Clusters of calcium release channels harness the Isingphase transition to confine their elementaryintracellular signalsAnna V. Maltseva,1,2, Victor A. Maltsevb,1, and Michael D. Sternb,1,3

aDepartment of Mathematics, University of Bristol, Bristol BS8 1TH, United Kingdom; and bLaboratory of Cardiovascular Science, National Institute onAging, NIH, Baltimore, MD 21224

Edited by Richard W. Aldrich, The University of Texas at Austin, Austin, TX, and approved May 31, 2017 (received for review January 26, 2017)

Intracellular Ca signals represent a universal mechanism of cell function.Messages carried by Ca are local, rapid, and powerful enough to bedelivered over the thermal noise. A higher signal-to-noise ratio isachieved by a cooperative action of Ca release channels such asIP3 receptors or ryanodine receptors arranged in clusters (release units)containing a few to several hundred release channels. The channelssynchronize their openings via Ca-induced Ca release, generating high-amplitude local Ca signals known as puffs in neurons and sparks inmuscle cells. Despite the positive feedback nature of the activation, Casignals are strictly confined in time and space by an unexplainedtermination mechanism. Here we show that the collective transition ofrelease channels from an open to a closed state is identical to the phasetransition associated with the reversal of magnetic field in an Isingferromagnet. Our simple quantitative criterion closely predicts the Castore depletion level required for spark termination for each clustersize. We further formulate exact requirements that a cluster of releasechannels should satisfy in any cell type for our mapping to the Isingmodel and the associated formula to remain valid. Thus, we describedeterministically the behavior of a system on a coarser scale (releaseunit) that is randomon a finer scale (release channels), bridging the gapbetween scales. Our results provide exact mapping of a nanoscale bi-ological signaling model to an interacting particle system in statisticalphysics, making the extensive mathematical apparatus available toquantitative biology.

calcium | heart | excitation–contraction coupling | ryanodine receptor |calcium spark

Clusters of release channels play a fundamental role in bio-logical systems as elementary signal generating, processing,

and amplifying units, like transistors in electronics. The timescale for termination of the rapid high-amplitude signals gener-ated by such units, together with scales of other elementarycellular processes such as driving molecular motors and re-fractory periods of action potentials, determines the emergentscale of crucial life functions such as heartbeat, muscle motion,and brain activities.As molecules interact the signal termination is an emerging

property of the channel cluster, rather than a property of an in-dividual channel. This nanoscale system of interacting moleculesthat generates elementary biological signals is particularly interestingbecause this is where physics meets biology. Despite extensiveexperimental and numerical studies on the nanoscale, includingstructures of individual molecules and their locations (1–4), theuniversal links of synchronized intracellular signals to physics remainmainly unknown and conceptual understanding remains elusive.The crucial importance of local Ca signaling amplified by Ca-

induced Ca release (CICR) (5) for regulation of cardiac musclecontraction was theoretically predicted in 1992 [local controltheory (6)] and further validated by the experimental discoveryof Ca sparks (7). Sparks are generated by release channel clus-ters (Ca release units, CRUs) in the form of localized Ca releasesfrom junctional sarcoplasmic reticulum (SR). The sparks arestrictly confined in time and space to roughly 20–40 ms by 2 μm

by a powerful termination mechanism. Stochastic attrition, localSR depletion, coupled gating, channel inactivation, and luminalregulation have been suggested as potential mechanisms of sparktermination (for a review see ref. 8). Since 2002 the Sobie et al.(9) model of the effect of mass action (decreasing Ca flux withCa spark duration and loss of Ca in the SR) has been a commondriving element in virtually all of the spark models and central tothe termination mechanism via interrupting CICR (10) in “per-nicious attrition” (11) or “induction decay” phenomenon (12)(i.e., the process opposed to CICR that facilitates channel clo-sure within the CRU). The model formulations have improvedfurther via superresolution modeling of calcium release (4), butthe only “feature” that seems to have disappeared from allcurrent models is the need for ryanodine receptor (RyR) in-activation. Thus, the current consensus in the field on the sparktermination problem is that SR depletion decreases the RyRchannel current, and therefore RyR interactions, to the pointthat CICR cannot be sustained any longer, culminating in sparktermination. At the same time, previous studies have not pro-vided a mathematical understanding of this emerging behavior.Here we apply statistical mechanics to describe quantitatively and

deterministically the collective behavior of release channels withinCRUs during spark termination. Specifically, we show that thechannel synchronization that corresponds to termination of signalsgenerated by a CRU is described by the same equation (and thusworks on the same principle) as the phase transition associated withthe change of magnetic field in an Ising ferromagnet.

Significance

Living organisms are built on a hierarchy of levels, startingfrom macromolecules and clusters of molecules, to organelles,cells, and tissues, with interacting organs finally forming theentire organism. At the lowest of these levels life depends onindividual molecules synchronizing their states to generaterobust intracellular signals over the thermal noise. Here weapproach the problem via statistical mechanics to describequantitatively and deterministically this first emerging level oflife. We discover that the synchronization that corresponds totermination of local Ca signals generated by clusters of Ca re-lease channels is governed by the same equations as the phasetransition associated with the reversal of magnetic field in aclassical Ising ferromagnet.

Author contributions: A.V.M., V.A.M., and M.D.S. designed research, performed research,analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1A.V.M., V.A.M., and M.D.S. contributed equally to this work.2Present address: School of Mathematical Sciences, Queen Mary University of London,London E1 4NS, United Kingdom.

3To whom correspondence should be addressed. Email: SternMi@mail.nih.gov.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1701409114/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.1701409114 PNAS | July 18, 2017 | vol. 114 | no. 29 | 7525–7530

BIOPH

YSICSAND

COMPU

TATIONALBIOLO

GY

Dow

nloa

ded

by g

uest

on

Dec

embe

r 31

, 202

0

The Ising model describes ferromagnetism using a square gridof interacting binary (±1) random variables representing atomicspin. An analytic solution of the 2D case of this model was foundby Onsager (13). The spins “want” to be aligned, and in the ab-sence of magnetic field they can synchronize to either a +1 or −1state. The application of a magnetic field breaks this symmetry,and the spins synchronize according to the sign of the magneticfield culminating in the phase transition. Thus, the magnetic fieldindicates which state the spins “prefer.” As described above, Carelease channels also “want” to be aligned in either a closed or openstate. An open release channel tries to open its neighbors via CICR(5), and similarly a closed release channel favors a closed state of itsneighbors (12).Following this analogy, we construct an exact mapping between

two systems in two dimensions: a system of release channels withina CRU and the Ising model of interacting spins within a ferro-magnet. Our mapping allows us to describe both systems by thesame mathematical formulations. We take advantage of the ex-tensive research on Ising models (since 1920; ref. 14); specifically,our criterion for spark termination in a CRU is the same as thewell-known fact that the Ising model undergoes a phase transitionwhen the external magnetic field reverses. This criterion closelypredicts the depletion level required for spark termination foreach CRU size. We demonstrate this mechanism using numericalmodel simulations of Ca sparks over a wide range of CRU sizesfrom 25 to 169 release channels.

ResultsWe linked a system of release channels within a CRU and theIsing model of interacting spins within a ferromagnet via thesame mathematical formulations. Here we formulate four gen-eral conditions that a system of release channels should satisfyfor our mapping between the two systems to hold. When theseconditions are satisfied the RyR interactions are quantitativelyidentical to the interactions in a carefully constructed version ofthe Ising model (Methods):

i) The channels are arranged in a lattice structure.ii) Each channel can be in an open or closed state, correspond-

ing to the plus and minus states of the Ising model.iii) In the absence of interactions, the Ca profile resulting from

one channel opening is roughly stable in time, correspondingto time invariance of channel interactions (achieved, e.g., bythe joint action of buffer and diffusion out of the channelcluster), and roughly the same for any channel in the cluster,corresponding to translation invariance.

iv) The channel opening rate as a function of Ca can be de-scribed by an exponential and the closing rate by a constant.

We vividly demonstrate the above principles by mapping arecent numerical model of sparks in cardiac cells (10) (referredhereafter as the Stern model) (Fig. 1) to the Ising model. RyRsin cardiac cells are indeed positioned in a crystal-like square gridand generate sparks by their synchronous opening (Movies S1and S2, for different grid sizes). The closing rate of RyR is shownto be approximately constant in experimental studies (12) and ithas been set to a constant in the original Stern model (10). RyRopening rate is described by an exponential λeγ[Ca] (red line inFig. 2A), and Ca profiles are roughly stable in time (Fig. 2B).Please note that the exponential in Fig. 2A fits experimental datawell only in our range of interest from 10 to 100 μM (sparktermination range); beyond 100 μM the data points show satu-ration, but at the low resting Ca of 0.1 μM the opening rate isexpected to be substantially lower than that approximated by λ(to match the known resting spark rate). To make sure that ourIsing model operates within the realistic exponential fit, we es-timated [Ca] at channels in closed states (Fig. 1C, Bottom) using

Time, ms

0

20

40

60

80

100

120

0 10 20 30 40 50 60 70 80

B

C

sRyR nepo 77 ,sm 47.5sRyR nepo 7 ,sm 89.2

22.70 ms, 32 open RyRs 31.36 ms, 13 open RyRs 37.62 ms, 4 open RyRs

sRyR nepo 72 ,sm 46.3sRyR nepo 1 ,sm 83.2

48.78 ms, 0 open RyRs

Spark Ac�va�on

Spark Termina�on

Number ofopen RyRs

SR [Ca] (CaSR), mM

0102030405060708090

0 10 20 30 40 50 60 70 80

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50 60 70 80

Termina�on

A

[Ca] at closed channels, mM

Maximum

Mean

Minimum

Cell membrane

L type Ca channel

Ca

Junctional SR

Ca-induced-Ca-release15 nm dyadic cle�

100-300 nm Ca release unit size

CaSR

L type Ca channels

Interac�ngrelease channelsin a square grid

30 nm

Release channels

Ca CaCaCa

Free-SR & Cytosol

Fig. 1. The Stern numerical model describes collective behavior of RyRensemble during spark activation and termination. (A) A schematic rep-resentation of a CRU in cardiac cells. (B) An example of Ca spark gener-ated by a CRU featuring 9 × 9 RyRs separated by 30 nm. The sequence ofthe RyR ensemble states is shown along with their instant local [Ca] on agrid with 10 × 10 nm computational voxels in the cleft. White up-arrowsindicate open channels. Green down-arrows indicate closed channels (seealso Movies S1 and S2, for different grid sizes). [Ca] is coded by red shades,saturated at 30 μM. (C ) Dynamics of open number of RyRs and SR [Ca](CaSR) during the spark. Termination span of ∼40 ms is shown by a blueshadow. (C, Bottom) At each time sample, we collected information about[Ca] at all closed channels (i.e., ready to open) and report here respectivemaximum, mean, and minimum values.

7526 | www.pnas.org/cgi/doi/10.1073/pnas.1701409114 Maltsev et al.

Dow

nloa

ded

by g

uest

on

Dec

embe

r 31

, 202

0

numerical simulations and found that on average this [Ca] staysbelow 80 μM (i.e., within our fit in Fig. 2A).We next read off the Ca profile ψ that results from the opening

of one RyR (Fig. 2C) and we let the interaction profile ϕ(r) =ψ(Ur)/ψ(U) in our version of the Ising model be a normalized andscaled version of ψ (U is the distance between channels). The Cadiffusion out of the CRU corresponds to imposing a negativeboundary condition. Using iv we can construct a dynamic for theIsing model that satisfies detailed balance. This means that theMarkov chain is reversible and implies convergence to the explicitlyknown equilibrium (Gibbs) measure given in Eq. 2 (Methods).From our exact mapping we derive a formula for the analog of

the magnetic field h (Methods), which reflects an imbalance inthe probability and indicates whether release channels “prefer”an open or a closed state:

h=12β

ln�λ

C

�+ 2π

Zs>.5

ϕðsÞds.

Here β = γψ is the analog of inverse temperature in the classicalIsing model, C = 117 s−1 is the channel closing rate, and λ and γdetermine the opening rate as λeγ[Ca] (Fig. 2A). The transition isexpected at h just above 0, equivalent to reversal of polarity inferromagnetism. Because C, λ, γ are constants, but ϕ (and ψ)and its integral over neighboring channels depend on the grid sizeand CaSR ([Ca] in SR), we explored h vs. these two parametersand found that h reverses at lower CaSR levels in larger grids(Fig. 2D).

We tested Ising model predictions in in silico experimentsusing the Stern model, in which CaSR was clamped at variouslevels. As CaSR increases from 0.1 to 0.15 mM the system un-dergoes a sharp transition from terminating sparks to sparks thatdo not terminate (Fig. 3A). At higher CaSR levels, the local [Ca]remains high, despite a negative boundary effect (Fig. 3B).We further examined statistics of the termination time transi-

tion for different grid sizes and compared them to the Ising modelprediction. As CaSR increases, the termination time indeed un-dergoes a sharp increase (>10,000 ms) due to a phase transitionfor all grid sizes tested (Fig. 4A). The phase transition onsets arewell-described by an exponential function (Fig. 4B) with steeperexponential rise for larger grids (Fig. 4C). The CaSR at which hreverses corresponds closely to the onset of phase transition foreach grid size, specifically about 30% increase in termination timein our numerical simulations (Fig. 4D). This criterion remainseffective for physiological conditions, when CaSR is not fixed.Namely, if a spark reaches the h reversal it terminates; otherwise,it becomes metastable (Fig. 4 E and F and Movies S3 and S4).

DiscussionIn the present study we approach the long-standing problem of Caspark termination by an unexpected (for biologists and physiolo-gists) application of statistical mechanics to the nanoscale biological

A

C D

B

Fig. 2. Construction of an exact mapping between a CRU described by theStern model and the Ising model of interacting spins. (A) The exponentialrelation of RyR opening rate vs. [Ca] in the cleft. All previous models fit apower function to original data obtained in lipid bilayers. Here we fit anexponential (red line) to the same data points (original data and power fitare reproduced from ref. 12, with permission from Elsevier; www.sciencedirect.com/science/journal/00222828). Thus, we replaced the quadratic opening ratein the original Stern model with the exponential opening rate from this fit. (B)Representative [Ca](t) when one RyR is open in the center of the grid: at theopen RyR and its closest neighbor. (C) A steady-state spatial [Ca] profile whenone RyR is open in the center of a 9 × 9 grid (similar profiles for other gridsizes are not shown). (D) Plots of h as a function of CaSR for five different gridsizes. (D, Inset) The Ising model predicts phase transition as h reverses atdifferent CaSR for each grid size; the transition requires lower CaSR levels forlarger grids.

# op

en R

yRs(

CaSR

clam

p)

100

00 80

Time, ms4020 60

StableSparksCaSR0.025..0.125 mM

Phase transi�on

Metastable sparks CaSR 0.15..0.25 mM

50

0.9 ms 20.3 ms 40.1 ms 62.8 ms

A

B

0.8 ms 20.4 ms 40.2 ms 60.1 ms

CaSR=0.1 mM (Below h reversal, terminated spark)

CaSR=0.15 mM (Above h reversal, non-terminated spark)

Fig. 3. In silico CaSR clamp experiments validating Ising model predictionsfor spark termination. (A) Evolution of a 9 × 9 RyR ensemble at various CaSRlevels after all RyRs are set in the open state at time 0. Sparks do not ter-minate at CaSR above 0.15 mM. The sharp transition in the numerical modelbehavior is in line with the Ising model prediction of the phase transitionabove 0.12 mM for the 9 × 9 grid. (B) [Ca] profiles in the dyadic cleft forterminated and nonterminated sparks at CaSR levels higher and lower thanthe phase transition. [Ca] is coded by red shades, saturating at 50 μM.

Maltsev et al. PNAS | July 18, 2017 | vol. 114 | no. 29 | 7527

BIOPH

YSICSAND

COMPU

TATIONALBIOLO

GY

Dow

nloa

ded

by g

uest

on

Dec

embe

r 31

, 202

0

signaling. We discovered that the spark termination process ismathematically isomorphic (governed by the same equations) tothe phase transition associated with the reversal of magnetic fieldin a classical Ising ferromagnet. Because this phase transition inferromagnets is well-understood, we used this knowledge to for-mulate a criterion for the phase transition determining the sparktermination in the CRU system. Our numerical model simulationshave validated the theoretical prediction of the spark termination:They clearly show that sparks terminate when Ca SR depletionreaches the predicted level of the phase transition for each CRUsize. Conversely, sparks do not terminate when SR depletion doesnot reach the phase transition level.Abnormal sparks that do not promptly terminate [dubbed meta-

stable sparks (10), embers (15), or “leaky” releases (16)] have beengenerated by numerical model simulations (10, 17, 18) and observedexperimentally (19). In cardiac muscle, failure of Ca release to ter-minate leads to explosive Ca waves, which cause life-threateningarrhythmia (20, 21). Our finding of the release termination failureat higher CaSR levels (Figs. 3 and 4 and Movie S4) is in line with

experimental observation of long-lasting Ca releases (19) and Cawaves (22) under Ca overload or rapid intra-SR Ca diffusion andreuptake (10, 18), facilitating CICR among RyRs and CRUs. Recentnumerical studies demonstrated that conditions favoring long-lastingsparks actually include a rather complex (and often counterintuitive)interplay of SR Ca loading, number of functional RyRs, and RyRgating kinetics (10, 17, 18). These conditions can now be recast andexplained in the terms of the phase transition criterion formulated inthe present study. Thus, our approach could be helpful in effectivepredicting and directing drug actions to avoid the metastable sparkregime and to normalize cardiac rhythm.One important immediate advantage of the CRU to Ising mapping

is that it provides a much more efficient computational means forevaluating the behavior of models of Ca release channel clusters. In-deed, although the Markov chain formulation has an analytic solutionfor its steady state, the number of states is very large for a reasonablesize cluster of release channels. For example, the number of states is2̂ (number of channels), that is, 2169 ∼ 1051 (for an RyR cluster of13 × 13). Thus, computing the dynamics for all states in the fullMarkovian representation using the analytic solution to Markov ma-trix involves taking exponentials of huge matrices, which is impractical.Moreover, the full CRUmodel also includes the dynamics of diffusionof Ca within the cleft, which is not explicitly represented in theMarkovmodel. The only practical way to study such a complex system is bynumerical simulations (10, 12), which still require many hours ofcomputing on powerful computer clusters. The analytic representationusing statistical mechanics suggested in the present study is muchmorecompact and efficient. It allows the evaluation of the model behaviorwithin milliseconds of computing time for a given channel interactionprofile and CaSR level. For example, our rough evaluation of phasetransitions for five grid sizes required 23,300 simulation runs (Fig. 4 BandD, red symbols), whereas the same transitions were found virtuallyinstantly by using Eq. 10 (Fig. 4D, blue symbols).Our study extends the domain of applicability of statistical me-

chanics, which traditionally describes systems with a numerous (orinfinite) number of elements. Here we show that this relatively smallbiological system with as few as 25 molecules is able to functionusing the smoothed-out phase transition associated with finite sys-tems (Fig. 4 and Movie S2). Although larger clusters, such as of169 molecules, generate stronger local signals, they are “harder” toterminate (i.e., lower CaSR levels are required, Fig. 4A). Super-resolution imaging data on CRU ultrastructure show that actualRyR clusters in cardiac cells exhibit complex shapes and varioussizes, with some CRUs being an incompletely filled grid of channels(3, 4) that could help reach a balance between signal strength andtermination. In cardiac pacemaker cells, where diastolic Ca releasesare synchronized via local propagation (23, 24), bigger RyR clustersare mixed with smaller “connecting” clusters (25). The Ising modelwith some sites missing would be a dilute Ising model, which iscurrently an object of active research in physics (26).The equivalence of an Ising and CRU model can tell us the

limiting probability of any configuration of open/closed releasechannels. It will be given by the equilibrium measure for the cor-responding Ising model and will depend only on the length of thecontour. With this tool one can quantify the extent of terminationfailure in terms of the equilibrium measure. Such an application ofinteracting particle systems bridging the gap between scales of in-dividual molecules and their collective behavior has been a long-standing problem in biology (27).Ca puffs generated by IP3 receptors (IP3r’s) are generally accepted

to be collective events in which clustered channels are mutually ac-tivated by CICR. Their termination mechanism remains uncertain(28). Our present model cannot be directly applied to puffs, becauseunlike the RyR the IP3r is strongly inactivated by high calcium,meaning that our condition iv only holds for a small part of thecalcium range. There is considerable uncertainty about the mecha-nisms of IP3r opening and closing in puffs (28–30), so it is prematureto speculate on details of any possible extension of the Ising paradigm

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

9,000

10,000

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

0

20

40

60

80

100

120

140

160

180

200

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

0

0.005

0.01

0.015

0.02

0.025

0.03

0 50 100 150 200

A

B

CTermina�on

�me, ms

Termina�on�me (Tt), ms

CaSR, mM

Phasetransi�ons

13x1311x11 9x9 7x7 5x5

5x5

13x1311x11

9x97x7

Tt= A*exp(CaSR/ΔCaSR)+Tt,inf

CaSR, mM

0

0.05

0.1

0.15

0.2

0.25

0.3

0 50 100 150 200

Number of release channels

CaSR_h_reversal

CaSR_30%_Tt_increase

Number of release channels

D

Phase transi�on spanΔCaSR, mM

Transi�on onset, mM

13x1311x119x9

7x7

5x5

13x1311x11

9x9

7x7

5x5

0

50

100

150

200

0 10 20 30 40 50 60 70 800

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

20 30 40 50 60

A metastable spark

A stable spark

CaSR, mM A metastable spark

A stable spark

Time, msTime, ms

h reversal

EFN open RyRs

>

h reversal atCaSR≈ 0.084 mM

Fig. 4. Results of statistical analysis of our in silico experiments testing Isingmodel predictions for various grid sizes. (A) Median termination times (Tt) plottedas a function of clamped CaSR. Each data point was obtained from 100 simulationruns. (B) The data set of A, but at a smaller scale. The transition onsets are closelydescribed by an exponential (shown at the plot). (C) The transitions are sharper forlarger grids as occurring within smaller ranges of CaSR (ΔCaSR). (D) h reversal(CaSR_h reversal) in our Ising model closely predicts the onset of the phase tran-sition in our model simulations. The transition onset in the simulations is estimatedas a 30% increase in the median Tt (CaSR_30%_Tt_increase) calculated using re-spective exponential fits in B. (E and F) Open RyRs and CaSR dynamics in repre-sentative examples of stable and metastable sparks (13 × 13 RyR grid). Themetastable spark was generated by increasing SR Ca refiling rate (TAUFILL wasdecreased from 6.5 ms to 1.5 ms). Inset shows a narrow margin for CaSR thatdetermines spark termination fate (see Movies S3 and S4 for more details).

7528 | www.pnas.org/cgi/doi/10.1073/pnas.1701409114 Maltsev et al.

Dow

nloa

ded

by g

uest

on

Dec

embe

r 31

, 202

0

to puffs. However, in 2009 Smith and Parker (31) resolved individualopenings and closings of IP3r’s during a puff that showed a tendencyfor IP3r’s to close abruptly and collectively (“square” puffs). Wiltgenet al. (28) examined further the “square puff” phenomenon andshowed unambiguously that IP3r’s in a decaying puff do not behaveindependently, but tend to close synchronously. This implies somekind of “closing signal” coordinating the various channels. The natureof this signal remains unknown. However, the collective inactivationmay be suggestive of some kind of phase transition, and a “closingsignal” could be accommodated in an Ising-like model.

MethodsBrief Introduction to the Ising Model. The Ising model we will work withconsists of binary random variables (i.e., taking values±1) called spins positioned ona 2D finite gridΛ (e.g., section 3.3.5 in ref. 32). A configuration of spins is a functionσ that assigns 1 or−1 to each point x ∈Λ. The configuration spaceΩ is the set of allpossible assignments of spins to points in Λ, that is, all possible functions σ   : Λ →{1, −1}. An interaction profile ϕ: R→R is a function with ϕðxÞ→ 0 rapidly as x→∞and ϕ > 0. We choose ϕ so that ϕ(1) = 1. We furthermore place our finite grid Λinside of a bigger grid Λb (b for boundary) and let σðxÞ = −1 for any x ∈Λb=Λ. Inthis way we impose a −1 boundary condition on Λ. HereΛb/Λmust “frame”Λ andits thickness has to be at least as wide as the effective interaction range, which inour case will be around 5. To be precise, ifΛ is an n bym grid,Λb will be an n+ 10by m + 10 grid with Λ situated in the middle of Λb. The Hamiltonian is

HðσÞ=−X

x, y∈Λb

ϕðjx − yjÞσðxÞσðyÞ−hXx∈Λb

σðxÞ. [1]

Here the first sum is over Λb instead of Λ. This is necessary to ensure theinteraction with the boundary.

In physics, h is the magnetic field. The Hamiltonian can be interpreted as theenergy of the system. The equilibrium measure (Gibbs measure) is given by

πðσÞ= Z−1e−βHðσÞ. [2]

The normalization constant Z is well-defined because our lattice Λ is finite,and we will not need to know it explicitly for our analysis. Here β is theinverse temperature. (For further information on the general Ising model, ofwhich this is an instance, cf. sections 2.1 and 2.2 in ref. 33.)

Dynamic Ising: Detailed Balance and the Transition Rates. Let Λ be a 2D integergrid of a finite size. Recall thatΩ is the configuration space and let σ: Λ→ {1,−1}be an element of Ω. One can introduce a dynamic on spin configurations sothat the configuration space Ω becomes the state space for a Markov chainwith a transition matrix P. We introduce the notation σx to mean

σx =�

σðyÞ for   y ≠ x− σðyÞ for   y = x

,

that is, σx coincides with σ everywhere except at x, where the spin is reversed.To obtain a Glauber-like dynamic for the Ising model, it suffices to choose aspin uniformly at random at each time increment and to give the probabilitythat it flips, that is, to give Pðσ→ σxÞ.

The condition on P that guarantees that π as in Eq. 2 is indeed theequilibrium measure for the Markov chain is called detailed balance, and itstates that the Markov chain is reversible with respect to π (cf. equation1.30 and proposition 1.19 in ref. 32). The equation for detailed balance is thefollowing: for all σ ∈ Ω and x ∈ Λ we have that

Pðσ→ σxÞe−βHðσÞ = Pðσx → σÞe−βHðσx Þ. [3]

This is equivalent to

Pðσ→ σxÞPðσx → σÞ= eβHðσÞ−βHðσ

x Þ

= e−2β

�Py∈∧b

ϕðjx−yjÞσðxÞσðyÞ+hσðxÞ�= e

−2βðxÞ�P

y∈∧bϕðjx−yjÞσðyÞ+h

�.

[4]

The detailed balance equations will be satisfied for a wide variety of rates P,so we can choose P to be most appropriate to our CRU model. Because weknow that the release channel opening rate is an exponential whereas theclosing rate is a constant, we look for P so that the transition from −1 to 1 isexponential and the transition from 1 to −1 is a constant. This indeed can beachieved simultaneously with the detailed balance condition. If σðxÞ=−1we

let Pðσ→ σxÞ=Ce2β

� Py∈Λb

ϕðjx−yjÞσðyÞ+h�, yielding that Pðσx → σÞ=C to satisfy

detailed balance. Thus, the Markov chain is given as follows. We pick alocation x uniformly at random and define the transition matrix P to be

Pðσ, σxÞ=8<:Ce

2β

�Py∈Λb

ϕðjx−yjÞσðyÞ+h�

for  σðxÞ=−1C for  σðxÞ= 1

9=;. [5]

Here time is continuous and the above are transition rates. In our numericalmodel, time is discrete and we take Δt = 0.05 ms. The transition matrix withthe discretized time becomes

Pðσ, σxÞ=8<:ΔtCe

2β

�Py∈Λb

ϕðjx−yjÞσðyÞ+h�

  for  σðxÞ=−1ΔtC for  σðxÞ= 1

9=; [6]

andwe ensure thatΔt is small enough so that all transition probabilities are smallerthan 1. Letting also Pðσ, σÞ= 1− Pðσ, σxÞ ensures that P is indeed stochastic.

The CRU as an Ising Model. A numerical model of the CRU consists of a squaregrid of calcium release channels Λ and each release channel can be open orclosed. We assign 1 to each open and −1 to each closed release channel, thusobtaining a configuration σ   : Λ → {1, −1}. We introduce the constant U torepresent the spatial distance between nearest release channels. In ournumerical model U = 30 nm.

We let ψ be the 1D slice of the time-stable spatial calcium profile resultingfrom the opening of one release channel. This is sufficient to contain all ofthe information about the calcium profile because ψ is rotationally sym-metric. We obtain ψ from our numerical simulation. However, ψ is an im-mediate result of the environment, including current, diffusion, and bufferand is not an emergent property. We interpret it as a scaled interactionprofile and let ϕ in Eq. 1 be given as ϕðrÞ=ψðUrÞ=ψðUÞ, where Ur is thedistance to the open release channel. The multiplication by U accounts forthe fact that the release channels are U units apart whereas spins are 1 unitapart. The division by ψðUÞ is a choice of scaling for the interaction profilefunction ϕ. With this scaling we have ϕð1Þ= 1. We choose this scaling for ϕ sothat at the nearest neighbors its value matches the classic Ising model, whereeach spin interacts with four neighbors with a strength of 1.

The distance between CRUs is assumed to be too large for calcium fromone CRU to influence another. However, calcium is diffusing out of the CRUand in this way the release channels in the CRU interact with the outside. Themodel would be identical if the CRU were surrounded by release channelsthat are always closed. In this way, the boundary condition of the CRU modelis equivalent to a negative boundary condition of the Ising model.

We will compute the analogs of inverse temperature β and the magneticfield h in our CRU model as functions of initial model parameters. They playthe exact same role in the mathematical description of our CRU model asthey do in the Ising model even though they do not carry the same physicalmeaning. We will note that β is an increasing function of the concentrationof Ca inside the junctional SR and we vary the SR Ca in our numerical modelto test the predictions of the CRU Ising model.

Relating [Ca] and the Ising Hamiltonian. Let us introduce the set SðxÞdfs∈R : s=jx − yj≠ 0  for  some  y ∈Z2g. We can rewrite both the local [Ca] at x (we denoteit [Ca](x)) and the exponent in the −1 to 1 transition in P in terms of a sumover SðxÞ. Given a configuration of open and closed release channels σ and agiven release channel at a point x, let NUs be the number of open RyRs at adistance Us from x. If the release channel at x is closed, we can approximate[Ca] at x by

½Ca�ðxÞ=Xs∈SðxÞ

ψðUsÞNUs. [7]

We similarly rewrite P. We introduce the following notation: Ts(x) = total numberof spins at distance s from x; Ls(x) = number of −1 spins at distance s from x;Ns(x) = number of +1 spins at distance s from x; and we have Ns(x) + Ls(x) = Ts(x).

Henceforth in this section let us fix a site x ∈ Λ and suppress the de-pendence on x in Ts, Ls, Ns, and S for ease of notation. Then we can rewritethe expression in the exponent of the Ising −1 to +1 transition probability inEq. 5 in the following way:

Xy∈Λb

ϕðjx − yjÞσðyÞ=Xs∈S

ϕðsÞðNs − LsÞ=Xs∈S

ϕðsÞð2Ns − TsÞ

= 2Xs∈S

ϕðsÞNs −Xs∈S

ϕðsÞTs ≈ 2Xs∈S

ϕðsÞNs − 2πZs>.5

ϕðsÞ  ds.[8]

In the last approximate equality, we have replacedPs∈S

ϕðsÞTs by 2πRs>.5ϕðsÞds

where the factor of 2π is due to the fact thatPs∈S

ϕðsÞTs is approximately a 2D

Maltsev et al. PNAS | July 18, 2017 | vol. 114 | no. 29 | 7529

BIOPH

YSICSAND

COMPU

TATIONALBIOLO

GY

Dow

nloa

ded

by g

uest

on

Dec

embe

r 31

, 202

0

integral of a rotationally symmetric function. We observe that the first termin the final expression in Eq. 8 is a scalar multiple of the total calcium [Ca] (x)as given in Eq. 7.

Crucial Parameters and the Spark Termination Criterion. We want to solve forthe analogs of h and β in the CRU model. We again fix a site x ∈ Λ andsuppress the dependence on x in [Ca] and S for ease of notation. From ex-perimental data we fit the exponential λeγ[Ca] to the Ising transition ratefrom −1 to +1 in Eq. 5:

λeγ½Ca� =Ce2β

�Py∈Λb

ϕðjx−yjÞσðyÞ+h�.

Then, we replace the left-hand side using Eq. 7 and the right-hand side usingthe expression derived in Eq. 8 to obtain

λeγP

s∈SψðUsÞNUs =Ce

2β

�2P

s∈SϕðsÞNs−2π

Zs>.5

ϕðsÞds+h�

=Ce−4βπ

Zs>.5

ϕðsÞ+2βhe4βð

Ps∈S

ϕðsÞNsÞ.[9]

Because we wish the above equality to hold for any configuration, we mustequate the coefficients of

Ps∈SϕðsÞNs to obtain β = γψ(U)/4.

Next we equate the coefficients in front of e4β�P

s∈ SϕðsÞNs

�to obtain

λ=Ce−4βπ

Zs>.5

ϕðsÞds+2βh

yielding that

h=12β

ln�λ

C

�+ 2π

Zs>.5

ϕðsÞds. [10]

Because h is the analog of the magnetic field in the CRU model, the emer-gent behavior of release channels can be predicted based on h. Duringtermination all of the release channels begin in an open state (analogous to+1). The Ca diffusion out of CRU is equivalent to a negative boundarycondition. We can hence deduce the signal termination criterion: If h < 0,then the spark will terminate and this termination is mathematically iden-tical to reversal of polarity in ferromagnetism. Mathematically, this phasetransition follows from the Lee–Yang theorem. However, if h > 0, the sparkwill be metastable.

ACKNOWLEDGMENTS. This work was supported by the Intramural ResearchProgram of the National Institute on Aging, NIH and Leverhulme Trust EarlyCareer Fellowship ECF 2013-613 (to A.V.M.).

1. Peng W, et al. (2016) Structural basis for the gating mechanism of the type 2 ryano-dine receptor RyR2. Science 354:aah5324.

2. Soeller C, Crossman D, Gilbert R, Cannell MB (2007) Analysis of ryanodine receptorclusters in rat and human cardiac myocytes. Proc Natl Acad Sci USA 104:14958–14963.

3. Baddeley D, et al. (2009) Optical single-channel resolution imaging of the ryanodinereceptor distribution in rat cardiac myocytes. Proc Natl Acad Sci USA 106:22275–22280.

4. Walker MA, et al. (2014) Superresolution modeling of calcium release in the heart.Biophys J 107:3018–3029.

5. Fabiato A (1983) Calcium-induced release of calcium from the cardiac sarcoplasmicreticulum. Am J Physiol 245:C1–C14.

6. Stern MD (1992) Theory of excitation-contraction coupling in cardiac muscle. BiophysJ 63:497–517.

7. Cheng H, Lederer WJ, Cannell MB (1993) Calcium sparks: Elementary events un-derlying excitation-contraction coupling in heart muscle. Science 262:740–744.

8. Cheng H, Lederer WJ (2008) Calcium sparks. Physiol Rev 88:1491–1545.9. Sobie EA, Dilly KW, dos Santos Cruz J, Lederer WJ, Jafri MS (2002) Termination of

cardiac Ca(2+) sparks: An investigative mathematical model of calcium-induced cal-cium release. Biophys J 83:59–78.

10. Stern MD, Ríos E, Maltsev VA (2013) Life and death of a cardiac calcium spark. J GenPhysiol 142:257–274.

11. Gillespie D, Fill M (2013) Pernicious attrition and inter-RyR2 CICR current control incardiac muscle. J Mol Cell Cardiol 58:53–58.

12. Laver DR, Kong CH, Imtiaz MS, Cannell MB (2013) Termination of calcium-inducedcalcium release by induction decay: An emergent property of stochastic channelgating and molecular scale architecture. J Mol Cell Cardiol 54:98–100.

13. Onsager L (1944) Crystal statistics. I. A two-dimensional model with an order-disordertransition. Phys Rev 65:117–149.

14. Lenz W (1920) Beiträge zum verständnis der magnetischen eigenschaften in festenkörpern. Physik Z 21:613–615.

15. Csernoch L (2007) Sparks and embers of skeletal muscle: The exciting events of con-tractile activation. Pflugers Arch 454:869–878.

16. Lehnart SE, et al. (2008) Leaky Ca2+ release channel/ryanodine receptor 2 causesseizures and sudden cardiac death in mice. J Clin Invest 118:2230–2245.

17. Song Z, Karma A, Weiss JN, Qu Z (2016) Long-lasting sparks: Multi-metastabilityand release competition in the calcium release unit network. PLoS Comput Biol12:e1004671.

18. Sato D, Shannon TR, Bers DM (2016) Sarcoplasmic reticulum structure and functionalproperties that promote long-lasting calcium sparks. Biophys J 110:382–390.

19. Zima AV, Picht E, Bers DM, Blatter LA (2008) Partial inhibition of sarcoplasmic re-ticulum Ca release evokes long-lasting Ca release events in ventricular myocytes: Roleof luminal Ca in termination of Ca release. Biophys J 94:1867–1879.

20. Sobie EA, Lederer WJ (2012) Dynamic local changes in sarcoplasmic reticulum calcium:Physiological and pathophysiological roles. J Mol Cell Cardiol 52:304–311.

21. Qu Z, Weiss JN (2015) Mechanisms of ventricular arrhythmias: From molecular fluc-tuations to electrical turbulence. Annu Rev Physiol 77:29–55.

22. Ter Keurs HE, Boyden PA (2007) Calcium and arrhythmogenesis. Physiol Rev 87:457–506.

23. Maltsev AV, et al. (2011) Synchronization of stochastic Ca2+ release units creates arhythmic Ca2+ clock in cardiac pacemaker cells. Biophys J 100:271–283.

24. Maltsev AV, Yaniv Y, Stern MD, Lakatta EG, Maltsev VA (2013) RyR-NCX-SERCA localcross-talk ensures pacemaker cell function at rest and during the fight-or-flight reflex.Circ Res 113:e94–e100.

25. Stern MD, et al. (2014) Hierarchical clustering of ryanodine receptors enables emer-gence of a calcium clock in sinoatrial node cells. J Gen Physiol 143:577–604.

26. Bodineau T, Graham B, Wouts M (2013) Metastability in the dilute Ising model.Probab Theory Relat Fields 157:955–1009.

27. Qu Z, Garfinkel A, Weiss JN, Nivala M (2011) Multi-scale modeling in biology: How tobridge the gaps between scales? Prog Biophys Mol Biol 107:21–31.

28. Wiltgen SM, Dickinson GD, Swaminathan D, Parker I (2014) Termination of calciumpuffs and coupled closings of inositol trisphosphate receptor channels. Cell Calcium56:157–168.

29. Shuai J, Pearson JE, Foskett JK, Mak DO, Parker I (2007) A kinetic model of single andclustered IP3 receptors in the absence of Ca2+ feedback. Biophys J 93:1151–1162.

30. Ullah G, Parker I, Mak DO, Pearson JE (2012) Multi-scale data-driven modeling andobservation of calcium puffs. Cell Calcium 52:152–160.

31. Smith IF, Parker I (2009) Imaging the quantal substructure of single IP3R channelactivity during Ca2+ puffs in intact mammalian cells. Proc Natl Acad Sci USA 106:6404–6409.

32. Levin DA, Peres Y, Wilmer EL (2008) Markov Chains and Mixing Times (AmericanMathematical Society, Providence, RI), 1st Ed.

33. Martinelli F (1999) Lectures on glauber dynamics for discrete spin models. Lectures onProbability Theory and Statistics. Lecture Notes in Mathematics, ed Bernard P(Springer, Berlin), Vol 1717, pp 93–191.

7530 | www.pnas.org/cgi/doi/10.1073/pnas.1701409114 Maltsev et al.

Dow

nloa

ded

by g

uest

on

Dec

embe

r 31

, 202

0