2010HartmanSobieSmith10(CaSparksAndHomeostasis)

Spontaneous Ca2! sparks and Ca2! homeostasis in a minimal model ofpermeabilized ventricular myocytes

Jana M. Hartman,1 Eric A. Sobie,2 and Gregory D. Smith1

1Department of Applied Science, The College of William and Mary, Williamsburg, Virginia; and 2Department ofPharmacology and Systems Therapeutics, Mount Sinai School of Medicine, New York, New York

Submitted 23 March 2010; accepted in final form 13 September 2010

Hartman JM, Sobie EA, Smith GD. Spontaneous Ca2! sparks andCa2! homeostasis in a minimal model of permeabilized ventricular myo-cytes. Am J Physiol Heart Circ Physiol 299: H1996–H2008, 2010. Firstpublished September 17, 2010; doi:10.1152/ajpheart.00293.2010.—Manyissues remain unresolved concerning how local, subcellular Ca2!

signals interact with bulk cellular concentrations to maintain ho-meostasis in health and disease. To aid in the interpretation of dataobtained in quiescent ventricular myocytes, we present here a minimalwhole cell model that accounts for both localized (subcellular) andglobal (cellular) aspects of Ca2! signaling. Using a minimal formu-lation of the distribution of local [Ca2!] associated with a largenumber of Ca2!-release sites, the model simulates both randomspontaneous Ca2! sparks and the changes in myoplasmic and sarco-plasmic reticulum (SR) [Ca2!] that result from the balance betweenstochastic release and reuptake into the SR. Ca2!-release sites arecomposed of clusters of two-state ryanodine receptors (RyRs) thatexhibit activation by local cytosolic [Ca2!] but no inactivation orregulation by luminal Ca2!. Decreasing RyR open probability in themodel causes a decrease in aggregate release flux and an increase inSR [Ca2!], regardless of whether RyR inhibition is mediated by adecrease in RyR open dwell time or an increase in RyR closed dwelltime. The same balance of stochastic release and reuptake can beachieved, however, by either high-frequency/short-duration or low-frequency/long-duration Ca2! sparks. The results are well correlatedwith recent experimental observations using pharmacological RyRinhibitors and clarify those aspects of the release-reuptake balancethat are inherent to the coupling between local and global Ca2!

signals and those aspects that depend on molecular-level details. Themodel of Ca2! sparks and homeostasis presented here can be a usefultool for understanding changes in cardiac Ca2! release resulting fromdrugs, mutations, or acquired diseases.

calcium signaling; ryanodine receptor; leak; Markov chain; tetracaine

INTRACELLULAR Ca2! is a ubiquitous biological signal thatserves diverse functions in many cell types. In individual cells,information can be conveyed by both “global,” or cell-wide,changes in [Ca2!] and by “local,” subcellular Ca2! signals.Local signals are frequently caused by the release of Ca2! fromintracellular stores, primarily the endoplasmic reticular (ER)/sarcoplasmic reticulum (SR). Local release events occurthrough closely packed clusters of release channels, inositol1,4,5-trisphosphate (IP3) receptors or ryanodine receptors(RyRs), and are observable experimentally as Ca2! sparks (7)or puffs (51). When one or several of the channels in a releasesite are open, the [Ca2!] experienced by clustered channels isdramatically different from the [Ca2!] in the bulk cytosol.

In cardiac myocytes and many other cell types, local and globalCa2! signals are closely linked to one another. Local signals

frequently form the building blocks from which global signals arebuilt (32), and, conversely, changes in bulk [Ca2!] in the myo-plasm or SR can influence the frequency, amplitude, and kineticsof local events. In ventricular myocytes, for instance, propagatingCa2! waves emerge when spontaneous Ca2! sparks trigger ad-ditional sparks in a regenerative fashion (5). Similarly, changes inchannel gating at the level of individual RyRs can immediatelyaffect the production of local events and, over time, influence bulkmyoplasmic and SR [Ca2!]. An increase in RyR opening willcause a gradual decrease in SR [Ca2!] (46), whereas inhibition ofRyR opening, over time, will lead to elevated SR [Ca2!] (21).This principle has been elegantly demonstrated by Lukyanenkoand coworkers (35) in quiescent rat ventricular myocytes treatedwith caffeine (to sensitize RyRs) or tetracaine (to inhibit RyRs).

In heart cells, changes in Ca2! signaling due to altered RyRactivity are currently receiving considerable attention because ofclose links to disease (13, 48). In particular, catecholaminergicpolymorphic ventricular tachycardia (CPVT), an inherited disor-der associated with a dramatic increase in arrhythmia risk, resultsfrom mutations in either RyRs or calsequestrin, a SR Ca2! bufferprotein that associates with and modulates RyRs (20, 33). Exper-iments in vitro have shown that CPVT-causing mutations usuallyincrease the open probability of the RyR, resulting in a hyperac-tive or “leaky” channel (12, 28, 31). Studies (1, 36) have alsosuggested that leaky RyRs are characteristic of several experimen-tal heart failure models. Thus, a quantitative understanding of howchanges in RyR gating influence local and global Ca2! responsescan provide insights into disease pathophysiology and can poten-tially suggest novel therapies.

The difference in spatial scales between local and global Ca2!

signals, however, creates significant challenges for the develop-ment of mechanistic mathematical models. In particular, gating ofRyRs depends on both myoplasmic and SR [Ca2!] (30), andconcentrations within clusters during local events can be dramat-ically different from the bulk concentrations. In addition, becauseof the relatively small number of RyRs responsible for Ca2!

sparks (6), the stochastic gating of these channels must be con-sidered when simulating local events. Previous studies (9, 27, 42,44) have used Monte Carlo simulation methods to investigate thestochastic triggering of Ca2! sparks, but these have generallytreated myoplasmic and bulk SR [Ca2!] as fixed boundary con-ditions. Conversely, modeling studies (4, 10, 40) focusing oncellular Ca2! transients have usually used representations of SRCa2! release that do not account for the stochastic nature of thelocal events. Attempting to simulate Ca2! signaling at both spatialscales simultaneously is a daunting prospect because the stochas-tic behavior of thousands of local events must be considered todetermine the effects on the bulk concentrations. As a result, onlya few studies (16, 17, 49, 50) have attempted to capture bothphenomena.

Address for reprint requests and other correspondence: G. D. Smith, Dept.of Applied Science, The College of William and Mary, McGlothlin-StreetHall, Rm. 305, Williamsburg, VA 23187 (e-mail: [email protected]).

Am J Physiol Heart Circ Physiol 299: H1996–H2008, 2010.First published September 17, 2010; doi:10.1152/ajpheart.00293.2010.

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In this report, we introduce a computationally efficientminimal model of the coupling between local and global Ca2!

signals in permeabilized ventricular myocytes. The modelaccounts for the random generation and termination of spon-taneous Ca2! sparks, the resulting changes in myoplasmic andSR [Ca2!], and the feedback of these changes on spark fre-quency. We make few assumptions about the factors influenc-ing RyR gating, which allows us to distinguish between thoseresults that are inherent to the coupling between local andglobal signals and those that are specific to the RyR gatingscheme. To validate the model, we considered experimentsrecently performed by Zima et al. (52), who showed thattetracaine, a RyR inhibitor, caused an initial suppression ofCa2! sparks followed by an increase in SR [Ca2!] and a partialrecovery of spark frequency. Surprisingly, these authors foundthat prolonged exposure to tetracaine led to an increase in Ca2!

spark duration (see Fig. 1C in Ref. 52). The simulationspresented here recapitulate these experimental results, suggest-ing that the observed increase in spark duration directly resultsfrom the interplay between RyR inhibition and the resultingchanges in SR [Ca2!]. More broadly, this model provides apowerful yet minimal framework for understanding how mu-tations, posttranslational modifications, or drugs can alter dia-stolic SR Ca2! release in ventricular myocytes.

METHODS

Model formulation. The minimal whole cell model of local andglobal Ca2! responses developed here takes into account stochasticCa2!-release site dynamics as well as the balance of release andreuptake fluxes leading to Ca2! homeostasis in quiescent ventricularmyocytes (Fig. 1). The model assumes that RyR Ca2! channels areclustered on the ER/SR membrane in release sites composed of 10–30channels. All channels in a given release site experience the samelocal [Ca2!] (myoplasmic and SR), but these “domain” [Ca2!] areheterogenous throughout the cell, i.e., different release sites experi-ence different domain [Ca2!]. Similar to previous work by Hinch andcolleagues (16, 25, 26), we assume that when the number of openchannels in a Ca2!-release site changes, the local [Ca2!] rapidlyequilibrates in a manner that balances the fluxes into and out of thespatially restricted domains. In our model formulation, a large numberof stochastically gating Ca2!-release sites are coupled to the bulkmyoplasmic and SR [Ca2!] in a manner that allows spontaneous Ca2!

sparks to change the balance of Ca2! release, or “leak,” and reuptakeby sarco(endo)plasmic reticulum Ca2!-ATPase (SERCA) pumps. Thebulk myoplasmic and SR [Ca2!] determine the relationship betweenthe number of open channels in a Ca2!-release site and the resultingdomain [Ca2!], and, consequently, changes in these bulk concentra-tions influence the dynamics of spontaneous sparks. This minimal yetrealistic representation of bidirectional coupling between local andglobal aspects of Ca2! handling is a novel aspect of our modelformulation that has not been emphasized in previous work.

Ca2!-release site model. In our model formulation, Ca2!-releasesites are composed of N coupled Markov chains representing individ-ual RyRs. For simplicity, we used a RyR model with two states,

closed (C) and open (O), but the model formulation can be generalizedfor channel models with more states (18, 19). Each RyR opens at arate that depends on the local myoplasmic domain (i.e., the diadicsubspace) [Ca2!], denoted by cmyo

d , and closes with a Ca2!-indepen-dent transition rate:

C ºkoc

kco!Cmyod "2

O (1)

where we assume cooperative Ca2! binding. kco(cmyod )2 and koc are

transition rates (in units of reciprocal time), and kco is an associationrate constant (with units of concentration"2·time"1).

We assume that the local [Ca2!] experienced by the Ca2!-regula-tory site of each channel depends only on the number of openchannels at the Ca2! release site [NO(t)] and the bulk Ca2! concen-trations (cmyo and csr), as described below. Because the channels areidentical and indistinguishable, the state space for the N channelCa2!-release site includes N ! 1 states (0 ! NO ! N), as follows:

0 ºkoc

Nkco!Cmyod, 0 "2

1 º2koc

!N"1"kco!Cmyod, 1 "2

· · · º!N"1"koc

2kco!Cmyod, N"2"2

N " 1 ºNkoc

kco!Cmyod, N"1"2

N(2)

where the states (0, 1, . . . , N " 1, N) indicate the possible numbersof open channels and cmyo

d,n is the local myoplasmic domain [Ca2!] thatapplies when there are n open channels. The infinitesimal generatormatrix, denoted by Q # (qij), that corresponds to Eq. 2 is tridiagonal:

Q ##! Nkco!cmyo

d, 0 "2 0 · · · 0 0 0

koc ! !N " 1"kco!cmyod, 1 "2 · · · 0 0 0

É É É Ì É É É0 0 0 · · · !N " 1"koc ! kco!cmyo

d, N"1"2

0 0 0 · · · 0 Nkoc !

$ (3)

Fig. 1. The minimal whole cell model represents two bulk compartments: thesarcoplasmic reticulum (SR; csr) and the myoplasm (cmyo). The membrane ofthe SR includes multiple release sites, each of which includes a local Ca2!

domain on both sides of the membrane. The domain [Ca2!] (cmyod and csr

d ) areinfluenced by fluxes that involve bulk [Ca2!] (myoplasmic and SR) and thestate of the release site. Fluxes include diffusion from myoplasmic domains tothe bulk myoplasm (Jmyo), diffusion from the bulk SR to the luminal domains(Jsr), sarco(endo)plasmic reticulum Ca2!-ATPase (SERCA) pump flux thatresequesters Ca2! into the SR (Jpump), a passive leak from the SR to themyoplasm (Jleak), and fluxes across the plasma membrane (Jpm).

H1997Ca2! SPARKS AND HOMEOSTASIS IN PERMEABILIZED MYOCYTES

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where qij is the transition rate from state i to state j and the diamonds(!) indicate a diagonal entry leading to a row sum of zero.

Once the local [Ca2!] (cmyod,n ) that apply for any given number of

open channels are specified, all of the statistical properties of theCa2!-release site model can be determined using Eq. 3. In particular,the time evolution of the probability distribution of the number ofopen channels in the release site can be found by solving the followingordinary differential equation (ODE) initial value problem:

d$

dt# $Q (4)

where the row vector ! # ($0, $1, . . . , $N), $i(t) is the probabilityof finding a release site in state i, and !(0) is the initial condition. Thenumerical solution of Eq. 4 can also be interpreted as providing theprobability distribution for the state of a release site randomly sam-pled at time t from a large population that was initially prepared witha distribution of states given by !(0).

Myoplasmic and SR domain Ca2!. As shown schematically in Fig.1, the domain [Ca2!] for each release site (cmyo

d and csrd ) are coupled

to the bulk compartments via the myoplasmic and SR fluxes (Jmyo andJsr) and coupled to one another through the release flux (Jrel). Weassume these fluxes take the following form:

Jreln # vrel%n!csr

d, n " cmyod, n " (5)

Jmyon # vmyo!cmyo

d, n " cmyo" (6)

Jsrn # vsr!csr " csr

d, n" (7)

where cmyo and csr are the bulk myoplasmic and SR concentrations,%n # n/N indicates the fraction of channels at any given Ca2!-releasesite that are open, and vrel is the maximum release rate. Because theparameter vmyo is related to the exponential time constant for thedecay of elevated myoplasmic domain [Ca2!] to the myoplasmic bulk[Ca2!] when an open release site closes, we will refer to vmyo as therate of myoplasmic domain collapse (37). Similarly, the parameter vsr

is the rate of luminal domain recovery that determines the exponentialtime constant for the relaxation of depleted junctional SR [Ca2!] asthis compartment is refilled via Ca2! translocation from the bulkSR (27).

Recall that Eqs. 2 and 3 require the specification of the local [Ca2!]that applies for any given number n of open channels (cmyo

d,n ). Assum-ing the dynamics of domain Ca2! are fast compared with the gatingof RyRs, these domain [Ca2!] are found by balancing the fluxes Jrel

n #Jmyo

n and Jsrn # Jrel

n . Solving these equations simultaneously for cmyod,n

and csrd,n yields the following:

cmyod, n #

vmyo

vmyo & vsrcmyo &

vsr

vmyo & vsrcsr (8)

csrd, n #

vmyo

vmyo & vsrcmyo &

vsr

vmyo & vsrcsr (9)

where

vmyo #%nvrelvmyo

%nvrel & vmyo, vsr #

%nvrelvsr

%nvrel & vsr, and %n #

n

N(10)

The solid circles in Fig. 2 show the myoplasmic and SR domainconcentrations given by Eqs. 8 and 9 as functions of the number ofopen channels (NO # n). Note that as the number of open channelsincreases, cmyo

d,n increases and csrd,n decreases, but always cmyo

d,n & csrd,n.

The open circles in Fig. 2 show how an increase in the bulk SR [Ca2!](csr) influences the relationship between the number of open channelsin a Ca2!-release site and the resulting cytosolic (cmyo

d,n ) and luminal(csr

d,n) domain [Ca2!]. In particular, note that an increase in the bulkSR [Ca2!] (csr) leads to an increase in cytosolic domain [Ca2!] (cmyo

d,n )provided one or more RyRs are open (n ' 1 in Eq. 8).

Conventional Monte Carlo simulation. As shown in Fig. 1, the bulkmyoplasmic and SR [Ca2!] are affected by the SERCA pump flux(Jpump), a passive leak from the SR to the myoplasm that is indepen-dent of release site activity (Jleak), as well as the fluxes from themyoplasmic domains to the bulk myoplasm (Jmyo) and from the bulkSR to the SR domains (Jsr). In a conventional Monte Carlo simulationinvolving a large number of Ca2!-release sites, concentration balanceequations for the myoplasmic and SR [Ca2!] consistent with Fig. 1are written as follows:

dcmyo

dt# Jmyo

T & Jleak " Jpump & Jpm (11)

dcsr

dt#

1

(sr!"Jsr

T " Jleak & Jpump" (12)

where Jpm is plasma membrance flux, 'sr # Vsr/Vmyo, Vmyo and Vsr

are the effective myoplasmic and SR volumes (i.e., accounting forCa2!-buffering capacity), and Jmyo

T and JsrT are total fluxes obtained

summing over all release sites. Under the assumption of fast domainCa2!, these fluxes can be expressed as a sum over all possible releasesite states as follows:

JmyoT # %

n#0

N

fnvmyoT !cmyo

d, n " cmyo" (13)

JsrT # %

n#0

N

fnvsrT!csr " csr

d, n" (14)

where fn(t) are random variables denoting the fraction of release siteswith n open channels, 0 ! fn ! 1, (n # 0

N fn # 1, cmyod,n and csr

d,n aregiven by Eqs. 8–10, and the rate constants vmyo

T and vsrT are propor-

tional to vmyo and vsr but scaled by the total number of release sites.In a conventional Monte Carlo approach to simulation of a whole cellmodel of local and global Ca2! dynamics, the ODEs for bulkmyoplasmic and SR Ca2! are integrated while bidirectionally coupled(through fn in Eqs. 11–14) to a stochastic simulation of a large finitenumber of Markov chains (Eq. 2), each one of which represents aCa2!-release site.

Accelerated simulation assuming a large population of releasesites. In our model formulation, we avoided Monte Carlo simulationof a large number of Markov chains by assuming that the number ofrelease sites (and associated domains) is large enough that the prob-ability distribution of release site states is well approximated by !(t)solving Eq. 4. In a naive application of this approach, Eqs. 11 and 12would be integrated simultaneously with Eq. 4 with the substitution of$n for fn in Eqs. 13 and 14. Unfortunately, this approach is invalidbecause the assumption of rapid Ca2! domain formation and collapseis a singular limit of the corresponding whole cell model formulationin which ODEs are used to solve for the dynamics of domain Ca2!

Fig. 2. A: myoplasmic domain [Ca2!] (cmyod,n in Eq. 8) as a function of the

number of open channels (NO # n). B: SR domain [Ca2!] (csrd,n in Eq. 9). Solid

and open circles correspond to different bulk SR [Ca2!] (csr # 342 and 1,112)M, respectively). Myoplasmic [Ca2!] is cmyo # 0.1 )M in both cases.

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(see Supplemental Material).1 Instead, our model formulation is basedon the following numerical solution of concentration balance equa-tions for the total myoplasmic (cmyo) and SR (csr) [Ca2!]:

dcmyo

dt# Jrel

T & Jleak " Jpump & Jpm (15)

dcsr

dt#

1

(sr!"Jrel

T " Jleak & Jpump" (16)

where the total release flux JrelT is given by the following:

JrelT # %

n#0

N

$n%nvrelT !csr

d, n " cmyod, n " (17)

where %n # n/N, cmyod,n and csr

d,n are given by Eqs. 8–10, $n is theprobability that a randomly sampled release site has n open channels,and, as mentioned above, ! # ($0, $1, . . . , $N) is found byintegrating Eq. 4. The total myoplasmic (cmyo) and SR (csr) [Ca2!]that solve Eqs. 15 and 16 are the following sums of the bulk anddomain concentrations weighted by effective volume ratios:

cmyo # cmyo & )myod cmyo

d (18)

csr # csr &)sr

d

(srcsr

d (19)

In these definitions, c!myod and c!sr

d are the average myoplasmic and SRdomain [Ca2!] that would be obtained upon randomly samplingrelease sites from within the cell, that is:

cmyod # %

n#0

N

$n cmyod, n (20)

csrd # %

n#0

N

$n csrd, n (21)

The effective volume ratios that appear in Eqs. 18 and 19 are given by*myo

d # Vmyod,T /Vmyo, *sr

d # Vsrd,T/Vmyo, and 'sr # Vsr/Vmyo, where

Vmyo and Vsr are the effective myoplasmic and SR volumes and Vmyod,T

and Vsrd,T are the effective volumes of the aggregated myoplasmic and

SR domains, respectively.Because the experimental observations that are of primary rele-

vance to this report involve permeabilized cells (52), the plasmamembrane flux (Jpm) was chosen to be as follows:

Jpm # kpm!cext " cmyo" (22)

where kpm was chosen to be large enough to “clamp” the bulkmyoplasmic [Ca2!] (cmyo) at the level of the extracellular bath (cext #0.1 )M in the standard parameter set). Even so, the total myoplasmic[Ca2!] (cmyo, Eq. 18) that solves Eq. 15 is not fixed, because thisconcentration includes Ca2! in the myoplasmic domains. The fluxesbetween the bulk myoplasm and bulk SR that occur in Eqs. 15 and 16include Ca2! reuptake by SERCA pumps (Jpump):

Jpump #vpump cmyo

2

kpump2 & cmyo

2 (23)

and a passive leakage flux (Jleak):

Jleak # vleak!csr " cmyo" (24)

This minimal model of the relationship between Ca2! sparks andCa2! homeostasis was implemented in Matlab (The Mathworks)running on a 1.67-GHz Power PC with 1-GB memory. The modelODEs are stiff and were integrated using Matlab’s built-in functionode15s using an adaptive time step and relative and absolute toler-ances of 10"3 and 10"6.

Summary and significance of the model. Although minimal innature, the whole cell model of local and global Ca2! signaling thatis the focus of this report accounts for the changes in myoplasmic andSR [Ca2!] mediated by the balance of stochastic release and reuptakeby the SR and the feedback of myoplasmic and SR [Ca2!] on sparkfrequency. As discussed in the Introduction, previously publishedmodels of Ca2! signaling in cardiac myocytes that include thestochastic release of SR Ca2! either have not included bidirectionalcoupling between local Ca2! release and global Ca2! homeostasis or,because of the computational challenge of the required Monte Carlosimulations, have not emphasized the phenomenon. To our knowl-edge, this is the first systematic modeling study of the relationshipbetween RyR kinetics, spontaneous and stochastic release of SRCa2!, and the resulting balance of bulk [Ca2!] in permeabilizedventricular myocytes. It is also the first model of Ca2! sparks andhomeostasis that bypasses Monte Carlo simulation by assuming botha large number of Ca2!-release sites and rapid Ca2! domain dynam-ics, resulting in a minimal formulation that facilitates parameterstudies.

The minimal whole cell model of local and global Ca2! signalingthat is the focus of this report includes N ! 3 ODEs and severalalgebraic relations. Two ODEs are concentration balance equationsfor the total myoplasmic (cmyo) and SR (csr) [Ca2!] (Eqs. 15 and 16).The additional N ! 1 ODEs (Eq. 4) account for the dynamics of alarge number of Ca2!-release sites, each composed of N two-stateRyRs (Eqs. 1 and 2). Algebraic relations include the fluxes (Eq. 17and Eqs. 22–24) that appear in the concentration balance equations aswell as the assumed relationship between myoplasmic (cmyo

d,n ) and SR(csr

d,n) domain [Ca2!] and the number of open channels (Eqs. 8–10).Note that the fluxes Jrel

T , Jpump, and Jleak are functions of cmyo and csr,which are functions of cmyo, csr, and !. The algebraic relationshipbetween these quantities is found by inverting Eqs. 18 and 19 aftersubstitution of Eqs. 8, 9, 20, and 21 (see Supplemental Material).

Although our model formulation assumes a large population ofCa2!-release sites, we do not have to specify a precise number. To seethis, note that the domain concentrations cmyo

d,n and csrd,n do not depend

on the number of release sites in the cell (M) when the rate constantsare defined by vrel # vrel

T /M, vmyo # vmyoT /M, and vsr # vsr

T/M (Eqs. 8and 9). Because the rate vrel

T that appears in Eq. 17 does not correspondto release through one release site but rather the entire population, itis convenient to specify cmyo

d,n and csrd,n using Eqs. 8–10 with the

replacement of vrelT , vmyo

T , and vsrT for vrel, vmyo, and vsr.

The minimal model of local and global Ca2! signaling includes 14parameters, far fewer than most mathematical models of Ca2! han-dling in cardiac myocytes (see the Supplemental Table in the Supple-mental Material). Some parameters [such as the effective volumeratios 'sr, *sr

d , and *myod and the SERCA pump maximum rate (vpump)

and dissociation constant (kpump)] were either chosen to be consistentwith previous work (49, 50) or do not require extensive considerationbecause model responses to changes in these parameters are obviousand intuitive. Because the ventricular myocyte is assumed to bepermeabilized, the precise value of the parameter kpm is unimportantso long as there is rapid equilibration of bulk myoplasmic Ca2! withthe extracellular [Ca2!] (cext). The assumed number of RyRs in eachrelease site (N # 10) was chosen to be consistent with estimates of thenumber of channels activated during a Ca2! spark (7, 15, 43). This isa smaller number of RyRs than previously reported in electronmicroscopic studies performed a decade ago (11, 34) but is consistentwith more recent estimates based on superresolution optical tech-niques and three-dimensional electron tomography (2, 23). The mostimportant of the model parameters [the kinetic parameters for thestochastic gating of the two-state RyR (kco and koc) and the rateconstants for Ca2! release vrel

T , myoplasmic domain collapse vmyoT , and

luminal domain recovery vsrT] are more difficult to constrain, and,

consequently, these parameters are the focus of numerous sensitivitystudies (see below).

1 Supplemental Material for this article is available online at the AmericanJournal of Physiology-Heart and Circulatory Physiology website.



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The following aspects of the model behavior suggest that ourstandard parameter set is physiologically realistic. At 100 nM cyto-solic [Ca2!] (cmyo), the average duration of a spontaneous Ca2!-release event is on the order of 20 ms, similar to the observed rise timeof Ca2! sparks (6, 7). The Ca2! spark rate with cmyo # 100 nM is0.043 sparks/s per release site (+1 spark every 23 s). Assuming20,000 release sites in a ventricular myocyte, this corresponds to 860sparks/s per cell, that is, 86 sparks/s in a fast two-dimensionalconfocal frame scan that samples 10% of the cell volume. This valueis consistent with an experimental study (3) performed in intact cellsthat reported spontaneous spark rates of 1–4 , 10"5 )m"2·ms"1,which corresponds to 30–120 sparks/s assuming a cross-sectional areaof 100 , 30 )m. Consistent with experiment, an increase in myo-plasmic [Ca2!] in the model leads to an increase in the spontaneousCa2! spark rate.

RESULTS

RyR open probability and spontaneous Ca2! sparks. Theminimal model of local and global Ca2! signaling that is thefocus of this report simulates stochastic Ca2! release byclusters of RyRs and the resulting whole cell Ca2! homeostasisin quiescent ventricular myocytes. The modeling formalism(described in Summary and significance of the model) waschosen to be as simple as possible while still provide mecha-nistic insights into the perturbation of SR Ca2! leak that resultsfrom pharmacological agents, mutations, or posttranslationalmodifications of the RyR, as may occur in disease states. Forexample, tetracaine, a potent local anesthetic that allostericallyblocks Ca2!-release channels, reduces the open probability ofRyRs in planar lipid bilayer experiments (52) by increasing themean closed dwell time of channels (21). Because the meanclosed time of the two-state RyR model is given by -C #1/kco(cmyo

d )2 (Eq. 1), we simulated the application of tetracaine topermeabilized ventricular myocytes by decreasing the rate con-stant kco, which influences the stochastic dynamics of the Ca2!-release sites (Eq. 2) in the minimal whole cell model. We wantedto understand how the simulated application of tetracaine influ-ences the dynamics of Ca2! sparks and homeostasis in thepermeabilized ventricular myocyte model.

Figure 3 shows a summary of 60 numerical calculations ofthe stationary dynamics of the minimal whole cell modelperformed using different values of the RyR Ca2! activationrate constant kco. The circles show the result of two particularsimulations: one corresponding to the standard parameter val-ues (kco # 4.5 )M"2·s"1; solid circles) and the other corre-sponding to the simulated application of tetracaine (kco # 0.5)M"2·s"1; open circles). In the latter case, the value of kco waschosen so that the single channel probablity (Popen), given bythe following:

Popen #!cmyo

d "2

!cmyod "2 & K2 where K2 #

koc

kco(25)

decreased by 88% upon the action of tetracaine, consistent withexperiments in which 0.7 mM tetracaine was applied (52).

Figure 3A shows that the simulated application of tetracaineled to increased bulk SR [Ca2!] in the whole cell model (csr #342 to 1,112 )M; compare solid and open circles). As ex-pected, steady-state bulk SR [Ca2!] increased as RyR openprobability decreased, due to a decrease in the total release flux(Eq. 17). Because SERCA pump flux (Eq. 23) is independentof bulk SR [Ca2!], maximum bulk SR [Ca2!] (csr) asymptot-

ically approaches 2.5 mM when association rate constant kco isvery small. When a nonspecific passive leak was not includedin the model, csr increased further (not shown). Results similarto those shown in Fig. 3A can be obtained without a passiveleak by extending the SERCA pump model to include bothforward and reverse modes (39).

Figure 3B shows that during the simulated application oftetracaine, the fraction of open channels in a randomly sampledrelease site (fO) was reduced by 79%, less than the reduction inthe single channel open probability given by Eq. 25 (88%).This result indicates that elevated bulk SR [Ca2!] and theinteraction between RyRs combine to attenuate the decrease inchannel activity occurring during the simulated application oftetracaine. That is, increased SR [Ca2!] increases the drivingforce during stochastic Ca2!-release events and elevates myo-plasmic domain [Ca2!] (compare solid and open circles in Fig.2B). The fraction of open channels can be calculated as fO #E[NO]/N, where E[NO] is the expected value of the number ofopen channels in a randomly sampled release site:

E&NO' # %n#0

N

n$n (26)

Figure 3, C and D, shows the frequency and duration ofspontaneous Ca2! sparks occurring in the whole cell model asa function of the parameter kco. The presence or absence ofCa2! sparks is assessed by calculating the following Ca2!

spark score:

Score #1

N

Var&NO'E&NO' (27)

where

Fig. 3. Steady-state responses of the minimal whole cell model of local andglobal Ca2! signaling as a function of the Ca2! activation rate constant (kco)in the single channel ryanodine receptor (RyR) model (Eq. 1). A: bulk SR[Ca2!] (csr, Eq. 18). B: fraction of open channels in the whole cell model (fO,solid line) as well as single channel RyR open probability (PO, Eq. 25, dashedline). C: spark frequency. D: mean spark duration. Solid circles indicatestandard parameter values; open circles indicate the reduction in kco corre-sponding to the simulated addition of tetracaine.



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Var&NO' # %n#0

N

!n " E&NO'"2$n (28)

The Ca2! spark score takes values from 0 to 1, and a score of0.2 or higher indicates robust sparks (38). The solid lines inFig. 3, C and D, show that Ca2! sparks were observed whenthe RyR Ca2! activation rate constant (kco) was between 0.04and 322 )M"2·s"1, a range spanning four orders of magnitude.

With spark initiation defined as a release site reaching athreshold number of open channels (NO # 4 to 5 transition) andspark termination defined as all channels closing (NO # 1 to 0transition), spark frequency and mean duration were calculatedusing the matrix analytic method described in Ref. 18. Thesolid and open circles in Fig. 3, C and D, show that thesimulated application of tetracaine decreased Ca2! spark fre-quency but increased mean Ca2! spark duration, consistentwith experimental observations (52). While bulk SR [Ca2!](csr) and spark frequency are monotone functions of RyR openprobability (decreasing and increasing, respectively), meanspark duration is a biphasic function of RyR open probability(first increasing and then decreasing).

Spark frequency and duration upon the application oftetracaine. Figure 4A, left, shows representative Ca2!-releaseevents exhibited by a Ca2!-release site in the standard wholecell simulation. Spontaneous Ca2! sparks were simulated us-ing Gillespie’s method (14). The parameters used correspondto the solid circles shown in Fig. 3 (kco # 4.5 )M"2·s"1) andresulted in robust Ca2! sparks (score # 0.51) for the bulkconcentrations (cmyo and csr) of the equilibrated whole cellmodel. Note the high frequency of spontaneous release events,including five sparks (NO ' 5) and a large number of smallerrelease events. These small release events, termed “Ca2!

quarks” (32), would not be detectable with standard confocal

microscopy and would therefore contribute to “invisible” SRCa2! leak (43). Figure 4A, right, shows an expanded version ofthe first spark (asterisk in Fig. 4A, left), which has a duration(17.5 ms) close to the mean spark duration with these standardparameters (18.3 ms; solid circle, Fig. 3D).

Figure 4B shows representative Ca2!-release events duringthe simulated addition of tetracaine. The parameters usedcorrespond to the open circles shown in Fig. 3 (kco # 0.5)M"2·s"1). While robust Ca2! sparks were observed (score #0.51), the simulated application of tetracaine significantly re-duced spark frequency; one spark and three quarks wereobserved. While the mean spark duration was 26.8 ms (opencircle, Fig. 3D), Fig. 4B, right, shows an expanded version ofthe observed spark, which was over 90 ms in duration. Con-sistent with experimental observations (52), such long-durationsparks are not infrequent during the simulated addition oftetracaine, despite the fact that they almost never occurred withthe standard parameter set (see below).

To confirm that this decreased spark frequency and in-creased mean spark duration are due to overloading of bulk SR[Ca2!], Fig. 4C shows a control simulation using the singlechannel RyR parameters shown in Fig. 4B (kco # 0.5)M"2·s"1) with bulk SR [Ca2!] “clamped” at the value shownin Fig. 4A (csr # 342 )M). The resulting simulation showedonly a few release events, none with more than three channelsopen (score # 0.11). We conclude that the overloading of SR[Ca2!] that occurs when RyR open probability is decreased isrequired for the presence of prolonged sparks in the whole cellmodel.

As mentioned above, the simulated addition of tetracaineresulted in Ca2! sparks whose duration tended to be longerthan that observed with the standard parameters (compare Fig.4, A, right, and B, right). To further quantify this effect, Fig. 5Ashows the numerically calculated distribution of spark dura-tions for the standard (solid line) and tetracaine (dashed line)parameter sets. While the mode of these distributions wasnearly identical (standard: 4.8 ms and tetracaine: 4.6 ms), in thecase of tetracaine the distribution extended further to the right,consistent with the higher probability of long sparks (Fig. 4B,right). Integrating the results shown in Fig. 5A led to cumula-tive probability distributions (Fig. 5B) that showed that 21.5%of the sparks in the tetracaine simulations but only 9.4% of thesparks in the standard simulations were longer than 40 ms(compare solid and open circles).

Magnitude of Ca2! release due to spontaneous sparks. Asdiscussed above, the SR [Ca2!] overload induced by tetracaineled to higher myoplasmic domain [Ca2!] (compare open andsolid circles in Fig. 2A), higher SR domain [Ca2!] (Fig. 2B),and higher release flux (Jrel

T , Eq. 17) for any given number ofopen channels. The resulting changes in the dynamics ofCa2!-mediated coupling of RyRs led to a decrease in sparkfrequency and an increase in spark duration (Fig. 3, C and D).Nevertheless, the solid and open circles in Fig. 5C show thatthe application of tetracaine decreases the aggregate releaseflux in the whole cell model. In fact, the solid line in Fig. 5Cshows that the aggregate release flux is a monotone increasingfunction of RyR open probability, despite the fact that the SR[Ca2!] is monotone decreasing (Fig. 3A).

During experimental observations of spontaneous Ca2! re-lease, small-amplitude events may not be detectable. Thus, it isof interest to dissect the aggregate release flux of the whole cell

Fig. 4. Representative Ca2! sparks in the minimal whole cell model underthree different steady-state conditions. Left, time and amplitude of Ca2! quarksand sparks exhibited by a representative release site; right, Ca2!-release events(*) in more detail. A: standard parameters corresponding to solid circles in Fig.3 (kco # 4.5 )M"2·s"1, resulting in spark score # 0.51). B: simulatedapplication of tetracaine corresponding to open circles in Fig. 3 (kco # 0.5)M"2·s"1, score # 0.59). C: control simulation with RyR parameters as in B(kco # 0.5 )M"2·s"1) and bulk SR [Ca2!] as in A (csr # 342 )M, resulting inscore # 0.11). The durations of the Ca2!-release events shown at the right are17.54 ms (A), 91.02 ms (B), and 3.76 ms (C). Fluctuations in the number ofopen channels before termination of the Ca2!-release event are always present(A–C) but are most obvious in long-duration sparks (B).



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model to determine the fraction of spontaneous release thatoccurs via release sites that have few open channels. Thedotted and dashed lines in Fig. 5C show that the release fluxmediated by release sites with one or two open channels is amonotone increasing function of the RyR Ca2! activation rateconstant kco. Figure 5D shows aggregate flux Jrel

T jointly dis-tributed with the number of open channels for both the tetra-caine and standard parameter sets. Both distributions werebimodal; a peak was observed at NO # 1 as well as NO # 6 or7. Tetracaine suppressed the proportion of Ca2! releasedthrough sites with seven or fewer open channels (NO ! 7),whereas release mediated by sites with eight or more openchannels (NO ' 8) increased slightly in the presence oftetracaine. These observations are consistent with the increasedprobability of long-duration Ca2! sparks observed upon theapplication of tetracaine (Fig. 5A).

Because the detectability of sparks recorded with fluorescentdyes is primarily determined by spark amplitude (i.e., inte-grated Ca2! release) (8, 41), Fig. 6A shows a summary ofMonte Carlo simulations analyzing how spark amplitude andduration are jointly influenced by the simulated application oftetracaine (compare solid and open circles). Here, spark eventswere defined as beginning with a NO # 0 ¡ 1 transition,whereas spark amplitude was the integrated stochastic releaseflux (Eq. 5) before spark termination via a NO # 1¡ 0transition. Figure 5, B and C, shows the cumulative probabilitydistributions of spark amplitude and duration, respectively.Sparks had both larger amplitude and extended duration whentetracaine was applied (consistent with Fig. 5A). Nevertheless,the decrease in spark frequency in the presence of tetracaine

led to an overall decrease in the aggregate release flux dis-cussed above (Fig. 5C).

Figure 6D shows the percentage of undetected spark events(and the “hidden” Ca2!-release flux mediated by undetectedsparks) as a function of a detection threshold on spark ampli-tude. The vertical dashed lines in Fig. 6, B and D, indicate asensitive detection threshold equivalent to the amount of Ca2!

released by an average quark; in the standard simulation, this isa single channel release event with a duration of 1.1 ms. Withthis sensitive detection threshold, 36% of release events are notobserved; most of these release events are quarks, brief singlechannel openings through which &1% of the stochastic Ca2!

release occurs. Because the simulated application of tetracaineled to increased SR load and greater release flux for any givennumber of open RyRs (Fig. 2), the tetracaine condition led tofewer hidden events (15%) and decreased hidden release(&0.1%). The solid and dashed lines in Fig. 6D show that thepercentage of hidden release events and hidden release fluxwere both increasing functions of detection threshold. For therange of possible detection thresholds shown, the percentage ofhidden events decreased by two- to threefold upon the appli-cation of tetracaine.

Transient effects upon the application of tetracaine. Whilethe above simulations focused on steady-state dynamics of thewhole cell model, Fig. 7 shows transient effects upon bulk SR[Ca2!] (csr), mean spark duration, and spark frequency thatoccurred during the simulated application and washout oftetracaine. Consistent with experimental observations (21, 52),the initial application of tetracaine caused spark frequency todecrease; the mean spark duration during this phase (5.0 ms)was much shorter than the baseline value (18.3 ms). However,this reduced spontaneous Ca2! release caused a slow increasein SR load that ultimately increased the mean spark duration to26.8 ms, consistent with the steady-state results (Fig. 3D).

Figure 7 also shows that upon the simulated washout oftetracaine (right arrow), there was a transient increase in sparkfrequency (maximum of +4 times the baseline value) and arapid depletion of SR [Ca2!] from elevated to baseline values.For a short period of time, the mean spark duration was quitelarge (see asterisk); however, the value attained is not relevantbecause it is greater than duration of the phase itself (400 ms).Shortly after this burst of spark activity, the mean sparkduration returned to baseline.

Model parameters and Ca2! homeostasis. Figure 8 shows asummary of 2,500 calculations of the stationary dynamics ofthe whole cell model as a function of vsr

T and vmyoT . These

parameters control the rate of “diffusion” or translocation ofCa2! between the different cellular subspaces represented inthe minimal model, for example, from the individual myoplas-mic domains (diadic subspaces) to the cytoplasm and from thebulk SR to the luminal domains (network to junctional SR).These domain rate constants also influence the strength of theCa2!-mediated coupling between RyRs and the extent of SRdomain depletion during sparks (recall Eqs. 8–10). Because vsr

T

and vmyoT are difficult to constrain via experiments, they are

good choices for a parameter study designed to determine theireffect on experimentally observable quantities such as steady-state bulk SR [Ca2!] and mean spark frequency and duration.

Figure 8A shows that bulk SR [Ca2!] is an increasingfunction of rate of myoplasmic domain collapse (vmyo

T ) and adecreasing function of the rate of SR domain recovery (vsr

T).

Fig. 5. A and B: probability density (A) and cumulative probability (B) of sparkduration for the standard (solid line) and tetracaine-based (dashed line) param-eter sets. Shown are the probabilities of observing a spark of 40 ms or longerin the standard (solid circles) and tetracaine (open circles) cases. C: aggregateflux (Jrel

T , solid line) through release sites in the whole cell model as a functionof the Ca2! activation rate constant (kco) in the single channel RyR model (Eq.1). The flux due to release sites with a small number of open channels is alsoshown (NO # 1 and and NO # 1 or 2). Open and solid circles indicate thetetracaine and standard parameter sets, respectively. D: release site flux (Jrel

T )jointly distributed with the number of open channels (NO) for the tetracaine(open bars) and standard (solid bars) parameter sets.



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When vmyoT is very large or vsr

T is very small, RyRs becomedecoupled such that most openings fail to trigger neighboringchannels, and the reduced leak causes an increase in bulk SR[Ca2!]. Figure 8, B and D, shows that the fraction of openchannels and mean spark duration were much more sensitive tovmyo

T than to vsrT. When vmyo

T is small, sparks are extremely longbecause subspace [Ca2!] remains elevated after RyRs close(cf. Refs. 27 and 37). Fast SR refilling alone is not sufficient toinduce long-duration Ca2! sparks of the sort observed upon thesimulated application of tetracaine (cf. Fig. 4B).

Robust Ca2! sparks were observed in the whole cell modeleven when the domain rate constants were ranged over severalorders of magnitude (Fig. 9A). This robust Ca2! spark behavioris a consequence of the homeostatic changes in bulk SR [Ca2!]

accounted for in our model formulation; when these simula-tions were repeated with bulk myoplasmic and SR Ca2!

concentrations fixed at baseline values (cmyo # 0.1)M and csr # 342 )M), the range of domain rate constantsleading to Ca2! sparks was considerably smaller (Fig. 9B).

Figure 10 shows stationary dynamics of the whole cellmodel as a function of bulk myoplasmic [Ca2!] (cmyo) andmaximum rate of the SERCA pump (vpump), two parametersthat can be easily manipulated in experiments. Bulk SR [Ca2!]was (as expected) an increasing function of vpump; however, theSR Ca2! load was a biphasic function of cmyo (Fig. 10A). The

Fig. 6. A: Monte Carlo sampling of blip/spark duration andamplitude (integrated Ca2! release) for the standard (solidcircles) and tetracaine-based (open circles) parameter sets.Each blip or spark was initiated by a NO # 0 ¡ 1 transition (cf.Fig. 5, where NO # 4 ¡ 5 was interpreted as a spark initiationevent). B and C: cumulative probability distribution of sparkamplitude (B) and duration (C) for the standard (solid line) andtetracaine-based (dashed line) parameter sets. The amplitude ofrelease is expressed in units of femtocoulombs under theassumption of a total myoplasmic volume of 2 , 10"5 )l and2 , 104 release sites per cell (see Supplemental Material).D: percentage of “hidden” (undetected) spark events and per-centage of Ca2!-release flux mediated by hidden sparks as afunction of the spark amplitude detection threshold.

Fig. 7. Transient effects upon bulk SR [Ca2!] (csr), mean spark duration, andspark frequency during the simulated application and washout of tetracaine(arrows). The Ca2! activation rate constant was decreased from kco # 4.5 to0.5 )M"2·s"1 (left arrow) and later restored to its original value (right arrow).*The mean spark duration was not applicable at the indicated time (see text).

Fig. 8. Effect of myoplasmic domain collapse (vmyoT ) and SR domain recovery

(vsrT) on bulk SR [Ca2!] (A), fraction of open channels (B), spark frequency (C),

and mean spark duration (D) in the minimal whole cell model. *Standardparameters. White indicates that mean spark duration was not calculatedbecause the spark score was &0.2.



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fraction of open channels and mean spark duration were bothincreasing functions of SERCA pump activity (Fig. 10, B andD). Spark frequency was not sensitive to SERCA activity butwas a rapidly increasing function of bulk myoplasmic [Ca2!](Fig. 10C). Spark duration was a biphasic function of cmyo

(Fig. 10D), consistent with the biphasic effect of this parameteron SR load (Fig. 10A). Bulk myoplasmic [Ca2!] (cmyo) signif-icantly changed the functional dependence of the stationarydynamics of the whole cell model on the RyR Ca2! activationrate constant (kco). On the other hand, when the application oftetracaine was simulated by comparing kco values that corre-spond to a decrease in RyR activity from fO # 9.6 , 10"4 to2.0 , 10"4 (Table 1), we observed increased SR load, de-creased spark frequency, and increased spark duration for cmyo

in the range of 0.1–0.5 )M.RyR inhibition mechanism and spark duration. We modeled

the action of tetracaine as a decrease in the Ca2! activation rate

constant kco, which reduces the open probability (Eq. 25) of theRyR model (Eq. 1) by increasing the mean closed dwell time,-C # 1/kco(cmyo

d )2. However, the open probability of the RyRcan also be reduced by increasing the rate constant koc, therebydecreasing the mean open dwell time (-O # 1/koc). Such achange would be analogous to the pharmacological action ofthe antiarrhythmic agent flecainide, which has been shown toreduce the dwell time of RyR open states (47).

Figure 11, C and D, shows Ca2! spark frequency and meanspark duration as a function of the RyR kinetic constants kco

and koc when these spark statistics are well defined (score '0.2). The contours dividing the plane into the areas wheresparks are present (gray) and absent (white) follow constantK # koc/kco, indicating that sparks occur provided the singlechannel RyR open probability (Eq. 25) is neither too low or toohigh. If RyR parameters were changed from the standardvalues (asterisk), the resulting change in SR [Ca2!] alsodepend only on RyR open probability (Fig. 11A). SR [Ca2!]was too high for sparks in the top left white region and too lowfor sparks in the bottom right (cf. Fig. 3A). In contrast, sparkfrequency and mean spark duration strongly depended onwhether a pharmacological perturbation of RyR kinetics isassumed to affect kco, koc, or both (diamonds). A reduction inkco (increase in -C), analogous to effect of tetracaine, led tofewer sparks but an increase in spark durations. Conversely, anincrease in koc (decrease in -O), analogous to the effect offlecainide, caused more sparks with decreased mean duration.Examples of Monte-Carlo simulations under these differentconditions are shown in Fig. 12. Similarly, we can assume thatlow-dose caffeine leads to an increase kco accompanied by asmaller decrease in koc (! symbol) (29). The model predictedthat this would cause a decrease in SR [Ca2!], an increase inspark frequency, and little change in spark duration, roughlyconsistent with the results observed by Lukyanenko et al. (35).

These results illustrate a fundamental point about the inter-play between local and global Ca2! signals during pharmaco-logical interventions designed to manipulate spontaneous cel-

Fig. 9. Spark score as a function of the rate of myoplasmic domain collapse(vmyo

T ) and SR domain recovery (vsrT ). Robust sparks correspond to a score of

.0.2. *Standard parameters. A: in the minimal whole cell model of quiescentventricular myocytes, bulk SR [Ca2!] is an increasing function of vmyo

T and adecreasing function of vsr

T (Fig. 8A) and sparks were observed for a wide rangeof values of the rate constants. B: when these calculations were repeated withbulk myoplasmic and SR [Ca2!] fixed at baseline values (cmyo # 0.1 )M andcsr # 342 )M), the range of domain rate constants leading robust sparks wasmuch smaller.

Fig. 10. Effect of cmyo and vpump on bulk SR [Ca2!] (A),fraction of open channels (B), spark frequency (C), and meanspark duration (D) in the minimal whole cell model. *Standardparameters. White indicates that mean spark duration was notcalculated because the spark score was &0.2.



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lular responses. Local (microscopic) aspects of cell response,such as spark frequency and duration, are highly dependent onthe molecular details, that is, precisely how of the kinetics ofRyR stochastic gating kinetics have been modified. Global(macroscopic) aspects of the cell response, such as steady-statebulk SR [Ca2!], are less dependent on the kinetic details of theperturbation but remain determined by equilibrium quantitiessuch as the dissociation constant for Ca2! binding to the RyR.

DISCUSSION

This report presents a minimal whole cell model of local andglobal Ca2! signals in quiescent ventricular myocytes. Themodeling formalism accounts for the effect of random sponta-neous Ca2! sparks, changes in bulk myoplasmic and SR[Ca2!] mediated by the balance of stochastic release andreuptake by the SR, and feedback of myoplasmic and SR[Ca2!] on spark frequency. The functional organization of themodel (Fig. 1) is similar to previously published Monte Carlomodels of local control (17). The assumptions made hereregarding the rapid equilibration of domain Ca2! are similar toassumptions made in previously published local control modelsthat represent the stochastic dynamics of a large number of

Ca2!-release sites (16, 25), but these previous studies have notmade a distinction between domain and bulk SR Ca2! as donein our minimal whole cell model.

To our knowledge, this is the first theoretical study of therelationship between RyR kinetics, spontaneous and stochasticCa2! release, and the resulting balance of bulk myoplasmicand SR [Ca2!] in quiescent ventricular myocytes. Because ofthe computational challenge of large-scale simulations, a tra-ditional Monte Carlo approach is not well suited to investigatethese phenomena. The whole cell modeling approach intro-duced here bypasses Monte Carlo simulation by assuming alarge number of Ca2!-release sites and rapid Ca2! domaindynamics, resulting in a minimal formulation that facilitatesparameter studies.

The minimal model of local and global Ca2! responses inquiescent ventricular myocytes presented here is able to reca-pitulate recent experiments by Zima et al. (52) showing thattetracaine, an inhibitor of RyRs, causes a transient suppressionof Ca2! sparks that is followed by an increase in bulk SR[Ca2!], partial recovery of spark frequency, and an increase inCa2! spark duration (Fig. 11). Conversely, if flecainide isassumed to decrease RyR mean open time while not affecting

Table 1. The three mechanisms of RyR inhibition have different consequences on mean spark duration and spark frequency

Standard Parameters Tetracaine (-C1) Flecainide (-O2) Dual Mechanism (-C1 and -O2)

Corresponding figure Fig. 12A Fig. 12B Fig. 12C Fig. 12Dkco, )M"2 " s"1 4.5 0.5 4.5 1.5koc, s"1 500 500 4,500 1,500Fraction of open channels 9.6 , 10"4 2.0 , 10"4 * *Score 0.51 0.59 * *Sarcoplasmic reticulum [Ca2!], )M 342 1112 * *Duration, ms 18.2 26.7 2.97 8.90Frequency, sparks/s 0.043 6.5 , 10"4 0.053 0.0059

Shown are results for the whole cell model with standard parameters and modified Ca2! activation rate constant (kco), which represents the application oftetracaine and led to an increase in ryanodine receptor (RyR) open time (cf. solid and open circles in Fig. 3, respectively). When the same level of RyR inhibitionwas modeled as a decrease in open dwell time (-O2), as may occur upon the application of flecainide, or a dual mechanism (-C1 and -O2), the mean sparkduration changed. *Results identical to the tetracaine case.

Fig. 11. Effect of rate constants kco and koc in the singlechannel RyR model (Eq. 1) on bulk SR [Ca2!] (A), fraction ofopen channels (B), spark frequency (C), and mean sparkduration (D). *Standard parameters. Open diamonds indicatevarious changes in the open and closed dwell times of the RyR(cf. tetracaine, flecainide, and dual mechanism in Table 1 andFig. 12). Open squares indicate that an increase in kco wasaccompanied by a smaller decrease in koc corresponding to alow dose of caffeine. The steady-state SR load (csr) andfraction of open channels (fO) are functions of the RyR Ca2!-binding constant K # koc/kco, which was constant along thelines parallel to the diagonal band for which spark frequencyand duration are well defined (dashed lines, score . 0.2).



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closed times, the model predicts that this drug will cause anincrease in spark frequency and decrease in spark duration,similar to recent experimental observations (24). However, itshould be noted that this study found no change in steady-stateSR [Ca2!], suggesting that the effects of flecainide on RyRgating are somewhat more complex.

More broadly, these examples illustrate how the modelprovides insights into the relationships between RyR gating,Ca2! spark characteristics, and balance of bulk myoplasmicand SR [Ca2!] (Figs. 3–5). For example, simulations showingthat mean spark duration is a biphasic function of the RyRCa2! activation rate constant kco suggest that the increase inspark duration observed after the application of tetracaine maybe concentration dependent (Fig. 3D). Despite the fact thatspark duration is biphasic, a dissection of model responsessuggests that the release flux is a monotone function of RyRopen probability. The model also predicts that tetracaine sup-presses the hidden flux mediated by Ca2!-release events belowdetection threshold more strongly than observable releaseevents (Fig. 6D). Parameter experiments exploring the effect ofdifferent mechanisms of RyR inhibition (Table 1 and Fig. 12)indicate that the whole cell Ca2! balance is largely determinedby RyR open probability (a function of K # koc/kco). Con-versely, the frequency and duration of Ca2! sparks are sensi-tive to RyR open and closed dwell times, which are determinedby kco and koc independently (see Fig. 11).

The Ca2!-release site model used here is quite simple andassumes “instantaneous mean field coupling” (38) of N # 10two-state RyRs. When model simulations were performed withrelease sites composed of larger numbers of channels, qualita-tively similar results were observed (supplemental Fig. S1).Our whole cell modeling approach can be applied with singlechannel models of arbitrary complexity, so long as the statespace of the resulting Ca2!-release site model is not so largethat integrating Eq. 4 is impractical. Our use of a two-state RyR

in this first study of the relationship between Ca2! sparks andhomeostasis allows us to focus on aspects of the cellularresponse to perturbations that are likely to be fundamental andgeneral. A more complicated RyR model would call intoquestion the specific details of how we (arbitrarily) supposepharmacological perturbations influence the kinetic constantsthat determine RyR stochastic gating, assumptions that couldeasily influence the whole cell response and our conclusions.We find it intriguing that this model of local and global Ca2!

responses in quiescent ventricular myocytes is able to repro-duce changes in spark frequency and duration caused by theapplication of both tetracaine and flecainide, despite the factthat the RyR model used does not include regulatory processessuch as Ca2!-dependent inactivation and/or sensitization bySR [Ca2!]. The effect of these well-established aspects of RyRCa2! regulation is beyond the scope of this work.

The Ca2! activation process in the RyR model is mediatedby myoplasmic domain Ca2! (cmyo

d , Eq. 9), and it would bepossible to augment the model to include the sensitization ofRyRs by SR domain Ca2! (csr

d , Eq. 9). However, the assump-tion of instantaneous coupling of RyRs (i.e., rapid equilibrationof myoplasmic and SR domain Ca2!) may not work well whenluminal Ca2! plays a role in spark termination (27, 42). Acomputational study of the contribution of luminal RyR regu-lation to the bidirectional coupling of spontaneous Ca2! re-lease and Ca2! homeostasis will likely require more subtlemathematical formulations, similar to the probability densityand moment closure techniques that accelerate “local control”simulations of high-gain graded Ca2! release in voltage-clamped cardiac myocytes (49, 50). These representations ofheterogenous domain Ca2! concentrations associated with alarge number of Ca2!-release sites remain valid even when thedynamics of SR domain Ca2! are not fast compared withchannel gating. Indeed, the model presented here can beviewed as a reduction of the moment closure formulation (50)that is valid when SR Ca2! domains rapidly equilibrate withmyoplasmic domain and bulk SR Ca2!, in which case there isa negligible variance of SR domain [Ca2!] for any givenCa2!-release site state.

In preliminary work, we studied whole cell responses usingRyR models that included both Ca2! activation and inactiva-tion by myoplasmic domain Ca2!, for example, the followingthree-state model:

C ºkoc

kco!cmyod "2

O ºkro

kor!cmyod "2

R (29)

which includes a long-lived closed state (R). As expected,release sites composed of such channels exhibit Ca2! sparksand homeostasis similar to that observed for the two-state RyRwhen the dissociation constant Kinact # kro/kor is large (seesupplemental Fig. S3). While fast Ca2! inactivation (large kro

and kor with fixed Kinact) is associated with decreased sparkduration and increased spark frequency, extremely fast Ca2!

inactivation can preclude sparks (supplemental Fig. S3, C andD). Provided Ca2! inactivation is sufficiently fast, decreasingKinact leads to increased SR Ca2! load, decreased RyR activity,increased spark frequency, and decreased spark duration (sup-plemental Fig. S3, A–D). Interestingly, decreasing the Ca2!

activation rate constant (kco) in the three-state model with Ca2!

inactivation (Eq. 29) to simulate the application of tetracaine

Fig. 12. Representative Ca2! sparks in the minimal whole cell model under thefour different steady-state conditions shown in Table 1. A and B: stochasticCa2!-release events in the whole cell model with standard parameters (A) andmodified Ca2! activation rate constant kco, which represents the application oftetracaine and led to an increase in RyR open time (B). When the same levelof RyR inhibition was modeled as a decrease in open dwell time, as may occurupon the application of flecainide (-O2; C), or a dual mechanism (-C1 and-O2; D), the mean spark duration changed.



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may lead to longer or shorter duration sparks depending onwhether the rate of Ca2! inactivation is slow or fast, respec-tively, despite the fact that neither RyR activity nor SR load arestrongly affected by the rate of Ca2! inactivation in eithercondition (see supplemental Fig. S4). This sensitivity of thestochastic dynamics of Ca2! release to the rate of Ca2!

inactivation is consistent with results obtained using Ca2!-release site models that do not account for Ca2! homeostasis(18). However, the significance of these observations is uncleargiven recent experimental results showing that, at even 50 )Mmyoplasmic [Ca2!], inactivation is unable to suppress SRCa2! release in permeabilized myocytes (45).

While the simulated spark duration histograms shown in Fig.5A are unimodal, the experimentally observed spark durationhistogram shows two peaks in the presence of tetracaine,suggesting two distinct populations of sparks (52). Zima andcoworkers (52) suggested that the prolonged sparks occurred atrelease sites with highly interconnected junctional SR and highrefilling rates. Our simulations shown in Fig. 8 address thisidea by investigating the effects of a larger domain refilling rate(vsr

T). Consistent with the hypothesis of Zima et al. (52), anincrease in vsr

T from 10 to 50 s"1 caused a slight increase inbaseline Ca2! spark duration as well as a greater percentincrease upon the application of tetracaine (57% vs. 47%, notshown). This suggests that the long sparks observed upon theapplication of tetracaine (52) may indeed be associated withfast rather than slow SR refilling.

Finally, it is intriguing that robust Ca2! sparks were ob-served in the whole cell model even when the domain rateconstants were ranged over several orders of magnitude (Fig.8). Because the range of domain rate constants leading to Ca2!

sparks is considerably smaller when bulk myoplasmic and SR[Ca2!] are fixed at baseline values (compare Fig. 9, A and B),we conclude that this robust Ca2! spark behavior is a conse-quence of homeostatic changes in bulk SR [Ca2!] accountedfor in our model formulation. The fact that the feedback ofspontaneous SR leak on spark frequency extends the regimewhere spontaneous sparks occur is potentially physiologicallyrelevant. Speaking teleologically, the homeostatic mechanismsappear to encourage SR leak mediated by Ca2! sparks anddiscourage the alternatives: SR leak mediated by quarks ortonically active release sites. These observations underscorethe importance of accounting for global Ca2! balance inmodels of localized Ca2!-release events.

ACKNOWLEDGMENTS

Some of the results of this study have previously appeared in abstractform (22).

GRANTS

This work was supported in part by National Science Foundation (NSF)Grant 0443843 (to G. D. Smith) and a Howard Hughes Medical Institute grantthrough the Undergraduate Biological Sciences Education Program to TheCollege of William and Mary. This work was performed in part usingcomputational facilities at The College of William and Mary provided with theassistance of the NSF, Virginia Port Authority, Sun Microsystems, andVirginia’s Commonwealth Technology Research Fund.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

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AJP: Heart and Circulatory Physiology — Supporting Material

Spontaneous Calcium Sparks and Calcium Homeostasis in a Minimal Model of

Permeabilized Ventricular Myocytes.

Jana M. Hartman, Eric A. Sobie, and Gregory D. Smith

Supporting Material — Figure S1

100

102

104

c srss (µ

M)

A

10 7

10 4

10 1

f O

B

10 2 100 102

10 4

10 2

100

Freq

(spa

rks/

s)

kco (µM 2s 1)

CCCC

10 2 100 102

20

60

100

Dur

atio

n (m

s)

kco (µM 2s 1)

D

Figure S1: Steady-state responses of minimal whole cell model of local and global Ca2+

signaling as a function of the Ca2+-activation rate constant (kco) when release sites arecomposed of clusters of N = 10 (blue line), N = 20 (green line), and N = 30 (red line)two-state RyRs. Mean spark duration calculated assuming a threshold for spark initiationof N/2. See caption of Fig. 3.


100

102

104

c srss (µ

M)

AAA

10 7

10 4

10 1

f O

BBB

10 2 100 102

10 4

10 2

100

Freq

(spa

rks/

s)

kco (µM 2s 1)

CCC

10 2 100 102

10

20

30D

urat

ion

(ms)

kco (µM 2s 1)

DDD

Figure S2: Effect of the bulk myoplasmic [Ca2+] (cmyo) on spontaneous sparks in the equi-librated whole cell model. The blue curve is a reproduction of the results of Fig. 3 usingthe standard value (cmyo = 0.1 µM). Increasing the [Ca2+] in the bath (cext in Eq. 22) leadsto increased cmyo (green, 0.2; red, 0.5 µM) leads to increased RyR activity (fO, panel B)and spark frequency (C). Filled and open circles indicate RyR Ca2+-activation rate constant(kco) in ‘standard’ and ‘tetracaine’ cases. When cmyo = 0.2 and 0.5 µM, the kco values arechosen to give a decrease in RyR activity—from fO = 9.6 × 10−4 (standard) to 2.0 × 10−4

(tetracaine)—-identical to that observed when cmyo = 0.1 µM (see panel B and Table 1).

Supporting Material — Figure S3K in

act (µ

M)

Csrss (µM)

A

1

10

100

340

360

380

400

fO

B

0.0006

0.0008

0.001

K inac

t (µ

M)

kor (µM 2 s 1)

Freq (sparks/s)

C

10 2 100 1021

10

100

0.01

0.1

1

kor (µM 2 s 1)

Duration (ms)

D

10 2 100 1025

10

15

20

Figure S3: Whole cell responses as a function of the dissociation constant for Ca2+ inac-tivation (Kinact =

�kro/kor) and the association rate constant for inactivating Ca2+ (kor)

when release sites are composed of 10 three-state RyRs (Eq. 29) with a closed, open, andrefractory state (C ↔ O ↔ R).


k oc (µ

M2 s

1 )

Csrss (µM)

A

0.1

1

10

500

1000

1500

2000

fO

B

0.0001

0.0005

0.001

k oc (µ

M2 s

1 )

kor (µM 2 s 1)

Freq (sparks/s)

C

10 2 100 102

1

10

0.0001

0.001

0.01

0.1

1

kor (µM 2 s 1)

Duration (ms)

D

10 2 100 1020

5

10

15

20

Figure S4: Whole cell responses as a function of the association rate constant for Ca2+

activation (kco) and inactivation (kor) when release sites are composed of 10 three-stateRyRs (Eq. 29) with a closed, open, and refractory state (C ↔ O ↔ R). The dissociationconstant for Ca2+ inactivation is fixed at Kinact =

�kro/kor = 50 µM. The solid white lines

indicate the standard parameter value for the association rate constant for Ca2+ activation(kco = 4.5 µM−2s−1); the dashed white lines designate the reduction in kco corresponding tothe simulated addition of tetracaine (kco = 0.5 µM−2s−1).

Supporting Material — Table S1

Parameter Definition Valueλsr bulk SR volume fraction 1/6Λd

srtotal luminal domain volume fraction 1/30

Λd

myototal myoplasmic domain volume fraction 1/30

vT

relrelease flux rate 10 s−1

vT

myomyoplasmic domain collapse rate 100 s−1

vT

srluminal domain recovery rate 10 s−1

vleak RyR-independent SR leak rate 0.001 s−1

vpump maximum SERCA pump rate 5 µMs−1

kpump SERCA pump dissociation constant 0.1 µMcext extracellular bath [Ca2+] for permeabilized cells 0.1 µMkpm plasma membrane rate constant 105 s−1

kco RyR Ca2+ association rate constant 4.5 µM−2s−1

koc RyR Ca2+ dissociation rate constant 500 s−1

N number of channels per release site 10–30

Table S1: Standard parameters for the minimal model of local and global Ca2+ signaling.Application of tetracaine is simulated by changing kco from 4.5 to 0.5 µM−2s−1, whichcorresponds to a 88% reduction in Popen (Eq. 25) when cd

myo= 0.1 µM (9.0× 10−4 to 7.9×

10−4). Note that the plasma membrane flux (Eq. 22) assumes a permeabilized membrane.For a non-permeabilized cell a larger value of cext is used in conjunction with a smaller valueof kpm (corresponding to extrusion of Ca2+ via ATP-dependent pumps). By identifyingJin = kpmcmyo in this case we have Jpm = Jin − kpmcmyo.

Supporting Material — Model Formulation

In our model formulation, the fluxes Jpump (Eq. 23), Jleak (Eq. 24), and Jpm (Eq. 22) arefunctions of the bulk concentrations (cmyo and csr), not the total concentrations (cmyo andcsr) defined in Eqs. 18 and 19 and simulated using Eqs. 15–16. The bulk concentrations canbe obtained from the total concentrations by inverting

cmyo = cmyo + Λd

myo

N�

n=0

�πn

�vT

myo

vTmyo

+ vTsr

cmyo +vT

sr

vTmyo

+ vTsr

csr

��(S.1)

csr = csr +Λd

sr

λsr

N�

n=0

�πn

�vT

myo

vTmyo

+ vTsr

cmyo +vT

sr

vTmyo

+ vTsr

csr

��, (S.2)

where

vmyo =γnvT

relvT

myo

γnvT

rel+ vT

myo

, vsr =γnvT

relvT

sr

γnvT

rel+ vT

sr

, and γn =n

N. (S.3)

These expressions that are derived by combining Eqs. 18 and 19 with Eqs. 8 and 9 (with thereplacement discussed in the previous paragraph) and using the definitions of c d

myoand c d

sr

(Eqs. 20 and 21). Rearranging Eqs. S.1–S.3 gives

cmyo = α11cmyo + α12csr (S.4)

csr = α21cmyo + α22csr (S.5)

where

α11 = 1 + Λd

myo

N�

n=0

�πn

vT

myo

vTmyo

+ vTsr

�(S.6)

α12 = Λd

myo

N�

n=0

�πn

vT

sr

vTmyo

+ vTsr

�(S.7)

α21 =Λd

sr

λsr

N�

n=0

�πn

vT

myo

vTmyo

+ vTsr

�(S.8)

α22 = 1 +Λd

sr

λsr

N�

n=0

�πn

vT

sr

vTmyo

+ vTsr

�. (S.9)

Using Cramer’s rule we have

cmyo =α22cmyo − α12csr

Dcsr =

α11csr − α21cmyo

D(S.10)

where D = α11α22 − α12α21. When the concentration balance equations for cmyo and csr

(Eqs. 15–16) are integrated, the fluxes Jpump (Eq. 23), Jleak (Eq. 24), and Jpm (Eq. 22) areevaluated using cmyo and csr obtained from Eq. S.10 and Eqs. S.6–S.9.

Supporting Material — Stochastic Methods

Monte Carlo simulation of Ca2+

release events

The minimal Ca2+ release site model used here (Eq. 2) is a continuous-time discrete-stateMarkov chain constructed under the assumption that RyRs are instantaneously coupledvia rapidly equilibrating myoplasmic and SR domain Ca2+ concentrations (cd, n

myoand cd, n

srin

Eqs. 8–10). When the whole cell model is at steady state, the fluxes JT

rel, Jleak, Jpump, and Jpm

appearing in Eqs. 15 and 16 are balanced and the bulk myoplasmic and SR Ca2+ concentra-tions are stationary (dcmyo/dt = 0 and dcsr/dt = 0). After equilibration of bulk myoplasmicand SR Ca2+ in the whole cell model, the Markov chain representing the stochastic dynamicsof spontaneous Ca2+ release of any individual Ca2+ release site is time-homogeneous, i.e.,the transition rates of the Q-matrix (Eq. 3) are constant.

Stochastic Ca2+ release events (i.e., spontaneous Ca2+ quarks and sparks) of the equili-brated whole cell model were simulated using Gillespie’s algorithm (Gillespie 1976), a nu-merical method with no intrinsic time step. Given the current state i, Gillespie’s algorithmrequires the non-zero off-diagonal elements of the ith row of the Q-matrix, that is, the ratesof all allowed transitions from state i to state j. The dwell time in state i is determinedby generating an exponentially distributed pseudorandom number with mean 1/

�j �=i

qij.When there is more than one destination state j, a pseudorandom number Y is producedthat is uniformly distributed on an interval of length

�j �=i

qij. This interval is partitionedinto sub-intervals of length qi1, qi2, · · · , qiM (not including qii) and the i→ j transition occursif Y falls on the partition associated with j. Trajectories of the Ca2+ release site’s state (i.e.,the number of open channels NO as a function of time) were produced by repeating thesesteps (e.g., Fig. 4).

The stochastic method described above produces an instantiation of the stochastic processdescribed by the Markov chain Ca2+ release site model. When individual trajectories of Ca2+

release events are not required, but rather the dynamics of the probability distribution π(t),Eq. 4 can be solved using conventional methods for integrating ODEs (e.g., the first-orderEuler’s method or the fourth-order Runge-Kutta method). See references (Stewart 1994,Smith 2002, Groff et al. 2008) for further reading on the numerical solution of Markovchains.

Matrix analytic calculation of spark duration

We define spark duration as the length of a time interval beginning when a specified number(κ) of channels open (NO = κ− 1→ κ) and ending when all channels close (NO = 1→ 0).In Fig. 3D and Fig. 5A and B, we choose κ = 5 to exclude small puff/spark events.

While the distribution of spark durations may be estimated through Monte Carlo sim-ulation as described above, it is equivalent and far more efficient to directly calculate theabsorption time of the terminating Markov chain (DeRemigio and Smith 2005, Groff and

Smith 2008),

QA =

�01×1 01×N

tN×1 TN×N

�

=

0 0 0 · · · 0 0 0koc � (N − 1)kco(cd, 1

myo)2 · · · 0 0 0

......

.... . .

......

...0 0 0 · · · (N − 1)koc � kco(cd, N−1

myo)2

0 0 0 · · · 0 Nkoc �

.

This N +1×N +1 generator matrix QA is identical to Q (Eq. 3) except that the first row iszeroed out, because it corresponds to the absorbing state where all channels are closed (NO= 0). The N×N matrix T (the lower right block of Q) defines the dynamics of the transientstates where 1 ≤ NO ≤ N . The N×1 column vector t = (koc, 0, · · · , 0)T gives the transitionrates from the transient to absorbing state(s). The vector t has only one non-zero element,because the only way to enter the state NO = 0 is via a transition from NO = 1 → 0 at ratekoc. The diamonds (�) indicate a diagonal entry leading to a row sum of zero; consequently,t = −Te where e is a N × 1 column vector of ones.

The statistics of spark duration can be found by analyzing the the absorbing Markovchain QA. For example, the matrix T is invertible and the ij entry of the opposite of thisinverse, [−T−1]ij, is the expected amount of time spent in state j before absorption, given theinitial state of the chain was NO = i (Asmussen and Bladt 1997, Latouche and Ramaswami1999). Furthermore, if φ is a vector of initial probabilities of the Ca2+ release site being instates NO = 1 through N , we can write Xφ to indicate the random variable giving the timeuntil absorption into state NO = 0. The probability density function for spark duration isphase-type distributed (Neuts 1981) with parameters φ and T , that is,

fXφ(x) = φexT t, (S.11)

where exT is a matrix exponential. In Fig. 5A, where spark initiation is defined as a NO =4 → 5 transition, the probability density of spark durations is calculated using Eq. S.11 withinitial distribution φ1×N+1 = (01×5, 1,01×N−5). The cumulative probability distributionsshown in Fig. 5A are given by

FXφ(x) = 1− φexT e, (S.12)

which can be derived by integrating Eq. S.11. In Fig. 3D, the expected amount of time untilabsorption into state NO = 0 is the positive scalar

E[Xφ] =

� ∞

0

xφexT tdx = −φT−1e. (S.13)

Matrix analytic calculation of spark frequency

Our analysis of spark frequency utilized a matrix analytic formula presented in AppendixB of Groff and Smith 2008. The formula is derived by noting that spark termination is a

NO = 1 → 0 transition and, consequently, the initiation of all sparks is preceded by anNO = 0→ 1 transition with steady-state probability flux,

α = π0q01 = π0Nkco(cmyo)2, (S.14)

where q01 is an element of the release site generator matrix Q (Eq. 3). In this expression,π0 is the element corresponding to NO = 0 in the stationary distribution π that satisfiesπQ = 0 subject to πe = 1 (i.e., the steady-state of Eq. 4). However, not all NO = 0 → 1transitions indicate the beginning of spark events, because the Markov process may proceedto the state NO = κ = 5 (spark initiation), or it may return to NO = 0 having never madea NO = κ− 1→ κ transition (NO = 4→ 5). To account for these possibilities, we partitionthe generator matrix as follows,

Q =

q00 q01 0 0 0q10 q11 q1� 0 00 q�1 Q�� q�5 00 0 q5� q55 q5•

0 0 0 q•5 Q••

, (S.15)

where the indices correspond to single states (0, 1, and 5) or sets of states (�, NO = 2, 3, or4; •, NO ≥ 6); and the square matrices on the diagonal are of size 1 (q00, q11, q55), 3 (Q��)and N − 5 (Q••); and the remaining blocks are row (q1�, , q�5, q5•) or column (q�1, q5�, q•5)vectors, each with one nonzero element. The probability flux of spark initiation (i.e., thespark frequency) is given by γ5 in the vector-matrix product

(γ0 γ5) = (α 0�)

�−

�q11 q1�

q�1 Q��

�−1 �q10 00 q�5

��.

where 0� is a 1 × 3 row vector of zeros, α is given by Eq. S.14. Expanding this expressiongives

(γ0 γκ) = (α 0 0 0)

−

q11 q12 0 0q21 q22 q23 00 q32 q33 q34

0 0 q43 q44

−1

q10 00 00 00 q45

where the term in brackets can be identified as a 4×2 stochastic matrix (a matrix with non-negative elements and rows that sum to one) giving the probability of first arriving in NO = 0or 5 (the two columns) for initial states NO = 1, 2, 3, 4 (the four rows). Multiplication by(α 0 0 0) results in a scaled version of the first row of this stochastic matrix and γ5 is thesecond element of the resulting 1× 2 row vector.

Spark amplitude

Note that total release flux JT

rel(Eq. 17) has units of µM s−1 and corresponds to release

through M release sites, where M is a large but unspecified number. Consistent with Eq. 5,the stochastic release through an individual release site is given by

Jrel = vrel

NO

N

�cd

sr− cd

myo

�,

where NO, cd

sr, and cd

myoare random functions of time that are piecewise constant and right

continuous. Consistent with Eqs. 8–10, the domain concentrations cd

srand cd

myoare the

following instantaneous functions of NO

cd

myo=

vT

myo

vTmyo

+ vTsr

cmyo +vT

sr

vTmyo

+ vTsr

csr (S.16)

cd

sr=

vT

myo

vTmyo

+ vTsr

cmyo +vT

sr

vTmyo

+ vTsr

csr (S.17)

where

vT

myo=

γvT

relvT

myo

γvT

rel+ vT

myo

, vT

sr=

γvT

relvT

sr

γvT

rel+ vT

sr

, and γ =NON

. (S.18)

While cd

srand cd

myodo not depend on the number of release sites in the cell (M), the stochastic

Ca2+ release flux Jrel is inversely proportional to M given the relationship between theunitary and total release rate constants, vrel = vT

rel/M .

As discussed above, the spark duration Xφ with initial probabilities φ is phase-typedistributed PH (φ, T ) (Latouche and Ramaswami 1999, Neuts 1981). The amplitude of anindividual spark is a random variable (Y ) corresponding to the integrated release flux

Y =

�Xφ

0

Jrel (NO(t)) dt, (S.19)

that is, the release accumulated between spark initiation (t = 0) and spark termination (t =Xφ). It can be shown (Kulkarni 1989) that Y has a phase type distribution PH (φ, R) whereR = (rij) and rij = tij/Jrel(i). Thus, the probability density and cumulative probabilitydistribution of spark amplitudes are given by Eqs. S.11 and S.12 with the substitution ofR for T . Unfortunately, there is no closed form matrix analytic expression for the jointdistribution for spark duration (X) and amplitude (Y ) (Kulkarni 1989). For this reasonMonte Carlo simulation was performed to generate the scatter plot in Fig. 6A (500 trials)and the marginal distributions of spark amplitude and duration (Fig. 6B and C; 10,000trials).

By considering the relationship between the the total release flux JT

reland the total

myoplasmic [Ca2+] concentration in Eq. 15, we see that integrating the single site release fluxJrel = JT

rel/M over the duration of a spark event will result in a measure of spark amplitude

that has units of µM. Furthermore, the reference volume for this concentration increase is thebulk myoplasm and all the myoplasmic domains in aggregate. Assuming a total myoplasmicvolume of 2 × 10−5 µL (Williams et al. 2007) and M = 2 × 104 release sites per cell, anintegrated total release flux of 1 µM corresponds to 10−6 M · 2× 10−11 L ÷ 2× 104 = 10−21

moles Ca2+ ions (i.e., about 600). This is equivalent to 2 · 96485 C/mole · 10−21 mole =1.93 × 10−16 C or about 0.2 femtocoulomb (fC). The mean spark duration of 20 ms inthe standard condition (Fig. 6A, filled circles) results in an integrated release amplitude ofapproximately Y = 20 µM or 4 fC (Eq. S.19).

References

1. Stewart, W. 1994. Introduction to the Numerical Solution of Markov Chains. Prince-ton University Press, Princeton.

2. Smith, G. D. 2002. Modeling the stochastic gating of ion channels. In ComputationalCell Biology. C. Fall, E. Marland, J. Wagner, and J. Tyson, editors. Springer-Verlag.291–325.

3. Groff, J. R., H. DeRemigio, and G. D. Smith. 2010. Markov chain models of ionchannels and the collective gating of Ca2+ release sites. In Stochastic Methods inNeuroscience. C. Laing, and L. Gabriel, editors. Oxford University Press. 29–64.

4. Asmussen, S., and M. Bladt. 1997. Renewal theory and queueing algorithms formatrix-exponential distributions. In Matrix-analytic methods in stochastic models.S. Chakravarthy, and A. Alfa, editors. Marcel Dekker, Inc., New York. 313–341.

5. Latouche, G., and V. Ramaswami. 1999. Introduction to Matrix Analytic Methods inStochastic Modeling. Society for Industrial and Applied Mathematics, Philadelphia,PA.

6. Neuts, M. F. 1981. Matrix-Geometric Solutions in Stochastic Models: an AlgorthmicApproach. Dover Publications, Inc.

7. Kulkarni, V. G. 1989. A new class of multivariate phase type distributions. OperationsResearch. 37:151–158.

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