Vnsr/!nsr and Vmyo = Vmyo/!myo.In the Monte Carlo model used to introduce the moment-based whole cell model, the

trigger Ca2+ flux into each of the N diadic spaces through DHPR channels (Jndhpr in Eq. S11)

is given by

Jndhpr = !

Am

zFIndhpr (S33)

where Am = Cm!myo/Vmyo. The inward Ca2+ current (Indhpr " 0) is given by

Indhpr = #n

dhpr

P Tdhpr

N

!

zFV

V!

" !

cndse

V/V! ! cext

eV/V! ! 1

"

(S34)

where V! = RT/zF , P Tdhpr is the total (whole cell) permeability of the L-type Ca2+ channels,

and #ndhpr is a random variable that is 0 when the L-type Ca2+ channel associated with the n-

th CaRU is closed and 1 when this channel is open. Thus, the quantities J0dhpr = JT,0

dhpr/N and

J1dhpr = JT,1

dhpr/N required to evaluate cnds,0 (Eq. S9) and cn

ds,1 (Eq. S10) are defined through

JT,0dhpr =

AmP TdhprV

V!

!

cext

eV/V! ! 1

"

JT,1dhpr =

AmP TdhprV

V!

!

eV/V!

1 ! eV/V!

"

,

consistent with Eq. S11. In the moment-based whole cell simulations that are the focus ofthis paper, the total flux through L-type Ca2+ channels is given by

JTdhpr = !AmP T

dhpr

V

V!

!

cndse

V/V! ! cext

eV/V! ! 1

"

(S35)

and the quantities JT,0dhpr and JT,1

dhpr are used to evaluate cids,0 (Eq. S29) and ci

ds,1 (Eq. S30).

Summary of model formulation

To summarize, the whole cell model of Ca2+ cycling that is the focus of this paper utilizesconcentration balance equations for myoplasmic and network SR [Ca2+] (Eqs. S1 and S2)and a moment-based description of heterogenous diadic subspace and jSR [Ca2+] (Eq. S23)that assumes rapid equilibration of diadic subspace Ca2+ but accounts for slow dynamics ofjSR [Ca2+] associated with a large population of CaRUs. The fluxes that involve the diadicsubspaces and jSR compartments are expressed in terms of the moments of jSR [Ca2+](Eqs. S25–S30). Importantly, the moment-based whole cell model discussed above accountsfor heterogeneity of jSR Ca2+. To assess the functional consequences of this heterogeneityin simulations of Ca2+ alternans, the standard model discussed above is compared to a no-variance/fast-jSR model simplification that assumes rapid refilling of the jSR compartments

and quasi-static equilibrium between the junctional and network SR. Using Eqs. S3–S5, thisassumed balance between release (Jn

ryr) and refill (Jnrefill) fluxes implies

Ei[cjsr] =

µi1

µi0

=vT

refillcnsr + !iryrv

Tryrc

ids,0

vTrefill + !i

ryrvTryr(1 ! ci

ds,1)(S36)

In this simplification Eq. S23 is solved for q = 0 and Eq. S36 is used when the first moment(µi

1) is required. Here jSR [Ca2+] is ‘slaved’ in a CaRU state-dependent fashion to the bulkconcentrations cnsr and cmyo, the later via ci

ds,0 and cids,1 (Eqs. S9 and S10).

JRyR

JDHPR J

efflux

JSERCA

ATP

ADP

Jrefill

Jleak

JNCX

Jinc

ds

cjsr

cmyo

cnsr

n

n

n

n

cext

n

n

Figure S1: Diagram of model components and fluxes. Each Ca2+ release unit consists of tworestricted compartments (the diadic subspace and jSR with [Ca2+] denoted by cds and cjsr,respectively), a two-state L-type Ca2+ channel (DHPR), and a 6-state RyR “megachannel.”The t-tubular [Ca2+] is denoted by cext and the fluxes Jn

dhpr, Jnryr, Jn

efflux, Jnrefill, Jin, Jncx,

Jserca, and Jleak are described in the Model Formulation of the Supplemental Material.

A

!40 !20 0 20 40

Test potential (mV)

0

0.2

0.4

0.6

0.8

1

cm

yo

tot

(µ

M)

B

!40 !20 0 20 40

Test potential (mV)

500

600

700

800

900

1000

1100

csr

tot (µ

M)

Figure S2: High-gain Ca2+ release in the moment-based whole cell model is graded with 100ms test membrane potentials in a simulated voltage clamp protocol. Both the maximumtotal myoplasmic [Ca2+] (A, Eq. 4) and minimum total SR [Ca2+] (B, Eq. 4) during therelease event are shown. Dashed line indicates the steady-state Ca2+ concentrations prior tothe voltage pulse.

0.9 (mM)

1.0 (mM)1.1 (mM)

1.2 (mM)

!40 !20 0 20 40Test potential (mV)

0

5

10

15

20

25

30

Ga

in

Figure S3: EC coupling gain in the moment-based whole cell model as a function of the testpotential, computed as the ratio between the integrated Ca2+ release and trigger fluxes overthe duration of the 100 ms voltage pulse. The gain increases with increased steady-state SR[Ca2+] prior to the depolarizing pulse (compare filled circles).

C

0 0.5 1 1.5 2

Time (s)

0.85

0.9

0.95

1

1.05

1.1

RyR

clo

se

d

D

0 0.5 1 1.5 2

Time (s)

0

0.05

0.1

0.15

RyR

op

en

Figure S4: A and B: The distribution of jSR [Ca2+] conditioned on the RyR megachannelbeing closed/open (states C1–C5/O in Eq. S12). For clarity, these distributions are normalizedand the legend indicates the fraction of CaRUs in these aggregate states. The dark greenand blue distributions correspond to the alternating response with a large myoplasmic Ca2+

transient at the onset of the voltage pulse as indicated by (a) and (b) in Fig. 2C . The lightgreen and blue distributions correspond to the smaller Ca2+ transient (Fig. 2C (c) and (d)).C and D: The fraction of closed/open RyR megachannels as a function of time on alternatingcycles. Green and blue circles correspond to the time when the distributions in panels Aand B are calculated.

Rn = r(L

n)

A

700 800 900 1000 1100 1200 1300 1400

Ln (µM)

0

0.5

1

1.5

2

Rn (µ

M)

All Ca2+

resequestered

Part of the Ca2+

extruded

Ln+1

= u(Rn)

B

0 0.5 1 1.5 2

Rn (µM)

800

900

1000

1100

1200

Ln+

1 (µ

M)

Figure S5: Load-release and release-uptake curves computed by integrating the momentclosure model equations over one cycle for various initial SR loads under the assumptionthat at the beginning of the n-th cycle E[cjsr(tn)] = cnsr(tn) and Var[cjsr(tn)] = 0. A: Load-release curve which shows a monotonic increase in the release as a function of the SR load.B: The release-uptake function shows that when the release event is small all Ca2+ goinginto the myoplasm is resequestered by the SERCA pump, causing an increase in the SR loadat consecutive cycles. On the other hand, when a large release occurs, part of the Ca2+ isextruded by the NCX decreasing the amount of Ca2+ returning to the SR. The open squareindicates the location of the fixed point (cf. Fig. 3).

A

k+,standard

ryr

0.75X

1.25X

800 1000 1200 1400 1600

Ln,L

n+1 (µM)

0

0.5

1

1.5

Rn (µ

M)

B

k+,standard

ryr,*

0.25X

2X

800 1000 1200 1400 1600

Ln,L

n+1 (µM)

0

0.5

1

1.5

Rn (µ

M)

C

vrefill

standard

0.67X

33X

700 800 900 1000 1100 1200

Ln,L

n+1 (µM)

0

0.5

1

1.5

Rn (µ

M)

Figure S6: Additional examples of the e!ect of various model parameters on the load-release(solid lines) and release-uptake (dashed lines) curves computed from the moment-based wholecell model. Filled circles indicate period-1 or -2 oscillatory responses during 1 Hz stimulation.A: Cytosolic Ca2+ activation rate constant of RyR megachannel (k+

ryr) at standard value,0.75X, and 1.25X. B: Rate constant for luminal Ca2+-regulation of RyR megachannel (k+

ryr,!)at standard value, 0.25X, and 2X. C: Network-to-junctional SR Ca2+ transfer rate (vrefill)at standard value, 0.67X, and 33X.

A

freq. = 0.5 Hz

freq. = 1 Hz

freq. = 2 Hz

700 800 900 1000 1100 1200

Load: Ln (µM)

0.2

0.4

0.6

0.8

1

1.2

Rele

ase: R

n (µ

M)

B LR standard model

RU standard model

LR no!variance/fast!jSR

RU no!variance/fast!jSR

1000 1500 2000 2500

Load: Ln (µM)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Rele

ase: R

n (µ

M)

Figure S7: A: Load-release (thick solid line) and release-uptake (thin solid lines) curvescomputed from the moment-based whole cell model and values obtained from ongoing sim-ulations (symbols) for three di!erent stimulation frequencies. The 1 Hz load-release curveis indistinguishable from the 0.5 and 2 Hz curves (not shown). The family of release-uptakecurves have essentially the same shape but are shifted rightwards toward higher SR loadsat higher stimulation frequencies. B: Load-release (LR) (solid line) and release-uptake (RU)(dashed lines) curves for the standard model as well as the no-variance/fast-jSR model sim-plification, the latter of which that assumes rapid refilling of the jSR compartments andquasi-static equilibrium between junctional and network SR (Eq. S36). The load-release andrelease-uptake curves as calculated from the no-variance/fast-jSR simplification do not agreewith the curves calculated from the standard model. This indicates that the assumption ofrapid equilibrium between junctional and network SR Ca2+ is debilitating.

A

standard model

no!variance/fast!jSR

10!1

100

101

Stimulus Frequency (Hz)

0

0.5

1

1.5

ma

x(c

myo

tot

) (µ

M)

B

10!1

100

101

Stimulus Frequency (Hz)

500

1000

1500

2000

min

(csr

tot )

(µM

)

Figure S8: Black filled circles and solid lines show frequency-dependence of Ca2+ alternansexhibited by the moment-based model (replotted from Fig. 1C and D). Panels A and Bshows total myoplasmic and network SR [Ca2+], respectively (Eq. 4). This result of thestandard model is compared to the no-variance/fast-jSR model simplification that assumesrapid refilling of the jSR compartments and quasi-static equilibrium between junctional andnetwork SR (gray filled circles, Eq. S36).

Parameter Definition ValueVnsr network SR volume 3.15 ! 10!7 µLVmyo myoplasmic volume 2.15 ! 10!5 µLV T

ds = NVds total diadic subspace volume 2 ! 10!8 µLV T

jsr = NVjsr total junctional SR volume 2.45 ! 10!8 µLCm membrane capacitance 1.534 ! 10!4 µF!ds subspace bu!ering factor 0.5!jsr junctional SR bu!ering factor 0.065!nsr network SR bu!ering factor 1.0!myo myoplasmic bu!ering factor 0.05vT

refill = "Tjsr/#refill junctional SR refilling rate 0.027 s!1 *

vTefflux = "T

ds/#efflux diadic subspace e"ux rate 22.1 s!1 *F Faraday’s constant 96480 coul mol!1

R gas constant 8314 mJ mol!1 K!1

T absolute temperature 310Kcext extracellular Ca2+ concentration 1.8 mM[Na+]ext extracellular Na+ concentration 140 mM[Na+]myo intracellular Na+ concentration 10 mM *

Table S1: Model parameters including volume fractions, Ca2+ bu!ering factors, rate con-stants for exchange between restricted domains and the bulk, physical constants, and fixedionic concentrations. Here and in Tables S2 and S3, asterisks indicate parameters that wheremodified from prior work (10). Note that "T

jsr is the ratio of the e!ective volumes of the jSRand myoplasm given by "T

jsr = (V Tjsr/!jsr)/(Vmyo/!myo) = 8.8! 10!4; consequently, the time

constant for jSR refilling is #refill = "Tjsr/v

Trefill = 32 ms. A similar calculation shows that

the time constant for diadic subspace Ca2+ is much faster (#efflux = "Tds/v

Tefflux = 4 µs).

Parameter Definition ValuevT

ryr = Nvryr total RyR cluster release rate 0.9 s!1

P Tdhpr = NPdhpr total DHPR permeability 3.5 ! 10!8 cm s!1 *

V !dhpr DHPR activation threshold "10 mV

!dhpr DHPR activation parameter 6.24 mVk+

dhpr maximum rate of DHPR opening 556 s!1

k!

dhpr closing rate of DHPR opening 5000 s!1

k!

dhpr rate of DHPR closing 5000 s!1

k+ryr rate of RyR activation 4000 µM!1s!1 *

k!

ryr rate of RyR deactivation 16000 s!1 *k+

ryr," rate of RyR opening 40 µM!1s!1

k!

ryr," rate of RyR closing 500 s!1

" cooperativity factor 2# cooperativity factor 2

Table S2: Ca2+ release unit parameters including rate constants for the L-type Ca2+ channel(Eqs. S13 and S14) and RyR megachannel (Eq. S12).

Parameter Definition ValueKfs forward half-saturation constant for SERCA pump 0.17 µMKrs reverse half-saturation constant 2042.4 µM *!fs forward cooperativity constant 0.75!rs reverse cooperativity constant 0.75vserca maximum SERCA pump rate 21.5 µM s!1 *I0ncx magnitude of Na+-Ca2+ exchange current 2250 µA µF!1 *

Kncx,n Na+ half saturation constant 87.5 ! 103µMKncx,c Ca2+ half saturation constant 1.38 ! 103µMksat

ncx saturation factor 0.1!ncx voltage dependence of Na+-Ca2+ exchange 0.35vleak SR Ca2+ leak rate constant 2.4 ! 10!6 s!1

gin maximum conductance of background Ca2+ influx 9.6 ! 10!5 mS µF!1

Table S3: Model parameters: Na+-Ca2+ exchange current, SERCA pumps, and backgroundCa2+ influx.